Physics Reports vol.362

Instructions to authors Aims and scope Physics Reports keeps the active physicist up-to-date on developments in a wide r...

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Instructions to authors Aims and scope Physics Reports keeps the active physicist up-to-date on developments in a wide range of topics by publishing timely reviews which are more extensive than just literature surveys but normally less than a full monograph. Each Report deals with one specific subject. These reviews are specialist in nature but contain enough introductory material to make the main points intelligible to a non-specialist. The reader will not only be able to distinguish important developments and trends but will also find a sufficient number of references to the original literature. Submission In principle, papers are written and submitted on the invitation of one of the Editors, although the Editors would be glad to receive suggestions. Proposals for review articles (approximately 500–1000 words) should be sent by the authors to one of the Editors listed below. The Editor will evaluate proposals on the basis of timeliness and relevance and inform the authors as soon as possible. All submitted papers are subject to a refereeing process. Editors J.V. ALLABY (Experimental high-energy physics), EP Division, CERN, CH-1211 Geneva 23, Switzerland. E-mail: [email protected] D.D. AWSCHALOM (Experimental condensed matter physics), Department of Physics, University of California, Santa Barbara, CA 93106, USA. E-mail: [email protected] J.A. BAGGER (High-energy physics), Department of Physics & Astronomy, The Johns Hopkins University, 3400 North Charles Street, Baltimore MD 21218, USA. E-mail: [email protected] C.W.J. BEENAKKER (Mesoscopic physics), Instituut–Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands. E-mail: [email protected] E. BREZIN (Statistical physics and field theory), Laboratoire de Physique Theórique, Ecole Normale Superieure, 24 rue Lhomond, 75231 Paris Cedex, France. E-mail: [email protected] G.E. BROWN (Nuclear physics), Institute for Theoretical Physics, State University of New York at Stony Brook, Stony Brook, NY 11974, USA. E-mail: [email protected] D.K. CAMPBELL (Non-linear dynamics), Dean, College of Engineering, Boston University, 44 Cummington Street, Boston, MA 02215, USA. E-mail: [email protected] G. COMSA (Surfaces and thin films), Institut fur . Physikalische und Theoretische Chemie, Universit.at Bonn, Wegelerstrasse 12, D-53115 Bonn, Germany. E-mail: [email protected] J. EICHLER (Atomic and molecular physics), Hahn-Meitner-Institut Berlin, Abteilung Theoretische Physik, Glienicker Strasse 100, 14109 Berlin, Germany. E-mail: [email protected] M.P. KAMIONKOWSKI (Astrophysics), Theoretical Astrophysics 130-33, California Institute of Technology, 1200 East California Blvd., Pasadena, CA 91125, USA. E-mail: [email protected]

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Instructions to authors

M.L. KLEIN (Soft condensed matter physics), Department of Chemistry, University of Pennsylvania, Philadelphia, PA 19104-6323, USA. E-mail: [email protected] A.A. MARADUDIN (Condensed matter physics), Department of Physics and Astronomy, University of California, Irvine, CA 92697-4575, USA. E-mail: [email protected] D.L. MILLS (Condensed matter physics), Department of Physics and Astronomy, University of California, Irvine, CA 92697-4575, USA. E-mail: [email protected] R. PETRONZIO (High-energy physics), Dipartimento di Fisica, Universita" di Roma – Tor Vergata, Via della Ricerca Scientifica, 1, I-00133 Rome, Italy. E-mail: [email protected] S. PEYERIMHOFF (Molecular physics), Institute of Physical and Theoretical Chemistry, Wegelerstrasse 12, D-53115 Bonn, Germany. E-mail: [email protected] I. PROCACCIA (Statistical mechanics), Department of Chemical Physics, Weizmann Institute of Science, Rehovot 76100, Israel. E-mail: [email protected] E. SACKMANN (Biological physics), Physik-Department E22 (Biophysics Lab.), Technische Universit.at Munchen, . D-85747 Garching, Germany. E-mail: [email protected] A. SCHWIMMER (High-energy physics), Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 76100, Israel. E-mail: [email protected] R.N. SUDAN (Plasma physics), Laboratory of Plasma Studies, Cornell University, 369 Upson Hall, Ithaca, NY 14853-7501, USA. E-mail: [email protected] W. WEISE (Physics of hadrons and nuclei), Institut fur . Theoretische Physik, Physik Department, Technische Universit.at Munchen, . James Franck Strae, D-85748 Garching, Germany. E-mail: [email protected] Manuscript style guidelines Papers should be written in correct English. Authors with insufficient command of the English language should seek linguistic advice. Manuscripts should be typed on one side of the paper, with double line spacing and a wide margin. The character size should be sufficiently large that all subscripts and superscripts in mathematical expressions are clearly legible. Please note that manuscripts should be accompanied by separate sheets containing: the title, authors’ names and addresses, abstract, PACS codes and keywords, a table of contents, and a list of figure captions and tables. – Address: The name, complete postal address, e-mail address, telephone and fax number of the corresponding author should be indicated on the manuscript. – Abstract: A short informative abstract not exceeding approximately 150 words is required. – PACS codes/keywords: Please supply one or more PACS-1999 classification codes and up to 4 keywords of your own choice for indexing purposes. PACS is available online from our homepage (http://www.elsevier.com/locate/physrep). References. The list of references may be organized according to the number system or the nameyear (Harvard) system. Number system: [1] M.J. Ablowitz, D.J. Kaup, A.C. Newell and H. Segur, The inverse scattering transform – Fourier analysis for nonlinear problems, Studies in Applied Mathematics 53 (1974) 249–315. [2] M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).


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[3] B. Ziegler, in: New Vistas in Electro-nuclear Physics, eds E.L. Tomusiak, H.S. Kaplan and E.T. Dressler (Plenum, New York, 1986) p. 293. A reference should not contain more than one article. Harvard system:

Ablowitz, M.J., D.J. Kaup, A.C. Newell and H. Segur, 1974. The inverse scattering transform – Fourier analysis for nonlinear problems, Studies in Applied Mathematics 53, 249–315. Abramowitz, M. and I. Stegun, 1965, Handbook of Mathematical Functions (Dover, New York). Ziegler, B., 1986, in: New Vistas in Electro-nuclear Physics, eds E.L. Tomusiak, H.S. Kaplan and E.T. Dressler (Plenum, New York) p. 293. Ranking of references. The references in Physics Reports are ranked: crucial references are indicated by three asterisks, very important ones with two, and important references with one. Please indicate in your final version the ranking of the references with the asterisk system. Please use the asterisks sparingly: certainly not more than 15% of all references should be placed in either of the three categories. Formulas. Formulas should be typed or unambiguously written. Special care should be taken of those symbols which might cause confusion. Unusual symbols should be identified in the margin the first time they occur.

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Physics Reports 362 (2002) 1–62

Two-dimensional turbulence: a physicist approach Patrick Tabeling Laboratoire de Physique Statistique, Ecole Normale Superieure, 24 rue Lhomond, 75231 Paris, France Received June 2001; editor: I: Procaccia Contents 1. 2. 3. 4.

Introduction Exact results Coherent structures Statistics of the coherent structures in decaying two-dimensional turbulence 5. Equilibrium states of two-dimensional turbulence 6. Inverse energy cascade

2 4 8 16 22 29

7. Dispersion of pairs in the inverse cascade 8. Condensed states 9. Enstrophy cascade 10. Coherent structures vs. cascades 11. Conclusions 12. Uncited references Acknowledgements References

40 42 45 52 56 57 57 57

Abstract Much progress has been made on two-dimensional turbulence, these last two decades, but still, a number of fundamental questions remain unanswered. The objective of the present review is to collect and organize the available information on the subject, emphasizing on aspects accessible to experiment, and outlining contributions made on simple 9ow con:gurations. Whenever possible, open questions are made explicit. Various subjects are presented: coherent structures, statistical theories, inverse cascade of energy, condensed states, Richardson law, direct cascade of enstrophy, and the inter-play between cascades and coherent structures. The review o=ers a physicist’s view on two-dimensional turbulence in the sense that experimental facts play an important role in the presentation, technical issues are described without much detail, sometimes in an oversimpli:ed form, and physical arguments are given whenever possible. I hope this bias does not jeopardize the interest of the presentation for whoever wishes to visit c 2002 Elsevier Science B.V. All rights reserved. the fascinating world of Flatland. PACS: 47.27.Ak; 47.27.Eq; 05.45.−a

c 2002 Elsevier Science B.V. All rights reserved. 0370-1573/02/$ - see front matter PII: S 0 3 7 0 - 1 5 7 3 ( 0 1 ) 0 0 0 6 4 - 3

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P. Tabeling / Physics Reports 362 (2002) 1–62

1. Introduction Several motivations or prospects may lead to get involved in studies on two-dimensional turbulence. One motivation, probably shared by all the investigators, is that hydrodynamic turbulence, despite its practical interest, its omnipresence, and the series of assaults it has been submitted to for decades, is still with us as the most important unsolved problem of classical mechanics. ‘Unsolved’ means that, for the simplest cases we may conceive, we are unable to infer the statistical characteristics of a turbulent system from :rst principles. It is not a matter of being unable to perform a detailed calculation. The structure of the solution is at the moment beyond the reach of existing theoretical approaches. Inspiration has not lacked for devising clever theoretical methods, but for long, success did not come and in this respect, hydrodynamic turbulence looks as a cemetery for a number of ideas that turned out to be outrageously successful in other areas of condensed matter physics. At the moment, the situation perhaps evolves, further to recent breakthroughs made on the turbulent dispersion problem. In particular, we now have, for the passive dispersion problem, a complete knowledge of turbulent solutions for particular (idealized) cases. An issue is to what extent these ideas may be used to tackle the hydrodynamic turbulent problem. The least one can say is that the issue is open at the moment. Another motivation is related to geophysical 9ows. Flows in the atmosphere and in the ocean develop in thin rotating strati:ed layers, and it is known that rotation, strati:cation and con:nement are eEcient vectors conveying two-dimensionality. Experimental devices developed in laboratories typically use these ingredients to prepare acceptably two-dimensional 9ows. The extent to which two-dimensional turbulence represents atmospheric or oceanographic turbulence is however not straightforward to circumvent, and this question has been discussed in several places [77,29,118]. In the simplest cases, pure two-dimensional equations must be amended by the addition of an extra-term, characterizing the e=ect of Coriolis forces, to represent physically relevent situations. This term generates waves which radiate energy, a mechanism absent in pure two-dimensional systems. In more realistic cases, topography, thermodynamics, strati:cation, must be incorporated in the analysis, and indeed full two-dimensional approximation hardly encompasses the variety of phenomena generated by these additional terms. Nonetheless, purely two-dimensional 9ows provide useful guidance to understand some important aspects of geophysical 9ows. A remarkable (although rather isolated) example is the trajectories of the tropical hurricanes. For long, two-dimensional pointwise vorticity models have been used, leading to fairly acceptable predictions. In the oceans, the concept of coherent structures, discovered in early studies of two-dimensional turbulent 9ows, is frequently substantiated. A context where concepts of two-dimensional turbulence appear as particularly relevant is the dispersion of chemical species in the atmosphere, and tracers in the oceans. The structure of the ozone concentration :eld around the polar vortex can be accounted for if :lamentary processes, similar to those prevailing in two-dimensional turbulent 9ows, are assumed to take place. For this particular aspect, evoking the presence of waves do not provide clues while two-dimensional turbulence o=ers an interesting framework, allowing to construct acceptable representations [71]. A third motivation is linked to the three-dimensional turbulence problem. The recent period has shown the existence of a common conceptual framework between two- and three-dimensional turbulence. Phenomena, such as cascades, coherent structures, dissipative processes, :lamentation mechanisms take place in both systems. On the other hand, turbulence is simpler to represent


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and easier to compute in two than in three dimensions and physical experiments provide much more information when the 9ow is thoroughly visualized that when it is probed with a single sensor. Therefore, operating in Flatland may be instructive to test ideas or theories, prior to considering them in the three-dimensional world. Note that three-dimensional turbulence is not three dimensional in all respects; an example is the worms, which populate the dissipative range: they are long tubes and thus can be tentatively viewed as to two-dimensional objects. One may also add that two-dimensional turbulence is helpful for understanding turbulence in general. Turbulence is a general phenomenon, involving nonlinear dynamics and broad range of scales, including as particular cases two- and three-dimensional 9uid turbulence, but also super9uid turbulence. Nonlinear ShrIodinger turbulence, scalar turbulence of any sort, i.e. passive or active, low or high Prandtl numbers, ‘burgulence’ (i.e. the particular turbulence produced in Burgers equation), to mention but a few. Across these subjects, a common conceptual framework exists, and two-dimensional turbulence can obviously be used to understand several aspects of the general problem. If we trace back the history, it seems that it took a long time to realize that 9ows can be turbulent in two dimensions. For long, two-dimensional turbulence was thought impossible, frozen by too many conservation laws. Batchelor, in his in9uential textbook on turbulence [8], overlooked the possibility that turbulent cascades may exist in 2D. In the presentations of turbulence, it was not uncommon to stress that two-dimensional 9ows are inhospitable to turbulent regimes. The situation changed in the sixties, further to two contributions, given independantly by Kraichnan and Batchelor. Batchelor reconsidered his views on two-dimensional turbulence, and proposed that cascades may develop in the plane, in a way analogous to the direct energy cascade in three dimensions, with the enstrophy playing the role of the energy [9]. Kraichnan [67] discovered the double cascade process and explained that turbulence may be compatible with the presence of several conservation laws. These contributions stimulated research in the :eld, by suggesting that something can be learnt on turbulence in general from investigations led on two-dimensional systems. Still, the development of the :eld was considered as diEcult, since it was generally considered that turbulent two-dimensional 9ows could not be achieved in the laboratory, and it was uncertain that the development of the computers could provide, within a reasonable range of time, reliable and accurate data on such systems. Now the situation has drastically changed, since on both of these aspects, we witnessed considerable developments. Numerics exploded in the seventies, and succeeded to produce a considerable 9ux of reliable data. Several phenomena were discovered, among them the coherent structures, which deeply modi:ed our view on turbulence. For long the available data in Flatland was computationally based; this is no more true today. The experimental e=ort, starting in the seventies with pioneering experiments, could be pushed forward, thanks to the availability of image processing techniques. In some cases, the physical experiment could o=er conclusions diEcult to draw out from numerics. We now are in a situation where the :eld of two-dimensional turbulence has reached a sort of maturity, in the sense that numerical and experimental techniques are in position to convey valuable and relevant information concerning most of the questions we may ask on the problem. This is not true as soon as complications, such as wave propagation, or topographic e=ects are added. A source of progress is certainly expected from the theory, which at the moment, is

4


unable to answer the simplest questions. In several cases, the theory seems to be stuck, like a bee in a honey pot, and conceptual breakthroughs appear as the only chance to move forward. Speci:c issues such as the role of coherent structures in turbulent systems, the intermittency e=ects in the inverse cancade, the ergodic issue in free systems, are example of problems which seem to raise considerable diEculties. In this context, it seemed to me that it could be interesting to review the knowledge accumulated during the last decades on the subject. At the moment, the results are spread in various papers and for those who wish to get in touch with the :eld, it would take time to collect the most relevant information. Two-dimensional turbulence has already been reviewed by several authors. Kraichnan and Montgomery [69] emphasized on statistical theories, giving a complete account of the theoretical e=orts made until 1980. A more recent review on the subject, written in a pleasant pedagogical form, is given by Miller et al. [89]. Lesieur [77] reviewed di=erent subjects, emphasizing on spectral descriptions. Frisch [43] wrote a swift review, which emphasizes many of the far-reaching issues still open in the :eld. A short account on recent progress in two-dimensional turbulence has been given by Nelkin [94]. The presentation I am proposing here can be viewed as an extension and an update of Frisch’s review. Some material was unavailable at that time, at theoretical, numerical and physical levels and I hope the present review will bring useful complements. As a physicist, I will put emphasis on the experiment, and on the theoretical e=orts dedicated to explaining phenomena within the reach of the observation. This certainly tends to con:ne the view on the subject. Two-dimensional turbulence is a :eld of investigation for theorists, who need to develop their own approach without being necessarily constrained by the experimental facts. I decided to warn the reader of this personal bias by making an announcement at the :rst line of the paper, i.e. in the title. I will not make any description of closure models such as EDQNM, or test :eld model, despite their practical interest. The reader will :nd a presentation of these aspects in [77]. Also cascade models, such as shell model, will not be discussed. I hope these biases and limitations do not outrageously reduce the scope of the review, and that the present paper will be of interest for a broad range of scientists interested in turbulence, for whatever reason. 2. Exact results There exist several exact results, important to know for understanding two-dimensional turbulent 9ows. They are quoted in papers by Batchelor [9], reviews by Rose and Sulem [121], Kraichnan and Montgomery [69], or books by Lesieur [77], and Frisch [43]. One starts with the Navier–Stokes equations which read: Du @u 1 (1) = + (u )u = − ∇p + fext + Ou ; Dt @t ∇ · u=0

(2)

in which u is the velocity, p the pressure, fext is the external forcing, and is the kinematic viscosity. One usually takes a system of coordinates (x; y) for the plane of 9ow and z for the coordinate normal to the plane.


5

In the inviscid case ( = 0), and without forcing (fext = 0), these equations are the Euler equations; in two dimensions, it has been possible to prove that the Euler equations are regular in a bounded domain, and, if the vorticity is initially bounded, the solution exists and is unique (see [121]). This implies in particular that there are no singularities at :nite time for the velocity, a possibility which, at the moment, is not ruled out in three dimensions. By taking the curl of these equations, one gets an equation for the vorticity ! = (∇xu)z in the form D! @! = + u ! = g + O! ; (3) Dt @t where g is the projection of the curl of the forcing, along the axis normal to the plane. Let us stress that this equation holds in two dimensions, and not in three. In three dimensions, the governing equation for the vorticity has a di=erent form: an additional term, called vortex stretching, must be added. Sticking to the two-dimensional case, one :nds that, in the inviscid limit ( = 0), and with a curl free forcing (g = 0), the vorticity is conserved along the 9uid particle trajectories, a result known as Helmhotz theorem. In two-dimensional inviscid 9ows, the 9uid particles keep their vorticity for ever. In three-dimensional turbulence, it is not so: the additional vortex stretching term in the vorticity equation ampli:es, in the average, vorticity along the trajectories, leading to the formation of small intense :laments. Freely evolving two-dimensional turbulence cannot generate small intense vortices. This di=erence, between two and three dimensions, is fundamental to underline. A quantity of fundamental interest in turbulence is the kinetic energy per unit of mass, de:ned by E = 12 u2 ;

(4)

where the brackets mean statistical averaging, i.e. the average of a large number of independant realizations. It is usually considered that in homogeneous or spatially periodic systems, taking spatial averages is equivalent to taking statistical average. This may not be always true, but we do not address this issue in this section. In 2D, and in homogeneous or periodic systems, the evolution equation for the energy reads: dE 1 du2 (5) = = − Z ; dt 2 dt where Z = !2 is the enstrophy. This quantity is governed by the following equation (still in homogeneous or periodic systems): DZ (6) = − (∇!)2 : Dt The right-hand term involves vorticity gradients. In two-dimensional turbulence, vorticity patches, distorted by the background velocity, generate thin :laments, and thus work at enhancing the vorticity gradients. These gradients increase up to a level where viscosity come into play to smooth out the :eld. A stationary state may be reached, such that the dissipation (∇!)2 becomes a constant, independent of the viscosity. This de:nes a ‘dissipation anomaly’, similar to the three-dimensional case [65,95], but dealing with the enstrophy, instead of the energy. The concept of dissipation anomaly is fundamental in turbulence, and it turns out to operate

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both in two and three dimensions. In two dimensions, the enstrophy is forced to decrease with time, while in three-dimensional turbulence, it vigorously increases with time as long as viscous e=ects remain unimportant. Using the energy equation, one infers that in two-dimensional homogeneous turbulence, the energy is essentially conserved as viscosity tends to zero (from the positive side). This again contrasts with three-dimensional turbulence, where, according to Kolmogorov law, the energy decreases at a constant rate independant of viscosity. This di=erence between the decay properties of the di=erent integrals led Kraichnan and Montgomery [69] to introduce a distinction between robust (such as the energy) and fragile (like the enstrophy) invariants. One consequence (probably the most striking) of this discussion is that two-dimensional systems are unable to dissipate energy at small scales. Said di=erently, there is no viscous sink of energy at small scale in 2D turbulence. As a consequence, there cannot be a direct energy cascade in two dimensions. Where does the energy go? We will see later that it goes to large scales, and eventually gets dissipated by friction on walls in contact with the 9uid. The situation is di=erent for the enstrophy. Enstrophy can dissipate at small scales as the viscosity tends to zero, generating the dissipation anomaly mentioned above. Therefore, there can be a direct enstrophy cascade. According to these remarks, it is often considered that an analogy can be made between two and three dimensions if one treats the enstrophy as the energy, and the vorticity of the velocity. This analogy is helpful for a quick inspection of the phenomena; it is nonetheless misleading in many respects, since, as we will see, the energy and enstrophy cascades have di=erent qualitative properties. In the past, it seemed to be an impression that, on considering the inviscid limit, it could be argued that two-dimensional turbulence cannot exist. The argument was based on the remark that in the inviscid case, two-dimensional 9ows are constrained by an in:nity of invariants. Quantities such as Zn = !n

(7)

are conserved for any n, in statistically homogeneous systems. There is also an in:nite number of invariants in three dimensions, since, according to Kelvin’s theorem, the circulation along any closed loop is conserved. However, since the loops adopt increasingly intricate shapes with time, it may be suggested that such invariants will not yield e=ective constraints [100]. In two dimensions, there are many more conservation laws, which moreover can be expressed as constraints along :xed boundaries. It is thus legitimate to ask whether, in the huge phase spaces representing two-dimensional 9ows, there remain enough degrees of freedom to sustain turbulent regimes. In the labyrinth of constraints, is it certain that even These could :nd a pathway leading to two-dimensional turbulence? The answer is known: the study of (inviscid) point vortex systems shows that the number of degrees of freedom linearly increases with the number of vortices; therefore, from the viewpoint of the constraints, chaotic and turbulent regimes can easily develop in such systems, provided the number of vortices is large. Moreover, in a number of situations, viscosity destroys almost all the invariants, releasing most of the constraints the system would be subjected to if the 9ow were inviscid. The main issues raised by the existence of an in:nite number of invariants deal with the cascades. Are cascades compatible with the abundance of invariants in two dimensions? Can two-dimensional systems be turbulent in the same way as in three dimensions? We will see later that the answer to this question is yes,


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i.e. cascades develop in two dimensions, so that, from this respect, there is no point contrasting two- and three-dimensional systems. An interesting limiting case of two-dimensional systems (already referred to) is when the vorticity is point-wise. This approximation is extremely helpful for understanding the dynamics of two-dimensional systems, turbulent or not, with coherent structures. Pointwise approximation has been shown to be acceptable, for some range of time, in a number of situations. The equations governing systems of point vortices can be found in text-books, and their dynamics has been remarkably discussed by Aref [3]. Many situations can be solved exactly. From the equations, one can prove that dipoles propagate, pairs of like sign vortices undergo rotational motion, three vortices systems are solvable, and chaos may appear as soon as the number of point vortices is larger than or equal to four. These results are useful for analyzing the dynamics of coherent structures in turbulent systems. The case of many vortices may receive a rigorous statistical treatment, using the tools of statistical mechanics, a topic we will return to later. There exist also interesting exact results, obtained for forced turbulence, with prescribed forcings, and possessing virtues of isotropy and homogeneity. The :rst one corresponds to the case where turbulence is forced at large scales compared to the scales we are considering. It reads 4 dS2 DL (r) − 2 = − r ; (8) dr 3 where DL is the longitudinal vorticity correlation, de:ned by DL (r) = (uL (x) − uL (x + r))(!(x) − !(x + r))2

(9)

in which uL represents the velocity component along the separation vector r; S2 (r) is the structure function of order two de:ned by S2 (r) = (!(x) − !(x + r))2

(10)

and is the enstrophy dissipation rate, de:ned by = (∇!)2 :

(11)

It is the analog of Corrsin–Yaglom [90] relation for passive tracers. Another relation corresponds to the inverse situation, i.e. the forcing holds on a scale much smaller than those we address. In this case, one can derive an equation constraining the third order structure function [90]. This equation reads: F3 (r) = 32 :

(12)

where S3 (r) is the longitudinal third order structure function, de:ned by F3 (r) = (uL (x) − uL (x + r))3

(13)

and is the dissipation, de:ned by = (!)2 :

(14)

This is the two-dimensional analog of the Kolmogorov relation [90]. Compared to the threedimensional case, there is a change in the prefactor ( 32 instead of − 45 [103]); more importantly this prefactor is positive, indicating that in two dimensions, the energy transfers proceed, in the

8


average, from small to large scales. It turns out that the above relations (the one on enstrophy 9uxes, and the other on energy transfers), have played a limited role in our understanding of turbulence; perhaps their de:ciency is to provide little information on the system when coherent structures (we will come back to this notion in the next section) are present, i.e. in situations which focused most of the research e=orts along the years [96]. 3. Coherent structures Two-dimensional turbulence has provided a remarkable context for the study of ‘coherent structures’ and the interplay with the classical cascade theories. In the two-dimensional context, the very :rst observations of coherent structures, inferred from direct inspections of numerically computed vorticity :elds, traces back to the seventies. In two-dimensional jets, Aref and Siggia [2] noticed that remarkable vortical structures survived for long periods of time, in conditions where turbulence can be considered as fully developed (see Fig. 1). These numerical observations, appearing after the Brown and Roshko [17] experiment, shed new light on the intimate structure of turbulence, and one may say that today, research in turbulence is still challenged by these early observations. Later, in a turbulent wake produced in a soap :lm. Couder [23] observed structures — vortices and dipoles — keeping their coherence for long periods of time, and apparently forming the basic elements of the wake (see Fig. 2). These observations were largely con:rmed by the subsequent numerical and physical studies carried out on the subject [25,85]. McWilliams gave evidence of the formation of vortices, whose lifetimes are hundred times their internal enstrophy, a period exceeding by far any expectation based on traditional ideas of turbulence. Legras et al. [72] showed several long-lived structures, including a superb long-lived tripole in a forced two dimensional turbulent system (see Figs. 3 and 4). Coherent structures can often be visualized directly on the vorticity :eld, as long-lived objects of generally circular topology, with a simple structure, ‘clearly’ distinguishable from the background within which they evolve. Coherent structure is nonetheless a loose concept and attempts have been made to provide sharp de:nitions, useful for their systematic identi:cation prior to the characterization of their properties. There are several issues raised in proposing a

Fig. 1. Simulation of a two-dimensional jet showing the formation of long-lived coherent structures. Fig. 2. Two-dimensional Von Karman street, formed behind a cylinder dragged in a soap :lm.


9

Fig. 3. Typical vorticity :eld of a free-decay experiment, obtained several vortex turn-around times after the decay process starts. Fig. 4. Tripole obtained in a simulation of freely decaying turbulence.

de:nition of coherent structures. For single vortices, it may be diEcult to assume the internal pro:le be smooth since in some situations — during vortex merging for instance — vortices develop a complex intricate structure and their pro:le can be extremely corrugated, especially at high Reynolds numbers. Proposing a vorticity threshold criterium for identifying structures may also raise diEculties at high Reynolds numbers, since Helmoltz theorem stipulates the 9uid particles keep their vorticity level wherever they move. McWilliams analyzed these issues and proposed an empirical procedure of identi:cation of the vortices [86]. The vortex census proposed by him includes several steps: 1. identify extremas (using a threshold criterion), 2. identify interior and boundary regions (de:ning a threshold for the boundary and check it has a circular topology), 3. eliminate redundancy (checking the structures are separate), 4. test vortex shape (there should be no strong departure from axisymmetry). Another way of identifying structures is based on the use of Weiss’s criterion. This criterion is discussed in detail by Basdevant et al. [6]. Weiss criterion is based on the remark that the evolution equation for the vorticity gradient satis:es the following equation: d ∇! + A∇! = 0 dt

(15)

10


in which A is the velocity gradient tensor. The eigen values associated to A are calculated as the roots of the following equation: 2 = 14 (2 − !2 ) = 4Q ;

(16)

where is the strain. Then assuming the velocity gradient tensor varies slowly with respect to the vorticity gradient, Weiss derived the following criterion: in regions where Q is positive, strain dominates; the vorticity gradient is stretched along one eigenvector, and compressed along the other. In regions where Q is negative, rotation dominates, and the vorticity gradients are merely subjected to a local rotation. Thus, selecting strongly negative Q allows capturing regions occupied by stable strong vortices, and this o=ers a way for identifying coherent structures, which has the advantage of incorporating some dynamics. In Ref. [6], the hypothesis on which Weiss’s criterion relies is assessed; the relevance of the criterion for analyzing the dynamics of the 9ow is shown, although the criterion is strictly valid only near the vortex core and the hyperbolic points. Progress made to propose a more rigorous criterion, useful for handling dispersion phenomena, has recently been o=ered by Hua and Klein [55]. One may also mention a method, linked to the Weiss criterion, but based on the pressure, for identifying coherent structures [70]. The method is based on the remark that the Laplacian of the pressure is proportional to the Weiss discriminant Q. Therefore, by looking at the pressure :eld, one may identify regions dominated either by strong rotation or by high strains. To check internal coherence, an attractive method is to determine, in the frame of reference of the supposedly coherent structure [97], the scatter plot (also called the coherence plot), i.e. the relation between the vorticity ! and the stream function . For a stationary solution, vorticity is a function (not necessarily single valued) of the stream function. Thus, the scatter plot shows a well de:ned line for a stationary structure, and randomly distributed points if the structure undergoes strong temporal changes. The existence of well de:ned lines, which do not evolve for long periods of times, provides a direct visualization of the internal coherence of the so-called ‘coherent structures’. An example of a scatter plot, obtained in a decaying experiment [10], is shown in Fig. 5. Tracking such curves in a turbulent :eld may be a method for identifying coherent structures; however, since one must work in a frame of reference moving and rotating with the structure at hand, this technique is diEcult to implement in practice. Another technique for identifying coherent structures, based on wavelet decomposition, has been advocated by Farge [35] these last years. It consists of decomposing the :eld into orthonormal wavelet coeEcients, and selecting coeEcients larger than some threshold. Coherent structures correspond to the selected coeEcients, the rest forming the background. Interesting properties have been shown. In the cases considered, the background vorticity is found to be normally distributed, o=ering a favorable situation for theoretical analysis. Also much of the statistics of the :eld is captured by only a few wavelets, which may be interesting for computational purposes. Determining the stability of isolated vortical structures is a central issue and a diEcult theoretical problem even for the simplest morphologies. There are some mathematical results, partly reviewed by Dritschel [30]. Circular patches of uniform vorticity are nonlinearly stable; this is important to mention, because the stability of isolated circular vortices opens a pathway for the formation of coherent structures in two-dimensional turbulence. It also marks a di=erence


11

Fig. 5. Scatter plot (or ‘coherence plot’) obtained in a decaying experiment by Benzi et al. The presence of well de:ned branches reveal the existence of coherent structures, characterized by the same !– relationship. Fig. 6. Several phases of the axisymmetrization process, transforming an initially elliptical vortex into a circular one.

between two and three dimensions, since the three dimensional equivalent of a two-dimensional vortex — the Burgers vortex — , is observed to be unstable with respect to three dimensional perturbations. In two dimensions, vortices are intrinsically stable, and the emergence of coherent structures is easier than in 3D. Elliptical vorticity patches, however, are not stable. The physical mechanism at work is better seen on initially elliptical vortices, as shown in Fig. 6 [57]. Due to the di=erence in rotation rate between the center and the edges, the edges of the vortex is subjected to :lamentation, and after a while, the initially elliptical vortex adopts a circular shape. The overall picture of the monopolar stability has been con:rmed experimentally [40]. More complicated morphologies are mostly out of reach of theoretical analysis, and one must rely on experimental evidence to determine whether a structure is stable or not. In this framework, isolated dipoles and tripoles (such as the one of Fig. 2) are reputed to be stable while pairs of like sign vortices are unstable against perturbations leading to vortex merging. The internal vorticity pro:les of the coherent structures are mostly controlled by the initial conditions and therefore can hardly be universal (Fig. 7 shows such example of such a pro:le, extracted from [85]). In contrast, the shapes of the coherent structures (more speci:cally the vortices) tend to be circular after a few turnaround times, under the action of the aforementioned axisymmetrization process [87]. The dynamics of a small number of vortices has been studied analytically, numerically and experimentally for several decades. A classi:cation of vortex interactions, involving up to four elements, has been o=ered, in view of providing information helpful for the description of larger vortex systems, prone to sustain fully turbulent regimes [30]. In many cases (i.e. as distances between vortex cores exceeds by far the vortex sizes), the dynamics of vortex patches can be described by using point vortex approximation. The point vortex approximation has been used in a number of cases to study basic phenomena, such as the inverse cascade of energy and the free

12


Fig. 7. An example of an averaged pro:le of a set of vortices, with negative vorticity, in a freely decaying system. Fig. 8. A dipole formed in a strati:ed 9uid system, loosing 9uorescein behind it as it moves forward.

decay of turbulence [126,10]. The simplest structure involving more than one vortex is the Lamb dipole; the pointwise vortex approximation can be used to estimate their propagation velocity; however, by using this approximation, one does not accurately reproduce the dynamics of the saddle point, located at the rear of the dipolar structure, and across which leaks of vorticity or dye (if the vortices carry dye) can occur [134]. Several aspects of the dipolar structure have been investigated in detail in Refs. [38,39] (Fig. 8). Clearly, some vortex interactions cannot be represented by using pointwise vortices. An example is the merging of like sign vortices, which plays a fundamental role in the free decay of two-dimensional turbulence, since it drives the formation of increasingly large structures, the most striking characteristics of freely decaying regimes. Vortex merging has been studied along di=erent approaches in the past, and we may say today it is a subtle, rather well documented process. It is subtle because it involves a broad range of scales: as early numerical simulations suggested, vorticity :laments, rapidly lost by the action of viscosity, must be produced for merging to occur. The phenomenon thus involves an interplay between small and large scales. In the particular case of the merging of two identical vortices, it is remarkable that we now have an exact solution of the Navier–Stokes equations [136]; from a mathematical point of view, one thus may consider the problem is solved. However, it is not straightforward to dig out the physical content of the solution, and it is worth considering several concepts devised along the years, for discussing the phenomenon on a physical basis. First, why do vortices merge? Melander et al. [87] propose an instability mechanism, they called “axisymmetrization process” (because the merging will axisymmetrize the system formed by the vortex pair), driven by vorticity :lament formation. They showed that whenever two like-sign vortices are separated


13

Fig. 9. The six steps of the merging process shown in the paper of Melander et al. (1988). Fig. 10. Experiment, carried out in a plasma, showing the detail of the merging process.

by less than a critical distance, they merge. Far from each other, they do not interact; as they get close, they strain each other, developing thin :lamentation along their edges, and as a result — for energetic or mechanical reasons — the cores get closer and closer, eventually forming a single structure (see Figs. 9 and 10). Merging is completed by the smoothing action of viscosity. The notion of critical distance is not straightforward to understand, but it represents well the observations (see Fig. 11). For equal vortices, of circular shapes and uniform vorticity, the critical distance is estimated to be 1.6 time the radius of each vortex. In practice however, merging of unequal vortices is more frequent than the one of equal vortices. One may also mention an appealing description, based on statistical premises, viewing vortex merging as the expression of a maximum entropy principle. I will come back to this theory later. On the experimental side, we now have detailed analysis of the merging process: by using magnetically con:ned columns of electrons, Driscoll et al. [36] con:rmed many of the features discussed above, in particular the existence of a critical distance; this notion appears through a plot showing that the merging time tends to diverge as the intervortical distance approaches a critical value, consistent with earlier estimates. Another process involved in the decay phase is vortex break-up, which causes the elimination of the weakest vortices, under the action of the strain exerted by the others. This process is also rather well documented. Idealized cases, with isolated patches of uniform vorticity have been worked out by Moore and Sa=man [93] and Kida [63]. Legras and Dritschel [73] showed

14


Fig. 11. Experiment showing the divergence of merging time as the intervortical distance increases (by Driscoll et al., 1991). Fig. 12. Numerical experiment showing that a vortex, subjected to a strain, undergoes distortion, and, above some threshold, break-up.

that vortex breaking is controlled by a critical ratio of the strain rate over the core vorticity of the vortex, and this ratio was found weakly dependent on the vorticity pro:le and, provided the Reynolds number is large, independent of the viscosity (Mariotti et al. [80]) (see Fig. 12). The main lines of the analysis have been con:rmed in physical experiments [109,108,132] (see Fig. 13). Recent progress on the subject has been achieved by Legras et al. [74]. One may also mention an analytical study of the e=ect of the background strain on a population of vortices, well con:rmed numerically by Jimenez et al. [58]. Other processes are involved in turbulent systems, as pointed out by McWilliams [86]: lateral viscous spreading of the vorticity, translation through mutual advection, and higher order interactions between vortices; all contribute to reinforce the diversity of two-dimensional dynamics, and the reader may refer to the references listed in the present review, which include descriptions of these phenomena (Figs. 14 and 15 depict two steps of the decay of a population of 1000 vortices showing the formation of increasingly large structures, according to McWilliams [86]).


15

Fig. 13. Experiment, carried out in a strati:ed 9uid, showing the break-up process.

Fig. 14.

Fig. 15.

16


4. Statistics of the coherent structures in decaying two-dimensional turbulence So much emphasis has been put on the free decay of two-dimensional turbulence that it sometimes stands as the two-dimensional turbulence problem. This is unjusti:ed in my opinion; we will see in this review that freely decaying systems de:ne a speci:c physical situation, which shares common features but also displays strong di=erences with the forced case. The main di=erence comes from the fact that coherent structures spontaneously pop up in freely decaying systems, whereas they can be inhibited or destroyed in forced systems. In three dimensions, as far as statistics are concerned, there is essentially no distinction between forced and free situations; in two dimensions, this is not so, and in general, a distinction must be made. Brie9y stated, the problem of the free decay of two-dimensional turbulence is to determine how abundant populations of vortices freely evolves with time at large Reynolds numbers. The vortices may initially be spread at random in the plane, or form a perfect crystal, have Gaussian pro:les or be of uniform vorticity. In all the documented cases, the system evolves towards a state where coherent structures dominate the large scales of the 9ow, and apparently universal features show up. The crude aspects of the process were known in the sixties, but the :rst detailed analysis was performed in 1990, by McWilliams [86]: McWilliams succeeded to disentangle the few elementary mechanisms at work: he showed that in the decay phase, the weakest vortices are destroyed by the break-up process, while the strongest ones merge with other partners, of weaker or comparable strength; these mechanisms generate a re:nement of the vorticity distribution (see Fig. 16). The vorticity distribution thus sharpens as the system decays, but in this process, the most probable vorticity level was observed to remain constant. Driven by a sequence of merging events, the system gradually and unavoidably evolves towards a state where vortices become fewer and larger. Throughout the decay regime, thin vorticity :laments

Fig. 16. Vorticity distributions plotted at various times, from t = 0 and 40; the distributions change, but the location of the maximum does not appreciably shift during the decay process (same paper as above).


17

keep being produced, either during merging events, or in the break-up process, developing a vorticity background. At moderate Reynolds number (such as the one of Ref. [86]), they rapidly disappear from the landscape, but at larger Reynolds number, they may nucleate additional small vortices, driven by Kelvin–Helmholtz type instability. Eventually, in current numerical and physical experiments, one typically reaches a state where a pair of large counter-rotating vortices are left in the system, forming the so-called “:nal dipole”. The reader may refer to Ref. [86] for more quantitative aspects. The problem of the free decay of turbulence is rich, and several issues are embedded in it. What is the structure of the small scales, i.e. those smaller than the initial injection length? Are general cascade concepts relevant to discuss them? What is the ultimate state of the 9ow, appearing at the end of the decay process? Here, I restrict myself to presenting the dynamical evolution of the large scales of the system, i.e. those dominated by coherent structures, a problem on which much has been learnt these last years. The other issues will be discussed later. The early theoretical analysis of the decay problem, proposing a statistical description, can be found in a remarkable paper by Batchelor [9]; the paper contains the important result that the decay process must be selfsimilar in time, a result which in9uenced all the theoretical approaches developed on the problem. The theory, which does not particularly specify the range of scale it addresses, is expressed in the spectral space. Batchelor’s approach has been reinterpreted in the real space, and for the large scale range, by Carnevale et al. [21]; this interpretation is interesting to consider for o=ering a consistent presentation of the free-decay process. If one constructs an expression for the number of vortices per unit of area, one must write the following expression: N = f(E; t) ; where t is time and E is the energy per unit mass and area, the only invariant surviving, at large Reynolds numbers, in neutral (i.e. with zero mean vorticity) populations [98]. Dimensional analysis further leads to the result that the density of vortices decreases as ∼ E −1 t −2 [9]. The same argument applied to the vortex size, a, and the intervortex spacing, r shows they must be proportional to time, implying the system expands. This approach has been considered as a cornerstone for a long time. However, with the advent of powerful computers, unacceptable deviations appeared and we had to move towards other paradigms. The :rst deviations were revealed by McWilliams [86] analysis, mentioned in the beginning of the section. The paper is a numerical study of a population of one thousand Gaussian vortices, located at random in a square with periodic boundary conditions. The number of vortices, the size and the intervortical distances were measured and the results are shown on Figs. 17 and 18. One could see well de:ned power laws, but the exponents were found clearly incompatible with Batchelor’s theory. The exponent for the vortex density for instance, was found equal to −0:7, a value incompatible with Batchelor’s expectation (leading to −2); a similar comment can be made for the size (growing as t 0:2 instead of t) and the mean separation (increasing as t 0:4 instead of t). To account for this observation, Carnevale et al. [21] o=ered the idea that another invariant must be dug out, and this gave rise to the ‘universal decay theory’ [101]. More speci:cally,

18


Fig. 17. Decay law for the number of vortices, found by McWilliams [86]; the results show evidence for the existence of a power law, with an exponent close to −0:70. Fig. 18. Power laws obtained for the vortex density , size a, intervortical distance r and extremal vorticity .

they proposed to take the extremum vorticity of the system !ext as the missing ghost invariant. There is no rigorous justi:cation for this assumption, but a physical plausibility, supported by numerical observation. Indeed in the inviscid limit, the maximum vorticity level, as any other levels, is conserved. But in the problem we address here, it is not clear the Euler equations provide an acceptable approximation. The constancy of !ext is physically justi:ed by the fact that the strong vortices undergo two types of situations; either they wander in the plane or they merge; in none of these situations, their internal circulation has an opportunity to substantially decrease. Therefore, it is plausible, on a physical basis, to consider the extremum vorticity as a constant. This argument is supported by the observation that the most probable vorticity level remains (approximately) constant during the decay phase [86]. The next step is to estimate various quantities of the problem. The following equations are written 2 Z ∼ !ext a2 ;

(17)

2 a4 E ∼ !ext

(18)

in which Z is the enstrophy per unit area and E is the energy per unit mass and area. The estimate for the energy neglects logarithmic contributions; this delicate omission is discussed in some detail in Ref. [139]. Thus, using the above relations, along with the constant extremum vorticity assumption, and further hypothesing power laws, Carnevale et al. [21] obtained the following laws for the density of vortices , the vortex radius a, the mean separation between vortices r, the velocity u of a vortex, the total enstrophy Z, and the kurtosis Ku of the vorticity


distribution: −2

∼l

−!

t T

;

a∼l

!=4

t T

19

;

√ u∼ E ; r ∼ l(t=T )!=2 ; −!=2 !=2 t t −2 ; Ku ∼ (19) Z ∼T T T in which length l and time scale T are de:ned by √ −1 −1 l = !ext E; T = !ext : (20) The exponent ! is not determined by this theory, but, once it is :xed, all the power law exponents of the problem are set. Numerical studies, either of the full Navier–Stokes equations or of point-vortex models, have consistently obtained power laws, with values of ! ranging between 0.71 and 0.75 [21,139,86]. Concerning the physical experiments, early investigations [24,53,46] revealed the essential qualitative features of the decay process, but could not provide precise determinations of whatever exponent involved in the problem. Owing to the progress made in digital image processing, the situation changed in the last decade and now, accurate measurements are currently performed on two-dimensional decaying systems [102]. One may refer to a study of decay regimes performed in electromagnetically driven 9ows, which yielded reliable data, in good quantitative agreement with the numerical simulation [47]. Examples of such data are shown in Figs. 19 –21. In this case, the decay process stretches over a period of time hardly exceeding 25 s, representing 10 initial turnaround times. Despite this limitation, scaling regimes take place rather neatly, and the related exponents could be quite accurately determined. Agreement with the aforementioned numerical studies was obtained, and this reinforced the consistency and robustness of the “universal decay theory” framework. ! thus appeared as a crucial exponent and several theoretical assaults have been given to determine its value [116,49,133,1]. Pomeau proposed a derivation that yields ! = 1, arguing for lowering corrections [116]. On the other hand, using a probabilistic method to describe the motion of vortices in an external strain-rotation :eld, it has been suggested that the value of ! depends on initial conditions [49]. In a related context, the 2D ballistic agglomeration of hard spheres with a size–mass relation mimicking the energy conservation rule for vortices, the value ! = 0:8 is derived under mean-:eld assumptions [133]. Further, in another possibly related context, that of Ginzburg–Landau vortex turbulence, the value ! = 34 has been proposed [1]. Recently, progress was made on the characterization of the decay process [47]. The mean square displacements v2 of the vortices, the mean free path , the collision time $, have been measured in a physical experiment, and the following power laws have been found (see Figs. 22 and 23):

v2 ∼ t

with v ∼ 1:3 ± 0:1;

$ ∼ t 0:57±0:1 : (21) ∼ t 0:45±0:1 ; The vortices thus do not undergo Brownian motion in the plane, but possess a hyperdi=usive motion, with an exponent equal to 1.3. On physical grounds, it is clear that this dispersion law,

20


Fig. 19. Three steps of the decay process observed in a physical experiment, in a thin layer of electrolyte, where the vortices are driven by electromagnetical forces.

and the related exponent (we call ) are linked to the decay problem. The authors of Ref. [47] argue that a relation between the mean square displacement exponent and the decay exponent, !, holds; it is expressed by the following formula: = 1 + 34 ! :

(22)

This law is obtained by using standard relations of kinetic gas theory (assumed to be applicable for the problem at hand). In Ref. [47], the above relation appears well supported by the


21

100

N(t* ) 10

N r

10

a

1

1

t* [s]

2

10

3

4

5

6

7 8 9 10

t* [s]

Fig. 20. Number of vortices with time obtained in the physical experiment. Fig. 21. Additional data, showing the vortex size, and the intervortical distance with time, in the physical experiment.

10

3τ 10

σv 2

λt r

1

λ 1

0.1 1

t* [s]

0

1

t* [s]

0

Fig. 22. Temporal evolution of the squared averaged distance between vortices, obtained in a physical experiment. Fig. 23. Temporal evolution of various quantities, characterizing the evolution of a system of vortices: collision time, mean free paths.

experiment. According to this, both the dispersion and the decay problem seem to be the two faces of the same coin and one may ask which of them is the more amenable to theoretical understanding. The presentation emphasizes on exponents, since their existence was implicitly admitted in most of the recent work made on the problem. However, one may consider that the situation is not fully settled; recent work revealed log periodic oscillations on the temporal decay of the number of vortices [48]; the authors further argue complex exponents must be used to represent the underlying processes. On the other hand, numerical results obtained by Dritschel [31], using

22


vortex surgery technique, challenged the existence of power law behavior. There are several consequences of this result, one of them being that power laws may break down at extremely large Reynolds number. On the experimental side, the “universal decay theory” is generally considered as :tting well the experimental observations; however, one must mention a recent work [135] which suggests that in the experiment, the walls may have a sizeable e=ect in the determination of the exponents. Concerning the decay problem, we are thus left at the present time with an elegant phenomenological theory (“universal decay theory”) which turns out to represent consistent sets of numerical and experimental observations. In this framework, despite several attempts, the fundamental exponent governing the process is still unexplained. On the other hand, moving further away from phenomenology would probably be desirable, but at the moment, solving the decay problem from :rst principles appears as a formidable challenge theorists do not seem to rush out to tackle. 5. Equilibrium states of two-dimensional turbulence As we discussed above, freely decaying two-dimensional turbulence evolves towards a state where eventually large structures arise, coexisting with a featureless turbulent background, formed by a collection of short lived vortex :laments. This has been shown for the turbulence decay problem, involving initially abundant vortex populations, and this is also true for systems initially composed of a few structures. Examples are displayed on Figs. 24 –26, or can be found in [144,2,24,84 –86,11,134,53,137]. One sees systems undergoing complicated evolutions, until eventually a steady state takes place, characterized by the presence of a few structures. The :nal patterns are not universal — their shapes sensitively depend on the boundaries and the initial conditions. The question is whether theory can account for them. From the observations, it is appealing to consider that the :nal state is a “vorticity melasse”, resulting from the mixing of the many elementary vorticity patches de:ning the initial state; along this line of thought, one is tempted to develop an analogy with thermodynamics, so as to interpret the :nal state as an equilibrium state, which would maximize some entropy. This intuitive idea is, however, not straightforward to substantiate. It has been :rst formalized by Onsager, for point vortices systems, i.e. in a context where the analogy could be worked out rigorously [107]. The Hamiltonian structure of the problem and the presence of a Liouville theorem (given by Helmhotz theorem) allowed to calculate equilibrium states, in the same way as in statistical physics. For the problem at hand, the canonical variables are the cartesian coordinates of the positions of the point vortices. Onsager found various possibilities, depending on the initial conditions. Among them, negative temperature states, which arise as a consequence of the :niteness of the phase space; those states are in form of clusters of like signed vortices, coexisting with regions where weak vortices are ‘free to roam practically at random’. In this problem, entropy is gained by randomizing weak vorticity regions, and clusters are formed so as to conserve the total energy. This theory brought an illuminating way to account for the spontaneous formation of coherent structures in 2D 9ows, as repeatedly revealed experimentally and numerically. Onsager’s theory conveyed new concepts in hydrodynamics and inspired, for four decades, a considerable number of contributions [69]. The theory was further


23

Fig. 24. Evolution of a system of vortices obtained by Matthaeus et al. (1991), showing the formation of the “:nal dipole”. Fig. 25. Evolution of a vorticity strip towards the formation of a chain of vortices (after Sommeria et al., 1991).

extended to magnetohydrodynamic 9ows and plasmas, and attempts were made to use it for the three-dimensional problem. This interest is probably motivated by the possible relevance of this approach to hydrodynamic turbulence but also — as quoted in Kraichnan and Montgomery’s review [69] — by the fact that we deal with a “bizarre and instructive statistical mechanics”, which makes it interesting in its own right. There is a number of theoretical issues involved in Onsager’s theory, related to the ergodicity assumption, the existence of a critical point, the pair collapses, the high and low wave-number limits, and they are reviewed in detail in Ref. [69]; also we may mention an excellent, more recent, pedagogical review by Miller et al. [89]; we comment here on the point vortices approximation, which a=ects the practical interest of the theory — I mean its relevance to explain observations. As noted by Pomeau [115] point vortex systems may provide a theoretical framework for super9uid turbulence, where vorticity is quantized into :laments, 1 A in diameter, with a prescribed circulation, but for ordinary 9uids, the discrete approximation is constraining in the sense that it precludes the possibility to compare the theory with the experiment. The problem

24


Fig. 26. Evolution of a vorticity pattern towards the formation of a dipole, in a periodic box (after Segre et al.).

of working with discrete vortices is that one has an in:nite number of possibilities to construct a given initial continuous vorticity :eld [92]; as a consequence, by using Onsager’s theory, one gets an in:nity of possible equilibrium states; there is thus no way to make a quantitative contact between this approach and ordinary 9uids, and a qualitative comparison is in practice extremely limited. The desire to cross the bridge between discrete and continuous vorticity :elds has stimulated a number of works, well reviewed by Kraichnan and Montgomery Refs. [69] and Miller et al. [89] and more recently by Brands [16] and Segre et al. [127]. Assuming negative temperature, and a two-level initial vorticity :eld, Montgomery and Joyce [91] proposed to de:ne the vorticity as a local average of the pointwise vorticity :eld, under the assumptions that the vortex number density slowly varies with the position. The set of equilibrium states found under this assumption is governed by the equation ∇2 + %2 sinh(&‘a ) = 0 ;

(23)

where is the equilibrium stream function, % is a constant and & is a free parameter, corresponding to a temperature. Interestingly, this theory leads to a universal relation between the locally averaged vorticity and the stream-function, in form of a sinh function. This sinh function turns out to :t well the actual ultimate states observed in a number of experiments. Another approach has been to work in discrete truncated Fourier spaces. The truncated system is Hamiltonian and has a Liouville theorem, so that one may calculate statistical equilibrium states, using Fourier coeEcients as the conjugate variables. In this approach, the equilibrium state is found as the gravest mode (i.e. with the lowest wave-number) compatible with the boundary conditions. Formally, at equilibrium, the vorticity is found proportional to the stream-function. This proposal has been called the ‘minimum enstrophy principle’ since, in this problem, the equilibrium state turns out to minimize the enstrophy at constant energy [76]. A related concept is the ‘selective decay’, stipulating that freely evolving two-dimensional systems decrease their


25

enstrophy at a rate much higher than the energy. The theory is appealing in its simplicity, and qualitatively reproduces well the trends observed in decaying systems. However, the linear relationship between vorticity and stream-function is generally not observed. In many cases, sinh functions or hyperbolic functions characterize the scatter plot of the :nal states. The theory can also lead to unphysical predictions. The minimum enstrophy approach is discussed in several places and the reader can refer to Kraichnan–Montgomery’s review [69] to have a detailed account of this work. Ten years ago, a statistical theory, tackling the discretization problem in a di=erent way, has been worked out [88,120]. The key step has been to introduce a mesoscopic scale over which the stream function could be treated as uniform, and a local determination of the statistical ensembles could be achieved. In this context, the authors [88,120] calculated analytically the maximum entropy states, under the constraints of conserving the energy and the vorticity distribution, i.e. only the invariants which survive after coarse graining. The statistical theory provides, as a result, a relation between the averaged vorticity :eld !(x) and the stream function (x) which fully de:ne the equilibrium state. At variance with the previous theories, there is no free parameter in the solution; except in special cases, the form of the relation vorticity — stream function (the so-called scatter plot), requires an iterative procedure to be determined. Here, we present the essential steps of the calculation leading to the result. The :nal equilibrium for a distribution of discrete vorticity levels ai of vorticity, is found by maximizing a mixing entropy de:ned by S=− pi (x) ln(pi (x)) dx ; i

where pi (x) is the probability of :nding the level ai at the location x. By introducing the Lagrange parameters & and )i , respectively, related to the conservation of the energy E and initial distribution of vorticity Ai , and prescribing that the :rst order variation of S vanishes, one :nds that the equilibrium vorticity :eld !(x) and stream function (x) are given by exp(−&ai (x) − )i ) ai pi (x) = ai !(x) = ; (24) Z( ) i i Z( ) being the normalization factor of the probability distribution. To compute the equilibrium spatial vorticity distribution, one needs to solve the following set of equations: 1 ! dx ; (25) E= 2 Ai = pi (x) dx ; (26) != − O ;

(27)

associated with the boundary condition = 0. In the above equations, the set of Ai , prescribed by the initial conditions, is known. It is possible to :nd the structure of the solutions to this system, in con:gurations of great simplicity. For instance, the case of two-dimensional 9ows in a closed rectangular domain (with

26


Fig. 27. Final states obtained in a plasma experimented by Fine et al. They are in the form of regular lattices of vortices. The dissipation is low enough to investigate thousands of turnaround times, i.e. more than the most conservative numerical simulations.

periodic or rigid boundary conditions), with initial uniform vorticity can be solved. The solution is [131] ) sinh(* + * ) != − O = 1 + ) cosh(* + * ) in which all the parameters, i.e. *; * ; ) can be calculated using the initial conditions (actually the task is not simple). There is no free parameter. It is also worth noting that the sinh Poisson is recovered in the limit of dilute vorticity. It has thus been suggested the sinh Poisson is a particular case of a more general framework. Nonetheless, the resolution of the system in more realistic cases, along with the determination of the constants in situations such as the previous one, are not straightforward and for years, the numerical pathway for obtaining equilibrium states was ineEcient; this diEculty was solved by the advent of ingenious numerical procedures [140]; the procedure devised by Whitaker and Turkington [140] is iterative and involves the following steps: 1. Compute a set of Lagrange parameters from the vorticity :eld and the related stream function form (25) and (26). 2. Determine the corresponding probability distribution according to Eq. (24). 3. Solve (27) for the new calculated vorticity :eld. 4. Return to :rst step. The procedure is iterated until the convergence of all parameters is completed. The reader may refer to [59] for examples of applications (other examples can be found in [26,89]). In practice, the result is insensitive to the precise choice of n. This theory could be confronted by experiments; it turned out to provide, in several case, predictions agreeing, even at some quantitative level, with the observation: for example, the cases of the jet and the mixing layer, yields good consistency between the theory and the simulation [130,131]. The theory predicts the existence of phase transitions, involving symmetry breakings, as the energy or geometrical factors are varied. Some of them have been found experimentally, consistently with the theory, in a two-dimensional circular mixing layer, [27]. The theory predicts the merging of two-like sign vortices, which is observed, provided they are closer than some critical distance (we will come back to this point later); some deviations have been pointed out in a series of experiments using plasmas [56], while acceptable agreement was shown in another case [82]. It would, however, be incorrect to consider that agreement is the rule. In several cases, it appeared that the theory misses some important features found experimentally. In [37], some observed states — such as vortex crystals of Fig. 27 — appear incompatible with the theory, as worked out in the original papers. An experimental study, done on the decay of


27

Fig. 28. Final states obtained in a physical experiment, and detailed comparison with the theoretical predictions of Robert-Sommeria.

systems of vortices, and comparing the prediction with the observation on a quantitative basis, obtained discrepancies between theory and experiment — in some cases at qualitative level [82,60] (see Fig. 28). A striking confrontation has been carried out by Segre and Kida [127]. Oscillatory states have been found as :nal states in a number of situations (examples are shown in Fig. 29). This is incompatible with the theory, which expects the :nal state to be stationary in all cases; in the same study, it is shown that the :nal state is not uniquely determined by the vorticity distribution and the energy level: two di=erent two-levels vorticity patterns, with approximately the same energy have been observed to lead to di=erent :nal states; this is also incompatible with the theory. Whether we like it or not, the existence of situations where the theory fails to predict the ultimate state obviously generates a number of questions. There are technical issues, but the most critical question is ergodicity. Is is legitimate to assume ergodicity? Does the system investigate all the accessible phase space with equal probability, or can it be trapped for long in a part of it? Is the notion of equilibrium state relevant for interpreting experiments? These points have been discussed in several papers in considerably more detail and in more technical terms than herein (see [69,89,115,16]). The issue can be illustrated by the two like-sign vortices problem; in this case, statistical theory (I could say all statistical theories) predict the vortices to merge, whereas experiments show that above a critical distance, the vortices remain separated for ever (‘ever’ means thousands of turn-around times); in the framework we consider here, we may interpret this observation as an inhibition to reach an equilibrium state. Why can it be so? Pomeau [115] pointed out that the origin of randomness in 2D 9ow is far weaker than in ordinary statistical systems: as the system evolves, at :nite Reynolds number, the dynamical role of the vorticity :laments gets weaker, whereas the large structures tend to form stable circular regions. The small scale 9uctuations thus tend to collapse and consequently, the possibility to reach an equilibrium state, as in ordinary statistical mechanics, becomes questionable. One may ask whether increasing the Reynolds number favors ergodicity. At the moment, there is no evidence it can

28


Fig. 29. Oscillatory states found as the :nal states in simulations initiated with two-levels vorticity patterns. The right :gures are the vorticity distributions and the scatter plots. The nonstationary nature of the state is made visible by the thickness of the curve in the scatter plot.

be so. It appears that in two-dimensional systems, there is no mechanism ensuring ergodicity in general, and assuming ergodicity seems to be in too many cases a misleading premise. A discussion on ergodicity from a more dynamical prospect has been given by Weiss and McWilliams [138]; these authors underline the existence of dynamical pathways, which control the decay phase. After the decay phase is achieved, the 9uctuations have declined, and the system ceases to evolve. According to this view, the :nal dipole is seen as the completion of a selfsimilar process, rather than an equilibrium state resulting from the mixing of the vorticity patches. The relevance of this approach has been underlined in a physical experiment [82]. Note :nally that one can :nd cases where ergodicity does not hold [138]. Several criteria have been o=ered to determine the validity of the ergodicity premise [92,16]. One may perhaps propose another one, based on physical considerations, and expressed in terms


29

of a dimensionless parameter P de:ned by P = E=L2 Z0 ; where L is the box size con:ning the 9ow and Z0 is the initial enstrophy; this parameter is the natural dimensionless quantity one may introduce to discuss the problem (see for instance [115,89,19]). As P is small the system must gain much entropy to reach an equilibrium state, and this involves substantial mixing, diEcult to achieve in two dimensions. On the other hand, when P is high, the system does not need to gain much entropy to reach equilibrium, because the energy level is high, and the number of accessible states is low; thus in this case, moderate mixing may be suEcient for the system to approach an equilibrium state and we may expect that statistical theory provides useful predictions. These rough ideas have been illustrated for vortex arrays [83]. Just to :x the ideas, one :nds that deviations appear as P gets smaller than 5 × 10−3 ; this may de:ne a cross-over between two situations, one favorable for equilibrium states to be reached, the other unfavorable. A possible paradox of this situation is that statistical theory works better when the entropy tends to become irrelevant. Recently, several modi:cations have been brought to improve and extend the original theoretical framework. The idea is to de:ne bubbles within which ergodicity holds and outside of which the 9ow is deterministic [20]. A similar idea was proposed earlier by Driscoll, in view of proposing a more realistic version of minimum enstrophy theory [56]. This adjustment allows to improve the consistency between theory and experiment, at the expense of introducing a substantial complication. To conclude, one may say that at the moment, the statistical approaches have not been proven yet to o=er a more than qualitative framework for the interpretation of experimental observations. It seems that we are in a situation where unresolved issues are hanging, and the theory has to face formidable diEculties to make further progress. Do these theories correctly indicate the direction of the arrow of time? This is not ascertained in general. Nonetheless, the conceptual framework is appealing and certainly deserves stronger interest than now in the 9uid dynamics community. 6. Inverse energy cascade The concept of the inverse energy cascade traces back to Kraichnan [67] who :rst proposed that energy and enstrophy can cascade in two dimensions, a possibility often ruled out before him. In two dimensions, one has the conservation of the vorticity of each 9uid parcel in the inviscid limit. These conservation laws imply the existence of two quadratic invariants: the energy and the enstrophy. These two constraints led Kraichnan to propose the existence of two di=erent inertial ranges for two-dimensional turbulence [67]: one with constant energy 9ux extending from the injection scale toward larger scales and one with constant enstrophy 9ux extending from the injection scale down to the viscous scale. The :rst one is known as the inverse energy cascade, and the other as the enstrophy cascade. The argument leading to this double cascade picture is better given in the Fourier space because the relation between energy and enstrophy is straightforward; it unfortunately does not :nd a simple expression in the real space, making it hard to explain physically in accurate terms.

30


The argument can be found in several places [77] and I restrict myself to provide the context. Let us consider a triad of wave numbers k; p; q, associated to energy densities E(k); E(p); E(q), and enstrophy densities Z(k); Z(p); Z(q); the interaction conserves the energy and the enstrophy, and one must have the following relations: E(k) = E(p) + E(q) ; k 2 E(k) = p2 E(p) + q2 E(q) :

(28)

From the study of the above equations, it can be shown that energy is transferred preferentially towards small wave-numbers while enstrophy is transferred preferentially to large wave-numbers [99]. The argument must be repeated to successive triads to draw out the conclusion that energy 9ows towards large scale while enstrophy 9ows towards small scales (see Ref. [77] for a detailed presentation). This is the double cascade process proposed by Kraichnan. In this description, the only invariant involving the vorticity is the enstrophy. It is assumed that the others, such as the set of Zn introduced in Section 2, are irrelevant, but to-date no justi:cation is provided for this assumption. This assumption is satis:ed in the low wave-number range, simply by the fact that the conservation laws dealing with vorticity are irrelevant in this range. The situation in the high wave-number range has been discussed by Eyink [33]. In this section, I focus on the inverse energy cascade. For the inverse cascade, Kraichnan predicted the existence of a selfsimilar range of scales in which the energy spectrum scales as k −5=3 . Its form reads E(k) = C 2=3 k −5=3 :

(29)

C

In which is the Kolmogorov–Kraichnan constant. In Kraichnan’s view, this cascade cannot be stationary: a sink of energy at large scales is required to reach a stationary state and, in a pure two-dimensional context, there is no candidate. Kraichnan conjectured that in :nite systems, the energy will condensate in the lowest accessible mode, in a process similar to the ‘Einstein–Bose condensation’. We will come to this process later. It remains that, as far as the inverse cascade is concerned, Kraichnan’s view leads to considering it as a transient state, and in this respect, this energy cascade clearly di=ers from the three-dimensional case. It has been realized recently that there are physically realizable ways to provide an energy sink at large scales. By working with thin 9uid layers over 9at plates, or using magnetic :elds, one introduces a friction e=ect which can remove the energy on large scales. The friction e=ect, external to the two-dimensional framework, can be parametrized by adding a term proportional to the velocity in the Navier–Stokes equations, in the form −&u, where & is a parameter characterizing the amplitude, is the 9uid density and u is the local velocity; in the geophysical context, such an additional term is called the ‘Rayleigh friction’. By using Kolmogorov type estimates, and comparing the nonlinear terms and the linear friction in this equation, one generates a new scale, whose expression reads: lD = (=&3 )1=2 :

(30)

This scale — we may call it the ‘dissipative scale’ in a sense similar to the Kolmogorov scale in three dimensions — represents a balance between the transfers across the cascade and the transfers towards the energy sink. At scales smaller than lD , the energy is essentially transferred


31

Fig. 30. Energy spectrum by Smith and Yakhot, for the inverse cascade.

through a cascade process while at scales larger than lD , it is essentially burned. Therefore, we have here a way to extract energy at large scales, and this stabilizes the cascade. The inverse cascade, as a transient process, was soon observed in several numerical simulations. The early observations were made by Lilly [78] were suggestive, and further simulation of Frisch and Sulem [42], led to :rst convincing con:rmations of the spectral law. Later work consistently con:rmed these observations; in particular one may mention the simulation of Smith and Yakhot [141,142], who displayed the expected spectral law over two decades, along with several crucial characteristics of the phenomenon, such as the constancy of the energy transfer in the inertial range (see (31)) (Figs. 30 and 31). From these studies, the Kraichnan Kolmogorov constant could be measured, and the best estimate to-date, provided by Smith and Yakhot, leads to C ≈ 7 :

(31)

In the physical experiments, the strategy has been to inject at the smallest available scale so as to favor the development of a broad inertial range. The :rst observation of the inverse cascade in such conditions traces back to Sommeria [129]; in this experiment, two-dimensionality is maintained by applying a magnetic :eld. He observed Kolmogrov type spectra by probing the velocity :eld along a line of electrodes, therefore, supporting Kraichnan’s conjecture [129] (see Fig. 32). In this case, at variance with the aforementioned numerical studies, the cascade, stabilized by a magnetohydrodynamic friction is a stationary process. Recently, Paret et al. [111], by using thin strati:ed layers of electromagnetically forced electrolytes, observed an inverse energy cascade with a neat k −5=3 scaling (see Fig. 33); the measurement of the full

32


Fig. 31. Energy transfer by Smith and Yakhot, in the inverse cascade.

velocity :eld moreover allowed to check statistical isotropy (see Fig. 34) and homogeneity. They measured a Kolmogorov constant consistent with numerical estimates. Here again, the cascade was made stationary by the action of a friction exerted onto the 9uid by the bottom plate. Finally, it is interesting to mention a recent observation of the inverse cascade, using soap :lms [122]. In this experiment, the authors could observe the entire process, i.e. both the inverse energy and the direct enstrophy cascades (see Fig. 35). These observations were further extended by a 20482 numerical simulation by Bo=eta et al. [12], in which a linear forcing is added so as to force statistical stationarity. In particular, the Kolmogorov relation mentioned in Section 2 S3 (r) = 32 r could be accurately obtained. This is shown in Fig. 36.


33

(a)

E(k,t)

10

final

1

transient

0.1

initial

0.1

-1

k / 2π (cm )

1

Fig. 32. Energy spectrum determined along a linear array of electrodes in a magnetohydrodynamic experiment. Fig. 33. Energy spectrum obtained in a physical experiment where the 9ow is driven by electromagnetic forces.

Fig. 34. Two-dimensional energy spectrum obtained in a physical experiment where the 9ow is driven by electromagnetic forces.

34


Fig. 35. Two-dimensional energy spectrum obtained in a soap :lm. Fig. 36. Kolmogorov relation observed in the inverse energy cascade.

The physical mechanism involved in the inverse cascade deserves detailed analysis, because it substantiates a phenomenon which, in three dimensions is diEcult to grasp. A typical vorticity :eld is displayed in Fig. 37(a) together with the corresponding stream function :eld [Fig. 37(b)]. The recirculation zones on the stream function map display a broad range of sizes but the vortices of the vorticity :eld are rather small in size. This is con:rmed by the distribution of vortex sizes displayed in Fig. 38. The vortex sizes are essentially con:ned around the injection scale. This point was previously recognized by Maltrud and Vallis [79]. Since there is no vorticity patch greater than the injection scale, the conservation laws constraining the vorticity :eld take the form of a tautology, in the range of scale where the inverse cascade grows: zero equals zero. This is the way the system gets rid of the in:nity of invariants which might have severely constrained it. The absence of vortices on scales above the injection scale is certainly the most striking di=erence between decaying and forced two-dimensional turbulence. It is sometimes considered that the inverse cascade is made of a sequence of merging of like sign vortices as is the case in decaying turbulence. However, would it be so, the distribution of vortex sizes would be broad, at variance with the observation. Such merging events actually may occur, but they are rare. The inverse cascade is rather an aggregation process, driving the formation of large clusters of like sign vortices, which sustain large eddies conveying the energy towards larger scales. This picture is illustrated by the inspection of the stream function [Fig. 37(b)]: closed streamlines de:ne large scale regions containing several like sign vortices (for example, the :ve “white” vortices in the upper-right corner of Fig. 37(a) are all enclosed within closed streamlines). Dynamically, the aggregation progress is observed during transient regimes. Fig. 39 displays, at 4 di=erent times in the transient regime, the regions of positive (white) and negative (black)


Fig. 37. Typical vorticity and streamfunction :elds in the inverse cascade.

35

36


1 04

Number

1000

100 li

10

1 0

0.5

1

1.5

2

2.5

3

3.5

4

Size

Fig. 38. Distribution of the vortex size in the inverse cascade.

Fig. 39. Evolution of the vorticity :eld after the energy has been switched on, showing transfers towards larger scales in the inverse cascade.


37

l

ε

2l

4l

Fig. 40. Schematic view of the inverse cascade, represented in a way similar to Richardson cascade, the actual aggregation process replacing the hypothetical break-up event. In this picture, the vortices have the same sign; to represent the whole 9ow, a similar picture must be added, representing the vortices of the other sign. Fig. 41. PDFs of the longitudinal velocity increments at various scales in the inertial range.

vorticity: it can be seen that, as the cascade builds up, the vorticity tends to seggregate and form larger and larger patches of the same sign. We thus may propose the following picture for the inverse cascade. Vortices are continuously nucleated by the forcing; these vortices have sizes on the injection scale and they have a :nite lifetime, equal to 1=&, i.e. controlled by the friction against the bottom wall. Soon after they are formed, they wander in the plane, get distorted and strained by the action of neighboring vortices, and thus tend, in the average, to aggregate with other vortices, so as to conserve energy. The aggregation process is gradual. Small clusters are rapidly formed, and clusters of large sizes take time to be formed. In the internal clock of the vortices, it becomes extremely diEcult to form new clusters beyond 1=&, since the vortices have consumed most of their kinetic energy. The maximum cluster size is thus limited by the :nite lifetime of the vortices. Here the cascade process can be substantiated by a selfsimilar build-up of clusters of increasing sizes. Kolmogorov arguments may be used in this context, with the understanding that the physical scale corresponds to the eddy size, i.e. the vortex cluster dimension, and the hypothetical eddy breaking is replaced by the observed vortex aggregation process. This description is illustrated as a cartoon in Fig. 40. The search for deviations from Kolmogorov scaling in the inverse cascade is recent. The reason is that for several decades, numerical simulations could produce only unsteady cascades, and in such a context, analyzing high order moments was uncertain. The study of Ref. [111] and the numerical work of Bofetta et al. [12] could grasp this problem and provide reliable data on intermittency e=ects. Intermittency can be characterized using several methods: it can be signaled by a nonlinear dependence of the scaling exponents n on the order n, by a scale dependence of the normalized moments of the PDF or by a dependence of the PDF shapes on the separation r. The plots of Figs. 41– 43 review the three aspects, both in the experiment of Ref. [111] and the numerical work of Ref. [12].

38


Fig. 42. PDFs of the transverse velocity increments at various scales in the inertial range. Fig. 43. PDFs of the longitudinal and transverse velocity increments at various scales in the inertial range.

1 05

5

4

n=6

1000

n=5

100

n=4

10

n=3

10

ζn / ζ3

H ||2n ( r )

4 3 2

n=2

1

1

li

0.1

0

1

l = r / li

10

0

2

4

6

8

10

12

14

n

Fig. 44. Hyper9atnesses of the longitudinal velocity increments at various orders. Fig. 45. Structure function exponents of the longitudinal velocity.

Distributions of the longitudinal and transverse velocity increments, displayed above, show they are close to Gaussian at all scales. They are slightly asymmetric, as they should be, if energy transfers are at work but this is a weak e=ect. A recent estimate for the skewness of the pdfs of the longitudinal velocity increments, leads to values on the order of 0.03; also, hyper9atnesses of the distributions are almost scale independent and have values close to the Gaussian limit, as shown in Fig. 44.


39

16

y (cm)

14 12 10 8 6 6

8

10

12

14

16

x (cm) Fig. 46. Deviations from Gaussianity in the inverse cascade, according to Bofetta et al. (1999). Fig. 47. Two particles released in the 9ow, in the inverse cascade regime.

The longitudinal structure function exponents of order n n are determined by the following expression: Sn(L) (r) = |/vL (r)|n ∼ r n : In the above expression, /vL (r) is the increment along the direction of the vector r; in Ref. [?], the symbol || is used rather than letter L to refer to longitudinal quantities. The exponents n have thus been measured both in the experiment and the simulations, and some data are displayed in Fig. 45. They hardly depart from the Kolmogorov line. We may thus conclude there is no substantial intermittency in the inverse cascade. The proximity to Gaussianity raised a hope that the problem be more amenable to theoretical analysis than the three-dimensional problem. However, as pointed out by Bofetta et al. [12], ‘the quasiGaussian behavior is moot, as deviations are intrinsically entangled to the dynamical process of inverse energy cascade’ (Fig. 46). These authors measured the odd moments of the distributions of the velocity increments, and obtained that they increase quite vigorously with the order, which indicates that the departure from Gaussianity cannot be a small parameter. One may also mention a recent work given by Yakhot [143], who takes advantage of the quasiGaussianity of the distributions to construct a representation of the pressure terms, and provide a modelling of the inverse cascade. To conclude, the inverse cascade presents, compared to the direct one, several features which perhaps suggest it may be an easier problem to tackle: in particular, there is no dissipative anomaly, i.e. the viscosity can be taken as zero without harm. Moreover, the observed statistics is close to Gaussianity; this property may tentatively be exploited to devise a perturbative scheme, based on the smallness of the skewness of the velocity increment distributions. Nonetheless, several attempts have shown the diEculty of the problem remains formidable, and at the

40


moment, the absence of intermittency in the inverse cascade does not seem much easier to explain than the presence of intermittency in the standard three-dimensional cascade.

7. Dispersion of pairs in the inverse cascade The dispersion of pairs is on the border of the scope of this paper since it bears issues related to the problem of passive scalar dispersion. However, it enlightens, from the Lagrangian view point some of the aspects of the inverse cascade, and this is the reason why I include it in this review. The subject starts with the empirical proposal of Richardson [119], leading to a mean squared separation growing as the third power of time; this result has long served as a backbone for the analysis of dispersion processes in the atmosphere and the ocean [90,114]. Richardson law has further been reinterpreted in the framework of Kolmogorov theory [105,106], and the concept of e=ective di=usivity on which it relies reassessed by Batchelor [7]. Batchelor and Richardson approaches lead to the same scaling law for the pair mean squared separation, but provide strongly di=erent expressions for the underlying distributions [50]. Kraichnan further reanalyzed the problem in the context of LHDI closure approximation [66] and more recently, a reinterpretation of the t 3 law, based on Levy walks, was proposed by Shlesinger et al. [125]. Richardson law has nonetheless received little experimental support for long, owing to the diEculty of performing Lagrangian measurements in turbulent 9ows. Existing experimental data, bearing on limited statistics and weakly controlled 9ows, show exponents lying in the range 2–3 [90,114]. The law has :rst been convincingly observed in a two-dimensional inverse cascade [5]. Recently, progress has been made, and we now have detailed information on crucial quantities, such as the pairwise separation distributions, and the Lagrangian correlation. At the moment, this data can be found in several studies; I focus here on results obtained in the physical experiment of Jullien et al. [61]. According to this work, the way the pairs are dispersed does not evoke a Brownian process (see Fig. 47): the released particles stay close to each other for some time, then separate out violently, wandering increasingly far from each other. At variance with Brownian motion, the pair separation is not a progressive process, but rather involves sequences of quiet periods and sudden bursts. Also, inspection of the pair trajectories suggests that internal correlations may persist. The distribution of the separations are shown in Fig. 48, in a rescaled form. One sees long tails, in form of stretched exponentials. The following :t was proposed for the distributions q(s; t): q(s; t) = p(s; t) = A exp(−*s& )

(32)

with * ∼ 2:6 and & = 0:50. The separation velocity distributions also deserve a discussion since they are the Lagrangian counterpart of the velocity increment distributions of Fig. 42, which, as we previously saw, are close to being normal. So far they have not been published. They are displayed in Figs. 49 and 50, in a rescaled form.


41

1000 100

σ

2

10 1 0.1 0.01 0.001 1

10 time (s)

100

Fig. 48. Variance of the separation for a pair of particles; note that numerical evidence for the t-cube law was obtained earlier, in a pioneering work by Babiano et al. (1990).

1

1

0.1

0.8

0.01

0.6

R(t,ττ)/R(t,0)

p(δv)*σ(δv)

Fig. 49. Rescaled pdfs of the separations, using Richardson scaling for the abcissa, and forcing the distributions to have their variance equal to 1.

0.001

0.2

0.0001 1 0-5 -10

0.4

0 -5

0

5

10

δv/σ(δv)

15

20

25

-1

-0.8

-0.6

τ/t

-0.4

-0.2

0

Fig. 50. Pdfs of the velocity of separation of pairs released in the 9ows, rescaled in the same way as the previous :gure; each symbol corresponds to a given time. Temporal selfsimilarity applies in a restricted range of values of the separation velocity. Deviations from selfsimilarity are visible at large velocity separations. Fig. 51. Lagrangian correlations of the separation for pairs of particles, rescaled as indicated in the text.

They are strongly asymmetric and have long stretched exponential tails. They hardly share anything common with Gaussian curves. Their asymmetry can be understood by the fact that in the average, the pairs separate, so that the probability to have positive separation velocity should be higher than negative ones. Their overall shape contrasts with the plots of Fig. 42, and illustrate the extent to which Eulerian and Lagrangian statistics may di=er.

42


Another aspect of the problem is the persistence of correlations. We introduce here the Lagrangian correlation of separations de:ned by R(t; $) = /r(0):/r($) in which /r(t) is the position of the separation vector at time 0, for a pair released in the system at time — t. The renormalized Lagrangian correlation of the separations R(t; $)=R(t; 0) are shown in Fig. 51, at di=erent times, in a temporal range covering the inertial domain of the inverse cascade. Here, despite turbulence is stationary, we do not expect the Lagrangian correlations to be time independent quantities, owing to the fact that in the Lagrangian framework, the pair separation is a transient process. This makes a crucial di=erence between the corresponding eulerian quantities, which are time independent in stationary turbulence. As time grows, we e=ectively :nd the correlation curve broadens, and the time beyond which the particles decorrelate raises up. However, the set of curves collapses well onto a single curve, by renormalizing their maximum to unity and using the rescaled time $=t, where t is the time spent since they have been released. This is shown in Fig. 51. This suggests that the general form for the Lagrangian correlation function reads: R(t; $)=R(t; 0) = f($=t);

−t 6 $ 6 0 ;

where f is a dimensionless function. This result can be obtained by applying Kolmogorov arguments to Lagrangian statistics. There is, therefore, in this problem, a single underlying correlation function, for all the inertial domain. The corresponding physical Lagrangian correlation time $c , estimated from Fig. 51, is $c ≈ 0:60t which underlines the persistence of correlations throughout the separation process. At any time, the pairs remember more than half of their history, considering their history starts once they are released. Here is thus some of the available data on the pair dispersion in the inverse cascade. In the dispersion process, temporal selfsimilarity, persistence of internal correlations, and nonGaussianity prevails at all stages. The Richardson dispersion problem and the inverse cascade are intimately related, and probably solving one will allow to get clues for the other. For the Richardson problem, we do not understand the origin of nonGaussianity beyond the remark that, due to the presence of long range correlations, there is no reason to expect the central limit theorem to apply. 8. Condensed states In Section 6, the situations we described corresponded to cases where the dissipation scale ld is smaller than the box size con:ning the 9ow. What happens when ld exceeds the system size? In this case, a new phenomenon takes place, leading to the formation of a condensed state, characterized by a scale comparable to the system size. As we already pointed out, this e=ect was anticipated by Kraichnan [67]; in his paper, he compared the phenomenon to the Bose–Einstein condensation, a terminology which remains. The analogy is justi:ed: in Bose– Einstein condensation, below a certain temperature, the system collapses onto a fundamental


43

Fig. 52. Vorticity :eld in the condensed regime, showing the formation of a peak of vorticity, calculated by Smith et al. (1994). Fig. 53. Streamfunction in the condensed regime, showing that the 9ow is dominated by large scale eddies, according to Smith et al. (1994).

state, compatible with the boundaries of the system. In the phenomenon we discuss here, in Fourier space, the energy piles up onto the smallest wave-number, compatible with the boundaries. The phenomenon is intuitive as soon as one accepts that energy is continuously dragged towards large scales. In such a situation, the way the energy is eventually burned depends on the system at hand. In practice, in physical experiments, energy is dissipated by friction along the walls limiting the system. We now have several observations of the condensation process. The phenomenon has been :rst observed in physical experiments by Sommeria [129], and later by Paret et al. [112]; numerically, the observation has been made by Hossain et al. (see Ref. [54]) and more recently by Smith et al. [141,142]. We show typical numerical and physical results in Figs. 52–55. In the physical experiments, carried out in square boxes, the condensed regimes are dominated by a global rotation, whose sign erratically 9uctuates in time. With periodic boundary conditions, Smith and Yakhot [141,142] observed the formation of a dipole, whose size is a fraction of the spatial period of the system. To analyze the system, it is interesting to look at the energy spectra. A typical energy spectrum for the condensate regime is shown in Fig. 56: the spectrum displays a sharp bump at large scales. One may perhaps speculate that the tendency to form a k −3 spectrum on the 9ank of the bump has some link with an enstrophy cascade [104]. In the condensation regime, there is an interesting phenomenon of vorticity intensi:cation, :rst identi:ed numerically by Smith and Yakhot [141,142], and later con:rmed experimentally by Paret et al. [112]. The instantaneous vorticity :elds display a single central vortex with a small intense core around which the mean global rotation is driven. A vorticity pro:le along a

44


Fig. 54. Vorticity :eld observed in a physical experiment in the condensation regime; a localized vortex sits at the center of the box; this has been observed by Paret et al. (1996).

10 8 10

6

-3 1

2

E ( k ) / E

ω / ω rms

4

0

0.1

-2 -4 -6 0

2

4

6 x / li

8

10

0.01 0.01

0.1

1

10

k / Ki

Fig. 55. Vorticity pro:le showing the peak vorticity in the condensation regime (Paret et al., 1996). Fig. 56. Energy spectrum in the condensation regime, obtained in a physical experiment (Paret et al., 1996).


45

Fig. 57. Vorticity :eld in the condensation regime (Dubos et al., 2001).

line passing through this vortex is displayed in Fig. 55, reveals this peculiar structure. It can be seen that the central vortex is much stronger than the 9uctuations on the sides. The existence of a sharp peak of vorticity is even clearer in the numerical simulation (see Fig. 52). Such a pattern is not understood at the moment. Concerning the statistical characterization of the condensed regime, some progress has been made recently [32]. As we discuss above, in this regime, the realizations of the vorticity :elds are extremely nonhomogeneous: rare, intense, peaked vortical patches coexist with low vorticity background (taking a loose de:nition of intermittency, it is often said that in such a case, the vorticity :eld is ‘intermittent’; however, we will not use this word here, since the intermittency we refer to in this review involves a dependence on scale, not just a spottiness of the quantity at hand). From the point of view of turbulence, and more speci:cally of the scaling, it has been shown recently that there is no substantial intermittency in this regime. This is displayed in Fig. 58, where we see the velocity increment distributions of a condensed state, whose vorticity :eld is represented in Fig. 57. At variance with the inverse cascade regime, the distributions are nonGaussian, but they collapse fairly well at all scales, showing the system is nonintermittent (from the scaling point of view). 9. Enstrophy cascade As mentioned previously, the existence of the enstrophy cascade was proposed by Kraichnan [67], as part of a double cascade process, taking place in a forced two-dimensional system. At about the same time, in the context of freely-decaying turbulence, Batchelor [9] proposed that

46


Fig. 58. Pdfs of the velocity increments in the condensation regime (Dubos et al., 2001). Fig. 59. Lines of isovorticity in a freely decaying turbulence regime, where, for some range of time, an enstrophy cascade has been observed.

enstrophy may cascade towards small scales, in a way analogous to energy in three dimensions, in the sense that the transfer develops at constant rate across the scales. In a range of scale de:ned as the inertial range of the problem, the energy and the enstrophy spectrum, dimensionally constrained, have the following expression: E(k) = C 2=3 k −3 ;

(33)

Z(k) = C 2=3 k −1 ;

(34)

where C is the Kraichnan Batchelor constant and is the enstrophy transfer rate. On physical grounds, the process prevailing in the cascade is essentially the elongation of the vorticity patches, already mentioned in Section 1. The process is displayed in Figs. 59 – 61. Weak vorticity blobs, embedded in a random large scale strain, are elongated, and, since, for a perfect 9uid, the vorticity patches conserve area, one must have a compression in the other direction, leading to the formation of thin vorticity :laments, inducing the steepening of the vorticity gradients (see Fig. 59). In this process, the enstrophy transfer rate is controlled by the strain. The nonpassive nature of the vorticity :eld re9ects the fact that vorticity tends to locally decrease the strain. The process stops as viscosity comes into play, leading to irreversibly di=using vorticity. Di=usion by viscosity is the dissipative mechanism at work for this cascade; it is associated to a scale ld , de:ned by the following expression: ld = −1=6 1=2 ;

(35)


47

Fig. 60. Vorticity :eld in a decay experiment, showing the :lamentation process.

where is the kinematic viscosity of the 9uid. One may infer from this result that the total number of degrees of freedom of the process scales as the Reynolds number, yielding a computationally more advantageous situation than the three dimensional energy cascade (which requires Re9=4 degrees of freedom). There are several issues involved in the enstrophy cascade. A :rst one, discussed early by Kraichnan [68] is the nonlocality of the cascade. In the enstrophy cascade, and within the inertial range, the turn-around time is equal to the rms value of the vorticity, and is therefore the same at all scales. This property has a disturbing consequence: it implies there is no internal mechanism ensuring statistical independence between large and small scales. As coined by Herring and McWilliams [52], large and small scale ‘may become coherent with impunity’; the fact that small scales may directly be a=ected by large scales means the cascade is ‘nonlocal’ (in Fourier space). Physically, nonlocality originates in the fact that the large scale strain controls the formation of small scales, by continuously shaping vorticity patches into thin :laments, without being mediated by a succession or events across which decorrelation could take place. In the enstrophy cascade, the large scales are directly coupled to the small scales, i.e. the interactions cannot be assumed to be local in scale. It is conceivable that nonlocality allows nonuniversal e=ects to contaminate so much the inertial range that classical theory would be of

48


Fig. 61. Vorticity :eld in a physical experiment, where a k −3 spectrum has been observed.

little practical interest; we will see later that the contamination is limited, and in particular it is possible to de:ne universal quantities such as the Kraichnan–Batchelor constant. Another issue is the presence of unpleasant divergences in the classical theory: for an energy spectrum decaying as k −3 , the transfer rate is a logarithmic function of the wave-number, which diverges as the injection wave-number tends to zero, or — equivalently — as the Reynolds number tends to in:nity; this is a concern for the internal consistency of the theory. Such divergences are called “infrared divergences”. In order to remedy this diEculty, Kraichnan proposed a modi:ed form for the spectrum: E(k) = C 2=3 k −3 (Ln(k=k0 ))1=3 ;

(36)

where k0 is the injection wave-number. With such a spectrum, the divergence on the enstrophy transfer rate is suppressed; this approach can be viewed as a one-loop approximation of a more general theory which remains to be written. Theoretical progress on the enstrophy cascade has been made recently, and one may consider that theory now sits on much more solid grounds than in the past. The theories I present below use structure functions of the vorticity, de:ned by the following relations: Sp (r) = (!(x + r) − !(x))p


49

in which x and r are vectors, and r is the modulus of r. The brackets mean statistical average. The exponent of order p of the structure functions — called !p is de:ned by Sp (r) ≈ r !p : Eyink [33] obtained exact inequalities for the set of scaling exponents !p . Assuming the large scale velocity :eld be smooth, he showed that the structure function exponents for the vorticity :eld are smaller or equal to zero for p ¿ 3. The analysis of Falkovich and Lebedev [34] showed that the structure functions Sp have logarithmic behavior, i.e. apart from logarithmic correction, the exponents !p are essentially zero. It is interesting to note that this result has been obtained by using Lagrangian approach. The analysis showed that at variance with the energy cascade in three dimensions, the two-dimensional enstrophy cascade is nonintermittent and classical estimates “a la Kolmogorov” apply within logarithmic accuracy. The logarithmic behavior of the structure function could be found in the Batchelor theory on passive scalars, but the merit of Falkovitch and Lebedev has been to establish this result on rigorous grounds, without needing to postulate a number of quantities are uncorrelated. One may mention that di=erent schemes were proposed in the nineties, as alternatives to the aforementioned theoretical approaches. It is worth mentioning the conformal proposal [117], presented in the original paper as the theory of two-dimensional turbulence; this work did not provide outstanding clues, probably because the relevance of conformal theory to two-dimensional turbulence is limited. Other approaches, based on structural premises, were developed in the same period and before. Sa=mann [123] proposed a dynamical scenario for the formation of vorticity :laments, leading to a k −4 spectrum. More recently, Mo=att, emphasizing the importance of spiral like structures, proposed a k −11=3 spectrum. So far, these appealing descriptions have not been compared in detail with the experiment. It remains to assess the extent to which these structural approaches may describe the enstrophy cascade. Concerning the comparison between theory and experiment, it turned out that the advent of large computers revealed puzzling deviations between the observation and the Batchelor– Kraichnan spectrum, especially in decaying systems [10,15,72]. A variety of energy spectral slopes, between −3 and −6 was observed, depending on the range of time under consideration, and the initial conditions. We will come back to these studies later, in a section dedicated to the relation between coherent structures and spectra. Here we focus on situations where coherent vortices are destroyed, and this leads us to concentrate on forced cases. To disrupt long-lived structures, one may adjust the forcing so as vortical structures of circular topology do not survive more than a few turn around times. This is a delicate and empirical task, and in practice, one tries di=erent protocols and check, a posteriori, from the inspection of the vorticity :eld, that long-lived structures are no more there. In this context high resolution simulations [52,14,79] convincingly showed that classical behavior takes place; Herring et al. [52] were able to obtain di=erent regimes, with and without coherent structures, and could conclude that as soon as long-lived structures, and could conclude that as soon as long-lived structures disappear from the :eld, classical behavior is restored. In the same spirit, Borue [13] obtained remarkable agreement with the classical theory. Figs. 62 and 63 display an enstrophy 9ux, constant over two decades of wave number, and a spectrum, agreeing with Kraichnan expectation, and suggesting logarithmic corrections may be within the reach of the measurement. The following estimate of

50


Fig. 62. Compensated spectra, obtained by multiplying the energy spectra by k 3 , and for various Reynolds numbers. Fig. 63. Energy 9uxes in the spectral space, showing a constant transfer rate holds, directed towards small scales.

C was proposed: C ≈ 1:4 :

(37)

This is probably the best estimate to date. On the (physical) experimental side, early attempts were made to observe the enstrophy cascade by measuring velocity 9uctuations behind a grid towed in a mercury tank, in the presence of a strong magnetic :eld, which is expected to favor two-dimensionality. k −3 spectra were observed but interpretations, emphasizing on the role of the anisotropic Joule dissipation, or the role of the walls, were further proposed [128]; this makes the connection of this observation with the enstrophy cascade uncertain. In a recent period, soap :lm experiments provided evidences that enstrophy cascades may develop, consistently with classical expectations, in real systems [45,62,81,122]. More recently, Paret et al. [112], using electromagnetically driven 9ows in a strati:ed 9uid layers, obtained neat k −3 spectra, in a system shown to be homogeneous and isotropic. The Kraichnan Batchelor constant was found consistent with numerical estimates. These experiments are displayed in Figs. 64 and 65. It is instructive to point out the actual relation between the vorticity and the velocity :eld, using experimental data obtained in a context where an enstrophy cascade develops. Fig. 66 shows a cut of the velocity pro:les corresponding to a vorticity :eld similar to Fig. 56. One shows the overall structure of the pro:les displays variations over lengths on the order of 3 cm, i.e. a fraction of the box size. This substantiates the injection scale. Nonetheless, there are a few steep, localized gradients on this pro:le, signalling the presence of strong vorticity :laments. This is visible in Fig. 66. In the theory, we usually neglect these steep regions to estimate the strain, in a way somewhat similar to mean :eld approximation. One may worry that this raises an uncomfortable situation but the success of the theories discussed above, indicates this approximation is probably the :rst step of a selfconsistent approach. Results on higher order structure functions, and distributions of vorticity increments were obtained in a physical experiment carried out by Paret et al. [113], which revealed the absence of intermittency in the enstrophy cascade, in good agreement with the expectations of Falkovich et al. [34] and Eyink [33]. Fig. 67 shows a set of :ve distributions of the vorticity increments,


51

Fig. 64. Energy spectrum obtained in a physical experiment, displaying the expected k 3 spectrum. Fig. 65. Two-dimensional energy spectrum, showing isotropy, in a physical experiment.

Fig. 66. Vorticity pro:le, taken at a given time, in the enstrophy cascade regime. Fig. 67. Pdfs of the vorticity increments, in a physical experiment.

obtained for di=erent inertial scales. As usual, in order to analyze shapes, the pdfs have been renormalized to impose their variance be equal to unity. The shapes of the pdfs are not exactly the same, but it is diEcult to extract a systematic trend with the scale. Within experimental error, they seem to collapse onto a single curve, indicating absence of anomalous scaling, we usually associate with the absence of small scale intermittency; the analysis of the structure functions of the vorticity shown in Fig. 68 con:rms this statement. Fig. 68 represents a series of vorticity

52


S1 0( r ) / 1 05 10

2

2 S8 ( r ) / 1 04 S6 ( r ) / 1 03 S4 ( r ) / 1 0 S2 ( r ) / 3

100 1

2

3 4 r ( c m)

5

6 7

Fig. 68. Structure functions of the vorticity increments, in a physical experiment. Fig. 69. Spectra in a freely decaying turbulent regime, at various times.

structure functions Sp (r), obtained in such conditions, emphasizing the inertial domain, i.e. with r varying between 1 and 10 cm. The structure functions vary weakly with the scale, indicating the exponents are close to zero. The corresponding values fall in the range −0:05; 0:15, for p varying between 2 and 10; owing to experimental uncertainty, this is indistinguishable from zero. The result is compatible with the classical theory, for which the exponents are predicted to be zero at all orders. Concerning logarithmic deviations, such as those proposed by the theory [68,34], the experiment could not draw out any :rm conclusion. To conclude on the enstrophy cascade, one may say the theory is at a well advanced stage, even if at the moment, quantities such as the pdfs of the vorticity increments have not been determined theoretically. It seems the overall structure of the problem is documented, perhaps understood, and confrontation between theory and experiment leads to acceptable agreement. In this context, a number of observations made in soap :lms [81], revealing departures from classical behaviour, are challenging; they call for theoretical understanding, not necessarily in a strict two-dimensional framework. 10. Coherent structures vs. cascades As we mentioned previously, two-dimensional turbulence has provided a remarkable context for the study of “coherent structures” and the interplay with the classical cascade theories. Describing the full statistics of a system including coherent structures probably represents at the moment one of the most challenging problems raised by two-dimensional 9ows. A prototype situation is the free decay of turbulence, whose large scales were previously discussed. Coherent structures are also observed in some forced systems, but the phenomenology is more compli-


53

Fig. 70. Spectra in a freely decaying turbulent regime, for various times.

cated, and I focus here on freely decaying systems, on which most of the information available on the subject has been collected. In freely decaying turbulence, one expects, from the transposition of Kolmogorov ideas, the development of energy spectra decreasing like k −3 . This has been shown by Batchelor [9]. Extending Kolmogorov approach [64] to the free decay problem, Batchelor predicted an energy spectrum in the form: E(k) ∼ t −2 k −3 :

(38)

As we mentioned earlier, substantial deviations from this spectral law were consistently observed in a number of numerical experiments (Figs. 69 and 70). The :rst observations of Lilly suggested k −3 but soon after, using improved resolution, Herring et al. (1974) [51] found a k −4 energy spectrum. As noted by Lesieur [77], contours of the vorticity plot already indicated the presence of what has been called later the coherent structures. Deviations from Batchelor spectral laws were further con:rmed in a number of numerical simulations. In particular, a variety of slopes, typically lying between −3 and −6, depending on the range of time under consideration, and the initial conditions were obtained [124,72]. Recent work by Chasnov [22] indicates the same trends, although in his case, a proposal for the existence of a critical Reynolds number, at which the classical exponents are obtained, is made. We give here a few examples of the results. To account for this diversity, Brachet et al. [15] proposed to split the decay regime into two steps: the :rst one is inviscid; it leads to the formation of isolated :laments, and is governed by the k −4 spectral law, proposed by Sa=man [123]. In a second step, the :laments get packed and k −3 spectrum is recovered. This work o=ered an observation of the classical slope for the enstrophy cascade, along with an elegant explanation for the establishment of steeper slopes. However, this description did not encompass the considerable diversity of the spectral laws obtained in the simulations on decaying regimes. At about the same time, the relation between

54


the presence of coherent structures and the spectral slopes was analyzed, and eventually the main origin of the de:ciency of the classical theory could be :gured out. From the dynamical viewpoint, it is now clear that coherent vortices, with their stable topology, inhibit the transfers towards small scales, and thus tend to break the selfsimilarity of the process. In this view, coherent structures can be viewed as ‘laminar drops in a turbulent background’ as quoted in [124], or in more culinary terms, as lumps in a soup. On the other hand, the coherent structures may have complex internal structures, with sharp gradients, giving contributions to the spectrum, at all wave-numbers. Since the coherent structures tend to keep their internal vorticity at high Reynolds number, this possibility can be controlled by the initial conditions. If initial conditions include isolated cusps or fronts of vorticity with a stable topology, the contribution of the coherent structures to the spectrum can indeed be substantial throughout the decay process (note that it would be unlikely or physically hardly conceivable that vorticity cusps arise spontaneously in the decay process). This underlines the importance of the initial conditions in the spectral properties of decaying systems. The dynamical inhibition of the enstrophy cascade and possible contributions of the vortices to the spectrum are presently considered as the main causes for deviations from classical laws observed in decaying systems. The contribution of the coherent structures on the spectral law obviously depends on their density and spatial distribution. One may stress that even if their density is small, since they retain most of the energy of the system, their contribution to the energy spectrum can be substantial. More speci:cally, it has been proposed that coherent structures are selfsimilar in space, i.e. their size distribution is selfsimilar [124]. Pushing the analysis further, a relation between the distributions of sizes of the coherent structures and the spectral slope has been o=ered by Santangelo et al. [124]. If the probability distribution of vortices as a function of their radius has the form p(a) ∼ a−* then the spectral law is E(k) ∼ k *−6 : In this context, the spectral exponent is found controlled by the size distribution of the coherent structures, which depends on the initial conditions. In an ordinary cascade, the distribution of vortex sizes would not have any power law structure. With coherent structures, the distribution is a power law in scale, and the exponent is sensitive to the initial conditions. The possibility that the vortices have a selfsimilar structure has been further supported by numerical studies based on contour surgery technique, showing the presence of coherent structures at practically all scales of interest [30]. However, it is not clear that selfsimilarity of the coherent structures is a genuine property of freely decaying systems. The vorticity :eld presented in Section 2 does not o=er any evidence of selfsimilarity. The existence of selfsimilarity certainly depends on the way the forcing is made, and on the range of time we consider, and the few available counter-examples leave an impression that the existence of selfsimilarity cannot be taken as a general feature of decaying two-dimensional systems. Attempts have been further made to remove the coherent structures from the :eld and analyze the remaining part, i.e. that forming the background :lling up the intervortical region. Although the procedure of removal incorporates some subjectiveness, the trend appears that the


55

background display properties interpretable in terms of classical cascade theories. In particular, k −3 spectra are typically observed. This leads to the generally well accepted picture that decaying two-dimensional turbulence is a superposition of a classical background, with universal properties, and coherent structures, strongly dependent of the initial conditions. In three dimensions, the coherent structures currently mentioned are located on the lower border of the spectrum, and they hardly a=ect the inertial range because vorticity intensi:es along the energy cascade. In two dimensions, there is no such intensi:cation process, and coherent structures may contaminate the spectrum without much e=ort. Coherent structures may thus deserve stronger attention in two than in three dimensions. The matter is completely di=erent as soon as coherent structures are present at all scales. In this case, they are in position to control the statistical characteristics of the 9ow, in two and three dimensions as well, and this possibility is sometimes pushed forward as an alternative to the current description of turbulent systems [35]. A concept often used in two-dimensional studies is intermittency, and a confusion can be made between di=erent de:nitions of intermittency. The intermittency usually referred to in turbulence studies (at least in three dimensions) is clearly presented in Frisch textbook [43]. To de:ne the word, one operates in one dimension, and consider the 9atness factor, de:ned by the quantity F=

(u(x + r) − u(x))4 ; (u(x + r) − u(x))2 2

where u is the variable at hand, x the coordinate, and the brackets are statistical averages. If this factor is scale dependent, with a power law, the system is called “intermittent”. In the direct energy cascade in three dimensions, the exponent is negative. The 9ow thus becomes more and more intermittent or spotty as we go to smaller and smaller scales. In the inverse energy cascade, the only intermittency we can think of is large scale intermittency; this would imply a positive exponent for the power law of the 9atness factor. This de:nition is neat, and :ts well with the conceptual framework developed over the last decade, which emphasizes on anomalous exponents. Another de:nition for intermittency is often used in two-dimensional studies. It refers to the fact that the vorticity :eld is typically low in the intervortical region, and high within the coherent structures. This is a loose de:nition of intermittency, used to characterize the spottiness of the vorticity :eld rather than an anomalous scaling behavior. It would be more appropriate perhaps to call this intermittency ‘spottiness’, in order to avoid confusion. Intermittency is a special case of spottiness, since when a direct cascade is intermittent, the system becomes more and more spotty as we probe smaller and smaller scales. On the other hand, one may have situations where the system is spotty and nonintermittent (in the sense discussed above). The condensed states described in Section 8 o=er an example of such a situation: here we have a spotty vorticity :eld but no anomalous scaling in the vorticity statistics. The 9atness factor may be high in this case, but still the system is nonintermittent, because this factor does not vary with the scale. There have been only a few attempts :guring out a framework for describing turbulent systems with coherent structures. It is certainly a diEcult task. Any decomposition of the :eld into two parts faces the problem of handling complicated correlations. As stressed by M Farge, we are not in a situation where the two components occupy well separate ranges of scale, as

56


is often the case in condensed matter physics. One can say at the moment, the problem is a formidable theoretical challenge. 11. Conclusions Let us summarize a few points made in the review: • It is well established that cascades develop in two-dimensional systems; these cascades

require coherent structures to be disrupted — a rather loose concept —. These last years, a number of detailed studies of the enstrophy and inverse energy cascades have been made available, and now, one can say that the two-point statistics of these cascades are rather well documented. Concerning the enstrophy cascade, it is interesting to mention that theory is at a well advanced stage; most of the characteristics of the cascade seem to fall under the grasp of the theory. • In contrast, the inverse cascade is much less understood. We now know that this cascade is nonintermittent (and thus, Kolmogorov theory fully applies), but, so far, no hint has been found for explaining the absence of intermittency. The statistics of the velocity increments is close to Gaussian, but this fact seems hard to exploit theoretically. Concerning the pair dispersion problem in the inverse cascade, it appears that Kolmogorov theory successfully captures several aspects of the problem, in particular the selfsimilar properties. When the dissipation length of the inverse cascade is larger than the system size, a ‘condensation’ regime takes place. In this regime, interesting phenomena arise, such as the formation of point vortices like structures. Theoretical understanding of these phenomena stands, at the moment, at a preliminary stage. • When coherent structures are present, the 9ow dynamics may thoroughly di=er from classical expectations, i.e. those obtained by using Kolmogorov type arguments. Nonclassical exponents have been repeatedly observed along the years, and it is now established they originate in the presence of long-lived structures. The in9uence of the coherent structure is both kinematical (by contributing to the vorticity :eld) and dynamical (by inhibiting the development of the enstrophy cascade). The absence of universality, the strong dependence with the initial conditions,: : : are characteristics which are challenging for the construction of a general theory of these systems. On the other hand, coherent structures are important to incorporate in our vision of two-dimensional systems, since they spontaneously arise, trap much energy, and (by de:nition) are long-lived. A situation, perhaps more favorable to theoretical analysis, is the free decay of two-dimensional turbulence. In this problem, power laws are observed, with apparently robust and possibly universal exponents. An elegant approach, mostly phenomenological, provides deep insight into the problem. A (hopefully) helpful link between this problem and the dispersion problem has been pointed out. • Statistical theories, initiated by Onsager, represent an appealing approach to twodimensional turbulence; it has given rise to an impressive amount of work. These last years, a method for handling continuous vorticity :eld has been proposed. Also, new observations have been made available, so that theoretical analysis could be confronted with experiment. Although the theory proved to be successful in a few cases, it turned out to


57

fail in a number of situations, at quantitative and qualitative levels. The main issue seems whether two-dimensional turbulent systems possess ergodic properties justifying a search for equilibrium states. I hope to have shown, in this review, that two-dimensional turbulence encompasses a rich variety of phenomena; in three dimensions, there is no life for turbulence outside the direct energy cascade, while in two dimensions, there exists several types of cascades, and turbulence, in the ordinary sense, may exist without cascade at all. In this respect, the two-dimensional world is surprisingly more wealthy than the three-dimensional one. On another aspect, two-dimensional turbulence may be more pleasant to teach than three-dimensional turbulence, in the sense that the phenomena can be visualized in all detail. On the other hand, the phenomenology is more complicated, and the diversity of landscapes in Flatland may appear as a bit disorientating; however, thanks to progress made and in the recent period, visibility has been improved, and it certainly is an easier task for the traveller to :nd his way. 12. Uncited references [4,28,41,44,75] Acknowledgements The author is grateful to M.C. Jullien, A. Babiano, C Basdevant, T. Dubos and M. Farge for valuable discussions related to this work. He is particularly indebted to Y. Pomeau and B. Legras for comments on the :rst version of the manuscript, and I. Procaccia and K.R. Sreenivasan for additional remarks on the latest versions. This work has been supported by Centre National de la Recherche Scienti:que, Ecole Normale SupWerieure, UniversitWes Paris 6 and Paris 7. References [1] G. Huber, P. AlstrHm, Universal decay of vortex density in two dimensions, Phys. A 195 (1993) 448. [2] H. Aref, E.D. Siggia, Vortex dynamics of the two-dimensional turbulent shear layer, J. Fluid Mech. 100 (1980) 705. [3] H. Aref, Integrable, chaotic and turbulent vortex motion in two-dimensional 9ows, Annu. Rev. Fluid Mech. 15 345. [4] A. Babiano, C. Basdevant, R. Sadourny, Structure functions and dispersion laws in two-dimensional turbulence, J. Atmos. Sci. 42 (1985) 941–949. [5] A. Babiano, C. Basdevant, P. Le Roy, R. Sadourny, Relative dispersion in two-dimensional turbulence, J. Fluid Mech. 214 (1990) 535–557. [6] C. Basdevant, T. Philipovitch, On the validity of the Weiss criterion in two-dimensional turbulence, Physica D 73 (1994) 17–30. [7] G.K. Batchelor, Di=usion in :eld of homogeneous turbulence II: the relative motion of particles, Proc. Camb. Phil. Soc. 48 (1952) 345–362. [8] G.K. Batchelor, Theory of homogeneous turbulence, Cambridge University Press, Cambridge, 1956. [9] G.K. Batchelor, Computation of the energy spectrum in homogeneous two-dimensional turbulence, Phys. Fluids 12 (Suppl. II) (1969) 233–239.

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[97] This is not easy to do. In the general case, one must remove the translational velocity and the mean rotation of the candidate. [98] In the case of nonneutral populations, other invariants should in principle be introduced, such as the total vorticity; in the special case of circular boundaries, angular momentum conservation must be added. We shall, however, restrict ourselves to the case of neutral populations, as is traditionally done for this problem. [99] The argument is not straightforward, and requires some calculation. Take a triad with p ¡ k ¡ q; on using the conservation equations for the energy and the enstrophy, one shows that more enstrophy is transferred on q than on p, and more energy is transferred on p than on q. By iterating the argument, one obtains the result that at large Reynolds number, enstrophy is essentially transferred towards large wave-numbers while energy is essentially transferred towards small wave-numbers. A detailed reasoning is displayed in Lesieur’s book, and in the paper by Kraichnan. [100] This remark has been made by Kraichnan, 1967. [101] The authors seem to have no responsibility in the name given later to their theory. The name can be critized: for instance, “Universal” is perhaps a bit too strong, although several experiments, performed in di=erent conditions, suggest the exponents are ‘universal’, i.e. they are the same for a broad range of di=erent initial conditions. Also, the word ‘theory’ is perhaps slightly in accurate, if we consider that a true theory should be constructed from :rst principles, which is not the case for ‘universal decay theory’. Nonetheless, the name ‘universal decay theory’ is useful; it allows to refer to a particular framework, using only three words, and without being misleading. [102] There exists some confusion concerning an experimental work performed in thin layers of electrolytes, dedicated to studying the problem of the free decay of turbulence. A :rst series of experiments was conducted in nonstrati:ed electrolytic layers; they are reported in Cardoso et al. ([18]); this work revealed discrepancies between universal decay theory and experiments. The origin of the discrepancy is due to the presence of three-dimensional 9ows which develop within the thin layer. Later, the authors suppressed these 9ows by working with strati:ed layers (see Ref. [110]). The study of the free decay, using strati:ed layers, is given by Hansen et al. [47], 7261–7271; in such conditions, the authors found good agreement between universal decay theory and experiment. The second series of experiment is relevant to two-dimensional 9ows, while the :rst one, contaminated by three-dimensional e=ects, is probably more diEcult to interpret. The situation is clear; it has unfortunately been made confusing by the publication of preliminary measurements (Experimental study of freely decaying two-dimensional turbulence, P. Tabeling, S. Burkhart, O. Cardoso, H. Willaime, Phys. Rev. Lett., 67, 3772 (1991)), using nonstrati:ed layers, and in which the authors claimed that experiments agree with universal decay theory; at that time, the errors bars were too large to draw out such a conclusion, for nonstrati:ed layers. [103] The general formula in d dimensions is 12=d(d + 2). [104] The speculation is that there is some large scale vorticity, produced by the condensation mechanism; these vorticity patches may undergo :lamentation and give rise to an enstrophy cascade; this would favor the formation of a k −3 spectrum. [105] A.M. Obukhov, Dokl. Akad. USSR 32 (1941) 22. [106] A.M. Obukhov, Izv. Akad. Nauk. USSR, Ser. Georgr. i. Geo:z. 5 (1941) 453. [107] L. Onsager, Statistical hydrodynamics, Nuevo Cimento Suppl. 6 (1949) 279–287. [108] O. Paireau, Dynamique d’un tourbillon cisaille, interaction chimie-melange: tudes experimentales, thse de doctorat, Universite Pierre et Marie curie, Paris, 1997. [109] O. Paireau, P. Tabeling, B. Legras, Vortex subjected to a shear: an experimental study, J. Fluid. Mech. 351 (1997) 1. [110] J. Paret, D. Marteau, O. Paireau, P. Tabeling, Are 9ows electromagnetically forced in thin strati:ed layer two-dimensional?, Phys. Fluids 9 (1997) 3102–3104. [111] J. Paret, P. Tabeling, Experimental observation of the two-dimensional inverse energy cascade, Phys. Rev. Lett. 79 (1997) 4162–4165. [112] J. Paret, P. Tabeling, Intermittency in the 2D inverse cascade of energy: experimental observations, Phys. Fluids 10 (1998) 3126–3136. [113] J. Paret, C. Jullien, P. Tabeling, Statistics of the two-dimensional enstrophy cascade, Phys. Rev. Lett. 83 (1999) 3418.

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Physics Reports 362 (2002) 63–192

The quantum damped harmonic oscillator Chung-In Uma , Kyu-Hwang Yeonb , Thomas F. Georgec; ∗ a

Department of Physics, College of Science, Korea University, Seoul 136-701, South Korea School of Physics, Korea Institute for Advanced Study, Seoul, 130-012, South Korea b Department of Physics, Chungbuk National University, Cheonju, 306-763, South Korea c O)ce of the Chancellor=Departments of Chemistry and Physics & Astronomy, University of Wisconsin-Stevens Point, Stevens Point, WI 54481-3897, USA Received August 2001; editor: T:F: Gallagher Contents 1. Introduction 2. Chronological survey 3. Quantum damped harmonic oscillator. I. Propagator method 3.1. Classical case 3.2. Path integral and propagator 3.3. Wavefunction and energy expectation values 3.4. Uncertainty relation and transition amplitudes 3.5. Coherent states 3.6. Applications 4. Quantum damped harmonic oscillator. II. Coupled oscillators 4.1. Classical case 4.2. Propagator of coupled harmonic oscillators 4.3. Propagator of coupled driven harmonic oscillators

65 66 78 78 79 82 85 89 93 103 103 104 106

4.4. Energy expectation values of coupled harmonic oscillators 4.5. Propagator of coupled damped driven harmonic oscillator chains 5. Harmonic oscillator with time-dependent frequency and external force 5.1. Propagator 5.2. Wavefunctions 5.3. Energy expectation values 5.4. Uncertainty product and coherent states 5.5. Applications 6. Time-dependent bound and unbound quadratic hamiltonian system: dynamical invariant method 6.1. Time-dependent bound quadratic Hamiltonian 6.2. Time-dependent unbound quadratic Hamiltonian

∗

Corresponding author. Fax: +1-715-346-2561. E-mail address: [email protected] (T.F. George). c 2002 Elsevier Science B.V. All rights reserved. 0370-1573/02/$ - see front matter PII: S 0 3 7 0 - 1 5 7 3 ( 0 1 ) 0 0 0 7 7 - 1

108 109 114 114 116 119 121 122 128 128 133

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C.-I. Um et al. / Physics Reports 362 (2002) 63–192

7. Quantum damped harmonic oscillator. III. Dynamical invariant and second quantization method 7.1. Quantum-mechanical treatment for the damped driven harmonic oscillator 7.2. Harmonic oscillator with timedependent frequency 7.3. Harmonic oscillator with exponentially decaying mass 7.4. Applications

137 137 140 144 152

8. Linear canonical and unitary transformations on general Hamiltonian systems 8.1. Linear canonical transformations for classical and quantum systems 8.2. Quantum treatment for special types of linear canonical transformations 8.3. Applications 9. Summary Acknowledgements References

165 166 171 175 185 187 187

Abstract Starting with the quantization of the Caldirola–Kanai Hamiltonian, various phenomenological methods to treat the damped harmonic oscillator as a dissipative system are reviewed in detail. We show that the path integral method yields the exact quantum theory of the Caldirola–Kanai Hamiltonian without violation of Heisenberg’s uncertainty principle. Through the dynamical invariant and second quantization methods together with the path integral, we also present systematically the exact quantum theories for the various dissipative harmonic oscillators, bound and unbound quadratic Hamiltonian systems, and the relation between the canonical and unitary transformations for the classical and quantum dissipative c 2002 Elsevier Science B.V. All rights reserved. systems. PACS: 03.65.Ge; 03.65.Ca; 03.65.−w


65

1. Introduction Since Bateman proposed the time-dependent Hamiltonian in the classical context [1] for the description of dissipative systems, there has been much attention paid to the quantum-mechanical treatments of nonconservative and nonlinear systems. Irreversible and dissipative properties that appear in most physical phenomena in our everyday world have been an unrelievable burden to science, especially physics, for a long time. One experiences that in observing our surrounding world, the most physical phenomena are irreversible. DiHusion and dissipation are the standard examples in which mechanical energy is transformed into heat due to frictional force. Dissipation can be observed easily from interactions between the system of interest and another surrounding system, with energy Iow through an irreversible process. To investigate the quantum-mechanical description of dissipating systems, there may be two types of treatments: One is found in the interactions between two systems via an irreversible energy Iow [2,3], and the other is a phenomenological treatment under the assumption of a nonconservative forces [4,5]. In studying nonconservative systems, it is essential to introduce a time-dependent Hamiltonian which describes the damped oscillation, i.e., the Caldirola–Kanai Hamiltonian. This was discovered Jrst by Caldirola [6] and rederived independently by Kanai [7] via Bateman’s dual Hamiltonian and afterward by several others [8–10]. In dealing with the SchrKodinger picture, there are two models: one is deJned by a SchrKodinger–Langevin equation, and the other is described by a SchrKodinger equation with the time-dependent Caldirola–Kanai Hamiltonian. The SchrKodinger–Langevin equation was derived from the Heisenberg equations of motion by Kostin [11], the SchrKodinger variational principle by Razavy [12,13], and the stochastic method by Nelson [14]. In the case of the SchrKodinger equation, there are signiJcant diMculties in obtaining the quantum-mechanical solutions for the Caldirola–Kanai Hamiltonian. Quantization with this Hamiltonian violates the uncertainty relations. The uncertainty relations Jnally vanish as time goes to inJnity. Brittin [15] was the Jrst to address this problem, followed by others [16 –18]. All these kinds of quantum-mechanical solutions violate one of the fundamental laws of quantum mechanics. A given solution guarantees the uncertainty relation and then violates one of the other fundamental laws, for example, the commutation law. Although Svin’in [19] obtained artiJcially the quantum-mechanical solution of the damped harmonic oscillator, which preserves the uncertainty relation, the theory does not obey the uncertainty relation in the absence of noise. To avoid these diMculties, Dedene [20], in the framework of geometric quantization [21– 24], developed a complex symplectic formulation for the damped harmonic oscillator, which is based on the complex dynamical variables given by Dekker [25 –28]. Dedene’s Hamiltonian is closely connected with the time-dependent Bateman’s dual Hamiltonian through a simple complex canonical transformation. However, this theory does not satisfy the uncertainty relations. Bopp [29] introduced the separation of the physical system and its artiJcial adjoint, although his approach seems to miss some of the subtleties associated with the uncertainty relation. The evaluation of the propagator is one of the methods for Jnding the quantum-mechanical solutions for a given system. Tikochinsky [30] showed the way to obtain the exact propagator for a given quadratic Hamiltonian as an initial value problem. He evaluated the exact propagator for the Hamiltonian with constant coeMcients and the Hamiltonians associated with the

66


damped harmonic oscillator. On the other hand, Cheng [31] evaluated the propagator for the Caldirola–Kanai Hamiltonian as the quantum dissipative system by Montroll’s method [32] and also the propagator for the damped harmonic oscillator beyond and at the caustics with the help of the HorvPathy–Feynman formula [33]. He also obtained the time-dependent wavefunction corresponding to the propagator. However, although both of them evaluated the correct propagator for the Caldirola–Kanai Hamiltonian, they did not obtain the exact wavefunction and thus did not prove the uncertainty relation. In 1987, the present authors [34] evaluated the exact propagator for the Caldirola–Kanai Hamiltonian with an external driving force as the damped driven harmonic oscillator through the path integral method. The present authors also obtained the time-dependent wavefunction, uncertainty relation and transition amplitudes. This theory guarantees not only Heisenberg’s uncertainty principle, but also the other fundamental laws in quantum mechanics (hereafter referred to as the Um–Yeon solution). In this article we review the phenomenological approach of the exact quantum theories, which are based on the present authors’ works, for various time-dependent Hamiltonians as a quantum dissipative system in comparison with other theories, through the path integral method. To ease the complexity of the problem, we will concentrate on one-dimensional problems at absolute zero temperature. We make use of the propagator method to deal with the theories in Sections 3–5, while the theories in Sections 6 and 7 are treated through the dynamical invariant and second quantization methods. In Section 2, we give a chronological survey of the theories of the quantum damped harmonic oscillator developed from the early 1930s to the mid-1980s. In Section 3, using the Caldirola–Kanai Hamiltonian with an external driving force for the damped harmonic oscillator as the quantum dissipative system, we evaluate the exact propagator, wavefunction, uncertainty relation, transition amplitudes and coherent states by the Feynman path integral method and applications are provided. We extend the propagator method given in Section 3 to obtain the quantum-mechanical solutions for the coupled, coupled driven and coupled damped driven harmonic oscillator in Section 4, and we investigate a forced harmonic oscillator with a time-dependent frequency and evaluate the quantum-mechanical solutions in Section 5. In Section 6 we obtain the exact quantum theory of a general time-dependent bound and unbound quadratic Hamiltonian and Jnd the relation between the quantum-mechanical solutions and the dynamical invariant. We rederive the exact quantum theory for the damped harmonic oscillator obtained in Section 3 and the harmonic oscillator with an exponentially decaying mass through the dynamical invariant and second quantization methods, and we apply these results to several problems in Section 7. The canonical transformation which transforms the SchrKodinger equations for a damped harmonic oscillator with the Caldirola–Kanai Hamiltonian into the SchrKodinger equation for an undamped harmonic oscillator are discussed in Section 8, and Jnally in Section 9 we give a summary. 2. Chronological survey In this section, we survey chronologically the various approaches for the linearly damped harmonic oscillators from the year 1931 to the mid-1980s, following some parts of Dekker’s argument [5], and the historical approaches of the theories through the Lagrangian and Hamiltonian formalisms [2,3].


67

In 1931, Bateman [1] presented the Jrst investigation of the so-called dual or mirror-image Hamiltonian. His Hamiltonian consists of two diHerent time-dependent Hamiltonians: One represents the simple one-dimensional damped harmonic oscillator. The energy dissipated by the oscillator is completely absorbed at the same time by the mirror-image oscillator, and thus the energy of the total system is constant. The other describes the harmonic oscillator with an exponentially growing mass, that can be reproduced as the so-called Caldirola–Kanai Hamiltonian. Bateman’s dual Hamiltonian is given as H = ppR − (xpR − xp) R + !2 xxR ;

(2.1)

where xR is the mirror variable corresponding to the variable x. The Lagrangian corresponding to Eq. (2.1) is L = x˙xR˙ − 2 xxR + (xxR˙ − x˙x) R ;

(2.2)

and from Eq. (2.2), we can obtain two equations of motion for the variables x and x: R xK + x˙ + 2 x = 0 ;

(2.3)

xRK − xR˙ + 2 xR = 0 :

(2.4)

Hence, xR˙ = d x=d(−t). Eq. (2.4) represents clearly the time reversal process of Eq. (2.3) and called the equation of motion for the image-mirror oscillator of Eq. (2.3). The canonical momenta for this dual system can be obtained from Eq. (2.2): p = xR˙ − x; R

pR = x˙ + x :

(2.5)

The dynamical variables x; p and x; R pR are the operators that should satisfy the commutation relations [x; p] = i˝;

[x; R p] R = i˝ :

(2.6)

In Eq. (2.5) this does not imply a nonzero commutator between the position x and momentum x. ˙ Therefore, the quantum theory based on Bateman’s dual Hamiltonian can be expected to face diMculties. Since the Hamiltonian of Eq. (2.1) is Hermitian, we may introduce the creation and annihilation operators; their inverse transformations and conjugates, constituted of the variables x; p, and x; R p, R can be easily obtained. With the use of these operators and the Baker–HausdorH relation [35,36], we may reduce Eq. (2.1) to a simpler form and then Jnd the eigenstates and eigenvalues for the Hamiltonian of Eq. (2.1). However, the time-dependent uncertainty product obtained in this method, 2 2 2 ˝ (Tpx Tx)2 = e−4t 1 + 4 sin4 !t ; (2.7) 2 ! ! tends obviously to zero. On the other hand, Eq. (2.7) violates Heisenberg’s principle for = 0, regardless of how small the frictional coeMcient is. Therefore, Bateman’s dual Hamiltonian describes classical mechanics correctly, but does not follow the fundamental principles of quantum physics. This problem is closely connected with the treatment of the eigenstates and eigenvalues.

68


Caldirola [6,37] developed a generalized quantum theory of a nonconservative system in 1941. Starting with the Hamiltonian formalism, Caldirola built up the quantum theory for a linear dissipative system. The equation of motion of a single particle system subjected to generalized nonconservative force Q can be written as d @T @T @V − =− + Q(q) ; (2.8) dt @q˙ @q @q where the potential V is only a function of q. Let us perform a nonlinear transformation on time as a canonical variable, t ∗ = (t) ; dt = (t) dt ∗ ; t

(t) = e

0

(t) dt

;

(2.9)

and assume the rth component of Q to have the form s

@T = − (t) ark q˙k ; Qr = − (t) @q˙r

(2.10)

k=1

where (t) is an arbitrary function, and ark are constants. The Lagrangian equation and Hamiltonian can be expressed as d @L∗ @L∗ − =0 ; (2.11) dt ∗ @q˙∗ @q L∗ = T ∗ − V ∗ ; H∗ =

p∗2 + V∗ : 2m

(2.12) (2.13)

We may construct the SchrKodinger equation for a nonconservative system from the classical Hamiltonian through the operators H ∗ and p∗ . From Eq. (2.13) we have the SchrKodinger equation ˝2 ˜ 2 @ ∗ ∗ H = i˝ ∗ = − ∇ + V : (2.14) @t 2m Transforming again the nonlinear time t ∗ into the ordinary time t in Eq. (2.14), a single-particle SchrKodinger equation can be obtained as @ ˝2 −0 t ˜ 2 0 t i˝ (2.15) = − e ∇ +e V ; @t 2m where (t) is taken as a time-independent constant 0 . Eq. (2.15) is known as the Caldirola– Kanai Hamiltonian, which yields the linear dissipative equation of motion, Eq. (2.3).


69

The commutation and uncertainty relation in this formalism are given as [x; p∗ ] = i˝ ; TxTpx∗ ¿

(2.16)

˝

: (2.17) 2 In Eq. (2.17) the uncertainty relation holds only for a coordinate variable x and the corresponding conjugate momentum p∗ . However, the conjugate momentum is obviously diHerent from the ordinary kinetic momentum p. The commutation relation and uncertainty product for the coordinate variable x and the corresponding kinetic momentum p become [x; p] = i˝e0 t ;

(2.18)

˝ TxTpx ¿ e−0 t : (2.19) 2 As time goes to inJnity, the uncertainty relation vanishes, and thus this formalism violates the fundamental principle in quantum physics. In 1948, Kanai [7] derived the Caldirola–Kanai Hamiltonian from Bateman’s dual Hamiltonian by application of the canonical transformation. Kanai transformed the dual Hamiltonian into the Hamiltonian that describes the undamped oscillator,

H = PPR + !2 X XR ;

(2.20)

through the canonical generator as F = xPet + xRPR e−t :

(2.21)

R PR obey the commutation relations, from Eq. (2.6). To simplify The new variables x; P and x; Eq. (2.20), we treat these new variables in the complex plane and introduce another canonical transformation 1 R + i (P − P) R ; X = (Q + Q) 2 2! 1 R ; R + i! (Q − Q) P = (P + P) (2.22) 2 2 R QR also obey where XR and PR are given by the Hermitian conjugates of Eq. (2.22). P; Q and P; the commutation relations from Eq. (2.6). Substitution of Eq. (2.22) into Eq. (2.20) yields two independent identical oscillators. Both are of the same form, and thus we consider only one of them given by H (P; Q) = 12 (P 2 + !2 Q2 ) :

(2.23)

We Jrst transform H (P; Q) into H (y; ) through the canonical generator [38] as F = Q + 12 Q2

(2.24)

and then introduce again the fourth canonical generator to transform H (y; ) into H (y; ): F = ye−t :

(2.25)

70


The transformation leads to the Caldirola–Kanai Hamiltonian H = 12 e−2t 2 + 12 e2t 2 y2 :

(2.26)

This Hamiltonian is identical to Eq. (2.15) and leads to the linear dissipative equation of motion, Eq. (2.3). The Hamiltonian of Eq. (2.26) describes the energy of the dissipative system. From Hamilton’s equation, x˙ = e−2t ;

(2.27)

and the mechanical momentum of the damped oscillator, p = x, ˙ the commutation relation becomes [x; p] = i˝e−2t :

(2.28)

In this construct, Heisenberg’s uncertainty principle is clearly violated. The eigenstates of the SchrKodinger equation constructed from the stationary Caldirola–Kanai Hamiltonian may be expressed by the pseudostationary eigenstates [38– 41]

! 1=4 1 1=2 2 1 1 2t x n (x; t) = − exp i n + !t + − (! + i)e t ˝ 2n n! 2 2 2˝

! t : (2.29) ×Hn e x ˝ Regarding the Hermite polynomial in Eq. (2.29), the inclusion of et term in the argument makes this quite diHerent from the case of a simple harmonic oscillator. It is interesting to compare these eigenstates with the SKussen–Hasse–Albrecht results [42– 44] when excluding et in the pseudostationary states, both are identical to each other. The pseudostationary states may be used to evaluate the expectation values. By evaluating the quantum Iuctuations for the proper mechanical momentum and canonical momentum, one Jnds the uncertainty relations: 1 2 ˝2 2 2 (TP TX ) = 2 n + ; (2.30) ! 2 2 2 1 2 2 −4t ˝

(TPX TX ) = e n+ : (2.31) !2 2 The uncertainty product, Eq. (2.30), for the proper mechanical momentum is conserved in the course of time, but Eq. (2.31) for the canonical momentum tends to zero as time goes to inJnity. On the other hand, the Hamiltonian (2.26) as a generator of motion does not obey the fundamental principle in quantum mechanics. The Caldirola–Kanai Hamiltonian can be treated in an electrical model. This model was proposed Jrst by Pryce [45]. Stevens [45] adopted the Pryce model and showed that it is possible to introduce damping into the quantum description of a harmonic oscillator that is made of a LC circuit coupled to a semi-inJnite transmission line. These kinds of approaches have been investigated by many other authors [46 –55], who have chosen a chain of coupled harmonic


71

oscillators as the heat reservoir. The coupled oscillators must be in the limit of inJnitely many oscillators. On the other hand, the damped equation of motion for a LC circuit, d2 Q dQ Q +R + =0 ; 2 dt dt C can be derived from the Hamiltonian L

(2.32)

P2 AQ2 + eRt=L ; (2.33) 2AL 2C where the momentum P and coordinate Q correspond to the current and charge, respectively, in the LC circuit. Eqs. (2.32) and (2.33) have the same form as those of Eqs. (2.3) and (2.27). The quantum mechanical treatment of the Hamiltonian of Eq. (2.33) by Svin’in [10] preserves the uncertainty relation artiJcially as time goes to inJnity as 2 ˝ ˝

˝2 2 2 [Tpx (∞)TX (∞)] = ¿ coth : (2.34) 2 2kB T 4 H = e−Rt=L

Within the framework of geometric quantization, Dedene [20] has proposed a complex symplectic formulation of the damped harmonic oscillators. This theory is based on the complex dynamical variables given by Dekker [25 –28,56 –58]. Dedene introduced the canonical transformation 1 1 z = √ (pR − i!x); z˜ = √ (pR + i!x) R ; (2.35) 2! 2! where z˜ means a formal complex mirror conjugation. The dynamical variables z and izR correspond to a canonically paired coordinate and momentum [z˜ = zR∗ = zR† ]. Writing Eq. (2.35) in terms of the mechanical variables p and pR obtained from the Lagrangian given by Eq. (2.1), one obtains 1 1 z = √ [p + (% − i!)x]; z˜ = √ [pR − (% − i!)x] R ; (2.36) 2! 2! and the inverse of Eq. (2.35) together with Eqs. (2.36) leads to 1 1 x = √ (z − z ∗ ); p = √ [(! − i%)z + (! + i%)z ∗ ] : 2! 2!

(2.37)

One should notice the diHerence between Eqs. (2.35) and (2.37). In this formalism, the fundamental commutation relation which corresponds to Eq. (2.3) changes to [z; z˜ ] = ˝;

[z˜† ; z † ] = ˝ :

(2.38)

Substitution of Eq. (2.35) in the Bateman dual Hamiltonian yields Dedene’s Hamiltonian H = H + H∗ ;

(2.39)

H = (! − i%)z z˜ :

(2.40)

72


In the commutation relation, Eq. (2.38), z˜ is not the Hermitian conjugate of z, but demands an additional mirror conjugation. In agreement with Dedene [27], z and z˜ are the annihilation and creation operators as generalized Hermitian. Allowing for the usual Weyl symmetrization [33], the Hamiltonian of Eq. (2.39) can be separated into two uncorrelated parts, i.e., H+ and H− , which are mixtures of the physical oscillator and its mirror image, in corporating a nonzero separation constant, &: H = H− + H+ ;

(2.41)

H− = !(zz ˜ + 12 ˝') − i%(zz ˜ + 12 ˝&) ; H+ = !(z † z˜† + 12 ˝') + i%(z † z˜† + 12 ˝&) ;

(2.42)

where ' = 1 and 0 exhibit the Weyl and normal ordering [35], respectively. The eigenvalues of the Hamiltonian are given by Hn(±) = (n + 12 ')˝! ± i(n + 12 &)˝%;

n = 0; 1; 2; : : : :

(2.43)

Taking ' = 0 and & = 2, Eq. (2.43) is reduced to the eigenvalue expression, and the choice of ' = 1 and & = 0 yields Bopp’s spectrum [29]. The equation of motion can be expressed in the conventional commutator form, and thus the general functions F+ (z † ; z˜† ) and F− (z; z) ˜ can be written as sums of the factorizing terms F+ F− given by F(z; z ∗ ; z; ˜ z˜∗ ). Then one can obtain the mean value of the correct equation of motion. Through this procedure one can obtain the position and momentum spread of the damped oscillator. The uncertainty product obtained in the complex symplectic Hamiltonian is identical to Eq. (2.7). Therefore, Heisenberg’s principle is apparently violated again. The main Iaw in the complex Hamiltonian comes from the incorrect fundamental commutation relation. On the other hand, the physical oscillator and its mirror mathematical adjoint do not commute with each other. Bopp oHered a hint to solve this problem by proposing to write from Eqs. (2.39) and (2.40) z˙ = − iHz˜∗ ;

z˙∗ = − iH∗z˜ :

(2.44)

Then the commutation relation Eq. (2.38) becomes [a; a† ] = 1 ; (2.45) √ √ where a = z= ˝ and a† = z † = ˝. Using the inverse transformation to the real coordinate and momentum for the damped oscillator, Eq. (2.45) transforms into [x; p] = i˝ :

(2.46)

From Eq. (2.44) one obtains i a˙ = − Ha∗ ; ˝

i a˙∗ = H∗a ; ˝

(2.47)


73

and Bopp’s Hamiltonian is given by the complex nonHermitian generator [H∗ = H] together with its eigenvalues: H = ˝(! − i%)a∗ a ; Hn = n˝(! − i%);

(2.48) n = 0; 1; 2; : : : :

(2.49)

For % = 0, Eqs. (2.48) and (2.49) are reduced to those of the simple harmonic oscillator [12,59]. Bopp introduced the pseudo-density operator w as the projection operator w = |) )| and derived the general density matrix in terms of the coherent states |) as

2 (2.50) *n; m = P()0 )*(0) nm d )0 ; where P()) is a quasi-probability density and *(0) nm is the density operator. Eq. (2.50) satisJes the master equation. Using Eq. (2.50), one obtains the expectation value for the uncertainty relation:

˝ −2t ˝ 2 2 2 (TX ) = 1 + sin 2!t + 2 2 sin !t + (2.51) e (1 + e−2t ) ; 2

! ! 2!

˝! −2t ˝ 2 2 2 2 (TPX ) = 1 − sin 2!t + 2 2 sin !t + (2.52) e (1 + e−2t ) : 2 ! ! 2! The Jrst parts in Eqs. (2.51) and (2.52) are exactly the same as those of Eq. (2.7), but have extra terms, and thus Eqs. (2.51) and (2.52) guarantee the correct uncertainty product. The evaluation of the propagator can be a convenient method for Jnding the quantummechanical solution for a given system. Especially, the propagator for a given quadratic Hamiltonian can be expressed exactly as the path integral [60,61]. Tikochinsky [30] has shown how to solve this problem as an initial value problem. We consider the quadratic Hamiltonian given by H (x; p; t) = )0 + )1 x + )2 x2 + )3 p + )4 p2 + )5 xp + )5∗ px ; where )i = )i (t). The Hamiltonian of Eq. (2.53) satisJes the SchrKodinger equation @K i˝ = HK : @t If the Hamiltonian is quadratic, then the propagator [62] can be written as

i K(x; t) = F(t) exp Sc (x; t) ; ˝

(2.53) (2.54)

(2.55)

where F(t) is a factor depending only on the time interval, and Sc (x; t) is the classical action. Substituting Eq. (2.55) into Eq. (2.54) under the assumption @S @S + H x; ; t = 0 ; (2.56) @t @t together with S(x; t) = a(t) + b(t)x + c(t)x2 ;

(2.57)

74


one obtains coupled Jrst-order diHerential equations for a(t); b(t) and c(t), and )i (t) from the Hamilton–Jacobi equation. When the )i is not time dependent, the time-dependent solutions for a(t); b(t) and c(t) are given by a(t) = At − (B − C cos t)=( sin t) + x ()3 + x Re )5 )=22)4 ;

(2.58)

b(t) = [ − x 2 + 21(cos t − 1) − )3 sin t]=(2)4 sin t) ;

(2.59)

c(t) = [ − 2 Re )3 sin t + cos t]=(4)4 sin t) ;

(2.60)

where

= 2[)3 )4 − (Re )5 )2 ]1=2 ; 1 = ()1 )4 − )3 Re )5 ) ;

(2.61)

A = [412 + ()32 − 4)0 )4 ) 2 ]=(4)4 2 ) ;

(2.62)

B = 1(x 2 + 21)=()4 2 ) ;

(2.63)

C = [(x 2 + 21)2 + 412 ]=(4)4 2 ) ;

(2.64)

F(t) = exp [iIm )5 t][ =4i˝)4 sin t]1=2 :

(2.65)

For the simple harmonic oscillator, we get )0 = )1 = )3 = )5 = 0, )2 = m!2 =2 and )4 = 1=2m, and then Eqs. (2.58) – (2.60) and (2.65) become the well-known result [41]. For convenience let us write the Caldirola–Kanai Hamiltonian, Eq. (2.15) as p2 1 (2.66) + e2t m!2 x2 ; 2m 2 which yields the linearly damped equation, (2.3). In this case we have )0 = )1 = )3 = )5 = 0, )2 = e2t m!2 =2 and )4 = e−2t =2m. Then we obtain H(t) = e−2t

a(t) = 12 m x2 cos t=sin t + 12 x2 ;

(2.67)

b(t) = − m x exp(t)=sin t ;

(2.68)

c(t) = 12 me2t ( cos t − sin t)=(sin t) ;

(2.69)

F(t) = et=2 [m =(2i˝ sin )]1=2 ;

(2.70)

= (!2 − 2 )1=2 :

Setting = 0, Eqs. (2.67) – (2.70) are reduced to those of the simple harmonic oscillator. We will determine the validity of such expressions in the next section. Bateman’s Hamiltonian was introduced to derive the equations of motion of the damped harmonic oscillator by Feshbach [63,64]. Feshbach’s Hamiltonian may be written as 1 H = px py + (ypy − xpx ) + m 2 xy ; (2.71) m


75

which is a two-dimensional problem. One of the corresponding equations of motion is given by Eq. (2.3), and the other is the time reversed equation of motion: yK − 2y˙ + !2 y = 0 : Through the following transformations: 1 1 x = √ (x1 + x2 ); y = √ (x1 − x2 ) ; 2 2 the Hamiltonian of Eq. (2.71) can be expressed in the simpliJed form 2 2 p1 p2 1 1 H= + m 2 x12 − + m 2 x22 − (x1 p2 + x2 p1 ) : 2m 2 2m 2

(2.72) (2.73)

(2.74)

This Hamiltonian is the diHerence of two simple harmonic oscillators with extra coupled terms between x and p. Performing the exact same procedures as we have done before under the assumption @S @S @S ; =0 (2.75) + H x1 ; x2 ; @t @x1 @x2 with S = a + bx1 + cx2 + dx12 + ex22 + fx1 x2 ;

(2.76)

one can determine the time-dependent coeMcients in the classical action: a(t) = 12 m (x12 − x22 ) cos t=sin t ; b(t) = − m (x1 cosh t − x2 sinh t)=sin t ; c(t) = − m (x1 sinh t − x2 cosh t)=sin t ; d(t) = 12 m cos t=sin t ; F(t) = m =(2˝ sin t) :

(2.77)

It is easy to show that as tends to zero, the propagator is reduced to the product of two propagators for the simple harmonic oscillators. In this theory it is instructive to note that one may bypass the equations of motion and obtain the classical action and propagator directly as a solution of an initial value problem. Cheng [31] evaluated the propagator for the Caldirola–Kanai Hamiltonian as the modiJed Feynman path integral in conJguration phase space via Montroll’s method [32]. Furthermore he has shown that the propagator for the damped harmonic oscillator can also be evaluated beyond and at caustics with the help of the HorvPathy–Feynman formula [33]. The propagator can be expressed as a path integral in phase space

1 K(q ; q ; T ) = exp (pq˙ − H (p; q) dt) DpDq ˝

∞

∞ N −1 dp dpk dqk ··· ; (2.78) = lim 1=2 N →∞ −∞ (2˝)1=2 −∞ (2˝) k=1

76


where H (p; q) is the Hamiltonian of the system considered and DpDq is the two-dimensional path diHerential measure in phase space. Here let us set T = t − t ; ' = (t − t )=N; rj = r(t + j'); r = r(t ) and r = r(t ). To deal with the quantum-mechanical diMculties in the interpretation of this propagator [8,40], Cheng introduced the Caldirola–Kanai Hamiltonian, Eq. (2.15), for the dissipative system p2 H (p; q; t) = e−t (2.79) + et V (q) ; 2m where V (q) is the potential energy. With the help of the Lagrangian propagator, Eq. (2.78) may be expressed as

K(s ; s ; T ) = lim (i)N=2 (met =2˝')1=2 exp{i[(s2 + s2 ) − '2 !2 s2 ]} L→∞

×

∞

−∞

···

∞

−∞

N −1

exp i

'

2 2

(1 + e − ! '

)sk2

k=1

−2

N −1 k=0

'=2

e

sk sk+1

N −1

dsk ;

k=1

(2.80)

where dqk and sk are given by dqk = (2˝'=m)1=2 e−tk =2 dsk ; sk = qk (m=2˝')1=2 etk =2 : According to Montroll’s method [32], the multiple integral in Eq. (2.80) is transformed into the Gaussian integral

∞

∞ N T T ··· exp[i(S AS + 2b S)] dsk = (i)N=2 (det A)−1=2 exp(−ibT A−1 b) : (2.81) −∞

−∞

Comparison of Eqs. (2.80)  a1 −d 0  −d a2 −d   0 −d a3   .. .. A =  ... . .   0 0 0   0 0 0 0 0 0

k=1

and 0 0 −d .. . 0 0 0

(2.81) yields  ··· 0 0 0 0 ··· 0 0 0 0   ··· 0 0 0 0   ..  .. .. .. .. . . .  . .  · · · −d aN −3 −d 0   ··· 0 −d aN −2 0  ··· 0 0 −d aN −1

(2.82)

with ak = 1 + e' − !k2 ';

and

d = e−'=2 ;

and b1 = − s e'=2 = − cq '−1=2 et1=2 ; bk = 0;

(k = 2; 3; : : : ; N − 2) ;

bN −1 = − s e'=2 = − cq e−1=2 e(t

+')=2

:

(2.83)


77

Assuming the factor exp[ − i'2 !2 s2 ] in Eq. (2.80) to be one as ' → 0, one has only to evaluate the limit values of (' det A) and [(s2 + s2 ) − bT A−1 b]. Substituting these limit values into Eq. (2.80), one obtains the propagator for the damped harmonic oscillator described by the Caldirola–Kanai Hamiltonian as 1=2 m (t +t )=2) m e K[q ; q ; T ] = exp − et q2 − et q2 2i˝ sin t 4i˝

im

t 2 t 2 (t +t )=2 ×exp [(e q + e q ) cos T − 2e qq ] : (2.84) 2˝ sin t For the quadratic Lagrangian, the propagator has the form of Eq. (2.55) [62,65]. The preexponential path integral (2.55) is given by in Eq. t

im F(t ; t ) = exp (8˙2 − !2 82 )et dt D 8(t) (2.85) 2˝ t for the damped harmonic oscillator with 8 = 8 = 0. Using the transformation 9 = 8et=2 , one obtains

t 2 im 2 2 F(t ; t ) = exp (9˙ − 9 ) dt D9(t) ; (2.86) 2˝ t where 9 = 9 = 0. Following the argument of HorvPathy [33] one gets the propagator for the damped harmonic oscillator beyond caustics: 2

(t +t )=2 m e i 1

T K[q ; q ; T ] = exp − + Ent 2˝|sin T | 2 2

m im

t 2 t 2 ×exp − (e q − e q ) exp 4i˝ 2˝ sin T (2.87) ×[(et q2 + et q2 ) cos T − 2 exp[(t + t )=2]q q ] ; where Ent( T=) stands for the largest integer which is less than or equal to T=. At caustics for T = n, the phase factor of F(t ; t ), i.e., e−in=2 , is a jump in phase at every half-period [66 – 68]. Using Eqs. (2.55) and (2.87) and choosing tR = t − =2 , one obtains the propagator at caustics of the damped harmonic oscillator as in K[q ; q ; T = n= ] = exp − exp[(t + t )=4]&[et =2 q − (−)n et =2 q ] : (2.88) 2 In case of a bound system, the propagator is expressed in terms of the time-dependent wavefunction n (x; t) [62,69,70] as ∞ n (q ; t )n∗ (q ; t ) : (2.89) K(q ; t ; q ; t ) = n=0

Comparison of Eqs. (2.89) and (2.84) yields n (q; t) = Nn exp(iEn t= ˝) exp[ − (m =2˝)(1 + i=2 )et q2 ]Hn [(m = ˝)1=2 ]et=2 q] ;

(2.90)

78


with the energy eigenvalues En = [(n + 12 ) + 14 ]˝;

n = 0; 1; 2; : : : ;

(2.91)

where the normalization factor is Nn = (m =˝)1=4 (2n n!)−1=2 and Hn (X ) is the nth-order Hermite polynomial. Eqs. (2.90) and (2.91) are equivalent to those of Kerner [38] and Hasse [40]. The propagator, Eq. (2.84), has also been evaluated by Khandekar and Lawande [69,70] and Jannussis et al. [71]. It should be noted that the derivation of this propagator is carried out by the method generalized by Cheng [72]. However, Cheng did not show whether or not this derivation guarantees the uncertainty relation. The correct derivation, i.e., the Um-Yeon solution, will be shown in the next section. In this section, we have reviewed the previous various theories for the dissipation of the linear damped harmonic oscillator in quantum mechanics. Besides these theories, there are many other theories such as the modiJed Bopp–Dekker master equation for the reduced density operator in the treatment of the quantum optics oscillator [73–81], the nonlinear frictional quantum theory in a Gaussian approximation due to Hasse [82], the Hamilton–Jacobi formalism [12,13,82–87], the quantization of the novel Hamiltonian in the SchrKodinger-Razavy variation procedure [4,11], and so on. We should also point out that all theories for this linear damped harmonic oscillator are not perfect. If they satisfy one of the fundamental principles, then they do not guarantee the others in quantum mechanics. In the next section, we will present what we believe to be a more elegant quantum theory for the damped harmonic oscillator. 3. Quantum damped harmonic oscillator. I. Propagator method Although the Feynman path integral formulation [62] oHers a general approach for treating quantum-mechanical systems, only a few time-dependent SchrKodinger equations can be solved exactly. One can obtain the time-dependent SchrKodinger equation for the damped harmonic oscillator by replacing the momentum with (˝=i)(@=@x) in the Caldirola–Kanai Hamiltonian [6,7]. However, as we have in the previous section, the question is whether or not one can obtain the correct quantum mechanical solution [5]. Most of the previous theories violate the uncertainty relation. This diMculty is critically reviewed by Dodonov–Man’ko [88], Greenberger [89], and Cervero and Villaroel [90]. To treat a generalized Hamiltonian for the dissipative system, we introduce the Caldirola– Kanai Hamiltonian with an external driving force for the damped harmonic oscillator. In this section we will discuss the quantum-mechanical solution (Um–Yeon solution) for this dissipative system via the propagator method developed by the present authors [34,91]. 3.1. Classical case We introduce the Caldirola–Kanai Hamiltonian with a time-dependent external driving force f(t), deJned as the time-dependent damped driven harmonic oscillator (DDHO) as H = e−t p2 =2m + et ( 12 m!02 x2 − xf(t)) :

(3.1)


79

Hamilton’s equations of motion for Eq. (3.1) are x˙ = e−t p=m ;

(3.2)

p˙ = − e+t (m!02 x − f(t)) :

(3.3)

Eqs. (3.1) and (3.2) yield the Lagrangian L = et ( 12 mx˙2 − 12 m!02 x2 + xf(t)) ;

(3.4)

and the corresponding equation of motion is xK + x˙ + !02 = f(t)=m :

(3.5)

Eq. (3.1) can be considered as the Hamiltonian of a quantum damped driven harmonic oscillator, which bears analogy to that of a classical damped driven harmonic oscillator. The classical solution of Eq. (3.5) is

t

t f(s) −s=2 −t=2 −t x(t) = Ae cos(!t + ) + e cos !t cos 2!t dt cos !s ds (3.6) e m with ! = (!02 − 2 =4)1=2 :

(3.7)

The mechanical energy can be expressed as E = e−2t p2 =2m + 12 m!02 x2 :

(3.8)

Here, the energy expression in Eq. (3.8) is not equal to the Hamiltonian itself. 3.2. Path integral and propagator In the path integral formulation, the solution of the SchrKodinger equation is given by the path-dependent integral equation with the propagator K as

(x; t) = K(x; t; x0 ; 0)(x0 ; 0) d x0 ; (3.9) which provides the wavefunction (x; t) at time t in terms of the wavefunction (x0 ; 0) at time t = 0. The Hamiltonian of a damped free particle and the corresponding Lagrangian are given as H = e−t p2 =2m ;

(3.10)

L = et 21 mx˙2 :

(3.11)

Combining Eqs. (3.10) and (3.11) with Eq. (2.52), one obtains the damped free particle propagator: 2 met=2 imet=2 K(x; t; x0 ; 0) = exp (3.12) (x − x0 )2 : 4i˝ sinh 12 t 4˝ sinh 12 t

80


If the Lagrangian is quadratic, the pre-exponential path integral yields Eq. (2.83). Thus the path can be expressed as a damped Fourier sine series with a fundamental period t:

y(t ) =

N

an e−t =2 sin

n

n=1

t

t ;

t = 'N :

(3.13)

Substitution of Eq. (3.13) into Eq. (2.83) within the limiting process for N → ∞ and ' → 0 yields the propagator of the damped quadratic Hamiltonian system as 1=2 m!et=2 i K(x; t; x0 ; 0) = exp Sc (x; x0 ; t) : (3.14) 2i˝ sin !t ˝ The classical action of the DDHO Hamiltonian is

Sc = et ( 12 mx˙2 − 12 m!02 x2 + xf(t )) dt :

(3.15)

In Eq. (3.15) for small !0 , the kinetic energy is dominant with the Lagrangian acting like that of a damped free particle, so that one may take the propagator for DDHO as having the following form: K(x; t; x0 ; 0) = F(t) exp[ − a(et x − x0 )2 ] :

(3.16)

Here, a is a time-dependent function of a damped free particle. The propagator for DDHO can be expressed as

1=2 t m!0 2 t=2 m!0 K(x; t; x0 ; 0) = exp − a(t)e x + b(t)e x + c(t) : (3.17) ˝ ˝ Changing the variables t and x into m! 1=2 !0 0 x; p = t; y = 2i ˝

(3.18)

the propagator is then expressed as

2 ) 2 K(y; p; y0 ; 0) = exp − a(p) exp i p y + b(p) exp i p y + c(p) : !0 !0

(3.19)

Eq. (3.19) must satisfy the SchrKodinger equation, Eq. (2.51). Substitution of Eqs. (3.1) and (3.19) into Eq. (2.51) determines the time-dependent coeMcients in Eq. (3.17): a(t) = 1 + 12 cot(2p) with

1=2 2 = −1 ; 4!02

2(m˝!03 )1=2 B − b(t) = sin(2 p) sin(2p)

) ; 1=i 4!0

(3.20)

(3.21) p

f(p) exp i p sin(2p) dp ; !0

(3.22)


81

and 1 B2 − 21p + log[sin(2p)] 2 2 2

p

1 2 + sin2 (2p) (m˝!03 )1=2

c(t) = −

2 2 f(p) exp p sin(2p) dp !0

p f(p) exp i p sin(2p) dp dp + log c0 ; !0 p

2B 2 sin (2 p) (m˝!03 )1=2 1=2 m!2 B=i x0 ; ˝!0 m! 1=2 m − i x02 : log c0 = log 2i˝ 4˝ To simplify the expression, one can write the propagator, Eq. (3.17), as im 2 ˜ 2 ˜ ˜ K(x; t; x0 ; 0) = F(t) exp (ax ˜ + bx0 2 + 2cx ˜ 0 x + 2dx + 2ex ˜ 0 − f) ; 2˝ −

2

(3.23) (3.24) (3.25)

(3.26)

where the new time-dependent coeMcients are a˜ = (− 12 + ! cot !t)et ;

(3.27)

b˜ = ( 12 + ! cot !t) ;

(3.28)

! t=2 e ; sin !t ;(t) t=2 d˜ = e ; m sin !t (t) e˜ = ; m sin !t

t f(t )et =2 sin !t dt ; ;(t) =

0t (t) = f(t )et =2 sin !(t − t ) dt ; c˜ = −

0

∇(t) f˜ = 2 ; 2m !

t ∇(t) = 0

0

(3.29) (3.30) (3.31) (3.32) (3.33) (3.34)

t

f(t )f(S) exp[(S + t )=2] sin !(t − t ) sin !S dS dt ;

m!et=2 F(t) = 2i˝ sin !t

(3.35)

1=2

:

(3.36)

All coeMcients except for a; ˜ b˜ and c˜ depend on the external driven force. Setting f(t) = 0 or = 0, Eq. (3.26) is reduced to the propagator of the damped harmonic oscillator or the

82


forced harmonic oscillator. When f(t) = = 0, Eq. (3.26) becomes the familiar propagator of the simple harmonic oscillator. 3.3. Wavefunction and energy expectation values The Hamiltonian of DDHO, Eq. (3.1), reduces to quadratic form at time t = 0: p2 1 f(0) 2 f2 (0) 2 H= − : + m!0 x − 2m 2 m!02 2m!02 The corresponding wavefunction (x; 0) and energy eigenvalue are given by

f(0) 1 2 f(0) 2 n (x; 0) = N0 Hn )0 x − exp − )0 x − ; 2 m!02 m!02 f2 (0) 1 En = n + ˝! − ; 2 2m!02 where Hn (x) is the Hermite polynomial of order n and the coeMcients are m! 1=2 )01=2 ; N0 = n √ : )0 = ˝ (2 n! )1=2

(3.37)

(3.38) (3.39)

(3.40)

For the convenience of calculation of other quantities, one sets f(0) = 0, and then Eqs. (3.38) and (3.39) reduce to n (x; 0) = N0 Hn ()0 x) exp(− 12 )0 x2 ) ;

(3.41)

En = (n + 12 )˝!0 :

(3.42)

From Eq. (3.9) together with Eqs. (3.1), (3.26) and (3.41), one obtains the wavefunction in the form

∞ n (x; t) = K(x; t; x0 ; 0)n (x0 ; 0) d x0 −∞

1 1 −1 ˜ = N n 1=2 exp −i n + cot + cot !t + f 2 2! (2 n!) ×exp[ − (Ax2 + 2Bx)]Hn [D(x − E)] :

Here, the time-dependent coeMcients are given by m! 1=4 exp[ − ((t)∗ :

(3.60)

1 E nn = n + 2

˝!

−t

85

2

2

(3.58)

Except for the second oH-diagonal elements (En+2; n and En−2; n ), the diagonal and Jrst oHdiagonal elements are involved in the external driving force, i.e., f(t). 3.4. Uncertainty relation and transition amplitudes The main Iaw of the other theories is the uncertainty relation. As t goes to inJnity, the uncertainty relation vanishes. To evaluate the uncertainty relation, in a similar way to that used to obtain Eqs. (3.53) and (3.54), one can calculate the expectation values of x mn and @=@x mn :

∞ x mn = m∗ (x; t)xn (x; t) d x −∞

n + 1 1=2 Re B exp[i cot −1 (=2! + cot !t)]&m; n+1 − &m; n Re A Re A 1 n 1=2 + exp[ − i cot −1 (=2! + cot !t)]&m; n−1 ; 2 Re A

1 = 2

and

@ @x

mn

=

∞

−∞

m∗ (x; t)

(3.61)

@ n (x; t) d x @x

A = [2(n + 1)]1=2 exp[i cot−1 (=2! + cot !t)]&m; n+1 − 2(B + AE)&m; n D A 1=2 − (2n) − D exp[ − i cot −1 (=2! + cot !t)]&m; n−1 : D

(3.62)

Using Eqs. (3.53) and (3.54), the uncertainty relation in the various states can be evaluated: 2 2 2 2 1=2 [(Tx)2 (Tp)2 ]1=2 n+2; n = [(x − x )(p − p )]n+2; n ˝ = [(n + 2)(n + 1)]1=2 − cot !t =(t)2 sin2 ! 2 2! + + cot !t − i exp[2i cot −1 (=2! + cot !t)] 2!

= [(n + 2)(n + 1)]1=2 9(t) ;

(3.63)

86

C.-I. Um et al. / Physics Reports 362 (2002) 63–192 2

[(Tx)

=(t) sin !t − cot !t =(t)2 sin2 !t 2! (m˝!)1=2 −;(t) − cot !t (t) + + cot !t − i + 2! sin !t 2! 2 1 + n 1− − cot !t =(t)2 sin2 !t + + cot !t 2 2! 2! + 2i − cot !t =(t)2 sin2 !t + + cot !t 2! 2! 1=2 1=2 [2(n + 1)]1=2 (t) 1 −1 ×exp[i cot (=2! + cot !t)] − n (m˝!)1=2 =(t) sin !t 2

(Tp)2 ]1=2 n+1; n

= ˝ (2n)1=2

×exp[i cot −1 (=2! + cot !t)] ;

2

[(Tx)

(Tp)2 ]1=2 n; n =

(3.64)

2 1=2 1 2 2 ˝ 1+ n+ − cot !t =(t) sin !t + ; + cot !t 2 2! 2! (3.65)

and 1=2 ∗ [(Tx)2 (T)2 ]1=2 n−2; n = [n(n − 1)] 9(t) :

(3.66)

Taking the complex conjugate and changing (n + 1) into n in Eq. (3.64), one can easily obtain the uncertainty in the (n − 1; n) state. Using the wavefunction Eq. (3.43), one can compute the transition amplitudes amn for the damped driven harmonic oscillator from a state |m to a state |n :

∞ amn = d xm (x; 0)n (x; t) −∞

1=2 m!= ˝ 1 1 = m+n exp t − )e−i! t ;

(3.101)

f3 ('(! ); >) = E0 )(! )!02 sin >[1 + f1 ('(! ); >)] exp(−ikd cos >) :

(3.102)

where Taking the real part of the external driving force, the classical equation of motion for the dipole moment of the adsorbed molecule can be expressed as 2 xK + x˙ + !m x = f(t)=m :

(3.103)

Here the dipole moment C is Qx where Q is the average dipole charge and m is the mass of the molecule. The form of Eq. (3.103) is the same as that of the usual damped driven harmonic oscillator, i.e., Eq. (3.5). In Eq. (3.103) f(t) is given by f(t) = g(! ; >) cos(! t − 0 ) ; g(! ; >) =

2E0 m!02 F(! ) sin > 2 [g1 (! ; >) + g22 (! ; >)]1=2 ; Qg3 (! ; >)

g1 (! ; >) = cos > cos(kd cos >){|'(! )|2 cos > + R(! ; >)['1 (! ) cos 8 + '2 (! ) sin 8]} + R(! ; >) cos > sin(kd cos >)['1 (! ) sin 8 − '2 (! ) cos 8] ; g2 (! ; >) = R(! ; >) cos > cos(kd cos >)['2 (! ) cos 8 − '1 (! ) sin 8] − R(! ; >) sin(kd cos >){R(! ; >) + cos >['1 (! ) cos 8 + '2 (! ) sin 8]} ;

g3 (! ; >) = R2 (! ; >) + |'(! )|2 cos2 > + 2R(! ; >) cos >['1 (! ) cos 8 + '2 (! ) sin 8] ; 0 = tan−1 [g2 ('(! ); >)=g1 ('(! ); >)] ;

96


8 = 12 tan−1 {'2 (! )=['1 (! ) − sin2 >]} ; R(! ; >) = {['1 (! ) − sin2 >]2 + '2 (! )}1=4 :

(3.104)

The classical Lagrangian and Hamiltonian corresponding to the equation of motion, Eq. (3.103) were already given by Eqs. (3.4) and (3.1). It is straightforward to write the propagator as

1=2 m!et=2 K(x; t; x0 ; 0) = exp[A1 (t)x2 − A2 (t)x] exp[ − A3 (t)x02 + A4 (t)x0 ] : (3.105) 2i sin(!t) The coeMcients are given by A(! ; >) = g(! ; >)=b1 (! ) ;

b1 (! ) = [(! )2 + (!2 − ! 2 + 2 =4)2 ]1=2 ;

1 = tan−1 [! =(!2 − ! 2 + 2 =4)] ; m! t e − cot(!t) ; A1 (t) = 2i˝ 2!

! A(! ; >)! t A2 (t) = e − cot(!t) cos(! t − 0 − 1 ) − sin(! t − 0 − 1 ) ; i˝ 2! ! m! A3 (t) = + cot(!t) ; 2i˝ 2! A4 (t) = −

m!x A(! ; >)! t=2 e cos(! t − 0 − 1 ) + et=2 : i˝ sin(!t) i˝ sin(!t)

(3.106)

The wave function of the dipole moment at time t can be obtained through Eq. (3.9) as

1 D(t) 1=2 −1 √ exp −i n + cot [9(t)] − B3 (t) n (x; t) = 2 2n n! ×exp[ − B1 (t)x2 + B2 (t)x]Hn (D(t)[x − E(t)]) ;

where

+ cot(!t) ; 2!

m!et 1=2 ; D(t) = ˝=1 (t)

9(t) =

E(t) =

A(! ; >) cos(! t − 0 − 1 ) ; m

B1 (t) = 12 D2 (t)[1 + i=2 (t)] ; B2 (t) = D2 (t)E(t)[1 + i=3 (t)] ;

(3.107)


97

B3 (t) = 12 D2 (t)E 2 (t)[1 + i9(t)] ; =1 (t) = sin2 (!t)[1 + 92 (t)] ; =2 (t) = 9(t) + =1 (t)[=! − 9(t)] ; =3 (t) = =2 (t) − =1 (t)! =! tan(! t − 0 − 1 ) :

(3.108)

To obtain the energy expectation values, we can add the contribution of the external force, xf(t), to the mechanical energy operator of the system, Eq. (3.51), which gives Eop = −

˝2

2m

e−2t

@2 1 2 2 + m!m x − xf(t) : 2 @x 2

(3.109)

One can conJrm that Eq. (3.109) is not identical to the Hamiltonian, Eq. (3.1), and Hˆ does not represent the total energy of the system, but rather is the generator of motion of an energy-dissipative open system [123]. It is straightforward to evaluate the matrix elements of the energy operator: 2 " m!m Eop m; n = { (n + 1)(n + 2)( 12 − 2%1 B1 )&m; n+2 2D2 " + 2(n + 1)(DE + 2%1 %2 B1 − D%3 )&m; n+1 + [(n + 12 ) + 2(2n + 1)%1 %4 B1 √ + D2 E 2 − %1 %22 − 2D2 E%3 ]&m; n + 2n(DE − 2%1 %2 %4 − D%3 )&m; n−1 " + n(n − 1)( 12 − 2%1 %42 D2 )&m; n−2 } exp{i(m − n) cot−1 [9(t)]} ; (3.110) where the coeMcients are

2 ! %1 (t) = ; !m = 1 D 2 %3 (t) =

%2 (t) = D(B2 − 2B1 E) ;

A(! ; >)b1 (! ) cos(! t − 0 ); 2 m!m

%4 (t) = D2 − B1 :

(3.111)

In energy expectation values, Eq. (3.110), only the diagonal element E nn and four oH-diagonal elements E n±1; n and E n±2; n have nonzero values. The matrix elements of the dipole moment are expressed through a similar procedure as " √ Q C m; n = exp[i(m − n) cot−1 (9)][ 2(n + 1)&m; n+1 + 2n&m; n−1 + 2DE&m; n ] : (3.112) 2D This equation contains the diagonal element which is closely related to the parity problem of the wave function. The selection rule for the dipole transition between diHerent states is Tm = ± 1. The explicit forms of the dipole matrix elements are

(n + 1)˝et 1=2 i!t C n+1; n = Q e + sin(!t) 2m! 2! = C ∗n; n+1 ;

98


C n; n =

QA(! ; >) cos(! t − 0 − 1 ) m

C n; n = 0

(t = 0) :

(t = 0) ; (3.113)

In this section we have modeled a molecule adsorbed on a dielectric solid surface as a damped harmonic oscillator driven by a time-dependent electric Jeld, consisting of a primary and a secondary Jeld. As we mentioned earlier, the Hamiltonian of the model system is not identical to the mechanical energy operator, showing a general characteristic of the Caldirola– Kanai Hamiltonian. The propagator, Eq. (3.105), has the same structure as that of Gerry [124] for f(t) = 0. The wavefunction of Eq. (3.107) has no deJnite parity, which can easily be seen from the matrix elements of x and x2 , " √ m(t)|x|n(t) = (n + 1)u(t)&m; n+1 + nu∗ (t)&m; n−1 + E(t)&m; n ; (3.114) " " 2 m(t)|x2 |n(t) = (n + 1)(n + 2)u2 (t)&m; n+2 + n(n − 1)u∗ (t)&m; n−2 " + [(2n + 1)|u(t)|2 + E 2 (t)]&m; n + 2 (n + 1)u(t)E(t)&m; n+1 √ + 2 nu∗ (t)E(t)&m; n−1 : (3.115) If m (x; t) and n (x; t) have deJnite parity and both have the same even or odd parity,

x mn must always be zero, while if they have opposite parities, x2 m; n must vanish. However,

Eqs. (3.114) and (3.115) do not satisfy this rule. These properties of the wavefunction are not due to the decay characteristic of the system, but to the interaction between the system and driving force, because the wavefunction of a damped (not driven) harmonic oscillator has a deJnite parity [105]. Because of the absence of deJnite parity of the wavefunction, the diagonal element of the dipole moment takes the form given in Eq. (3.113), in contrast to a simple harmonic oscillator. The real part of the oH-diagonal elements of the dipole moment is the same as that of the classical case [125]. The expectation values of the mechanical energy, E and other quantities do not stay constant in time because of the time-dependence of |(x; t)|2 . The diagonal parts of E and H may be approximated as 1 g2 (! ; >) ; (3.116) ˝!e−t + Eop n; n ∼ n + 2 − ! 2 + ! ) 2 2m(!m Hˆ n; n = et Eop n; n : (3.117)

The Jrst term in Eq. (3.116) shows the decay of the quantum state, and the second term represents the energy absorption from the external Jeld. These processes are related to the lineshape in optical phenomena. When there is no driving force, Eqs. (3.116) and (3.117) reduce to Eq. (3.58) and that of a damped harmonic oscillator, which was not calculated in the previous section [40]. Hence, we can conJrm that for a damped harmonic oscillator, the expectation values of H stay constant in time and those of E do not, while for a damped driven oscillator the expectation of both operators vary in time. For the simple harmonic oscillator, of course, the energy and Hamiltonian operator are identical, and their expectation values remain stationary.


99

Application of the results in this section to optical phenomena on a dielectric surface, to Jnding the coherent states [97,126] and to the time-dependent invariant [127] will be presented in other sections. 3.6.2. Charged particle in an in:nite square-well potential with a constant electric :eld The inJnite square-well potential problem is one of the simplest bound-state problems in quantum mechanics. The propagator can be obtained not only by the path integral method but also through the introduction of the image-point method equivalent to a sum over classical paths [128] and a suitably chosen point canonical transformation [129 –131]. This is interesting in regard to the quantization of the charged particle in an inJnite square well potential in the presence of a time-dependent electric Jeld [132,133]. Consider the potential of a charged particle with a constant electric Jeld [134,135] as 0 for t ¡ t ; V (x; t) = (3.118) −q'(t)x for t ¿ t ; where the applied Jeld '(t) is 0 for t ¡ t ; '(t) = ' for t ¿ t : The Lagrangian for the system is 1 M x˙2 for t ¡ t ; L(x; t) = 21 2 2 M x˙ + q'(t)x for t ¿ t ;

(3.119)

(3.120)

where M is a mass of the charged particle. Using the integral equation, Eq. (3.43), and the deJnition of the propagator, Eq. (2.86), in the path integral method, one obtains #

M iM (x − x0 )2 K0 (x; t; x0 ; t0 ) = ; t ¡ t ; (3.121) exp 2i˝(t − t0 ) ˝2(t − t0 ) #

M i M (x − x0 )2 K0 (x; t; x0 ; t0 ) = exp 2i˝(t − t0 ) ˝ 2(t − t0 ) 1 q2 ' 2 + q'(t − t0 )(x + x0 ) − (3.122) (t − t0 )3 ; t ¡ t0 ; 2 24M where the electric Jeld is applied at the classical point (x ; t ). Here, Eqs. (3.121) and (3.122) correspond to the propagators for a free particle and a charged particle in a constant electric Jeld, respectively. It is easy to evaluate the wavefunction at t ¡ t and t ¿ t from Eqs. (3.121) and (3.122): 1 ˝ 2 (3.123) (x; t0 ) − √ exp i k t0 ; t ¡ t ; 2M 2 1 ˝ 2 √ (x; t0 ) − exp i (3.124) k t0 − ikx ; t ¿ t : 2M 2

100


For the case t0 ¡ t and t ¡ t, Eq. (2.86) can be expressed as  −1

∞ N (x ; t ) i  K(x; t; x0 ; t0 ) = dx lim d xj exp S0 (x ; t ; x0 ; t0 N →∞ ˝ −∞ (x0 ; t0 ) j=1

× lim

N →∞

=

∞

−∞

 i d xk exp Se (x; t; x ; t )  ˝ (x ; t )

−1 N (x; t)

j=1

Ke (x; t; x ; t )K0 (x ; t ; x0 ; t0 ) d x :

(3.125)

With the use of Eq. (3.43) and Eqs. (3.123) – (3.125), the wavefunction is expressed as ( 1 q' → (x; t) = √ exp i k + (t − t ) x ˝ 2

iq2 '2 iq' i ˝k 2 3 2 ×exp − (t − t ) − k(t − t ) − (t − t ) ; 6˝M 2M 2M ( 1 q' ← (x; t) = √ exp i −k + (t − t ) x ˝ 2

iq' iq2 '2 i ˝k 2 3 2 ×exp − (t − t ) + k(t − t ) − (t − t ) ; 6˝M 2M 2M

t ¡ t ;

(3.126)

t ¿ t :

(3.127)

The substitution of Eq. (3.122) and (3.124) in Eq. (3.43) also yields Eqs. (3.126) – (3.127). If the electric Jeld is in the negative x-direction and the charge is positive, the second exponential term in Eq. (3.122) becomes negative. Then the wavefunction in Eqs. (3.126) and (3.127) is

( 1 q' iq2 '2 3 iq' 2 i˝k 2 (x; t) = √ exp i k − t x exp − (3.128) t + kt − t : ˝ 6˝M 2M 2M 2 It can be easily proved that the propagator of Eq. (3.121) is composed of the wavefunction of Eqs. (3.123) and (3.124), and their complex conjugates, #

M i M (x − x0 )2 K0 (x; t; x0 ; t0 ) = exp 2i˝(t − t0 ) ˝ 2(t − t0 )

˝2 2 1 ∞ = d k exp[ik(x − x0 )] exp −i k (t − t0 ) 2 −∞ M

∞ = d kk∗ (x0 ; t0 )k (x; t); −∞

(3.129)


101

and the propagator of Eq. (3.122) is represented by Eq. (3.128) and its complex conjugate form #

M i M (x − x0 )2 1 K0 (x; t; x0 ; t0 ) = exp + q'(t − t0 )(x + x0 ) 2i˝(t − t0 ) ˝ 2(t − t0 ) 2 q2 ' 2 3 − (t − t0 ) 24M

iq' 1 ∞ iq' 2 2 = d k exp ik(x − x0 ) − (tx − t0 x0 ) + k (t − t02 ) 2 −∞ ˝ 2M i˝ 2 iq2 '2 3 − k (t − t0 ) − (t − t03 ) 2M 6˝ M

∞ = d kk∗ (x0 ; t0 )k (x; t) : (3.130) −∞

The propagator of Eq. (3.122) has another representation as 1=3

∞ E 2M 2=3 2M (q')−1=3 dEAi∗ − q' x+ K(x; t; x0 ; t0 ) = ˝2 ˝2 q' −∞ 1=3

2M i E − q' x + exp '(t − t ) ×Ai − 0 0 ˝2 q' ˝

∞ = dEE∗ (x0 ; t0 )E (x; t) ; −∞

(3.131)

where Ai (x) is the Airy function. For the above equation, one can think the wavefunction E (x; t) as 1=3 E 2M 2=3 2M i −1=6 E (x; t) = (q') Ai − q' x+ exp − Et : (3.132) ˝2 ˝2 q' ˝ Eq. (3.132) does not reduce to Eq. (3.123) or Eq. (3.124) when ' → 0 and thus is not the wavefunction of the system. Eq. (3.132) is a wavefunction for a potential which is linear in position, whereas the system is a free space at t ¡ t and a linear potential at t ¡ t. Thus the potential is time dependent at t = t , and the wave function at t ¡ t reduces to the wavefunction of the free particle at t ¡ t if ' → 0. The particle movement is restricted between 0 and L, and there is an inJnite number of paths to connect the points (x0 ; t0 ) and (x; t) in the inJnite square-well potential. In consideration of the image points, the particle will behave as if it moves at each image point xr rL + x for r even; (3.133) xr = (r + 1)L − x0 for r odd; where r is a number of reIections in the classical path. For r reIections of the particle on some classical path, the propagator corresponding to that path is multiplied by (−)r . Then the

102


propagator can be constructed in the inJnite square-well potential with a constant electric Jeld as K(x; t; x0 ; t0 ) =

∞

(−1)r K(x; t; x0 ; t0 ) :

(3.134)

r=−∞

With the help of Eq. (3.128), one can express Eq. (3.131) as ∞ 1 K(x; t; x0 ; t0 ) = d k∗ (2lL + x0 ; t0 )k (x; t) 2 l=−∞

=

n n 2 q' 2 sin x+ (t − t02 ) sin x0 L −∞ L 2M L ∞

i˝2 n2 iq2 ' 3 ×exp − (t − t ) − (t − t ) : 0 0 2ML2 6˝

(3.135)

Applying the electric Jeld ' at time t ¿ 0, the propagator of Eq. (3.134) becomes ∞ n n 2 q' 2 K(x; t; x0 ; 0) = sin x+ sin t x0 L L 2M L n=1

i˝2 n2 iq2 '2 3 q' t − ×exp − t +i t 2 2ML 6˝ML ˝

:

From Eqs. (3.136) and (3.43), one can obtain the wavefunction at t ¿ 0 as q' 2 q' 2 n i˝2 n2 iq2 '2 3 n (x; t) = x+ exp − t− sin t t +i t : L L 2M 2ML2 6˝ML ˝

(3.136)

(3.137)

The general wavefunction corresponding to Eqs. (3.126) and (3.127) is given as (x; t) = A→ (x; t) + B← (x; t) ( q' 2 q' 2 = C sin k x + t + D cos k x + t 2M 2M q' i˝2 n2 iq2 '2 3 t− ×exp − t +i t : 2ML2 6˝ML ˝

(3.138)

This wavefunction satisJes the boundary condition. At x = 0 or L, Eq. (3.137) becomes zero and must reduce to the simplest form, i.e., box quantization for ' = 0. One can obtain the uncertainty product from Eqs. (3.138) and (3.63), n˝ (Tx · Tp)nn = √ : 12

(3.139)

Since total energy of the system is E = 12 M x˙2 − q'x ;

(3.140)


103

from Eq. (3.128) the energy expectation value of the system is given as E n =

˝ 2 2 n2

2ML2

+

q'tp2 q'L − ; 2M 2

(3.141)

where tp = tm − tm for tm ¡ t ¡ tm . In this section, using the image point path integral method, the wavefunction, propagator, uncertainty product and energy expectation values are derived. The wavefunction is Jnite in time which is an integral multiple of the special time inversely proportional to the size of the well. The energy expectation value of the system corresponds to the quantum states that are changing with time. The exact quantum mechanical solutions of the Caldirola–Kanai Hamiltonian have been obtained through the propagator method. Any assumptions and artiJcial conditions are not introduced in the derivation. The mechanical energy operator [Eq. (3.51)] is not identical to the Hamiltonian operator [Eq. (3.1)], but is rather the generator of the motion of an energy dissipative open system. This Um–Yeon solution guarantees that the fundamental laws in quantum mechanics, and especially Heisenberg’s uncertainty principle are preserved. The theory is successfully applied to a molecular system adsorbed on a dielectric surface and a charged particle in an inJnite square-well potential with a constant electric Jeld. 4. Quantum damped harmonic oscillator. II. Coupled oscillators A number of situations such as superconducting quantum interference devices [136], quantum nondemolition measurements [137,138], magnetic hydrodynamics [139], etc., can be described by driven coupled harmonic oscillators. In this section we will extend the propagator method developed in the previous section to investigate coupled harmonic oscillators [CHO], coupled driven harmonic oscillators (CDHO) and coupled damped driven harmonic oscillators (CDDHO) [140,141]. 4.1. Classical case One may consider a system of two harmonic oscillators coupled together by means of another spring. Let us assume that masses and spring constants of the oscillators are all the same, and that the forces f1 (t) and f2 (t) are exerted on the two oscillators, and their displacements are x1 and x2 . Then the Hamiltonian for the coupled driven harmonic oscillators can be written as H=

1 (p2 + p22 ) + m!2 (x12 − x1 x2 + x2 ) − f1 (t)x1 − f2 (t)x2 ; 2m 1

(4.1)

where !2 = k=m. Hamilton’s equations of motion are then x˙1 = p1 =m ;

(4.2)

x˙2 = p2 =m ;

(4.3)

104


p˙ 1 = m!2 (x2 − 2x1 ) + f1 (t) ;

(4.4)

p˙ 2 = m!2 (x1 − 2x2 ) + f2 (t) :

(4.5)

Eqs. (4.2) – (4.5) yield the Lagrangian L = (p1 x˙1 + p2 x˙2 ) − H =

m 2 (x˙ + x˙22 ) − m!2 (x12 − x1 x2 + x22 ) + f1 (t)x1 + f2 (t)x2 2 1

(4.6)

with the corresponding equations of motion xK1 + !2 (2x1 − x2 ) = f1 (t)=m ;

(4.7)

xK2 + !2 (2x2 − x1 ) = f2 (t)=m :

(4.8)

The classical solutions of Eqs. (4.7) and (4.8) are given as √ √ x1 (t) = A sin(!t) + B cos(!t) + C sin( 3!t) + D cos( 3!t)

t J dJ dKei!(2J−K−t) [f1 (K) + f2 (K)] ; +

(4.9)

√ √ x2 (t) = A sin(!t) + B cos(!t) − C sin( 3!t) − D cos( 3!t)

t J + dJ dKei!(2J−K−t) [f1 (K) − f2 (K)] :

(4.10)

4.2. Propagator of coupled harmonic oscillators In the path integral formalism, the propagator and solution of SchrKodinger’s equation are given by Eqs. (2.52) and (3.9), respectively. From the Lagrangian, Eq. (4.6), the classical action becomes S(x1 ; x2 ; x1 ; x2 ; t) = Sc (x1 ; x2 ; x1 ; x2 ; t)

t 2 + dJ {y˙ 21 (J) + y˙ 22 (J) − 2!2 [y12 (J) − y1 (J)y2 (J) + y22 (J)]} ; (4.11) m 0 where Sc is the classical action of the non-driven coupled harmonic oscillators and yi is the deviation of xi (t) from its classical path xci given as yi = xi − xci (i = 1; 2). From Eq. (2.52) the multiplicative function, F(t), is given in the form

0

t 2 2 2 2 2 F(t) = Dx(t) exp (im=2˝) dt[y˙ 1 + y˙ 2 − 2! (y1 − y1 y2 + y2 )] : (4.12) 0

0

It is easy to show that F(t) has the same form for CHO and CDHO. Therefore the propagator depends only on the classical action in both cases. However, it is not straightforward to evaluate


105

the multiplicative function. Introducing new variables, one may separate the action into two functionals with just the same variables. Then the multiplicative function becomes a product of two multiplicative functions of the simple harmonic oscillators. Hence, the propagator of CDHO can be written as 1=2 √ 3 m! √ K(x1 ; x2 ; t; x1 ; x2 ; 0) = ei Sc = ˝ : (4.13) 2i˝ sin(!t) sin( 3!t) To evaluate the exact propagator expressed by Eq. (4.13), one should Jrst obtain the classical action for CHO:

t m 2 2 2 2 Sc = dJ ) − m!2 (xc1 − xc1 xc2 + xc2 ) (xc1 + xc2 2 0

t m m t = (xc1 x˙c1 + xc2 x˙c2 )|0 − dJ xc1 [xKc1 + !2 (2xc1 − xc2 )] 2 2 0

t m − dJ xc2 [xKc2 + !2 (2xc2 − xc1 )] 2 0 m = [xc1 (t)x˙c1 (t) + xc2 (t)x˙c2 (t) − xc1 (0)x˙c1 − xc2 (0)x˙c2 )] ; (4.14) 2 where xc1 and x˙c1 are the classical path and velocity, and the second and third terms become zero because of Eqs. (4.7) and (4.8). To obtain the exact expression of the classical action, one can solve Eqs. (4.7) and (4.8) for f1 (t) = f2 (t) = 0. After some calculation, the classical action becomes √ √ m! 2 Sc = {(x1 + x22 + x12 + x22 )[cot(!t) + 3 cot( 3!t)]} 4 √ √ + 2(x1 x2 + x1 x2 )[cot(!t) − 3 cot( 3!t)] √ √ − 2(x1 x1 + x2 x2 )[1=sin(!t) + 3=sin( 3!t)]} √ √ + 2(x1 x2 + x2 x1 )[ − 1=sin(!t) + 3=sin( 3!t)] : (4.15) Then the propagator for CHO can be obtained as √ m! √ K(x1 ; x2 ; x1 ; x2 ; 0) = [ 3=sin(!t) sin( 3!t)]1=2 2i˝

√ √ ×exp{(im!=4˝)[(x12 + x22 + x12 + x22 )[cot(!t) + 3 cot( 3t)] √ √ + 2(x1 x2 + x1 x2 )[cot(!t) − 3 cot( 3!t)] √ √ − 2(x1 x2 + x1 x2 )[1=sin(!t) + 3=sin( 3!t)] √ √ + 2(x1 x2 + x1 x2 )[ − 1=sin(!t) + 3=sin( 3!t)]} :

(4.16)

106


4.3. Propagator of coupled driven harmonic oscillators The derivation of the propagator for CDHO is very similar to the method in Section 3. The propagator of CDHO is written in the form K(x1 ; x2 ; t; x1 ; x2 ; 0) = exp[a(t)x12 + b(t)x1 x2 + c(t)x22 + d(t)x1 + g(t)x2 + h(t)] :

(4.17)

Here the time-dependent coeMcients a(t); b(t), etc. should be determined. Eq. (4.17) together with Eq. (4.1) should satisfy the SchrKodinger equation, Eq. (2.51). Substitution of Eq. (4.17) in Eq. (2.51) gives the time-dependent coeMcients: i˝ a(t) ˙ = (4.18) [4a2 (t) + c2 (t)] + m!2 =i˝ ; 2m ˙ = i˝ [4b2 (t) + c2 (t)] + m!2 =i˝ ; b(t) (4.19) 2m 2i˝ c(t) ˙ = (4.20) [a(t)c(t) + b(t)c(t)] − m!2 =i˝ ; m

˙ = i˝ [2a(t)d(t) + c(t)g(t)] + i f1 (t) ; d(t) (4.21) m ˝

i i˝ g(t) ˙ = [2b(t)g(t) + c(t)d(t)] + f2 (t) ; (4.22) m ˝ ˙ = i˝ [d2 (t) + g2 (t) + 2a(t) + 2b(t)] : h(t) (4.23) 2m To solve these equations, one should make use of the transformation of variables. Through a lengthy but not diMcult calculation, we obtain the solutions as √ √ i!m a(t) = b(t) = [cot(!t + >1 ) + 3 cot( 3!t + >2 )] ; (4.24) 4˝ √ √ i!m c(t) = b(t) = [cot(!t + >1 ) − 3 cot( 3!t + >2 )] ; (4.25) 2˝

t dJ[f1 (J) + f2 (J)] sin(!t) d(t) = [i=2˝ sin(!t)] 0

√

+ [i=2˝ sin( 3t)]

0

t

√ dJ[f1 (J) − f2 (J)] sin( 3!J)

√ + [)=2 sin(!t)] + [1=2 sin( 3t)] ;

t dJ[f1 (J) + f2 (J)] sin(!t) g(t) = [i=2˝ sin(!t)]

(4.26)

0

t √ √ − [i=2˝ sin( 3t)] dJ[f1 (J) − f2 (J)] sin( 3!J) 0

√ + [)=2 sin(!t)] − [1=2 sin( 3t)] ;

(4.27)


107

√ √ i˝ [)2 cot(!t) + (12 = 3) cot( 3!t)] 4m!

t − [)=m! sin(!t)] dJ[f1 (J) + f2 (J)] sin[!(t − J)]

h(t) = −

0

√

√

− [1= 3m! sin( 3!t)]

+ [1=4i˝m! sin(!t)] √

0

t

t

0

dJ

√ dJ[f1 (J) − f2 (J)] sin[ 3!(t − J)]

t

0

√

+ [1=4 3i˝m! sin( 3!t)]

0

t

dK[f1 (J) + f2 (J)][f1 (K) + f2 (K)] sin[!(t − J)] sin(!t)

dJ

t

dK[f1 (J) − f2 (J)][f1 (K) − f2 (K)]

0

√ √ √ ×sin[ 3!(t − J)] sin( 3!K) − ln[sin(!t) sin( 3!t)] + & ;

(4.28)

where ); 1 and & are constants to be determined by comparisons among Eqs. (4.24) – (4.28). Substitution of these results into Eq. (4.17) gives the propagator for the CDHO: √ m! √ { 3=[sin(!t) sin( 3!t)]}1=2 K(x1 ; x2 ; t; x1 ; x2 ; 0) = 2i˝

√ √ im! ×exp (x12 + x22 + x12 + x22 )[cot(!t) + 3 cot( 3!t)] 4˝ √ √ + 2(x1 x2 + x1 x2 )[cot(!t) − 3 cot( 3!t)] √ √ − 2(x1 x1 + x2 x2 ){1=sin(!t) + 3=sin( 3!t)} √ √ + 2(x1 x2 + x1 x2 )[ − 1=sin(!t) + 3=sin( 3!t)]

t 2x1 + [1=sin(!t)] dJ[f1 (J) + f2 (J)] sin(!J) m! 0

t √ √ + [1=sin( 3!t)] dJ[f1 (J) − f2 (J)] sin( 3!J) 2x2 + m!

0

[1=sin(!t)] √

− [1=sin( 3!t)]

4x + 1 m!

t

0

t

0

√

√

+ [1=sin( 3!t)]

0

t

dJ[f1 (J) − f2 (J)] sin( 3!J)

[1=sin(!t)]

dJ[f1 (J) + f2 (J)] sin(!J)

0

t

dJ[f1 (J) + f2 (J)] sin[(!(t − J))] √

dJ[f1 (J) − f2 (J)] sin( 3!J)

108


4x + 2 m!

[1=sin(!t)]

√

− [1=sin( 3!t)]

t

0

− [1=m2 !2 sin(!t)]

t

0

dJ[f1 (J) + f2 (J)] sin[!(t − J)] √

dJ[f1 (J) − f2 (J)] sin( 3!J)

0

t

dJ

×sin[!(t − J)]sin(!K) √

2

2

√

t

0

− [1= 3m ! sin( 3!t)] √

√

0

t

dK[f1 (J) + f2 (J)][f1 (K) + f2 (K)]

dJ

0

t

dK[f1 (J) − f2 (J)][f1 (K) − f2 (K)]

×sin[ 3!(t − J)] sin( 3!K)

:

(4.29)

4.4. Energy expectation values of coupled harmonic oscillators When setting f1 (t) = f2 (t) = 0, the Hamiltonian of CDHO becomes the Hamiltonian of CHO: H = 12 (p12 + p22 ) + m!2 (x12 − x1 x2 + x22 ) :

(4.30)

Using Eqs. (2.51) and (3.9) with Eq. (4.30), the SchrKodinger equation becomes i˝(@=@t)(x1 ; x2 ; t) = Hop (x1 ; x2 ; t) :

(4.31)

Since Eq. (4.31) can be separated into its time and coordinate parts, we can write K(t) = e−iHop t=˝ ;

(4.32)

Hop |l; n = Eln |l; n :

(4.33)

Here, the states |l; n are the complete set of eigenfunctions of Hop . Since a function with the states |l; n can be expressed by ln (x1 ; x2 ) = x1 ; x2 |l; n , the propagator at t ¿ 0 becomes K(x1 ; x2 ; t; x1 ; x2 ; 0) = x1 ; x2 |e−iHop t=˝ |x1 ; x2

= x1 ; x2 |ln ln|eiHop t=˝ |l ; n l ; n |x1 ; x2

l

=

n

l

n

l

n

ln (x1 ; x2 )e−iEln t=˝ ∗ln (x1 ; x2 ) :

(4.34)

Eq. (4.34) should be same as Eq. (4.16). Setting x1 = x1 and x2 = x2 in Eq. (4.16) and integrating over x1 and x2 , we obtain d x1 d x2 ∗ln (x1 ; x2 )e−iHln t=˝ ln (x1 ; x2 ) = e−iEln t=˝ ; (4.35) l

n


and

109

√ m! √ { 3=[sin(!t) sin( 3!t)]}1=2 2i˝

√ im! 2 2 ×exp [(x1 + x2 ) − 3(x1 − x2 ) ][cot(!t) − 1=sin(!t)] 2i˝ √ = − 12 [sin(!t=2) sin( 3=2!t)]−1 :

d x1 d x2

(4.36)

Hence one has √ e−iEln t=˝ = − 12 [sin(!t=2) sin( 3=2!t)]−1 l

n

√

√

= [e−i!t=2 =(1 − e−i!t )][e−i 3!t=w =(1 − e−i 3!t )]

∞ ∞ √ 1 1 exp −i!t l + + 3 n+ : = 2 2

(4.37)

l=0 n=0

Therefore, the form of the expectation values of CHO becomes √ Eln = [(l + 12 ) + 3(n + 12 )]˝! :

(4.38)

The forms of the propagators [Eqs. (4.16) and (4.29)] for CHO and CDHO obtained through the path integral method are new. Setting f(t) = 0, Eq. (4.29) is reduced to Eq. (4.16). Although CDHO is a nonconservative system, the quantum mechanical problem for the momentum operator does not appear because the canonical momentum is equal to the kinetic momentum [34,91]. The energy expectation value, Eq. (4.38), for CHO is the sum of two energy expectation values corresponding to the quantum states of two oscillators. One may easily surmise that the wavefunction for CHO will be given by the multiplication of two wavefunctions for two oscillators. In the case of CDHO, one cannot apply Eq. (4.29) to obtain the energy expectation values because this equation cannot be expressed in the form of Eq. (4.34) and the energy operator is not equal to the Hamiltonian operator in a nonconservative system. Here the wavefunctions and energy expectation values for CDHO are not evaluated. 4.5. Propagator of coupled damped driven harmonic oscillator chains We introduce N harmonic oscillators that are coupled together by N + 1 springs and assume that the masses of the oscillators, spring constants and decay constants are all the same for simplicity [142]. Let each of their displacements be xj , whereby the Hamiltonian for N -CDDHO can be written as N N 1 −t 2 2 t H =e pj =2m + m!0 e [xj2 − xj (xj+1 + xj−1 )] 2 j=1

−

N j=1

mfj (t)xj et ;

j=1

x0 = xN +1 = 0 ;

(4.39)

110


where fj (t) is a time-dependent external force exerted on the jth oscillator. Hamilton’s equation of motion is the same as Eqs. (4.2) – (4.5): x˙j = e−t pj =m;

(j = 1; 2; : : : ; N ) ;

(4.40)

p˙ j = 12 m!02 et [xj+1 + xj−1 − 2xj ] :

(4.41)

From Eqs. (4.39) – (4.41), the Lagrangian becomes N

1 L = met [x˙2j − !02 [xj2 − (xj+1 + xj−1 )] + fj (t)xj ] 2

(4.42)

j=1

with the corresponding equation of motion xKj + x˙j + !02 (− 12 xj−1 + xj − 12 xj+1 ) = fj (t) ;

(4.43)

which can be expressed in matrix form as XK + x˙ + !02 Wx = f(t) :

(4.44)

Here, x and f(t) are column vectors with N components and the matrix W is given by ) ) ) 1 ) − 1=2 0 0 · · · · 0 ) ) ) −1=2 1 −1=2 0 · · · · 0 )) ) ) 0 −1=2 1 −1=2 · · · · 0 )) ) ) 0 0 −1=2 1 · · · · 0 )) ) ) · · · · · · · · )) : W =) · (4.45) ) · ) · · · · · · · · ) ) ) 0 · · · · −1=2 1 −1=2 0 )) ) ) 0 · · · · 0 −1=2 1 1=2 )) ) ) 0 · · · · 0 0 −1=2 1 ) The classical solutions of Eq. (4.44) are −t

xj = e

N

Qjk ck ei!k t ;

(4.46)

k=1

x = e−t Qc(t) :

(4.47)

Here, the matrix Q is composed of the eigenvectors of the matrix W , and c(t) is a vector with the elements {ck ei!k t }. The factor wk , which includes the eigenvalues of W , is given as !k = (%k !02 − 2 =4)1=2 :

(4.48)

The Lagrangian, Eq. (4.42) consists of a quadratic form of the 2N variable set {xj ; x˙j }. Thus the propagator can be written in the form of Eq. (2.52) [62],

1 K(x; t; x t ) = F(t; t ) exp Sc (X; t; X ; t ) : (4.49) ˝


111

The multiplicative function F(t; t ) has the same form as Eq. (2.82),

∞ n−1 N 1 dyik F(t; t ) = lim n→∞ 0 A0 Aj ×exp

i h˝t

j=1 k=1 t

t

ds(m=2) exps [y(s) ˙ T y(s) − !02 y(s)Wy(s)] ;

(4.50)

where y is the deviation of x from its classical path xc given by yi = xj − xcj (j = 1; 2; : : : ; N ). It is convenient to change the variable {yj } into {zj } by z = QT y :

(4.51)

Then the multiplicative function can be expressed as  

∞ n−1 N i t m  d zj 1 F(t; t ) = lim J exp ds es (z˙2j − !j2 zj2 ) ; n→∞  ˝ t  A0 Aj 2 0 j=1

(4.52)

j=1

where J is the Jacobian and becomes unity. As we have mentioned in the derivation of Eq. (4.13), the action can be separated into the N functionals with just the same variables in the path integral, and thus F(t; t ) can be written as the multiplication of N path integrals, each of them being a path integral of the damped harmonic oscillator. Therefore, the multiplicative function can be expressed readily as

F(t; t ) =

N

[m!j e(t+t )=2 =2i˝ sin !j (t − t )]1:2 :

(4.53)

j=1

Then the propagator of CDDHO becomes

m! 1=2 t im exp (x02 − et x2 ) K(x; t; x0 ; 0) = + 2 sin !t 4 4˝ 2! 2 t 2 t=2 + ; [(x e + x0 ) cos !t − 2e x0 x] sin !t

(4.54)

where ! = [!02 − 2 =4]1=2 :

(4.55)

It is well known that if the Lagrangian is a quadratic form of x and x , then the action is also a quadratic form of the variables of x and x . Thus, as we have done in Eq. (4.17) or Eq. (3.19), we can write the propagator

K(x; t; x ; t ) = exp[xT A(t; t )X + xT B(t; t )x + x T C(t; t )x

+ xT D(t; t ) + x T E(t; t ) + G(t; t )] :

(4.56)

Eq. (4.56) should satisfy the SchrKodinger equation, Eq. (2.51), and its complex conjugate. Substituting Eqs. (4.29) and (4.56) into Eq. (2.41) and its complex conjugate, one obtains a

112


time-dependent system of nonlinear matrix diHerential equations: im!02 dA 2i˝ −t T = e A A− W ; (4.57) dt m 2h 2i˝ −t T d (4.58) (Bx + D) = e A (Bx + D) − met {f(t)}=i˝ ; dt m 2i˝ −t d T (4.59) (x Cx + Ex + G) = e [(Bx + D)T (Bx + D) + 2TA] ; dt m im!02 dC 2i˝ −t T = e C C+ W ; (4.60) dt m 2h 2˝ d (4.61) (Bx + E) = e−t C T (Bx + E) + met {f(t)}=i˝ ; dt im ˝ −t d T (4.62) (x Ax + Dx + G) = e [(Bx + E)T (Bx + E) + 2TC] : dt 2mi All the matrices and vectors in Eqs. (4.57) – (4.62) are functions of (t; t ), and {f(t)} is a vector with components fj (t). Diagonalization of the matrix W and comparison of the results with the known result, Eq. (4.54) yields the solutions of the basic diHerential equations, A(t; t ) = QU (A) QT ; where

(4.63)

im t im t e !j cot !j (t − t ) − e &j; k ; 2˝ 4˝ and similarly we get

Uj;(A) k

B(t; t ) = QU (B) ;

(4.64) (4.65)

(t+t )=2 =i˝ sin !j (t − t )]&j; k ; Uj;(B) k = [m!j e

(4.66)

C(t; t ) = QU (C) QT ;

im t im t (C) Uj; k = e !j cot !j (t − t ) + e &j; k ; 2˝ 4˝ D(t; t ) = − QV (t; t ); E(t; t ) = QV (t; t ) ;

t t =2 −p=2 Vj (t ; t) = ime = ˝ sin !j (t − t ) dp sin !j (p − t)e fj (p) ; t

t t im 1 G(t; t ) = dq dp sin !j (t − p) 2˝ !j sin !j (t − t ) t t j

(4.67)

(4.68) (4.69) (4.70)

×sin !j (t − q)e−(p+q)=2 fj (p)fj (q)

t

t −1 −(p+q)=2 + !j dp dq sin !j (q − p)e fj (p)fj (q) t

t

+ ln[sin−1=2 !j (t − t )] + (t + t )N=4 :

(4.71)


113

Finally, we obtain the propagator for CDDHO as   1=2    N   N=2  (t+t )=2  =2] !j =sin !j (t − t ) K(x; t; x ; t ) = [me     j=1   N N N  im  Qlj Qkj {[et !j cot !j (t − t ) − et =2]xl xk  2˝ j=1 k=1 l=1

+ [et !j cot !j (t − t ) + et =2 ]xl xk } +

N N

Qkj {[2e(t+t )=2 !j =sin !j (t − t )]xj xk }

j=1 k=1

−

N N

2Qkj =sin !j (t − t )

et=2

j=1 k=1

t =2

− e

+

t

t

N

−p=2

dp sin !j (p − t)e

1 !j sin !j (t − t )

j=1

t

t

dq

t

t

t

t

dp sin !j (p − t )e−p=2 fj (p) xk

fj (p)

xk

dp sin !j (t − p)

×sin !j (t − q)e−(p+q)=2 fj (p)fj (q)

+ !j−1

t

t

dp

t

t

dq sin !j (q − p)e−(p+q)=2 fj (p)fj (q)

   : 

(4.72)

In the simplest cases of two and three coupled harmonic oscillators, the matrices W2 and W3 are ) ) ) 1 ) ) −1=2 0 )) ) ) 1 ) 1=2 ) 1 −1=2 )) : W2 = )) ; W3 = )) −1=2 (4.73) −1=2 1 ) ) 0 −1=2 1 ) Since the eigenvalues of W2 and the corresponding eigenvectors are 1=2, 3=2 and {2−1=2 ; 2−1=2 }, {2−1=2 ; −2−1=2 }, respectively, the frequency !2 and matrix Q2 are given as !2(1) = (!02 − 2 =4)1=2 ; !2(2) = (!02 − 2 =4)1=2 ; ) −1=2 ) −1=2 ) )2 2 ) ; Q2 = )) −1=2 2 −2−1=2 )

(4.74)

where the subscripts on ! and Q represent the number of coupled oscillators. In the same way, the eigenvalues of W3 are {1; 1 + 2−1=2 ; 1 − 2−1=2 } and the corresponding eigenvectors

114


{−2−1=2 ; 0; 2−1=2 }, {1=2; −2−1=2 ; 1=2}, with w3 and the matrix Q3 given by

!3(1) = (!02 − 2 =4)1=2 ;

!3(2) = [!02 (1 + 2−1=2 ) − 2 =4]1=2 ;

!3(3) = [!02 (1 − 2−1=2 ) − 2 =4]1=2 ; ) −1=2 ) ) −2 1=2 1=2 )) ) Q3 = )) 0 −2−1=2 2−1=2 )) : ) 2−1=2 1=2 1=2 )

(4.75)

The exact propagator for the coupled damped driven harmonic oscillators has been obtained via the path integral method. The propagator of CDDHO can be applied to various Jelds of physics, such as the generation and detection of squeezed and correlated light [143,144], the theory of lasers and masers [145 –147], the parametric excitation of squeezed states in time-varying quantum chains [148,149], the nonstationary Casimir eHect [150], and so on. The exact propagators for CHO, CDHO and CDDHO obtained in this section are new. The energy expectation values for the CHO are given by the sum of two energy expectation values corresponding to the quantum states of two oscillators. Calculations of the wave function, energy eigenvalues and uncertainty product of CDDHO are in progress. 5. Harmonic oscillator with time-dependent frequency and external force In general, the solutions of the SchrKodinger equation for explicit time-dependent systems have been investigated by many authors [105,151–154]. Camiz et al. [155] and Khandekar and Lawande [69,70] have obtained the wavefunctions of a time-dependent harmonic oscillator with or without an inverse quadratic potential. In this section we will study the propagator, wavefunction, energy expectation values, uncertainty relation and coherent states for a forced harmonic oscillator with a time-dependent frequency through the path integral method [156,157], with the delta-pulse excitation of the medium as an example [158]. 5.1. Propagator We Jrst consider a forced harmonic oscillator with time-dependent frequency !(t), whose Hamiltonian is of the form p2 m H= (5.1) + !2 (t)x2 − f(t)x ; 2m 2 where f(t) is an external driving force. The corresponding Lagrangian is L = 12 mx˙2 − 12 m!2 (t)x2 + f(t)x ;

(5.2)

with the classical equation of motion 1 d2 x + !2 (t)x = f(t) : (5.3) m dt 2 For the case !(t) = !0 , the solution of Eq. (5.3) represents harmonic motion; otherwise, it is not easy to evaluate the exact solution.


115

Since the Lagrangian is quadratic, we can assume the propagator to have the form [34,140,159]

K(x; t; x ; t ) = exp[a(t; t )x2 + b(t; t )xx + c(t; t )x 2 + g(t; t )x + h(t; t )x + d(t; t )] :

(5.4)

Since the propagator and the solution of the SchrKodinger equation are given by Eqs. (2.51) and (3.9), substitution of Eq. (5.4) into (3.9) and its complex conjugate yields the diHerential equations for the time-dependent coeMcients in Eq. (5.4): da 2i˝ 2 m 2 ! (t) ; (5.5) = a + dt m 2i˝ d 2i˝ i (5.6) (bx + g) = a(bx + g) + f(t) ; dt m ˝ d i˝ i˝ (5.7) (cx 2 + hx + d) = a + (bx + g)2 ; dt m 2m dc 2˝ 2 im = c + !(t )2 ; (5.8) dt mi 2˝ d 2˝ f(t ) (bx + h) = ; (5.9) c(bx + h) + dt mi i˝ d ˝ ˝ (ax2 + gx + d) = (bx + h)2 + c ; (5.10) dt 2im mi while it is not straightforward, it is not diMcult to solve these diHerential equations. The transformation of variables into complex variables yields the real and imaginary parts of the differential equations, which can be solved easily. Performing this procedure, one can obtain the time-dependent coeMcients as

im 8˙ a(t; t ) = (5.11) + ˙ cot( − ) ; 2˝ 8

im 8˙ − + ˙ cot( − ) ; (5.12) c(t; t ) = 2˝ 8 " im ˙˙ b(t; t ) = − ; (5.13) ˝ sin( − ) "

t i ˙ f(s) sin[(s) − (s)] ; ds " (5.14) g(t; t ) = ˝ sin( − ) t (s) ˙ "

t i ˙ f(s) ds " (5.15) sin[ − (s)] ; h(t; t ) = ˝ sin( − ) t ˙

(s)

t t f(s) f(p) i " d(t; t ) = − ds − (s)] dp " sin[ sin[(p) − ] ; (5.16) 2m˝ sin( − ) t (p) ˙ (s) ˙ t where the relation between 8(t) and (t) is given by q(t) = 8(t)ei(t) ; and the function q(t) satisJes the diHerential equation d 2 q(t) + !2 (t)q(t) = 0 : dt 2

(5.17) (5.18)

116


Here, 8(t) and (t) are real quantities. From Eqs. (5.17) and (5.18), the real and imaginary parts of the diHerential equation are given as 8(t) K − 8(t)(t) ˙ 2 + !2 (t)8(t) = 0 ;

(5.19)

28(t) ˙ (t) ˙ + 8(t)(t) K =0 ;

(5.20)

8(t)2 (t) ˙ = ;

(5.21)

where the constant is a time-independent quantity. Substitution of Eqs. (5.11) – (5.16) into Eq. (5.4) gives the propagator for the forced timedependent harmonic oscillator as

m(˙) ˙ 1=2 K(x; t; x ; t ) = 2i˝ sin( − )

im 8˙ 2 8˙ 2 ×exp x − x 2˝ 8 8 1 im 2 2 ( x ˙ + ˙ x ) cos( − ) − 2 ˙˙ xx ×exp 2˝ sin( − ) " t " 2 ˙ 2 ˙ t f(s) f(s) + ds " ds " sin[(s) − ] + sin[ − (s)] x x m m (s) ˙ (s) ˙ t t

t f(s) f(p) 1 t − 2 ds " dp " : (5.22) sin[ − (s)] sin[(p) − ] m t (s) ˙ (p) ˙ t Here, the unprimed and the primed variables denote the quantities which are functions of the times t and t , respectively. In the case of !(t) = !0 (just a real constant), one has 8(t) = 1 and (t) = !0 t, and then the propagator (5.22), reduces to the usual expression for a forced harmonic oscillator. 5.2. Wavefunctions In order to derive the wavefunction, we rewrite the propagator in another form:

1=2

t 2" f(s) m(˙) ˙ 1=2 im 8˙ 2 K(x; t; x ; t ) = exp x ˙ ds " cos[ − (s)] x + 2i˝ sin( − ) 2˝ 8 M (s) ˙

t 1 f(s) 8˙ 2 2 ˙ x ds " cos[ − (s)] − x + 8 m (s) ˙  2

 " f(s) im 1 t cot( − ) x ˙ − ds " sin[ − (s)] ×exp  2˝ m (s) ˙


+

1

1 m

˙ x −

1 ˙ x − m

×exp

i 2˝m

1 x ˙ − m

2 sin( − )

t

f(s) ds " sin[ − (s)] (s) ˙

f(s) ds " sin[ − (s)] (s) ˙

t

cot( − )

t

+ cot( − )

2 − sin( − )

f(s) ds " sin[ − (s)]  (s) ˙

+

2  

"

im 1 − ˝ sin( − ) 1

t

117

t

t

t

t

2 f(s) sin[ − (s)] ds " (s) ˙

2 f(p) dp " sin[ − (p)] (p) ˙

ds sin( − (s))

t

f(p) dp " sin[ − (p)] (p) ˙

f(s) ds " sin( − (s)) (s) ˙

t

t

f(p) dp " sin[(p) − ] (p) ˙

(5.23)

=

m 1=2 e−i(− ) (˙ )1=4 ()1=2 ˝ 1 − e−2i(− )

t f(s) im 8˙ 2 2 " ×exp x ˙ ds " cos[ − (s)] x + 2˝ 8 m (s) ˙

1

t f(s) 8˙ 2 2 x + ˙ x ds " cos[ − (s)] 8 m (s) ˙   2

t  " f(s) m 1 x ˙ − ds " sin[ − (s)] ×exp  2˝  m (s) ˙

−

+

1

1 ˙ x − m

t

2   f(s)  ds " sin[ − (s)]  (s) ˙

118


×exp

−m

˝[1 − e−2i(− ) ]

1

"

1 x ˙ − m

t

t

f(s) ds " sin[ − (s)] (s) ˙ 2

f(s) sin[ − (s)] ds " (s) ˙

" 1 t f(s) sin[ − (s)] −2 x ˙ − ds " m (s) ˙ 1

1 t f(s) ˙ x − ds " e−i>(t) e−i>(t ) ; sin[ − (s)] × m (s) ˙ +

where

1 ˙ x − m

(5.24)

2 f(s) ds " sin[ − (s)] (s) ˙ 2 t f(s) + cot( − ) ds " sin[ − (s)] (s) ˙

t

t f(s) f(p) 1 +2 ds " dp " sin[ − (s)] sin[ − (p)] sin( − ) (s) ˙ (p) ˙

t

t f(s) f(p) − ds " dp " (5.25) sin[ − (s)] sin[(p) − ] : (s) ˙ (p) ˙ t t

1 {cot( − ) >(t ) − >(t) = 2˝m

2

t

We can obtain directly the wavefunction from the deJnition of the propagator, Eq. (2.86), and from Mehler’s formula [160] ∞ exp[ − (X 2 + Y 2 − 2XY )=(1 − Z 2 )] Zn −(X 2 +Y 2 ) √ = exp (5.26) Hn (X )Hn (Y ) ; 2n n! 1 − Z2 n=0

where

1 t f(s) m " X= x ˙ + ds " sin[ − (s)] ; ˝ m (s) ˙

f(s) m " 1 t Y= x ˙ − ds " sin[ − (s)] ; ˝ m (s) ˙

Z = e−i(− ) :

(5.27) (5.28) (5.29)

Substitution of Eqs. (5.26) – (5.29) into Eq. (5.24) and comparison of the result with Eq. (2.86) give the wavefunction as n (x; t) = exp{i[>(t) − (n + 12 )(t)]}n (x; t) ;

(5.30)


where

119

1=2 1=2

t f(s) 1 ˙ im 8˙ 2" exp x ˙ ds " cos[ − (s)] n (x; t) = n x− 2 n! ˝ 2˝ 8 m (s) ˙  2 

t m "  f(s) 1 x ˙ − ds " cos[ − (s)] ×exp  2˝  m (s) ˙

×Hn

m ˝

"

1 x ˙ − m

t

f(s) ds " : sin[ − (s)] (s) ˙

(5.31)

The wavefunction, Eq. (5.30), is merely a unitary transformation of n (x; t), where n (x; t) satisJes all the properties associated with n (x; t);

∗ d xm n = m | n = d x∗n n − &m; n (5.32) and the expectation value of the operator O is

∗ m | O | n = d x m O n = d x∗n On :

(5.33)

5.3. Energy expectation values The Hamiltonian of Eq. (5.1) and Lagrangian of Eq. (5.2) represent the time-dependent energy. Therefore, one must derive a time-invariant energy operator. Let 1(t) be a particular solution of Eq. (5.3), and making use of Eqs. (5.19) and (5.21), one can express the energy as m 2 @ 88˙ ˝2 82 @2 2 2 Eop = − + (8˙ + 8˙ )x − 2x + 1 2m @x2 2 2 @x m m ˙ ˝ @ + m8x) ˙ 2: + (18˙ − 81)(i ˙ + m8˙2 1x + 8˙2 12 + (18˙ − 81) @x 2 2 Eq. (5.31) can be simpliJed to the form

1=2 & 2 2 2 n (x; t) = n √ eiCx +i%x e−1=2& (x−1) Hn [&(x − 1)] 2 n!

1=2 & 2 eAx +Bx Hn [&(x − 1)] ; = n √ 2 n! where the new coeMcients are m 1=2 ; &= ˝ m 8˙ C(t) = ; 2˝ 8

(5.34)

(5.35)

(5.36) (5.37)

120


%(t) =

1" ˙ ˝

1 1(t) = " m ˙

t

f(s) ds " cos[ − (s)] ; (s) ˙ t

(5.38)

f(s) ds " sin[ − (s)] ; (s) ˙

(5.39)

A = − iC − &2 =2 ;

(5.40)

B = i% + 1&2 :

(5.41)

Here, 1(t) is the particular solution of Eq. (5.3). In the evaluation of the energy expectation values, Emn , it is straightforward to calculate the following: √ 1 √ x | n = √ [ n + 1 | n + 1 + n | n − 1 ] ; (5.42) 2& " 1 " (5.43) x2 | n = 2 [ (n + 2)(n + 1) | n + 2 + (2n + 1) | n + n(n − 1) | n − 2 ] ; 2& A √ ˝ ˝ A &+ 2n | n − 1 ; (5.44) p | n = 2(n + 1) | n + B | n + i& i i &

2 √ AB √ " 2 2 2A (n + 2)(n + 1) | n + 2

+ 2 2 n + 1 | n

p | n = −˝ &2 & √ AB √ A2 + 2 A + 2 (2n + 1) | n + 2 2 n | n − 1

+ &B & & 2 " A 2 +2 + 2A + & n(n − 1) | n ; (5.45) &2 B √ ˝ A" √ (n + 2)(n + 1) | n + 2

+ n + 1 | n + 1

xp | n = i &2 2&

A + 2 (2n + 1) + n | n

& " B √ A +√ n | n − 1 + +1 n(n + 1) | n − 2 ; (5.46) &2 2& px | n = xp | n + | n : Substitution of the above quantities into Eq. as ˝ 2 En; n = En = (8 )(2n ˙ + 1) = ˝ n + 2

(5.47) (5.34) gives directly the energy expectation values 1 2

:

Therefore, the energy expectation value is a time-independent quantity.

(5.48)


121

5.4. Uncertainty product and coherent states Since the uncertainty product is deJned in Eq. (3.63), from Eqs. (5.42) – (5.46) we obtain 1=2 8˙2 1 (TxTp)n; n = 1 + 2 2 ˝; n+ (5.49) 2 ˙ 8 1=2 " 8˙2 (n + 2)(n + 1)˝ ; (5.50) (TxTp)n+2; n = 1 + 2 2 ˙ 8 1=2 " 8˙2 n(n − 1)˝ : (5.51) (TxTp)n; n+2 = 1 + 2 2 ˙ 8 Since the minimum uncertainty of Eq. (5.49) is larger than ˝=2, one needs the minimum uncertainty states. First, to obtain the coherent states, we can construct a creation operator a† and annihilation operator a for the time-dependent harmonic oscillator. The conditions for a coherent state are given in Section 3. From Eqs. (5.42) and (5.44) we obtain 1=2 8˙ p m˙ † a = 1+i x−i ; (5.52) 2˝ 8˙ m˙ 1=2 8˙ p m˙ 1−i x+i : (5.53) a= 2˝ 8˙ m˙ The representation (x; p) in terms of (a† ; a) is ˝ 1=2 † (a + a) ; x= 2m˙ 8˙ 8˙ ˝m˙ 1=2 −i a : p= + i a† + 2 8˙ 8˙

(5.54) (5.55)

Here, the commutation relations are preserved: [x; p] = i˝ ;

(5.56)

[a; a† ] = 1 :

(5.57)

Since the coherent state can be deJned as the eigenstate of the non-Hermitian operator a [106], i.e., Eq. (3.74), one can get the x representation of the coherent state from Eq. (5.53): 1=2 8˙ m˙ 2m ˙ −1 + i x | ) = N exp x2 + )x : (5.58) 2˝ 8˙ ˝ The constant N should satisfy Eq. (3.77). The eigenvector of the operator a in the coordinate representation | x becomes 1=2 1=2 8˙ m˙ m˙ 2m ˙ 1 1 −1 + i exp x2 + )x − | ) |2 − )2 : (5.59) x | ) = ˝ 2˝ 8˙ ˝ 2 2

122


Next, we show that a coherent state is a minimum uncertainty state. Making use of Eqs. (5.53), (5.54) and (3.70), we evaluate the expectation values of x; p; x2 and p2 in the state | ) : ˝ 1=2 ∗ ) | x | ) = () + )) ; (5.60) 2m˙ m˝˙ 1=2 i i ∗ ) | p | ) = −i ) ; (5.61) +i ) + 2 8˙ 8˙ ˝ ) | x2 | ) = ) | a+2 + a2 + aa+ + a+ a | )

2m˙ ˝ = (5.62) ()∗2 + )2 + 2))∗ + 1) ; 2m˙ 2 2 2 ˝ m ˙ 8 ˙ 8 ˙ 8 ˙ ) | p2 | ) = − i )2 + + 1 (2))∗ + 1) : (5.63) + i )∗2 + 2 8˙ 8˙ 8˙ Then the uncertainty product is TxTp = [() | x2 | ) − ) | x | ) )() | p2 | ) − ) | p | ) ]1=2 2 1=2 ˝ 8˙ = 1+ : 2 8˙

(5.64)

This is the minimum value allowed by Eq. (5.49). Setting !(t) = !0 gives the results for the forced harmonic oscillator. For the case where f(t) = 0 and !(t) = !0 , all results reduce to those of the simple harmonic oscillator. The Hamiltonian, Lagrangian and mechanical energy have the units of energy but are not time independent. For this reason, we used the time-independent operators and derived the energy operator from the classical equation of motion and used it to calculate the energy expectation values. This energy operator is very similar to the Ermakov–Lewis invariant operator [151,152]. The uncertainty product is time-dependent, but consistent with Heisenberg’s uncertainty principle. The coherent states have also been constructed. 5.5. Applications 5.5.1. Mode of the electromagnetic :eld in a resonator :lled with a time-dependent dielectric medium There are two general time-dependent systems among the various physical systems. One is formed through its own environmental conditions, and the other is formed due to existence of external forces described in terms of a time-dependent electromagnetic Jeld. We will study the propagator, wavefunctions and coherent states in the quantization of an electromagnetic Jeld inside a resonator Jlled with a dielectric medium acted upon by a time-dependent electromagnetic Jeld [158]. Consider the so-called “one-dimensional electromagnetics” where the linearly polarized, mutually perpendicular electric and magnetic Jelds depend only on the single space variable 9 and


time t. Then Maxwell’s equation becomes @E 1 @B 1 @B 1 @D 4 = ; = + J; @9 c @t C(t) @9 c @t c

D = j(t)E :

Introducing the usual vector potential as 1 @A @A E=− ; B=− ; c @t @9

123

(5.65)

(5.66)

we obtain the second-order diHerential equation

1 @ @A 1 @2 A 4 j − (t) = j(9; t) ; c2 @t @t C(t) @92 c which coincides with Euler’s equation @L @L @L @ @ − + =0 ; @t @(@A=@t) @9 @(@A=@t) @A

(5.67)

(5.68)

for the Lagrangian density

1 1 j(t) @A 2 1 @A 2 − + JA : L= 2 8 c @t C(t) @9 c The canonically conjugated density is @L j(t) @A 1 = = =− D; 2 @(@A=@t) 4c @t 4c

(5.69)

(5.70)

and thus the Hamiltonian is given by

1 @A D2 1 1 @A 2 − JA : − L= W = + @t 8 j(t) C(t) @9 c

(5.71)

One may assume the Jeld to occupy an ideal resonator with the walls at the points 9 = 0 and 9 = L. Then the mode is decomposed into

3=2 n 2 D(9; t) = c nx n (t) sin (5.72) 9 ; L L n

1=2 n 8L Pn (t) (5.73) A(9; t) = sin 9 : n L n Substitution of Eqs. (5.72) and (5.73) into Eq. (5.71) yields the Hamiltonian of a countable set of noninteracting time-dependent forced harmonic oscillators:

L cn 2 x2 1 P2 n n H (t) = W (t; 9) d9 = + fn (t)Pn ; (5.74) + 2 C(t) L j(t) 0 n where

fn (t) = −

0

L

8L c2 n2

1=2

j(9; t) sin

n

L

9 d9 :

(5.75)

124


To quantize the Hamiltonian, Eq. (5.74), one may treat the variable Pn as a generalized momentum and variable x n as a generalized coordinate to satisfy the commutation relations [161–163]. Then this situation corresponds to the harmonic oscillator with a time-dependent frequency. It is suMcient to consider the function C(t) = const: = 1. Since all modes in Eq. (5.74) are uncoupled, one may omit the number of modes and consider only a single-mode Hamiltonian Hˆ = 12 [pˆ 2 + !2 (t)xˆ2 ] + f(t)pˆ : !2 (t) =

(5.76)

[cn]2 : L2 j(t)

(5.77)

Here, one should recognize that the Hamiltonian is coupled to an external time-dependent current represented by the function f(t), not through the coordinate, but through the momentum. Since the Hamiltonian, Eq. (5.74), is quadratic, one can assume the propagator to have the form of Eq. (5.4). Through the same process used in the derivation of Eqs. (5.22) – (5.24), we obtain the propagator as 1=2

)]1=2 [ (t) ˙ (t ˙ i 8(t) ˙ 8(t ˙ ) 2 2 K(x; t : x ; t ) = exp x − x 2˝sin (t; t ) 2˝ 8(t) 8(t ) i 2 cos (t; t )[(t)x ˙ + (t ˙ )x2 ] − 2[(t) ˙ (t ˙ )]1=2 xx 2˝sin (t; t )

1=2

t d (t) ˙ − 2x dJf(J) sin (J; t ) dJ (J) ˙ t

t d (t ˙ ) 1=2 dJf(J) sin (t; J) − 2x dJ (J) ˙ t t

t d sin (t; J) d sin (s; t ) −2 dJf(J) dsf(s) : 1=2 1=2 dJ ((J)) ds ((s)) ˙ ˙ t t

+

(5.78)

If we consider a complex solution of Eq. (5.18) satisfying the additional condition qq ˙ ∗ − q˙∗ q = 2i ;

(5.79)

which is equivalent to the choice = 1 in relation (5.21), we then have a non-Hermitian operator [106,164], i.e., the linear integral of motion

−1=2 ˆ = i(2˝) A(t) q(t)pˆ − q(t) ˙ xˆ + dJf(J)q(J) ˙

t −1=2 i(t) i(J)−i(t) = i(2˝) e 8(t)pˆ − (8˙ + i8) ˙ xˆ + dJf(J)(8˙ + i8)e ˙ ; (5.80) t

and this operator satisJes the boson creation–annihilation operator relation †

ˆ [A(t); Aˆ (t)] = 1 :

(5.81)


125

Thus one can construct an overcomplete set of coherent states ) (x; t), satisfying both the SchrKodinger equation and the equation ˆ A(t) ) (x; t) = )) (x; t) ;

(5.82)

where ) is an arbitrary complex number. Through the same treatment as in Section 3.5 and a special case of general formulas for coherent states of multidimensional quadratic systems [165], we Jnd the coherent state in the x-representation as

˙ 1=4 ˙ i8˙ 2 1 1 ) (x; t) = exp − 1− x − i(t) − )2 e−2i ˝ 2˝ 8˙ 2 2

1=2

t i(t) √ d 1 2 ˙ e " − |)| + 2 e−i(t) x 2˝ ) − i dJf(J) 2 ˝ dJ (J) ˙ t

1=2

t d sin (t; J) 2 −i(t) " + e ) dJf(J) ˝ dJ (J) ˙ t

t d sin (t; J) d e−i(t; s) i t " " dJf(J) dsf(s) : (5.83) − ˝ t dJ dJ (J) ˙ (s) ˙ t The expectation values of x and p in these states are

1=2

t 2˝ sin (t; J) d " )|x|) = Re[)e−i(t) ] − dJf(J) ; ˙ dJ (J) ˙ (t) ˙ t

(5.84)

d )|x|) : (5.85) dt The variances (TX )2 and (TP)2 depend neither on ) nor on f(t), but are completely determined by the function q(t): ˝ ˝ (TX )2 = |q(t)|2 = ; (5.86) 2 2˙ 2 8 ˙ ˝ ˝ ˙ |2 = ˙ 1 + : (5.87) (TP)2 = |q(t) 2 2 ˙ )|p|) =

Thus the uncertainty product 2 2 8 ˙ ˝ 1+ (TX )2 (TP)2 = 4 ˙

(5.88)

varies in time and is greater than the minimal possible value ˝2 =4 provided 8˙ = 0. This is explained be fact that the state, (5.83) is really a correlated coherent state with nonzero correlated coeMcient [166] 8˙ xˆpˆ + pˆ xˆ =2 − xˆ pˆ

r= = 2 : (5.89) 1=2 (Fx Fp ) [8˙ + ] ˙ 1=2

126


The correlated coherent states minimize the SchrKodinger–Robertson generalized uncertainty relation [167], (5.90) (TX )2 (TP)2 [1 − r 2 ] ¿ ˝2 =4 : It is well known [99] that the coherent states form a generating function for the Fock states: ∞ )n |) = |n ; (5.91) (n!)1=2 n=0 ∗ Aˆ Aˆ |n = n|n : (5.92) Since the exponential function of a quadratic form is the generating function for the Hermite polynomials [160], we can obtain the formula x|n = n (x; t) = [2n n!]−1=2 e−in(t) ×Hn

˙ ˝

1=2

0 (x; t)

t

x+

t

d sin (t; J) dJf(J) ; dJ {(t) ˙ (J) ˙ }1=2

(5.93)

where 0 (x; t) is given by Eq. (5.83) with ) = 0. The functions of Eq. (5.93) are also eigenstates of the operator [153,154] ˆ = ˝ [Aˆ † Aˆ + 1=2] E(t) 1 ˙ xˆpˆ + pˆ x] ˆ = {82 pˆ 2 + [8˙2 + 82 ˙2 ]xˆ2 − 88[ 2 + 2Re F(t)e−i(t) [8[pˆ − i˙x] ˆ + 8˙x] ˆ + |F(t)|2 } ; where

F(t) =

t

t

dJf(J)[8˙ + i8]e ˙ i(J) ;

(5.94) (5.95)

and the functions 8(t) and (t) ˙ satisfy Eqs. (5.18) – (5.21) with the given positive constant , ! 1 ˆ n = 0; 1; 2 : : : : (5.96) E(t) n (x; t) = ˝ n + 2 n ; ˆ The operator E(t) is an integral of the motion coinciding with the usual energy operator in the case of the time-independent functions and f. Therefore, it can be called the generalized energy operator. With the use of the eigenstates, Eq. (5.93), one can evaluate the uncertainty products 1=2 8˙2 [TxTp]n; n = ˝ 1 + 2 2 [n + 1=2] ; (5.97) ˙ 8 1=2 8˙2 [(n + 2)(n + 1)]1=2 ; (5.98) [TxTp]n+2; n = ˝ 1 + 2 2 ˙ 8 1=2 8˙2 [(n − 1)n]1=2 ; (5.99) [TxTp]n; n+2 = ˝ 1 + 2 2 ˙ 8 where all other elements, except for the above three, are zero.


127

As an example, we introduce the simplest time dependencies of the eHective frequency and force in the form of delta impulses: !2 (t) = 2 + W&(t); W ¿ 0 ; f0 ; |t | ¡ a=2 ; a → 0; f → ∞; af → F = const : f(t) = 0; |t | ¿ a=2 :

(5.100) (5.101)

The dependence of Eq. (5.100) represents the fast ionization of the medium, i.e., the plasma window [168]. The Hamiltonian of this system has the form pˆ 2 mxˆ 2 Hˆ = (5.102) + [ + W&(t)] + f(t)pˆ : 2m 2 Following the procedure in the previous calculations and taking into consideration the results given in Ref. [168], one gets the linear integral of motion for Eq. (5.102) as pˆ i t W xˆ 1 i t 2 A(t) = 1=2 [e − sin t] − [i e − W cos t] + &(t); t ¿ 0 ; (5.103) p0

x0 2 where m 1=2 p0 = [˝m ]1=2 ; x0 = [˝=m ]1=2 ; &(t) = (i F − FW=2) ; ˝

F = lim f0 a : (5.104) a→0 f0 →∞

ˆ and its Hermitian conjugate satisfy the commutation One can easily verify that the operator A(t) † ˆ Aˆ ] = 1. relation, [A; The ground state of the quantum oscillator transforms to a correlated coherent state, Eq. (5.83), after &-kicking the frequency with the variances, Eqs. (5.86) and (5.87) equal to

W2 2 ˝ W 2 (Tx) = 1 + 2 sin t − sin 2 t ; (5.105) 2m

1 W2 2 W 2 1 + 2 cos t + sin 2 t : (5.106) (Tp) = 2m

The correlation coeMcient of Eq. (5.89) is nonzero after &-kicking and is equal to r = 1 − {1 + [W 4 =4 4 ]sin2 2 t + [W 2 = 2 ]cos2 t + [W 3 =2 3 ]sin 4 t }−1 :

(5.107)

The squeezing coeMcient k = m (Tx=Tp) is not equal to unity and is given by the formula 1=2 1 + W 2 = 2 sin2 t − W= sin 2 t k= : (5.108) 1 + W 2 = 2 sin2 t + W= sin 2 t The exact propagator, wavefunction, energy expectation values, uncertainty relations and coherent states for a harmonic oscillator with a time-dependent frequency and an external driving time-dependent force have been evaluated rigorously. As an example, the exact propagator, coherent states and Fock states have been obtained for the quantum oscillator with time-dependent frequency linearly coupled via momentum with an external current.

128


6. Time-dependent bound and unbound quadratic Hamiltonian system: dynamical invariant method As we have seen in the previous sections, much attention has been paid to the quantum mechanical solution of the oscillator system with a time-dependent Hamiltonian [169 –173]. In this section we investigate the exact quantum theory of a general time-dependent bound and unbound quadratic Hamiltonian system [174,175] through the dynamical invariant and path integral method and Jnd the relation between the quantum mechanical solution and the dynamical invariant [176 –182]. 6.1. Time-dependent bound quadratic Hamiltonian 6.1.1. Classical invariant One may consider a system with a time-dependent quadratic Hamiltonian of the type H = 12 [A(t)p2 + B(t)(xp + px) + C(t)x2 ] ;

(6.1)

where x and p are the canonical coordinate and its conjugate momentum, A(t) is a nonzero time-dependent function, and B(t) and C(t) are time-dependent functions of arbitrary form, which are continuously diHerentiable with respect to time. From Hamilton’s equations of motion, the classical equation of motion becomes ˙ ˙ A(t)B(t) A(t) 2 ˙ xK − − B (t) − B(t) x=0 : (6.2) x˙ + A(t)C(t) + A(t) A(t) Introducing the new variable q as x = q exp B(t) dt ; Eq. (6.2) can be simpliJed to ˙ A(t) qK + 2B(t) − q˙ + A(t)C(t)q = 0 : A(t)

(6.3)

(6.4)

However, the solution with a general form for arbitrary time-dependent coeMcients is not known. For simplicity let us write Eq. (6.2) as xK + =(t)x˙ + 9(t)x = 0 :

(6.5)

A general solution of Eq. (6.5) can be expressed in the form x = 8(t)ei(t) ;

(6.6)

where the functions 8(t) and (t) must be determined from Eq. (6.5). Substitution of Eq. (6.6) into Eq. (6.5) yields the real and imaginary parts of Eq. (6.5) as 8K − 8˙2 + =(t)8˙ + 9(t)8 = 0 ;

(6.7)

8K + 28˙˙ + 9(t)8˙ = 0 :

(6.8)


129

The Jrst invariant quantity (t) can be found from Eq. (6.8) in the form 82 ˙

= : (6.9) A(t) Eq. (6.9) is a time-invariant quantity with an auxiliary condition given by Eq. (6.5). If is not equal to zero, then is not constant and the position has the form of a complex function of time. Since the particle of the system will pass through more than two points on the trajectory, the motion of the system will be found in some restricted region. If is equal to zero, the motion of the system will be unbound. It is possible to Jnd another classical invariant quantity with an auxiliary condition given by Eq. (6.5). Assume that this invariant quantity I (t) is given as ! I (t) = 12 )(t)p2 + 21(t)xp + &(t)x2 ; (6.10) where )(t); 1(t) and &(t) are all real time-dependent functions. From Hamilton’s equations of motion, the time derivative of I (t) becomes dI @I = + [I; H ] : (6.11) dt @t Combining Eqs. (6.1) and (6.10) with Eq. (6.11), we obtain the coupled diHerential equations of )(t); 1(t) and &(t). The solution of these equations yields ) = 82 (t) ;

(6.12)

1 B(t) 2 8 − 88˙ ; (6.13) A(t) A(t)

2 B(t) 1 &= (6.14) 8− 8˙ + 2 ; A(t) A(t) 8 together with the subsidiary conditions, i.e., Eqs. (6.7) and (6.8). Finally, we obtain the invariant quantity as

2

2 B(t) 1 I (t) = : (6.15) x + 8− 8˙ x + 8p 8 A(t) A(t) 1=

Note that I (t) is always positive in the unbound system, and negative in the bound system. 6.1.2. Propagator and wavefunction The propagator for a quadratic Hamiltonian has the following form [62]:

1=2 1 @2 Sc K(x; t; x ; t ) = e(i=˝)Sc 2i˝ @x@x i 2 2 = exp [a(t; t )x + b(t; t )xx + c(t; t )x d(t; t )] : (6.16) ˝ Instead of Eq. (6.16), however, one may introduce the deJnition of the propagator for the bound system given by Eq. (2.86): ∗ K(x; t; x ; t ) = (6.17) n (x; t) n (x ; t ) : n

130


For the unbound system, the propagator is given as

K(x; t; x ; t ) = d k k (x; t) k∗ (x ; t ) :

(6.18)

The above two propagators, Eqs. (6.17) and (6.18) must satisfy the SchrKodinger equation, Eq. (2.51), and its complex conjugate, i.e., @K i˝ = H (x; p; t)K ; (6.19) @t @K ∗ −i˝ (6.20) = H + (x; p; t)K ∗ : @t Substituting Eqs. (6.1) and (6.16) into Eq. (6.19) and solving the three coupled diHerential equations, we obtain i 1 q˙ i 1 B(t) a(t) = − ; (6.21) 2˝ A(t) q 2˝ A(t) A(t) b0 b(t) = ; (6.22) q

t i˝ A(s) (6.23) ds + ln q−1=2 + d0 : c(t) = x2 + d(t) − b20 x2 2 q2 (s) From the auxiliary condition, i.e., the classical solution, if 8(t)ei(t) is that solution, then 8(t)e−i(t) is also a solution of that system. The classical solution can be written as q(t; t ) = 88 sin( − ) ;

(6.24)

where 8 = 8(t) and 8 = 8 (t), and so on. With the use of Eq. (6.24) together with Eqs. (6.7) and (6.8), the coeMcients of the propagator are given by

1 8˙ B a(t) = ; (6.25) + ˙ cot( − ) − 2A 8 A

1=2 ˙˙ 1 b(t) = ; (6.26) AA sin( − )

B 1 8˙ (6.27) c(t) = − + ˙ cot( − ) + ; A 8 A 1=2

1=2 1=2 1=2 2S 1 @ ˙ ˙ c e(i=˝)d(t) = = (6.28) 2i˝ @x@x 2i˝ sin( − )A1=2 A1=2 and substitution of Eqs. (6.25) – (6.28) into Eq. (6.16) yields the propagator of this system as 1=2

˙1=2 ˙1=2 i 8˙ K(x; t; x ; t ) = exp + ˙ cot( − ) − B x2 2˝A 8 2i˝ sin( − )A1=2 A1=2

1=2

i i 8˙ ˙ ˙ xx − + ˙ cot( − ) + B x2 + + : (6.29) 2˝A 8 ˝ AA sin( − )


Introducing the new variables

1=2 ˙ x; X= ˝A

1=2 ˙ Y= x ; ˝A

Z = e−i(− ) ;

131

(6.30) (6.31) (6.32)

we can express the propagator, Eq. (6.29), in the simple form 1=2

1=2 1 ˙ ˙ 1 2 2 √ e−i(− ) e(X +Y )=2 K(x; t; x ; t ) = 2 ˝ AA 1−Z −(X 2 + Y 2 ) + 2XYZ ×exp : 1 − Z2

(6.33)

Comparing Eq. (6.33) with Mehler’s formula [160], Eq. (5.26), we Jnd an explicit form of the wavefunction of the system:

1 ˙ 1=4 1 1=2 −i[(1=2)+n] 1 ˙ 1=2 e × Hn x n (x; t) = ˝ A 2n n! ˝A

1 8˙ ˙ − i − B x2 : (6.34) ×exp − 2˝A 8 Equation (6.34) is the wavefunction of the bound system with the auxiliary condition of the classical solution. 6.1.3. Uncertainty product and energy expectation values For convenience, let us write the uncertainty product of Eq. (3.63) in another form as (TxTp)m; n = {[(m|x2 |n − m|x|n 2 )∗ (m|x2 |n − m|x|n 2 )]1=2 ×[(m|p2 |n − m|p|n 2 )∗ (m|p2 |n − m|p|n 2 )]}1=2 :

Using Eq. (6.34) and performing the integral over x yields √ √ m|x|n = n + 1C&m; n+1 + nC? &m; n−1 ; √ √ m|p|n = n + 1K&m; n+1 + nK? &m; n−1 ; " m|x2 |n = (n + 1)(n + 2)C2 &m; n+1 + (2n + 1)CC∗ &m; n " 2 + n(n + 1)C∗ &m; n−2 ; " m|p2 |n = (n + 1)(n + 2)K2 &m; n+2 + (2n + 1)KK∗ &m; n " 2 + n(n + 1)K∗ &m; n−2 ;

(6.35) (6.36) (6.37)

(6.38)

(6.39)

132


" 1 m| (xp + px)|n = (n + 1)(n + 2)CK&m; n+2 2

∗ " CK + C∗ K + (2n + 1) &m; n + n(n − 1)C∗ K∗ &m; n−2 ; 2

where C(t) and K(t) are

˝A(t) 1=2 i e ; C = C(t) = 2˙

1=2

˝ 8˙ K = K(t) = − B(t) + i˙ ei : 2A(t)˙ 8 Then one can obtain the uncertainty products for various states from Eq. (6.35): " (TxTp)n+2; n = (n + 1)(n + 2)|C||K|

2 1=2 ˝" 1 8˙ = (n + 1)(n + 2) 1 + 2 − B(t) ; 2 ˙ 8

2 1=2 ˝ 1 8˙ (TxTp)n+1; n = (n + 1) 1 + 2 − B(t) ; 2 ˙ 8

2 1=2 1 1 8˙ (TxTp)n; n = n + ˝ 1+ 2 − B(t) : 2 ˙ 8

(6.40)

(6.41) (6.42)

(6.43) (6.44) (6.45)

Note that the diagonal element of uncertainty product in the ground state is larger than the minimum uncertainty value, ˝=2. The quantum invariant operator corresponding to Eq. (6.15), i.e., the classical quantity, can be deJned as 2 2 1 1 8

2 2 2 2 I= + 2 B(t)8 − 8˙ x + ˙ + px) + 8 p ; (6.46) [B(t)8 − 8](xp 2 82 A (t) A(t) where x and p are the position and momentum operators, respectively. The function I (t) satisfy the quantum condition that corresponds to the classical condition, Eq. (6.11), dI @I 1 + [I; H ] = 0 ; = i˝ dt @t with the Hamiltonian given by Eq. (6.1). Substitution of Eqs. (6.38) – (6.40) into Eq. gives the expectation values 2 1 8 ˙ ˝ m|I |n = n + &m; n 2 A 1 ˝ &m; n : = n+ 2

should (6.47) (6.47)

(6.48)

An explicit time-dependent invariant for the classical time-dependent quadratic Hamiltonian system is obtained. Though the system is not closed, one can aMrm whether or not the system


133

is bound by using this invariant quantity. The solution of the equation for a simple harmonic oscillator is a linear combination of ei!0 t and e−i!0 t , and thus (t) = !0 t and 8 = const:, where the invariant quantity for the harmonic oscillator is m!0 82 . Therefore the system is bound. The propagator and wavefunction have been obtained, where the wavefunction has been expressed in terms of the classical solution. In the evaluation of the uncertainty product and expectation values, we have used the wavefunction together with the invariant operator, which is inferred from its classical counterpart. The expectation values of the quantum mechanical invariant operator I also satisfy the uncertainty product. 6.2. Time-dependent unbound quadratic Hamiltonian 6.2.1. Classical invariants For convenience we express the Hamiltonian of Eq. (6.1) as H = 12 [A(t)p2 + B(t)(pq + qp) + C(t)q2 ] ;

(6.49)

where q and p are the canonical coordinate and its conjugate momentum, respectively. The time-dependent coeMcients are exactly the same as those of Eq. (6.1). The classical equation of motion is given by qK + =(t)q˙ + 9(t)q = 0 :

(6.50)

Let the general solution of Eq. (6.50) be of the form q = C1 *(t) exp[(t)] + C2 *(t) exp[ − (t)] ;

(6.51)

where C1 and C2 are constants. In Eq. (6.51) the two terms are linearly independent and the functions *(t) and (t) can be determined from Eq. (6.50), where *(t) is real. Substituting Eq. (6.51) into Eq. (6.50) we obtain *K + *˙2 + =(t)*˙ + 9(t)* + *˙ + 2*˙K + =(t)* ˙ =0 ;

(6.52)

˙ =0 : *K + *˙2 + =(t)*˙ + 9(t)* − *K − 2*˙˙ − =(t)*

(6.53)

For *(t) and (t) to satisfy Eqs. (6.52) and (6.53) simultaneously, the following diHerential equations must be satisJed: *K + *˙2 + =(t)*˙ + 9(t)* = 0 ;

(6.54)

*˙ + 2*˙K + =(t)* ˙ =0 :

(6.55)

The coeMcients =(t) and 9(t) in Eq. (6.50) are real, and the function q∗ (t) in Eq. (6.51) is also a solution of Eq. (6.50). Thus the two diHerential equations, Eqs. (6.54) and (6.55), have the same form by replacing (t) with ∗ (t), where we see that is a real or pure imaginary function: (t) = ∗ (t)

(6.56)

(t) = − ∗ (t) :

(6.57)

or

134


The diHerential Eq. (6.55) oHers the invariant (t): %2 ˙

(t) = : A(t) One can Jnd another classical time-invariant quantity from Eq. (6.50) given by I (q; p; t) = 12 [)(t)p2 + 21(t)qp + &(t)q2 ] ;

(6.58) (6.59)

where )(t); 1(t) and &(t) are real time-dependent functions. Combining Eqs. (6.49) and (6.59) with Eq. (6.11), we can determine these functions, as done previously, as follows: )(t) = *2 (t) ; 1 B(t) 2 * (t) − *(t)*(t) ˙ ; A(t) A(t) 1

2 1 *(t) − *(t) ˙ + 2 : &(t) = A(t) A(t) * (t) Finally, we can obtain the invariant quantity as 2 2

B(t) 1 + I (q; p; t) = *− *˙ q + *p : * A(t) A(t) The coeMcient matrix of the quadratic variable in Eq. (6.63) is  2 

2 B(t) 1 B(t) 1 *− *˙ *   * q + A(t) * − A(t) *˙ A(t) A(t)  : W=   B(t) 1 2 * *− *˙ * A(t) A(t) The eigenvalues of W in Eq. (6.64) are given by  2 2 2 1 2

B(t) 1 % + + %= %− %˙ 2 % A(t) A(t) 5  6 2 2 2 2  6

B(t) 1 ± 7 %2 + + + 4 2 : %− %˙  % A(t) A(t) 1(t) =

(6.60) (6.61) (6.62)

(6.63)

(6.64)

(6.65)

When is real, one of the eigenvalues of W is positive and the other is negative, so that the invariant quantity I (q; p; t) shows a hyperbola in phase space, which represents unbound motion. On the other hand, when is imaginary and 2 *˙ 2 A (t) + + B(t) 6 ˙ ; * the two eigenvalues of W are positive, and thus I (q; p; t) becomes an increasing or decreasing ellipse in phase space. If one of the two eigenvalues is not inJnite, its motion will become bound. When is imaginary and 2 *˙ 2 A (t) + + B(t) ¿ ˙ ; *


135

all eigenvalues are negative real numbers or complex numbers, so that the motion will not be determined. 6.2.2. Propagator Consider a quantum unbound system with as real. Taking the propagator in the form of Eq. (6.16) and using Eqs. (6.19) and (6.20), we obtain the coupled diHerential equations for the coeMcients, a(t), b(t) and c(t). Since both r˙ and are real in an unbound system, according to Eq. (6.51), the solution of the diHerential equation, Eq. (6.50), can be expressed as q = ** sinh(r − r ) ;

(6.66)

* = *(t); * = *(t ),

and so forth. Solving the coupled diHerential equations for the coefwhere Jcients together with Eq. (6.66) one can obtain the time-dependent coeMcients in Eq. (6.16) for the unbound quadratic Hamiltonian:

i %˙ i a(t; t ) = + ctgh(r − r ) − B(t) ; (6.67) ˝A(t) % 2˝A(t) b(t; t ) =

i

;

˝ %% sinh(r − r )

i % i − + ˙ ctgh(r − r ) − B(t ) ; c(t; t ) = 2˝A(t) % 2˝

˙1=2 ˙1=2 1 d(t; t ) = ln 2 2i˝A1=2 A1=2 sinh(r − r )

:

(6.68) (6.69) (6.70)

Substituting Eqs. (6.67) – (6.70) into Eq. (6.16), we obtain the propagator for the system with the unbound time-dependent quadratic Hamiltonian as 1=2 ˙1=2 ˙1=2 K(q; t; q ; t ) = 2i˝A1=2 A1=2 sinh(r − r )

i %˙ i ×exp + ctgh(r − r ) − B(t) q2 2˝A(t) % 2˝A(t)

i

qq ×exp ˝ %% sinh(r − r )

i % i − + ˙ ctgh(r − r ) − B(t ) q 2 : (6.71) ×exp 2˝A(t) % 2˝ 6.2.3. Examples It is well known that the Hamiltonians for a free particle, damped free particle, overdamped and underdamped harmonic oscillator, and negative harmonic oscillator belong to the unbound quadratic Hamiltonian systems. Consider the damped free particle, whose Hamiltonian is given by p2 H = exp[ − 1t] : (6.72) 2m

136


The classical equation of motion and its general solution are qK + 1q˙ = 0 ;

1 1 1 1 q = C1 exp − t + t + C2 exp − t − t : 2 2 2 2

(6.73) (6.74)

Comparison of this solution with Eq. (6.51) gives * = exp[ − 1t=2] and = 1t=2. Then the propagator (6.71) is reduced to that of the damped free particle,

1=2

m i 1m(q − q )2 K(q; t; q ; t ) = exp : 2i˝(exp[ − 1t ] − exp[ − 1t]) ˝ 2(exp[ − 1t ] − exp[ − 1t]) (6.75) Eq. (6.75) is the same result as in previous work [118]. The Hamiltonian for the overdamped and underdamped harmonic oscillator is p2 m H = e−1t + e1t !02 q2 : 2m 2 The classical equation of motion is qK + 1q˙ + !02 q2 = 0 ;

(6.76) (6.77)

with !0 and 1 related as 12 − !02 ¿ 0 : (6.78) !2 = 4 The general solution of Eq. (6.77) becomes

1 1 (6.79) q = C1 exp − t + !t + C2 exp − t − !t : 2 2 The propagator for this case can be easily found to have the form

! exp [(1=2)(t + t )] 1=2 K(q; t; q ; t ) = 2 isinh (!T )

! im 1 2 1 1 2 ×exp q exp − t − q exp − t + 2˝ 2 2 2 sinh(!T )

1 × (q2 exp[1t] + q2 exp[ − 1t ])cosh(1T ) − 2exp ; (6.80) (t + t ) qq 2 where T = t − t . For the negative harmonic oscillator, the Hamiltonian, classical equation of motion and its general solution are given by p2 m 2 2 H= − ! q ; (6.81) 2m 2 qK − !2 q = 0 ; (6.82) q = C1 exp[!t] + C2 exp[ − !t] :

(6.83)


137

With * = 1 and = !t, the propagator of the negative harmonic oscillator has the form

1=2

m! im! 2 2 × {(q + q )cosh(!T ) − 2qq } : exp (6.84) K(q; t; q ; t ) = 2i˝sinh(!T ) 2˝ Two time-invariant quantities with an auxiliary condition have been obtained. One of these invariants can be used to determine whether or not the system is unbound. The propagator for the unbound system with a quadratic Hamiltonian enables one to Jnd the propagators for several unbound systems. The propagator, wavefunction and expectation values have been evaluated explicitly where the expectation value of the quantum mechanical invariant obeys the uncertainty relation for a time-dependent bound quadratic Hamiltonian. Several examples of the propagator for the unbound case are shown. The theory oHers a relation between the wavefunction and the invariants which determines whether or not the system is unbound. 7. Quantum damped harmonic oscillator. III. Dynamical invariant and second quantization method In the treatment of a time-dependent quantum system, the dynamical invariant method has been introduced by Lewis and Riesenfeld [176,177] to obtain the exact quantum mechanical solutions. The invariants have received primary concern because of their use in discussing physical problems [69,70,183–185] and their possibility in applications of classical and quantum physics [180,186,187]. In this section, we Jrst evaluate the dynamical invariant quantity for the damped driven harmonic oscillator and then obtain the creation and annihilation operators to represent the invariant quantity. We make use of these results to evaluate the propagator, wavefunction, energy eigenvalues and uncertainty products. We will extend this method to obtain the quantum-mechanical solutions of the harmonic oscillators with an exponentially decaying mass and time-dependent frequency. 7.1. Quantum-mechanical treatment for the damped driven harmonic oscillator Consider an external driving force with a tail, g(t), with the form [188]

t g(t) = dt )e−)(t−t ) f(t ) ; −∞

(7.1)

where f(t) is an instantaneous force at a given time t, and )e−)(t−t ) can be expressed in terms of the &-function as &) (t − t ) (t ¿ t ) −)(t−t ) lim )e = : (7.2) 0 (t ¡ t ) )→∞ As the time t increases, the contributions at early times may be negligible. Then the Hamiltonian of the damped harmonic oscillator with the above force has the form [189]

ˆ2 −t p t 1 2 2 H (p; ˆ q; ˆ t) = e (7.3) +e m!0 qˆ − g(t)qˆ ; 2m 2

138


where q and p are the canonical coordinate and its conjugate momentum, and !0 is the frequency of a simple harmonic oscillator. Eq. (7.3) is exactly the same as Eq. (3.1), i.e., the so-called Caldirola–Kanai Hamiltonian. The classical Lagrangian and corresponding equation of motion are given as

1 2 t 1 2 2 L(q; ˙ q; t) = e (7.4) mq˙ − m!0 qˆ + g(t)q ; 2 2 g(t) qK + q˙ + !02 q = : (7.5) m Through the same procedure as in Section 6.1, we can obtain an invariant operator, I (p; q; t), that satisJes Hamilton’s equation, Eq. (6.47), with the assumption of a quadratic form for p and q as

2 2 1 1 t t Iˆ = e m2 !02 − (qˆ − q0 )2 + e− (pˆ − p0 ) + m(qˆ − q0 ) ; (7.6) 2 4 2 where p0 (t) = mert q0 (t), and q0 (t) is the solution of the diHerential equation g(t) qK0 + q˙0 + !02 q0 (t) = : m From Eq. (7.7) , q0 (t) may be regarded as the particular solution of Eq. (7.5),

t

t ) dt dt e−(t−t )=2 e−)(t −t ) sin[!d (t − t )]f(t ) ; q0 (t) = m!d t0 t0

(7.7)

(7.8)

where !d (!02 − r 2 =4)1=2 : Note that the inIuence of the past is involved in the time-dependent function q0 (t): The dynamical invariant operator can be written in terms of annihilation and creation operators as follows: 1=2 ( 1 t a(t) ˆ = q ˆ me ! + i − %(t)] + i p ˆ ; (7.9) [ d t 2˝m!d e 2 1=2 ( 1 t a(t) ˆ †= me ! − i [ q ˆ − %(t)] − i p ˆ ; (7.10) d 2˝m!d et 2 where

1=2 2 q˙0 (t) q˙0 (t) + i 1 − : %(t) = q0 (t) + 2 2 !0 2!0 4!0

(7.11)

These operators satisfy the commutation relation, [a; a† ] = 1. The dynamical invariant operator, I (q; p; t), can be represented in terms of a and a† as Iˆ = ˝!d (aˆ† aˆ + 12 ) :

(7.12)

Using the eigenstates of the invariant operators, one can easily obtain the exact wavefunctions satisfying the SchrKodinger equation. From Eq. (7.12) the nth excited state of the invariant


139

operator is represented as

m!d et 1=4 met !d m!d et 1=2 1 2 n (q; t) = √ exp − (q − q0 ) (q − q0 ) Hn ˝ 2˝ ˝ n!2n 2 i met q ˙ : (7.13) ×exp − (q − q0 )2 − 2q˙0 (q − q0 ) + 02 ˝ 2 2 2!0 Since the solution of the SchrKodinger equation diHers only by a time-dependent phase factor from the eigenstate of the invariant operator [177], we can write directly the wavefunction, n (x; t), as (i=˝)R(t) n (q; t) = n (q; t)e

:

(7.14)

Using Eqs. (7.3), (7.13) and (7.14) along with the SchrKodinger equation, we obtain the timedependent phase factor R(t) as

t R(t) = dt [L0 (t ) − (n + 12 )˝!d − W(t )] ; (7.15) t0

where L0 (t) = et [ 12 mq˙20 (t) − 12 m!02 q02 (t) + g(t)q0 (t)] ;

m2 et 2 !02 q˙0 (t) q˙0 (t) − − q0 (t)q˙0 (t) − W(t) = g(t) : 2 m 2!02

(7.16) (7.17)

Then the exact form of the wavefunction is given by

1 m!d et 1=4 met !d 2 exp − (q − q0 ) e−i(n+1=2)wd (t−t ) n (q; t) = √ ˝ 2˝ n!2n 2 q ˙ m!d et 1=2 i met ×Hn (q − q0 ) exp − (q − q0 )2 − 2q˙0 (q − q0 ) + 02 ˝ ˝ 2 2 2!0 t i dt [L0 (t ) − W(t )] : (7.18) ×exp h t0 An exact closed form of the propagator can be obtained readily from the expansion formula, Eq. (6.17) [62], as ∞ ∗ K(q; t; q ; t ) = n (q; t) n (q ; t ) n=0

m!d (et et )1=2 = 2i˝ sin !d (t − t ) + et

1=2

!d cot(t − t ) +

exp

im 2˝

(q − q0 )2 2

et !d cot !d (t − t ) − (q − q0 )2 2

140


im!d (et et )1=2 × exp − (q − q0 )(q − q0 ) ˝ sin !d (t − t ) im t [e q˙0 (q − q0 ) − et q˙0 (q − q0 )] × exp ˝

t i m t 2 t 2 (e q˙0 − e q˙0 ) − dt [L0 (t ) − W(t )] : × exp − h 4!02 t0

(7.19)

With the use of the wavefunction of Eq. (7.18), we can obtain the uncertainty products at various states through the same procedure performed in Section 6: ˝" (TqTp)n+2n = (n + 1)(n + 2) ; (7.20) 2 1=4  2  1=2  t 1 ˝ 8m!d e 2 n+ q0 (t) − cos !d t + sin !d t (TqTp)n+1n =  2 2  ˝(n + 1)

×

  

8 t m!d e ˝(n + 1)

1=2

2

q0 (t) + sin !d t

+ cos2 !d t

1=4  

;

(7.21)

˝ (TqTp)n; n = (2n + 1): (7.22) 2 Note that the oH-diagonal elements of the uncertainty products, (Tx · Tp)n±1; n , are governed by the past in terms of q0 (t) and p0 (t), and the products (Tx · Tp)n±2; n and (Tx · Tp)n; n have the same form as the simple harmonic oscillator.

7.2. Harmonic oscillator with time-dependent frequency The generalization of the relation between dynamical invariant and solution of the SchrKodinger equation for the time-dependent oscillator oHers wide applications to various Jelds, such as radiation Jelds [189], squeezed states [190 –194] and quantum optics [195 –199]. In this section we will investigate the quantum solutions of the harmonic oscillator with a time-dependent frequency via the dynamical invariant and second quantization method [200,201]. 7.2.1. Classical case As we already treated the classical cases in Section 6, we can here adopt the same procedures. The Hamiltonian of the harmonic oscillator with time-dependent frequency is given by p2 1 (7.23) + m!2 (t)q2 ; 2m 2 where !2 (t) is a real positive function, and the classical equation of motion for the Hamiltonian is H=

qK + !(t)2 q = 0 :

(7.24)


141

The solution of Eq. (7.24) can be written in the form q = p(t)ei(t) ;

(7.25)

where *(t) and (t) are determinable from Eq. (7.24). These two functions are real and depend only on time. Substitution of Eq. (7.25) into Eq. (7.24) yields the real and imaginary diHerential equations as *K − *˙2 + !(t)2 * = 0 ;

(7.26)

*K + 2*˙˙ = 0 :

(7.27)

From the imaginary part of the diHerential equations, Eq. (7.27), one invarient quantity can be found as

= m*2 ˙

(7.28)

with an auxiliary condition given by the classical solution. Combining Eq. (7.28) with Eq. (7.26), we can write

2 *K − 2 3 + !(t)2 * = 0 : (7.29) m* Another time-invariant quantity can be obtained from Eq. (6.11). Through a similar calculation together with Eqs. (6.11) and (7.23), we obtain this classical invariant quantity as

1 2 2 2 2 I= q + (*p − m *q) ˙ : (7.30) 2 *2 The invariant quantities and I are the measure of the bound system. If is real, Eq. (7.30) is an elliptic equation in phase space . Thus, as the values of q and p in the system are limited in the same region, it is a bound system. However, if is imaginary, Eq. (7.30) is a hyperbola in phase space, and q can take any value in phase space, hence making the system unbound. 7.2.2. Quantum-mechanical treatment In order to obtain the eigenfunctions and eigenvalues of the invariant operator, one can re-express the quantum dynamical invariant quantity in terms of creation and annihilation operators. To do this, one deJnes the annihilation and creation operators a and a† by the relations

1

a= √ ˙ q + i(*p − m*q) 2m˝ *

1 *˙ =" m˙ 1 − q + ip ; (7.31) *˙ 2m˝˙

1

a =√ ˙ q − i(*p − m*q) 2m˝ *

1 *˙ =" m˙ 1 + q − ip ; *˙ 2m˝˙ †

(7.32)

142


with the auxiliary conditions of Eqs. (7.26) and (7.27). These operators satisfy the commutation relation [a; a† ] = 1. Then the invariant quantity of Eq. (7.30) can be expressed in terms of a and a† as 1 † I = ˝ aa + ; (7.33) 2 and the eigenstates and eigenvalue spectrum of the invariant operator are given by a† a|n ¿ = n|n ¿ ; % = ˝(n + 12 );

n = 0; 1; 2; : : : ; n = 0; 1; 2; : : : :

(7.34) (7.35)

Applying the annihilation operator to the ground state and then solving this equation, we obtain the normalized ground state:

1=4

m˙ m˙ exp − (1 − i** ˙ )q ˙ 2 : (7.36) u0 = ˝ 2˝ The excited-state eigenfunctions are given by 1 un = √ (a† )n u0 : n!

(7.37)

The explicit form of the eigenstates un is related to the solution of the diHerential equation for the Hermite polynomial of order n. The normalized eigenfunction of an excited state n can be expressed as

* 2 1 1=2 m˙ 1=4 m˙ m˙ 1=2 un (q; t) = n exp − 1−i q Hn q : (7.38) 2 n! ˝ 2˝ * ˙ ˝ As mentioned earlier in Section 7.1, the relation between the eigenstates of the invariant operator and wavefunction of the SchrKodinger equation is given in Eq. (7.14). We can Jnd the phase factor as )n = − ( 12 + n) : Then the exact wavefunction of the nth state of the system is given by

* 2 1 1=2 m˙ 1=4 −(1=2+n) m˙ m˙ 1=2 e exp − 1−i q Hn q : (q; t) = n 2 n! ˝ 2˝ * ˙ ˝

(7.39)

(7.40)

With the help of Mehler’s formula, Eq. (5.26), the propagator of the system can be expressed as 1=2 "

m ˙ im *˙ 2 *˙ 2 K(q; t; q ; t ) = exp q − q 2i˝ sin ( − ) 2˝ * * 1 1 2 2 + ; (7.41) [(q ˙ + ˙ q ) cos ( − ) − 2 ˙ qq ] sin ( − )


143

where * = *(t ). Eqs. (7.24) and (7.41) are the same as the previous results, i.e., Eqs. (5.3) and (5.22) at f(t) = 0, which were derived by the propagator method [156]. 7.2.3. Squeezing operators In order to obtain the uncertainty products, one can make use of Eq. (6.35). Through similar calculation in Eqs. (6.36) – (6.40), we obtain the diagonal element,

˝ *˙ 1=2 (TqTp)n; n = (2n + 1) 1 + 2 : (7.42) 2 * ˙ The minimum uncertainty of Eq. (7.42) is larger than ˝=2, and the coherent states of the system [202] are not minimum uncertainty states. To evaluate the minimum uncertainty state, we can represent the Hamiltonian of Eq. (7.23) in terms of a and a† as H=

˝

4

[)2 + )∗ )†2 + 1{a; a† }] ;

(7.43)

where {a; a† } = aa† + a† a ;

(7.44)

) = m[*˙2 + !(t)2 *2 − *2 ˙2 ] − 2im**˙˙ ;

(7.45)

1 = m[*˙2 + !(t)2 *2 + *2 ˙2 ] :

(7.46)

and To diagonalize the Hamiltonian, we introduce new creation and annihilation operators [203], b = Ca + Ka† ; b† = C∗ a† + K∗ a ;

(7.47)

with the condition |C|2 − |K|2 = 1. If b and b† obey the relation [H; b] = − kb; the Hamiltonian of Eq. (7.43) can be diagonalized in some space. The transformation constants C and K are given by ) C= " ; (7.48) 2k(1 − k) 1−k K= " ; (7.49) 2k(1 − k) k = 2!(t) :

(7.50)

In this case the Hamiltonian and its eigenvalues are H = ˝!(t)(b† b + 12 ) ; %n = ˝!(t)(n + 12 );

(7.51) n = 0; 1; 2; : : : :

(7.52)

144


Performing the same procedure as for the wavefunction of Eqs. (7.36) and (7.37), the normalized ground state and excited eigenstates are given by

1=4 m˙ 2 0 = e−[m!(t)=2˝]q ; (7.53) ˝ 1 n = √ (b† )n u0 n!

1 1=2 m!(t) 1=4 −[m!(t)=2˝]q2 m!(t) 1=2 = n e Hn q : (7.54) 2 n! ˝ ˝ Note that Eq. (7.54) is not a solution of the SchrKodinger equation. This equation along with Eq. (6.35) yields the diagonal elements of the uncertainty product as (TqTp)n; n = ˝(n + 12 ) :

(7.55)

Since the minimum uncertainty of Eq. (7.55) is ˝=2, the minimum uncertainty state is an eigenstate of the Hamiltonian of the system. The eigencoherent state of the Hamiltonian of the system is the squeezed state of the system. In the case of negative !2 (t), the corresponding system is unbound. Then the creation and annihilation operators do not transform to a diHerent set of operators. Hence, the system gives no minimum uncertainty. 7.3. Harmonic oscillator with exponentially decaying mass In the previous section we have studied the harmonic oscillator with time-dependent frequency. However, the harmonic oscillator with a time-dependent mass [204 –208] is as important as that with a time-dependent frequency. In this section we will review the quantum-mechanical treatment of the harmonic oscillator with an exponentially decaying mass [209,210]. 7.3.1. Invariant quantity One can consider the harmonic oscillator with an exponentially decaying mass m(t) = m0 (t)e−)t in which the parameter ) is a positive real constant. Since the physical momentum is m0 qe ˙ −)t ; the equation of motion and corresponding classical Hamiltonian are given by q(t) K − )q(t) ˙ + !02 e)t q(t) = 0 ; H (p; q; t) =

p2 )t 1 e + m0 !02 q2 : 2m0 2

(7.56) (7.57)

Transforming the time scale into s = e)t , Eq. (7.56) becomes !02 d 2 q(s) + q(s) = 0 : ds2 )2 s The solution of Eq. (7.58) is given as q(t) = e)t=2 [C3 J1 (z) + C4 Y1 (z)] ;

(7.58) (7.59)


145

where z = (2!0 =))e)t=2 , J1 and Y1 are the Jrst and second Bessel functions of the order of 1, respectively, and C3 and C4 are the integration constants. This solution can be represented in complex form as q(t) = r(t)[C5 ei>(t) + C6 e−i>(t) ] ;

(7.60)

r(t) = e)t=2 [J12 (z) + Y12 (z)]1=2 ;

(7.61)

>(t) = arctan [Y1 (z)=J1 (z)] :

(7.62)

where

Substitution of Eq. (7.60) into Eq. (7.56) gives the invariant quantity as −)t ˙

= m0 r 2 (t)>(t)e

= m0 !0 e)t=2 [J1 (z)Y0 (z) − J0 (z)Y1 (z)] :

(7.63)

To Jnd another invariant quantity I , we may adopt the form I = 12 [A(t)p2 + 2B(t)pq + C(t)q2 ] :

(7.64)

Combining Eqs. (7.64) and (7.57) with Eq. (6.47), we obtain the diHerential equations for the time-dependent coeMcients of I , whose solution is A(t) = e)t [J12 (z) + Y12 (z)] ;

(7.65)

B(t) = − m0 !0 e)t=2 [J0 (z)J1 (z) + Y0 (z)Y1 (z)] ;

(7.66)

C(t) =

2[J12 (z)

1 {m2 !2 [J0 (z)J1 (z) + Y0 (z)Y1 (z)] + e−)t 2 } : + Y12 (z)] 0 0

(7.67)

Finally, we get the invariant quantity I (p; q; t) as 1 I (t) = {e−)t 2 q2 + {e)t=2 [J12 (z) + Y12 (z)]p 2[J12 (z) + Y12 (z)] − m0 !0 [J0 (z)J1 (z) + Y0 (z)Y1 (z)]q}2 } ;

(7.68)

which can be diagonalized by the transformation matrix cos (t) sin (t) T= ; −sin (t) cos (t) where

(t) = arctan − − 1+

(7.69)

e)t=2 [J12 (z) + Y12 (z)] &2 2m0 !0 [J0 (z)J1 (z) + Y0 (z)Y1 (z)] e)t

4m20 !02

J12 (z)

Y12 (z)

+ J0 (z)J1 (z) + Y0 (z)Y1 (z)

2

1=2  

&2 (t)



;

(7.70)

146


Fig. 6. Time variation of the angle (t) between the q-axis and major axis of a rosette-shaped orbit in phase space, where the solid and dotted curves coveraged to !0 = 1:1 and 1:2, respectively.

&(t) = 1 −

e−)t {e−)t 2 − m20 !02 [J0 (z)J1 (z) + Y0 (z)Y1 (z)]2 } : J12 (z) + Y12 (z)

(7.71)

The time-dependent quantity (t) is the angle between the q-axis and major axis of a rosetteshaped orbit in phase space, where its time variations are illustrated in Fig. 6. The solid and dotted curves correspond to the cases of !0 = 1:1 and 1.2, respectively. From the form of the dynamical invariant orbit, one may conJrm that the harmonic oscillator with an exponentially decaying mass is a bound system. The annihilation and creation operators, a and a† , can be deJned from Eq. (7.68) as done in the previous section as 2 1=2 [J1 (z) + Y12 (z)] 1 a(t) = [e−)t=2

2 2˝

J1 (z) + Y12 (z) − im0 !0 (J0 (z)J1 (z) + Y0 (z)Y1 (z))q] + ie)t=2 p ; (7.72)

[J12 (z) + Y12 (z)] a (t) = 2˝

†

1=2

1 [e−)t=2

+ Y12 (z) + im0 !0 (J0 (z)J1 (z) + Y0 (z)Y1 (z))q] − ie)t=2 p : J12 (z)

(7.73)

Then the dynamical invariant operator of the system is represented in Fock space as I (t) = ˝ [a(t)a† (t) + 12 ] :

(7.74)


147

7.3.2. Wavefunctions for the system To derive the exact wavefunctions for the system, we apply the annihilation operator to the ground state u0 . The ground state is given by 1=4 m0 !0 e−)t=2 [J1 (z)Y0 (z) − J0 (z)Y1 (z)] u0 (q; t) = ˝[J12 (z) + Y12 (z)] m0 !0 e−)t=2 [J1 (z)Y0 (z) − J0 (z)Y1 (z) ×exp − 2˝[J12 (z) + Y12 (z)] 2 − i(J0 (z)J1 (z) + Y0 (z)Y1 (z))]q : (7.75) Using Eq. (7.37), the explicit form of the nth eigenstate becomes 1 un (q; t) = √ [a† (t)]n u0 (q; t) n! 1=4

˙ 1 r(t) ˙ m(t)>(t) m(t) ˙ =√ >(t) − i exp − q2 ˝ 2˝ r(t) 2n n! #  ˙ m(t)>(t)  q : ×Hn  ˝

(7.76)

According to Eq. (7.14), the solution of the SchrKodinger equation diHers only by a time-dependent phase factor from of the invariant operator. phase factor is given by

This8 the eigenstates 1 2!0 2!0 #(t) = − n + arctan[Y1 (z)=J1 (z)] − arctan Y1 J1 : (7.77) 2 ) ) Taking into account Eq. (7.77) the exact wavefunction becomes 1=4 1 m0 !0 e−)t=2 [J1 (z)Y0 (z) − J0 (z)Y1 (z)] n (q; t) = √ ˝[J12 (z) + Y12 (z)] 2n n! m0 !0 e−)t=2 ×exp − {[J1 (z)Y0 (z) − J0 (z)Y1 (z)] 2˝[J12 (z) + Y12 (z)] − i[J0 (z)J1 (z) + Y0 (z)Y1 (z)]} q2 1 ×exp −i n + [arctan(Y1 (z)=J1 (z))] − arctan Y1 ×Hn

2

2!0 )

8

J1

2!0 )

m0 !0 e−)t=2 [J1 (z)Y0 (z) − J0 (z)Y1 (z)] ˝[J12 (z) + Y12 (z)]

1=2

q

:

Figs. 7 and 8 illustrate the variations of |0 (x; t)|2 and |1 (x; t)|2 , respectively.

(7.78)

148


Fig. 7. |0 (x; t)|2 as a function the coordinate (q) and time (t).

Fig. 8. |1 (x; t)|2 as a function the coordinate (q) and time (t).

In the case of a bound system, the propagator can be constructed in terms of the timedependent wave function of Eq. (7.78) from the deJnition of Eq. (6.17). With the use of Mehler’s formula, Eq. (5.26), we can write K(q; t; q ; t ) = F(t; t ) exp{D(t; t )q2 + E(t; t )q2 + G(t; t )qq } ;

(7.79)


where F(t; t ) =

149

m ! 1=2 0 0 2i˝

1=4 e−)t=2 e−)t =2 [J1 (z)Y0 (z) − J0 (z)Y1 (z)][J1 (z )Y0 (z ) − J0 (z )Y1 (z )] × ; (7.80) [Y1 (z)J1 (z ) − J1 (z)Y1 (z )]2 im0 !0 e−)t=2 D(t; t ) = [J0 (z)J1 (z) + Y0 (z)Y1 (z)] 2˝[Y12 (z) + J12 (z)] [J1 (z )Y0 (z ) − J0 (z )Y1 (z )][Y1 (z)J1 (z ) + J1 (z)Y1 (z )] ; (7.81) + [Y1 (z)J1 (z ) − J1 (z)Y1 (z )] im0 !0 e−)t =2 − [J0 (z )J1 (z ) + Y0 (z )Y1 (z )] E(t; t ) = 2˝[Y12 (z ) + J12 (z )] [J1 (z )Y0 (z ) − J0 (z )Y1 (z )][Y1 (z)J1 (z ) + J1 (z)Y1 (z )] ; (7.82) + [Y1 (z)J1 (z ) − J1 (z)Y1 (z )] im0 !0 e−)(t+t )=2 [J1 (z)Y0 (z) − J0 (z)Y1 (z)][J1 (z )Y0 (z ) − J0 (z )Y1 (z )] G(t; t ) = : ˝ [Y1 (z)J1 (z ) − J1 (z)Y1 (z )] (7.83)

The uncertainty products for various states can be evaluated from Eqs. (6.35) – (6.40). Using the wave function, Eq. (7.78), the average values of q and p vanish, and thus the quantum Iuctuations become Fx ≡ n|q2 |n − n|q|n 2 e)t=2 [J12 (z) + Y12 (z)] 1 ˝ = n+ ; 2 m0 !0 [J1 (z)Y0 (z) − J0 (z)Y1 (z)] 1 m0 !0 e−)t=2 2 2 ˝ 2 Fp ≡ n|q |n − n|q|n = n + 2 [J1 (z) + Y12 (z)] ×{[J1 (z)Y0 (z) − J0 (z)Y1 (z)] − i[J0 (z)J1 (z) + Y0 (z)Y1 (z)]} :

(7.84)

(7.85)

The variations of Fx and Fp are illustrated in Figs. 9 and 10. Substitution of Eqs. (7.84) and (7.85) in Eq. (6.35) yields the diagonal elements of the uncertainty products: (TqTp)n; n = ˝(n + 12 )[1 + f(t)]1=4 ; where

J0 (z)J1 (z) + Y0 (z)Y1 (z) f(t) = J1 (z)Y0 (z) − J0 (z)Y1 (z)

(7.86)

:

(7.87)

The variations of the function f(t) are illustrated in Fig. 11. As ) goes to zero, the variable mass is reduced to m0 and the function f(t) goes to zero. Therefore, the uncertainty products are identical to those of a simple harmonic oscillator.

150


Fig. 9. Variations of the quantum Iuctuation (Fx )2 with the parameter ) and time (t).

Fig. 10. Variations of the quantumn Iuctuation (Fp )2 with the parameter ) and time (t).

The minimum value of Eq. (7.86) is larger than ˝=2, As done in Section 7.2.3 to obtain the minimum uncertainty state, we can introduce the new the creation and annihilation operators deJned by b(t) = Ca(t) + Ka† (t) ;

(7.88)

b† (t) = C∗ a† (t) + K∗ a(t) ;

(7.89)


151

Fig. 11. Variations of f(t) contained in the uncertainty with ) and time (t).

where C and K satisfy the relation |C|2 − |K|2 = 1. Through a similar procedure as in Section 7.2.3, we obtain the minimum uncertainty functions as 1=4 ∗ 1 |C − K∗ | n m0 !0 e−)t=2 [J1 (z)Y0 (z) − J0 (z)Y1 (z)] n (q; t) = √ |C − K| ˝|C − K|2 [J12 (z) + Y12 (z)] 2n n! m0 !0 e−)t=2 ((1 − CK∗ + KC∗ ) ×exp − 2˝|C − K|2 [J12 (z) + Y12 (z)] 2 2 ×[J1 (z)Y0 (z) − J0 (z)Y1 (z)] − i|C − K| [J0 (z)J1 (z) + Y0 (z)Y1 (z)])q ×Hn

m0 !0 e−)t=2 [J1 (z)Y0 (z) − J0 (z)Y1 (z)] ˝|C − K|2 [J12 (z) + Y12 (z)]

1=2

q

:

(7.90)

Using Eqs. (7.90) and (6.35), we may obtain the diagonal elements of the uncertainty product as 1 (TqTp)n; n = ˝ n + 2

2 1=4 J (z)J (z) + Y (z)Y (z) 0 1 0 1 × 1 + −|C − K|2 + i(CK∗ − KC∗ ) : (7.91) J1 (z)Y0 (z) − J0 (z)Y1 (z)

152


For Eq. (7.91) to be minimum, C and K must have the form k C= √ ; (7.92) k 2−1 1 K= √ ei;(t) ; (7.93) k 2−1 where k is a real and positive constant. The condition of C and K for the minimum uncertainty is given by 1=2 J1 (z)Y0 (z) − J0 (z)Y1 (z) J1 (z)Y0 (z) − J0 (z)Y1 (z) 2 2 k= 1(t) ± 1 (t) − 1 ; J0 (z)J1 (z) + Y0 (z)Y1 (z) J0 (z)J1 (z) + Y0 (z)Y1 (z) (7.94) J0 (z)J1 (z) + Y0 (z)Y1 (z) cos ;(t) + sin ;(t) ; J1 (z)Y0 (z) − J0 (z)Y1 (z) J0 (z)J1 (z) + Y0 (z)Y1 (z) ;(t) = arctan J1 (z)Y0 (z) − J0 (z)Y1 (z)  1=2  2   J0 (z)J1 (z) + Y0 (z)Y1 (z) + 1 − 12 (t) [12 (t) − 1]−1 ±1(t)   J1 (z)Y0 (z) − J0 (z)Y1 (z) 1(t) =

and

J0 (z)J1 (z) + Y0 (z)Y1 (z) J1 (z)Y0 (z) − J0 (z)Y1 (z)

2

J0 (z)J1 (z) + Y0 (z)Y1 (z) 6 1 (t) 6 J1 (z)Y0 (z) − J0 (z)Y1 (z) 2

2

+1 :

(7.95)

(7.96)

(7.97)

Here, the minimum uncertainty is a function of one continuous parameter in the Jnite region. The behavior of nonconservative systems can sometimes be modeled by means of a timedependent mass. The immediate manifestation of a variable mass occurs in the case of a particle which is disintegrating and losing mass. By using annihilation and creation operators a(t) and a† (t), the invariant quantity I (p; q; t) can be expressed in terms of number states. From these states one can obtain the exact solution of the SchrKodinger equation and propagator. The minimum value of the uncertainty product is larger than ˝=2. To obtain the minimum uncertainty state, one can introduce the new operators b(t) and b† (t) construct the new invariant operator I (t) = ˝ [b† (t)b(t)+1=2], and then evaluate the eigenstates of the invariant operator I which are the minimum uncertainty states with the minimum uncertainty condition, which gives (Tq · Tp)n; n = ˝(n + 1=2). 7.4. Applications 7.4.1. Susceptibility for identical atoms subjected to an external force with a tail Since the interaction of electromagnetic radiation with atoms was discussed by Einstein, it has evoked continuing interest from many physicists [211–213]. A gas of atoms in a cavity can be regarded as a dielectric medium, and the interaction of radiation with atoms has traditionally been treated by means of damped harmonic oscillators with a time-dependent external


153

force [214]. This treatment gives a good explanation of the anomalous index of refraction and absorption of light. Consider a gas of one-electron atoms subjected to a time-dependent external force inIuenced by the past, i.e., Eq. (7.1). We can obtain the quantum-mechanical expression for the susceptibility, index of refraction and extinction coeMcient [188]. By using the wavefunction, Eq. (7.18), the average dipole moment of an atom at time t can be expressed as

∞ d(t) = − dq n∗ (q; t)eq n (q; t) : (7.98) −∞

The macroscopic polarization of the gas is P(t) = N d(t)=V :

(7.99)

Suppose that one turns on an oscillating electric Jeld, E(t) = E0 cos !t at time t = 0. From Eq. (7.98), the electric dipole moment of the atom parallel to the q-axis can be obtained as )e2 E0 d(t) = {C! ei!t + C!∗ e−i!t + C!d e−=2+i!dt + C!∗ d e−=2−i!dt − C) e−)t } ; (7.100) 2m!d where the coeMcients C! , C!d and C) are !d [)(!02 − !2 ) − !2 ] − i!d ![) + (!02 − !2 )] ; C! = ()2 + !2 )[(!02 − !2 )2 + 2 !2 ] C!d = C) =

2)!d − i(2!2 − )) !d) − i()=2 − )2 ) + ; ()2 + !2 )(!02 − ) + )2 ) 2()2 + !2 )[(=2)2 + !2 − !j2 − i!d ]

()2

+

2)!d : − ) + )2 )

(7.101)

!2 )(!02

Here, the electric Jeld is assumed to be suMciently weak that the atomic populations suHer negligible disturbance from their thermal equilibrium. The macroscopic polarization of the gas for N identical atoms in a volume V is given by Eq. (7.99): )e2 E0 N P(t) = {C$ ei!t + C!∗ e−i!t + C!d e−=2+i!d t 2m!d V + C!d e−=2−i!d t − C) e−)t } : (7.102) Assuming that the electric Jeld and polarization can be Fourier analyzed into frequency components, E(8) and P(8), we obtain the following:

∞

∞ )e2 E0 N i(8+0!)t ∗ P(8) = C! dte + C! dtei(8−!)t 12m!d V 0 0

∞ + C!d dte−=2+i(8+!d )t 0

+ C!∗ d E(8) =

E0 4

0

∞

0 ∞

−=2−i(8+!d )t

dte

dtei(8+!)t +

0

∞

− C)

∞

0

dtei(8−!)t

(−)+i8)t

dte

:

;

(7.103) (7.104)

154


From the deJnition of the frequency-dependent susceptibility, we can write e2 N )2 (!d2 − !2 ) − !2 ) + [)2 + )(!d2 − !2 )] : (!) = 3'0 mV ()2 + !2 )[(!02 − !2 )2 + 2 !2 ]

(7.105)

Since the real Jeld must give rise to a real polarization, the susceptibility must satisfy the relation (−!) = ∗ (!). Using this expression for the susceptibility, the refractive index n and extinction coeMcient Z are given as )2 (!d2 − !2 ) − !2 ) e2 N n2 − Z = 1 + ; (7.106) 3'0 mV ()2 + !2 )[(!02 − !2 )2 + 2 !2 ] ![)2 + )(!02 − !2 )] e2 N 2nZ = : (7.107) 3'0 mV ()2 + !2 )[(!02 − !2 )2 + 2 !2 ] Using the imaginary part of the susceptibility, one can get a Jne-grained transition rate 1=J as i![)2 + )(!d2 − !2 )] 1 e2 NE02 : (7.108) = J 6˝m ()2 + !2 )[(!02 − !2 )2 + 2 !2 ] Here, (!) belongs to a class of functions known overall as the response functions, which measure the response of the atoms to a stimulus in the form of an applied electric Jeld. The susceptibility, Eq. (7.105), satisJes the Kramer–Kronig relations, i.e., the real and imaginary parts of the susceptibility are very intimately connected. The inIuence of the past depends on the parameter ). Therefore, in the limit of ) going to inJnity, the susceptibility becomes e2 N (!02 − !2 ) + i! lim (!) = : (7.109) )→∞ 3'0 mV (!02 − !2 )2 + 2 !2 The expression of Eq. (7.109) is exactly the classical case [190]. Figs. 12 and 13 illustrate the variations with frequency of the real and imaginary parts of the susceptibility. The transmission of an electromagnetic wave through an atomic gas is governed by a refractive index n and extinction coeMcient Z. As ) goes to inJnity, n; Z and the Jne-grained transition rate J−1 are reduced to those of the classical cases: (!02 − !2 ) e2 N ; (7.110) lim n2 − Z2 = 1 + )→∞ 3'0 mV (!02 − !2 )2 + 2 !2 e2 N ! lim 2nZ = ; (7.111) )→∞ 3'0 mV (!02 − !2 )2 + 2 !2 1 e2 NE02 ! lim = : (7.112) 2 )→∞ J 6˝m (!0 − !2 )2 + 2 !2 Figs. 14 and 15 display the variations with the frequency of the refractive index and the extinction coeMcient. The results obtained in this section should be very useful for various problem in quantum optics and atomic and molecular physics. 7.4.2. Minimum uncertainty function for the bound quadratic Hamiltonian system The diagonal elements of the uncertainty relation in the quadratic Hamiltonian system, Eq. (6.45), are larger than ˝=2, i.e., the minimum uncertainty value. Therefore, a coherent


155

Fig. 12. Variations of the real part of the susceptibility with frequency and the parameter ). The solid, dashed and dotted lines correspond to the cases of ) = 1000; 300 and 100, respectively.

Fig. 13. Variations of the imaginary part of the susceptibilty with frequency and the parameter ). The solid , dashed and dotted lines correspond to the cases of ) = 1000; 300 and 100, respectively.

state of the system is not a minimum uncertainty state. To obtain the minimum uncertainty state [215], we introduce the new creation and annihilation operators deJned as b = Ca + Ka† ; b† = C∗ a† + K∗ a ;

(7.113)

for a pair of c numbers obeying |C|2 −|K|2 = 1: The above canonical transformation, Eq. (7.113), which keeps the commutator invariant, is a unitary transformation. The properties of b and b†

156


Fig. 14. Variations of the refractive index with frequency and the parameter ). The solid, dashed and dotted lines correspond to the cases of ) = 1000; 300 and 100, respectively.

Fig. 15. Variations of the extinction coeMcient K with frequency and the parameter ). The solid , dashed and dotted lines correspond to the cases of ) = 1000; 300 and 100, respectively.

are the same as those of b and b† in Eq. (7.47). Performing the same procedures as in Section 7.2.3, the wavefunction for the nth excited state is expressed as # 1=4 ∗ 1 1=2 ˙ C − K∗ ˙ Hn q n= 2n n! ˝A|C − K|2 |C − K| ˝|C − K|2 A

q2 *˙ 2 ∗ ∗ |C − K| + i(CK − KC )˙ ˙ − i B − : (7.114) ×exp − 2˝A|C − K|2 *


157

Evaluating the diagonal elements of the uncertainty product through Eqs. (6.35) and (7.114), we obtain

2 1=2 1 1 *˙ (TqTp)n; n = n + ˝ 1+ |C − K|2 + i(CK∗ − KC∗ ) B− : (7.115) 2 ˙ * The conditions of C and K for the minimum uncertainty are given as C= √

k ; k −1

1 ei> ; k −1 1 2 ) ± )2 − (4= ˙2 )(B − (*=*)) ˙

K= √ k=

2B

;

1  2 2 2 − a2 + 4 (4= ˙ )(B − ( *=*)) ˙ ± | ) | (4= ˙ )(B − ( *=*)) ˙  ; > = tan−1  )2 − 4 

4 ˙2

B−

*˙ *

2

6 )2 6

4 ˙2

B−

*˙ *

2

+4 :

(7.116)

The new operators b(t) and b† (t) determine the new invariant operator I (t), which has the form I (t) = [b† (t)b(t) + 1=2]: Therefore, the uncertainty products for the states of Eq. (7.114) are expressed as (TqTp)n; n = n + 1=2˝: 7.4.3. Damped harmonic oscillator with modi:ed time-dependent frequency Consider the damped harmonic oscillator with a time-dependent frequency described by the modiJed Caldirala–Kanai Hamiltonian [216,217] given by

2 p2 m f(t) H = f(t) e−t !2 + − x2 ; (7.117) + et 2m 2 4f(t) 2f(t)3 where f(t) is a dimensionless time-dependent function and has the value f(t)t=0 = 1. Through similar procedures as in the previous sections, we can obtain the exact propagator as  1=2

t =2(t+t ) m!e im!   9 K(x; t; x ; t ) = exp cot ! f(t) dt − et x2 t 2 ˝ 2!f(t) t 2i˝ sin ! t f(t) dt

t 2e=2(t+t ) xx + cot ! − f(t) dt + et x2 : (7.118) 9 2!f(t) sin(! tt f(t) dt) t To evaluate the wavefunction, we make use of Eq. (3.9) together with the wavefunction of a simple harmonic oscillator at time t = 0 (since the Hamiltonian, Eq. (7.117), indeed reduces to

158


that of a simple harmonic oscillator at time t = 0). Then the wavefunction is given as # 1=2

t m!0 = ˝ e=4t √ f(t) dt + (x; t) = exp −i(n + 1=2) cot !=!0 cot ! 9 2!0 2n n! t 2

×eAx Hn (Dx) ;

where

(7.119)

t t 4 2 sin ! f(t) dt + f(t) dt + 1 ; 9 = sin 2! 2! 16!2 !02 0 0 t m!0 et m! et 2 +i 9 cot ! f(t) dt − A=− 2˝ 92 2˝ 92 2!f(t) 0

t f(t) dt + ; − cot ! 2! 0 m!0 et D2 ; Re A = − D= ; !2 = !02 − : 2 ˝ 9 2 4 2

as

(7.120)

(7.121) (7.122)

By using Eqs. (6.35) and (7.119), we can evaluate the quantities for the uncertainty products

x mn =

∞

−∞

∗ m (x; t)x n (x; t) d x

√ √ n + 1 i>(t) n e &m; n+1 + √ e−i>(t) &m; n−1 = √

2D

2D

∗

= C&m; n+1 + C &m; n−1 ;

∞ ˝ @ ∗ m|p|n = n (x; t) d x m (x; t) i @x −∞ √ √ ∗ √ √ 2A˝ i>(t) 2A˝ = n + 1 −i e &m; n+1 + n −i e−i>(t) &m; n−1 D D = 8&m; n+1 + 8∗ &m; n−1; m|x2 |n =

"

(n + 2)(n + 1)C2 &m; n+2 + (2n + 1)CC∗ &m; n +

(7.124) "

n(n + 1)C2 &m; n−2;

" " m|p2 |n = (n + 2)(n + 1)82 &m; n+2 + (2n + 1)88∗ &m; n + n(n + 1)82 &m; n−2; ) ) " )1 ) Im A ) m ) (xp + px))) n = (n + 2)(n + 1)C8&m; n+2 + ˝ 2 (2n + 1)&m; n 2 D

+

"

n(n + 1)C∗ 8∗ &m; n−2;

(7.123)

(7.125) (7.126)

(7.127)


where

159

t ! >(t) = cot cot ! f(t) dt + ; (7.128) !0 2!0 0 # i>(t) e ˝ 9e−1=2t ei>(t) ; (7.129) C(t) = √ = 2m! 2D 0 √ 2A i>(t) 8(t) = −i˝ e D

t 2m!0 1 1 ! 2 =i 9 cot ! f(t) dt − exp t 1 − i ˝ 9 2 !0 2!f(t) 0 t f(t) dt + − cot ! 2! 0 m!0 ˝ 1 1=2t −1 = 1(t)ei[cot F(t)+>(t)] ; (7.130) e 2 9 t t ! 2 − cot ! 9 cot ! f(t) dt − f(t) dt + ; (7.131) F(t) = !0 2$f(t) 2! 0 0 " 1(t) = 1 + F2 (t) : (7.132) −1

With the use of Eqs. (7.123) – (7.127), the uncertainty products for various states are given by 1=2 2 2 2 2 [(Tx)2 (Tp)2 ]1=2 n+2; n = [(x − x )(p p )]n+2; n

= =

"

(n + 2)(n + 1)|C| |8|

˝"

2

(n + 2)(n + 1)1(t) ;

(7.133)

˝ [(Tx)2 (Tp)2 ]1=2 (7.134) n+1; n = (n + 1)1(t) ; 2 ˝ (7.135) [(Tx)2 (Tp)2 ]1=2 n; n = (n + 1)1(t) : 2 Changing (n + 1) to n and (n + 2) to n in Eqs. (7.134) and (7.135), respectively, one can obtain the uncertainties in the (n − 1; n) state and (n − 2; n) state. The coherent states can be constructed by introducing the annihilation operator a and creation operator a† as done in Section 3.5. We can Jnd again the eigenvectors of the operator a; Eq. (3.90), given in the coordinate representation |x as

1 8 2 ) 1 2 1 C∗ 2 1 x|) = exp (7.136) x + x − |)| − ) ; 2i˝ C C 2 2 C (2CC∗ )1=4 m! 1=4 0 (2CC∗ )−1=4 = 91=2 e−(J=4)t ; ˝

160


Fig. 16. (Tx)2 for the (0; 0) state as a function of !t for various values of =! with !== = 1.

i8 m!0 1 t e [1 − iF(t)] ; =− 2˝C 2˝ 92 C∗ C = e−2i>(t) :

(7.137)

From this result we Jnd ˝ (Tx)2 = CC∗ = 92 e−t ; 2m!0 (Tp)2 = 88∗ =

m!0 ˝ −2 t 2 9 e 1 (t) ; 2

and thus the uncertainty product becomes ˝ (Tx)(Tp) = |C| |8| = 1(t) : 2

(7.138) (7.139)

(7.140)

Taking = 0 and f(t) = 1, all the formulas which we have derived are reduced to those of the simple harmonic oscillator. The propagator, Eq. (7.118), and wavefunction, Eq. (7.119), do not have forms similar to those of Cheng [218] and others [53], but rather have new forms. Figs. 16 –18 represent the behaviors of (Tx)2 , (Tp)2 and (Tx · Tp) as a function of !t at various values of =! and !== for F(t) = e=t at = 0. The amplitude of (Tx)2 decreases exponentially, while that of (Tp)2 increases exponentially. The coherent states for the damped harmonic oscillator with a time-dependent frequency described by the modiJed Caldirola–Kanai Hamiltonian satisfy the renowned properties of coherent states.


Fig. 17. (Tp)2 for the (0; 0) state as a function of !t for various values of =! with !== = 1.

Fig. 18. Tx · Tp for the (0; 0) state versus !t for various values of =! with !== = 5.

161

162


7.4.4. Power-law suppressed harmonic oscillator As mentioned earlier, there are many kinds of time-dependent harmonic oscillators. Replacing the exponential factor exp(−)t) with the power-law factor (a=t)) in the Caldirola–Kanai Hamiltonian, one obtains the quadratic Hamiltonian a ) p2 t ) 1 H (p; q; t) = (7.141) + m!2 x2 ; t 2m a 2 where a is a coeMcient with the dimension of time and ) is a time-independent exponent. A power-law suppressed harmonic oscillator [219] is deJned as Eq. (7.141). From Hamilton’s equations of motion, the classical equation of motion is given by ) xK + x˙ + !2 x = 0 : (7.142) t In Eq. (7.142) the dissipative term includes the constant ) and thus depends on the magnitude of ). The dissipation may signiJcantly inIuence the motion of the system. As t goes to inJnity, Eq. (7.142) reduces to the ordinary harmonic oscillator. The general solution of Eq. (7.142) can be obtained as x(t) = C1 t (1−))=2 J|1−)|=2 (!t) + C2 t (1−))=2 Y|1−)|=2 (!t) ; (7.143) where J and Y are the Jrst and second kind Bessel functions. One may express Eq. (7.143) in terms of the Hankel function of the Jrst kind as

1=2 K 2 2 −1 J|K| (!t) x(t) = t [J|K| (!t) + Y|K| (!t)] exp i tan Y|K| (!t) ≡ MK (t)ei>K (t) ;

(7.144) where K is equal to (1 − ))=2. As we have done before, substitution of Eq. (7.144) in Eq. (7.142) yields the invariant quantity M 2 >˙K

= K A mt ) MK2 >˙K = : (7.145) )) Comparing Eq. (6.1) with Eq. (7.141), Eq. (7.141) does not include B(t)[x; p]+ . Another classical invariant quantity I (t) can be obtained as 2

2 mt ) ˙ 1 I (t) = x + MK p − ) M K x : (7.146) 2 MK ) Making use of the deJnition of the propagator and Mehlers formula [137], the propagator corresponding to Eq. (7.141) is given by  1=2 1=2 ∞ −i(>K −>K )n ˙K >˙K 1 > −i(>K −>K )  −1=2(X 2 +Y 2 ) e  K(x; t; x ; t ) = e e ˝ AA 2n n! n=0 i M˙ K 2 i M˙ K 2 ×Hn (X )Hn (Y ) exp x − x ; (7.147) 2˝A MK 2˝A MK


163

where A = A(t) and A = A(t ), and A comparison of Eq. (7.147) with Eq. (6.17) gives the exact wavefunction for the time-dependent power-law suppressed harmonic oscillator:  1=4 1=2  ) ) ˙ ˙ mt >K mt >K 1 e−(n+(1=2))>K Hn  x n (x; t) = √ ) n ˝ a ˝a) 2 n! M˙ K 2 mt ) ˙ ×exp − x : (7.148) >K (t) − i 2˝a) MK From the deJnition of the uncertainty product, Eqs. (6.35) and (7.148), one can easily derive the uncertainty products. " (TxTp)n+2; n = (n + 1)(n + 2)|C| |K|  1=2 # # 2 " 2 ˝A ˝  M˙ K = (n + 1)(n + 2) + >˙K  (7.149) ˙ ˙ M 2 A> K 2>K K (TxTp)n+1; n = (n + 1)|C| |K|  2 1=2 1 M˙ K  ˝ = (n + 1) 1+ 2 ; 2 >˙K MK

(7.150)

along with the diagonal elements of the uncertainty product given by (TxTp)n; n = (2n + 1)|C| |K|  2 1=2 ˙ 1 1 MK  ˝ 1 + 2 = n+ ; 2 >˙K MK where C and K are given by 1=2 ˝A C = C(t) ei>K ; 2 ˙ 2> K 1=2 ˝ M˙ K + i>˙K ei>K : K = K(t) 2 M ˙ K 2A>

(7.151)

(7.152) (7.153)

K

The energy expectation values can be obtained through Eqs. (7.141) and (7.148) as    2   ˙ ) 1 ˝ a  M K + >˙2K  + !2 : En; n = n +  2 2>˙K t  MK

(7.154)

The quantum invariant operator I (t) corresponding to the Hamiltonian, Eq. (7.141), can be written directly as  2  2   1 M˙ K

Iˆ(t) = x2 + MK p − : (7.155) x  2  MK A

164


Introducing the annihilation and creation operators as # A 1 ˙ M˙ K a(t) ˆ = >K − i x + ip ; MK 2˝>˙K A # ˙ A 1 M K aˆ† (t) = >˙K + i x − ip ; MK 2˝>˙K A

(7.156) (7.157)

the quantum invariant operator ∧I (t) can be represented as Iˆ(t) = ˝ (aˆ† aˆ + 12 ) ;

(7.158)

where these operators satisfy the commutation relation, [a; a† ] = 1. Owing to a† a|n = n|n , the invariant operator can be written as I = ˝ (n + 12 ) :

(7.159)

Applying the operator a to the ground state, one may solve this equation. The nth eigenstate of the invariant operator ∧I (t) becomes 1 un (x; t) = √ (aˆ† )n uo n! #  1=4 ˙K 1 >˙K > 2 ˙ ˙ Hn  =√ x e−(1=2˝A)(>K −i(M K =MK ))x : ˝A 2n n! ˝A

(7.160) (7.161)

The solution of the SchrKodinger equation diHers from the eigenstate of the invariant operator

∧I (t) by only a phase factor, identiJed as exp[ − i(n + 1=2)>K ] in comparison with Eqs. (7.147)

and (7.148). The time variations of the uncertainty products [Eq. (7.151)] in the (0,0) state are illustrated in Fig. 19. The uncertainty products do not depend on the coeMcients a and ). Fig. 20 represents the time variations of the energy expectation values as a function of K and a at various (n; n) states. The energy decreases rapidly and then tends to zero as time goes to inJnity, and decreases in proportion to the energy liberated from the system to the reservoir by the dissipation. Therefore, the energy expectation value of the system does not reduce to that of the simple harmonic oscillator, but approaches zero. This means that the ground-state energy of the system may be smaller than the ground-state energy of a simple harmonic oscillator. Exact wavefunctions, propagators and uncertainty products for (1) the damped harmonic oscillator governed by a time-dependent external force with a tail, (2) the harmonic oscillator with an exponentially decaying mass, (3) the damped harmonic oscillator with time-dependent frequency described by the modiJed Caldirola–Kanai Hamiltonian, and (4) the power-law suppressed harmonic oscillator have been explicitly evaluated by the dynamical invariant and second quantization methods. The theory has been applied to obtain the frequency-dependent susceptibility, index of refraction and extinction coeMcients of N -identical atoms. The minimum uncertainty states and the conditions are expressed for the two harmonic oscillators and bound quadratic Hamiltonian systems.


165

Fig. 19. Time dependence of the uncertainty relation in the (0; 0) state for K = 0, a = 10 and ! = 1.

Fig. 20. Energy expectation values in the (0; 0) state for ! = 1 and various values of K and a.

8. Linear canonical and unitary transformations on general Hamiltonian systems It is well known that a simple unitary transformation of the time variable transforms the SchrKodinger equation for a damped harmonic oscillator with the Caldirola–Kanai Hamiltonian [Eq. (3.1)] into a SchrKodinger equation for an undamped harmonic oscillator [220]. Fusimi [221] Jrst introduced the transformation to investigate a time-dependent Hamiltonian such as a forced harmonic oscillator. In this section we review the relations for the canonical transformations in classical mechanics and the unitary transformations in quantum mechanics [222–228], and provide the applications of this theory to the time-dependent quadratic Hamiltonian [229] and a harmonic plus inverse harmonic potential with time-dependent mass and frequency [230].

166


8.1. Linear canonical transformations for classical and quantum systems Let us Jrst review the linear canonical transformation for two classical systems and Jnd the Hamiltonian and Lagrangian which satisfy the suMcient condition that the transformed variables are canonical. We consider a system with the Hamiltonian given by p2 H (q; p) = + V (q) (8.1) 2m and Lagrangian 1 (8.2) L(q; q) ˙ = mq˙2 − V (q) : 2 The classical equation of motion of the system is dpk (q; q; ˙ t) @V (q) =− ; (8.3) dt @q where the kinetic momentum pk is deJned by pk = mq˙ and, in general, is not the same as the canonical momentum. We introduce the linear transformation of the canonical variables (q; p) to new variable (Q; P) given as Q = e1(t) q ;

(8.4)

P = e−1(t) p − )(t)e1(t) q :

(8.5)

Here, )(t) and 1(t) are real and diHerentiable functions of t. Eqs. (8.4) and (8.5) are not necessary canonical transformations between the new variables (Q; P) and the variables (q; p), but are merely time-dependent linear relations. If (Q; P) are to be canonical coordinates, there should exist a new Hamiltonian where (Q; P) is determined only by the old Hamiltonian and the transformation, Eqs. (8.4) and (8.5). The new Hamilton’s equation must be equivalent to the old one. For all trajectories in phase space, their variables must satisfy the relation dF(Q; P; t) P Q˙ − HQ (Q; P; t) = pq˙ − H (q; p; t) + ; (8.6) dt where the generating function, F(Q; P; t), is a time-dependent function in phase space and must be found. Since the new coordinates (Q; P) in Eq. (8.6) are independent, the coeMcients (P; p) ˙ P˙ are given as of Q; @q @F(Q; P; t) P−p = ; (8.7) @Q @Q @q @F(Q; P; t) −p = (8.8) @P @P and @q @F(Q; P; t) : (8.9) HQ(Q; P; t) = H (q; p; t) − p − @t @t Combining Eqs. (8.4) and (8.5) with Eqs. (8.7) and (8.8), we obtain the generating function ) (8.10) F(Q; P; t) = − Q2 : 2


167

Similarly, we obtain the new Hamiltonian and Lagrangian as 2 1 21 )2 21 p 21 ) ˙ ˙ PQ + e (8.11) HQ (Q; P; t) = e + 1+e + 2)1 + )˙ Q2 + V (Q) ; 2m m 2 m ˙ −21 + ))QQ˙ + 1 (m1˙ 2 e−21 − ))Q ˙ t) = e−21 m Q˙ 2 − (m1e LQ (Q; Q; ˙ 2 − V (Q) : (8.12) 2 2 The relation between the canonical momentum and kinetic momentum can be obtained from the Lagrangian as ˙ −21 + ))Q : P = e−21 Pk − (m1e

(8.13)

The diHerentiation of the inverse canonical transformation, Eqs. (8.4) and (8.5), gives the relation between the kinetic momenta of the new and old systems as ˙ + e−1 Pk : pk = − m1q

(8.14)

The mechanical energy of the new coordinate system is diHerent from the Hamiltonian of Eq. (8.11), though the mechanical energy is the same as the Hamiltonian of Eq. (1), which is given as 1 P 2 ˙ ) 21 21 E = e41 e PQ + (8.15) + 1+ e (m1˙ + )e21 )2 Q2 + V (Q) ; 2m m 2m where Eq. (8.15) includes ) but does not really depend on ). One can make use of Feynman’s path integral to investigate the quantum mechanical relation between two systems connected canonically [62,231,232]. A dynamical integral equation linearly connecting to the initial wavefunction to yield the Jnal wavefunction is given by Eqs. (2.75) and (3.9), i.e., the deJnitions of the propagator and wavefunction. To obtain the integral factor and the SchrKodinger equation, let us write t1 = t; t2 = t + ' and x2 = Q; x1 = Q + 9. Then the integral equation, Eq. (3.9) and the propagator, Eq. (2.75), become

∞ (Q; t + ') = K(Q; t + '; Q + 9; t)(Q + 9; t) d9 ; (8.16) −∞

1 9 i −21 m 92 9 −21 ˙ K(Q; t + '; Q + 9; t) = e + (m1e + )) × Q+ exp Aj (t + '=2) ˝ 2 '2 ' 2 1 ˙ 2 −21 9 2 9 + (m1 e − )) ˙ Q+ −V Q+ ' : (8.17) 2 2 2 For large 9, the integrand oscillates very rapidly and makes Therefore, the " no contribution. " main contribution to the integral comes from the values − '˝=m 6 9 6 '˝=m. Substituting Eq. (8.17) into Eq. (8.16) with the expansion in ' and comparing the linear terms in ', we can obtain the integral factor and SchrKodingers equation as 2i˝' Q Aj (t) = e1(t) ; (8.18) m @(Q; t) (8.19) i˝ = Hˆ Q (Q; t) ; @t

168

where


@ i ˝ ) 21 ˙ Hˆ Q = −e 1+e − 2Q +1 2m @Q2 2 m @Q 1 21 )2 ˙ + e + 2)1 + )˙ Q2 + V (Q) : 2 m 21

˝ 2 @2

(8.20)

This Hamiltonian operator has the same form as the classical Hamiltonian of Eq. (8.11), whose canonical variables are replaced by their corresponding quantum operators. Using Eqs. (8.2), (2.75) and (3.9), the SchrKodinger equation and integral factor of the old system are obtained as @ (q; t) ˝2 @2 (q; t) + V (q) (q; t) ; (8.21) =− @t 2m @q2 2i˝ Aj (t) = : (8.22) m Here, one can conJrm that the quantum Hamiltonian has the same form as the classical Hamiltonian, Eq. (8.1), whose canonical variables are replaced:by their corresponding quantum operators. The Wronskian determinant of {qj } and {Qj } is Nj −1 e1j . With the use of Eqs. (8.2), (8.12), (2.75), (8.18) and (8.22), we can obtain the relation between the propagators of the new and the old system as t2

∞ i m 2 ˙ −21 + ))QQ˙ K(Q2 ; t2 ; Q1 ; t1 ) = lim exp dt e−21 Q˙ − (m1e N →∞ −∞ ˝ t1 2 N −1 dQj 1 ˙ 2 −21 2 − ))Q ˙ − V (Q) × + (m1 e 2 Aj (t) i˝

j=1

= lim

N →∞

∞

i exp ˝ −∞

t2

t1

N −1 dqj m 2 d ) 21 2 dt q˙ − e q − V (q) × 2 dt 2 AQ j (t)

i i 212 2 211 2 = exp − )2 e q2 exp )1 e q1 × K(q2 ; t2 ; q1 ; t1 ) ; 2˝ 2˝

where )2 = )2 (t); 12 = 12 (t), etc. [233]. Consider the unitary operator connecting two quantum systems given as

i i 2 ˆ U (q; ˆ p; ˆ t) = exp − )qˆ exp − 1(qˆpˆ + pˆ q) ˆ : 2˝ 2˝

j=1

(8.23)

(8.24)

Using Eq. (8.24), one may evaluate the quantum operator relations corresponding to the classical canonical relations, Eqs. (8.4) and (8.5), as † Qˆ = Uˆ qÛˆ = e1 qˆ ;

(8.25)

† Pˆ = Uˆ pˆ Uˆ = e−1 pˆ − )e1 qˆ :

(8.26)


169

The inverse forms of Eqs. (8.25) and (8.26) can be easily obtained by using the unitary operator as ˆ P; ˆ P; ˆ P; ˆ t) = Uˆ (q( ˆ t); p( ˆ t); t) Uˆ Q (Q; ˆ Q; ˆ Q;

2 i i −21 ˆ 2 ˆ ˆ ˆ ˆ ˆ = exp − )e Q × exp − 1[(QP + P Q) + 2)Q ] : 2˝ 2˝

(8.27)

Eqs. (8.25) and (8.26) each have the same form as Eqs. (8.4) and (8.5). Their canonical variables are replaced by their corresponding quantum operators. We can prove that U † (x)(x) is the solution of Eq. (8.19) in some x-space if (x) is a solution of Eq. (8.21): | = Uˆ | :

(8.28)

The operation of q| or Q| on Eq. (8.28) yields the relation between the wavefunctions in each space as +

+

ˆ P; ˆ P; ˆ t)q| = Uˆ (Q; ˆ t)(e−1 Q) ; (q; t) = Uˆ (Q;

(8.29)

or ˆ P; ˆ t)Q| = Uˆ (q; (Q; t) = Uˆ (Q; ˆ p; ˆ t) (e1 q) ;

(8.30)

where the unitary transformation operator, Uˆ q (q; ˆ p; ˆ t), corresponding to that of the new coordinate, Eq. (8.27), is given by ˆ q; ˆ q; Uˆ q (q; ˆ p; ˆ t) = Uˆ (Q( ˆ p; ˆ t); P( ˆ p; ˆ t); t)

i i 21 2 2 = exp − )e [qˆ ] × exp − 1(qˆpˆ + pˆ q) ˆ − 2)q : 2˝ 2˝

(8.31)

Utilization of Eq. (8.28) yields the relation between the quantum average of the position and the momentum operators in the old and new spaces as +

|qˆ| = |Uˆ qÛˆ | = e−1 |Qˆ | ;

(8.32)

ˆ | = |pˆ | = e1 |(pˆ + )q) ˆ | = e1 |(Pˆ + )Q) | Pˆ | ;

(8.33)

|Qˆ | = e1 |qˆ| ;

(8.34)

|Pˆ | = e−1 |pˆ | − )e1 |qˆ| :

(8.35)

Eqs. (8.32) – (8.35) have the same form as Eqs. (8.4) – (8.5) and the inverse of Eqs. (8.4) – (8.5), respectively. From these, one can conJrm that the products of the quantum average between two systems can be realized from their corresponding canonical relations, and from Eqs. (8.19), (8.21) and (8.28) the two systems are related by the canonical transformation and form distinct

170


quantum spaces. In q-space each operator is given by qˆ = q ; pˆ =

˝ @

i @q ˆ Q = e1 q ;

(8.36) ;

(8.37) (8.38)

and ˝ @ ˝ @ Pˆ = e−1 − )e1 q = : i @q i @Q In Q-space Qˆ = Q ; ˝ @ ; Pˆ = i @Q

(8.39) (8.40) (8.41)

and (8.42) qˆ = e−1 Q ; ˝ @ ˝ @ pˆ = e1 + )Q = : (8.43) i @Q i @q From Eqs. (8.32) – (8.35), the relation between the quantum uncertainty for the old and new system [Eq. (6.35)] can be evaluated as (TqTp)2 = (TPTQ)2 + )2 (TQ)4 ˆ | − 2|Qˆ | |Pˆ | ] ; + )(TQ)2 [|(Qˆ Pˆ + Pˆ Q)

(8.44)

or (TPTQ)2 = (TqTp)2 + )2 E 41 (Tq)4 − )e21 (Tq)2 [ |(pˆ qˆ + qˆp) ˆ | − 2 |qˆ| |pˆ | ] :

(8.45)

It is obvious that if the old system satisJes Heisenberg’s principle, the new system also does. For ) = 0, the quantum uncertainties for the two systems are the same. The kinetic momentum is equal to the canonical momentum in the old system, and thus pk = p. One may designate the momentum operator as the canonical momentum operator to distinguish it from the kinetic momentum operators. However, the kinetic momentum is not equal to the canonical momentum in the new system. From Eq. (8.13), the kinetic momentum operator in the new system is Pˆ k = e21 Pˆ + (m1ˆ + )e21 )Qˆ : (8.46) Making use of Eqs. (8.34) and (8.35), the relation between the quantum averages of the kinetic momentum operators for both systems becomes ˙ 1 |qˆ| : |Pˆ | = e1 |pˆ k | + m1e (8.47)


171

From Eqs. (8.34) – (8.35) and (8.47), the relation between the quantum uncertainties of the position and the kinetic momentum operators for both systems can be obtained as 2 (TQTPk )2 = e41 {(Tpk Tq)2 + m2 1˙ (Tq)4 2 ˙ + m1(Tq) [ |(pˆ k qˆ + qˆpˆ k )| − 2 |qˆ| |pˆ k | ]}

or

(8.48)

2 ˙ 21 )(TQ)4 (TqTpk )2 = e−41 {(TQTPk )2 − (m2 1˙ + 2m1)e

˙ 21 (TQ)2 [|(Qˆ Pˆ + Pˆ Q) ˆ | − 2|Qˆ | |Pˆ | ]} : − m1e

(8.49)

Though Eqs. (8.48) and (8.49) contain )(t), the quantum uncertainty does not depends on ) but only on 1. Therefore, although the old system satisJes the uncertainty principle, the new system may not satisfy it in some cases if 1 ¡ 0. Furthermore, the uncertainty of the new system goes to zero. The mechanical energy operator for the new system can be deJned from Eq. (8.15) as pˆ 2 1 ˙ ) 21 21 ˆ ˆ ˆ Eq = e41 1+ e e (P Q + Qˆ P) + 2m 2 m 2 1 ˆ : + (8.50) (m1˙ + )e21 )2 Qˆ + V (Q) 2m Assuming that the potential energy to have a quadratic form of the position, the relation between the quantum averages of the mechanical energy operator for both systems becomes ˙ 2 m 1 21 2 |Eˆ Q | + 1˙ |q | + |(qˆpˆ + p˙ q) |Eˆ Q | = e ˆ |

2 2 ˙ ˙ −21 )|Eˆ | − 2m1˙|(Qˆ Pˆ + Pˆ Q) ˆ | ; = e−21 |Eˆ Q | − m1(2) + m1e (8.51) where Eˆ is the mechanical energy of the old system. Although ) appears in Eq. (8.51), this equation does not depend on ) but only on 1. For 1 = 0, the quantum average, Eq. (8.51), for the new system is the same as for the old system. 8.2. Quantum treatment for special types of linear canonical transformations Consider the special case where )(t) is set to zero, so that Eqs. (8.4) and (8.5) become Q = e1(t) ;

(8.52)

P = e−1(t) :

(8.53)

From the above equations, one can verify that the two systems are connected by the timedependent scale canonical transformation. From Eqs. (8.19) and (8.20), the SchrKodinger equation of this new system becomes 2 2 @(Q; t) @(Q; t) i˝ ˙ 21 ˝ @ (Q; t) i˝ − 1 2Q (8.54) =−e + 1 + V (Q)(Q; t) : @t 2m @Q2 2 @Q

172


Once one knows the solution of Eq. (8.19) together with Eq. (8.30), then one can Jnd the solution of the new SchrKodinger equation. For example, the relation between the general solutions of the classical equation of motion for the harmonic and damped harmonic oscillators with the same frequency can be expressed as Q = e−10 t q ; and the unitary operator connecting them is

i ˆ ˆ ˆ ˆ ˆ ˆ U (Q; P; t) = exp − 10 t(QP + P Q) ; 2˝

(8.55) (8.56)

where the uppercase and the lowercase letters mean the damped harmonic oscillator, and the harmonic oscillator, respectively. From Eq. (8.30) the SchrKodinger solution of the damped harmonic oscillator can be evaluated as ˆ P; ˆ P; ˆ P; ˆ t) = U (Q; ˆ t) HO (Q; ˆ t) DHO (Q; (m!= ˝)1=4 10 t=2 −210 t(m!=2˝)Q2 m! 10 t √ e e × Hn e Q : (8.57) = ˝ 1=4 2n n! The damped harmonic oscillator may be represented by another Hamiltonian as ˆ2 m!2 ˆ 2 −10 t P ˆ ˆ (8.58) HDHO (Q; P; t) = e + e1 0 t Q : 2m 2 Eq. (8.58) is also the damped harmonic oscillator Hamiltonian operator corresponding to the classical case of the harmonic oscillator, with the harmonic potential having the frequency (!2 − 102 =4)1=2 , through Eqs. (8.4) and (8.5) if 1(t) = − 10 t=2 and ) = (m10 e10 t )=2 in Eq. (8.11). From Eqs. (8.4) and (8.5), it is clear that the transformation giving this Hamiltonian system is not the scale transformation. Although this Hamiltonian gives a diHerent SchrKodinger equation than Eq. (8.54), the quantum average of any physical operator obtained through the solution of the SchrKodinger equation is the same for both Hamiltonians. For the scale transformation, since ) = 0, Eqs. (8.44) and (8.45) are equal to each other, i.e., the relation between the quantum uncertainty for both systems becomes (TqTp) = (TPTQ) :

(8.59)

For the damped harmonic oscillator, canonically connecting with the harmonic oscillators by the scale transformation and using Eqs. (8.59) and (8.48), one can obtain the uncertainties of the system using the canonical and kinetic momentum operators, respectively, as (TPTQ)DHO = (n + 12 )˝ ; 1=2 102 1 −210 t ˝: 1+ n+ (TQTPk )DHO = e 4w2 2

(8.60) (8.61)

The uncertainty of Eq. (8.61) vanishes as time goes to inJnity. However, the uncertainty of Eq. (8.60) is constant as time goes to inJnity. This means that the uncertainty does exist even though the oscillations have stopped, and is in agreement with Eq. (3.65) except for the oscillations.


173

For the damped harmonic oscillator expressed by the Hamiltonian of Eq. (8.59), the relation between the uncertainties of the position and momentum operators becomes 1=2 12 1 (TQTP)DHO = ˝ 1 + 02 n+ ; (8.62) 4w 2 and the uncertainty for the position and kinetic momentum is given as 1=2 12 1 n+ : (TQTPk )DHO = e−10 t ˝ 1 + 02 4w 2

(8.63)

The uncertainties of the position and canonical momentum operators for the diHerent Hamiltonian operators of the damped harmonic oscillator are constant in time but diHerent depending on the choice of the Hamiltonian. However, the uncertainties for the kinetic momentum operators for the two damped harmonic oscillators are not constant in time, but independent of the Hamiltonians chosen. By Eq. (8.51), the quantum average of the mechanical energy operator for the damped harmonic oscillator can be obtained as 2 1 1 −1 t 0 0 ˝! n + Eˆ DHO = e 1+ : (8.64) 8w2 2 Since Eqs. (8.55) and (8.58) have the same 1(t), the quantum average of the mechanical energy operator using the solution of the SchrKodinger equation with the Hamiltonian of Eq. (8.58) is also Eq. (8.64). Note that the mechanical energy operator, Eq. (8.64), vanishes as times goes to inJnity. This fact agrees precisely with that of Eqs. (3.58), (3.66) and (3.67). From the above general results, one can treat the canonical transformation where the equation of motion is unique. If 1(t) is zero, the canonical transformation equations, Eqs. (8.4) and (8.5), become Q=q ;

(8.65)

P = p − )(t)q :

(8.66)

The above equations show us that one position has innumerable counterpart canonical momenta of the system due to the arbitrariness of )(t). In this case the Hamiltonian of Eq. (8.11) becomes p2 ) 1 )2 HQ (Q; P; t) = (8.67) + PQ + + )˙ Q2 + V (Q) : 2m m 2 m In Eq. (8.67) depending on )(t), there are innumerable Hamiltonians and Lagrangians that give rise to one dynamical equation for the system. This means gauge invariance such that the classical equation of motion is independent of the choice of )(t) in Eq. (8.67). Therefore, Eqs. (8.65) and (8.66) are the gauge transformation. There are innumerable canonical momenta in the system, but the kinetic momentum is invariant with respect to the choice of )(t). The kinetic momentum for the transformation system becomes pk (t) = mq˙ = )(t)q + P :

(8.68)

174


If 1(t) = 0, Eqs. (8.25) and (8.26) are reduced to Eqs. (8.65) and (8.66). This means that there are innumerable momentum operators, i.e., innumerable quantum spaces which are dependent on the gauge chosen. In this case the Hamiltonian of Eq. (8.20) becomes @ 1 )2 i˝ ) ˝2 @ ˆ HQ = − − 2Q (8.69) +1 + + )˙ Q2 + V (Q) : 2m @Q2 2m @Q 2 m Eq. (8.69) gives innumerable SchrKodinger equations for a single system. From Eqs. (8.34) and (8.35) with 1(t) = 0, the quantum average of q is invariant for any space, but the momentum operator is not. From Eq. (8.68) one can deJne the kinetic momentum as Pˆ k (t) = )(t)qˆ + Pˆ :

(8.70)

This also is invariant. Since the mechanical energy operator does not depend on the particular gauge chosen, the quantum average of the functions for the position and kinetic momentum operators are invariant for any solution of the innumerable SchrKodinger equations, just as for the classical case [88]. On the other hand, if gauge invariance holds in classical mechanical treatments, then it holds also for quantum mechanical treatments. From Eq. (8.45), the relation of the canonical momentum’s uncertainty for the gauge transformation becomes (TQTP)2 = (TqTp)2 + )2 (Tq)4 − )(Tq)2 [ |pˆ qˆ + qˆpˆ | − 2 |qˆ| |pˆ | ] : (8.71) The uncertainty, Eq. (8.71), does not violate Heisenberg’s principle, but depends on the gauge chosen. From Eqs. (8.48) and (8.49), the relation of the kinetic momentum uncertainty for the gauge transformation becomes (TQTPk ) = (TqTpk ) :

(8.72)

It does not depend on the gauge chosen. In the above two equations, Eqs. (8.71) and (8.72), the uncertainties of the canonical and kinetic momentum operators for the harmonic oscillator after the gauge transformation become, respectively, 1=2 )2 1 (TQTP)HO = ˝ 1 + 2 2 n+ ; (8.73) mw 2 (TQTPk )HO = ˝(n + 12 ) :

(8.74)

Under the gauge transformations of Eqs. (8.65) and (8.66), the Hamiltonian for the damped harmonic oscillator of Eq. (8.58) becomes 2 2 ˆ2 2 −10 t P −10 t ) 10 t m! −10 t ) ˆ ˆ ˆ ˆ ˆ ˆ HGDHO (Q; P; t) = e (8.75) Qˆ : +e (P Q + QP) + e +e 2m 2m 2 2m The relation of the canonical momentum operator’s uncertainty between two systems for this gauge transformation can be obtained as

2 ˝2 ) )10 1 2 2 2 (TQTP)GDHO = (TQTP)DHO + 2 × n+ + : (8.76) m 2 (! − 102 =4) m2


175

Substitution of Eq. (8.60) into Eq. (8.76) yields the canonical momentum operator’s uncertainty corresponding to the Hamiltonian of Eq. (8.75) connecting with the transformation

1=2 ˝2 )2 )10 1 2 (TQTP)GDHO = 1 + 2 + n+ : (8.77) m 2 (! − 102 =4) m2 It is natural that the uncertainty for the kinetic momentum operator with the Hamiltonian, Eq. (8.75), connecting the transformation, will be given by Eq. (8.63). The uncertainties of the position and momentum for the system vary with the choice of gauge, but do not violate Heisenbeng’s principle, while on the other hand, the uncertainties for the position and kinetic momentum for the system might not satisfy Heisenberg’s principle but are invariant with respect to the choice of gauge. 8.3. Applications 8.3.1. Canonical and unitary transformation for a general time-dependent quadratic Hamiltonian system In Section 8.1 we have developed the relations for the canonical transformation in classical mechanics and the unitary transformation in quantum mechanics. In this section we will apply this theory to a general quadratic Hamiltonian system [229]. We will skip some calculations which already have been done in Section 6 for the general quadratic Hamiltonian system. Consider the classical Hamiltonian of a general time-dependent quadratic system [see Eq. (6.1)] as H (p; q; t) = 12 {A(t)p2 + 2B(t)qp + C(t)q2 } + D(t)q + E(t)p + F(t) :

(8.78)

From Hamilton’s equations of motion, one can obtain the nonhomogeneous time-dependent diHerential equation [see Eq. (6.2)] as qK + =(t)q˙ + =(t)q = (t) :

(8.79)

There are bound and unbound systems in the solution of Eq. (8.79). Here, we only choose the bound system. If the particular solution is qpa , the general solution of Eq. (8.79) can be written as q(t) = C1 8(t) exp{i>(t)} + C2 8(t) exp{−i>(t)} + qpa :

(8.80)

As we have done earlier in Section 6.1, we can Jnd the Jrst integral and a classical invariant quantity I given as 82 >˙

= = const ; (8.81) A 2 B 8˙ 1 2 2 (q − q0 ) + (q − q0 ) + 8(p − p0 ) ; (8.82) I (p; q; t) = 8− 2 82 A A where q0 and p0 are the time-dependent forms of the classical canonical coordinate and momentum. Since Eq. (8.82) is an ellipse in q- and p-space for the constant coeMcients, the system is bound.

176


Consider the other Hamiltonian that gives the same classical solution of Eq. (8.79). One can introduce a canonical transformation from (q; p) to new canonical variables (Q; P), Q=q ;

(8.83)

P = p − G(t)q ;

(8.84)

which are the same as Eqs. (8.65) and (8.66), respectively. There are numerous canonical momenta of the form of Eq. (8.84). One can introduce the time-dependent generating function [see Eq. (8.10)] F(P; Q; t) = − 12 G(t)Q2 : Then the new Hamiltonian [223], which gives the same equation of motion, becomes @q @F(P; Q; t) H (P; Q; t) = H (p; q; t) − p − : @t @t Combining Eqs. (8.78) and (8.85) with Eq. (8.86), the new Hamiltonian becomes ˙ }Q 2 H (Q; P; t) = 1 A(t)P 2 + 1 {A(t)G(t)2 + C(t) + 2B(t)G(t) + G(t) 2

(8.85) (8.86)

2

+ {A(t)G(t) + B(t)}QP + [D(t) + E(t)G(t)]Q + E(t)P + F(t) :

(8.87)

Since Q = q, only one classical solution can be found from numerous diHerent Hamiltonians. The quadratic invariant quantity of the new Hamiltonian corresponding to Eq. (8.82) is 2 1 2 B 8˙ 2 I (P; Q; t) = (Q − Q0 ) + (Q − Q0 ) + 8(P − P0 ) : (8.88) 8 + G8 − 2 82 A A The quantum Hamiltonian corresponding to Eq. (8.78) through the replacement of q and p by the operators qˆ and pˆ is ˆ + C(t)qˆ2 ] + D[t]qˆ + E(t)pˆ + F(t) : (8.89) Hˆ (p; ˆ q; ˆ t) = 12 [A(t)pˆ 2 + B(t)(pˆ qˆ + qˆp) This Hamiltonian should satisfy the SchrKodinger equation, Eq. (8.19). The quantum invariant operator I has a quadratic form of qˆ and pˆ as 2 1 2 B 8 ˙ I (p; ˆ q; ˆ t) = (qˆ − q0 )2 + (qˆ − q0 ) + 8(pˆ − p0 ) : (8.90) 8− 2 82 A A Eq. (8.90) has the same form as the classical invariant quantity whose canonical variables are replaced by the quantum operators. The invariant operator can be expressed by the annihilation and creation operators [see Eqs. (7.9) – (7.12)] as A 1=2 1 ˙ 8˙ aˆ = >+i B− (qˆ − q0 ) + i(pˆ − p0 ) ; (8.91) A 8 2˝>˙ and aˆ† =

A 2˝>˙

1=2

1 ˙ 8˙ >−i B− (qˆ − q0 ) − i(pˆ − p0 ) : A 8

If [q; ˆ p] ˆ = i˝, then we have [a; ˆ aˆ† ] = 1.

(8.92)


177

To Jnd the numerous Hamiltonians which give only one classical equation of motion but numerous canonical momenta, we introduce the unitary operator whose characteristics are equal to the canonical transformation in classical mechanics, i 2 ˆ U (p; ˆ q; ˆ t) = exp − G(t)qˆ ; (8.93) 2˝ i † 2 ˆ U (p; ˆ q; ˆ t) = exp G(t)qˆ ; (8.94) 2˝ where G(t) is the same as the generating function of Eq. (8.85). Using these unitary operators, the new operators Qˆ and Pˆ are deJned from the operators qˆ and pˆ as † Qˆ = Uˆ qÛˆ = qˆ ; (8.95) † Pˆ = Uˆ qÛˆ = pˆ + G(t)qˆ :

(8.96)

Here, Eqs. (8.95) and (8.96) correspond to the classical canonical transformation, Eqs. (8.83) and (8.84). These unitary operators transform the SchrKodinger equation of the Jrst Hamiltonian system, @ sˆ0 = Hˆ − i ; (8.97) @t into the SchrKodinger operator @ s1 = H (Q; P ¡ t) − i ; (8.98) @t namely, † (8.99) Sˆ 1 = Uˆ Sˆ 0 Uˆ : Then the new Hamiltonian can be expressed as ˆ P; ˙ }Qˆ 2 ˆ t) = 1 A(t)Pˆ 2 + 1 {A(t)G(t)2 + C(t) + 2B(t)G(t) + G(t) H (Q; 2

+

2

1 2 {A(t)G(t)

ˆ + B(t)}(Pˆ Qˆ + Qˆ P)

+ [D(t) + E(t)G(t)]Qˆ + E(t)Pˆ + F(t) :

(8.100)

The Hamiltonian of Eq. (8.10) has the same form as the classical Hamiltonian, Eq. (8.87), whose canonical variables are replaced by quantum operators. Here one can conJrm that there are numerous quantum Hamiltonians and SchrKodinger equations for a single system. The new quantum invariant operator Iˆ is represented in the quadratic form of Qˆ and Pˆ as 2 2 1

B 8 ˙ 2 ˆ t) = ˆ Q; Iˆ (P; (Qˆ − Q0 ) + (Qˆ − Q0 ) + 8(Pˆ − P0 ) : 8 + G(t)8 − 2 82 A A (8.101) Eq. (8.101) can also be simpliJed by introducing the new creation and annihilation operators, b† and b as † (8.102) Iˆ = ˝ bˆ bˆ + 12 ;

178

where


1=2 A 1 8 ˙ ˙ > + i AG + b − (Qˆ − Q0 ) + i(Pˆ − P0 ) ; bˆ = ˙ A 8 > 2 ˝ † A 1=2 1 ˙ 8˙ ˆ ˆ ˆ (Q − Q0 ) − i(P − P0 ) : > − i AG + b − b = A 8 2˝>˙ The spectrum and eigenfunctions of the invariant operator Iˆ are given as IÛn (q; t) = %Un (q; t) ;

(8.103) (8.104)

(8.105)

(8.106) % = ˝ (n + 12 ); n = 0; 1; 2; : : : : 1=4 " q2 >˙ >˙ 2 (−1)n √ Un (q; t) = exp − (8.107) q0 − & + 'q × Hn ( &r (q − q0 )) ; ˝A 2˝A 2 n! 8˙ 1 ˙ &= = &r + i&i ; (8.108) >+i B− ˝A 8 p0 : (8.109) ' = &q0 + i ˝ Taking into consideration the phase factor [see Eqs. (7.13) – (7.18)], we can write the solution of the SchrKodinger equation as 1=4 >˙ >˙ 2 '2 1 1 exp − > q − + i; − i n + n (q; t) = √ 2˝A 0 2& 2 n!2n ˝A " & ip0 2 × Hn ( &r (q − q0 )) ; q − q0 − (8.110) ×exp − 2 &˝ d; F 1 8˙ ˙ 0 × q˙ − 8˙ q0 + >q ˙ 0 − E2 : q˙0 − q0 + >q (8.111) =− − 0 dt ˝ 2˝ A 8 8 The wavefunction of the SchrKodinger equation corresponding to the new Hamiltonian, Eq. (8.101) can be obtained through a similar process as 2 ˙ 1=4 ˙ ' 1 > > 1 b exp − > Q2 − b + i;b − i n + n (Q; t) = √ 2˝A 0 2& 2 n!2n ˝A " & ip0 2 × Hn ( &r (Q − Q0 )) ; Q − Q0 − (8.112) ×exp − 2 &˝ where

1 ˙ 8˙ > + i GA + B − = &r + i&bi ; &b = ˝A 8 p0 ; 'b = &b Q0 + i ˝ d;b F 1 8 ˙ 8 ˙ 2 ˙ 0 × Q˙ − Q0 + >Q ˙ 0 −E Q˙ − + >Q : =− − dt ˝ 2˝A 8Q0 8

(8.113) (8.114) (8.115)


179

For the bound system, the propagator is deJned by Eq. (6.17), and with the help of Mehler’s formula, Eqs. (5.26), and (8.110) one can obtain the propagator as 

1=2 ˙ 1=2 >˙ 1 > 2  exp{i(; − ; )} k(q; t; q ; t ) =  2i˝ sin(> − > )81=2 81=2

8˙ 2 8˙ 2 2 2 ×exp i B− q + p0 q0 − B − q0 − p0 q0 − p0 − p0 8 0 8˙

p02 i i 8˙ 2 ˙ − p0 (q − q0 ) − ×exp (q − q0 ) > cot(> − > ) − B + 2˝A 8 ˝ 2˝&

p02 8˙ −i i 2 ˙ ×exp (q − q0 ) > cot(> − > ) − B + − p0 (q − q0 ) − 2˝A 8 ˝ 2˝&∗    i >˙>˙ 1=2  (q − q )(q − q )=sin(> − > ) ; (8.116) ×exp − 0 0  ˝ AA 

where the prime means the quantities at time t . In the same way, one can obtain the propagator for the new Hamiltonian as

1=2 1=2 1=2 >˙ >˙ k(Q; t; Q ; t ) = 2i˝ sin(> − > )81=2 81=2 p02 p02 2 2 ×exp −P0 − P0 − − exp{i(;b − ;b )} 2˝&b 2˝&∗b

8˙ 8˙ 2 2 B + AG − Q0 + p0 Q0 − B + A G − Q0 − P0 Q0 ×exp i 8 8

8˙ i 1 2 ˙ ×exp − P0 (Q − Q0 ) (Q − Q0 ) > cot(> − > ) − B − AG + ˝ 2A 8

8˙ i 1 2 ˙ ×exp − (Q − Q0 ) > cot(> − > ) − B − A G + ˝ 2A 8   1=2  i  ˙ ˙ i >> (Q − Q )(Q − Q )=sin(> − > ) : ×exp − P0 (Q − Q0 ) − 0 0  ˝  ˝ AA

(8.117) Note that propagators for diHerent types of Hamiltonians for a single system do not have the same form.

180


The momentum operators of the quantum system correspond to the canonical momenta in the classical system, and thus there are numerous momentum operators and Hamiltonians for a quantum treatment of the system. To Jnd their quantum averages, i.e., the expectation values, let us deJne the kinetic momentum operator, which corresponds to the classical kinetic momentum as pˆ k ≡ qˆ = A(t)pˆ + B(t)qˆ + E(t) :

(8.118)

According to Eq. (8.118), the commutation relation, which is quite diHerent from the commutation relation given by the canonical coordinate and momentum, is found as [q; ˆ pˆ k ] = i˝A(t) : ˆ and pˆ in the To obtain the expectation values, it is convenient to express pˆ k ; q, lowering and raising operators for the Jrst Hamiltonian as 1=2

A˝ 8˙ 8˙ ˙ ˙ pˆ k = qˆ = − i> aˆ + + i> aˆ† + Ap0 + Bq0 + E ; 8 8 2>˙ 1=2 ˝A˙ qˆ = (aˆ + aˆ† ) + q0 ; ˙ 2> i A˝ 1=2 ˙ 8˙ 8˙ i A˝ 1=2 ˙ pˆ = − >−i B− >+i B− aˆ + aˆ† + p0 ; A 2>˙ 8 A 2>˙ 8 and for the second Hamiltonian as 1=2

† A˝ 8˙ 8˙ ˙ ˙ ˆ ˆ ˆ − i> b + Pk = Q = + i> bˆ + Ap0 + (B + AG)Q0 + E ; ˙ 8 8 2> 1=2 † ˝˙A Qˆ = (bˆ + bˆ ) + Q0 ; 2>˙ i A˝ 1=2 ˙ 8˙ pˆ = − > + i AG + B − bˆ A 2>˙ 8 † i A˝ 1=2 ˙ 8˙ bˆ + p0 : > − i AG + B − + A 2>˙ 8

(8.119) form of (8.120) (8.121) (8.122)

(8.123) (8.124)

(8.125)

Let the quantum eigenstates of the two Hamiltonians be |n and |nb . From Eqs. (8.120) – (8.125) we Jnd the relations n|f(q; p; t)|n = nb |f(Q; P; t)|nb ;

(8.126)

n|f(q; pk ; t)|n = nb |f(Q; Pk ; t)|nb :

(8.127)

and We point out that there are numerous classical Hamiltonians and corresponding canonical momenta for the system. There is only one classical solution. Therefore, f(q; pk; t) is Jxed regardless of the selection of the Hamiltonian, but f(q; p; t) is diHerent depending on the Hamiltonian. Like the classical cases, although there are numerous SchrKodinger equations and their solutions,


181

the quantum average f(q; ˆ pˆ k ; t) is the same for all states of each diHerent Hamiltonian, but f(q; ˆ p; ˆ t) depends on the states of the selected Hamiltonian. The commutation relations for the original (q; ˆ p) ˆ and the unitary transformed coordinate and ˆ ˆ ˆ ˆ = i˝. Therefore, the uncertainty product of momentum (Q; P) are given as [q; ˆ p] ˆ = i˝ and [Q; P] ˆ P) ˆ is greater than ˝=2. Through Eqs. (8.120) – (8.125), we obtain the uncertainty (q; ˆ p) ˆ and (Q; products: ; 1=2 8˙ 2 1 1 n|TqTp|n = ˝ 1+ 2 B− n+ ; (8.128) 2 8 >˙ 1=4 1 ˝ 8˙ 2 n + 1|TpTq|n = (n + 1) 1 + B− 2 8 >˙2   2 1=4  1 2A 1=2 2p0 8˙ √ + × 1+ B−   8 n + 1 >˙ ˝>˙ ×

  2>˙ 1=2  ˝A

1=2 

2p0 −1  n+1

√

;

(8.129)

1=2 1 8˙ 2 n + 2|TpTq|n = (n + 2)(n + 1) × 1 + B− ; 2 8 >˙2 ; 1=2 8˙ 2 1 1 ˝ 1 + 2 AG + B − nb |TQTP |nb = nb + ; 2 8 >˙ 1=4 1 ˝ 2 nb + 1|TPTQ|nb = (nb + 1) 1 + (EG + B − 88) ˙ 2 >˙2   2 1=4 1=2  1 2A 2p 8˙ √ 0 + × 1+ AG + B −   8 n + 1 >˙ ˝>˙

˝"

×

nb + 2|TPTQ|nb =

  2>˙ 1=2  ˝A

˝"

2

1=2 

2p0 −1  n+1

√

;

1 (nb + 2)(nb + 1) × 1 + >˙2

(8.130) (8.131)

(8.132)

8˙ AG + B − 8

2 1=2

:

(8.133)

ˆ P) ˙ for any state. From Eq. (8.120), The uncertainty product of (q; ˆ p) ˆ is diHerent from that of (Q; the uncertainty of position and kinetic momentum does not satisfy Heisenberg’s principle for

182


ˆ Pˆ k ) is given as the case of |A(t)| ¡ 1: The uncertainty (q; ˆ pˆ k ) and (Q; n|TqTpk |n = nb |TQTPk |nb

 2 1=2 8˙  1 ˝|A| 1 + = n+ :

2

˙ >8

(8.134)

This result is the same for all Hamiltonian systems. Replacing the classical canonical variable by a quantum mechanical operator, one can uniquely obtain the quantum-mechanical Hamiltonian from any classical Hamiltonian system. Also, there are numerous kinds of classical Hamiltonians for one classical equation of motion, and any Hamiltonian for them can be selected as a quantum Hamiltonian through the replacement of canonical variables by the quantum operators. 8.3.2. Wavefunction of a harmonic plus an inverse harmonic potential with time-dependent mass and frequency Generally, it is not easy to calculate the exact quantum-mechanical solution for the simple harmonic oscillator with various perturbative potentials. An asymmetrical quantum septic harmonic oscillator [234], time-dependent harmonic plus inverse harmonic potential [235], harmonic oscillator with time-dependent mass and frequency and a perturbative potential [236], time-dependent harmonic oscillator with and without a singular perturbation [237], SchrKodinger’s equation for an attractive r −6 potential [238], coherent states for the Kepler motion [239], etc. have been investigated by various methods. In this section we apply the canonical and unitary transformation to evaluate the exact wavefunction for a harmonic plus an inverse harmonic potential with time-dependent mass and frequency [230,240 –243]. Consider the harmonic plus inverse harmonic potential of the type a2 v(x; t) = a1 (t)x2 + 2 ; (8.135) x where a1 and a2 are time-dependent and time-independent coeMcients, respectively. The Hamiltonian of the harmonic plus inverse harmonic potential with time-dependent mass and frequency becomes H=

p M (t)!(t)2 q2 1 : + + 2M (t) 2 2M (t)q2

(8.136)

Taking M (t) = met and !(t) = !0 and neglecting the third term, we can reduce Eq. (8.136) to the Caldirola–Kanai Hamiltonian. For a time-dependent canonical transformation, we introduce the generating function G(q; P; t) = qP[M (t)]1=2 −

M (t)(t)q2 ; 4

(8.137)

where (t) is given by (t) =

d[InM (t)] : dt

(8.138)


183

From Eq. (8.137), the new canonical variables and Hamiltonian become @G(q; P; t) (8.139) = q[M (t)]1=2 ; Q= @p @G(q; P; t) [M (t)]1=2 (t)Q P= = p[M (t)]1=2 − ; (8.140) @p 2 @G(q; P; t) HN = + H (t) : (8.141) @t The new Hamiltonian is given by 1

2 (t)Q2 1 ; (8.142) HN = P 2 + + 2 2 2Q2 2 (t) (t) ˙

2 (t) = !2 (t) − − : (8.143) 4 2 where (t) is the new frequency. One can conJrm that the commutation relation [Q; P] = [q; p] holds for both coordinates. The invariant quantity for the Hamiltonian can be constructed by Ermakov’s technique [244]. To derive the invariant for Eq. (8.142), let us consider the equation of motion K + 2 (t)Q(t) = 1 : Q(t) (8.144) Q3 (t) If *(t) is the solution of Eq. (8.144), then this equation can be expressed by 1 QK 1 − 4 *= 3 : (8.145) *K − Q Q * ˙ we obtain Multiplying Eq. (8.145) by (Q*˙ − Q*), 2 2 1 * Q d 1 d ˙ 2=− + : [(Q*˙ − Q*)] 2 dt 2 dt Q * Then the invariant, Eq. (8.146), can be written as 2 2 1 * Q I (t) = (*Q˙ − *Q) ˙ 2+ + : 2 Q *

(8.146)

(8.147)

This invariant coincides with that of Ref. [237,240] obtained by using the Lie algebra method. Let us introduce another transformation, *(t) = x(t)M 1=2 (t) ;

(8.148)

where x(t) is a function of time to be determined. Substitution of Eqs. (8.139), (8.143) and (8.146) into Eq. (8.144) yields 1 qK + (t)q˙ + !2 (t)q = 2 : (8.149) M (t)q3 If x(t) is some classical solution of Eq. (8.149), the invariant, Eq. (8.147), becomes 2 1 x q 2 I (t) = M 2 (t)(qx˙ − xq) ˙ 2+ + : (8.150) 2 q x

184


We thus say that Eq. (8.149) constitutes an Ermakov system, or the Hamiltonian given by Eq. (8.136) that has an invariant in the form of Eq. (8.150). For M (t) = 1, Eq. (8.150) is reduced to Eq. (8.146). In this case, the generating function, Eq. (8.137), corresponds to the identity transformation, and the structure of the Hamiltonian corresponding to Eq. (8.144) turns out to be

1 2 1 2 2 R H= pR + (t)* − 2 ; (8.151) 2 * where *˙ = p, R and this equation is analogous to the form of Eq. (8.142). The invariant, Eq. (8.147), maps H into HR and vice versa. Furthermore, when M (t) is time-dependent and the third term is neglected, one can recover the well-known time-dependent harmonic oscillator. The invariant I (t) satisJes Hamilton’s equation, Eq. (6.11) [176,177], and the eigenstates n (Q; t) of the invariant I (t) are assumed to form a complete orthogonal set with time-dependent eigenvalues %n : In (Q; t) = %n n (Q; t) : Let us consider the unitary transformation

i*Q ˙ 2 n (Q; t) = exp − n (Q; t) ; 2˝*

(8.152) (8.153)

through which we then obtain the new invariant I (t)[I = UIU + ]: I n (Q; t) = %n n (Q; t) ; 1 Q 2 1 * 2 ˝2 2 @2 + + : I =− * 2 @Q2 2 * 2 Q Taking F = Q=*, Eq. (8.154) is transformed into 2 2 F2 ˝ @ 1 − + + 2 n (F) = %n n (F) : 2 @F2 2 2F

(8.154) (8.155)

(8.156)

This represents an ordinary one-dimensional SchrKodinger equation with potential V (F) = F2 =2 + 1=2F2 whose solution [177] is given as 2 1=4

(2a+1)=4 4 The bilinear i=1 = + mi ) i part of LQCD then yields the determinantal factor i (D Det =

Nf

Det(D= + mi ) ;

(1.6)

i=1

which for any vectorial (non-chiral) theory can be shown to be positive for any A (x) (see Section 8) [6 –10]. If the integrand in the path integral expression for one correlation function is greater than the integrand in another correlation function for all A (x), then the positivity of the path integration measure guarantees that this feature persists for the integrated values. A rigorous inequality between the two correlation functions for all possible (euclidean) locations of the external currents will then follow. For the particular case of two-point functions an inequality of the form 0|Ja (x)Ja† (y)|0 ¿ 0|Jb (x)Jb† (y)|0

(1.7)

implies, via Eq. (1.4), the reversed inequality for the lowest mass physical states with the quantum numbers of Ja (x); Jb (x): (0) m(0) a 6 mb :

(1.8)

Most of the inequalities (1.7) involve the pseudoscalar currents (Ja = > i 5 j ) and the corresponding mass inequalities (1.8) the pion (i.e. the lowest pseudoscalar states in the u> 5 d; u> 5 u − d> 5 d; d> 5 u channels) [6]: m(0) (any meson) ¿ m$ ;

(1.9a)

m(0) (any baryon) ¿ m$ ;

(1.9b)

m$+ ¿ m$0 :

(1.9c)

The e8orts to obtain inequalities in the Hamiltonian approach follow a similar general pattern. Rather than attempting to solve the QCD SchrModinger equation (1.5) for a particular channel, relations are sought between baryonic and=or mesonic sectors of di8erent 3avors Bijk ; Mi—>, and di8erent spins. Flavor enters the Lagrangian (1.1) only via the bilinear, local mass term. Comparison of masses (or other features) of mesons or baryons di8ering just by 3avor may be easier than ab initio computations. The additive form of the mass term implies a relationship between the Hamiltonians obtained by restricting the full HQCD to di8erent 3avor sectors. Using the variational principle for the ground state masses and assuming 5avor symmetric ground state wave functions, these relations imply the inequalities (0) (0) 1 m(0) ij ¿ 2 (mi)> + mj —> ) ; (J with m(0) ij the ground state mass in the mesonic sector Mi—>

(1.10) PC

)

.

S. Nussinov, M.A. Lampert / Physics Reports 362 (2002) 193 – 301

197

a Physical color singlet states can be achieved in a variety of ways: the mesonic— > i ja ; baryonic— jabc ia jb kc ; exotic—jcde jabc ia jb kd le (or hybrid— > a b Gr r(ab) ) con=gurations; or glueballs—G r Gr ; frst (or drst ) G r G s G t [Here G = G represents the chromo-electromagnetic antisymmetric =eld tensor. We use abc(rst) for triplet (octet) color indices; r(ab) are the Gell–Mann matrices; and ijkl refer to 3avors. We reserve 01 for spinor indices, and 2 for vector Lorentz indices.] These con=gurations (mesonic, baryonic, etc.) di8er dynamically by having di8erent “color networks”. However, the di8erent (non-glueball or hybrid) sectors contain quarks and=or antiquarks which are sources or sinks of chromoelectric 3ux of the same universal strength. This suggests that the QCD Hamiltonians in the di8erent sectors may be related and mass relations of the type [11]

mbaryon ¿ 32 mmeson

(1.11)

can be obtained. The rigorous inequalities (1.9a) and (1.9b) derived via the euclidean path integral formulation amount to the well-known fact that the pion is the lightest hadron. These inequalities have, however, profound implications for the phase structure of QCD. Eq. (1.9a) implies no spontaneous breaking of vectorial global (isospin) symmetries in QCD. Eq. (1.9b), along with the ’t Hooft anomaly matching condition, implies that the axial global 3avor symmetry must be spontaneously broken. It is important to note that the basic feature of positivity of the determinant factor (1.6) and the functional path integral measure d(A) is common to all gauge, QCD-like vectorial theories with Dirac fermions. This has far-reaching implications for composite models for quarks and leptons. A basic puzzle facing such models is the smallness of the masses of the composite quarks and leptons: me mu md MeV in comparison with 3p ¿ TeV, the compositeness (“preonic”) scale. A natural mechanism for protecting (almost) massless composite fermions is an unbroken chiral symmetry. ’t Hooft [12] and others [13], using the anomaly matching constraint, formulated some necessary conditions for such a realization of an underlying global chiral symmetry in the spectrum of the theory. Together with the mass inequalities, these conditions rule out all vectorial composite models for which fermion–boson mass inequalities like Eq. (1.9b) [or (1.11)] can be proven. In this work we will mention, at one stage or another, most of the papers written on QCD inequalities, or on the related subject of inequalities in potential models. Particular attention will be paid to the seminal works of Weingarten [6], Vafa and Witten [7–9], and Witten [10]—all utilizing the euclidean path integral approach. Weingarten proved the inequalities (1.9a) and (1.9b) and pointed out the relevance of (1.9b) to spontaneous chiral symmetry breaking (SSB) in QCD. Vafa and Witten directly used the measure positivity to prove that parity and global vectorial symmetries like isospin do not break spontaneously [7,8]. Finally Witten [10] proved (1.9c) and the inter3avor relation (1.10) which holds for the case of pseudoscalars, with no need for the 3avor symmetry assumption. To date, QCD inequalities have been mentioned in approximately 600 papers. Most authors were concerned with symmetry breaking patterns, the motivation for the Vafa–Witten paper [7], which is cited most often. Here we equally emphasize the other facet of the inequalities, which constitute useful, testable, constraints on observed (and yet to be discovered) hadrons. This is why we elaborate on the baryon–meson mass inequalities and related inequalities for the exotic sector despite the fact that we have not been able to prove them via the rigorous euclidean path integral approach; and on the inequalities between mesons of di8erent 3avors, which cannot be justi=ed without the additional speci=c assumption of 3avor symmetric mesonic wave function(al)s.

198


In general we will follow a didactic rather than a chronological approach. We start in Section 2 by deriving Eq. (1.10) using a simple potential model [14] and the 3avor symmetry assumption. This is followed up in Section 3 by a potential model derivation of the baryon–meson mass inequalities (1.11) [11,15,16], and a discussion of baryon–baryon mass inequalities in Section 4. In Sections 5 and 6 and Appendix C we show that these relations may be valid far beyond the simple potential model [11,17]. The elegant analysis of Lieb [18] of the potential model approach is also reproduced in some detail in Appendix B. Section 7 concludes the =rst part of this review which utilizes the Hamiltonian variational approach by verifying that the inequalities indeed hold in the approximately forty cases where they can be tested, and by presenting lower bounds to the masses of new, yet to be discovered, mesonic and baryonic states. The middle part of the review focuses on the rigorous euclidean path integral approach. We start in Section 8 by proving the positivity of the measure and the ensuing inequalities (1:9), stating that the pseudoscalar pion is the lightest meson. We illustrate the power of these inequalities in Section 9 + − ) where m(0 ¿ mu(0d> ) is used as a shortcut to motivate the Vafa–Witten theorem on non-breaking of ud> isospin (which is then presented in some detail). We proceed in Section 10 with Weingarten’s proof of the pion–nucleon mass inequality [Eq. (1.9b)] and in Section 11 indicate how it can be utilized to prove SSB and discuss its implication for composite models of quarks and leptons. Section 12 presents Witten’s proofs of Eq. (1.9c) and of the 3avor mass inequality for pseudoscalars. We present also an alternate derivation of Eq. (1:9c) [19]. The Vafa–Witten argument [8] for non-spontaneous breaking of parity in QCD is presented in Section 13. Section 14 is concerned with the inequalities in the large Nc limit of QCD [20]. Section 15 is devoted to the glueball sector [21], and Section 16 discusses inequalities in the continuum meson–meson sector and for exotic states; in particular we also discuss extensions to two-point functions involving local quark combinations of a quartic degree [22,23]. In Section 17 we discuss extensions to =nite temperature, =nite chemical potential, and external electromagnetic =elds. In Section 18 we discuss the constraints implied by QCD for the > chiral Lagrangian approach, and also discuss the utilization of QCD inequalities to constrain the QQ potential V (R) in heavy quarkonium states, quark mass ratios, and weak matrix elements. Finally Section 19 discusses extensions beyond two-point functions. Towards the end of the review we adopt a more heuristic approach, applying QCD inequality-like relations to four and =ve particle states in Appendix F and to electromagnetic e8ects on scattering lengths in Appendix G. We discuss some applications of related inequalities in atomic, chemical, and biological contexts in Appendix H. Also in Section 14, we use the large N (planar) limit to extend the inter3avor meson mass inequalities which were rigorously proven only for the pseudoscalar channel [10] to other cases. Sections 14–16, and also Appendices C–H constitute mostly new, unpublished material. Section 20 includes a short summary. We also present two new conjectures concerning the possible utilization of the ferromagnetic character of the QCD euclidean Lagrangian, and a possible monotonic behavior of mass ratios with the number of quark 3avors Nf , as well as inequalities for quantities other than hadronic masses. 2. Derivation of avor mass relations in a simple potential model We =rst discuss inequalities in a simple potential “toy model” [14,17,18,24 –26], which contains some features of the full-3edged QCD problem. Speci=cally the interactions—represented here by


199

the potentials—are 3avor independent. Flavor dependence is manifested only via the masses in the kinetic term. Let us consider a two-body system described by the Hamiltonian H12 = T1 + T2 + V12 :

(2.1)

For a non-relativistic SchrModinger equation, the kinetic terms are T1 =

p ˜ 12 ; 2m1

T2 =

p ˜ 22 2m2

(2.2)

with m1; 2 the masses of particles 1 and 2. We assume that the potential V depends only on the relative coordinate ˜r =˜r1 −˜r2 (translational invariance), and we also take V = V (|˜r|) = V (r) to ensure rotational invariance. We can separate the motion of the center of mass: ˜ ˜

= e i P ·R

r) 12 (˜

;

˜R = m1˜r1 + m2˜r2 ; m1 + m 2 ˜r = ˜r1 − ˜r2 ; and write H12 as H12 =

˜P 2 ˜P 2 p ˜2 + + V12 (r) ≡ + h12 ; 2M 2 2M

(2.3)

˜ 2; p ˜ =p ˜ 1 −p ˜ 2 ; M = m1 + m2 ; and = m1 m2 =(m1 + m2 ). For the subsequent discussion with ˜P = p ˜ 1 +p we will specialize to ˜P = 0, i.e. to the center of mass system (CMS). We will be interested in the bound states of h12 satisfying h12

12

= j12

12

;

p ˜2 p ˜2 p ˜2 p ˜2 p ˜2 + V12 (r) = + + V12 (r) ≡ + + V (r) ; (2.4) 2 2m1 2m2 2m1 2m2 2 (˜r) = 1) state. (We assume that such bound states exist.) with 12 = 12 (˜r) a normalized ( d 3 r 12 We can next consider two additional systems with identical potentials V11 (r) = V22 (r) = V (r), but made of two particles with the same mass m1 (or m2 ) 1 1 + V (r) ; ˜2 + h11 = p 2m1 2m1 1 1 2 h22 = p + V (r) : (2.5) ˜ + 2m2 2m2 h12 =

To mimic the QCD problem, with quark–antiquark bound states, we still take the particles as non-identical so as to avoid issues of statistics.

200


(0) (0) Let j(0) 12 ; j11 ; j22 be the ground state energies for the three Hamiltonians

hij

(0) r) ij (˜

= j(0) ij

(0) r) ij (˜

:

(2.6)

We wish to derive the relation (0) 1 (0) j(0) 12 ¿ 2 (j11 + j22 ) :

(2.7)

Eqs. (2.4) and (2.5) imply the operator identity h12 = 12 (h11 + h22 ) :

(2.8)

Let us take the diagonal matrix element of both sides of this equation with wave function of h12 . We then have

(0) (0) 12 |h12 | 12

1 = j(0) 12 = 2 (

(0) (0) 12 |h11 | 12

+

(0) (0) 12 |h22 | 12 )

:

(0) 12 ,

the ground-state (2.9)

By the variational principle [27] each of the expectation values on the right-hand side exceeds (0) (0) (0) = 11 and 22 , j(0) 11 (or j22 ), respectively, which are minima of |h11 | and |h22 | with respectively. Thus Eq. (2.7) is obtained. The previous discussion has been carried out in the CMS frame, ˜Ptotal = 0. In this frame, upon adding the rest masses m1 + m2 to j(0) 12 , etc., inequality (2.7) translates into an inequality for the total masses of the bound states: m(0) ¿ 12 (m(0) + m(0) ): 12> 11> 22>

(2.10)

Also in this frame total angular momenta are the spins of the composites. The following observations will be useful for the discussion of Section 5: (1) The addition of the two Hamiltonians h11 and h22 and their comparison with h12 may appear to present some formal diQculties. In non-relativistic physics we have a super-selection rule for di8erent masses. Hence strictly speaking h11 ; h22 ; and h12 operate in di8erent Hilbert spaces consisting of states with the (1; 1), (2; 2), and (1; 2) pairs of particles, respectively. However, the crucial observation is that these are identical spaces. All the Hamiltonians h11 ; h22 ; h12 (or any other hij ) which are written in terms of ˜r and p ˜ can, therefore, be made to operate on the common generic Hilbert space of wave functions (˜r). The mass dependence is explicit in hij . It happens to be additive yielding Eq. (2.8), a meaningful and useful operator identity. (2) The above generic space can be block diagonalized by using the symmetries of the Hamiltonians hij . Thus for the case of central potentials, we consider separately subspaces with given total angular momentum l and solve hij

(l) ij

= jij(l)

(l) ij

:

(2.11)

The projection operator Pl used in Pl hij (Pl )† ≡ hij(l) can be expressed in terms of ˜r and p ˜ only, and is independent of the masses. Thus Pl simultaneously projects into the l subspace every term in Eq. (2.8): (l) (l) (l) h12 = 12 (h11 + h22 ):

(2.12)


201

Using the same arguments as above, we can deduce that l(0) 1 l(0) jl(0) 12 ¿ 2 (j11 + j22 )

(2.13)

holds for the ground states in each l-wave separately. (3) To the extent that heavy quarkonium systems can be treated via a non-relativistic SchrModinger equation approximation, with negligible spin e8ects, the above derivation would indeed directly (0) suggest mass relations such as m(0) ¿ 12 (m(0) cc> + mbb> ). cb> (4) Inequalities (2.7) and (2.13) are obtained by taking just one particular matrix element of the operator relation (2.8). Evidently the latter contains far more information. In particular one may wonder if the inequalities hold for excited states in each given l channel (i.e. for radial excitations). (1) 1 (1) The simple generalization, e.g. j(1) 12 ¿ 2 (j11 + j22 ) for the =rst radially excited state is, in gen(1) (1) (1) ; 22 and 12 should minimize the expectation values eral, incorrect. The point is that 11 (1) should |H11 | , |H22 | and |H12 | respectively, subject to di7erent constraints: 11 (0) be orthogonal to 11 , etc. and the three ground state wave functions are in general di8erent. However, such constraints can be avoided if we consider the two-dimensional space v2 spanned by the =rst and second excited states. The variational principal (for hij ) tells us that (1) (0) |hij | (0) + (1) |hij | (1) = j(0) ij + jij is the minimal value that tr v2 hij can achieve when we (1) consider all possible two-dimensional subspaces j(0) ij + jij = min v2 tr hij . More generally we have (1) (n−1) =minvn tr hij , for the sum of the ground state and the =rst n−1 excited states: j(0) ij +jij +· · ·+jij where the trace is now to be minimized over all n-dimensional subspaces vn . Considering then the operator relation h12 = 12 (h11 + h22 ) and taking the trace of both sides in the space vn1; 2 which minimizes tr vn h12 , we conclude that (1) (n−1) (1) (n−1) (1) (n−1) j(0) ¿ 12 [(j(0) ) + (j(0) )] : 12 + j12 + · · · + j12 11 + j11 + · · · + j11 22 + j22 + · · · + j22 (2.14)

with the summation extending up to any one of the radially excited states. (5) It is amusing to see how the inequalities work in familiar cases. For the Coulomb potential the ground state bindings are mi mj 1 1 (0) 2 02 ; Bij = ij 0 = 2 2 mi + m j 1 mi 2 0 ; Bii(0) = 2 2 1 mj 2 (0) 0 : = (2.15) Bjj 2 2 (0) ) so that the masses of the corresponding states satisfy the desired Obviously, Bij(0) 6 12 (Bii(0) + Bjj inequality (0) (0) (0) 1 m(0) ij = mi + mj − Bij ¿ 2 (mii + mjj ) :

(2.16)

202


Next, consider the harmonic oscillator potential, 12 Kr 2 , as a prototype of a con=ning potential. The ground state energies, measured with respect to the minimum of the potential well, are

1 ˝ (0) ˝ K ˝ 1 (0) jij = !ij = : K = + 2 2 ij 2 mi mj (0) 2 (0) 2 1 2 We therefore have (j(0) ij ) = 2 [(jii ) + (jjj ) ] which implies by simple algebra that (0) 1 (0) j(0) ij ¿ 2 (jii + jjj ) ;

(2.17)

and upon adding the rest masses: (0) (0) 1 m(0) ij ¿ 2 (mii + mjj ) :

(2.18)

We note in passing that the energies for these two systems happen to have quantization rules of the form j(n) = f(n)j(0) , where f(n) is independent of , the reduced mass. 1 Speci=cally, f(n) = 1=(n + 1)2 and (2n + 1) for the two cases. Hence all the above inequalities happen to (n) hold separately for each of the excited states jij(n) , jii(n) , and jjj . (6) While the above discussion was in the framework of a non-relativistic SchrModinger equation, the result (2.17) holds in a far more general context. The particular form of the kinetic energy Ti = p ˜ 2 =2mi did not play any role in deriving the operatorrelation (2.8). Hence Ti could have an arbitrary p dependence. In particular, we can take Ti = p ˜ 2 + m2i , the expression appropriate for relativistic motion. The key ingredient was that the potentials V12 ; V11 , and V22 are all the same, i.e. that we have “3avor independent interactions”. However, besides the requirement of translational (and preferably also rotational) invariance, the “potential” is not restricted. Thus, V need not depend on ˜r alone, but could have arbitrary dependence on ˜r and p ˜ . The only aspect we have to preserve is that 1 hij = Ti + Tj + Vij so that hij = 2 (hii + hjj ). Non-relativistic quark models have potentials with some explicit 3avor (quark mass) dependence, e.g. in the “hyper=ne interaction” [28]: VijHF =

(i · j ) (i · j )>3 (˜r) : mi mj

(2.19)

In Section 5 we show that in the full-3edged theory we still have an operator relation Hi—> + Hk l> = Hil> + Hk —> ;

(2.20)

with Hij being the QCD Hamiltonian restricted to a particular 3avor sector. With i = k; j = l the last relation reduces to Hi—> + Hj)> = Hi)> + Hj—> ; which is very reminiscent of Eq. (2.8) and, to the extent that the wave functions in 3avor, again leads to the same conclusion. 1

(2.21) i—>

are symmetric

This feature stems from the fact that these potentials have a power law behavior, r 0 , in which case j(n) = f(n) is clearly manifest in the semiclassical WKB limit.


203

One can avoid the mass dependent ˜s1 ·˜s2 or ˜s · ˜L interactions by considering the “centers of mass” of the various multiplets, e.g. (3m@ +m$ )=4, representing the mass values prior to hyper=ne splittings and applying the inequalities to these combinations [15,16]. We note that the hyper=ne interaction is strongly attractive for the spin singlet pseudoscalar meson 2 −2 −1 case. Since furthermore m− 1 + m2 ¿ 2(m1 m2 ) , this extra binding enhances the inequality (2.7) derived for the spin independent part of the interaction. Indeed, the only case for which relation (2.7) was proved by utilizing the euclidean correlation function approach, is that of the pseudoscalar mesons [10].

3. Baryon–meson mass inequalities in the gluon exchange model The simple “quark counting rules” relating meson and baryon total cross sections [29,30] and masses [31] were early indications for the relevance of the quark model [32,33]. If the mass of a ground state hadron is just the sum of the masses of its quark constituents, then (0) (0) (0) 2m(0) ijk = mi—> + mj k> + mk )> ;

(3.1)

(0) with m(0) ijk the mass of the ground state baryon consisting of quarks qi ; qj ; qk ; and mi—> the mass of the lowest lying qi q>j meson. The following discussion motivates a related QCD inequality [11,15,18,34] (0) (0) (0) 2m(0) ijk ¿ mi—> + mj k> + mk )> :

(3.2)

The particular variant mN ¿ m$ was derived by Weingarten [6] from the correlation function inequalities, and will be discussed in Section 10. If qq (qq) > interactions are generated via one gluon exchange, then the color structure is V12 ˙ ˜1 · ˜2 , with ˜ a vector consisting of the N 2 − 1 Gell–Mann matrices of the fundamental SU(N ) representation. In a meson the quarks q1 q>2 couple to a singlet and, using · · · to indicate expectation values, 0 = (˜1 + ˜2 )2 meson = 2˜2 + 2˜1 · ˜2 meson ;

(3.3)

so that (3.4) ˜1 · ˜2 M = −˜2 : 2 Here ˜2 = Nn=0−1 (n )2 , is a =xed diagonal N × N matrix proportional to the unit matrix. The SU(N ) baryon is constructed as the color singlet, completely antisymmetric combination of N quarks ja1 :::aN qa1 :::aN : Thus

(3.5)

N 2 N 0= i = N ˜2 + i j B = N ˜2 + N (N − 1)1 · 2 B ; i

B

i=j

(3.6)

204


where we used the fact that all the N (N − 1) expectation values i · j B are equal. Thus using Eq. (3.4) we =nd that 1 ˜2 1 1 · 2 M : 1 · 2 B = − (3.7) = (N − 1) (N − 1) This implies that the strength of the one gluon exchange interaction in the meson qi q>j is, for QCD (i.e. N =3), precisely twice the corresponding gluon exchange interaction for a qi qj pair in a baryon. Since 1 ·2 B = 12 1 ·2 M is just an overall relation of the color factors, we have the corresponding ratio of the pairwise interactions in the meson and in the baryon: Vi—>(· · ·)M = 2Vij (· · ·)B ;

(3.8)

as long as the qi q>j are in the same angular momentum, spin, radial state, etc. as the qi qj in the baryon (or in the appropriate mixture of states). For N = 3 and a general two-body interaction, we write the Hamiltonian describing the baryon as HB = Hijk = Ti + Tj + Tk + VijB + VjkB + VkiB ;

(3.9)

with Ti the kinetic, single-particle operators. The qi qj qk baryonic system can be partitioned into (2 + 1) subsystems in three di8erent ways: ((ij); k); (i; (jk)); (j; (ki)). Let us consider the three mesonic qq> systems corresponding to these partitionings: Mi—>; Mjk>; Mk )>, composed of (qi q>j ); (qj q>k ); (qk q>i ). The Hamiltonians describing these mesons are Hi—> = Ti + Tj + ViM —> ; Hjk> = Tj + Tk + VjMk> ; Hk )> = Tk + Ti + VkM)> :

(3.10)

B From ViM —> = 2Vij , etc. it is easy to verify the key relation

2Hijk = Hi—> + Hjk> + Hk )>

(3.11)

between the Hamiltonian describing the baryon Bijk and the three mesonic Hamiltonians corresponding to its diquark subsystems. Let us next take the expectation value of the last operator relation in (0) the normalized ground state ijk of the baryon at rest [to simplify the notation we suppress Lorentz PC (J ) quantum numbers]. The left hand side yields 2

(0) (0) ijk |Hijk | ijk

= 2m(0) ijk ;

(3.12)

with m(0) ijk the mass of the ground state baryon. The right hand side is

(0) (0) ijk |Hi—>| ijk

+

(0) (0) ijk |Hj k> | ijk

+

(0) (0) ijk |Hk )>| ijk

:

(3.13)

(0) (0) |Hi—>| ijk , The matrix elements of the two-body operators in the three-body wave function, e.g. ijk are evaluated by considering Hi—> as a three-body operator which is just the identity for quark k. (k) Thus we view ijk , for =xed coordinates of the quark qk , as a two-body (qi qj ) wave function ˜ , (k)

(k)

and compute ˜ i—> |Hi—>| ˜ i—> . Finally, this is integrated over all values of the coordinates of qk .

(i—) >


205

(k) (0) The key observation is that ˜ (i—)> , the two-body wave function prescribed by ijk , is, in general, (0) di8erent from (i—)> , the ground state wave function of the meson qi q—>. By the variational principle the latter wave function minimizes the expectation value of Hi—>, the mesonic Hamiltonian: (0) (0) min ˜ |Hi—>| ˜ = ˜ |Hi—>| ˜ = m(0) ; (3.14) ˜

i—>

i—>

i—>

i—>

i—>

i—>

where m(0) i—> , the energy of the state at rest, is the mass of the ground state qi q—> meson. Hence we have (k) (k) ˜ i—> |Hi—>| ˜ i—> ¿ m(0) (3.15a) i—> ; and likewise (i) (i) ˜ jk> |Hjk>| ˜ jk> ¿ m(0) ; j k> ( j)

(3.15b)

( j)

˜ k )> |Hk )>| ˜ k )> ¿ m(0) k )> :

(3.15c)

Since each of these inequalities persists after the normalized integration over the coordinates of the third quark [e.g. qk for Eq. (3.15a)], we have,

(0) (0) (0) ijk |Hi—>| ijk ¿ mi—> ;

etc :

(3.16)

Equating (3.12) and (3.13), the matrix elements of the left hand side and right hand side of the original operator relation, we arrive at the desired inequality (3.12) [11,15]. The following remarks elaborate on these inequalities and the conditions for their applicability. (1) We have not displayed the J P quantum numbers of the baryonic ijk or the mesonic i—> states. As indicated by the above construction of “trial” mesonic functions from the baryon wave function, P state (or in general, in a mixture of J P states) prescribed by the original i—> must be in the J (0) baryonic wave function ijk . ++ Consider =rst the A baryon. In the approximation where L = 0 components in the ground state wave function are ignored, the state with Jz (A++ ) = 3=2 consists of three up quarks with parallel spins: u ↑ u ↑ u ↑, with the arrow indicating spin direction. Each of the i(0) —> will be, in this case, u ↑ u> ↑, in a spin triplet state @ or !. The inequality reads mA++ ¿ 32 m@ (or m! )

(ijk = u ↑ u ↑ u ↑) :

(3.17)

Likewise, mC− ¿ 32 mD

(ijk = s ↑ s ↑ s ↑)

(3.18a)

2mE0 (1533) ¿ 2mK ∗ + mD

(ijk = s ↑ s ↑ u ↑)

(3.18b)

2mF+ (1384) ¿ 2mK ∗ + m@

(ijk = u ↑ u ↑ s ↑) :

(3.18c)

The situation is di8erent for the J = 12 (i.e. S = 12 in this L = 0 approximation) baryons such as the nucleon (see Appendix D). In this case the diquark systems are, on average, with equal probability in the S =0 and 12 states, as can be readily veri=ed from (s1 +s2 +s3 )2 =sN2 = 34 . [Strictly speaking, the uu diquark is pure triplet and the ud diquarks are in a triplet (singlet) state with probabilities 14 ( 34 ), respectively.]

206


To see the e8ect of having a mixed rather than a pure trial mesonic state, we revert back to (k) Eq. (3.15a). Instead of a single matrix element of Hi—> in a speci=c ˜ (i—)> state, we have, in general, a weighted sum of matrix elements corresponding to the mixture of two-body states generated from the three-body baryonic wave function: (0) (0) (k) |Hi—>| ijk = cn2 i(k) (3.19) ijk —(n) > |Hi—>| i—(n) > : n

Here n labels the di8erent (J P ) states and cn2 are the normalized weights ( n cn2 = 1). The variational argument can be applied separately for each of the (trial) ˜ (n) states and to their matrix elements ˜ (n) |Hi—>| ˜ (n) to obtain (0) (0) |Hi—>| ijk ¿ cn2 mi—(n) : (3.20) ijk > n

2 = For the speci=c case of the nucleon the weights of the qq singlet (triplet) con=guration are cs(qq)=1 1 2 cs(qq)=1=2 = 2 . The corresponding qq> states are the @(!) and the $, and the inequality (3.20) (plus similar ones for the matrix elements of Hjk> and Hk )>) yields:

2mN ¿ 32 (m$ + m@ ) :

(3.21)

Likewise by considering the 3 hyperon we =nd 2m3 ¿ 12 [2m$ + 3mK + mK ∗ ] :

(3.22)

(2) A simpli=ed version of the inequalities applied just to the spin-averaged (“center of mass”) multiplets was suggested by Richard [15]. Note that unlike for the case of the meson–meson mass relation in Section 2, the baryon–meson mass inequalities do apply even when we have mass dependent qi qj (or qi q—>) interactions such as the hyper=ne and=or spin orbit interactions generated by one gluon exchange. The point is that the same 3avor (mass) combinations (ij); (jk); (ki) appear in both the diquark subsystems and in the mesons Mi—>; Mjk>; and Mk )>. Hence, there is no need to consider only the above simpli=ed version. (3) The speci=cation of the combinations of spins and 3avors of the mesons appearing in these inequalities has used the notation and level assignments of the non-relativistic quark model. This is, however, mainly done for convenience and does not detract from the degree of rigor of the derivation. Thus let us consider the comparison between the polarized A++ and @ masses. Assume that the A++ ground state has important L = 0 components. Then we could have a uu S = 1 diquark with L = 2, or S = 0 with L = 1; : : : ; and we would then simply need to construct, from the A++ wave functional, the corresponding trial wave functional for the lowest J = 1 ud> state. The latter is still the @—no matter what orbital or other components it may have. (4) The above discussion utilized the non-relativistic quark model with a one gluon exchange potential. We show next (and in Section 6) that the inequalities are valid in a vastly larger domain [11]. First we note that any vertex or propagator insertions into the one gluon exchange diagram leave the 1 · 2 color structure intact (see Fig. 1). Such insertions generate a running coupling constant 0(q2 ) or 0(r). Even if the non-perturbative e8ect of the propagator insertions can generate a con=ning potential [35,36] V ∼ r, this potential still has the 1 · 2 color structure, and the derivation of the inequalities still applies.


207

Fig. 1. Vertex and propagator insertion of the one gluon exchange diagram with the 1 · 2 structure intact. Fig. 2. The general two quark interaction is a sum of 1 · 2 (octet exchange) and 11 · 12 (singlet exchange) interactions.

(5) The most general two-body interaction [due to any number of gluon and qq> exchanges; see Fig. 2] in the qq or qq> system, has, from the t-channel point of view, a 3> ⊗ 3 = 8 ⊕ 1 color structure. The octet part corresponds to the 1 · 2 structure; the singlet to 1 · 1. Thus we need not assume that the important interactions are due to one gluon exchange. Rather, we have to assume that the octet, 1 · 2 part of the full two-body interaction is dominant. We note that in the large Nc limit we expect the 1 · 2 part to dominate over the 1 · 1 part. (6) The trilinear gluon couplings are a potential obstacle to having the operator relation (3.11) in a general setting. These couplings could yield genuinely nonseparable interactions in the nucleon of the type shown in Fig. 3. However, it turns out that the trigluon diagram vanishes in the baryon state. The relevant color factor is

frst raa sbb tcc jabc ja b c ; with rst = 1; : : : ; 8; abc(a b c ) = 1; : : : ; 3 the adjoint and fundamental color indices, respectively. This color factor vanishes since an exchange r → s; a → b; a → b changes the sign of the summand. This cancellation is not modi=ed by any further dressing of this diagram with more gluon exchanges. (7) The above discussion did not require any speci=c form of the one-body kinetic terms Ti . We could have the non-relativistic Ti =mi +pi2 =2mi , a relativistic Ti = pi2 + m2i form, or, by considering the operator relation (3.11) as a matrix in spinor space, also a Dirac Ti = P=i + mi form. (8) Just as for the meson–meson relation, we could use the variational principle in the space of the lowest n states to obtain a relation between masses of radially excited baryons and mesons analogous to Eq. (2.14). (9) A diquark in a baryon cannot annihilate into gluons. This is not the case, however, for the uu> or dd> in a meson. These annihilations are avoided by choosing I = 1 [i.e. @ rather than ! in Eq. (3.17)] combinations. (10) A simple intuitive feeling for the deviation from equality expected in the baryon–meson relation has been o8ered by Cohen and Lipkin [37] (in a paper which predated the QCD inequalities). The point is that the diquark systems in the baryon are not at rest but recoil against the third

208


Fig. 3. A non-separable three body interaction in the baryon due to the three gluon vertex which is shown, however, to vanish.

quark with a typical momentum p ˜ of a few hundred MeV=c. Thus the trial meson wave functions correspond to a meson moving with p ˜ = 0. Consequently 2mijk ¿ Ei—> + Ejk> + Ek )> ≈ m2ij + p ˜ k2 + m2jk + p ˜ i2 + m2ki + p ˜ j2 ¿ mij + mjk + mki : A more comprehensive investigation of the pattern of deviations from equality as a function of the quark masses, and in particular for logarithmic interquark potentials was performed by Imbo [38]. (11) The baryon–meson relation can be easily extended via Eq. (3.7) to any number of colors N : 1 (0) m(0) mia )>b : (3.23) Bi1 :::iN ¿ N −1 a=b

4. Baryon–baryon mass inequalities The meson–meson mass inequalities above (see Section 2) follow in potential models only if the various two-body quark–antiquark potentials are independent of the quark masses. Baryon–baryon mass inequalities motivated by similar convexity arguments were suggested [17] originally to also hold when all two-body quark–quark interactions are 3avor-independent. It turns out however, that inequalities of the form E (0) (m; m; m) + E (0) (m; M; M ) 6 2E (0) (m; m; M )

(4.1)


209

do not hold in general [18,25,39]. As will be clearly indicated in Section 5, the key requirement for proving either meson–meson or baryon–baryon mass inequalities is the 3avor symmetry of the ground state wave function. In the two-body, pure potential case, the relevant, relative coordinate wave function (0) (˜r); ˜r =˜r1 −˜r2 is guaranteed to have the ˜r1 ↔ ˜r2 “3avor” symmetry. This however is not the case for three-body systems: H = T1 (˜ p1 ) + T2 (˜ p2 ) + T3 (˜ p3 ) + V12 (˜r1 − ˜r2 ) + V23 (˜r2 − ˜r3 ) + V31 (˜r3 − ˜r1 ) :

(4.2)

Indeed even for 3avor-independent potentials V12 = V23 = V31 , and simple non-relativistic kinetic terms Ti = p ˜ i2 =2mi , the ground state wave function (0) (˜r1 ;˜r2 ;˜r3 ) is not (3avor) symmetric under interchange of 1 ↔ 2, etc. As various counter examples show [18,39] (particularly the simplest one due to Lieb [18] which we present in Appendix A), for certain, rather “extreme” potentials Vij (r) = V (r), the kinematic asymetry due to the di8erent quark masses (m1 = m3 = M; m2 = m) is strongly enhanced so that Eq. (4.1) fails. This notwithstanding, the elegant work of Lieb [18] has shown that the conjectured equation (4.1) does hold for a wide class of two-body potentials V (˜ri ;˜rj ) for which the operator L1 (˜x; ˜y) = e−1V (˜x;˜y)

(4.3)

is positive semide=nite. The latter is equivalent, for V = V (˜x − ˜y) and L1 = L1 (˜x − ˜y), to a positive semide=nite Fourier transform of L1 . A suQcient condition for this is that V (˜ri ;˜rj )=V (˜ri −˜rj )=V (r) satis=es V (r) ¿ 0

(monotonically increasing) ;

(4.4a)

V (r) 6 0

(convex) ;

(4.4b)

and V (r) ¿ 0 :

(4.4c)

For heavy quarks (where potentials can be derived via the Wilson loop construction), Eqs. (4.4a), (4.4b) and the positivity of L1 can indeed be proven, as we will show in Section 18. All potentials used in quark model phenomenology to date satisfy all of Eqs. (4:4) and (4:3). We present in Appendix B Lieb’s proof of Eq. (4.1) for positive semide=nite exp[ − 1Vij (˜x; ˜y)]. We do this in some detail, even elaborating somewhat beyond the original concise paper, since the baryon–baryon inequalities are indeed born out by data and since we showed recently that L1 is positive. Also, Lieb’s line of argument invoking the full three-particle Green’s function serves as a “bridge” between the Hamiltonian, variational, largely potential model motivated, =rst part of our review; and the more formal Lagrangian correlator inequalities proved via the path integral representation which is the approach of the second part. 5. Relating masses in di%erent avor sectors In the following we investigate the inter3avor mass relations (2.18) in a lattice Hamiltonian formulation of QCD. The conservation of quark 3avors allows us to break the QCD Hilbert space into 3avor sectors. Each 3avor sector (U; D; S; C; : : :) consists of a net number U of up (u) quarks (U is negative for

210


u> excess), D of down (d) quarks, etc. with U + D + · · · = 0 (mod 3). We will be interested in the following “low” sectors: (1) M (0) : The 3avor vacuum U = D = · · · = 0. It consists of states with an arbitrary number of gluons and (ql q>l ) pairs. (2) M(i—)> ; i = j: The meson sector, with an excess of one quark 3avor of type i and a di8erent antiquark 3avor —> together with any number of gluons and (ql q>l ) pairs. (3) B(ijk) : The baryon sector with a net excess of three quarks, qi ; qj ; and qk . (4) M(i—k> l)> (i = j; i = l; k = j; k = l): The exotic meson sector with a net excess of two quark 3avors and two (di8erent) antiquark 3avors. We will also discuss in Section 16 and Appendix F other sectors such as the pentaquark and hybrid sectors. We can now de=ne Hi—> to be the QCD Hamiltonian restricted to the sector Mi—>; Hijk the Hamiltonian restricted to Bijk , etc. The meson spectrum will be given by Hi—>|i—> = mi—>|i—> ;

(5.1)

with a wave functional belonging to the Mi—> sector. Similar equations hold in other sectors. We show in Appendix C that [17] Hi—> + Hk l> = Hil> + Hk —> :

(5.2)

Particle masses are given by the SchrModinger equation, e.g. Hij |ij = (mij + A0 )|ij ;

(5.3)

with A0 an additive, common constant representing the vacuum energy, i.e. the lowest energy obtainable for functionals in M (0) . To ensure that A0 is =nite we restrict ourselves to =nite lattices so (0) that A0 = V j(0) vac , with V the volume and jvac the vacuum energy density. In writing Eq. (5.3) we assumed that the systems have no net translational motion. 2 Additional symmetries (rotations, parity, and for certain states, charge conjugations) could be used to project any desired state of given J PC quantum numbers. To simplify notation, the J PC labels are omitted. The lattice formulation reduces the symmetry from the full rotation to the cubic subgroup. However, the operator relations most likely also hold in the continuum limit where the full rotation symmetry is regained. The |i—> are wave functionals: —> Ai(conf (5.4) |i—> = ) |(conf ) : con=gurations

Even for a =nite lattice the (discrete) summation includes in=nitely many con=gurations characterized by the locations and spinors of all quarks and=or antiquarks and by Eñ2;ñ+nˆ at all links where the latter are constrained by Gauss’ law [Eq. (C.4)] (recall that arbitrarily high SU(3)C representations are a priori allowed). The speci=c 3avor (i—) > dependence enters only into the probability amplitudes —> Ai(conf for =nding a given con=guration in |i—>. The generic con=gurations used in Appendix C ) To ensure ˜P = 0, i.e. translational invariance, we need to sum over all locations of the centroid of the wave functional. For =nite lattices of size L, only P 6 1=L seems achievable, yet for periodic boundary conditions a discrete version of translational invariance persists. 2


211

have been de=ned in such a way so as to make operator relations such as Eq. (5.2) manifestly true. |i—> is normalized: —> 2 i—>|i—> = |Ai(conf (5.5) )| = 1 : con=gurations

A key observation is that the variational principle applies to wave functional solutions of the SchrModinger equation (5.3) much in the same way as it does to wave functions. In particular, the PC ground state |i(0) —> in any speci=c channel (say a mesonic i —> channel with given J ) is given by the requirement that: (0) A0 + mi—> = i(0) —> |Hi—>|i—> = min|Hi—>| ;

(5.6)

with the minimum sought in the space of all (normalized) |i—> functionals with the given quantum numbers: | = Aconf |(conf (mesonic)) : (5.7) conf

We would like next to argue that we can set i = l; j = k in Eq. (5.2) and still obtain a meaningful operator relation Hi—> + Hj)> = Hi)> + Hj—> :

(5.8)

In the light (ud) quark sector |mu − md |3QCD , and since 0EM 0QCD , isospin is a good 3avor symmetry. The I = 1|uu> − dd> + gluons + pairs states do not then mix with the I = 0; M (0) sector, so these states should then be used in deriving the inequalities. For heavier states cc; > bb> there is a > strong “Zweig rule” [32,40,41] suppression of cc; > bb annihilation. Hence, with the possible exception of Mss>, mesonic sectors Mi)> distinct from M (0) can be de=ned for which the i = l; j = k version of Eq. (5.2), namely Eq. (5.8), holds. Unfortunately, we cannot use this relation to obtain mass inequalities without further assumptions. In general, i(0) —> , the ground state wave functional for (for example) a heavy quark qi and a light antiquark q>j , may be di7erent from j(0) )> , even though i—> and j )> are related by charge 3 conjugation. Indeed one of the inequalities, namely 2mK ∗ 6 mD + m@ , is not manifest. Symmetry r = ˜ri − ˜rj , is automatically guaranteed in the simple potential model where i(0) —> depends only on ˜ the relative coordinate of qi q>j . It is also plausible in a large Nc limit where the important degrees of freedom are the gluonic ones. In the following we will assume i ↔ j symmetry. (0) By taking the expectation value of the left hand side of Eq. (5.8) in i(0) —> , we have i—> |Hi—> + (0) Hj)>|i(0) —> = 2mi—> + 2A0 , and the variational argument for deriving the inequalities proceeds exactly as in Section 2. After cancelling a common 2A0 term representing vacuum energy we obtain (0) (0) 2m(0) i—> ¿ mi)> + mj —> :

(5.9)

We also have the relations for the sum of the =rst n excited states in any M ij (J PC ) channel [Eq. (2.18)]. For the baryonic sector we readily =nd a relation Hijr + Hklr = Hilr + Hjkr : 3

This important point (missed in Ref. [17]) will be further elaborated in Section 20.

(5.10)

212


Eq. (5.10) is analogous to relation (5.2) and is obtained by adding a common “spectator” quark r = i; j; k; l. Assuming a 3avor symmetric baryon ground state, we obtain from variants of Eq. (5.10) with some of the 3avor indices set equal to each other, analog convexity relations for baryonic states (0) (0) 1 m(0) iij ¿ 2 [miii + mijj ]

(5.11a)

(0) (0) (0) (0) (0) (0) 1 m(0) ijk ¿ 6 [miij + miik + mjji + mjjk + mkki + mkkj ] :

(5.11b)

It should be emphasized [42] that for the baryonic case we have two relative coordinates, and 3avor symmetry of the wave function is not guaranteed even in the framework of a potential model. Indeed, as discussed in Section 4 and Appendix A, for certain potentials one can show that the inequalities such as Eq. (5.8) are violated—though as elaborated in Appendix B, they do hold [18] for the class of potentials which are of interest in QCD. 6. Baryon–meson inequalities in a non-perturbative approach Inequalities relating masses of baryons and mesons were motivated in Section 3 via a potential model. It was suggested there that such inequalities may persist beyond the (dressed) one gluon exchange approximation. In this section we will employ a non-perturbative framework which still allows the application of variational arguments and the derivation of the baryon–meson mass inequalities [20]. It has been argued that con=nement in QCD is the electric analog of the Meissner e8ect in a superconductor. The non-perturbative QCD vacuum develops a condensate of color monopole pairs and=or of large loops of magnetic 3ux [43– 45]. More recently, lattice and other approaches have made this much more concrete, particularly in the context of “center vortices” [46,47]. In this vacuum, the chromoelectric 3ux emitted from a quark and ending on an antiquark is localized along a thin tube [48]—the analog of the magnetic vortex in the ordinary superconductor. In the Nc → ∞ limit the chromoelectric 3ux tube may become in=nitely thin [49] and the original dual string model [50 –52] for hadrons could emerge. Also in the strong coupling limit of Hamiltonian lattice QCD a single set of minimally excited links of minimum total length connects the quarks and antiquarks in a single hadron. A more general approximation for the meson wave functional is given by a sum over con=gurations of strings [or chains of lattice links, see Fig. 4(a)] connecting q and q> (indicated by the symbol below), with a probability amplitude for each con=guration | M12 = A( )| : (6.1) 1

2

The end points 1; 2 can also vary. The kinetic (B2 and > D ) parts of the QCD Hamiltonian move the string and the fermions at the end points, respectively; while the “potential” E 2 term gives a diagonal contribution proportional to the length (number of links) of the string multiplied by the string tension (weighted by the values of the second Casimir operator for the representation residing on each link).


213

Fig. 4. (a) The color string picture for a meson. (b) The color network for a qqq baryon with one junction.

Fig. 5. Illustration of how a given Y con=guration in a baryonic wave functional yields con=gurations for three mesonic trial wave functionals. The relation 2H123 = H12> + H23> + H31> is illustrated by the fact that the B2 term depicted by the small string distortion, the > D terms depicted by the motion of the end quarks, and the E 2 (corresponding to the weighted string length) all appear twice.

In the same approximation, the baryon’s wave functional is a sum over con=gurations where the three E vortex lines (symbolized for convenience by Y in the following equation, though the vortices need not be straight) emanating from quarks 1, 2, and 3, join together at a common “junction point” x [see Fig. 4(b)]: | B123 = A(Y )|Y : (6.2) Y

Just as in the case of the non-relativistic potential model, we would like to extract trial wave functionals for the ground states of the mesons q1 q>2 , q2 q>3 , and q3 q>1 from the ground state baryon wave functional. This is achieved in the following way (see Fig. 5). We consider quarks 1 and 2, together with the string connecting them in the baryon as a possible con=guration in the wave functional of the meson M12>. To this end we need to color conjugate the quark q>2 and reverse the

214


chromoelectric 3ux in the section x − 2. A detailed analysis using a lattice formulation indicates this can be done for the case of minimal 3ux strings, without e8ecting the matrix elements of the Hamiltonian. Likewise quarks 2 and 3 with the string 2−x −3 are viewed as a possible con=guration for the meson M23> and similarly for M31>. This suggests the operator relation 2Hijk = Hi—> + Hjk> + Hk )> ;

(6.3)

with ijk the 3avors of q1 q2 q3 . Indeed by applying Hi—>; Hjk> and Hk )> to each of the above mesonic con=gurations we see that each part of the baryonic Hamiltonian Hijk is encountered twice. Thus the kinetic term and mass term (D= 1 + m1 ) associated with the motion of q1 (=qi ) occurs in both Hi—> and Hk )>, and so does the kinetic and potential energy associated with the motion, and total length, of the string bit connecting quark qi with the junction point x. A similar argument can be applied to the 2 − x and 3 − x string bits. Let us next take the expectation value of Eq. (6.3) in the ground state wave functional of the baryon at rest. As in Section 3, we obtain on one hand simply 2m(0) ijk and on the other hand the sum of the expectation values of Hi—>; Hjk> and Hk )> in the mesonic wave functionals extracted, in the manner described above, from the baryon’s ground state wave functional. In analogy to =xing the coordinates of the “third quark” (say qk ) in the potential model discussion of Section 3, we =x in each case the string bit (x − qk for Hi—>) which should be ignored in order to form a mesonic (Mi—>) trial wave functional. The expectation values of Hi—> in these trial wave functionals t = A(Y )|Y (6.4) | ijk Y

are then integrated with the weight implied by the original baryonic amplitude A(Y ) over the “ignored” section as well. Note that because of the special character of the baryonic state, the con=gurations in the trial wave functional (6.4), used here for the three mesonic qi q>j ; qj q>k ; qk q>i ground states, are in fact correlated. Speci=cally, the strings (or 3ux lines) associated with all three of these meson states are forced to have a common junction point x (which again is integrated over at the end). This extra constraint only reinforces the general result of the variational principle, namely that

(0) (0) (0) ijk |Hi—>| ijk ¿ mi—>

;

(6.5a)

(0) (0) (0) ijk |Hj k> | ijk ¿ mj k>

;

(6.5b)

(0) (0) (0) ijk |Hk )>| ijk ¿ mk )>

:

(6.5c)

Speci=cally these inequalities state that the mesonic trial wave functionals extracted from the subsystems of the baryon’s ground state wave functional are not optimized so as to minimize the energy (0) is constructed so as to minimize the expectation value of of the mesonic subsystems. Rather ijk Hijk and

(0) (0) ijk |Hijk | ijk

= m(0) ijk :

(6.6)


215

Fig. 6. The triangular inequalities and relevant vectors for the comparison of the potential energies in the strong coupling limit of the baryonic Y con=guration and mesonic subsystems. Fig. 7. Some matrix elements con=guration’ |HQCD | con=guration on a small 3 × 3 dimensional lattice. The upper entry in each box is the matrix element with 1=a the lattice energy scale. The lower entry indicates the terms in the Hamiltonian contributing to this particular matrix element.

Thus we have from Eq. (6.3) the required inequality (0) (0) (0) 2m(0) ijk ¿ mi—> + mj k> + mk )> :

(6.7)

It is amusing to see how this inequality is realized for the strong coupling limit. The 3ux then proceeds in straight lines in both the mesons and the baryons so as to minimize the total string length. This yields the potentials V12 = |˜r1 − ˜r2 |; V23 = |˜r2 − ˜r3 |; V31 = |˜r3 − ˜r1 | for the meson and the genuine three-body interaction [53] V123 = min (|˜r1 − ˜x| + |˜r2 − ˜x| + |˜r3 − ˜x|) x

(with the total length of the Y-shaped string con=guration minimized over the choice of function point ˜x) for the baryon. The inequality reduces in this case simply to 2V123 ¿ V12 + V23 + V31 which is just a sum of three triangular inequalities: |˜r1 − ˜x| + |˜r2 − ˜x| ¿ |˜r1 − ˜r2 | etc. (see Fig. 6). Since the potentials are linear for both systems, the virial theorem implies that the total energy is given by 2V , with V the expectation value of the potential energy, and 2E123 ¿ E12 + E23 + E31 . 4 We have so far considered the approximation in which the quarks are connected by a single set of minimally excited links on the lattice or by a single Y-type con=guration for the baryons. 4

The virial theorem (with massless quarks and linear potentials) was used in [54] to motivate the equipartition of the light cone momentum between quarks and gluons, and more recently in connection with the mass dependence of Bose–Einstein correlations in multi-particle =nal states [55].

216


Fig. 8. Con=gurations in the generalized baryonic wave functional which still allow separation into three mesonic subsystems.

We still have many paths of di8erent lengths. All of these con=gurations can be generated by repeated application of the kinetic (B2 ) term in the Hamiltonian which generates a closed 3ux line around a plaquette and shifts the initial 3ux line as indicated in Fig. 7. The problem of =nding the mesonic or baryonic ground state wave functionals, even in this approximation, is intractable, and at best we could hope for some numerical results. We would like to point out, however, that the baryon–meson inequalities can be proven in an even broader context when qq> pair creation, bifurcation of 3ux lines, and excitation of links to higher SU(3)C representations are all allowed. If we consider the procedure for extracting mesonic trial wave functionals from the baryonic wave functional, we can pinpoint the crucial ingredient for deriving Eq. (6.3). It is that the network of links in any con=guration in the baryonic wave functional includes only one junction point x, where we can separate the network into three “patches” P1 ; P2 ; and P3 connected to the external quarks qi ; qj ; and qk respectively (see Fig. 8). We consider P1 P> 2 , where P> 2 is the patch with all 3ux 3ows reversed and qj → q>j , as a con=guration in the trial meson functional | Mi—> with an amplitude A(P1 ; P2 ; P3 ); (P3 =xed) inferred from the baryonic wave functional. Likewise P2 ; P> 3 and P3 ; P> 1 o8er trial wave functionals for the q2 q>3 (qj q>k ) and q3 q>1 (qk q>i ) mesonic systems. We can easily verify that the operator relation 2H123 = H12> + H23> + H31> still holds for this class of baryonic and corresponding mesonic functionals. The kinetic and potential parts of the full QCD Hamiltonian operate on each patch P1 ; P2 ; P3 separately and those contributions are counted twice in H12> + H23> + H31>. The variational argument is applicable in the larger class of baryonic and mesonic trial wave functionals and it yields the required baryon–meson mass inequalities 2mijk ¿ mi—> + mjk> + mk )>.


217

The fact that the same inequality, Eq. (3.2), motivated by the color factor for the perturbative one-gluon exchange, can be rederived in a strong coupling, string-like limit (and some generalizations thereof), suggests that these inequalities may indeed be a true consequence of the full-3edged QCD theory. Indeed a particular relation involving the nucleon and pion masses has been rigorously proved by Weingarten [6], who utilized the correlation function techniques, and is reproduced in Section 10. 5 Finally, we note that a construction of mesonic trial wave functionals from the baryon’s ground state functional (i1 :::iN ) can be carried out for SU(N ) with general N as well. The con=gurations contributing to (i1 :::iN ) consist of N strings joined at a common junction point. It can be formally separated into N (N − 1)=2 mesonic subsystems with each part of the baryonic Hamiltonian counted (N − 1) times (since a segment or patch Pia connecting the quark q1a and the junction x appear in the (N − 1)qia q>ib ; a = b systems), and we have: Hia ib> ; (N − 1)Hi1 :::iN = ia =ib

leading to the baryon–meson inequality (3.23) for general SU(N ). The case of Nc = 2 is rather special. The lightest diquark baryon is in fact a 0++ mesonic state. The inequality mB ¿ m$ is in this case most likely an equality: +

−

(0 ) m(0 qq = mqq>

)

:

(6.8)

Indeed the gluon couplings inside the meson and “baryon” are the same here, so that to all orders in perturbation theory we expect the S-wave qq (or qq) > states to be degenerate. To obey the generalized Pauli principle, the S-wave spin and color singlet qq state ought to be a “3avor” antisymmetric ud combination which is to be compared with the u 5 d> pion. The pseudoreality of the SU(2) group implies that in the non-perturbative string or 3ux tube picture, the 3ux emitted by one quark in the bosonic diquark can end on the other quark. There is no junction point in this case, the wave functionals and Hamiltonians for the 0− q 5 q> and 0+ qq systems are identical, and Eq. (6.8) follows. 7. Comparison of the inequalities with hadronic masses We proceed next to list the various inequalities and compare them with available particle data [58]. First consider the meson–meson relations. As emphasized above, these inequalities do rely on an additional assumption of a 3avor-symmetric wave function. In all testable cases (with one possible exception) these inequalities are satis=ed. The inequalities could then be used as a rule of thumb to restrict the masses of as yet undiscovered new particles. > ub, > and cb. > 6 Because of There are six relevant mi—>; m0i = m0j 3avor combinations: us; > uc; > sc; > sb; the small violation of I -spin symmetry (via radiative electromagnetic corrections and the e8ect of |mu − md | 4–5 MeV) we have not separately considered members of an I -spin multiplet (obtained 5

Despite many e8orts [56,57] to extend the correlation function technique to more detailed, 3avor-dependent baryon–meson and baryon–baryon inequalities, no fully convincing results have been obtained. 6 Since the decay width of the top quark exceeds a few GeV, it decays before t u, > etc. states can form.

218


by u → d substitutions), but rather averaged over these states. Also in order to minimize the e8ects of the annihilation channels we consistently choose the I = 1 ud> rather than the I = 0 uu> states (e.g. @ and not !; a2 and not f0 , etc.). The comparison with measured masses is summarized in Table 1, which indicates the speci=c particles and masses relevant for each inequality. In some cases all three masses in Eq. (2.18) are J PC fairly well known or reliably estimated. In other cases lower bounds are predicted for certain Mi(0) . —> All masses are listed in MeV. For the pseudoscalars the inequalities have been derived by Witten [10] using the euclidean Green’s function method to be discussed at length in the second part of this review, and indeed hold with a fairly wide margin. The light pseudoscalar mesons $; K; L can also be viewed, to a good approxima(0) tion, as pseudo-Goldstone bosons. Current algebra methods [59] then lead to m2psij ˙ (m(0) qi + mqj ). Hence in this approximation 2m2psij = m2psii + m2psjj and 2mpsij ¿ mpsii + mpsjj is guaranteed. 7 For the I = 0 pseudoscalars the strong coupling to the gluonic M 0 channel (which in particular accounts for the massive√L ) [62,63] suggests strong mixing between the two I = 0 states that are > and the ss> state. Hence if we assume that ss> can be expressed made of light quarks, 1= 2(uu> + dd), , then the reasonable expectation [64] that the eigenstates L; L are the SU(3) as |ss > = 0|L + 1|L √ √ √ √ > > s) 3avor octet (1= 6)(uu+d > d−2s s) > and singlet (1= 3)(uu+d > d+s > implies 0=− 2=3; 1= 1=3 and −

ms(0s>

)

= (0L| + 1L |)H (0|L + 1|L ) = 02 mL + 12 mL : : : −

−

(7.1) −

−

−

and yields ms(0s> ) ≈ 706 MeV. This indeed satis=es ms(0s> ) + mu(0d> ) 6 2ms(0d> ) [i.e. ms(0s> ) + m$ 6 2mK ] − with a large margin. In general Eq. (7.1) implies 2mK ¿ m$ + ms(0s> ) as long as 12 =02 ¡ 1:5, a constraint satis=ed by all mixing schemes suggested for the 0− mesons [65]. The above discussion neglects the fact that the L mass is unusually high precisely because of the anomaly and instanton e8ects, i.e. the coupling to the M (0) sector. This means that the Zweig rule (0) (0) is not applicable and the derivation of m(0) Indeed we cannot choose 0 ss> + mud> 6 2msd> is suspect. √ > and 1 in (7.1) so as to make the coeQcients of both (uu> + dd)= 2 and M (0) (i.e. pure glue and pairs) states vanish simultaneously. We would like to thank H.J. Lipkin for pointing these diQculties out to us. − In the relation 2mu(0b> ) ¿ muu> + mbb> we use the mass of the vectorial state, mN , instead of the as yet unknown mLb . Following Weingarten we show in Section 8 that, when mixings with M 0 are neglected (which seems justi=ed by asymptotic freedom in the case of heavy quarks), the lowest state in any Mi—> sector is indeed pseudoscalar and mN ¿ mLb , so that the original inequality is a fortiori satis=ed. The vector meson mass inequalities hold with smaller margins than the corresponding inequalities for the pseudoscalars. This re3ects the hyper=ne splittings, which, as indicated in the conclusion of Section 2, tend to weaken (enhance) the inequality for the vector (pseudoscalar) mesons, respectively. We =nd it impressive that despite the very large spin splittings, e.g. m@ − m$ 650 MeV 5m$ , the inequality may still marginally hold for the vector mesons. For the speci=c case of K ∗ ; @, and D, the inequality seems to fail, although only within O(1%) of the widths of the states considered. 7

The last argument is reminiscent of that made earlier for harmonic potentials. It was noticed [60] that the “dual” Lovelace–Shapiro–Veneziano formula [61] for $–$ scattering has a remarkable tendency to conform to soft pion theorems. Considering the harmonic string origin of “dual” amplitudes, these may perhaps be related features.


219

Table 1 Meson–meson inequalities us> sector: mus> ¿ 12 (muu> + mss>) Pseudoscalar Vector Tensor Axial Scalar

mK ¿ 12 (m$ + 23 mL + 13 mL ) mK ∗ ¿ 12 (m@ + mD ) mK2∗ ¿ 12 (ma2 + mf2 ) mK1 ¿ 12 (ma1 + mf1 ) mK0∗ ¿ 12 (ma0 + mf0 )

495:009 ¿ 411:08a 893:14 ¿ 894:7 1427:7 ¿ 1296:55 1273 ¿ 1256:0 1429 ¿ 982

uc> sector: muc> ¿ 12 (muu> + mcc>) Pseudoscalar Vector Tensor Axial Scalar

mD ¿ 12 (m$ + mLc ) mD∗ ¿ 12 (m@ + mJ= ) mD2∗ ¿ 12 (ma2 + mc2 ) mD0 ¿ 12 (ma1 + mc1 ) 1 mD0+ ¿ 12 (ma1 + mc0 )

1867:7 ¿ 1558:9 2008:9 ¿ 1933:4 2458:9 ¿ 2437:1 2422:2 ¿ 2370 mD+ ¿ 2200:4

ub> sector: mub> ¿ 12 (muu> + mbb>) Pseudoscalar Vector Tensor

mB ¿ 12 (m$ + mN ) mB∗ ¿ 12 (m@ + mN ) mB∗2 ¿ 12 (ma2 + mb2 )

5279:0 ¿ 4799:2 5324:9 ¿ 5115:2 5739 ¿ 5615:65

0

mB0 ¿ 12 (ma1 + mb1 ) 1 mB0+ ¿ 12 (ma0 + mb0 )

mB0 ¿ 5560:95 1 mB0+ ¿ 5421:6

sc> sector: msc> ¿ 12 (mss> + mcc>) Pseudoscalar Vector Tensor Axial Scalar

mDs ¿ 12 ( 23 mL + 13 mL + mLc ) mDs∗ ¿ 12 (mD + mJ= ) mDs2+ ¿ 12 (mf2 + mc2 ) mDs1+ ¿ 12 (mf1 + mc1 ) mD0+ ¿ 12 (mf0 + mc0 )

1968:5 ¿ 1832:0 2112:4 ¿ 2058:15 2573:5 ¿ 2540 2468 ¿ 2396:2 mD0+ ¿ 1709

sb> sector: msb> ¿ 12 (mss> + mbb>) Pseudoscalar Vector Tensor Axial Scalar

mB0s ¿ 12 ( 23 mL + 13 mL + mN ) mB∗s ¿ 12 (mD + mN ) mB2+ ¿ 12 (mf2 + mb2 ) mB1+ ¿ 12 (mf1 + mb1 ) mB0+ ¿ 12 (mf0 + mb0 )

5369:3 ¿ 5072:2 5416:3 ¿ 5239:89 mB2+ ¿ 5730 mB1+ ¿ 5659 mB0+ ¿ 5421:9

cb> sector: mcb> ¿ 12 (mcc> + mbb>) Pseudoscalar Vector Tensor Axial Scalar

mB± ¿ 12 (mLc + mN ) c mBc1− ¿ 12 (mJ= + mN ) mBc2+ ¿ 12 (mc2 + mb2 ) mBc1+ ¿ 12 (mc1 + mb1 ) mBc0+ ¿ 12 (mc0 + mb0 )

6400 ¿ 6219:6 mBc1− ¿ 6278:6 mBc2+ ¿ 6734:7 mBc1+ ¿ 6701:2 mBc0+ ¿ 6638:6

Axial Scalar

a

See however the discussion following Eq. (7.2) qualifying the applicability of the inequality.

The inequalities for the tensor mesons can be tested in the sectors not involving the b quark. In the b quark sectors we can use the inequalities to predict lower bounds for the masses of the Btensor particles. The Ds2+ (2573) is not known for certain to be a 2++ state, but this is the quark model

220


prediction. The inequalities for the axial 1++ mesons all hold at the level of a few percent. Finally, the inequalities for scalars can be tested (and veri=ed) only for the case of the uc> and ub> 3avor combinations. 8 The deviations from equality >m mij − (1=2)(mii + mjj ) = >= (7.2) m mij are often small [O(1−2%)]. This may be traced to the variational principle. Since |Hii | ; |Hjj | are extremized at = ii(0) ; = jj(0) , respectively, the deviations Ai = ij(0) |Hii | ij(0) − ii(0) |Hii | ii(0) and Aj = ij(0) |Hjj | ij(0) − jj(0) |Hjj | jj(0) are quadratic in the wave function shifts, e.g. >m (> )2 ≈ ( The >

(0) ij

−

(0) 2 ii )

:

(7.3)

in turn are expected, on the basis of =rst order perturbative estimates, to be >

(Hij − Hii ) >H (0) Am Am

(0)

=

>H Am

(0)

;

(7.4)

(1) with Am 1 GeV the typical splitting m(0) ij − mij between the ground and =rst excited states in the speci=c channel considered. >H is solely due to the quark mass di8erences. Taking i = u; j = s estimate (7.5) >H m2s + p2 − m2u + p2 ;

with p2 the average (momentum)2 in the wave functions. If we then use the bare quark masses m0u 0; m0s 150 MeV and p2 (300 MeV)2 then |>H | m2s =2 p2 50 MeV, and Eqs. (7.2) – (7.5) yield > 1%. For heavy–light quark combinations the di8erence in the kinetic parts of the Hamiltonian are larger. However, most of the masses are given in this case by the heavy quark mass itself, so that fractional deviations again remain small. Approximate mass equalities 2mii = mii + mij or 2m2ij = m2ii + m2ij were suggested quite a while back on the basis of the SU(3) (Gell–Mann–Ne’eman) 3avor symmetry [64]. The underlying assumption was that the 3avor symmetry breaking part of the Hamiltonian behaves like H 8 , the neutral isoscalar member of an octet (a feature which is manifestly true for the QCD Hamiltonian), and that mass splittings can be obtained by using D(0) |H 8 |D(0) with D(0) the SU(3) symmetric wave functions. The present discussion suggests that these relations can be transformed into inequalities for the linear mass combinations. When the particles in question are broad resonances, the mass inequalities need not apply. This is not just due to the technical diQculty of precisely de=ning, for example, m@ (di8erent methods based on the $$ mass distributions or the Argand diagrams for phase shifts yield somewhat di8erent − − − ¿ m(0)1 + m(0)1 values for the masses of resonances [58]). Strictly speaking, the inequality 2m(0)1 us> uu> ss> holds only for the lowest 1− states in the Mus>; Muu>; Mss> sectors which are K$; $$; and (neglecting Zweig rule violations, i.e. mixing with the gluonic M 0 sector) K K> P-wave states at threshold. The inequality would then appear to degenerate into the trivial statement on the kinematical threshold: 8

Falk [66] tried using heavy quark symmetries to =x some of the unknown J P . The QCD inequalities nicely complement this program.


221

Table 2 Baryon–meson inequalities J P = 3=2+ mA (uuu) ¿ (3=2)m@ mF (suu) ¿ (1=2)(m@ + 2mK ∗ ) mE (ssu) ¿ (1=2)(mD + 2mK ∗ ) mC− (sss) ¿ (3=2)mD mFc++ (cuu) ¿ (1=2)(m@ + 2mD∗ ) mEc (csu) ¿ (1=2)(mK ∗ + mD∗ + mDs∗ ) mCc0 (css) ¿ (1=2)(mD + 2mD∗ ) mEcc (ccu) ¿ (1=2)(mJ= + 2mD∗ ) mCcc+ (ccs) ¿ (1=2)(mJ= + 2mDs∗ ) ++ (ccc) ¿ (3=2)(mJ= ) mCccc mFb (buu) ¿ (1=2)(m@ + 2mB∗ ) mEb (bsu) ¿ (1=2)(mK ∗ + mB∗ + mB∗s ) mCb (bss) ¿ (1=2)(mD + 2mB∗s ) mEbb (bbu) ¿ (1=2)(mN + 2mB∗ ) mCbb (bbs) ¿ (1=2)(mN + 2mB∗s ) mCbbb (bbb) ¿ (3=2)(mN )

1232:0 ¿ 1155 1384:6 ¿ 1278 1533:4 ¿ 1402:85 1672:45 ¿ 1529:12 2519:4 ¿ 2394 mEc ¿ 2507:2 mCc0 ¿ 2518:6 mEcc ¿ 3557:3 mCcc+ ¿ 3660:8 ++ ¿ 4645:32 mCccc mFb ¿ 5710 mEb ¿ 5793:7 mCb ¿ 5879:0 mEbb ¿ 10055:1 mCbb ¿ 10099:5 mCbbb ¿ 14190:56

J P = 1=2+ mN (uud) ¿ (3=4)(m$ + m@ ) mF (suu) ¿ (1=4)(2m@ + 3mK + mK ∗ ) mE (uss) ¿ (1=4)(2mD + 3mK + mK ∗ ) mFc (cuu) ¿ (1=4)(2m@ + 3mD + mD∗ ) mCc (ssc) ¿ (1=4)(2mD + 3mDs + mDs∗ ) mEc (ccu) ¿ (1=4)(2mJ= + 3mD + mD∗ ) mCcc+ (ccs) ¿ (1=4)(2mJ= + 3mDs + mDs∗ ) mFb (buu) ¿ (1=4)(2m@ + 3mB + mB∗ ) m3 (uds) ¿ (1=4)(2m$ + 3mK + mK ∗ ) m3+c (udc) ¿ (1=4)(2m$ + 3mD + mD∗ ) m30 (udb) ¿ (1=4)(2m$ + 3mB + mB∗ ) b mEc (cus) ¿ (1=4)[(mK + mK ∗ ) + (mDs + mD ) + (mDs∗ + mD∗ )] mEb (bus) ¿ (1=4)[(mK + mK ∗ ) + (mBs + mB ) + (mB∗s + mB∗ )] mbcs (bcs) ¿ (1=4)[(mDs + mDs∗ ) + (mBs + mB ) + (mBs∗ + mB∗ )]

938:919 ¿ 681 1193:15 ¿ 980 1318:1 ¿ 1104:25 2452:9 ¿ 2288 2704 ¿ 2514:1 mEc ¿ 3451:5 mCcc+ ¿ 3552:9 mFb ¿ 5675:5 1115:683 ¿ 663:56 2284:9 ¿ 1973:0 5624 ¿ 5359:5 2469:0 ¿ 2336:4 mEb ¿ 5682:7 mbcs ¿ 6355:9

mK + m$ 6 ( 12 )(2m$ + 2mK ). As we will show in Section 16, the operator relation enables us to go beyond this and deduce relations for the phase shifts. To the extent that the threshold physics is completely dominated by narrow K ∗ ; @; and D resonances the mass inequality 2mK ∗ ¿ m@ + mD can be regained from the phase shift inequalities. In the large Nc limit, resonance widths and Zweig rule violations vanish like 1=Nc and 1=Nc2 , respectively [67] and the distinction between the relation (5.2) and the approximate version (5.8) is lost. Also the 3avor symmetry assumption may be on better footing as the gluonic degrees of freedom dominate. We proceed next to the baryon–meson inequalities, with the results listed in Table 2. The speci=c 3avor-spin combinations appearing there are =xed in the manner explained in some detail

222


Table 3 Baryon–baryon inequalities for the octet (Gell–Mann–Okubo) 3m3 + 13 (mF− + mF+ + mF0 ) ¿ mp + mn + mE− + mE0

4731:6 ¿ 4514:1

for the decuplet mF ¿ (1=2)(mA + mE ) mE ¿ (1=2)(mC + mF )

1384:6 ¿ 1382:7 1533:4 ¿ 1528:5

in Appendix D. At the present the J P of 3+ c and Ec are not established and the choice made here (J P = 1=2+ , rather than 3=2+ ) is essential for the inequalities to be satis=ed. In general the baryon– meson inequalities are satis=ed with a higher margin than the meson–meson or baryon-baryon inequalities. This could be attributed to the weak basis for the inter3avor inequalities whose derivation required, beyond the rigorous operator relations, also 3avor symmetric ground state wave functions (or functionals). Also the mesonic wave functions for the di8erent 3avors may indeed be very similar to each other [see Eqs. (7.2) – (7.5)], whereas the baryonic wave functions are intrinsically di8erent from the mesonic ones. In the simple potential model picture the diquark subsystems are not at rest. Also the Y -con=guration in the baryon’s functionals are di8erent from the con=gurations in the corresponding mesonic functionals. The baryonic ground state wave functions are therefore a poorer approximation for the mesonic wave functions than the wave functions of mesons with di8erent 3avors. Baryon–baryon inequalities are listed in Table 3. From Eqs. (5:11) we have also an inequality version of the equal spacing for baryons in the decuplet (5.11a) and of the Gell–Mann–Okubo (GMO) mass formula for the baryon octet [64]. For the latter we use √ the inequality (5.11b): muds ¿ (1=6)(mF+ + mF− + mp + mn + mE + mE0 ). Identifying uds with (1= 2)(30 + F0 ), this becomes 3m30 + 3mF0 ¿ (mF+ + mF− + mp + mn + mE + mE0 ), which in the limit where I -spin splittings are neglected becomes the GMO relation. 9 While both the linearity in the decuplet and the GMO formula are very accurate it is gratifying that the small deviations are consistent with the inequalities. We should emphasize however that the 3avor symmetry of the three quark baryonic ground state wave functions which underlies these inequalities is strictly an additional assumption. For spin independent quark–quark potentials V (|˜r|), the inequalities hold however for the fairly large class of potentials discussed in Appendix B. We have not listed most of the many mass relations involving radially excited states, Eq. (2.14). While we have several known radially excited states in the heavy quark QQ> = cc; > bb> systems − − (particularly in the 1 channel), there are very few known radial excitations in the Qq> or qq> systems to allow useful comparisons. An exception is the case of radial excitations in the D(cq) > system, for which the continuous experimental e8ort at Fermilab [68] keeps providing a relatively elaborate charmed meson spectrum. We have therefore indicated the relevant charm radial excitations in Table 4.

9

The inequality listed in Table 3 di8ers from the one in the text by

2 (mF+ 3

+ mF− − 2mF0 ) 1 MeV.


223

Table 4 Inequalities involving charm meson radial excitations (1) (0) (1) (0) (1) 1 m(0) D + mD ¿ 2 [mLc + mLc + m$ + m$ ] (0) (1) (0) (1) (0) 1 mD∗ + mD∗ ¿ 2 [mJ= + mJ= + m@ + m(1) @ ]

Pseudoscalar Vector

a m(1) D ¿ 2138:2 4645:9 ¿ 4508:95

a

We can predict a lower bound on the mass of the as yet undiscovered D(1) , since the other masses in the inequality are known.

8. QCD inequalities for correlation functions of quark bilinears The set of all euclidean correlation functions, i.e. ordinary Green’s functions continued to the euclidean domain, contains the complete information on any =eld theory [69]. In particular, two-point euclidean correlation functions are closely related to the spectrum of the theory, and indeed play a key role in most attempts to compute the spectrum of QCD. Following Weingarten [6], we will prove in this section inequalities for mesonic correlation functions. Let Fa (x; y) = 0|Ja (x)Ja† (y)|0 ;

(8.1)

with Ja (x) a general, local, gauge invariant (i.e. color singlet) operator with the index a indicating Lorentz and=or 3avor indices, be such a two-point function. Note that in the euclidean case all x − y intervals are spacelike. The usual time ordering stating that the creation operator Ja† (x) should act prior to the annihilation Ja (x) is redundant. With an eye to the original Minkowski con=guration, we will still assume that x0 − y0 ¿ 0. In particular, we could use rotational (Lorentz) invariance to make x0 − y0 = |x − y| = t by choosing y =(0; ˜0); x =(t; ˜0). Inserting a complete set of physical energy–momentum eigenstates and using the (euclidean) time translation operator e−Ht Ja (0)eHt =Ja (t), we obtain a spectral function representation [70] ∞ Fa (x − y) = d2 a (2 )e−|x−y| (8.2) 0

for the correlation function. The spectral function is given by |0|Ja |n|2 >(pn2 − 2 ) ; a (2 ) =

(8.3)

n

with pn the four momentum of the state n. The asymptotic behavior of Fa (x − y) (as |x − y| → ∞) is controlled by the state of lowest mass contributing in Eq. (8.3): Fa (x − y) ∼ e−0 |x−y| ;

(8.4)

where ∼ means equality up to a residue factor a and powers of |x − y|−1 . Here 0 could correspond 2 to a physical two particle threshold. It could also be an isolated contribution p2 (na ) = [m(0) a ] of a particle with the quantum numbers of the current Ja , i.e. a state for which na |Ja† |0 = 0. In this case, (0)

Fa (x − y) ∼ e−ma

|x − y |

a

:

(8.5)

224


We note that due to color con=nement in QCD the =nite energy physical spectrum consists of color singlet states and therefore it is suQcient to consider only correlation functions of color singlet operators. 10 Eq. (8.5) is used in QCD lattice Monte Carlo calculations of the spectrum [3,4]. An exponential form is =tted to the large (many lattice spacings) distance behavior of the appropriate correlation function. The latter is estimated by numerical sampling via the functional path integral expression for the correlation functions: d[A (x)] d[ (x) d[ > (x)]Ja (x)Ja† (y)e−S(x) † ; (8.6) 0|Ja (x)Ja (y)|0 = d[A (x)] d[ (x)] d[ > (x)]e−S(x) with the action > (D=A + mi ) SQCD = d 4 xLQCD = d 4 x i

i

a a 2 + tr(F )

(8.7)

i

and D=A = (9 + ia Aa ). For the purpose of lattice calculations, and also for the derivation of the QCD inequalities, we should eliminate the Grassman variables (x) and > (x) (corresponding to the anti-commuting =elds) in the functional integral. Since i and > i appear only in the bilinear kinetic and mass term in the action, this integration can be done in closed form [4]. In the partition function Nf in the denominator of Eq. (8.6) this results simply in the additional factor i=1 Det(D=A + mi ), which modi=es the integration weight from d(A) = e−SYM (A) d[A (x)] : : : ; a a 2 ) is the pure Yang–Mills part of the QCD action, to where SYM (A) = tr(F d(A) = e−SYM (A) Det(D=A + mi )d[A (x)] :

(8.8)

The same change occurs in the numerator of Eq. (8.6) if Ja are functions of gluonic (F ) degrees of freedom only. The currents of interest—for the non-glueball part of the QCD spectrum—are, however, bilinears a of quark =elds, of the form JOi—> = > i O ja . We use the scalar, pseudoscalar, vector, and pseudovector currents and bilinear expressions with extra D derivatives: a Jsi—> = > i

ja ;

a Jvi—> = > i

ja ;

a > JTij = > i D=

a Jpsi—> = > i

5 ja ;

a i—> Jpv = >i

5 ja ;

ja ;

(8.9) +

0− ;

1− ;

+

+

1 ; and 2 mesons made up of quark 3avors in order to extract the spectrum of the 0 ; qi and q>j . In all of these cases we need to contract the extra ( > ) at (x) and ( > ) at (y). One chain of consecutive contractions can involve both (x) and > (y). This gives a “connected” 3avor structure with the quark qi propagating from y to x and antiquarks q>j from x to y [see Fig. 9(a)]; or, if i = j, we could also have the qi (x)q>i (x) and qi (y)q>i (y) in two separate contractions, yielding a (3avor) 10

We will not address the suggestion [71] that the inequalities mandate speci=c patterns of color symmetry breaking.


225

Fig. 9. Flavor connected (a) and 3avor disconnected (b) contraction contributing to two-point correlation functions of quark bilinears.

disconnected structure [see Fig. 9(b)]. The expression for the two-point correlation function then becomes j i d(A) tr[OSA (x; y)OSA (y; x)] d(A) (8.10a) for the 3avor connected case, or d(A) tr[OSAi (x; x)] tr[OSAi (y; y)] d(A)

(8.10b)

for the 3avor disconnected case. In Eqs. (8:10) SAi (x; y) is the full euclidean fermionic propagator (for 3avor qi ) in the background =eld A (x); O is the Dirac matrix in the expression for the current (8.9); and the trace refers to both spinor and color indices (which are suppressed). The quark propagator solves the equation (D=A + mi )SAi (x; y) = >(x − y) and formally is given by SAi (x; y) = x|(D=A + mi )−1 |y

(8.11)

with D= = D (A) involving the covariant derivatives D (A) = 9 + ia Aa in the given background =eld con=guration A (x). The euclidean D= is purely antihermitian and the are all hermitian and satisfy { ; } = 2> .

226


It is well known that the singular behavior of tr[ 5 SA (x; y)] when x → y [69,72,73], or, alternatively some subtleties in the fermionic integration [74], may induce an anomaly ∼ F F˜ in the 3avor disconnected parts for the pseudoscalar case. Thus for much of the following discussion we will take i = j and avoid altogether the 3avor disconnected contribution. The euclidean pure Yang–Mills action LE = E 2 + B2 is real so long as we do not have an iPE · B term, and hence exp[ − SYM (A)] is positive. An important element for the derivation of the inequalities is the positivity of the determinantal factor Det(D=A + mi ), and hence also of the complete integration measure d(A) in Eq. (8.8), for any given A (x) con=guration [6,7]. To show the positivity of the determinant factor, let us consider the eigenmodes A (A ) of the hermitian operator iD=A satisfying iD=A

A

= (A)

A

(8.12)

with real eigenvalues (A). The D=A = − 5 D=A

5

5

anticommutation

5 5

:

=−

implies the relation (8.13)

This in turn forces all non-zero eigenvalues of D=A to appear in complex conjugate pairs. Indeed from (8.12) and (8.13) iD=A (

5 )

=

2 =A 5 5 iD

= − 5 (iD=A )

= −(A)

5

;

(8.14)

so that 5 is the eigen function corresponding to the eigenvalues −(A). Thus we have explicitly positive determinants Det(D=A + mi ) = [i(A) + mi ][ − i(A) + mi ] = [2 (A) + m2i ] ¿ 0 (8.15)

and hence d(A) ¿ 0

(8.16)

for all A (x). It is important to notice that the above argument applies only for vectorial, non-chiral models with the quarks being Dirac fermions. If 5 is not a distinct new spinor (as is the case for chiral fermions) then the positivity argument breaks down. The measure positivity allows us to prove inequalities between correlation functions if we can show that the corresponding integrands in Eq. (8.10a) satisfy, for any A (x), the same inequality. Thus, let us assume that by simple algebra one can prove tr[Oa SAi (x; y)Oa SAj (y; x)] ¿ tr[Ob SAi (x; y)Ob SAj (y; x)] :

(8.17)

The same inequality continues to hold after we integrate over the normalized, positive d(A), yielding 0|Ja (x)Ja† (y)|0 ¿ 0|Jb (x)Jb† (y)|0

(8.18)

Ja(b) (x) = > i (x)Oa(b) i (x) :

(8.19)

with


We will next show that a relation like Eq. (8.17) holds with Oa = Eq. (8.11) de=ning the propagator SA (x; y). We then have 5 SA (x; y) 5

5.

227

To this end we use

= x| 5 (D=A + m)−1 5 |y = x|(−D=A + m)−1 |y

= x|[(D=A + m)−1 ]† |y = SA† (y; x) ;

(8.20)

where the last dagger refers to the conjugate matrix in color and spinor space. It is instructive to see how the same result is obtained in various more explicit expressions for the propagator. Thus consider the contribution of any fermionic path on the lattice to the propagator SAi (x; y) in the hopping parameter expansion [75]. It is [4] U (ñ;ñ + )(a ˆ + b ) ; (8.21) ñ;ˆ

where the product extends over any set of links connecting the initial point y to the =nal point x. Multiplying by 5 on the left and right and commuting the 5 factor through, we have ñ;ñ+ˆ U (ñ;ñ + )(a ˆ − b ) = ñ+;˜ n + ;˜ ˆ n)(a − b )]† (the euclidean are hermitian), which is the same as ˆ n [U (˜ the contribution of the reversed path of links (connecting x to y) to [SAi (y; x)]† . Summing over all fermionic paths, we reconstruct SAi (x; y) and SAi (y; x) and Eq. (8.20) is satis=ed. The key observation now is that if SAi (x; y) = SAj (x; y)

(8.22)

then Eq. (8.20) implies that the integrand in the pseudoscalar correlation function is a positive de=nite sum of squares: tr[ 5 SAi (x; y) 5 SAj (x; y)] = tr[(SAi )† (y; x)SAj (y; x)] = |(SAi )aa ; QQ |2 ¿ 0 ; (8.23) a;a Q;Q

with the sum extending over color (a) and spinor (Q) indices. The integrand for any correlation functions of the other, non-pseudoscalar currents in Eq. (8.9) involves again sums of bilinear products of the same matrix elements (SAi )aa ; QQ , but in general with alternating signs. Thus, up to an overall constant, the integrand for the pseudoscalar correlation functions is larger than all the other integrands. This then yields the desired result: 0|Jpsi—>(x)(Jpsi—>)† (y)|0 ¿ 0|JOi—>(x)JOi—>(y)|0 ;

(8.24)

with JO any non-pseudoscalar current. The requirement SAi (x; y) = SAj (x; y), with i = j, is made in order to avoid the 3avor disconnected contribution (8.10b) which could invalidate the derivation. It can be satis=ed in the I -spin limit— with mu − md 3QCD and EM e8ects neglected—by taking i = u; j = d. Since SAu (x; y) and SAd (x; y) then satisfy identical equations, SAu (x; y) = SAd (x; y) as required. The lowest mass particle in this > 5 d channel is then the pion. The inequality (8.18) between the correlation functions implies, by u going to the asymptotic |x − y| → ∞ region and utilizing the asymptotic behavior (8.4), the reverse inequality between the lowest mass particles in the corresponding channels: (0) m(0) a 6 mb :

(8.25)

228


Thus we obtain the result that the pion is the lightest meson in the ud> channels m$ 6 m@ (1− ) ; m$ 6 ma1 (1+ );

(8.26a) ma2 (2+ ) ;

m$ 6 m (0+ ) :

(8.26b) (8.26c)

We note that these level orderings are indeed expected in any qq> potential model [28] where the singlet S-wave state—i.e. the $ meson—is the lowest lying state. Could the various renormalizations which are required in order to render the correlation functions i and the local products > (x)O j (x) =nite invalidate the derivation? The regularization required in order to make the path integral d[A (x)] well-de=ned will not in general interfere with the derivation of the inequalities. Thus we could use a lattice discretization with any spacing a and with any action—the minimal Wilson action or with terms in the adjoint or higher representation. The inequalities will continue to be satis=ed at every step of the procedure of letting a → 0 and V , the volume of the lattice, to in=nity. Thus, to the extent that any sensible regularization exists, the inequalities will be satis=ed. Instead of the local (non-pseudoscalar) current J O (x) = q(x)Oq(x), > we could use a non-local gauge invariant version [6] x O A d x qj (x ) = q>i (x)OUc qj (x ): (8.27) Jc (x; x ) = q>i (x)Oexp i x

It creates an extended state—a quark at x, antiquark at x , and a connecting 3ux string along a curve c. The path-ordered Wilson line factor Uc is a unitary matrix. Likewise, we annihilate via JCO (y; y ) the =nal system with q>i qj at y; y with a connecting C line integral. The corresponding correlation function O † O (8.28) 0|JC (x; x )[JC (y; y )] |0 = d[(A)] tr[OUc SA (x; y)Uc† O† SA (y ; x )] can readily be shown, using a Schwartz inequality, to satisfy |0|JCO (x; x )[JCO (y; y )]† |0|2 6 0|Jp (x)Jp† (y)|00|Jp (x )Jp† (y )|0 ;

(8.29)

from which it follows that the mass of any particle created by JcO (x; x ) still satis=es m ¿ m$ . 11 The derivation of the inequalities applies regardless of the choice of “gauge =xing” which limits the allowed A (x) (or Uñ;ñ+ˆ in the lattice version) which should be summed over, but does not a8ect the positivity of the measure. Also, the derivation is not sensitive to the issue of how we put the fermions on the lattice. We can use Kogut–Susskind fermions [76] [which would correspond to a = 0; b = 1 in Eq. (8.21)], Wilson fermions [77] (a = 1; b = 1), or any intermediate procedure. 12 11

The advantage of the more general construction using a nonlocal JcO is that states of any spin can be created. Since the derivation of the inequalities may formally apply to any dimension and any shape of lattice, it should also hold for the new domain wall fermions [78]. It should be noted, however, that in order to ensure the positivity of the determinant in case of Wilson lattice fermions, we need an even number of degenerate 3avors. 12


229

The inequalities are independent of the issue of quark and color con=nement. In fact the inequalities also apply in non-con=ning vectorial theories like QED and in the short distance |x − y| → 0 limit where perturbative QCD is adequate. They would amount to the statement that the pseudoscalar spin singlet “para-positronium” type state is the lightest (this happens in QED even with i = j = electron since the e8ects of the F F˜ anomaly are weak for me = 0). Likewise, the euclidean pseudoscalar correlation function dominates, as x → 0 (or q → ∞ in momentum space), all the other correlation functions in the zero and one gluon approximation respectively. These inequalities =rst derived by Weingarten are the simplest of the exact QCD inequalities and yet, as will be made clear in the next section, are extremely useful. The positivity of the contribution of each A (x) (or Uñ;ñ+ˆ ) con=guration to the path integral de=ning the pseudoscalar propagator 0|Jps (x)Jps† (y)|0 is a novel feature speci=c to QCD-type theories. It is quite distinct from the positivity (by unitarity) of the contribution of each physical intermediate state |0|Ja |n|2 >(pn2 − 2 ) to the spectral function which holds for an arbitrary current Ja in any theory, vectorial or otherwise. We cannot in general appeal to both types of positivity at the same time, since, in order to have the correct physical states, we have to sum over all the background A (x) con=gurations =rst. Otherwise, we do not respect even the translational invariance used in order to derive Eq. (8.3). Anishetty and Wyler [80] and Hsu [81] noticed that the measure positivity and ensuing inequalities hold also for SU(2) chiral gauge theories with an even number of 3avors (required in order to avoid global anomalies [82]). Indeed in this case, we can de=ne i

=

i

+(

i+Nf =2 C

) ;

i = 1; : : : ; Nf =2

to be Nf =2 vectorial, interacting Dirac =elds. 9. QCD inequalities and the non-breaking of global vectorial symmetries The symmetry structure of a =eld theory, e.g. whether it spontaneously breaks vectorial and=or axial global symmetries present in the original Lagrangian, is closely tied in with the zero mass sector. Thus spontaneous breaking of global symmetries implies, via the Goldstone theorem [83], the existence of massless, scalar Goldstone bosons. Likewise spontaneous breaking of axial global symmetries implies Nambu–Goldstone massless pseudoscalars [84]. Finally, an unbroken axial symmetry may require for its realization massless fermions in the physical spectrum (or a parity doubled spectrum). Evidently mass relations such as Eqs. (1:9) between baryons (fermions) and scalar and=or pseudoscalar masses could restrict some of these possibilities and dictate the patterns of global symmetry realization in QCD and other vectorial theories. As a =rst illustration we will use the inequality (8:26c)—due to Weingarten—to motivate the Vafa–Witten theorem. The theorem states that in QCD (and in vectorial theories in general) global vectorial symmetries do not break down spontaneously. This will then be supplemented by some of the more rigorous discussion in the original Vafa–Witten paper. To be speci=c let us =rst consider a non-abelian global vectorial symmetry such as the SU(2) isospin symmetry for QCD. This symmetry is generated by + I = d 3 x u† (x) d (x) ;

230


I− =

1 I = 2 3

d3 x

†

d (x) u (x)

d 3 x[

;

†

u (x) u (x)

−

†

d (x) d (x)]

:

(9.1)

(0) In the limit of m(0) u =md and no electromagnetic interactions it is an exact symmetry of HQCD . If the symmetry breaks down spontaneously (either completely or to a U(1) subgroup generated by I3 ) we expect a massless Goldstone scalar 0+ . The essence of the argument against spontaneous breaking ud> (0) is that such massless scalars can be ruled out if m(0) u = md = 0. Alternatively an exponential fallo8 (0) as |x − y| → ∞ of all correlation functions can be proven for m(0) u = md ¿ 0. This con3icts with + the power fallo8 expected due to the intermediate zero mass 0ud> state contributing via Eq. (8.5) to the spectral representation of the scalar correlation function. Let us present =rst the more heuristic argument which is analogous to the =rst argument of Vafa + − ) ¿ mu(0d> ) states that the lowest scalar state in the ud> channel and Witten. The inequality (8.26c) m(0 ud> must have a mass equal to or larger than the mass of the lowest ud> pseudoscalar. If the I + symmetry + − ) = 0 and hence mu(0d> ) must vanish as well. If the axial global is spontaneously broken, then m(0 ud> symmetry is not spontaneously broken, such a vanishing could only be accidental and hence most − implausible. We next make a more precise and speci=c argument why mu(0d> ) should not vanish. (0) In the limit m(0) u = md = 0 the QCD Lagrangian possesses an extra global SU(2) symmetry generated by the axial analogs of Eq. (9.1): + (9.2) I5 = d 3 x u† (x) 5 d (x); etc:

The spontaneous breaking of I5+ could then yield a massless pseudoscalar ud> Nambu–Goldstone + − ) boson and m(0 ¿ mu(0d> ) would then be trivially satis=ed as 0 ¿ 0. However, following Vafa and ud> −

(0) (0 Witten we keep m(0) u =md = 0. No exact chiral symmetry then exists and mud> + ) m(0 ud>

)

should not vanish. The

− ¿ mu(0d> )

Weingarten inequality implies ¿ 0, which as we argue next, negates the spontaneous breaking of I + symmetry. (0) As we approach the m(0) u = md = 0 limit, the lowest lying pseudoscalar particle becomes a pseudo-Goldstone particle whose (mass)2 is given by [59,85] m2 f2 = (m(0) + m(0) ) > (9.3) $

$

u

d

(0) and vanishes linearly with the explicit chiral symmetry breaking (m(0) u + md ) in the Lagrangian. Similar manipulations utilizing the analog of the soft pion theorem and current algebra yielding Eq. (9.3) suggest an analogous relation for the mass of the tentative 0+ Goldstone boson : ud> (0) 2 2 (0) 2 2 m f = (m − m )( > u − > d ) 6 m f : (9.4)

m2$

d

u

u +

d

$

$

If f = 0, i.e. the Goldstone 0 boson does not decouple, then Eqs. (9.3) and (9.4) are inconsistent + − ) (0) (0) (0) with the QCD inequality m(0 ¿ mu(0d> ) so long as m(0) u − md mu + md , which we can maintain ud> (0) even as we approach m(0) u = md = 0. The inequality (8.26c) was derived in Section 8 by using SAu (x; y) = SAd (x; y). If, however, I -spin (i.e. u ↔ d) symmetry breaks spontaneously, this last equality could be violated as well! Is our


231

argument then circular, and we proved I -spin symmetry only after assuming it at an intermediate stage? 13 We do not think that this is the case. The rationale behind our argument was alluded to in Section 8 above. Spontaneous symmetry breaking is a collective long-range phenomenon manifest in the V → ∞ limit. The inequalities are proved =rst for =nite lattices. The proof is valid for any volume V (and also for any UV momentum cuto8), no matter how large. We thus expect the inequalities to hold in the V → ∞ limit. Vafa and Witten present also a more sophisticated and compelling argument. They show that the propagator SAA (x; y) of a massive quark between two regions of size A and around x and y is bound for any A (x) by SAA (x; y; m0 ) 6

e−m0 |x−y| e2m0 A m0 A4

(0) (m(0) u = md = m0 ) :

(9.5)

A A The square of this bound applies to tr[OSA (x; y; m0 )OSA (y; x; m0 )]. Using the weighted normalized averaging with d(A); d(A) ¿ 0 in Eq. (8.10a), the same bound is also derived for the “smeared” correlation functions for any currents bilinear in the quark =elds: 2

0|JAa (x)[JAa (y)]† |0 6

e−2m0 |x−y| e4m0 A : m20 A8

(9.6)

The =nite smearing around x and y does not a8ect the asymptotic |x − y|A behavior, which is still given by Eq. (8.5). Eq. (9.6) then implies that the lowest state in any mesonic channel qi q>j has a mass satisfying m(0) ij ¿ 2m0

(0) (0) (0) (0) (or m(0) ij ¿ mi + mj for mi = mj ) :

(9.7)

Thus for non-vanishing bare quark masses there are no massless bosons (and by a simple extension (0) also no massless fermions) in the physical spectrum. In particular for m(0) i = mj = 0 we could have no scalar Goldstone bosons. We now review the argument in detail. It is easy to show, =rst, that the propagator for a colored scalar in a background =eld A is always maximized by the free propagator x|K0−1 |y; K0 =−9 9 + m20 [87]. The latter can be cast in the form of a path integral expression [88] with a positive de=nite integrand 1 ∞ dT x|e−(1=2)TK0 |y D0 (x; y; m0 ) = x|K0−1 |y = 2 0 x(T )=y T dX 2 1 ∞ 1 = dT d x (t) exp − dt + m2 T : (9.8) 2 0 2 0 dt x(0)=x The e8ect of any external gauge =eld is to introduce just the additional path-ordered “phase factor” for each path connecting x(0) = x and x(T ) = y: x(T ) a : (9.9) P exp i d x (t)A a x(0)

13

A similar argument was made recently [86] in a more forceful manner in connection with the second Vafa–Witten theorem, regarding the non-breaking of the discrete parity symmetry in QCD.

232


This extra unitary matrix, when integrated over the positive measure in Eq. (9.8), decreases the functional integral d(A). Consequently DA (x; y; m0 ) 6 D0 (x; y; m0 ) (and in particular DA (x; y; m0 ) 6 exp[ − m0 |x − y|]). The motion in Eq. (9.8) is formally that of a particle in four space dimensions and a proper time. Vafa and Witten utilize ananalogous expression for the fermionic propagator ∞ A −1 SA = 0|(D=A + m0 ) |1 = dQ e−mQ 0|e−iQ(−iD) |1 ; (9.10) 0

where D=A is interpreted as the Dirac Hamiltonian in a (4 + 1) formulation, and 0 and 1 are smeared states of spatial extent A around x and y, respectively. The norms 0|0 = 1|1 = A−4 (norm f = max f is required for the purpose of the subsequent discussion) are =nite, and unitarity of exp[iQ(iD)] implies 0|exp[iQ(iDA )]|1 6 A−4 :

(9.11)

The minimal distance between the supports of 0 and 1 is R = |x − y| − 2A. Since the motion of the Dirac particle in 4 + 1 Minkowski space is causal, the minimal “time” required for propagation between 0 and 1 is Qmin = R. Using Eq. (9.11) and the lower limit Qmin = R in Eq. (9.10) we obtain the desired bound (9.6) for the smeared propagator ∞ e−m0 |x−y| e2m0 A SAA 6 dQ e−m0 Q A−4 = : (9.12) m0 A4 R The smearing of the states 0 and 1 can be made gauge invariant by a re=ned procedure (analogous to that used by Schwinger [69]) without a8ecting the basic argument. For the abelian case the current, e.g. JB0 = i > i i carries no net baryon number. However, Vafa and Witten argue that if spontaneous baryon number violation is to occur we will have a vacuum expectation value of some baryon number carrying operator such as T (x) = Bijk (x) 3 (x). This, in turn, would imply that TA (x)TA (y) cannot have an exponential fallo8. Such an exponential fallo8 is implied by Eq. (9.5) and T (x)T (y) d(A)[SA (x; y)]6 . The need for smearing the fermionic propagator arises from the existence of zero modes of the Dirac operator DA [89] in any external B =eld Bz (x; y), with Bz (x; y) d x dy ¿ D0 (h=e). This lowest Landau level corresponds to the classical spiraling motion of particles along B with a Larmor radius r B−1=2 . If B is aligned with (x − y) the particles originating in x do not diverge geometrically with a resultant |x − y|−2 factor in the four-dimensional propagator. Rather, particles can be funneled from x to a spot of size r 2 B−1 at y (see Fig. 10) yielding an enhanced propagator SA Be−m0 |x−y| (instead of SA e−m0 |x−y| =|x − y|2 ). The explicit B dependence prevents the derivation of the e−m0 |x−y| bound directly for SA (x; y) band requires the smearing over a region of geometric size A2 . Once BA−2 , further focusing of the motion to regions smaller than A2 will not enhance the 0 → 1 propagation and the uniform (B independent) bound Eq. (9.6) is obtained. The same type of spiraling motion also occurs for a charged scalar particle in a magnetic =eld. However, in this case the motion has no corresponding zero modes. The contribution of the spiraling trajectories to the path integral expression for the bosonic propagator DA (x; y) will decay as ∼ e−AET at large proper times and is therefore unimportant. Hence, DA (x; y) can be bound without utilizing the smearing procedure.


233

Fig. 10. The focusing e8ect of a parallel B =eld on charged particles propagating from x to y and its limited e8ect on propagation between regions of size A once B ¿ 1=A2 .

At =rst sight Eq. (9.7) seems somewhat surprising. In QED we have positronium states with nonzero binding and thus we might expect Eq. (9.7) to be violated in a weak coupling limit. However, recall that m0 refers to the bare mass. If we have an electron and positron (or quark and antiquark), then besides the attractive interaction between e+ and e− there are the self-energies of e+ and e− Due to the vector nature of the gauge interaction this self-interaction of a smeared (or otherwise regularized) charge distribution is repulsive. Since 1 @1 (˜r)K(˜r − ˜r )@1 (˜r ) + @2 (˜r)K(˜r − ˜r )@2 (˜r ) d˜r d˜r @1 (˜r)K(˜r − ˜r )@2 (˜r ) 6 2 (9.13) for any positive kernel K, the positive contribution due to self-interactions exceeds the coulombic binding and Eq. (9.7) does hold. Scalar self-interactions can be attractive and hence this reasoning is speci=c to gauge theories [7]. Indeed for scalar interactions the fermions have an e8ective mass m0 → gD + m0 in a D background =eld. By choosing D −m0 =g, we can generate propagators DD which do not have the characteristic e−m0 |x−y| behavior and the above proof of the Vafa-Witten theorem fails. For a 5 Yukawa coupling, m0 → m0 + ig 5 , and we even lose the positivity of the determinantal factor and of the measure. 10. Baryon–meson mass inequalities from correlation functions We next extend the discussion of Section 8 to baryonic correlation functions of currents trilinear in quark =elds: † B Fijk = 0|Bijk (x)Bijk (y)|0

Bijk =

j i k a (x) b (x) c (x)O01

(10.1a) ;

(10.1b)

234


with O01 indicating some matrix in spinor space and the j01 color factor. Eq. (8.5) implies that B the asymptotic behavior of Fijk (x; y) is prescribed by the lowest-lying baryon in this channel (say the nucleon for i = j = u; k = d, coupling to overall spin 12 ): B Fuud (x; y) ≈ e−mN |x−y| ;

|x − y| → ∞ :

(10.2) B

Also in analogy with Eq. (8.10a) we have a path integral representation of F (x; y) B Fijk (x; y) = d(A)SAi (x; y)aa SAj (x; y)bb SAk (x; y)cc Oabc Oa b c :

(10.3)

The strategy for deriving the inequalities is to compare this trilinear expression with the positive de=nite quadratic expression for the correlation function of the pseudoscalar in the mi = mj limit $ Fij (x; y) = d(A) |SAi (x; y)aa |2 : (10.4) aa

If an inequality of the form B (x; y) 6 [Fij$ (x; y)]1=p Fijk

(10.5)

can be derived, with (1=p) ¿ 12 , then Eqs. (10.2) and (8.5) imply the inequality mN ¿ (1=p)m$ :

(10.6)

1 2

[If (1=p) ¡ the last relation still allows for m$ ¿ 2mN , in which case the two nucleon threshold − is the lowest physical state in the Ju(0d> ) channel and the inequality derived from Eq. (10.5) becomes the trivial mN ¿ (2=p)mN .] The baryonic correlation function in Eq. (10.3) is readily bound by 3=2 B i 2 |SA (x; y)aa | : (10.7) Fijk (x; y) 6 d(A) aa

In order to bound the right hand side by [Fij$ (x; y)]p , Weingarten [6] appeals to the HMolder inequality [90] 1=p [(p−1)=p] p p=(p−1) dfg 6 d|g| ; (10.8) d|f| valid for p ¿ 1, and chooses (n−3)=(n−2)

i 2 |SA (x; y)aa | f=

g=

aa

3 −(n−3)=(n−2) |SAi (x; y)aa |2

2

(10.9)

aa

so that dfg is the right hand side in Eq. (10.7). For p = (n − 2)=(n − 3) the HMolder inequality (10.8) implies (n−3)=(n−2)  n=2 1=(n−2)   B Fijk d(A) 6 d(A) |SAi (x; y)aa |2 |SAi (x; y)aa |2 : (10.10)   aa

aa


235

Fig. 11. (a) The unique contraction when all 3avor indices are distinct. (b) An alternate contraction when we have permutable quarks in the two currents.

Note that the =rst term in curly brackets is simply Fij$ (x; y) which is then raised to a 1=p= (n − 3)=(n − 2) power. Let the number of light degenerate 3avors be larger than n which we take to be even. The second term in the square brackets in Eq. (10.10), in which the integrand is raised to the n=2 power, then has a direct physical interpretation. It is the two-point correlation function of products of n=2 pseudoscalar currents with all 3avors il ; jm di8erent: K(x; y) = 0|[Jips (x)Jips (x) · · · Jips (x)][Jips (y)Jips (y) · · · Jips (y)]† |0 : 1 j> 2 j> 1 j> 2 j> n=2 j> n=2 j> 1

2

n=2

1

2

n=2

(10.11)

There is no possibility of contracting a quark and an antiquark emanating from x (or from y), and we also have a unique pattern of contracting quark lines from x with those from y [see Fig. 11(a)] so as to form n=2 loops with separate color and spinor traces. i 2 and altogether we have Each of these traces then yields a factor aa |SA (x; y)aa | i 2 n=2 ( aa |SA (x; y)aa | ) as required. If we now make the rather weak assumption that the (lattice regularized) correlation function K(x; y) is =nite, we deduce from Eq. (10.10), that, up to a numerical constant, Eq. (10.5) is indeed valid. Finally, we conclude from Eq. (10.6) that mN ¿

n−3 m$ ; n2

(10.12)

with n the number of light degenerate quark 3avors, which should be even. Thus we can obtain mN ¿ (3=4)m$ if n=Nf =6. This large number of light degenerate 3avors required is associated with the particular derivation. Thus, if we assume only two light (u and d) 3avors we can still construct Eq. (10.11) by taking all i1 : : : in=2 = u and j>1 : : : j>n=2 = d and avoiding any 3avor “disconnected” xx or yy contractions. We have, however, several contractions between x and y, yielding not only {tr[SA† (x; y)SA (x; y)]}n=2 but also terms like tr(SA† SA SA† SA ) [see Fig. 11].

236


We could improve the bound (10.12) if, instead of K(x; y) 6 constant, we appeal to the asymptotic behavior (0)

K(x; y) e−mn=2 |x−y| ; m(0) n=2

with the lowest mass state created by the action of from Eq. (10.10) would then read n−3 1 mN ¿ m$ + m(0) n=2 : n−2 n−2

(10.13) (Jupsd> )n=2

on the vacuum. The bound derived (10.14)

+ quantum numbers then the If there were no bound states in the exotic channel with $1+ : : : $n=2

lowest state is at threshold and m(0) n=2 = (n=2)m$ . In this case Eq. (10.14) becomes mN ¿ (3=2) m$ ;

(10.15)

the result suggested by the more heuristic discussion in Sections 3 and 5. To our knowledge the nonexistence of exotic $+ $+ -type bound states has not been proved. This can be done in the large Nc limit (as will be discussed in Section 14), in which case Nc mN ¿ m $ : (10.16) 2 Weingarten [6] makes the simple observation that if SA (x; y) can be uniformly bound by an A (x) independent constant, SA (x; y) 6 constant ;

(10.17)

then from Eqs. (10.3) and (10.4) one can directly show that F B (x; y) 6 F $ (x; y) ;

(10.18)

implying mN ¿ m$ :

(10.19)

We can utilize for Eq. (10.17) the Vafa–Witten bound on SA (x; y) [Eq. (9.5)]. The requisite smearing of the points x and y into two regions of size A around x and y, respectively, was shown not to e8ect the asymptotic |x − y| → ∞ behavior and the ensuing mass relations. Weingarten [6] has independently motivated the bound (10.17) by considerations of lattice QCD: (D=)lattice + m0 = m0 + R + iI (with R; I being Hermitian, and R having a non-negative spectrum [91]). Thus for m0 ¿ 0; D= + m0 has no non-vanishing eigenvalues which implies that SA (x; y) = x|(D=A + m0 )−1 |y is regular and bounded for all A con=gurations. For m0 = 0; D=A does in general have zero modes which could be important for the issue of spontaneous breaking of chiral symmetry. 11. Mass inequalities and SSB in QCD and vectorial theories The question of whether global axial symmetries are spontaneously broken in QCD (or in other (0) =eld theories) is of utmost importance. In the m(0) u = md = 0 limit, QCD has the global axial SU(2) − + symmetry generated by I5 ; I5 , and I53 . If we can show that this symmetry is spontaneously broken


237

(i.e. SSB occurs), then QCD is guaranteed to reproduce Goldstone pions and the successful rich phenomenology of soft pion theorems and current algebra [59] developed in the late 1960s. Conventionally, SSB is achieved via formation of a qq> condensate in the QCD vacuum, > = 0. > is the order parameter for a phase transition from an axial SU(2) symmetric phase at high temperature to the broken phase at low temperature [92]. > (at zero temperature) serves, along with F 2 , as one of the non-perturbative inputs in the QCD sum rules [93,94]. Many attempts have been made to prove > = 0 in lattice QCD, to estimate its magnitude, and its =nite temperature behavior. Evidently > = d(A) > A ; (11.1) and > A was shown by Banks and Casher [95] to equal $@(0), with @() the spectral (eigenvalue) density of the Dirac operator iD=A A = A . These authors also argued that the same set of “3uxon” con=gurations of A (x) are responsible for both con=nement and the requisite dense set of zeroes of D=A . This supports earlier, more heuristic arguments [96] suggesting that con=nement, or even binding, of a massless fermion in a vectorial theory necessarily leads to SSB. The issue of SSB is also crucial for the case of composite models for quarks and leptons [97–99]. In such theories one assumes that the axial symmetry is not broken. The existence of practically massless fermions (me ; mu ; md a few MeV and even mb = 5 GeV are very small when compared with the compositeness scale 3p ¿ TeV [99,100]) in the physical spectrum is then believed to be a manifestation of these unbroken axial symmetries. However, the baryon–meson mass inequalities such as mN ¿ m$ provide an additional strong argument for SSB in QCD and similar vectorial theories. (0) Let us assume that the axial isospin symmetry (which holds in the limit m(0) u = md = 0) is not broken spontaneously, i.e. I50 |0 = 0. This symmetry can then be realized linearly in the massive spectrum by having degenerate parity doublets. However, one can show via the ’t Hooft anomaly matching condition [12] that we must also have at least one I = 12 massless physical spin- 12 fermion (the nucleon). Anomaly constraints have been discussed extensively [12,13,101]. Following Ref. [101], we consider the vertex O (q1 ; q2 ; q3 = −q1 − q2 ) = d x1 d x2 ei(q1 x1 +q2 x2 ) 0|T [J (x1 )J (x2 )J (x3 )]|0 (11.2) of three identical 3avor currents, which, in terms of chiral right and left combinations are a J = > i [A+(ij) (1 + 5 ) + A−(ij) (1 − 5 )] aj ; with A± matrices in 3avor space. We take tr A+ = tr A− = 0 to avoid the SU(Nc ) gluons 9 J = 0c (tr A+ − tr A− ) tr F F> = 0 :

5

(11.3)

anomaly involving (11.4)

It was shown [72,73,102] that O satis=es the anomalous Ward identity Nc (11.5) q3 O = 2 tr(A3+ − A3− ) : $ The right hand side is given by the lowest triangular graph involving the massless fundamental fermions of the theory.

238


Equation (11.5) is true to all orders [101,103] and most likely is a genuine non-perturbative result. Equation (11.5) implies that some invariant amplitudes in the decomposition of O have a pole at (q3 )2 = 0 with a prescribed residue, given by “the anomaly”, i.e. the right hand side of Eq. (11.5). Such a singularity can arise either by (a) having zero mass (pseudo-) scalar boson states with appropriate non vanishing n0 |J+ |0 matrix elements; or by (b) having a multiplet of massless spin 12 physical fermions (“nucleons”) such that in this basis i J = > N [B+(ij) (1 + 5 ) + B−(ij) (1 − 5 )] Nj ; (11.6) with a “matched anomaly” 3 3 − B− ) = tr(A3+ − A3− ) : tr(B+

(11.7)

Possibility (a) implies Q|0 = 0 with Q the charge d 3 xJ0 (x), i.e. a spontaneous breaking, which we assumed does not occur. This leaves us then with possibility (b), i.e. the need to have massless fermions satisfying Eq. (11.7). The last algebraic anomaly condition constrains the various composite models of quarks and leptons where the latter are the massless physical states Ni [12,99]. Let us now appeal to a new element, namely to the nucleon–pion (or fermion–boson) mass inequalities mN ¿ m$

(mF ¿ mB ) :

(11.8)

Possibility (b) of massless fermions can then be ruled out. Eq. (11.8) implies that if the nucleons are massless so are the pions. However, unless the matrix element 0|J5 |$ f$ q accidentally vanishes and the pion “decouples”, we then have spontaneous breaking of the axial symmetry which is precisely the possibility that we were trying to exclude. It has been shown [104] that in the Nc → ∞ limit, that anomaly matching via possibility (b) is ruled out. Amusingly in this limit we have, as indicated in Section 14 below, the inequality (10.16): (0) MN ¿ (Nc =2)m$ . It excludes, for in=nite Nc (and some very small non-vanishing m(0) u and md and consequently =nite m$ ) even =nite baryon masses, so that alternative (b) cannot a fortiori be realized. It is interesting to see how the inequality (11.8) is satis=ed if we introduce a small explicit (0) breaking of the axial symmetry via m(0) u = md = 0. In this case we have no strict arguments for the vanishing of either mN and=or m$ .√ From Eq. (9.3), we have m$ ∼ m0 . If the nucleon mass is generated exclusively via the explicit chiral symmetry breaking term m0 > q q , then this term and mN > N N both have to 3ip sign under the discrete exp[i$Q5 ] transformation. This, along with the mass dependence √ in the triangle anomaly diagram for quarks and for nucleons, suggests that mN am0 ¡ m$ ∼ mo for small m0 , violating the inequality mN ¿ m$ . This in turn rules out the possibility of explicit chiral symmetry breaking only and SSB is again indicated. An alternate approach to proving SSB [105] would be to start from the Weingarten inequality (8.24) for the pseudoscalar and scalar two-point correlation functions: J ij = > 5 j ; J ij = > j ; i = u; j = d ; (11.9) ps

i

s

i

and show that the inequality is strict 0|Jpsi—>(x)Jpsi—>(y)|0 − 0|Jsi—>(x)Jsi—>(y)|0 ¿ 0 :

(11.10)


239

If chiral symmetry remains unbroken (Q5 |0 = 0), we can show, by utilizing e−i$Q5 Jsij ei$Q5 = Jpsij that 0|Jpsi—>(x)Jpsi—>(y)|0 = 0|Jsi—>(x)Jsi—>(y)|0 :

(11.11)

Thus, proving Eq. (11.10) would amount to proving SSB. The fermionic propagator in a background =eld can in general be decomposed in terms of the Dirac O matrices: SF (x; y; A) = s(x; y; A)I + +

5 a

v

(x; y; A) + t (x; y; A)

(x; y; A) +

5 p(x; y; A)

:

(11.12)

Substituting this last expression into Eq. (8.10a) with O = 5 and I , and utilizing Eq. (8.20), we =nd that v and a contribute equally to (Jps Jps ) and (Js Js ). However, s; t ; and p contribute |s|2 + |t |2 + |p|2 to the =rst correlation and −|s|2 − |t |2 − |p|2 to the second. Thus, if we can pinpoint any set of gauge =eld con=gurations for which (11.13) d(A)(|s|2 + |t |2 + |p|2 ) ¿ 0 then those con=gurations could generate SSB by making the strict inequality hold. Note that the positivity of expression (11.13) and of d(A) insures that the e8ect of such con=gurations cannot be cancelled by some other con=gurations. The relation for the topologically invariant 1 4 4 d 4 xF (x)F˜ (x) (11.14) d xp(x; x; A) = d x tr[S(x; x; A) 5 ] = m0 suggests that if we have an instanton–anti-instanton [106,107] gas, the contribution of the region near any pseudoparticle is d(A)|p(x; x; A)|2 ¿ 0 : (11.15) d 4 xp(x; x; A) ±1 and Indeed, it has been speculated by several authors [107,108] that instantons may be the source of SSB. The detailed quark propagator structure in Eq. (11.12) was utilized in [109,110]. One may try showing directly the existence of a zero mass state in the QCD spectrum in the (0) m(0) u = md = 0 limit. Thus, if the representation for the pseudoscalar propagator ud> ud> Jps (x)Jps (y) = d(A) tr[SA† (x; y)SA (x; y)] as a sum of positive de=nite contributions could be used to show a power fallo8 of the correlation function, then the existence of massless hadronic states with 0− quantum numbers (massless pions or massless nucleons) would follow. The above program attempts to achieve the opposite goal as compared with the original work of Vafa and Witten [7] described in Section 9, where upper bounds on the euclidean correlation function were found. It has not been realized in the form described above. Vafa and Witten have,

240


however, succeeded by combining the measure positivity and various index theorems to prove a closely related result [9] which we will brie3y describe next. They considered the k-point function S(x1 ; : : : ; xk ) = 0| > 1

>

2 (x1 ) 2 3 (x2 ) : : :

>

k 1 (xk )|0

:

(11.16)

With all fermionic 3avors distinct, there is a unique, cyclic pattern of contractions. The k-point function then has the path integral representation S(x1 ; : : : ; xk ) = d(A) tr[SA (x1 ; x2 ) : : : SA (xk ; x1 )] : (11.17) Integrating over all xi and using translational invariance de=nes 4 S(k) = d x1 : : : d 4 xk −1 S(x1 ; x2 ; : : : ; xk −1 ; xk = 0) 1 = V

4

d x1 : : :

d 4 xk S(x1 ; x2 ) : : : S(xk −1 ; xk ) :

(11.18)

The idea is to prove that (after appropriate regularization) S(k) ¿ Ck S0 (k) ; with

S0 (k) = 4d(R)

(11.19) d4 p (2$)4

1 p2

(k=2)

(11.20)

[here d(R) is the dimension of the (color) representation of the fermions], the corresponding expression for the free massless fermions. This implies that for k ¿ 4; S(k) has, like S0 (k), infrared divergences which in terms of the physical states can be understood only if we had zero mass hadronic states. S(k) can be written, using Eqs. (11.17) and (11.18), as S(k) = d(A)SA (k) ; (11.21a) 1 k 1 1 4 4 d x1 : : : d xk tr[SA (x1 ; x2 ) : : : SA (xk ; x1 )] = tr ; (11.21b) SA (k) = V V DA where the last trace and operator multiplication refer not only to color and spinor space but x space as well. To prove Eq. (11.19), it is clearly suQcient to show that SA (k) ¿ Ck S0 (k)

(11.22)

holds for each A con=guration separately. This will survive after the integration over the positive measure, yielding Eq. (11.19). SA (k) can be expressed in terms of a Dirichlet sum over the eigenvalues of DA : iDA = SA (k) =

∞ 1 −k ; V i=1 i

(11.23) (11.24)


241

with 1 ; 2 ; : : : ; N the eigenvalues in ascending order. Vafa and Witten then prove that the lowest eigenvalue 1 scales with the size of the box considered as C 1 6 = CV −1=4 ; (11.25) L with C independent of the gauge =eld. This implies, for even k, that S(k) ¿ C −k V (k −4)=4 and the required infrared divergence ensues when k ¿ 4 (the limiting case of k = 4 can also be handled by a more re=ned discussion). We will not present here the proof of Eq. (11.25) which involves topological considerations for tracing out the 3ow of eigenvalues under gauge deformations and refer the reader to the original paper [9]. All the above discussion, while strongly suggesting SSB in QCD, fails to demonstrate that > is indeed non-zero. Recently Stern [111] suggested a novel pattern of SSB in QCD with the pseudoscalars still being the Nambu-Goldstone bosons associated with this spontaneous breaking, but where > = 0. Indeed, as Kogan, Kovner, and Shifman [112] noted, there could be some residual “custodial” discrete ZN axial symmetry which allows only higher order ( > )N parameters to have non-vanishing VEVs. The new scheme is phenomenologically interesting. In particular, since (0) (0) (0) (0) now m2$ =O(m2q ), larger values of m(0) u ; md , and a more symmetric mu ; md ; ms mass pattern would be implied. However, as pointed out by Kogan et al. [112], the inequality CA (x) ≡ 0|J;† A (x)J; A (0)|0 6 0|Jps† (x)Jps (0)| ≡ Cps (x) ;

(11.26)

can be judiciously utilized to rule out this pattern. Both correlators have their asymptotic behavior controlled by the lightest 0− pion states, i.e. CA (x); Cps (x) e−m$ |x| for x → ∞. However, the vacuum-to-pion axial matrix element has the conventional form 0|Ja |$b = iq F$ >ab ; with F$ scaling like 3QCD and not vanishing in the mq → 0 limit; whereas the vacuum-to-pion pseudoscalar matrix element behaves in the new scheme as 0|Jpsa |$b = >ab O(mq ) : If M O(3QCD ) is the mass of the =rst massive (non-pion) state in the 0− channel, then for 1 −1 , we have [recalling that the pion intermediate state still dominates C (x)] m− ps $ xM Cps (x) m2q e−m$ |x| m2$ e−m$ |x| : On the other hand, the derivative form of the axial coupling implies e−m$ |x| ; x2 and the inequality (11.26) is violated. The analog of Eq. (11.8) holds also for composite models based on an underlying vectorial gauge interaction [18]. This interaction should con=ne the massless preons at a scale (3p )−1 , leaving us with the quarks, leptons, Higgs particles, and conceivably also W ± ; Z 0 (and even the photon in some models!) as the low lying physical states. Eq. (11.8) suggests that global axial symmetry, assumed to protect the (almost!) massless fermions from acquiring large masses ∼ 3p , does spontaneously break down. The analog of (11.8) here is CA (x)

242


mF ¿ mH and the fact that we have no Higgs particles lighter than the lightest quarks and leptons is bothersome. (A scenario with only three Higgs states, which disappeared in the process of SU(2) × U (1) breaking, appears to con3ict the existence of several fermionic generations.) Finally, we note − − that the m(1 ) ¿ m(0 ) inequality suggests that any composite vector particle should be massive, and gauge symmetries, and in particular the exact U (1)EM (with m 6 10−20 eV! [113]) should not arise “accidentally” due to the existence of almost zero mass vectorial bound states. All the constraints stemming from mass inequalities do not apply to chiral preonic theories [114 –116], and=or preonic theories with fermions and bosons with Yukawa couplings [118]. In this case the measure positivity d(A) is lost (inequalities cannot be proven in supersymmetric models). Also in chiral models the pseudoscalar current > 5 , which was used extensively above, vanishes identically. The mass inequalities, together with the study of the anomaly constraints, shifted the focus of research for composite models from the early work on vectorial models [18,117] to chiral and=or fermion-boson composite models [114,116,119,120] suggested earlier. 14 It was observed [123] that in gauge and scalar (not pseudoscalar!) =eld theories, one can prove the inequality: 0|D† (x)D(x)D† (0)D(0)|00| > (x)

5

(x) > (0)

5

(0)|0 ¿ |0|D† (x) > (x)D(0) (0)|0|2 ; (11.27)

by integrating over ; > . Using the positivity of the d[A ] d[D]e−S[A ; D] integration and of the pseudoscalar correlator, the desired relation follows readily as a Schwartz-type inequality. However, we cannot now infer the mass inequality mF ¿ (1=2)(m$ + m ) between the lightest particles in the D† ; > 5 , and D† D channels, simply because the D† D(x)D† D(0) correlator always has the constant (0|D† D|0)2 contribution due to the vacuum state, which cannot be avoided without spoiling the positivity. This negates the claim made by Nussinov in [19]. Hsu [81] suggested that QCD, and in particular the Vafa–Witten inequalities along with the measure positivity for chiral SU(2) with an even number of 3avors, can be used to exclude the possibility that such strongly interacting chiral theories underlie the standard EW model [124,125]. Since his analysis relies on Vafa–Witten upper bounds for both fermionic and scalar propagators, one needs to choose a regularization point where the D4 coupling vanishes (otherwise the quadratic integral de=ning SD [A (x; 0)] cannot be performed). For a long period it was not clear, in view of diQculties encountered in putting the theory on the lattice [126], if consistent chiral gauge theories could be de=ned. However, the recent domain wall fermions [78] and overlap formalism [127] put the (lattice) regularization of fermionic theories with gauge interactions on a much more solid foundation. These developments also allowed for lattice calculations incorporating chiral symmetries in a more explicit manner [129,130]. Most likely proper regularization of chiral gauge theories [128,131] will also soon be feasible. Chiral composite models for quarks and leptons may revive. We will not pursue here the possibility that this novel approach could serve to “rigorize” the derivation of (some) of the QCD inequalities. 14

Aharony et al. [121] found that QCD-type inequalities were still useful in SUSY theories, though in the mHiggsino → ∞ limit (see Appendix F). Nishino [122] has shown, using the Vafa–Witten inequalities, that in SUSY theories parity is conserved.


243

12. Inequalities between masses of pseudoscalar mesons In this section we conclude the discussion of mass inequalities in the non-exotic qqq (baryonic) and qq> (mesonic) channels, by proving that [10]: mi(0—> with mi(0—>

−

−

)

)

− − 1 ¿ [mi(0)> ) + mj(0—> ) ] ; 2

(12.1a)

the mass of the lowest lying pseudoscalar meson in the qi q>j channel; and [10,19]

m$+ ¿ m$0 :

(12.1b)

The derivation of both inequalities relies on the positivity of the integrand in the functional integral expressing correlation functions of pseudoscalar currents. We have, however, to make the additional assumption (a) that the qq> annihilation channels generating the 3avor disconnected part [Eq. (8.10b)] make negligible contributions. For the derivation of Eq. (12.1b) we could assume instead (b) the validity of the soft pion expression [132] for (m2$+ − m2$− ) in terms of the (axial) vector spectral functions [10]. In order to derive Eq. (12.1a) we compare the three correlation functions 0|Jip—> (x)[Jip—> (y)]† |0 = d(A) tr{[SAi (x; y)]† SAj (x; y)} ; (12.2a) 0|Jip)> (x)[Jip)> (y)]† |0 0|Jjp—>(x)[Jjp—>(y)]† |0

= =

d(A) tr{[SAi (x; y)]† SAi (x; y)} ;

(12.2b)

d(A) tr{[SAj (x; y)]† SAj (x; y)} ;

(12.2c)

where we have left out, following our assumption (a), the 3avor disconnected contribution, e.g. d(A) tr[ 5 SAi (x; x)] tr[ 5 SAi (y; y)] in Eq. (12.2b). The expressions in Eqs. (12.2b) and (12.2c) have j i (or SA0a ) with A (x) and 0; a the spinor, color indices the form of perfect squares of vectors SA0a j i · SA0a . The Schwartz viewed as a joint index. Eq. (12.2a) is the corresponding scalar product SA0a i 2 j 2 i j 2 inequality (S ) (S ) ¿ |S · S | therefore implies that |0|Jip—> (x)[Jip—> (y)]† |0|2 6 0|Jip)> (x)[Jip)> (y)]† |00|Jjp—>(x)[Jjp—>(y)]† |0

(12.3)

and Eq. (12.1a) immediately follows from the last inequality by using Eq. (8.5) and going to the |x − y| → ∞ limit, where (0− )

0|Jip—> (x)[Jip—> (y)]† |0 e−mi—>

|x − y |

:

(12.4)

Eq. (12.3) is the lowest in a hierarchy of inequalities stating that all the principle minors of the matrix Mij = S i · S j are positive. We have noted in our earlier more heuristic discussion of the inequalities the need for making a “Zweig rule” assumption (a). It amounts to suppressing the annihilation of qi q>i into gluons and allows us to consider the mesonic sector Mi)> = qi q>i to be distinct from M (0) , the 3avor vacuum sector. This is valid particularly in the heavy 3avor QQ> sector.

244


− The inequalities are also automatically satis=ed in a soft pion limit with [mij(0 ) ]2 (f$ )−2 > (0) ×[m(0) i + mj ] [85]. We recall that in the actual comparison with particle data in Section 7, the pseudoscalar inequalities were indeed satis=ed with a larger margin than the other inequalities. We next prove Eq. (12.1b) along similar lines [19]. We compare the correlators p p (12.5a) J$+ (x)J$− (y) = d(A) tr{[SAu (x; y)]† SAd (x; y)} ;

and 1 = 2

J$p0 (x)J$p0 (y)

d(A) tr{[SAu (x; y)]† SAu (x; y) + [SAd (x; y)]† SAd (x; y)} ;

(12.5b)

with J$p+ = > u

5 d

1 J$p0 = √ ( > u 2

; 5 u

(12.6a) − >d

5 d)

;

(12.6b)

i.e. the pseudoscalar currents with the quantum numbers of the $+ ; $0 mesons respectively. We can take into account electromagnetic e8ects, which are the source of the $+ − $0 mass di8erence, 15 by considering the gauge group to be SU(3)C × U (1)EM . The path integral measure then becomes d(A) =

−SYM (AC ) −SEM (AEM ) d[AC ] d[AEM e ]e

Nf

Det[D=(AC ; AEM ) + mj ] ;

(12.7)

j=1

and the propagator in the joint external =elds is SAj (x; y) = x|[D=(AC ; AEM ) + mj ]−1 |y :

(12.8)

Since the EM interaction is vectorial, the arguments leading to the measure positivity, d(A) ¿ 0, and the positive de=nite form of the integrand for the pseudoscalar correlation functions remains intact. Comparing Eqs. (12.5a) and (12.5b) we can then show, from a2 + b2 ¿ 2a · b, that J$p0 (x)J$p0 (y) ¿ J$p+ (x)J$p− (y) ;

(12.9)

from which m$+ ¿ m$0 readily follows. The starting point for the original derivation of m$+ ¿ m$0 by Witten [10] is the soft pion–current algebra relation [132] 4 d k e2 2 2 m$ + − m $ 0 = 2 [V3 (k)V3 (−k) − A3 (k)A3 (−k)] ; (12.10) f$ k2 expressing the $+ − $0 mass di8erence as an integral over the di8erence of the momentum space vector and axial vector (isovector) correlation functions. The latter are Fourier transforms of the 15

It was realized early on that the Cottingham formula expressing the EM contributions to the mass di8erences, converges (0) to the correct $+ − $0 mass di8erence [133,134], and fails for the K + − K 0 and n − p di8erence—for which m(0) d − mu makes the dominant contribution.


245

euclidean space correlation functions which can be expressed via Eq. (8.10a), so that we have V3 (k)V3 (−k) − A3 (k)A3 (−k) 2 d 4 x eikx d 4 y e−iky d(A) tr[ SA (x; y) SA (y; x) − 5 SA (x; y) 5 SA (y; x)] = V 2 4 ikx 4 −iky d xe d ye d(A) tr[ SA (x; y) − 5 SA (x; y) 5 ]SA (y; x) : (12.11) = V To this lowest, 0EM order, there are no 3avor disconnected contributions to the isovector correlation function. The factor of 2 comes from V = > T 3 with tr[(T 3 )2 ] = 2. We note that in the present case—unlike in all previous inequalities—the actual sign of the expression in (12.11) is crucial. A minus sign coming from the fermion loop has been cancelled by the fact that the euclidean currents are V = i > (and A i > 5 ) and the i2 supplies the extra E E E minus sign. [ M → i is essential since the euclidean are hermitian and satisfy { ; } = > , M whereas { M ; } = g .] The di8erence [ SA (x; y) − 5 SA (x; y) 5 ] occurring in Eq. (12.11) is simply EA (x; y) , with EA (x; y) = x|m0 (−DA2 + m20 )−1 |y, the 5 even part of SA (x; y) = x|(DA + m0 )−1 |y = EA (x; y) + OA (x; y). The odd OA (x; y) behaves like a product of an odd number of matrices. Thus tr( EA OA ) = 0, and Eq. (12.11) can be rewritten as 2 d 4 x eikx d 4 y e−iky V3 (k)V3 (−k) − A3 (k)A3 (−k) = V d(A) tr[ EA (x; y) EA (y; x)] : (12.12) Viewing M ≡ eikx = eikx ( )00 as a matrix in spinor (00 ) and coordinate space jointly, we can rewrite this last expression as 2 d(A) tr[M EA M∗ EA ] ; V3 (k)V3 (−k) − A3 (k)A3 (−k) = (12.13) V where the last trace and matrix multiplications refer also to the coordinates x; y as indices. The operator EA is positive de=nite, as can be seen by going to the basis de=ned by iDA |A = A |A . In this basis m20 |(M )|2 2 tr[M EA M∗ EA ] = (A + m20 )(A2 + m20 ) A ;A is manifestly positive. The measure positivity then implies that d(A) tr[M EA M∗ EA ] is also positive (non-negative). We =nally arrive at V3 (k)V3 (−k) − A3 (k)A3 (−k) ¿ 0 ;

(12.14)

and m$+ ¿ m$0 follows from Eq. (12.10). Eq. (12.14) complements the asymptotic chiral symmetry [135] V3 (k)V3 (−k) A3 (k)A3 (−k), as k → ∞. Such an asymptotic equality is indeed required for the d 4 k integration in Eq. (12.10) to converge. It motivated the spectral function sum rules of Weinberg [136] some time ago.

246


Using unsubtracted dispersion relations for the di8erence of V3 (k)V3 (−k) and A3 (k)A3 (−k), the inequalities (12.14) imply d2 [V (2 ) − A (2 )](2 + k 2 )−1 ¿ 0 ; (12.15) with V (A ) the vector (axial vector) spectral functions, for all k 2 ¿ 0. The con=guration space analog of the inequality A (x)A (0) 6 V (x)V (0)

(12.16)

has been discussed at length [137] but no convincing proof exists at present. One diQculty is that the one pion contribution to the “longitudinal” part of the axial correlator CA (x) e−m$ |x| dominates at large distances and an appropriate transverse projection of A A is required. (This is not the case in momentum space since the large pion pole contribution at small k is suppressed by k k “derivative coupling” factors.) E8orts to prove such an inequality are motivated not only by the experimental fact that the lightest hadrons in the 1+ , 1− sectors satisfy ma1 ¿ m@ . Precision tests of the weak interactions indicate that the “S parameter” [138,139] S≡

d [FV (k 2 ) − FA (k 2 )]|k 2 =0 ; d k2

(12.17)

with FV ; FA the covariant (transverse) parts of A (k)A (−k) and V (k)V (−k), is negative. The second moment of the conjectured inequality (12.16) will imply that S ¿ 0 in all vectorial theories such as the original technicolor models [140,141], thereby ruling those out as viable mechanisms for dynamical breaking of the EW symmetry of the Standard Model. The previous derivation of m$+ ¿ m$0 , based on Eq. (12.9), did not utilize the soft pion limit (0) (0) (0) (tantamount to letting m(0) u + md → 0), but only the weaker assumption of isospin |mu − md | → 0. [In particular isospin was utilized to argue that there are no disconnected purely gluonic intermediate 0 state contributions to 0|J p($ ) (y)|n:] At =rst sight the previous derivation appears to lead to a stronger result. This however is not the case [142]. The point is that there are intermediate states with one photon which make 0|( > u

5 u

− >d

5 d )|multigluon

state + = 0 ;

(12.18)

and which contribute to the same order O(0EM ). (Because of charge conjugation parity and color neutrality, the lowest perturbative state of this type consists of at least three gluons and a photon.) This contribution could be neglected if we appeal to the analog of the Zweig rule hypothesis (a) used above. Alternatively, we could go to the soft $0 limit in which case the couplings of the Goldstone particle to the photon and any neutral hadronic system [and the “multigluon state” in Eq. (12.18) in particular] must vanish. 3 Since the EM interaction conserves JA , soft pion theorems imply that the EM contributions to masses of neutral Goldstone particle vanish to all orders. Thus, if m$0 (0 = 0) = 0 then also m$0 (0 = 0) = 0. The result m2$+ ¿ m2$0 , and its analog in other vectorial gauge theories, is therefore of great importance. If m2$+ 6 m2$0 = 0, the $+ becomes tachyonic. A condensate of charged pions could then form, leading potentially to spontaneous violations of EM charge conservation. Similar results


247

like m$(T+) ¿ m$(T0 ) could help =x the pattern of symmetry breaking or “vacuum alignment” [143,144] in other vectorial theories such as technicolor [145,146]. The inequality m$+ ¿ m$0 is one aspect of the general property mentioned in Section 9 [see in particular Eq. (9.7)] that vectorial interactions make positive contributions to the masses of physical particles. We can motivate this result (see Section 9) by considering the EM self-interactions and mutual interactions between arbitrary charge distributions (representing e.g. the extended constituent quarks and including e8ects of qq> pairs as well). This leads to the conjecture that the EM contributions to hadronic masses are always positive. In order to extract genuine EM contributions to hadron masses we need to form \I = 2 mass combinations such as >2 [F] = mF+ + mF− − 2mF0 ; >2 [@] = 2(m@+ − m@0 ) ;

(12.19)

(0) (0) in which the e8ects of (m(0) u − md ) = 0 have been cancelled to =rst order. (Next order [(mu − (0) md )=3QCD ]2 e8ects are smaller than the observed splitting.) In terms of a naive quark model the F isotriplets are xuu; xud, and xdd states with x = s; c; b : : : ; some heavy quark. The EM contribution to >2 (F) comes from mutual qi ; qj ; i = j interactions and we =nd 2 >2 0|Qu − Qd | @n (˜r)@n (˜r )=|˜r − ˜r | ¿ 0 ; (12.20)

with @n referring to the density of the light u or d quark, which in the I -spin symmetric limit are equal. Our conjecture is then that this positivity is not an artifact of the simple model but would persist in the full-3edged theory. Experimentally [58], we =nd >2 [F(1190)] = 1:5 ± 0:18 MeV ;

(12.21a)

>2 [F(1380)] = 2:6 ± 2:1 MeV ;

(12.21b)

>2 [Fc (2455)] = 2:0 ± 1:6 MeV ;

(12.21c)

>2 [@] = −0:2 ± 1:8 MeV :

(12.21d)

The positivity of >2 [F(1190)] and >2 [F(1380)] are statistically very signi=cant (in view of the fairly small width). Estimating m@+ − m@0 is diQcult because of the large widths O@ 150 MeV. 16 We also note that the positivity of the \I = 2 energy shift should apply not only to the ground state, but also to any excited states. 13. The absence of spontaneous parity violation in QCD Vafa and Witten [8] argued that QCD inequalities technique imply that there is no spontaneous breaking of parity symmetry in vectorial theories. Such a breaking should manifest via a nonzero 16

Also for the special case of the vectorial @+ @0 @− multiplet there is a relatively important one photon annihilation contribution, which the Coulomb interaction based argument presented above misses.

248


vacuum expectation value of some parity odd operator A. The simplest candidates for such operators are the local quantities constructed from the gauge =elds only ˜ = j F F ; j01 D F D F01 ; : : : A = FF

(13.1)

Alternatively we could use quantities involving fermions such as A = > 5 , with 5 = j . The distinguishing feature of any parity odd operator is that it includes an odd number of j tensors. While the euclidean matrices, =elds A , and metric g are hermitian and real, j , which transforms like d x1 d x2 d x3 d x4 , picks up a factor of i and becomes purely imaginary. Consequently any Aeuc corresponding to a parity odd operator is imaginary. If A = 0, then from ordinary =rst order perturbation theory it follows that a (hypothetical) theory with a modi=ed Lagrangian L = LQCD ( = 0) + d 3 xA (13.2) has, for small , a shifted vacuum energy density E() = E(0) + A ;

(13.3)

with A = A0 still computed for the = 0 vacuum. For a with an appropriate sign, A ¡ 0 and E() ¡ E(0)—just as the energy of a spontaneously magnetized ferromagnet is lowered by adding an external ˜B =eld anti-parallel to the magnetization ˜. The euclidean path integral representation of the vacuum energy density for L is 4 1 E() = − ln d[A ] d[ ] d[ > ]e− d xL : (13.4) V We will next show that E() ¿ E(0), negating the possibility that E() ¡ E(0) and forbidding A = 0. If A is of the type indicated in Eq. (13.1), the d[ ] d[ > ] integration can be carried out and 1 1 iA (13.5) E() = − ln d(A)e ¿ − ln d(A) = E(0) ; V V since the oscillatory factor eiA always decreases the integral. The same is true if A = > 5 = 0. Adding A to the Lagrangian is equivalent to introducing complex masses in the determinantal factor Det(D = + mi ) (e.g. mi → mi + i) which again destroys the positivity of the determinantal factor, i reduces Z(), and increases E(), so that Eq. (13.5) holds. The fact that Z() is minimal for = 0 allows also the proof that of all the di8erent QCD ˜ term to the Lagrangian), P vacua (which can be generated by adding the topological density PFF the one with the lowest vacuum energy is that with P = 0. (Also P = $ was excluded by similar arguments [147], con=rming suggestions of dilute instanton calculations [107] and other considerations [63].)


249

14. QCD inequalities and the large Nc limit We have commented on several occasions on the inequalities in the Nc → ∞ limit. We will next show [20] that in this limit we have indeed the stronger version of the baryon–meson inequality mB ¿

Nc m$ : 2

(14.1)

We will also indicate that the inter3avor mesonic mass inequalities could be extended to nonpseudoscalars. The Schwartz inequality implies that the baryonic correlation function B(x)B(y) = d(A) tr[OSA (x; y)Nc O] with SA (x; y) the common fermion propagator, is smaller than { d(A)(tr[SA† (x; y) SA (x; y)])Nc }1=2 . This last expression represents the correlation function of products of Nc pseudoscalar currents, 0|[J ps (x)]Nc [J ps (y)]Nc |0. We can show that in the large Nc limit this joint correlation function e8ectively factors into a product of the form [J ps (x)J ps (y)]Nc . Speci=cally, gluon exchanges between di8erent qq> bubbles do not modify by more than O(1), the energy Nc m$ , of the system viewed as separately propagating Nc qq> pairs. Indeed, the color trace counting argument [67,148] indicates that the interaction energy between any (qq)(q > q) > pair of bubbles is Nc 2 O(1=Nc ). Since we have ( 2 ) pairs, which can interact via planar diagrams of gluon exchanges, the total e8ect is O(1). The interaction of triplets (or a higher number k) of bubbles is O(1=Nc4 ), or O(1=Nc(2k −2) ). Since we have ( Nkc ) 6 Nck , the interaction of the k-plets is even less important. Thus in the large |x −y| limit, d(A) tr[SA† (x; y)SA (x; y)]Nc e−Nc m$ |x−y| , and since B(x)B(y) e−mB |x−y| , Eq. (14.1) follows. To extend the inter3avor mass relations consider the following three correlation functions of four 3avor currents (see Fig. 12): F1 = 0|J$+ (x1 )JK − (x2 )J$− (y1 )JK + (y2 )|0 ; F2 = 0|JK − (x1 )JK + (x2 )JK − (y1 )JK + (y2 )|0 ; F3 = 0|J$− (x1 )J$+ (x2 )J$− (y1 )J$+ (y2 )|0 ;

Fig. 12. The planar duality diagrams representing the euclidean correlations F1 , F2 ; and F3 . For F2 we use only the speci=c contraction with uu(s > s) > exchanged in the Q(T ) channel, respectively.

250


with the euclidean points x1 ; x2 ; y1 ; y2 forming a rectangle of height T and width Q in the, for example, 3– 4 plane. In the large Nc limit, quark annihilation is suppressed and we have the planar quark diagrams indicating the possible contractions. In the case of F2 we speci=cally choose the contraction with ss> exchanged in the vertical (T ) direction, so that in all three cases we have uu> exchanged in the horizontal (Q) direction. We can use the euclidean Hamiltonian to evolve the system in T , e.g. F1 = 0|J$+ (x1 )JK − (x2 )|nn|J$− (x1 )JK + (x2 )|0e−En T n

=

|0|J$− (x1 )JK + (x2 )|n|2 e−En T ;

(14.2)

n

with 0|J$− (x1 )JK + (x2 )|n independent, by translational invariance, of the speci=c common T value of x1 ; x2 . We can also evolve the system in Q to =nd analogous expressions: F1 = 0|JK + (x2 )JK − (y2 )|n n |J$+ (x2 )J$− (y2 )|0e−En Q ; (14.3a) n

F2 =

|0|JK + (x2 )JK − (y2 )|n |2 e−En Q ;

(14.3b)

|0|J$+ (x2 )J$− (y2 )|n |2 e−En Q ;

(14.3c)

n

F3 =

n

where, due to the choice of F2 , we have the same states |n (with uu > 3avor) in all three cases. From the Schwartz inequality we have |F1 |2 6 F2 F3

(14.4)

which, by going back to the representation (14.2) for F1 , and similar expressions for F2 and F3 , yields inequalities between the masses of the lowest intermediate states for the three cases 1 (0) (0) m(0) us> ¿ [muu + mss> ] ; 2

(14.5)

since by construction the intermediate “T -channel” states for F2 are ss> type states (and uu> for F3 ). As emphasized earlier in the large Nc limit the muu>; mss> sectors are distinct from m(0) , the 3avor vacuum, and from each other. To get perfect squares in Eqs. (14.3b) and (14.3c), we need to have identical currents at x1 ; x2 and y1 ; y2 . Thus the quantum numbers of these uu; > ss> (and us) > states are those of J$ J$ ; C = +; even G-parity, 0++ states. Similar unitarity motivated inequalities were suggested earlier in the context of dual models and were applied to intercepts of Regge trajectories on which 0++ states lie [149]. Also it has been conjectured that in the large Nc limit QCD approaches a dual, string-like model with linearly rising parallel trajectories. The present derivation pinpoints, however, the crucial element for deriving Eq. (14.5). It is that in the large Nc limit we can meaningfully separate the various planar contributions to the correlation function and each evolves separately under the appropriate Hi)> (or Hi—>) Hamiltonian.


251

We note that the last argument did not utilize the measure positivity in the euclidean path integral, but rather the unitarity based, more general, spectral positivity. QCD inequalities have also been applied in the large Nc limit in (1 + 1) dimensions [150].

15. QCD inequalities for glueballs QCD predicts “glueball” states consisting of gluonic degrees of freedom and no quark 3avors. Experimental evidence for such states exists at present [151]—but mixing with the many qq> and=or qqq > q> states in the 1–1:5 GeV region (where the glueball spectrum is estimated to start [152]) complicates the analysis. a a ˜ 2 + ˜B2 and is larger than all The local density F 2 = F F becomes, in the euclidean domain, E ˜ · ˜B. Thus, as pointed out by Muzinich and Nair [21,153], we ˜ → iE other F bilinears such as FF 2 2 have for any A (x), FA (x)FA (y) ¿ F˜ A FA (x)F˜ A FA (y). Due to measure positivity this translates into an inequality between the corresponding correlation functions 2 2 0|F (x)F (y)|0 = d(A)FA2 (x)FA2 (y) ¿

˜ ˜ FF(y)|0 : d(A)F˜ A FA (x)F˜ A FA (y) = 0|FF(x)

(15.1)

˜ create from the vacuum 0++ and 0−+ states, respectively. Thus, the standard argument F 2 (x); FF(x) suggests the inequality ++

m(0 gb

)

−+

(0 6 mgb

)

(15.2)

between the masses of the lowest glueball states in the respective channels. Con=nement and the existence of a mass gap in the gluonic sector prevents interpretation of Eq. (15.2) as a statement about gluon thresholds. Any polynomial in the component of F , used to create a glueball state of arbitrary J PC , is bound by an appropriate power of F 2 . Thus Eq. (15.2) can be generalized to ++

m(0 gb

)

6 (mass of any glueball) :

(15.3)

All these correlation function inequalities are automatically satis=ed if F 2 has a nonvanishing vacuum expectation value: 0|F 2 |0 = 0. In this case, correlation functions involving F 2 [or (F 2 )n ] have constant terms as |x − y| → ∞, corresponding to the obvious zero mass 0++ “state”: the vacuum, and Eq. (15.3) need not hold. In Ref. [154], West suggested that the inequalities (15.2) and (15.3) can be recovered if we use the time derivative of the correlator: d 0|F 2 (0)F 2 (t; ˜0)|0 = 0|F 2 (0)[H; F 2 (t; ˜0)]|0 dt = 0|F 2 (0)HF 2 (t; ˜0)|0 ;

(15.4)

252


˜ and the corresponding expression for FF. The o8ending constant term in the intermediate state summation would be absent in the new inequalities. The idea then is to prove that ˜0)|0 : ˜ ˜ 0|FF(0)HFF(t; ˜0)|0 ¿ 0|FF(0)H FF(t;

(15.5)

This could be accomplished by using the positivity of the Hamiltonian for each A (x) background con=guration separately in a path integral representation of the last two correlators. There are however subtleties in this argument which West tried to address. In general, while the Hamiltonian is positive when operating on physical states, the latter states are obtained only after the path integral has been carried out, and the pointwise positivity of H for every A (x) con=guration is not a priori guaranteed. It is the case for the pure Yang-Mills theory ˜ 2 + ˜B2 ); 17 however, in this case H |0 = 0, which has been assumed in the above if H = d x(E argument, holds only after an appropriate subtraction has been made. Such a common subtraction will then modify Eq. (15.5) to 0|FF(0)HFF(t; ˜0)|0 − c0|FF(0)FF(t; ˜0)|0 ˜0)|0 − c0|FF(0) ˜0)|0 ; ˜ ˜ ˜ ˜ ¿ 0|FF(0)H FF(t; FF(t;

(15.6)

with c positive. Due to Eq. (15.1) this may invalidate the desired inequality. In passing we remark that 0|F 2 |0 = 0 has been conjectured to be the driving mechanism for con=nement. Its value controls the bag constant [155], the string tension in extended hadronic states [156], and enters QCD sum rules [93]. The glueball sector can be divided into “even” and “odd” parts consisting of states with quantum numbers of two or three gluons with appropriate orbital and spin angular momentum. Thus the J PC states 0++ ; 2++ ; 0−+ ; 2−+ belong to the even part and 1− − to the odd. If we adopt a simple “constituent” description of the lowest even (odd) states in terms of two (three) gluons bound by one gluon exchange potentials, then the arguments of Section 3 leading to the baryon-meson mass inequality mB ¿ (3=2)mM can be repeated here to prove the inequality 3 (0) m(0) 3g ¿ 2 m2g

(15.7)

between the masses of the lowest odd and even states. Just as in the meson (baryon) 3 ⊗ 3> → > we have in the 2g (3g) state 8 ⊗ 8> → 1(8 ⊗ 8> → 8) > and we can repeat the argument 1(3 ⊗ 3> → 3), by replacing 1 · 2 in the fundamental representation by 31 · 32 in the adjoint representation of SU(3)C . Evidently, a constituent valence model for glueballs is on a much weaker footing than that for mesons and baryons. In particular, the latter (but not the former!) can be justi=ed in the large Nc limit. If the even glueball spectrum starts at 1.5 –2 GeV, then Eq. (15.7) suggests that the lightest 1− − glueball may be near the J= ; states. Its putative mixing with the latter could then modify the perturbative approach to J= → 3g decays and=or other matrix elements [157]. ˜ 2 + ˜B2 form, and the euclidean Hamiltonian Actually it is the Lagrangian which has this positive de=nite E euclidean 2 2 ˜ ˜ then has the form (B − E ). A non-vanishing 0|H |0 VEV then amounts to the non-trivial dynamical assumption of the dual Meissner e8ect in the QCD vacuum—namely the preponderance of large magnetic 3uctuations. 17


253

16. QCD inequalities in the exotic sector So far we have focused on correlation functions of qa q>a bilinear and jabc qa qb qc trilinear currents. In the following we utilize J ex (x)J ex (y) with quartic “currents”: J ex ∼ qi (x)q>j (x)qk (x)q>l (x) ;

(16.1)

to probe the “exotic” Mi—k> l> sector. We also address the possibility that the lowest lying intermediate state in J (x)J (y) is not a stable one particle state, but rather a two particle threshold at m1 + m2 . This last issue has been encountered already for non-exotic currents. Thus, the lowest lying state in the vector 1− ud> channel is not the @(760) but the $$ threshold at ≈ 280 MeV. Since this happens because m@ ¿ 2m$ , the Weingarten inequality m@ ¿ m$ is not invalidated. A general important observation is that the correlation function inequalities are derived for all |x − y|. By varying |x − y| we e8ectively scan the spectrum since we keep changing the relative weight of di8erent (2 ) regions in Eqs. (8.2) and (8.3). In the |x − y| → 0 limit the high 2 region dominates. The perturbative expressions for the correlation functions are adequate and indeed conform to the inequalities. In the other extreme, |x − y| → ∞, the threshold region dominates and we have, e.g., VV ∼ exp{−2m$ |x − y|}. However, following lattice calculations of hadronic masses, we can consider “intermediate times” during which the local u(x) > d(x) state, for example, evolved into a true @; qq> bound state, but has not yet evolved into the =nal decayed form of two pions (see Fig. 13). In the range O@−1 |x − 1 † y|m− @ ; 0|J (x)J (0)|0 ∼ exp{−m@ |x − y|} is a valid approximation from which meaningful mass values (and mass inequalities) could be extracted for the @ resonance. In this section we focus on the threshold regions and the |x − y| → ∞ domain in the correlator inequalities. Since near threshold the kinetic energies are much smaller than the masses of the stable particles, nonrelativistic kinematics, and some aspects of a potential model (in the $$; $K, and KK channels) may be applicable. The inequalities m(0) ¿ 12 (m(0) + m(0) ) often become in the continuum limit trivial equalities mi + ij> ii> j j> 1 1 mj = 2 (2mi + 2mj ) [or mK + m$ = 2 (2mK + 2m$ ) when we have con=ned i = u; j = s> quarks and a meson–meson continuum]. If we consider systems con=ned to a sphere of radius R, we expect the various inequalities to be satis=ed with a margin of order \E 1=2mR2 , the level splitting for a system of size R. For the continuum, R → ∞, and \E vanishes. However, the relevant quantities become the phase shifts. We wish to interpret the inequalities as nontrivial statements about the latter [20,158]. To this end, let use consider the “tr vn h12 ” inequalities (2.14): N n=0

Eij(n) ¿

N

1 (n) (E + Ejj(n) ) ; 2 n=0 ii

(16.2)

with the sum extending over the =rst N excited states. If we put our system in a large box, then each time a “bound state” is generated (by increasing the strength of the attractive potential), the phase shift > in the relevant channel changes by $. Levinson’s theorem [159] suggests replacing the

254


Fig. 13. The evolution of an initial ud> 1− state on di8erent time scales.

A discrete sum Nn=0 by 1=$ d>, and Eq. (16.2) then becomes 1 A 11 A d>Eij (>) ¿ d> [Eii (>) + Ejj (>)] : $ 2$

(16.3)

Evidently (16.3) reverts back to (16.2) in the narrow resonance approximation where d>=dE = 0 only near the resonances at E (n) . Near threshold > = ka, with k the center of mass momentum and a the scattering length. Using NR kinematics for kij (E) from Eq. (16.3) restricted to the threshold region, we can obtain the relation [20]

1 1 1 1 1 1 1 (16.4) + 2 ¿ + 2 2 : a2ij m2i mj a2ii m2i ajj mj Scattering data for $$ and $K can be obtained by considering $N → 2$N and KN → K$N scattering and extrapolating to the one pion exchange pole [160,161]. This extrapolation is much more diQcult for the K exchange case, making a direct test of (16.4) diQcult. However, the existence of the 0++


255

state in the K K> system slightly below threshold is expected to enhance aK K> so that (16.4) would most likely hold. 18 Consider next the exotic current with 0++ quantum numbers: ps > J> = Jips —> (x)Jk l> (x) = i (x)

>

5 j (x) k (x) 5 l (x)

:

(16.5)

If all 3avors are distinct, we have only one possible contraction, illustrated in Fig. 11(a), contributing to J> J> which, using Eq. (8.20), can be written as † (16.6) 0|J> (x)J> (y)|0 = d(A) tr{[SAi (x; y)]† SAj (x; y)} tr{[SAk (x; y)]† SAl (x; y)} : (0) (0) (0) If m(0) i = mj ; mk = ml , then the integrand in Eq. (16.6) becomes the product of two perfect squares and J> J> maximizes all other exotic correlation functions. Espriu et al. [22] speculate that this is associated with the fact that the lowest exotic qqq > q> states found in bag model calculations ++ [162] are 0 states. Even more dramatic results follow [22] if we take all four, distinct, 3avors, i; j; k; and l to be degenerate. In this case we have J> J> = d(A){tr[SA (x; y)† SA (x; y)]}2 , which by the Schwartz inequality is larger than the square of the pseudoscalar propagator:

J> (x)J> (y) ¿ |J ps J ps |2 :

(16.7) + +

$ = u>

While the last inequality is consistent with having a bound > state in the $ $ ($= u> 5 d; 5d ) channel, it does not require such a state. The “>” state could simply be a $$ threshold state and the inequality would be trivially satis=ed as

m($$ ) threshold 6 m$ + m$ :

(16.8)

We would like to interpret this, in analogy with our above discussion, as a statement that the low energy (threshold) interacts attractively. This in turn enhances the density of threshold states relative to the non-interacting, free case, and enhances the long distance euclidean correlators. The connection between an attractive potential in the i—> channel and 0|Jij† (x)Jij (y)|0 can be directly seen in a potential model limit. With one of the particles in=nitely heavy, the ij correlator is 18

The inequality JK + K − (x)JK + K − (y)J$+ $ − (x)J$+ $ − (y) ¿ |JK + $ + (x)JK − $ − (y)|2 ;

with JK + K − =JK + JK − ; JK + = > s (x) 5 u (x), etc. is readily derived, if we introduce additional degenerate 3avors u ; s ; mu = mu ; ms = ms . As |x − y| → ∞ the various correlation functions are dominated by the respective meson–meson thresholds. If we neglect the interaction between the propagating mesons, the above inequality becomes a trivial equality since each of the two-point functions factorizes, e.g. 0|J$+ (x)JK − (x)J$+ (y)JK − (y)|0 = f$ fK D$ (x; y)DK (x; y) with f$ = 0|J$ |$ and D$ the pion propagator. In this long distance, low energy limit, we can e8ectively treat the pions as elementary with interactions $$ $4 ; KK K 4 ; and $K $2 K 2 . To =rst order in the s theseinteractions modify the two-point correlation function as follows: J1 (x)J2 (x)J1† (y)J2† (y)=F12 F22 Dm1 (x; y)Dm2 (x; y)−12 d 4 xDm1 (x; z)Dm2 (x; z)Dm1 (z; y)Dm2 (z; y). The second term represents the e8ect of one m1 m2 collision. By going to momentum space the z integration can be done and an inequality between the s is obtained [20]. Unfortunately, it contains, besides the masses, also a subtraction point 0 and will not be reproduced here. Note that the minus sign in front of the =rst order euclidean perturbative contribution re3ects the expansion of e−Ht . Thus a negative 12 —corresponding to an attractive $$ interaction—enhances the joint propagation.

256


essentially that of the other particle moving in the attractive potential. The path integral expression for such a propagation is 2 Q d x 1 e− L dQ = exp − − V (x ) dQ ; (16.9) 2 dQ paths

paths

with Q a “proper time” and x (0)=x; x (Q)=y. Evidently an attractive potential (V ¡ 0) will enhance the positive contributions of the individual paths. This is indeed expected from the interpretation of the free euclidean propagator as a di8usion kernel: the probability of the di8using particle to return to the origin is clearly enhanced by an attractive potential. A more rigorous and systematic approach directly relating euclidean lattice correlators to phase shifts and scattering lengths was suggested by LMuscher [163]. It was utilized by Gupta et al. [158] to prove that the $–$ scattering length is positive. The idea in LMuscher’s approach, and of Neuberger’s [164] suggestion of a calculational lattice method for determining f$ , is to use the =nite size corrections to the correlators. The latter are the con=guration space analog of the 1=R energy shifts discussed here. We refer the reader to the original works for further details. We have argued that m($$ ) 6 (m$ + m$ ) can be ful=lled by an interacting $+ $+ threshold state if the scattering length is attractive (positive). The $ and $ can only interact via gluon exchanges since they are composed of di8erent quarks. The interesting fact that this interaction is attractive is in accord with an old result [165] that Van der Waals forces between systems of identical polarizabilities are always attractive. Indeed the Casimir–Polder two-photon exchange interaction: VCP (˜r) =

1 (1) (2) (2) (1) [ − 23(0E(1) 0E(2) + 0M 0M ) + 7(0E(1) 0M + 0E(2) 0M )] 4$r 7

remains attractive so long as the ratios of electric and magnetic polarizabilities 0E =0M for the two neutral systems (1) and (2) in question are similar. Our above result generalizes this attractive nature of the two-photon exchange to the full non-perturbative case, and applies also for nonabelian gauge interactions. (If, like in QCD, the theory is con=ning, then there is a mass gap in the pure glue sector, and we expect an exponential rather than power law fallo8 of the interaction. The attractive nature does however persist.) We note that two independent lines of argument suggest that the two-photon=gluon exchange interaction is attractive. The =rst non-relativistic, second order perturbation theory argument applies if systems (1) and (2) considered are in their respective ground states. The second relativistic =eld-theoretic argument uses t-channel dispersion [79] and the positivity of the corresponding spectral functions when particles (1) and (2) are the same. Amusingly the conditions required for proving > k l> pseudoscalars are the inequality (16.7) simultaneously conform to both arguments, since the ij; indeed the lowest states in their channels, and taking mi = mk ; mj = ml makes them (dynamically) identical. The attractive interaction manifests also in the fact [166] that contrary to some lore, two halves of a conducting spherical shell attract rather than repel. In the real world the $+ $+ scattering length is repulsive [167]. Indeed, with u=u ; d=d we have the additioinal contraction of Fig. 11(b). It has one fermion loop and makes hence a contribution


of the opposite sign: J> J> = d(A){[tr(SA† SA )]2 − tr[(SA† SA )† (SA† SA )]} ;

257

(16.10)

so that Eq. (16.7) can no longer be proven. Indeed at short distance, when the mesons’ wave functions overlap, we expect a repulsive Pauli e8ect which is re3ected in the minus sign in Eq. (16.10). We have not been able to show (except in the large NC limit) that this extra negative term reverses the sign in Eq. (16.7), so that a$+ $+ = a$$ (I = 2) is repulsive and no $+ $+ exotic bound state exists. 17. QCD inequalities for 2nite temperature and 2nite chemical potential Recently there has been much interest in QCD at =nite temperature and =nite baryon density, i.e. =nite chemical potential. This interest is partially motivated by the desire to better understand compact neutron = (strange) quark stars, and by the prospect that heavy ion collisions at the relativistic heavy ion collider (RHIC) can indicate the expected phase transition. In the following we would like to comment on the possible relevance of QCD inequalities in these cases. First, we note that all the correlator inequalities are maintained for T ¿ 0. In the euclidean formulation, introduction of =nite temperature is simply equivalent to the imposing periodicity 1=1=T in the time direction [168]. The restriction of the gauge =eld con=gurations in the euclidean path integral to such periodic con=gurations clearly does not spoil the positivity of the measure. Also the relation 5 SFA (x; 0) 5 = SF†A (x; 0) and the ensuing positivity of the fermionic determinant and the integrand in the pseudoscalar correlators are maintained at =nite temperature—and therefore so are all the correlator inequalities. Precisely because of the periodicity in time, we cannot use the Hamiltonian and its asymptotic e−mt behavior in order to infer bounds on masses smaller than T =1=1. Therefore we cannot attempt, when T = 0, to prove that the axial global symmetry spontaneously breaks down, as this feature is closely tied to the massless pseudoscalar Nambu-Goldstone bosons. Indeed, numerous theoretical arguments [169,170] and lattice simulations have virtually proven that in QCD there is in fact a phase transition corresponding to axial symmetry restoration (and quark decon=nement) at a temperature Tc 3QCD . The exact nature of this phase transition and its dependence on Nf , the number of quark 3avors, was for a long time unclear.19;20 The QCD inequalities technique may be even less useful in discussing transient, varying T phenomena, such as disoriented chiral condensates [173], suggested to occur in domains of cooling quark gluon plasma. The introduction of a =nite chemical potential, i.e. consideration of QCD in a background of uniform baryon density, has a much more drastic e8ect. It amounts to changing the fermionic part 19 It is generally believed to be weakly =rst order [171], and could be Nf dependent. We will later make a conjecture that Tc monotonically decreases with Nf . 20 T. Cohen [172] utilized the QCD inequalities to suggest that not only is SU(3) axial symmetry restored above Tc , but also U(1)A . Since the latter is broken via the QCD anomaly and not spontaneously, this result is rather surprising. Indeed as noted by Cohen the proof involves a technical assumption which may fail.

258


of the euclidean Lagrangian to L() =

Nf

[ > i D

i

+ >i

0 i

+ mi > i i ] ;

(17.1)

i=1

so that the new = 0 propagator no longer satis=es 5 SF 5

=

5

1 D= + 0 + m

5

=

1 −D= −

0+m

5 SF 5

= SF† ;

= SF† , but rather (17.2)

and hence the positivity of the fermionic determinant can no longer be inferred. This not only prevents the proof of the QCD inequalities, but also excludes the utilization of lattice numerical simulations in which the statistics of occurance of lattice gauge con=gurations prescribes their (positive!) weights. The various dramatic speculations concerning the high phase (involving parity breaking and “color superconductivity” [174]) are therefore not excluded. The special case of Nc = 2, i.e. SU(2) gauge theory, is a notable exception, and one can show measure positivity in this case even for = 0. This result, which has been utilized for some time in lattice simulations [175] is readily proven by using 5 CI2 SF 5 CI2

= SF∗ :

(17.3)

In Eq. (17.3) C is i 0 2 and I2 is the generator of the SU(2) color isospin, and it holds for arbitrary . This feature, which is due to the pseudoreality of SU(2), is essentially the same one used by Anishetty and Wyler [80] and Hsu [81] to extend the inequalities to chiral SU(2). It has been used by Kogut et al. [175] in order to prove that the correlator of the 0+ , I = 0 (i.e. antisymmetric in 3avor) combination M

=

†

CI2

5

can serve, for = 0, as an upper bound for any other correlator. This implies that for = 0 the 0+ diquark is the lightest boson. This complements the claim that for = 0 and Nc = 2, the 0+ diquark is degenerate with the 0− > 5 pion (see the end of Section 6), and nicely =ts with the unique patterns of symmetry breaking suspected in this case [175]. External electric and magnetic =elds modify the hadronic spectrum. Indeed lattice calculations utilized external magnetic =elds to probe, via the ˜ · ˜B interactions, the hadronic magnetic moments. Also very strong =elds (the analogs of supercritical =elds in superconductors) can modify the phase structure of spontaneously broken gauge theories. In the context of QCD and other vectorial theories the measure postivity is clearly maintained in the presence of the external ˜B =eld. However, QED=QCD inequalities for binding energies, such as BE (parapositronium) ¿ BE (orthopositronium), or m@ ¿ m$ may be modi=ed since the vector (triplet) state has a magnetic moment, and for a suQciently strong external ˜B =eld (|˜B| & 32QCD ), the state with ˜ antiparallel to ˜B may become lower than that of the singlet pion. In this connection we recall the amusing suggestion [176] that strong enough magnetic =elds, existing in appropriate astrophysical environments, can reverse mn ¿ mp to mp ¿ mn with an inverted 1 decay!


259

Unlike the e8ect of high temperature and=or chemical potentials, we believe that strong magnetic =elds do not cause decon=nement or chiral symmetry restoration. Indeed we expect [177] that in strong magnetic =eld (B32 ) the motion of the quarks becomes one dimensional, along the ˜B =eld lines. It is well known that in such cases con=nement only gets stronger (even U(1) theories con=ne!), and so should SSB. It is an interesting, open conjecture which we would like to make here, that even moderate ˜B =elds tend to enhance SSB, e.g. by enhancing the density near = 0 of the Dirac operator eigenvalues and thus, according to the Banks–Casher criterion, the > condensate.

3 potentials, quark masses, and weak transitions 18. QCD inequalities for QQ The main application of the techniques developed above is to obtain inequalities between directly measurable quantities such as hadron masses. However, there are several calculational approaches to QCD such as the potential model for heavy quarkonia [178–181], chiral perturbation theory for the low lying mesonic sector [182], and QCD sum rules [93,94]. Each of these schemes depends on a few input parameters and makes many predictions. Applying the techniques of QCD inequalities to these input parameters could therefore yield a very large body of suggestive results. The area in which most research along these lines has been done (in a large measure prior to, and independent from, the introduction of QCD inequalities in 1983) is that of potential models. The point is that general properties of the potential such as convexity of V (r) or other features, yield a wealth of information concerning the level ordering, thanks to the work of Martin and collaborators [24,183,184]; Lieb [18]; Fulton and Feldman [185]; and others. > systems is lower than An early important observation was that the P-wave excitation in cc> (or bb) the =rst radial excitation, and in general Enr +1;l ¿ Enr ;l+1 . This deviation from the famous degeneracy in the pure Coulombic case is related to the fact that the QCD potential VQCD (r) does not satisfy ∇2 VQCD = 0 but rather ∇2 VQCD ¿ 0 [185]. Also the assumption of a monotonically increasing, dV=dr ¿ 0, convex potential, d 2 V=dr 2 ¡ 0 allowed Baumgartner, Grosse and Martin [184] (BGM) to prove En; l 6 En−1; l+2 . In addition if d 3 V=dr 3 ¿ 0 (or if e−V has a positive Fourier transform), then the baryon mass relations Eqs. (5.11a) and (5.11b) can be proved (see [18]). These relations, which prescribe the sign of deviation from linearity of masses in the decuplet or from the Gell–Mann–Okubo relation in the octet, were proven above only under the explicit additional assumption of 3avor symmetric wave functions, a point emphasized by Richard and Taxil [25], and by Martin et al. [186]. It is not clear at the present what are all the model independent statements about VQQ > (R) that can be proven. We would like to mention, however, the very elegant result of Bachas [187] on the convexity R−r R+r 1 V +V (18.1) V (R) ¿ 2 2 2 of V (R). Together with earlier work by Simon and Ya8e [188] which shows a monotonically increasing V (R), this promotes the BGM level ordering into a QCD theorem, which is con=rmed in the cc> system (and partially tested in the bb> system).

260


Fig. 14. The original Wilson loop (W ), its parts W1 and W2 , and the re3ection paths used in proving Eq. (18.3).

Let us next sketch the proof. The static potential is de=nted in terms of a rectangular Wilson loop W [77] in the (t;˜r) plane of height T → ∞ and width R: 1 V (R) = lim − lntr U (W ) ; (18.2) T →∞ T with U (W ) the ordered product of Ulinks around the loop W . We have the path integral representation d(U ) tr U (W ) links ; (18.3) tr U (W ) = links d(U ) with d(U ) = d[U ] exp(−1=g2 ) tr Up the positive lattice measure. Let us denote by W1 ; DW2 the parts of the original path W to the left and to the right of a hyperplane perpendicular to # ), with the plane of W indicated by the broken line in Fig. 14. We can write W = W1 (DW 2


261

# , the line reversed version of the re3ected path DW2 . Obviously U (x) DW ˜ = U † (x) and thus 2 tr U (W ) = tr[U (W1 )U † (DW2 )]. We can compare tr U (W ) with the corresponding expressions for the symmetric Wilson loops of size R + r; R − r obtained by joining W1 and DW1 or W2 and DW2 . By dividing the d(U ) integration into variables to the left of, right of, and on the re3ection plane, the expectation values of tr(U (W1 )U (DW1 )) and tr(U (W2 )U (DW2 )) can be written as perfect squares, e.g. #1 )) = tr(U (W1 )U (DW #2 )) = tr(U (W2 )U (DW

don tr

dR U (W1 )

don tr

†

dR U (W1 )

†

dL U (W2 )

dL U (W2 )

:

Whereas the original tr U (W ) of Eq. (18.3) can be written as † #2 )) = don tr tr(U (W1 )U (DW : dR U (W1 ) dL U (W2 ) #2 )tr U (W1 )U (DW #1 ) ¿ |tr U (W2 )U (DW #1 )|2 . The The Schwartz inequality implies tr U (W2 )U (DW de=nition of V (R), Eq. (18.2), then readily yields the desired convexity (18.1). Similar arguments were applied [189] to the Eichten–Feinberg [140] representation of the tensor and spin–spin potentials. The inequalities obtained are basically in accord with expectations from scalar long range con=ning potentials [190]. The masses of the di8erent quark 3avors in the QCD Lagrangian (or Hamiltonian) H = H0 + Nf (0) > i i are the only explicit dimensional parameters. “Dimensional transmutation” generates, i=1 mi however, an additional scale 3QCD (≈ a few hundred MeV). Thus, unlike QED with rBohr (me )−1 (0) and Coulombic binding me , we expect that scaling all quark masses m(0) changes physi → cmi ical masses or inverses of physical lengths by less than a factor c. As we next argue, the simple linear dependence of HQCD on m(0) indeed restricts the variation of mass parameters as a function i (0) of mi . In quantum mechanics the ground state energy E (0) () is a convex function of any set of parameters ˜ , and ˜ = that the Hamiltonian depends upon linearly [191]. If H (˜) = H0 + i Hi = H0 + ˜ · H (0) ˜ (0) (0) 0˜ + (1 − 0)˜, then E () 6 0E (˜) + (1 − 0)E (˜). [This result follows immediately from H (˜) = 0H (˜) + (1 − 0)H (˜) by taking expectation values in (0) (˜), the ground state of H (˜), and using the variational principle.] PC We wish to apply this result to the masses m(0) ij (J ) of ground state mesons with di8erent 3avor and and Lorentz quantum numbers. We are hindered by the fact that we are not free to vary m(0) i by the need to subtract the vacuum energy. However, in a large Nc approximation where qi q>i do not annihilate and e8ects of closed quark loops are neglected, it is meaningful to describe each c (0) sector mij by a separate Hamiltonian Hi—> = HQCD + mi > i i + m—> > j jc . The knowledge of the masses of the 0− − ; 1− − ; 2++ ; : : : 3avor multiplets can then be used to constrain ratios of quark masses like

262


(0) (0) (0) R = (m(0) c − ms )=(ms − mu ) [192]. The result obtained is consistent with the estimates of Gasser 21 and Leutwyler [182], and corresponds to a lower bound m(0) s ¿ 80–100 MeV. Over the last twenty years there has been an ongoing e8ort to address in a systematic way the low energy sector of hadronic physics, and in particular the sector containing light quarks only via chiral perturbation theory (PT). The idea is to incorporate SSB, the Goldstone pions and current algebra, and the ensuing low energy theorems via e8ective Lagrangians [195,196] which will manifest the desired symmetries and which are written in terms of the pionic =elds only. Then one uses these chiral Lagrangians to perform a systematic expansion in the external momenta, and=or the pion mass divided by some “hadronic scale” usually taken to be 4$f$ . These e8ective Lagrangians contain a series of terms L1 ; L2 ; : : : ranked according to the number of derivatives appearing in each term. One then can also systematically compute higher loop processes. A particularly simple and elegant form of such an e8ective Lagrangian is the Skyrme model [197], which even incorporates the nucleon as a soliton state. In principle all the terms in the e8ective Lagrangian are computable from QCD. In practice they are often =xed by =tting some low energy data. In any event, QCD inequalities can constrain the range of these parameters. While no systematic program of this kind has been completed, some steps have been taken by Comellas et al. [198]. By expressing the currents in the inequality relating vector and pseudoscalar currents as a PT expansion in the pionic =eld, bounds on parameters in the e8ective Lagrangian were obtained, and are well satis=ed. Over the last decade the heavy quark approximation was often used in connection with Qq> or Qqq systems. This is based on the realization that in the in=nite mQ limit the heavy quark simply becomes a static color source leading to some universality relations, heavy quark symmetry, and a systematic expansion in inverse powers of mQ [199 –201]. The Witten pseudoscalar mass inequalities > ¿ m(q; q)+m(Q; > become trivial in this limit where the QQ> system is essentially such as 2m(q; Q) > Q) 2 Coulombic, with in=nite binding 0s mQ =r. Despite an interesting e8ort on the part of Guralnik and Manohar [202], it is not clear [203] how this can be amended to yield useful inequalities. Precise information on non-perturbative QCD parameters of weak decay is of particular importance and may decide the fate of the Standard Model with the three generation KM scheme [204]. It is worth pointing out some QCD motivated inequalities between such matrix elements [205]. Let us consider the K → $$ decay. In addition to the standard left-handed four fermion operators [say A = s(x) > > Lu(x)u(x) Ld(x), with L = 1 − 5 the left projection operator], we can extract from “penguin diagrams” mixed terms of the form B = s(x) > > Lu(x)u(x) Rd(x), with R = 1 + 5 . After using soft pion and current algebra techniques to reduce one pion we need to evaluate K|A(or B)|$ matrix elements. These latter matrix elements can be related to the asymptotic limit when |x − y| and |y − z| → ∞ of the three-point function 0|s(x) > > 5 u(x)A(y)u(z) 5 d(z)|0. The latter has the path integral form K|A|$ = d(A) tr[ 5 SA (x; y) (1 − 5 )SA (y; z) 5 SA (z; y) (1 − 5 )SA (y; x)] ;

21 It is amusing to note that the relatively “large” j =j ratio (as compared with previous theoretical estimates [193]) recently found in high precision experiments on CP violating kaon decays [194] do indeed suggest a relatively low m(0) s 80 MeV. The fact that we have so far been able to obtain only the above modest lower bound, rather than, say, (0) m(0) value s ¿ 190 MeV, makes it easier to accomodate the measured j =j in the standard model. Also the smaller ms would favor the stability of strange quark matter [193,194].


263

Fig. 15. The unique “8” pattern of contraction relevant to the K|B|$ ¿ K|A|$ inequality.

corresponding to the unique contraction in Fig. 15. A similar expression with the second 1 − 5 → 1+ 5 applies when A → B. We have assumed a 3avor symmetric limit and used the same propagator SA for all the quarks. Using 5 SA (x; y) 5 = S † (y; x), the hermiticity of the euclidean , and ( 5 )† = − 5 , we can show that the mixed L–R expression (case B) has a path integral with an integrand which is an absolute square. It is therefore larger than the expression for the “pure” L–L case (A) and allows the conclusion that K|B|$ ¿ K|A|$ : While the soft pion limit and 3avor symmetry assumed in the derivation considerably weakens this result we do still =nd it interesting and suggestive. 22 19. QCD inequalities beyond the two-point functions Most of the preceding sections, and of the work on QCD inequalities to date, has focused on euclidean two-point functions. These correlation functions are suQcient for obtaining hadronic masses via the spectral representation. An obvious advantage of two-point functions is that the path integrals expressing them [Eqs. (8.10a) and (8.10b)] contain products of quark propagators SA (x; y) between the same points x and y. Thus it is easy to prove positivity of certain combinations, e.g. tr[SA† (x; y)SA (x; y)] and the ensuing inequalities between the integrands of the path integrals for various two-point correlation functions. These algebraic inequalities which are true “pointwise” for each external A (x) con=guration do survive the path integration with the positive normalized measure d(A). Can we =nd inequalities also for three-, four-, and higher point euclidean correlation functions? Consider a generic four-point correlation function Ja (x)Jb (u)Jc (y)Jd (v), with a; b; c; d referring to Lorentz and 3avor quantum numbers. By appropriate contraction we obtain an expression for 22

The B matrix element with the “8”-type construction is one of the contributions to j . Since QCD inequalities are generally not operative when we have 3avor disconnected contractions as in the contribution of the “eye” of Fig. 16, the sign of this very important quantity cannot be =xed by QCD inequalities alone.

264


Fig. 16. The “eye” contraction relevant to the K 0 |A|0 matrix element.

correlation functions of the following general form, where for simplicity we assumed degenerate quark 3avors and hence the same propagator SA : Ja (x)Jb (u)Jc (y)Jd (v) = d(A) tr[Oa SA (x; u)Ob SA (u; y)Oc SA (y; v)Od SA (v; x)] : (19.1) All propagators SA in the path integral refer to di7erent pairs of points and it is not obvious how to construct positive de=nite combinations for each external A (x) con=guration separately. However, if we wish to study hadronic properties beyond the mass spectrum such as couplings, scattering amplitudes, weak interaction matrix elements, or wave functions, we need to consider more than just two-point correlation functions [207]. In particular correlation functions of the form given in Eq. (19.1) have been used in order to study the charge distribution of mesonic states. Let the current Ja† = > i (x)Oa j (x) ;

(19.2)


265

with x = (−T; ˜0), create at the origin, at some remote past instant, a quark and antiquark of 3avors i; —> in a speci=c Lorentz state. As the system evolves under the QCD Hamiltonian it settles into the wave function of the ground state meson in this channel. Speci=cally, we can infer from the spectral expression Eq. (8.3) that after time T the components of excited states in the wave function of the system are suppressed relative to the ground state amplitude by e−T \m , with \m the mass gap between the ground state and the =rst excited state. If we wish to =nd the relative separation of the quarks in the ground state, we can probe the system again with two external currents Jb (u)Jd (v) at u = (0;˜r1 ); v = (0;˜r2 ), with ˜r1; 2 = (˜R ±˜r)=2 and ˜r the relative separation. The currents Jb ; Jd should refer to the 3avors i and j of the quarks, respectively J b = > i Ob i ;

Jd = > j Od

j

;

(19.3)

and we will take Ob = Od = O. Finally, in order to obtain the complete gauge invariant correlation function the two quarks are propagated back to the origin where they are annihilated at y = (T; ˜0) by the current Jc =Ja† . Various attempts have been made to measure, via lattice Monte-Carlo calculations, charge distributions (or form factors) for the ground state mesons [208,209]. In the following we will focus on the particular case of the pion, i.e. using Ja† = > u

5 d (x)

Jb = > u O u ; we de=ne

F(t;˜r) =

= Jc ;

Jd = > d O

d

;

˜R ˜R ˜ r + −˜ r + † Ja (T; ˜0)Jd 0; d R 0 Ja (−T; ˜0)Jb 0; 0 : 2 2 3˜

(19.4)

This function has the following general properties [105]: 1. F(T;˜r) 6 F(T; ˜0), i.e. the “charge density” is maximal at the origin. 2. The Fourier transform (i.e. the “form factor”) L(T; p ˜ ) = ei˜p·˜r F(T;˜r)

(19.5)

is positive for all T and p ˜ values L(T; p ˜)¿0 :

(19.6)

Evidently (2) implies (1): F(t;˜r) is also the Fourier transform of the positive L(t; p ˜ ), and hence has 3 ˜ at the origin (˜r = 0). its maximum value L(t; p ˜)d p In order to prove (2) let us consider the four-point correlation function F = d(A) tr[ 5 SAi (x; u)OSAi (u; y) 5 SAi (y; u)OSAi (u; x)] =

˜ Ai (y; u)]† SAi (y; u)OSAi (u; x)} ; d(A) tr{[SAi (u; x)]† O[S

where we have used the “charge conjugation” property O˜ =

5O 5

= ±O = ±O† :

5 SA (x; y) 5

(19.7) = SA† (y; x). O˜ is de=ned as (19.8)

266


Fig. 17. Illustration of how as ˜r → 0 and u → v (and the diamond-like con=guration degenerates into a vertical line), we tend to get, due to the smoothness of the A con=guration, similar propagators, and hence a monotonic decrease of r|) with |˜r|. $ (|˜

Using u; v = (0;˜r1; 2 ) we take the Fourier transform with respect to ˜r =˜r1 −˜r2 , assigning momentum p ˜ (−˜ p) to the quark (antiquark) at ˜r1 (˜r2 ). Exchanging the order of the d(A) and d˜r1 d˜r2 integration the Fourier transform can be written as L(T; p ˜ ) = d(A) d˜r1 d˜r2 ei˜p·(˜r1 −˜r2 ) tr{S(y; u)OS(u; x)[S(y; v)OS(v; x)]† } =

2 i˜ p·˜r1 S(y; u)OS(u; x) ; d(A) tr d˜r1 e

(19.9)

and the manifest positivity of the integrand yields the desired result L(T; p ˜ ) ¿ 0. While F(t;˜r) and L(t; p ˜ ) are well-de=ned, gauge invariant, and, in principle, measurable quantities, the suggestive interpretations as “charge density” and “form factor” are more heuristic. In particular it is not evident from Eq. (19.4) why F(T;˜r) is positive de=nite as T → ∞, which should be the case if we interpret F(T → ∞;˜r) as | $ (˜r)|2 , with $ the “pion wave function”. Recall, however, that, as emphasized in the conclusion of Section 8, the positivity of norms [and spectral weight functions in Eq. (8.3)] emerges only after the d(A) integration has been performed and need not be manifest for each external A (x) con=guration separately. As ˜r2 approaches ˜r1 , the propagators S(y; v) and S(y; u) [and likewise S(x; u); S(x; v)] in Eq. (19.1) connect one vertex to two nearby points. For a typical smooth A (x) con=guration we expect S(y; v) → S(y; u) and the mixed products will gradually become squares (see Fig. 17). This suggests that F(T;˜r) is not only maximum at ˜r = 0 but is also monotonically increasing towards ˜r = 0. This property does not hold for all gauge con=gurations, but only after averaging, since we expect smooth A (x) con=gurations to dominate in d(A). Thus a proof of the conjectured monotonicity of F(T;˜r) requires understanding the correlation between A (x) along neighboring paths


267

and would depend on the speci=c form of the Yang–Mills action SYM . This “Ferromagnetic” action ˜ and ˜B prefers small of the gauge =eld A over a small region [gradients of A yield E variations 4 ˜2 2 and SYM = d x(E + ˜B )]. If we identify F(T = ∞;˜r) with the pion’s charge density, then it is amusing to note that the monotonicity of the ground state wave function can be proven in nonrelativistic quark models [210] when the potential is purely attractive. The fairly simple argument employs the spherical rearrangement technique [211], showing that given any trial wave function we always lower the energy by “rearranging” its values into a radially monotonically decreasing set. The present discussion of four-point correlation functions applies only to pseudoscalar currents J5 (x)J5 (y). Indeed it is precisely for the spin singlet S-wave “pion” state that all components of the nonrelativistic quark model potential—the con=ning linear part, the Coulomb force at shorter range, and the very short range, hyper=ne color magnetic interactions—are attractive. 20. Summary and suggested future developments We have presented above many inequalities for hadronic masses and analyzed the possible theoretical and phenomenological aspects. Many results follow essentially from the positivity of the measure in the functional path integral for euclidean correlation functions in QCD (or other vectorial theories). This may be on occasion complemented by fairly mild assumptions on the number of light degenerate 3avors or the Zweig rule (large Nc ) suppression of qq> annihilation. These results include the Weingarten mass relations, the Vafa–Witten theorem, m$+ ¿ m$0 , and Witten’s inter3avor relation for pseudoscalars. The fact that so many results, which have far-reaching implications, can be proven with such minimal input, is truly fascinating. We believe, however (and will try to make slightly more concrete conjectures later), that many more results would follow if we appeal to the speci=c form of the QCD action and in particular to its “ferromagnetic” character. A large class of meson–meson and baryon–baryon mass relations follow from the 3avor independence (apart from the explicit mass terms) of HQCD . This allows us to prove operator relations for HQCD restricted to di8erent 3avor sectors, which from the mass relations follow—though only for 3avor symmetric wave functions. Under fairly general assumptions we can also use Hamiltonian variational techniques to prove detailed baryon–meson inequalities. All of the above together with some general level ordering rules for the di8erent J PC states (again motivated by QCD inequalities applied to qq> potentials) can serve as extremely useful “Hund like” rules of thumb in the hadronic domain. This in turn can restrict the masses (or J PC quantum numbers) of new 3avor combinations or radial excitations. In many cases the inequalities are more sophisticated versions of relations suggested by a naive quark model. Eventually we hope that much of the vast information available on hadronic parameters (including scattering, wave functions, etc.) will be constrained by such inequalities. On the more theoretical side the generic properties of QCD and all vector QCD-like theories should be analyzed. The constraints imposed by the inequalities on composite models of quarks and leptons, together with the anomaly matching conditions, are particularly interesting. The fermion– (0) boson inequalities, m(0) F ¿ mB , exclude the protection of small masses for composite quarks and leptons via an unbroken global axial symmetry, in all cases when we have vectorial, QCD-like,

268


underlying dynamics. Such constraints are avoided by going to chiral gauge models or (supersymmetric) models with scalars where the measure positivity d(A) is lost. (In passing we note that large classes of scalar and scalar plus fermion theories are ruled out by “triviality” diQculties [212].) We have often referred to the euclidean correlation function approach to deriving the QCD inequalities as “rigorous” and to that based on the Hamiltonian variational approach as more “heuristic”. It should be emphasized that this does not re3ect any true objective distinction. Clearly the Hamiltonian and Lagrangian approaches to classical mechanics, and to quantum =eld theory, are equivalent and equally rigorous (or not) depending on the practitioner. The Hamiltonian variational approach allows inputting various approximations and=or physics intuition much more readily, however. These inputs can motivate certain restrictions on the wave function(al)s allowed. An example is our restriction in the nonperturbative derivation of the baryon–meson inequalities (Section 6) to baryonic con=gurations with only one junction point (see Fig. 4). Clearly any restrictions on the wave function(al)s pushes the energy of the ground state higher. Hence, in particular, our restricted baryon is not the true ground state baryon. If, however, we can assume that any network containing some extra junction (and in this case, also extra anti-junction) points corresponds to “massive” components in the wave function(al) (with essentially extra virtual baryon–antibaryon pairs), then such components are likely to be small in the true ground state, and we can therefore fairly safely neglect them. Similar comments apply to the neglect of possible sextet internal 3uxons in our ansatz functionals for the quadri- and pentaquarks (see Appendix F). Much of the intuition for hadronic physics stems from naive quark models. In these, the baryon– meson wave functionals are approximated by qcon qcon qcon and q>con qcon wave functions. Here qcon refers to “constituent quarks”, namely some e8ective quasiparticles obtained when some short distance modes are integrated. Clearly this concept would be on =rmer ground if we could indeed show (0) that the mass of the constituent u and d quarks—which in the m(0) u = md = 0 axial SU(2) symmetry limit is purely dynamical—is indeed generated by physics operating on distance scales smaller than the radius of the baryon or meson. 23 Because the constituent quarks are extended, complex entitities, the “potentials” acting between them are in general rather complex—with spin-, 3avor-, and possibly energy-dependence and nonlocality. Still, the naive expectations for potentials generated via gluon exchange seem to lead to very successful predictions. Overall the naive quark model, which predated QCD by almost a decade, still provides more insight and results than any of the more sophisticated e8ective Lagrangian approaches. Justifying its usage from =rst principles would therefore constitute a major triumph and is certainly a worthwhile e8ort. In our discussion of the QCD inequalities we have encountered on several occasions [e.g. in discussing the pion mass and wave function, the inter3avor relations, and the positivity of EM and more general vectorial interaction energies (see Sections 9 and 12 and Appendix G)] a remarkable coincidence between the “naive” quark model and rigorous QCD inequalities. Thus the e8orts to extract constraints on the interquark potentials from QCD, and then to utilize this rather broad class of potentials allowed to derive level ordering theorems and=or baryon–baryon mass relations (see Appendix B), is quite worthwhile.

23

The celebrated example of the extended yet most useful Cooper pair quasiparticles indicates that this may not be a necessary condition.


269

Even when we have mass dependent qcon q>con interactions, the system does retain its symmetry under qicon ↔ q>con j , and one may thus wonder if just this single feature, abstracted from the constituent quark model, is suQcient to supply the 3avor symmetry assumption which was the missing link in our proof of meson the inter3avor relations in Section 5. Unfortunately this is not the case. Essentially (0) the operation i(0) ↔ > j in Section 5 corresponds to exchanging the “fundamental”, pointlike, 3avor carrying, entities inside the extended constitutent quarks. For suQciently heavy quarks the “cloud” of > and possibly ss> and gluons) are universal and 3avor-independent. light degrees of freedom (uu; > dd; However, the clouds around s and u quarks may di8er signi=cantly. Hence the above symmetry assumption cannot be justi=ed in this case when one quark is heavy, and indeed the inequality 2mK ∗ 6 m@ + mD , which relies on an s(0) ↔ u(0) exchange, does marginally fail. Clearly the c ↔ u and b → u exchanges are even less justi=ed, yet since the hyper=ne mass splittings are very small (≈ 1=mc or 1=mb ), the naive, spin-dependent, potential model derivation of the inequalities holds. The rigor—and corresponding paucity of the euclidean correlation function QCD inequalities— re3ects, in our view, the lack of intuition as to which Aa (x) =eld con=gurations are more important in the functional path integral. 24 This makes justifying the approximation of keeping certain =eld con=gurations more diQcult, and is more of a handicap than a virtue. By developing such an intuition akin to that inspired by the quark model or strong coupling approximation (which suggested the more important components of wave function(al)s in the variational approach), we could obtain many more heuristic, yet very important and useful inequalities. In delineating directions for future research on the subject of this review, an obvious goal is to try and =nd more rigorous proofs for likely correct inequalities, whose present derivations are lacking. These inequalities would include the detailed, 3avor-dependent baryon–meson mass inequalities; relations between masses of di8erent mesons such as ma1 (1+ ) ¿ m@ (1− ); and mass relations for glueballs such as m0++ gb 6 (mass of any glueball). We believe, however, that there are additional, promising, richer avenues for further research. These include two general (although unfortunately not completely well de=ned) conjectures that we would like to make next, and the application of the QCD inequality techniques to observables other than hadron masses. 20.1. A conjecture on the Nf dependence of the QCD inequalities +

−

) We have seen that many of the QCD inequalities such as mN ¿ m$ and m(0 ¿ mu(0d> ) are related ud> to observed symmetry patterns, namely the spontaneously broken global axial and the conserved vectorial symmetries. The correlator inequalities hold for arbitrary Nf . Indeed for Nf degenerate quarks, Nf only appears in the positive determinantal factor via [Det(D= + m)]Nf . In particular, all the inequalities hold in the “quenched approximation” in which Nf = 0 and the determinantal factor disappears altogether. It is well known that in the quenched limit, lattice QCD calculations simplify enormously and one has (within this approximation!) quite reliable results.

24

This is clearly a subjective statement, as such intuition might have been developed by lattice gauge calculations. Indeed the positivity of contributions of =eld con=gurations to the pseudoscalar propagators was known to practitioners in this =eld prior to the advent of QCD inequalities.

270


The QCD inequalities, for example mN ¿ m$ and m@ ¿ m$ , are often satis=ed by a large margin. If we had, however, a systematic trend for the inequalities to become weaker in the sense, say, that mN =m$ , m@ =m$ decrease as Nf increases, that could furnish very useful information such as mN mN 6 : (20.1) m$ m$ expt

quenched

We would like to conjecture that this is indeed the case. Our motivation is the likely decrease with increasing Nf of the “strength” of the SSB as manifested via the magnitude of qq > (measured in an appropriate way). Indeed increasing Nf weakens—via enhanced screening—the qq> gluon exchange interactions and consequently the expected value of qq. > Therefore we expect that the ratio mN =m$ , which re3ects in some sense SSB, should decrease as well. Consider > A = tr SF A , with > A evaluated for a particular gauge background A. We can write 1 m = tr SF A = ; (20.2) 2 (i (A) + m) (A) + m2 j j j j where in the last expression we paired together the 5 conjugate, non-vanishing eigenvalues ±ij of D=. For the m → 0 limit of interest, the Lorentzians become >-functions and hence we have qq > A = 0 ↔ lim @A () = 0 ; →0

and so is @(), the density obtained after the 1=Z d(A) averaging. Let d0 (A) = d[A (x)] exp[ − SYM (A )] be the Nf = 0 measure. The conjectured decrease of > when Nf = 0 → Nf = 0 amounts to [l2 (A) + m2 ] [j2 (A) + m2 ]−1 6 [l2 (A) + m2 ] [j2 (A) + m2 ]−1 ; l

j

0

l

0

j

0

(20.3) 2 with f0 indicating averaging with the d0 (A) measure. Since the factors l (l + m2 ) and j (j2 + m2 )−1 appearing in the conjectured inequality (20.3) are, respectively, monotonically increasing (decreasing) with each j , the conjecture is very suggestive. Indeed for any two positive functions of one variable f(x) and g(x) which are monotonically increasing (decreasing), fg ¿ fg. However, as pointed out to us by Kenneth [213], this inequality does not generally hold for functions of many variables, and measures d(1 ; : : : ; N ) can be constructed for which the conjectured inequality (20.3) is reversed. This inequality is therefore heuristic and depends on additional assumptions. Recalling our comment on functions of one variable, the inequality would apply if there were one dominant variable in the measure d[A(x)]. Color con=nement and asymptotic freedom suggest that the overall scale R of the A (x) 3uctuations could serve in such a role. The enhanced weight of larger 3uctuations coupled with the expected smallness of the corresponding i (A) indicates that this may indeed be the case. Changing Nf changes the 1 function and hence, for =xed g2 , also 3QCD . This in turn implies that the mass=length scales used in two lattice calculations with di8erent Nf values should be appropriately changed to allow for meaningful comparisons.


271

√ In general hadron masses, , with the string constant (≈ the coeQcient of the linear potential in heavy quarkonia), and qq > 1=3 scale linearly with 3QCD and comparing ratios of such quantities for di8erent Nf values is straightforward. However, m$+ ≈ f$ m$ ≈ 3QCD m0 , with m0 the (u; d) bare quark mass. Hence instead of Eq. (20.1) we should use m N f$ mN f$ 6 : (20.4) m2 m2 $

$

quenched

In conclusion we recall yet another heuristic supporting argument for the conjectured decrease 3 of qq=(3 > QCD ) , i.e. of the quark condensate, with Nf . It is the restoration of Q5 symmetries in supersymmetric QCD theories [214] at moderate Nf =Nc ratios, for which asymptotic freedom (and even con=nement) may still hold. Since realk QCD is not supersymmetric, and further, since QCD inequalities may not apply for SUSY theories, the signi=cance of this in the present context is not clear. 20.2. Conjectured inequalities related to the “ferromagnetic” nature of the QCD action The monotonic pion (con=guration space) wave function was motivated in the conclusion of Section 19 above by the ferromagnetic nature of the Yang–Mills action. In some ferromagnetic spin systems this feature is embodied in a rigorous set of GriQth’s inequalities [215] for the spin correlations. The analogous A correlations are not gauge invariant but some gauge invariant versions can be conjectured, of which we will mention just a few. An E1a E2a (x) excitation at 25 x (with 1 and 2 ˜ a , the color electric =eld) and an E1a E2a (y) excitation at y correspond intuitively spatial indices of E to parallel “spins” (the corresponding plaquettes at x and y in the euclidean lattice are indeed parallel). Likewise B1a B2a (x) and B1a B2a (y) are “parallel”. However, E1a (x)B2a (x) and E2a (y)B1a (y) are not “parallel”. The preference of parallel con=gurations and the stronger positive correlation between such con=gurations suggests therefore that

E1a (y)E2a (y)E1a (x)E2a (x) ¿ E1a (y)B2a (y)E2a (x)B1a (x) ;

B1a (y)B2a (y)B1a (x)B2a (x) ¿ E1a (y)B2a (y)E2a (x)B1a (x) ; and in particular E1 E2 (x)E1 E2 (y)B1 B2 (x)B1 B2 (y) ¿ |E1 B2 (x)E2 B1 (y)|2 :

(20.5)

which has been suggested by Muzinich and Nair [21,153] along with many other inequalities. Since E1 E2 (or B1 B2 ) and E1 B2 acting on the vacuum creates 2++ and 2−+ states, respectively, the last inequalities suggest that the lowest lying 2++ glueball is lighter than the lowest 2−+ state: (0) m(0) 2++ 6 m2−+ :

(20.6)

We believe that Eq. (20.5), many other Muzinich–Nair relations (some of which involve J dependence of masses as well), and other relations yet to be discovered are true, and furnish tests not just of the measure positivity in QCD, but rather of the ferromagnetic nature of its action. 25

i.e. the state of the system obtained immediately after E1a (x)E2a (x) operates on the vacuum. Here a = 1 : : : 8 is a color index, and 1 and 2 indicate the components (E1 = F10 , etc.).

272


20.3. Inequalities for quantities other than hadronic masses The inequalities between two-point correlation functions and the variational Hamiltonian techniques naturally tend to yield inequalities between masses, rather than between other hadronic observables. We have seen, however, in Sections 16 and 19, and at the end of Section 18, other applications of the QCD inequalities techniques involving scattering lengths, form factors, and weak matrix elements. In this last subsection we would like to consider yet another quantity, namely high energy hadronic cross sections. These attracted much attention in connection with “Regge pole” exchanges in the crossed t-channel. In particular, the approximate “quark counting” suggested relation $N ≈ 23 NN was one of the early indicators for the relevance of the quark model. In retrospect asymptotic cross sections may re3ect physics which is quite di8erent from that controlling meson and baryon masses. Hence while the naive quark counting relation mB ≈ 32 mM has been “transformed” here to the baryon–meson mass inequalities, it is not clear that we have analogous inequalities for hadronic cross sections: NN & 32 MN :

(20.7)

However, this is precisely what we would like to conjecture here (along with the more detailed 3avor dependent variants). We have attempted to motivate such a conjecture by using intersecting chromoelectric 3uxons as a model for high energy cross sections [216] and the stringy trial wave functionals of Section 6. We note that if C1 E12 d 3 x and C2 E22 d 3 x roughly re3ect the masses of hadrons 1 and 2 extending ˜1 · E ˜ 2 d 3 x generates the Born scattering amplitude, we expect some relation of over C1 ; C2 , and E 26 the form 12 .

C m1 m2 : 34QCD

Clearly the case of the Goldstone, almost massless pion is again very special. 27 The arguments of Section 14 could be formally extended to the exchanges pertinent to high energy cross sections. This would then suggest relations like ($$ )(NN ) ¿ ($N )2 , which unfortunately would be diQcult to test. Physics similar to that motivating the QCD inequalities could be used at many other length scales, and not only for underlying composite models. Thus some variant of the Schwartz inequality for propagators (12.4) may have implications for the frequency of occurrence of like-sex non-identical twins, and the more heuristic inter3avor inequalities of Section 5, can suggest many inequalities between bindings of polar molecules. We discuss these applications in Appendix H. We believe that much more can and will be done on the subject of inequalities, both in and out of QCD.

26

12 ∼ m1 m2 is expected if the total cross section is dominated by a tensor particle exchange, which in turn dominates (in the sense of vector meson dominance) the graviton couplings [31]. 27 The smallness of the pion mass relative to, for example, the @, stems in quark models from large short range hyper=ne attraction. It is quite surprising that this hardly results in a smaller sized pion and in $N 6 @N (the “rho-pi puzzle” [217]).


273

Note added in proof Several papers question aspects of the theorem regarding the absence of spontaneous parity breaking [243]. In a most recent paper, Ji shows that the theorem does not apply [244]. E() of Eq. (13.4) above indeed increases with real . However, this is the case both when the parity violating operator A develops a VeV and when A = 0, i.e., both when parity is or is not broken spontaneously. Acknowledgements It is often diQcult to trace the precise inception of any project. The motivation of S. Nussinov to prove a baryon–meson inequality, using lattice techniques, stemmed from a well de=ned origin. It was the summary talk in a lattice workshop (held in the summer of 1982 at Saclay) by the late Claude Itzykson. Itzykson, a pioneer of lattice QCD and one of the =nest mathematical physicists of our generation, commented on the lack of rigorous results in QCD. Indeed even now, seventeen years later, there are preciously few such results. The work of S.N. on QCD inequalities, and this review in particular, has been carried out at Tel Aviv University; Brookhaven National Laboratory; Los Alamos National Laboratory; Universities of Maryland, Pennsylvania, and Minnesota; MIT; and at Boston University, SUNY Stonybrook, and the University of South Carolina. The material in Section 16 and a portion of Section 14 is based on joint work with B. Sathiapalan, and Section 19 and a portion of Section 11 on work with M. Spiegelglas. S.N. would like to acknowledge the crucial help of E. Lieb at a very early stage, and his contribution to Section 19; the friendly and helpful correspondance with D. Weingarten and discussions with E. Witten; the encouragement of J. Sucher and S.L. Glashow; and the interest of the late Y. Dothan, W. Greenberg, R. Mohapatra, and J. Pati. The critical comments of A. Casher, E. Lieb, A. Martin, J.-M. Richard, P. Taxil, and E. Witten made S.N. realize some shortcomings of his earlier work. During the writing of the review S.N. enjoyed discussions with M. Cornwall, S. Elitzur, Y. Hosotani, O. Kenneth, H.J. Lipkin, A. Martin, I.J. Muzinich, V.P. Nair, and M. Shifman. We are particularly grateful to B. Svetitsky for many helpful comments and advice, and to him and to B. Sathiapalan for reading earlier versions of the manuscript. S.N. would like to acknowledge the very warm hospitality of the Institute for Theoretical Physics at the University of Minnesota, and in recent years the University of South Carolina, which helped him very much in writing the review. M.L. would like to thank S.N. for introducing her to the subject of QCD inequalities and for asking her to help with this review. The work of M.L. was supported by the Israel Science Foundation under Grant No. 255=96-1. Finally, S.N. would like to acknowledge a grant from the Israel Academy of Sciences. Appendix A. Lieb’s counterexample to Eq. (4.1) Following Lieb [18] we prove that for the case of m = ∞ and two-body potentials which are in=nite square wells: ∞; r ¿ r0 ; Vij (˜ri − ˜rj ) = Vi:s:w: (r) = (A.1) 0; r 6 r0 ;

274


the conjectured “convexity” relation E (0) (m; m; m) + E (0) (m; M; M ) 6 2E (0) (m; m; M )

(A.2)

is violated. To achieve minimal (and indeed even =nite!) ground state energies, the wave functions have to vanish when |˜ri − ˜rj | ¿ r0 for any quark pair. For heavy quarks this can be done with no kinetic energy penalty. Thus let us =x ˜r1 = 0 in all three wave functions (0) (m; m; m); (0) (m; m; M ); and (0) (m; M; M ), thereby disposing also of the overall translational degrees of freedom. For (0) (m; m; m) we could take all ˜ri = 0, achieving the minimal possible energy: E (0) (m; m; m) = 0. In (0) (m; m; M ) it is clearly advantageous to put also ˜r2 = ˜r1 = 0 so that the condition |˜r3 − ˜r2 | 6 r0 is satis=ed automatically once |˜r3 − ˜r1 | 6 r0 . Thus =nding E (0) (m; m; M ) reduces to =nding the minimal energy of the one-body Hamiltonian p ˜ 2 =2M + 2V (˜r), which for V = Vi:s:w: is actually the same as 2 h≡p ˜ =2M + V (˜r). On the other hand, E (0) (m; M; M ) is the minimal energy of p32 =2M + V (˜r3 )] + V (˜r2 − ˜r3 ) : (A.3) hˆ = [˜ p22 =2M + V (˜r2 )] + [˜ If V (˜r2 − ˜r3 ) were absent, hˆ separates into two one-body in=nite square well problems and ˆ = 2E (0) (h) : E (0) (h) However, since V23 ¿ 0 we have ˆ ≡ E (0) (m; M; M ) ¿ 2E (0) (h) = 2E (0) (m; m; M ) ; E (0) (h)

(A.4)

(0) r1 mMM (˜

and the desired inequality (A.2) is violated. Indeed = 0;˜r2 ;˜r3 ) has to vanish now in all of the region |˜r2 − ˜r3 | ¿ r0 in addition to the vanishing for r2 ¿ r0 ; r3 ¿ r0 . Incorporating this extra constraint reduces the allowed six-dimensional volume in (˜r2 ;˜r3 ) space from (4$r03 =3)2 to a fraction thereof. Obviously this will increase the kinetic energy from 2E (0) (h) to (2 + 0)E (0) (h), with 0 ≈ 1. By continuity we therefore expect Eq. (A.2) to fail already when we approach the limit m → ∞ and V (r) → Vi:s:w: (r), e.g. via V (r) = cn (r=r0 )n ;

n→∞:

(A.5)

n

Indeed for V (r) = r ; n ¿ 4, a counter-example to Eq. (A.2) was produced =rst [39]. In passing we note that exp[ − 1Vi:s:w: (r)] = V(r − r0 ) does not have a positive Fourier transform. This should be the case since we have, as indicated in the following appendix, Lieb’s theorem that the desired inequality holds when exp[ − 1V ] is positive semide=nite. Appendix B. Discussion of Lieb’s results for three-body Hamiltonians Following Lieb [18], we prove the inequality (A.2) when the potentials are 3avor independent: V (m; m) = V (M; M ) = V (m; M ) ≡ V ; and exp[ − 1V (˜x − ˜y)] positive semide=nite. The propagation over (imaginary) euclidean time 1 of the three-body system is given by X |e−1H | X , with X = (x1 ; x2 ; x3 )initial ; X = (x1 ; x2 ; x3 )=nal . We compare three systems with Ha = T1 (x1 ) + T2 (x2 ) + T2 (x3 ) + V2 (x1 ; x2 ) + V2 (x1 ; x3 ) + V1 (x2 ; x3 ) ;


275

Hb = T1 (x1 ) + T2 (x2 ) + T3 (x3 ) + V3 (x1 ; x2 ) + V2 (x1 ; x3 ) + V1 (x2 ; x3 ) ; Hc = T1 (x1 ) + T3 (x2 ) + T3 (x3 ) + V3 (x1 ; x2 ) + V3 (x1 ; x3 ) + V1 (x2 ; x3 ) : For X =

X ,

X |e

(B.1)

X |e−1H |X

Z1 (x) = is dominated, for 1 → ∞, by the lowest energy state 2 −1En |X = |X |n| e → |X |0|2 e−1E0 ;

−1H

(B.2)

n

where we use the completeness sum over energy eigenstates. Thus to prove the desired inequality E (0) (a) + E (0) (c) 6 2E (0) (b) it suQces to show that Z1 (a)Z1 (c) ¿ Z21 (b) :

(B.3)

In general the path integral, and Z1 in particular, is obtained by dividing the total evolution into many consecutive evolutions over small time steps, so that Z1 = lim Z1 (N ) = [(e−1T=N e−1V=N )N ] ; N →∞

with T and V the total one-body kinetic and two-body potential parts of the Hamiltonian H = T + V . Clearly it is suQcient to prove the inequality for each N . We next insert a complete set of X -space states between each pair of e−1T=N e−1V=N factors. This gives Z1 as a path integral. Each such path consists of three (spatially) closed polygonal paths X˜ 1 ; X˜ 2 ; X˜ 3 , with X˜ 1 consisting of the N + 1 points X˜ 1 (0) = X1 ; X1 (1=N ); X1 (21=N ); : : : ; X1 (N1=N ) = X1 (1) = X1 ; and likewise for X˜ 2 and X˜ 3 (see Fig. 21). Since each factor X1 (j)X2 (j)X3 (j)|e−1T=N e−1V=N |X1 (j + 1)X2 (j + 1)X3 (j + 1)

(B.4)

involves an evolution over an in=nitesimal “time” 1=N , the non-commutativity of the kinetic and potential parts of H is neglected. The kinetic single particle operators contribute to ZN1 (a) a factor of Fa = F1 (X˜ 1 )F2 (X˜ 2 )F2 (X˜ 3 ) ; and to ZN1 (b) and ZN1 (c) Fb = F1 (X˜ 1 )F2 (X˜ 2 )F3 (X˜ 3 ) ; Fc = F1 (X˜ 1 )F3 (X˜ 2 )F3 (X˜ 3 ) ; with Fi (X ) corresponding to Ti in Eq. (B.1) above. The potential two-body operators contribute to ZN1 a product of three terms depending on the three pairs of paths. Speci=cally, the contributions for Ha ; Hb , and Hc are ZN1 (a) : G2N (X˜ 1 ; X˜ 2 )G2N (X˜ 1 ; X˜ 3 )G1N (X˜ 2 ; X˜ 3 ) ; ZN1 (b) : G2N (X˜ 1 ; X˜ 2 )G3N (X˜ 1 ; X˜ 3 )G1N (X˜ 2 ; X˜ 3 ) ; ZN1 (c) : G3N (X˜ 1 ; X˜ 2 )G3N (X˜ 1 ; X˜ 3 )G1N (X˜ 2 ; X˜ 3 ) ;

276


where the Gi (X˜ k ; X˜ l ) are generated from Vi (Xk ; Xl ) via Gi (X˜ k ; X˜ l ) =

N

exp[ − (1=N )Vi (Xk (1j=N ); Xl (1j=N ))] :

j=1

For the particular case of V1 , the fact that e−1V1 =N is positive semide=nite ensures that the N -fold tensor product de=ning G1 (X˜ 2 ; X˜ 3 ) is also positive semide=nite. Collecting all terms and separating the d 3(N −1) X1 integrations over the N − 1 intermediate points X12 ; : : : ; X1N along the polygonal path X˜ 1 we have N 3(N −1) 3(N −1) ˜ X1 F1 (X 1 ) d X2 d 3(N −1) X3 Z1 (a) = d ×G2N (X˜ 1 ; X˜ 2 )F2 (X˜ 2 )G2N (X˜ 1 ; X˜ 3 )F2 (X˜ 3 )G1N (X˜ 2 ; X˜ 3 ) N 3(N −1) 3(N −1) ˜ X1 F1 (X 1 ) d X2 d 3(N −1) X3 Z1 (b) = d G2N (X˜ 1 ; X˜ 2 )F2 (X˜ 2 )G3N (X˜ 1 ; X˜ 3 )F3 (X˜ 3 )G1N (X˜ 2 ; X˜ 3 ) N 3(N −1) 3(N − 1) Z1 (c) = d X1 F1 (X˜ 1 ) d X2 d 3(N −1) X3 ×G3N (X˜ 1 ; X˜ 2 )F3 (X˜ 2 )G3N (X˜ 1 ; X˜ 3 )F3 (X˜ 3 )G1N (X˜ 2 ; X˜ 3 ) : De=ning next the N × N matrices V2N (X˜ 1 ; X˜ 2 ) = F1 (X˜ 1 )G2N (X˜ 1 ; X˜ 2 )F2 (X˜ 2 ) ; N ˜ ˜ V3 (X 1 ; X 2 ) = F1 (X˜ 1 )G3N (X˜ 1 ; X˜ 2 )F3 (X˜ 2 ) ;

(B.5)

(B.6)

we can write Eq. (B.5) in the concise form (ommiting the N in V N ) ZN1 (a) = tr(V2T · G1 · V2 ) ; ZN1 (b) = tr(V2T · G1 · V3 ) ; ZN1 (c) = tr(V3T · G1 · V3 ) :

(B.7)

We note that the positivity of F1 (X˜ 1 ) was implicitly assumed in taking the square root in Eq. (B.6). For the case of interest with T1 = p ˜ 12 =(2m1 ); X1 |e−1T |X1 is the probability for returning to the initial point in a Gaussian random walk after a time 1. F N (X1 ) is the probability that this happens for a speci=c path X1 with N − 1 speci=c intermediate steps, and hence is clearly positive. This positivity of the “heat kernel” can be generalized to other relativistic forms of the kinetic energy, for example T1 = p ˜ 12 + m21 . To complete the proof we note that Eqs. (B:7) de=ne a “scalar product” of V2 with itself, V2 and V3 , and V3 with itself. This can be most clearly seen by transforming to the basis in which G1


277

is diagonal with all diagonal elements real and positive. This change of basis leaves tr(V2T G1 V3 ) invariant but casts it in the form (V2 )0n (G1 )nn (V3 )n0 ; 0; n

which is bilinear in V2 and V3 and positive whenever two identical “V -vectors” (of length [3(N −1)]2 ) are used. Hence these scalar products satisfy the Schwartz inequality which is precisely the desired result (B.3). It has been noted by Lieb that the assumption of having only two-body potentials is rather restrictive and we can allow also genuine three-body potentials V (X1 ; X2 ; X3 ) as long as e−1V is positive semide=nite as a function of X2 ; X3 . Appendix C. Proof of Eq. (5.2) We prove Eq. (5.2) using a concrete basis of states in which it holds for all matrix elements. We will use the Hamiltonian version of QCD with a spatial cubic lattic (sites ñ, unit vectors Lˆ along the positive x; y; z axes). (See also Fig. 18) The QCD Hamiltonian is written as [4,218] g2 1 HQCD = tr(Eñ;ñ+Lˆ)2 + 2 tr(UUU † U † + h:c:) 2 2g plaquettes

ñ;L¿0 ˆ

+

Nf

†

i

(ñ)Uñ;ñ+Lˆ i (ñ + L) ˆ + h:c: +

i=1 ñ;L¿0 ˆ

Nf i=1

mi > i (ñ) i (ñ) + · · · :

(C.1)

ñ

It depends on the SU(3) “connection” matrices Uñ;ñ+Lˆ which live on lattice links between ñ and ñ + L, ˆ the canonically conjugate generators Eñ;ñ+Lˆ, and the fermionic spinors i (ñ) [and conjugate > (ñ)] at each lattice site. The =rst three terms are analagous to the E ˜ 2 ; ˜B2 and > (x)D= (x) terms in i the continuum Hamiltonian. We adopt the “strong coupling” basis in which all tr(Eñ;ñ+Lˆ)2 are diagonal. At each lattice site we specify the spinors i (ñ). The mass term is Nf

†

i ñ

(ñ) 0 mi i (ñ) :

(C.2)

i=1

˜ 3ux line going clockwise (or anti-clockwise) Each of the plaquette terms creates a minimal, closed, E around an elementary plaquette. This will, in general, change tr(Eñ;ñ+Lˆ)2 on each of the four adjoining links and hence gives rise to non-diagonal elements in the representation considered here. Also the third [ > (x)D= (x)] term changes the lattice con=guration by allowing the quark to “hop” to a neighboring site (dragging along a 3ux line), and by creating or annihilating ql q>l pairs at neighboring sites. We illustrate some of the matrix elements of HQCD on a small 3 × 3 two-dimensional lattice in Fig. 7. Each of the columns and rows is labeled by a complete lattice con=guration. In the matrix we indicate both the actual value of the matrix elements (above the dotted line) and the speci=c

278


Fig. 18. The three polygonal paths X˜ 1 , X˜ 2 , X˜ 3 consisting of the overall periodic propagation of the particles at X1 , X2 , X3 from “t”=1 = 0 to “t”=1 with Xi (1) = Xi (0).

terms of HQCD contributing to it (below the dotted line). We omit additional four-fold spinor indices for each of the occupied sites. Local gauge invariance n) i (˜

→ V (ñ) i (ñ) ;

> (ñ) → > (ñ)V † (ñ) ; i i Uñ;ñ+Lˆ → V (ñ)Uñ;ñ+LˆV † (ñ + L) ˆ ; Eñ;ñ+Lˆ → V (ñ)Eñ;ñ+LˆV † (ñ + L) ˆ

(C.3)

is respected by all terms of HQCD . It constrains the physically allowed states via Gauss’ equation Eñ;ñ+Lˆ = >(˜r − ˜rl )Ql ; (C.4) ±Lˆ

l

stating that the sum of all outgoing Eñ;ñ+Lˆ 3ux lines [which generate V (ñ)] vanishes at all lattice sites ñ except those ˜rl with external color charges Ql . (The Ql are due to quarks. Up to 2Nf quarks and 2Nf antiquarks are allowed at each site by Fermi statistics.) It is precisely via this Gauss condition that the speci=c (mesonic, baryonic, etc.) sector becomes relevant. Fig. 7 pertains to the Mud> sector. The simple con=gurations in Fig. 7 have just one u and one d> in the lattice. Eq. (C.4) then simpli=es into Eñ;ñ+Lˆ = >(˜r − ˜ru ) + >(˜r − ˜rd )† ; (C.5) +Lˆ

† > where ru(d) > are the locations of u(d), and ; are the corresponding color matrices.


279

Since HQCD commutes with the gauge transformations, the evolution of the state on the lattice, allowing for creation of qq> pairs at one lattice point and their subsequent separation and possible annihilation with quarks and antiquarks originating from other qq, > maintains the Gauss condition (C.4) [though in general not its simple “valence” version (C.5)]. Since the Gauss law constraints are common to all the mesonic sectors of di8erent 3avors, they do not interfere with the suggested operator relation Eq. (5.2). We start by showing that (5.2) is satis=ed when we switch o8 the quark creation and annihilation terms and then work our way gradually to the general case. (1) In the simple case where Mi—> consists of just the qi q>j quarks and not other pairs, it is easy to see that all the non-mass terms in HQCD have identical matrix elements in all Mij> sectors. Also the Gauss law constraints are the same. The matrix elements depend only on the U ’s and E’s and the generic quarks or antiquarks but not on the speci=c quark 3avors. Taking for concreteness ijkl = udsc, we have 3avor dependence only in the mass terms ( C denotes a charge conjugated spinor of an antiquark): Humass = mu d>

†

(˜rq )

0

(˜rq ) + md (

C †

= mc Hcmass s>

†

(˜rq )

0

(˜rq ) + ms (

C †

= mu Humass s>

†

(˜rq )

0

(˜rq ) + ms (

C †

= mc Hcmass d>

†

(˜rq )

0

(˜rq ) + md (

C †

) (˜rq>)

) (˜rq>) ) (˜rq>) ) (˜rq>)

0 0 0 0

C

(˜rq>) ;

(C.6a)

C

(˜rq>) ;

(C.6b)

C

(˜rq>) ;

(C.6c)

C

(˜rq>) :

(C.6d)

Since the same four terms appear in Hud> + Hcs> as in Hus> + Hcd> , Eq. (5.2) is valid in this valence approximation: Hus> + Hcd> = Hud> + Hcs> (subspace with two quarks) :

(C.7) > (2) The above argument is not a8ected by the creation=annihilation terms of all pairs xx> = uu; > dd; ss; > cc. > The created quarks or antiquarks cannot be Pauli blocked or annihilated by the valence quarks or antiquarks. Their e8ect, just like that of gauge =elds, is common to Hud> ; Hcs>, etc. (3) Finally, we turn on also pair creation=annihilation of the valence 3avors. We would then generate con=gurations such as qiv (u1 u> 1 ) : : : (xp x>p )q>vj in which u1 is at the same lattice site, and same spin and color state as the valence quark qiv . If qiv = u, such con=gurations should, by the Pauli principle, be disallowed. Also u> 1 could subsequently annihilate the valence quark. Both e8ects occur in Mud> and Mus> but not in Mcs> and Mcd> . To avoid a possible diQculty in deriving Eq. (5.2) we will de>ne the con=guration corresponding to uv (u1 u 1 ) : : : (xp x>p )d> v

in Mud>

[or

uv (u1 u 1 ) : : : (xp x>p )s>v in Mus>]

(C.8)

cv (c1 c>1 ) : : : (xp x>p )d> v

in Mcd>

[or

cv (c1 c>1 ) : : : (xp x>p )s>v in Mcs>] :

(C.9)

to be This, rather than the de=nition with a common set of pairs, ensures that con=gurations excluded in Mud> or Mus> will be excluded in Mcd> or Mcs>; and that for any subsequent u> 1 uv annihilation induced by Hud> or Hus> there will be a corresponding annihilation of c>1 cv in Hcd> or Hcs>. The diagonal mass terms which contributed to Hud> + Hcs> and Hcd> + Hus> in such con=gurations will still be the same.

280


In each case we have 3mu ; 3mc ; md and ms terms and a common set of non-uscd quark masses due to other pair creation. The construction of corresponding con=gurations is readily generalized to the case of arbitrary number of pairs with the 3avors of the valence quark. A generic state in Mi—> is (p> v ) (p> v ) q>j )q>j

(pv ) (pv ) qi (qi(1) q>(1) q>i )(xx) > nx (qj(1) q>(1) i ) : : : (qi j ) : : : (qj

:

(C.10)

> etc. Altogether we The superscripts pv ; nx ; : : : simply count the number of pairs of type (qi q>i ), (x; x), have pv pairs of the valence quark 3avor, p> v pairs of the valence antiquark 3avor, and additional xx> pairs. With the speci=c relation (5.2) in mind we will explicitly exhibit among those xx> pairs the (qk q>k ) and (ql q>l ) pairs: Mi—> = qi (qi q>i )pv (qj q>j )p> v q>j (qk q>k )p1 (ql q>l )p2 (xx) > nx : : :

(C.11)

with x = qi ; qj ; qk ; ql . The corresponding states in the other three mesonic sectors will be taken as Mk l> = qk (qk q>k )pv (ql q>l )p> v q>l (qi q>i )p1 (qj q>j )p2 (xx) > nx : : : ; Mil> = qi (qi q>i )pv (ql q>l )p> v q>l (qk q>k )p1 (qj q>j )p2 (xx) > nx : : : ; Mk —> = qk (qk q>k )pv (qj q>j )p> v q>j (qi q>i )p1 (ql q>l )p2 (xx) > nx : : : :

(C.12)

All of these con=gurations have the same total number of quarks and antiquarks: N = 2 + 2pv + 2p> v + 2p1 + 2p2 + 2 x=ijkl nx . Also the corresponding quarks (or antiquarks) in Mi—>; Mk l>, etc. are taken to be at the same locations with identical spinor and color states. The Gauss constraints, the antisymmetrization e8ects, and (except for mass contributions) also all matrix elements are the same for Hi—>; Hk l>; Hil> and Hk —>. Finally, we have in Hi—> + Hk l> the same mass terms as in Hil> + Hk —>. [Altogether there are 2p1 + 2pv + 1 terms with mi as a coeQcient, the same number with coeQcient mk ; and 2p2 +2p> v +1 terms with coe8ecients mj (ml ).] All of these terms make identical contributions to both sides of Eq. (5.2), which is therefore correct. Viewing states with equal of numbers of (say) cc> and uu> pairs as “corresponding” states does not imply that heavy and low mass pairs occur with equal probability in physical states. This issue is determined by solving H (ij) |ij = E (ij) |ij and heavy pairs are in fact strongly suppressed [219]. This is analagous to the relative separation rqq> in the example of Section 2, where rcc> and ruu> correspond to the same “generic” degree of freedom r. While solving the SchrModinger equations with mc mu leads to very di8erent expectation values of r rJ= r! ; the operator relation Hcc> + Huu> = 2Hcu> still holds for the non-relativistic Hamiltonians. Appendix D. Enumeration and speci2cation of the baryon–meson inequalities Altogether we have 42 di8erent 3avor-spin baryonic “ground state” combinations consisting of udscb quarks, when we do not distinguish between members of I -spin multiplets and assume no orbital, radial, or gluonic excitations. (Electromagnetic mass splittings were discussed in Section 12). We consider =rst the J P = (3=2)+ “A-like” baryons. In baryons made of three identical quarks bbb; ccc, sss (C− ), and uuu = ddd (A++ ; A− ), each diquark combination is in a spin triplet with


281

J P = (3=2)+ . Also, in all other (3=2)+ states all quark pair subsystems, and hence the corresponding mesonic systems qi qj (→ qi q>j ); qj qk (→ qj q>k ); qk qi (→ qk q>i ), must be, in the L = 0 approximation adopted (see comment (3) in Section 3), in the triplet, S = 1 state. Hence only vector meson masses appear on the right hand side of the inequalities. If we list all mesons in lexicographic order with b ¿ c ¿ s ¿ u; d, we have the following (3=2)+ states: uuu(A); suu(F∗ ); ssu (E∗ ); sss(C− ); cuu(Fc∗ ); csu (Ec∗ ), css(Cc∗ ); the doubly charmed baryons ccu and ccs; the ccc state; and buu(Fb∗ ); bsu(Eb∗ ), and bss(Cb∗ ). There are also the heavier states bcc; bbu; bbs; bbc, and =nally bbb, which are unlikely to be discovered soon. For the sake of completeness, we list in Table 2 all the relevant inequalities. The =rst =ve can already be tested and are satis=ed with a reasonable margin in all cases. The next eight inequalities constitute lower bounds predicted for as yet undiscovered baryonic states, which hopefully can be veri=ed in the near future. The remaining three will presumably be veri=ed only in the distant future. We move next to the J P =( 12 )+ baryons. Let us =rst focus on the case where we have two identical quarks, namely duu (p), suu(F+ )(or sud with I = 1), ssu(E− ), cuu(Fc++ ); css(Ec ); ccu; ccs; buu; bss, and bcc. In all these “yx1 x2 - type” states the identical 3avor quarks x1 x2 must be in a triplet state, namely (˜sx1 +˜sx2 )2 = 1 · (1 + 1) = 2 :

(D.1)

Writing next the total baryon spin as (˜sx1 +˜sx2 +˜sy )2 = (sx1 + sx2 )2 + 2˜sx1 ·˜sy + 2˜sx2 ·˜sy + (sy )2 = sB2

(D.2a)

or, using Eq. (D.1) and sB = 1=2: 2 + 2(˜sx1 ·˜sy +˜sx2 ·˜sy ) +

3 4

=

3 4

;

(D.2b)

i.e. ˜sx1 ·˜sy B + ˜sx2 ·˜sy B = −1 ;

(D.2c)

with B denoting the expectation value in the baryon state. Since we have x1 ↔ x2 symmetry of the above expectation values, we =nally conclude that ˜sx1 ·˜sy B = ˜sxx ·˜sy B = − 12 :

(D.3)

Hence each of the yxi (→ yx>i ) subsystems must be a mixture of 34 singlet and 14 triplet states for which ˜s · ˜s = − 34 and + 14 , respectively. This then implies, by collecting all the terms in (x1 x>2 ); (yx>1 ); (x2 y), > that the baryon–meson inequalities read as follows: m@ 1 3 + (3m$ + m@ ) = (m@ + m$ ) 2 4 4 m@ 1 + (3mK + mK ∗ ) mF ¿ 2 4 mD 1 mE ¿ + (3mK + mK ∗ ) 2 4 mn ¿

282


m@ 1 + (3mD + mD∗ ) 2 4 m@ 1 + (3mB + mB∗ ) : (D.4) m Fb ¿ 2 4 We have similar relations for the doubly strange baryons: mD 1 mEc = mcss (1=2)+ ¿ + (3mDs + mDs∗ ) 2 4 mD 1 + + (3mBs + mB∗s ) ; mEb = mbss (1=2) ¿ (D.5) 2 4 and the doubly charmed baryons (hopefully to soon be found in FNAL experiments): mJ= 1 mccu ( 12 )+ ¿ + (3mD + mD∗ ) ; 2 4 mJ= 1 1 + mccs ( 2 ) ¿ (D.6) + (3mDs + mDs∗ ) : 2 4 The speci=c numerical values for all of the above and a few other x1 x2 y states are listed in Table 2. Among the remaining ( 12 )+ baryonic states composed of three di8erent quark 3avors, we have three “3-type” states in which the light quark ud subsystem coupled to I = 0 (and hence to sud = Iud = 0): mFc ¿

s $%&' ud = 3; I =0; s=0

c $%&' ud = 3c ; I =0; s=0

b $%&' ud = 3b :

(D.7)

I =0; s=0

Using (˜sud ) ≡ (˜su +˜sd )2 = 0 and (˜s3 ) ≡ (˜sx +˜sud )2 =

3 4

;

we can now deduce, using also u ↔ d exchange symmetry, that ˜sx ·˜su = ˜sx ·˜sd = 0 :

(D.8)

> subsystem we have the opposite triplet– Eq. (D.8) implies that in each xu(→ xu) > and xd(→ xd) singlet mixture as in the previous case of Eq. (D.3), namely 14 singlet and 34 triplet. Thus collecting > $) subsystems, we =nally have: these and the term corresponding to the singlet ud → ud(≈ m$ 1 + (3mK + mK ∗ ) ; m3 ¿ 2 4 m$ 1 + (3mD + mD∗ ) ; m 3c ¿ 2 4 m$ 1 + (3mB + mB∗ ) : (D.9) m 3b ¿ 2 4 The remaining baryonic states which consist of three di8erent 3avor quarks csu; bsu; bcu, and bcs are either of the “3-type” in which the lighter two quarks couple to S = 0 subsystems, or of the “F-type” in which the lighter two quarks couple to S = 1. Because the hyper=ne interaction


283

[˙ (˜si · ˜sj )=(mi mj )] is bigger for the lighter quark system we expect that the 3-type states will be lighter than the corresponding F-type states. This will be particularly so for the su subsystem in csu or bsu, which have been observed or are likely to be observed sooner than the bcu and bcs states. When we specify the spin q of the su subsystem to be sq1 q2 = 1 (or sq1 q2 = 0), and follow the previous discussion, we conclude that, for csu for example: ˜sc ·˜su + ˜sc ·˜ss = −1

(for sud = 1)

(D.10a)

(for sud = 0) :

(D.10b)

or ˜sc ·˜su + ˜sc ·˜ss = 0

However, unlike in the previous cases, we cannot deduce that ˜sQ ·˜su B = ˜sQ ·˜sd B without appealing to the approximate Gell–Mann–Ne’eman SU(3) u ↔ s symmetry. If we do this nonetheless, as a =rst approximation, we obtain 1 (mDs + mD ) 3(mDs∗ + mD∗ ) mK + + ; (D.11a) mcus (3) ¿ 2 4 2 2 1 (mDs∗ + mD∗ ) 3(mDs + mD ) m∗K mcus (F) ¿ + + : (D.11b) 2 4 2 2 In the real case, we expect the spin-averaged variant, consisting of the symmetric combination of Eqs. (D.11a) and (D.11b) (this is the value we list in Table 2, with the left hand side called mcus ): 1 (mK + mK ∗ ) (mD + mD∗ ) (mDs + mDs∗ ) [mcus (3) + mcus (F)] ¿ + + ; (D.12) 2 4 4 4 and in general we expect that the right hand sides of (D.11a) and (D.11b) to be, respectively, an over- (under-) estimate of the combined mesonic masses. Speci=cally, we expect that Eqs. (D.11a) and (D.11b) to be satis=ed with a somewhat smaller (larger) than usual margin. The di8erence between the right hand side in Eq. (D.11a) and Eq. (D.11b) is (mK ∗ − mK ) 1 (mD − mD∗ ) (mDs − mDs∗ ) + ≈ 130 MeV : (D.13) + 2 2 2 2 We will not pursue this issue further here. Appendix E. Application to quadronium It has been suggested that the puzzling narrow resonance found in heavy ion collisions is in some sense an e+ e− e+ e− bound state [220]. The need to achieve very strong bindings (BE me 0:5 MeV) in this normally weakly (electromagnetically) coupled system is a serious problem of the quadronium hypothesis. While the experimental motivations are questioned, we =nd that there is a nice, simple, application of the variational technique [221]. It con=rms the suggestion that we cannot really have a two-body dominated interaction in this system, and that new, unusual, four-body interactions are called for. The following inequality on the binding energy of quadronium in terms of the binding of positro> 2) > be the wave function of quadronium in its nium serves to sharpen the issue. Let Q = 0 (112

284


ground state. The Hamiltonian is HQ = T1 + T1> + T2 + T2> + V11> + V12> + V21> + V22> + V12 + V1>2> :

(E.1)

Clearly, we expect the repulsive electron–electron (V12 ) and positron–positron (V1>2>) interactions to lower the quadronium binding. Hence 0

j0B (Q) 6 j˜B (Q) ;

(E.2)

0 with j˜B (Q) the binding of a =ctitious Hamiltonian H˜ Q , which is the original HQ with these repulsive interactions eliminated:

H˜ Q = T1 + T1> + T2 + T2> + V11> + V12> + V21> + V22> :

(E.3)

If we compare H˜ Q with the two-body Hamiltonians, H11> = T1 + T1> + V11>;

H12> = T1 + T2> + V12> ;

H21> = T2 + T1> + V21>;

H22> = T2 + T2> + V22> ;

we see that 2H˜ Q = H11> + H12> + H21> + H22> ;

(E.4)

where H11> is obtained from H11> by doubling the interaction: H11> = T1 + T1> + 2V11>, etc. This can be achieved for the one-photon exchange by doubling the e8ective coupling strength 0 → 0 = 20. 0 Eq. (E.4) can next be used in the by now familiar way to put an upper bound on j˜B (Q), and thus, via Eq. (E.2), on j0B (Q) as well. The bound is j0B (Q) 6 2jB0 (P) ; with jB0 (P) the binding of positronium in which the strength of the interaction has been doubled by 0 → 20. Thus we have 0

j0B (Q) 6 j˜B (Q) 6 2jB0 (P) :

(E.5)

The fact that the real positronium spectrum conforms so nicely to QED predictions with radiative corrections included strongly suggests that if 0 = 1=137 is scaled up to 0 = 2=237 we can still treat the e+ e− e+ e− system with the perturbatively calculated potential. In this case m m j0B (P) (0 )2 = 202 = 2 Ry = 27 eV ; 4 2 0 0 and hence jB (Q) 6 2jB (P) 6 54 eV, and the binding falls short by about 104 of the required 0:5 MeV binding. Appendix F. QCD inequalities for exotic novel hadron states Most known hadrons are qi q>j mesons and qi qj qk baryons. We refer to (non-glueball) states belonging to neither of the above two categories as “exotic”. Such states have been searched for and discussed for more than 30 years. “Duality” arguments predating QCD [60] suggested “qqq > q” > states coupling mainly to baryon–antibaryon as the most likely exotic states. Within QCD hybrids


285

(qq+gluon) > arise naturally [222]. Also qqs > s> “bag states” were suggested as candidates for the K K> threshold states [223–225]. Considerations of hyper=ne chromomagnetic interactions, VHF ≈

(˜i · ˜j )(˜i · ˜j ) ij

mi mj

V (rij ) ;

generated via one gluon exchange between the quarks qi and qj suggested a particular “hexaquark” H=uuddss [223] below the 33 threshold which would be strong interaction stable. 28 More recently similar considerations [228,229] singled out the speci=c “pentaquark” P = csuud > as a more likely strong interaction stable, new, exotic, bound state. Both states have been experimentally searched for with inconclusive results [230 –232] in almost all cases. The exotic states can be separated into disjoint color singlet hadrons. 29 Indeed we have seen in Section 16 that for degenerate quark 3avors (and most likely in other cases as well) there is an attractive interaction in the qi q>j qk q>l channels. The question of whether these residual color forces between color singlet states can lead to weakly bound states, in analogy to the nuclear forces in the deuteron and other nuclei, is of some interest. However we believe that it is a detailed, delicate, issue that QCD inequalities cannot decide (much in the same way that the deuteron’s binding requires detailed calculation). Here we would like to focus on the question whether genuine multiquark states—referred to in the literature qualitatively as “single bag states”—which have signi=cantly higher than nuclear bindings, exist. The arguments for bound hexa- and pentaquark states utilized the ad hoc assumption that in all multiquark hadronic states the quarks are in the same “universal” bag as the baryons and mesons, and have the same hyper=ne interactions. Here we will utilize the complementary, strong coupling picture, where the quarks in the exotic states are connected via electric 3ux lines into a more complex connected color network than the 3ux line between q and q> for mesons and the “Y” network with one junction point for the baryons. Let us =rst focus on quadriquarks. In the limit when two quarks (or antiquarks) are heavy, one can show directly that a novel overall singlet system which cannot be broken into color singlet clusters will form [233]. In a qi qj Q> k Q> l state, the two heavy quarks always bind coulombically into 1 a color triplet (Q> k Q> l )3 diquark with binding and size proportional to mQ and m− Q , respectively. The small (Q> k Q> l )3 subsystem acts e8ectively as a heavy quark. Together with the light qi qj it will then form, as a 3Q analog, the qi qj Q> k Q> l quadriquark. Because of the small numerical coeQcient in the coulombic energy, EB = 12 (0S =2)2 mQ =2, the latter does not exceed 3QCD 0:2 GeV even for mQ = mB 5 GeV. Hence the question of the stability of the novel pattern of color coupling—which in a strong coupling chromoelectric 3ux tube picture corresponds to the connected color network illustrated in Fig. 19—remains open.

28 The existence of a 33 bound state could encourage the further conjecture [226,227] that higher 3N = uN dN sN “strangelets” exist and that at large densities strange quark matter is the most stable, which if true has fascinating astrophysical rami=cations. 29 With the exception of genuine “quantum number exotics”, for example J PC = 0± or 1± states that cannot decay into mesons.

286


Fig. 19. The color string network for a quadriquark with two light quarks and two heavy quarks. Fig. 20. The color string network for a quadriquark (solid line) plus its doubled network with 3ux lines reversed (dashed line).

In the following we would like to use variational QCD inequalities to address this problem. This we do by following the derivation in the same strong coupling, 3ux tube approximation used for the baryon–meson mass inequalities in Section 6 above. Thus, let us take for the quadriquark wave functional: A./ | ./ ; (F.1) |qq Q> Q> = ./

with the possible generalizations maintaining the two junction points (1) and (2) (see the discussion in Section 6). We have used ./ to symbolize the graph of Fig. 19. As indicated in Fig. 20 we can extract from |qq Q> Q> —and the wave functional of an anti-quadriquark superposed on it with 3ux lines reversed—trial wave functionals for the four mesons QQ> ; Qq> ; Qq> , and qq> . In the above Hilbert space, we then have the operator relation 2HQQ q>q> = HQQ> + HQq> + Hqq> + Hq Q> ;

(F.2)

from which we obtain, via the variational principle, the mass inequality (0) (0) (0) (0) 2m(0) QQ q>q> ¿ mQQ> + mQq> + mqq> + mQ q> :

(F.3)

For the case Q; Q = c, qq = u; d, this imposes lower bounds on the (hypothetical) quadronium mass by the known charmonium, charmed meson, and light meson masses. Speci=cally for ccu>u> > the inequality reads: (or ccd> d), 1 m(0) ccuu ¿ [mJ= + m@ + (1 + 0)mD∗ + (1 − 0)mD ] ; 2

(F.4)

where mcc> =mJ= and muu> =m@ follows from the generalized Pauli principle as discussed previously. 30 The weights of the D and D∗ depend on the total spin of the ccu>u> system (see Appendix D). For example, consider J P = 1+ quantum numbers. In this case the quadriquark → DD is forbidden since the two D mesons can couple only to 0+ ; 1− ; 2+ etc. The lowest state available for decay is then 30

The identical uu or cc quarks are in the lowest L = 0 state, and hence the uu> or cc> in the corresponding meson should be in the triplet state.


287

DD∗ . Performing a calculation similar to those in Appendix D we =nd that 0 = 0 is indeed the correct weight and the inequality reads + 1 m(0) ccuu (1 ) ¿ 2 (mJ= + m@ + mD∗ + mD ) ; + m(0) ccuu (1 ) ¿ 3870:7 MeV ≈ mD∗ + mD :

(F.5)

Our previous comparisons of many baryon–meson inequalities with data indicate that these inequalities are satis=ed with a margin of 150 –300 MeV. Cohen and Lipkin [37] and Imbo [38] suggest heuristic arguments for this margin. Such a margin should a fortiori be valid for the quadriquark inequalities. 31 Thus we expect that the ccu>u> state actually lies well above the D∗ D threshold and > In this case we could choose ud> to be therefore is unstable. The situation is very di8erent for ccu>d. in the spin singlet state. This is consistent with the cc spin triplet diquark and the u>d> spin singlet anti-diquark, with relative zero orbital angular momenta, adding up to an overall J P =1+ . The analog of Eq. (F.5) is then (1+ ) ¿ 12 (mJ= + m$ + 32 mD∗ + 12 mD ) ; m(0) ccud> (1+ ) ¿ 3590:6 MeV ≈ mD∗ + mD − 300 MeV ; m(0) ccud>

(F.6)

A bound ccud(1+ ) state with m(0) 6 mD + mD∗ would satisfy the inequality with a margin of ccu>d> approximately 300 MeV, and is therefore quite possible. The analogs in the Q = b system + 1 ∗ m(0) bbuu (1 ) ¿ 2 (mN + m@ + mB + mB ) ; + m(0) bbuu (1 ) ¿ 10416:3 MeV ;

(F.7)

and 1 m(0) (1+ ) ¿ (mN + m$ + 32 mB∗ + 12 mB ) ; bbud> 2 (1+ ) ¿ 10113:3 MeV m(0) bbud>

(F.8)

allow bindings in both 3avor combinations, though again with a considerably higher margin in the bbu>d> case. > In the Q> Q> qq system the Q> Q> and qq could also In baryons each quark pair must couple to a 3. couple to a color sextet. Thus the segment connecting the junction points (1) and (2) in Fig. 19 would then carry a sextet 3ux — a possibility thatour discussion above neglected. This is justi=ed in the strong couplinglimit. The potential energy g2 d 3 xE 2 then dominates, and the quadratic Casimir > Nonetheless, operator (to which E 2 is proportional) of the sextet is 10=3 times larger than that of 3. the need to exclude the sextet 3ux, and the related fact that there is no alternative derivation in the weak coupling, one gluon exchange limit, puts the present inequalities on a weaker footing than that of the baryon–meson mass inequalities. We believe however the enhanced likelihood for a ccu>d> quadriquark state suggested by the inequalities. 31

Here we have for some of the Qq> mesons worse trial wave functionals as compared with those extracted from baryons, namely mesonic strings consisting of three segments going through the two junction points.

288


Fig. 21. (a) The color network for the putative pentaquark state with three junction points (1), (2), (3). (b) The string picture of the trial baryon + meson states obtained from that of the pentaquark by omitting the (2) – (3) string bit, and reversing the 3ux direction in the (2) –de and (3) –Q> f sections.

It is amusing to note that an alternative, purely hadronic approach to the problem of a bound DD∗ state also suggests that D0 D∗+ (or a ccu>d> composition) is more likely to be bound than D0 D0∗ . Thus following Tornqvist [234] (who coined the name “deusons” for these deuteron-like extended mesonic states), let us consider the one pion exchange interaction in the D∗ D channel. The resulting Yukawa force has, due to √the D$ threshold—D∗ proximity (i.e. mD∗ − mD − m$ ≡ jm$ ), an anomalously long range 1= 2jm$ 3–7 fm, 32 and could generate a D∗ D bound state. However, because of simple I -spin Clebsch–Gordan coeQcients, the potential generated by the $+ exchange in the D∗+ D0 ↔ D∗0 D+ system is twice as strong as that due to the $0 exchange in the D∗0 D0 ↔ D∗0 D0 system, and a D∗+ D0 bound state is more likely. Let us next consider pentaquarks in the same strong coupling, 3ux string approximation. The corresponding novel connected color network is illustrated in Fig. 21. This “network” contains three junction points. At point (1), two light quark 3uxes are coupled to a 3> 3ux, and at point (2) the same holds for the 3uxes emanating from the other two light quarks. Finally at the third “anti-junction” > couple to point the three 3> 3uxes originating from points (1), (2), and from the heavy antiquark Q, a singlet. As in the case of the quadriquark we could have coupled the 3uxons in either the =rst vertex (1) or the second vertex (2) to a color sextet 3ux, which again is neglected in the strong coupling limit. For pentaquarks, we furthermore have an alternative (similar) network obtained by exchanging one of the u quarks from the uu in vertex (1) with the d quark in vertex (2)—which represents a di8erent, though not necessarily orthogonal, color coupling con=guration. By reversing the 3ux lines on the segments (2)-d and (3)-Q> indicated by the double arrows in Fig. 21, and ignoring the intermediate 3uxon between (2) and (3), we can extract from a “stringy” pentaquark wave functional trial wave functionals for a charmed baryon (cuu in this speci=c case) and an sd> meson. The full Hamiltonian for the quark and 3uxon system is ˜ 2 + (˜B)2 ] ; H= [(E) HD i + (F.9) quark

3ux segment

˜ 2 represents the “potential energy” density where HDi are the Dirac Hamiltonians for the quark, (E) 2 ˜ for the 3uxons, and (B) plays the role of kinetic energy for moving and distorting the string segments and the ∇i in HDi “moves” the tip quarks. 32

Because of the P-wave D∗ → D$ vertex this is compensated by a reduced e8ective coupling.


289

Omitting (the generally positive contribution of) the (2)–(3) 3ux segment will only reduce the energy. This and the fact that the trial con=gurations of the baryons and mesons thus extracted do not optimize the wave functionals of the meson and baryon at rest, strongly suggest the inequality mP ¿ msd> + mcuu :

(F.10)

Because of the Pauli principle the two u quarks in cuu which are color antisymmetric and 3avor symmetric must be spin symmetric and hence couple to Suu = 1. Since these subcon=gurations originate from the speci=c pentaquark state for which the four light quarks’ spin has to add to a total spin S = 0, we must also have Ssd> = 1. This means that we have to identify the sd> meson with K ∗ (880) and the baryon with Fc (2445) with a total mass of mFc + mK ∗ = 3334 MeV, which is way ( 430 MeV) above the threshold at 2906:8 MeV. In general, however, the pentaquark state consists of a superposition of various 3avor assignments at the tips of the same color network. The right-hand side of Eq. (F.10) should therefore contain, in general, some weighted average so that mP ¿ 0[msd(S=1) + mccu(S=1) ] + 1[msu + mcud ] > + [mud> + mcsu ] + >[muu(S=1) + mcsd(S=1) ] ;

(F.11)

with 0 + 1 + + > = 1 and 0; 1; ; > ¿ 0. The =rst K ∗ + Fc con=guration was discussed above. The spins in the last con=guration are =xed by the same consideration as those used in the case of sd> − cuu separation discussed above. The corresponding total mass m@ + mEc = 3221:3 MeV is 315 MeV above the Ds − p threshold. In the remaining two con=gurations we will have in general admixtures of the lighter pseudoscalar $ and K states, and of the lighter 3c baryon. The lightest combination is m$ + mEc = 2605:2 MeV : (The other alternative including the kaon, mK + m3c = 2778:6 MeV is only 128 MeV below threshold.) We need an unlikely large admixture of the particular $ + Ec con=guration to bring the right-hand side of Eq. (F.11) below 2906:8 MeV, the Ds − p threshold. The baryon–meson inequalities are generally satis=ed with a substantial margin of 150 MeV. If this is also the case for the relation in Eq. (F.11), then it appears that the present inequalities cannot allow for a stable pentaquark bound state. It should be emphasized that the present approach, assuming that con=nement via the speci=c mechanism of color 3ux tubes (or strings) is the dominant aspect, is practically orthogonal to the one-gluon exchange potential approach. Indeed in the latter, con=nement is assumed from the outset by simply postulating that all the quarks in penta- and hexaquark states are inside the same universal bag as that of the baryons. Yet con=nement forces play here a minimal role in the pentaquark binding. Rather, the overriding consideration is to maximize the overall hyper=ne interaction even if that involves having large components of the wave function with qq pairs in a color sextet state.

290


It is therefore logically consistent that the inequalities derived in the framework of one approach con3ict with a prediction of a bound pentaquark in another framework. Nonetheless, the inequalities do suggest that if the pentaquark is not found, then neither QCD nor the naive quark models which so nicely explain the baryonic and mesonic spectra are at fault, but rather a lack of proper treatment of the basic con=nement mechanism for pentaquark and=or hexaquark con=gurations. One other class of exotics are hybrid states containing an extra gluon such as ( > i j )8 G and ( i j k )8 G. The (· · ·)8 notation indicates that the quarks in the hadron couple to a color octet, which then forms the color singlet hadron with an additional gluon. To probe this sector, we should consider correlators of the form H (x; 0) = 0|[

a r † a r >b >b i (x) j (x)ab Gr (x)] [ i (0) j (0)a b Gr (0)]|0

:

(F.12)

If we represent the gluons via a =eld strength, then H (x; 0) can be written schematically as H (x; 0) = d(A)F (x)F (0)SAi (x; 0)SAj (x; 0) ; (F.13) and hence a Schwartz-type inequality implies 2 2 |H |2 6 d(A)F (x)F (0) d(A)SAi (x; 0)† SAi (x; 0) d(A)SAj (x; 0)† SAj (x; 0) :

(F.14)

This would imply, if the diQculty with the vacuum expectation value 0|F 2 |0 could somehow be surmounted, that mhybrid ¿ 12 mglueball + m$ 0:8 GeV ;

(F.15)

which is a rather weak bound, from a phenomenological point of view. A similar inequality in the context of SUSY models (in the msquark → ∞ limit) 33 has proven useful in assessing the symmetry (phase) structure of that theory. Speci=cally, the possibility that massless, singlet, fermionic composites of qi q>j and a gluino may manifest unbroken chiral symmetries, has been ruled out. This was done by proving a QCD inequality between the mass of this state (the “hybridino”) and the mass of the pion [121]: −

[mhybrid = m(0) (qi q>j g)] ˜ ¿ [mq(0i q>j ) = m$ ] :

(F.16)

Since unlike for gluons with non-linear self-couplings one can de=ne here the gluino propagator, the derivation of the last inequalities along lines paralleling those of Weingarten’s in deriving mN ¿ m$ (see Section 10), is fairly straightforward. The hybridino–hybridino correlator (written in a concise, index-free form) can be bound by Schwartz inequalities: † Jhyb (x)Jhyb (0) = dS S > S 33

From the discussion at the end of Section 11, a decoupling of the squarks and elimination of their Yukawa couplings is indeed required for the following derivation.


6

d(|S |2 )1=2 (|S > |2 )1=2 (|S |2 )1=2

6

d|S |2

6

291

2

d|S |

1=2 1=2

d|S > |2 |S |2 2

d|S > |

1=2

1=2

2

d|S |

1=2

:

(F.17)

The =rst and second factors in the last expression are the square root of a pion propagator. Hence up to a constant † Jhyb (x)Jhyb (0) 6 e−m$ |x| ;

(F.18)

and Eq. (F.16) follows.

Appendix G. QCD inequalities between EM corrections to nuclear scattering In this appendix we combine the results of Section 12 on the essentially positive nature of the EM contribution to the energies, and the approach of Section 16, in order to apply “electromagnetic QCD type inequalities” to scattering states. To this end we =rst elaborate on some of the material of Section 16. We have seen in Section 16 that QCD arguments leading to relations of the form B(aa) + B(bb) 6 2B(ab)

(G.1)

between binding in channels (aa), (bb), and (ab) are useful even if there are no bound states in those channels. Thus, imagine that the relative coordinate ˜r =˜r1 −˜r2 of the particle pair is con=ned to a sphere |˜r| 6 R, with R an infrared cuto8 larger than the relevant scales in the problem. Let j0n; l and jn; l denote the levels in various l channels without and with (respectively) the interparticle interactions turned on. Both sets of levels become dense as R → ∞. [In particular, the free energies are j0n; l = (kn;0 l )2 =2m with kn;0 l R ≡ x0n; l (n$=2 + l$=2 + · · ·), the nth zero of jl (x).] Yet a careful study of the shifts \(n; l) ≡ −jn; l + j0n; l reveals the full information on the phase shifts in various channels. The key observation is that the binding energy inequalities hold in general even when the R cuto8 is placed, since the relevant QCD or nuclear interactions are short range. These imply then that \n; l (aa) + \n; l (bb) 6 2\n; l (ab). [For the unperturbed part we have trivially, from the de=nition of reduced masses, j0n; l (aa) + j0n; l (bb) ≡ 2j0n; l (ab).] Therefore the density of levels dnl (ab)=d k in the (ab) channel is larger than the average of these densities in the (aa) and (bb) channels. Levinson’s theorem implies that the phase shift >(ab) l (k) serves as a “level counter” in the continuum limit. Hence d>l (ab) dnl (ab) : dk dk

(G.2)

292


However, in the k → 0 limit the last quantity is the scattering length for l = 0. The inequality d>(aa) d>(bb) d>(ab) + 62 ; dk dk dk yields in the k → 0 limit the desired inequality between scattering lengths a(aa) + a(bb) 6 2a(ab) :

(G.3)

(G.4)

We would next like to argue that the statement on the \I = 2 energy shifts of NN continuum states, for example \BE(pp) + \BE(nn) 6 2\BE(pn)I =1

(G.5)

does imply the corresponding inequality for the respective scattering lengths a(pp) + a(nn) 6 2a(pn) :

(G.6)

The notation \BE(pn)I =1 in Eq. (G.5) refers to the shift from the idealized \BE(pn)I =1 of some (0) continuum level in the I = 1 (3avor symmetrized) np state for the case when 0EM = 0 and m(0) u = md . In this exact I -spin limit, all (pn)I =1 ; (pp); and (nn) states would be the same, and hence so would be phase shifts, scattering lengths, etc. The \I =2 combination 2\BE(pn)I =1 −\BE(nn)−\BE(pp) and the similar combination of phase shifts, level densities, and scattering lengths are e8ected only by electromagnetism—i.e. by 0EM = 0. The general feature of positivity of such electromagnetic self energies can therefore be applied. This is so since the positivity applies not only for ground states but to any set of corresponding states (0) which have equal charge density in the 0EM = 0; m(0) u = md limit. This then implies the positivity of all these combinations and in particular the desired relation (G.6). In applying the suggested scattering length inequality (G.6), we face a “technical” diQculty. The long-range Coulombic interaction yields a (logarithmically) in=nite Coulombic phase. In order to extract a meaningful result for a(pp), one must subtract this Coulomb phase shift. Our inequality is based precisely on the positive nature of the Coulombic self-interaction. Thus subtraction of even a part of this interaction could, in principle, invalidate the derivation of the inequality. Indeed, the very approach to continuum phase shifts by quantization in a =nite sphere of radius R is jeopardized by the 1=r in=nite range potential. We can, however, avoid the Coulombic pp phase shifts and still have a meaningful relation. This relies on the observation that screened versions of the Coulomb potential (i.e. Yukawa potential) still yield positive EM self-interactions [speci=cally the momentum space representation (2 + k 2 )−1 is positive]. We can choose the cuto8 −1 to be of the order of the sum of the radii of the two nucleons. Thus the long-range Coulombic phase shift will be screened away, yet all other manifestations of EM interactions which occur during the nucleons’ overlap will be maintained. These include the EM interactions between quarks in the two di8erent nucleons or between their mesonic charge clouds, or more subtle indirect EM e8ects such as the $+ − $0 mass di8erence and the resulting di8erent ranges for $+ ; $0 exchange potentials [235]. It is the latter e8ects (occurring also in the np and nn systems) which we wish to consider. We cannot prove that the above cuto8 procedure is indeed equivalent to Coulomb phase subtraction. However, any cuto8 [or even arbitrary positive superpositions of potentials with di8erent cuto8s, such as d2 ()e−r =r] can be used. Hence for scattering lengths computed with this


293

cuto8, Eq. (G.6) holds a() (pp) + a() (nn) 6 2a() (pn) :

(G.7)

We therefore believe that this inequality applies to the quoted “nuclear parts”. Recent measurements indicate that the suggested inequality is indeed satis=ed with a wide margin. In principle, the original general relation (G.3) contains much more information than that used to derive the scattering length inequality by considering only l = 0 and going to the k → 0 limit. Thus many other inequalities for other scattering parameters in all l waves can be derived for the two nucleon system, although these are not as useful and cannot be readily compared with data. Inequalities analagous to (G.6) should hold for any I = 12 isospin multiplet. This in particular we should have a(3 He3 He) + a(3 H3 H) 6 2a(3 He3 H) :

(G.8)

The I = 2 part can be extracted also for scattering of two particles which are members of di8erent I -spin doublets. This suggests relations of the form a(p3 H) + a(n3 He) ¿ a(p3 He) + a(n3 H) ; a(K + n) + a(K 0 p) ¿ a(K + p) + a(K 0 n) ; a(K 0 n) + a(K − p) ¿ a(K 0 p) + a(K − n) ;

(G.9)

and many more. Since many of these relations involve unstable nuclear isotopes, testing them requires the new radioactive beam facilities. Finally we note that purely EM \I = 2 combinations of masses or scattering lengths can be found for nuclei of higher I -spin. Appendix H. QCD-like inequalities in atomic, chemical, and biological contexts In this last appendix we present various inequalities inspired by analogies with the binding energy and correlator inequalities. H.1. Mass relations between compounds of di7erent isotopes Di8erent isotopes provide an almost ideal example of “3avor independent” interactions. Thus, let (1) (2) (1) (2) there be n1 stable isotopes of a speci=c atom (Z (1) ; A(1) 1 ) : : : (Z ; An1 ) and n2 of another (Z ; A1 ) : : : (Z (2) ; A(2) n2 ). In the adiabatic (Born–Oppenheimer) approximation [236], it is clear that the interatomic interactions are independent of the speci=c isotopes Z (1) A(1) and Z (2) A(2) chosen. The i j arguments in Section 19 then imply that the binding energies Bij of the n1 n2 possible compounds thus formed are a convex function of a reduced mass sampled at the values ij =

(2) (2) m(Z (1) A(1) i )m(Z Aj ) (2) (2) m(Z (1) A(1) i ) + m(Z Aj )

;

(H.1)

294


with m(Z; A) the nuclear masses. This includes in particular the analog of the inter3avor mass inequalities 34 BE(x; x) + BE(y; y) ¿ 2BE(x; y), e.g., BE(H2 ) + BE(D2 ) ¿ 2BE(HD) ; BE(16 O2 ) + BE(18 O2 ) ¿ 2BE(16 O18 O) :

(H.2)

We have not investigated the availability of data verifying these many true mass inequalities. The 3avor (i.e. isotopic) independence arguments can be extended to more complex atomic compounds like XXZ; XZZ, etc. and the “convexity” relations (e.g., the analogs of those conjectured for baryons in QCD) are likely to hold. This will indeed be the case if the two- and three-body interactions between the nuclei satisfy the condition of positive exp{V } utilized in Lieb’s proof in Appendix B. We are indebted to Phil Allen [237], for pointing out to us this rather nice test of QCD-like inequalities. H.2. Conjectured inequalities for chemical bindings Atoms with closed shells or (n; l) subshells, constituting one Slater determinant of all possible ml ; ms states, are singlets of any relevant quantum numbers. This suggests using such states, say (Ne|Z = 10), as vacuum states upon which we can build “particle” states X = (Na|Z = 11) or “hole” states X> = (F|Z = 9). Likewise another noble gas “vacuum”, say (Ar|Z = 18), yields Y = (K|Z = 19), Y> = (Cl|Z = 17). If we use this identi=cation of vacuums, particles, and antiparticles, we might be tempted to conjecture, in “analogy” with Eq. (2.7), that BE(X X> ) + BE(Y Y> ) ¿ BE(X Y> ) + BE(Y X> ) :

(H.3)

This yields, for example, BE(NaF) + BE(KCl) ¿ BE(NaCl) + BE(KF) ; BE(MgO) + BE(CaS) ¿ BE(MgS) + BE(CaO) ;

(H.4)

etc. We might, however, consider half-=lled shells, e.g., C, Si, etc. with an equal number of electrons and holes to be the correct atomic analog of the vacuum, in which case Eq. (H.3) translates into BE(LiF) + BE(NaCl) ¿ BE(LiCl) + BE(NaF) ;

(H.5)

etc. H.3. Mass inequalities in nuclear physics For a while it seemed [238] that true QCD inequalities could show that even-even N = Z states are—barring Coulomb e8ects—the most tightly bound. The third order Garvey–Kelson [239] di8erence relations connect masses of many isotopes. However, there is no obvious pattern of deviations from the relations, and we have not found a simple motivation for any such pattern. 34

The BE refers to the true molecular ground states or to sums over the =rst n states. The fact that Bose=Fermi statistics can imply even=odd Js in the rotation band of homoisotopic compounds has a negligible e8ect and can be corrected for.


295

H.4. A biological analog for inequalities between correlators The pseudoscalar mass inequalities of Section 12 re3ect simple Schwartz inequalities for correlators, i.e. for weighted bilinears in quark propagators. Generically the latter have gauge interaction induced “phases”. These cancel in the tr(Si Si† ) combinations appearing in the particular case of the pseudoscalar propagators. 35 In an extremely wide variety of circumstances we may encounter joint propagation of two equal or two di8erent entities. Quantum phases are typically irrelevant and we can assign a positive probability for the propagation of “A” from “P1 ” to “P2 ” for any given set of relevant in3uencing “factors” An —the analog of the background gauge =elds in the QCD case: P{A(at P1 ) → A(at P2 )}|{An } :

(H.6)

The overall probability of A “propagating” from P1 to P2 is then given by a “functional” (path integrated) averaging over the distribution of the {An } factors: P{A(at 1) → A(at 2)} = d{A1 : : : An }P{A(at 1) → A(at 2)}|{An } ; (H.7) with a normalized ( d{A} = 1), positive measure of {A1 : : : An }. The probability of the joint propagation of A(1); B(1) → A(2); B(2) is given by the corresponding weighted average of the bilinear product of propagators: A(1) → A(2) = d{Ai }P{A(1) → A(2)}|{An } · P{B(1) → B(2)}|{An } : P (H.8) B(1) → B(2) Likewise, the probability of joint propagation of two A objects from (1) to (2) (or two B objects) is given by a similar expression involving squares of propagators: A(1) → A(2) (H.9) P = d{Ai }[P{A(1) → A(2)}|{An } ]2 ; A(1) → A(2) and

P

B(1) → B(2) B(1) → B(2)

=

d{Ai }[P{B(1) → B(2)}|{An } ]2 :

The Schwartz inequality readily implies the =nal desired relation: 2 A(1) → A(2) B(1) → B(2) A(1) → A(2) ¿ P : P ·P B(1) → B(2) B(1) → B(2) A(1) → A(2)

(H.10)

(H.11)

There are certainly innumerable applications of the above relation in all areas of science technology and life sciences (most of which were very likely well known for some time). In the last page of this review we consider one particular, somewhat intriguing, and exotic application concerning the sex of non-identical twins. 35

Schwartz inequalities have often been used in other contexts of particle physics. See, e.g., Ref. [240].

296


The conception of non-identical twins can be viewed as the joint propagation in the same (or relatively similar 36 ) “background” of two sperms. These have two Y chromosoms for male twins, two X chromosoms for female twins and X; Y chromosoms for male female twins. Hence we expect that for non-identical twins, 37 P(male; male)P(female; female) ¿ [P(female; male)]2 :

(H.12)

Clearly the probability of any speci=c single (or double) conception need not be accurately re3ected in the percentages at birth. We could however still derive the same inequalities if we consider not merely the propagation from “inception” to “conception” but also the subsequent nine month long “time-like” propagation to birth. The extent to which the inequality (H.12) is satis=ed, i.e. the deviation of the ratio of the RHS and LHS from unity, can serve as a measure of the overall degree of non-parallelism of the male and female birth “vectors”, i.e. as a crude measure of the total importance of “variables” known, or as yet unknown, in determining the sex of the fetus, which is a quantity of considerable interest. A priori [241] there could be some unique rather surprising e8ect which would disqualify our proof and possibly reverse the sign in Eq. (H.12). This would be the case if there were a direct interaction between the propagating elements—in this case the fetuses of the two twins. The biblical story of the rather unrestful pregnancy of Rebekah [242] with Esau and Jacob (a clear case of non-identical twins!), suggest that the competition between equal sex brothers (and possibly sisters) may extend to the prenatal stages. Clearly if it is too strong (and in particular stronger than that between same sex o8spring) it could tend to disrupt (MM) and (FF) births, thus tending to reverse the sign of the e8ect considered. Hopefully this is not the case. We have not attempted to verify any of the above non-QCD mass inequalities. In particular, testing the last twin inequalities may require rather extensive statistical analysis. 38 We hope that the inequalities will eventually be tested. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]

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If the two eggs are fertilized in subsequent encounters, or somewhat separated spatially. P(F; M ) refers to half the probability of mixed-sex twins—say in which the female was born =rst. 38 Construction of a clean sample of non-identical twins (non-biased by sex consideration!) is a highly demanding task. 37

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Physics Reports 362 (2002) 303 – 407 www.elsevier.com/locate/physrep

The role of the in nuclear physics Giorgio Cattapanb; c , L'(dia S. Ferreiraa; ∗ a

Centro de F sica das Interacco˜ es Fundamentais (CFIF), Departamento de F sica Instituto Superior T ecnico, Avenida Rovisco Pais, 1096 Lisboa, Portugal b Dipartimento di Fisica dell’ Universit)a, I-35131 Padova, Italy c Istituto Nazionale di Fisica Nucleare, I-35131 Padova, Italy Received September 2001; editor: G:E: Brown

Contents 1. Introduction 2. The free 2.1. The as a resonance 2.2. Isospin mass splittings in the multiplet and hadron structure 2.3. Deviation from spherical symmetry: the E2=M 1 ratio 3. Theoretical models for the N – interaction 3.1. Basic meson-exchange models for the N interaction 3.2. Meson–baryon couplings 3.3. The coupled-channel approach to the N system 3.4. Relativistic and unitarity corrections in the N coupled-channel approach

305 307 307 314 318 326 327 330 352 357

3.5. QCD-inspired models 4. Contents of in nuclei 4.1. Light nuclei: the percentage of in the nuclear wave function 4.2. Mesonic-exchange currents and 4.3. propagation in nuclei 4.4. The in nuclear matter 5. Conclusions and outlook Appendix A. Dispersion relation constraints A.1. Preliminaries A.2. Fixed-variable dispersion relations Appendix B. The N scattering matrix and electromagnetic corrections References

360 363 363 365 370 384 388 390 390 391 397 401

Abstract We review the properties of , and the role it plays in Nuclear Physics. We try to assess what has been ascertained up to now, and what remains to be clari>ed about this hadron, which represents the outstanding structure in the medium-energy pion–nucleon interaction. We consider the free >rst, with particular emphasis on those aspects which may give information on the internal hadron structure, such as the isospin mass splittings in the multiplet, and the electromagnetic >ngerprints of deviation from spherical symmetry. ∗

Corresponding author. E-mail address: [email protected] (L.S. Ferreira).

c 2002 Elsevier Science B.V. All rights reserved. 0370-1573/02/$ - see front matter PII: S 0 3 7 0 - 1 5 7 3 ( 0 1 ) 0 0 0 9 3 - X

304

G. Cattapan, L.S. Ferreira / Physics Reports 362 (2002) 303 – 407

We then discuss the N – interaction, both under the point of view of meson-exchange models, and under a microscopic quark-model perspective. In so doing, we review the present status of the meson–nucleon and meson– coupling constants and form factors, whose knowledge is a prerequisite for a description of nuclei in terms of eEective hadronic degrees of freedom. We discuss also the coupled-channel approach to the NN –N system, and how it compares with relativistic treatments grounded on Quantum Field Theory. We then consider light nuclei, where the percentage of components in the nuclear wave function, and contributions to meson-exchange currents can be studied in a quantitative way, with a minimum of uncertainties due to nuclear-structure eEects. Finally, we discuss propagation in >nite nuclei and nuclear matter. To this end, we try to give an update presentation of classical subjects, such as the eEective interaction in the nuclear environment, medium eEects and quenching phenomena, where -hole excitations and nuclear-structure eEects c 2002 Elsevier Science B.V. All rights reserved. are strictly intertwined. PACS: 13.40.−f; 13.75.Gx; 14.20.Gk; 21.30.−x; 21.65.+f; 25.20.Lj; 25.40.Kv; 25.80.Hp Keywords: Nucleon resonances; Meson–baryon interactions; Photon and meson induced reactions; Quenching phenomena; Nuclear matter


305

1. Introduction The excitation of the resonance is the most striking phenomenon in N scattering for pion kinetic energies below 300 MeV. This is clearly exhibited by the phase shift behavior, with the P33 partial wave in the J = T = 32 channel playing by far the dominant role, and a weak s-wave interaction even near threshold. The cross section for forming this resonance is at the unitarity limit of 200 mb, testifying its very strong coupling to the N system. When regarded as an excited state in the baryon spectrum, the has the smallest mass gap with respect to the nucleon, about 300 MeV. Because of these features, one may expect that the can have a prominent role in clarifying items in low-energy Hadron Physics and in highlighting non-nucleonic degrees of freedom in Nuclear Physics. In low-energy hadron physics, the N – mass diEerence is one of the basic observables quark models are required to reproduce [1]. Moreover, the analysis of the mass splittings within the multiplet, and of observables such as the quadrupole moment and the E1=M 2 ratio in pion photo-production can give information about quantum chromo dynamics (QCD) in the low-energy regime, since these experimental data are strictly connected to quark–quark tensor color interactions and=or the eEective coupling of constituent quarks with the Goldstone bosons associated with the breaking of chiral symmetry [2,3]. The resonance, considered as a baryon with its own spin and isospin quantum numbers, has by now a long history as an “exotic” degree of freedom in Nuclear Physics. Its coupling to the other hadronic degrees of freedom, i.e. nucleons and mesons, can be described through eEective Lagrangians, whose structure is dictated by symmetry considerations. As a matter of fact, in low-energy nuclear physics it has become customary to mimic the usual meson-theoretic approach to the nuclear-force problem. One looks at the N → transition or at the interaction between two ’s as due to the exchange of mesons; a non-relativistic reduction of the corresponding Feynmann amplitudes then provides the required baryon–baryon potentials. In so doing, the meson–baryon coupling strengths come into play as free parameters, to be determined in reproducing the experimental data. This means that the task of extracting signals of non-nucleonic degrees of freedom from experiments is unavoidably intertwined with the determination of the “best” values for the various meson–baryon coupling constants. Hopefully, one would like to get more information on the strengths from >rst-principle arguments, namely from a QCD description of hadron structure. As is well-known, in spite of great progresses in QCD lattice calculations, this is still beyond our present possibilities. Even in the framework of more phenomenological quark models, a clear and unquestionable calculation of the hadron–hadron interaction is still missing, since this entails the solution of a non-trivial few-body problem. It is not surprising therefore, that for many years one has limited oneself to look for relations between the various coupling strengths based essentially on symmetry assumptions. A noteworthy example is the SU (4) quark model, where nucleon and are members of the same multiplet, the latter being a spin–isospin excitation of the former with no change in the orbital motion of the constituent quarks; as a consequence, the coupling strengths of and mesons to nucleon and can only diEer by purely geometric factors. Similar considerations apply to the strong-coupling model, a generalization of the celebrated Chew–Low model for the N system. Baryons are described as static sources interacting through meson exchange, with a common scale parameter for their mutual interaction; for → ∞, a dynamical symmetry emerges, which dictates well-de>ned relations among the coupling strengths.

306


When looking for signals of excitation in nuclear phenomena, it is natural to start from the two-nucleon system. Thus, one may try to see whether an explicit inclusion of degrees of freedom improves our understanding of the nuclear force. It is by now well known that intermediate propagation can substantially reduce contributions from the eEective meson in One-Boson-Exchange models of the NN interaction. More generally, to achieve a sounded description of NN bound-state and scattering observables, one needs a non-perturbative approach. This leads to the NN –N– coupled channel problem, which has been the subject of many investigations since the 1970s, with increasing progress in the number of coupled channels taken into account, and in the realistic nature of the employed baryon–baryon potentials [4,5]. From a fundamental point of view, one has to ascertain to what extent this essentially non-relativistic scheme catches the essential features of what is in itself a relativistic problem, the description of exchange forces mediated by meson >elds. If the is taken as a constituent of the atomic nucleus, one may wonder how much it contributes to the total nuclear wave function. Warnings of caution has been raised in the past about this issue [6]. As a matter of fact, the “particle” is in a sense an artifact, which mimics a resonant structure in the N continuum. If it is described as a Rarita–Schwinger >eld through the direct product of a spin 1 vector state and a spin 12 state, one has to deal with non-physical spin- 12 components [7]. Their contributions can be dismissed on the energy shell by imposing additional constraints on the Rarita–Schwinger >eld. In nuclei, however, propagation occurs oE-shell, and unphysical degrees of freedom are still possible. These ambiguities are masked, but not in principle solved, by the non-relativistic approximations necessary in order to match physics with the wave functions of nuclear-structure theory. In spite of these problems, the idea of the resonance as a possible constituent of the nucleus is a robust one, as testi>ed by the success of the -hole model in reproducing photon- or pion-induced scattering cross sections [8]. To reduce the ambiguities inherent to any model-dependent analysis, it is natural to look for the in few-nucleon systems, and=or in processes where this degree of freedom is excited through the simple and well-known electromagnetic or weak interactions. In the former case, indeed, one can exploit the presently available Faddeev or variational techniques to get realistic nuclear wave functions; in the latter, one is naturally led to consider the electromagnetic and weak currents which in the nuclear system couple to the external electroweak >eld. The can contribute both to the one-body current through direct coupling to the photon, and to the two-body currents, much in the same way as it comes into play in the intermediate NN interaction. In nuclear matter, or in heavier nuclei, more eEective descriptions are required, in order to keep computations to a tractable level. In particular, short-range NN and N correlations are described by means of eEective forces, in the spirit of Landau theory of quantum Fermi liquids. Great eEorts have been devoted to gain information on the proper values to be assigned to the corresponding eEective strength parameters [9]. This can be done either by looking at nuclear-structure and reaction data, or, on the theoretical side, through many-body calculations. The main issue is to disentangle contributions arising from excitation in the nuclear medium from eEects due to the presence of higher-order “ordinary” con>gurations in the nuclear wave function. The detailed structure of the N force can be probed exploiting the diEerent ways by which diEerent probes couple to nuclear targets. Pions and photons interact with nucleons and ’s through longitudinal and transverse couplings [8], respectively, whereas both types of interactions intervene in charge-exchange reactions [10]. The formation, propagation and >nal decay of the resonance can be thus explored through diEerent windows, which emphasize diEerent facets of the same basic


307

dynamical mechanism. The most striking diEerence between nuclear and electromagnetic probes is that with the former one observes a systematic downward shift of the resonance position with respect to the proton target, whereas the latter only lead to a broadening of the peak. To ascertain the physical meaning of this fact, the speci>city of the experimental probe is crucial. Under this point of view, charge-exchange reactions turned out to have a privileged role. For the same reason, exclusive reactions are expected to convey more information than simpler inclusive experiments, even if they are obviously more demanding for the theoretical analysis. The excitation of -hole states can be regarded as a polarization eEect of the nuclear medium. Together with N -hole excitations, it has been advocated to explain why several observable quantities are quenched with respect to the value they ought to have in a simple single-particle description of the nuclear response. Well-known examples are the quenching of the nuclear magnetic moment and of the axial coupling constant in nuclei with a closed-shell core plus (or minus) one nucleon [9,11], and the missing strength in the giant Gamow–Teller resonance, excited through intermediate-energy charge-exchange reactions [9,10]. Here also the main problem is to clarify the respective role played by -hole excitations, and by the mixing of higher-order con>gurations involving nucleonic degrees of freedom only. In this paper, we try to give an overall insight into the present understanding of the under a nuclear physicist point of view, updating what can be found in some previous excellent reviews [12,13]. Needless to say, it is practically impossible in a single paper to cover all the items which would deserve attention. For instance, we shall not consider production in high-energy heavy-ion reactions, nor shall we dwell on production reactions due to hadron probes, or on absorption processes in the resonance region. We shall rather follow the scheme outlined above. Thus, in Section 2 we discuss the free , with particular emphasis on the determination of the resonance parameters within the baryon spectrum, and on the relation between its isospin mass splittings and deviation from spherical symmetry and our present views of hadron structure. Theoretical models for the N – interaction are discussed in Section 3. There, we devote considerable attention to the coupling strengths among the eEective hadron >elds playing the main role in nuclear physics, namely among nucleons, ’s, and the and mesons. The coupled-channel N – problem is also considered, both from a non-relativistic and a relativistic point of view. Finally, modern approaches based upon QCD-motivated quark models are brie?y discussed. The particle as a constituent of the nucleus is the subject of Section 4, both for light and heavier nuclei. In addition to isobaric contributions to electromagnetic currents, “classical” topics such as Landau parameters, medium eEects in nuclear reactions, and quenching phenomena are updated. At the end, the merging of the coupled N – system in nuclear matter is discussed. Conclusions and outlook are given in Section 5. 2. The free 2.1. The as a resonance Under an empirical point of view the baryon can be perceived as a strong, isolated resonance in N scattering. In elastic ± p, + n, and charge-exchange −p going into 0 n cross sections, one can see the presence of a pronounced maximum for a pion lab energy around 190 MeV, which may be assigned to an intermediate state with mass M ∼ 1230 MeV. In Fig. 1, the total

308


(a)

(b)

(c)

(d)

Fig. 1. Total −p and elastic + p cross sections as functions of the pion lab kinetic energy T . Data from Ref. [14].

cross sections for ∓ p scattering are plotted as functions of the beam energy; they show explicitly an excited state with charge 2e and 0, respectively. If charge independence is assumed, these two states can be assigned to an isospin multiplet. The isospin value is determined from the ratio of the total cross sections for isospin-related reactions, such as elastic ± p scattering and the charge-exchange reaction −p → 0 n. As is well-known near the resonance one has (+ p → + p) : (−p → 0 n) : (−p → −p) ∼ 9 : 2 : 1 ; which is consistent with a total isospin T = 32 . In fact four charge states have been identi>ed for the isobar, that is − ; 0 ; + ; ++ . These considerations can be put on a more quantitative level through a partial-wave analysis of the scattering data. In principle, the quantum numbers of the resonance can be guessed through rather simple quantum–mechanical considerations of –N scattering. Assuming that the range R of the N interaction is >xed by the pion mass (R ∼ 1=m ), for pion energies in the resonance region (T ¡ 300 MeV) the usual semi-classical relation l ∼ qR between the angular and linear momentum of the pion gives that the s and p waves must dominate the scattering process. If the spin-free and spin-?ip amplitudes f(q; ) and g(q; ) are expanded into partial waves up to l = 1 one has (see Eqs. (B.13), and (B.15) – (B.16)), f(q; )

1 [a0; 1=2 + (2a1; 3=2 + a1; 1=2 ) cos ] ; q

(2.1)

g(q; )

1 (a1; 3=2 − a1; 1=2 ) sin : q

(2.2)


309

Fig. 2. DiEerential cross sections for N charge exchange at the resonance, E =190 MeV, from diEerent measurements. Figure from Ref. [15].

Here, we have labeled the various partial-wave amplitudes by means of the orbital and total angular momentum quantum numbers l and j, respectively, and assumed purely elastic scattering, so that Eq. (B.17) is written fl; j =

1 1 1 (exp 2il; j − 1) = exp il; j sin l; j ≡ al; j : 2iq q q

(2.3)

Using the standard expression for the unpolarized diEerential cross section d = |f(q; )|2 + |g(q; )|2 ; d it is a trivial task to show that the angular distribution can be written in this approximation as a second-degree polynomial in cos , 1 d 2 (A0 + A1 cos + A2 cos2 ) ; d q

(2.4)

where the coeScients A0 ; A1 and A2 depend upon the partial-wave amplitudes al; j according to A0 = |a0; 1=2 |2 + |a1; 3=2 − a1; 1=2 |2 ;

(2.5)

A1 = 2 Re[a? 0; 1=2 (2a1; 3=2 + a1; 1=2 )] ;

(2.6)

A2 = 3|a1; 3=2 |2 + 6 Re[a? 1; 3=2 a1; 1=2 ] ;

(2.7)

so that A2 only contains contributions from the p wave. At resonance the angular distribution in the CM system has a parabolic dependence upon cos , with a remarkable consistency among diEerent data sets. This is clearly seen for example in Fig. 2, where data for –N charge exchange at 190 MeV are reported. The diEerential cross sections at resonance can be reproduced with A0 ∼ 1; A1 ∼ 0, and A2 ∼ 3, a result consistent with a dominant l = 1; j = 32 contribution, with the corresponding phase shift 1; 3=2 passing through =2. A con>rmation of this assignment can be obtained by looking

310


at the value of the total cross section at resonance, where the dominant contribution ought to be given by the jth component Tj ∼ 2 (2j + 1)(#j + 1) ; q which, for j = 32 and a purely elastic process (#j =1) gives T 8=q2 . This estimate is in reasonable agreement with the experimental values shown in Fig. 1. The actual, quantitative extraction of the resonance parameters, energy and width, is obviously far less simple than the above illustrative considerations. This is even more true, when the coupling to the possible reaction channels is taken into account, and a simultaneous identi>cation of the various baryon resonances is attempted, in order to map the excitation spectrum of the nucleon. The determination of resonance parameters is generally a two-step process. One has >rst to express cross section and polarization data in terms of partial-wave amplitudes. For elastic scattering this is accomplished through partial-wave analysis; for inelastic data one has to resort to isobar models, where the reaction process is described as a coherent superposition of quasi-two-body channels. The second step involves the extraction of resonance parameters from the partial wave amplitudes, and has been accomplished through several approaches, which diEer in the way of handling the coupling among channels, the implementation of unitarity, and the very parameterization of the scattering matrix. Benchmarks in partial wave analysis of N scattering have been the solutions provided by the Karlsrhue–Helsinki (KH) [16 –18] and Carnegie–Mellon–Berkeley (CMB) [19 –21] groups. These studies emphasized the crucial role played by dispersion relations in constraining the partial wave amplitudes extracted from the data. As the quality of the data base improved in time, new >ts to the elastic N data have been pursued by the Virginia Polytechnic Institute and State University group (the VPI group) [22,23]. They also advocated both forward dispersion relations and dispersion relations at >xed momentum transfer t to constrain the solution. We shall give some hints about this, when dealing with the much debated question of the optimal value for the NN coupling constant. A general prescription which has to be satis>ed in the extraction of the resonance parameters is unitarity. The KH and VPI approaches focused on the elastic N channel. The former accounted for all inelasticity through absorption parameters #l; j . In the analysis, one has to take into account electromagnetic eEects, so as to de>ne purely hadronic partial wave amplitudes. How this has been accomplished by KH is summarized in Appendix B. In the VPI approach, one accounts for the overall inelasticity introducing an eEective channel, and unitarity is implemented through a coupled-channel K-matrix formalism [14]. The elastic scattering amplitude in each partial wave is then given by Te =

Im Ce K˜ 1 − Ce K˜

(2.8)

with K˜ expressed in terms of the 2 × 2 K-matrix according to K˜ = Ke +

Ci K02 : (1 − Ci Ki )

(2.9)

The functions Ce and Ci are the elastic and inelastic Chew–Mandelstam functions, obtained by integrating phase-space factors over the appropriate unitarity cuts. Extensive energy-dependent, and


311

energy-independent (i.e. single-energy) analysis have been performed by VPI, whose results are updated and can be found in the VPI repository [23]. Other approaches exploited formalisms which allow for many channels. Thus, Manley and Saleski [24] factorized the S-matrix into background and resonant contributions SB and SR , respectively, S = SRT SB SR

(2.10)

with SB expressed in terms of a background K-matrix in the usual way, namely 1 + iKB SB = ; (2.11) 1 − iKB and the resonant component written as the product of N factors, where N is the number of resonances being parameterized, SR = S11=2 S21=2 : : : SN1=2 :

(2.12)

Each factor Sk1=2 is constructed in such a way that its square can be written in terms of a transition matrix, namely Sk =1+2iTk , which is parameterized in the standard multichannel Breit–Wigner form. As a consequence, near an isolated resonance one simply has S ∼ SR , if background contributions are negligible (SB 1). This approach allows for a direct extraction of resonance parameters, and has been used starting from partial-wave amplitudes for N → N scattering and from isobar-model amplitudes for N → N reactions [24]. A multichannel model, devised to subsume both unitarity and analyticity requirements, has been developed several years ago by the Carnegie Mellon–Berkeley group [19 –21], and recently revived by T.-S.H. Lee and collaborators [25]. The basic assumption of this model is that a physical (asymptotic) channel a can be mapped into another asymptotic channel b through a set of intermediate resonant states i = 1; 2; : : : ; R. Both physical and intermediate states can be of two-body (N; #N; : : :) or quasi-two-body (; N; : : :) nature, i.e. one of the two particles in a given channel can be itself a resonance. The coupling of resonance i to an asymptotic state a is described by √a strength parameter +ai and a form factor fa (s), depending upon the total center-of-mass energy s, so that the multichannel T -matrix can be written Tab =

R

fa (s)

a (s)+ai Gij (s)+jb

b (s)fb (s);

a; b = 1; : : : ; M :

(2.13)

i; j=1

Here, a is the √ phase-space factor for channel a. For a two-body channel this means that one simply has a = pa = s, whereas for quasi-two-body channels a more complex recipe has to be employed, in order to satisfy the unitarity condition Im(Tab ) =

M

? Tac Tcb :

(2.14)

c=1

Finally, Gij is the full (dressed) propagator, satisfying the Schwinger–Dyson equation Gij = Gij0 + Gik0 .kl Glj ;

(2.15)

kl

where the bare propagator Gij0 = ij (s − s0; i )−1 is diagonal, with non-vanishing elements associated to the propagation of the ith resonance. The integral equation (2.15) can be reduced to a set of R × R

312


algebraic equations assuming a factorized form for the self-energy .ij , .ij =

M

+ci /c (s)+cj :

(2.16)

c=1

The physical meaning of (2.16) is obvious. Resonance i can decay into an asymptotic channel c, whose particles can coalesce again to form resonance j. The channel propagators /c have to be parameterized so that the unitarity prescription (2.14) is satis>ed. Here, we shall limit ourselves to outline how this is accomplished for stable two-particle channels, in order to show how analyticity can be explicitly introduced in the formalism. For more details the reader is referred to the relevant c literature [25]. If sth is the threshold for channel c, one writes ∞ 1 /c = fc? (s )gc (s ; s)fc (s) ds ; (2.17) sthc with

c (s ) c (s ) 1 −P gc (s ; s) = ; s − s + i0 s − s0

(2.18)

P the principal-value part of the propagator, and s0 plays the role of a subtraction point. Insertion of Eq. (2.18) into (2.17) immediately gives Im(/c (s)) = fc2 (s)c (s) ; Re(/c (s)) = Re(/c (s0 )) +

(2.19) c s − sth

∞

c sth

Im(/c (s )) ds : (s − s)(s − s0 )

(2.20)

The former of these relations clearly sets the discontinuity of the propagator across the right-hand, unitarity cut. The latter represents a subtracted dispersion relation very similar in form to what can be established for N scattering on the ground of axiomatic >eld-theoretic or S-matrix considerations. The subtraction point s0 can be chosen such that the resulting scattering matrix has convenient analyticity properties. Thus, for N scattering one can choose s0 so that the scattering amplitude in N the complex s plane has a pole on the left-hand side, and a branch cut from sth = (m + M )2 to +∞, as it should be (see Appendix A). This type of parameterization allows also to simulate the left-hand singularities due to t- and u-channel mechanisms. That these singularities must be there is dictated by crossing symmetry; in potential-scattering language, they arise because the interactions at work can be always associated to the exchange of quanta in the crossed channels. These singularities can be easily simulated introducing bare states with mass s0; i below the N threshold, and additional free propagators Gij0 . This prescription introduces non-resonant contributions into the scattering amplitude. One advantage of the CMB approach, is that the separable structure of the transition amplitude, and its dependence upon s only, make it easy to determine the complex poles of the scattering amplitude in the complex energy plane. To >nalize our discussion on the identi>cation of resonance parameters, we would like to mention that one can evaluate the K matrix in terms of some eEective Lagrangian, thus reducing drastically the number of involved parameters, since background and resonance amplitudes are obtained from the same set of Feynmann graphs [26]. Moreover, aspects of the hadron dynamics, such as chiral


313

Table 1 (1232) Breit–Wigner mass and width, and pole position, in MeV. To be consistent with the Breit–Wigner parameterization E = M − (i=2)2 , the pole position is given as Re(Epole ) and −2 Im(Epole ). The values recommended by PDG are given in boldface Reference

M

2

Manley [24] CMB [20] HUohler [16] Arndt [23] HUohler [28] PDG [27]

1231 ± 1 1232 ± 3 1233 ± 2 1233

118 ± 4 120 ± 5 116 ± 5 114

1232

120

Re(Epole )

−2 Im(Epole )

1210 ± 1

100 ± 2

1211 1209 1210

100 100 100

symmetry, can be taken into account in a straightforward way. The main drawback of a K-matrix, eEective-Lagrangian approach is that analyticity of the resulting scattering amplitude is no longer guaranteed. At the same time, because of the complicated functional form of the driving potential V , to restore analyticity by imposing dispersion-relation constraints is far from trivial. As we have seen, analyticity constraints can be rather easily introduced in a T -matrix parameterization of the CMB type. The overall >t of the scattering data may show large discrepancies in diEerent resonances, depending upon the particular ansatz which has been employed. A remarkable example is given by the S11 (1535) resonance, seen in the T = 1=2; J = 1=2 s-wave channel of the N system, whose parameters largely diEer in the various analysis [25], in spite of its high-con>dence (4?) status in the PDG tables [27]. For strongly excited and well-de>ned resonances, such the P33 (1232) isobar, on the other hand, there is a substantial consensus about the value of the identifying parameters. For the reader’s convenience, we report in Table 1 the Breit–Wigner mass and width for mixed charges, given by Refs. [16,20,24] and [23; 28], together with the PDG recommended values [27]. In addition, we provide also the position of the corresponding pole of the scattering amplitude in the complex energy plane. This information has become more and more popular in recent years [27], since the P33 N partial wave gets large background contributions from the nucleon pole term (a point to which we shall come back again when dealing with the determination of the NN coupling constant). Whereas the conventional Breit–Wigner parameters strictly speaking pertain to the resonance and to the large background in N scattering, the pole position is strictly associated to the resonant state. As it can be seen from this table, the pole position remains stable near 1210 − i50 MeV in all the analysis. This is true even when the mass assignment deviates from the PDG recommended value, ranging in the literature from 1210 up to 1241 MeV [29]. This can be understood in the light of the background eEects mentioned before, and may be particularly relevant when data coming from diEerent experiments are analyzed. Indeed, even for an isolated, narrow multichannel resonance the S-matrix has to contain a (Breit–Wigner) resonance factor and a slowly varying background term, S(E)=SB (E)T SR (E)SB (E). If data from a production experiment are analyzed, the initial factor SB (E) ought to be replaced by other energy-dependent factors, thereby introducing some model dependence. That the pole position and not the Breit–Wigner mass is the quantity which can be most rigorously associated with a resonance has been stressed by HUohler some years ago [28].

314


Fig. 3. Data on + d and − d total cross sections as functions of the pion lab kinetic energy T from Ref. [31]. The full line represents the >t obtained with Eq. (2.23) for the -mass splitting.

2.2. Isospin mass splittings in the multiplet and hadron structure The mass values for the various charge states of the resonance have been extracted from diEerent data sets. The ++ and 0 masses have been determined from partial-wave analysis of N scattering data [18,30,31], and have the most precise values, whereas for the + it has been obtained from pion photoproduction data [27]. The − mass, on the other hand, has never been determined. Various relations can be found involving the masses within the T = 32 multiplet; they may help either in determining the − mass from the experimentally known masses, or in testing the degree of charge-independence violation in the system. A classical example is provided by scattering on the deuteron. Since the deuteron is an isosinglet (Td = 0), the neutral and charged pion cross sections must be equal in the limit of charge-independent interactions. This symmetry is reduced to the invariance with respect to the replacement T3 → −T3 (charge symmetry) by the p–n and ± –0 mass diEerences. Even with this less restrictive assumption, one has that the + –d total cross section + d ought to be equal to the − –d one − d . Needless to say, electromagnetic eEects have to be removed before any meaningful comparison with the experimental data may be done. This is a feasible although non-trivial task, as we outline in Appendix B. The deuteron being a loosely bound system, one can expect that the total cross section is dominated by the contributions due to scattering from a free neutron and a free proton. In the resonance region, therefore, one can write with an obvious notation + d (!) = ++ (!) + 13 + (!) ;

(2.21)

− d (!) = 13 0 (!) + − (!) :

(2.22)

These expressions show that, if the masses associated to the various charge states are diEerent, the total ± d cross sections cannot be the same any longer. This must be re?ected in − d (!)−+ d (!), which diEers from zero across the resonance region. This is shown in Fig. 3, where the result of a classical experiment by Pedroni et al. is exhibited [31]. The solid curve corresponds to a >t with


315

Table 2 Additive quantum numbers of the light and strange quarks Quark

d

u

s

Q

− 13

+ 23

− 13

Tz

− 12

+ 12

0

0

0

−1

S Table 3 Quark structure of nucleon and p

n

++

+

0

−

(uud)

(udd)

(uuu)

(uud)

(udd)

(ddd)

respect to the parameter D = M− − M++ + 13 (M0 − M+ ) :

(2.23)

Once rescattering eEects, as well as corrections due to the target nucleon motion have been taken into account, one >nds D = 4:6 ± 0:2 MeV :

(2.24)

Eqs. (2.23) and (2.24) allow one to >x the − mass from the already known masses. In a sense, they allow one to study the − charge state through the scattering of negative pions from the neutron bound in the deuteron. The mass splitting within an isospin multiplet can be described by a simple mass formula quadratic in Tz , which has been established many years ago by Weinberg and Treiman [32]: M = a − bTz + cTz2 :

(2.25)

Substituting Tz = − 12 and 12 in this relation, one can >x the linear coeScient bN for the nucleon isodoublet, given the n–p mass diEerence Mn − Mp = bN = 1:29 MeV :

(2.26)

Similarly for the 0 (Tz = −1=2) and + (Tz = 1=2) states one has M0 − M+ = b 1:38 MeV :

(2.27)

If, on the other hand, one uses the Weinberg–Treiman relation in Eq. (2.23), one gets D=(10=3)b = 4:6 MeV, in good agreement with the result (2.24). The equality between the linear coeScients bN and b in Eq. (2.25) >nds a natural explanation in the constituent quark model of baryons. According to this model, both the nucleon and the are bound systems of three u and d quarks, which have eEective (constituent) masses mu md ∼ 350 MeV. For the reader’s convenience we report in Table 2 the spin-?avor assignments of the u, d, and s quarks, and in Table 3 the quark structure of the nucleon and .

316


In order to ful>ll the Fermi statistics for ++ , the color degree of freedom was assigned to quarks. The established baryons are then color-singlet (qqq) states [33], and their state functions must be antisymmetric under interchange of any two equal-mass quarks (such as the u and d quarks in the limit of isospin symmetry). As a consequence, one may write |qqq A = |color A × |space; spin; >avor S ;

(2.28)

where the subscripts A and S denote antisymmetry or symmetry with respect to interchange of any two of the equal mass quarks. For “ordinary” baryons, only the isospin and strangeness ?avors are relevant; these three ?avors enjoy an approximate SU (3) symmetry, which can be combined with the SU (2) spin symmetry to give an overall SU (6) spin-?avor symmetry. In this limit the six basic states are u ↑, u ↓; : : : ; s ↓, where the symbols ↑ and ↓ denote spin up and spin down states, respectively. Group theory then shows that baryons can be classi>ed into various SU (6) and SU (3) multiplets. + The four charge states belong to a J = 32 spin decuplet, with strangeness S = 0, together with the .± , .0 , S = −1 states, the ;± with S = −2, and the − , S = −3 particle. Looking at Table 3, one sees that the 0 and the neutron have the same (udd) quark structure, apart the diEerent way the quark spins are coupled. The same applies to the (uud) + state and to the proton. In the SU (6) limit, therefore, one expects the n–p mass splitting to be the same as the mass splitting between the + and 0 states. This is indeed approximately veri>ed in Eqs. (2.26) and (2.27). Actually, in this limit, the mass splittings can be simply ascribed to the diEerent masses of the down and up quarks, i.e. M 0 − M+ = Mn − M p = md − mu :

(2.29)

The observed isospin splitting in the nucleon and systems, therefore, may provide indications about the corresponding property at the quark level, and the degree of accuracy the quark model can reach in describing hadron properties. Simple quark model considerations can be used to derive a variety of relations for baryon isomultiplet splittings within the same SU (3) multiplet. In particular, one can >nd relations which constrain the unknown − mass, and allows one to understand the electromagnetic mass splittings of baryons as a result of quark dynamics. A >rst, basic result is suggested by the Weinberg–Treiman relation (2.25), which states that the mass splitting within an isomultiplet is given by a mass formula quadratic in Tz ; as a consequence, the third derivative of M with respect to Tz must vanish. Applying standard >nite-diEerence formulas [34] one >nds D3 ≡ M++ − 3M+ + 3M0 − M− = 0 :

(2.30)

This result is actually corroborated by the quark model. To understand how this comes out, one has >rst to identify the terms in the quark-model Hamiltonian where quark masses may come into play [35]. There is obviously the kinetic energy of the quarks. There will then be a one-body contribution Kqi , plus a two-body term Kqi qj taking into account the quark–quark central components of the interaction, such as an eEective con>ning potential. This contributions can be subsumed into the kinetic energies, a fact justi>ed on the ground of the virial theorem T = (r=2) dV=dr . One has then to consider the isospin breaking eEects. These can be classi>ed into (a) the mass diEerence q between the up and down quarks; (b) the quark–quark Coulomb interaction; (c) the strong hyper>ne splitting due to the gluon-exchange interaction acting dominantly on pairs in an S-wave state; and (d) the spin-dependent hyper>ne contribution of electromagnetic origin. For all these contributions


317

one can derive simple scaling laws with the quark masses, on the (reasonable) assumption that the relevant quantities do not vary from one baryon to another within the same multiplet. Let us consider, for instance, the Coulomb interaction energy 1 em ; (2.31) VEij = ne structure constant, and Qi the quark charge. If 1=rij does not vary where < ∼ 137 appreciably within a multiplet one can parameterize this quantity as VEijem =aQi Qj , with a a universal constant. As for the hyper>ne strong and electromagnetic interaction energies, they both can be written in the form

VEijHFx = >x ×

|?ij (0)|2 i · j ; mi mj

(2.32) HF

where >x depends upon the strong coupling constant for VEij strong , and is given by > = −2ne contributions reduces itself to the evaluation of the mean value i · j . This can be accomplished through simple considerations. First, one can use the value of the total spin squared S2 to show that (2.34) i · j = ∓3 i¡j 1 2

for J = and 32 , respectively. From Eq. (2.28) one sees that two like quarks in the S-wave orbital of a ground-state baryon must be in a spin-symmetric state, namely, they have to couple to spin 1, which implies · = 1. The same is true for quarks in the ?avor decuplet with J = 32 , since their spins must be aligned together. Finally, in octet baryons one has a pair of like quarks, and two qi qj pairs, with i = j and · = −2. In the light of the above considerations, the various baryon isospin violating mass shifts can be evaluated almost straightforwardly. For instance, the p–n mass diEerence turns out to be

1 c 4 a 1 1 + : (2.35) − − Mp − Mn = q + Ku − Kd + Kuu − Kdd + + b 3 m2u m2d 9 m2u m2d Notice that, if isospin violating interactions are ignored, one simply gets Mp − Mn mu − md . A similar calculation shows that Eq. (2.30) is actually satis>ed. The accuracy of this relation has been investigated by Jenkins and Lebed [36], combining a perturbative treatment of ?avor-breaking eEects with the 1=Nc expansion. In this approach, the completely symmetric spin-?avor SU (2F) representation (see Eq. (2.28)) for the lowest-lying baryons is decomposed into a tower of baryon states of increasing angular momentum J = 12 ; 32 ; : : : ; Nc =2 for arbitrary Nc . The physical baryons can be identi>ed with states at the top of the ?avor representations, since in this case the number of strange quarks is of order 1, and not O(Nc ). Jenkins and Lebed estimated the accuracy of Eq. (2.30) to be

318


of order 0 0 =Nc3 , where 0 and 0 0 represent the strong and electromagnetic isospin-symmetry violating parameter, respectively. With their parameters this implies D3 ∼ 10−3 MeV. A check of the quality of the above constraints on the masses can be obtained by combining the linear relation (2.23) with Eq. (2.30) to get 3D 9D ; M− − M++ = : (2.36) M 0 − M+ = 10 10 Using the value (2.24) for D, and the Breit–Wigner values 1233:6 and 1230:9 for the 0 and ++ masses [18,27], one has M+ 1232:2 MeV;

M− 1235:0 MeV :

(2.37)

Overall, this value of the + mass compares fairly well with the values given in PDG (M+ = 1231:2–1231:8). The above considerations can be extended to include the masses of strange particles. For instance, one can consider the constraint [36] (M++ + M− ) − (M+ + M0 ) = 2.2? ≡ 2(M.?+ + M.?− − 2M.?0 )

(2.38)

with .? ≡ .(1385). This relation is expected to be accurate up to 3 × 10−5 [36]. The consistency of this relation constraining the masses with Eqs. (2.23) and (2.30), as well as with the known Breit–Wigner masses, has been investigated in [29]. Since the value of .2? extracted from data is not accurate enough, this quantity has been estimated using the experimental value of .2 ≡ M.+ + M.− − 2M.0 , and a dynamical quark model calculation of the diEerence .2? − .2 . The set of masses obtained in such a way turned out to be inconsistent with the values given by Eqs. (2.23), (2.24), and (2.30). Even if such a conclusion has to be taken with some caution, because of the essential role played by dynamical, quark model calculations, it is worthwhile to note that this diSculty is overcome if the pole positions of the resonances are used, instead of the Breit–Wigner masses. 2.3. Deviation from spherical symmetry: the E2=M 1 ratio A wealth of information on the can be gained from its excitation through electromagnetic 1 probes. This is quite understandable in view of the small value of the coupling strength < 137 . The isobar excitation can be obtained either by means of real photons or by exchange of virtual photons in electron scattering. In the latter case, to a good degree of approximation one can think of the electron interacting with the target nucleon through exchange of a single photon of four momenta (!; q) with ! = |q|, as depicted in Fig. 4, while, for real photons, one obviously has ! = |q|. It would be impossible here to consider the subject in all its details; as a matter of fact the reader can >nd excellent reviews in the literature [2]. Here, we limit ourselves to some basic facts illustrating how the existence and features of the isobar are seen through the electromagnetic window; we then consider in some detail a topic which is receiving a lot of attention by several research groups, the electric–quadrupole=magnetic-dipole ratio E2=M 1. The excitation of the is clearly perceived in pion photoproduction processes, which complement what one can learn in elastic pion–nucleon scattering. Thus, both +p → 0 p and +p → + n reactions exhibit a pronounced resonance behavior for photon energies around 300 MeV, as seen in Fig. 5.


319

Fig. 4. electroproduction through exchange of a virtual photon of four momenta Q = k1 − k1 ≡ (!; q) between the scattered electron and the nuclear target. Fig. 5. Total pion photoproduction cross sections from Ref. [8].

The general structure of the photoproduction transition amplitude can be >xed on the ground of general symmetry arguments. Let us consider a photon of four momenta (!; k) impinging on a nucleon, leading to a pion of four momenta (q0 = m2 + q2 ; q) and isospin state b in the >nal state. Since both the initial particles and the >nal nucleon have two spin states, there are eight degrees of freedom to characterize the transition amplitude. Parity conservation reduces these eight amplitudes to four complex ones. Therefore the number of physical independent observables to be measured at any photon energy and pion scattering angle is seven, once a common phase factor has been >xed. Lorentz and Gauge invariance dictate the structure of the S-matrix to be [37]

4 (b) b (q)N (p )|S − 1|+(k)N (p) = i(p + q − p − k)u(p ) (2.39) A (s; t)M u(p) : =1

p

denote the nucleon four momenta in the initial and >nal state, respectively, and u(p) Here, p and and u(p X ) represent the corresponding spinors in the spin and isospin space. The coeScients M are purely geometric quantities, to be constructed with the Dirac matrices so as to satisfy invariance requirements; the whole information on the dynamics of the process is contained therefore in the invariant amplitudes A(b) (s; t), whose dependence upon the external kinematics is best expressed in terms of the Mandelstam variables s = (p + k)2 and t = (q − k)2 . Note that Eq. (2.39) states that for each pion isotopic state there are four scalar functions of energy and angle, hence 12 functions to describe the four physical processes +p → + n;

+p → 0 p ;

(2.40)

+n → −p;

+n → 0 n :

(2.41)

There are diEerent possible sets of amplitudes [38– 40] to describe the above photoproduction processes. The photoproduction amplitudes can be analyzed under the point of view of their isospin

320


and spin structure. As for the former, one can proceed in analogy to the pion–nucleon case, thereby obtaining in terms of the Pauli operators for the nucleon isospin Bi (0) 1 (− ) A(b) = A(+) b3 + 2 A [Bb ; B3 ] + A Bb :

(2.42)

The physical amplitudes for the four photoproduction charge channels can be written as linear combinations of the A(±) and A(0) amplitudes [37]. More interesting on physical grounds is the de>nition of invariant amplitudes referring to the isospin states of the >nal N system, namely A(1=2) = A(+) + 2A(−) ;

A(3=2) = A(+) − A(−) :

(2.43)

As for the spin structure of the scattering matrix, in a relativistic framework it is best exhibited through the helicity formalism of Jacob and Wick, which allows for an elegant expression of the observables in single pion photoproduction [38]. To introduce a decomposition into electric and magnetic multipoles, however, it is maybe more convenient to re-write the S-matrix in terms of two-component Pauli spinors Ci and Cf† for the target and >nal nucleon [39]. This can be accomplished by de>ning the operator F according to

4 M √ (q)|u(p ) A M u(p)|+(k) = Cf† FCi ; (2.44) 4 s =1

where M is the nucleon mass, and isospin dependence has been omitted for the sake of simplicity. The overall F can be written again in the N center-of-mass frame in terms of four complex amplitudes Fi , depending on total energy, scattering angle, and the invariants constructed with nucleon spin , photon polarization vector , and kˆ ≡ k=|k| and qˆ ≡ q=|q|, which de>ne the scattering plane. One has ˆ qˆ · ) + iF4 ( · q)( ˆ ˆ qˆ · ) : F = iF1 · + F2 ( · q)( · (kˆ × )) + iF3 ( · k)(

(2.45)

Eq. (2.45) is the classical Chew–Goldberger–Low–Nambu (CGLN) representation for the photoproduction scattering matrix [40]. In terms of F the diEerential photoproduction cross sections are given by |q| † d = | C FCi |2 : (2.46) d ! f Finally, rotational invariance leads to a decomposition of the four quantities Fi (s; x) (x ≡ cos ≡ ˆ in terms of derivatives of the Legendre polynomials, kˆ · q) ∞ ∞ [lMl+ + El+ ]Pl+1 (x) + [(l + 1)Ml− + El− ]Pl−1 (x) ; F1 = l=0

F2 =

∞

l=2

[(l + 1)Ml+ + lMl− ]Pl (x) ;

l=1

F3 =

∞ l=1

F4 =

∞ l=2

[El+ − Ml+ ]Pl+1 (x) +

∞

[El− + Ml− ]Pl−1 (x) ;

l=3

[Ml+ − El+ − Ml− − El− ]Pl (x) :

(2.47)


321

Here, the usual notation l± is employed to denote the total angular momentum J = |l ± 1=2| of the N system. The coeScients El± are the electric multipole amplitudes, while the quantities Ml± represent the magnetic multipole contributions. All the observable quantities can be expressed in terms of these amplitudes. For instance, inserting Eqs. (2.47) into the CGLN representation (2.45), using (2.46) and integrating over the scattering angle one obtains for the total cross section ∞ 2|q| tot = {l(l + 1)2 [|Ml+ |2 + |E(l+1)− |2 ] + l2 (l + 1)[|Ml− |2 + |E(l−1)+ |2 ]} : (2.48) ! l=1

One may ask at this point what information can be extracted from the data about the role played by the various multipole amplitudes. As discussed in Ref. [8], it is well established by now that the photoproduction cross sections of Fig. 5 are dominated by the E0+ and M1+ dipole contributions. The former produces s-wave pions with the >nal N system in a J = 12 state, the latter is responsible for the production of p-wave pions. In this regime Eq. (2.48) becomes, 4|q| [|E0+ |2 + 2|M1+ |2 ] ≡ (E0+ ) + (M1+ ) ; (2.49) ! and the two dipole contributions are enough to reproduce the experimental cross sections to a reasonable degree of accuracy. The non-resonant E0+ component is non-negligible across the whole resonance region for the +p → + n reaction, whereas the +p → 0 p cross section is dominated by the magnetic dipole amplitude. This can be nicely explained in simple classical terms [8], considering the classical dipole moment −(m =M )eD, where D is the distance between the neutral pion and the N center of mass. One can expect at threshold and in the resonance region that (E0+ ) scales like (m =M )2 , representing a simple center-of-mass correction. In the limit (m =M ) → 0, the electric dipole contributions to neutral pion photoproduction are completely quenched. The basic theoretical framework for the description of photoproduction can be obtained on the ground of simple isobaric models. We shall not go into the details here, since they can be found in several excellent textbooks [8] and review papers [2]. We limit ourselves to recall that the resonant 3 3 ; channel can be described in terms of the excitation, propagation and subsequent decay of an 2 2 intermediate particle, much in the same way as in N scattering processes. Schematic models in the dipole approximation focus their attention on the leading features of photoproduction processes with outgoing s- and p-wave pions. Here, we consider in more detail what can be ascertained about the role played by smaller multipole amplitudes, and in particular by 3=2 the electric quadrupole amplitude E1+ , because of its relevance to the physics of the isobar in the context of quark models of hadrons. To understand this point, we recall that SU (6) quark models predict this quantity to be zero, since the nucleon and wave functions factorize into a spin and a purely s-wave space part, so that the matrix element of the electric quadrupole operator vanishes identically and the +N → excitation is a pure M 1 transition, the well-known Becchi–Morpurgo selection rule [41]). Since the mid 1970s, however, de Rujula, Georgi and Glashow recognized that gluon-exchange could lead to hyper>ne interquark interactions, with the subsequent presence of a d-state admixture in the baryon ground state wave function. These tensor eEects, which are responsible also of the hyper>ne contributions to the baryon interaction energy of Eq. (2.32), induce a small violation of the Becchi–Morpurgo selection rule. In chiral quark models of the nucleon most of the E2 strength comes from tensor correlations between the pion cloud and the quark bag, or from meson exchange currents between the quarks. In Skyrme models, both the nucleon and tot

322


states are obtained from the same soliton con>guration via an adiabatic SU (2) rotation, and are spherically symmetric. The charge distribution in the spin–isospin stretched state, however, may have a non-vanishing electric quadrupole component, leading to an E2=M 1 ratio diEerent from zero. The small value of this observable can be expected on the ground of power-counting arguments in the number of colors Nc . As a matter of fact, it can be shown that E2=M 1 scales as Nc−2 [42]. Theoretical calculations of the E2=M 1 ratio performed with diEerent models of the nucleon have led to rather diEerent predictions. Constituent quark models have given −2% ¡ E2=M 1 ¡ 0, depending upon the strength of the hyper>ne interaction and the bag radius [43– 46]. Larger negative values in the range −6% ¡ E2=M 1 ¡−2:5% have been obtained with Skyrme models [42], while chiral bag models have led to −3% ¡ E2=M 1 ¡ − 2% [47]. Preliminary results have been also obtained with lattice QCD calculations [48], even if with a large uncertainty, that is E2=M 1 = (+3 ± 9)%. Finally, values around −3:5% have been predicted in a quark model with exchange currents [3]. A precise measurement of the E2=M 1 ratio, therefore, represents an important testing ground for microscopic models of the nucleon, and gives information about small but physically important components of the baryon wave function. The determination of the electric and magnetic multipole amplitudes is a demanding task under the experimental point of view. As we have seen before, one has to deal with eight degrees of freedom, i.e. the real and imaginary parts of the four helicity or CGLN amplitudes of Eq. (2.45). A kinematically complete experiment would require therefore at least eight independent observables to specify the multipole amplitudes to any order in l [49]. Such an ambitious goal is still beyond our present possibilities, and one has to rely on the diEerential cross section, and the three single polarization observables ., Pf , and Pi , which are usually referred to as photon asymmetry, recoil nucleon polarization, and target asymmetry, respectively. The photon (or beam) polarization is de>ned for linearly polarized photons, whose polarization can be perpendicular or parallel to the production plane. If the corresponding cross sections are denoted by ⊥ and , respectively, its operative de>nition is d⊥ − d : (2.50) d⊥ + d For linearly polarized photons on unpolarized targets the diEerential cross section is expressed in terms of the unpolarized cross section d0 ()=d and . by the simple formula d(; /) d0 () = [1 − .() cos(2/)] ; (2.51) d d where / is the azimuthal angle of the emerging pion with respect to the beam direction. Similarly, the emerging or incoming nucleon can have two polarization states Pnf =±1 and Pni =±1 along the perpendicular to the production plane, and one can de>ne .=

Pf =

dPnf =1 − dPnf =−1 dPnf =1 + dPnf

; =−1

Pi =

dPni =1 − dPni =−1 : dPni =1 + dPni =−1

(2.52)

The extraction of the T = 32 M 1 and E2 components from a >t to the multipole expansion of the above observables is prone to the Donnachie’s ambiguity [50], that is the presence of higher partial-wave (l ¿ 2) strength in lower partial waves. One can try to overcome this problem by representing the higher partial waves by the corresponding Born contributions, and by constraining the analysis through as many observables as possible, coming from the simultaneous measurement of diEerent


323

reaction data. Information coming from N scattering is essential in these analysis. In particular, below the two-pion production threshold one can resort to the Fermi–Watson theorem [51], which expresses the phases of the complex multipole amplitudes directly in terms of the N scattering phase shifts Tl± , ElT± = |ElT± | exp iTl± + n;

MlT± = |MlT± | exp iTl± + n ;

(2.53)

where T = 12 ; 32 . The multipole amplitudes can be extracted from the experimental database by means of two diEerent, basic methods, the energy-independent and the energy dependent approach. In the former, one investigates each given energy through a standard C2 procedure, where the >tting parameters are the real and imaginary parts of the multipole amplitudes. Below the two-pion production threshold the Fermi–Watson theorem (2.53) allows one to reduce the number of >tting parameters by a factor of two, since only the absolute value of the amplitudes needs to be determined from the data. In the energy-dependent method, on the other hand, one simultaneously considers the data at all energies. In this case, the energy dependence of the data must be taken into account either through a suitable parameterization, or by means of dispersion relations. Which one of the two approaches has to be preferred has been the subject of a long debate in the literature, as we shall see when dealing with the determination of the pion–nucleon coupling constant. It is fair to say, here, that maybe the right answer depends upon the situation one is considering. If the data refer to both cross sections and polarization observables at each energy, and are closely spaced in energy, as it happens in a complete, dedicated experiment, one may regard the energy-independent approach as the best one. For widely spaced data, on the other hand, and when only a few polarization observables are available, the energy-dependent approach may be preferable, because continuity is built in from the very beginning, and systematic errors tend to cancel out. The same approach can be also useful when the general structure of the resonances is already known, and one wants to extract small partial-wave amplitudes. 3=2 The determination of the E2=M 1 ratio requires the measurement of the small E1+ amplitude with respect to the dominant magnetic dipole amplitude, in presence of a large background contribution to the former. High-quality data coming from coincidence experiments are therefore welcome. Up to a few years ago medium-energy electron accelerators were pulsed, with limited photon ?uxes for these experiments. The situation has drastically changed with the advent of cw-beam facilities, such as the Laser Electron Gamma Source (LEGS) at BNL, or the Mainz Microtron MAMI. This explains the ?ourishing of new determinations of the E2=M 1 ratio in the last years. The LEGS collaboration performed high-precision measurements of p(˜+; 0 ), p(˜+; + ) and p(˜+; +) cross sections and beam asymmetries in the energy region 209 ¡ E+ ¡ 333 MeV [52]. The reactions have been analyzed simultaneously, with a dispersion calculation of Compton scattering as a constraint on the photopion amplitudes. For each isospin channel the various multipole amplitudes Ml± (M = E; M ) have been parameterized in the form l± l± ) + AR TN ; Ml± = AB (1 + iTN

(2.54)

l± is the corresponding partial-wave transition amplitude for N scattering, and the backwhere TN ground term AB contains the contributions of the Born graphs. The physical meaning of this parameterization can be easily understood below the 2 production threshold. In this case, writing the

324


l± N T -matrix in terms of the pion–nucleon scattering phase shifts, TN = sin l± exp il± , Eq. (2.54) becomes

Ml± = (AB cos l± + AR sin l± ) exp il± ;

(2.55)

and the Fermi–Watson theorem Eq. (2.53) is satis>ed. Eq. (2.54) represents a K-matrix-like unitarization to parameterize the photopion multipoles. Note that at the K-matrix pole associated with the 3=2 3=2 resonance the corresponding phase is obviously equal to =2; Re E1+ = Re M1+ = 0, and one has 3=2 Im E1+ E 3=2 E2 AR = = : = 1+ 3=2 3=2 M 1 M1+ AR Im M1+

(2.56)

3=2 3=2 Thus, the ratio Im E1+ =Im M1+ , which is usually compared to theoretical calculations, is essentially given by the ratio of the coeScients AR obtained in the >tting procedure. The LEGS collaboration quoted a percentage,

E2 = [ − 3:0 ± 0:3stat+syst ± 0:2model ]% ; (2.57) M1 where the last error takes into account the uncertainties arising from multipole truncation, the choice of the N phase-shift solution, and the evaluation of the Compton dispersion integrals. A diEerent approach has been followed by the MAMI collaboration [53,54]. They determined the diEerential cross sections and beam asymmetries for the reaction p(˜+; p)0 and exploited the parallel part of the diEerential cross section d =d , when the pion is detected in the plane de>ned by the photon polarization and the beam momentum, to extract E2=M 1. Indeed, in the s- and p-wave approximation, this quantity in analogy to the Eq. (2.4) is given by d q = [A + B cos + C cos2 ] d ! with the coeScients A ; B , and C given by

(2.58)

A = |E0+ |2 + |3E1+ − M1+ + M1− |2 ; B = 2 Re[E0+ (3E1+ + M1+ − M1− )? ] ; C = 12 Re[E1+ (M1+ − M1− )? ] :

(2.59)

Central to the analysis of the MAMI collaboration is the ratio of the third and >rst coeScient in the parameterization (2.58) of the cross section, R≡

1 C Re[E1+ (M1+ − M1− )? ] = : 12 A |E0+ |2 + |3E1+ − M1+ + M1− |2

(2.60)

3=2 2 At resonance, where Re(M1+ − M13=2 − ) = 0, neglecting |E0+ | ; E1+ and M1− with respect to M1+ , and 1 the isospin 2 contributions to E1+ and M1+ , one gets

R

3=2 Im E1+

3=2 Im M1+

:

(2.61)


325

On the ground of a mild energy dependence of R, the MAMI collaboration extracted the E2=M 1 ratio from a >t of cross section and beam asymmetry data for the reaction p(˜+; p)0 between 270 and 420 MeV; obtaining [53] E2 (2.62) = (−2:5 ± 0:2stat ± 0:2sys )% : M1 The basic approximation Eq. (2.61) has been questioned in Ref. [55] and by the VPI group [56]. The former used the Mainz data set to extract the E2=M 1 ratio without the above approximation by means of their eEective Lagrangian approach [57], obtaining E2=M 1 = (3:2 ± 0:25)% as the recommended value. The latter, used their multipole amplitudes, obtained through a combination of energy-dependent and energy-independent analysis [58], getting the much lower value E2=M 1 = (−1:5 ± 0:5)%. A short summary of the debate among the three groups can be found in [54]. Here, we limit ourselves to note that a new, more careful evaluation of the E2=M 1 ratio through Eq. (2.60), with inclusion of data from a simultaneous measurement of the p(˜+; + )n reaction, has been recently published by the MAMI collaboration [54]. They performed both energy-dependent and energy-independent multipole analysis of their data set, the energy dependence being constrained through >xed-t dispersion relations [59]. The >nal result of this new analysis is E2=M 1 = (−2:5 ± 0:1stat ± 0:2sys ), substantially con>rming their previous assignment. The dependence of the E2=M 1 ratio upon the four momenta transfer can be studied through electroproduction experiments. Recently, high-precision measurements have been made at the MIT-Bates Linear Accelerator at Q2 = 0:126 GeV=c2 for the reaction p(e; e p)0 [60], yielding the value E2 (2.63) = (−2:1 ± 0:2stat+sys ± 2:0model )% : M1 It is worthwhile to stress the large uncertainty in the >nal result due to some unavoidable model dependence in the analysis of data. This uncertain state of aEairs is quite understandable, in view of the many uncertainties implicit in the determination of E2=M 1. In the >rst place, the intensive experimental activity in the last several years has shown that the result is sensitive to the structure of the database being considered. On the other hand, in absence of a >rm dynamical scheme for the N interaction, one is forced to implement unitarity through some phenomenological method, such as the K-matrix approach, which unavoidably introduces some model dependence. Various dynamical models of the N system have been proposed in the past to remove or at least to reduce the ambiguities in the extraction of the +N ↔ transition amplitudes from photoproduction or electroproduction data [61], with particular attention to a clearer separation between background and resonance contributions to the amplitudes. Due to our limited understanding of the N dynamics starting from fundamental principles, however, some degree of model dependence is always present. From this point of view, an interesting and original analysis has been performed in Ref. [62]. The essential point of this paper is the separation of a resonant contribution from the total transition amplitude, that implies the introduction of a component into the N scattering state, which vanishes in the asymptotic region. The explicit form of the wave function, therefore, depends upon the model used to de>ne the isobar contribution, and can be always changed by means of a unitary transformation, much in the same way as one can always pass from one representation to another in quantum mechanics. The argument is illustrated through a model calculation, where the pion production multipoles consist of three terms, a background contribution, a “bare” resonant multipole, and a vertex-renormalization term, where the

326


3=2 3=2 Fig. 6. Multipole M1+ or E1+ pion photoproduction amplitudes.

produced pion interacts with the nucleon to give rise to resonance propagation before being emitted in the >nal state. This is diagrammatically illustrated in Fig. 6. One can show that, once the model is speci>ed, one can construct a whole class of unitary equivalent models which, while providing the same >t to the data, predict however diEerent resonance amplitudes. Had these conclusions a general validity, they would be a strong challenge to theoreticians. Indeed, a meaningful comparison of the E2=M 1 ratio with the results of QCD-inspired models would require a unique, consistent treatment of the dynamics. Needless to say, these considerations are also relevant with reference to the old question of the role played by isobar components in nuclei. In principle, information on the deviation from spherical symmetry can be also obtained by looking at other observable quantities. A noticeable example is the quadrupole moment Q. For the nucleon, angular momentum selection rules imply that it has to be zero even in presence of d-state admixtures due to inter-quark tensor forces. This is not true for the , because of its higher spin, and actually quark models lead to the prediction Q ∼ −0:09 fm2 [44]. Unfortunately, the experimental determination of the quadrupole moment would require the elastic scattering of photons on this unstable particle. The situation is less unsatisfactory as far as the magnetic moment F is concerned. In this case, one can estimate its value from radiative scattering on proton targets. However, a model-independent analysis of the data is again impossible, since a dynamical model of the N scattering amplitude is required. This explains the large uncertainty in our present knowledge of this quantity (3:7 ¡ F ¡ 7:5 nuclear magnetons according to PDG), so that a vis-[a-vis comparison between experiments and quark-model predictions is not possible. 3. Theoretical models for the N – interaction The embedding of the isobar into the nuclear environment requires the meson–baryon–baryon (N, N, : : :) vertices as basic building blocks. Once these coupling Lagrangians have been given, standard non-relativistic reduction techniques provide the various transition potentials one needs in nuclear-structure and nuclear-reaction models. Whereas the basic structure of the couplings can be determined on the ground of general invariance requirements, one is left with the problem of providing information about their strengths, and with the oE-shell freedom associated with the vertex form factors. A direct experimental determination of the strengths is obviously hampered by the unstable nature of the intervening particles. One is therefore forced to rely on relations between meson–NN and meson–N strengths following from simple quark-model or strong-coupling considerations. In spite of the ?ourishing of quark-model approaches to meson and baryon structure, these relations still retain their value, and we shall review them, stressing the essential role played in their derivation by general algebraic properties rather than by detailed dynamical assumptions. In this perspective, the knowledge of the NN and NN coupling strengths is a necessary prerequisite, apart from the obvious basic nature of these quantities. For this reason, we discuss the present


327

status of their determination >rst, devoting particular attention to the NN coupling constant, which has been the subject of great debate in the nineties. In so doing, we emphasize the role played by dispersion relation constraints in the analysis of the experimental data. We then touch the problem of the oE-shell behavior of the meson–baryon–baryon vertices, where more explicit dynamical models are required, whether one starts from an eEective approach in terms of hadron degrees of freedom, or from more microscopic quark models. The comparison of the predictions based on the assumed model N interactions with nuclear phenomenology entails the solution of a coupled-channel (CC) problem. One is then confronted with two questions. On the one hand, one has to ascertain to what extent the solution of these CC equations—which are essentially non-relativistic—can simulate higher-order contributions of covariant >eld-theoretic descriptions of hadron dynamics. This is just one aspect of the much more general problem of the role played by relativistic eEects in nuclear physics. On the other, one has to embed the CC N system in the nuclear environment. The latter question has received the more satisfactory, albeit not conclusive, answer in two limiting situations, for few-nucleon (A 6 3) nuclei, and for nuclear matter. In the second half of this chapter we consider some aspects of these complex topics, deferring the analysis of isobar excitations in nuclear matter to the last part of the paper. 3.1. Basic meson-exchange models for the N interaction An eEective description of the as an elementary particle can be given in quantum >eld theory through the Rarita–Schwinger formalism. As is well-known, this can be accomplished without ambiguities only for an on-shell , since its oE-shell propagation and coupling to other hadrons is aEected by the presence of unphysical spin 12 degrees of freedom [64,65]. Fortunately, in most applications of low- and medium-energy nuclear physics one can limit to a non-relativistic description, which mimics the meson-theoretic approach to the NN problem. To see how this comes out let us recall that pion–nucleon coupling can be described either through the pseudoscalar (PS) interaction Lagrangian ˜ LPS = −g X (x)i+5 ˜ (x) · (x) ;

(3.1)

or by the pseudovector (PV) coupling Lagrangian LPV =

fNN X ˜ : (x)+F +5 ˜ (x) · 9F (x) m

(3.2)

˜ represents the isovector Here, m is the pion mass, (x) is the spinor >eld for the nucleon, and (x) pion >eld, with ˜ the usual Pauli vector in isospin space. In the non-relativistic limit one gets from these interaction Lagrangians the eEective coupling Hamiltonian HNN = −

fNN ˜ ; ( · ∇)(˜ · ) m

(3.3)

provided that the PS and PV coupling constants are related by fNN g = m 2M

(3.4)

328


(a)

(b)

(c)

(d)

(e)

(f)

Fig. 7. Processes with NN , N and vertices. The full and double lines represent the nucleon and , respectively, and the dashed lines the exchange of diEerent mesons.

with the usual Pauli matrix-vector, and M the nucleon mass. In the static approximation the eEective Hamiltonian (3.3) leads to the well-known NN potential in momentum space [8] Vs (NN ) = −

2 (1 · q)(2 · q) fNN ˜1 · ˜2 : 2 m q2 + m2

(3.5)

Time-retardation eEects can be taken into account considering the full Feynmann propagator for the exchanged pion, bringing the energy-dependence ! of the pion in the potential, that is V (NN ) =

2 (1 · q)(2 · q) fNN ˜1 · ˜2 : m2 !2 − q2 − m2

(3.6)

The N coupling can be described in a similar way, starting from the Lagrangian LN =

fN (i m

F ˜ T N

· 9F ˜ + h:c:) ;

(3.7)

˜ a transition operator connecting isospin 1 where F is the Rarita–Schwinger >eld for the , and T 2 and 32 states [12,66]. According to the Wigner–Eckart theorem, its matrix elements between isospin eigenstates can be simply de>ned as the Clebsch–Gordan coeScients 32 tz |T | 12 tzN = 1 12 tzN )| 32 tz :

(3.8)

Lagrangian (3.7) in the non-relativistic limit leads to the eEective coupling Hamiltonian HN = −

fN + ˜ + · ˜ S · ∇T m

(3.9)

˜ The various coupling and transition with S a transition spin operator de>ned in the same way as T. potentials between NN and N states can be now written down on the ground of straightforward perturbation theory. Thus, the NN → VN transition corresponding to Fig. 7 is given by [8,12] V (NN → VN ) =

fNN fN (S+ 1 · q)(2 · q) ˜ + T · ˜2 : 2 m !2 − q2 − m2 1

(3.10)


329

This expression can be immediately obtained from the NN potential (3.6) by appropriately replacing ˜+ the coupling constants, and substituting the operators 1 and ˜1 with S+ 1 and T1 , respectively. Similarly, the direct and exchange interactions V (N → N) and V (N → VN ) become fNN f (1 · q)(2 · q) ˜1 · ˜ 2 (3.11) V (N → N) = m2 !2 − q2 − m2 and 2 (S+ fN 1 · q)(S2 · q) ˜ + ˜ T · T2 ; (3.12) m2 !2 − q2 − m2 1 where the spin–isospin transition operators and ˜ in V (N → N) refer to the vertex. They are uniquely de>ned once their reduced matrix elements between spin or isospin 32 states are >xed. √ ˜ 3 = 2 15. One can take [8] 32 32 = 32 2 The potentials given above describe the simplest one-pion exchange interaction between the hadrons. At the next level of complexity in the number of exchanged mesons one has the two-pion exchange contributions [8,67]. For the NN case, symmetry considerations severely restrict the exchanged quantum numbers. Indeed one can have only the scalar–isoscalar exchange with J = 0+ and T = 0 spin and isospin quantum numbers, or the vector–isovector exchange with J = 1− ; T = 1. In the vector–isovector channel a prominent role is played by the vector meson meson, a proper two-meson resonance with a physical decay width. It can be described in an eEective way by the relativistic Lagrangian gT X (x)FI (x)9I ˜ · ˜ F (x) (3.13) L = −gV X (x)+F (x)˜ · ˜ F (x) + 2M with ˜ F (x) a four-vector in con>guration space and a vector in isospin space associated, to the meson. In the non-relativistic limit one gets the spin- and isospin-dependent -exchange potential 2 fNN (1 × q) · (2 × q) V = ˜1 · ˜2 ; (3.14) m2 !2 − q2 − m2

V (N → VN ) =

with its characteristic transverse vector coupling × q with respect to the longitudinal coupling · q appearing in the scalar -exchange interaction. The NN coupling constant in (3.14) is related to the vector and tensor coupling strengths gV and gT by fNN gV (1 + gT =gV ) : = (3.15) m 2M Making use of the identity (1 × q) · (2 × q) = 1 · 2 |q|2 − (1 · q)(2 · q) : Eq. (3.14) can be written in the form 2 fNN 2 1 1 2 2 V = − 2 2 · |q| − [( · q)( · q) − · |q| ] ˜1 · ˜2 ; 1 2 1 2 1 2 m q + m2 3 3

(3.16)

(3.17)

which shows that the Pauli operators i and Bj enter into the spin–isospin-dependent -exchange potential in the same way as they come into play in the -exchange NN interaction. By the way, this result exhibits the well-known fact that -exchange produces a tensor NN potential which partially counteracts the -exchange contribution.

330


These considerations can be immediately extended to the N system. Owing to the spin and isospin quantum numbers of the intervening hadrons one has that -exchange can contribute also ˜ to the N interaction. Thus, replacing the Pauli spin and isospin operators and ˜ with S and T, respectively, one gets for the exchange N → VN transition V (N → VN ) =

2 fN (S+ 1 × q) · (S2 × q) ˜ + ˜ T1 · T2 : 2 m !2 − q2 − m2

(3.18)

Similarly, the direct interaction V (N → N) can be written as V (N → N) =

fNN f (1 × q) · (2 × q) ˜1 · ˜ 2 : m2 !2 − q2 − m2

(3.19)

3.2. Meson–baryon couplings In the Lagrangians and potentials described in the previous section masses and coupling constants appear as phenomenological parameters. Here, we shall brie?y consider the coupling strengths, as they can be determined on the ground of experimental analysis or simple theoretical models, starting from the pion–nucleon coupling, which, because of its basic nature in nuclear physics, has been the subject of very many investigations in the last forty years. This quantity can be determined through diEerent routes, starting from N scattering data, NN forward dispersion relations, and=or NN partial-wave analysis. This topic has been the subject of great debate in recent years, therefore it is worthwhile to consider it in some detail. 3.2.1. The NN coupling constant It is impossible to do full justice to the enormous amount of work which has been devoted to the determination of the NN coupling constant. The >rst estimate of this parameter has been presumably given in 1950, on the ground of photoproduction data for charged mesons [68], when N and NN scattering experiments were still in their infancy, and far less accurate than photoproduction experiments. The Kroll–Ruderman Theorem [69], which states that the pion photoproduction amplitude at threshold is just proportional to the NN coupling strength, allowed Bernardini and Goldwasser 2 [70] to produce the value fNN =4 = 0:065 in the mid-1950s. Under a >eld-theoretic point of view, a major achievement at those days has been represented by the static Chew–Low model [71–73]. There, the interaction between a recoilless nucleon and the pion >eld is described by the pseudovector eEective Hamiltonian (3.3). The whole dynamical content of the theory can be subsumed into the Low equation, which expresses the N transition amplitude in terms of the scattering and production amplitudes starting from the same initial state or ending on the same >nal one. Under a dispersion-theoretic point of view, the Low equation essentially expresses the requirements dictated by unitarity, crossing and analyticity. Note, however, that in general the scattering amplitudes appear in the Low equation both on- and oE-the energy shell. As a consequence of the static limit, in the Chew model this equation can be written only in terms of physical scattering amplitudes. In a fully relativistic theory, this is not possible, and proper dispersion relation techniques have to be employed. In spite of the drastic approximations inherent to the Chew–Low model, it turned out to be quite successful in describing the low-energy P-wave 2 pion–nucleon scattering, providing the value fNN =4 = 0:08 for the pion–nucleon coupling [74].


331

The very >rst application of forward dispersion relation to N data to determine the pion–nucleon coupling is due to Haber–Schaim [75] and Davidon and Goldberger [76]. As explained in more detail in Appendix A, the essential ingredient of these analysis is the dispersion relation that relates the total cross section ± for ± N scattering to the forward N isospin odd transition amplitude T (−) (!), evaluated at incoming-pion kinetic energy !, namely ∞ !2 Re T (−) (!) k(! ) − (! ) − + (! ) − 2P (!N2 − !2 ) d! 2 2 − ! 2 ! 4 ! ! m !N2 − !2 ∞ k(! ) 2 = − 4f + (− (! ) − + (! )) d! ; (3.20) 2 42 ! m where k(!) = !2 − m2 . The pole corresponding to the nucleon propagation in the intermediate state is located at !N ≡ −m2 =2M . The use of dispersion relations represented a great achievement, since at those times a >rm dynamical scheme for strong interactions was lacking, and one had to look for constraints which were independent upon detailed assumptions about the dynamics of strongly interacting particles. Under this point of view, dispersion relations represented an ideal tool, based as they were upon general requirements only, such as analyticity hopefully related to causality, unitarity and crossing symmetry. Under a practical point of view, the above relation states that the quantity on the left-hand side must be a linear function of the pion energy squared, so that an extrapolation to the nucleon pole !N could give the N coupling strength. The result of these pioneering applications of dispersion relation constraints (DRC) was f2± NN =4 = 0:08 ± 0:01 for the coupling in its pseudovector form. At the beginning of the 1980s Koch and Pietarinen [18], and Kroll [77] obtained what has been considered for several years the textbook values of the ± N and 0 N coupling constants, respectively. The former used dispersion relations at >xed momentum transfer t, as functions of the variable 1 1 I≡ (3.21) (s − u) = (2s − 2m2 − 2M 2 + t) ; 4M 4M to constrain the ± N transition amplitudes. The quantity I exhibits clearly the crossing-symmetric properties of the formalism since one has I ↔ −I when s ↔ u. At the same time it simply reduces to the pion kinetic energy ! at forward direction t = 0. We give in Appendix A the detailed derivation of the crucial relation, here we limit ourselves to quote the result, i.e., ∞ Im B+ (I ; t) Im B− (I ; t) dI I (IB ± I) ∓Re B± (I; t) ± P + I0 I ∓ I I ± I I g2 ± NN ˜ t)(IB ± I) : + B(0; (3.22) M The quantities B± represent the invariant amplitudes for elastic ± N scattering, I0 ≡ m + t=4M , and IB = (t − 2m2 )=4M is the location of the nucleon pole, corresponding to s = M 2 or u = M 2 , depending upon the momentum transfer t. Eq. (3.22) represents the generalization to t = 0 of Eq. (3.20). Its value at the unphysical point I = IB , corresponding to the single-particle intermediate state with nucleon propagation, gives the required pion–nucleon coupling constant. Koch and Pietarinen used Eq. (3.22) in their partial-wave analysis (PWA) of pion–nucleon scattering data. Needless to say, the long-range, isospin-breaking electromagnetic eEects have to be =

332


taken into account before Eq. (3.22) can be applied. How this is achieved is brie?y reviewed in Appendix B. To use the scattering parameters coming from the PWA in Eq. (3.22), one has to guarantee that partial-wave and >xed-t analysis agree with each other and with the experimental data. This has been achieved by Koch and Pietarinen by requiring the consistency between the transition amplitudes from the PWA and those appearing in the >xed-t dispersion relations. As the result of their analysis, they obtained the value g2 ± NN = 14:28 ± 0:18 4 for the pseudoscalar coupling constant, which corresponds to

f2± NN m± 2 g2 ± NN = = 0:079 ± 0:001 4 2Mp 4

(3.23)

(3.24)

for pseudovector coupling. The 0 N coupling has been determined by Kroll [77] by means of an elegant application of forward dispersion relations to NN scattering. We shall brie?y illustrate how the method works, following essentially Ref. [8]. In the present case it is convenient to choose the center-of-mass momentum squared | |2 as variable to write down the dispersion relation, since one has s = 4(M 2 + | |2 );

t = 0;

u = −4| |2 ≡ −4z :

(3.25)

According to the general philosophy of dispersion theory, to establish the dispersion relation let us >rst identify the possible intermediate states dictated by unitarity in the direct (s) and crossed (u) channels. There is obviously the right-hand scattering cut starting at z = 0 (s = 4M 2 ), plus the cuts starting at u = n2 m2 (n = 2; 3; : : : ; ), corresponding to multi-pion exchange in the u (N NX → N NX ) channel. Because of Eq. (3.25), these cuts map into the left-hand cuts from −n2 m2 =4 in the z-plane. To these contributions one has to add the pole terms associated to one-pion intermediate states. To be de>nite, let us consider pp scattering, in which case only 0 propagation is possible in the crossed channels, both describing pX p → pX p elastic scattering as shown in Fig. 8. There is no single-particle intermediate state in the direct channel, on the other hand the u-channel has the intermediate 0 propagation, which leads to the pole term in the scattering matrix Tu(p) ∼

g2 0 NN g2 0 NN = − : u − m2 4z + m2

(3.26)

A similar contribution is present also in the t-channel, Tt(p) ∼

g2 0 NN g2 0 NN = − : t − m2 m2

(3.27)

Non-relativistically, these amplitudes can be simply interpreted in terms of the crossed and direct one-pion-exchange Born graphs (see Fig. 9). Indeed, for the direct process, in which the two protons emerge with momenta and − , one has T (d) ∼ −

g2 0 NN ; q2 + m2

(3.28)


333

Fig. 8. Intermediate 0 propagation in ppX → ppX scattering. Fig. 9. Direct and exchange Born graphs for 0 exchange in pp scattering.

where q = − is the momentum transfer, whereas for the crossed graph, where the role of the emerging protons is exchanged and the momentum transfer is + , one gets T (e) ∼ −

g2 0 NN : ( + )2 + m2

(3.29)

Taking into account that for forward scattering the momentum transfer q vanishes and = , one immediately gets Eqs. (3.27) and (3.26). Since the two Born terms have to be put together with a minus sign in between, because of the fermionic nature of the nucleons, one >nally gets the pole contribution to the forward pp amplitude Rz T (pole) = (3.30) z + m2 =4 representing a pole at z = −m2 =4, with residue R proportional to the 0 pp coupling constant. All the above analyticity properties can be summarized into the dispersion relation in the z-plane ∞ 2 z −m Im T (z ) z Im T (z ) Rz + d z dz ; + (3.31) Re T (z) = Re T (0) + P z + m2 =4 −∞ z (z − z) 0 z (z − z) where a subtraction in z = 0 has been performed. One can include all the singularities in the unphysical region (z ¡ 0) in a unique integral from −∞ to 0, and re-write Eq. (3.31) as ∞ z Im T (z ) z 0 Im T (z ) Re T (z) − Re T (0) − P (3.32) dz = d z ≡ (z) : 0 z (z − z) −∞ z (z − z) The function (z) clearly collects all the contributions from one- and multi-pion propagation in the pX p system in the unphysical region of the pp scattering process, and is referred to as the discrepancy function [78,79]. It is a measure of what is lacking in the information coming from the physical region, in order to satisfy the dispersion relation. Stated in another way, the unphysical region can be regarded as re?ecting the physics of the pX p channel, mostly in a kinematic domain far below the physical pX p threshold. In the spin-singlet (S =0) channel, the dispersion relation (3.32) is particularly eScient in isolating the pion pole, owing to strong selection rules. In fact, with reference to the N NX system, one has that parity is given by J =(−1)L+1 , L being the orbital angular momentum, whereas for the exchange one has J = (−1)L = (−1)J . It follows that, since J = L in the singlet channel, two-pion exchange is forbidden by parity conservation. Other selection rules follow from G-parity conservation.

334


Since G is related to the charge-conjugation operator C and the total isospin by G = C exp(iTy ), one has, for the corresponding eigenvalue G = (−1)L+S+T for the N NX system, and G = (−1)n for a system of n pions. As a consequence, an even value of L + S + T entails the exchange of an even number of pions, while an odd value must correspond to the exchange of an odd number of pions. For forward pp scattering (T = 1) in the singlet channel, one >nds again that two-pion exchange is forbidden, while one-pion exchange can occur in the unnatural parity channel 0− . Since, as remarked before, for pp scattering one-pion exchange can occur only through a neutral pion, one has a pole located at z = −m20 =4 according to Eq. (3.30), corresponding to the extrapolated laboratory kinetic energy m2 0 2 0 = z = − −10 MeV : TLab M 2M Two-pion exchange being totally forbidden, the next singularity, corresponding to three-pion exchange, is a cut starting at TLab = −9m20 =2M −90 MeV. Thus, the eEects of one-pion exchange, and the associated pole are expected to be well exhibited in the singlet discrepancy function with respect to other exchange mechanisms in forward pp scattering. These facts have been exploited by Kroll to extract the 0 pp coupling from pp scattering data. The quantities on the left-hand side of Eq. (3.32) can be all >xed by experiment, the real part of the forward amplitude Re T (z) from a phase-shift analysis or by Coulomb interference, and Im T (z) from the optical theorem. It is worthwhile to note, here, that the singlet discrepancy function s is a rapidly varying function in the extrapolation region, since it vanishes by de>nition at TLab = 0, and becomes in>nite at the 0 meson pole near TLab . For this reason it is more convenient to work with the reduced discrepancy ˜ function s 0 | TLab + |TLab s ; (3.33) ˜ s ≡ TLab which allows a smooth extrapolation to the pion pole. The result of Kroll’s analysis has been g2 0 NN = 14:52 ± 0:40 (3.34) 4 for the pseudoscalar coupling, corresponding to

f20 NN m± 2 g2 0 NN = = 0:080 ± 0:002 : (3.35) 4 2Mp 4 for the pseudovector coupling constant. Eqs. (3.24) and (3.35) have been accepted as the standard values for charged- and neutral-pion coupling to nucleons for several years. They are consistent with charge independence, or with small charge independence breaking (CIB) eEects. This general consensus was shattered in 1983 by the Nijmegen group, that found smaller values for the couplings, on the ground of their energy-dependent partial-wave analysis of the low-energy pp scattering data [80]. The basic philosophy of the Nijmegen approach is to take into account as closely as possible the energy variation of the phase shifts in a smooth way, while taking advantage of the present knowledge of the tail in the pp interaction. The Nijmegen group has devoted a lot of eEorts to re>ne their analysis; in particular, they extended the database to include both pp and np scattering [81], as well as N NX data [82]. The eEects of vertex structure and the in?uence of the backward np cross section data were studied in particular detail, and found, according to the authors, to be negligible [83]. The result of all this work can be


335

summarized in the “recommended” values f20 NN = 0:0745 ± 0:0006; 4

g2 0 NN = 13:47 ± 0:11 ; 4

(3.36)

f2± NN = 0:0748 ± 0:0003; 4

g2 ± NN = 13:52 ± 0:05 4

(3.37)

for the neutral- and charged-pion coupling constants. Independently from de Swart and collaborators, the VPI group started from their partial-wave solution of the ± N scattering data [14], using the >xed-t dispersion relation (3.22). They searched for an optimal value of g± NN [22]. To explore the sensitivity of the >tting procedure to the value of the coupling, the solutions were generated for a grid of N coupling constants and isoscalar scattering lengths (g2 =4; a(+) ), looking for a minimum of C2 in the C2 (g2 =4) plane. The result has been f2± NN = 0:076 ± 0:001; 4

g2 ± NN = 13:75 ± 0:15 : 4

(3.38)

Up to this point, the new determinations of the pion–nucleon coupling constant seemed to point to lower values for these parameters than usually assumed in the past. A con?icting result, however, has been recently obtained by the Uppsala collaboration, in a dedicated experiment on backward np scattering [84]. It is expected that backward np cross section ought to be sensitive to the pion– nucleon coupling as remarked by Chew [85] in the 1950s. It is worthwhile here to brie?y recall Chew’s physically transparent and elegant argument. Let us consider two nucleons with CM momenta and before and after the collision. The direct Born amplitude for the exchange of a neutral pion is given by Eq. (3.28)) with q2 = 2| |2 (1 − cos ), where represents the scattering angle in the CM system. This amplitude is clearly singular for cos = 1 + m2 =(2| |2 ). In particular, contributes to the pole in the forward direction for pp scattering, as we have already seen when discussing the discrepancy function. If, on the other hand, the analysis refers to np scattering and ± exchange, so that the incoming proton emerges as a neutron and vice versa, one has the crossed Born graph, with the corresponding Born amplitude TBc = −

g2 ± NN ; 2| |2 (1 + cos ) + m2

(3.39)

singular in the backward direction, namely for cos = −(1 + m2 =(2| |2 )) ≡ cos P . These considerations prompted Chew to suggest an extrapolation of the backward np cross section to the unphysical point cos P in order to determine the charged-pion–nucleon coupling constant. Since the cross section scales as |TBc |2 , it is singular in the backward non-physical region of the scattering angle. To perform the extrapolation, it is more convenient to look at the smooth function y(x) ≡

1 2 d(x) x d gR4

(3.40)

where x ≡ q2 − m2 , and try an expansion in powers of x, so that y(0) gives the >rst term of the expansion, a0 = (g± NN =gR )4 , with the constant gR >xing a reference scale.

336


In the Uppsala analysis a modi>cation of the Chew method was employed, to analyze the diEer◦ ◦ ential cross sections measured at 162 MeV, in the angular range 72 6 CM 6 180 , >nding g2 ± NN f2± NN = 0:0803 ± 0:0014; = 14:52 ± 0:26 ; (3.41) 4 4 in very good agreement with the Koch–Pietarinen and Kroll determinations, and at variance with the Nijmegen–VPI conclusions. These results have been recently con>rmed by new analysis at 96 MeV [86]. In the light of the above results, it is fair to say that one is here confronted with two diEerent philosophies. The Nijmegen–VPI approach stresses the smooth, energy-averaged features of the data, and points to achieve, through an energy-dependent analysis an overall view of the N and NN interactions. The Uppsala group emphasizes the crucial role played by an accurate analysis of a selected data set, where all details of the procedure are subjected to rigorous scrutiny. Whatever the adopted attitude might be, one obviously expects that diEerent methods, if equally valid, converge sooner or later to consistent results, a goal which presently has not yet been achieved. One may ask, in any case, what consequences that 5% diEerence in the value of g2 =4 would have for our understanding of nuclear phenomena. A >rst constraint for gNN is given by the Goldberger–Treiman relation [87]. A well-known consequence of chiral symmetry, it relates the strong-interaction constant gNN to quantities characterizing weak interactions, the weak axial coupling gA and the pion decay constant f , M ; (3.42) gNN = gA f where both gA and gNN are evaluated at zero momentum transfer q2 = 0, and not at the pion pole q2 = m2 . As the experimental determination of f has improved in the years, Eq. (3.42) has been veri>ed with increasing accuracy, up to the present 2% level [88], without form-factor eEects. A decrease of gNN would improve the agreement with the Goldberger–Treiman relation even more. In some non-linear -models this could even overshoot the goal, owing to the presence of additional quark-mass terms. Another simple relation between the pion–nucleon coupling and directly observable quantities is provided by the Kroll–Ruderman theorem [69]. It relates fNN to the pion photo-production amplitude in the long-wavelength limit, and can most easily obtained by considering the static isovector Hamiltonian (3.3) in the photon >eld A and making the minimal substitution, ∇ → ∇ ∓ ieA, where ∓ applies for positive or negative pions, respectively. After Fourier analysis of the electromagnetic >eld, one gets when the photon momentum k approaches zero fNN ± (q)|T |+(k) ∼ ±e · B∓ ; (3.43) m where describes the photon polarization, and the isospin operator B∓ obviously refers to the target nucleon. In virtue of Eq. (3.43), the pion photoproduction cross sections at threshold are proportional 2 to fNN . These cross sections are presently known with a ∼ 2% accuracy, so that a 5% change in 2 fNN would imply a change of about 2.5 standard deviations in the cross section values in the same direction. This means an overall discrepancy of about 3:5 standard deviations, which would not be easy to accommodate in our present understanding of photo-reaction phenomena. In principle, a good testing ground of pion physics is represented by the deuteron. In particular, the deuteron quadrupole moment and the asymptotic D=S ratio are known to satisfy a linear relation


337

Fig. 10. NN scattering processes in the s (NN → NN ) and t (N NX → N NX ) channels.

2 for any reasonable local NN potential [89]. This allows one to infer that a 2% variation in fNN would lead to a 1% change in the D=S ratio, with the quadrupole moment >xed at its experimental value. The problem here is that diEerent experimental determinations of D=S do not yet agree to the required precision, in spite of the 2% accuracy claimed for this quantity. Other consequences of a smaller value for fNN would emerge in the tensor=vector coupling ratio in NN coupling, and, obviously, in the value of meson– couplings, when they are related to fNN through quark model considerations. These are the next subjects to be discussed.

3.2.2. Other meson–nucleon couplings The determination of the NN coupling constant is intimately related to the evaluation of the two-pion-exchange contribution to the NN interaction. The dispersion-theoretic approach to nuclear forces has been vigorously developed in the 1970s, leading to the >rst quantitative model of the nuclear potential at intermediate NN distances, as exhaustively described in Refs. [90,91]. This approach enforces in a consistent way the constraints due to analyticity and unitarity, as well as to the experimental information about N scattering and annihilation processes. Let us brie?y recall how one proceeds. The NN scattering amplitude in the s-channel is related, via crossing symmetry, to the N NX → amplitude in the t-channel, as graphically depicted in Fig. 10. The reason for this indirect way of proceeding is that unitarity in the s-channel would require the knowledge of all possible processes that could occur in NN scattering, whereas in the crossed channel one naturally exhibits through unitarity intermediate states with one, two, or any number of pions, which can be regarded as meson exchanges in the other channel. One thus obtains a decomposition of the NN scattering amplitude into terms corresponding to the exchange of an increasing number of pions. In particular, once the 2 unitarity contribution is exhibited in the N NX → amplitude, one can have information about the two-pion-exchange term M2 (s; t) in the NN transition amplitude, through analytic continuation of the annihilation amplitude. For two intermediate pions in the relative p-wave state, the exchanged quantum numbers are J = 1− , T = 1, because of Bose statistics, corresponding to the quantum numbers of the resonance. Since the N NX → amplitude can be related, via dispersion theory, to the (p-wave) and N amplitudes, one can link the two-pion component of the NN interaction to and N physics.

338


Assuming Mandelstam analyticity in the variables s and t, M2 (s; t) can be written, ignoring spin and isospin degrees of freedom [8,90,91] 1 ∞ #2 (s; t ) dt ; M2 (s; t) = (3.44) 4m2 t − t where the spectral function #2 (s; t) can be extracted from the p-wave N NX → helicity amplitudes. At low energy (s 4M 2 ), in the static approximation t −|q|2 , with q the NN three-momentum transfer, Eq. (3.44) can be written, 1 ∞ #2 (t ) M2 (|q|) = dt : (3.45) 4m2 t + |q|2 After a Fourier transform, the latter can be identi>ed with a superposition of Yukawa potentials weighted by the spectral function #2 [8]. These considerations can be extended to the spin–isospin-dependent part of the -exchange NN potential (3.14), obtained from the eEective NN Lagrangian (3.13). In analogy to Eq. (3.14) one can write V = −(1 × q) · (2 × q)F (|q|)˜1 · ˜2 with 1 F (|q|) ≡

∞

4m2

#2 (t ) dt : t + |q|2

(3.46)

(3.47)

If two-pion exchange is approximated with the propagation of a narrow resonance, so that, consistently with dispersion theory, the spectral function #2 is simply given by a pole term 2 fNN #2 (t) = (t − m2 ) ; m2

Eq. (3.46) reproduces the -exchange potential (3.14) in the static approximation. This suggests the de>nition 2 fNN 1 ∞ = #2 (t ) dt (3.48) m2 4m2 for the eEective NN coupling constant which takes into account the dispersive eEects due to the >nite width of the meson and the 2 continuum. It is related to the vector and tensor couplings gV and gT by (see Eq. (3.15)) 2 2 fNN 2 (1 + C ) = g : V m2 4M 2

The above considerations show that the overall strength of the NN coupling in the nucleon–nucleon potential can be most eSciently measured by the quantity [92] 2 GNN g2 (3.49) ≡ V (1 + C )2 : 4 4 The best source of information about NN coupling is represented by the electromagnetic form factors of nucleons. Let us recall that the matrix element between nucleon states of the electromagnetic


339

current JF as a function of the invariant momentum transfer squared t = (p − p)2 , is given, on the ground of relativistic invariance, by [93] i I FI (p − p) F2 (t) u(p) N (p )|JF (0)|N (p) = eu(p ) +F F1 (t) + (3.50) 2M with p and p the initial and >nal four momenta of the nucleon, respectively. The Lorentz scalars F1 (t) and F2 (t) are the Dirac and Pauli form factors, associated to the charge and anomalous magnetic moments of the nucleon Cp and Cn . For zero momentum transfer they are normalized according to F1(p) (0) = 1; F1(n) (0) = 0;

F2(p) (0) = Cp ; F2(n) (0) = Cn :

(3.51)

In terms of the exchange of well-de>ned isospin quantum numbers the e.m. current can be more conveniently decomposed into isoscalar and isovector components, with the corresponding de>nition of the isoscalar and isovector form factors Fi(s) (t) = 12 (Fi(p) (t) + Fi(n) (t));

Fi(v) (t) = 12 (Fi(p) (t) − Fi(n) (t)) ;

(3.52)

(i = 1; 2). Because of Eq. (3.51) their normalization for t = 0 is F1(s) (0) = F1(v) (0) = 12 ;

F2(s) (0) = 12 (Cp + Cn );

F2(v) (0) = 12 (Cp − Cn ) :

(3.53)

A dynamical theory for the nucleon e.m. form factors can be developed on the assumption that one-photon exchange dominates electron–nucleon scattering, which is quite reasonable in view of the smallness of the >ne-structure constant nite value, in order to exclude higher resonances. HP choose tmax =50m2 ∼ 1 GeV2 in their calculations. Their results have been essentially con>rmed by Grein, who combined forward dispersion relations with the information coming from the HP partial-wave N NX amplitudes to analyze NN and N NX forward scattering data [101]. He obtained gV2 =4 = 0:55; gT =gV = 6:0. Presently, the recommended values are [102] gV2 = 0:55 ± 0:06; 4

C =

gT = 6:1 ± 0:6 : gV

(3.58)

The analysis of electromagnetic nucleon form factors has been reconsidered in Ref. [95], imposing new constraints given by low-energy neutron–atom and electron–proton scattering data, which establish the nucleon mean square radius and proton charge distribution. Constraints from perturbative QCD where also considered, which >x the behaviour of the form factors at large momentum transfers. The form factor parameterization has then three pole terms of the form (3.56), plus the contribution as given by HP, and an extra term enforcing the QCD asymptotic behavior, i.e. 50m2 Im Fi (t ) m2V gi 1 QCD (v) Re Fi (t) = P dt + : (3.59) + ci F˜ i 2 4m2 t −t fV m V − t V The analysis performed with this parameterization turned out again to be consistent with the values (3.58) for the NN coupling constants. It is worthwhile to compare GNN , determined from dispersion analysis, with the values given in modern NN potentials based upon >eld-theoretic approaches. If the standard values (3.58) are


341

inserted into Eq. (3.49) one gets 2 GNN 28 ; 4 whereas the VDM value of C yields 2 GNN 12 : 4 In the full Bonn potential, on the other hand, one has [103] gV2 =4 = 0:84, C = 6:1, thereby obtaining 2 GNN 42 ; 4 while the Bonn B potential, which parameterizes the full Bonn through one-boson exchange, gives, with gV2 =4 = 0:90, C = 6:1 [67],

2 GNN 45 : 4 Clearly, the eEective -coupling parameter in the Bonn potentials is much closer to the value dictated by dispersive analysis than to the VDM determination. One arrives at the same conclusion for the Nijmegen soft-core interaction, once the corresponding coupling constant gV2 =4 = 0:795, with C = 4:221, is renormalized by the overall form factor exp(−|q|2 =N2 ), evaluated at the -meson pole −|q|2 = t = m2 . With N = 964:52 MeV and m = 770 MeV one >nds [92] 2 GNN 41 : 4 Finally, let us observe that these results allow for a direct comparison between the eEective -meson coupling strength fNN and the meson–nucleon coupling constant fNN . Inserting the values (3.58) 2 into Eq. (3.15) one >nds, with fNN =4 0:08, 2 2 fNN fNN 2:0 : m2 m2

(3.60)

Evidence for a strong NN coupling C ∼ 6 from NN scattering observables has been discussed in Ref. [92]. In Fig. 11 the predictions for the 01 mixing parameter by the full Bonn, Bonn B, Paris, and Nijmegen potentials are compared with the experimental data. Predictions by the Reid potential which has a weak NN coupling and by a model without -exchange contributions are also given. One can see that the “strong ” interactions are in good agreement with the experimental points, whereas the results of “weak ” calculations severely overestimate the data in the energy region between 200 and 300 MeV, where second-order contributions from the overall tensor force are negligible [92]. One may conclude that, as far as the NN coupling is concerned, there is presently a substantial agreement between the results coming from NN studies, and the dispersive analysis of the nucleon electromagnetic form factors. Up to now, we have discussed NN coupling in terms of hadronic degrees of freedom. From the point of view of QCD this is an eEective description of the underlying dynamics, one is left when quark and gluon degrees of freedom have been integrated over. An ab initio determination of the NN coupling parameters, starting from QCD, would obviously represent a major goal. Similarly

342


Fig. 11. Predictions for the 01 mixing parameter by the full Bonn, Bonn B, Paris, Nijmegen (solid lines) and Reid (dashed line) potentials compared with a model without -exchange contributions (dotted line) and the experimental data. Figure taken from Ref. [92].

to other basic quantities appearing in low- and intermediate-energy Nuclear Physics, this would entail the solution of an extremely diScult many-body problem, where quarks and gluons move in a highly non-trivial vacuum, with non-perturbative eEects playing a dominant role. As a consequence, only a few exploratory calculations of the NN coupling strengths have appeared up to now, where non-perturbative QCD eEects have been taken into account in a more or less approximate way. Thus, the NN coupling has been considered already several years ago in a two-phase Skyrme model, where a little quark bag is surrounded by a pionic cloud [104]. If the baryon charge is assumed equally distributed between the quarks and the pionic cloud, one has that the tensor-to-vector ratio turns out to be about twice the VDM value, for a quark core radius of about 0:5 fm in the nucleon, as required by low-energy phenomenology. Some QCD-inspired calculations of the NN coupling parameters have been recently attempted in the framework of Sum Rules, where the non-perturbative eEects of the vacuum are described through various quark, gluon, or mixed quark–gluon condensates. In [105] the meson is treated as an external >eld, coupled to the quarks through a vector-type eEective Lagrangian. A value C = 3:6, consistent with the VDM hypothesis, has been obtained. A more recent calculation [106], employing the operator-product expansion on the light cone to produce a microscopic wave function, obtains QCD sum rules for the -nucleon coupling parameters, which yield values for gV2 =4 and C in good agreement with Eq. (3.58). 3.2.3. Meson– couplings The coupling constants involving the resonance are less well-known under the experimental point of view, given the unstable nature of the particle. The determination of the parameters characterizing the resonance, i.e. its mass M and width 2 , has been discussed in Section 2.1. In terms of these parameters the J = 3=2+ , T = 3=2 partial-wave amplitude is given by the relativistic


343

Breit–Wigner formula [102] f13=2 + =

M 2 (s) 1 ; q M2 − s − iM 2 (s)

(3.61)

where q is the momentum of the emerging pion. Relativistic kinematics dictates its dependence upon energy and masses of the emerging particles to be [107] s2 + M 4 + m4 − 2sM 2 − 2sm2 − 2M 2 m2 √ : q= 2 s √ The width 2 (s) at the resonance peak s = M can be related to the N coupling constant GN as follows [102,107]: 2 M 2 2 (M 2 ) GN = 2 3 : 4 q(M )

(3.62)

This is the coupling strength used in SU (3) >ts, where the is a member of the baryon decuplet. One may look for the relation between this strength and the coupling constant fN appearing in the eEective relativistic Lagrangian LN (3.7). This can be accomplished through a straightforward, but somewhat lengthy calculation. To simplify the notation, let us refer to the ++ → +p case. In the reference frame with the at rest the two-particle >nal state can be characterized by the value |q| and the angular variables (; /) of the emerging-pion momentum, plus the helicity of the >nal nucleon. On the ground of rotational invariance one expects that the decay matrix element for the baryon → baryon transition T (|q|; ; /; Sz ) factorizes into purely geometric factors, times a quantity subsuming the dynamical aspects of the system. As a matter of fact one has [107] Tfi ≡ T (|q|; ; /; Sz ) = 2 exp[i/(Sz − )]dS3=2 z ()V3=2 () ;

(3.63)

z where dS3=2 z () are the usual reduced rotation matrices, and S the third component of the spin. The vertex V3=2 () is determined once Eq. (3.63) is compared with the matrix element of the Lagrangian (3.7) between the initial and >nal N state. In momentum space one gets the quantity (fN =m )u(M X )qF uF , where uX and uF are the Dirac and Rarita–Schwinger spinors for the proton and the , respectively. The latter is simply obtained as the coupling of a spin-(1=2) spinor u(M ) and a spin-one four vector 0F , with the proper Clebsch–Gordan coeScients. For the ++ → + p vertex one simply has uF (+3=2) = u(M ; 1=2)0F (+1). The Dirac invariant u(M X )u(M ) can be evaluated through standard techniques [107], and gives a mass factor ((M + M )2 − m2 )1=2 , while the evaluation of the scalar product qF 0F in the c.m. system allows the identi>cation of V3=2 (). One has 1 fN (M + M )2 − m2 q : (3.64) V3=2 () = √ 6 m

For unpolarized particles, the decay width can be obtained from the decay amplitude Eq. (3.63) integrating over the angular variables, averaging |Tfi |2 over the initial spin states and summing over the >nal spin states, namely [107] q 1 √ 2 = (3.65) d |Tfi |2 : 322 M s

344


(a)

(b)

(c)

(d)

Fig. 12. Direct and crossed nucleon and pole contributions to the scattering matrix.

Inserting V3=2 () given by Eq. (3.64), one obtains the standard decay width M 2 (M2 ) =

2 (M + M )2 − m2 1 q(M2 )3 fN : 6 M 4 m2

(3.66)

Comparing Eq. (3.62) with Eq. (3.66), one >nally gets 2 2 fN GN m2 : =6 4 (M + M )2 − m2 4

(3.67)

2 According to Eq. (3.62), the value of GN is obtained from the determination of the resonance 2 parameters, >xed through a >t of N scattering data, giving GN =4 = 14:54 from + p scattering, − 2 2 and GN =4 = 14:39 from p [102]. Hence, taking for GN =4 an indicative value of 14.5, one has from Eq. (3.67) 2 fN 0:362 (3.68) 4 for the N coupling constant appearing in the eEective relativistic Lagrangian (3.7). Most applications in Nuclear Physics need a reasonable non-relativistic treatment of the as an eEective particle, to be matched with non-relativistic wave functions and eEective operators. As outlined at the beginning of this Section, this can be accomplished through the eEective static Hamiltonian (3.9), which allows a straightforward evaluation of the various N transition potentials. This is indeed possible, as shown many years ago by Sugawara and von Hippel [66] by the non-relativistic reduction of the Feynmann amplitude corresponding to the Born OPE graph, which entails a judicious dropping of terms of order M − M m : ∼ M + M M + m

The coupling constant fN appearing in (3.9) can be >xed by requiring that the non-relativistic isobar model is able to reproduce the N scattering data in the same extent as the static Chew–Low model. In this sense fN can be expressed in terms of the N coupling constant. To see how this comes out, let us consider p-wave N scattering, as proceeding through the direct and crossed Born graphs with intermediate nucleon and contributions shown in Fig. 12. It is more convenient to


345

use a formalism based upon a K-matrix approach. In this way unitarity of the S-matrix is fully guaranteed by the Cayley relation between the S- and K-matrix, see Eq. (2.11). The same does not apply for an approximate T -matrix including only the pole contributions and is crucial for a meaningful comparison with the Chew–Low parameterizations of the scattering data. The required K-matrix is 2 fNN ( · q )( · q) ( · q)( · q ) K = K N + K = |q| B˜b Bã + Bã B˜b 4m2 −! ! 2 (S · q )(S† · q) ˜ ˜ + (S · q)(S† · q ) ˜ ˜ + fN T bT a + T aT b : (3.69) + 4m2 ! − ! ! + ! Here, ! is the pion energy, and ! ≡ M − M represents the nucleon– mass shift. The K-matrix given by Eq. (3.69) satis>es crossing symmetry, as one can verify through the substitutions ! ↔ −!, q ↔ −q and a ↔ b, and possesses the proper pole at the N– mass diEerence ! . Invariance with respect to rotations in ordinary and isospin space can be now exploited, for a projection onto the partial-wave eigenchannels. In particular, for J = T = 3=2 one has 2 2 2 fN 1 fN 1 |q|3 4fNN + + : (3.70) K33 (!) = tan 33 = 3 4m2 ! ! − ! 9 ! + ! The last term, comes from the -crossed graph, and contributes no more than 2% near threshold, so that at low-energy one can write 2 2 fN 1 |q|3 4fNN tan 33 + : (3.71) 3 4m2 ! ! − ! This result has to be compared with the Chew–Low eEective-range approximation for the J =T =3=2 phase shift, which is obtained from the Chew–Low equation keeping only the pole and one-pion intermediate-state contributions. One gets for the 33 phase shift [108] 2 ! |q|3 fNN ; (3.72) tan 33 2 m ! 3 ! − ! which fully exhibits the resonant behavior. Clearly, to have Eq. (3.71) coinciding with the 2 2 2 Chew–Low result (3.72), one needs fN = 4fNN . Taking for fNN =4 the indicative value 0:08 one has 2 f2 fN (3.73) = 4 NN 0:32 ; 4 4 which is not too far from the value (3.68), extracted from the experimental position and width by means of the relativistic Breit–Wigner formula. Many relations among meson–baryon couplings can be obtained by considering the strong-coupling limit of static >eld-theoretic models of hadrons [109,110], where recoilless sources interact through the exchange of diEerent mesons. Although obtained many years ago, these results still compare well with the experimental data, and represent a beautiful example of how one can exploit the group-theoretical symmetries embedded in phenomenological models of strong interactions. As we have seen, in the simplest case the Chew–Low equation relates the elastic scattering amplitude to all the reaction and production amplitudes allowed by unitarity, in a way that crossing is satis>ed. If several types, ; : : : , of mesons are exchanged between diEerent baryonic sources B; B ; : : : , the

346


Chew–Low equation can be straightforwardly generalized by replacing the transition amplitude T (!) B B with a set of amplitudes T>< (!). Disregarding the contributions with two or more mesons in the intermediate states, these transition amplitudes satisfy the generalized Chew–Low equation [109]

B B B B B B B B V V V V < < > > BB (!) = 2 + T>< ! + MB − MB −! + MB − MB B

B B B B B B B B T+> (!p )∗ T+< (!p ) (!p )∗ T+> (!p ) T+< : (3.74) + + ! − !p + MB − MB −! − !p + MB + MB B ;+;p

Here, is the renormalized strength parameter, and Veld and the baryon states. For a given ]) + 2 ([V> ; [V< ; M ]]) + O ! ! 2 where M is a diagonal matrix with MB as diagonal element. Now, unitarity implies that the scattering amplitude must be >nite in the strong-coupling limit. From Eq. (3.75) one then has the commutation rules [V< ; V> ] = 0

(3.76)

for the interaction terms, which can be regarded as dynamical constraints. Other commutation rules can be obtained under the assumption of invariance under some symmetry group. For the Symmetric Pseudoscalar-Meson theory the symmetry group is SU (2)J ⊗SU (2)T , implying invariance with respect to spin and isospin rotations. The generators of the associated Lie algebra are, in the spherical representation, the usual highering and lowering spin and isospin operators J± and T± , and the third components Jz and Tz , with standard commutation rules (CR), which must be supplemented with the ones for the meson sources VFI . They can be written down immediately, by requiring that the meson currents also transform like the regular representation under SU (2)J ⊗ SU (2)T [J± ; VFI ] = (1 ∓ F)(2 ± F)VF±1I ; [Jz ; VFI ] = FVFI : (3.77) Here, in the meson terms VFI the >rst index refers to spin, and the second one to isospin, so that the CR in isospin space can be obtained from Eq. (3.77) simply interchanging the role of F and I. Eqs. (3.76) and (3.77) actually de>ne the Lie algebra associated to the whole symmetry group [SU (2)J ⊗ SU (2)T ] × T9 of the strong-coupling Symmetric theory, where T9 is the nine-parameter Abelian group. Its presence here can be understood on physical grounds, by noting that the mesons are described by isovector operators, and interact in p-wave only with the baryons. Relations (3.76) and (3.77) have been solved by Singh [110], with particular reference to the simplest irreducible representation of the symmetry group, which has the spin–isospin content J = T = 1=2; 3=2; : : : : The explicit realization of the commutation rules implies that the baryon–baryon–meson vertices are essentially given by a product of two Clebsch–Gordan coeScients times a universal coupling


347

constant N, so that for two baryons with spin and isospin J = T and J = T , respectively, one has in an obvious notation [4] 2J + 1 NJ 1Mm | J M + m J 1MT mT | J MT + mT : NBB = (3.78) 2J + 1 Since in the strong-coupling limit the baryon masses belong to a “rotational” band in spin and isospin space [109,110] M (J; T ) = M0 + M1 J (J + 1) + M2 T (T + 1) ;

(3.79)

one can identify the lowest baryons of the series with the nucleon and the , respectively. Thus, the various meson–baryon coupling constants diEer from one another simply by geometric factors, and in particular one >nds 2 2 9 fNN fN = ; 4 2 4

(3.80)

2 2 which, for fNN =4 0:08 gives fN =4 = 0:36, in surprisingly good agreement with the experimental value (3.68). The simple relation (3.80) between the NN and N couplings is due to the fact that the physics contained in static >eld-theoretic models can be expressed completely by group-theoretic arguments in the strong-coupling limit. A similar simpli>cation can be achieved in quark models of hadrons, provided that simple baryon wavefunctions are employed, so as to exploit again the invariance under a suitable symmetry group. This has been clearly shown several years ago by Brown and Weise for the SU (4) quark model [12], in which the nucleon and the belong to the same spin–isospin multiplet, so that here we limit ourselves to recall some of the main points. In this model the ++ isobar, can be described as a bound state of three isospin “up” quarks with their spin “up” u ↑, as detailed in Table 3. The other baryon states can be generated from the ++ wave function by employing step-down operators in spin and isospin space, expressed in terms of quark creation and annihilation operators. Thus, the spin down operator s(−) is given by

s (− ) =

3 i=1

[a†u↓ (i)au↑ (i) + a†d↓ (i)ad↑ (i)] :

(3.81)

The analogous operator in isospin space t (−) can be obtained by interchanging the role of spin and isospin in Eq. (3.81). Operating with s(−) and t (−) upon | 3=2 3=2 one can produce all the required normalized and symmetric baryon wave functions. Under a mathematical point of view this leads to analyze the representations of the group SU (4) ⊃ SU (2) ⊗ SU (2). However, one can proceed also in a more heuristic way, by introducing the usual non-relativistic coupling Eq. (3.3) at the quark level, so as to write HQQ = −

3 fQQ ˜ ; ((i) · ∇)((i) ˜ · ) m i=1

(3.82)

where the pion >eld ˜ is regarded as an external c-number classical >eld. The qq coupling constant can be related to the “macroscopic” NN coupling strength by requiring the expectation value of

348


(3.82) in the quark-model proton state |P1=21=2 to be equal to the usual meson–proton coupling operator. One has 3 11 fQQ 1 1 fNN ∇ z /3 = P (z (i)B3 (i)) P ∇ z /3 : (3.83) 22 m 22 m i=1

The matrix element with respect to quark degrees of freedom can be evaluated as in simple shell-model calculations, and one gets [12] fQQ = 35 fNN :

(3.84)

Similarly, one can >nd out the relation between fN and fNN just equating the matrix element of the meson–quark coupling operator between the quark and proton states |1=21=2 and |P1=21=2 to the corresponding “macroscopic” matrix element, leading to [12] 2 2 fN 72 fNN = ; (3.85) 4 25 4 2 2 which, for fNN =4 0:08 gives fN =4 0:23, a value somewhat smaller than the experimental one. The above “simple-minded” quark model has been widely used, in spite of its moderate success in reproducing the experimental N coupling, because it can be easily extended to other meson– baryon–baryon couplings. Thus, the strength is related to the NN coupling constant by [12] f = 45 fNN :

(3.86)

Similar considerations allow one to relate the fN coupling strength to the already known coupling parameters. With reference to this point we recall that -exchange plays a crucial role in reducing the dependence of the N transition potential upon the regularizing cut-oE parameter [4,111,112]. To understand how the required relation comes out, let us recall the -exchange N – interaction (3.18), where the spin and isospin dependence has the same structure as in the meson–nucleon coupling. As a consequence, in the SU (4) quark model, with eEective coupling to external c-number meson >elds, the only diEerence among the various meson–baryon–baryon vertices amounts to purely geometric factors, i.e. Clebsh–Gordon coeScients, the reduced matrix elements remaining always the same. One can expect, for instance, that the ratio between fN and fN is the same as the ratio between fNN and fNN , namely [4,13,111] fNN fN : (3.87) fN = fNN Similar arguments lead >nally to establish that the analogue of Eq. (3.86) is f = 45 fNN :

(3.88)

One may ask what value for fN has to be inserted in Eq. (3.87), to >x fN . Should one use the experimental value (3.68), extracted from the width? Or, to be consistent with the SU (4) quark model, should one use Eq. (3.85), in spite of its moderate agreement with the experimental information? With reference to this point, as already observed by Green several years ago [4], it is worthwhile to recall that the geometric relations (3.87) and (3.88) can be obtained also in the framework of the strong-coupling model, which is able to give the remarkable value 0.36 for the N coupling constant. Certainly, from a fundamental point of view, a derivation from more realistic


349

quark models of hadrons, or, hopefully, from QCD would be much more preferable. To the best of our knowledge, in spite of the progresses accomplished in this >eld, a satisfactory calculation of this type has yet to be performed. 3.2.4. Form factor eDects and correlated exchange of mesons The strengths discussed above exhaust the information one needs at the meson–hadron vertices when the external particles are on their mass or energy shell. In few- and many-body calculations, however, oE-shell situations occur, which require the introduction of vertex form factors. Originally introduced to make loop integrations convergent, these form factors take into account the intrinsic composite nature of strong-interacting particles, which have to be regarded as collective excitations of the underlying QCD degrees of freedom. The actual derivation of form factors from a dynamic theory represents an extremely diScult task. Therefore, they have been generally parameterized through simple, phenomenological functions. A consistent treatment of form factors would in principle require their dependence upon all the four momenta of the involved particles, with a parameterization valid in all the kinematic domains explored in the various reaction processes where they can be involved [113]. In practice, one limits oneself to the dependence from the four momenta of the exchanged particle. Thus, for the baryon–baryon interactions considered in the previous section, one simply replaces the strengths feld-theoretical models, as advocated by the Bonn group [67,103]. The latter approach has been considered in Refs. [124,125]. Correlated exchange is of particular relevance here, since these contributions in the channel with pionic quantum numbers can provide a strong tensor-force component in the NN interaction, able to counterbalance the eEects of a soft NN form factor. In Ref. [125] correlated exchange has been included by relating the NN → NN scattering amplitude in the s-channel to the N NX → N NX T -matrix in the t-channel via crossing symmetry. These amplitudes can be expressed, through a dispersion relation, in terms of the unitarity contributions due to N NX → transitions. The corresponding T -matrix TN NX → is given by the direct NX N annihilation, described by the driving term VN NX → , followed by , propagation and rescattering, described by the Green function G and the amplitude T→ , that is [125] TN NX → = VN NX → + VN NX → G T→ :

(3.90)

The driving term is given by N and exchange, plus an ! pole term, whereas T→ is obtained starting from an explicit meson-exchange model, involving physical , , and ! mesons [126]. A good >t to the two-nucleon observables can be obtained with a NN cut-oE NNN of less than 1 GeV. To be consistent with these >ndings, one ought to obtain a comparable reduction of NNN in a dynamical model of the NN vertex, based on a similar meson-exchange description of hadron interactions. This has indeed been shown by the JUulich group [127,128], by dressing the bare NN vertex with contributions, where three-pion states are described by and pairs, as depicted in Fig. 15. The corresponding eEective T amplitude can be evaluated through a meson-exchange model in the crossed, annihilation channel, where t ¿ 0. The NN form factor in the s channel, where the momentum transfer t is spacelike, is >nally obtained by means of a dispersion integral. The resulting NN form factor can be >tted by a standard monopole form with a cut-oE NNN ∼ 0:8 GeV. The lesson to be learned from the above example is that a proper microscopic treatment of hadron dynamics can lead to a softer form-factor than the one suggested by a more naive calculation based on purely phenomenological functions. This is in particular true, when the composite nature of meson–baryon–baryon vertices is taken into account. Thus, in Ref. [129], the bare NN and N vertices have been dressed by N rescattering through the non-pole part of the N T -matrix, as seen in Fig. 16. The N transition matrix has been obtained starting from the meson-exchange model developed in [130]. Softer dressed form factors are found in respect to the bare ones, with cut-oE masses between 500 and 700 MeV, the result depending upon the treatment of the underlying reaction dynamics. Similarly, small NNN and NN are obtained, when composite models are used for the corresponding vertices [131,132]. This “softening” of the form factors has been carefully studied in a model

352


Fig. 16. Dressing of the NN or N vertex due to N rescattering. The squared box represents the oE-shell N transition matrix T .

calculation by Liu et al. [133]. They considered Feynmann graphs where the vertices have some speci>c topological structure, and examined the eEects of intermediate particle propagation and >nite momentum-space cut-oEs for the constituent sub-vertices on the overall vertex function. It turned out that the non-locality inherent to composite vertices produced smaller cut-oE masses, when they were parameterized through simple functional forms such as Eq. (3.89). In the framework of an eEective theory of hadrons, based on meson and baryon degrees of freedom, the actual shape of x the coupling strengths of the to the various mesons from the experimental information. As a matter of fact, coupling constants and cut-oE parameters are still to be regarded as semi-phenomenological quantities to be >xed in reproducing NN bound-state and scattering data. When the degrees of freedom are explicitly taken into account, one is naturally lead to face a coupled-channel problem, as graphically depicted in Fig. 7. The >rst attempt to solve the coupled NN –N problem was in con>guration space, much in the same way as one gets coupled SchrUodinger equations in low-energy nuclear and atomic physics [4]. In such a case one thinks of the wave function as endowed with both NN and N components ? = a?NN + b?N

(3.91)

and de>nes a suitable Hamiltonian with nucleon and kinetic terms plus the N mass diEerence, and all the relevant baryon–baryon interactions, described in Section 3.1. Projecting onto the available wave function components one gets (−∇2 =2FN + VNN − E)?NN = −VN ?N ; (−∇2 =2F + V + M − M − E)?N = −VN ?NN ;

(3.92)


353

where, FN and F represent the NN and N reduced masses and E is the total energy of the system. Only NN → N and N → N transitions were considered, and are represented by the potentials VN and V . Eqs. (3.92) do not pose any new problem with respect to ordinary coupled-channel equations, if the is considered as an elementary particle and does not decay. This is not true, however. Owing to the unstable nature of the isobar, its mass M has to be supplemented with an energy-dependent width, namely i2 (E0 ) ; (3.93) 2 where the internal energy E0 of the isobar is obtained from the total energy by subtracting the relative-motion term q2 =2F , E0 = E + M − (q2 =2F ). In con>guration space q2 has to be regarded as a diEerential operator, so that Eqs. (3.92) are strictly speaking non-local. This non-locality becomes harmless in momentum space, where the SchrUodinger Eqs. (3:92) can be replaced by the Lippmann–Schwinger set M = M (0) −

TNN = VNN + VNN GN TNN + VN G TN ; TN = VN + VN GN TNN + V G TN

(3.94)

with boundary conditions given through the free Green functions GN and G . In general, if states also are taken into account, one will have a larger set of coupled equations, which can be written in the compact form T = V + V ; (3.95) +

where ; + can be any of the two-particle states with either the nucleon or the . In such a case, the additional transition potentials V (NN → ), V (N → ), and interaction terms V ( → ) of Fig. 7 have obviously to be considered. The Eqs. (3.95) are the basic equations to be solved in order to >t phenomenological models for the transition potentials to the two-body data. At the same time they provide on- and oE-shell t-matrices to be employed in few- and many-body calculations. Eqs. (3:95) use static potentials V , and disregard time-retardation eEects characteristic of a relativistic formalism. The question naturally arises to what extent the non-relativistic approach is justi>ed. The diEerence between the two approaches can be clearly perceived by a comparison of the perturbative expansion of Eqs. (3.95) with the corresponding Feynmann graphs. The second iteration of the basic NN and N potentials gives the box contributions to NN scattering exhibited in Fig. 13(a) – (d). Crossed exchange graphs, such as those given in Fig. 13(e) – (h) are therefore disregarded in the CC approach. Several papers have been devoted since the 1970s to assess the role played by these graphs in the coupled NN –N problem. The >rst systematic investigation has been performed by Smith and Pandharipande [136]. They considered all the eight uncorrelated fourth-order Feynmann graphs one can have for NN scattering in a relativistic >eld-theory with NN , N or intermediate propagation, as seen in Fig. 13. If intermediate states with positive-energy baryons only are retained, each relativistic graph corresponds to six possible time-ordered diagrams. Thus, graphs (a) – (f) in Fig. 17 represent the time-ordered diagrams one obtains from the box Feynmann graph with intermediate N propagation. Clearly, only the four graphs with sequential meson exchange can be reproduced by twice-iterated transition potentials, whereas the two “stretched”

354


(a)

(c)

(e)

(g)

(i)

(k)

(b)

(d)

(f)

(h)

(j)

(l)

Fig. 17. Time-ordered diagrams with positive-energy intermediate N states.

graphs (e) and (f), with two mesons in ?ight in the intermediate state, as well as the remaining crossed diagrams cannot have any correspondence in a standard coupled-channel calculation. Smith and Pandharipande studied the diEerence between iterated Born contributions, and the result of the whole series of time-ordered graphs, for non-relativistic, static nucleons and isobars. They found that the twice-iterated OBE potentials can approximate reasonably well the whole series of time-ordered graphs if scalar–isoscalar mesons are considered. Things are however diEerent for the pion, because the crossed graphs give an isospin-dependent contribution with opposite sign with respect to the box graphs. Physically this is due to the fact that, after emission of the >rst pion, the baryons can emit a second pion while moving in a diEerent total isospin state, a possibility which is not allowed by iterated instantaneous potentials. Therefore they subdivided the crossed contributions into a piece having the same isospin dependence as the box graphs, plus a remainder. The overall sum of the box and crossed contributions with the same isospin factor was overestimated by only about 10 –15% by the second Born approximation in the coupled-channel approach, while there were strong cancellations among the crossed terms with the “unpleasant” isospin dependence, thereby giving strong support to standard coupled-channel calculations. The general validity of these results has been subsequently questioned by the Stony Brook group, on the ground of a comparison of iterated static potentials with calculations based upon a covariant, dispersion-theoretic approach to the NN interaction [137]. By considering the iterative box graphs, they showed that static pion-ranged transition potentials have to be supplemented with shorter-range terms. Moreover, it turned out that both the N – mass diEerence and relativistic eEects play an important role in the N system, since they reduce the isobar box contributions by a factor between 2 and 3. Needless to say, the evaluation of non-iterative, “stretched” and crossed, contributions is much more involved than the iterative ones, because the meson energies occur in all propagators. One can then understand the quest for reasonable approximations, avoiding the direct calculation of the non-iterative diagrams. The whole set of two-pion exchange diagrams has been considered by the Bonn–JUulich group, using time-ordered perturbation theory [138–140]. They showed that the sum of all graphs with N intermediate states can be written [138–140] 2 TN = 2(BN + CN ) + 1 · 2 (BN − CN ) : 3

(3.96)


355

where BN and CN represent the box and crossed contributions, respectively. The salient feature of this result is the opposite sign of the terms depending upon the isospin degrees of freedom of the external nucleons, i.e. the isovector-exchange pieces. If BN CN the last term cancels out; (0) moreover, it can happen that the twice-iterated transition-potential contribution BN overestimates the (0) exact amplitude BN by a factor of two, namely BN 2BN BN + CN , so that one can replace Eq. (3.96) with (0) : TN 2BN

(3.97)

The exact contribution is thus replaced by the isoscalar part of the twice-iterated transition potentials, provided that the assumptions about the relative weight of Born, box and crossed contributions detailed above are satis>ed. This has to be contrasted with the coupled-channel treatments, where also the isovector part is retained and one writes (0) : TN (2 + 23 1 · 2 )BN

(3.98)

Therefore, in the static approximation, retaining a pion-ranged potential for the isovector part of the amplitude also, cannot be considered in general a good approximation, and may lead to a vanishing contribution in isospin-zero states [137–140]. The approximation (3.97) has been carefully tested in Refs. [138–140]. It works fairly well for the 1 S0 partial wave in NN scattering; in other partial waves, however, things are not so simple. The exact calculations con>rm many of the conclusions of the Stony–Brook analysis [137]. In higher partial waves (L ¿ 1) the twice-iterated pion-range transition potentials grossly overestimate the exact contributions with N intermediate states; this can be understood on physical grounds, since the realistic contributions are of shorter range and hence much more suppressed for L ¿ 1 than static contributions with pion range. As for the isovector term, it turns out to be possibly small only when all intermediate (NN , N, and ) states are simultaneously taken into account. These results show that the use of static approximations in coupled-channel calculations require a careful vis-[a-vis comparison with relativistic treatments, to ascertain the possible presence of delicate cancellation eEects. Even with simpli>ed baryon-baryon transition potentials, the solution of the N CC problem remains a big task. This can be easily perceived, when rotational and isospin invariance are exploited to give the transition amplitudes in a partial-wave representation. The lowest NN , N, and T = 0; 1; 2 channels are given in Table 4 in the usual spectroscopic notation. The NN scattering observables are not very sensitive to the N and channels. Moreover, the T =2 and some T =1 states such as the 3 S1 are not allowed from >rst principles in the NN sector. This implies that any information on these channels has to come from other indirect processes [141], like elastic d scattering or polarization observables, since the unstable cannot be used as a scattering probe. In fact, in Ref. [141] some N channels were indispensable to reproduce consistently the cross sections and the vector analyzing power in d elastic scattering. When N couplings are explicitly taken into account in the NN interaction, bound-state and scattering data ought to be re->tted starting from a consistent solution of Eq. (3:95). To this end, the diagonal NN potential has to be suitably renormalized by subtracting the pure iterations of the OBE and transition potentials, in order to avoid double-counting when solving the coupled-channel equations [4]. In many calculations, however, this has not been done, but N transition potentials M have been simply added to some well-established phenomenological or OBE model potential VNN .

356


Table 4 Partial waves

2S+1

LJ for the N system up to L = 6 in the T = 0; 1 channels and up to L = 2 in the T = 2 channels NN

T =0

S 1 –3 D 1 D2 3 D 3 –3 G 3 3 G4 3 G 5 –3 I5 1 P1 1 F3 1 H5

T =1

S0 D2 1 G4 3 P0 3 P1 3 P 2 –3 F 2 3 F3 3 F 4 –3 H 4 3 H5

N

3

S1 3 D 1 7 D 1 7 G 1 D 2 7 D2 7 G2 7 S3 3 D 3 7 D 3 3 G 3 7 G 3 7 I3 7 D 4 3 G 4 7 G 4 7 I4 7 D 5 3 G 5 7 G 5 3 I5 7 I5 1 P1 5 P1 5 F1 5 P3 1 F3 5 F3 5 H3 5 F5 1 H5 5 H5

3

3

1

1

3

5

D0 S2 3 D 2 5 D 2 5 G 2 5 D 4 3 G 4 5 G 4 5 I4 3 P0 3 P1 5 P1 5 F 1 3 P2 5 P2 3 F 2 5 F2 5 P3 3 F3 5 F 3 5 H3 3 F4 5 F4 3 H 4 5 H4 5 F5 3 H5 5 H 5 3 S1 3 D 1 5 D 1 5

1

S0 5 D 0 S2 1 D 2 5 D 2 5 G 2 5 D 4 1 G 4 5 G 4 5 I4 3 P0 7 F0 3 P1 7 F1 3 P2 7 P2 3 F2 7 F2 7 H2 7 P3 3 F3 7 F3 7 H3 7 P4 3 F4 7 F4 3 H4 7 H 4 7 F5 3 H5 7 H5 5 D1 5

5

T =2

D0 S1 3 D 1 5 D 1 5 S2 3 D 2 5 D 2 3 D3 5 D 3 5 D4 3 P0 3 P1 5 P1 3 P2 5 P2 5 P3 3

3

S1 3 D 1 7 D 1 D 2 7 D2 7 S3 3 D 3 7 D 3 7 D4 3

1

P1 5 P1 P2 5 P3 5

M The resulting ansatz is obliged to be phase-equivalent to VNN at a certain well-de>ned energy E0 , that is M VNN ≡ VNN − VN G (E = E0 )VN :

(3.99)

This procedure is clearly reminiscent of what in often done to represent a local interaction through a >nite-rank potential. Its practical applicability strongly depends upon the sensitivity of the approximation when one moves away from the energy E0 . It has been used together with the Paris potential, by the Hannover group, to develop their force model for the N interaction [142–144]. A rather elaborate approach to the CC problem, with few- and many-body applications in mind, has been developed by Haidenbauer et al. [145]. Owing to retardation eEects in the exchange of mesons, interactions derived from a covariant meson–baryon >eld theory are intrinsically non-local and energy-dependent, unless severe approximations are introduced. Whereas non-locality does not represent a big problem, especially in momentum space, energy dependence can make many-body


357

calculations quite cumbersome. However, the energy dependence can be systematically removed by resorting to folded-diagram techniques. This method, originally developed in the framework of nuclear shell-model theory [146], has been adapted to the NN problem by Johnson many years ago [147], to derive instantaneous, energy-independent potentials from energy-dependent exchange forces. The basic idea consists in requiring the matrix elements of a meson-baryon interaction in the state space of mesons, nucleons and ’s to be the same as the matrix elements of an eEective potential in the restricted space of nucleons and ’s only. This is accomplished order by order in the time-ordered perturbative expansion of the time-evolution operator. As a consequence, this method can be naturally applied to the Bonn interactions, which are obtained through non-covariant, Bloch–Horowitz perturbation theory. Technically, one integrates over all the time variables in the original time-ordered Feynmann graphs, with the exception of the time at which the resulting eEective potential acts. This formalism has been applied in [145] to nucleons and isobars interacting through the exchange of ; ; ; ! and mesons. Both NN ↔ N, NN ↔ , and N ↔ oE-diagonal one-meson exchange transition potentials have been considered. As for the diagonal potentials, the direct one-meson-exchange interaction V (N → N) has been taken into account, whereas the exchange contribution V (N → N ) has been ignored. The diagonal interaction has been omitted altogether. Finally, for the nucleon–nucleon potential, uncorrelated two-meson exchange graphs, with box and crossed contributions, were included. With the above approximations the CC equations (3.95) were solved for the deuteron and NN scattering, thus >tting the meson–nucleon coupling constants and cut-oE masses, >nding minor deviations with respect to the parameters of the full Bonn potential [148]. Thanks to a more satisfactory treatment of the N threshold, the coupled-channel model provides an overall improvement in reproducing the low-energy phase shifts, with respect to the single-channel calculation. The Argonne group [149] starts from a bare phenomenological NN interaction, the Urbana model [150], and adds extra N and transition operators, with a consistent >tting of low-energy two-nucleon observables. The transition potentials that involve the isobar have long range OPEP terms, and the short and intermediate regions representing the exchange of the and ! mesons are described semi-phenomenologically by a combination of generalized spin and tensor operators and a Woods–Saxon shape at short distances. Beside the N and coupling constants it has no more free parameters then the ones already present in the v14 model. There is a central repulsion in the N and channels and some of the box-diagrams generated by two pion exchange process were also approximately included. In the N and sectors, the intermediate and short range regions are represented only by central operators, that produce a repulsive core comparable in size to the ones in the NN channels. The Argonne v28 potential is then a more complete interaction, easy to use in con>guration space, however degrees of freedom are taken into account in a very phenomenological way. Possible improvements require theoretical guidance, since NN data show little sensitivity to excitations. It was used in three-nucleon [151] and in nuclear matter [5,152] calculations, and strong repulsive eEects were found, as it will be discussed in the following sections. 3.4. Relativistic and unitarity corrections in the N coupled-channel approach With increasing energy, it is expected that relativistic eEects may play a more and more important role in the description of the coupled NN –N– system. Relativistic requirements can be taken

358


into account in a minimal way through relativistic propagation. As we shall see later, however, a consistent relativistic treatment requires the solution of coupled Bethe–Salpeter equations. At the same time, the opening of inelastic thresholds would require a more satisfactory treatment of unitarity requirements than the one oEered by two-body-type formalisms. This is particularly true for production processes, which proceed mainly through intermediate formation and dominate in the energy region between the threshold ( 300 MeV) and 1 GeV in the laboratory system. Great theoretical activity has been devoted in the 1970s and 1980s to the development of three-body unitary models for the NN –NN system, in which the degrees of freedom could be accommodated in a natural way. All these approaches aim at an extension of the non-relativistic Faddeev theory to the relativistic domain, so as to include eEects due to pion emission or absorption. This is not a trivial task, as can be easily appreciated taking into account that the Faddeev equations can be regarded as a clever resummation of the multiple-scattering series, which requires deep modi>cations when the number of particles can change [153]. Theories with pion emission=absorption satisfying three-body unitarity have been developed in the past along several diEerent routes. The NN and NN space can be allowed to communicate through formation and decay of the particle, which is introduced in the formalism from the very beginning [144]. Alternatively, one can employ diagrammatic or projection-operator techniques to derive Faddeev-like equations, coupling all the scattering, production and absorption processes one can have in the coupled NN –NN system [154 –156]. The isobar is then generated dynamically as a consequence of the N interaction. At the end, one arrives at coupled equations very similar in structure in all approaches. As for the relativistic corrections, unitary three-body formalisms have been developed by means of quasi-potential techniques, which yield covariant, three-dimensional equations strictly resembling the Lippmann–Schwinger and Faddeev ones of non-relativistic scattering theory, at the price of some restrictions on the treatment of the particle propagation in the intermediate states [157,158]. These formalisms, together with phenomenological two-body interactions designed to reproduce two-body observables, have been employed to study pion production in NN scattering, NN ↔ d processes, and d elastic scattering with allowance for the coupling between the various reaction channels. It is really impossible to do full justice here to this enormous amount of work. It has been reviewed in the book by Garcilazo and Mizutani, where all the relevant references as well as a detailed comparison between models and experimental data can be found [159]. Here, we limit ourselves to a brief discussion of the approach pursued by Tjon and collaborators [160,161], which, being based upon the Bethe–Salpeter equation (BSE), is in our opinion very close in spirit to the papers considered in the previous sections. As is well-known, the BSE is a four-dimensional, integral equation for the transition amplitude, or the wave function, whose kernel contains the contributions from all two-particle irreducible diagrams, the ones which cannot be divided into disjoint pieces by drawing a line which cuts no boson line, and each fermion line only once. The exchange Born graph and the crossed fourth-order diagram are noticeable examples. As a consequence, the BSE sums up all the two-particle irreducible diagrams one can have in the considered >eld-theory [162]. In spite of its apparent similarity to the non-relativistic Lippmann–Schwinger equation, its solution represents a challenging problem. The construction of the driving term is in itself as diScult as solving the full scattering problem. As a matter of fact, one generally limits oneself to the ladder approximation, retaining the Born exchange graph only in the driving term.


359

Fig. 18. Diagrammatic representation of the BS equation for coupled-channel NN –N scattering. The dashed lines represent the exchange of various mesons.

The ladder BS equation has been applied by Tjon to the coupled N system, 3 i 4 Vij (p0 p; q0 q)G0j (q0 q)Tj1 (q0 q; 0k) ; Ti1 (p0 p; 0k) = Vi1 (p0 p; 0k) − 3 d q 4 j=1

(3.100)

where the incoming and outgoing four momenta are k ≡ (k1 − k2 )=2 = (0; k) and p ≡ (p1 − p2 )=2 = (p0 ; p), respectively. The former is assumed to be on shell, whereas q ≡ (q1 − q2 )=2 = (q0 ; q) represents the intermediate relative four-momentum as seen in Fig. 18. The indices i; j = 1; 2; 3 label the NN; N, and channels, respectively, and the two-body propagators G0j are given in terms of factors of the form 1 1 G0 (q0 q) = ; (3.101) 2 2 2 2 (E + q0 ) − q − M + i0 (E − q0 ) − q2 − M 2 + i0 times projection operators for Dirac (N ) or Rarita–Schwinger () spinors. The energies and masses appearing in (3.101) correspond to the particles propagating in the speci>c channels. To take into account the onset of pion production processes, in the early calculations the mass has been given a negative imaginary part, with the requirement that the latter vanishes below the pion production threshold [160]. One is thus operating in a two-body framework, three-body unitarity being taken into account only in an eEective way. It may be expected then, and it has been actually veri>ed, that this may lead to problems near the production threshold. A rather re>ned treatment of the driving forces characterizes this approach. The NN interaction V11 is described through one-boson-exchange of , , #, 0, and mesons, the NN and NN couplings being given by the relativistic Lagrangians (3.1) and (3.13), respectively. The transition interaction V12 ≡ VN and its inverse are given by one-boson-exchange graphs. The N vertex is derived from the interaction Lagrangian (3.7), while the eEective Lagrangian fN X I F 5 ˜ + LN = i + + T · (9I ˜ F − 9F ˜ I ) + h:c : (3.102) m is employed for the N vertex. The calculations show that exchange in the N coupling reduces the strongly attractive pionic interaction by contributing a short-range repulsive term, con>rming what comes out from non-relativistic coupled-channel calculations [4]. In particular, the resonance-like structures in the 1 D2

360


and 3 F3 waves, which in the past have given origin to many speculations about possible dibaryon resonances, turn out weaker than one could obtain without -exchange contributions. The eEects of intermediate states in the T = 1 NN channels turn out to be small, and the inelasticities are dominated by N coupling. Pion production from states is also small. A remarkable point is the strong sensitivity of the inelastic parameters to the treatment of the propagator, which in non-relativistic approaches is always treated in a more or less approximate, phenomenological way. At a >rst stage, the momentum dependence of the width has been introduced in an eEective way. Calculations clearly exhibit the consequences of such an approximation. For a >xed complex mass one gets a systematic overprediction of inelasticity for the peripheral waves with a rather steep onset of inelasticity near the pion production threshold. To overcome these limitations, the BSE model has been extended to comply with three-body unitarity by dressing the nucleon and isobar propagators [161] via the Dyson equations GN = G0 + G0 .N GN ; G = G0 + G0 . G ;

(3.103)

which are solved in the lowest-order approximation for the self-energy ., identi>ed with the bubble diagram. For NN scattering, the >eld-theoretic treatment of the propagator considerably improves the threshold behavior of the inelasticities, with a much more smooth behavior at threshold. It turns out, however, that propagator dressing Eq. (3.103) in the NN sector introduces considerable attraction in the low partial waves, with a detrimental eEect on the overall >t of the phase shifts. This might indicate that a more exhaustive treatment of coupling to other channels, like for example the d one, is required. As is well-known, channel-coupling eEects have been one of the hot topics in unitary three-body models for the NN –NN system [159]. 3.5. QCD-inspired models We have seen that the meson-exchange picture can at most provide semi-phenomenological models of the N interaction, where coupling strengths and cut-oE parameters have to be regarded as quantities to be determined >tting the experimental observables. This task is made even more diScult by the unstable nature of the , which has up to now hampered the attempts at getting a detailed knowledge of the N interaction. In particular, the short-range part of the baryon–baryon potential remains largely elusive, being more strictly related to the internal structure of the strongly interacting particles, and requires detailed analysis of reaction processes other than NN scattering. A noteworthy example is provided by Ref. [141], where total and diEerential cross sections, and the tensor analyzing power for d elastic scattering in the resonance region have been analyzed, looking at eEects due to N rescattering in the intermediate state. The corresponding contributions have been added to the amplitudes obtained through a Faddeev calculation for the NN system, and the N scattering amplitude has been parameterized so as to respect unitarity bounds. It turned out that calculations could be brought in good agreement with the experimental data only if short-range interactions were included in the 5 S2 and 5 P3 N channels. Clearly, a detailed mapping of the N interaction through this route would require much more analysis and accurate experimental data than presently


361

available. These problems would be overcome if the N interaction could be derived from the underlying QCD Lagrangian, thereby relating its structure to the fundamental quark and gluon degrees of freedom. This is not yet possible in the highly non-perturbative regime relevant to Nuclear Physics, and one has to resort to models, which take into account in an approximate way the salient non-perturbative features of QCD, i.e. con>nement and spontaneous breaking of chiral symmetry (CSB). In QCD gluons, as gauge bosons associated to a non-abelian symmetry, carry a color charge and can couple to one another; as a consequence, the quark–quark force becomes extremely strong at low energies and prevents colored objects from propagating far away from the interaction region. In quark models, this long-range phenomenon is generally taken into account introducing a linear or quadratic con>ning quark–quark potential. Spontaneous CSB can explain the large diEerence in value between current quark masses appearing in the QCD Lagrangian, of the order of a few MeV, and the constituent quark masses of quark models, of the order of 13 of the nucleon mass M . Indeed, if the current masses were rigorously zero, the QCD Lagrangian would be chiral invariant. Chiral symmetry, however, is broken by the QCD vacuum because of non-perturbative, short-range eEects. The consequences of spontaneous CSB are twofold. Quarks acquire a dynamical mass, which can be as large as M=3 at low energies and low momenta; at the same time, Goldstone chiral >elds appear, which couple directly to quarks. In some hybrid quark models these Goldstone bosons are identi>ed with meson >elds, and phenomenological meson-quark–quark couplings are introduced. Needless to say, these meson >elds are regarded as elementary, and no eEort is made to a description in terms of quark and gluon degrees of freedom. A hybrid quark model of the type outlined above has been proposed by the TUubingen–Salamanca group some years ago [163], and subsequently used to derive the N interaction from a Resonating Group Model (RGM) of the two-baryon system [164]. Quarks are coupled to a pseudoscalar chiral >eld, which is identi>ed with the pion in order to be consistent with the well-established picture of the long-range part of the NN force. In addition, a scalar Goldstone boson is introduced, reminiscent of the boson of phenomenological OBE models. The former leads to a one-meson-exchange inter-quark potential with the usual spin–isospin and tensor components, the latter to a central exchange force of the Yukawa type [163]. These interactions are supplemented with a quadratic con>ning potential. Perturbative QCD eEects are, as usual, taken into account through the De R[ujula– Georgi–Glashow OGE interaction [165]. In standard quark models of hadrons OGE is advocated to account for the hyper>ne mass splitting within the same multiplet. Here, it concurs, together with the OPE quark–quark potential, to >x the –N mass diEerence to its experimental value. In this model hadrons are described as colorless clusters of three quarks, and for each state of total spin S and isospin T the two-baryon wave function is given by the RGM ansatz

ST st st ?BB (q4 q5 q6 )]ST C(R)} ; = A{[/ (q1 q2 q3 )/

(3.104)

where A is the total antisymmetrization operator among the quark degrees of freedom, R is the relative coordinate between the two quarks, and the immaterial total center-of-mass wave function has been omitted. In principle, C(R) ought to be determined solving the Hill–Wheeler RGM equations. In Ref. [163], however, the N potential has been simply evaluated in the Born–Oppenheimer approximation, evaluating the total potential energy of the six-quark system between two-cluster states. One then de>nes the N interaction as the diEerence between this expectation value for

362


a given R and its value for R → ∞. In so doing, the is treated as a stable particle, whose properties are modi>ed by a nearby nucleon, because the two baryons can exchange constituent quarks and virtual bosons between each other. The coupling to the N continuum has to be taken into account separately, through the resonance width, in a way appropriate to any particular problem. Once the model parameters have been >xed >tting the NN bound-state and scattering observables, and the N mass diEerence, the N interaction is univocally determined. It exhibits a short-range behavior diEerent from what is predicted by meson-exchange models, with a hard core in several partial waves. This can be explained by Pauli correlation eEects between the constituent quarks, much in the same way as one predicts a short-range core in the eEective elds. The implementation of this program led to Chiral Perturbation Theory, where external momenta low with respect to the chiral scale NC ∼ 1 GeV, and the inverse of the nucleon mass are used as small expansion parameters. These techniques have been already used successfully in the low-energy N sector [169], and extended to deal with the NN problem [170 –172]. When going at higher energies, problems arise because of the appearance of nucleon resonances. This is particularly true for the , owing to its strong coupling to the N system, and the small mass diEerence with respect to the nucleon. Recently, new formulations of eEective chiral >eld-theory have been developed, which explicitly include the resonance as a (3=2; 3=2) >eld, and treat the N mass splitting as a small expansion parameter [173,174]. Presently, these formalism are used to study the chiral dynamics of the N system in the resonance region. It would be outside the scope of this paper to consider in detail these developments; we limit ourselves to observe that they give indications that the experimental N phase shifts can be reasonably reproduced with an eEective N coupling constant fairly close to the quark-model value. The extension of this ambitious program to the N system would certainly be a very interesting goal.


363

4. Contents of in nuclei 4.1. Light nuclei: the percentage of in the nuclear wave function A natural question at this point is how one can “see” the eEects of the in >nite nuclei. Few-nucleon systems (A 6 4) are very good candidates to look for these eEects, since in this case one can resort to microscopic calculations, with a minimum of model approximations. One can determine, for instance, the content of in nuclei by analyzing the weight of the components in the wave function, with respect to the nucleonic ones. This corresponds to evaluating percentage P . The simplest case is represented by the deuteron, even if isospin conservation excludes N components, and one has to look for the possible presence of con>gurations. Pioneering calculations have been performed at the beginning of the 1970s by ArenhUovel and collaborators [175], using static transition potentials with -exchange only. The isobar con>gurations were generated through an impulse approximation of the form ?N G VN ?NN :

(4.1)

A -percentage P ∼ 1% was found. These results, however, can be considered at most qualitative, because of the impulse approximation, and of their strong dependence on the cut-oE required to regularize the -exchange transition potential at the origin. A major improvement was given by the introduction of -exchange [112], and by the advent of CC calculations [176 –178]. The overall interaction due to and exchange is much weaker at short distances, leading to a mild cut-oE dependence, whereas CC calculations, summing up the NN – couplings to all orders, oEer more reliable results. A percentage around 0.8% was found. A modern example of CC approach is the model of Dymarz and Khanna [179 –183], where a non-relativistic version of the Bonn potential has been employed for the NN interaction, supplemented with static transition potentials to channels of the form (3.10), (3.11). The meson–baryon coupling strengths were chosen according to quark and=or strong-coupling models, and the full interaction was re>tted to the two-body observables. A probability of component P ∼ 0:4% has been obtained in deuteron wave function. Three-body systems, 3 He and 3 H, oEer the possibility of exploring both single- and multiple- components. The embedding of the coupled-channel NN –N system in few-body calculations clearly implies highly non trivial problems. For A=3 nuclei this was achieved by the Hannover group through a generalization of the Faddeev approach, where baryons can exist in two spin and isospin states, corresponding to the nucleon and the isobar excitation [142,184]. For phenomenological applications, they modi>ed the Paris interaction and described the NN → N transitions via and exchange [142,143,185]. The probability for a con>guration with a single in the nuclear wave function was found about 2.5%, with an underbinding of about 0:8 MeV for the triton binding energy. These calculations, originally restricted to one- con>gurations only, have been subsequently extended so as to include and components [151]. It was found that the repulsive dispersive effects and the total -induced three-body attractive contributions to the triton binding energy almost completely cancel. Since the Hannover NN -force model was found to be somewhat defective in reproducing the two-body data, similar calculations have been performed with the Argonne v28 interaction [151]. Again dispersive and three-body eEects were found to cancel to a large extent. The main diEerence between the Hannover and v28 potentials is a reduced three-body force contribution. The two calculations predict diEerent probabilities; the Hannover model gives P ∼ 4%, whereas

364


the v28 produces a more comfortable value P ∼ 1%. Finally, eEects were always found to be negligible. It would be interesting to compare these results with N potentials derived from an underlying >eld theory, like the ones of Ref. [145]. To the best of our knowledge, this has not been attempted up to now. The generalization of the Faddeev formalism to A ¿ 3 nuclei is extremely diScult. A much more ?exible approach is represented by the variational method, which has met great successes in the last several years in describing the properties of nuclei up to A = 8 starting from realistic phenomenological potentials [186,187]. As is well known, a suitable parameterized trial function is used to calculate an upper bound to the energy which is minimized, and the lowest value taken as the approximate ground-state energy. The quality of the calculation depends on the starting function, therefore it is useful to consider trial functions with reasonable correlations built in. It is expected that the correlations brought by the couplings are re?ected in the wave function, therefore the same operators appearing in the NN interaction are used to construct the correlated state. In principle, if the NN interaction includes all transitions involving excitation, like for example the v28 potential [149], this procedure would generate a correlated wave function with components. This is however extremely diScult to achieve in practice, due to the very large number of channels and parameters which should be adjusted variationally for each nucleus. An alternative procedure is to use transition correlations which describe two-body bound-state and scattering wave functions, for an interaction with degrees of freedom, and assume that these correlations are relatively A independent, since they appear to be short-ranged [188]. As a consequence, the NN components of the correlated wave function are required to be proportional to the projected NN channels of the full two-body wave function, calculated with an interaction with transitions. The N and channels are obtained from the action of transition correlation operators that simulate the transitions in the assumed two-body interaction, so that one writes [188]

TR ? = S (1 + Uij ) ?N ; (4.2) i¡j

where ?N contains only nucleonic degrees of freedom, S is a symmetrizer taking into account the non-commutative nature of the intervening operators, and the pair transition correlation operator UijTR contain N, N and components. In practice, they are constructed with the spin and isospin ˜ i , times radial functions to be determined by >tting Pauli and transition operators i , Si , ˜i , and T the solution of the CC NN –N problem. This approach is therefore intrinsically non-perturbative. To grasp what a perturbative treatment of the isobar would be in this context, let us go back to Eqs. (3.92), and solve it with respect to ?N . Disregarding the interaction, one immediately gets up to lowest-order in VN Eq. (4.1). The total wave function is therefore in this approximation ? (1 + G VN )?NN :

(4.3)

If the kinetic-energy terms are disregarded in the free Green function G , one sees that the >rst-order perturbation theory result (4.3) is equivalent to transition correlation operators U TR of the form VN ; (4.4) U TR M − M namely, U TR is simply given by the N transition potentials scaled by the inverse of the N – mass diEerence.


365

Correlated wave functions of the type (4.2) have been used to calculate contributions to electroweak observables and radiative and weak capture cross sections at very low energy [188,189]. The respective transition matrix elements between initial and >nal states are calculated expanding the wave functions into a pure nucleonic part ?N plus corrections containing the excitations generated by Uij , i.e. Uij ?N + · · · : (4.5) ? = ?N + i¡j

We shall go back to this topic in the next section. 4.2. Mesonic-exchange currents and To explore the structure of the nuclear wave function, a privileged source of information is given by the coupling with an external electromagnetic probe. Below threshold for meson production one can explain many aspects of nuclear physics in terms of meson-exchange nuclear forces. Since the interaction in presence of an external electromagnetic >eld can be determined according to the minimal electromagnetic coupling ∇ → ∇ − (ie=c)A, one gets speci>c forms for the meson exchange currents (MEC), in terms of the same operators responsible for the NN interaction [11] These currents add to the electric and magnetic convection currents in the nucleus, and can explain part of the discrepancy between experiments and calculations for magnetic moments, form factors and radiative capture in few-body systems. In deuteron electrodisintegration by backscattered electrons, in particular, they provide the largest part of the total cross section [8,187]. It is impossible here to account for the enormous amount of work done in this >eld, and we will refer the reader to the excellent review papers in the literature [11,187], limiting ourselves to quote few relevant results to the present topic. The calculation of magnetic moments, form factors and radiative capture cross sections in few-body systems requires the evaluation of matrix elements of the current operator j between the proper initial and >nal states. The current can be expressed as the sum of one-, two- and many-body terms, that operate on the nucleon and degrees of freedom, j = j(1) + j(2) + · · · :

(4.6)

In the impulse approximation (IA) only the one-body contributions are kept. Current conservation implies a relation between j and the nuclear Hamiltonian H = T + VNN expressed by the continuity equation ∇ · j + i[H; ] ˆ =0

(4.7)

where ˆ is the charge-density operator. Eq. (4.7) can be immediately separated into continuity equations for the one- and two-body components, ˆ =0 ; ∇ · j(1) + i[T; ]

(4.8)

∇ · j(2) + i[VNN ; ] ˆ =0 ;

(4.9)

involving the kinetic energy T and the nucleon–nucleon potential VNN , respectively. From symmetry requirements and conservation laws, the currents may be written in terms of longitudinal and

366


(a)

(c)

(b)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

Fig. 19. Feynmann representation of hadronic currents associated to diEerent meson exchanges: one-body currents without and with excitation (a), (b); one-body current (c); two-body seagull components with and exchange (d), (e); mesonic exchange contributions (f), (g); the transverse + and !+ components (h), (i); pair graphs with and ! exchange (j), (k). The wavy lines represent the photon.

transverse operators [190]. Eqs. (4.8) and (4.9) constrain only the longitudinal part of the currents. In particular, the electric form factors have to be the same in the one- and two-body currents, as well as in the charge-density operator . ˆ A further constraint is imposed by Eq. (4.9), since the longitudinal two-body current has to be consistent not only with ˆ but also with the assumed NN interaction. In the literature this consistency is normally achieved in two ways; one either uses the same type of meson–baryon vertices in j(2) and VNN [179 –183], or one phenomenologically modi>es the two-body currents so as to be consistent with a phenomenological VNN [11]. This constrained part of the two-body current can therefore be considered model-independent, since it contains the same meson exchanges terms, and no extra parameters than the ones already present in the NN interaction. The currents contain contributions due not only to nucleons, but to isobar excitations as well, (i) (i) j(i) = j(i) NN + jN + j

(i = 1; 2) :

(4.10)

The longitudinal and transverse components of the one-body part j(1) arise from direct couplings of nucleons or ’s, respectively [191], with the probing electromagnetic >eld, as shown in Fig. 19(a) – (c), and are expressed in terms of the free baryon electric and magnetic form factors, which take into account the >nite size of the baryon. For the two-body currents j(2) possible processes are the seagull diagrams shown in Fig. 19(d) and (e) where the probing photon is attached to a vertex of a - or -exchange graph, and contributions


367

Fig. 20. Exchange currents involving excitation of an intermediate . The dashed lines represent the exchange of or mesons. Table 5 Magnetic moment of the deuteron from various calculations: (1) and (2), Ref. [181]; Bonn, Ref. [103]; Paris, Ref. [192]; Argonne v14, Ref. [193]; Argonne v18, Ref. [202] VNN

FdNN

FdNN +MEC

FdNN +

FdNN ++MEC

(1) (2) Bonn Paris v14 v18

0.8513 0.8459 0.852 0.847 0.8453 0.847

0.8544 0.8537 0.860 0.859 0.8638 0.871

0.8656 0.8683

0.8687 0.8761

Exp.

0.857406(1)

where the photon directly couples to the exchanged meson, as in Fig. 19(f) and (g). In addition, there are purely transverse contributions, which cannot be related to the assumed interaction through the continuity equations. Examples of these model-dependent terms are the + and !+ graphs depicted in Fig. 19(h) and (i). Other transverse terms are given by the pair graph of Fig. 19(j), usually combined with the seagull term, and the ! pair diagram, which is the only ! contribution, since this isoscalar meson cannot directly couple to the electromagnetic >eld. If the NN interaction does not explicitly contain transitions, two-body currents involving the isobar will also fall in the model-dependent category. The exchange current operators containing can be divided into two classes shown in Fig. 20. In graphs 20(a) and (b), the nucleon resonance is excited by the probing electromagnetic >eld. In the second type 20(c) and (d) the resonance is already present in the nuclear wave function and couples elastically to the external photon. The latter contributions are model-independent, once the excitation and two-meson exchange are explicitly introduced in the nuclear force model. The determination of nuclear magnetic moments amounts essentially to the evaluation of the matrix element of the current j between initial and >nal states. The >rst column of Table 5 reports the deuteron magnetic moment calculated in impulse approximation for several realistic NN interactions. It is immediately perceived that all these calculations cannot account for the experimental result, and part of the diEerence among them is due to the diEerence in the D-state percentage. The second column shows the non-negligible eEect of the MEC quoted above. The explicit inclusion of the through a coupled-channel calculation gives the >gures reported in the >rst two rows [181], using a Bonn-type NN interaction, a static potential re>tted to the two-body observables, and the meson–baryon coupling strengths given by the quark and

368


strong-coupling model, respectively. In the calculation with the Argonne v14 interaction [193] the exchange currents involve the intermediate excitation of , which was found to give a very small contribution, due to cancellations between the - and -exchange terms that involve the isobar. The coupled-channel calculations of Ref. [181], on the other hand, seems to indicate a non-negligible importance of the state in the deuteron, which amounts in this model to 0.36%. In fact, this component in the deuteron form factor accounts to a large extent for the shift of the diEraction minimum observed in the Rosenbluth structure function for elastic electron–deuteron scattering [183]. Similarly, the isobaric degrees of freedom seem to play a signi>cant role at large momentum transfers in 2 H (e; e )pn reaction at backward angles [182]. The calculations employing the v14 potential and based on a conserved current, on the other hand, obtain a fairly good agreement with the experimental data without explicit isobar components in the wave function [193]. One must observe, in any case, that no de>nite conclusion about the role in electromagnetic currents can be drawn, until more detailed information about the meson–baryon interaction is available. The considerations made above have all been developed considering non-relativistic current operators. The inclusion of relativistic eEects like retardation eEects, pair N NX contributions, and others, has been the subject of several investigations in the literature [194 –199]. A consistent calculation of observables requires relativistic equations, like the Bethe–Salpeter one, to obtain the wave function. Calculations along these lines have either included meson-exchange contributions in some quasi-potential approximation [197], or have been limited to the nucleon-propagator singularities in equal-time [198] or ladder approximation [199]. A good description of the experimental data for the Fd has been obtained [199], with less satisfactory results for the quadrupole moment, where probably additional eEects including the isobar have to be taken into account. The introduction of these contributions would imply a relativistic coupled-channel framework, such as the one developed by Tjon and co-workers [160,161], plus a covariant description of exchange currents. This seems still to be done. The static properties and the charge and magnetic form factors for 3 He and 3 H have been calculated with Faddeev techniques by the Hannover group [185]. Both one- and two-body currents are obtained as non-relativistic limits of the standard Feynmann amplitudes, with explicit contributions, and satisfy the continuity equation only approximately. The good agreement with the experimental electromagnetic form factors is obtained mainly from the inclusion of selected relativistic corrections, and the use of the Dirac nucleon form factor in place of the Sachs form factor. Subsequent calculations based upon a slightly diEerent force model have given rather small eEects of the contributions to the electromagnetic properties of three-nucleon systems [200]. The in?uence of the isobar upon the electroweak properties of very light nuclei has been studied in detail by means of the variational method. Indeed, within this approach the three-nucleon electromagnetic properties, the Gamow–Teller matrix element in tritium >-decay, and the low-energy neutron radiative capture and proton weak capture on 3 He have been calculated both without and with degrees of freedom [188,189]. In particular, the electromagnetic structure of trinucleons has been studied [189] with the v18 two-nucleon [202] and Urbana IX [203] three-nucleon interaction; the latter consists of a long-range term due to excitation of an intermediate , plus a short-range repulsive phenomenological contribution, which simulates the dispersive eEects arising when integrating out degrees of freedom. The isobar components in the wave function have been introduced through the transition-correlation-operator method described in Section 4.1. As we have seen, these operators correspond to transition potentials with a long-range OPE tail and a phenomenological


(1)

(2)

(d)

(e)

(f)

(g)

(c)

(d)

(e)

(f)

(g)

(j)

(k)

(l)

(a)

(b)

(c)

(h)

(i)

(j)

(a)

(b)

(i)

(h)

369

Fig. 21. Diagrammatic representation of operators included in the one-body (a) and two-body (b) currents in the variational calculations of Ref. [189]. The dashed and crossed-dashed lines represent transition-operators insertions. Table 6 Magnetic moments of three-nucleon systems: impulse approximation IA, purely nucleonic current contribution j(3N ), eEect of one-body currents j (1) (), full calculation and experimental result. From Ref. [189] F

IA

j(3N )

j(3N ) + j (1) ()

Full

Exp.

F(3 H) F(3 He)

2.571 −1:757

2.961 −2:077

2.971 −2:089

2.994 −2:112

2.979 −2:127

short-range part constrained by two-body observables. A perturbative treatment of components in the wave function is simpler, but it may lead to a substantial overprediction of their importance since they tend to be too large at short distances. Model-independent two-body currents consistent with the NN interaction have been determined via the continuity equation, supplemented with the model-dependent contributions with explicit excitation. The initial and >nal total wave functions ?i and ?f in ?f |j|?i determining the three-nucleon electromagnetic form factors were given in terms of the pure nucleonic wave functions ?N by Eq. (4.2). If ?f and ?i are expanded according to Eqs. (4.5) one gets the contributions to the form factor exhibited in Figs. 21(a) and (b), for the one- and two-body currents, respectively. Connected three-body terms are neglected. According to this analysis, the contributions associated with components are small, but help in bringing the calculated magnetic moments and electromagnetic form factors for 3 He and 3 H in agreement with the experimental data. In Table 6 the eEect of excitation in the currents for trinucleon magnetic moments are presented. The >rst column represents the result obtained in impulse approximation (IA), whereas the second (j(3N )) gives the outcome of a calculation including purely nucleonic two- and three-body currents.

370


In j(3N ) + j (1) () the contributions of one-body currents with excitation corresponding to Fig. 21(a) have been added. Finally, the result of the full calculation is reported in the last column. In the case of 3 He the currents seem to help the agreement. The low-energy radiative and weak capture cross sections on 3 He have been also evaluated in this model [188], >nding some improvement in the comparison with the data. Similar calculations have been performed in Ref. [201], where correlated Hyperspherical Harmonic wave functions with isobar admixtures were used to evaluate cross section and polarization observables for the radiative capture reactions 2 H(ñ; +)3 H, and 2 H(˜ p; +)3 He at low energies, as well as the energy dependence of the astrophysical S-factor. Again, an overall satisfactory agreement with the observed data was found. With the advances in theoretical approaches to the nuclear few- and many-body problem, one may expect that increasingly sophisticated microscopic calculations will clarify the intertwined role of the and meson-exchange currents in larger and larger nuclei. There are already indications that, including the simplest exchange isobaric contributions, can give important corrections to the magnetic dipole moment in p-shell (A=4–16) nuclei [204]. There, complete 0˝! and (0+2)˝! shell model calculations have been performed. Since the considered mass region partially overlaps with that accessible to modern ab initio Green-Function–Monte-Carlo or variational calculations [187], an analysis able to ascertain to what extent these results are sensitive to the employment of model wave functions would be welcome. The results of all these analysis always depend to some extent on the uncertainties in the N and interactions, and on the parameters appearing in the meson–baryon and photon–meson vertices. Thus, any >nal conclusion may be drawn in this cautious perspective. However, a general result is for sure on the ground of modern CC or variational calculations, that the eEects due to components in the nuclear wave function are signi>cantly smaller than the ones obtained using perturbation theory. This is particularly true in reactions such as the radiative or weak captures on 3 He at very low energy, where the small overlap between the main components of the bound-state wave function strongly quenches the nucleonic part of the one-body current operator. Another general feature of the considered processes is the strong sensitivity of observables to the tensor eEects in the isovector component of the two-body currents, which can substantially change for diEerent interactions. 4.3. propagation in nuclei 4.3.1. EDective interaction The excitation of represents the dominant mechanism for pion photo- and electroproduction on nucleons [8,205], much in the same way as this resonance is the prominent feature for low- and intermediate-energy N scattering [2,8]. It is strongly excited and propagates as a quasi-particle in the nuclear medium, as can be clearly seen from the energy dependence of the total - and +-nucleus cross sections illustrated in the example of Fig. 22. For photo-induced reactions, the cross sections re?ect the position and strength of the free throughout the periodic table, as seen in Fig. 23. In fact, the experimental cross section peaks at the same position as the free nucleon one (+N ). Similar considerations apply to inclusive electron–nucleus scattering, as exempli>ed in Fig. 24, where the (e; e ) cross section for 12 C is compared with the sum of the free nucleon cross sections. Here also one observes a substantial broadening but no shift of the resonance peak with respect to the free nucleon case.


371

Fig. 22. Pion- and photon-12 C total cross sections in the resonance region. The full lines represent the free N and +N cross sections times the mass number A. From Ref. [8].

Fig. 23. Total photonuclear cross section per nucleon for diEerent targets in comparison with the average single nucleon total photoabsorption cross section (dashed line). From Ref. [206]. ◦

Fig. 24. Inclusive 12 C(e; e ) cross section for incident electrons of 620 MeV at = 60 . The short-dashed line represents the sum of the free nucleon cross sections. The dotted and long-dashed lines give theoretical estimates of non-resonant contributions and kinematical eEects, respectively, whereas the full line is the outcome of a full -hole model calculation. From Ref. [207].

In -nucleus scattering, on the other hand, one has a damping of the resonance and a marked downward shift of its position. These features show that the survives even in a strongly interacting environment, and can therefore be treated as a quasi-particle at the same level as the nucleon. This is the basis of the -hole model [208–210], where the diEerences between pion and photon-induced

372


(a)

(b)

Fig. 25. (a) One-pion exchange -hole interaction; (b) intermediate → N decay responsible for the decay width.

reactions are naturally explained in terms of the diEerent coupling of the two probes with the nuclear medium. As is well known, in the -hole model one assumes that nucleons and ’s move in a mean->eld, and excitations of the nuclear medium are described in terms of -hole states strongly coupled to the pion >eld. The nuclear Hamiltonian is the sum of a single-particle Hamiltonian for non-interacting nucleons and ’s H0 , plus the NN and N coupling terms not included in the mean >eld, H = H0 + H + HNN + HN ;

(4.11)

where H is the free pion Hamiltonian. The N Hamiltonian couples the nucleon and by absorption or emission of a pion as given in Eq. (3.9). It naturally introduces an OPE N → N interaction (3.11), which provides the longest-range driving mechanism of the -hole force of Fig. 25(a), and the possibility of → N decay illustrated in Fig. 25(b). The decay width is introduced in propagation. Owing to the interaction of the with the surrounding medium, its width 2 acquires a non-trivial energy dependence, and can be written in terms of the free width, plus corrections associated to elastic broadening, Pauli quenching, and coupling to absorption [8], 2 (E) = 2free + 2el + 2Pauli + 2abs :

(4.12)

The absorption cross section can amount up to one third of the total one. The overall eEect of the medium on can be subsumed into a complex optical potential, whose real part contains binding eEects in the mean >eld, dispersive shifts associated to absorption, and short-range correlations, while the imaginary component takes into account the presence of the absorptive couplings. The detailed behavior of in nuclei depends upon the structure of the -hole interaction. For the longest-range part the latter can be determined by the and exchange contributions (3.11), (3.18). In the spirit of Landau–Fermi-liquid theory, the eEect of short-range correlations are described regarding nucleons near the Fermi surface kF as quasi-particles, endowed with an eEective mass M ∗ due to the surrounding cloud of N -hole excitations [211]. This leads to the residual contact -hole interaction [8] V(corr) h ; h =

2 fN ˜ †2 : g S1 · S†2 T˜ 1 · T 2 m

(4.13)


373

For the sake of simplicity, we have considered only the B channel. The overall residual interaction can be decomposed into a spin-longitudinal and a spin-transverse part taking into account the analogue of Eq. (3.16) for S. One gets V h ; h (q) =

2 fN ˆ †2 · q) ˆ {W (LO) (!; q)(S1 · q)(S m2

˜1 · T ˜ †2 ; ˆ · (S†2 × q)} ˆ T + W (TR) (!; q)(S1 × q)

(4.14)

where qˆ ≡ q=|q|. The longitudinal and transverse coeScients W (LO) (!; q) and W (TR) (!; q) depend : upon the transferred energy and momentum as well as upon the Landau parameter g W (LO) (!; q) ≡ g +

W

(TR)

q2 ; !2 − |q|2 − m2 + i0

2 fN m2 q2 : (!; q) ≡ g + 2 2 2 m fN ! − |q|2 − m2 + i0

(4.15) (4.16)

The residual interaction (4.14) has by now become the “standard model” for the description of -hole correlations in nuclei. In practical applications one often uses the quark or strong-coupling 2 2 relation (3.87), together with (3.60) to write (fN =m2 ) × (m2 =fN ) 2. In a coupled treatment of the N system Eq. (4.14) has to be supplemented with a NN ↔ N transition interaction. For exchange this is provided by Eq. (3.10), whereas short-range correlations can be subsumed into the Fermi–Landau type contact force fNN fN corr ˜ †2 + h:c : VNN gN 1 · S†2 ˜1 · T (4.17) ↔N = m2 , g , and A basic question is obviously what can be said about the various Landau parameters gNN gN , characterizing the short-range behavior of the N - and -hole excitations in the nuclear medium. In particular, the actual value of gN strongly in?uences the role of the isobar in quenching the , more quenching nuclear response to an isovector spin probe, as we shall see below; larger gN eEects are attributable to isobar currents. Big eEorts have been devoted during the 1980s to this problem, as extensively discussed in the review paper by Towner [9], to which we refer the reader for more details. Here, we limit ourselves to touch some essential points of this fascinating but, admittedly, intricate problem. On the ground of quark-model and=or chiral symmetry arguments it has been proposed that the Landau parameters might satisfy the universality relation [212–214] gNN = gN = g ≡ g :

(4.18)

A simple estimate can be inferred for g by the minimal requirement that the short-range correlations cancel completely the -function term in the OPE potential [8,9], which gives the well-known value g = 13 . Under a phenomenological point of view, the Landau–Migdal interaction in the N -hole sector reproduces well the collective states in both light and heavy nuclei, where it is probed in the Landau limit ! → 0, q = k2 − k1 → 0, with the values of the initial and >nal nucleon momenta k1 ; k2 ∼ pF . One >nds [215,216] g 0:7–0:8 ;

(4.19)

374


Fig. 26. Direct (a) and exchange (b) NN → N transition potentials with particle-hole couplings, and higher-order ‘induced’ contribution (c). The shaded area represents the sum of RPA particle-hole interactions.

(a)

(b)

(c)

(d)

(e)

(f)

Fig. 27. Graphical representation of coupled equations to obtain the eEective particle-hole interaction with transition. Direct (a), exchange (b), and ‘induced’ (c) – (f) contributions.

a result indicating that the short-range spin–isospin interaction is much more repulsive than one could expect on the ground of a simple picture of non-overlapping nucleons. On the theoretical side, several attempts have been made to evaluate the Landau parameters microscopically, starting from Brueckner–Bethe theory [217–220]. In these approaches one >rst evaluate the Brueckner G-matrix; the basic contributions to the -hole interaction are then given by the direct and exchange graphs shown in Fig. 26(a) and (b). It turned out that satisfactory results could be obtained only by including medium polarization eEects [9], described in Fig. 26(c); omission of these contributions would otherwise lead to too much attraction in the scalar–isoscalar channel and to instability of nuclear matter with respect to small density ?uctuations. This implies that the overall particle-hole interaction Fph has to be obtained from the Brueckner G-matrix Gph through the self-consistent solution of the non-linear coupled equations Fph = Gph + Finduced Fph

(4.20)

as graphically depicted in Fig. 27. Calculations by the JUulich [219] and Tokyo [220] groups gave . Both found, in any case, that core-polarization eEects boost the somewhat diEerent results for gN Landau parameters, with g = gN , as shown in Table 7. Recently, the Landau parameters g and gN have been determined through careful analysis of exclusive charge-exchange, quasifree decay, and 2p emission reactions [221,222]. As we shall discuss more extensively below, the 12 C(3 He; t+ )12 C(g:s:) process selectively probes the residual interaction in the longitudinal channel, and in a kinematic domain where the longitudinal response function attains its maximum value. One can expect, therefore, that the cross sections are particularly . Indeed, the energy spectra for the above sensitive to variations g in the Landau parameter g


375

Table 7 Landau parameters g and gN obtained with diEerent models: I, bare G-matrix; II, bare G-matrix + ‘induced’ interaction; III, bare G-matrix + induced interaction + relativistic eEects Model

g

gN

JUulich

I II III

0.49 0.58 0.56

0.35 0.56 0.68

Tokyo

I II

0.52 0.61

0.35 0.45

reaction exhibit an energy shift, which scales linearly with g [221]. The experimental data can be reproduced only by a value of g 0:33, as required by minimal short-range correlations. As , the quasifree decay of and 2p emission, induced by pion absorption and charge-exchange for gN 3 ( He; t) reactions, have been analyzed in the framework of the -hole model [222]. Since the coupling interaction is minimally aEected by oE-shell ambiguities in the quasifree decay of the , one can employ these processes to study distortion eEects on the wave functions of the outgoing pion and nucleon. The N → NN transition in 2p emission reactions has been described by the + + g model, and found to be dominated by the Landau–Migdal term. The experimental data for both the (+ ; pp) and (3 He; tpp) processes could be reproduced with gN in the range 0.25 – 0.35.

4.3.2. Medium eDects in direct and charge-exchange reactions In the -hole model pion- and photon-induced reactions are described in terms of excitation of the by the external probe, followed by propagation and decay of the resonance. The diEerence in behavior of the respective cross sections can be simply ascribed to the longitudinal nature of the pion coupling Eq. (3.9) with respect to the transverse +N coupling, f+N † H+N ≡ S · ( × k)Tz† : (4.21) m The vectors k and are the momentum and polarization vector of the impinging photon. Indeed, the doorway -hole state excited by the photon is followed by coherent propagation via multiple scattering, until the >nal photon is created and escapes from the nucleus. The spin-transverse (S† × k) · coupling of the photon, and the longitudinal S · q coupling of the pion then combine to produce a (q × k) · dependence in the transition amplitude. In in>nite nuclear matter, where momentum conservation implies that k must be parallel to q, one has that the coherent propagation is prohibited. The same is true in a >nite nucleus, apart from minor corrections due to propagation in the non-forward direction. One may conclude that in elastic scattering of photons the probe only measures the broadening of the resonance due to Fermi motion and binding in the nuclear mean >eld. This situation has to be contrasted with what happens in elastic -nucleus scattering, where coherence eEects are responsible for the observed shift and damping of the resonance. The diEerent behavior of the cross section in pion- and photon-induced reactions is therefore a direct consequence of the diEerent spin structure of the N and +N couplings. excitations also takes place in charge-exchange reactions, where a downward shift of the resonance peak position ∼ 70 MeV is observed in nuclear targets, with respect to the single proton

376


Fig. 28. Experimental diEerential cross section at =0o for charge-exchange reactions on 12 C compared with the p(p; n)++ reaction at 800 MeV. From Ref. [223]. Fig. 29. Triton spectra seen in (3 He; t) reactions at 2 GeV projectile energy for diEerent nuclei. From Ref. [224].

target [10,225]. This is exhibited in Fig. 28, where the experimental zero-degree spectra for the inclusive p(p; n)++ and 12 C(p; n) reactions are reported. The same trend can be observed in (3 He; t) charge-exchange reactions on a variety of nuclei, as shown in Fig. 29, where the triton spectra at 2 GeV incident energy are plotted versus the kinetic energy T of the outgoing triton. In terms of the excitation energy ! ≡ E3 He − T , one has again a downward shift of ∼ 70 MeV with respect to the free p → transition. This situation is strongly reminiscent of what is observed in pion–nucleus scattering. A more careful analysis of the actual displacements of the resonance peak positions, reveals a systematic diEerence between proton and 3 He-induced processes. Thus, in the former case the resonance is located at ! 365 MeV for the proton target, whereas it appears at ! 295 MeV for A ¿ 12 targets. In the latter, on the other hand, one has resonance energies of 325 and 255 MeV for scattering processes on protons and A ¿ 12 nuclei, respectively. This apparent shift of around 40 MeV between proton and 3 He induced reactions can be simply explained in terms of the composite structure of the 3 He projectile. In this case the probability that the triton survives the scattering process is rapidly decreasing with increasing momentum transfer, and the cross section gets greater contributions at low excitation energies. This explanation is con>rmed by an analysis of the dependence of 3 He–t form factor upon the excitation energy ! and momentum transfer q [226,227]. For a detailed discussion of form-factor eEects for composite projectiles see Ref. [225].


(a)

377 (b)

Fig. 30. Experimental neutron (a) and triton (b) zero degree spectra for the reactions 12 C(p; n) at E = 800 MeV and 12 C(3 He; t) at 2 GeV, respectively. The theoretical curves for the cross section represent the longitudinal LO, transverse TR components, and the full calculation with and without particle-hole correlations of Ref. [223].

It is by now ascertained that, from the 70 MeV shift observed between proton and nuclear targets, 40 MeV can be explained in terms of the Fermi motion of the nucleons and ’s in the nuclear mean >eld [228,229]. The “interesting” part of the eEect is the remaining 30 MeV, which re?ects the diEerent kinematic domain explored in charge-exchange reactions and photon- and pion-induced processes. Indeed, in real photon scattering theinvolved energy and momentum transfers are related by ! = q, whereas pion scattering implies ! = q2 + m2 . In charge-exchange reactions, on the other hand, the target is probed by the virtual pion- and -meson >elds of the projectile=ejectile system. From a kinematic point of view, these meson >elds must satisfy the energy-momentum relation ! ¡ q, thereby exploring the response function of the target in a (!; q) region inaccessible to photoor pion-induced reactions. The reaction mechanism at the energies of interest can be described in the framework of distorted wave impulse approximation (DWIA), which means that one can think of an eEective projectile– nucleon target–baryon interaction transferring the energy ! and momentum q to the target. This excites N - and -hole states in the NN and N sectors. The evolution of the nuclear system depends then upon the full residual interaction, which consists of the four couplings Vp h ; ph , where p; p can be either a nucleon or a , and contains both spin-longitudinal and spin-transverse components. Consequently, charge-exchange reactions provide a mixed spin-longitudinal (LO) and spin-transverse (TR) probe, whereas pion and photon scattering only probe one of these aspects. The interplay between the two, LO versus TR, couplings, and the decay properties of in nuclei have been investigated in a beautiful series of papers by Udagawa, Osterfeld et al. [221–223,230]. In particular, it has been shown that the downward shift in the peak position is mainly due to the strongly attractive -exchange interaction in the LO channel. This can be seen, for instance, in Figs. 30, where the zero-degree neutron and triton spectra are shown for the inclusive 12 C(p; n) and 12 C(3 He; t) reactions, respectively. In the uncorrelated calculations the residual p-h interaction has been switched oE. Clearly, p-h correlations improve the agreement with the experimental data. The failure of the model in reproducing the data in the low-energy region of both spectra and in the high-energy tail of the (p; n) cross section can be reasonably attributed to the presence of background contributions. A noteworthy feature of these results is the very good agreement observed in the higher ! part of the 12 C(3 He; t) spectrum. At variance with the 12 C(p; n) reaction, there is no background due to projectile excitation on the high-energy tail of the resonance for the 12 C(3 He; t)

378


(a)

(b)

Fig. 31. Angular distributions for coherent pion production in 12 C(3 He; t+ )12 C(g:s:) at 2 GeV (a) and for elastic pion scattering on 12 C at 120 MeV (b). The solid line represents the LO component, and the dashed curve the full calculation with particle-hole excitations. Figure from Ref. [221].

process, since the probability that the excited projectile can decay into a triton plus a free pion is small. Even more interesting is the comparison between the LO and TR responses exhibited in the >gures. For both reactions one sees that the LO cross section is strongly shifted downwards, whereas the TR one is not. The origin of this diEerent behavior can be clari>ed by looking at a multipole decomposition of the cross sections [223]. It turns out that the contributions of unnatural-parity states are lowered in the excitation energy by about 60 MeV with respect to those of the natural parity excitations. Looking at the behavior of the residual interaction as a function of momentum transfer, one has a longitudinal component with a singularity at qpole = !2 − m2 , repulsive below the pole and attractive above it. In charge-exchange reactions the in the LO channel is excited by -exchange with a transferred momentum q ¿ qpole . The overall eEect is an attractive energy shift for all multipoles in the LO channel, with a consequent downward shift of the -peak position. The above considerations can be extended to exclusive processes, for which data are accumulating through coincidence experiments [231–233]. Of particular interest here is coherent pion production via 12 C(3 He; t+ )12 C(g:s:) scattering [221]. The angular distribution for this reaction at 2 GeV is displayed in the upper part of Fig. 31, in comparison with -12 C elastic scattering. Clearly, the LO channel is selectively populated by the process, which allows an experimental separation of LO and TR responses. The strict proportionality between the production and elastic cross sections shows that the same mechanisms are at work in the two coherent processes. Actually, the (3 He; t) kinematics


379

forces the virtual pions to propagate in the nuclear medium along the direction of the momentum transfer q. The eEective vertices at work in the longitudinal channel are S† · q, acting as an excitation operator, and S · q as de-excitation operator, producing the outgoing pion. As in elastic pion ◦ scattering, the angular distribution must therefore scale as (qq cos )2 , with a peak at = 0 . In 12 3 + 12 other words, the coherent pion production in the C( He; t ) C(g:s:) reaction can be regarded as a virtual pion scattering oE the target nucleus, where the initial oE-mass-shell pion is converted, through a multiple scattering process, into an on-mass-shell particle, which is >nally emitted from the nucleus. As already noticed at the beginning of the section, this is in contrast with pion photoproduction reactions. There, due to TR excitation operator S† × q, the diEerential cross section scales ◦ as (qq sin )2 , and therefore peaks at ∼ 90 , as con>rmed by the experimental data. Information on the spin structure of nuclear correlations can be obtained from spin observables [10]. One can measure the forward polarization transfer coeScients Dzz and Dxx , for spin transfer along and perpendicular to the beam axis, respectively. They are related to the longitudinal and transverse components of the strength function Eq. (4.34) by [234] ◦

Dxx (0 ) = ◦

Dzz (0 ) =

−|SLO |2 ; |SLO |2 + 2|STR |2

(4.22)

|SLO |2 − 2|STR |2 : |SLO |2 + 2|STR |2

(4.23)

Calculations of these observables [223] show that they are indeed sensitive to the degree of N −1 correlations in the nuclear system, the uncorrelated results being far more structureless than the correlated ones. It has to be observed, however, that attempts to reproduce the tensor analyzing power in the 12 C(˜d; 2p) reaction measured at Saturne [235] have obtained up to now only a moderate success [223]. 4.3.3. Quenching phenomena The excitation of -hole states has been advocated as an important ingredient in explaining several low-energy nuclear phenomena, where the value of observable quantities is quenched with respect to what one can predict on the ground of models involving nucleonic degrees of freedom only. The basic physical picture behind this point of view is that a nucleon can polarize the nuclear medium, much in the same way as a magnetic dipole induces spin alignment in a surrounding medium [8]. Actually one can envisage two basic polarization mechanisms. One is due to the tensor force between a valence nucleon and the core, which is very similar to a dipole–dipole interaction, and is intimately related to exchange. The other mechanism is linked to spin–isospin transitions, due to virtual -hole excitations. By their very nature, these quenching phenomena simultaneously involve the dynamics of and non-trivial many-body problems. A classical example of this subject is given by the screening of the -nucleon coupling in the presence of virtual -hole states. In close analogy to what is done in considering -hole contributions to the pion self-energy in nuclear matter, one can envisage the excitation and subsequent decay of -hole states in between the pion and the nucleon, as illustrated in Fig. 32. This produces a renormalization of the eEective -NN coupling constant by a factor 1 2≡ ; (4.24) 1 + g C

380


Fig. 32. Renormalization of the NN vertex by -hole excitations. The wiggly line represents the eEective -hole interaction minus one-pion exchange. = g where g ≡ g N comes from the vertex eEect described in Fig. 32 by the shaded area. The -hole susceptibility C is related to the -hole contribution to the pion self-energy, and is given by [8] 2 8 fN ; (4.25) C = 2 9 m M − M with the nuclear density. As is well known, the renormalization factor (4.24) has been estimated to be of the order of 30 –35% in nuclear matter [11]. It counteracts the enhancing eEects of nucleon- and -hole excitations on the pion propagator, thereby preventing medium eEects on the pion exchange interaction from being too large at normal nuclear density. Signals of this quenching are rather clearly seen in the -nucleus forward dispersion relation, once the pole contributions from low-lying pion-like states are lumped up into a unique, eEective coupling constant [236]. In the 1990s, medium eEects on hadron properties, and their dependence upon the density and=or temperature of the nuclear environment have attracted increasing attention, in order to identify the >ngerprints of the partial restoration of chiral symmetry, due to change of the QCD chiral condensate [237]. Other quenching eEects are observed in electromagnetic or weak transitions in nuclei. Two wellknown examples are provided by the magnetic moment operator, and by the Gamow–Teller (GT ) transitions in >-decay [9,12]. If one considers a valence nucleon around a closed-shell core in the independent-particle model, the nuclear magnetic moment is the one of the valence particle, namely

= gl l + 12 gs ;

(4.26)

with l the orbital angular momentum and gl , gs the orbital and spin g-factors, respectively. Because of the interaction of the valence nucleon with the core, however, the magnetic moment operator has to be interpreted as an eEective one, which implies that Eq. (4.26) must be replaced by [9,238] eE = gleE l + 12 gseE + 12 gp [Y2 ][1] ;

(4.27)

where the last term represents an induced one-body tensor operator, obviously absent in the bare magnetic moment. The eEective orbital and spin g-factors are renormalized with respect to the bare ones, i.e. gleE = gl + gl , gseE = gs + gs . Similarly, the one-body operator for GT transitions in the non-relativistic limit is simply given by gA (4.28) A± = B± ; 2


◦

Fig. 33. Neutron spectra at = 0 From Ref. [10].

381

for the (p; n) reaction on various targets, and incoming protons of 200 MeV.

where gA is the axial-vector coupling constant, and the ± signs refer to >∓ decays. For a free nucleon one has gA 1:26. The observed transition strengths in closed-shell-plus-one, or minus one, nuclei, on the other hand, can be reproduced only with an eEective value gAeE of the coupling, 20% lower than the free one. The origin of the above quenching eEects can be traced back to two major contributions. Transition operators are necessarily evaluated in a truncated model space, so that they have to be regarded as eEective quantities, evaluated perturbatively in the eEective interaction expansion. A second contribution to the quenching arises from processes in which the external electromagnetic or weak probe couples to the nuclear system in presence of meson exchange. These are the MEC eEects already discussed in Section 4.2. They can excite a nucleon to a state, which is subsequently de-excited by the external probe. In the non-relativistic limit, these contributions can be re-interpreted in the language of the -hole model. Indeed, in the MEC approach one gets two-body operators O2 , which have to be evaluated between states of the valence nucleon and core. The leading contributions to O2 can be factorized into the product of a one-body spin–isospin operator, times a meson-exchange transition potential. One thus recovers the -hole screening mechanism already considered in Fig. 32. This mechanism may be operative both in the quenching of gA and in the renormalization of the spin g-factor. A simple estimate is again given by Eq. (4.24) [11]. In the light of the previous discussion it is clear that a careful assessment of the role played by -hole excitations in quenching phenomena requires the simultaneous evaluation of higher-order perturbative contributions coming from “ordinary” nuclear-structure eEects. Second-order core polarization processes driven by the strong tensor interaction between the valence nucleon and core have a major role in renormalizing both gA and gs . We shall not go into the details of this topic here, referring the reader to previous reviews [9,239], and prefer to consider in more detail quenching phenomena, as seen from a nuclear-reaction perspective. Let us >rst consider what is observed in charge-exchange (p; n) reactions at forward angles shown in Fig. 33. A single, prominent peak is seen on top of a large, structureless background, with the exception of the even–even 40 Ca nucleus. This peak can be interpreted as a collective spin–isospin oscillation, in which the excess neutrons coherently change the direction of their spins and isospins

382


without changing their orbital motion [240]. At the same time, shell model and Pauli principle suggest that a collective spin–isospin excitation ought to be prohibited in a spin saturated nucleus such as 40 Ca. For heavier nuclei, neutron excess increases, a larger neutron fraction can participate in the oscillation, with a corresponding increase in the cross section, in agreement with the trend of the experimental data. To understand better the connection between this phenomenology and quenching eEects, let us recall that the energies of interest here (100 ¡ E ¡ 800 MeV) lead to a one-step, direct mechanism. The inelastic transition is given in DWIA in terms of an eEective projectile-target interaction. For larger energies than the Fermi energy in the target (0F ∼ 37 MeV), one can expect that the renormalized NN interaction in the medium approaches the free NN transition matrix [242]. This theoretical expectation has been implemented by Love and Franey [243,244], to determine the eEective interaction directly from the phase-shift analysis of NN scattering data. The eEective interaction has been written as a combination of central, spin–orbit and tensor isoscalar- and isospin-dependent terms, requiring that the corresponding antisymmetrized NN Born T -matrix reproduces the free empirical one. For high energy and very low momentum transfer, this analysis shows an eEective interaction dominated by a central scalar–isoscalar term, plus a central spin–isospin contribution with a characteristic (1 · 2 )(˜1 · ˜2 ) dependence. On the ground of these results one can expect that the forward (p; n) cross section at high energy can be considerably simpli>ed, and that the nuclear transition can be described essentially in terms of spin and isospin operators. As a matter of fact, for q ∼ 0 one can write [245] d d

GT

◦

( = 0 ) =

M kf NB |TB (q = 0)|2 B+ (GT ) ; 2 ki

(4.29)

where TB (q = 0) is the forward transition amplitude associated to the central spin–isospin term in the eEective interaction, NB a distortion factor, and 2 A (4.30) j B± (j)|i B± (GT ) ≡ f| j=1 represent the GT B-values for the i → f nuclear transition, well-known from >-decay theory. It is by now a textbook result in Nuclear Physics that the GT strengths B± , when summed over the possible >nal states, satisfy the Ikeda sum rule [246] .GT ≡ B+ (GT ) − B− (GT ) = 3(N − Z) : (4.31) f

f

This result is model independent, under the only assumption that the ˜ operators refer to point-like nucleons. If N Z, B− referring to >+ decay is strongly suppressed owing to the Pauli principle, and one can simply write .GT B+ (GT ) = 3(N − Z) : (4.32) f

Eqs. (4.29) and (4.32) state that the forward (p; n) cross section for heavy nuclei, once integrated over the >nal states, directly counts the number of excess neutrons participating in the process, and allow to determine the overall GT strength. When the experimental forward (p; n) cross section is


383

converted into the GT sum-rule strength, however, one >nds that only about 60% of the expected strength is located in the excitation energy region where the major GT peaks occur [247]. These results are strongly reminiscent of the quenching of the axial coupling constant mentioned above, and prompted several physicists to look for an explanation in terms of the same screening mechanisms. The nuclear response to the exciting probe would then be a mixture of nuclear-structure con>guration mixings and -hole excitations in the spin–isospin channels. Indeed, if this is the case, one might expect that

eE 2 exp .GT gA = 0:6 ; (4.33) gA .GT which gives for gAeE a value about 20% lower than gA , in fair agreement with the experimental indications. The exact role played by the various, possible mechanisms, “conventional” nuclear-structure eEects versus -hole excitations, in quenching the GT strength has been the subject of many investigations, often leading to con?icting results. On one hand, one may argue that a strong repulsive eEective force between - and N -hole states can remove strength from the low-lying excitation spectrum, enhancing the transitions to the high-lying -hole states. In spite of the large energy gap between N - and -hole states, ∼ 300 MeV, the N – mass diEerence, this might be possible, due to the large number of -hole con>gurations not inhibited by the Pauli principle [248]. On the other hand, ordinary con>guration mixing, where 1p–1h states couple to high-lying 2p–2h states, could also shift the GT strength into the energy region far beyond the GT resonance. Clearly, the relative weight of -hole and conventional nuclear-structure mechanisms depends in a crucial way upon the actual values of the corresponding coupling strengths. For doubly-closed shell nuclei, the excitation energies of the low-lying GT states, and the fragmentation of the strength between them and the GT resonance can be calculated in the framework of the Random Phase Approximation (RPA). The solution of the RPA equations is equivalent to summing up to all orders processes where particle-hole states are created, annihilated, and propagate through the action of the residual interaction. Once the excitation energies EI and the corresponding states I have been determined, the response of the nuclear many-body system to a probing one-body >eld F can be determined from the strength function [249] (! − I)|I|F|0 |2 ; (4.34) SF (!) ≡ I

where |0 is the ground state. The method is not adequate to describe the width of the GT resonance, an aspect of primary importance for its connections with the quenching of the GT strength in (p; n) reactions. At high excitation energies of relevance, 2p–2h excitations can participate in damping the resonance, and may be taken into account in the framework of the second RPA (SRPA) [250], or via two-phonon states [251]. These approaches lead to some fragmentation of the strength, with tails in the high-energy region containing up to 30 – 40% of the total integrated strength, relative to the sum-rule value 3(N − Z). The results and considerations outlined above show that only a delicate balance of data analysis and theoretical calculations can provide reliable information on the actual extent and nature of the damping mechanism of the GT strength. From the experimental point of view, a naive procedure for background subtraction leads to a large underestimation of the GT strength [10]. This explains why detailed theoretical models, employing large-basis RPA calculations and a careful treatment of

384


Fig. 34. Diagrammatic representation a BBG equation with transitions. The indices ; + specify the channels NN , N and , respectively.

the continuum, are necessary to shed some light on the role of the various quenching mechanisms in GT transitions. Several microscopic or semi-microscopic analysis of (p; n) reactions on doubly closed-shell targets suggested that part of the strength is indeed shifted from the low-energy part of the spectrum to the high-energy tail because of the mixing of higher-order con>gurations with 1p–1h states, whereas the quenching of the strength due to isobars cannot amount to more than 10 –20%, in order not to destroy the agreement with the experimental data [10]. These conclusions can be contrasted with a recent RPA calculation of contributions for the GT strength, which con>rms the role of -hole con>gurations in quenching [252]. Presently, large-scale shell model calculations can provide a reliable description of the nuclear excitation spectrum. Great progresses in this >eld were achieved in the nineties, with full-scale calculations for pf-shell nuclei, through the development of more eScient computational schemes for direct disorganization [253–256], or by Monte-Carlo techniques [257–259]. These approaches made possible to compare the outcome of 0˝! computations in the full major shell with the results of truncated calculations. It turned out that in general the full calculation recovers much more quenching than the truncated ones, with a reduction factor close to 2. However, to obtain agreement with the Ikeda sum rule, even the exact results need a renormalization of the axial coupling constant gA by a factor ∼(0:77)2 , quite consistent with Eq. (4.33). On the ground of these analysis, in line with previous results for the sd [260] and p shells [261], the quenching of gA appears as a genuine in-medium eEect, once shell correlations have been properly taken into account. How much excitations are responsible for this is still to be checked. 4.4. The in nuclear matter The dynamics of the resonance has been investigated in the Nuclear Matter regime, where translational invariance allows single-particle plane-wave states, thereby simplifying the treatment of the many-body problem. In spite of being an “ideal” system only realized in the interior region of heavy nuclei it is a natural lab to test microscopic interactions. Calculations have been generally performed within the non-relativistic Bethe–Brueckner–Goldstone (BBG), or relativistic Dirac–Brueckner (DB) approaches. The usual BBG theory, at the two-hole-level approximation, or DB are extended to accommodate the coupled-channel nature of the NN –N– system, and the ladder series for the reaction G-matrix is summed up with excitations in intermediate states, propagating as “virtual” particles. The corresponding coupled-channel BBG equations are illustrated in Fig. 34, and can be


385

written as, k1 k2 eld or simply considering it constant. Unfortunately, U is far less known than UN , both under the experimental and the theoretical point of view, but it can have an important role in changing the trend of EOS. It was shown [262,263] that the repulsive eEects on saturation due to excitations, can be largely counteracted by the action of a realistic attractive potential. This is illustrated in Fig. 35 where the saturation curve of nuclear matter is determined with diEerent choices for U , ranging from a constant attractive potential, to a density and momentum-dependent form with the same structure as UN . Features of propagation in >nite nuclei are described within the -hole model with a >eld similar to the one felt by the nucleon. The calculation of U from the v28 showed a strongly repulsive mean >eld [262,263], mainly due to the projection of the interaction in states not seen in NN scattering. The magnitude of U is than strongly dependent on the model used for interaction between nucleons and ’s.


387

A realistic description of propagation in nuclei should account also for decay. Therefore, in addition to the average binding eEects described by U , coupling to the N state and modi>cations of the width due to the presence of the surrounding medium should be considered. To lowest order this can be accomplished through the “bubble” graph of Fig. 25, as already outlined with reference to the -hole model. Self-energy eEects have been studied in [265] under the assumption that the self-energy contribution .0 (k ; E) corresponding to Fig. 25 has a weak momentum dependence, and essentially depends upon the single-particle energy only. Binding eEects are included through the self-consistent nucleon potential UN (k) in the non-relativistic nucleon propagator. This approximation amounts to assuming an additional constant potential due to the modi>cation of the decay channel in the nuclear medium, and is consistent with the original description of the as an elementary particle with a modi>ed mass. At low density this additional contribution vanishes. The self-energy mass shift is determined self-consistently, together with Eqs. (4.35), (4.36) and (3.47), as solution of the Dyson equation 0 (k ) = M − M +

k2 + Re[.(k ; 0 )] ; 2M

(4.38)

where the total isobar self-energy is given by .(k ; 0 ) = .0 (0; 00 ) + U (k ):

(4.39)

Here, 00 is the solution of Eq. (4.38) with k = 0. It has been veri>ed that U is largely unaEected by the inclusion of the decay channel self-energy in the self-consistent procedure [265], the decay channel contribution giving an additional repulsion of about 20 MeV, without modifying however the general trend of the potential, especially around the saturation point. It should be stressed, at this point, that many aspects of behavior in nuclear matter deserve further investigation. First, one might ask what is the role of three-body diagrams with intermediate states, which ought to give the main contribution of three-body correlations to the binding energy. It has been pointed out already several years ago that contributions of n-body ring diagrams do not converge with increasing n when degrees of freedom are explicitly included, presumably because of the large coupling of the pion to particle-hole and isobar-hole states, so that suitable regrouping of terms in the perturbation expansion is required [266]. Coupled-channel two-hole-line calculations with the continuous choice for the average potential implicitly include some three-body contributions with intermediate isobar excitation through the self-consistent procedure [5]. These contributions turn out to be repulsive. The attractive part of the three-body force needs certainly to be included in order to bring the saturation curve in agreement with the experimental trend. This will require three-hole-line calculations with degrees of freedom explicitly built in. The percentage of ’s in the nuclear medium can be obtained from the defect parameter which can be regarded as the expansion parameter of the BBG theory. When the contributions are added, one has three defect parameters, the usual one UNN , expressing the content of two-particle-two-hole components in the nuclear ground state, and two parameters UN and U , giving an indication of the percentage of virtual particles in the ground state C = (UN + U )=(1 + U), where U is the sum of all the three parameters. The defect parameters are determined from U< = IIX< where and I are the nuclear density and the statistical weight of the channel, respectively, and the bar indicates

388


an average over initial two nucleon and states in the Fermi sea of the quantities d 3 q Gtted reproducing the P33 N phase shift behavior, by demanding that the full dressed propagator has a pole at the physical mass M − i2 =2. Only positive-energy contributions have been considered for the single-particle states, and both the Bethe–Salpeter and the BG equations have been solved after a three-dimensional Thompson reduction. At a >rst stage, the self-energy was taken as a spin-averaged quantity (which is essentially equivalent to a non-relativistic approximation), whereas the nucleon was described in a fully relativistic way, with the large scalar and vector components characterizing relativistic nuclear models [7]. Subsequently, a full Lorentz representation has been employed for both the nucleon and the isobar self-energies, thereby treating both particles in essentially the same way [65]. The results for nuclear matter near saturation also show a strong repulsion. The eEective mass is almost constant as a function of density, and the self-energy is not very much momentum dependent, an indication that the properties of the isobar seem hardly aEected by the medium in this type of relativistic models. 5. Conclusions and outlook We have seen that >ngerprints of the appear in many nuclear phenomena. It contributes at the 1–2% level to the total wave function of two- and three-nucleon systems, and has to be taken into account in the transverse, model-dependent part of two-body electromagnetic currents in order to improve the agreement between calculated and experimental electromagnetic form factors. To a good extent, this baryon behaves as a quasi-particle in the nuclear medium, as testi>ed by the success of the -hole model in explaining photon- and pion-induced reaction data, the diEerence between position and width of the resonance with respect to the free case being attributable to medium eEects. The comes into play in renormalizing the eEective pion–nucleon coupling, as well as the weak axial-vector coupling constant gA . With reference to this point, modern large-scale shell model calculations give


389

clear evidence that the quenching of gA can be regarded as a universal phenomenon in nuclei, although the relative weight of -hole excitations and “standard” con>guration mixing is still to be completely clari>ed. The mechanism of excitation and propagation in charge-exchange reactions, and the role played by longitudinal and transverse couplings is now better understood, thanks to more and more re>ned nuclear-structure and reaction calculations. In spite of these successes, much remain to be done to better understand the physics of the . From a fundamental point of view, its parameters, the isospin mass splitting within the associated multiplet, and the long-standing issue of the quadrupole moment have to be assessed in the broader framework of QCD or QCD-motivated quark models. Under this point of view, some basic issues, such as the role played by the residual OGE qq interaction, seem still to be clari>ed. Usually, these color-magnetic forces are invoked to remove the degeneracy within the same SU (6) multiplet, in particular between nucleon and , and, through their dependence upon the quark masses, concur to determine the >ne structure of the isospin multiplets, as discussed in Section 2.2. Moreover, their tensor component is responsible for the D-state admixture in the hadron wave functions, leading to non-vanishing quadrupole moments. Simple quark models, however, meet some diSculties in reproducing the level orderings of positive and negative-parity excitations in the baryon spectra, and several modi>cations have been proposed in the literature, which give diEerent emphasis to OGE contributions. Thus, we have seen that in hybrid quark models an eEective meson–quark interaction is simply added to OGE [271,272], whereas in the Goldstone–Boson-Exchange quark model chiral symmetry arguments are pursued to their extreme consequences, and no room is left for the standard OGE potential [168,273]. These diEerent perspectives on the hadron structure have direct implications on the interpretation of isospin mass splittings and quadrupole moments, where plays a distinguished role in virtue of its prominent position in the baryon spectrum and its large coupling to the N system. From a more technical point of view, the necessity of high-quality, few-body calculations of hadron spectra is emerging as more and more compelling, in order to assess the reliability of diEerent quark models. This is exempli>ed by predictions of the N – mass splitting for diEerent qq interactions; a good agreement with the experimental value, obtained through an approximate evaluation of the three-quark baryon wave function, may be completely lost in exact calculations [167]. Another topic which will certainly require further investigations is the derivation of the meson– baryon–baryon coupling strengths and oE-shell form factors from the underlying QCD, or from some sensible approximation to the full QCD dynamics. As a matter of fact, the relations among the various strengths based upon symmetry arguments reviewed in Section 3:1:2 can be at most of guiding value; similarly, one needs some hint from microscopic models about the proper values to be given to the various cut-oE parameters appearing in the strong (N; N; : : :) form factors. In the last several years there has been a ?ourishing activity in this direction, ranging from lattice calculations and Nambu–Jona–Lasinio-type models, to non-perturbative approaches based on Schwinger–Dyson formalism [274]. The relative merits and limitations of all these attempts have still to be assessed on the ground of a detailed comparison with the experimental data. A more profound knowledge of the couplings and a deeper information on the short-range behavior from experiments or non-perturbative QCD calculations would open the way to a complete treatment of the interactions between nucleons and ’s. A >eld which is presently enjoying an uprise of experimental and theoretical activity is the study of the electromagnetic structure of the nucleon and its excitations. As a matter of fact, the baryon

390


spectrum alone is not enough to distinguish between the many quark models proposed in the literature. Other observables, such as the elastic and transition electromagnetic form factors, have to be considered in order to constrain theoretical calculations. Here also new quark models of baryons compete with the traditional ones in trying to get a better description of data. An example is given by the Hypercentral Constituent Quark Model, where baryons are regarded as true three-body bound states, to be described through hyperspherical coordinates, and the driving interaction is represented by a three-body potential depending upon the hyperradius only [277,278]. From the experimental point of view, thanks to new, high-performance machines like the Thomas JeEerson Accelerator Facility, the baryon structure and electromagnetic transitions are explored in new kinematic domains with high-luminosity beams. Theoreticians are therefore confronted with the demanding task of providing models able to bridge the gap between the low-energy region where quark models of hadrons may perform well, and high-energy, deep-inelastic processes where a perturbative approach to QCD is possible [279]. A direct solution of QCD equations being beyond our present capabilities, one will have to resort to ingenious approximation schemes. A promising approach is represented by light-cone >eld theory, which allows for a particularly simple Fock representation of the vacuum state, so that more and more complex con>gurations can be introduced into the baryon wave functions, when needed, in the framework of a Hamiltonian description of the dynamics. Long-range phenomena associated to spontaneous symmetry breaking, on the other hand, can be associated to the vacuum zero modes [280]. The isobar will again play a paramount role in testing new theoretical models, and might give new hints on the energy domain where constituent quark models break down, and perturbative QCD applies. In particular, the behavior of the N transition form factor at momentum transfers of several GeV=c2 , may still have signatures of the non-perturbative QCD regime. As we have seen, the understanding of the physical processes involving the is not completed. Many open problems still remain to be solved. The new experimental facilities might shed new light and give new hints to theoreticians on this fascinating issue.

Appendix A. Dispersion relation constraints A.1. Preliminaries In the dispersion approach to scattering it is intended to explore the analytic properties of the S-matrix regarded as a function of the kinematic variables (s; t; u) to give a complete description of the process. The basic requirements are [282–284] (a) analyticity, (b) unitarity, and (c) crossing symmetry. Analyticity means that the scattering amplitude can be safely continued from the physical region, where all singularities corresponding to the physical processes lie, into the complex energy (s) or=and momentum-transfer (t) or u plane. Actually, one requires what is called “minimal analyticity, i.e. the only singularities that S(s; t; u) has in the complex plane of the Mandelstam variables are those dictated by unitarity.


391

Due to unitarity, expressing ?ux conservation, the S-matrix has only poles associated to the propagation of a single on-mass-shell particle, if allowed by conservation laws, plus branch points at the scattering cuts associated to multiparticle propagation in intermediate states. The discontinuities across the cuts are >xed by the unitarity relation. Crossing symmetry allows one to look at the transition amplitudes for diEerent processes as boundary values of the same analytic function in diEerent regions of the kinematic variables. However, the T -matrix can contain kinematic factors with diEerent singularities in the various crossed channels. This would make the analytic continuation from one channel to another impossible. To avoid this problem, invariant amplitudes have to depend in a symmetric way upon the Mandelstam variables, so as to exhibit in all channels only the singularities due to unitarity. This is simple for spin-less particles, but requires non-trivial algebraic relations when some of the particles have spin [107,282]. According to the Mandelstam hypothesis, the various scattering amplitudes for the processes in the s-, t-, and u-channel are related to each other, as boundary values of the same analytical function, determined once the leading intermediate states have been speci>ed. A.2. Fixed-variable dispersion relations The analytic properties of the scattering amplitude can be summarized by a dispersion relation. By this terminology one means a Hilbert integral transform that expresses the real part of the transition amplitude through a contour integral in the complex energy plane. There, one assumes analyticity for the scattering amplitude, apart from singularities dictated by the unitarity condition, which are located on the real axis, so that the integration contour is (A.1). Applying the Cauchy theorem one gets 1 T (s ; t) T (s; t) = ; (A.1) ds 2i C s −s where T (s; t) is the scattering amplitude with all kinematic factors removed. For spinless particles this factor is just a phase-space quantity, whereas for particles with spin non-trivial transformations are required to get invariant amplitudes whose singularities only depend upon the dynamics. The contour C in the complex-energy plane is chosen to avoid all the singularities of T dictated by unitarity requirements in the various channels that divide the physical region into pieces. The scattering matrix is analytical in them, and can be analytically continued from one piece to another, as the same regular function of the variable s. In order to avoid the singularities one is obliged to go through paths connecting the various pieces of analyticity, like the integration contour in Fig. A.1. Alternatively, one can ascribe an in>nitesimal imaginary part i0 to the various singularities, thereby shifting them out of the real physical region, no longer divided into disconnected pieces. One can thus integrate over the real axis in the Cauchy integral (A.1), and write 1 ∞ Im T (s ; t) 1 s0 Im T (s ; t) T (s ; t) 1 T (s; t) = + + (A.2) ds ds ds : s0 s − s − i0 −∞ s − s − i0 2i 2 s − s Here s0 and s0 are the lower and upper limits of the singular regions on the real axis, and 2 represents the curved part of the contour C. The Schwartz re?ection principle T (z ∗ ) = [T (z)]∗ has been used to write T (s + i0; t) − T (s − i0; t) = 2i Im T (s; t) :

392


Fig. A.1. The Cauchy contour for the dispersion integral in the energy plane of the N scattering amplitude.

The actual nature of the singularities can be identi>ed through unitarity, 1 dLips(s; k)Tk (s; t ) ; Im T (s; t) = 2

(A.3)

k

where < and > denote the initial and >nal channels. The sum is over all channels which are physically allowed, and dLips(s; k) represents the appropriate Lorentz-invariant phase-space diEerential. The simplest singularities are associated to the propagation of single-particle states in the s-channel allowed by conservation laws. In particular, for the case of N → N scattering in the s-channel, a nucleon can propagate as an intermediate on-shell particle. It can be shown that the single-particle contribution to Eq. (A.3) is [107,282] 1 Im T (s; t)(1) = (M 2 − s)>|T † |jm; X jm; X |T |< : (A.4) 2 m A partial-wave decomposition has been performed to exhibit the intrinsic spin j of the intermediate particle of mass M . The matrix element jm; X |T |< , evaluated at s = M 2 , measures the strength of the coupling between the incoming particles and the intermediate one, and is de>ned as a coupling constant G<X . In these terms Eq. (A.4) can be rewritten as Im T (s; t)(1) = G<X GX> (M 2 − s) :

(A.5)

When Eq. (A.5) is inserted into the dispersion relation (A.2), and the transition amplitude is assumed to vanish for |s| → ∞ one gets for example in − N scattering G2 1 ∞ Im T (s ; t) 1 s0 Im T (s ; t) T (s; t) = 2 (A.6) + ds + ds : M − s s0 s − s − i0 −∞ s − s − i0 where G denotes the −p → n coupling constant. The pole term coming from the u-channel is absent in this example. In addition to the single-particle states, the unitarity relation (A.3) contributes with cuts in the s- and u-channels, due to the propagation of two or more on-shell particles. For N scattering one


393

has a branch point at s0 = u0 = (m + M )2 . The singularities in the u variable can be transferred into the complex s-plane in virtue of the Mandelstam identity m2i = 2m2 + 2M 2 ; (A.7) s+t+u= i

so that, one has in the s-plane a right-hand cut starting at s0 = (m + M )2 , and a left-hand cut starting at s0 = −t + (M − m )2 , coming from unitarity in the u-channel. Using the well-known identity s

1 1 =P + i(s − s) − s − i0 s −s

one >nally gets the >xed-t dispersion relation ∞ s 0 Im T (s ; t) G2 1 Im T (s ; t) 1 Re T (s; t) = 2 + P ds ds : + P M − s s0 s − s −∞ s − s

(A.8)

(A.9)

In practice, one actually has only one principal-value integral according to the energy region considered. Thus, for s0 ¡ s ¡ + ∞ only the >rst one gets a singularity, whereas the second gets it for −∞ ¡ s ¡ s0 . For real s lying in between s0 and s0 ; T (s; t) is real. Similar considerations in general apply for the crossed u-channel amplitude Tu . For elastic N scattering, however, one can choose the s-channel so as to have no single-particle contributions in the u-channel. Thus, in our example, if the elastic −p process is in the s-channel, the crossed u-channel is + p → + p, and no single-particle propagation is allowed. From the Mandelstam hypothesis the amplitudes are boundary values of the same function and one can write [107], T (s; t; u) = Tu (u; t; s)

(A.10)

for any given t. Using Eqs. (A.10) and (A.7), the dispersion relation (A.9) can be >nally written in the form ∞ ∞ Im T (s ; t) 1 Im Tu (s ; t) 1 G2 ds P + ds (A.11) + P Re T = 2 M − s s0 s − s s0 s + t + s − i m2i with s0 = (m + M )2 . The above derivation can be repeated for the crossed amplitude Tu . Eqs. (A.11) and the equivalent for Tu represent the required >xed-t dispersion relations for the amplitudes. Crossing symmetric and antisymmetric combinations of these amplitudes can also be de>ned, T (±) ≡ 12 [T (s; t; u) ± Tu (s; t; u)] = 12 [T (s; t; u) ± T (u; t; s)] ;

(A.12)

which obviously satisfy T (±) (u; t; s) = ±T (±) (s; t; u) : The whole formalism can be made more transparent by introducing the variable [107] 1 1 2s − I≡ m2i + t : (s − u) = 4M 4M i

(A.13)

(A.14)

394


For forward scattering (t = 0); I gives the incoming pion energy in the laboratory system ! = (s − M 2 − m2 )=2M . It is straightforward to show that Eqs. (A.11), its equivalent for Tu , and (A.12) lead to ∞ 1 1 1 (± ) (± ) (± ) dI ; Re T (I; t) = Tpole + P Im T (I ; t) (A.15) ± I0 I − I I + I where I0 = m + t=4M depends upon the squared momentum transfer t, and the pole terms (+) Tpole =

IB G2 ; 2M (IB − I)(IB + I)

(− ) Tpole =

I G2 2M (IB − I)(IB + I)

(A.16)

are singular at the nucleon pole IB = (t − 2m2 )=4M , corresponding to s = M 2 or u = M 2 . When spin degrees of freedom are included, suitable invariant transition amplitudes that exhibit the isospin degrees of freedom have to be extracted from the S-matrix, in order to avoid the singularities associated to purely kinematic factors. For N scattering, let Tba (q ; q) ≡ b (q )|T |a (q)

(A.17)

be the invariant amplitude for a transition from a pion of isospin a and four momenta q, to a pion with isospin b and four momenta q . It can be decomposed into isospin even and odd components as follows [8]: Tba (q ; q) = T (+) ba + 12 [Bb ; Ba ]T (−) ;

(A.18)

where the matrices T (±) are related to the scattering amplitudes in the total-isospin representation by, T (+) = 13 (T 1=2 + 2T 3=2 );

T (−) = 13 (T 1=2 − T 3=2 ) :

(A.19)

Taking into account the eEect of the incoming and outgoing nucleon, it can be proved that the most general parity-conserving form of the on-shell T -matrix is T (±) = u [A(±) (s; t; u) + 12 (q= + q= )B(±) (s; t; u)]u ;

(A.20)

where u and u denote the incoming and outgoing nucleon spinors, and the Feynmann notation q= ≡ q · + has been employed. The scalar functions A(±) (s; t; u) and B(±) (s; t; u) are the so-called invariant amplitudes, free of kinematic singularities. Obviously, they can be expressed in terms of invariant amplitudes for the s- and u-channel much in the same way as the transition amplitudes T (±) for scalar particles are related to T and Tu , namely A(±) = 12 [A(s; t; u) ± Au (s; t; u)]N → + N

(A.21)

and similarly for B(±) . Since A(s; t; u) = A(u; t; s);

B(s; t; u) = −B(u; t; s)

(A.22)

with the minus sign from the replacement of q + q into −q − q in passing from the s- to the u channel, one gets the crossing properties, A(±) (u; t; s) = ±A(±) (s; t; u);

B(±) (u; t; s) = ∓B(±) (s; t; u) :

(A.23)


395

The singularity structure of the invariant amplitudes is only determined by unitarity, therefore, one can write down immediately >xed-t dispersion relations similar to Eq. (A.15), ∞ 1 1 1 (± ) (± ) ± dI ; Re A (I; t) = P Im A (I ; t) (A.24) I0 I − I I + I ∞ 1 1 1 (± ) (± ) (± ) ∓ dI ; Re B (I; t) = Bpole + P Im B (I ; t) (A.25) I0 I − I I + I where the pole terms (+) = Bpole

I g2 ; M (IB − I)(IB + I)

(− ) Bpole =

IB g2 ; M (IB − I)(IB + I)

(A.26)

are now contained only in the isospin odd amplitudes B(±) . To make the present notation consistent with the Born contribution given by the pseudoscalar >eld-theoretic √ Lagrangian (3.1). The coupling constant g has been de>ned for the p → p0 vertex, with g = G= 2. The pole term appears only in the B± amplitudes as Eqs. (A.24) and (A.25) show. In order to extract the coupling constant from the experimental data, it is more convenient to separate out explicitly the I = 0 contribution in Eq. (A.25) giving, ∞ dI I 1 1 (− ) (− ) (− ) ˜ t) Re B (I; t) = Bpole + P Im B (I ; t) + B(0; (A.27) − I0 I − I I + I I with ˜ t) ≡ 2 P B(0;

∞

I0

Im B(−) (I ; t) dI : I

The dispersion relation for B(+) can be identically re-written in the form ∞ 1 dI I 1 (+) (+) (+) + Im B (I ; t) : Re B (I; t) = Bpole + P I0 I − I I + I I

(A.28)

(A.29)

Combining both Eqs. (A.27) and (A.29) one immediately gets, ∞ Im B+ (I ; t) Im B− (I ; t) dI I (IB ± I) ∓Re B± (I; t) ± P + I0 I ∓ I I ± I I =

g2 ˜ t)(IB ± I) ; + B(0; M

(A.30)

with B± = B(+) ∓ B(−) . Eq. (A.30) can be used as a linear relation in the symmetric variable I to extrapolate the experimental data for elastic ± N scattering processes up to the pion pole I = IB [18,22]. The comparison of the pseudoscalar meson–baryon vertex functions and the T -matrix in the Born approximation gives then the strength of the interaction g. The above procedure simpli>es somewhat for forward scattering (t =0). In this case it is customary to write the dispersion relations in the laboratory system, where the nucleon remains at rest. Since I equals the incoming pion energy !, and the nucleon pole IB is located at !N ≡ −m2 =2M ,

396


Eqs. (A.24) and (A.25) become, ∞ 1 1 1 (± ) (± ) ± d! ; Im A (! ) Re A (!) = P m ! − ! ! + ! ∞ 1 1 1 (± ) (± ) (± ) d! ; Im B (! ) Re B (!) = Bpole + P ∓ m ! − ! ! + ! with (± ) Bpole

1 1 g2 ∓ : ≡ M !N − ! ! N + !

(A.31) (A.32)

(A.33)

Owing to the normalization of the nucleon spinors, and the property pF ; u(pN ; )+F u(pN ; ) = M the transition amplitude (A.20) is diagonal in the spin quantum numbers as expected for forward scattering, and simply given by T (±) (!) = A(±) (!) + !B(±) (!) ;

(A.34)

since the nucleon momentum pN is zero. Inserting (A.31) and (A.32) into this result one gets for the isospin odd amplitude ∞ Im T (−) (! ) 1 g2 2!N ! 2! (− ) P + d! ; (A.35) Re T (!) = (Re T− − Re T+ ) = 2 − ! 2 2 M !N2 − !2 ! m where the shorthand notation T± ≡ T± N ≡ T (± N → ± N ) has been employed. This equation can be related to the experimental data by resorting to the optical theorem, k(!)± (!) Im T± (!) = (A.36) 4 with k(!) = !2 − m2 . Using the trivial identity 1 1 !2 ; = + !2 − !2 !2 !2 (!2 − !2 ) one arrives at the remarkable result ∞ !2 k(! ) − (! ) − + (! ) Re T (−) (!) − 2P (!N2 − !2 ) d! 2 2 − ! 2 ! 4 ! ! m m 2 !2 − !2 ∞ k(! ) =− g2 + N 2 (− (! ) − + (! )) d! ; 2 M 4 ! m

(A.37)

which states that the complicated quantity on the left is a linear function of !2 , and can be determined from the measured total cross sections, and a phase shift analysis for the forward amplitude T (−) (!). The coupling strength g2 can then be determined again by a linear extrapolation to the nucleon pole !2 = !N2 . These extrapolations, based upon the >xed-t relations (A.30) or upon the forward dispersion relation (A.37), clearly represent the physicist’s way of determining the residue of the scattering amplitude in correspondence to its essential singularity at I = IB (or ! = !N for forward scattering in the laboratory system). The crucial physics coming into play through unitarity is that


397

this residue represents, apart from mass factors, the coupling between the nucleon and the exchanged meson. Eq. (A.37) has been used, in conjunction with phase-shift analysis, to obtain the >rst reliable determination of the N coupling constant based upon dispersion theory, yielding the well-known result f2± =4 = 0:08 for the charged meson–nucleon pseudovector coupling [75]. Similar considerations apply for the isospin even amplitude T (+) (!). However, whereas in Eq. (A.37) the diEerence between the total cross sections appears, the integrals can be assumed well-behaved with increasing !, since + N (!) − N (!), for T (+) (!) the sum of the cross sections comes into play. As a consequence, a subtraction is required to make the corresponding integral convergent. One generally takes ! = m as the subtraction point. Finally, it is worthwhile to note that one of the >rst applications of forward dispersion relations was to the unraveling of the energy dependence of the pion–nucleon phase shifts. As a matter of fact, the angular distribution data in the 1950s could be reproduced within the experimental errors by six diEerent partial-wave solutions. By insisting that any set obtained should satisfy the dispersion relations, one could discard four of the six possible solutions [285]. Of the two remaining sets, one was characterized by a large phase shift in the J = T = 3=2 state (the Fermi solution), whereas the other had a large phase shift in the J =3=2; T =1=2 state (the Yang solution). By a clever dispersion analysis Davidon and Goldberger were able to show that the Fermi set could be univocally selected [76]. Thus, dispersion relations played a crucial role since the very beginning of physics! Appendix B. The N scattering matrix and electromagnetic corrections Let us consider a pion of four momenta q impinging on a nucleon of four momentum p. If q and p denote the four momenta of the two particles emerging from the scattering process, the non-trivial part of the S-matrix is related to the invariant transition matrix T by [8] N (p )b (q )|S − 1|N (p)a (q) = (2)4 4 (p + q − p − q) ×u(p )Tba (s; t; u)u(p) ;

(B.1)

where u(p) and u(p X ) are the spinors (de>ned in both spin and isospin space) describing the initial and >nal nucleon, respectively, and Tba ≡ b (q )|T |a (q) :

(B.2)

The pion isospin is here given in cartesian representation. The invariance with respect to rotations in isospin space can be exhibited by decomposing T into components of well-de>ned total isospin (1=2 or 3=2) through the projectors Pˆ 3 = 13 (2 + t˜ · B); ˜ 2

Pˆ 1=2 = 1 − Pˆ 1=2 = 13 (1 − t˜ · B) ˜ ;

(B.3)

to write T = T (1=2) Pˆ 1=2 + T (3=2) Pˆ 3=2 :

(B.4)

According to Eq. (B.2), this operator has to be evaluated between cartesian pion states. The identities b |Pˆ 1=2 |a = 13 Bb Ba = 13 (ba + 12 [Bb ; Ba ])

(B.5)

398


and b |Pˆ 3=2 |a = ba − 13 Bb Ba = 13 (2ba − 12 [Bb ; Ba ])

(B.6)

then yield immediately Tba = T (+) ba + 12 [Bb ; Ba ]T (−)

(B.7)

with T (±) de>ned according to Eq. (A.19). This is the decomposition of the T -matrix into isospin even and odd components we already considered in Appendix A. For the extraction of the singularityfree amplitudes A(±) and B(±) is convenient to discuss crossing symmetry, and write dispersion relations for the N system. However, to exploit rotational invariance, so as to introduce partial-wave amplitudes, it is more convenient to write the transition amplitude in terms of the spin non-?ip and spin-?ip components. To achieve this end the invariant T -matrix is written as the representative kernel of a suitable operator T between two-component nucleon Pauli spinors in spin and isospin space, namely M √ u(p )(q )|T |(q) u(p) = C† TC : 4 s

(B.8)

Here T is a function of the >nal and initial pion three momenta q and q for an on-shell pion, whereas it is still an operator in the space of the (spin and isospin) nucleon degrees of freedom. The normalization is such that the diEerential cross section is given by d = |C† TC|2 : (B.9) d The kernel operator T can be expressed in terms of the invariant amplitudes A(s; t; u) and B(s; t; u) through a straightforward but lengthy calculation. One gets [8,107] ˆ T(q ; q) = f1 (s; ) + f2 (s; )( · qˆ )( · q)

(B.10)

with √ 1 √ (E + M )[A + ( s − M )B] ; 8 s √ 1 f2 (s; ) = √ (E − M )[ − A + ( s + M )B] : 8 s f1 (s; ) =

(B.11) (B.12)

The variable is the scattering angle in the N center-of-mass system, qˆ ≡ q=|q|, and E represents 2 2 2 the nucleon relativistic energy, namely E = p + M = q + M 2 . One can now use the identity ˆ = qˆ · qˆ − i · (qˆ ∧ qˆ ) = cos − i · (qˆ ∧ qˆ ) ( · qˆ )( · q) to obtain the required result, namely T(q ; q) = f(s; ) + ig(s; ) · nˆ ;

(B.13)

where the spin-free and spin-?ip amplitudes are related to f1 and f2 by f(s; ) = f1 (s; ) + f2 (s; ) cos ;

g(s; ) = −f2 (s; ) sin

(B.14)


399

and nˆ ≡ qˆ ∧ qˆ =|qˆ ∧ qˆ | is the usual unit vector perpendicular to the scattering plane. Rotational invariance >nally leads to the well-known partial-wave expansions f(s; ) =

∞

[(l + 1)fl+ (s) + lfl− (s)]Pl (cos ) ;

l=0

g(s; ) = sin

∞

(B.15)

[fl+ (s) − fl− (s)]Pl (cos )

;

(B.16)

l=1

for the non-?ip and spin-?ip amplitudes, with coeScients only depending upon the total CM scattering energy s. As usual, Pl (x) represents the derivative of the Legendre polynomial Pl (x) with respect to x, and l± are associated to the total angular momentum eigenvalues J ± 1=2, respectively. The unitarity of the S-matrix >nally implies that the partial-wave amplitudes fl± (s) can be written in terms of phase and inelasticity parameters, f l± =

1 (#l± exp 2il± − 1) : 2iq

(B.17)

Needless to say, the isospin decomposition (B.4) and the partial-wave expansion (B.15) and (B.16) could be combined to introduce inelasticities and phase shifts for each total isospin and angular momentum state (I; J ). The above developments apply in presence of short-range, strong interactions only. The eEects of the electromagnetic interactions modify this scenario for at least two reasons. First, the partial-wave expansion does not converge any more, because of the long-range tail of the Coulomb potential; second, isospin symmetry is explicitly broken by the electromagnetic force. Moreover, problems associated with the zero mass of the photons arise, so that the diEerential cross section has to be multiplied by a factor which takes into account soft photon emission [102], namely d (B.18) = h(s; t; VE; )[|f(s; t; )|2 + |g(s; t; )|2 ] ; d where is a >ctitious photon mass acting as a regularizing parameter, VE the energy resolution, and h(s; t; VE; ) a factor which takes into account the undetected photons of total energy less than VE. For non-forward scattering one >nds that the spin-free and spin-?ip amplitudes f and g vanish in the → 0 limit, infrared catastrophe. It is not possible here to consider these questions in detail, and we refer the reader to the relevant literature [102,286,287]. Even apart from this problem, the additivity of the interactions does not obviously re?ect itself in the additivity of the amplitudes, so that the treatment of electromagnetic eEects always imply some model dependence. What is generally done is to deFne the nuclear partial-wave amplitudes through

fN (s; ) ≡ f(s; ) − fC (s; ) =

∞

[(l + 1)fl+ (s) + lfl− (s)]Pl (cos ) ;

(B.19)

l=0

gN (s; ) ≡ g(s; ) − gC (s; ) =

∞

[fl+ (s) − fl− (s)]Pl (cos ) ;

(B.20)

l=1

where the Coulomb amplitudes fC and gC are obtained from Feynmann graphs with only photon exchange, and eventually meson and proton form factors. Nuclear phase shifts and inelasticities can

400


be de>ned by factoring out the appropriate Coulomb phases. This is particularly simple for elastic + N scattering, which involves a pure total isospin state (T = 3=2). One has

2il #l± e

2il±

−1

(B.21) 2iq with the Coulomb phase l derived from fC and gC (the divergence of the partial-wave expansion for the Coulomb amplitudes being in this context irrelevant). Thus, for scattering of a point charged meson oE a point-like nucleon one has the text-book expression f l± = e

l = arg2(l + 1 + i#)

(B.22)

in terms of the 2 function, with # the usual Sommerfeld parameter. If the >nite extension of the strong-interacting hadrons is taken into account, a dynamical treatment of the photon–hadron vertex is required [107], and the Coulomb amplitudes and phase-shifts are expressed in terms of dispersion integrals [18]. This is not the whole story, however, since the pure hadronic phases and inelasticities are not yet those appearing in Eq. (B.21). We will limit ourselves to outline the procedure followed by Tromborg et al. [288–290], which has been employed in the classical determination of the ± N coupling constant by Koch and Pietarinen. For elastic + N scattering one de>nes (H )l± = − ˜l± (B.23) l±

and (#H )l± = #l± + #˜l±

(B.24)

as hadronic parameters, with corrections ˜l± and #˜l± given in tabulated form [288,289]. The situation is more complicated for elastic − N scattering and the charge-exchange reaction −p → 0 n, because of the mixing of the T = 1=2 and T = 3=2 states. In these cases one has >rst to de>ne isospin amplitudes fl1± and fl3± for the T = 1=2 and 3=2 channel, respectively, plus a mixing amplitude fl13± , arising from electromagnetic eEects which break isospin symmetry. These amplitudes can be then parameterized in terms of inelasticities and phase parameters. For elastic − N scattering one has √ fl± = 13 (2fl1± + fl3± − 2 2fl13± )e−2il± (B.25) with flI± given by an expression similar to Eq. (B.21), namely 1 I 2iIl± (# e − 1) ; flI± = 2iq l±

(B.26)

whereas a more complex expression is needed for the mixing amplitude fl13± , 1

3

i(l± +l± ) + i#˜13 2 √ (#13 l± )e : (B.27) 2 l± 3 2iq Only at this stage one can de>ne the hadronic parameters. For the diagonal (T = 1=2; 3=2) amplitudes one has in analogy to the Eqs. (B.23) and (B.24), 1 3 (1H )l± = 1l± + 23 ˜l± ; (3H )l± = 3l± + 13 ˜l± (B.28)

fl13± =

and (#IH )l± = #Il± + #˜l±

(B.29)


401

For the charge-exchange case extra kinematic factors enter into play, owing to the diEerent masses in the initial and >nal channel. It is worthwhile to observe that in low-energy N scattering the − inelasticities #˜Il± and #13 l± are mainly due to the coupling to the p → n+ reaction. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37]

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CONTENTS VOLUME 362 P. Tabeling. Two-dimensional turbulence: a physicist approach C.-I. Um, K.-H. Yeon, T.F. George. The quantum damped harmonic oscillator

1 63

L.S. Ferreira, G. Cattapan. The role of the D in nuclear physics

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Contents of volume 362

409

Forthcoming issues

410

PII: S 0 3 7 0 - 1 5 7 3 ( 0 2 ) 0 0 0 5 7 - 1

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FORTHCOMING ISSUES* T. Yamazaki, N. Morita, R. Hayano, E. Widmann, J. Eades. Antiprotonic helium J.K. Basu, M.K. Sanyal. Ordering and growth of Langmuir–Blodgett films: X-ray scattering studies G.E. Brown, M. Rho. On the manifestation of chiral symmetry in nuclei and dense nuclear matter C.A.A. de Carvalho, H.M. Nussenzveig. Time delay C. Chandre, H.R. Jauslin. Renormalization-group analysis for the transition to chaos in Hamiltonian systems S. Stenholm. Heuristic field theory of Bose–Einstein condensates G. Baur, K. Hencken, D. Trautmann, S. Sadovsky, Yu. Kharlov. Coherent gg and gA interactions in very peripheral collisions at relativistic ion colliders R. Durrer, M. Kunz, A. Melchiorri. Cosmic structure formation with topological defects T. Peitzmann, M.H. Thoma. Direct photons from relativistic heavy-ion collisions H. Rafii-Tabar, A Chirazi. Multi-scale computational modelling of solidification phenomena

*The full text of articles in press is available from ScienceDirect at http://www.sciencedirect.com. PII: S 0 3 7 0 - 1 5 7 3 ( 0 2 ) 0 0 0 5 8 - 3

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