Physics Reports vol.315

Physics Reports 315 (1999) 1}284 Looking forward: frontiers in theoretical science. Memorial symposium in honor of Rich...

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Physics Reports 315 (1999) 1}284

Looking forward: frontiers in theoretical science. Memorial symposium in honor of Richard Slansky, Los Alamos NM, 20}21 May 1998 editors Fred Cooper, Geo!rey B. West Communicated by A. Schwimmer, R. Petronzio Contents Preface M. Jacob, Multiparticle production C.M. Bender, The complex pendulum N. Dombey, A. Calogeracos, Seventy years of the Klein paradox F. Cooper, Inclusive dilepton production at RHIC: a "eld theory approach based on a nonequilibrium chiral phase transition A.S. Goldhaber, Dual con"nement of grand uni"ed monopoles? G. Chapline, Is theoretical physics the same thing as mathematics? J.H. Schwarz, From superstrings to M theory A.B. Balantekin, Neutrino propagation in matter

Elsevier Science B.V.

3 7 27 41

59 83 95 107 123

T. Pengpan, P. Ramond, M(ysterious) patterns in SO(9) R.W. Haymaker, Con"nement studies in lattice QCD C.-I. Tan, Di!ractive production at collider energies and factorization E. Timmermans, P. Tommasini, M. Hussein, A. Kerman, Feshbach resonances in atomic Bose}Einstein condensates J.N. Ginocchio, A relativistic symmetry in nuclei J. Patera, R. Twarock, Quasicrystal Lie algebras and their generalizations A.C. Hayes, Nuclear structure issues determining neutrino-nucleus cross sections J.E. Mandula, The gluon propagator

137 153 175

199 231 241 257 273

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Preface On May 20 and 21, 1998 a Symposium was held in Los Alamos, New Mexico to honor the memory of Richard Slansky, Director of the Theoretical Division at Los Alamos National Laboratory, who died unexpectedly from a brain aneurism. Dick was only 58 years old and in the prime of life. The shock and tragedy of his death has left a gaping hole in the lives of many who have lost a dear friend and colleague. We miss him terribly and mourn the loss. As a "tting tribute to Dick, the Symposium entitled: ¸ooking Forward: Frontiers in ¹heoretical Science brought together many who had known him at various stages of his career. It consisted of two closely connected parts. The "rst day was concerned more with Dick's contribution as a science administrator, particularly in fostering a climate of "rst-rate research at Los Alamos. The second day was primarily devoted to Dick's many interests in the area of High Energy and Elementary Particle Physics. Dick would probably not have approved of this separation since he did not like to think in us vs. them terms; throughout his years as part of management he always thought of himself as a scientist. Indeed, one of his most admirable qualities was his remarkable knowledge of the cutting edge of a wide range of exciting "elds. He was, however, a pragmatic man and, as such, will surely forgive us. It is this second day's activities that is recorded in this issue of Physics Reports. This is particularly "tting since Dick served as a long time Editor (1985}1998) of the journal and many of those assembled for the occasion had, at some time, been encouraged by Dick to contribute. Although Dick's major interest during his career was model building and Grand Uni"ed Theories, his interests were extremely broad. For example, when the issue of where to build the Superconducting Supercollider was in the forefront of attention, Dick single-handedly constructed a site atlas discussing the terrain of candidate sites. This included detailed calculations of problems of radiation and geology. In addition, he personally viewed several of the sites from a helicopter. In his early career he worked on various models for multiparticle production and was always interested in having experimental collaborators, such as Tom Ludlam, to con"rm his ideas. In many ways he was a frustrated experimentalist; this was re#ected throughout his career by an intense interest in the latest phenomenology and experimental results as well as in an active interest in the search for fractional charge and free quarks. Early on while at Yale, he developed a keen interest in group theory fostered by Feza Gursey and it is through his work in this area that he was best known. He wrote a famous Physics Reports classifying group representations relevant for model building that became a best-selling bible for model builders. In collaboration with Bob Moody and George Patera he extended this in a book which included the classi"cation of all possible Kac-Moody algebras. In addition to these important contributions, he has left his mark as one of the originators, together with Murray Gell-Mann and Pierre Ramond, of the much-cited see-saw mechanism for the generation of neutrino mass, which has become a major area of intense interest in the last few years. As an administrator, Dick was a champion for the preservation of excellence in the Theoretical Division at Los Alamos. Because of his broad interests and easy-going personality he was outstanding at being able to bridge the two cultures of fundamental basic research and applied programmatic research which coexisted in the Division. This was a particularly di$cult task in the post cold-war environment, especially when fundamental physics was under threat and funding was jeopardized. The talks on the "rst day of this symposium testi"ed to the success he had not

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Preface

only in keeping the Division strong but in attracting some of the best young researchers to Los Alamos during his tenure as Division Director. It is in this spirit that this Special Issue of Physics Reports, covering a broad spectrum of particle physics, is dedicated to the memory of our beloved colleague and friend, Dick Slansky, who is sorely missed by many of his colleagues not just at the Laboratory but world-wide. Fred Cooper Geo!rey B. West

Physics Reports 315 (1999) 7 } 25

Multiparticle production M. Jacob CERN, DSU Division, CH-1211 Geneva 23, Switzerland

Abstract A collaboration with R. Slansky on the analysis of multiparticle production in the early 1970s is recalled, with a short review of the situation at that time, as seen by him in Physics Reports. Present approach to particle production is illustrated by a particular example namely the humpback inclusive distribution in jet fragmentation. 1999 Elsevier Science B.V. All rights reserved. PACS: 01.60.#q

1. Foreword Less than two months ago I retired from CERN, as one has to at the age of 65. This was the occasion for several events and I am much indebted for all the work which some of my friends invested into this celebration. However, there was a strong element of sadness in this overall joyful happening. Two close friends, who had been my junior collaborators in the past, at a 10-year interval, and who had both long expressed the wish to be associated with such an a!air when its time would come, had both met with an untimely death by the time it occurred. One of them is Claude Itzikson, with whom I worked in the early 1960s, and the other one is Dick Slansky, with whom I worked in the early 1970s. I very much enjoyed my collaborations with them and I was later very happy to see them both rising to research heights where I would have had di$culties following them. They both remained close friends over several decades and their passing away was a great pain for me. We are here to commemorate the memory of Dick Slansky and I would like to recall the spirit in which we worked together, already long ago. 2. A brief historical survey It started at Yale in the fall of 1971, when Dick was an assistant professor there and where I was visiting for a semester. Dick had come to Yale after being a postdoc at Caltech and he stayed there 0370-1573/99/$ - see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 9 ) 0 0 0 2 8 - 9

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M. Jacob / Physics Reports 315 (1999) 7 } 25

until he moved to Los Alamos in 1974, after having been promoted to associate professor. Dick was a westerner at heart and he was very happy to move to New Mexico. It was however at an extremely hard time for him since his "rst wife was then dying of cancer. Let us go back to Yale in the fall of 1971. At that time, Bubble chambers were producing a large amount of data and their phenomenological analysis was raising many challenges. With incident beams in the 15}30 GeV range one could study collisions in which several secondary particles were produced. They were shot forward in the laboratory system and one was already calling that jets. The mean charged particle multiplicity was of the order of 5, slowly rising with incident energy. The low multiplicity was partly due to the fact that a leading hadron was often keeping a considerable fraction of its incident energy. This was referred to as the leading particle e!ect. In the case of p}p collisions, a "nal state proton would thus keep on the average half of the incident energy. Yet a typical collision saw the formation of several secondaries, mainly n mesons. Feynman had already proposed his scaling property for inclusive spectra. Yang had advocated a limiting fragmentation approach. Multiperipheral models, on the one hand, and statistical models, on the other hand, were both claiming some elements of the truth, if not the ultimate truth, but many features of the actual data were already showing important departures from either one of these two extreme approaches, for a review see [1]. Talking with experimentalists at CERN, I had collected interest in the matter. Coming to Yale, I had some recent data and some questions with me. I remember vividly Dick coming to my o$ce to ask me what I was interested in. The next day we were collaborating. Within a month we had what we thought were some interesting results and we started to discuss their implications with experimental colleagues and, in particular with T. Ludlam and R. Adair, at Yale, and with R. Panvini, on the telephone, at Brookhaven. The prevailing attitude at that time was to start from an symptotic model, assumed to apply at very high energy, and to "nd out some of its predictions in the existing data. We took a complementary attitude, namely to see what was needed to reproduce the data in their already great variety. We developed a simple model, "tting reasonably well the available data. Most of them had to do with inclusive distributions, namely the longitudinal and transverse momentum distributions for a single secondary particle, summing over everything else. Fig. 1a shows such a distribution (n- in pp collision by R. Panvini et al.) and the way we could reproduce it, and actually predict it from other pieces of data [2]. Also shown (Fig. 1b) are the predictions which we had for Fermilab and CERN-ISR energies, with the rise in the central region which was later veri"ed. The value of the longitudinal momentum is, as usually done, normalized to its maximum accessible value. This de"nes the Feynman variable x, according to which Feynman had advocated that the data should scale, namely show eventually no variation with energy [1,3]. Fig. 1c shows other inclusive distributions and the calculated yields [2]. I cannot resist showing in Fig. 2 some ISR results [3] which provided in the early seventies a textbook illustration of the scaling property. The variable is here the pseudorapidity of the

This re#ects the situation as we were aware of it when we started to collaborate in 1971. The Feynman scaling variable is de"ned as x"2p /(s, where p is the longitudinal momentum and (s is the centre * * of mass energy. The rapidity is de"ned as ln(E#p )/(E!p ) and approximated as ln x(s/M for high energy E<M. * * Here M is the transverse mass M "(M#p. The pseudo rapidity is de"ned as n" ln(p#p )/(p!p ). It is very 2 * * close to the rapidity for energetic secondaries and simply relates to the production angle.


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Fig. 1. (a) Inclusive p-distribution in p}p collisions at 28.5 GeV together with the prediction of the fragmentation model. (b) Approach to scaling as predicted in the model; (c) Inclusive distributions in p}p collisions at 30 GeV as calculated in the model for di!erent types of secondary particles.

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Fig. 1. Continued.

charged particles produced in pp collisions with two incident beams of 15.4 and 26.7 GeV, respectively, and also in an asymmetric beam con"guration with a beam of 15.4 GeV and the other of 26.7 GeV. The data show clearly that the fragments of one proton have a pseudo-rapidity (in practice energy) distribution which is independent of the energy with which the fragmenting proton has been hit, provided that it has been hit hard enough. The transverse distributions were known to be sharply cut o! at large transverse momentum. This left for production only the so-called longitudinal phase space. Typical distributions available at that time, in 1971, are shown in Fig. 3a. It was only by the fall of 1972, with the advent of the ISR data, that inclusive distributions stopped showing an exponential fall-o! with transverse momentum, as production rates could be measured beyond 1 GeV and found surprisingly high. The observed yields were much larger than simple extrapolations would suggest, as shown in Fig. 3b. This was the "rst evidence for hard scattering among quarks and gluons (partons at that time) and


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Fig. 2. Feynman scaling as beautifully illustrated by the early ISR data.

it was quickly recognized that this was the case [4]. This behaviour led eventually to the beautiful and predicted jet results at the CERN p}pbar collider and now at the Fermilab Tevatron. Fig. 4 shows this two jet structure as clearly seen with CERN p}pbar data, in the early eighties [5]. This can be considered as a modern view of the Rutherford experiment whereby hard point like constituents within the colliding particles scatter at wide angle and materialize into jets of hadrons. Quantitative predictions could come when these processes could be analyzed in the framework of perturbative QCD [6]. I could present predictions for inclusive jet yields at the CERN p}pbar collider during a key note address at a symposium in Los Alamos in 1979, and they were veri"ed in 1983. This was my "rst visit to Dick here, which was to be followed by several others. Dick also came to CERN and in particular for a six month stay in 1977}1978. His main interest had then shifted to the applications of Group Theory to model building and to the study of higher symmetries. He wrote a great Physics Reports article along that line in 1981 [6]. This paper was so much in Dick's style. He had `taken the bull by the hornsa, fully mastered sophisticated techniques in Group Theory and written a very clear and comprehensive article in which the model builder could "nd all that was needed. But, back to particle production in 1971, the transverse momentum data were usually "tted to a Gaussian shape and one knew that all there was to be seen in practice was at very low transverse momentum. This was accordingly incorporated in any approach. The rare large transverse

Results from UA (1). For a review of collider physics, see [5].

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Fig. 3. (a) Transverse momentum distributions for di!erent secondary particle as measured at the ISR (data points) and at 24 GeV on a "xed target (dashed curves); (b) Large transverse momentum production at the ISR.


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Fig. 4. Jet production at the CERN p}pbar collider. Result from UA(1).

momentum events correspond to a largely independent hadronization with, in that case, a similar strong transverse momentum cut-o! with respect to the jet axis. Our model related the sharp fall o! of the transverse momentum distribution to the sharp fall o! also observed for the longitudinal momentum distribution of the secondary particles. We could reproduce inclusive yields (Fig. 1). We had the luck to come up with predictions which got veri"ed and it even continued when two-particle correlation data became available [7]. We were very excited even though we realized well that we had very little quantum dynamics in our model. We had mainly kinematics and some enforced speci"c features. We were actually turning the question around asking: are the data telling us something just interesting or something much deeper? How di$cult are they to reproduce? Two months later we were calculating many inclusive distributions, meeting with perhaps a too easy success with them at typically 15}30 GeV. We received the help of C.C. Wu, a graduate student of Dick, who eventually wrote a thesis along these lines. As I moved to Fermilab, Ed Berger, from Argonne, started collaborating with us. We had success with inclusive spectra and also with the Bubble chamber

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two particle correlation data which were just becoming available [7]. We had also some success with the "rst Fermilab results.

3. What were we doing? To put things in a nutshell, we were analyzing particle production in terms of that of clusters, with invariant mass of typically 1}2 GeV, which were excited from the incident particles and fragmenting isotropically into the observed "nal particles. Dick coined the word `Novaa for these clusters. It was a very good name translating the ephemeral #are of pions produced as the colliding particles were excited. But `clustera is the term which eventually remained, probably because `Novaa was too much associated with other features of the model which eventually had to be dropped, like their di!ractive production and isotropic decay up to high masses which the model kept. In any case, the key feature was that light particles such as pions, which are those dominantly produced, had to have, on average, the same velocity, or more speci"cally the same rapidity, as the rather massive clusters which they were originating from. As a result the inclusive spectra were falling rather steeply with momentum, in very good agreement with the data over several orders of magnitude, and one could "nd a rationale and easy "ts to the inclusive distribution of all the other secondaries. This is shown in Fig. 1. The leading particle e!ect was automatically inforced by the quasi-elastic production process which was assumed to dominate. On top of that, since a proton, as a member of a cluster, is produced with the same average rapidity as the pion fragments, it takes a centre of mass momentum typically 3 times larger than that of the pions. The invariant distribution of pions originating from a Nova of mass M is [2] f (x)"R e\V+) ,

(1)

where K is the mean momentum of the pions in the Nova rest frame, a quantity "xed by the sharply cut o! transverse momentum distribution, which was reproduced at the same time. R is here a normalization factor which was speci"ed in the model. With a typical mass of 2 GeV, one readily gets an e\V distribution, already in good agreement with the surprisingly fast (at that time) fall o! of the inclusive pion distributions in p}p collisions (Fig. 1). Scaling and the approach to scaling for secondaries produced at large x (typically 0.1(x(0.6), in the so-called fragmentation region, was associated with the dominantly di!ractive production of the lower mass Novas. This has stood the test of time. One could understand quantitatively how scaling was achieved at relatively low energies for pions and only at much higher energies for heavy particles. It had to be approached from below in the central region. In order to keep a compact and simple model, we bravely extended this di!ractive mechanism to higher and higher mass Novas still with globally isotropic decays. Since we advocated a step-bystep fragementation mechanism, isotropy was not speci"c to low masses. Considering di!ractive production we were led to assume dominance of single over double excitation. We advocated that one or two cluster(s) was all there was to be in practice in the 15}30 GeV range, even though production of more many clusters could eventually prevail at much higher energy [2]. Even though the di!ractive production of clusters was not a priori justi"ed at higher mass, it kept an a priori correct asymptotic behaviour for inclusive spectra when simply, and bravely, extended. It gave


15

eventually a rapidity plateau approached from below. We thus kept it, up to the limit of phase space. It is actually natural and easy that way to obtain eventually a properly normalized rapidity plateau, and this was certainly good enough to reproduce the 15}30 GeV data, which hardly showed any sign of it. A cluster with isotropic decay distributes its products over typically two units of rapidity and, at such energies, there was little more available anyway for the larger multiplicity con"gurations. In large multiplicity con"gurations, secondaries were all in a central rapidity cluster with longitudinal and transverse momenta of the same order in the center of mass. However, with the advent of the Fermilab and then of the "rst CERN-ISR results, the rapidity interval was suddenly extended from 4 to 8 units in 1972}1973, exposing at long last the central rapidity plateau and allowing correlation measurements at central rapidity, or among particles emitted with relatively little energy in the center of mass [3]. Fig. 5 shows the inclusive rapidity distributions for several types of secondaries as they soon became available at the ISR [3], with the extending rapidity plateau long advocated by Feynman and which we had in our model. One could thus at long last separate a fragmentation region, covering about 2 units of rapidity, from a central region with its extending rapidity plateau slowly rising with energy. What we had been advocating and modeling was that there were only fragmentation regions at 15}30 GeV. Extending that without any change to ISR energy, the rapidity plateau was populated with the numerous fragments of massive Novas. This mechanism, which we had singled out in our model, and which was adequate at 15}30 GeV, turned out to be invalid for the central rapidity region appearing at ISR energies, despite the fact that we could still have correctly predicted the inclusive yield there. We started to be wrong with correlations. Fig. 6a shows the rapidity correlation among two charged secondaries (mainly pions) as measured at Fermilab and at the ISR [3]. Whereas at Fermilab (at that time) one still saw mainly a big clustering in the middle, at the ISR, one could see a developing ridge, showing that secondaries were indeed clustered in rapidity but in the same way over an extending rapidity range. One had to conclude that, now, di!erent lower mass clusters were being produced over the central rapidity range. Fig. 6b shows the central two-body correlation, extending over typically two units of rapidity, as expected from an isotropic cluster decay, but with a value independent of energy, as if the increase in collision energy was not extending the mass range (and the fragment multiplicity) of the clusters but rather resulted in the production of more and more of them, distributed rather uniformly over the central rapidity plateau. The naive extension of our model was totally failing on that whereas the data were far more in line with short range (in rapidity) order, as long advocated in the multiperipheral approach [8]. In that case one expects asymptotically a uniform spread of the secondaries over the full rapidity range. This was not holding for individual particles but it was holding for clusters. Electronic detectors were pouring out data at a much faster rate than the Bubble chambers. The properties of these central clusters were quickly analyzed. Their mean multiplicity is of the order of 3}4. Their mean invariant mass does not exceed 1.5 GeV. In present vernacular one is led to say that these clusters correspond to the hadronization of quark and gluon systems when one reaches the limit of validity of perturbative QCD. This is indeed the energy which it takes to pull a quark apart over a distance which corresponds to the penetration length of color into the vacuum. The typical cut-o! value for the invariant mass of a parton system which is used in jet fragmentation model is of the same order.

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Fig. 5. Inclusive rapidity distributions for di!erent types of secondaries as measured at the ISR, with the extending rapidity plateau, and compared with 24 GeV "xed target results.

Back at CERN, I got much involved with the analysis of all kinds of ISR data and corresponded regularly with Dick who had become very knowledgeable about all features of multi particle dynamics. In 1973}1974, he wrote in particular a review paper summarizing very well all that was then known [9]. At the same time, his beautiful work along that line caught the attention of P. Carruthers as he was forming a high-energy group of theorists at Los Alamos. P. Carruthers was then also much interested in multiparticle production [10]. This is this review paper which I now use to discuss particle production as seen by Dick at that time. He takes the reader through a very large amount of data extracting clearly all the important features. There is in particular one section which is called `how to build a successful modela in which he distinguishes very well what is most signi"cant.


17

Fig. 6. (A) Two body rapidity correlations among charged pions at Fermilab and at the ISR, with evidence for the extending ridge. The di!erent plots correspond to (a) 102 GeV, (b) 400 GeV, (c) 11 against 11 GeV at ISR and (d) 31 against 31 GeV; (B) Two body correlations at central rapidity at the ISR. One secondary is taken at zero rapidity. Open dots are for 23 GeV and full dots are for 63 GeV in the centre of mass.

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4. Particle production circa 1973}1974 Dick "rst remarked that there are many models, from multiperipheral ones which tend to spread uniformly secondaries over the available rapidity range to those which emphasize the formation of heavy states, as the Nova model did. However, a successful model with inclusive data does not necessarily implies correct physics. He stressed that one should rather emphasize the complementarity between models and try to extract general features from the data. Instead of seeing a low invariant mass system always as a chain (with secondary equally spread in rapidity) one can see it as a cluster. On the other hand, a large rapidity gap in a multiperipheral chain, as frequently the case for lower multiplicities, leads to a di!ractive process. A `successfula model for particle production had to have a strong damping at large transverse momentum. We have good reasons to understand that for hadrons of "nite size (about 1 fm). Since what actually matters is the transverse mass, the lightest secondaries (pions) are automatically favored. The secondaries are in practice limited to longitudinal phase space (Fig. 2). Yet, they do not appear to take all the energy which they could have. The distribution are also sharply cut o! in x. The model has to have `weak inelasticitya, as associated with the leading particle e!ect. This is however not enough. The answer, as already said, is that the light pions have actually on the average the same rapidity as that of a much more massive object from which they originate. The rapidity of a fast particle (E
but it becomes in"nite as NP2\ (see Fig. 7). The path passes many turning points as it spirals clockwise from x . [The nth turning point lies at the angle (3N!2!4n)/2Np \

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C.M. Bender / Physics Reports 315 (1999) 27}40

Fig. 5. Classical paths in the complex-x plane for the N"3 oscillator. As the paths get larger, they approach a limiting shape that resembles a cardioid. We have plotted the rescaled paths.

Fig. 6. Classical paths for the case N"2.5. These paths do not intersect; the graph shows the projection of the parts of the path that lie on di!erent sheets of the Riemann surface. As the size of the paths increases a limiting cardioid appears on the principal sheet of the Riemann surface. On the remaining sheets of the surface the path exhibits a knot-like topological structure.

(x corresponds to n"0).] As N approaches 2 from below, when the classical trajectory passes \ a new turning point, there is a corresponding additional merging of the quantum energy levels as shown in Fig. 1. This correspondence becomes exact in the limit NP2\ and is a manifestation of Ehrenfest's theorem.


35

Fig. 7. Classical paths in the complex-x plane for N"1.8, 1.85 and 1.9. These paths are not periodic. The paths spiral outward to in"nity. As NP2 from below, the number of turns in the spiral increases. The lack of periodic orbits corresponds to a broken PT-symmetry.

4. Applications of complex Hamiltonians There appear to be many applications of complex, PT-invariant Hamiltonians in physics. Hamiltonians having an imaginary external "eld have been introduced recently to study delocalization transitions in condensed matter systems, such as vortex #ux-line depinning in type-II superconductors [7], or even to study population biology [8]. In these cases, initially real eigenvalues bifurcate into the complex plane due to the increasing external "eld, indicating the unbinding of vortices or the growth of populations. We believe that one can also induce dynamic delocalization by tuning a physical parameter (here N) in a self-interacting theory. The PT-symmetric Hamiltonian in Eq. (1) may be generalized to include a mass term mx. The massive case is more elaborate than the massless case; phase transitions appear at N"0 and at N"1 as well as at N"2 (see Fig. 8) [2]. Replacing the condition of Hermiticity by the signi"cantly weaker constraint of PT-symmetry allows one to construct new kinds of quasi-exactly solvable quantum theories [9]. An example of a quasi-exactly solvable Hamiltonian is H"p!x#2iax#(a!2b)x#2i(ab!J)x ,

(10)

where J is an integer and a and b are arbitrary parameters. A plot of the spectrum for various values of a and b is given in Fig. 9.

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Fig. 8. Energy levels of the Hamiltonian H"p#mx!(ix), as a function of the parameter N. There are now phase changes at N"0, N"1, and N"2.

Fig. 9. The spectrum for the QES Hamiltonian (10) plotted as a function of b for the case J"3 and a"0. For b' the QES eigenvalues are real and are the three lowest eigenvalues of the spectrum. When b goes below , two of the QES eigenvalues become complex and the third moves into the midst of the non-QES spectrum. We believe that the non-QES spectrum is entirely real throughout the (a, b) plane.


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Fig. 10. The spectrum for the Hamiltonian p#x(ix)C plotted as a function of e.

There are many classes of Hamiltonians for which one can construct complex deformations. For example, one can deform the potential x) by replacing it by x)(ix)C. For e positive the spectrum of the Hamiltonian is real. However, for e negative PT-symmetry is broken and the spectrum becomes complex. As e approaches !K, the eigenvalues gradually disappear into the complex plane until at e"!K there are no eigenvalues at all. In Figs. 10 and 11 we display the energy levels of the Hamiltonian H"p#x(ix)C (case K"2) as a function of the parameter e. This "gure is similar to Fig. 1, but now there are four regions: When e50, the spectrum is real and positive and it rises monotonically with increasing e. The lower bound e"0 of this PT-symmetric region corresponds to the pure quartic anharmonic oscillator, whose Hamiltonian is given by H"p#x. When !1(e(0, PT-symmetry is spontaneously broken. There are a "nite number of positive real eigenvalues and an in"nite number of complex conjugate pairs of eigenvalues; as a function of e the eigenvalues pinch o! in pairs and move o! into the complex plane. By the time e"!1 only eight real eigenvalues remain; these eigenvalues are continuous at e"1. Just as e approaches !1 the entire spectrum reemerges from the complex plane and becomes real. (Note that at e"!1 the entire spectrum agrees with the entire spectrum in Fig. 1 at e"1.) This reemergence is di$cult to see in Fig. 10 but is much clearer in Fig. 11 in which the vicinity of e"!1 is blown up. Just below e"!1, the eigenvalues once again begin to pinch o! and disappear in pairs into the complex plane. However, this pairing is di!erent from the pairing in the region !1(e(0. Above e"!1 the lower member of a pinching pair is even and the upper member is odd (that is, E and E combine, E and E combine, and so on); below e"!1 this pattern reverses (that is, E combines with E , E combines with E , and so on). As e decreases from !1 to !2, the number of real eigenvalues continues to decrease until the only real eigenvalue is the ground-state energy. Then, as e approaches !2>, the ground-state energy diverges logarithmically. For e4!2 there are no real eigenvalues. In Fig. 12 we display the energy levels of the Hamiltonian H"p#x(ix)C (case

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Fig. 11. A blow-up of Fig. 10 in the vicinity of the transition at e"!1. Just above e"!1 the entire spectrum reemerges from the complex plane, and just below e"!1 it continues to disappear into the complex plane. The spectrum is entirely real at e"!1.

Fig. 12. Energy levels of the Hamiltonian p#x(ix)C as a function of the parameter e. This "gure is similar to Fig. 10, but now there are "ve regions: When e50, the spectrum is real and positive and it rises monotonically with increasing e. The lower bound e"0 of this PT-symmetric region corresponds to the pure sextic anharmonic oscillator, whose Hamiltonian is given by H"p#x. The other four regions are !1(e(0, !2(e(!1, !3(e(!2, and e(!3. The PT-symmetry is spontaneously broken when e is negative, and the number of real eigenvalues decreases as e becomes more negative. However, at the boundaries e"!1 and e"!2 there is a complete positive real spectrum. When e"!1, the eigenspectrum is identical to the eigenspectrum in Fig. 10 at e"1. For e4!3 there are no real eigenvalues.


39

K"3) as a function of the parameter e. This "gure is similar to Fig. 10, but now there are "ve regions in the parameter e. Quantum "eld theories analogous to the quantum-mechanical theory in Eq. (1) have astonishing properties. The Lagrangian L"(

)#m !g(i ), (N real) possesses PT invariance, the fundamental symmetry of local self-interacting scalar quantum "eld theory [10]. While the Hamiltonian for this theory is complex, the spectrum appears to be positive de"nite. Also, as L is explicitly not parity invariant, the expectation value of the "eld 1 2 is nonzero even when N"4 [11]. Thus, in principle, one can calculate directly (using the Schwinger}Dyson equations, for example [12]) the (positive real) Higgs mass in a renormalizable theory, such as !g or ig , in which symmetry breaking occurs naturally (without introducing a symmetry-breaking parameter). Replacing conventional g or g theories by !g or ig theories reverses signs in the beta functions. Thus, theories that are not asymptotically free become asymptotically free and theories lacking stable critical points develop such points. We believe that !g in four dimensions is nontrivial. Furthermore, PT-symmetric massless electrodynamics has a nontrivial stable critical value of the "ne-structure constant a [13]. We have examined supersymmetric PT-invariant Lagrangians [14] and "nd that the breaking of parity symmetry does not induce a breaking of the apparently robust global supersymmetry. We have investigated the strong-coupling limit of PT-symmetric quantum "eld theories [15]; the correlated limit in which the bare coupling constants g and !m both tend to in"nity with the renormalized mass M held "xed and "nite is dominated by solitons. (In parity-symmetric theories the corresponding limit, called the Ising limit, is dominated by instantons.)

Acknowledgements I thank my coauthors S. Boettcher, H. Jones, P. Meisinger, and K. Milton for their contributions to this research and D. Bessis, A. Wightman, and Y. Zarmi for useful conversations. This work was supported by the US Department of Energy.

References [1] [2] [3] [4] [5] [6]

D. Bessis, private discussion. C.M. Bender, S. Boettcher, Phys. Rev. Lett. 80 (1998) 5243. C.M. Bender, S. Boettcher, P.N. Meisinger, J. Math. Phys., to appear. I. Herbst, Commun. Math. Phys. 64 (1979) 279. C.M. Bender, A. Turbiner, Phys. Lett. A 173 (1993) 442. C.M. Bender, S.A. Orszag, Advanced Mathematical Methods for Scientists and Engineers, McGraw-Hill, New York, 1978. [7] N. Hatano, D.R. Nelson, Phys. Rev. Lett. 77 (1996) 570, and Phys. Rev. B 56 (1997) 8651. [8] D.R. Nelson, N.M. Shnerb, condmat/9708071.

Ultimately we may de"ne ? as PCT, the fundamental symmetry of the world. There is no analog of the C operator in quantum mechanical systems having one degree of freedom and in scalar "eld theories.

40 [9] [10] [11] [12] [13] [14] [15]

C.M. Bender / Physics Reports 315 (1999) 27}40 C.M. Bender, R.F. Streater, C.M. Bender, C.M. Bender, C.M. Bender, C.M. Bender, C.M. Bender,

S. Boettcher, J. Phys. A: Math. Gen. 31 (1998) L273. A.S. Wightman, PCT, Spin & Statistics, and all that, Benjamin, New York, 1964. K.A. Milton, Phys. Rev. D 55 (1997) R3255. K.A. Milton, submitted for publication. K.A. Milton, J. Phys. A: Math. Gen. 32 (1999) L87. K.A. Milton, Phys. Rev. D 57 (1998) 3595. S. Boettcher, H.F. Jones, P.N. Meisinger, Phys. Rev. D, submitted for publication.


Seventy years of the Klein paradox N. Dombey *, A. Calogeracos Centre for Theoretical Physics, University of Sussex, Falmer, Brighton BN1 9QJ, UK Low Temperature Laboratory, Helsinki University of Technology, PO Box 2200, FIN-02015 HUT, Finland

Abstract The Klein paradox is examined. Its explanation in terms of electron}positron production is reassessed. It is shown that a potential well or barrier in the Dirac equation can produce positron or electron emission spontaneously if the potential is strong enough. The vacuum charge and lifetime of the well/barrier are calculated. If the well is wide enough, a seemingly constant current is emitted. These phenomena are transient whereas the tunnelling "rst calculated by Klein is time-independent. Furthermore, tunnelling without exponential suppression occurs when an electron is incident on a high barrier, even when it is not high enough to radiate. Klein tunnelling is therefore a property of relativistic wave equations and not necessarily connected to particle emission. The Coulomb potential is investigated in this context: it is shown that a heavy nucleus of su$ciently large Z will bind positrons. Correspondingly, it is expected that as Z increases the Coulomb barrier will become increasingly transparent to positrons. This is an example of Klein tunnelling. 1999 Elsevier Science B.V. All rights reserved. PACS: 03.65.Pm; 11.10.Kk; 14.60.Cd Keywords: Klein paradox; Quantum tunnelling; Positron production

1. Introduction to the paradox(es) It is an honour to be back at Los Alamos to commemorate Dick Slansky. He had a special interest in quantum mechanics and so I think it appropriate to discuss here some work I have been doing with Calogeracos on one of the oldest problems associated with the Dirac equation: the

* Corresponding author. Talk given by N. Dombey. Permanent address: NCA Research Associates, PO Box 61147, Maroussi 151 22, Greece. E-mail address: [email protected] (N. Dombey) 0370-1573/99/$ - see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 9 ) 0 0 0 2 3 - X

42

N. Dombey, A. Calogeracos / Physics Reports 315 (1999) 41}58

Fig. 1. An electron of energy E scattering o! a potential step of height result that electrons should be able to tunnel through high potential steps. In 1928 Klein [1] submitted a paper for publication in which he calculated the re#ection and transmission of electrons of energy E, mass m and momentum k incident on the potential step (5)


43

With this choice of p, both R , ¹ are positive and satisfy R #¹ "1. So is the result of Eq. (2) 1 1 1 1 paradoxical? The general consensus is that it is for as the potential step

E#m k

i e IVh(!x) k E#m

i!1
#

1!i E#m (2p u (E, x)" 0 2k 1#i

#

E#m 2k

(2i # i#1

i e NVh(x) "p" E#m!
(22)

We write "p" rather than p in these equations since the group velocity is negative for x'0 (cf. Eq. (17)). The factors (2p come from the energy normalisation factors in Eq. (13). We need to evaluate the currents corresponding to the solutions of Eqs. (21) and (22). According to our conventions a "c c "!p so V V W j ,!uR (E, x)p u (E, x)"(2i/p)/(i#1) , * * W *

(23)


j ,!uR (E, x)p u (E, x)"!(2i/p)/(i#1) . 0 0 W 0

47

(24)

3.2. The dexnition of the vacuum and the vacuum expectation value of the current Now expand the wavefunction t in terms of creation and annihilation operators which refer to our left- and right-travelling solutions:

t(x, t)" dE +a (E)u (E, x)e\ #R#a (E)u (E, x)e\ #R#bR (E)v (E, x)e #R#bR (E)v (E, x)e #R, * * 0 0 * * 0 0 (25)

with tR given by the Hermitian conjugate expansion. The creation and annihilation operators in Eq. (25) satisfy the usual anticommutation relations +a (E), aR (E),"id(E!E), etc. Again * * the (2p factors in Eqs. (21) and (22) ensure that these anticommutation relations are consistent. We must now determine the appropriate vacuum state in the presence of the step. States described by wavefunctions u (E, x) and v (E, x) correspond to (positive energy) electrons and * * positrons, respectively, coming from the left. Hence with respect to an observer to the left (of the step) such states should be absent from the vacuum state, so a (E)"02"0, b (E)"02"0 . (26) * * Wavefunctions u (E, x) for E'm#< describe for an observer to the right, electrons incident from 0 the right. These are not present in the vacuum state hence a (E)"02"0, E'm#< . (27) 0 Wavefunctions v (E, x) describe, again with respect to an observer to the right, positrons incident 0 from the right; again b (E)"02"0 . (28) 0 The wavefunctions that play the crucial role in the Klein problem belong to the set u (E, x) for 0 m(E(
48


!10"aR (E)a (E)"02uR (E, x)p u (E, x) 0 0 0 W 0 #10"a (E)aR (E)"02uR (E, x)p u (E, x), . (31) 0 0 0 W 0 The "rst term in Eq. (31) vanishes due to (26). The second term becomes uR (E, x)p u (E, x)d(E!E) * W * if we use the anticommutation relations and Eq. (26). The third term yields !uR (E, x)p u (E, x)d(E!E) using Eq. (29) and the fourth term vanishes using the anticommuta0 W 0 tion relations (i.e. the exclusion principle; the state "02 already contains an electron in the state u hence we get zero when we operate on it with aR ). One energy integration is performed 0 0 immediately using the d function. The "nal result is

1 4i(E) 1 dE 10" j"02" dE (!j #j )"! * 0 2p (1#i(E)) 2 or

1 10" j"02"! dE ¹ (E) , 1 2p

(32)

where the energy integration is over the Klein range m(E(
4. Scattering by a square barrier We now turn our attention to a square barrier in place of the Klein step. Consider the square barrier

tan pa"

(m!E)(E#

tan pa"!

(m#E)(E# (37)

(38)

where now the well momentum is given by p"(E# > I #cR(k, 0)u (k, x)e #R#cR(k, 0)u (k, x)e #R, \ \ > \ , , (40) # b (0)u (x)e\ #HR# dR(0)u (x)e\ #HR . H H H H H H Operators aR, a create and annihilate travelling electrons; cR, c are the corresponding ones for positrons. Operators b (bR) annihilate (create) bound electrons whereas d (dR) annihilate (create) H H H H bound positrons. The use of the (-) in Eq. (40) is dictated by the sign of the exponential and conforms to current literature. The Hermitian conjugate expansion is

tR(x, t)" +aR(k, 0)uR (k, x)e #R#aR(k, 0)uR (k, x)e #R > > I #c (k, 0)uR (k, x)e\ #R#c (k, 0)uR (k, x)e\ #R, \ \ ,> ,\ # bR(0)uR(x)e #HR# d (0)u (x)e #HR , (41) H H H H H H where we took into account the reality of u . The standard anticommutation relations are H obeyed

+a (k, t), aR (k, t,"d d +b (t), bR(t),"d , IIY G H GH +c (k, t), cR (k, t,"d d +d (t), dR(t),"d . IIY G H GH

(42)

We work in the Heisenberg picture throughout: The time dependence is carried by operators whereas state vectors are time-independent. However, basis ket vectors (and in particular the vacuum) are time-dependent. The vacuum "02 is de"ned by a (k)"02"c (k)"02"b "02"d "02"0 . G G

(43)


51

The total charge is de"ned by (according to our conventions the electron charge is !1)

1 Q(t)" dx o(x, t)"! dx [tR(x, t), t(x, t)] . 2

(44)

Substituting (40), (41) and using (42) we get Q"Q #Q ,

(45)

where the normal-ordered or particle charge Q is given by Q " +!aR(k, t)a (k, t)#cR(k, t)c (k, t)!aR(k, t)a (k, t) I ,> ,\ # cR(k, t)c (k, t)! bRb # dRd H H H H H H and the vacuum charge Q by

(46)

1 Q " + N (E'0)! N (E(0), , (47) 2 I I where N (E'0) is the number of positive energy states and N (E(0) is the number of negative energy states. Given de"nition (43) of the vacuum we immediately get 10"Q"02"Q (48) so the vacuum charge turns out to be the spectral asymmetry of the Hamiltonian. It is important to note that we would not have obtained the connection between Q and the spectral asymmetry had we not identi"ed bound states with E(0 as positrons (Zeldovich and Popov [15] use a di!erent de"nition) or not used the commutator in Eq. (44). Note also that the total charge Q is always conserved. There are pitfalls when Eq. (47) for the vacuum charge Q is applied to actual systems and it is easy to obtain incorrect results. One way of doing things properly is to enclose the system in a box, impose periodic boundary conditions and take the ¸PR limit right at the end of the calculation (see [14] for details). Now back to the delta function potential. For j just larger than p/2, Q "#1 because of the presence of the positron and so the vacuum charge Q must now equal !1 to conserve charge. As the potential is increased further, j will reach p. Here E"!m which is the condition for supercriticality: the bound positron reaches the continuum and becomes free. This is the well-known scenario of spontaneous positron production "rst discussed [15,16] over 25 years ago. Note that at supercriticality j"p, the even bound state disappears and the "rst odd state appears. We can continue to increase j and count positrons: the total number of positrons produced for a given j is the number of times E has crossed E"0; that is Q "Int[(j/p)#(1/2)] ,

(49)

52


where Int[x] denotes the integer part of x. The more interesting quantity for us is the number of supercritical positrons Q : the number of states which have crossed E"!m. This is given by 1 Q "Int[j/p] . (50) 1 Note that for any j there is at most one bound positron state. 5.3. Wide well We consider the general case of \

S[u ]# J u ! SH[u ]# J u > > > \ \ \

" du exp i(S[u ]#J u?)1u , i"o"u , i2,e 5 (? , (2.18) ? ? ? ? where 1u , i"o"u , i2 is the density matrix de"ning the initial state. We use the matrix notation u u?" > , a"1, 2 , (2.19) u \ with a corresponding two-component source vector

J > , a"1, 2 . J \ On this matrix space there is an inde"nite metric J?"

c "diag(#1,!1)"c?@ , ?@ so that, for example,

(2.20)

(2.21)

J?c u@"J u !J u . ?@ > > \ \ From the path integral we get the following matrix Green's function:

(2.22)

G?@(t, t)"d=/dJ (t)dJ (t)" , ? @ ( G(t, t),G (t, t)"i Tr+ou(t)u (t), , G(t, t),G (t, t)"$i Tr+ou (t)u(t), , G(t, t)"i Tr+oT[u(t)u (t)], "H(t, t)G (t, t)#H(t, t)G (t, t) , G(t, t)"i Tr+oTH[u(t)u (t)], "H(t, t)G (t, t)#H(t, t)G (t, t) . We notice that G "G(t, t) and G H"G(t, t). $ $ We also will need the relationships

(2.23)

G (t, t)"iH(t!t)[U(t),UM (t)] "H(t!t)[G(t, t)!G(t, t)] , ! G (t, t)"!iH(t!t)[U(t),UM (t)] "H(t!t)[G(t, t)!G(t, t)] , ! as well as the relations between the Green's functions

(2.25)

G (t, t)"G (t, t)!G(t, t)"!G H(t, t)#G(t, t) , $ $ G (t, t)"G (t, t)!G(t, t)"!G H(t, t)#G(t, t) . $ $

(2.24)

(2.26)

(2.27)

64

F. Cooper / Physics Reports 315 (1999) 59}81

2.2.1. Large-N approximation The method for reducing the number of degrees of freedom in the Heisenberg picture is the large-N approximation [5]. If we have an N-component scalar "eld with Lagrangian ¸I [U]"(R U )(RIU )!(j/8N)(U U !(2Nk/j)) , G G G I G we can rewrite this as

(2.28)

s ¸I [U, s]"!U (䊐#s)U #(N/j)s #k , G G 2

(2.29)

where i"1,2, N and s"!k#(j/2N)U U . (2.30) G G If k'0, a spontaneous symmetry is breaking at the classical level. At this minimum the O(N) symmetry is spontaneously broken, s"0 and there are N!1 massless modes. Small oscillations in the remaining i"N (radial) direction describe a massive mode with bare mass equal to (2k"(jv . The generating functional for all graphs is given by [6]

Z[ j, K]" d ds exp+iS[ , s]#i [ j #Ks], .

(2.31)

Perform the Gaussian integral over the "eld :

Z[ j, K]" ds exp+iNS [s, j, K], ,

(2.32)

where

1 1 s i S " dx jG\[s] j#Ks# s #k # Tr ln G\[s] , 2 j 2 2 G\[s](x, y),+䊐#s,d(x!y) .

(2.33)

Because of the N in the exponent one is allowed to perform the integral over s by stationary phase. This leads to an expansion of Z in powers of 1/N the lowest term (stationary-phase point) is related to the previous Gaussian (Hartree) approximation. The e!ective action of the leading order is S [ , s]"S [ , s]#(i /2) Tr ln G\[s] . Varying the action leads to the mean "eld equations

(2.34)

+䊐#s, "0 , s"!k#(j/2N)( #(1/i)G(x, x; s)) .

(2.35)

We notice that this is the same equation found in the Gaussian approximation with m(t) being identi"ed with s.


65

3. Dynamical evolution of a non-equilibrium chiral phase transition One important question for RHIC experiments is } can one produce disoriented chiral condensates (DCCs) in a relativistic heavy ion collision? Recently, Bjorken, Rajagopal and Wilczek and others proposed that a nonequilibrium chiral phase transition such as a quench might lead to regions of DCCs [7]. The model Rajagopal and Wilczek considered was the O(4) linear sigma model in a tree-level approximation, where a quench was assumed. Two de"ciencies of that model were its classical nature (it could not describe p}p scattering), and the quench was put in by hand. Our approach [8] instead was to look at the quantum theory in an approximation that captures the phase structure as well as the low-energy pion dynamics. We also used the natural expansion of an expanding plasma to cool the plasma and built into our approximation boost invariant kinematics which result from a hydrodynamic picture where the original plasma is highly Lorentz contracted. In the linear sigma model treated in leading order in the 1/N expansion the theory has a chiral phase transition at around 160 MeV and we choose the parameters of this theory to give a reasonable "t to the correct low-energy scattering data. We obtain natural quenching for certain initial conditions as a result of the expansion process. 3.1. Review of the linear p model The Lagrangian for the O(4) p model is ¸"RU ) RU!j(U ) U!v)#Hp .

(3.1)

The mesons form an O(4) vector U"(p, p ) . G As we discussed earlier in our discussion of the large-N approximation we introduce s"j(U ) U!v) and use the equivalent Lagrangian ¸ "! (䊐#s) #s/4j#sv#Hp . G G

(3.2)

The leading order in 1/N e!ective action which we obtain by integrating out the "eld and keeping the stationary phase contribution to the s integration is

i C[U, s]" dx ¸ (U, s, H)# N tr ln G\ , 2

(3.3)

G\(x, y)"i[䊐#s(x)]d(x!y) . This results in the equations of motion [䊐#s(x)]p "0, G

[䊐#s(x)]p"H ,

(3.4)

66


and the constraint or gap equation s"!jv#j(p#p ) p)#jNG (x, x) . (3.5) We will introduce #uid proper time and rapidity variables to implement the kinematic constraint of boost invariance: 1 g, log(t!z/t#z) . 2

q,(t!z),

To implement boost invariance we assume that mean (expectation) values of the "elds U and s are functions of q only: q\R qR U (q)#s(q) U (q)"Hd , O O G G G s(q)"j(!v#U(q)#NG (x, x; q, q)) . (3.6) G To calculate the Green's function G (x, y; q, q) we "rst determine the auxiliary quantum "eld (x, q) (q\R qR !q\ R!R #s(x)) (x, q)"0 . O O E , G (x, y; q,q),1¹+ (x, q) (y, q),2 . (3.7) We expand the quantum "eld in an orthonormal basis:

1

(g, x , q), [dk](exp(ikx) fk(q)ak#h.c.) , , q where kx,k g#k x , [dk],dk dk /(2p). The mode functions and s obey E , , E , f$k#((k/q)#k #s(q)#(1/4q)) fk"0 . E , 1 s(q)"j !v#U(q)# N [dk]" fk (q)"(1#2nk) . G q

(3.8) (3.9)

when s goes negative, the low-momentum modes with

((k#1/4)/q)#k ("s" E , grow exponentially. These growing modes then feed back into the s equation and get damped. Low momentum growing modes lead to the possibility of DCCs as well as a modi"cation of the low momentum distribution of particles. To "x the parameters of this mode we use the PCAC relation R AG (x),f mpG(x)"HpG(x) , I I p p and the de"nition of the broken symmetry vacuum, s p "mp "H , p p "f "92.5 MeV , p

m"!jv#jf #jN p p

K

[dk]

1 2(k#m p

.

(3.10)


67

The mass renormalized gap equation is

1 j . s(q)!m"!jf #jU(q)# N [dk] " fk(q)" (1#2nk)! p p G q 2(k#m

(3.11)

j is chosen to "t low energy scattering data. We choose our initial data (at q "1) so that the system is in local thermal equilibrium in a comoving frame n "1/e@#I !1 , I

(3.12)

where b "1/¹ and E"((k/q)#k #s(q ). I E , The initial value of s is determined by the equilibrium gap equation for an initial temperature of 200 MeV and is 0.7 fm\ and the initial value of p is just H/s . The phase transition in this model occurs at a critical temperature of 160 MeV. To get into the unstable domain, we then introduce #uctuations in the time derivative of the classical "eld. For q "1 fm there is a narrow range of initial values that lead to the growth of instabilities 0.25("p"(1.3. The results of numerical simulations described in [8] for the order parameter s are shown in Fig. 1. Fig. 1 displays the results of the numerical simulation for the evolution of s ((3.8)}(3.9)). We display the auxiliary "eld s in units of fm\, the classical "elds U in units of fm\ and the proper time in units of fm (1 fm\"197 MeV) for two simulations, one with an instability (p" "!1) and O one without (p" "0). O We notice that for both initial conditions, the system eventually settles down to the broken symmetry vacuum result as a result of the expansion. We also considered a radial expansion and obtained similar results [9]. In the radial case, the outstate was reached earlier, but the number of oscillations where s became negative was similar. To determine the single particle inclusive pion spectrum we go to an adiabatic basis and introduce an interpolating number operator which

Fig. 1. Proper time evolution of the s "eld for two di!erent initial values of p.

68


interpolates from the initial number operator to the out number operator. Introduce mode functions f which are "rst order in an adiabatic expansion of the mode equation I dy /dt"u , f "e\ WIO/(2u , I I I I 1

(g, x , q), [dk](exp(ikx)f (q)a (q)#h.c.) . , q

(3.13) (3.14)

In terms of the initial distribution of particles n (k) and b we have n (q),f (k , k , q)"1aR(q)a (q)2"n (k)#"b(k, q)"(1#2n (k)) , (3.15) I E , I I where b(k, q)"i( f (Rf /Rq)!(Rf /Rq)f ), n (q) is the interpolating number density. The distribution I I I I I of particles is f (k , k , q)"dN/p dx dk dg dk . E , , , E Changing variables from (g, k ) to (z, y) at a "xed q we have E dN dN " " p dz dx Jf (k , k , q) E , E , dk p dy dk ,

(3.16)

"A dk f (k , k , q)" f (k , k , q)kI dp . , E E , E , I

(3.17)

To compare our "eld theory calculation with some standard phenomenological approach, we considered a hydrodynamic calculation with boost invariant kinematics and determined the spectrum assuming that at hadronization the pions where at the breakup temperature ¹"m (as p well as ¹"1.4m ), with the distribution given by the Cooper}Frye}Schonberg formula [11] p dN dN " " g(x, k)kI dp . (3.18) E I dk p dk dy , Here g(x, k) is the single-particle relativistic phase-space distribution function. When there is local thermal equilibrium of pions at a comoving temperature ¹ (q) one has g(x, k)"g +exp[kIu /¹ ]!1,\ . (3.19) p I The comparison is shown in Figs. 2 and 3. We therefore "nd that a non-equilibrium phase transition taking place during a time evolving quark}gluon or hadronic plasma can lead to an enhancement of the low-momentum distribution of pions.

3.2. Determination of the ewective equation of state Equation of state is obtained in the frame where the energy momentum tensor is diagonal } we are already in that boost invariant frame ¹ "diag+e, p , p , . IJ E ,


69

Fig. 2. Single-particle transverse momentum distribution for p"!1 initial conditions compared to a local equilibrium hydrodynamical calculation with boost invariance. Fig. 3. Single-particle transverse momentum distribution for p"0 initial conditions compared to a local equilibrium hydrodynamical calculation with boost invariance.

Fig. 4. Equation of state p/e as a function of q for the massless p model where we start from a quench.

When we have massless goldstone pions in the p model (H"0) then s goes to zero at large times. In the spatially homogenous case 1¹ 2"e, 1¹ 2"pd . GH GH The equation of state becomes p"e/3 at late times even though the "nal particle spectrum is far from thermal equilibrium as is seen in Fig. 4.

70


3.3. Dephasing and looking for DCCs As we have shown in [10], dephasing justi"es the replacement of the exact Gaussian o by its diagonal elements. At large-N or in mean "eld theory the density matrix is a product of Gaussians in space: I 1u "o "u 2"(2pm)\exp+!(p/8m)(u !u )!(1/8m)(u #u ), . (3.20) I I I I I I I I I I After a short while because of dephasing, the Gaussian distribution o! the diagonal u "u is I I strongly suppressed: m /p +( /2kn(k));m . I I I This is shown in Fig. 5. We "nd no support for `SchroK dinger cata states in which quantum interference e!ects between the two classically allowed macroscopic states at v and !v can be observed. An ensemble may be regarded as a classical probability distribution over classically distinct outcomes. The particle creation e!ects in the time-dependent mean "eld give rise to strong suppression of quantum interference e!ects and mediate the quantum to classical transition of the ensemble. If we project the density matrix onto an adiabatic number basis, we can reconstruct classical "eld con"gurations from the diagonal density by replacing the "eld operator a(k) by a(k)P[n(k)]e (I , with n(k) obtained by throwing dice on the density matrix and being randomly chosen between 0( (2p. Typical "eld con"gurations as a function of r (averaging over angles) are shown in Fig. 6.

Fig. 5. The Gaussian o for k"0.4 from Ref. [10] illustrating the strong suppression of o!-diagonal components due to dephasing. Fig. 6. Four typical "eld con"gurations drawn from the same classical distribution of probabilities.


71

4. Inclusive dilepton production and Schwinger's closed time-path formalism Schwinger's CTP formalism is designed to allow one to calculate expectation values of operators in the initial density matrix. One quantity we are interested in for obtaining an e!ective hydrodynamics is the expectation value of the energy momentum tensor 1in"¹IJ(x)"in2,(e#p)uIuJ!pgIJ ,

(4.1)

where ¹IJ(x) is the "eld theory energy momentum tensor. Also by Fourier transforming this energy momentum tensor and looking in a comoving frame, we can ask how much energy is in the `freea part of various components and de"ne an equivalent number of quanta by dividing by u for each I species. If we consider the inclusive production of electron}positron pairs the probability amplitude is 1e\(k, s)e>(k, s)X " i2"1X"bd "i2 . IQ IY QY The inclusive distribution function for dileptons: (E /m)(E /m)(dN/[dk][dk]),1i"d>b>bd "i2 . I I IY QY I Q I Q IY QY Using the relations between b, d to W and the free òuta "elds we obtain

1i " dx dx dx dx e IV\V+u> W(x ),+W>(x )u , I Q I Q ;e IYV\V+v> W(x ),+W>(x )v , " i2 . IY QY IY QY Now using the weak asymptotic condition [12] that W" "ZW , (4.2) R inside of matrix elements as well as the equation of motion of the spinors and the identity

R dF "F(t )!F(t ) . dt R We obtain

(4.3)

1e\(k, s)e>(k, s)X " P P 2

"iZ\ dx dx e IV>IYV u D 1X"T+W(x )WM (x ),"P P 2 DM v . I Q IY QY

(4.4)

Squaring this amplitude and summing over X we obtain

dN E E I I (k, k; s, s)" dx dx dx dx e IV\Ve IYV\Vv D u D IY QY V I Q V m m [dk][dk] 1P P "TH+W(x )WM (x ),T+W(x )WM (x ),"P P 2 DM u DM v . V I Q V IY QY

(4.5)

72


The matrix element involved here, 1P P "TH+W(x )WM (x ),T+W(x )WM (x ),"P P 2 (4.6) is precisely the type of Green's function that is obtained from the generating functional of Schwinger's CTP formalism. The Lagrangian we will use to determine this four-point function is the O(4) linear p model # electrodynamics. This Lagrangian has three pieces: The mesons form an O(4) vector U"(p , p). This strongly G interacting Lagrangian is given by ¸ "! (䊐#s) #s/4j#sv#Hp , G G s"j(U ) U!v) .

(4.7)

To this we add the free lepton and photon Lagrangian ¸ "!F FIJ!(1/2a)(R ) A)#WM [ic RI!m]W . IJ I The interaction of the photons with the pion plasma and the leptons is given by e ¸ [ , A , W, WM ]" ( # )A AI#e ( R ! R )AI!eWM cIWA #LA . I I I I G I 2

(4.8)

(4.9)

If we treat the electromagnetic interactions perturbatively in e and the pions in the mean "eld approximation we obtain the graph shown in Fig. 7. The inverse propagators in the LSZ representation lop o! the external legs and put the leptons on mass shell. One is left with (E /m)(E /m)(dN/dkdk)"M (k, s; ks)=IJ(k, k) , I IY IJ where

(4.10)

MIJ(k, s; ks)"v (k, s)cIu(k, s)u (k, s)cJv(k, s) , = (k, k),= #= #= IJ IJ IJ IJ

Fig. 7. Leading contribution from the plasma to dilepton production. The four fermion graph is to be evaluated using the matrix CTP Green's functions.


73

"ie dy dy dz dz e I>IYX\W[D(y , y )PNH(y , z )D (z , z ) IN HJ #D(y , y )PNH (y , z )D (z , z )#D(y , y )PNH (y , z )D(z , z )] . IN HJ IN HJ If we want the invariant mass distribution function when M"q,

q"k#k ,

we obtain dN/dq"2 qdN/dM dq"R (q)=IJ(q) , IJ where

(4.11)

[dk] [dk] d(q!k!k)¸M (k, k) IJ E E I IY 1 2p 4m 2m " 1! 1# (qIqJ!gIJq) . (2p) 3 q q

R (q), IJ

(4.12)

If we were doing an ordinary perturbation theory calculation analytically we could use the translational invariance of the polarization tensor

P (y , z )" [dq]e\ OW\XP (q) IJ IJ

(4.13)

and the representation of the free photon propagator in Feynman gauge

g IJ D (z , z )" [dk]e\ IX\X $IJ k#ie

(4.14)

to obtain = (k, k)"!ie IJ

(2p)d(0) P (q) IJ q

"!iee\-channel M &2m , q (300 MeV. p , This would be visible by CERES if k"60 MeV . , In our calculations we have ignored the possible important e!ects of direct two body scattering in the plasma which arise only in next order in the 1/N approach. A similar enhancement seen by Boyanovsky et al. [15] in photon spectrum. Another problem for us is that our result is in#uenced by large "nite time corrections which are apparent for the free pion gas.


81

Acknowledgements The work presented here was done in collaboration with Emil Mottola, Yuval Kluger, Volker Koch, Juan Pablo Paz Ben Svetitsky, Judah Eisenberg, Paul Anderson and John Dawson. This work was supported by the Department of Energy.

References [1] P.A.M. Dirac, Proc. Camb. Phil. Soc. 26 (1930) 376; A.K. Kerman, S.E. Koonin, Ann. Phys. 100 (1976) l332; R. Jackiw, A.K. Kerman, Phys. Lett. A 71 (1979) 158; F. Cooper, S.-Y. Pi, P. Stancio!, Phys. Rev. D 34 (1986) 3831. [2] Y. Kluger, V. Koch, J. Randrup, X.N. Wang, Phys. Rev. D 57 (1998) 280. [3] J. Schwinger, J. Math. Phys. 2 (1961) 407; K.T. Mahanthappa, Phys. Rev. 126 (1962) 329; P.M. Bakshi, K.T. Mahanthappa, J. Math. Phys. 4 (1963) 1; 4 (1963) 12; L.V. Keldysh, Zh. Eksp. Teo. Fiz. 47 (1964) 1515; [Sov. Phys. JETP 20 (1965) 1018]. [4] G. Zhou, Z. Su, B. Hao, L. Yu, Phys. Rep. 118 (1985) 1; R.D. Jordan, Phys. Rev. D 33 (1986) 44; E. Calzetta, B.L. Hu, Phys. Rev. D 35 (1987) 495. [5] K. Wilson, Phys. Rev. D 7 (1973) 2911; J. Cornwall, R. Jackiw, E. Tomboulis, Phys. Rev. D 10 (1974) 2424; S. Coleman, R. Jackiw, H.D. Politzer, Phys. Rev. D 10 (1974) 2491. [6] R. Root, Phys. Rev. D 11 (1975) 831; C.M. Bender, F. Cooper, G.S. Guralnik, Ann. Phys. 109 (1977) 165; C.M. Bender, F. Cooper, Ann. Phys. 160 (1985) 323. [7] J.D. Bjorken, Int. J. Mod. Phys. A 7 (1992) 4189; J.P. Blaizot, A. Krzywicki, Phys. Rev. D 46 (1992) 246; K. Rajagopal, F. Wilczek, Nucl. Phys. B 404 (1993) 577. [8] F. Cooper, Y. Kluger, E. Mottola, J.P. Paz, Phys. Rev. D 51 (1995) 2377; Y. Kluger, F. Cooper, E. Mottola, J.P. Paz, A. Kovner, Nucl. Phys. A 590 (1995) 581c; F. Cooper, Y. Kluger, E. Mottola, Phys. Rev. C 54 (1996) 3298. [9] M.A. Lampert, J.F. Dawson, F. Cooper, Phys. Rev. D 54 (1996) 2213. [10] S. Habib, Y. Kluger, E. Mottola, J.P. Paz, Phys. Rev. Lett. 76 (1996) 4660; F. Cooper, S. Habib, Y. Kluger, E. Mottola, Phys. Rev. D 55 (1997) 6471. [11] F. Cooper, G. Frye, E. Schonberg, Phys. Rev. D 11 (1975) 192. [12] H. Lehmann, K. Symanzik, W. Zimmerman, Nuovo Cimento 1 (1955) 205. [13] L.D. McLerran, T. Toimela, Phys. Rev. D 31 (1985) 545. [14] P.V. Ruuskanen, in: E.H.H. Gutbrod, J. Rafelski (Eds.), Particle Production in Highly Excited Matter, Plenum Press, New York, 1993, p. 593. [15] D. Boyanovsky, H.J. de Vega, R. Holman, S. Kumar, Phys. Rev. D 56 (1997) 5233.


Dual con"nement of grand uni"ed monopoles? Alfred Schar! Goldhaber Institute for Theoretical Physics, State University of New York, Stony Brook, NY 11794-3840, USA

Abstract A simple formal computation, and a variation on an old thought experiment, both indicate that QCD with light quarks may con"ne fundamental color magnetic charges, giving an explicit as well as elegant resolution to the `global colora paradox, strengthening Vachaspati's SU(5) electric}magnetic duality, opening new lines of inquiry for monopoles in cosmology, and suggesting a class of geometrically large QCD excitations } loops of Z(3) color magnetic #ux entwined with light-quark current. The proposal may be directly testable in lattice gauge theory or supersymmetric Yang}Mills theory. Recent results in deeply inelastic electron scattering, and future experiments both there and in high-energy collisions of nuclei, could give evidence on the existence of Z(3) loops. If con"rmed, they would represent a consistent realization of the bold concept underlying the Slansky}Goldman}Shaw `glowa model } phenomena besides standard meson}baryon physics manifest at long distance scales } but without that model's isolable fractional electric charges. 1999 Elsevier Science B.V. All rights reserved. PACS: 12.38.Av; 14.80.Hv; 11.27.#d Keywords: Quantum chromodynamics; Magnetic monopoles; Cosmic strings

1. Introduction A quarter century ago, the simultaneous and independent discoveries by Gross and Wilczek [1] and by Politzer [2] that quantum chromodynamics [QCD] is asymptotically free made this theory instantly what it still is } the unique candidate theory for describing the structure and interactions of baryons, as well as the mesons produced when baryons collide. For length scales below 0.1 fm and energy scales above 1 GeV, phenomena may be described accurately by perturbative techniques in terms of elementary quarks and gluons. At longer distances and lower energies, the most useful degrees of freedom become the baryons and mesons themselves, while the connection between these two regimes is less well determined because of the calculational di$culty associated with nonperturbative QCD. Nevertheless, a variety of experimental and theoretical approaches have produced so many successes that it would seem natural to assume there is little or no room for 0370-1573/99/$ - see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 9 ) 0 0 0 1 4 - 9

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A.S. Goldhaber / Physics Reports 315 (1999) 83}94

surprising new phenomena in QCD. That is especially plausible for the perturbative regime, so if there are surprises lurking they most likely will be found at long-distance scales. A rare if not unique proposal in this direction, which constitutes perhaps the boldest enterprise with Richard Slansky's name on it, is the paper of Slansky, Goldman, and Shaw [SGS] [3] suggesting explicit departures from naive QCD expectations at long-distance scales. Their work was stimulated by experimental indications of isolated electric charge with value or that of an electron charge. As Gordon Shaw explains [4], assuming that the manifest SU(3) gauge symmetry of QCD is reduced by a Higgs mechanism at very long-distance scales to SO(3), one may envision isolated SO(3) singlets made of two quarks, and therefore carrying fractional electric charge. While seeking such objects in the laboratory remains a worthwhile challenge, there are serious grounds to be hesitant about the explicit SGS proposal. The reason is that a Higgs mechanism is easy to formulate in a regime where the gauge coupling of the theory is weak, as in the standard model of electroweak interactions, but becomes very hard to interpret if the coupling is strong, which inevitably would be true for QCD on the scale they had in mind. There was little choice about this for SGS, because any shorter distance scale or higherenergy scale with such phenomena would have been prohibited by existing theoretical and experimental knowledge. Even now there is no experimental con"rmation of fractional electric charge, though searches continue [4], nor of any other long-range QCD e!ect. This paper presents a proposal which has signi"cant features in common with SGS, namely, new long-range phenomena beyond ordinary hadron dynamics (including con"nement of fundamental magnetic monopoles), but does not imply fractional electric charge. If the proposal turns out to have merit, then it could well be viewed as a vindication of the essence of SGS. Certainly the intellectual structure they developed was an important in#uence on my thinking. The presentation involves both `pusha and `pulla heuristic arguments, i.e., reasons to suspect the existence of new phenomena as well as appealing consequences which would follow if they occurred, but does not include a proof that they are inevitable. Because of the wide range of application, there likely will be a number of ways to test the proposal, including several outlined later. The focus begins with particles dual to quarks, namely, fundamental magnetic monopoles carrying both ordinary and color magnetic charge. Before plunging into the proposal, we should review some long-range e!ects which already are expected, resulting from hybridization between di!erent scales. If a region of space is su$ciently hot, then the temperature ¹ sets a scale which invokes asymptotic freedom, and thus allows one to describe the properties as if dealing with a gas of free quarks and gluons, commonly known as the `quark}gluon plasmaa. Clearly long-distance correlations in this regime should look quite di!erent from those at ¹"0. Thus, long-distance phenomena are altered in a predictable way, but in a high-energy rather than low-energy regime. Something similar should happen if, for example, compression of cold nuclear matter, as in the interior of a neutron star, were to produce baryon density much greater than that in normal nuclei. This time the high density gives a short-distance and therefore high-energy scale which implies asymptotic freedom and a change in long-distance correlations. Neither of these e!ects would be a surprise. Something a bit closer was the proposal of Ja!e [5] that a six-quark system of strangeness S"2 might be stable against decay to two K particles. This in turn raises a possibility discussed by Witten [6] that electrically neutral strange matter might be stable or metastable. However, even this e!ect if it occurred would be a consequence of relatively


85

short-range interactions. Something still closer to what follows is the suggestion [7] that nuclei might have stable or metastable toroidal forms, where a quark-containing tube of color-magnetic #ux is bent into a closed curve, a structure which might exhibit rigidity and incompressibility as well as tension.

2. Heuristic arguments for monopole con5nement 2.1. Gauge invariance Though the existence of a magnetic monopole has yet to be con"rmed by experimental observation, even as a concept this object repeatedly has played the role of an intellectual aqua regia, exposing profound aspects of structure in physical systems. The most noted example is Dirac's realization [8] that the existence of isolable monopoles would require the quantization of electric charge q and magnetic charge g, through the quantum condition that the product of q with g is proportional to an integer. Dirac monopoles ìnserteda into QED give a model for con"nement, because emanating from an elementary pole inside a Type II superconductor with its electron-pair condensate must be two strings of superconductor-quantized magnetic #ux, each terminating only on an antipole. If one pair of pole and antipole were slowly separated, clearly the two strings coming out of the pole would terminate on the antipole, implying a con"ning string tension holding the two together. By analogy, if the vacuum of QCD without light quarks comprised a monopole condensate [9,10], this then would con"ne heavy quarks. However, if there are elementary quarks light on the scale of K , then there is no con"nement of heavy quarks: Instead pair creation of light quarks allows /!" heavy-quark-containing mesons to be separated with no further energy cost. A general argument of 't Hooft [11] for QCD without light quarks shows that either heavy elementary quarks or heavy fundamental monopoles should be con"ned, but not both. If this argument still applied in the presence of light quarks, then monopole con"nement would be a triviality. In any case this makes it clear that there would be nothing obviously inconsistent about such con"nement. To understand why it might be expected, let us examine in the more familiar superconductor case the issue of screening, and what charges can or cannot be screened. In electrodynamics it is useful to distinguish two di!erent kinds of conserved charge, local or Gauss-law charge and Aharonov}Bohm [AB] or Lorentz-force charge [12]. Although the local charge of an electron-quasiparticle is completely screened inside a superconductor, the AB charge cannot be screened [13,14], because of the reciprocity requirement that an AB phase of p must occur whether a quasiparticle is di!racted around a #uxon (i.e., a superconductor quantum of #ux) or a #uxon is di!racted around a quasiparticle. If the e!ect on the #uxon is to be described by a local interaction, evidently the AB charge is not screened. Now let us look at the same issue for a fundamental monopole in QCD. The monopole may be characterized as the source of a Dirac string carrying color magnetic #ux which would produce an AB phase 2p/3 for a fundamental quark di!racted around it. Of course, in addition to this color #ux there must be an ordinary magnetic #ux in the string yielding a phase 4p/3 mod(2p). The fractional color #ux in the string implies that there must be a net nonzero color magnetic #ux coming out of the pole. A monopole whose Dirac string would carry full 2p color #ux has no such

86


consequence, because that could be exactly compensated by an àdjointa monopole made from a classical con"guration of purely SU(3) gauge "elds. Thus, in the sense just de"ned, adjoint monopoles can be screened but fundamental monopoles cannot (a suggestive analogue to what happens with adjoint gluons and fundamental quarks). In the absence of dynamical quarks, this lack of screening may not matter, because vacuum #uctuations in the form of loops carrying #ux 2p/3 occur easily on arbitrarily large length scales, so that the magnetic charge is not de"nable as an eigenvalue. In this sense, it may be screened just like electric charge in a normal metal, i.e., with mean value zero but such large #uctuations that it is not de"ned as a sharp quantum observable. Thus the monopole charge, rather than being screened or compensated, may be hidden in much the same way a needle becomes invisible inside a haystack. Here is another perspective: In the theory with only adjoint "elds, such as those of the gluons, the gauge symmetry is SU(3)/Z(3), so that an arbitrarily thin tube of Z(3) magnetic #ux would be invisible even by the AB e!ect to all elementary excitations, hence could not excite the vacuum, and therefore need not carry an observable energy per unit length, as would have to be true for an observable string. On the other hand, once quarks are present, there could be a nonvanishing string tension for loops of Z(3) color magnetic #ux, so that geometrically large quantum #uctuations of these loops should be suppressed. A reason for suspecting this is that now the AB e!ect would make even the thinnest tube visible for those quark trajectories which surround the tube. Thus, the fractional color magnetic charge could become sharp, meaning that an observable color magnetic "eld, con"ned to a tube of "xed radius, emanates from the monopole out to in"nity. More formally, because now there are particles in the fundamental representation of SU(3), the full gauge symmetry applies, and so a nonzero Z(3) color magnetic #ux out of a fundamental monopole is at least potentially observable. 2.2. Ideal experiment Here is a thought experiment suggesting the same conclusion. Imagine a hadron such as a proton at rest near an SU(5) monopole, with its ordinary as well as color magnetic charge. If a deeply inelastic electron scattering sends a quark out of the proton with very high momentum parallel to that of the incident electron, then the quark's evolution in the beam direction can be described perturbatively for a time proportional to that momentum, which means the ordinary magnetic "eld of the monopole will de#ect it in such a way that only a fraction of a quantum of angular momentum will be transferred to the quark. This is inconsistent with conservation and quantization of angular momentum [15]. The analysis also can be carried out in the rest frame of the fastest "nal hadron. In this frame a perturbative computation is accurate for a "xed time of order 1 fm/c. However, the magnetic "eld of the monopole #ashes by the quark in a much shorter time because of the Lorentz contraction of the "eld con"guration, and therefore again there is a de"nite, but fractional transfer to the quark of angular momentum projected along the beam direction. One way to restore consistency is to assume that there is also a spherically symmetric color magnetic "eld, so that the combined "elds always transfer an integer number of angular momentum units to the quark. However, that assumption directly contradicts the most basic understanding of QCD, which requires a mass gap for color-carrying excitations, so that a longrange, `classicala, isotropic, color-magnetic "eld is impossible. How can these two requirements, of


87

nonscreening and yet no isotropic long-range "eld, be reconciled? An obvious if not unique way to avoid the dilemma is by èscape into asymptotic freedoma: Color "elds make sense in the high-energy, short-distance, perturbative regime, so if the magnetic #ux comes out in a tube with radius of scale 4K\ it is consistent with knowledge about the low-energy behavior of the theory, /!" and at the same time satis"es the requirement of nonscreening. Evidently such a tube must have a "nite tension, so that the energy of a pole}antipole pair connected by the tube must rise linearly with separation, and this implies con"nement of fundamental monopoles. From the viewpoint of the deeply inelastic `thought experimenta, why shouldn't the monopole con"nement argument apply even if there are no dynamical quarks? In that case heavy èxternala quark sources certainly are con"ned, and the failure of angular momentum quantization for a single pole}quark pair is acceptable, as there is never a single isolated quark moving in the "eld of the monopole. While the above arguments might be appealing, they surely do not constitute a proof of monopole con"nement. The reason is that even with light quarks it may be that large loops of magnetic #ux, at least of a certain cross sectional radius, still have arbitrarily low energy, in which case they would be part of the vacuum structure rather than physical excitations. Then the net #ux out of a monopole again would be hidden by vacuum #uctuations. However, because the quarks would be sensitive to arbitrarily thin tubes even with Z(3) #ux, there is now a much stronger constraint, from below as well as above, on the acceptable radii for #ux tubes with very low energy. Both because the thought experiment was the germinating element in my own thinking on this subject, and because more careful examination could tend either to strengthen or to weaken the argument, it seems worthwhile to focus more explicitly on the wave function evolution entailed by this process. In the presence of gauge "elds, the conventional (nongauge-invariant) momentum of an object whose charges couple to these "elds becomes unde"ned. Thus, in the directions transverse to the very high momentum of the struck quark, it makes no sense to think about the momentum of the quark by itself. However, the correlated wave function of the quark and the associated slower remnants might have a well-de"ned wave function in transverse momentum p , 2 a wave function which initially would be strongly peaked at p +0 and azimuthal angular 2 momentum about the beam direction also zero. Then the essential idea is that, absent any contribution from color magnetic "elds, the only e!ect feeding some change in this azimuthal angular momentum would be coming from the scattering of the fast quark on the ordinary magnetic "eld of the monopole. A fractional value for this angular momentum transfer gives the conclusion that something is inconsistent about this picture, and leads by elimination of alternatives to the inference that an observable string of color magnetic #ux emanates from the monopole. If we accept that inference, how do we "nd consistency restored? As an example, imagine that the observable string comes out of the monopole in the direction parallel to the fast quark momentum. We still may use gauge invariance to place the Dirac string of ordinary plus color #ux anywhere we like, and thus may choose it along the observable color string. In this case, for all except those trajectories which penetrate the observable, "nite-thickness string, the e!ective "eld is just that of a pole which is one end of a solenoid with ordinary magnetic #ux such that a quark going around it acquires a fractional phase 2p/3 mod 2p. Evidently, mesons or baryons generated by fragmentation of the fast quark will always have integer azimuthal angular momentum, but nevertheless the initial e!ect of fast passage of the quark by the monopole will be to generate a fractional change in the net

88


angular momentum of the entire system interacting with the pole, something now allowed because an observable string with fractional magnetic #ux is present.

3. Consequences and applications Now let us look at how fundamental monopole con"nement would reorient perspectives on a variety of issues. 3.1. Paradox of `global colora A number of authors addressed the problem of generalizing a collective}coordinate quantization technique, accepted as describing the electric or `dyona charge of an SU(2) monopole, to the case of the SU(5) monopole [16}19]. They found that for the unbroken SU(3) of color the dyon charge of an isolated pole is not de"ned } an e!ect reminiscent of spontaneous symmetry breaking as in ferromagnetism. Evidently if monopoles with fundamental color charge are con"ned, this problem simply disappears. A more general and straightforward comment is that, with or without monopole con"nement, the paradox is ill-posed, because the collective}coordinate method has been used to quantize zero modes of the monopole placed in a perturbative QCD vacuum, which de"nitely is an incorrect description of the lowest-energy degrees of freedom on length scales large compared to K\ . Thus, while monopole con"nement eliminates the problem at the very beginning, the /!" signi"cance of that resolution perhaps is diminished because there might well not be such a problem if the right vacuum were understood well enough to be implemented for the analysis. 3.2. Electric}magnetic duality in a grand unixed model Recently Vachaspati [20,21] has described a remarkable duality of SU(5), clearly relevant for any grand uni"ed theory. The fundamental monopole is part of a family of tightly bound states, with magnetic charges 1, 2, 3, 4, and 6 times the fundamental charge. These "ve states can be identi"ed as dual partners of the "ve fundamental fermions in SU(5), three quarks, a lepton, and a neutrino. There is a possibly deep or possibly just technical issue, that the charge-2 state should be identi"ed with an antiquark. There are two other di$culties. First, the monopoles appear to be spinless, while the fermions of course have spin-. This problem arose already with the original Montonen} Olive proposal of duality between monopoles and gauge bosons [22,23], and eventually found two resolutions. One is to introduce supersymmetry, so that both monopoles and the dual elementary particles come in families with the same range of spins [24}28]. The other, acknowledging the possibility in principle of making a perfect correspondence through supersymmetry, is to be satis"ed with what might be called `virtual dualitya } a symmetry applied to all properties except spin. Whichever approach one prefers, with respect to this issue Vachaspati's system is in the same category as the older examples. The other di$culty [20,21] is that the e!ective long-range couplings of the monopoles and their dual partners are identical, except that quarks are con"ned, whereas previous discussions suggested that the colored monopoles are not. For this reason Vachaspati considered introducing the con"nement essentially by hand. The argument above that the monopoles which nominally carry


89

nontrivial Z(3) color magnetic charge are automatically con"ned gives a way to perfect Vachaspati's duality, lending additional interest to pursuing it further. One side note: Con"nement could lead to loosely bound `baryonsa, but these then could collapse to the tightly bound `leptonsa already identi"ed in Vachaspati's scheme. Clearly, this is di!erent from the separate baryon and lepton conservation laws which apply at low energy scales, but as one expects those laws not to hold for particles on energy scales approaching the monopole masses this may well be a consistent result. Monopole evolution in cosmology Monopoles formed on a mass scale signi"cantly higher than the mass scale for in#ation would have disappeared during in#ation [29]; indeed, that is one of the attractive features of in#ationary models. However, lighter monopoles would need some other mechanism to explain why we do not see abundant evidence of their existence today. Many such mechanisms have been proposed, up to quite recent times. One possibility is that the dynamics at some intermediate era between monopole formation and the present would make the poles unstable, allowing them to disappear, even though any remnant which did survive would be stable now [30,31]. If monopoles were created at some early epoch and not swept away meanwhile, then the only way to explain their scarcity today would be by con"nement, exactly the phenomenon discussed here. How would that work? If monopoles were formed above the QCD phase transition expected at a temperature of order K , then con"nement below that transition would result in attachment /!" of Z(3) strings to each pole, either a single outgoing 2n/3 string, or two outgoing !2p/3 strings, with the opposite arrangement for antipoles. A pair connected by a single string likely would have disappeared by now, thanks to dissipative forces leading to gradual collapse and annihilation. On the other hand, a large loop with alternating negative and positive #ux connecting alternating pole and antipole could be much more durable. This kind of `cosmic necklacea, with the poles as `beadsa, was suggested by Berezinskii and Vilenkin [32] as a possible source of the highest-energy component of the cosmic-ray spectrum, through occasional annihilations of poles and antipoles, which might for example slowly drift together by sliding along the string. The evolution of networks of such strings is an interesting and nontrivial problem, which could be studied once the basic couplings associated with string crossings were determined. In particular, in principle a `fusiona of three strings converging together should be possible, which would allow three monopoles to be connected to each other in a dual version of a baryon. However, if this could happen easily then the problem of too many monopoles would be restored, so a crucial question is whether there is a substantial inhibition of such fusion.

4. New phenomena in QCD at accessible scales 4.1. Theoretical aspects Even though it is consideration of heavy, fundamental monopoles and their interactions which has led here to the suggestion that they would be held together by Z(3) color #ux strings, that statement clearly has a consequence for phenomena at much lower scales than the monopole mass. It means that even in the absence of such poles QCD must support excitations consisting of loops

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of color magnetic #ux, with the mass of a loop being proportional to its circumference. The loops would be unstable against shrinkage, but would give an interesting and nontrivial structure of QCD excitations on a length scale large compared to 1/K . This is reminiscent of the /!" Slansky}Goldman}Shaw proposal to explain experimental reports of fractional electric charge [3]. As mentioned earlier, they noted that if a Higgs mechanism at energy scales below, or length scales above, the scale associated with K could operate to reduce SU(3) of color to SO(3) of `glowa, /!" then diquarks could exist in isolation, and of course would carry fractional electric charge. Shortly after, Lazarides, Sha", and Trower [LST] [33] observed that such a Higgs e!ect automatically would imply con"nement of fundamental monopoles exactly like that argued above. As was also mentioned earlier, there is no natural starting point from which the phenomena of this particular Higgs mechanism could be deduced in a perturbative framework, the only recognized way to do it. This criticism applies equally to the deduction by LST. Of course, the fact that a conceivable route to a particular result turns out to be rocky and uncertain does not mean the result itself is necessarily wrong, only one still lacks evidence that it is right. Here the issue has been approached from the other end, and fundamental monopole con"nement derived. This does not necessarily imply the isolability of fractional electric charge or the screening of some QCD color-electric "elds, but it certainly does say there must be a new feature of QCD at large length scales, namely, loops of color-magnetic #ux, just as indicated by LST. Without light quarks, heavy quark con"nement implies loops of color-electric #ux, so familiar pictures would not be changed so enormously, just `dualizeda. This means that the change in structure of QCD as the mass of light quarks passes from above to below K would be quite subtle: Above there would be /!" at least metastable color-electric but substantial suppression of color-magnetic strings (more accurately, very low magnetic string tension), and below something more like the opposite would be true. The meaning of con"nement or non-con"nement needs a bit more attention. In terms of a four-dimensional euclidean path integral, con"nement is associated with exponential suppression, with the area of an appropriate loop appearing in the exponent, as opposed to e!ects associated with widely separated "nite-mass excitations, in which case only the length (perimeter) of the loop appears. For QCD with dynamical quarks, su$ciently large loops must exhibit a perimeter law, but the coe$cient of the perimeter term itself falls exponentially with quark mass because the tunneling leading to quark pair creation is exponentially suppressed. Thus a visible transition on a "nite lattice from area to perimeter law occurs at some "nite mass, presumably of order K , and should be rather smooth. For the proposed monopole con"nement, with mono/!" poles expected to be extraordinarily massive, the breaking of strings by monopole pair creation should be impossibly rare for observation on any "nite lattice. If the magnetic strings only exist for "nite quark mass, it becomes a delicate question exactly how the string tension depends on that mass. However, again one would expect a smooth transition, with the maximum tension approached for quark mass below K . This leads to the amusing conclusion that fundamental /!" dyons carrying both monopole and quark charges might exhibit an e!ective con"nement with very weak dependence on quark mass. If all this were con"rmed, it would be a vindication of the essential claim of SGS for nontrivial manifestations of fundamental QCD degrees of freedom at large length scales. These colormagnetic-#ux-loop excitations presumably should be an important class of what have been called `glueballsa, which likely would be drastically di!erent in character from what one would "nd in


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QCD without light quarks, and would NOT be pure glue, as the light quarks must be an essential part of their structure. As already stated, the fact that con"nement of fundamental monopoles would be equivalent to the existence of Z(3) magnetic #ux strings means that there is a way to test this proposal in familiar energy regimes of QCD. In particular, as lattice calculations grow steadily better at taking account of light quark degrees of freedom, it should become possible to study this issue on the lattice and obtain credible results. The best way to formulate the problem might be to insist a` la Wu and Yang [34] that along a straight line between monopole and antimonopole there is a gaugematching between vector potentials outside and inside the smallest plaquettes surrounding that line, involving one unit of Z(3) color #ux, and one Aharonov}Bohm unit of ordinary magnetic #ux. This means a phase of 2p associated with those plaquettes for u quarks encircling them, but 0 for d quarks. Of course, in all other respects there is a standard Dirac electromagnetic monopole vector potential for the pole}antipole system. All this implies, as stated earlier, that there must be a net Z(3) color magnetic #ux between monopole and antimonopole. If that #ux were observable and not hidden, then monopole con"nement would follow, and would be signaled by an area law for exponential suppression of monopole loops in the path integral, associated with the product of the separation between pole and antipole and the Euclidean time duration of that separation. There might be an analytic approach to determining whether or not con"nement occurs, a!orded by recent progress in studying supersymmetric nonabelian gauge theories [27,28,35,36]. In these theories it is often possible to make precise conjectures about the particle spectrum, and to verify the conjectures not by a direct proof but rather by subjecting the proposed forms to many di!erent consistency checks, and "nding that all are passed. To do this for our problem would require starting with at least an SU(5) theory (including a hypermultiplet containing quarks and leptons), and following an elaborate sequence of Higgs mechanisms to break the manifest symmetry down to SU(3) ;;(1) . This is surely much more complicated than anything which has been done so far with such systems, but might nevertheless be manageable. 4.2. Experimental aspects As physics is an experimental science, it surely is worth considering how the new kind of structure proposed here might be accessible to experimental observation. Up to this point in the paper, the main speculation has been the unproved proposal that Z(3) #ux loops may exist. To connect that with experiment entails more speculation. Conventional hadron collisions are not promising. First of all, any frequently occurring peculiar phenomena in such processes would have been noted already. Secondly, because Z(3) strings cannot break by creation of light-quark pairs, their coupling to conventional hadrons should be weak. This implies that they would not be generated easily in typical collisions. What couplings would be possible? Because u and d quark vacuum currents would circulate oppositely around the string, there should be a o -meson magnetic coupling `contact terma } i.e., only acting on sources which themselves overlap geometrically with the string. Thus a contracting string could release energy through emission of o mesons, but as these are rather massive there would be a poor match, between the likely small energy release in the contraction from one loop energy eigenstate to a lower one and the large mass of the emitted particle. All this implies a quite substantial lifetime for a large loop before collapse.

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Slow decay almost invariably goes together with low production rates, and helps to explain why even if they can exist Z(3) loops would not have leapt to our attention. Recently, experiments on deeply inelastic electron}proton scattering [37,38] have been interpreted as indicating a `hard-pomerona contribution to the reaction [39}41]. By familiar reasoning of Regge duality, such an e!ect should be associated with a new class of glueball excitations [39]. Could these new glueballs be the magnetic loops proposed here? If so, then it would not be strange if processes described by the hard pomeron also produced the loops. Perhaps detailed exclusive or semi-inclusive studies of such events would reveal structure related to the loops, formed as geometrically large and therefore high-energy excitations. It then becomes interesting to consider what kind of signal such an object would generate, but that is not easy to determine. All features of closed-string dynamics, many still obscure despite all the years of string studies, would appear to be relevant for the behavior of these Z(3) loops. Thus some caution is needed in guessing what should happen in these scattering processes. This time of course the coupling leading to production would be electromagnetic, but again would involve a contact interaction which at lowest order in momentum transfer would be to the anapole or toroidal moment of the #ux loop. This suggests that at the moment of appearance the loop would be quite small in size, but then could expand. If such primitive thinking covers the main features, then it becomes possible to suppose that in a suitable frame boosted along the beam direction there would be a fairly large isotropic ensemble of pions. Because of the decay energy mismatch mentioned earlier, the pions might be quite limited in their range of momenta. Such an e!ect could be quite striking, and very di!erent from typical results of deeply inelastic scattering. A di!erent picture, analogous to bremsstrahlung of photons, would be that with small probability virtual Z(3) loops exist in the neighborhood of the incident proton, and these are made real by the absorption of the highly virtual photon. To avoid enormous form-factor suppression, in a suitable Breit frame the initial and "nal momenta of the loop would both have to be large, implying Lorentz contraction which compensates for spatial oscillation of the phase factor in position space, an e!ect discussed some time ago for elastic scattering on deuterons [42]. If deeply inelastic electron}proton scattering gives an indirect hint of new long-distance dynamics in QCD, plus the potential to provide more direct evidence, then very high-energy nucleus}nucleus collisions at least have the possibility of generating such evidence in processes with large rate. By heating substantial volumes above the QCD phase-transition temperature, such collisions could permit formation of Z(3) loops, and if so their slow decay during the cooling process would give a characteristic signal, providing important evidence that quark}gluon plasma had formed. If the idea of slow decay is right, this would allow a loop to escape from the dense, highly excited formation region, and then to populate a small volume in pion momentum space with a large number of particles. While these thoughts about experimental signals are vague and sketchy, it may well be possible with further study to make more precise statements. What seems likely to be unchanged is the fact that a geometrically large object of high coherence, which decays slowly in small energy steps, will produce a signal di!erent from any more familiar system, including a large nucleus. [Of course, if the objects turned out to be at least stable against small perturbations, the signal would be even more striking.]


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5. Conclusions In the title the phenomenon proposed here is referred to as `dual con"nementa. Of course, this makes sense because familiar color-electric con"nement of heavy quarks is replaced by colormagnetic con"nement of heavy monopoles, but if the proposal is correct as stated then something deeper is at work. Usual discussions posit a duality between superconducting screening of one kind of charge and con"nement of the other. Here, however, the screening of color-electric charge is more powerful even than that by a superconductor, because for every heavy quark there is an attached light antiquark, exactly screening not only the local charge but also the Aharonov}Bohm charge. Total screening of the color electric charge carried by heavy quarks is the remnant of the heavy-quark con"nement which exists without light quarks. Thus the duality would be one between con"nement of fundamental color-magnetic monopoles and total screening of heavy quark color charge, not inconsistent with the familiar version but nevertheless clearly di!erent. If found, such a duality therefore would be something new. It is enticing to think that physics research is now at a stage where within a short time there might be direct evidence from a variety of directions on whether Z(3) strings occur in nature. Lattice gauge theory or supersymmetric gauge theory could give information, as could deeply inelastic electron scattering or high-energy nucleus}nucleus scattering. A positive answer would provide a "rm foundation for the theoretical and cosmological applications explored above. Perhaps even more satisfying if this happened would be the realization of a remarkable new consequence of QCD. This suggests a further challenge: Are there any other possible ways in which QCD could really give us a surprise? Not easy or obvious, but surely worth looking!

Acknowledgements This study was supported in part by the National Science Foundation. I have bene"ted over a period of time from conversations with Martin Bucher, Georgi Dvali, Edward Shuryak, Mikhail Stephanov, and Tanmay Vachaspati. Richard Slansky was a valued friend and colleague from student days on } about 40 years. Although we never collaborated on a paper, it was always a pleasure to discuss with him, and to experience his intelligent and discriminating enthusiasm for physics. His courage in facing physical challenges (in all senses!) was inspiring. Truly the word `glowa was as descriptive of his luminous personality as of the beautiful SGS idea. References [1] [2] [3] [4] [5] [6] [7] [8]

D.J. Gross, F. Wilczek, Phys. Rev. Lett. 30 (1983) 1343. H.D. Politzer, Phys. Rev. Lett. 30 (1983) 1346. R. Slansky, T. Goldman, G.L. Shaw, Phys. Rev. Lett. 47 (1981) 887. G.L. Shaw, unpublished (1998). R. Ja!e, Phys. Rev. Lett. 38 (1977) 195; 38 (1977) 617 (Erratum). E. Witten, Phys. Rev. D 30 (1984) 272. L. Castillejo, A.S. Goldhaber, A.D. Jackson, M.B. Johnson, Ann. Phys. 172 (1986) 371. P.A.M. Dirac, Proc. Roy. Soc. London, Ser A 133 (1931) 60.

94 [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] [38] [39] [40] [41] [42]

A.S. Goldhaber / Physics Reports 315 (1999) 83}94 S. Mandelstam, Phys. Rep. C 23 (1976) 245. G. 't Hooft, Nucl. Phys. B 138 (1978) 1. G. 't Hooft, Nucl. Phys. B 138 (1978) 1. A.S. Goldhaber, S.A. Kivelson, Phys. Lett. B 255 (1991) 445. B. Reznik, Y. Aharonov, Phys. Rev. D 40 (1989) 4178. A.S. Goldhaber, R. MacKenzie, F. Wilczek, Mod. Phys. Lett. A 4 (1989) 21. A.S. Goldhaber, Phys. Rev. B 140 (1965) 1407. A. Abouelsaood, Nucl. Phys. B 226 (1983) 309. A.P. Balachandran, G. Marmo, N. Mukunda, J.S. Nilsson, E.C.G. Sudarshan, F. Zaccaria, Phys. Rev. Lett. 50 (1983) 1553. P. Nelson, A. Manohar, Phys. Rev. Lett. 50 (1983) 943. P. Nelson, S. Coleman, Nucl. Phys. B 237 (1984) 1. T. Vachaspati, Phys. Rev. Lett. 76 (1996) 188. H. Liu, G.D. Starkman, T. Vachaspati, Phys. Rev. Lett. 78 (1997) 1223. C. Montonen, D. Olive, Phys. Lett. B 72 (1977) 117. P. Goddard, J. Nuyts, D. Olive, Nucl. Phys. B 125 (1977) 1. E. Witten, D. Olive, Phys. Lett. B 78 (1978) 97. H. Osborn, Phys. Lett. B 83 (1979) 321. A. Sen, Int. J. Mod. Phys. A 9 (1994) 3707. N. Seiberg, E. Witten, Nucl. Phys. B 426 (1994) 19; B 430 (1994) 485 (Erratum). N. Seiberg, E. Witten, Nucl. Phys. B 431 (1994) 484. A.H. Guth, Phys. Rev. D 23 (1981) 347. P. Langacker, S.-Y. Pi, Phys. Rev. Lett. 45 (1980) 1. G. Dvali, H. Liu, T. Vachaspati, Phys. Rev. Lett. 80 (1998) 2281. V. Berezinskii, A. Vilenkin, Phys. Rev. Lett. 79 (1997) 5202. G. Lazarides, Q. Sha", W.P. Trower, Phys. Rev. Lett. 49 (1982) 1756. T.T. Wu, C.N. Yang, Phys. Rev. D 12 (1975) 3845. K. Intriligator, N. Seiberg, Nucl. Phys. Proc. (Suppl.) 45BC (1996) 1. M. Shifman, Prog. Part. Nucl. Phys. 39 (1997) 1. C. Adlo! et al., Nucl. Phys. B 497 (1997) 3. J. Breitweg et al., Phys. Lett. B 407 (1997) 432. A. Donnachie, P.V. Landsho!, Phys. Lett. B 437 (1998) 408. D. Haidt, Proceedings of the Workshop on DIS Chicago, April 1997, AIP, 1997. N.N. Nikolaev, B.G. Zakharov, V.R. Zoller, JETP Lett. 66 (1997) 138. H. Cheng, T.T. Wu, Phys. Rev. D 6 (1972) 2637.

Physics Reports 315 (1999) 95 } 105

Is theoretical physics the same thing as mathematics?夽 George Chapline Lawrence Livermore National Laboratory, Livermore, CA 94551, USA

Abstract The growing realization that the fundamental mathematical structure underlying superstring models is closely related to Langlands' program for the uni"cation of mathematics suggests that the relationship between theoretical physics and mathematics is more intimate than previously thought. We show that quantum mechanics can be interpreted as a canonical method for solving pattern recognition problems, which suggests that mathematics is really just a re#ection of the fundamental laws of physics. 1999 Elsevier Science B.V. All rights reserved. PACS: 03.65.!w

1. Introduction One of the perennial mysteries of theoretical physics is why the laws of physics should so often have an elegant mathematical formulation } a circumstance often referred to as the ùnreasonable e!ectiveness of mathematicsa [5]. In fact, natural phenomena very often exhibit regularities that from the mathematical point of view seem to involve especially unique and beautiful mathematical structures. A good example of this is provided by the apparently strong likelihood [1] that the parity violation observed in the weak interactions has its origins in the naturally chiral nature of a supersymmetric E gauge theory in 10 dimensions. Indeed the implied involvement in elementary particle physics of the exceptional Lie group E provides a connection between fundamental physics and various remarkable mathematical structures including octonians, selfdual lattices, perfect error correcting codes, Kummer surfaces, and non-standard Euclidean

夽

Based on a talk given at the Dick Slansky Memorial Symposium, May 21, 1998. E-mail address: [email protected] (G. Chapline) 0370-1573/99/$ - see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 9 ) 0 0 0 1 5 - 0

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G. Chapline / Physics Reports 315 (1999) 95 } 105

spacetimes. Furthermore, the discovery of the anomaly structure of superstrings [2] implies [3] that the appearance of E in elementary particle physics is closely related to the theory of automorphic forms, which in turn suggests that the search for the fundamental laws in physics may be related to Langland's program for the uni"cation of mathematics. Apparent connections of the mathematical structure underlying superstring theories with the Monster sporadic group [4] and with number theory provide further evidence for a relationship between fundamental theoretical physics and Langlands' idea that there are fundamental structures related to automorphic forms which unify the various branches of mathematics. What is one to make of all this? My answer is that theoretical physics and mathematics are fundamentally the same thing. An obvious objection that one could easily imagine would immediately occur to anyone contemplating the idea that mathematics and theoretical physics are the same is that a great deal of mathematics was, or at least might have been, created by human beings to solve problems that have nothing to do with physics. For example, the di!usion equation might have been discovered not by a theoretical physicist seeking to describe the #ow of heat in a solid, but by an arbitrageur seeking to protect his employer's "nancial position in some asset. Indeed, it is often claimed now a days that mathematics is just a social}cultural}historical construction [6]. Arrayed against this view of mathematics as just a cultural phenomenon, though, is the powerful argument that the physical world as an embodiment of mathematical structures existed long before there were societies or even humanoids. There is little doubt, for example, that the planets were tracing out almost perfect ellipses long before there were any humans to observe the planets. Of course, with the exception of some isolated instances, such as the invention of the calculus, one might well question whether on the whole the mathematical regularities discovered in nature by theoretical physicists have any profound signi"cance for mathematics itself. In particular, a seemingly formidable obstacle to the idea that theoretical physics and mathematics are same is the fact that the core of theoretical physics } namely, quantum mechanics } has not played an important role in the development of pure mathematics. It is true that a number of "rst rate pure mathematicians have been fascinated with the formalism of quantum mechanics, and have made signi"cant contributions to the development of quantum mechanics. It is also noteworthy that the study of von Neumann algebras led to some surprising advances in knot theory [7] and supersymmetric quantum mechanics has been applied to topology [8]. Nonetheless on the whole the signi"cance of ordinary quantum mechanics for pure mathematics itself has remained obscure. Furthermore, and in my opinion, this is perhaps the most damaging criticism of the proposition that theoretical physics and mathematics are the same, ordinary quantum mechanics has not yet to this day found any practical applications unrelated to the microscopic properties of physical systems. On the other hand, there are some recent developments which promise to make it more di$cult if not impossible to take the position that quantum mechanics is just applied mathematics. First of all, it has been noticed that phase space quantization can be thought of as the geometric quantization of the =(R) group [9,10]. Secondly, the discovery that quantum computers could in principle be used to factor large numbers [11] and quickly search databases [12] o!er the hope that quantum mechanics will, in the not too distant future, be used to solve practical problems completely unrelated to microscopic physics. These developments suggest that, quite apart from its traditional role of providing a description for microscopic physical phenomena, quantum mechanics can also be given a purely mathematical interpretation. Our main objective in this paper will be


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to make a plausible case that indeed quantum mechanics might actually have been developed as a purely mathematical theory. As a bonus our argument will lead us to a provocative way of viewing the relationship between mathematics, theoretical physics, and neurobiology. Our basic argument is that quantum mechanics can be regarded as a fundamental theory of distributed parallel information processing and pattern recognition. It is worth noting in this connection that both Shor's [11] and Glover's [12] quantum mechanical algorithms depend on the unique ability of quantum mechanics to carry out parallel computations, and that Glover's database search algorithm can also be thought of as a pattern recognition algorithm. That quantum mechanics might be regarded as an underlying theory for conventional pattern recognition techniques, though, is to the author's knowledge a completely new idea, and will be the focus of this paper. However, some of the more sophisticated practitioners of the art of pattern recognition may have at least subliminally recognized that there might be such a connection. One of the fundamental problems of pattern recognition is feature vector quantization [13]; but despite an enormous e!ort the practical methods that have been developed to solve this problem remain largely ad hoc. Quantum mechanics, on the other hand, o!ers the possibility of a natural approach to vector quantization. Not unrelated to the problem of vector quantization is the problem of providing a physical de"nition of information [14]. Again quantum mechanics provides a natural solution to this problem; namely, the information that can be carried by a physical system is precisely determined by the dimension of Hilbert space [15]. Quantum mechanics also provides a natural theory of information #ow in a distributed system [16], and a rigorous bound on the rate of transmission of information [17]. It might be noted that in a sense these relatively recent developments concerning the physical properties of information were anticipated in the 1920s, when it was recognized that quantum mechanics could solve the long-standing problem of calculating the entropy constant of a gas. The rule of thumb that was eventually developed [18] was to divide phase space in `cellsa each of whose volume was equal to D, where f is the number of degrees of freedom. Later a rigorous way to quantize phase space, based on the Born}Jordan quantum mechanics, was developed by Wigner [19] and Moyal [20]. Because of its fundamental connection with information theory one might guess that the Wigner}Moyal formulation of quantum mechanics would be a good place to start when looking for a far-reaching interpretation of quantum mechanics as a theory of pattern recognition. In fact, we will show in the following that when the phase space to be quantized is a curved surface, Wigner}Moyal-like quantization not only leads us to a new way of formulating of quantum mechanics, but also a remarkable insight into why quantum computers should be so useful for pattern recognition. In the next section we review the procedure introduced in Ref. [10] for quantizing the parameterization of a curved surface. This generalization of the Wigner}Moyal phase space quantization formalism can also be given the `physicala interpretation as the Weyl quantization of a complete holographic representation of the surface. In particular, we note that in a paraxial ray-type approximation for the electromagnetic "eld the usual mode variables for the electromagnetic "eld can be replaced by "eld variables E(x) and E>(x) whose variation represents the transverse structure of a hologram. These "eld variables depend on both the position on the hologram and the orientation of the illuminating laser beam. The great advantage of using these variables is that quantization amounts to replacing the classical variables E(x) and E>(x) with

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ordinary creation and annihilation operators. This procedure is most interesting in the case of Riemann surfaces with non-trivial topology. We point out that representing a Riemann surface holographically amounts to a pedestrian version of a mathematically elegant characterization of a Riemann surface in terms of its Jacobian variety and associated theta functions. Applied to Riemann surfaces the end result of our holographic approach to quantization is a natural representation of the phase space for quantum systems with a "nite dimensional Hilbert space in terms of theta functions. While this representation is not in itself a new result } indeed it is equivalent to using the well known generalized coherent states for an SU(N) Lie algebra } our way of relating this representation to holograms of a Riemann surface points us in the direction of a new model for quantum mechanics. In Section 3 we make use of the connection between holographic representations of Riemann surfaces and "nite dimensional Hilbert spaces to formulate what is in e!ect an entirely new way of looking at quantum dynamics. Speci"cally we consider the problem of changing the shape of the Riemann surface so that if the relative phase of the illuminating laser beam at di!erent locations on the surface is varied, the holographic representation for the surface is unchanged. In the case of classical holograms this feedback control problem is equivalent to the classical principle of least action. However, when photon noise is introduced, we make use of results originally derived by Dyson [21] in the context of optimizing the performance of active optical systems to show that the problem becomes equivalent to the multi-channel quantum mechanical scattering theory of Newton and Jost. In Section 4 we discuss how this result may be related to a theory underlying superstring models, and suggest why it may lead to an explanation of the ùnreasonable e!ectiveness of mathematicsa.

2. Quantum holography of Riemann surfaces As a generalization of the basic problem of quantizing a #at two-dimensional phase space parameterized by classical momentum and position variables p and q we now turn to the problem of quantizing the parameterization of a curved two-dimensional surface. In the case of a Riemann surface the curved surface can be represented by a collection of #at sheets glued together along branch cuts. Thus in this case the problem of quantizing parameterizations of the surface would appear to be very similar to the Moyal problem of quantizing #at p, q phase space, except that now the phase space quantizations on each sheet must be matched along the branch cuts. One might guess that such a system could be quantized by introducing a set of p, q variables with an index j which represented which sheet one was on. The usual Weyl operators a"p#iq and a>"p!iq are now replaced with sets +a , and +a>,. Thus the original Weyl}Heisenberg group will be H H replaced by a Lie group generated by operators of the form , , aHa, aHa and aa>, a a> . H H H H H H States playing much the same role as the coherent states that play such an important role in quantum optics [22] will be generated by the operators D(a)"exp(aa>!aHa)

(1)


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which are analogous to the displacement operators D(m, g). These operators obey the multiplication rule D(a)D(b)"e ' ?@M D(a#b) .

(2)

Replacing the Weyl}Heisenberg algebra with an arbitrary Lie algebra leads to the generalized coherent states of Perelomov and Gilmore [23,24]. For these generalized coherent states the space parameterized by a and aH is no longer the complex plane but a symmetric space G/H. These symmetric spaces have a natural sympletic structure [24] and provide a canonical phase space structure for any quantum system with a "nite-dimensional Hilbert space whose Hamiltonian acts simultaneously on at least two variables, as well as classical systems whose dynamics is described by Lax-pair-type equations [25]. Multi-phase solutions of the KdV equation can also be described using these states [26]. For these systems the symmetric space phase space structure is in fact similar to the p, q phase space of elementary mechanics. For example, in the SU(N) case in terms of the variables z"a(sinh aa)/(aa , the metric of the space parameterized by the a variables is proportional to dzdzH, and the Poisson bracket takes the simple form

Rf Rg Rg Rf ! . (3) + f, g,"!i Rz RzH RzH Rz H H H H Evidently, the variables z and zH play essentially the same role as the p and q variables in ordinary mechanics, and therefore we expect that the formalism introduced in Section 2 can be used to de"ne operators on the 2N-dimensional phase space parameterized by a and aH and introduce a Wignerlike distribution function on this space (in a way completely analogous to the way Wigner distributions are used in quantum optics, cf. [22]). It is interesting to note that when G/H is a complex torus then the coherent states generated by the D(a) can be identi"ed with the theta functions that play such an important role in the theory of the Jacobian varieties associated with Riemann surfaces [27]. Indeed the condition that the generalized coherent states be single valued on the torus means that the phase factor Im(abH) in Eq. (45) must be equal to 2p times an integer when a and b correspond to periods of the torus. Remarkably this is just the condition that a complex torus be an abelian variety; i.e. the Jacobian of a Riemann surface. Thus, it appears possible to regard the Jacobian of a Riemann surface as being very similar to the usual p, q phase space and to use theta functions to de"ne a quantization of this space in a way completely analogous to the way Wigner and Moyal used two-dimensional Fourier transforms to quantize p, q phase space. Since according to the classical Torelli theorem a Riemann surface can be reconstructed from its Jacobian and associated theta functions [27], one might assume that applying the Moyal formalism to the Jacobian of the Riemann surface e!ectively solves the problem of quantizing the geometry of a Riemann surface. We will now argue that what is involved here is essentially the assumption that quantizing the Riemann surface is equivalent to quantizing holographic representations of the surface. As a quick reminder all information concerning an arbitrarily curved surface in three-dimensions can be encoded onto a #at plane or the surface of a sphere by recording photographically or

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otherwise the interference of light scattered o! the surface with a reference beam. Of course, this is only useful if the illuminating and reference beams are su$ciently temporally and spatially coherent. In addition, if the surface is reentrant then the interference pattern must be recorded for various orientations of the illuminating beam in order to capture the shape of the entire surface. For the purpose of physically describing these holograms one may introduce a slowly varying ènvelopea electric "eld whose rms magnitude corresponds to the intensity of the interference pattern. To this end we write the vector potential on the recording surface as a function of position h on the hologram in the form (for simplicity and because we later want to make contact with the theory of inverse scattering we will assume that our holograms are recorded on a large spherical surface parameterized by two angles that we collectively call h) A(h)"!(i/k)E(h, t) exp(ikz)#h.c. ,

(4)

where z is a coordinate for the direction perpendicular to the surface of the hologram and the factor exp(ikz) represents the rapidly varying phase of the reference and scattered beams. It is straightforward to quantize these "elds by substituting expression (47) into the standard radiation gauge Lagrangian for the electromagnetic "eld. One "nds the following commutation relations for the electric "eld operators on the recording surface [E (h), E>(h)]"2pud d(h!h) . G H GH

(5)

The electric "eld E(h, t) receives contributions from various points on the surface. In the dipole approximation the contribution from each little patch of surface is determined by the cross product between the vector pointing from the patch of surface to the point x and the direction of the oscillating polarization induced in the surface patch by the illuminating beam. Since these two vectors are curl-free on the surface the sum of contributions over the surface has the character of an inner product of two harmonic di!erentials

(6) (u , u )" u u . 1 Now an inner product for harmonic di!erentials of the form (6) plays an important role in the theory of Riemann surfaces; speci"cally an inner product of the form (6) allows one to pick out from the space of all complex tori those tori that are abelian varieties; i.e. those complex tori that can be regarded as the Jacobians of a Riemann surface. Thus we see that the famous Torelli theorem has a `physicala interpretation in terms of holography. In one obviously important respect though the holographic representation of a Riemann surface using the electric "eld E(h, t) di!ers from the representation discussed at the beginning of this section that involved N-copies of the Weyl}Heisenberg group; namely, the holographic representation involves an in"nite number of annihilation and creation operators corresponding to the electric "eld intensity at any of the in"nite number of positions on the hologram, whereas the representation based on multiple copies of the Weyl}Heisenberg group involved only a "nite number of such operators. This discrepancy can be traced to the fact that the quantization problem most closely related to ordinary p, q phase space quantization assumes that the shape of the surface is "xed. Under such circumstances the electric "eld vectors at di!erent x points are not independent


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for a given orientation of the illuminating beam. Indeed upon re#ection it is clear that the only way to obtain algebraically independent annihilation and creation operators is to illuminate the di!erent handles of the Riemann surface in distinct ways. An aesthetic choice for these distinct illuminations would be to choose the di!erent illuminating beams in such a way that the polarizations induced on the surface by the di!erent beams can be identi"ed with a canonical basis for the "rst cohomology group. There are 2g 1-forms in such a canonical basis; one for each of the 2g 1-cycles associated with the g handles of the surface. Armed with such canonical illuminations we can construct a set of 2g holograms, from which the Riemann surface can be completely reconstructed. Corresponding to these 2g algebraically independent holograms are a set of 2g independent annihilation and creation operators, viz. E (x ) and E>(x ) for j"1,2, 2g, thus H H con"rming our previous guess concerning the structure of the quantized phase space that corresponds to a Riemann surface. The representation of this phase space in terms of generalized coherent states shows that we are dealing with an N-state system where N"2g. In the next section we will develop a model for the quantum dynamics of this system, based on our holographic representation for the phase space.

3. Adaptive optics model for quantum mechanics In Section 2 we have seen that the properties of theta functions allow us to de"ne a natural Wigner}Moyal-like quantization for Riemann surfaces, and that this natural quantization corresponds to the phase space of a quantum system whose Hilbert space is "nite dimensional; i.e. an `N-statea system. We now turn to the quantum dynamics of such a system, and show that viewing the quantization of the phase space from the point of view of holography of a Riemann surface leads to an interesting and novel interpretation for the quantum dynamics. This new interpretation for quantum dynamics can actually be summarized very succinctly: if the phase of the illuminating light beam is varied as a function of time and position, a small patch of the surface should move in such a way so as to exactly compensate for the changes in the e!ective path length of the beam illuminating that patch of surface. This formulation of quantum dynamics bears an obvious similarity to the engineering problem of adaptive optics, where one is interested in changing the shape of a mirror to compensate for the degradation of focal plane images by atmospheric turbulence. Indeed, we will make generous use of results from the theory of active optical systems in our formulation of quantum dynamics. It should be kept in mind, though, that there are close parallels with the theory of the KdV equation (cf. [26]), which could perhaps have been used as the basis for the following development. We begin by writing down equations which describe an interplay between deformations in the shape of a Riemann surface and changes in the intensity of light on a set of holograms that mimic the feedback control of an active optical system. The "rst equation states that we have a feedback control system that adjusts the absolute position X(p, t) of the surface with su$cient accuracy so that the intensity of light on the hologram at time t and position x is a linear function of the error e(p, t)"*X(p, t)#a(p, t); i.e.

I(h, t)"I (h)# dp B(x, p)e(p, t) , 1

(7)

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where I(h, t) is the recorded intensity of light on the hologram at time t and position h, and I (h) is the recorded intensity on the hologram when the Riemann surface is illuminated with a `standarda beam with no arti"cially imposed variations in phase with respect to position or time. In writing Eq. (7) we are neglecting the light propagation time between the Riemann surface and the hologram. Also we will be particularly interested in the following in the situation where I (x)"E>(x) ' E (x), where E (x) is the electric at position x on the hologram when the polarization H H H induced on the Riemann surface by the illuminating beam corresponds to one of the 2g harmonic 1-forms in a canonical basis for the "rst cohomology group of the surface. The second equation relates the deformation of the Riemann surface at point p, i.e. *X(p, t), produced by the feedback control system to the intensity of light on the hologram:

*X(p, t)" dX

R

\

dt A(p, h, t)I(h, t) .

(8)

When photon noise is neglected, i.e. when we regard the hologram as a classical object, then Eqs. (7) and (8) have the solution (in vector notation) e"[1!AB]\a .

(9)

Eq. (9) shows that when the negative feedback is strong the error e(p, t) is reduced to a small fraction of the perturbation in path length. That is, the position of the surface just tracks locally the change in optical path length. Upon re#ection one soon realizes that this just the classical principle of least action! What is perhaps most remarkable about this setup though is that the problem of changing the shape of the Riemann surface to compensate for changes in the optical path of the illuminating beam becomes equivalent to solving the SchroK dinger equation when the e!ects of photon noise on the hologram are taken into account. This situation is qualitatively di!erent from the classical case because the negative feedback in Eq. (9) will amplify noise, and so there is now a limit to how strong the negative feedback can be made. It is not hard to show that in the presence of photon noise the two point correlation function for the errors averaged over a time long compared to the characteristic time for photon number #uctuations has the form (again in operator notation) 1e e 2"[1!A B ]\[1!A B ]\+; #A A d I , ,

(10)

where ; is the average of a a over the same time and d "d(p !p )d(t !t ). In contrast with the classical case an optimal choice for the feedback matrix A is somewhat arbitrary. However if, following Dyson, we take as the criterion for optimizing the system that a quadratic function of the feedback errors should be minimized, then it can be shown [21] that the optimal feedback matrix A(p, h, t) can be expressed in the form A"KB2I\ ,

(11)

where K(p , p , t !t ) is a causal matrix satisfying the non-linear operator equation K#K2#K(B2I\B)K2#;"0 .

(12)


103

This equation is a matrix generalization of the Gelfand}Levitan equation, and is essentially equivalent to the multi-channel inverse scattering theory of Newton and Jost [28]. In our formulation of their theory time plays the role of their position coordinate r, and their discrete channel labels a"1,2, N are replaced by an index j"1,2, 2g running over the harmonic 1-forms u in a canonical basis for the cohomology of a Riemann surface. Therefore the matrices H A and K are "nite matrices describing surface deformations whose support coincides with the support of harmonic functions f such that u "df . H H H It is worth noting that in our formulation of quantum mechanics the dynamical behavior of the system is entirely determined by the pair correlation matrix U. This is a consequence of the linearity of the fundamental Eqs. (7) and (8) and exactly mirrors the fundamental theorem of quantum computation [29,30], which states that any quantum computation can be e!ected by quantum logic gate which consists of a unitary operator acting on an arbitrary choice of two input variables. Thus, the pattern recognition capabilities of quantum computers are intimately related to the pattern recognition capabilities of adaptive optics systems. It is amusing to note in this connection that our quantized version of Dyson's algorithm is formally similar to Hop"eld's scheme for `collective computationa using spins [31]. 4. Mathematics as theoretical physics? The meaning of mathematics and its relation to physics have been controversial subjects for millennia among mathematicians and philosophers. It has not been our purpose here to review these historical discussions. Instead our intention has been to add a new ingredient to these discussions that may in the course of time help to resolve some of the long-standing philosophical issues. This new ingredient has two di!erent but related aspects. The "rst aspect is that, prompted by the discovery of potentially useful applications for quantum computers, we have found a mathematical interpretation of quantum mechanics as a theory of pattern recognition. The second aspect is that our àdaptive opticsa model for quantum mechanics is an invitation to relate our formulation of quantum mechanics to a quantum theory of membranes, which is a putative candidate for the mathematical theory underlying superstring models [32]. Quantum #uctuations in a hologram due to photon noise give rise via the feedback control system introduced in Section 3 to quantum #uctuations in the shape of the surface, and this may perhaps be interpreted by saying that the classical geometry of the Riemann surface has been replaced with quantum geometry. Thus quantum holography might also be interpreted as a quantum theory of membranes. In addition as noted in Section 2 our holographic formulation of quantum kinematics is closely related to the classical theory of integrable dynamical systems. Now it happens that certain kinds of membranes correspond to the large N limit of integrable systems whose Lax-pair dynamics is similar to SU(N) quantum dynamics [33]. Furthermore, it has independently been suggested [34] that the same kind of Wigner}Moyal quantization as we have employed could be used to quantize these large N models for membranes. Therefore it appears that our holographic formulation of quantum mechanics may indeed be pointing us in the direction of a fundamental structure for theoretical physics. Although the exact nature of the mathematical structure underlying superstring models remains shrouded in mist, let us we accept for the moment that our formulation of quantum mechanics is

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actually closely related to the quantum theory of membranes being sought as the theory underlying superstring models. This allows us to immediately make a tentative yet intriguingly plausible conjecture as to the nature of the fundamental relationship between mathematics and theoretical physics. Namely, we are led to suggest that the fundamental connecting link between mathematics and theoretical physics is the pattern recognition capabilities of the human brain. On one hand, it seems quite reasonable to assume that the human brain's aptitude for mathematical reasoning evolved from the mammalian brain's more primitive function as a pattern recognition engine. On the other hand, the pattern recognition capabilities of the mammalian brain may be closely related to our formulation of quantum mechanics. Of course, given our present state of ignorance concerning how the mammalian brain actually functions this last assertion is highly speculative. However, the general idea that a quantum mechanical theory of information #ow (cf. [7]) can be looked upon as a model for the type of distributed information processing carried out in the brain has a lot to recommend it. For example, one of the fundamental heuristics of distributed information processing networks is that minimization of energy consumption requires the use of time division multiplexing for communication between the processors [35], and it would be natural to identify the local internal time in such networks as a quantum phase.

References [1] G. Chapline, R. Slansky, Dimensional reduction and #avor chirality, Nucl. Phys. B 209 (1982) 461. [2] M. Green, J.H. Schwarz, Anomaly cancellations in supersymmetric D"10 superstring theory, Phys. Lett. B 144 (1984) 117. [3] A. Schellekens, N. Warner, Anomolies and Modular Invariance in String Theory, Phys. Lett. B 177 (1985) 317. [4] G. Chapline, Uni"cation of elementary particle physics in 26 dimensions, Phys. Lett. B 158 (1985) 393; The Monster sporadic group and a theory underlying superstring models, in: Proceedings of the Strings '96 Conference, Institute for Theoretical Physics, 1996. [5] E.P. Wigner, The unreasonable e!ectiveness of mathematics in the natural sciences. Comm. Pure Appl. Math. 13 (February 1960). [6] R. Hersh, What is Mathematics, Really? Oxford University Press, Oxford, 1997. [7] L.H. Kau!man, On Knots, Princeton University Press, 1987. [8] E. Witten, Supersymmetry and Morse theory, J. Di!erential Geom. 17 (1982) 661; Topological Quantum Field Theory 117 (1988) 353. [9] T. Dereli, A. Vercin, =(R) covariance of the Weyl}Wigner}Groenewold}Moyal quantization, J. Math. Phys. 38 (1997) 5515. [10] G. Chapline, A. Granik, Moyal quantization, holography, and the quantum geometry of surfaces, Chaos, Solitons and Fractals, to be published. [11] P.W. Shor, Algorithms for quantum computation: discrete log and factoring, in Proceedings of the 35th Annual Symposium on Foundations of Computer Science, IEEE Press, 1994. [12] L.K. Glover, Quantum mechanics helps in searching for a needle in a haystack, Phys. Rev. Lett. 79 (1997) 325. [13] T. Kohonen, Self-Organization and Associative Memory, Springer, Berlin, 1988. [14] B. Schumacher, Information from quantum measurements, in: W.H. Zurek (Ed.), Complexity, Entropy and the Physics of Information, Addison-Wesley, Reading, MA, 1990. [15] A.S. Kholevo, Bounds for the quantity of information transmitted by a quantum communication channel, Problems of Information Transfer 9 (1973) 3. [16] G. Chapline, Information #ow in quantum mechanics, in: T.D. Black et al. (Eds.), Santa Fe Workshop on Foundations of Quantum Mechanics, World Scienti"c, Singapore, 1988.

G. Chapline / Physics Reports 315 (1999) 95 } 105 [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35]

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C.M. Caves, P.D. Drummond, Quantum limits on bosonic communication rates, Rev. Modern Phys. 66 (1994) 481. E. Fermi, Notes on Thermodynamics and Statistics, University of Chicago Press, Chicago, 1966. E. Wigner, On the quantum correction for thermodynamic equilibrium, Phys. Rev. 40 (1932) 749. J.E. Moyal, Quantum mechanics as a statistical theory, Proc. Camb. Phil. Soc. 45 (1949) 99. F.J. Dyson, Photon noise and atmospheric noise in active optical systems, J. Optical Soc. Am. 65 (1975) 551. D.F. Walls, G.J. Milburn, Quantum Optics, Springer, Berlin, 1994. A.M. Perelomov, Coherent states for an arbitrary Lie group, Commun. Math. Phys. 26 (1972) 222. W. Zang, D.H. Feng, R. Gilmore, Coherent states: theory and applications, Rev. Mod. Phys. 62 (1990) 867. M. Adler, P. van Moerbeke, Completely integrable systems, Lie algebras, and curves, Adv. Math. 38 (1980) 267. A.C. Newell, Solitons in Mathematics and Physics, SIAM, 185. H.M. Farkas, I. Kra, Riemann Surfaces, Springer, Berlin, 1992. R.G. Newton, R. Jost, The construction of potentials from the S-matrix for systems of di!erential equations, Nuovo Cimento 1 (1955) 590. D. Deutsch, A. Barenco, A. Ekert, Universality in quantum computation, Proc. R. Soc. London A 449 (1995) 669. S. Lloyd, Almost any quantum logic gate is universal, Phys. Rev. Lett. 75 (1995) 346. J.J. Hop"eld, Neurons with graded response have collective computational properties like those of two-state neurons, Proc. Natl. Acad. Sci. B 209 (1984) 3088. P. Horava, E. Witten, Heterotic and Type I string dynamics from eleven dimensions, Nucl. Phys. B 460 (1996) 506. E.G. Floratos, G.K. Leontaris, Integrability of self-dual membranes in (4#1) dimensions and the Toda lattice, Phys. Lett. B 223 (1989) 153. C. Castro, A Moyal quantization of the continuous Toda "eld, Phys. Lett. B 413 (1997) 53. G. Chapline, Sentient networks, in: Proceedings of the First International Conference on Multisource}Multisensor Fusion, CREA Press Athens, Georgia, 1998.


From superstrings to M theory夽 John H. Schwarz California Institute of Technology, Pasadena, CA 91125, USA

Abstract In the strong coupling limit Type IIA superstring theory develops an 11th dimension that is not apparent in perturbation theory. This suggests the existence of a consistent 11D quantum theory, called M theory, which is approximated by 11D supergravity at low energies. In this review we describe some of the evidence for this picture and some of its implications. 1999 Elsevier Science B.V. All rights reserved. PACS: 11.25.!w; 11.25.Sq; 04.50.#h Keywords: String theory; M theory; Supersymmetry; Superstrings

1. Introduction Superstring theory is currently undergoing a period of rapid development in which important advances in understanding are being achieved. The purpose of this review is to describe a portion of this story to physicists who are not already experts in this "eld. The focus will be on explaining why there can be an 11D vacuum, even though there are only 10 dimensions in perturbative superstring theory. The nonperturbative extension of superstring theory that allows for an 11th dimension has been named M theory. The letter M is intended to be #exible in its interpretation. It could stand for magic, mystery, or meta to re#ect our current state of incomplete understanding. Those who think that (2D) supermembranes (the M2-brane) are fundamental may regard M as standing for membrane. An approach called Matrix theory is another possibility. And, of course, some view M theory as the mother of all theories. Superstring theory "rst achieved widespread acceptance during the xrst superstring revolution in 1984}1985. There were three main developments at this time. The "rst was the discovery of an 夽

Work supported in part by the U.S. Dept. of Energy under Grant No. DE-FG03-92-ER40701. For a more detailed review see Ref. [1]. 0370-1573/99/$ - see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 9 ) 0 0 0 1 6 - 2

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J.H. Schwarz / Physics Reports 315 (1999) 107}121

anomaly cancellation mechanism [2], which showed that supersymmetric gauge theories can be consistent in 10 dimensions provided they are coupled to supergravity (as in type I superstring theory) and the gauge group is either SO(32) or E ;E . Any other group necessarily would give uncancelled gauge anomalies and hence inconsistency at the quantum level. The second development was the discovery of two new superstring theories } called heterotic string theories } with precisely these gauge groups [3]. The third development was the realization that the E ;E heterotic string theory admits solutions in which six of the space dimensions form a Calabi}Yau space, and that this results in a 4D e!ective theory at low energies with many qualitatively realistic features [4]. Unfortunately, there are very many Calabi}Yau spaces and a whole range of additional choices that can be made (orbifolds, Wilson loops, etc.). Thus there is an enormous variety of possibilities, none of which stands out as particularly special. In any case, after the "rst superstring revolution subsided, we had "ve distinct superstring theories with consistent weak coupling perturbation expansions, each in 10 dimensions. Three of them, type I theory and the two heterotic theories, have N"1 supersymmetry in the 10D sense. Since the minimal 10D spinor is simultaneously Majorana and Weyl, this corresponds to 16 conserved supercharges. The other two theories, called types IIA and IIB, have N"2 supersymmetry (32 supercharges) [5]. In the IIA case the two spinors have opposite handedness so that the spectrum is left}right symmetric (nonchiral). In the IIB case the two spinors have the same handedness and the spectrum is chiral. The understanding of these "ve superstring theories was developed in the ensuing years. In each case it became clear, and was largely proved, that there are consistent perturbation expansions of on-shell scattering amplitudes. In four of the "ve cases (heterotic and type II) the fundamental strings are oriented and unbreakable. As a result, these theories have particularly simple perturbation expansions. Speci"cally, there is a unique Feynman diagram at each order of the loop expansion. The Feynman diagrams depict string world sheets, and therefore they are 2D surfaces. For these four theories the unique ¸-loop diagram is a closed orientable genus-¸ Riemann surface, which can be visualized as a sphere with ¸ handles. External (incoming or outgoing) particles are represented by N points (or `puncturesa) on the Riemann surface. A given diagram represents a well-de"ned integral of dimension 6¸#2N!6. This integral has no ultraviolet divergences, even though the spectrum contains states of arbitrarily high spin (including a massless graviton). From the viewpoint of point-particle contributions, string and supersymmetry properties are responsible for incredible cancellations. Type I superstrings are unoriented and breakable. As a result, the perturbation expansion is more complicated for this theory, and the various worldsheet diagrams at a given order (determined by the Euler number) have to be combined properly to cancel divergences and anomalies [6]. An important discovery that was made between the two superstring revolutions is called T duality [7]. This is a property of string theories that can be understood within the context of perturbation theory. (The discoveries associated with the second superstring revolution are mostly nonperturbative.) T duality shows that spacetime geometry, as probed by strings, has some surprising properties (sometimes referred to as quantum geometry). The basic idea can be illustrated

A discussion with Richard Slansky helped to convince us that E ;E would work.


109

by the simplest example. This entails considering one spatial dimension to form a circle (denoted S). Then the 10D geometry is R;S. T duality identi"es this string compacti"cation with one of a second string theory also on R;S. However, if the radii of the circles in the two cases are denoted R and R , then R R "a . (1) Here a"l is the universal Regge slope parameter, and l is the fundamental string length scale (for both string theories). The tension of a fundamental string is given by ¹"2pm"1/2pa , where we have introduced a fundamental string mass scale

(2)

m "(2pl )\ . (3) Note that T duality implies that shrinking the circle to zero in one theory corresponds to decompacti"cation of the dual theory. Compacti"cation on a circle of radius R implies that momenta in that direction are quantized, p"n/R. (These are called Kaluza}Klein excitations.) These momenta appear as masses for states that are massless from the higher-dimensional viewpoint. String theories also have a second class of excitations, called winding modes. Namely, a string wound m times around the circle has energy E"2pR ) m ) ¹"mR/a. Eq. (1) shows that the winding modes and Kaluza}Klein excitations are interchanged under T duality. What does T duality imply for our "ve superstring theories? The IIA and IIB theories are T dual [8]. So compactifying the nonchiral IIA theory on a circle of radius R and letting RP0 gives the chiral IIB theory in 10 dimensions! This means, in particular, that they should not be regarded as distinct theories. The radius R is actually a vev of a scalar "eld, which arises as an internal component of the 10D metric tensor. Thus types IIA and IIB theories in 10D are two limiting points in a continuous moduli space of quantum vacua. The two heterotic theories are also T dual, though there are technical details involving Wilson loops, which we will not explain here. T duality applied to type I theory gives a dual description, which is sometimes called I'. Names IA and IB have also been introduced by some authors. For the remainder of this paper, we will restrict attention to theories with maximal supersymmetry (32 conserved supercharges). This is su$cient to describe the basic ideas of M theory. Of course, it suppresses many fascinating and important issues and discoveries. In this way, we will keep the presentation from becoming too long or too technical. The main focus will be to ask what happens when we go beyond perturbation theory and allow the coupling strength to become large in type II theories. The answer in the IIA case, as we will see, is that another spatial dimension appears.

2. M theory In the 1970s and 1980s various supersymmetry and supergravity theories were constructed. (See [9], for example.) In particular, supersymmetry representation theory showed that 10 is the largest spacetime dimension in which there can be a matter theory (with spins 41) in which supersymmetry is realized linearly. A realization of this is 10D super Yang}Mills theory, which has 16

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supercharges [10]. This is a pretty (i.e., very symmetrical) classical "eld theory, but at the quantum level it is both nonrenormalizable and anomalous for any nonabelian gauge group. However, as we indicated earlier, both problems can be overcome for suitable gauge groups (SO(32) or E ;E ) when the Yang}Mills theory is embedded in a type I or heterotic string theory. The largest possible spacetime dimension for a supergravity theory (with spins 42), on the other hand, is 11. Eleven-dimensional supergravity, which has 32 conserved supercharges, was constructed 20 years ago [11]. It has three kinds of "elds } the graviton "eld (with 44 polarizations), the gravitino "eld (with 128 polarizations), and a three-index gauge "eld C (with 84 polarizaIJM tions). These massless particles are referred to collectively as the supergraviton. 11D supergravity is also a pretty classical "eld theory, which has attracted a lot of attention over the years. It is not chiral, and therefore not subject to anomaly problems. It is also nonrenormalizable, and thus it cannot be a fundamental theory. Though it is di$cult to demonstrate explicitly that it is not "nite as a result of `miraculousa cancellations, we now know that this is not the case. However, we now believe that it is a low-energy e!ective description of M theory, which is a well-de"ned quantum theory [13]. This means, in particular, that higher dimension terms in the e!ective action for the supergravity "elds have uniquely determined coe$cients within the M theory setting, even though they are formally in"nite (and hence undetermined) within the supergravity context. Intriguing connections between type IIA string theory and 11D supergravity have been known for a long time. If one carries out dimensional reduction of 11D supergravity to 10D, one gets type IIA supergravity [14]. Dimensional reduction can be viewed as a compacti"cation on circle in which one drops all the Kaluza}Klein excitations. It is easy to show that this does not break any of the supersymmetries. The "eld equations of 11D supergravity admit a solution that describes a supermembrane. In other words, this solution has the property that the energy density is concentrated on a 2D surface. A 3D world-volume description of the dynamics of this supermembrane, quite analogous to the 2D world volume actions of superstrings, has been constructed [15]. The authors suggested that a consistent 11D quantum theory might be de"ned in terms of this membrane, in analogy to string theories in 10 dimensions. Another striking result was the discovery of double-dimensional reduction [16]. This is a dimensional reduction in which one compacti"es on a circle, wraps one dimension of the membrane around the circle and drops all Kaluza}Klein excitations for both the spacetime theory and the world-volume theory. The remarkable fact is that this gives (previously known) type IIA superstring world-volume action [17]. For many years these facts remained unexplained curiosities until they were reconsidered by Townsend [18] and by Witten [13]. The conclusion is that type IIA superstring theory really does have a circular 11th dimension in addition to the previously known 10 spacetime dimensions. This fact was not recognized earlier because the appearance of the 11th dimension is a nonperturbative phenomenon, not visible in perturbation theory.

Unless the spacetime has boundaries. The anomaly associated to a 10D boundary can be cancelled by introducing E supersymmetric gauge theory on the boundary [12]. Most experts now believe that M theory cannot be de"ned as a supermembrane theory.


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To explain the relation between M theory and type IIA string theory, a good approach is to identify the parameters that characterize each of them and to explain how they are related. Eleven-dimensional supergravity (and hence M theory, too) has no dimensionless parameters. As we have seen, there are no massless scalar "elds, whose vevs could give parameters. The only parameter is the 11D Newton constant, which raised to a suitable power (!1/9), gives the 11D Planck mass m . When M theory is compacti"ed on a circle (so that the spacetime geometry is R;S) another parameter is the radius R of the circle. Now consider the parameters of type IIA superstring theory. They are the string mass scale m , introduced earlier, and the dimensionless string coupling constant g . An important fact about all "ve superstring theories is that the coupling constant is not an arbitrary parameter. Rather, it is a dynamically determined vev of a scalar "eld, the dilaton, which is a supersymmetry partner of the graviton. With the usual conventions, one has g "1e(2. We can identify compacti"ed M theory with type IIA superstring theory by making the following correspondences: m"2pRm , g "2pRm . Using these one can derive other equivalent relations, such as

(4) (5)

g "(2pRm ) , (6) m "gm . (7) The latter implies that the 11D Planck length is shorter than the string length scale at weak coupling by a factor of (g ). Conventional string perturbation theory is an expansion in powers of g at "xed m . Eq. (5) shows that this is equivalent to an expansion about R"0. In particular, the strong coupling limit of type IIA superstring theory corresponds to decompacti"cation of the 11th dimension, so in a sense M theory is type IIA string theory at in"nite coupling. This explains why the 11th dimension was not discovered in studies of string perturbation theory. These relations encode some interesting facts. The fact relevant to Eq. (4) concerns the interpretation of the fundamental type IIA string. Earlier we discussed the old notion of double-dimensional reduction, which allowed one to derive the IIA superstring world-sheet action from the 11D supermembrane (or M2-brane) world-volume action. Now, we can make a stronger statement: The fundamental IIA string actually is an M2-brane of M theory with one of its dimensions wrapped around the circular spatial dimension. No truncation to zero modes is required. Denoting the string and membrane tensions (energy per unit volume) by ¹ and ¹ , one deduces that $ + ¹ "2pR¹ . (8) $ + However, ¹ "2pm and ¹ "2pm. Combining these relations gives Eq. (4). $ +

The E ;E heterotic string theory is also 11D at strong coupling [12].

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Type II superstring theories contain a variety of p-brane solutions that preserve half of the 32 supersymmetries. These are solutions in which the energy is concentrated on a p-dimensional spatial hypersurface. (The world volume has p#1 dimensions.) The corresponding solutions of supergravity theories were constructed by Horowitz and Strominger [19]. A large class of these p-brane excitations are called D-branes (or Dp-branes when we want to specify the dimension), whose tensions are given by [20] ¹ "2pmN>/g . (9) "N This dependence on the coupling constant is one of the characteristic features of a D-brane. It is to be contrasted with the more familiar g\ dependence of soliton masses (e.g., the 't Hooft}Polyakov monopole). Another characteristic feature of D-branes is that they carry a charge that couples to a gauge "eld in the RR sector of the theory. (Such "elds can be described as bispinors.) The particular RR gauge "elds that occur imply that even values of p occur in the IIA theory and odd values in the IIB theory. In particular, the D2-brane of type IIA theory corresponds to our friend the supermembrane of M theory, but now in a background geometry in which one of the transverse dimensions is a circle. The tensions check, because (using Eqs. (4) and (5)) ¹ "2pm/g "2pm"¹ . (10) " + The mass of the "rst Kaluza}Klein excitation of the 11D supergraviton is 1/R. Using Eq. (5), we see that this can be identi"ed with the D0-brane. More identi"cations of this type arise when we consider the magnetic dual of the M theory supermembrane. This turns out to be a "ve-brane, called the M5-brane. Its tension is ¹ "2pm. Wrapping one of its dimensions around the circle + gives the D4-brane, with tension ¹ "2pR¹ "2pm/g . (11) " + If, on the other hand, the M5-frame is not wrapped around the circle, one obtains the NS5-brane of the IIA theory with tension ¹ "¹ "2pm/g . (12) ,1 + This 5-brane, which is the magnetic dual of the fundamental IIA string, exhibits the conventional g\ solitonic dependence. To summarize, type IIA superstring theory is M theory compacti"ed on a circle of radius R"g l . M theory is believed to be a well-de"ned quantum theory in 11D, which is approximated at low energy by 11D supergravity. Its excitations are the massless supergraviton, the M2-brane, and the M5-brane. These account both for the (perturbative) fundamental string of the IIA theory and for many of its nonperturbative excitations. The identities that we have presented here are exact, because they are protected by supersymmetry.

In general, the magnetic dual of a p-brane in d dimensions is a (d!p!4)-brane.


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3. Type IIB superstring theory In the previous section we discussed type IIA superstring theory and its relationship to 11D M theory. In this section we consider type IIB superstring theory, which is the other maximally supersymmetric string theory with 32 conserved supercharges. It is also 10D, but unlike the IIA theory its two supercharges have the same handedness. Since the spectrum contains massless chiral "elds, one should check whether there are anomalies that break the gauge invariances } general coordinate invariance, local Lorentz invariance, and local supersymmetry. In fact, the UV "niteness of the string theory Feynman diagrams (and associated modular invariance) ensures that all anomalies must cancel. This was veri"ed also from a "eld theory viewpoint [21]. The low-energy e!ective theory that approximates type IIB superstring theory is type IIB supergravity [5,22], just as 11D supergravity approximates M theory. In each case the supergravity theory is only well de"ned as a classical "eld theory, but still it can teach us a lot. For example, it can be used to construct p-brane solutions and compute their tensions. Even though such solutions themselves are only approximate, supersymmetry considerations ensure that their tensions, which are related to the kinds of charges they carry, are exact. Another signi"cant fact about type IIB supergravity is that it possesses a global SL(2, R) symmetry. It is instructive to consider the bosonic spectrum and its SL(2, R) transformation properties. There are two scalar "elds } the dilation and an axion s, which are conveniently combined in a complex "eld o"s#ie\( .

(13)

The SL(2, R) symmetry transforms this "eld nonlinearly: oP(ao#b)/(co#d) ,

(14)

where a, b, c, d are real numbers satisfying ad!bc"1. However, in the quantum string theory this symmetry is broken to the discrete subgroup SL(2, Z) [23], which means that a, b, c, d are restricted to be integers. De"ning the vev of the o "eld to be 1o2"h/2p#i/g , (15) the SL(2, Z) symmetry transformation oPo#1 implies that h is an angular coordinate. More signi"cantly, in the special case h"0, the symmetry transformation oP!1/o takes g P1/g . This symmetry, called S duality, implies that the theory with coupling constant g is equivalent to coupling constant 1/g , so that the weak coupling expansion and the strong coupling expansion are identical! The bosonic spectrum also contains a pair of two-form potentials B and B, which transform IJ IJ as a doublet under SL(2, R) or SL(2, Z). In particular, the S duality transformation oP!1/o interchanges them. The remaining bosonic "elds are the graviton and a four-form potential C , IJMH with a self-dual "eld strength. They are invariant under SL(2). In the introductory section we indicated that types IIA and IIB superstring theories are T dual, meaning that if they are compacti"ed on circles of radii R and R one obtains equivalent theories for the identi"cation R R "l. Moreover, in Section 2 we saw that type IIA theory is actually M theory compacti"ed on a circle. The latter fact encodes nonperturbative information. It turns out to be very useful to combine these two facts and to consider the duality between M theory

114


compacti"ed on a torus (R;¹) and type IIB superstring theory compacti"ed on a circle (R;S). Recall that a torus can be described as the complex plane modded out by the equivalence relations z&z#w and z&z#w . Up to conformal equivalence, the periods can be taken to be 1 and q, with Im q'0. However, in this characterization q and q"(aq#b)/(cq#d), where a, b, c, d are integers satisfying ad!bc"1, describe equivalent tori. Thus, a torus is characterized by a modular parameter q and an SL(2, Z) modular group. The natural, and correct, conjecture at this point is that one should identify the modular parameter q of the M theory torus with the parameter o that characterizes type IIB vacuum [24,25]! Then the duality gives a geometrical explanation of the nonperturbative S duality symmetry of the IIB theory: the transformation oP!1/o, which sends g P1/g in the IIB theory, corresponds to interchanging the two cycles of the torus in the M theory description. To complete the story, we should relate the area of the M theory torus (A ) + to the radius of the IIB theory circle (R ). This is a simple consequence of formulas given above mA "(2pR )\ . (16) + Thus the limit R P0, at "xed o, corresponds to decompacti"cation of the M theory torus, while preserving its shape. Conversely, the limit A P0 corresponds to decompacti"cation of the IIB + theory circle. The duality can be explored further by matching the various p-branes in 9 dimensions that can be obtained from either the M theory or the IIB theory viewpoints [26]. When this is done, one "nds that everything matches nicely and that one deduces various relations among tensions, such as ¹ "(1/2p)(¹ ) . (17) + + This relation was used earlier when we asserted that ¹ "2pm and ¹ "2pm. + + Even more interesting is the fact that the IIB theory contains an in"nite family of strings labelled by a pair of relatively prime integers (p, q) [24]. These integers correspond to string charges that are sources of the gauge "elds B and B. The (1, 0) string can be identi"ed as the fundamental IIB IJ IJ string, while the (0, 1) string is the D-string. From this viewpoint, a (p, q) string can be regarded as a bound state of p fundamental strings and q D-strings [27]. These strings have a very simple interpretation in the dual M theory description. They correspond to an M2-brane with one of its cycles wrapped around a (p, q) cycle of the torus. The minimal length of such a cycle is proportional to "p#qq", and thus (using q"o) one "nds that the tension of a (p, q) string is given by ¹ "2p"p#qo"m . (18) NO The normalization has been chosen to give ¹ "2pm. Then (for h"0) ¹ "2pm/g , as expected. Note that decay is kinematically forbidden by charge conservation when p and q are relatively prime. When they have a common division n, the tension is the same as that of an n-string system. Whether or not there are threshold bound states is a nontrivial dynamical question, which has di!erent answers in di!erent settings. In this case there are no such bound states, which is why p and q should be relatively prime. Imagine that you lived in the 9D world that is described equivalently as M theory compacti"ed on a torus or as type IIB superstring theory compacti"ed on a circle. Suppose, moreover, you had very high-energy accelerators with which you were going to determine the `truea dimension of


115

spacetime. Would you conclude that 10 or 11 is the correct answer? If either A or R was very + large in Planck units there would be a natural choice, of course. But how could you decide otherwise? The answer is that either viewpoint is equally valid. What determines which choice you make is which of the massless "elds you regard as ìnternala components of the metric tensor and which ones you regards as matter "elds. Fields that are metric components in one description correspond to matter "elds in the dual one.

4. U dualities Maximal supergravity theories (ones with 32 conserved supercharges) typically have a noncompact global symmetry group G. For example, in the case of type IIB supergravity in 10 dimensions the group is SL(2, R). When one does dimensional reduction one "nds larger groups in lower dimensions. For example, N"8 supergravity in four dimensions has a noncompact E symmetry [28]. More generally, for D"11!d, 34d48, one "nds a maximally noncompact form of E , B denoted E . These are statements about classical "eld theory. The corresponding statement about BB superstring theory/M theory is that if we toroidally compactify M theory on R";¹B or type IIB superstring theory on R";¹B\, the resulting moduli space of theories is invariant under an in"nite discrete U duality group. The group, denoted E (Z), is a maximal discrete subgroup of the B noncompact E symmetry group of the corresponding supergravity theory [23]. An example that BB we will focus on below is E (Z)"SL(3, Z);SL(2, Z) . (19) The U duality groups are generated by the Weyl subgroup of E plus discrete shifts of axion-like BB "elds. The subgroup SL(d, Z)LE (Z) can be understood as the geometric duality (modular group) B of ¹B in the M theory picture. This generalizes the SL(2, Z) discussed in the preceding section. The subgroup SO(d!1, d!1; Z)LE (Z) is the T duality group of type IIB superstring theory B compacti"ed on ¹B\. These two subgroups intertwine nontrivially to generate the entire E (Z) B U duality group. Suppose we wish to focus on M theory and disregard type IIB superstring theory. Then we have a geometric understanding of the SL(d, Z) subgroup of E (Z) from considering M theory on B R\B;¹B. But what does the rest of E (Z) imply? To address this question it will su$ce to B consider the "rst nontrivial case to which it applies, which is d"3. In this case the U duality group is SL(3, Z);SL(2, Z). The "rst factor is geometric from the M theory viewpoint and nongeometric from the IIB viewpoint, whereas the second factor is geometric from the IIB viewpoint and nongeometric from the M theory viewpoint. So the question boils down to understanding the implication of the SL(2, Z) duality in the M theory construction. Speci"cally, we want to understand the nontrivial qP!1/q transformation. To keep the story as simple as possible, we will take the ¹ to be rectilinear with radii R , R , R (i.e., g &Rd ) and assume that C "0. Let us suppose that R corresponds to the èleventha GH G GH dimension that takes us to the IIA theory. Then we have IIA theory on a torus with radii R and R . The nongeometric duality of M theory is T duality of IIA theory. T duality gives a mapping to

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an equivalent point in the moduli space for which R PR"l/R "l/R R , i"1, 2 (20) G G G G with l unchanged. Note that we have used Eq. (4), reexpressed as l"R l. Under a T duality the string coupling constant also transforms. The rule is that the coupling of the e!ective theory (8D in this case) is invariant: 1/g"4pR R /g"4pR R /(g ) .

(21)

Thus, g "g l/R R . What does this imply for the radius of the 11th dimension R ? Using Eq. (5), R "g l PR "g l . Thus,

(22)

R "g l/R R "l/R R . However, the 11D Planck length also transforms, because

(24)

l"g lP(l )"g l implies that

(25)

(23)

(l )"g l/R R "l/R R R . (26) The perturbative IIA description is only applicable for R ;R , R . However, even though T duality was originally discovered in perturbation theory, it is supposed to be an exact nonperturbative property. Therefore, this duality mapping should be valid as an exact symmetry of M theory without any restriction on the radii. Another duality is an interchange of circles, such as R R . This corresponds to the nonperturbative S duality of the IIB theory, as we discussed earlier. Combining these dualities we obtain the desired nongeometric duality of M theory on ¹ [29]. It is given by R Pl/R R and cyclic permutations, accompanied by

(27)

lPl/R R R . Eqs. (27) and (28) have a nice interpretation. Eq. (27) implies that

(28)

1/R P(2pR )(2pR )¹ . (29) + Thus it interchanges Kaluza}Klein excitations with wrapped supermembrane excitations. It follows that these six 0-branes belong to the (3, 2) representation of the U-duality group. Eq. (28) implies that ¹ P(2pR )(2pR )(2pR )¹ . + +

(30)


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Therefore, it interchanges an unwrapped M2-brane with an M5-brane wrapped on the ¹. Thus these two 2-branes belong to the (1, 2) representation of the U-duality group. This basic nongeometric duality of M theory, combined with the geometric ones, generates the entire U duality group in every dimension. It is a property of quantum M theory that goes beyond what can be understood from the e!ective 11D supergravity, which is geometrical. This analysis has been extended to allow C O0 [30]. In this case there are indications that the torus should be considered to be noncommutative [31].

5. The D3-brane and N:4 gauge theory D-branes have a number of special properties, which make them especially interesting. By de"nition, they are branes on which strings can end } D stands for Dirichlet boundary conditions. The end of a string carries a charge, and the D-brane world-volume theory contains a ;(1) gauge "eld that carries the associated #ux. When n Dp-branes are coincident, or parallel and nearly coincident, the associated (p#1)-dimensional world-volume theory is a ;(n) gauge theory. The n gauge bosons AGH and their supersymmetry partners arise as the ground states of oriented strings I running from the ith Dp-brane to the jth Dp-brane. The diagonal elements, belonging to the Cartan subalgebra, are massless. The "eld AGH with iOj has a mass proportional to the separation of the ith I and jth branes. This separation is described by the vev of a corresponding scalar "eld in the world-volume theory. The ;(n) gauge theory associated with a stack of n Dp-branes has maximal supersymmetry (16 supercharges). The low-energy e!ective theory, when the brane separations are small compared to the string scale, is supersymmetric Yang}Mills theory. These theories can be constructed by dimensional reduction of 10D supersymmetric ;(n) gauge theory to p#1 dimensions. In fact, that is how they originally were constructed [10]. For p43, the low-energy e!ective theory is renormalizable and de"nes a consistent quantum theory. For p"4, 5 there is good evidence for the existence nongravitational quantum theories that reduce to the gauge theory in the infrared. For p56, it appears that there is no decoupled nongravitational quantum theory [32]. A case of particular interest, which we shall now focus on, is p"3. A stack of n D3-branes in type IIB superstring theory has a decoupled N"4, d"4 ;(n) gauge theory associated to it. This gauge theory has a number of special features. For one thing, due to boson}fermion cancellations, there are no ;< divergences at any order of perturbation theory. The beta function b(g) is identically zero, which implies that the theory is scale invariant (aside from scales introduced by vevs of the scalar "elds). In fact, N"4, d"4 gauge theories are conformally invariant. The conformal invariance combines with the supersymmetry to give a superconformal symmetry, which contains 32 fermionic generators. Half are the ordinary linearly realized supersymmetrics, and half are nonlinearly realized ones associated to the conformal symmetry. The name of the superconformal group in this case is SU(4"4). Another important property of N"4, d"4 gauge theories is electric}magnetic duality [33]. This extends to an SL(2, Z) group of dualities. To understand these it is necessary to include a vacuum angle h and de"ne a complex coupling 7+ q"h /2p#i 4p/g . (31) 7+ 7+

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Under SL(2, Z) transformations this coupling transforms in the usual nonlinear fashion (qP(aq#b)/(cq#d)) and the electric and magnetic "elds transform as a doublet. Note that the conformal invariance ensures that q is a meaningful scale-independent constant. Now consider the N"4 ;(n) gauge theory associated to a stack of n D3-branes in type IIB superstring theory. There is an obvious identi"cation, that turns out to be correct. Namely, the SL(2, Z) duality of the gauge theory is induced from that of the ambient type IIB superstring theory. In particular, the q parameter of the gauge theory is the value of the complex scalar "eld o of the string theory. This makes sense because o is constant in the "eld con"guration associated to a stack of D3-branes. The D3-branes themselves are invariant under SL(2, Z) transformations. Only the parameter q"o changes, but it is transformed to an equivalent value. All other "elds, such as BG , IJ which are not invariant, vanish in this case. As we have said, a fundamental (1, 0) string can end on a D3-brane. But by applying a suitable SL(2, Z) transformation, this con"guration is transformed to one in which a (p, q) string } with p and q relatively prime } ends on the D3-brane. The charge on the end of this string describes a dyon with electric charge p and magnetic q, with respect to the appropriate gauge "eld. More generally, for a stack of n D3-branes, any pair can be connected by a (p, q) string. The mass is proportional to the length of the string times its tension, which we saw is proportional to "p#qo". In this way one sees that the electrically charged particles, described by fundamental "elds, belong to in"nite SL(2, Z) multiplets. The other states are nonperturbative excitations of the gauge theory. The "eld con"gurations that describe them preserve half of the supersymmetry. As a result their masses saturate a BPS bound and are given exactly by the considerations described above. An interesting question, whose answer was unknown until recently, is whether N"4 gauge theories in four dimensions also admit nonperturbative excitations that preserve 1/4 of the supersymmetry. To explain the answer, it is necessary to "rst make a digression to consider three-string junctions. As we have seen, type IIB superstring theory contains an in"nite multiplet of strings labelled by a pair of relatively prime integers (p, q). Three strings, with charges (p , q ), i"1, 2, 3, can meet at G G a point provided that charge is conserved [34,35]. This means that p " q "0 , (32) G G if the three strings are all oriented inwards. (This is like momentum conservation in an ordinary Feynman diagram.) Such a con"guration is stable, and preserves 1/4 of the ambient supersymmetry provided that the tensions balance. It is easy to see how this can be achieved. If one regards the plane of the junction as a complex plane and orients the direction of a (p, q) string by the phase of p#qq, then Eqs. (18) and (32) ensure a force balance. The three-string junction has an interesting dual M theory interpretation. If one of the directions perpendicular to the plane of the junction is taken to be a circle, then we have a string junction in nine dimensions. This must have a dual interpretation in terms of M theory compacti"ed on a torus. We have already seen that a (p, q) string corresponds to an M2-brane with one of its cycles wrapped on a (p, q) cycle of the torus. So now we join three such cylindrical membranes together. Altogether we have a single smooth M2-brane forming a >, like a junction of pipes. The three arms are wrapped on (p , q ) cycles of the torus. This is only possible topologically when Eq. (32) is G G satis"ed.


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We can now describe a pretty construction of 1/4 BPS states in N"4 gauge theory, due to Bergman [36]. Such a state is described by a 3-string junction, with the three prongs terminating on three di!erent D3-branes. This is only possible for n53, which is a necessary condition for 1/4 BPS states. The mass of such a state is given by summing the lengths of each string segment weighted by its tension. This gives a result in agreement with the BPS formula. Clearly, this is just the beginning of a long story, since the simple picture we have described can be generalized to arbitrarily complicated string webs. So long as the web is in a plane, charges are conserved at the junctions, and all string segments are oriented in the way we have described, the con"guration will be 1/4 BPS. Remarkably, arbitrarily high spins can occur. There are simple rules for determining them [37]. When the web is nonplanar, supersymmetry is completely broken, and reliable mass calculations become di$cult. However, one should still be able to achieve a reliable qualitative understanding of such excitations. In general, there are regions of moduli space in which such nonsupersymmetric states are stable.

6. Conclusion In this brief review we have described some of the interesting advances in understanding superstring theory that have taken place in the past few years. Many others, such as studies of black hole entropy, have not even been mentioned. The emphasis has been on the nonperturbative appearance of an 11th dimension in type IIA superstring theory, as well as its implications when combined with superstring T dualities. In particular, we argued that there should be a consistent quantum vacuum, whose low-energy e!ective description is given by 11D supergravity. The relevant quantum theory } called M theory } has important features, such as the nongeometric U duality described in Section 4, that go beyond what can be understood within ordinary (nonrenormalizable) 11D supergravity. What we have described makes a convincing self-consistent picture, but it does not constitute a complete formulation of M theory. In the past two years there have been some major advances in that direction, which we will brie#y mention here. The "rst, which goes by the name of matrix theory [38], bases a formulation of M theory in #at 11D spacetime in terms of the supersymmetric quantum mechanics of N D0-branes in the large N limit. This proposal has been generalized to include an interpretation for "nite N. In that case Susskind has proposed an identi"cation with discrete light-cone quantization of M theory, in which there are N units of momentum along a null compact direction [39]. Both versions of matrix theory have passed all tests that have been carried out, some of which are very nontrivial. At times there appeared to be discrepancies, but these were all the result of subtle errors that have now been tracked down. The construction has a nice generalization to describe compacti"cation of M theory on a torus ¹L [40]. However, it does not seem to be useful for n'5 [32], and other compacti"cation manifolds are (at best) awkward to handle. Another shortcoming of this approach is that it treats the eleventh dimension di!erently from the other ones. Another proposal relating superstring and M theory backgrounds to large N limits of certain "eld theories has been put forward recently by Maldacena [41] and made more precise by others [42]. In this approach, there is a conjectured duality (i.e., equivalence) between a conformally invariant "eld theory (CFT) in n dimensions and type IIB superstring theory or M theory on an

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Anti-de-Sitter space (AdS) in n#1 dimensions. The remaining 9!n or 10!n dimensions form a compact space, the simplest cases being spheres. The three examples with unbroken supersymmetry are AdS ;S, AdS ;S, and AdS ;S. This approach is sometimes referred to as AdS/CFT duality. This is an extremely active and very promising subject. It has already taught us a great deal about the large N behavior of various gauge theories. As usual, the easiest theories to study are ones with a lot of supersymmetry, but it appears that in this approach supersymmetry breaking is more accessible than in previous ones. For example, it might someday be possible to construct the QCD string in terms of a dual AdS gravity theory, and use it to carry out numerical calculations of the hadron spectrum. Indeed, there have already been some preliminary steps in this direction [43]. Despite all of the successes that have been achieved in advancing our understanding of superstring theory and M theory, there clearly is still a long way to go. In particular, despite much e!ort and several imaginative proposals, we still do not have a convincing mechanism for ensuring the vanishing (or extreme smallness) of the cosmological constant for nonsupersymmetric vacua. Superstring theory is a "eld with very ambitious goals. The remarkable fact is that they still seem to be realistic. However, it may take a few more revolutions before they are attained.

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Neutrino propagation in matter夽 A.B. Balantekin * Institute for Nuclear Theory, University of Washington, Box 351550 Seattle, WA 98195-1550, USA Department of Astronomy, University of Washington, Box 351580 Seattle, WA 98195-1580, USA

Abstract The enhancement of neutrino oscillations in matter is brie#y reviewed. Exact and approximate solutions of the equations describing neutrino oscillations in matter are discussed. The role of stochasticity of the media that the neutrinos propagate through is elucidated. 1999 Elsevier Science B.V. All rights reserved. PACS: 14.60.Pq; 26.30.#k; 26.65.#t; 96.40.Tv Keywords: Neutrino oscillations; The MSW e!ect; Solar neutrinos; Supernova neutrinos

1. Introduction Particle and nuclear physicists devoted an increasingly intensive e!ort during the last few decades to searching for evidence of neutrino mass. Recent announcements by the Superkamiokande collaboration of the possible oscillation of atmospheric neutrinos [1] and very high statistics measurements of the solar neutrinos [2] brought us one-step closer to understanding the nature of neutrino mass and mixings. Experiments imply that neutrino mass is small and the seesaw mechanism [3], to the development of which Dick Slansky contributed, is perhaps the simplest model which leads to a small neutrino mass. If the neutrinos are massive and di!erent #avors mix they will oscillate as they propagate in vacuum [4]. Dense matter can signi"cantly amplify neutrino oscillations due to coherent forward scattering. This behavior is known as the Mikheyev}Smirnov}Wolfenstein (MSW) e!ect [5]. Matter e!ects may play an important role in the solar neutrino problem [6}8]; in transmission of

夽

Expanded version of a talk at the Slansky Memorial Symposium, Los Alamos, May 1998. * Permanent address. Department of Physics, University of Wisconsin, Madison, WI 53706, USA. E-mail address: [email protected] (A.B. Balantekin) 0370-1573/99/$ - see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 9 ) 0 0 0 1 7 - 4

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A.B. Balantekin / Physics Reports 315 (1999) 123}135

solar [8] and atmospheric neutrinos [9,10] through the Earth's core; and shock re-heating [11] and r-process nucleosynthesis [12] in core-collapse supernovae. If the neutrinos have magnetic moments matter e!ects may also enhance spin-#avor precession of neutrinos [13]. The equations of motion for the neutrinos in the MSW problem can be solved by direct numerical integration, which must be repeated many times when a broad range of mixing parameters are considered. This often is not very convenient; consequently various approximations are widely used. Exact or approximate analytic results allow a greater understanding of the e!ects of parameter changes. The purpose of this article is to present a review of the solutions of the neutrino propagation equations in matter. Recent experimental developments and astrophysical implications of the neutrino mass and mixings are beyond the scope of this article. Very rapid developments make a medium such as the World Wide Web more suitable for the former and the latter was recently reviewed elsewhere [14]. Recent experimental developments can be accessed through the special home page at SPIRES [15] and theoretical results at the Institute for Advanced Study [16] and the University of Pennsylvania [17]. An assessment of the Superkamiokande solar neutrino data was recently given by Bahcall et al. [6]. A number of recent reviews cover implications of recent results for neutrino properties [18].

2. Outline of the MSW e4ect The evolution of #avor eigenstates in matter is governed by the equation [5,20]

u(x) R W(x) " i

Rx W (x) (K I where

(K

!u(x)

W (x) , W (x) I

(1)

u(x)"(1/4E)(2(2 G N (x)E!dm cos 2h ) $ for the mixing of two active neutrino #avors and

(2)

u(x)"(1/4E)(2(2 G [N (x)!N (x)/2]E!dm cos 2h ) $ for the active-sterile mixing. In these equations

(3)

(K"(dm/4E) sin 2h , (4) dm,m!m is the vacuum mass-squared splitting, h is the vacuum mixing angle, G is the $ Fermi constant, and N (x) and N (x) are the number density of electrons and neutrons, respective ly, in the medium. In a number of cases adiabatic basis greatly simpli"es the problem. By making the change of basis

W (x) cos h(x) " W (x) sin h(x)

!sin h(x)

cos h(x)

W (x) , W (x) I

(5)


125

the #avor-basis Hamiltonian of Eq. (1) can be instantaneously diagonalized. The matter mixing angle in Eq. (5) is de"ned via sin 2h(x)"(K/((K#u(x))

(6)

cos 2h(x)"!u(x)/((K#u(x)) .

(7)

and

In the adiabatic basis the evolution equation takes the form

!(K#u(x) R W(x) i

" Rx W (x) i h(x)

!i h(x)

(K#u(x)

W (x) , W (x)

(8)

where prime denotes derivative with respect to x. Since the 2;2 `Hamiltoniana in Eq. (8) is an element of the SU(2) algebra, the resulting time-evolution operator is an element of the SU(2) group. Hence it can be written in the form [19]

;"

W (x) W (x)

!WH(x) , WH(x)

(9)

where W (x) and W (x) are solutions of Eq. (8) with the initial conditions W (x )"1 and W (x )"0. If the matter mixing angle, h(x), is changing very slowly (i.e., adiabatically) its derivatives in Eq. (8) can be set to zero. In this approximation the `Hamiltoniana in the adiabatic basis is diagonal and the system remains in one of the matter eigenstates. To calculate the electron neutrino survival probability Eq. (1) needs to be solved with the initial conditions W "1 and W "0. Using Eq. (9) the general solution satisfying these initial conditions I can be written as W (x)"cos h(x)[cos h W (x)!sin h WH(x)]#sin h(x)[cos h W (x)#sin h WH(x)] ,

(10)

where h is the initial matter angle. Once the neutrinos leave the dense matter (e.g. the Sun), the solutions of Eq. (8) are particularly simple. Inserting these into Eq. (10) we obtain the electron neutrino amplitude at a distance ¸ from the solar surface to be W (¸)"cos h [cos h W !sin h WH ]exp(i(dm/4E)¸) 1 1 # sin h [cos h W !sin h WH ]exp(!i(dm/4E)¸) , 1 1

(11)

where W and W are the values of W (x) and W (x) on the solar surface. The electron neutrino 1 1 survival probability averaged over the detector position, ¸, is then given by P(l Pl )"1"W (¸)"2 "# cos 2h cos 2h (1!2"W ") * 1 ! cos 2h sin 2h (W W #WH WH ) . 1 1 1 1

(12)

126


If the initial density is rather large, then cos 2h &!1 and sin 2h &0 and the last term in Eq. (12) is very small. Di!erent neutrinos arriving the detector carry di!erent phases if they are produced over an extended source. Even if the initial matter density is not very large, averaging over the source position makes the last term very small as these phases average to zero. The completely averaged result for the electron neutrino survival probability is then given by [21] P(l Pl )"# cos 2h 1cos 2h 2 (1!2P ) ,

(13)

where the hopping probability is P ""W " , 1

(14)

obtained by solving Eq. (8) with the initial conditions W (x )"1 and W (x )"0. Note that, since in the adiabatic limit W remains to be zero P "0. 1 3. Exact solutions Exact solutions for the neutrino propagation equations in matter exist for a limited class of density pro"les that satisfy an integrability condition called shape invariance [22]. To illustrate this integrability condition we introduce the operators AK "i R/Rx!u(x), \

AK "i R/Rx#u(x) . >

(15)

Using Eq. (15), Eq. (1) takes the form AK W (x)"(KW (x), I \

AK W (x)"(KW (x) . > I

(16)

The shape invariance condition can be expressed in terms of the operators de"ned in Eq. (15) [23] AK (a )AK (a )"AK (a )AK (a )#R(a ) . \ > > \

(17)

We also introduce a similarity transformation which formally replaces a by a : ¹K (a )O(a )¹K \(a )"O(a ) .

(18)

The MSW equations take a particularly simple form using the operators [24] BK "AK (a )¹K (a ), > >

BK "¹K \(a )AK (a ) , \ \

(19)

which satisfy the commutation relation [BK , BK ]"R(a ) , \ >

(20)

where a is de"ned using the identity R(a )"¹K (a )R(a )¹K \(a ) L L\

(21)


127

with n"1. Two additional commutation relations [BK BK , BK L ]"(R(a )#R(a )#2#R(a ))BK L , > \ > L >

(22)

[BK BK , BK \L]"(R(a )#R(a )#2#R(a ))BK \L > \ \ L \

(23)

and

can easily be proven by induction. Using the operators introduced in Eq. (19), Eq. (1) can be rewritten as BK BK W (x)"KW (x) . > \

(24)

Eqs. (22) and (23) suggest that BK and BK can be used as ladder operators to solve Eq. (24). > \ Introducing

W&exp !i u(x; a ) dx , \

(25)

one observes that AK (a )W"0"BK W . \ \ \ \

(26)

If the function L f (n)" R(a ) I I

(27)

can be analytically continued so that the condition f (k)"K

(28)

is satis"ed for a particular (in general, complex) value of k, then Eq. (22) implies that one solution of Eq. (24) is BK I W. Similarly, the wave function > \

W& exp #i u(x; a ) dx , >

(29)

satis"es the equation BK W"0 . > >

(30)

Then a second solution of Eq. (24) is given by BK \I\W. Hence for shape invariant electron \ > densities the exact electron neutrino amplitude can be written as [24]

W (x)"bBK I exp !i u(x; a ) dx #cBK \I\ exp #i u(x; a ) dx , \ >

(31)

128


where b and c are to be determined using the initial conditions W (x )"1 and W (x )"0. For the linear density pro"le N (x)"N !N (x!x ) , 0 where N is the resonant density: 2(2G N E"dm cos 2h , $ using the technique described above we can easily write down the hopping probability

(32)

P ""W (x )""exp(!pX) , where

(34)

dm sin 2h N . X" 4E cos 2h N This is the standard Landau}Zener result [21,25]. For the exponential density pro"le N (x)"N e\?V\V0 , where N is the resonant density given in Eq. (33), the hopping probability is [26] P "(e\pd\ F!e\pd)/(1!e\pd) , where we de"ned dm d" . 2Ea

(33)

(35)

(36)

(37)

(38)

4. Supersymmetric uniform approximation The coupled "rst-order equations for the #avor-basis wave functions can be decoupled to yield a second-order equation for only the electron neutrino propagation ! RW (x)/Rx![K#u(x)#i u(x)]W (x)"0 . (39) The large body of literature on the second-order di!erential equations of mathematical physics motivates using a semiclassical approximation for the solutions of Eq. (39). The standard semiclassical approximation gives the adiabatic evolution [27]. For a monotonically changing density pro"le supersymmetric uniform approximation yields [28]

i dm PH dr[f(r)!2f(r) cos 2h #1] , (40) P "exp(!pX), X" p 2E P where rH and r are the turning points (zeros) of the integrand. In this expression we introduced the scaled density f(r)"2(2G N (r)/dm/E , $

(41)


129

where N is the number density of electrons in the medium. By analytic continuation, this complex integral is primarily sensitive to densities near the resonance point. The validity of this approximate expression is illustrated in Fig. 1. As this "gure illustrates the approximation breaks down in the extreme non-adiabatic limit (i.e., as dmP0). Hence it is referred to as the quasi-adiabatic approximation. The near-exponential form of the density pro"le in [29] motivates an expansion of the electron number density scale height, r , in powers of density: N (r) (42) !r , " b NL , L N (r) L where prime denotes derivative with respect to r. In this expression a minus sign is introduced because we assumed that density pro"le decreases as r increases. (For an exponential density pro"le, N &e\?V, only the n"0 term is present). To help assess the appropriateness of such an expansion the density scale height for the Sun calculated using the Standard Solar Model density pro"le is plotted in Fig. 2. One observes that there is a signi"cant deviation from a simple exponential pro"le over the entire Sun. However the expansion of Eq. (42) needs to hold only in the

Fig. 1. The electron neutrino survival probability for the Sun [28]. The solid line is calculated using Eq. (40). The dashed line is the exact (numerical) result. The dotted line is the linear Landau}Zener result. In the top "gure, the lines are indistinguishable. An exponential density with parameters chosen to approximate the Sun was used [29]. Fig. 2. Electron number density scale height (cf. Eq. (42)) as a function of the radius for the Sun [29]. The dashed line is the exponential "t over the whole Sun. The shaded are indicates where the small angle MSW resonance takes place for neutrinos with energies 5(E(15 MeV.

130


MSW resonance region, indicated by the shaded area in the "gure. Real-time counting detectors such as Superkamiokande and Sudbury Neutrino Observatory, which can get information about energy spectra, are sensitive to neutrinos with energies greater than about 5 MeV. For the small angle solution (sin 2h&0.01 and dm"5;10\ eV), the resonance for a 5 MeV neutrino occurs at about 0.35R and for a 15 MeV neutrino at about 0.45R (the shaded area in the "gure). In that > > region the density pro"le is approximately exponential and one expects that it should be su$cient to keep only a few terms in the expansion in Eq. (42) to represent the density pro"le of the Standard Solar Model. Inserting the expansion of Eq. (42) into Eq. (40), and using an integral representation of the Legendre functions, one obtains [30]

dm L b dm L [P (cos 2h )!P (cos 2h )] , b (1!cos 2h )# X"! L> 2n#1 L\ 2E 2(2G E L $

(43)

where P is the Legendre polynomial of order n. The n"0 term in Eq. (43) represents the L contribution of the exponential density pro"le alone. Eq. (43) directly connects an expansion of the logarithm of the hopping probability in powers of 1/E to an expansion of the density scale height. That is, it provides a direct connection between N (r) and P (E ). Eq. (43) provides a quick and J J accurate alternative to numerical integration of the MSW equation for any monotonically changing density pro"le for a wide range of mixing parameters. The accuracy of the expansion of Eq. (43) is illustrated in Fig. 3 where the spectrum distortion for the small angle MSW solution is plotted. In this calculation we used the method of Ref. [31] and neglected backgrounds. The neutrino-deuterium charged-current cross-sections were calculated using the code of Bahcall and Lisi [32]. One observes that for the Sun, where the density pro"le is nearly exponential in the MSW resonance region, the "rst two terms in the expansion provide an excellent approximation to the neutrino survival probability.

5. Neutrino propagation in stochastic media In implementing the MSW solution to the solar neutrino problem one typically assumes that the electron density of the Sun is a monotonically decreasing function of the distance from the core and ignores potentially de-cohering e!ects [33]. To understand such e!ects one possibility is to study parametric changes in the density or the role of matter currents [34]. In this regard, Loreti and Balantekin [35] considered neutrino propagation in stochastic media. They studied the situation where the electron density in the medium has two components, one average component given by the Standard Solar Model or Supernova Model, etc, and one #uctuating component. Then the Hamiltonian in Eq. (1) takes the form HK "((!dm/4E) cos 2h#(1/(2)G (N (r)#NP (r)))p #((dm/4E) sin 2h)p , $ X V where one imposes for consistency 1NP (r)2"0

(44)

(45)


131

Fig. 3. (a) Spectrum distortion at SNO for the small-angle MSW solution (dm&5;10\ eV and sin 2h&0.01). The solid line is the exact numerical solution. The dashed, dot}dashed, and dotted lines result from values of n up to 0, 1, and 2 in Eq. (43). The error bars on the exact numerical result correspond to two and "ve years of data collection. The dot}dot}dot}dashed line is the spectrum without MSW oscillations, normalized to the same total rate as with MSW oscillations. Note that on the scale of this "gure the n"1 and 2 lines are not distinguishable from the exact answer. (b) The relative error arising from the use of Eq. (43).

and a two-body correlation function 1NP (r)NP (r)2"bN (r)N (r) exp(!"r!r"/q ) .

(46)

In the calculations of the Wisconsin group the #uctuations are typically taken to be subject to colored noise, i.e. higher-order correlations f 2"1NP (r )NP (r )22

(47)

are taken to be f "f f #f f #f f

(48)

and so on. Mean survival probability for the electron neutrino in the Sun is shown in Fig. 4 [36] where #uctuations are imposed on the average solar electron density given by the Bahcall}Pinsonneault model. One notes that for very large #uctuations complete #avor de-polarization should be achieved, i.e. the neutrino survival probability is 0.5, the same as the vacuum oscillation probability for long

132


Fig. 4. Mean electron neutrino survival probability in the sun with #uctuations. The average electron density is given by the Standard Solar Model of Bahcall and Pinsonneault [37] and sin 2h"0.01.

distances. To illustrate this behavior the results from the physically unrealistic case of 50% #uctuations are shown. Also the e!ect of the #uctuations is largest when the neutrino propagation in their absence is adiabatic. This scenario was applied to the neutrino convection in a corecollapse supernova where the adiabaticity condition is satis"ed [38]. Similar results were also obtained by other authors [39}42]. It may be possible to test solar matter density #uctuations at the BOREXINO detector currently under construction [43]. Propagation of a neutrino with a magnetic moment in a random magnetic moment has also been investigated [35,44]. Also if the magnetic "eld in a polarized medium has a domain structure with di!erent strength and direction in di!erent domains, the modi"cation of the potential felt by the neutrinos due polarized electrons will have a random character [45]. Using the formalism sketched above, it is possible to calculate not only the mean survival probability, but also the variance, p, of the #uctuations to get a feeling for the distribution of the survival probabilities [36] as illustrated in Fig. 5. In these calculations the correlation length q is taken to be very small, of the order of 10 km, to be consistent with the helioseismic observations of the sound speed [46]. In the opposite limit of very large correlation lengths are very interesting result is obtained [38], namely the averaged density matrix is given as an integral

1 dx exp[!x/(2b)]o( (r, x) , (49) lim 1o( (r)2" (2nb \ O reminiscent of the channel-coupling problem in nuclear physics [47]. Even though this limit is not appropriate to the solar #uctuations it may be applicable to a number of other astrophysical situations.


133

Fig. 5. Mean electron neutrino survival probability plus minus p in the sun with #uctuations. The average electron density is given by the Standard Solar Model of Bahcall and Pinsonneault and sin 2h"0.01. Panels (a), (b), (c), and (d) correspond to an average #uctuation of 1%, 2%, 4%, and 8%, respectively.

Acknowledgements This work was supported in part by the U.S. National Science Foundation Grant No. PHY9605140 at the University of Wisconsin, and in part by the University of Wisconsin Research Committee with funds granted by the Wisconsin Alumni Research Foundation. I thank Institute for Nuclear Theory and Department of Astronomy at the University of Washington for their hospitality and Department of Energy for partial support during the completion of this work.

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M(ysterious) patterns in SO(9) Teparksorn Pengpan*, Pierre Ramond Institute for Fundamental Theory, Department of Physics, University of Florida, Gainesville FL 32611, USA

Abstract The light-cone little group, SO(9), classi"es the massless degrees of freedom of 11-dimensional supergravity, with a triplet of representations. We observe that this triplet generalizes to four-fold in"nite families with the quantum numbers of massless higher spin states. Their mathematical structure stems from the three equivalent ways of embedding SO(9) into the exceptional group F . 1999 Elsevier Science B.V. All rights reserved. PACS: 0.2.20.Sv; 04.65.#e; 11.30.Pb; 12.60.Jv

1. N ⴝ 1 supergravity in eleven dimensions It has been recently pointed out that 11-dimensional supergravity is the local limit of a much bigger theory, called M-theory [1], that also contains in di!erent limits all known string theories in 10 dimensions. At present, it is still elusive, and only a partial formulation [2] exists in the literature. Since M-theory lives in 11 dimensions, its massive degrees of freedom must be expressible as multiplets of SO(10), the Lorentz little group of eleven dimensions and its massless degrees of freedom must form in representations of SO(9). Among those are the "elds of the local supergravity theory which reveal themselves in the local limit. While it is likely that some of the physical objects in M-theory are not local, one still expects that they would be expressible in terms of in"nite towers of representations of these little groups. There is a pervading lore against interacting theories that contain massless states of higher spin. It is based on several no-go theorems, formulated in terms of local "eld theory [3]. They state that relativistically invariant theories with a "nite number of local massless "elds of spin higher than

* Corresponding author. 0370-1573/99/$ - see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 9 ) 0 0 0 1 8 - 6

138

T. Pengpan, P. Ramond / Physics Reports 315 (1999) 137}152

two, and with a "nite number of derivatives in their interactions, do not exist. It follows that any such theory with an in"nite tower of "elds, and arbitrarily high derivative couplings escapes the no-go theorems and could conceivably exist. Even with supersymmetry, building such a theory seems like a hopeless task, and the many published attempts have met with partial success. A four-dimensional formulation [4] uses an in"nite-dimensional superalgebra, with the interesting feature that it necessarily contains a cosmological constant. Hence it seems that the lore against massless high-spin interacting theories is mainly based on the di$culties associated with their construction rather than on their impossibility. Since M-theory is most likely non-local, it may evade the no-go theorems, and could contain an in"nite number of "elds. It is therefore interesting to examine the SO(9) properties of 11-dimensional supergravity, whose massless states are local limit of M-theory. In the following, we would like to draw attention to a remarkable mathematical fact, which shows that the supergravity triplet of SO(9) representations is actually the tip of a mathematical iceberg. We will start by presenting group-theoretical evidence that the supergravity representations are the "rst of an in"nite family of massless states of higher spin. Then we will o!er a mathematical resolution in terms of embeddings of SO(9) into the exceptional group F , as well as some generalizations. Since there are no coincidences in the study of these highly constrained theories, it is tempting to muse that these extra higher-spin massless states represent the degrees of freedom of M-theory, even though we have not been able to obtain any dynamical evidence for this conjecture.

2. Group phenomenology of SO(9) The classical Lie group SO(9) plays an important dual role in the study of theories in ten and eleven dimensions, as the light-cone little group of Lorentz-invariant theories in 10 space and one time dimensions, and as the little group of massive representations of theories in 9 space and one time dimensions. The representations of SO(9) are best described in Dynkin's language, which Dick Slansky used to great e!ectiveness in particle physics [5]. As a rank 4 Lie algebra, it takes four positive integers to label its irreducible representations, in the form [a a a a ]. Its four basic representations are: E Vector, [1000], with nine components, < , G E Adjoint, [0100], with 36 components, B , GH

E Three-form, [0010], with 84 components, B , GHI

E Spinor, [0001], with 16 components, t . ? All representations with odd a are spinorial. The irreps of SO(9) are characterized by "ve generalized Dynkin indices I , wN, p"0, 2, 4, 6, 8 , (1) N where w are the weights in the representation. Thus I is the dimension of the irrep, and I is related to the quadratic Casimir invariant by C "36I /I .


139

Table 1 Irrep

[1001]

[2000]

[0010]

I I I I I

128 256 640 1792 5248

44 88 232 712 2440

84 168 408 1080 3000

N"1 supergravity in 11 dimension is a local "eld theory that contains three di!erent massless "elds, two bosonic that describe gravity and a three-form, and one Rarita}Schwinger spinor. Its physical degrees of freedom are classi"ed in terms of the light-cone little group, SO(9), E Graviton as a symmetric second-rank tensor, [2000], G , GH E Third-rank antisymmetric tensor, [0010], B , GHI

E Rarita}Schwinger spinor-vector, [1001], W . ?G Their group-theoretical properties are summarized in Table 1 [6]. We note that these indices, except for I , match between the fermion and the two bosons. As is well known, equality of the bosonic and fermionic dimensions is an indication of supersymmetry. On the light-cone, the supersymmetry algebra reduces to +Q , Q ,"d , (2) ? @ ?@ where the supersymmetric generators transform as the 16 spinor of SO(9). They split into creation and annihilation operators under the decomposition SO(9)MSO(6);SO(3), 16"(4, 2)#(4, 2)

(3)

and we obtain a Cli!ord algebra +QI , QI R,"1 ,

(4)

where QI transforms as (4, 2), and QI R as (4, 2). The states of the Sugra multiplet are then obtained by successive applications of the QI R on the vacuum state, to yield 128 bosons and 128 fermions +1, QI R,(QI R),2, (QI R), (QI R),"02 .

(5)

The equality between the number of bosons and fermions is manifest. All three irreps have the same quadratic Casimir invariant, since they have the same I /I ratio. Surprisingly, we have found that some higher spin representations of SO(9) also occur in triples with the same quadratic Casimir invariant, and show remarkable group-theoretical kinships with the supergravity triplet. The higher-spin triplets appear in four di!erent types, S}¹}¹, S}S}S, ¹}¹}S, and ¹}S}¹, where S describes fermionic (odd a ), and ¹ bosonic (even a ) degrees of freedom. The largest representation is listed "rst, and its dimension is equal to the sum of dimensions of the other two irreps of the triplet. Thus only triplet of the S}¹}¹ type display supersymmetry-like properties.

140


Table 2 Irrep

[2100]

[0110]

[1101]

I I I I I

910 3640 19 864 130 840 977 944

1650 6600 34 920 217 320 1 498 344

2560 10 240 54 784 348 160 2 466 304

2.1. S}T}T triples In Dynkinese, these triples are of the form [1#p#2r, n, p, 1#2q#2r][2#p#2q#2r, n, p, 2r][p, n, 1#p#2r, 2q] ,

(6)

labelled by four integers, n, p, q, r"1, 2,2; the sum of the Dynkin invariants I , and I , I , I , over the bosons match those of the fermion representation. All three have the same quadratic Casimir invariant. The simplest of this class is the supergravity multiplet which we have already discussed, and only the lowest of these triples has manifest supersymmetry. The number of fermions and bosons of each triplets are equal, and a multiple of 128, but their construction does not follow that of the supergravity multiplet as polynomials of QI R acting on some state. They appear to be supersymmetric without supersymmetry, the simplest being: E The supertriple, with n"1, p"q"r"0, contains [2100] #[0110] #[1101] , with group-theoretic numbers given by Table 2 and described by "elds of the form

(7)

h #A #W . (8) GHIJ GHIJK ?GHI Their index structure indicates the appearance of higher spin "elds. It is not possible to generate this triple by repeated use of the light-cone supersymmetry algebra acting on some "eld "j2, with dimension equal to 20 (1, QI R, (QI R),2, (QI R), (QI R))"j2 .

(9)

This would imply that "j2 appears twice in the triple, but the triple contains no duplicate representations of SO(6);SO(3) that add up to dimension equal to 20. These "elds appear in the Kronecker product of the supergravity triplet with the two-form [0100], [2000][0100]"[2100][2000]+[1010][0100], ,

(10)

[0010][0100]"[0110][0010]+[1002][1000][0110][1100][0002], ,

(11)

[1001][0100]"[1101][1001] +[2001][1101][1001][0101][0011][0001], .

(12)


141

Table 3 Irrep

[3010]

[1020]

[2011]

I I I I I

7700 46 200 384 360 3 938 760 46 646 664

12 012 72 072 585 624 5 748 792 64 127 736

19 712 118 272 969 984 9 687 552 110 529 792

Irrep

[4000]

[0012]

[1003]

I I I I I

450 2200 15 160 130 360 1 325 944

4158 20 328 131 784 1 016 520 8 839 560

4608 22 528 146 944 1 146 880 10 103 296

Table 4

A suitable product can be found that automatically traces out the extra representations (in the curly brackets), subtracts all traces, and all totally antisymmetric tensors. The states of this triple could be understood as some sort of bound state between the supergravity states and something having the Lorentz properties of a 2- or 7-form in the light cone little group. Whether this union can be consumated through actual dynamics remains to be seen. Although this feature usually associated with supersymmetry remain, it is not a supermultiplet. E The second tower of triples is obtained by multiplying all its representations by [1010], and performing suitable subtractions. This representation appears in the antisymmetric product of two second-rank antisymmetric tensor "elds. It could therefore be generated by applying two two-forms on the supergravity multiplet, resulting in a bound state between supergravity and two branes. The simplest with p"1, n"q"r"0, contains [3010] #[1020] #[2011] , (13) with group theory table (Table 3). E The third tower is more complicated, multiplying the fermion and the second boson by [0002], and the "rst boson by [2000]. The simplest in this series, with q"1 n"p"r"0, and contains (see Table 4) [4000] #[0012] #[1003] . (14) E The fourth in"nite tower is also twisted. The simplest of this series has r"1, with content [4002] #[0030] #[3003] and group-theory mugshot (Table 5).

(15)

142


Table 5 Irrep

[4002]

[0030]

[3003]

I I I I I

32 725 261 800 2 938 280 41 127 080 673 801 256

23 595 188 760 2 055 768 27 239 256 414 212 568

56 320 450 560 4 994 048 68 366 336 1 084 279 808

Table 6 Irrep

[2003]

[4001]

[0021]

I I I I I

18 480 117 040 1 010 992 10 640 944 128 166 448

5280 33 440 297 632 3 303 584 43 030 688

13 200 83 600 713 360 7 337 360 85 922 192

2.2. S}S}S triples Here all three representations are spinors [2#p#2r, n, p, 3#2q#2r][4#p#2q#2r, n, p, 1#2r][p, n, 2#p#2r, 1#2q] (16) and the dimension of the "rst is the sum of the other two. The simplest example is [2003][4001][0021] .

(17)

Its group-theory mugshot is given in Table 6. 2.3. T}T}S triples In this class, the dimension of the largest boson (listed "rst) is equal to that of the spinor and the second boson, [1#p#2r, n, p, 2#2q#2r][3#p#2q#2r, n, p, 2r][p, n, 1#p#2r, 1#2q] .

(18)

The lowest member of this class is [1002][3000][0011] with mugshot (Table 7).

(19)


143

Table 7

Irrep

[1002]

[3000]

[0011]

I I I I I

924 3080 13 400 68 216 3 82 328

156 520 2392 13 432 87 544

768 2560 11 008 54 784 2 99 776

Irrep

[2002]

[3001]

[0020]

I I I I I

3900 18 200 1 14 920 8 75 720 7 549 064

1920 8960 57 728 4 55 936 4 148 096

1980 9240 57 192 4 19 784 34 53 384

Table 8

2.4. T}S}T triples The last class contains the representations [2#p#2r, n, p, 2#2q#2r][3#p#2q#2r, n, p, 1#2r][p, n, 2#p#2r, 2q] .

(20)

Its lowest-lying member is [2002][3001][0020]

(21)

with mugshot (Table 8). There are several triples which only match dimensions and quadratic Casimir invariants; we found one made entirely of spinors [1033][7001][0305],

[7122][6008][4018]

(22)

with the dimension of the "rst equal to the sum of the other two, and all with the same quadratic Casimir, but their I do not match. 2.5. Basic operations It is possible to understand these di!erent triples in terms of four basic operations, which starting from the supergravity multiplet, generate all triples: E D : Increase the Dynkin labels all three irreps within a triple by [0100]. E D : Increase the Dynkin labels of all three irreps within a triple by [1010].

144


E D : Increase the Dynkin labels of the "rst and third irreps by [0001], the second by [1000]. E D : Increase the Dynkin labels of the "rst and second irreps by [1001], the third by [0010]. The D operations may be simplest to understand as they can be generated by applying representations that appear either as the light-cone 2-form [0100], or in its twice-antisymmetrized product, since ([0100][0100]) "[0100][1010] . (23) A light-cone 2-form may indicate a brane state, and these triples could then be understood as bound states of the supergravity "elds with these branes. The third and fourth operations are more complicated as they treat the di!erent members di!erently. However, starting from the supergravity multiplet, they generate all other triples, as shown in the diagram below, where the upward arrow denotes D , and the downward arrow denotes D :

It is clear that the supergravity multiplet sits at the beginning of a very intricate and beautiful complex of irreps of SO(9). Limited by the two dimensions of the paper, we have not shown the e!ect of the D operations which act uniformly on any of the triples in the picture. The whole pattern is summarized by the general form of the triples [1#a #a , a , a , 1#a #a ][2#a #a #a , a , a , a ] [a , a , 1#a #a , a ] , where a are non-negative integers. G

(24)

3. Mathematical origin of the triples So far we have only o!ered numerical evidence for the remarkable structure of the SO(9) representations. A recent paper by Gross et al. [7] and one of us (P.R.), unveils its mathematical


145

origin. The following is a watered-down version of its contents. It points to a construction of a more general character, but does not (yet) seem to shed light on its physical interpretation. The triples stem from the triality of SO(8), which is explicitly realized in F , and the three equivalent ways to embed SO(9) into F . That very triality is already familiar to particle physicists: the three equivalent ways to embed SU(2);;(1) in SU(3), called I-spin, ;-spin, and . 3.2. Flavoring of bare Pomeron We have proposed sometime ago that `baryon paira, together with other `heavy #avora production, provides an additional energy scale, s "eW, for soft Pomeron dynamics, and this e!ect can be responsible for the perturbative increase of the Pomeron intercept to be greater than unity, aP(0)&1#e, e'0. One must bear this additional energy scale in mind in working with a soft Pomeron [15]. That is, to fully justify using a Pomeron with an intercept aP(0)'1, one must restrict oneself to energies s's where heavy #avor production is no longer suppressed. Conversely, to extrapolate Pomeron exchange to low energies below s , a lowered è!ective trajectorya must be used. This feature of course is unimportant for total and elastic cross sections at Tevatron energies. However, it is important for di!ractive production since both m\ and M will sweep right through this energy scale at Tevatron energies. Flavoring becomes important whenever there is a further inclusion of e!ective degrees of freedom than that associated with light quarks. This can again be illustrated by a simple one-dimensional multiperipheral model. In addition to what is already contained in the Lee}Veneziano model, suppose that new particles can also be produced in a multiperipheral chain.

184

C.-I. Tan / Physics Reports 315 (1999) 175}198

Fig. 2. E!ect of #avoring factor R(s) when applied to a standard rising cross section: p "bsC, e"0.1 and b"16 mb, given by the solid curve. With R(y) given by Eq. (31), the dashed}dotted curve has e "0, j "1, and #avoring scale y "9, and the dotted curve corresponds to e "!0.04. Fig. 3. Renormalization factor due to #avoring alone, Z (m; s),R(m\)R(M), as a function of rapidity y"log m\ for various "xed center of mass energies. These curves correspond to parameters used for the solid line in Fig. 4.

Concentrating on the cylinder level, the partial cross sections will be labeled by two indices (12) p K(g/p!q!)2N>O(gN log s)N(gN log s)Os(\ , NO where q denotes the number of clusters of new particles produced. Upon summing over p and q, we obtain a `renormalizeda Pomeron trajectory a P "a P #e ,

(13)

where a P K1 and eK2gN. That is, in a non-perturbative QCD setting, the e!ective intercept of Pomeron is a dynamical quantity, re#ecting the e!ective degrees of freedom involved in nearforward particle production [15]. If the new degree of freedom involves particle production with high mass, the longitudinal phase space factor, instead of (log s)O, must be modi"ed. Consider the situation of producing one NNM bound state together with pions, i.e., p arbitrary and q"1 in Eq. (12). Instead of (log s)N>, each factor should be replaced by (log(s/m ))N>, where m is an e!ective mass for the NNM cluster. In terms of rapidity, the longitudinal phase space factor becomes (>!d)N>, where d can be thought of as a one-dimensional èxcluded volumea e!ect. For heavy-particle production, there will be an energy range over which keeping up to q"1 remains a valid approximation. Upon summing over p, one "nds that the additional contribution to the total cross section due to the production of one heavy-particle cluster is [13] p &p (>!d)(2gN)log(>!d)h(>!d), where a P K1. Note O the e!ective longitudinal phase space `threshold factora, h(>!d), and, initially, this term represents a small perturbation to the total cross section obtained previously (corresponding to q"0 in Eq. (12)), p . Over a rapidity range [d, d#d ], where d is the average rapidity required for producing another heavy-mass cluster, this is the only term needed for incorporating this new degree of freedom. As one moves to higher energies, `longitudinal phase space suppressiona


185

becomes less important and more and more heavy-particle clusters will be produced. Upon summing over q, we would obtain a new total cross section, described by a renormalized Pomeron, with a new intercept given by Eq. (13). We assume that, at Tevatron, the energy is high enough so that this kind of `thresholda e!ects is no longer important. How low an energy do we have to go before one encounter these e!ects? Let us try to answer this question by starting out from low energies. As we have stated earlier, for >'3}5, secondary trajectories become unimportant and using a Pomeron with aK1 becomes a useful approximation. However, as new #avor production becomes e!ective, the Pomeron trajectory will have to be renormalized. We can estimate for the relevant rapidity range when this becomes important as follows: y '2d #1q2 d . The "rst factor d is associated with leading

particle e!ect, i.e., for proton, this is primarily due to pion exchange. d is the minimum gap associated with one heavy-mass cluster production, e.g., nucleon}antinucleon pair production. We estimate d K2 and d K2}3, so that, with 1q2 K2, we expect the relevant #avoring rapidity

scale to be y K8}10. 4. Pomeron dominance hypothesis at Tevatron energies We shall "rst explore consequences of the observation that both total cross sections and elastic cross sections can be well described by a Pomeron pole exchange at Tevatron energies. Absorption correction, if required, seems to remain small. Since the singly di!ractive cross section, p, is a sizable part of the total, it must also grow as sC. This qualitative understanding can be quanti"ed in terms of a sum rule for `rapidity gapa cross sections. This in turn imposes a convergence condition on our unitarized Pomeron #ux, F P(m, t). ? To simplify the discussion, we shall "rst ignore transverse momentum distribution by treating the longitudinal phase space only. For instance, for singly di!raction dissociation, the longitudinal phase space can be speci"ed by two rapidities, y,log(m\) and y ,log M. The "rst variable

speci"es the rapidity gap associated with the detected leading proton (or antiproton), and the second variable speci"es the rapidity `spana of the missing mass distribution. At "xed s, they are constrained by y#y K>,log s (see Fig. 1), and we can speak of di!erential di!ractive cross

section dp/dy. We shall in what follows use +m\, M, and +y,log m\, y ,log M, inter changeably. Dependence on transverse degrees of freedom can be reintroduced without much e!ort after completing the main discussion. Consider the process a#bPc#X, where the number of particles in X is unspeci"ed. However, unlike the usual single-particle inclusive process, the superscript for X indicates that all particles in X must have rapidity less than that of the particle c, i.e., the detected particle c is the one in the "nal state with the largest rapidity value. Kinematically, a single-gap cross section is identical to the singly di!raction dissociation cross section discussed earlier. Under the assumption where all transverse motions are unimportant, one has y Ky ,> and the di!erential gap cross section, A ?

dp /dy, can also be considered as a function of y and y , with y#y K>.

Because the detected particle c has been singled out to be di!erent from all other particles, this is no longer an inclusive cross section, and it does not satisfy the usual inclusive sum rules. Upon integrating over the rapidity gap y and summing over particle type c, no multiplicity enhancement factor is introduced and one obtains simply the total cross section, i.e., a gap cross section satis"es

186


the following exact sum rule:

dp (14) dy ?@A, p " p "p . ?@A L ?@ dy A L A Interestingly, this allows an identi"cation of the total cross section as a sum over speci"c gap cross sections, p , each is `deriveda from a speci"c `leading particlea gap distribution. Since no ?@A restriction has been imposed on the nature of the `gap distributiona, e.g., particle c can have di!erent quantum numbers from a, the notion of a gap cross section is more general than a di!raction cross section. For di!raction dissociation, c must have same quantum numbers as the incident particle a. For de"niteness, we discuss leading particle gap distribution relative to the incoming particle a with a positive, large rapidity, and assume c"a. For pp collision, actual di!ractive cross section, p, is arrived at by taking into account contributions involving di!raction at both p and p vertices. It follows that the singly di!ractive dissociation cross section, p, is a part of p . ?@ ?@? Consider next our factorized ansatz, Eq. (3). For y and y large, it leads to a gap distribution,

dp /dy"e\WF (y)p (y ). If p (y )"gb eCW , it follows that contribution from each gap distribu?@A ?A @ @ @ tion is Regge behaved, p KbA eC7b , where the total Pomeron residue is a sum of `partial residuesa ?@A ? @ b " bA " dy FA (y)ge\>CW . (15) ? ? ? A A For above integral to converge, each #ux factor must grow slower than eCW. That is, FA (y)e\CWP0 as ? yPR. In a traditional Regge approach, the large rapidity gap behavior for each #ux factor is controlled by an appropriate Regge propagator, e?G>?H\W. Clearly, a standard triple-Pomeron behavior with aP'1 is inconsistent with the pole dominance hypothesis. Unitarity correction must supply enough damping to provide convergence. There is yet another way of expressing the consequence of the pole dominance hypothesis. Dividing each gap di!erential cross section by the total cross section, factorization of Pomeron leads to a `limiting distributiona: o (y,>)Po (y), oA (y). That is, the limit is independent of the A ? ?@ ? total rapidity, >, and the gap density is normalizable, dy o (y)" dy oA (y)" (bA /b )"1. A A ? ? ? ? This normalization condition for the gap distribution is precisely Eq. (15). Let us now restore the transverse distribution and concentrate on the di!ractive dissociation contribution, which can be identi"ed with the high M and high m\ limit of dp (t, m; M). In ?@? terms of m, M, and t, the di!erential cross section at large M under our factorizable ansatz takes on the following form, dp/dt dmKF P(m, t)p P (M), where p P (M)"gPPP(t)(M)Cb (0). It fol@ @ @ ?@ ? lows from Eq. (15) that F P(m, t) must satisfy the following bound: ? K (16) dm F PgPPP(t)(m, t)mC4 dt dt dm F P(m, t)gPPP(t)mC,b (b (0) . ? ? ? ? \ K Q \ The hypothesis of a Pomeron pole dominance for the total and elastic cross sections is of course only approximate. However, to the extend that absorptive corrections remain small at Tevatron energies, one "nds that a modi"ed Pomeron #ux factor must di!er from the `classicala Pomeron #ux at small m in such a way so that the upper bound in Eq. (16) is satis"ed. We shall refer to F P(m, t) as the ùnitarized Pomeron #uxa. How this can be accomplished via "nal-state screen? ing will be discussed next. Note both the similarity and the diwerence between Eq. (16) and the


187

`#ux normalizationa condition mentioned in the Introduction. Here, this convergent integral yields asymptotically a "nite number, b , and the ratio b /b (0) can be interpreted as the probability of ? ? ? having a di!ractive gap at high energies. 5. Final-state screening The best-known example for implementing the idea of `screeninga in high-energy hadronic collisions has been the èxpanding diska picture for rising total cross sections. Di!raction scattering as the shadow of inelastic production has been a well established mechanism for the occurrence of a forward peak. Analyses of data up to collider energies have revealed that the essential feature of non-di!ractive particle production can be understood in terms of a multiperipheral clusterproduction mechanism. In such a picture, the forward amplitude is predominantly absorptive and is dominated by the exchange of a `bare Pomerona. If the Pomeron intercept is greater than one, it forces further unitarity corrections as one moves to higher energies. For instance, saturation of the Froissart bound can be next understood through an eikonal mechanism, with the absorptive eikonal s(s, b) given by the bare Pomeron amplitude in the impact-parameter space. The main problem we are facing here is not so much on how to obtain a `most accuratea #ux factor F P(m, t) at very small m. We are concerned with a more di$cult conceptual problem of how ? to reconcile having a potentially large screening e!ect for di!raction dissociation processes and yet being able to maintain approximate pole dominance for elastic and total cross sections up to Tevatron energies. We shall show using an expanding disk picture that absorption works in such a way that inelastic scattering can only take place on the èdgea of disk. Therefore, once applied using "nal-state screening, the e!ect of initial-state absorption will be small, hence allowing us to maintain Pomeron pole factorization for elastic and total cross sections. 5.1. Expanding disk picture Let us brie#y review this picture which also serves to establish notations. At high energies, a near-forward amplitude can be expressed in an impact-parameter representation via a twodimensional Fourier transform,

¹(s, t),2is dbe q b fI (s, b),

fI (s, b)"(4ins)\ dqe\ b q¹(s, t) ,

(17)

where tK!q. Assume that the near-forward elastic amplitude at moderate energies can be described by a Born term, e.g., that given by a single Pomeron exchange where we shall approximate it to be purely absorptive. Let us denote the contribution from the Pomeron exchange to fI (s, b) as s(s, b). With aP(t)"1#e#aPt, and approximating b(t) by an exponential, we "nd s(s, b)K X(s)e\@ Q, X(s),s(s, 0)Kp (s)/4B(s) ,

(18)

This mechanism has been recognized before. It was used in Ref. [20] to explain why a `maximal odderona cannot be allowed in any hadronic scheme which admits an expanding-disk interpretation at high energies.

188


where B(s)"b #aP log s and p (s)"p sC. With e'0, the Born approximation would eventually violate s-channel unitarity at small b as s increases. A systematic procedure which in principle should restore unitarity is the Reggeon calculus. However, our current understanding of dispersion-unitarity relation is too qualitative to provide a de"nitive calculational scheme. The key ingredient of `screeninga correction is the recognition that the next-order correction to the Born term must have a negative sign. (The sign of double-Pomeron cut contribution.) In an impact-representation, Reggeon calculus assures us that the correction can be represented as fI (s, b)K!(1/2!)k(s)s(s, b) , (19) where k is positive. To go beyond this, one needs a model. A physically well-motivated model which should be meaningful at moderate energies and allows easy analytic treatment is the eikonal model. Writing fI (s, b)"fI (s, b)#fI (s, b)#fI (s, b)#2, the expansion alternates in sign, and with simple weights such that fI (s, b)"[1!e\IQQ@]/k, and

¹(s, t)"

2is k

dbe q b+1!e\IQQ@, .

(20)

Conventional eikonal model has k"1. We keep k41 here so as to allow the possibility that screening is ìmperfecta. Observe that the eikonal derived from the Pomeron exchange, s(s, b), is a monotonically decreasing function of b, taking on its maximum value X(s) at b"0, which increases with s due to e'0. The eikonal drops to zero at large b and is of the order 1 at a radius, b (s)K A (B(s)log kX(s)&log s. Within this radius, fI (s, b)"O(1) and it vanishes beyond. This is the èxpansion diska picture of high-energy scattering, leading to an asymptotic total cross section O(b (s)). A 5.2. Inelastic screening In order to discuss inelastic "nal-state screening, we follow the `shadowa scattering picture in which the `minimum biaseda events are predominantly `short-range ordereda in rapidity and the production amplitudes can be described by a multiperipheral cluster model. Substituting these into the right-hand side of an elastic unitary equation, Im ¹(s, 0)" "¹ ", one "nds that the resulting L L elastic amplitude is dominated by the exchange of a Regge pole, which we shall provisionally refer to as the `bare Pomerona. Next consider singly di!ractive events. We assume that the `missing massa component corresponds to no gap events, thus the distribution is again represented by a `bare Pomerona. However, for the gap distribution, one would insert the `bare Pomerona just generated into a production amplitude, thus leading to the classical triple-Pomeron formula. Extension of this scheme leads to a `perturbativea treatment for the total cross section in the number of bare Pomeron exchanges along a multiperipheral chain. Such a scheme was proposed long time ago [17], with the understanding that the picture could make sense at moderate energies, provided that the intercept of the Pomeron is one, a(0)K1, or less. However, with the acceptance of a Pomeron having an intercept greater than unity, this expansion must be embellished. Although it is still meaningful to have a gap expansion, one must improve the descriptions for parts of a production amplitude involving large rapidity gaps by taking into account absorptions for the


189

gap distribution. We propose that this `partial unitarizationa be done for each gap separately, thus maintaining the factorization property along each short-range ordered sector. This involves "nal-state screening, and, for singly di!raction dissociation, it corresponds to the inclusion of ènhanced Pomeron diagramsa in the triple-Regge region. To simplify the notation, we shall use the energy variable, m\, and the rapidity gap variable, y"log(m\), interchangeably. Let us express the total unitarized contribution to a gap cross section in terms of a ùnitarized #uxa factor, F(y, t),(eW/16p)"g (y, t)",(1/16pm)" f (m, t)", so that it B B reduces to the classical triple-Pomeron formula as its Born term. That is, the corresponding Born amplitude for f (m, t) is the `square-roota of the triple-Pomeron contribution to the classical B formula, fP (m, t)"bP(t)(m\)?PR\. Screening becomes important if large gap becomes favored, i.e., when e'0. Let us work in an impact representation, g(y , t),2idqe b qg (y , b). Consider an expansion B B B g (y, b)"s (y, b)#g (y, b)#g (y, b)#2 where, under the usual exponential approximation for B the t-dependence, the Fourier transform for the Born term is s (y, b)"(pB(y)/4B (y)) e\@ BW, with B B B (y)"b #aPy and pB(y)"p eCW. The key physics of absorption is B B B g (y, b)K!k s(y, b)s (y, b) . (21) B B The proportionality constant k can be di!erent from the constant k introduced earlier for the B elastic screening, either for kinematic or dynamic reasons, or both. For a generalized eikonal approximation, one has g (y, b)"s +1!k s# (k s)!2,"s (y,b)e\IBQW @. If we de"ne the B B r B B `"nal-state screeninga factor as the ratio between the unitarized #ux factor and the classical triple-Pomeron formula, F P(y, t)"S(y, t; X)F P(y, t), we then have ? ? S(y, t; X)"" f (y, t)/f (y, t)" . (22) B B We shall use this expression as a model for probing the physics of inelastic screening in an expanding disk picture. Let us examine this eikonal screening factor in the forward limit, t"0, where dbs (y, b)e\IBQW @ B . (23) S(y, 0; X)" dbs (y , b) B B Unlike the elastic situation, the integrand of the numerator is strongly suppressed both in the region of large b and in the region ìnsidea the expanding disk. The only signixcant contribution comes from a `ringa region near the edge of the expanding disk [20]. Since the value of the integrand is of O(1) there, one "nds that the numerator varies with energy only weakly. On the other hand, the denominator is simply pB(y), which increases as eCW. Therefore, this leads to an exponential cuto! in y, S(y, 0; X)&e\CW. This damping factor precisely cancels the m\C behavior from the classical triple-Pomeron formula at small m, leading to a unitarized Pomeron #ux factor.

It is possible also to apply an eikonal model to study initial-state screen for singly di!ractive dissociation cross section. This indeed has been performed before, however, without taking "nal-state screening into account [12]. The fact that inelastic absorption takes place at small impact parameter, with surviving scattering allowed only at the edge of the expanding disk has also been noted there. Since our "nal-state absorption would remove all scattering within the disk, applying an eikonal initial-state absorption procedure becomes unnecessary.

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To be more precise, let us work in a simpler representation for S(y, 0; X) by changing the variable bPz,e\@ W. With our Gaussian approximation, one "nds that

S(y, 0; X)" r

dz z P\e\VX

, (24) VIB6WPPW , where where r(y),B(y)/B (y). This expression can be expressed as +rx\PC(x, r)," B B VI 6WPPW C(x, r)"V dz zP\e\X is the incomplete Gamma function. In this representation, one easily veri"es that the screening factor has the desired properties: As k P0, screening is minimal and one B has S(y, 0; X)P1. On the other hand, for y large, X(y) increases so that S(y, 0; X)P[k X(y)]\, B as anticipated. Similarly, we "nd that the logarithmic width for the unitarized #ux D(y, t) at t"0 has increased . As k P0, from 2B to 2B where B "B +!r log dz zP\e\VX, B B VI 6WPPW B B B B P B PB . For y very large, B PB log k X(y)&b (y)Jy. This corresponds to a faster B B B B A shrinkage than that of ordinary Regge behavior. Averaging over t, one "nds at large di!ractive rapidity y, the "nal-state screening provides an average damping 1S 2Pe\CW"mC . (25) This leads to a unitarized Pomeron #ux, F P(m, t), which automatically satis"es the upper bound, ? Eq. (16), derived from the Pomeron pole dominance hypothesis [12].

6. Final recipe Having explained earlier the notion of #avoring and its e!ects both on Pomeron intercept and on its residues, we must build in this feature for the "nal-state screening. As we have shown in the last section, inelastic screening is primarily driven by the ùnitarity saturationa of the elastic eikonal (Eq. (18)). However, because of #avoring, screening sets in only when the Pomeron #avoring scale is reached. This picture is consistent with the fact that, while low-mass di!raction seems to be highly suppressed, high-mass di!raction remains strong at Tevatron energies. Since a Pomeron exchange enters as a Born term, i.e., the eikonal for either the elastic or the inelastic di!ractive production, #avoring can easily be incorporated if we multiply both s(y, b) and s (y, b) by a #avoring factor R(y). That is, if we adopt a generalized eikonal model for "nal-state B screening, the desired screening factor becomes S (m, t)"S(y, t; R(y)X(y), k ) (26) B where S is given by Eq. (22). We have also explicitly exhibited the dependence on the maximal value of the #avored elastic eikonal, RX, and on the e!ectiveness parameter k . B Let us now put all the necessary ingredients together and spell out the details for our proposed resolution to Dino's paradox. Our xnal recipe for the Pomeron contribution to single-di!raction dissociation cross section is (27) dp/dt dm"F P(m, t)pP (M) , @ ? where the unitarized #ux, F P, and the Pomeron-particle cross section, pP , are given in @ ? terms of their respective classical expressions by F P(m, t),Z (m, t)F P(m, t) and pP (M), @ ? B ?


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Z (M)p P (M, t). It follows that the total suppression factor is @ K Z(m, t; M)"Z (m, t)Z (M)"[S (m, t)R(m\)]R(M) (28) B K with the screening factor given by Eq. (26) and the #avoring factor given by Eq. (11). Finally, we point out that the integral constraint for the unitarized #ux (Eq. (5)), when written in terms of these suppression factors, becomes

dt dm S (m, t)R(m\)F P(m, t)g(t)mC"b (b (0) , ? ? ? \ where F P(m, t),(1/16p)b (t)(m\)?PR\. ? ?

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6.1. Phenomenological parameterizations Both the screening function and the #avoring function depend on the e!ective Pomeron intercept, and we shall adopt the following simple parameterization. The transition from a (0)"1#e to a(0)"1#e will occur over a rapidity range, (y, y). Let y ,(y#y) and j\,(y!y). Similarly, we also de"ne e ,(e#e ) and *,(e!e ). A convenient parameterization for e we shall adopt is e (y)"[e#* tanhj (y!y )] . (30) The combination [e!e (y)] can be written as (2e )[1#(s/s )H]\ where s "eW. Combining this with Eq. (11), we arrive at a simple parameterization for our #avoring function (31) R(s),(s /s)C >QQ H \. With a P K1, we have e K0, e K*Ke/2, and we expect that j K1}2 and y K8}10 are reasonable range for these parameters. To complete the speci"cation, we need to provide a more phenomenological description for the "nal-state screening factor. First, we shall approximate the screening factor by an exponential in t: (32) S (y, t)KS (y, 0)e BWR , where S (y, 0)"+rx\PC(x, r),, with x"k R(y)X(y) and r"r(y). The width, *B (y), can be ob B B tained by a corresponding substitution. Note that S (y, 0) depends on B(y), B (y), X(y), and k . B B Phenomenological studies allow us to approximate B(y)Kb #0.25y and B (y)Kb #0.25y, B B b Kb /2}2.3 GeV\. B The only quantity left to be speci"ed is the e!ectiveness parameter k . Since the physics of B "nal-state screening is that driven by a Pomeron with intercept greater than unity, the relevant rapidity scale is again y . Let us "x k "rst by requiring that screening is small for y(y , i.e., B S (y, 0)&1 as one moves down in rapidity from y to y . Similarly, we expect screening to approach its full strength as one moves past the #avoring threshold y . We thus "nd it economical to

By choosing e (0, it is possible to provide a global àveragea description mimicking `secondary trajectorya contributions for various low energy regions. In Ref. [13], acceptable estimates are e K!0.11 to !0.5.

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parameterize k (y)K(k /2)+1#tanh[j (y!y )], (33) B B and we expect y &y and j &j . This completes the speci"cation of our unitarization procedure. B 6.2. High-energy diwractive dissociation The most important new parameter we have introduced for understanding high-energy di!ractive production is the #avoring scale, s "eW. We have motivated by way of a simple model to show that a reasonable range for this scale is y K8}10. Quite independent of our estimate, it is possible to treat our proposed resolution phenomenologically and determine this #avoring scale from experimental data. It should be clear that one is not attempting to carry out a full-blown phenomenological analysis here. To do that, one must properly incorporate other triple-Regge contributions, e.g., the PPR-term for the low-y region, the nnP-term and/or the RRP-term for the low-y region, etc.,

particularly for (s4(s &100 GeV. What we hope to achieve is to provide a `caricaturea of the interesting physics involved in di!ractive production at collider energies through our introduction of the screening and the #avoring factors [13]. Let us begin by "rst examining what we should expect. Concentrate on the triple-Pomeron vertex g(0) measured at high energies. Let us for the moment assume that it has also been measured reliably at low energies, and let us denote it as g (0). Our #avoring analysis indicates that these two couplings are related by g(0)Ke\CWg (0) .

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With eK0.08}0.1 and y K8}10, using the value g (0)"0.364$0.025 mb [21], we expect a value of 0.12}0.18 mb. Denoting the overall multiplicative constant for our renormalized triple-Pomeron formula by K, (35) K,b(0)gPPP(0)b (0)/16p . @ ? With bK16 mb, we therefore expect K to lie between the range 0.15}0.25 mb. N We begin testing our renormalized triple-Pomeron formula by "rst turning o! the "nal-state screening, i.e., setting S "1. We determine the overall multiplicative constant K by normalizing the integrated p to the measured CDF (s"1800 GeV value. With e"0.1, j "1, this is done for a series of values for y "7, 8, 9, 10. We obtain respective values for K"0.24, 0.21, 0.18, 0.15, consistent with our #avoring expectation. As a further check on the sensibility of these values for

The published CDF p values at (s"546 and 1800 GeV are 7.89$0.33 and 9.46$0.44 mb, respectively. These values correspond to m "0.15. We shall restrict m(0.05 and t to be inside the extreme forward peak. For m "0.05,

we reduce these values by&8% while maintaining their relative ratio of 0.834. For t to be within the extreme forward di!raction peak we scale down the ISR di!ractive cross sections also by approximately 8%. This is appropriate for our determination of the triple-Pomeron coupling.


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Fig. 4. Fits to representative single-di!raction dissociation cross sections from ISR to Tevatron [6]. The solid line corresponds to e"0.08, e "!0.07, j "1, y "9, with a small amount of "nal-state screening, k "0.1. The dotted M and the dashed}dotted curves correspond to k "0.2 and k "0, i.e., no screening, respectively.

the #avoring scales, we "nd for the ratio o,p(546)/p(1800) the values 0.63, 0.65, 0.68, 0.72, respectively. This should be compared with the CDF result of 0.834. Next we consider screening. Note that screening would increase our values for K, which would lead to large values for g. Since we have already obtained values for triple-Pomeron coupling which are of the correct order of magnitude, the only conclusion we can draw is that, at Tevatron, screening cannot be too large. With our parameterization, we "nd that screening is rather small at Tevatron energies, with k K0.0}0.2. This comes as somewhat as a surprise! Clearly, screening will become important eventually at higher energies. After #avoring, the amount of screening required at Tevatron is apparently greatly reduced. Having shown that our renormalized triple-Pomeron formula does lead to sensible predictions for p at Tevatron, we can improve the "t by enhancing the PPR-term as well as RRP-terms which can become important. Instead of introducing a more involved phenomenological analysis, we simulate the desired low-energy e!ect by having e K!0.06 to !0.08. A remarkably good "t M results with e"0.08}0.09, y "9 and k K0}0.2 [13]. This is shown in Fig. 4. The ratio o ranges from 0.78 to 0.90, which is quite reasonable. The prediction for p at LHC is 12.6}14.8 md. Our "t leads to a triple-Pomeron coupling in the range of gPPP(0)K0.12}0.18 mb ,

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exactly as expected. Interestingly, the triple-Pomeron coupling quoted in Ref. [6] (g(0)" 0.69 mb) is actually a factor of 2 larger than the corresponding low-energy value [21]. Note that this di!erence of a factor of 5 correlates almost precisely with the #ux renormalization factor N(s)K5 at Tevatron energies. We believe, with care, the physics of #avoring and "nal-state screening can be tested independent of the speci"c parameterizations we have proposed here. In particular, because our unitarized Pomeron #ux approach retains factorization along the `missing massa link, unambiguous predictions can be made for other processes involving rapidity gaps.

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7. Predictions for other gap cross sections For both double Pomeron exchange (DPE) and doubly di!ractive (DD) processes, one is dealing with three rapidity variables which can become large. We will treat these two cases "rst before turning to more general situations. 7.1. Prediction for DPE cross sections For double Pomeron exchange (DPE), we are dealing with events with two large rapidity gaps. The "nal-state con"guration can be speci"ed by "ve variables, t , t , m , m , and M. For t and t small, one again has a constraint, m\Mm\Ks. Alternatively, we can work with rapidity variables, y ,log (m\), y ,log (m\), and y ,log M, with y #y #y K>"log s. The

appropriate DPE di!erential cross section can be written down, with no new free parameter. Let us introduce a renormalization factor dp dp "Z (y , t , y , y , t ) . ".#

dy dt dy dt dy dt dy dt One immediately "nds that, using Pomeron factorization for the missing mass variable,

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Z "[S (y , t )R(y )]R(y )[S (y , t )R(y )] ".#

"Z (m , t )Z (M)Z (m , t ) . (38) B B Alternatively, we can express this cross section in terms of singly di!ractive dissociation cross sections as

dp dp dp ?@ +R(y )p (y ),\ ?@ , ?@ " (39)

@? dy dt dy dt dy dt dy dt where p (y )"b b eCW . This clean prediction involves no new parameter, with the understanding @? @ ? that, when y is low, secondary terms must be added.

7.2. Prediction for DD cross sections For double-di!raction dissociation (DD), there are two large missing mass variables, M, M, separated by one large rapidity gap, y, and its associate momentum transfer variable t. Again, for t small, we have the constraint y #y #yK>. K K (y, t)p P(y ), where the classical The classical di!erential DD cross section is p P(y )FM P @ K ? K `gap distributiona function is FM P (t, y)"(1/16p)eC>?PRW. After taking care of both #avoring and "nal-state screening, one obtains for the renormalization factor Z (y , y, t, y )"R(M)[SM (y, t)R(y)]R(M),Z (M)ZM (m, t)Z (M) . (40) "" K K

B

A new screening factor, SM (y, t), has to be introduced because of the di!erence in the t-distribution associated with two factors of triple-Pomeron coupling. It can be obtained from S (y, t) by replacing B (y) by BM (y)"bM #aPt, where, by factorization, bM "2b !b (bM is the t-slope associated with the B B triple-Pomeron coupling).


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Alternatively, this cross section can again be expressed as a product of two single di!ractive cross sections

dp dp dp ?@ +ZM (y, t)p (y, t),\ ?@ , ?@ " (41) ?@ dy dt dy dt dy dt dy K K K K where p (y, t),(1/16p)"b (t)b (t)eC>?PRW". Other than the modi"cation from S to SM , this predic?@ ? @ tion is again given uniquely in terms of the single-di!raction dissociation cross sections. 7.3. Other gap cross sections We are now in the position to write down the general Pomeron contribution to the di!erential cross section with an arbitrary number of large rapidity gaps. For instance, generalizing the DPE process to an n-Pomeron exchange process, there will now be n large rapidity gaps, with n!1 short-range ordered missing mass distributions alternating between two gaps. The corresponding renormalization factor is ZL "Z (m , t )Z (M)ZM (m , t )Z (M)2Z (M )Z (m , t ) . (42) .# B B

L\ B L L Other generalizations are all straightforward. However, since these will unlikely be meaningful phenomenologically in the near future, we shall not discuss them here. It is, nevertheless, interesting to point out that, if any cross section does become meaningful experimentally, #avoring would dictate that it is most likely the classical triple-Regge formulas with aP(0)K1 that would be at work "rst.

8. Comments Let us brie#y recapitulate what we have accomplished. Given Pomeron as a pole, the total Pomeron contribution to a singly di!ractive dissociation cross section can in principle be expressed as (43) dp/dt dm"[S (s, t)][F P(m, t)][pP (M)] , @ ? F P(m, t)"S (m, t)FP (m, t) . (44) ? ? E The "rst term, S , represents initial-state screening correction. We have demonstrated that, with a Pomeron intercept greater than unity and with a pole approximation for total and elastic cross sections remaining valid, initial-state absorption cannot be large. We therefore can justify setting S K1 at Tevatron energies. E The "rst crucial step in our alternative resolution to the Dino's paradox lies in properly treating the "nal-state screening, S (m, t). We have explained in an expanding disk setting why a "nal-state screening can set in relatively early when compared with that for elastic and total cross sections. E We have stressed that the dynamics of a soft Pomeron in a non-perturbative QCD scheme requires taking into account the e!ect of `#avoringa, the notion that the e!ective degrees of freedom for Pomeron is suppressed at low energies. As a consequence, we "nd that FP (m, t)"R(m\)F P (m, t) and pP (M)"R(M)p P (M) where R is a `#avoringa factor. ? ? @ @

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It is perhaps worth contrasting what we have achieved with the #ux renormalization scheme of Goulianos [6]. By construction, the normalization factor N(s) is of the form which one would have obtained from an initial-state screening consideration. Although this breaks factorization, one might hope perhaps the scheme could be phenomenologically meaningful at Tevatron energies. Note that, for (s'22 GeV, the renormalization factor N(s) has an approximately factorizable form: N(s)&0.25sC"0.25(m\)C(M)C. It follows that the di!ractive di!erential cross section remains in a factorized form (45) m dp/dt dm&0.25[mC>F P (m, t)][(M)\Cp P (M, t)] . N N It can be shown that Eq. (45) leads to a di!ractive cross section p which, up to log s, is asymptotically constant. That is, the di!ractive dissociation contribution no longer corresponds to the part of total cross sections represented by the Pomeron exchange. This is not in accord with the basic hypothesis of Pomeron dominance for total and elastic cross sections at Tevatron energies. Our "nal resolution shares certain common features with that proposed by Schlein [8]. At a "xed m, Z (M)K1 as sPR so that it is possible to identify our renormalization factor

Z (m, t)"S (m, t)R(m\) with the #ux damping factor Z (m) of Schlein. In Ref. [8], it was emphaB 1 sized that the behavior of Z (m) can be separated into three regions. (i) (m , m ) where Z K1, (ii) 1 1 (m , m ) where Z drops from 1 to 0.4 smoothly, and (iii) (0, m ) where Z (m)P0 rapidly as mP0. The 1 1 boundaries of these regions are m &0.015 and m &10\. The "rst boundary m can be identi"ed with our energy scale, s &m\&eW. If we identify the boundary between region-(ii) and region-(iii) with our #avoring scale y by s\"e\W"m , one has y K9, which is consistent with our estimate. Since S (m, t)K1 for m's\ and R(m\) drops from R(1)KsC to 1 at s , their Z (m) behaves 1 qualitatively like our renormalization factor. If one indeed makes this connection, what had originally been a mystery for the origin of the scale, m , can now be related to the non-perturbative dynamics of Pomeron #avoring. It should be stressed that our discussion depends crucially on the notion of soft Pomeron being a factorizable Regge pole. This notion has always been controversial. Introduced more than 30 years ago, Pomeron was identi"ed as the leading Regge trajectory with quantum numbers of the vacuum with a(0)K1 in order to account for the near constancy of the low-energy hadronic total cross sections. However, as a Regge trajectory, it was unlike others which can be identi"ed by the

Ultimately, these two schemes can be di!erentiated by confronting experimental data. For our scheme, because of Pomeron pole dominance, it leads to a normalizable limiting gap distribution, o (y; >), i.e., o(y, t; >)Po (y, t). For ? ? y(y ;>, it is cut-o! in y at least as fast as o (y, t)Je\CW. In contrast, the #ux renormalization scheme, Eq. (45), leads to ? a gap distribution of the form o (y; >)Je\C7eCW, for 1;y;>. Test of these two alternatives for either the normaliz? ation and the y-distribution can, in principle, be carried out by comparing data at two Tevatron energies by focusing on the region of "xed small t and 0.025m50.002 (y (y(y ). Interestingly, both behaviors seem to provide acceptable

"ts based on data presented in Ref. [11]. There are also several di!erences between our result and that of Schlein. First, our renormalization factor Z(m, t; s) is t-dependent whereas Schlein's is not. At very small m, our suppression factor does not vanish as fast as that of Schlein: (Z (m)&m, whereas ours behaves as mC). Furthermore, since we have found that there is very little screening needed at 1 Tevatron energies, our slow cuto! might not set in until much higher energies so that it could indeed be possible to observe the m\C behavior for dp/dy dt at Tevatron energies.


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particles they interpolate. With the advent of QCD, the situation has improved, at least conceptually. Through large-N analyses and through other non-perturbative studies, it is natural to A expect Regge trajectories in QCD as manifestations of `string-likea excitations for bound states and resonances of quarks and gluons due to their long-range con"ning forces. Whereas ordinary meson trajectories can be thought of as òpen stringsa interpolating qq bound states, Pomeron corresponds to a `closed stringa con"guration associated with glueballs. However, the di$culty of identi"cation, presumably due to strong mixing with multi-quark states, has not helped the situation in practice. In a simpli"ed one-dimensional multiperipheral realization of large-N QCD, the non-Abelian gauge nature, nevertheless, managed to re-emerge through its topological structure [18]. The observation of `pole dominancea at collider energies has hastened the need to examine more closely various assumptions made for Regge hypothesis from a more fundamental viewpoint. It is our hope that by examining Dino's paradox carefully and by "nding an alternative resolution to the problem without deviating drastically from accepted guiding principles for hadron dynamics, Pomeron can continue to be understood as a Regge pole in a non-perturbative QCD setting. The resolution for this paradox could therefore lead to a re-examination of other interesting questions from a "rmer theoretical basis. For instance, to be able to relate quantities such as the Pomeron intercept to non-perturbative physics of color con"nement represents a theoretical challenge of great importance.

Acknowledgements I would like to thank K. Goulianos for "rst getting me interested in this problem during the Aspen Workshop on Non-perturbative QCD, June 1996. Intensive discussions with K. Goulianos, A. Capella, and A. Kaidalov at Rencontres de Moriond, March, 1997, have been extremely helpful. I am also grateful to P. Schlein for explaining to me details of their work and for his advice. I want to thank both K. Goulianos and P. Schlein for helping me to understand what I should or should not believe in various facets of di!ractive data! Lastly, I really appreciate the help from K. Orginos for both numerical analysis and the preparation for the "gures. This work is supported in part by the D.O.E. Grant CDE-FG02-91ER400688, Task A.

References [1] A. Donnachie, P.V. Landsho!, Phys. Lett. B 296 (1992) 227; J.R. Cuddel, K. Kang, S.K. Kim, Phys. Lett. B 395 (1997) 311; R.J.M. Covolan, J. Montanha, K. Goulianos, Phys. Lett. B 389 (1996) 176; M. Block, to be presented at VIIth Blois Workshop on Elastic and Di!ractive Scattering, Seoul, June 10}14, 1997. [2] G. Ingelman, P. Schlein, Phys. Lett. B 296 (1992) 227. [3] D. Silverman, C.-I. Tan, Relation between the multi-Regge model and the missing-mass spectrum, Phys. Rev. D 2 (1970) 233. [4] C. DeTar et al., Phys. Rev. Lett. 26 (1971) 675. [5] D. Horn, F. Zachariasen, Hadron Physics at Very High Energies, Benjamin, New York, 1973. [6] K. Goulianos, Phys. Lett. B 358 (1995) 379.

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[7] K. Goulianos, Proceedings of the Third Workshop on Small-x and Di!ractive Physics, Argonne National Laboratory, USA, September 1996; K. Goulianos, Proceedings of the Fifth International Workshop on Deep Inelastic Scattering and QCD (DIS-97), Chicago, USA, April 1997. [8] P. Schlein, Proceedings of the Third Workshop on Small-x and Di!ractive Physics, Argonne National Laboratory, USA, September 1996; P. Schlein, Proceedings of the Fifth International Workshop on Deep Inelastic Scattering and QCD (DIS-97), Chicago, USA, April 1997. [9] S. Erhan, P. Schlein, Saturation of the Pomeron #ux factor in the proton by damping small Pomeron momenta, Phys. Lett., submitted for publication. [10] A. Brandt et al., Measurements of single di!raction at (s"630 GeV; Implication for the Pomeron #ux factor, Nucl. Phys., submitted for publication. [11] K. Goulianos, Comments on the Erhan-Schlein model of damping the Pomeron #ux at small-x, hep-ph/9704454. [12] E. Gotsman, E.M. Levin, U. Maor, Phys. Rev. D 49 (1994) 4321. [13] T.K. Gaisser, C.-I. Tan, Phys. Rev. D 8 (1973) 3881; C.-I. Tan, Proceedings IX Rencontres de Moriond, Meribel, France, 1974. [14] J.W. Dash, S.T. Jones, Phys. Lett. B 157 (1985) 229. [15] C.-I. Tan, in: K. Goulianos (Ed.), Proceedings of Second International Conference on Elastic and Di!ractive Scattering, Editions Frantieres, Dreux, 1987, p. 347; C.-I. Tan, in: D. Schi!, J.T.V. Tran (Eds.), Proceedings of Nineteenth International Symposium on Multiparticle Dynamics, Arles, Editions Frontieres, Dreux, 1988, p. 361. [16] H. Harari, Phys. Rev. Lett. 20 (1968) 1395; P.G.O. Freund, Phys. Rev. Lett. 20 (1968) 235. [17] W. Frazer, D.R. Snider, C.-I. Tan, Phys. Rev. D 8 (1973) 3180. [18] H. Lee, Phys. Rev. Lett. 30 (1973) 719; G. Veneziano, Phys. Lett. B 43 (1973) 314; F. Low, Phys. Rev. D 12 (1975) 163. [19] A. Capella, U. Sukhatme, C.-I. Tan, J.T.V. Tran, Phys. Rep. 236 (1994) 225. [20] G. Finkelstein, H.M. Fried, K. Kang, C.-I. Tan, Phys. Lett. B 232 (1989) 257. [21] R.L. Cool, K. Goulianos, S.L. Segler, H. Sticker, S.N. White, Phys. Rev. Lett. 47 (1981) 701.


Feshbach resonances in atomic Bose}Einstein condensates Eddy Timmermans *, Paolo Tommasini, Mahir Hussein, Arthur Kerman T-4, Los Alamos National Laboratory, Los Alamos, NM 87545, USA Institute for Atomic and Molecular Physics, Harvard-Smithsonian Center for Astrophysics 60 Garden Street, Cambridge, MA 02138, USA Instituto de Fn& sica, Universidade de SaJ o Paulo, C.P. 66318, CEP 05315-970 SaJ o Paulo, Brazil Center for Theoretical Physics, Laboratory for Nuclear Science and Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

Abstract The low-energy Feshbach resonances recently observed in the inter-particle interactions of trapped ultra-cold atoms involve an intermediate quasi-bound molecule with a spin arrangement that di!ers from the trapped atom spins. Variations of the strength of an external magnetic "eld then alter the di!erence of the initial and intermediate state energies (i.e. the &detuning'). The e!ective scattering length that describes the low-energy binary collisions, similarly varies with the near-resonant magnetic "eld. Since the properties of the dilute atomic Bose}Einstein condensates (BECs) are extremely sensitive to the value of the scattering length, a &tunable' scattering length suggests highly interesting many-body studies. In this paper, we review the theory of the binary collision Feshbach resonances, and we discuss their e!ects on the many-body physics of the condensate. We point out that the Feshbach resonance physics in a condensate can be considerably richer than that of an altered scattering length: the Feshbach resonant atom}molecule coupling can create a second condensate component of molecules that coexists with the atomic condensate. Far o!-resonance, a stationary condensate does behave as a single condensate with e!ective binary collision scattering length. However, even in the o!-resonant limit, the dynamical response of the condensate mixture to a sudden change in the external magnetic "eld carries the signature of the molecular condensate's presence: experimentally observable oscillations of the number of atoms and molecules. We also discuss the stationary states of the near-resonant condensate system. We point out that the physics of a condensate that is adiabatically tuned through resonance depends on its history, i.e. whether the condensate starts out above or below resonance. Furthermore, we show that the density dependence of the many-body ground-state energy suggests the possibility of creating a dilute condensate system with the liquid-like property of a selfdetermined density. 1999 Elsevier Science B.V. All rights reserved.

* Corresponding author. E-mail address: [email protected] (E. Timmermans) 0370-1573/99/$ - see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 9 ) 0 0 0 2 5 - 3

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E. Timmermans et al. / Physics Reports 315 (1999) 199}230

PACS: 03.75.Fi; 05.30.Jp; 32.80.Pj; 67.90.#z Keywords: Bose}Einstein condensates; Feshbach resonance; Coherent matter wave dynamics

1. Introduction The recent observation of Feshbach resonances in the inter-particle interactions of a dilute Bose}Einstein condensate of Na-atoms by Ketterle's group at MIT [1] was an eagerly anticipated event. The signi"cance of this experimental breakthrough appears of singular importance as its consequences are far-reaching in two sub"elds of physics: (i) Atomic and Molecular Physics. Although predicted theoretically [46], technical di$culties had previously prevented the observation of the low-energy Feshbach resonances [2]. This situation abruptly changed when the experimental e!orts recently culminated in the observation of resonances in ultra-cold Na at MIT, in Rb by Heinzen's group at U.T. Austin [3] and by the Wieman-Cornell collaboration at JILA [4], as well as in Cesium by Chu's group at Stanford [5]. This cascade of results indicates that the "eld of atomic trapping and cooling has achieved the necessary amount of control and precision to carry out systematic studies of the resonances in a variety of atomic systems. (ii) BEC-physics. An important distinguishing feature of the MIT-experiment [1], is that the Feshbach resonances were observed in an atomic BEC-system [6}8]. The resonances were observed by varying an external magnetic "eld, thereby altering the &detuning' (de"ned as the di!erence between the initial and intermediate state energies). Similarly, the e!ective scattering length that describes the low-energy atom}atom interaction, varies with magnetic "eld and is consequently &tunable'. As all quantities of interest in the atomic BECs crucially depend on the scattering length, a tunable interaction suggests very interesting studies of the many-body behavior of condensate systems. Previous searches for the low-energy Feshbach resonances had been unsuccessful [2]. The di$culties that had to be overcome were multiple: the magnetic traps can only trap low "eld seeking states, the resonant magnetic "eld strengths are rather high and the margin of error on the predicted values for the resonant magnetic "elds were considerable due to the uncertainties in the interatomic potential curves. Conversely, the measured values of the resonant "elds and the &widths' will be of great help in re"ning the potential curves that characterize the inter-atomic interaction. This, in turn, has important applications in spectroscopy, high-frequency resolution measurements and atomic clocks. The Feshbach resonance has been listed as one of the prospective schemes to alter the e!ective inter-particle interactions of the cold-atom systems. A variable interaction strength is a highly unusual degree of freedom in experimental studies of many-body systems. Especially for the atomic condensate system, the &tunability' of this parameter suggests very interesting applications as virtually all observable quantities, as well as the stability of the system, sensitively depend on its value. This aspect which motivated much of the theoretical work [28}33] was emphasized in [1], as well as the Nature article that accompanied the original report [9]. Other schemes have been proposed to alter the inter-particle interactions, e.g. by means of external electrical "elds [10].


201

Among the many applications suggested by this novel degree of freedom, we name but a few: (i) Study of negative scattering length condensates. A dilute gas condensate of bosons that experience an e!ective mutual attraction is unstable in the absence of an external potential. A trapping potential can &stabilize' a negative scattering length condensate of limited number of bosons [11}15]. Once the population of the condensate exceeds this critical number, the condensate collapses. Although most theories agree on the collapse and the critical number of particles, the details and the mechanism of the collapse are still being debated in the current literature (for a discussion see, for example, Ref. [15]). Measurements taken by Hulet's group at Rice university with Li provide valuable data on this interesting system, but a study of the collapse at variable values of the interaction strength will give de"nitive tests of the theoretical predictions and yield much needed insight in the dynamics of the collapse. (ii) Study of the condensate phase separation. Overlapping condensates are unstable if the strength of the unlike boson interactions exceeds the geometric mean of the like-boson interaction strengths. Mutually repelling condensates then separate spatially and act as immiscible #uids. As the phase separation criterion depends solely on the interaction strengths, one could by varying one of the strengths [16}18], render the condensates miscible or immiscible. Whether the BEC-systems will have practical applications remains to be seen, but a two-#uid system that can be made miscible or immiscible at will, does suggest applications in areas such as data-storage, or perhaps even quantum computation. (iii) Study of Josephson oscillations. A mixture of same species condensates in di!erent internal states can exchange bosons coherently, for example by interacting with coherent near-resonant laser light [19}21]. Such coherent inter-condensate particle exchange processes are often referred to as inter-condensate &tunneling' processes because of the strong analogy with Josephson tunneling [22,23]. Interestingly, unlike the condensed matter Josephson junctions, the dilute condensate mixtures can actually probe the non-linear regime of the Josephson oscillations [19}22]. Like the DC Josephson junction, the number of bosons in each condensate oscillates when the values of the chemical potentials of the respective condensates di!er. For a single dilute condensate, the chemical potential is equal to the product of the interaction strength and the density. Thus, a sudden change of the interaction strength e!ects precisely such a chemical potential di!erence. (iv) Condensate Dynamics. While the near-equilibrium dynamics of dilute single condensate systems are well-understood, at least in the low-temperature limit, the far-from equilibrium condensate dynamics poses a problem that has not been satisfactorily resolved. A detailed understanding of the condensate formation, in particular of the formation time [24}27], will have important implications in a variety of "elds, such as "eld theory and early universe theories. Experimental studies with a variable interaction strength will give a "rm understanding of the role of the inter-particle interactions. Clearly, each of the above applications represents an exciting prospect. It is in this context, the creation of a tunable interaction strength, that the motivation for much of the previous research on Feshbach resonances has been situated [9,28}33]. However, we would urge caution in interpreting the e!ects of the Feshbach resonance on the condensate solely as altering the inter-boson interaction. We believe that the e!ects of the Feshbach resonance are considerably more profound. In particular, in this paper, we review some of our previous research results ([34,35], see also Ref. [36] for a brief review of the tunneling aspect) that show that in a near-resonant BEC, the Feshbach resonant atom-molecule coupling creates a second condensate component of quasibound molecules. The many-body dynamics predicts that the expectation value of the molecular

202


"eld does not vanish in a near-resonant magnetic "eld, 1tK 2O0: the hallmark of Bose condensa tion. Whether the presence of the molecular condensate predicts a behavior that di!ers from that of a single condensate with altered scattering length depends on the values of several relevant parameters: the detuning, the rate of change of the detuning (in a dynamical experiment with changing magnetic "eld), the lifetime of the quasi-bound molecules, limited } most importantly } by collisions with other atoms or molecules that change the molecular state. O!-resonance, the stationary condensate system does behave as a single condensate with a scattering length that has the value predicted by the binary collision description. However, even in the o!-resonant limit, the dynamical response of the condensate system to a sudden change of the external magnetic "eld, can di!er signi"cantly from that of a single condensate with e!ective scattering length. We "nd that, subsequent to a sudden change of the external magnetic "eld, the number of atoms and molecules that occupy the respective condensates oscillate. These oscillations are damped out on the time scale of the single molecule lifetime. The o!-resonant condensate lives much longer than the individual molecules, since the atomic condensate acts as a reservoir of atoms, continually replenishing the molecular condensate. The oscillations can then be observed by illuminating the condensate with light that is near resonant with a transition of the quasi-bound molecule. The oscillations in the molecular condensate population then reveal themselves as an oscillating intensity of the image. The oscillations, like the oscillating current observed in Josephson junctions, are caused by the coherent inter-condensate exchange of particles. Unlike the Josephson tunneling, the exchange involves boson pairs. This, as we shall show, has a profound e!ect on the stationary state properties of the BEC system. For instance, we "nd that, for a near-resonant detuning, the homogeneous ground state system is always unstable in the limit of ultra low atomic particle density. Unlike the negative scattering length condensate, this instability does not necessarily lead to collapse. At higher densities the near-resonant condensate system can be stabilized by the inter-particle interactions of the atoms and molecules. In that case, the many-body energy, as a function of the atomic particle density, goes through a minimum. The many-body system can "nd this minimum by decreasing its volume and, if given enough time, can relax to the state of minimal energy and self-determined density, a typical liquid-like property. This self-determined density would still be of the order of 10 cm\, so that these considerations suggest the possibility of creating the world's "rst rari"ed liquid! The paper is organized as follows. In Section 2, we review the binary collision theory of low-energy Feshbach resonances. In Section 3, we specialize to the magnetically controlled Feshbach resonances recently observed in atomic traps. In that section, we set up the Hamiltonian for the many-body problem and we argue that a second molecular condensate component is formed. A compelling argument follows from the equations that describe the many-body dynamics. We also discuss the e!ects of the most important destructive in#uence that the molecular condensate undergoes: collisions of the molecules with other atoms or molecules that change the vibrational state of the quasi-bound molecule. In Section 4, we investigate the stationary states of the near-resonant condensate system, neglecting, for the time being, the e!ects of particle loss. We point out that the state the near-resonant condensate system "nds itself in, depends on its history. If the system was brought near resonance by adiabatically increasing the detuning, its state di!ers from that of the system created by lowering the detuning. Furthermore, we show that, in contrast to the e!ective scattering length description, the condensate system does not have to collapse, as it is tuned adiabatically through resonance. Finally, we conclude in Section 6.


203

2. Low-energy Feshbach resonances 2.1. Introduction In this paper, we focus on the implications of a Feshbach resonance on the many-body behavior of Bose}Einstein condensed systems. To this purpose, we develop the necessary theoretical framework to describe the relevant binary collision physics. The following discussion does not aim at being comprehensive with regard to Feshbach resonance physics. For such treatment, we refer the reader to more speci"c articles [37}39]. By de"nition, Feshbach resonances involve intermediate states that are quasi-bound, so that they are sometimes referred to as closed-channel collisions. These intermediate states are not bound in the true sense of the word, as they acquire a "nite lifetime due to the interaction with continuum states of other channels (such as the channel of the incident projectile/target system). For example, in electron}atom and electron}ion scattering, the intermediate states generally decay by ejecting the electron captured in the intermediate state. These states are known as auto-ionization states. In the atom}atom scattering Feshbach resonances of interest here, the intermediate states are molecules with electronic and nuclear spins that have been rearranged from the spins of the colliding atoms by virtue of the hyper"ne interaction. The intermediate molecular states interact with the continuum states of the incident channel that are the scattering states of the single-channel atom}atom scattering problem. In the next section, we discuss these single-channel scattering states. 2.2. Low-energy potential scattering As the interactions of interest involve bosonic atoms at ultra-low translational energies, the collision physics reduces to the description of s-wave scattering. Speci"cally, the angular momentum potential barrier of &height' &( /M¸), where ¸ is the range of the interatomic interaction and M the mass of a single atom, prevents colliding atoms with relative motion of lower kinetic energy and non-vanishing angular momentum from entering the inter-atomic interaction region. The magnitude of the angular momentum potential barrier height is of order ( /M¸)"( /m a);(a /¸);(m /M)&10}100 mK, where m represents the electron mass, (m /M)&10\}10\, and where a denotes the usual Bohr radius (¸/a )&10. Thus, in cold atom samples of temperature below 1 mK, bosonic atoms undergo pure s-wave scattering. The atomic Bose}Einstein condensates have temperatures of the order of 1 lK. Before we proceed with the treatment of low-energy Feshbach resonances, we describe the solution to the single-channel scattering problem for the collision of two of such indistinguishable atoms. In the center-of-mass frame the problem reduces to an integration of the radial s-wave Schrodinger equation with the corresponding molecular potential. The resulting regular solution, u(r), where r denotes the internuclear distance and ®ular' means that lim u(r) is "nite, has to be P normalized. We normalize u(r) to u (r) by requiring its asymptotic behavior to be similar to that of , the zeroth-order spherical Bessel function, which is the regular solution to the free atom Schrodinger equation u (r)&sin(kr#d )/(kr), rPR , , where d is the s-wave phase shift.

(1)

204


For future reference, we relate u (r) to the s-wave components of scattering states that have been , normalized using alternative schemes. One useful normalization consists of separating the scattered wave into an incident plane wave of wave vector k and an outgoing spherical wave: uk(r)& exp(ik ) r)#f exp(ikr)/r, rPR ,

(2)

where f is the scattering amplitude. By comparing its asymptotic behavior to that of u , Eq. (1), we , "nd the usual expression for the s-wave scattering amplitude: f"[ exp(2id )!1]/2ik , (3) where k denotes the wave number, which is the magnitude of the k-vector. Furthermore, the s-wave component of uk is equal to [uk(r)] " exp(id )u (r) . (4) , An alternative normalization that we shall consider, requires the regular solution to the Schrodinger equation to be a superposition of incident and outgoing spherical waves: u>(r)& exp(!ikr)/r!S exp(ikr)/r, rPR ,

(5)

where the coe$cient of the outgoing wave, S, is the scattering (or S)-matrix for the single-channel s-wave scattering problem, S" exp(2id ). By comparing the asymptotic behaviors of u and u>, , we "nd that u>(r)"!2ik exp(id ) u (r) . (6) , The alkali atoms in the atomic-trap condensates interact through molecular potentials that support bound states. In accordance with Levinson's theorem, u (r) has then nodes in the , inter-atomic interaction region. The number of nodes is equal to the number of bound states of the corresponding potential. Furthermore, u (r) for the ultra-cold collision energies is essentially , independent of the energy. To see that, we start by noting that for the relevant collision energies the de Broglie wavelength, (2p/k), vastly exceeds the range ¸ of the inter-atomic potential. Outside the range of the inter-atomic interaction, r'¸, but well within the de Broglie wavelength, r(k\, ru(r) is approximately linear: ru(r)Jr!a, where a is the scattering length. By scaling u(r) to the normalized function u , which in this region of the internuclear distance, ¸(r(k\, takes , on the form u (r)+1#(d /kr), we see that d "!ka and , u (r)+1!(a/r) where ¸(r(k\ , (7) , independent of the collision energy. 2.3. Low-energy Feshbach resonances In describing the Feshbach resonant collision, we distinguish the channels of the continuum incident projectile/target state and the closed (molecular) channels to which it is coupled. To this purpose, we introduce the projection operators P and M that denote, respectively, the projections onto the Hilbert subspace of the incident channel and the subspace of the closed (molecular)


205

channels. These projection operators satisfy the usual projection operator properties 0"MP"PM , P"P and M"M .

(8)

The time-independent SchroK dinger equation, satis"ed by the total scattering state "W2 of the binary atom system, (E!H)"W2"0 ,

(9)

takes on the form of coupled equations: (E!H )P"W2"H M"W2 , .. .+ (E!H

)M"W2"H P"W2 , ++ +.

(10) (11)

where we use the notation PHP"H , etc. .. We may obtain the projection of the scattering state onto the Hilbert space of the quasi-bound molecules, M"W2, by formally inverting Eq. (11): M"W2"(E!H

)\H P"W2 . ++ +.

(12)

The substitution of Eq. (12) into the projection of the SchroK dinger equation onto the continuum channel (10) then yields an e!ective SchroK dinger equation for the continuum scattering state, (E!H )P"W2"0, with an e!ective Hamiltonian, 1 H , H "H #H +. .. .+ E!H ++

(13)

that exhibits a strong dependence on the energy, E, of the colliding particles. For the purpose of treating the scattering problem, it is, in fact, more instructive to start by inverting Eq. (10) for P"W2, by means of the propagator for outgoing waves, g>(E)"(E!H #ig)\ , . ..

(14)

where g is an in"nitesimally small positive number. This inversion leads to P"W2""u>2#g>(E)H M"W2 . N N .+

(15)

The u>-state in the above expression is a scattering state of the single-channel (P) scattering N problem, (E!H )"u 2"0. In addition, we choose the asymptotic boundary condition of the .. N scattering state so that u> is the superposition of incident and outgoing spherical waves shown in N Eq. (5) for the P-channel, lim u>(r)"exp(!ikr)/r!S exp(ikr)/r. Upon insertion of the P N expression for P"W2 from Eq. (15) into Eq. (11), we "nd the e!ective SchroK dinger equation satis"ed by M"W2: (E!H

)M"W2"H "u>2#H g>(E)H M"W2 . ++ +. N +. N .+

(16)

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Finally, substitution of the formal solution for M"W2 from Eq. (16), 1 H "u>2 , +. N !H g>(E)H ++ +. N .+ into the initial expression for P"W2 of Eq. (15) yields the following result: M"W2" E!H

(17)

1 H "u>2 . (18) +. N !H g>(E)H ++ +. N .+ The asymptotic dependence of P"W2 on the radial inter-nuclear distance r supplies the S-matrix that characterizes the low-energy collision with Feshbach resonance. At the end of this section, we determine S in this manner. P"W2""u>2#g>(E)H N N .+ E!H

2.4. Width The experimentally observed low-energy Feshbach resonances are narrow } each individual resonance is well-separated in frequency space from the other resonances. Near a particular resonance m, with a single intermediate molecular state "u 2 of speci"c ro-vibrational quantum K number, we may further simplify the expression (18) by keeping only a single diagonal matrix element in evaluating the energy denominator of Eq. (18). Speci"cally, we replace 1 1 P"u 2 1u " , K E!E #iC /2 K E!H !H g>(E)H K K ++ +. N .+ where the energy E and width C of the resonance are equal to K K E "Re1u "H #H g>(E)H "u 2 , K K ++ +. N .+ K C K"!Im1u "H g>(E)H "u 2 . K +. N .+ K 2

(19)

(20)

Furthermore, the coupling between the continuum and molecular states for these resonances is weak enough that we may evaluate "u 2 in the spirit of perturbation theory as the eigenstate of the K molecular potential, and approximate E by its eigenvalue. K In particular, the low-energy dependence of the width is of importance to the observed resonances and we will evaluate C (E) in detail. The expansion of g>(E) in continuum states "k2 K N gives with Eq. (14) the following expression:

"1u "H "k2" C K +. K"!Im "p "1u "H "k2"d(E!Ek) . K +. E!Ek#ig 2 k k

(21)

In this equation, the continuum states, "k2, are not plane waves, but properly normalized scattering states uk (introduced in Eq. (2)). We shall work in box normalization so that in coordinate space, [exp(ik ) r)#f exp(ikr)/r] 1r"k2"uk(r)/(X& , rPR , (X

(22)


207

where X represents the macroscopic volume to which the binary atom system is con"ned. At the ultra-low collision energies of interest, the matrix elements 1u "H "k2, are dominated by K .+ the s-wave component of the "k2-wave. Within the molecular interaction range, the amplitudes of the higher partial waves have all but vanished at these energies as the colliding particles lack the energy to overcome the angular momentum potential barrier. Furthermore, the low-energy s-wave, u (r) is essentially energy independent, so that 1u "H "k2 is not only independent of the direction , K .+ of the k-vector, but also of its magnitude:

exp(id ) exp(id ) a, dr u (r)HK u (r)" 1u "H "k2+ K +. , K +. (X (X

(23)

where the a-parameter denotes the integral over the relative internuclear position, a"dr u (r)HK u (r). Consequently, the width is proportional to the square of a and the K +. , remaining &phase space factor' k d(E!E )/X: I

1 C (E) K "pa d(E!Ek) Xk 2 or

M

C (E)"a k , K 2p #

(24)

where M is the mass of a single atom and k the wave number corresponding to the relative velocity # of a pair of atoms with total kinetic energy E in the center-of-mass frame. Note that the width depends on the energy of the colliding atoms through the phase space factor, which is a measure of the phase space volume available to the binary atom system after the collision. Evidently, this remark is of importance to the Feshbach resonances in the ultra-cold atoms systems, where the relative velocity of the interacting atoms rigorously vanishes. It is customary to introduce a &reduced width', c, which makes the k-dependence of the width explicit:

M , C (E)"2ck where c"a K 4p

(25)

where it is understood that k is the wave number corresponding to E. Under condensate conditions, kP0, and C (E)P0, although the value of the coupling constant, a, remains constant. K The corresponding phase shift also vanishes linearly with k, but the e!ective scattering length tends to a well-de"ned "nite value, as we shall see below. 2.5. Scattering matrix We evaluate the continuum scattering wave P"W2, 1 1u "H "u>2 . P"W2""u>2#g>(E)H "u 2 N N .+ K E!E #iC /2 K +. N K K

(26)

208


As discussed above, "u>2 is normalized by requiring its asymptotic r-dependence to take on the N form u>(r)"exp(!ikr)/r!exp(2id )exp(ikr)/r, where d is the s-wave phase shift. In the lowN energy regime, d "!ka, where a is the scattering length. Furthermore, we remarked in Eq. (6) that u>(r)"!2ik exp(id )u (r), where u is the regular solution to the SchroK dinger equation with N , , u (r)+1!r/a outside the range of the molecular potential. , In evaluating the state (26) in the asymptotic region of coordinate space, the following asymptotic expansion of the s-wave component of the g>-propagator is useful: N M exp(ikr) exp(id ) u (r) where rPR, (27) [g>(E; r, r)] P! , N 4p r

and where we replaced the mass in the usual expression for the propagator by the reduced mass, M/2. The asymptotic behavior of g>(E)H "u 2 of Eq. (26) is then given by N .+ K M exp(ikr) lim 1r"g>(E)H "u 2"! exp(id ) dr u (r)HK u (r) , .+ K N .+ K 4p r P M exp(ikr) "! exp(id )a . (28) 4p r

Finally, with 1u "H "u>2"!2ik exp(id )a, we obtain the desired asymptotic behavior: K +. N M exp(ikr) ak 1r"g>(E)H "u 21u "H "u>2"i exp(2id ) N .+ K K +. N 2p r

"i exp(2id )C (E) K

exp(ikr) . r

(29)

Consequently, we obtain the following expression for the asymptotic r-dependence of the scattering state of Eq. (26):

iC (E) exp(ikr) exp(!ikr) K ! 1! exp(2id ) . (30) 1r"P"W2& E!E #iC (E)/2 r r K K By identifying the factor in square brackets in Eq. (29), with the scattering matrix S, we obtain

C (E) K , (31) E!E #i C (E)/2 + K where we used that d "!ka. Note that the S-matrix is unitary, "S""1, as be"ts scattering without loss-channels. As a consequence, we can describe the scattering by means of an e!ective scattering length, S"exp(!2ia k), where a "a#a, with C K exp(!2ika)"1!i E!E #iC /2 K K E!E !iC /2 K K . " (32) E!E #iC /2 K K S"exp(!2ika) 1!i


209

Thus, we "nd that the e!ective scattering length for collision energy E is equal to

1 C ;[E!E ] K K a (E)"a# tan\ . (33) 2k (E!E )#C /4 K K In the ultra-low-energy limit appropriate for condensate systems, E and C (E) vanish and K E!E P!e, where we will refer to e, the energy of the molecular state relative to the continuum K level of the P-channel, as the detuning of the Feshbach resonance. In this limit, we need to expand Eq. (33) to lowest order in k. With C "2ck, E" k/M and tan\(x)+x if x;1, we "nd K c lim a (E)"a! . (34) e # Furthermore, in describing the weakly interacting many-body system, it is notationally more convenient to work with the interaction strength j than with the scattering length a. If the inter-particle scattering can be described in the Born approximation, then the j-parameter represents the zero-momentum Fourier component of the inter-particle interaction potential. The zero momentum Fourier component is related to the scattering length a, calculated in the same approximation as j"(4p /M)a. However, the Born approximation cannot be used in describing the low-energy binary atom collisions. In that case, the interaction strength is still proportional to the scattering length, although the latter has to be determined more accurately from the full potential scattering problem. In the same spirit, we may introduce an &e!ective' strength, j "(4p /M)a , to describe the binary atom interaction, related to j as j "j!(a/e) . (35) In Eq. (35) we made use of the expression for the reduced width, c"a(M/4p ). 3. Feshbach resonances in atomic condensate systems 3.1. Magnetically controlled, hyperxne-induced Feshbach resonance In each of the experiments, the low-energy Feshbach resonances were observed by studying the behavior of the ultra-cold-atom systems under variations of an external magnetic "eld. The resonance in the binary-atom interactions is caused by the hyper"ne interaction which #ips the electronic and nuclear spins of one of the colliding atoms, bringing the collision system from the continuum (P)-channel into a closed channel, M, of di!erent spin arrangement. The M-channel is closed by the external magnetic "eld which has raised the continuum level of the binary spin #ipped atom system. While in the M-channel, the colliding atoms reside in a quasi-bound molecular state m. A second hyper"ne-induced spin #ip breaks up the molecule, returning the system to the initial P-channel. If the energy of the intermediate quasi-bound molecule is equal to the continuum level of the P-channel, the above-described collision process is &on resonance'. Variations of the external

Strictly speaking, the usual scattering length is only de"ned in the limit EP0, but Eq. (33) with S"exp(!2ia k) gives the correct low energy scattering matrix.

210


magnetic "eld shift the relative energies of the M molecule and the P-continuum level and tune, or detune, the interacting atoms close to, or far from resonance. We now describe the physics of this collision in more detail. We "nd it instructive to discuss the collision "rst in the limit of high external magnetic "eld, although this is not the regime in which the experiments were conducted. A high magnetic "eld, B"Bz( , aligns the electronic (s) and nuclear (i) spins of the atoms. To be de"nite, we consider a system of ultra-cold Na atoms (i") with spin projections m "!, and m "!. The G Q interaction of two such atoms is described by the triplet potential, where &triplet' refers to the total electronic spin of the interacting atoms. In the triplet state, the valence electrons of the atoms behave as indistinguishable fermions and &avoid each other', thereby reducing the Coulomb repulsion of the electrons. In contrast, if the spins of the colliding atoms are arranged in a singlet state, the valence electrons do not avoid each other and the Coulomb repulsion generally reduces the depth of the inter-atomic potential, as compared to the triplet potential. Thus, the inter-atomic interaction depends on the magnitude of the total electronic spin, S, where S"s #s . In this case, the spins of the initial binary atom system are in a state "S 2""m ", m "!; Q G m "!, m "!2, with electronic spins arranged in a pure triplet state, S"1. Consequently, Q G the atoms interact through the molecular triplet potential. However, the binary atom hyper"ne interaction, < "(a / );[s ) i #s ) i ], does not commute with S and can #ip the electronic spins of a triplet state to a singlet con"guration. At large internuclear separation, this singlet channel corresponds to the binary atom system with a single spin-#ipped atom. Consequently, the continuum of the singlet channel lies an energy D above the continuum of the incident triplet channel, where D"B[2k #k ], and where k and k denote the electronic and nuclear magnetic , , moments. Under near-resonant conditions, the singlet potential supports a quasi-bound molecular state u (r)"S 2, of energy E near the continuum of the P-channel and total (electronic and nuclear) K HY K spin state "S 2. HY In this context, the atom}molecule coupling, H #H , is provided by the binary atom .+ +. hyper"ne interaction, < . The corresponding a-parameter that indicates the strength of this inter-channel coupling is the product of the spin matrix element and the overlap of the regular triplet wave function with the molecular singlet wave function,

a"1S "< "S 2; dr u (r)u (r) , K , HY

(36)

which characterizes the resonance. The observed resonances were created at intermediate values of the magnetic "eld for which the actual spin state of the individual atoms are not states of good m and m -quantum number. Instead G Q the single atom spin degrees of freedom occupy a state that diagonalize the spin-dependent part of the single-atom Hamiltonian:

a [2k s !k i ] , . H " s ) i #B ) 1

(37)

The "rst term on the right-hand side of Eq. (37) is the single-atom hyper"ne interaction, characterized by a , an energy that depends on the isotope (e.g. a "42.5 mK for Na, see, e.g. Ref. [40]). At zero magnetic "eld, B"0, the diagonalization yields the hyper"ne states of good &f ' quantum


211

number, where f represents the &total' single atom spin, f"s#i. At high magnetic "eld strengths BY r-integral, proportional to the Feshbach resonance a-parameter: (48) 1K, m"< "k, k2"(1/(X)dK k k a , >Y where we have used that exp(id )+1. Consequently, the Feshbach-resonant interactions, in second quantization, are described by

a a 1 [c( R dr tK R (r)tK (r)tK (r) . c( c( ]" HK " K ? ? +. (X k k (2 K k>kY ? k ? kY (2 Y

(49)

Similar to the &elastic' inter-atomic interactions, the low-energy conditions imply a coupling strength to the molecular channel that is independent of the momentum transfer. Consequently, the Feshbach-resonant interactions are also characterized by a single parameter: the atom}molecule coupling strength a. In accordance with these results, we generalize the Hamiltonian density for the many-body system, Eq. (44), to describe the many-atom/molecule system:

j

j # ? tK RtK tK #tK R ! #e# K tK R tK tK (r) H K "tK R ! ? ? ? K ? 2M 2 4M 2 K K K a #j tK RtK tK R tK # [tK R tK tK #tK tK RtK R] , ? ? K K (2 K ? ? K ? ?

(50)


215

where the position dependence of the operators is understood, and where j , j and j represent the ? K strengths of the atom}atom, molecule}molecule and atom}molecule interactions, respectively. In Eq. (50), we have assumed that the atoms and molecules do not experience external potentials. We note that the same Hamiltonian correctly reproduces the Feshbach-resonant behavior of a binary atom system. 3.3. Dynamics of the condensate mixture A new phenomenon occurs in the many-body physics: the appearance of a molecular condensate component, which follows from the many-body dynamics. A convenient starting point to describe this dynamics are the Heisenberg equations of motion for the atomic and molecular "eld operators: i tKQ (x, t)"[HK , tK (x, t)] , ? ? i tKQ (x, t)"[HK , tK (x, t)] . K K

(51)

With the following expressions for the relevant commutators, [tK R(r, t), tK (x, t)]"d d(r!x), and G H G H [tK (r, t), tK (r, t)]"0, i, j"a or m, we obtain the following coupled operator equations: H G

i tQK "! tK #j tK RtK tK #jtK R tK tK #(2atK tK R , ? ? ? ? ? K K ? K ? 2M ?

a i tQK "! tK #etK #j tK R tK tK #jtK RtK tK # tK RtK R , K K K K K K ? ? K (2 ? ? 4M K

(52)

where it is understood that all "eld operators depend on the same position. The operator equations (52), which provide an &exact' description of the many problem, are generally very di$cult to solve. However, for dilute condensates, we obtain a closed set of equations for the condensate "elds,

(r)"1tK (r)2 and (r)"1tK (r)2, by taking the expectation value of Eqs. (52). Furthermore, we K ? K ? assume that for the dilute gas systems considered here, we may take the condensed "elds to be totally coherent in the sense that the expectation value of the products is equal to the product of the expectation values, e.g. 1tK tK 2+ . This corresponds to a particular Gaussian trial wave ? ? ? functional in the Dirac time-dependent variational scheme [41]. We "nd

i

Q " ! #j " "#j" " #(2a H , ? ? ? K ? K ? 2M

a i

Q " ! #e#j " "#j" " #

. K K K ? K (2 ? 4M

(53)

These coupled non-linear equations replace the usual Gross}Pitaevskii equation that describes the time evolution of the dilute single condensate system [42,43]. Note that the -"eld has a source K term J so that the expectation value of the molecular "eld operator is forced to take on a ? "nite value when O0: the atom}molecule coupling creates a molecular condensate component ? in the presence of an atomic condensate.

216


The Gaussian trial wave function, which leads to an e!ective classical Hamiltonian density,

j

j H " H ! # ?" " # H ! #e# K" " # j" "" " ? ? K K ? K 2M 2 ? 4M 2 K #

a

[ H # H] , K ? (2 K ?

(54)

gives equations of motion identical to Eq. (53). 3.4. Ewects of particle loss The experimental lifetime of the m-molecules is not only determined by the hyper"ne-induced spin #ips, but also by collisions with other atoms, or even molecules. The importance of such three-body collisions is particularly pronounced as the recent experiments &resonate' on a molecular state m of high vibrational quantum number (e.g. l"14 for the MIT experiment). The created quasi-bound molecules are consequently fragile and a collision with a third particle, atom or molecule, likely causes the molecule to decay into a state of lower vibrational quantum number. Such collisions that &quench' the internal molecular state are a potential problem for molecular Bose condensation. They are also the most likely culprit for the particle loss that served as a signal to detect the Feshbach resonances in the MIT experiment. Particle loss in atomic traps is usually described to su$cient accuracy, by very simple rate equations n "n [!c n !c n ] , ? ? ?? ? ?K K (55) n "n [!c n !c n ] . K K K? ? KK K In the above equation, n and n represent the particle densities of atoms and molecules, n "" ", ? K G G where i"a or m, if all particle are Bose condensed. The c -coe$cients represent the rate GH coe$cients for collisions between particles i and j that change the internal state of the i-particle. Typical values for the alkali atoms are c &10\}10\ cm s\. The fragility of the loosely ?? bound alkali dimers is expressed by atom}molecule and molecule}molecule state changing collision rates that exceed the atom}atom rates by several orders of magnitude: 10\}10\ cm s\, where these numbers are estimates based on calculations with hydrogen molecules [44]. A &pure' molecular condensate of density 10 cm\ of such dimers is then not expected to survive longer than 10\ s. Nevertheless, this time scale might actually su$ce to study interesting molecular condensate physics. Furthermore, o!-resonance, the molecular condensate is signi"cantly smaller than the atomic condensate, which keeps &replenishing' the small molecular condensate with bosons. The lifetime of the condensate mixture is then equal to the molecular lifetime divided by the fraction of molecules, e.g. a condensate system with a 1 percent molecular condensate can survive 100 times longer than a single-molecule embedded in an atomic condensate of the same density. Of equal importance is the question whether, and how, the atomic/molecular condensate system reaches its equilibrium as the detuning e is altered by varying the magnetic "eld. From the two-"eld coupling in the equations of motion, Eq. (53), we expect, as we discuss below, that a sudden change


217

of the detuning is followed by oscillations in the atomic and molecular populations. From Eqs. (53), it would appear as if these oscillations persist inde"nitely, or, at least until the condensates have disappeared due to state changing collisions or other three-body recombination processes. However, we point out that a correct treatment of the e!ects of state-changing collisions on the condensate predicts that the number oscillations damp out on a time scale that is the lifetime of a single-molecule. As we noted above, the molecular condensate can survive for a much longer time and the condensate system can reach a &quasi-equilibrium' state on a time scale small compared to its lifetime. To build in the e!ects of state-changing collisions, we treat the loss-processes by considering the channels of all chemical reactions that remove particles from the atomic and molecular condensates. The elimination of the two-body collision channels in perturbation theory modi"es the equations of motion (53) in a predictable way: the interaction strengths become absorptive with an imaginary part that determines the loss-rates. In the -equation, for instance, the interaction K strengths are replaced by j Pj !i c /2 and jPj!i c /2. We de"ne the o!-resonant regime K K KK K? to correspond to values of the detuning that exceed, in absolute value, the molecular kinetic energy, as well as any of the single-particle interaction energies e<j n , jn , j n , jn and n ;n . In this K K K ? ? ? K ? regime, we can neglect the variations of an initially constant atomic density and the main e!ect of the atomic condensate is that of a coherent (i.e. one that preserves the phase information) reservoir of atoms for the molecular condensate. In that limit, the molecular "eld equation of motion is linear: (56) i

Q "[e(t)#jn !i c /2] #(a/(2) (t) , ? K ? K K where c corresponds to the molecule loss-rate: c / "c n . In the same o!-resonant limit, the K K K? ? atomic condensate "eld, to lowest order, propagates without feeling the e!ect of the molecular condensate: (t)+(n exp(!ij n t/ ). The solution to Eq. (56) for a detuning that is suddenly ? ? ? ? shifted from its initial value to e then gives the time dependent "eld (t): D K it it c t

(t)" exp ! (2j n ) #[ ! ]; exp ! (e #jn ) exp ! K , (57) K ?

? ?

D 2

where represents the initial value of the molecular "eld, " (t"0), and is the value that K the molecular "eld tends to at large times, t \ solution, "0. Although the all-molecule solution represents a third solution in the detuning ? interval !2(e(#2, the analysis of the next paragraph shows that this solution is unstable in this detuning region. The previously obtained o!-resonant limit (62), +!an/(2e or K x+!1/e, corresponds to x if e \ into Eq. (77) yields a set of linear equations in the a and a -coe$cients. The frequency at which > \ the system near the all-molecule state oscillates after it has been perturbed by breaking up a few molecules into atoms, follows from the requirement of a non-trivial solution to the linear set of equations in a and a : > \

u"(e/2)!(a(n) . (79) Thus, u is imaginary if "e"(2a(n, and the all-molecule system is unstable in the regime where both "x "(1 and "x "(1. > \

Fig. 4. Plot of the molecule fraction (N /N), as a function of the detuning, scaled by the &tunneling energy', a(n, for both K stationary state solutions, called x and x in the text to denote a molecular "eld of, respectively, positive and negative > \ sign. The atomic and molecular condensates were assumed to be &ideal' in this calculation, i.e. j"j "j "0. The ? K all-molecule branch (N /N)"1/2, of the x -state is shown in dotted line to indicate that the homogeneous x -branch is K > > unstable in this regime, as explained in the text.

226


Thus, at any value of the detuning, we "nd two (and not three) di!erent stationary states the system can occupy. Which state the system will "nd itself in, depends on its history. For instance, if the condensate was brought near-resonance by adiabatically increasing the detuning from &below resonance', the system is expected to reside in the x -state. In Fig. 4, we plot the molecule fraction > of both stationary states as a function of the detuning. Interestingly, the analysis of the next paragraph shows that the all-molecule branch of the x -solution is unstable with respect to > density #uctuations. In the "gure, we indicate the instability by plotting the unstable branch in a dotted line. 4.3. Considerations regarding the mechanical stability The previous analysis does not give insight into the stability of the homogeneous system with respect to position dependent #uctuations of the densities. That this is a major concern in describing the dynamics of these systems will become clear from the discussion below. A "rst example that illustrates the importance of position dependent "eld #uctuations, is the instability of the homogeneous all-molecule solution if e'2a(n. To see the instability, we allow for position dependent #uctuations and solve for plane wave -#uctuations around the all? molecule solution,

"exp(!iet/2 )[a exp(!i[k ) r!ut])#a exp(i[k ) r!ut])] . (80) ? > \ Upon insertion of this expression in the equations of motion (74), after restoring the kinetic energy term, we "nd that the resulting frequency,

k e !(a(n) , !

u" I 2M 2

(81)

does become imaginary for a "nite range of wave numbers if e'2a(n. Speci"cally, the modes with k vectors of magnitude (M(e!2a(n )( k((M(e#2a(n ) grow exponentially: the homogeneous x "#1-branch is unstable and the system spontaneously generates a "nite atomic > condensate. Consequently, we show the corresponding branch of the x -state in Fig. 4 in dotted > line, to indicate that the system will not remain in that state. Whether the other branches are stable, is a question that can be answered from a more general RPA-study, and will be the subject of further research. However, even without the knowledge of the excitation modes, we can make statements about the mechanical stability of the system by investigating the pressure, for example. Whereas we could omit the inter-particle interactions to get an accurate qualitative picture of the population dynamics, their e!ects in the mechanical stability of the homogeneous system are all-important and they have to be considered. Indeed, the ground state of the homogeneous near-resonant ideal gas-condensate is always unstable. This follows from the pressure, P, which we determine from the energy per atomic particle, e, P"n(de/dn): P"n "n

Re Re dx #n Rn Rx dn Re , Rn

(82)


227

where x represents the actual value of the scaled molecular "eld, which follows from Re/Rx"0. In the ideal gas condensate, e"e "(e/2)x#a(n(1!x)x, we "nd P "n(a(n/2)x (1!x) , (83) so that its pressure has the same sign as the molecular "eld. In the ground state system, for which x "x (0, P is negative and the homogeneous system is unstable. Furthermore, we expect the \ stationary state of positive molecular "eld, x , to be equally unstable, as its energy can be lowered > by spatially separating the atomic and molecular condensates. The inter-particle interactions can stabilize the homogeneous condensate system, but only at higher values for the density or in the o!-resonant regime. The pressure that follows from Eq. (82) can be written as j a(n j nx(1!x) . P" ? n# K n #jn n # K ? 2 2 K 2 ?

(84)

In this expression, we recognize the &elastic' inter-particle interaction contributions in addition to the atom-molecule Feshbach resonant coupling term that was the sole contribution to P . The inter-particle interaction contributions are proportional to n, j n/2, where j is an interaction P P strength of a magnitude that is representative of j , j and j. The a-contribution to the ground ? K state pressure, on the other hand, is of order !an if the system is near-resonance, e+0, and of order !an;(a(n/[2e]) in the o!-resonant regime. Near-resonance, the inter-particle interaction and atom-molecule coupling contributions are of the same order of magnitude if n+n "(a/j ). In the limit of vanishing density, n;n "(a/j ), the density-dependent contribuP P P P tion to e is dominated by the Feshbach-resonant interaction and the system behaves as the near-resonant ideal gas system. In contrast, at &high' densities, n#N

(10)

where u are the Dirac spinors u "(), u "(), E "(p#m, and N is the normalization. I \ N This wave function has de"nite helicity and hence is an eigenfunction of the helicity operator, ; W "2kW . N NI NI

(11)

The wave functions with k"$ are doublets for SK and, since SKI "; SK ; , the wave functions G G N G N WI "; W "2kW are doublets for SIK , the pseudospin. N NI N I G N I 5. Realistic mean 5elds A near equality in the magnitude of mean "elds, < +!< , is a universal feature of the 1 4 relativistic mean "eld approximation (RMA) of relativistic "eld theories with interacting nucleons and mesons [18] and relativistic theories with nucleons interacting via zero range interactions [19], as well as a consequence of QCD sum rules [20]. We shall discuss QCD sum rules in the next section. Recently, realistic relativistic mean "elds were shown to exhibit approximate pseudospin symmetry in both the energy spectra and wave functions [15,21,22]. In Fig. 2 we show the energy splittings between pseudospin doublets normalised by 2lI #1 as a function of the average binding #elI )/2. We see that the energy splitting for the same pseudo-orbital energy 1e2"(elI > \ angular momentum decreases as the radial quantum increases; that is, as the binding energy decreases. Also for the same binding energy, the energy splitting increases as the pseudo-orbital angular momentum increases. These features follow from the square well potential [6]. In Table 1 we tabulate some of the calculated energy splittings compared to the measured splittings. What is interesting is that the measured splittings are smaller than those calculated by relativistic mean

236

J.N. Ginocchio / Physics Reports 315 (1999) 231}240

Fig. 2. Pseudospin doublet energy splittings normalised by 2lI #1 as a function of the average binding energy 1e2 [15]. n is the radial quantum number of the state with the lower orbital quantum number.

Table 1 Pb pseudospin doublet energy splittings for RMA [15] compared to the experimental values lI

4 2 3 1

ps doublets Neutrons 0h !1f 1f !2p Protons 0g !1d 1d !2s

elI !elI (RMA) > \

elI !elI (EXP) > \

2.575 0.697

1.073 !0.328

4.333 1.247

1.791 0.351

"eld theory indicating that pseudospin symmetry breaking is overestimated by the relativistic mean "eld approximation. Pseudospin doublets will manifest themselves for deformed potentials as well when < +!< . 1 4 In Fig. 3 the single particle (s.p.) energies of the doublets are plotted versus the deformation. We see


237

Fig. 3. Single-particle (s.p.) energies for neutron pseudospin partners as a function of deformation b [21].

that for each value of pseudo asymptotic quantum [NI n KI ] numbers there is a quasi-degenerate pseudospin doublet with X"KI $. As mentioned in the last section the relativistic SU(2) pseudo-spin symmetry implies that the spatial wave function for the lower component of the Dirac wave functions will be equal in shape and magnitude for the two states in the doublet. For spherical nuclei the Dirac wave function for "(g lI [>lI s]HlI >, if lI [> s]HlI >), the two states in the doublet are W O > > K O > lI K O HlI > K lI lI "(g lI [> s]H \, if lI [> s]H \) where g, f are the radial wave W lI O \ lI \ K O \ lI K H \ K functions, >l are the spherical harmonics, s is a two-component Pauli spinor, and [2]H means coupled to angular momentum j. For a square well potential, the overall phase between the two amplitudes will be a minus sign [6] so we expect that, in the symmetry limit for realistic (r)"!flI (r). In Fig. 4 we see that, for realistic zero range potentials, potentials, flI > \ (r)+!flI (r) [15]. flI > \ These results are also valid for the relativistic mean "eld approximation to a nuclear "eld theory with meson exchanges [21]. In Fig. 5a the pseudospin doublets in the vicinity of the Fermi surface for neutrons and protons are shown. The upper (g) and lower components (f ) of the pseudospin doublets are also shown in Fig. 5b}d. While the upper components are very di!erent with di!erent nodal structure, the lower components are almost identical. We also note that the lower components are small with respect to the upper components, which is consistent with the non-relativistic shell model.

238


Fig. 4. Pb lower component wave functions f lI (r) (dash line), !f lI (r) (dot}dash line) for the (2s ,1d ) O \ O > pseudospin doublet (lI "1) as a function of the radius r [15].

Fig. 5. Pb: (a) energy spectrum; (b}d) upper (g) and lower (f ) components of the Dirac wave function for pseudospin doublets as a function of the radius r [21].


239

6. QCD sum rules Applying QCD sum rules in nuclear matter, the scalar and vector self-energies were determined to be [20] R "!4pp o /mm , Q , ,

(12) R "32po /m , T , where o is the nuclear density, and m the quark mass. p is the sigma term which arises from the , , breaking of chiral symmetry [23]. The ratio then becomes R /R "!p /8m . (13) Q T , For reasonable values of p and quark masses, this ratio is close to !1. The implication of these , results is that chiral symmetry breaking is responsible for the scalar "eld being approximately equal in magnitude to the vector "eld, thereby producing pseudospin symmetry.

7. Antinucleon spectrum The antinucleon states are obtained by charge conjugation, C, applied to the negative energy eigenstates of the Dirac Hamiltonian [16]. This leads to a spectrum which has quasi-degenerate spin doublets, not pseudospin doublets. This follows from the fact that, under charge conjugation,

s( CRSIK C" G G 0

0 "SK . (14) G s( G Thus in Eq. (14) the spin operator s( operates on the upper component and hence the spatial wave G functions for the upper components of the states in spin doublet will be very similar. Likewise for spherical nuclei, the pseudo-orbital angular momentum goes into the orbital angular momentum, and for axially deformed nuclei, pseudo-orbital projection goes into orbital projection along the body "xed z-axis. This symmetry in the antinucleon spectrum also follows from the fact that the antinucleon potentials are <M "CR< C"< , and <M "CR< C"!< . Thus <M +<M and the symmetry 1 1 1 4 4 4 1 4 of the Dirac Hamiltonian generated by Eq. (3) applies and spin doublets are produced in the antinucleon spectrum [17].

8. Summary We have shown that pseudospin symmetry is a broken SU(2) symmetry of the Dirac Hamiltonian which describes the motion of nucleons in realistic scalar and vector mean "eld potentials, < +!< . This symmetry predicts that the spatial wave functions of the lower components for 1 4 states in the doublet will be very similar in shape and size and this has been substantiated by

240


relativistic mean "eld approximations of relativistic nuclear "eld theories and relativistic nuclear Lagrangians with zero range interactions. This symmetry has been linked via QCD sum rules to chiral symmetry breaking in nuclei. Finally, the antinucleon spectrum is shown to have a spin symmetry rather than a pseudospin symmetry. Future applications of pseudospin symmetry will involve the testing of the wave function through the relationships between transitions within pseudospin doublets which follow from pseudospin symmetry [24].

Acknowledgements This work was supported by the United States Department of Energy.

References [1] R. Slansky, Phys. Rep. 11 (1974) 99. [2] R. Slansky, Phys. Rep. 79 (1981) 1. [3] S. Kass, R.V. Moody, J. Patera, R. Slansky, A$ne Lie Algebras, Weight Multiplicities, and Branching Rules, University of California Press, Berkeley, 1990. [4] K.T. Hecht, A. Adler, Nucl. Phys. A 137 (1969) 129. [5] A. Arima, M. Harvey, K. Shimizu, Phys. Lett. 30B (1969) 517. [6] J.N. Ginocchio, Phys. Rev. Lett. 78 (1997) 436. [7] J.N. Ginocchio, A. Leviatan, Phys. Lett. B 425 (1998) 1. [8] A. Bohr, I. Hamamoto, B.R. Mottelson, Phys. Scripta 26 (1982) 267. [9] T. Beuschel, A.L. Blokhin, J.P. Draayer, Nucl. Phys. A 619 (1997) 119. [10] J. Dudek, W. Nazarewicz, Z. Szymanski, G.A. Leander, Phys. Rev. Lett. 59 (1987) 1405. [11] W. Nazarewicz, P.J. Twin, P. Fallon, J.D. Garrett, Phys. Rev. Lett. 64 (1990) 1654. [12] F.S. Stephens et al., Phys. Rev. C 57 (1998) R1565. [13] B. Mottelson, Nucl. Phys. A 522 (1991) 1. [14] A.L. Blokhin, C. Bahri, J.P. Draayer, Phys. Rev. Lett. 74 (1995) 4149. [15] J.N. Ginocchio, D.G. Madland, Phys. Rev. C 57 (1998) 1167. [16] W. Greiner, B. MuK ller, J. Rafelski, Quantum Electrodynamics of Strong Fields, Springer, New York, 1985. [17] J.S. Bell, H. Ruegg, Nucl. Phys. B. 98 (1975) 151. [18] B.D. Serot, J.D. Walecka, The relativistic nuclear many-body problem, in: J.W. Negele, E. Vogt (Eds.), Advances in Nuclear Physics, vol. 16, Plenum Press, New York, 1986. [19] B.A. Nikolaus, T. Hoch, D.G. Madland, Phys. Rev. C 46 (1992) 1757. [20] T.D. Cohen, R.J. Furnstahl, K. Griegel, X. Jin, Prog. Part. Nucl. Phys. 35 (1995) 221. [21] G.A. Lalazissis, Y.K. Gambhir, J.P. Maharana, C.S. Warke, P. Ring, Phys. Rev. C 58 (1998) R45; LANL archives nucl-th/9806009. [22] J. Meng, K. Sugawara-Tanabe, S. Yamaji, P. Ring, A. Arima, Phys. Rev. C 58 (1998) R628. [23] T.P. Cheng, L.F. Li, Gauge Theory of Elementary Particle Physics, Oxford University Press, New York, 1984. [24] J.N. Ginocchio, Phys. Rev. C 59 (1999), in press; LANL archives nucl-th/9812025.


Quasicrystal Lie algebras and their generalizations Jir\ mH Patera , Reidun Twarock* Centre de Recherches Mathe& matiques, Universite& de Montre& al, Montre& al, Que& bec, Canada Arnold Sommerfeld Institut, TU-Clausthal, 38678 Clausthal, Germany

Abstract We review and extend the results about quasicrystal Lie algebras of Patera et al. [Phys. Lett. A 246 (1998) 209], which is a new family of in"nite dimensional Lie algebras over the real and complex number "elds, whose generators are in a one-to-one correspondence with the points of a one-dimensional quasicrystal. Some new properties of quasicrystal Lie algebras and further details on their representation theory are pointed out and the concept of generalized quasicrystal Lie algebras is presented. The latter allows to associate to the generators of the Lie algebra quasicrystal points of one-dimensional quasicrystals with acceptance windows symmetric around 0, which was not possible in the framework of Patera et al. 1999 Elsevier Science B.V. All rights reserved. PACS: 02.20.Sv; 61.44.Br Keywords: Lie algebras; Quasicrystals; Witt (Virasoro) algebra

1. Introduction Recent development in the theory of quasicrystals has led to the study of the so-called CUT AND also called model sets, or quasicrystals. The latter we will use here for simplicity. Such quasicrystals can be viewed as rather idealized models of physical quasicrystals studied in laboratories, or as a generalization of lattices. PROJECT POINT SETS,

* Corresponding author. E-mail addresses: [email protected] (J. Patera), [email protected] (R. Twarock) Work supported in part by the Natural Sciences and Engineering Research Council of Canada and by the Fonds FCAR of Quebec. Work supported by the Ministry for Science and Culture of Lower Saxony in the framework of the DorotheaErxleben Program. 0370-1573/99/$ - see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 9 ) 0 0 0 2 2 - 8

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J. Patera, R. Twarock / Physics Reports 315 (1999) 241}256

Quasicrystals share some important properties with lattices. Both are deterministic point sets so that the removal or the addition of a single point in either of them creates a defect in the structure. Both are also uniformly dense throughout the entire space and uniformly discrete, a property technically referred to as the Delone property. The crucial distinction between lattices and quasicrystals is the periodicity of the former in contrast to the complete lack of such periodicity in the latter. A quasicrystal, in spite of its determinism, does not contain a periodic subset. There is a further general class of common properties between lattices and quasicrystals which only recently were fully discovered [1,2]. These are the scaling symmetries. In lattices, such symmetries are rarely exploited because most of the pertinent results can be obtained using the periodicity, but in quasicrystals there is no choice but to use scaling symmetries, which represent an important tool in the description, generation, and theoretical handling of quasicrystals. There are di!erent types of quasicrystals. They can be split into families distinguished by the irrationality which is built into the coordinates of their points. The best studied among them are the quasicrystals involving quadratic irrationalities, given by the so-called Pisot numbers which are de"ned via solutions of certain algebraic equations. Among Pisot numbers related to quadratic equations, the ones by far most studied in connection with quasicrystals are the ones related to the equation x"x#1 and thus to its solutions q"(1#(5),

q"(1!(5) .

(1)

The presence of two solutions (1) allows for a new symmetry-like transformation in the theory, namely the interchange q q. It turns out that this transformation is of fundamental importance for the theory of quasicrystals. In physics, quasicrystals related to this type of irrationality were "rst observed in 1984 [3]. The aperiodic structures were recognized by the presence of localities with non-crystallographic 5-fold re#ection symmetry in the X-ray di!raction pattern. Note that irrationality q is the golden ratio known for a number of reasons since Antiquity and plays a fundamental role in nature, e.g. in the spiral formed by a shell and the curve of a fern [4}9]. The observation of the re#ection symmetry generating 5-fold rotation brought up a further relation between quasicrystals and lattices through the "nite Coxeter groups, which split into two types, the crystallographic and non-crystallographic ones. The former are "nite symmetry groups of lattices and it is known that in three-dimensional lattices one "nds 2-, 3-, 4-, and 6-fold symmetries. Correspondingly, in the general n-dimensional case, lattice symmetries are given by the appropriate Coxeter group. Furthermore, crystallographic Coxeter groups are in a 1}1correspondence with the semisimple "nite dimensional Lie algebras (over the complex number "eld) and play a fundamental role in physics. The non-crystallographic Coxeter groups exists (as irreducible groups) only in dimensions 2, 3, and 4. There are in"nitely many such groups in dimension 2, which are the dihedral groups well known in physics; in dimensions 3 and 4 there is precisely one non-crystallographic group in each of them, namely the groups H and H , both containing the 5-fold symmetry dihedral group H . The analogous properties of lattices and quasicrystals, particularly the link between crystallographic and non-crystallographic Coxeter groups, suggests the possibility of a link at the level of Lie theory. Are there Lie algebras related to non-crystallographic Coxeter groups of any kind? The


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answer for now is de"nitely negative, or at least there is no standard link between such groups and Lie algebras. Thus at best one may hope for some kind of non-traditional relation between them. In this article we show two constructions of Lie algebras built around one-dimensional quasicrystals. The Lie algebras we construct here resemble more to the Witt algebra than to the semisimple ones. In fact, our algebras may be perfect but are never semisimple.

2. Mathematical preliminaries In this section we want to introduce quasicrystals and recall their pertinent properties. Furthermore, we recall some information from Lie theory for later use. The quasicrystal points in our case are represented by numbers on the real axis, more precisely numbers of the form a#bq where a and b are integers. Thus the stage for our construction is the algebraic number "eld Q[q], although in most cases it su$ces to use its ring of integers Z[q]"Z#Zq. The de"nition of a quasicrystal below is based on the (Galois) automorphism, denoted by prime, which exists in Q[q] and hence also in Z[q] and interchanges q with q. In the general theory of quasicrystals in higher dimensions the automorphism is combined with the corresponding mapping of the bases of these spaces by what is called the star map. In one-dimension, we also call this automorphism a star map, even if it is here just the Galois automorphism in Q[q]. A point x becomes x according to x"a#b(5 x"a!b(5, a, b3Z . We stress here a remarkable property of the star map, which underlies the de"nition of a quasicrystal and is therefore crucial in our context: the star map is everywhere discontinuous. Two arbitrarily close points on the real line can have very distant images under the star map, as can be seen for example from the pairs of numbers 1 and 1#q\, with 1 and 1#q\"1!q after the mapping. It is known that Z[q] is a Euclidean domain, in particular Z[q] is a unique factorization domain. For a set FLZ[q], one has gcd+F,"(gcd+F,). The group of units of Z[q] consists of +$qI " k3Z,. It is also known that each prime p of Z of the form p,$2 (mod 5) remains prime in Z[q]. Each prime p of Z of the form p,$1 (mod 5) splits as a conjugated pair p"qq with qOq. We now introduce the de"nition for a one-dimensional quasicrystal [2,10}12]. Such an object is speci"ed by a bounded interval (r, t) (acceptance window) in R. A point x belongs to the quasicrystal provided its star map image x is in (r, t). De5nition 1. Let (r, t) be a bounded interval. The quasicrystal R((r, t)) is the point set R((r, t)) " : +x3Z[q] " x3(r, t), .

(2)

The interval (r, t) is said to be the acceptance window of R((r, t)). Note that in the de"nition one could take the interval to be open or closed, or even semi-open. The corresponding quasicrystals di!er in these cases by two or only one point. Therefore, for simplicity, we take the interval open whenever possible in order to avoid the exceptional points in

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the quasicrystal. We further note that the density of points in a quasicrystal is not in#uenced by the position of the acceptance window in R. The density is determined by the length "r!t" of the acceptance window. It is conventional to consider distances between adjacent points of a quasicrystal as tiles. Quasicrystals with the acceptance window satisfying "r!t""qI for some integer k are two-tile quasicrystals, otherwise there are three tiles in the quasicrystal [13]. A useful example is the quasicrystal R([0, 1]). The acceptance condition is given as 04x"a#qb"a!q\b41 which leads to a formula for the points of R([0, 1]) given by R([0, 1])"+[1#b/q]#qb " b3N,6+0, ,

(3)

where [A] denotes the integer part of A. The points of R([0, 1]) nearest to the origin are the following: 2,!q, 0, 1, 1#q, 2#2q, 2#3q, 3#4q, 4#5q, 4#6q, 5#7q, 5#8q,2

(4)

It is obvious from (3) that there are only three possible tiles in R([0, 1]), one tile occurring precisely once between 0 and 1. The other tiles are q or 1#q"q. We remark that rescaling of the boundaries by qI with k an integer, produces a rescaled version of the quasicrystal, i.e. the tiles are rescaled accordingly, but the tiling sequence remains the same. Furthermore, the tiling sequence is constrained by the fact that the ratio between adjacent tiles is 1 : q or 1 : 1 [13]. A further important feature of quasicrystals R([a, b]) is their re#ection symmetry around the point (a#b)/2. Thus, in the case of R([0, 1]) there is a re#ection symmetry around , and therefore its negative points are readily obtained from those greater than 0. Indeed, for every x3R([a, b]), one has y"(b#a)!x3R([a, b]) because x3[a, b]0y"b#a!x3[a, b]. Correspondingly, for R([!1, 0]) we obtain R([!1, 0])"+[n/q]#qn " n3N, which can be seen to be given by the points of R([0, 1]) re#ected through the origin. From the de"nition of quasicrystals given in (2) some properties typical of quasicrystals can be readily inferred: E Selfsimilarity: If the acceptance interval (r, t) is open, any "nite pattern P3R((r, t)) is repeated in"nitely often. Since (r, t) is assumed to be open, x#PL(r, t) if x is chosen such that x is small enough. E Absence of any periodic subset: A translational symmetry would mean that we have x#R((r, t))LR((r, t)), i.e. x#(r, t)L(r, t) and also for the closure of the interval x#[r, t]L[r, t]. However, the latter implies x"0 and thus x"0. E Central q-in#ation symmetry and quasiaddition: Instead of translational symmetries, we have for intervals (!r, r) symmetric around the origin qLx3R((!r, r)) with n3N whenever x3R((!r, r)). Furthermore, R((!r, r)) is invariant under quasiaddition xq!yq for x, y3R((!r, r)). The "rst can be seen from the fact that (q)Lx3(!r, r). The second follows from the fact that (q) and !q are both positive and add up to 1.


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We remark that the q-in#ation mentioned above is only one representative of an in"nite family of scaling symmetries [2].

3. Quasicrystal Lie algebras In this section we review the construction of a particular type of in"nite dimensional Lie algebras over the real and/or complex number "elds, called quasicrystal Lie algebras, which was "rst introduced in [1]. Its generators are in a one-to-one correspondence with points from a onedimensional quasicrystal. Like the quasicrystals introduced in (2) they depend crucially on the choice of the acceptance window, and they are therefore denoted by Q(X), where the acceptance window X is a bounded interval. There is a certain resemblance between the quasicrystal Lie algebras and the Witt algebra [14}16], particularly some of the rank-2 Witt algebras of Patera and Zassenhaus [17], because the latter have served as a building block for the quasicrystal Lie algebras. But despite this super"cial resemblance, there are substantial di!erences between the two types of Lie algebras, which stem from the quasicrystal set underlying the construction of quasicrystal Lie algebras. To anticipate only a few of the di!erences at this stage, we remark that quasicrystal Lie algebras do not allow for a central extension, so that we do not have, strictly speaking, a quasicrystal analog to the Virasoro algebra. However, quasicrystal Lie algebras contain an abundance of "nite dimensional subalgebras, which might open interesting possibilities for applications to physics. In order to de"ne quasicrystal Lie algebras Q(X) corresponding to a one-dimensional quasicrystal R(X) with bounded X according to Patera et al. [1], the acceptance interval has to be restricted to a positive or negative interval, i.e. 0 is not allowed to be an inner point. This restriction is necessary to establish the Lie algebra structure of the quasicrystal Lie algebras. Then we have the following. De5nition 2. Let F be any number "eld such that FMQ[q] and let aOb be real numbers such that 04ab(R. Let X be one of the intervals [a, b], (a, b], [a, b), or (a, b). The quasicrystal Lie algebra Q(X) over F is the F-span of its basis B(Q(X))"+¸ " n3R(X), , L with the commutation relations of the basis elements given by

(m!n)¸ if n#m3R(X) , L>K [¸ , ¸ ]" L K 0 if n#m , R(X) .

(5)

Whereas the antisymmetry of the commutators is obvious, the Jacobi identity has to be ensured by the condition that either XLRY or XLRX, i.e. it holds provided ab50 and requires the restriction of the acceptance intervals as mentioned before. We note that it is always possible to enlarge the algebra Q([a, b]) with ab'0 to Q([a, b]6+0,). This will be important for the representation theory. The commutation relations (5) resemble the standard Witt algebra [14}17], but in contrast to the latter they contain many commutators equal to 0, because the operation àdditiona is often not

246


compatible with the quasiperiodic structure of the quasicrystal, i.e. it does not follow from m, n3R(X) that one must have m#n3R(X). As an example consider some of the commutators in Q([0, 1]): [¸ ,¸ ]"!q¸ , [¸ ,¸ ]"(1#q)¸ , >O >O >O >O >O >O [¸ ,¸ ]"0, [¸ ,¸ ]"(2#3q)¸ , >O >O >O >O >O [¸ ,¸ ]"!(3#6q)¸ , [¸ ,¸ ]"!(4#7q)¸ . >O \\O >O \\O \O The behaviour of the corresponding quasicrystal points under addition is illustrated in Fig. 1. It is often more convenient to rewrite the commutation relations (5) using the characteristic function sX of the interval X: [¸ , ¸ ]"(m!n)sX(n#m)¸ . (6) L K L>K In real life quasicrystals are always "nite size objects, fragments of the ideal in"nite quasicrystals we have been considering throughout the paper so far. A "nite quasicrystal implies that also the window contains only a "nite number of points, forming another quasicrystal of "nite size. The star map is then a duality transformation between the two pictures of one (?) quasicrystal: one in the quasicrystal space, the other in the space of the acceptance window. We note here that it is also possible to discretize the acceptance window X of the quasicrystal R(X) by viewing it as another quasicrystal,now with an acceptance window X lying in the original quasicrystal space. The discretization of X to X leads to a restriction of R(X) to R(X ) which is B B a "nite subset of R(X), i.e. a "nite quasiperiodic set. The algebra corresponding to this set, i.e. Q(X ), B is "nite dimensional and is in one-to-one correspondence with points m3X which under the star map fall into the acceptance interval X. It takes the form (7) [¸ , ¸ ]"(m!n)sX(n#m)sX (n#m)¸ Y L>K L K with n, m3X and n, m3X and it is easily veri"ed that antisymmetry and the Jacobi identity hold, so that the object is again a Lie algebra. As an example, consider Fig. 2. The points in X correspond to generators of a "nite dimensional Lie algebra with "ve non-vanishing commutators given by [¸ , ¸ ]"(2!q)¸ , [¸ , ¸ ]"¸ , \>O O \>O \>O O \>O [¸ , ¸ ]"(1#q)¸ , [¸ , ¸ ]"(1!q)¸ , \>O O \>O \>O O \>O [¸ , ¸ ]"¸ . \>O O \>O Further examples may be obtained by changing the size of X and X.

Fig. 1. Addition of quasicrystal points corresponding to the Lie algebra in the example above.


247

Fig. 2. Illustration of projection method for "nite quasicrystal Lie algebras.

4. Properties of quasicrystal Lie algebras Here we illustrate some properties of quasicrystal Lie algebras following from De"nition 2.1. Some of them can be found also in [1], others are stated here for the "rst time. E The algebras Q(X) and Q(!X) are isomorphic under ¸ ¸ for any x3R(X). V \V The reason for this is that x3X is equivalent to !x3!X. E From now on we assume that the boundary points of X satisfy 04a(b. E Any Q([a, b]) with 2a5b is an Abelian Lie algebra. Any Q([a, b]) with 2a(b is non-Abelian. The commutator between ¸ and ¸ is zero if n#m is not in R([a, b]), i.e. if n#m are L K not in [a, b]; also, the commutator of an element with itself is zero. Hence all commutators vanish for 2a5b. Otherwise, there can always be found two points which lead to a non-trivial commutator. E From now on we discuss only the non-Abelian algebras unless otherwise stated. E The Lie algebra Q([a, b]) has a non-trivial center precisely if a'0. Its center is the Lie algebra Q((b!a, b]). The center contains elements ¸ with k#n'b for all n3[a, b]. Hence, k3(b!a, b]. I : +¸ " m3R LR(X), of Q(X) is E Let r " : inf R1 m. Then the centralizer Q(X) of a subset S " 1 K 1 KZ given as Q(X) "Q((b!r, b]), if the in"mum is attained, i.e. r " : min R1 m, and as 1 KZ Q(X) "Q([b!r, b]) otherwise. 1 With r as above, the centralizer contains ¸ with k#r'b (resp. k#r5b), so that k3(b!r, b] I (resp. k3[b!r, b]).

248


E For a subalgebra A " : +¸ " m3R([r, b]), a4r(b, of Q([a, b]) the normalizer A is given by K , A "Q([a, b]). For a subalgebra AI " : +¸ " m3R([r, l]), r5a, l(b, 2r'b, of Q([a, b]) the , K normalizer AI is given by AI "+¸ " m3R((b!r, b]),6+¸ , if a"0 and , , K AI "+¸ " m3R((b!r, b]), otherwise. , K For the case of A this follows from the fact that m#n5r if m3[r, b] and n3[a, b]. In the case , of AI notice "rst of all that the condition 2r'b is necessary to ensure that it is indeed a subalgebra. Then the claim follows from the condition: r4n#m4l or n#m5b for m3[r, l] and n3[a, b]. E In the case of the toy-model R([0, 1]), it can be shown that any n points of this quasicrystal never add up to 0. As a consequence, the lower or upper central series of the quasicrystal Lie algebra Q([0, 1]) never produces the generator ¸ . This can be seen by implementing (3). E The non-Abelian part of Q([a, b]) is indecomposable. The commutation of two Lie algebra generators corresponds on the quasicrystal level to the addition of primed indices in X. Because XLR, every point of it can be written as a sum of (in"nitely) many pairs of other points of X and hence Q([a, b]) is indecomposable. E The Lie algebras Q([a, b]) and Q([qa, qb]) are isomorphic, ¸ ¸ . More generally, non-Abelian V OV Lie algebras Q([a, b]) and Q([c, d]) are isomorphic precisely if a"qIc and b"qId for k3Z. It is a consequence of the fact that qZ[q]"Z[q]. E The algebra Q([c, b]) is an ideal of Q([a, b]) provided 04a(c(b. Consequently, there are no semi-simple quasicrystal Lie algebras. It follows from the fact that n#m5c if n3[c,b] and m3[a, b]. E The derived algebra of Q([a, b]) is Q((2a, b]). Only the algebras Q((0, b]) and Q((0, b)) are perfect. The algebras Q([a, b]) are solvable and nilpotent if a'0. The "rst claim is again a consequence of the fact that commutation of generators in the Lie algebra means addition of primed coe$cients in the acceptance window of the quasicrystal. Solvability and nilpotency hold because the upper and lower central series break after the primed indices leave the acceptance window under addition. This happens after a "nite number of steps for a'0. E The algebras Q(X) admit only a trivial central extension [15], i.e. the direct sum Q(X)Fc. It follows from the standard cocycle condition, if one uses the fact that in the quasicrystals admissible for the construction of quasicrystal Lie algebras, points are never centrally symmetric around 0, i.e. if n3R(X) one does not have !n3R(X).

5. Finite dimensional subalgebras of quasicrystal Lie algebras In this section we want to illustrate an important property of quasicrystal Lie algebras, which is its abundance of "nite dimensional subalgebras mentioned earlier. This structure could be crucial for applications of quasicrystal Lie algebras in physics, because quasicrystal Lie algebras substantially di!er in this respect from the standard Witt algebras as well as the rank-2 Witt algebras. The generators of the latter are indexed by the elements of Z[q] (see [17]), but the corresponding algebras have no indecomposable subalgebras of "nite dimension greater than 3. The phenomenon is described by the following.


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Lemma 3. The closure of any xnite set of generators from B(Q(X)) under the commutations (5) is a xnite dimensional subalgebra of Q(X). Proof. Let ¸ ,2, ¸ I3B(Q(X)) be a "nite set of basis elements. This implies x ,2, x 3X. Since V I V X is bounded, there is only a "nite number of linear combinations n x #n x #2#n x in I I X with n ,2, n 3ZY. Hence only "nitely many multiple commutators of the elements of the set I are non-zero. 䊐 As an example consider the subalgebra of Q([0, 1]) generated by the basis elements ¸ and >O ¸ . The closure of the elements ¸ and ¸ under the commutation relations is \\O >O \\O a 12-dimensional subalgebra, which we denote by D. Setting a"2#3q and b"!1!2q, we get the rest of the basis elements by repeated communications of ¸ "¸ , ¸ "¸ : ? >O @ \\O [¸ , ¸ ]"(b!a)¸ "(!3!5q)¸ , ? @ ?>@ >O [¸ , [¸ , ¸ ]]"b(b!a)¸ "(13#21q)¸ , ? ? @ ?>@ >O [¸ , [¸ , ¸ ]]"a(b!a)¸ "(!21!34q)¸ , @ ? @ ?>@ \O [¸ , [¸ , [¸ , ¸ ]]]"b(b!a)¸ "(34#55q)¸ , ? ? ? @ ?>@ >O [¸ , [¸ , [¸ , ¸ ]]]"2ab(b!a)¸ "(178#288q)¸ , ? ? ? @ ?>@ >O [¸ , [¸ , [¸ , ¸ ]]]"a(b!a)¸ "(!55!89q)¸ , @ @ ? @ ?>@ \\O [¸ , [¸ , [¸ , [¸ , ¸ ]]]]"b(b!a)(2a#b)¸ "(322#521q)¸ , ? ? ? ? @ ?>@ >O [¸ , [¸ , [¸ , [¸ , ¸ ]]]]"3ab(b!a)¸ "(699#1131q)¸ , @ ? ? ? @ ?>@ >O [¸ , [¸ , [¸ , [¸ , ¸ ]]]]"3ab(b!a)¸ "(699#1131q)¸ , ? @ @ ? @ ?>@ [¸ , [¸ , [¸ , [¸ , [¸ , ¸ ]]]]]"b(b!a)(2a#b)(3a#b)¸ "(5257#8506q)¸ . ? ? ? ? ? @ ?>@ >O D is solvable: Its upper central series consists of the derived algebra D and D " : [D , D ] generated by the following basis elements: D "+¸ , ¸ ,¸ ,¸ ,¸ ,¸ ,¸ ,¸ ,¸ ,¸ ,, >O >O >O >O >O >O >O \O \\O D "+¸ , ¸ ,. >O Furthermore, D is nilpotent: Its lower central series consists of D , E " : [D, D ],2, E " : [D, E ], given by E "+¸ ,¸ ,¸ ,¸ ,¸ ,¸ ,¸ ,¸ ,¸ ,, \\ \O >O >O >O >O >O >O E "+¸ ,¸ ,¸ ,¸ ,¸ ,¸ ,¸ ,, \\ >O >O >O >O >O E "+¸ , ¸ ,¸ ,¸ ,, >O >O >O E "+¸ ,. >O

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6. Representations of the quasicrystal Lie algebra Q(X) Here we provide information on the representation theory of quasicrystal Lie algebras Q(X). For simplicity, we "rst consider the Lie algebra Q([0, a]) and its representation in a space I X IK>Q X IZR . "sX(n#m#s)sX(m#s)(j#s)(j#m#s)d PL>K>Q With this, one obtains (¸ ¸ ) !(¸ ¸ ) "(sX(m#s)(j#m#s)!sX(n#s)(j#n#s)) L K PQ K L PQ ;sX(n#m#s)(j#s)d

PL>K>Q

.

s (n#m#s)(j#s) However, this is equal to (m!n)sX(n#m)d PL>K>Q X sX(n#m#s)O0, one has sX(n#m)"sX(m#s)"sX(n#s)"1.

because,

for

Note that representation (9) holds also for domains X"[a, b] with a'0, because the corresponding quasicrystal Lie algebra is a subalgebra of Q([0, b]). As in (9), it can be shown that the commutation relations are also satis"ed by , (¸ ) "sX(m#s#j)(j#s)d PK>Q>H K PQ>H

(10)


251

which resembles more closely to the corresponding results for the usual Witt algebra. To see this, consider the realization ¸ " : zK(zR ) acting on "s#j2 " : zQ>H, i.e. K X ¸ "s#j2"(s#j)"s#m#j2 K and its matrix form (¸ ) "(j#s)d (11) K PQ>H PK>Q>H and notice that (11) di!ers from (10) only by the factor sX(m#s#j). Correspondingly, a realization of quasicrystal Lie algebras in terms of di!erential operators can only be LOCAL in the sense that the operator realization depends on the vector of the representation space it is acting on. This dependence is then controlled by the function sX(m#s#j). In particular, we have ¸ "s#j2"zK(zR )sX(m#s#j)zQ>H"(s#j)sX(m#s#j)"s#m#j2 . K X Up to now it has been assumed that m and s (resp. s#j) are from the same quasicrystal, i.e. restricted by the same acceptance window. This assumption can be weakened by the introduction of another acceptance window X, such that ¸ zQ"zK(zR )sX (m#s)zQ"ssX (m#s)zK>Q Y K X Y again ful"lls the commutation relations of the quasicrystal Lie algebra. It can be shown that X is restricted by the following: Lemma 5. For given X"[a, b] the acceptance window X"[c, d] is constrained by the condition c5d!b, i.e. the largest X compatible with X is given by X"[d!b, d]. Proof. The left-hand side of (6) can be calculated } under the assumption that both X and X are simultaneously positive or negative } to give [¸ , ¸ ]zI"k(m!n)sX (m#n#k)zL>K>I . Y L K On the other hand,

(12)

[¸ , ¸ ]zI"(m!n)sX(m#n)¸ zI"k(m!n)sX(m#n)sX (m#n#k)zL>K>I . L K L>K Y To guarantee equality, we need sX (m#n#k)"1NsX(m#n)"1, which is violated if we have Y n#m'b and c#n#m4d, i.e. for c#b(d. 䊐

7. Generalized quasicrystal Lie algebras A crucial feature of quasicrystal Lie algebras is the one-to-one correspondence of their generators with the points of a suitably chosen one-dimensional quasicrystal. For technical reasons, the possible quasicrystals had to be restricted to those with an acceptance window of the form [a, b], (a, b], [a, b) or (a, b) with 04ab(R in order to guarantee that the quasicrystal Lie algebras are

252


indeed Lie algebras. The idea of this section is to generalize the concept of quasicrystal Lie algebras so as to admit quasicrystals with acceptance windows containing 0 as an inner point. In particular, the generators of generalized quasicrystal Lie algebras are in one-to-one correspondence with one-dimensional quasicrystals with acceptance windows of the form X"[!b,!a]6+0,6[a, b]. The advantage of this generalization is that it opens the possibility to use quasicrystal Lie algebras for modeling situations in physics which require such a type of central symmetry, e.g. the interaction of particles with opposite electric charge or particles and their antiparticles, by assigning to one of the particles the quasicrystal points associated with the positive acceptance interval and to the other one the quasicrystal points with negative acceptance interval. Several (related) generalizations of the concept of quasicrystal Lie algebras in the above sense are possible. All of them have in common that their generators are in a one-to-one correspondence with one-dimensional quasicrystals associated to an acceptance window X"X 6+0,6X with X "!X and that the commutation relations of the basis elements are de"ned separately for the four cases (n, k)3X ;X with i, j3+1, 2,. G H The prototype of a generalized quasicrystal Lie algebra is of the following form (Referred to as VERSION Ia): For n3X and k3X : [¸ , ¸ ]"!(k#n)+¸ sX(n!k)!¸ sX(k!n), . I\L L I L\I

(13)

For n3X and k3X : [¸ , ¸ ]"(k#n)+!¸ sX(n!k)#¸ sX(k!n), . I\L L I L\I

(14)

For n, k3X and n, k3X : s (!k!n), . [¸ , ¸ ]"(k!n)+¸ sX(n#k)!¸ \I\L X L I L>I

(15)

Similarly, in VERSION Ib, the sign on the right-hand side of the de"ning relations (13) and (14) can be simultaneously reversed without changing the Lie algebra structure of the object. VERSION Ia and Ib have the interesting property that a restriction of X to X or X , i.e. if sX is replaced by sXG for i"1, 2, respectively, leads to the fact that the last identity reproduces the commutation relations which de"ne the quasicrystal Lie algebras (6). Alternatively, the following VERSION IIa, which is obtained from VERSION Ia via sign changes in the right-hand side of the de"ning relations, can be used, because it also de"nes a Lie algebra structure: For n3X and k3X : [¸ , ¸ ]"!(k#n)+¸ sX(n!k)#¸ sX(k!n), . I\L L I L\I

(16)

For n3X and k3X : [¸ , ¸ ]"(k#n)+#¸ sX(n!k)#¸ sX(k!n), . I\L L I L\I

(17)

For n, k3X or n, k3X : s (!k!n), . [¸ , ¸ ]"(k!n)+¸ sX(n#k)#¸ \I\L X L I L>I

(18)


253

As before, a simultaneous sign #ip on the right-hand side of the "rst two identities (the resulting identities will be referred to as VERSION IIb) is consistent with the Lie algebra structure. A restriction of X to X or X in the last identity reproduces for X the commutation relations which de"ne the quasicrystal Lie algebras in [1], but for X only reproduces them up to a sign. The four generalized quasicrystal Lie algebras are of two principally di!erent types: E

VERSION

Ia and

VERSION

IIb are of a similar structure. For these algebras we have

[[¸ , ¸ ], ¸ ]"0 L I J for all possible choices of the coe$cients n, k and l. The upper central series thus brakes up already in the second step and the Lie algebras are not perfect. Hence, a universal central extension cannot be expected. E In the case of VERSION Ib and VERSION IIa, we "nd perfect Lie algebras with a structure similar to the quasicrystal Lie algebras. Here as well, there is no non-trivial central extension. In order to decide which of the four versions of the generalized quasicrystal Lie algebras is most suitable for applications, it is convenient to view them in another basis. With the notation R " : ¸ sX(n)!¸ sX(!n) , \L L L S " : i(¸ sX(n)#¸ sX(!n)) , K L \L one obtains the following identities: E For VERSION Ia: If n, m3X , i"1, 2 G [R , R ]"0 , L K [R , S ]"0 , L K [S , S ]"4(n!m)R . L K L>K If n3X and m3X with iOj, i, j3+1, 2, G H [R , R ]"0 , L K [R , S ]"0 , L K [S , S ]"4(n#m)R . L K L\K E For VERSION IIa: If n, m3X [R , R ]"0 , L K [R , S ]"0 , L K [S , S ]"4i(m!n)S L K L>K

(19) (20)

(21) (22) (23)

(24) (25) (26)

(27) (28) (29)

254


and with reversed sign in the last line if n, m3X . If n3X and m3X [R , R ]"0 , L K

(30)

[R , S ]"0 , L K

(31)

[S , S ]"!4i(n#m)S , L K L\K

(32)

and with reversed sign in the last line if n3X and m3X . E For VERSION Ib: If n, m3X , i"1, 2 G [R , R ]"4(n!m)R , L K L>K

(33)

[R , S ]"0 , L K

(34)

[S , S ]"0 . L K

(35)

If n3X and m3X with iOj, i, j3+1, 2, G H [R , R ]"4(n#m)R , L K L\K

(36)

[R , S ]"0 , L K

(37)

[S , S ]"0 . L K

(38)

E For VERSION IIb: If n, m3X [R , R ]"!4i(m!n)S , L K L>K

(39)

[R , S ]"0 , L K

(40)

[S , S ]"0 , L K

(41)

and with reversed sign in the "rst line if n, m3X . If n3X and m3X [R , R ]"!4i(n#m)S , L K L\K [R , S ]"0 , L K [S , S ]"0 , L K and with reversed sign in the "rst line if n3X and m3X .

(42) (43) (44)

Finally, we indicate matrix representations for the generalized quasicrystal Lie algebras. Since the representation theory for VERSION Ia and VERSION IIb is similar, we only present it for the case of VERSION Ia.


255

In this case, R and S are de"ned separately on the sets X and X as follows: L L

0 zL R " : , n3X , L 0 0

(45)

0 !z\L , n3X , R " : L 0 0

(46)

zL zLzR X , S " : (2i) L 0 zL

(47)

S " : (2i) L

n3X ,

z\L z\LzR X , n3X . 0 z\L

(48)

Notice, that the de"nition is such that S "S , because n!m3X implies m!n3X . L\K K\L Similarly, as a representative of the cases VERSION IIa and Ib, we show explicitly only matrix representations of VERSION IIa:

* 0 R " : , n3X , i3+1, 2, L G 0 0

(49)

is admissible for any freely chosen entry *. Furthermore, one has

0 0 S " : (4i) L 0 zLzR

, n3X ,

(50)

(51)

X

0 0 S " : (4i) , n3X . L 0 z\LzR X

In particular, one may choose * in (49) in such a way that the corresponding anticommutator of the generators R ([R , R ] " : R R #R R ) looks similar to the commutator-relation for the S . L L K > L K K L L For instance, choosing * to be zL for n3X and z\L for n3X leads to the following relations: [R , R ] "2R , n, m3X , i3+1, 2, , L K > L>K G

(52)

[R , R ] "2R , n, 3X , m3X , iOj3+1, 2, . L K > L\K G H

(53)

8. Conclusion and outlook Quasicrystal Lie algebras and their various generalizations discussed here might be a useful tool in physics in all those areas where the Witt and the Virasoro algebra play a crucial role. In particular, they might be useful due to their abundance of "nite dimensional subalgebras, which are not so extensively found in the case of the Witt or the Virasoro algebra.

256


One possible example for an application of quasicrystal Lie algebras might be the construction of a discrete quantum mechanics on a discretization of S along the lines of Dobrev et al. [18]. Due to the one-to-one-correspondence of quasicrystal Lie algebras with points of one-dimensional quasicrystals, they might be viewed as `quasicrystal analogsa to the Witt algebra. Because of the lack of a central extension, we do not have, strictly speaking, a `quasicrystal Virasoro algebraa. The generalized quasicrystal Lie algebras, which allow for acceptance windows symmetric about the origin, could be used in relation to physical models which have a built-in symmetry similar to a `charge conjugationa type of symmetry, e.g. models related to interactions of particles with opposite electric charges or particle anti-particle con"gurations.

Acknowledgements We would like to thank Dr. H. de Guise for a careful reading of the manuscript. R.T. is grateful for the hospitality extended to her during this work at the Centre de Recherches MatheH matiques, UniversiteH de MontreH al.

References [1] J. Patera, E. Pelantova, R. Twarock, Quasicrystal Lie algebras, Phys. Lett. A 246 (1998) 209}213. [2] Z. MasaH kovaH , J. Patera, E. PelantovaH , In#ation centers of the cut and project quasicrystals, J. Phys. A: Math. Gen. 31 (1998) 1443}1453. [3] D. Shechtman, I. Blech, D. Gratias, J. Cahn, Metallic phase with long-range order and no translational symmetry, Phys. Rev. Lett. 53 (1984) 1951}1953. [4] H. Huntley, The Divine Proportion: A Study In Mathematical Beauty, Dover Publ., New York, 1970. [5] M. Ghyka, The Geometry of Art and Life, Dover Publ., New York, 1977. [6] D.R. Hofstader, GoK del, Escher, Bach: an Eternal Golden Braid, Basic Books, New York, 1979. [7] B. Grunbaum, G. Shephard, Tilings and Patterns, W.H. Freeman & Co., New York, 1987. [8] R. Lawlor, Sacred Geometry } Philosophy and Practice, Thames & Hudson, New York, 1989. [9] J. Kappra!, Connections: The Geometric Bridge Between Art and Science, McGraw-Hill, New York, 1991. [10] R.V. Moody, J. Patera, Quasicrystals and icosians, J. Phys. A: Math. Gen. 26 (1993) 2829}2853. [11] L. Chen, R.V. Moody, J. Patera, Noncrystallographic root systems, in: J. Patera (Ed.), Quasicrystals and Discrete Geometry, Fields Institute Monograph Series, vol. 10, Amer. Math. Soc., Providence, RI, 1998. [12] J. Patera, Noncrystallographic root systems and quasicrystals, in: R.V. Moody (Ed.), Mathematics of Long Range Aperiodic Order, Kluwer, Dordrecht, 1997. [13] Z. MasaH kovaH , J. Patera, E. PelantovaH , Minimal distances in quasicrystals, J. Phys. A: Math. Gen. 31 (1998) 1539}1552. [14] R. Moody, A. Pianzola, Lie Algebras With Triangular Decompositions, Wiley-Interscience, New York, 1995. [15] V. Kac, A. Raina, Highest Weight Representations of In"nite Dimensional Lie Algebras, World Scienti"c, Singapore, 1987. [16] S. Kass, R. Moody, J. Patera, R. Slansky, A$ne Lie Algebras, Weight Multiplicities and Branching Rules, Univ. of Calif. Press, Los Angeles, 1990. [17] J. Patera, H. Zassenhaus, Higher rank Virasoro algebras, Comm. Math. Phys. 136 (1991) 1}14. [18] V.K. Dobrev, H.-D. Doebner, R. Twarock, A discrete, nonlinear q-SchroK dinger equation via Borel quantization and q-deformation of the Witt algebra, J. Phys. A: Math. Gen. 38 (1997) 1161}1182.


Nuclear structure issues determining neutrino-nucleus cross sections A.C. Hayes Los Alamos National Laboratory, Los Alamos, NM 87544, USA

Abstract We investigate the nuclear structure issues that determine the neutrino-nucleus cross sections of interest in nuclear, particle, and astrophysics. We discuss the uncertainties involved in calculating these cross sections, and the expected accuracy of model predictions. 1999 Elsevier Science B.V. All rights reserved. PACS: 25.30.Pt Keywords: Neutrino interactions; Nuclear structure; Shell model

1. Introduction Several of the searches for neutrino oscillations involve measuring the interaction of neutrinos with the nucleus. A primary example is the recently announced evidence [1] for neutrino oscillations at Super-Kamiokande, which comes from comparing the ratio of muons to electrons created by the scattering of atmospheric neutrinos from oxygen in the Cherenkov water detector. Another detector based on neutrino-nucleus scattering is the Sudbury Neutrino Observatory (SNO) which will measure the interaction of B solar neutrinos with deuterium in heavy water. Both the Liquid Scintillator Neutrino Detector (LSND) and KARMEN experiments involve neutrino scattering from carbon in mineral oil. The search for neutrino oscillations (l Pl and l Pl ) at the BooNE I I experiment at Fermi Lab will be a search for l CPe\N and l CPe>B quasi-elastic scattering. In addition to acting as a signal for neutrino oscillations, neutrino-nucleus reactions appear as background cross sections in these experiments and provide important checks on neutrino #ux or

E-mail address: [email protected] (A.C. Hayes) 0370-1573/99/$ - see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 9 ) 0 0 0 2 6 - 5

258

A.C. Hayes / Physics Reports 315 (1999) 257}271

detector e$ciency. Crucial to all these experiments is our understanding of neutrino-nucleus scattering and our ability to predict the cross sections to su$ciently high accuracy. Another "eld in which neutrino-nucleus scattering plays a signi"cant role is that of nucleosynthesis. As the core of a massive star collapses to form a neutron star, the #ux of neutrinos is so large that signi"cant nuclear spallation occurs, despite the small cross-sections. It has been pointed out by Woosley et al. [2] that neutrinos of all #avors excite nuclei to particle unbound states through the neutral current, and the A(l,lX)A reaction, or the l-process, may be an important process for nucleosynthesis of a number of elements. The temperature of the neutrinos is #avor-dependent and is ¹ I K O O&8 MeV and ¹ &4}5 MeV. This translates to transferring an average of 25 MeV J J J J SJ J of energy to the nucleus, with a Fermi}Dirac tail up to &80 MeV. Thus, an understanding of both the neutral-current excitation and the subsequent multi-particle breakup of the nucleus are needed. Neutrino-nucleus scattering also plays an important role in r-process nucleosynthesis, which is responsible for the formation of half of the elements with A'70. In a stellar environment where a neutron gas exists alongside nuclei, neutron capture becomes the dominant mode for synthesizing medium and heavy mass nuclei. Under stellar conditions where the neutron capture rate is fast compared to b-decay, the conditions for the rapid or r-process, the nucleosynthesis rate becomes proportional to the the b-decay rate. The r-process is thought to take place in the expanding &hot bubble' of a type II supernova, which would mean that the #ux of neutrinos is su$ciently intense to cause signi"cant neutrino-nucleus reactions. The competition between neutron capture and neutrino scattering can be used to determine the distance of the r-process site from the neutron star and to estimate the time scale for the process. Extracting information on the issues requires knowledge of the neutrino capture cross sections by the very neutron-rich nuclei lying along the r-process path. The physics determining the various neutrino-nucleus cross sections of interest to particle and astrophysics varies with neutrino energy and with the structure of the nuclei involved. In this article we discuss the main nuclear physics issues involved and examine the dependence of the predicted cross-sections on models of nuclear structure.

2. Formalism Neutrino absorption on the nucleus occurs through the charged-current of the weak interaction l!#(Z, A)P(ZG1, A)#l! ,

(1)

where the incoming #ux is a beam of either neutrinos or anti-neutrinos, and the leptons are either electrons or muons or their anti-particles. The expression for neutrino absorption on the nucleus in terms of nuclear structure matrix elements has been derived by O'Connell [3] and by Walecka [4].

> G d(cos(h)d(E !E #E !E )p E "M" , p(E )" J J 2p \ D

(2)

where G is the weak interaction coupling constant, G/( c)"1.16639;10\ GeV\, E !E is the mass di!erence between initial and "nal nuclear states, p and E are the neutrino momentum and J J


259

energy in the laboratory, p and E are the lepton momentum and energy, and cos(h)"(p ) p )/"p ""p " . (3) J J The matrix element "M" involves combinations of the "ve lepton traces listed in [3] with nuclear matrix elements of the seven multipole operators detailed in Donnelly and Haxton [5]. The seven operators, which are functions of the position coordinate and of the magnitude of the momentum transfer, q""q"""p !p " are J M (qx)"j (qx)> (X ) , (4) ( ( ( V D (qx)"M (qx) ) (1/q) , ( (( J J#1 1 M # M D (qx)" ! ) , ((> ((\ q ( 2J#1 2j#1

R (qx)"M (qx) ) r , ( (( J J#1 M # M )r, R (qx)" ! ((> ( 2J#1 2J#1 ((\

R(qx)" (

J#1 J )r, M # M ((> ((\ 2J#1 2J#1

1 X (qx)"j (qx)> (X )r ) , ( ( ( V q where M (qx) ) e"j (qx)[> (X ) e](. (* * * V The single particle matrix elements of the above operators, 1 j ""O("" j 2, have been tabulated by ? @ Donnelly and Haxton. Matrix elements between many-body nuclear states are given by 1J ""O(""J 2"R ? @1J ""[aR?, a @]( ""J 21 j ""O("" j 2 . (5) D H H D H H ? @ Here the matrix elements 1J ""[aR?, a @](""J 2 are the one-body density matrix elements which D H H describe the probability of removing a particle from an orbit j in the initial state J and creating @ a particle in the orbit j to form the "nal state J . These one-body density matrix elements contain @ D all the nuclear structure information determining the neutrino cross sections. If the neutrino #ux is known, the model dependence involved in determining the one-body densities matrix elements represents the uncertainty of the predicted neutrino-nucleus cross sections.

3. Atmospheric neutrinos The most recent evidence (1) for neutrino oscillations comes from the anomaly in the number of k- and e-type neutrinos reaching the Super-Kamiokande detector after being produced in the atmosphere by cosmic rays. The observed ratio of muons to electrons is about a factor of two less than expected. The addition of a signi"cant zenith angle dependence in the ratio of muon to electron events, where muon neutrino coming from larger distances (zenith angle 903) evidence

260


larger depletion, while muon neutrino coming from overhead show no loss, is strongly suggestive of neutrino oscillations. The evidence of neutrino disappearance is reported as the observed ratio of muon to electron events divided by the ratio of events calculated in a Monte Carlo simulation. The atmospheric neutrino #ux involves neutrinos in the 0.1}3.0 GeV range. These neutrinos interact in the water detector through a charge exchange on oxygen O(l, l). The neutrino}oxygen cross sections are calculated [6] using the relativistic Fermi-gas model (RFG) [7], which incorporates binding energy e!ects through an average separation energy. At the higher neutrino energies (E '1 GeV) the J RFG is expected to work reasonably well, although the accuracy of its predictions maybe unclear. Given the signi"cance of the nuclear cross sections there is a clear need to examine these and determine whether there is any room for model dependence. This has been done by Engel et al. [8] who investigated carefully the neutrino-nucleus cross sections for Kamiokande. They examined several phenomena beyond the scope of Fermi-gas models. These included the role of bound states and resonances in oxygen, the Coulomb interactions of the outgoing leptons and nucleons with the residual A"15 nucleus, and the two-body interaction between nucleons in O. None of these e!ects are accurately represented in the Monte Carlo simulations used to predict event rates at Kamiokande and IMB. Nonetheless, Engel et al. concluded that the neglected physics could not account for the anomalous k to e ratio, nor were these e!ects likely to change the absolute event rates by more than 10}15%. They therefore conclude that the k/e ratio is a robust measure of the anomaly. The agreement between the more detailed nuclear structure model for O and the Fermi gas model re#ects the neutrino energies involved. For the most part, the atmospheric neutrinos excite the nucleus to a region well above the giant resonances, minimizing the importance of details of the structure of the nucleus. The Fermi gas model provides a reasonable description of quasi-free scattering, and this excitation energy range dominates the cross section at these neutrino energies.

4. Solar neutrino detection Four of the "ve solar neutrino experiments involve detecting neutrino-nucleus reactions, namely, Homestake, SAGE, GALLEX, and SNO. At Kamiokande solar neutrinos are detected via neutrino}electron scattering. The neutrino-nucleus cross sections measured in these detectors have been examined in detail in [9]. For completeness we summarize the key issues involved here. The primary source of neutrinos from the sun is the proton}proton burning chain, with an additional weaker source of neutrinos from the CNO cycle. Table 1 (taken from Bachall [9]) lists the nuclear reactions involved in the solar neutrino #ux, with the corresponding neutrino energies and the calculated #uxes. 4.1. Cl(l , e\)Ar The experiment at the Homestake Mine in South Dakota detects solar neutrinos by the reaction Cl(l , e\)Ar using a 0.6 kton perchloroethylene (C Cl ) detector. The experiment counts the number of individual Ar atoms produced by the l interactions. With accurate knowledge of the expected cross section one can check the #ux of solar neutrinos reaching the detector.


261

Table 1 Neutrinos from the pp chain in the sun Reaction

Label

l energy (MeV)

Predicted #ux (10 cm\ s\)

p#pPH#e>#l p#e\#pPH#l Be#e\PLi#l

pp pep Be

6.0 0.014 0.47

BPBe#e>l He#pPHe#e>#l

B hep

40.42 1.442 (90%) 0.861 (1010%) 0.383 (15 418.77

5.8;10\ 8;10\

The threshold for the charge-exchange reaction on Cl is 0.814 MeV, which is above the maximum energy of the pp neutrinos (Table 1). The detector is sensitive primarily to Be, and B neutrinos, and also to the weaker #uxes of CNO(N, N, F), hep and pep neutrinos from the sun. Of these, only the B neutrinos have enough energy to excite states of Ar lying higher than the ground state, and approximately 77% of the event rate is expected to come from B neutrinos. Most of the uncertainty in the expected cross sections on Cl comes from the excited states contribution. The cross section to the ground state is known because it is determined by the measured half-life of Ar, which decays exclusively by electron capture to the Cl . The B b-decay spectrum involves neutrinos up to energies of about 15 MeV, so that many excited states of Ar can be populated. However, only those states below particle threshold (up to about 8.4 MeV of excitation) contribute to the production of Ag . Of these, the Fermi transition to the isobaric analog of Cl dominates. In the absence of small radiative corrections or isospin mixing, the Fermi transition rate is independent of nuclear structure and is dependent only by the matrix element of the isospin raising operator. Thus, this contribution to the event rate is known to high accuracy. The uncertainty in the total neutrino-nucleus cross section at Homestake is dominated by the cross section to the remaining excited states of Ar. For a pure Gamow}Teller transition, ¸"0, J"1, in the limit of small momentum transfer qP0, the matrix element entering the expression for the neutrino cross section can be expressed in terms of the B(GT) value from beta decay. (6) "M"P(1!cos(h))B(GT) , where B(GT) is the strength of the equivalent beta decay transition and is related to the beta-decay ft value by ft"(6146/B(GT)) s .

(7)

cos(h) is as de"ned in Eq. (3). For solar neutrinos only Fermi and Gamow}Teller transitions contribute signi"cantly to the cross section. All other transitions to excited states, which would involve ¸O0, are so-called forbidden and are expected to contribute little for an average neutrino energy of 7 MeV and as much as 10% for the end-point 15 MeV neutrinos. Thus, for the B neutrino spectrum only

262


Gamow}Teller contributions to the excited state cross section need to be considered. These have been determined [10] to within 3% accuracy from a measurement of the b-delayed protons from the b>-decay of Ca. Ca is the isobaric analog of Cl. If isospin is a good symmetry, the Cl(l , e\)Ar excited state cross sections are determined by the ft values of the Ca b> decays to the K isobaric analogues of the Ar levels. 4.2. Ga(l , e\)Ge At both SAGE and GALLEX solar neutrinos are detected through the neutrino absorption reaction Ga(l , e\)Ge in large (tens of tons) gallium detectors. The threshold for the charge exchange reaction on Ga is low, E "0.233 MeV. Thus, all low-energy neutrinos from both the pp and CNO chains can be detected. This unique feature of the gallium detectors make them the sole detectors sensitive to the pp neutrinos (E 40.42 MeV). Because of the neutrino energies NN and #uxes involved most of the capture cross section is to the ground state of Ga, though the excited state contributions represent about 88% of the total B part of the cross section. The ground state of Ge decays 100% to Ga by electron capture with a known half-life, so that the ground state to ground state neutrino capture cross section can be determined accurately. The "rst excited state of Ge lies at 0.175 MeV and a number of excited states can be populated by the higher energy neutrino #uxes. Of the excited states the largest nuclear matrix element is the Fermi transition to the isobaric analog 3/2$8.89 MeV) state (IAS). However, Champagne et al. [11] measured the decay properties of this state and found it to decay primarily by neutron emission with a negligible gamma-decay branch. They, therefore, concluded that there is very little contribution to the detection sensitivity of the Ga detectors from the neutrino population of the IAS. The Gamow}Teller transitions to other excited states of Ga are di$cult to calculate accurately. The transition matrix elements can be deduced from (p, n) measurements but the uncertainties remain large. The resulting uncertainty into total capture rate is &10%, and it is unlikely to be improved upon without a signi"cantly more accurate extraction of the GT transition matrix elements from (p, n) measurements. 4.3. d(l , e$ and d(l, l) The Sudbury Neutrino Observatory (SNO) will measure interactions of B and hep neutrinos with deuterium using a heavy water (D O) detector. SNO will measure both neutrino absorption by deuterons through the charged current l #dPp#p#e\ and neutrino-disintegration of the deuteron through the neutral current l#dPl#p#n .

(8)

(9)

The restriction to B and hep neutrinos is because the thresholds for these two reactions is 1.442 and 2.225 MeV, respectively. SNO will also be sensitive to neutrino}electron elastic scattering. These scattered electron are peaked strongly in the forward direction and can be distinguished for those electrons produced in the charge exchange reaction on deuterium.


263

A measurement of the shape of the recoil electron spectrum in the charged-current reaction will provide a direct measure of the spectrum for the incident neutrinos. The neutral current reaction is #avor blind and therefore gives a measure of the total neutrino #ux and a test of the solar models. The ratio of the charged-current to neutral-current events will allow a determination of whether electron neutrinos have `disappeareda and oscillated to other neutrino #avors. For both the charged-current test of the total #ux and the ratio of charged-current to neutralcurrent test of the electron neutrino #ux, the measured quantity has to be referenced to a theoretical expectation. The experiment will search for a departure form theory, which would signal either (or both) physics beyond the Standard Model of particle physics or the Standard Solar Model. Clearly, the expected theoretical cross sections have to be known to high accuracy. Bachall et al. [12] have calculated the expected "rst and second moments of the recoil electron's kinetic energy and the ratio of the number of charged to neutral-current events expected at SNO. They determined shifts in the SNO observables for various neutrino-oscillation scenarios. All Standard Model corrections to these estimates need to be examined. The most signi"cant of these include the role of meson-exchange currents and of radiative corrections in l#d scattering. Towner [13] has investigated the impact that radiative corrections in neutrino}deuterium scattering for the charged-current and neutral-current channels have on the observables to be measured at SNO. The calculations showed that the corrections are generally small and can be neglected. However, in the case where internal Bremsstrahlung photons emitted in the reaction l #dPp#p#e\#c are detected by the Cherenkov detectors, the ratio of the number of charged-current to neutral-current events seen at SNO was found to be shifted by about one standard deviation. Calculations for the meson-exchange corrections for neutrino}deuterium scattering have yet to be calculated.

5. Neutrino}Carbon scattering at LSND and KARMEN At both LSND and KARMEN the signal for neutrino oscillations involves neutrino interactions in mineral oil (CH ), and understanding neutrino scattering from carbon is important for these experiments. The neutrino source at both these experiments comes from the decay of pions produced in the beam stop. The vast majority of the pions decay at rest producing one muon neutrino, muon anti-neutrino and electron neutrino. Of these only the electron neutrino has enough energy to cause a nuclear charge-exchange reaction on carbon. However, at LSND 3.4% of the pions decay while in #ight and produce muon neutrinos of su$cient energy for such reactions. The oscillation of decay-in-#ight (DIF) muon neutrinos, (l Pl ), is detected at LSND by the I appearance of high-energy electrons from the l CPNe\ reaction. Extracting oscillation para meters from this search requires knowledge of the expected cross section. In addition, the measured l CPNk\ cross-section acts as a test of the l DIF #ux and of the detector e$ciency. The I I KARMEN experiment also has the e$ciency to measure the lCPBk> reaction. At both LSND and KARMEN the inclusive l CPXe\ and the exclusive l CPN cross-sections are mea sured for the electron neutrinos from the decay of the pion at rest. The Michel spectrum for these decay-at-rest (DAR) electron neutrinos is known, and the l C cross sections provide a nice constraint on the nuclear structure models used for carbon.

264


The DAR neutrino spectrum involves neutrino energies 0}52 MeV, with an average neutrino energy E & 32 MeV. The Q-value for the charge-exchange reaction (l , e\) on C is about J 17 MeV. The DAR inclusive cross section is then dominated by low multipoles (1>, 1\, 2\) and by excitation of the giant resonances. In the case of the l C cross section, the DIF muon I neutrino #ux involves an average neutrino energy of about E &150 MeV, but the #ux is J "nite up to E &250 MeV. Because of the mass of the muon and the mass di!erence between J C and N the Q-value for the (l , k\) reaction is close to 123 MeV. Calculations for this I cross section need to include both a good description of the giant resonance region (E &15}40 MeV of excitation in C) and of higher excitation energy regions (up to 80}100 MeV). V Furthermore, all multipoles j&0}5 make signi"cant contributions to the inclusive cross section. The di!erence in the momentum and energy transfers between the DAR (q&0.2 fm\) and the DIF (q&1.0 fm\) cross sections results in the latter being considerably more di$cult to calculate accurately. In calculating very low-energy or very high-energy neutrino-nucleus cross sections the choice of model space to describe the structure of the nucleus is usually straightforward. In the case of low-energy processes it is most important to provide a very detailed description of the nuclear wave functions for the initial and "nal states involved. Thus, in a shell model sense, one attempts to include all con"gurations, and full con"guration mixing, within a small number of shells. In contrast, for high neutrino energies the details of con"guration mixing are unlikely to be important, whereas the inclusion of highly excited particle}hole con"gurations crucial. As discussed in the case of atmospheric neutrinos, high-energy neutrino reactions are reasonably well described by Fermi-gas model calculations. However, the DIF l C cross section at LSND I poses an especially di$cult problem in that the neutrino energies involved are intermediate between these two situations. As discussed below, both details of con"guration mixing and the inclusion of particle}hole excitations across many shells play a signi"cant role in determining the cross section. The inclusive C(l , k\)X cross section measured at LSND was "rst calculated by the Caltech I group [14] using a sophisticated continuum RPA model. The calculated cross sections overestimated experiment by almost a factor of two, and suggested that the measured cross sections may be inconsistent with other observables for C. The key observables that need to be considered as checks on model calculations are k-capture, the DAR C(l , e\)X, (e, e), photoabsorption, and b-decay. These di!erent probes involve di!erent energy and momentum transfers and, thus, constrain di!erent aspect of the calculations. The disagreement between theory and experiment for the DIF l cross section caused a #urry of theoretical activity [15}17] in an e!ort to uncover I possible shortcomings of the RPA calculations. Studies of the e!ect of di!erent nuclear structure assumptions on the predicted neutrino-nucleus cross sections were quite revealing, and we discuss these in some detail below. In the simplest model, C consists of four neutrons and protons in the p-shell outside a closed He core. Excited states reached by the (l , k\) reactions would be simple particle}hole states built I on the ground state. However, the structure of both the C and the continuum states N involve considerably more sophisticated con"gurations. Since there is a limit on the size of the model space that can be included in any calculation, one is forced to make approximations in the hope that these incorporate all the essential physics for the problem under consideration. In the case of the neutrino cross sections on carbon the assumptions or approximations that are


265

most likely to a!ect the predicted cross sections are: 1. 2. 3. 4. 5.

the C ground state p-shell structure; the treatment of ground state correlations beyond the p-shell; the model space truncation, especially for the "nal states; con"guration mixing in the "nal states; and the size parameter entering the calculations. In the next few subsections we discuss the e!ect of each of these on the predicted cross section.

5.1. Ground state p-shell structure There are two simple, but almost opposite, starting points for describing the C . The "rst of these is to assume that C consists of the p-shell equivalent of three a-particles. In this model the g.s. is in an ¸"0 S"0 state, and has good SU(4) symmetry [44 4]. A description of C in terms of three tightly bound a-particles is a reasonable starting point. Indeed, the time-honoured Cohen}Kurath p-shell interaction predicts that this state makes up 78% of the C ground state wave function. The second starting point is to assume that the four protons and neutrons "ll the lowest available p-shell orbit, i.e., that C corresponds to a closed p -shell. This starting point is quite distinct from the assumption of three a's or from the Cohen}Kurath wave function. It involves a ground state that is 84% SO0. Table 2 summarizes the spin structure of 12C under di!erent model assumptions. The most serious problem that arises in assuming a closed p -shell is the overestimation of transitions within the p-shell. This is clearly demonstrated by considering the exclusive chargeexchange cross section to the 1> N , which is dominated by the pPp Gamow}Teller (GT) transition. The GT operator, pq , conserves SU(4) symmetry. However, there is no way to make > a ¹"1 1> state with [44 4] symmetry. Thus, in the 3-a model of C the GT transition to the N is forbidden. On the other hand, starting with a closed p -shell leads to a strong p Pp transition, which is a factor of six larger than that experiment. When one includes [14] RPA correlations the factor of six is reduced to a factor of four. The Cohen}Kurath interaction, which was "tted to p-shell nuclei with particular attention paid to GT transitions, provides a good description of this transition. The transition to the lowest 2> state of N is also overestimated by a factor of several when C is restricted to a closed p -shell. Table 2 Spin structure of the C ground state Model

S"0 (%)

S"1 (%)

S"2 (%)

% [44 4]

Three a-particles Cohen}Kurath interaction (p )

100 81 16

0 18 59

0 (2 25

100 78 6

266


In the case of particle}hole excitations out of the p-shell, the p-shell structure of the ground state does not a!ect the total sum-rule for a given operator. However, the very di!erent g.s. spin structures for the models discussed above means that the strength for a given operator will be distributed di!erently over the "nal state multipoles. The net result is that the predicted #ux averaged cross section will be di!erent for di!erent C p-shell wave functions. Table 3 shows the distribution of B(*J)"1J ""[r*, p] (""J 2 strength for the isovector 1 u D G dipole and octupole multipoles in C. We note that for the spin-independent multipoles the strength is independent of the p-shell structure of the ground state, while the spin-multipole strengths are model-dependent. Table 4 summarizes the 0 uP0 u and 0 uP1 u contributions to k-capture, (l , e\) DAR and (l , k\) DIF for the open-shell Cohen}Kurath and the closed p ground state wave functions. I The largest model dependence is seen in the GT transition to the N . However, even when one excludes the ground state cross section, there remains a signi"cant di!erence in the predicted inclusive k-capture rate and the (l , k\) DIF cross section. This is particularly true in the latter case, I where the two calculations di!er by 25%. Table 3 Distribution of isovector dipole and octupole 1 u strength for di!erent C ground states C model

Dipole 1\

SU(4) 3a-particles Closed p

11.6 11.6

SU(4)3a-particles Closed p

Spin dipole 0\ 3.9 5.6

Spin dipole 1\ 11.6 14.1

Spin dipole 2\

Total

19.3 15.0

46.2 46.2

Oct. 3\

Spin Oct. 2\

Spin Oct. 3\

Spin Oct. 4\

Total

67.7 67.7

48.4 16.1

67.7 56.4

87.0 130.5

270.8 270.8

Table 4 Model dependence in (0#1) u calculations for neutrino reactions k-capture rate (10 s\) Closed p

Open p

(l , e\) DAR 10\ cm Closed p

Open p

(l , k\) DIF I 10\ cm Closed p

Open p

pPp 1> 2> 3> pP(sd) 0\ 1\ 2\ 3\ 4\

37.46 1.10 *

6.41 0.30 0.08

86.25 0.24 *

9.92 0.04 0.01

4.35 2.42 *

0.87 0.92 0.24

4.15 21.35 12.15 0.07 0.06

3.05 18.0 13.10 0.06 0.05

0.01 3.11 3.79 0.0 0.0

0.03 3.09 4.04 0.0 0.0

0.09 6.79 3.75 1.27 1.43

0.07 4.85 4.13 1.33 1.14


267

5.2. Ground state correlations The restriction of the C ground state to a p-shell wave function, be it a closed p or the Cohen}Kurath wave function, leads to an overestimate of the di!erent neutrino cross sections. The one-body operators that determine the neutrino cross sections (Eqs. (4)) connect the ground state to 1p}1h excited states. When 2p}2h excitations are included in the ground state they lead to destructive interference, the so-called `backward-going diagramsa of RPA. The inclusion of these 2p}2h correlations, either in an RPA or a shell model calculation, reduces the neutrino cross sections by about a factor of two. In the shell model all 2p}2h con"guration allowed up to a given

u of excitation are normally treated on an equal footing. In contrast, in RPA calculations the 2p}2h correlations are restricted to the type "(h\, p )L SJ,¹; (h\, p )L S1J, ¹ : 002 ,

(10)

where the 2p}2h states are made up only from the coupling of two 1p}1h states of the spin, isospin J, ¹ that comprise the basis states of N. Here we are labeling the 1p}1h states by their excitation, n u, in an oscillator model. In Table 5 we compare the predictions of two calculations (17), which start with a closed p wave function for C, both which di!er in that the calculation labeled TDA does not include 2p}2h ground state correlations. Both calculations treat the excited states of N as 1p}1h states "N2""(h\, p)L SJ,¹2, n"0, 1, 2, 3, 4 .

(11)

It is clear that for all observables considered, (l , k\)DIF, (l , e\), k-capture and photoabsorption, I the inclusion of 2p}2h correlations reduces the predictions considerably (& a factor of two), and brings the calculated values in closer agreement with experiment.

Table 5 Inclusive cross sections involving a continuum of N states, including the ground state (l , k\) DIF I p;10\ cm

(l , e\) DAR p;10\ cm

k-capture K ;10 s\

Closed-p -shell TDA 0 u (0#1) u (0#1#2) u (0#1#2#3) u (0#1#2#3#4) u

5.62 18.21 25.61 30.36 31.85

52.29 59.53 57.52 57.59 57.62

35.56 72.98 75.41 75.23 75.21

Closed-p -shell#correlations 0 u (0#1) u (0#1#2) u (0#1#2#3) u (0#1#2#3#4) u

2.56 9.60 13.78 16.10 16.77

19.50 23.60 27.37 27.34 28.46

13.13 34.01 38.17 37.61 38.38

Photoabsorption p ;10\ cm

27.28 27.31

14.84 14.50

268


5.3. Model space truncation In calculating the inclusive neutrino cross sections one has to include all "nal states that contribute signi"cantly. The size of the required "nal state model space clearly depends on the neutrino energy. For the DAR neutrino spectrum (E (52 MeV) the calculated cross section J converges at 1 u of excitation, i.e., the inclusive cross section is almost entirely accounted for in term of pPp and pPsd transitions. In addition, multipoles higher than j"2 do not contribute signi"cantly to the cross section. For the (l , k\) DIF neutrino #ux (E (250 MeV) only 57% of I J the cross section is accounted for in a (0#1) u calculation. Approximately, 85% of the cross section is accounted for in a (0#1#2) u model space and about 97% in a (0#1#2#3) u calculation. All multipoles up to j"5 contribute signi"cantly to the DIF cross section. The need to include excitations up to 4 u of excitation while paying careful attention to the ground state structure of C makes the (l , k\) DIF cross section particularly di$cult to calculate accurately. I The size of the model space increases considerably as the number of shells included increases. At some stage it is no longer possible to diagonalize the Hamiltonian. 5.4. Final state conxguration mixing Up till now, we have only considered 1p}1h "nal states in N. However, in reality these 1p}1h con"gurations are strongly mixed with 2p}2h as well as more complicated con"gurations. The e!ect of including these con"gurations is to spread the 1p}1h strength in energy thus changing the #ux-averaged cross section. Detailed calculations [17] "nd that the DIF (l ,k\) cross section is I decreased by a few percent, while the (l , e\) DAR cross section, k-capture and photoabsorption cross section all increase by about 6}25%. The di!erent e!ect the inclusion of 2p}2h "nal state con"gurations has on the di!erent probes results from the fact that each sample has di!erent linear combinations of multipoles. 5.5. Nuclear size parameter A parameter that has to enter all the nuclear calculations for the neutrino calculations in the size of the nucleus. In shell model calculations this enters through the oscillator parameter. Changing the oscillator parameter changes the shape of the axial-vector and vector form factors entering the neutrino cross sections. The oscillator parameter is normally chosen to reproduce key observables, for example, the ground state charge radius. For the ground state of C this suggests an oscillator parameter of b"1.64 fm. This value of b has been used in all the neutrino calculations listed above for C. However, the shape of the measured (e,e) form factor to the 1\ and 2\¹"1 high-lying states of C suggests the need for a larger oscillator parameter, b"1.82 fm. The M2 contribution is signi"cant for all the neutrino}carbon reactions under discussions. Fig. 1 shows the M2 vector form factor for b"1.64 and 1.82 fm. For momentum transfers below the "rst maximum for the form factor the neutrino cross section increases as b increases. In contrast, momentum transfers beyond the "rst peak in the form factor result in the predicted cross section being decreased as b increases. For higher multipoles the "rst maximum in the form factor occurs at higher q, so that even at q&1 fm\ an increase in the oscillator parameter means an increase in the neutrino cross section.


269

Fig. 1. M2 vector form factor for the 1 u transition pP(sd). The shape of the form factor is shown for two choices of the oscillator parameter, b"1.64 and 1.82 fm. These two choices are suggested by the measured ground state charge form factor and the excited state (e,e) form factors, respectively. As discussed in the text and shown in Table 6, their e!ect on the neutrino reactions depends on the momentum transfer involved.

We have examined the dependence on the calculated (l , e\) DAR, (l , k\) DIF cross sections I and the k-capture rates on the assumed oscillator parameter by comparing the predictions for b"1.64 and 1.82 fm. Relative to the b"1.64 fm calculation the inclusive (l , e\) DAR cross section increased by 17%, the (l , k\) DIF cross section decreased by 9%, while the k-capture rate I increased by 14%. 5.6. Shell-model predictions In Table 6 we present the results of the most complete to date shell model results for the various neutrino reactions on carbon of interest. The results are presented as a function of the size of the model space, increasing from 0 u to 4 u. The 4 u calculation, which includes 2p}2h correlations in both the ground state of C and the "nal states of N predicts a value for the (l , k\) cross I section of 14.5;10\ cm for b"1.64 fm and 13.4;10\ cm for b"1.82 fm. This is to be compared with the experimental value of 12.3;10\ cm [18] and the RPA prediction of (18!20);10\ cm. The smaller prediction for this cross section in the shell model arises from a combination of the e!ects discussed above, namely, the ability to include in a shell model formalism an open p-shell and 2p}2h correlations in the "nal states. Both the (l , e\) cross section

270


Table 6 Inclusive cross sections involving a continuum of N states, but excluding the ground state (l , k\) DIF I p;10\ cm

(l , e\) DAR p;10\ cm

k-capture K ;10 s\

Photoabsorption p ;10\ cm

Shell-model b"1.64 0 u (0#1) u (0#1#2) u (0#1#2#3) u (0#1#2#3#4) u

1.16 8.09 11.95 13.74 14.16

0.27 4.85 4.75

0.69 22.06 24.55

14.75

Shell-model b"1.82 (0#1#2) u (0#1#2#3#4) u

8.06 12.89

5.56

28.0

18.17

Expt.

11.7

5.7(8) LSND Collab. 6.1(13) [19]

31.0

and the k-capture rate show agreement with the experiment for the larger value of b, but theory is somewhat low compared to experiment when the smaller oscillator parameter is used. A direct comparison between these latter reactions and the (l , k\) cross section can be misleading because I there is a very di!erent distribution of the strength over the di!erent multipoles in each case. 5.7. Theoretical uncertainties Of the cross sections discussed in this article, the theoretical DIF (l , k\) cross section on carbon I probably carries the largest uncertainty. For the solar neutrino cross sections, the neutrino energies are low enough to allow only Fermi and Gamow}Teller transitions. For the nuclei of interest, these can usually be determined from experimental constraints, e.g., beta-decay GT strengths. In the case of atmospheric neutrinos, the neutrino energies are high enough that the cross section is dominated by quasi-free scattering, thus minimizing the importance of details of nuclear structure. In contrast, the predictions for (l, l\) cross sections on carbon can be quite model dependent. Calculations restricted to a Tamm}Danco! approximation built on a closed p ground state overestimate all neutrino reactions by about a factor of 3. As one builds in more correlations into both the ground state and the "nal states the predictions of the calculations come closer to experiment. When these correlations are mostly included, the remaining di!erences between the predictions of the model calculations and the dependence of the predictions on nuclear size suggest that the theoretical uncertainties are of the order of 25}35%.

References [1] Super-Kamiokande Collaboration, Phys. Rev. Lett. 81 (1998) 1562. [2] S.E. Woosley, D.H. Hartmann, R.D. Ho!man, W.C. Haxton, Astrophys. J. 356 (1990) 272.


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[3] J.S. O'Connell, T.W. Donnelly, J.D. Walecka, Phys. Rev. C 6 (1972) 719. [4] J.D. Walecka, in: V.W. Hughes, C.S. Wu (Eds.), Muon Physics, Academic Press, New York. [5] T.W. Donnelly, W.C. Haxton, Atomic Data Nucl. Data Tables 23 (1979) 103; T.W. Donnelly, W.C. Haxton, Atomic Data Nucl. Data Tables 25 (1980) 1. [6] M. Nakahata et al., J. Phys. Soc. Jpn 55 (1986) 3786. [7] R.A. Smith, E.J. Moniz, Nucl. Phys. B 43 (1972) 605. [8] J. Engel, E. Kolbe, K. Langanke, P. Vogel, Phys. Rev D 48 (1993) 3048. [9] J.N. Bachall, em Neutrino Astrophysics, Cambridge University Press, Cambridge, 1998. [10] A. Garcia, E.G. Adelberger, P.V. Magnus, H.E. Swanson, O. Tengblad, ISOLDE Collaboration, D.M. Moltz, Phys. Rev. Lett. 67 (1991) 3654. [11] A.E. Champange, G.E. Dodge, R.T. Kouzes, M.M. Lowry, A.B. MacDonald, M.W. Roberson, Phys. Rev. C 38 (1987) 900. [12] J. Bachall, P.I. Krastev, E. Lisi, Phys. Rev. C 55 (1997) 494. [13] I.S. Towner, Phys. Rev. C 58 (1998) 1288. [14] K. Kolbe, K. Langanke, F.K. Thielemann, P. Vogel, Phys. Rev. C 52 (1995) 3437; K. Kolbe, K. Langanke, S. Krewald, Phys. Rev. C 49 (1994) 1122. [15] N. Auerbach, N. Van Giai, O.K. Vorov, Phys. Rev. C 56 (1997) R2368. [16] S.K. Singh, N.C. Mukhopadhyay, E. Oset, Phys. Rev. C 57 (1998) 2687. [17] I.S. Towner, A.C. Hayes, in preparation. [18] LSND collaboration using most recent evaluation of DIF #ux, private commun. [19] B.E. Bodmann et al., Phys. Lett. B 332 (1994) 251.


The gluon propagator Je!rey E. Mandula* Division of High Energy Physics, US Department of Energy, Washington, DC 20585, USA

Abstract We discuss the current state of what is known non-perturbatively about the gluon propagator in QCD, with emphasis on the information coming from lattice simulations. We review speci"cation of the lattice Landau gauge and the procedure for calculating the gluon propagator on the lattice. We also discuss some of the di$culties in non-perturbative calculations } especially Gribov copy issues. We trace the evolution of lattice simulations over the past dozen years, emphasizing how the improvement in computations has led not only to more precise determinations of the propagator, but has allowed more detailed information about it to be extracted. 1999 Published by Elsevier Science B.V. All rights reserved. PACS: 11.15.!q; 11.15.Ha; 14.70.Dk Keywords: Gluon; Propagator; Gauge; Lattice

1. Introduction In quantum "eld theory, the Green's functions carry all the information about the theory's physical and mathematical structure. Aside from the vacuum expectation values of "elds, the moduli which parametrize the phase structure of a "eld theory, the two-point functions are its most basic quantities. From this point of view, the gluon propagator may be thought of as the most basic quantity of QCD. Even without quarks, in a pure Yang-Mills theory, the gluon propagator is well de"ned. At short distances or equivalently large momentum transfers, because of asymptotic freedom we expect that perturbation theory should be su$cient to describe any Green's function. By contrast, at large distances or small momenta, there is no available analytic method to pin down the

* Corresponding author. E-mail address: [email protected] (J.E. Mandula) 0370-1573/99/$ - see front matter 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 9 ) 0 0 0 2 7 - 7

274

J.E. Mandula / Physics Reports 315 (1999) 273}284

behavior of the Green's functions. Furthermore, the fact that there are no asymptotic gluon states raises the possibility that the gluon propagator may be quite di!erent than those associated with stable particles. As a result of its centrality from the "eld theortic perspective, the infrared limit of the gluon propagator has been subject of calculation and speculation since QCD was accepted as the correct theory of the strong interactions in the early 1970s. Ideas about the structure of the gluon propagator have been informed by many attractive hypotheses and conceptual puzzles. Among the recurring themes are: E The relation of the gluon propagator, speci"cally its infrared behavior, to con"nement; E The behavior of Green's functions in a theory in which one expects that none of the quanta of the fundamental "elds are physical particles in that theory; E How the Green's functions express such general properties of "eld theory as spectral positivity; E How the absence of any physical gluon states can be compatible with any non-zero gluon propagator. Note that despite the absence of physical asymptotic gluon states, gluons are real } every bit as real as quarks. They are also observed in the same way } through the jets of hadrons that result when they are produced in high-energy collisions. Their presence was indirectly inferred from the original deep inelastic electron}proton scattering experiments performed at SLAC in the late 1960s and early 1970s. There they were needed to account for the fraction of the momentum of the proton that was not attributable to quarks, the proton's electrically charged constituents. This fraction, according to the so-called momentum sum rule, was close to 50%. Gluon jets were directly observed experimentally in experiments at DESY in the 1980s. The problem of having fundamental constituents that only occur con"ned inside hadrons is not a physical nor a conceptual one. It is purely a problem of reconciling this physical structure with our notions of how Green's functions behave in "eld theory. The plan of this paper is the following: In Section 2, we brie#y review the situation in the ultraviolet, where perturbation theory holds, and use this discussion to "x notation. In Section 3, we discuss some of the ideas that have been advanced regarding the behavior of the gluon propagator in the infrared. In Section 4, we discuss the procedure for calculating the gluon propagator on the lattice. We introduce the lattice Landau gauge, and discuss its implementation. We also discuss some of the di$culties in non-perturbative calculations } especially Gribov copy issues. In Section 5 we trace the evolution of lattice simulations over the past dozen years, emphasizing how the improvement in computations has led not only to more precise determinations of the propagator, but has allowed more detailed information about it to be extracted. In Section 6 we summarize what we now know, from non-perturbative lattice studies, about the gluon propagator, and also what is still obscure about it.

2. The gluon propagator in the ultraviolet Since in this discussion we shall be exclusively concerned with the gluon propagator, we restrict our considerations to pure Yang}Mills theory, with no quarks. The gluon propagator is the Fourier transform of the time-ordered matrix element of two gluon "elds A? (x), where for gauge I


275

group S;(N) the index a runs from 1 to N!1.

D?@ (q)"!i dxe OV 10"¹(A? (x)A@(0)"02. IJ I J

(1)

In covariant gauges the propagator has the kinematic form D?@ (q)"!id?@[(g !(q q /q))D(q)#a(q q /q)D*(q)] . IJ IJ I J I J The parameter a speci"es the gauge, and the Landau gauge is a"0. To zeroth order in perturbation theory, the propagator is the same as in QED,

(2)

D (q)"1/q . (3) To any "nite order in perturbation theory, this power dependence remains valid. It is possible that when all orders are summed, there could result an anomalous dimension D(q)&1/(q>A) .

(4)

What the value of c may be, or even if it is non-zero, is not known.

3. The gluon propagator in the infrared In the infrared, the situation regarding the gluon propagator is much murkier. It is instructive to summarize brie#y some of the more thoroughly studied ideas and speculations about the infrared behavior of the gluon propagator: 3.1. The gluon propagator explicitly displays conxnement If the con"nement potential is linear, it can be expressed in terms of the exchange of quanta whose momentum space propagator behaves like D(q)&1/q, (qP0) .

(5)

This behavior was hypothesized by Mandlestam [1] in the late 1970s, and it was made the characteristic of the phenomenological model studied for many years by Baker et al. [2]. More recently, by studying certain truncations of the Schwinger}Dyson equations, this behavior has been advocated by Brown and Pennington [3]. From the point of view that con"nement is a result of a sort of &&Dual Meissner E!ect'', the idea that the gluon propagator should express con"nement directly is in some sense rather heretical. 3.2. The propagator has a non-zero anomalous dimension Marenzoni et al. [4] carried out lattice simulations of the gluon propagator and interpreted their results, speci"cally the observation that the propagator fell o! quite di!erently than in perturbation

276


theory, in terms of an anomalous dimension. Their "ts to lattice data were consistent with the anomalous dimension being consistent with c+1. Their preferred functional form was 1

q ? Z m#q K

.

(6)

3.3. The propagator acquires an ewective mass The earliest lattice simulations of the gluon propagator in the Landau gauge, by Mandula and Ogilvie and by Gupta et al. [6] were interpreted in terms of a massive particle propagator. D(q)&1/(q#m) .

(7)

3.4. The propagator vanishes at vanishing momentum This behavior, which is often described as the propagator having a pair of poles with conjugate complex masses m"be! p was hypothesized by Gribov [7], in connection with his study of gauge copies. It has been advocated on a number of di!erent grounds, by Stingl [8], Cudell and Ross [9], Smekal et al. [10], Zwanziger [11], Namislowski [12], and Bernard et al. [13]. The gluon propagator in this scenario may take the explicit form in the infrared D(q)&q/(q#m) .

(8)

This form is an explicit realization of a theorem due to Zwanziger [11], namely that on the lattice, for any "nite spacing but in the in"nite volume limit, the gluon propagator must vanish at q"0. lim D(q"0)"0 , , where N is the number of sites on each side of the lattice.

(9)

3.5. The propagator takes its perturbative form The form for the gluon propagator incorporated into all QCD models used in simulations to design experimental detectors and interpret the results of collider experiments is simply the perturbative form D(q)"1/q .

(10)

4. The gluon propagator on the lattice Lattice simulations of the gluon propagator have been carried out since the late 1980s, and are still being pursued. The goal of these studies has remained to try to arrive at a de"nitive understanding of the gluon propagator's infrared behavior. More speci"cally, one wishes to have as


277

accurate a numerical determination of D(q) as one can, over the largest range of momenta, and for the numerical result to converge to an analytic picture of the gluon propagator. One hopes to articulate the character of D(q) such that it expresses physics of a theory without physical gluon asymptotic states. Each of the possibilities listed above is sensibly motivated, in that each incorporates some known or expected property of QCD. As with all lattice calculations, the quality of the results, measured by the statistical precision, the size of the lattice spacing, or the total volume of space}time simulated, have steadily improved over time. In order to appreciate what has and has not been learned, we must "rst review the proper de"nitions of operators and gauge conditions, and the sources of errors on the lattice. 4.1. The kinematics of discretization On a "nite lattice, momentum is a periodic discrete variable. Denoting the lattice spacing by a and the number of sites per side by N, each component of the momentum takes the values q "0, $2n/aN, $2(2n/aN), 2, $n/a . I The kinematic range of the dimensionless momentum,

(11)

aq,( aq aq I I I

(12)

aq3[0, 2p] .

(13)

is

The free propagator on the lattice is a periodic function of the lattice momentum and we can use it to de"ne a lattice corrected momentum. 1 1 D(q)" , . (14) (4/a) sin(aq /2)#m q( #m I The momentum so de"ned absorbs much of the lattice artifact errors in propagators. It has the kinematic range aq( 3[0, 4] .

(15)

For small momentum it approaches the ordinary dimensionless momentum. 4.2. The lattice Landau gauge On the lattice, where the basic variables are the unitary matrices ; (x) that express parallel K transport, the gauge potential may be de"ned as [5,14] ; (x)!;R(x) ; (x)!;R(x) I . I !Tr I A (x)" I I 6iag 2iag

(16)

278


The lattice Landau gauge condition could be de"ned in terms of it by D A (x), A (x)!A (x!k( )"0 . (17) I I I I I However, it is truer to the continuum situation to formulate it as a maximization condition Max ReTr;E (x) , I EV VI ;E (x),g(x); (x)g(x#k( )R . (18) I I The maximization condition implies the "nite di!erence one, and has the virtue that it excludes very unsmooth gauge con"gurations which would maximize the trace on some sites and minimize it on others, in the extreme case on alternating sites. The maximization condition, in this sense, carries the smoothness character of the continuum Landau gauge. Any gauge condition expressible as f (;)"0 can be implemented by following the Fade'evPopov procedure. This consists of writing the path integral of any quantity O(;) with a gauge invariant measure, multiplying by 1 in the form of a delta function of the gauge condition times the reciprocal of its Jacobian, the Fade'ev-Popov determinant, judiciously interchanging the order of the path integrals, and "nally reexpressing the result in terms of an un"xed measure again.

D;e\QO(;)" " D; Dg e\QO(;)D (;)d( f (;E)) D3 $. " Dg D;e\QO(;)D (;)d( f (;E)) $. " D;e\QO(;M ) .

(19)

Here, for "xed g, ;M is the gauge transform of ; to the f (;)"0 gauge: ;M [;]";E, f (;M )"0 .

(20)

In contrast to perturbation theory, where the evaluation of D gives rise to the introduction of $. ghost "elds, in simulations there is no need to compute the Fade'ev-Popov determinant. The correct adjustment to the measure is built into the simulation recipe: Perform the simulation without speci"cation of the gauge, but for each lattice con"guration, transform the link variables to the f (;)"0 gauge before evaluating and averaging the path integrand. 4.3. Gribov copies Gribov copies [7] are a serious conceptual problem in using lattice simulation to understand the behavior of Green's functions in a gauge theory. Their existence on the lattice was investigated starting in the early 1990s [15}17]. Expressing the gauge condition as a maximization condition at each site avoids some trivial lattice artifact copies, but all the analogues of the continuum Gribov copies are still present. From the "rst it was clear that the treatment of Gribov copies in


279

a simulation could have a substantial impact on its results for gauge propagators [15,18]. Because of the uncertainty they lend to simulations of gauge dependent quantities, Gribov copies have been studied by several groups, and studies of their impact on lattice simulations continue [19}21]. Variant gauge conditions and algorithms continue to be investigated as well [22}24]. A conceptual procedure for selecting one out of several copies is to take the maximization condition as an absolute maximum. The space of con"gurations which are absolute maxima is called the fundamental modular domain, and the rule that the path integral should be restricted to the fundamental modular domain removes copies in principal, except on the boundary of the domain [25}28]. Unfortunately, there is no practical procedure known for actually "nding the fundamental modular domain. All methods of gauge "xing are liable to end on copies. Experience in simulations has shown that there are typically many copies, and that quite di!erent con"gurations can have nearly the same value of the maximization functional. An inconvenience, though a serious one, is that gauge "xing is a notoriously slow process. Many procedures have been advocated for accelerating the process, and all work fairly well if the size of the lattice is not too large. However, for very large lattices gauge "xing seems to become much slower, and all acceleration methods seem to loose much of their e!ectiveness. The situation is in some sense as bad as it can be: the role of Gribov copies is not fully understood, there is no perfect recipe for dealing with them, yet they do seem to matter in that there is ample evidence from simulations that the manner in which they are treated sometimes has a signi"cant impact on the "nal result of a simulation.

5. Results from the lattice In this section we will describe the progress that has been made over the dozen or so years that the simulations of the gluon propagator have been carried out. 5.1. Earliest simulations The "rst lattice simulations of the gluon propagator were carried out by Mandula and Ogilvie [5] and by Gupta et al. [6] in 1987. With the computers available at that time, the statistical quality of their signals deteriorated very quickly with lattice time. Therefore, they expressed their results in terms of Euclidean lattice time at zero spacial momentum. The graphs of their results are shown in Figs. 1 and 2. Mandula and Ogilvie's results come from using a 4;10 lattice at b"5.8. The open circles are the time}time component of the D (qo "0o , t) propagator, which is #at in lattice Landau gauge, IJ while the "lled circles are the space}space components, which carry the dynamical information. The work of Gupta et al. [6] used the largest lattices that had been employed for lattice gauge simulations to that time, 18;42. They used b"6.2 as the lattice coupling, and employed the most powerful supercomputer that was available for lattice simulations, a CRAY at Los Alamos National Lab. The major conclusion from these analyses follow from the fact that the dynamical components of the propagator seem to fall linearly over an extended range, which, since the plots are on a semi-log scale, is the behavior expected for a massive particle. At the largest distances the lattice periodicity leads to an enhancement, clearly observable from Fig. 2.

280


Fig. 1. The zero spatial momentum gluon propagator vs. lattice time, from Mandula and Ogilvie [5].

Fig. 2. The zero spatial momentum gluon propagator vs. lattice time, from Gupta et al. [6].

Another salient feature evident from the "gures is that the data clearly follow a curve which is concave downwards for small lattice time. This is quite strange behavior, because it implies that propagator's spectral function is not positive de"nite. This is easy to see. The curvature of an arbitrary positive linear combination of straight exponential ("xed mass) decays is

d ln c e\KGR " G dt G

c e\KGR G G

c m e\KGR ! c m e\KGR G G G G G G c e\KGR G G

.

(21)


281

Fig. 3. The e!ective gluon mass vs. lattice time, from Bernard et al. [13].

For all positive c , this expression is positive as a consequence of the Schwartz inequality. G 5.2. The ewective gluon mass The time slice to time slice fallo! of the gluon propagator provides a natural de"nition of an e!ective gluon mass. D(q"0, t#1) m (t),!ln . D(q"0, t)

(22)

For such a de"nition to be useful requires more computational resources than were available in 1987, when the best that could be done was a global "t to determine the best average mass over the full extent in lattice time. With considerably greater computational resources, in 1993 Bernard et al. [13] and Marenzoni et al. [4] carried out simulations of the gluon propagator with su$cient precision to infer an e!ective mass value as a function of the lattice time. The results from Ref. [13] are shown in Fig. 3. Bernard et al. used 16;40 lattices at b"6.0. Evidently, the e!ective mass grows with (Euclidean) time, at least for small times where the lattice data are best. There is no conclusion to be drawn from this analysis about whether or not it levels out to a "xed asymptotic value. This is another demonstration that the propagator is not described by a positive spectral function. For any such, the e!ective mass would be a monotonically falling function of Euclidean time. 5.3. The most recent lattice results The state of gluon simulations continues to improve, as much more powerful computers have become available for such studies. Recently Leinweber et al. [29] have carried out a high statistics study simulation on a very large lattice, 32;64 sites, at b"6.0. They expressed their results in momentum space, Fig. 4.

282


Fig. 4. The momentum space gluon propagator scaled by q, from Leinweber et al. [29].

The "gure gives aqD(q), the momentum-space propagator scaled by aq, for all values of the lattice 4-momentum. The factor aq is the lattice corrected momentum. The dispersion in the values for moderate q, which is well outside their statistical errors, indicates that lattice artifacts are still present. This graph displays in yet a di!erent way the fact that the gluon propagator is not describable in terms of a positive spectral function. For a free, massless particle, the graph would be #at, and for a free massive particle, or a general propagator with a positive spectral density, the graph would be monotonically increasing and concave downwards everywhere. The graph is also incompatible with the Gribov form, D(q)"q/(q#m) .

(23)

This would also give a monotonically increasing curve. Finally, if there is an anomalous dimension, it is certainly very small.

6. Conclusions: What have we learned? The essential conceptual problem about the gluon propagator is that there are no asymptotic states associated with it. In a Lehmann}KaK llen representation, all the intermediate states that contribute are non-physical, that is, they lie in gauge-variant sectors of the full Hilbert space of the quantum "eld theory. It is just the unfamiliarity of this situation that has given rise to suggestions about the analytic structure of the gluon propagator ranging from its being an entire function to its having an essential singularity at the origin. The most striking observation about the gluon propagator, one that was seen from the earliest simulations, has held up in subsequent analyses. It is that the spectral function describing the gluon propagator is not positive de"nite. Further simulations have not yet given a de"nitive picture of the propagator's structure, but they have ruled out some of the plausible suggestions, including some inspired by the earlier simulations.


283

Simulations of the gluon propagator are no longer compatible with a &&complex mass'' form a% la Gribov. The point is not that one is certain that the gluon propagator is "nite at q"0, but rather that it seems that it cannot vanish as rapidly as q as qP0. The question of an anomalous dimension is still open, although a substantial one cannot be squared with the latest simulations, at least for large q. One should note though that in those simulations, the propagator falls much more steeply than 1/q for intermediate values of the momentum. A "nal puzzle is Zwanziger's theorem, speci"cally that even on the largest lattices there is no sign of the vanishing of the propagator at q"0. Here the problem may be analytic rather than computational. The statement of the theorem is that at q"0, all gluon Green's functions vanish as the lattice volume goes to in"nity. Even for an in"nite lattice, the theorem and its proof give no indication of what the rate of approach to 0 might be. It might be very weak, and might also strongly depend of the total lattice volume. If the goal of getting a de"nitive picture of the gluon propagator has not been fully realized as yet, the progress in the past 12 years has been impressively substantial.

Acknowledgements The occasion for this Symposium was the untimely death of Dick Slansky. The author wishes to express his appreciation to Fred Cooper and Geo! West for organizing this meeting to honor Dick's memory. He also wishes to thank them for encouraging this contribution to the Symposium } a review of a subject that Dick supported from its beginnings.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]

S. Mandelstam, Phys. Rev. D 20 (1979) 3223. M. Baker, J.S. Ball, F. Zachariasen, Nucl. Phys. B, 186 (1981) 531, 560. N. Brown, M.R. Pennington, Phys. Lett. B 202 (1988) 257; Phys. Rev. D 38 (1988) 2266, 39 (1989) 2723. P. Marenzoni, G. Martinelli, N. Stella, M. Testa, Phys. Lett. B 318 (1993) 511. J.E. Mandula, M. Ogilvie, Phys. Lett. B 185 (1987) 127. R. Gupta, G. Guralnik, G. Kilcup, A. Patel, S. Sharpe, T. Warnock, Phys. Rev. D 36 (1987) 2813. V.N. Gribov, Nucl. Phys. B 139 (1978) 1. M. Stingl, Phys. Rev. D 34 (1986) 3863. J.R. Cudell, D.A. Ross, Nucl. Phys. B 359 (1991) 247. L. v Smekal, A. Hauck, R. Alkofer, Phys. Rev. Lett. 79 (1997) 3591; Ann. Phys. 267 (1998) 1. D. Zwanziger, Nucl. Phys. B 364 (1991) 127; Phys Lett. B 257 (1991) 168. J.M. Namislowski, 1993, unpublished. C. Bernard, C. Parrinello, A. Soni, Nucl. Phys. B (Proc. Suppl.) 30 (1993) 535. L. Giusti, M.L. Paciello, S. Petrarca, B. Taglienti, M. Testa, Phys. Lett. B 432 (1998) 196. A. Nakamura, M. Plewnia, Phys. Lett. B 255 (1991) 274. Ph. de Forcrand, J.E. Hetrick, A. Nakamura, M. Plewnia, Nucl. Phys. B (Proc. Suppl.) 20 (1991) 194. A. Hulsebos, M.L. Laursen, J. Smit, A.J. van der Sijs, Nucl. Phys. B (Proc. Suppl.) 20 (1991) 199. P. Coddington, A. Hey, J. Mandula, M. Ogilvie, Phys. Lett. B 197 (1987) 191. S. Petrarca, Nucl. Phys. B (Proc. Suppl.) 26 (1992) 435. H. Suman, K. Schilling, Phys. Lett. B 373 (1996) 314; Nucl. Phys. B (Proc. Suppl.) 53 (1997) 850.

284 [21] [22] [23] [24] [25] [26] [27] [28] [29]

J.E. Mandula / Physics Reports 315 (1999) 273}284 A. Cucchieri, T. Mendes, Nucl. Phys. B (Proc. Suppl.) 63 (1998) 841; A. Cucchieri, Nucl. Phys. B 508 (1997) 353. D. Zwanziger, Nucl. Phys. B 412 (1994) 657. M. Mizutani, A. Nakamura, Nucl. Phys. B (Proc. Suppl.) 34 (1994) 253. P. van Baal, Nucl. Phys. B 367 (1992) 259. P. van Baal, hep-th/9711070, 1997. G. Dell'Antonio, D. Zwanziger, Commun. Math. Phys. 138 (1991) 291. P. van Baal, Nucl. Phys. B 417 (1994) 215; hep-th/9511119 (1995). A. Cucchieri, Nucl. Phys. B 521 (1998) 365. Leinweber, Skullerud, Williams, and Parrinello, Phys. Rev D (Rapid Commun.) 58 (1998) 031501-1; heplat/9811027, 1998.

D. Bailin, A. Love/Physics Reports 315 (1999) 285}408

ORBIFOLD COMPACTIFICATIONS OF STRING THEORY

D. BAILIN , A. LOVE Centre for Theoretical Physics, University of Sussex, Brighton BN1 9QJ, UK Department of Physics, Royal Holloway and Bedford New College, University of London, Egham, Surrey TW20-0EX, UK

AMSTERDAM } LAUSANNE } NEW YORK } OXFORD } SHANNON } TOKYO

285


Orbifold compacti"cations of string theory D. Bailin , A. Love Centre for Theoretical Physics, University of Sussex, Brighton BN1 9QJ, UK Department of Physics, Royal Holloway and Bedford New College, University of London, Egham, Surrey TW20 0EX, UK Received October 1998; editor: J.A. Bagger

Contents 1. Orbifold constructions 1.1. Introduction 1.2. Toroidal compacti"cations 1.3. Point groups and space groups 1.4. Orbifold compacti"cations 1.5. Matter content of orbifold models 1.6. Lattices 1.7. Asymmetric orbifolds 2. Orbifold model building 2.1. Introduction 2.2. Wilson lines 2.3. Modular invariance for toroidal compacti"cation 2.4. orbifold modular invariance 2.5. GSO projections 2.6. Modular invariant Z orbifold compacti"cations 2.7. Untwisted sector massless states 2.8. Twisted sector massless states 2.9. Anomalous ;(1) factors 2.10. Continuous Wilson lines 3. Yukawa couplings 3.1. Introduction 3.2. Vertex operators for orbifold compacti"cations 3.3. Space group selection rules 3.4. H-momentum conservation 3.5. Other selection rules 3.6. 3-point functions from conformal "eld theory

288 288 289 293 296 302 303 309 314 314 315 316 318 320 322 324 325 327 327 328 328 329 331 332 334 335

3.7. 3-point function for Z orbifold 3.8. B "eld backgrounds 3.9. Classical part of 4-point function from conformal "eld theory 3.10. Quantum part of the 4-point function 3.11. Factorisation of the 4-point function to 3-point functions 3.12. Yukawa couplings involving excited twisted sector states 3.13. Quark and lepton masses and mixing angles 4. KaK hler potentials and string loop threshold corrections to gauge coupling constants 4.1. Introduction 4.2. KaK hler potentials for moduli 4.3. KaK hler potentials for untwisted matter "elds 4.4. KaK hler potentials for twisted sector matter "elds 4.5. String loop threshold corrections to gauge coupling constants 4.6. Evaluation of string loop threshold corrections 4.7. Modular anomaly cancellation and threshold corrections to gauge coupling constants 4.8. Threshold corrections with reduced modular symmetry 4.9. Uni"cation of gauge coupling constants

0370-1573/99/$ - see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 8 ) 0 0 1 2 6 - 4

338 340 341 343 345 349 351

353 353 355 358 363 365 367

371 375 378

D. Bailin, A. Love / Physics Reports 315 (1999) 285}408 5. The e!ective potential and supersymmetry breaking 5.1. Introduction 5.2. Non-perturbative superpotential due to gaugino condensate(s) 5.3. E!ective potential

380 380 382 385

5.4. Supersymmetry breaking 5.5. Cosmological constant 5.6. A-terms and B-terms 5.7. Further considerations 6. Conclusions and outlook References

287 395 397 398 400 401 404

Abstract The compacti"cation of the heterotic string theory on a six-dimensional orbifold is attractive theoretically, since it permits the full determination of the emergent four-dimensional e!ective supergravity theory, including the gauge group and matter content, the superpotential and KaK hler potential, as well as the gauge kinetic function. This review attempts to survey all of these calculations, covering the construction of orbifolds which yield (four-dimensional space}time) supersymmetry; orbifold model building, including Wilson lines, and the modular symmetries associated with orbifold compacti"cations; the calculation of the Yukawa couplings, and their connection with quark and lepton masses and mixing; the calculation of the KaK hler potential and its string loop threshold corrections; and the determination of the non-perturbative e!ective potential for the moduli arising from hidden sector gaugino condensation, and its connection with supersymmetry breaking. We conclude with a brief discussion of the relevance of weakly coupled string theory in the light of recent developments on the strongly coupled theory. 1999 Elsevier Science B.V. All rights reserved. PACS: 11.25.-w; 12.10.-g; 12.60.Jv

288

D. Bailin, A. Love / Physics Reports 315 (1999) 285}408

1. Orbifold constructions 1.1. Introduction It is well known that the construction of a consistent quantum string theory is possible only for speci"c dimensionalities of the (target) space}time. For the bosonic string the required dimension is D"26, while for the superstring dimension D"10 is required. Thus from the outset we are forced to consider the `compacti"cationa of the (spatial) dimensions which are surplus to the d"4 dimensions of the world that we inhabit, if we are to have any chance of connecting the string theory with experimental (particle) physics. The string theory which is best placed to generate such a connection is the heterotic string [117], a theory of closed strings, in which the right-moving degrees of freedom of the superstring are adjoined to the twenty-six left-moving degrees of freedom of the bosonic string. To endow such a construction with a geometrical interpretation sixteen of the left-movers are compacti"ed by associating them with a 16-dimensional torus, with radii of order the Planck length (l &10\ m). Just as the compacti"cation of one dimension onto a circle in the (original) . "ve-dimensional Kaluza}Klein theory [135,141] generates a gauge boson, so here the compacti"cation generates gauge "elds, including some of a stringy origin which derive from the possibility of the string winding around the torus. In this way, the 16 left-movers generate an ìnternala gauge symmetry with the (rank 16) gauge group E ;E being consistent with the cancellation of gauge and gravitational anomalies which is essential for a satisfactory quantum theory [113]. Although this scenario explains in a satisfying way how a gauge symmetry can emerge from string theory, there are serious problems which remain. Firstly, there is the fact that the symmetry group E ;E is far larger than the (rank 4) SU(3);SU(2);;(1) gauge symmetry which we observe. Secondly, there remains a ten-dimensional space}time, six of whose dimensions must be compacti"ed before we even contemplate questions like gauge symmetries and matter generations. The orbifolds [79,80], which are the subject of this review are one method of compactifying the unobserved six dimensions. An orbifold is obtained when a six-dimensional torus (¹) is quotiented by a discrete (`pointa) group (P), as we shall see shortly. The identi"cation of points on ¹ under the action of the point group generates a "nite number of "xed points where the orbifold is singular. At all other points the orbifold is (Riemann) #at. It is for this reason that we are able to calculate rather easily all of the parameters and functions of the emergent supergravity theory: the gauge group and matter content; the Yukawa couplings and KaK hler potential, which determine the quark and lepton masses and mixing angles; the gauge kinetic function, including string loop threshold corrections, which in turn determine the uni"cation scale of the gauge coupling constants. We shall see also how modular invariance constrains the e!ective potential, and hence determines the actual value of the coupling constants at uni"cation, as well as the nature of the supersymmetry breaking mechanism. There are, of course, other methods of string compacti"cation including Calabi}Yau manifolds [43,115,116], free fermion models [139,3], and N"2 superconformal "eld theories [107,108,140], and (some) orbifold models are connected to some of these models [138,98,13,14,24]. However, none of the alternatives has so far been as fully worked out as the orbifold theories, and it is for this reason that we have focused upon them. If for no other reason, they illustrate the sort of predictive power which we should eventually like string theory to have (even if it should transpire that nature does not in fact utilize orbifolds!)


289

1.2. Toroidal compactixcations The construction of the ten-dimensional heterotic string has been fully described elsewhere (see, for example, [114,132,35]) and we need not review it here. As already noted, to have any chance of a realistic theory it is obviously essential that six of the (nine) spatial dimensions have to be compacti"ed to a su$ciently small scale as to be unobservable at current accelerators. The simplest way to do this is to compactify on a torus. This ensures that the simple linear string (wave) equations of motion are una!ected, since the torus is #at. We work in the light-cone gauge. Then there are eight transverse bosonic degrees of freedom denoted by XG(q,p) where i"1,2 labels the two transverse four-dimensional space}time coordinates, and XI(q,p) where k"3,2,8 labels the remaining six spatial degrees of freedom. (q, p with 0)p)p are the world sheet parameters.) XG and XI are split into left and right moving components in the standard manner XGI(q,p)"XGI(q!p)#XGI(q#p) . (1.1) 0 * In addition there are eight right-moving transverse fermionic degrees of freedom WG (q!p), 0 WI (q!p), and the 16 (internal) left-moving bosonic degrees of freedom X' (q#p) (I"1,2,16) 0 * which generate the E ;E gauge group of the ten-dimensional heterotic string. The (toroidal) compacti"cation of the six spatial coordinates XI(q,p) (k"3,2,8) does not a!ect the mode expansions of XG(q,p), WG (q!p), WI (q!p) or X' (q#p), so 0 0 * 1 1 i aG e\ LO\N# aG e\LO>N , (1.2) XG(q,p)"xG#pGq# n L n L 2 L$

WGI(q!p)" dGIe\ LO\N (R) 0 L P

(1.3)

or " bGIe\ PO\N (NS) (1.4) P PZ8> depending on whether the world-sheet fermion "eld obeys periodic (Ramond, R) or anti-periodic (Neveu}Schwarz, NS) boundary conditions t (q!p!p)"#t (q!p) (R) , 0 0 t (q!p!p)"!t (q!p) (NS) . 0 0 The mode expansion of the gauge degrees of freedom is i a' X' (q#p)"x' #p' (q#p)# L e\ LO>N * * * 2 n L$ with the momenta p' lying on the E ;E root lattice. * In an orthonormal basis, vectors on the E root lattice the form (n ,n ,2,n ) or (n #,n #,2,n #)

(1.5)

(1.6)

(1.7)

290


with n integers and G n "0 mod 2 . (1.8) G G There is an alternative formulation of these internal degrees of freedom which replaces the 16 bosonic left movers (X') compacti"ed on the E ;E lattice by 32 real fermionic left-movers (j,jM (A"1,2,16)) where j,jM may separately have either periodic (R) or antiperiodic (NS) boundary conditions. Then j" je\ LO>N (R) L L (1.9) " je\ PO>N (NS) , P PZ8> and similarly for the second set jM . (j,jM ) transform as the (16,1)#(1,16) representation of the maximal subgroup O(16);0(16) L E ;E . The compacti"cation of XI entails the identi"cation of the corresponding centre-of-mass coordinates xI with points which are separated by a lattice vector of the torus. Thus xI,xI#2p¸I ,

(1.10)

where the factor 2p is for convenience and the vector L with coordinates ¸I belongs to a sixdimensional lattice K

(1.11) K, r e " r 3Z , R R R R where e (t"3,2,8) are the basis vectors of the lattice. Then the closed string boundary conditions R for the coordinates XI may also be satis"ed when XI(q,p)"XI(q,0)#2p¸I

(1.12)

corresponding to the string winding around the torus. The compacti"cation also requires the quantization of the eigenvalues of the corresponding momentum operators pI. The eigenfunctions exp(i pIxI) are single-valued only if I pI¸I3Z . (1.13) I Thus, the momenta are quantized on the lattice KH which is dual to K

KH" m eH " m 3Z , R R R R where the basis vectors eH of KH satisfy eHIeI,eH ) e "d . R I R S R S RS

(1.14)


291

Then the generalized mode expansions are 1 i XI (q!p)"xI #p)(q!p)# aI e\ LO\N , 0 0 0 n L 2 L$ 1 i XI (q#p)"xI #pI (q#p)# aIe\ LO>N * * * n L 2 L$

(1.15) (1.16)

with pI ,(pI!2¸I) , (1.17) 0 pI ,(pI#2¸I) , (1.18) * xI"xI #xI , 0 * where p3KH and L3K. The mass formula for the right movers in ten-dimensional heterotic string theory, which derives from the constraint equations, yields the four-dimensional mass formula m"N(b)#pI pI !a(b) , (1.19) 0 0 0 where b"R, NS labels the boundary conditions of the fermionic right-movers, and the number operators N(b) is given by N(b)"N #N (b) , D

(1.20)

with N " (aG aG #aI aI) , (1.21) \L L \L L L N (R)" (ndG dG #ndI dI) , (1.22) $ \L L \L L L N (NS)" (rbG bG #rbI bI) . (1.23) $ \P P \P P P a(b) arises from the normal ordering of the operator ¸ in the Virasoro algebra and has the values a(R)"0 , (1.24) a(NS)" . (1.25) (Sums over i"1,2 and k"3,2,8 are implied by the repeated su$xes.) Similarly, the fourdimensional mass formula for the left movers is m"NI #pI pI #p' p' !1 , * * * * * where a sum over I"1,2,16 is also implied and NI " (aG aG #aI aI#a' a') . \L L \L L \L L L

(1.26)

(1.27)

292


If the fermionic formulation of the (left-moving) internal degrees of freedom is used the mass formula becomes (1.28) m(b,c)"NI (b,c)#pI pI !a (b,c) * * * when b,c"R,NS labels the (independent) boundary conditions for the two sets of real fermions j,jM , and NI (b,c)"NI #NI (b)#NI (c) , $ $ where NI " (aG aG #aI aI) , \L L \L L L NI (R)" n(j j#jM jM ) , $ \L L \L L L NI (NS)" r(j j#jM jM ) . $ \P P \P P P Similarly the normal ordering constant a (b,c)"a #a (b)#a (c) , $ $ where

(1.29)

(1.30) (1.31) (1.32)

(1.33)

a ", a (R)"!, a (NS)" . (1.34) $ $ The mass formulae (1.19), (1.26) and (1.28) all include contributions from momenta pI ,pI in the 0 * compacti"ed manifold, which, as we have shown in Eqs. (1.17) and (1.18), are quantized. As we shall see, the lattice K and hence its dual KH generically have some arbitrary scale factors R , the lengths R of the basis vectors e , and angles between basis vectors. So, except for certain isolated values of R these parameters, massless states, in particular, only arise when momenta and winding numbers on the compact manifold are zero pI "0"pI . (1.35) 0 * In fact, the particles we observe in nature must all derive from massless string states, since otherwise their masses would be of the order of the string scale (10 GeV). We may now see why the simple toroidal compacti"cation under consideration is unacceptable for phenomenological reasons. Let us consider a massless state, so m"0"m . (1.36) * 0 Suppose we "x the (massless) left-mover state; for example, we may use one of the a operators on \ the left-movers' ground state "02 , or use momentum p' on the E ;E lattice with p' p' "2. To * * * * each such left-moving state we may attach a massless right-moving state bG "02 (i"1,2) utilizing \ 0 the NS fermionic oscillators. Since i"1, 2 corresponds to the two transverse space}time dimensions, the overall string state transforms as a space}time vector or a space}time tensor, the latter case arising only if the left-moving state is aH "02 ( j"1,2). Alternatively, we may attach the \ *


293

(massless) Ramond groundstate "02 to the "xed left-moving state. This transforms as an eight0 component SO(8) chiral spinor, the opposite chirality spinor having been deleted by the GSO projection used in the superstring construction. This eight-component SO(8) chiral spinor may be decomposed into representations of SO(2);SO(6)LSO(8), the SO(2) corresponding to the two transverse space}time coordinates, and the SO(6) to the six compacti"ed coordinates. Then 8 "(#12);4#(!12)4 (1.37) * and it is clear that there are four space}time spinor particles of each chirality. Thus, if the (bosonic) string state constructed "rst was a vector particle, the fermionic state we have just constructed is four gauginos whereas if the bosonic state "rst constructed was a space}time tensor, the graviton, the fermionic state is four gravitinos. Evidently the toroidal compacti"cation under consideration leads inevitably to N"4 space}time supersymmetry, and hence to a non-chiral gauge symmetry. The observed cancellation of the gauge chiral anomaly within each generation of fermions strongly suggests (but does not conclusively prove) that the gauge symmetry is chiral, and hence that there can be at most N"1 space}time supersymmetry; N*2 supersymmetries automatically cancel chiral anomalies within each supermultiplet. 1.3. Point groups and space groups In the previous section we considered the compacti"cation of the ten-dimensional heterotic string in which the six left-movers and six right-movers XI ,XI (k"3,2,8) are compacti"ed onto 0 * the (same) torus ¹ generated by the lattice K, with the 16 left-movers X' compacti"ed on the * (self-dual) E ;E torus ¹#"#. This latter torus is generated by the root lattice of the group E ;E . A torus is de"ned by identifying points of the underlying space which di!er by a lattice vector l3C"2pK x,x#l .

(1.38)

This identi"cation is called `moddinga and in the six-dimensional toroidal case we write ¹"R/C .

(1.39)

We may generalize this process by identifying points on the torus which are related by the action of an isometry h. To be well-de"ned on the torus h must be an automorphism of the lattice, i.e. hl32pK if l32pK and preserve the scalar products he ) he "e ) e . R S R S The isometry group is called the point group (P) and an orbifold X is de"ned as X"¹/P;¹#"#/G ,

(1.40)

(1.41)

where G is the embedding of P in the gauge group E ;E . P and therefore G are discrete groups. Evidently the six-dimensional orbifold ¹/P may be obtained by identifying points of the underlying space (R) which are related by the action of the point group, up to a lattice vector l x,hx#l .

(1.42)

294


We may regard the right-hand side as the action of the pair (h,l) upon the point x, and the set of all such pairs S,+(h,l) " h3P,l32pK, ,

(1.43)

de"nes a group S, the space group, with the product de"ned in the obvious way by [(h ,l )(h ,l )]x,(h ,l )[(h ,l )x] . Thus we may also write ¹/P"R/S .

(1.44)

(1.45)

The solution of the string equations propagating on an orbifold are almost as straight forward as for a toroidal compacti"cation, since the orbifold is #at almost everywhere. The exceptions are the points of the torus which are left "xed by the point group. Modding out the point group identi"es di!erent lines on the torus passing through the "xed points, so that a conical singularity occurs and the orbifold is not locally isomorphic to R at such points. It follows from Eq. (1.42) that the "xed points satisfy x "hx #l (1.46) D D so if 1!h is singular there are "xed lines or tori, rather than isolated "xed points. The full de"nition of an orbifold compacti"cation requires the speci"cation of ¹ or equivalently the lattice C, the discrete point group P, and its embedding G in the gauge degrees of freedom. The elements h 3 P act upon the bonsonic coordinates XI(q,p) (k"3,2,8) of the string as SO(6) rotations. Possible choices of P are further restricted by the phenomenological requirement to obtain an N"1 space}time supersymmetric spectrum; no supersymmetry (N"0) might also be acceptable, but the conventional wisdom is that N"1 supersymmetry is preferred because of the solution to the technical hierarchy problem which it a!ords. To get N"1 supersymmetry the point group P must be a subgroup of SU(3) [43] PLSU(3) .

(1.47)

This may be seen by recalling that SO(6) is isomorphic to SU(4), so if P satis"es Eq. (1.47) there is a covariantly constant spinor on the six-dimensional orbifold, and it is this extra symmetry which generates the required supersymmetry. For the present we restrict our attention to the cases when the point group P is abelian. Then it must belong to the Cartan subalgebra of SO(6) associated with XI (k"3,2,8). We denote the generators of this subalgebra by M ,M ,M . Then in the complex basis de"ned by (1.48) Z,(1/(2)(X#iX) , Z,(1/(2)(X#iX) ,

(1.49)

Z,(1/(2)(X#iX)

(1.50)

the point group element h acts diagonally and may be written h"exp[2pi(v M#v M#v M)]

(1.51)


295

with 0)"v "(1 (i"1,2,3). The condition that Eq. (1.47) is satis"ed then gives G $v $v $v "0 (1.52) for some choice of signs; this may be seen by noting that the eigenvalues of h acting on a spinor are e p!T!T!T. The requirement that h acts crystallographically on the lattice C plus the condition (1.52) then leads to the conclusion [79,80] that P must either be Z with N"3,4,6,7,8,12 or Z ;Z with , + , N a multiple of M and N"2,3,4,6. Some of the point groups have two (inequivalent) embeddings in SO(6), i.e. they are realized by the inequivalent sets of v ,v ,v . The complete list is given in Tables 1 and 2. These results are the six-dimensional analogue of the famous result that crystals in three dimensions have only N"2,3,4,6-fold rotational symmetries, (augmented by the N"1 space}time supersymmetry requirement (1.52)). In all cases it is possible to "nd a lattice upon which P acts crystallographically, and in many cases there are several lattices for a given P. Often the massless spectrum and gauge group of the orbifold are independent of the choice of lattice, and are determined solely by P. However, we shall see in Section 2 that when the full space group, not just Table 1 Point group generators for Z L SU(3) orbifolds h"exp 2pi(v M#v M#v M) , Point group

(v , v , v )

Z Z Z !I Z !II Z Z !I Z !II Z !I Z !II

(1,1,!2) (1,1,!2) (1,1,!2) (1,2,!3) (1,2,!3) (1,2,!3) (1,3,!4) (1,4,!5) (1,5,!6)

Table 2 Point group generators for Z ;Z L SU(3) orbifolds h"exp 2pi(v M#v M#v M); u"exp 2pi(w M# + , w M#w M) Point group

(v ,v ,v )

(w ,w ,w )

Z ;Z Z ;Z Z ;Z Z ;Z Z ;Z !I Z ;Z !II Z ;Z Z ;Z

(1,0,!1) (1,0,!1) (1,0,!1) (1,0,!1) (1,0,!1) (1,0,!1) (1,0,!1) (1,0,!1)

(0,1,!1) (0,1,!1) (0,1,!1) (0,1,!1) (0,1,!1) (1,1,!2) (0,1,!1) (0,1,!1)

296


the point group, is embedded in the E ;E group then the orbifold properties, not surprisingly, do depend upon the lattice K. 1.4. Orbifold compactixcations The existence of the point group P means that there are additional ways, over and above the toroidal conditions (1.12), in which the closed string boundary conditions may be satis"ed. Let the Z point group be generated by an element h, so that the general element is hL (0)n)N!1). , (The generalization to Z ;Z generated by h,u is trivial.) Then the identi"cation (1.42) means that + , the closed string boundary conditions for the coordinates XI (k"3,2,8) may also be satis"ed when X(q,p)"(hL,l)X(q,0)"hLX(q,0)#l .

(1.53)

Evidently the ùntwisteda sector (n"0) corresponds to the toroidal compacti"cation discussed in the previous section. However, there are additional `twisteda sectors, satisfying Eq. (1.53), with nO0 , and these generate new string states which were not present in the toroidal compacti"cation. Before considering these new states, however, an immediate question arises: what feature of the orbifold removes the unwanted gaugino and gravitino states which we showed are a generic feature of toroidal compacti"cations, and which are present in the untwisted sector of the orbifold compacti"cation? We have explained that the de"nition of an orbifold requires the speci"cation of a discrete group G comprising the space group S and its embedding in the gauge degrees of freedom. Thus to each element of g3G there corresponds an operator g which implements the action of g on the Hilbert space. Because the orbifold is de"ned by modding out the action of G, it follows that physical states must be invariant under G. That is to say, they are eigenvectors of g with eigenvalue unity. Now consider the four gravitino states in the untwisted sector "02 aH "02 ( j"1,2) . (1.54) 0 \ * Since j"1, 2 corresponds to the transverse space}time coordinates which are una!ected by the point group transformations, it is clear that g acts trivially on the left-moving piece of the state. The right moving piece is the Ramond sector ground state, which is an SO(8) chiral spinor. The decomposition (1.37) is given explicitly by 8 "(,,,),(,,!,!)#(!,!,!,!),(!,!,,) , (1.55) 0 where the underlining indicates that all (three) permutations are included, and the individual entries are the eigenvalues of M, M, M, M respectively. The point group generator h is given by Eq. (1.51), and we see that acting on the "rst four states its eigenvalues are hM "exp[ip(v #v #v )],exp ip(v !v !v ),exp[ip(v !v !v )],exp [ip(v !v !v )] (1.56) with the second four states having complex conjugate eigenvalues. Condition (1.52) ensures that at least one of these states have hM "1. Suppose, for example, that v #v #v "0 .

(1.57)


297

(Similar arguments are easily constructed for the other possibilities.) Then the eigenvalues of the above four states are hM "1,exp(2piv ),exp(2piv ),exp(2piv ) . (1.58) So provided that v , v and v are all non-zero, the last three states all have hM O1. It follows that they are not invariant under the action of the point group, and are therefore not space-group invariant either. Thus three of the four gravitinos are deleted, as required if we are to obtain an N"1 space}time supersymmetric theory. On the other hand, if one of v is zero only two of the four gravitinos are deleted and we have at least N"2 supersymmetries surviving. It is for, this reason that Table 1 lists only point the nine group elements with v all non-zero. Similarly in Table 2 we list point group elements of the Z ;Z orbifolds which for a"1,2,3 have v and w not both zero. + , ? ? The twisted sectors of the orbifold string theory are de"ned by Eq. (1.53) with nO0. Let us consider the case of a Z orbifold and the n"1 twisted sector. The extension to n'1 and Z ;Z , , + is easily done. The "rst thing to note is that the modi"ed boundary conditions lead to a di!erent form of the various mode expansions. In this complex basis de"ned in (1.48)}(1.50), the mode expansion of the string world sheet is

1 1 i b? e\ L>T?O\N# bI ? e\ L\T?O>N (1.59) Z?"z? # D 2 n#v L>T? n!v L\T? ? ? L$ where a"1,2,3 labels the three complex planes. The fractional modings are needed to supply the phase factors exp(2piv ) acquired by Z? under the action of the point group. z? is a complex "xed ? D point, constructed from the real "xed points (1.46) analogously to (1.48)}(1.50). Evidently the full speci"cation of a twisted sector requires not only the point group element (h in this case) but also the particular "xed point (or torus) which appears in the zero mode part of the world sheet. Note too that the boundary conditions require that the momentum is zero, since h acts non-trivially in all planes; this is not necessarily the case in all twisted sectors of non-prime orbifolds. For example it is clear from Table 1 that in the h-sectors of the Z -orbifold the mode expansion of Z will have non-zero, but quantized, momentum. The complex conjugate mode expansion is 1 M 1 i bM ? ?e\ L\T?O\N# bI ? e\ L>T?O>N (1.60) ZM "z ? # L\T ? D 2 n#v L>T? n!v ? ? L$ which appear in Z? and ZM ? obey the commutation relations and operators b? ?, bI ? ?, bM ? ?, bIM L>T L\T L\T L>T? [b? ?,bM A A]"d?A(n#v )d , L>T K\T ? K>L [bI ? ?,bIM A A]"d?A(n!v )d . L\T K>T ? K>L Thus the b with n#v'0 are (proportional to) annihilation operators and the bM the L>T \L\T associated creation operators. Likewise the b with n#v(0 are creation operators and the L>T bM the associated annihilation operators. Similarly for bI and bIM . \L\T L\T L>T The point group also acts upon the right-mover fermionic degrees of freedom, so that in the h-twisted sector the boundary conditions are modi"ed: t? (q!p!p)"ep T?t? (q!p) (R) , 0 0 t? (q!p!p)"!ep T?t? (q!p) (NS) , 0 0

(1.61)

298


where the complex t? (a"1,2,3) are constructed from the tI just as the Z? are de"ned in terms of 0 0 the XI (k"3,4,2,8) in (1.48)}(1.50). Thus the modi"ed mode expansions are t? (q!p)" e? ?e\ L>T?O\N (R) 0 L>T L " c? ?e\ P>T?O\N (NS) P>T P

(1.62)

and tM ? (q!p)" e ? ?e\ L\T?O\N (R) 0 L\T P " c ? ?e\ P\T?O\N (NS) , P\T P

(1.63)

where , +e? ?,e @ @,"d?@d K>L L>T K\T +c? ?,c @ ,"d?@d . (1.64) P>T Q\T@ P>Q The space group may also be embedded in the gauge degrees of freedom, and in general, it must be, as we shall see. The element (h,l) of the space group is generally mapped on to (H,V) where H is an automorphism of the E ;E lattice and V is a shift on the lattice. In this section we only address the (compulsory) embedding of the point group elements (h,0) in the gauge group. The (optional) embedding of the lattice elements (1,l), Wilson lines, is discussed in Section 2.2. It is easiest to consider "rst the embedding using the fermionic formulation of the gauge degrees of freedom. The 16 real fermions j transform as the vector representation of O(16)LE . The simplest non-trivial embedding is achieved by picking an O(6) subgroup of O(16), in which the vector representation decomposes into a (six-dimensional) vector representation of SO(6) plus (ten) SO(6) singlets. We next form 3 complex fermions from the 6 real fermions, precisely as we did for the right-moving fermions tI (k"3,2,8), and then take the action of the point group on these 0 3 complex fermions to be precisely what it is on the three complex right-moving fermions t? ; the 0 other ten-fermions are untransformed. This is called the standard embedding [80]. Evidently the mode expansions of these three complex gauge fermions will be modi"ed precisely as are those of the complex fermionic right-movers. The second set of fermions (jM ) are left completely untransformed. This embedding amounts to a shift on the E ;E lattice when we use the bosonic formulation. To see why we need the relationship t'(q#p) ": exp(2iX' ): (1.65) * between the bosonic toroidal coordinates X' and the complex fermions. Then multiplying t by 0 a phase factor exp(2pi * 0

(2.95)


327

the twisted sectors with r "0 mod 3 contain 9 generations of quarks and leptons, together with MM associated states to make up 9 copies of the 27 of E . However, the other twisted sectors contain only exotic massless matter which can form fractionally charged colour singlet states. In this example, not all exotic matter can be con"ned by the extra SU(3) factor in the gauge group. A complete classi"cation of models in the absence of Wilson lines, their gauge groups and massless matter content, has been carried out for all Z orbifolds [137]. Potentially realistic models , with Wilson lines producing standard model gauge group and 3 generations of quarks and leptons have been obtained in the cases of Z as just discussed and Z orbifolds [51] though a complete classi"cation has not been carried out. It is worth noting that there is never any need to adjust the theory to be free of gauge (and gravitational) anomalies due to chiral fermions. Freedom from such anomalies comes as an automatic consequence of the modular invariance of the string theory [172]. 2.9. Anomalous ;(1) factors In the "rst instance, model building leads to theories with SU(3);SU(2);;N(1) gauge group with p'1. To reach the standard model, it is necessary for all but one of the ;(1) factors in the (observable) gauge group to be broken at a large scale. Frequently, one of the ;(1) factors is anomalous [76,12,75] with an anomaly arising from diagrams with 3 non-abelian gauge bosons, or one ;(1) gauge boson and two gravitons, as external legs. Then, at string one loop order a Fayet-Iliopoulos D-term is generated for this ;(1) factor, ; (1), and the corresponding D-term, D , in the Lagrangian takes the form g qG # qG " " , (2.96) D " G 192p G G whereas, for a non-anomalous ;(1), say ; (1), D " qG " " , (2.97) G G where qG and qG are the corresponding ;(1) charges of the scalar "elds . Since, in general, these G

carry not only the anomalous ;(1) charge but also other ;(1) charges, many ;(1) factors may be G broken in this way [52,104]. As a consequence of selection rules on the Yukawa couplings and non-renormalizable couplings in an orbifold theory (which we shall discuss later) the e!ective potential often possesses F #at directions. Then, spontaneous symmetry breaking may occur along such a direction, with ;(1) factors in the gauge group being broken at a very large scale. 2.10. Continuous Wilson lines The discussion of Wilson lines so far has assumed that the point group is embedded in the gauge group as a shift FH\FI(z !w )FM G>FM H\FM I I for zPw, where C are some coe$cients. For twist "elds, we can write GHI p (x,x )p (z ,z )& >IL p (3.128) (z z )"x!z "\FI,\FI, I,D I,D D D D I,D D for xPz , where the sum is over "xed points, the coe$cients > can be interpreted as Yukawa couplings, as will be seen shortly, and the conformal weights are given by h

1k k "hM " 1! I, I, 2N N

(3.129)

as in Eq. (3.104). The 4-point function can then be written in the form valid for xPz , Z "1p (0)p (x)p (1)p (z )2 \I,D I,D \I,D I,D (0)p (1)p (z )2;"x!z "\FI,\FI, . & >IL 1p D D D \I,D \I,D I,D D

(3.130)


347

Moreover, for conformal "elds A ,A and A , G H I 1A (z ,z )A (z ,z )A (z ,z )2"C (z !z )\FGH(z !z )\FM GH G H G H I GHI G H GH

(3.131)

with h "h #h !h GH G H I and similarly for hM . In the case of twist "elds, GH "z "\FI, 1p (0)p (1)p (z )2">HI, \I,D \I,D I,D DDD Consequently, Z takes the form for xPR,

(3.132)

(3.133)

>HI, . (3.134) Z +"x"\FI,\FI,"z "\FI, >I, DDD DDD D To complete the factorisation, we have to use the requirement that the u channel "xed points f summed over must be consistent with the space group selection rule, which takes the form (1!hI)( f#K)"hI(1!hI)( f #K)#(1!hI)( f #K) , where the action of the point group element in this complex plane is h"ep , .

(3.135)

(3.136)

The relation (3.135) is also correct if we interchange f and f , and there are similar relations with f and f replacing f and f . Aided by the space group selection rule we can show that v "h\I(1!hI)( f!f #K) (3.137) and v "!(1!hI)( f!f #K) . This allows Z to be written in the factorised form for xPR, and k/N(1!k/N, "x"I,I,\(C(1!k/N)) e\1I T e\1I T Z +c cos(kp/N)(C(1!2k/N)) D T T with "v " . SI (v )" 4p sin(2kp/N)

(3.138)

(3.139)

(3.140)

It remains to "x the normalisation constant in Eq. (3.139). This can be done by considering the s channel factorisation (Fig. 5) which gives the coupling for the annihilation of two twisted states into an untwisted state. To study these s channel couplings we need to Poisson resum e\1 so T T as to write Z in terms of momenta on the dual KH of the lattice K corresponding to the momenta of untwisted S channel states. Because the sum over v is over the coset (1!hI)( f !f #K) rather than K, it is necessary "rst to arrange for a sum over K by writing

pk ( f !f #q) v "!2ie pI,sin N

(3.141)

348


where q3K. Writing S in terms of q and using the Poisson resummation identity 1 exp(!p)(q# )2A\(q# )!2pid2(q# ) T= M N\T , (3.143) , "2 l (!1)I,\l, for I (1! l , , , , 1p (0)p l (1)p l (z )2 \I, \ , I> , 1q (0)q l (1)p l (z )2 I> , "2(1!I )(!1)I,\l, for I '1! l . \I, \ , (3.163) , , , 1p (0)p l (1)p l (z )2 \I, \ , I> , Taking account of the normalisation of the excited twist "elds discussed above and powers of !1 from the conformal "eld theory of the 3-point function, the Yukawa coupling ># involving excited states should be de"ned by 1 l " ># \(1!I )\(!1)l,\I,1q (0)q l (1)p l (z )2 I> , \I,\l,I>l, 2(N) , \I, \ ,

(3.164)

and consequently ># \I,\l,I>l, "( l, for I (1! l , 1p (0)p l (1)p l (z )2 , \I, , \I, \ , I> , ># \I,\l,I>l, "(\I, for I '1! l . (3.165) l 1p (0)p l (1)p l (z )2 , , , \I, \ , I> , There are thus twist-dependent suppression factors arising in the excited twisted sector Yukawa couplings relative to the Yukawa couplings amongst twisted sector ground states [21,22].


351

3.13. Quark and lepton masses and mixing angles The exponential suppressions [78,120,112] due to the moduli dependence of twisted sector Yukawa couplings can lead to a hierarchial quark and lepton mass matrix [127,54,55]. By utilising all the possible embeddings of the point group and all possible choices of Wilson lines a huge number of models can be obtained for each Z or Z ;Z orbifold. The strategy that has been , + , adopted [55] in exploring the possibilities for the quark and lepton masses (and weak mixing angles) has been to allow the quarks and leptons and Higgses to be assigned to arbitrary twisted sectors and arbitrary "xed points. In general, the Lagrangian terms ¸ for the quark masses take the form O d u (3.166) ¸ "(dM ,s ,bM ) M s #(u ,c ,tM ) M c #h.c. , * S O * B b t 0 0 where M and M are matrices deriving from couplings to Higgses H and H . In Eq. (3.166), B S (u ) ,(d ) ,(c ) ,(s ) ,(t ) and (b ) are the [SU(2)] doublet quark "elds, in terms of which the weak * * * * * * * current JI coupled to the = boson takes the form > d . (3.167) JI "(u ,c ,tM ) cII s > * b * On the other hand, in terms of the states u,d,c,s,t and b that diagonalise the quark mass matrix the weak current JI takes the form > d

JI "(u ,c ,tM ) cI< s > * b

,

(3.168)

* where the matrix < is the usual Kobayashi}Maskawa matrix

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