Physics Reports vol.359

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

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Instructions to authors

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


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Physics Reports 359 (2002) 1–168

Transverse polarisation of quarks in hadrons Vincenzo Baronea; b; ∗ , Alessandro Dragoc , Philip G. Ratcli,ed a Di.S.T.A., Universita del Piemonte Orientale “A. Avogadro”, 15100 Alessandria, Italy Dipartimento di Fisica Teorica, Universita di Torino, and INFN—Sezione di Torino, 10125 Torino, Italy c Dipartimento di Fisica, Universita di Ferrara and INFN—Sezione di Ferrara, 44100 Ferrara, Italy d Dipartimento di Scienze CC.FF.MM., Universita degli Studi dell’Insubria, sede di Como, 22100 Como, and INFN—Sezione di Milano, 20133 Milano, Italy b

Received May 2001; editor: W: Weise Contents 1. Introduction 1.1. History 1.2. Notation and terminology 1.3. Conventions 2. Longitudinal and transverse polarisation 2.1. Longitudinal polarisation 2.2. Transverse polarisation 2.3. Spin density matrix 3. Quark distributions in DIS 3.1. Deeply inelastic scattering 3.2. The parton model 3.3. Polarised DIS in the parton model 3.4. Transversely polarised targets 3.5. Transverse polarisation distributions of quarks in DIS 4. Systematics of quark distribution functions 4.1. The quark–quark correlation matrix 4.2. Leading-twist distribution functions 4.3. Probabilistic interpretation of distribution functions 4.4. Vector, axial and tensor charges

3 6 7 8 9 10 11 12 13 13 19 23 23 25 27 27 28 31 33

4.5. Quark–nucleon helicity amplitudes 4.6. The So,er inequality 4.7. Transverse motion of quarks 4.8. T -odd distributions 4.9. Twist-three distributions 4.10. Sum rules for the transversity distributions 5. Transversity distributions in quantum chromodynamics 5.1. The renormalisation group equations 5.2. QCD evolution at leading order 5.3. QCD evolution at next-to-leading order 5.4. Evolution of the transversity distributions 5.5. Evolution of the So,er inequality and general positivity constraints 6. Transversity in semi-inclusive leptoproduction 6.1. De?nitions and kinematics 6.2. The parton model 6.3. Systematics of fragmentation functions

34 36 37 40 42 45 45 46 48 52 57 61 64 64 67 69

∗ Corresponding author. Dipartimento di Fisica Teorica, UniversitBa di Torino and INFN, Sezione di Torino, 10125 Torino, Italy. E-mail address: [email protected] (V. Barone).

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

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V. Barone et al. / Physics Reports 359 (2002) 1–168 6.4. T -dependent fragmentation functions 6.5. Cross-sections and asymmetries in semi-inclusive leptoproduction 6.6. Semi-inclusive leptoproduction at twist-three 6.7. Factorisation in semi-inclusive leptoproduction 6.8. Two-hadron leptoproduction 6.9. Leptoproduction of spin-1 hadrons 6.10. Transversity in exclusive leptoproduction processes 7. Transversity in hadronic reactions 7.1. Double-spin transverse asymmetries 7.2. The Drell–Yan process 7.3. Factorisation in Drell–Yan processes 7.4. Single-spin transverse asymmetries 8. Model calculations of transverse polarisation distributions 8.1. Bag-like models 8.2. Chiral models 8.3. Light-cone models

73 77 82 83 90 96 97 98 98 99 104 113 118 119 127 133

8.4. Spectator models 8.5. Non-perturbative QCD calculations 8.6. Tensor charges: summary of results 9. Phenomenology of transversity 9.1. Transverse polarisation in hadron– hadron collisions 9.2. Transverse polarisation in lepton– nucleon collisions 9.3. Transverse polarisation in e+ e− collisions 10. Experimental perspectives 10.1. ‘N experiments 10.2. pp experiments 11. Conclusions Acknowledgements Appendix A. Sudakov decomposition of vectors Appendix B. Reference frames B.1. The ∗ N collinear frames B.2. The hN collinear frames Appendix C. Mellin moment identities References

137 137 139 141 141 147 154 155 155 156 157 158 158 159 159 160 161 161

Abstract We review the present state of knowledge regarding the transverse polarisation (or transversity) distributions of quarks. After some generalities on transverse polarisation, we formally de?ne the transversity distributions within the framework of a classi?cation of all leading-twist distribution functions. We describe the QCD evolution of transversity at leading and next-to-leading order. A comprehensive treatment of non-perturbative calculations of transversity distributions (within the framework of quark models, lattice QCD and QCD sum rules) is presented. The phenomenology of transversity (in particular, in Drell–Yan processes and semi-inclusive leptoproduction) is discussed in some detail. Finally, the prospects c 2002 Elsevier Science B.V. All rights reserved. for future measurements are outlined. PACS: 13.85.Qk; 12.38.Bx; 13.88.+e; 14.20.Dh Keywords: Transversity; Polarisation; Spin; QCD; Scattering

V. Barone et al. / Physics Reports 359 (2002) 1–168

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1. Introduction There has been, in the past, a common prejudice that all transverse spin e,ects should be suppressed at high energies. While there is some basis to such a belief, it is far from the entire truth and is certainly misleading as a general statement. The main point to bear in mind is the distinction between transverse polarisation itself and its measurable e>ects. As well-known, even the ultra-relativistic electrons and positrons of the LEP storage ring are signi?cantly polarised in the transverse plane [1] due to the Sokolov–Ternov e,ect [2]. Thus, the real problem is to identify processes sensitive to such polarisation: while this is not always easy, it is certainly not impossible. Historically, the ?rst extensive discussion of transverse spin e,ects in high-energy hadronic physics followed the discovery in 1976 that hyperons produced in pN interactions even at relatively high pT exhibit an anomalously large transverse polarisation [3]. 1 This result implies a non-zero imaginary part of the o,-diagonal elements of the fragmentation matrix of quarks into hyperons. It was soon pointed out that this is forbidden in leading-twist quantum chromodynamics (QCD), and arises only as a O(1=pT ) e,ect [6 –8]. It thus took a while to fully realise that transverse spin phenomena are sometimes unsuppressed and observable. 2 The subject of this report is the transverse polarisation of quarks. This is an elusive and diKcult to observe property that has escaped the attention of physicists for many years. Transverse polarisation of quarks is not, in fact, probed in the cleanest hard process, namely deeply inelastic scattering (DIS), but is measurable in other hard reactions, such as semi-inclusive leptoproduction or Drell–Yan dimuon production. At leading-twist level, the quark structure of hadrons is described by three distribution functions: the number density, or unpolarised distribution, f(x); the longitudinal polarisation, or helicity, distribution Lf(x); and the transverse polarisation, or transversity, distribution LT f(x). The ?rst two are well-known quantities: f(x) is the probability of ?nding a quark with a fraction x of the longitudinal momentum of the parent hadron, regardless of its spin orientation; Lf(x) measures the net helicity of a quark in a longitudinally polarised hadron, that is, the number density of quarks with momentum fraction x and spin parallel to that of the hadron minus the number density of quarks with the same momentum fraction but spin antiparallel. If we call f± (x) the number densities of quarks with helicity ±1, then we have f(x) = f+ (x) + f− (x);

(1.0.1a)

Lf(x) = f+ (x) − f− (x):

(1.0.1b)

The third distribution function, LT f(x), although less familiar, also has a very simple meaning. In a transversely polarised hadron LT f(x) is the number density of quarks with momentum fraction x and polarisation parallel to that of the hadron, minus the number density of quarks 1

An issue related to hadronic transverse spin, and investigated theoretically in the same period, is the g2 spin structure function [4,5]; we shall discuss its relation to transversity later. 2 This was pointed out in the pioneering paper of Ralston and Soper [9] on longitudinally and transversely polarised Drell–Yan processes, but the idea remained almost unnoticed for a decade, see below.

4


with the same momentum fraction and antiparallel polarisation, i.e., 3 LT f(x) = f↑ (x) − f↓ (x):

(1.0.2)

In a basis of transverse polarisation states LT f too has a probabilistic interpretation. In the helicity basis, in contrast, it has no simple meaning, being related to an o,-diagonal quark-hadron amplitude. Formally, quark distribution functions are light-cone Fourier transforms of connected matrix elements of certain quark-?eld bilinears. In particular, as we shall see in detail (see Section 4.2), LT f is given by (we take a hadron moving in the z direction and polarised along the x-axis) d− ixP+ − LT f(x) = PS | N (0)i1+ 5 (0; − ; 0⊥ )|PS : (1.0.3) e 4 In the parton model the quark ?elds appearing in (1.0.3) are free ?elds. In QCD they must be renormalised (see Section 5.1). This introduces a renormalisation-scale dependence into the parton distributions: f(x); Lf(x); LT f(x) → f(x; 2 ); Lf(x; 2 ); LT f(x; 2 );

(1.0.4)

which is governed by the Dokshitzer–Gribov–Lipatov–Altarelli–Parisi [10 –13] (DGLAP) equations (see Section 5). It is important to appreciate that LT f(x) is a leading-twist quantity. Hence it enjoys the same status as f(x) and Lf(x) and, a priori, there is no reason that it should be much smaller than its helicity counterpart. In fact, model calculations show that LT f(x) and Lf(x) are typically of the same order of magnitude, at least at low Q2 , where model pictures hold (see Section 8). The QCD evolution of LT f(x) and Lf(x) is, however, quite di,erent (see Section 5.4). In particular, at low x; LT f(x) turns out to be suppressed with respect to Lf(x). As we shall see, this behaviour has important consequences for some observables. Another peculiarity of LT f(x) is that it has no gluonic counterpart (in spin- 12 hadrons): gluon transversity distributions for nucleons do not exist (Section 4.5). Thus LT f(x) does not mix with gluons in its evolution, and evolves as a non-singlet quantity. One may wonder why the transverse polarisation distributions are so little known, if they are quantitatively comparable to the helicity distributions. No experimental information on LT f(x) is indeed available at present (see, however, Section 9.2.2, where mention is made of some preliminary data on pion leptoproduction that might involve LT f(x)). The reason has already been mentioned: transversity distributions are not observable in fully inclusive DIS, the process that has provided most of the information on the other distributions. Examining the operator structure in (1.0.3) one can see that LT f(x), in contrast to f(x) and Lf(x), which contain + and + 5 instead of i1+ 5 , is a chirally-odd quantity (see Fig. 1a). Now, fully inclusive DIS proceeds via the so-called handbag diagram which cannot Pip the chirality of the probed quark 3

Throughout this paper the subscripts ± will denote helicity whereas the subscripts ↑↓ will denote transverse polarisation.


5

Fig. 1. (a) Representation of the chirally odd distribution LT f(x). (b) A handbag diagram forbidden by chirality conservation.

(see Fig. 1b). In order to measure LT f the chirality must be Pipped twice, so one needs either two hadrons in the initial state (hadron–hadron collisions), or one hadron in the initial state and one in the ?nal state (semi-inclusive leptoproduction), and at least one of these two hadrons must be transversely polarised. The experimental study of these processes has just started and will provide in the near future a great wealth of data (Section 9). So far we have discussed the distributions f(x); Lf(x) and LT f(x). If quarks are perfectly collinear, these three quantities exhaust the information on the internal dynamics of hadrons. If we admit instead a ?nite quark transverse momentum k⊥ , the number of distribution functions increases (Section 4.7). At leading twist, assuming time-reversal invariance, there are six k⊥ -dependent distributions. Three of them, called in the Ja,e–Ji–Mulders clas2 ), g (x; k2 ) and h (x; k2 ), upon integration over k2 , yield si?cation scheme [14,15] f1 (x; k⊥ 1L 1 ⊥ ⊥ ⊥ f(x), Lf(x) and LT f(x), respectively. The remaining three distributions are new and disappear when the hadronic tensor is integrated over k⊥ , as is the case in DIS. Mulders has called 2 ), h⊥ (x; k2 ) and h⊥ (x; k2 ). If time-reversal invariance is not applied (for the them g1T (x; k⊥ 1L 1T ⊥ ⊥ physical motivation behind this, see Section 4.8), two more, T -odd, k⊥ -dependent distribution ⊥ (x; k2 ) and h⊥ (x; k2 ). At present the existence of these distributions functions appear [16]: f1T 1 ⊥ ⊥ is merely conjectural. To summarise, here is an overall list of the leading-twist quark distribution functions: k⊥ -dependent no k⊥ ⊥ ⊥ ⊥ ⊥ f; Lf; LT f; g1T ; h1L ; h1T ; f1T ;h : 1 T -odd At higher twist the proliferation of distribution functions continues [14,15]. Although, for the sake of completeness, we shall also briePy discuss the k⊥ -dependent and the twist-three distributions, most of our attention will be directed to LT f(x). Less space will be dedicated to the other transverse polarisation distributions, many of which have, at present, only an academic interest. In hadron production processes, which, as mentioned above, play an important rôle in the study of transversity, there appear other dynamical quantities: fragmentation functions. These are in a sense specular to distribution functions, and represent the probability for a quark in a given polarisation state to fragment into a hadron carrying some momentum fraction z. When

6


the quark is transversely polarised and so too is the produced hadron, the process is described by the leading-twist fragmentation function LT D(z), which is the analogue of LT f(x) (see Section 6.3). A T -odd fragmentation function, usually called H1⊥ (z), describes instead the production of unpolarised (or spinless) hadrons from transversely polarised quarks, and couples to LT f(x) in certain semi-inclusive processes of great relevance for the phenomenology of transversity (the emergence of LT f via its coupling to H1⊥ is known as the Collins effect [17]). The fragmentation of transversely polarised quarks will be described in detail in Sections 6 and 7. 1.1. History The transverse polarisation distributions were ?rst introduced in 1979 by Ralston and Soper in their seminal work on Drell–Yan production with polarised beams [9]. In that paper LT f(x) was called hT (x). This quantity was apparently forgotten for about a decade, until the beginning of nineties, when it was rediscovered by Artru and Mekh? [18], who called it L1 q(x) and studied its QCD evolution, and also by Ja,e and Ji [14,19], who renamed it h1 (x) in the framework of a general classi?cation of all leading-twist and higher-twist parton distribution functions. At about the same time, other important studies of the transverse polarisation distributions exploring the possibility of measuring them in hadron–hadron or lepton–hadron collisions were carried out by Cortes et al. [20], and by Ji [21]. The last few years have witnessed a great revival of interest in the transverse polarisation distributions. A major e,ort has been devoted to investigating their structure using more and more sophisticated model calculations and other non-perturbative tools (QCD sum rules, lattice QCD, etc.). Their QCD evolution has been calculated up to next-to-leading order (NLO). The related phenomenology has been explored in detail: many suggestions for measuring (or at least detecting) transverse polarisation distributions have been put forward and a number of predictions for observables containing LT f are now available. We can say that our theoretical knowledge of the transversity distributions is by now nearly comparable to that of the helicity distributions. What is really called for is an experimental study of the subject. On the experimental side, in fact, the history of transverse polarisation distributions is readily summarised: (almost) no measurements of LT f have been performed as yet. Probing quark transverse polarisation is among the goals of a number of ongoing or future experiments. At the Relativistic Heavy Ion Collider (RHIC) LT f can be extracted from the measurement of the double-spin transverse asymmetry in Drell–Yan dimuon production with two transversely polarised hadron beams [22] (Section 10.2). Another important class of reactions that can probe transverse polarisation distributions is semi-inclusive DIS. The HERMES collaboration at HERA [23] and the SMC collaboration at CERN [24] have recently presented results on single-spin transverse asymmetries, which could be related to the transverse polarisation distributions via the hypothetical Collins mechanism [17] (Section 9.2.2). The study of transversity in semi-inclusive DIS is one of the aims of the upgraded HERMES experiment and of the COMPASS experiment at the CERN SPS collider, which started taking data in 2001 [25]. It also represents a signi?cant part of other projects (see Section 10.1). We may therefore say that the experimental study of transverse polarisation distributions, which is right now only at the very beginning, promises to have an exciting future.


7

1.2. Notation and terminology Transverse polarisation of quarks is a relatively young and still unsettled subject, hence it is not surprising that the terminology is rather confused. Notation that has been used in the past for the transverse polarisation of quarks comprises hT (x) L1 q(x)

(Ralston and Soper); (Artru and Mekh?);

h1 (x) (Ja,e and Ji); The ?rst two forms are now obsolete while the third is still widely employed. This last was introduced by Ja,e and Ji in their classi?cation of all twist-two, twist-three and twist-four parton distribution functions. In the Ja,e–Ji scheme, f1 (x); g1 (x) and h1 (x) are the unpolarised, longitudinally polarised and transversely polarised distribution functions, respectively, with the subscript 1 denoting leading-twist quantities. The main disadvantage of this nomenclature is the use of g1 to denote a leading-twist distribution function whereas the same notation is universally adopted for one of the two polarised structure functions. This is a serious source of confusion. In the most recent literature the transverse polarisation distributions are often called q(x) or LT q(x): Both forms appear quite natural, as they emphasise the parallel between the longitudinal and the transverse polarisation distributions. In this report we shall use LT f, or LT q, to denote the transverse polarisation distributions, reserving q for the tensor charge (the ?rst moment of LT q). The Ja,e–Ji classi?cation scheme has been extended by Mulders and collaborators [15,16] to all twist-two and twist-three k⊥ -dependent distribution functions. The letters f, g and h denote unpolarised, longitudinally polarised, and transversely polarised quark distributions, respectively. A subscript 1 labels the leading-twist quantities. Subscripts L and T indicate that the parent hadron is longitudinally or transversely polarised. Finally, a superscript ⊥ signals the presence of transverse momenta with uncontracted Lorentz indices. In the present paper we adopt a hybrid terminology. We use the traditional notation for the k⊥ -integrated distribution functions: f(x), or q(x), for the number density, Lf(x), or Lq(x), for the helicity distributions, LT f(x), or LT q(x), for the transverse polarisation distributions, ⊥ ⊥ and Mulders’ notation for the additional k⊥ -dependent distribution functions: g1T , h⊥ 1L , h1T , f1T ⊥ and h1 . We make the same choice for the fragmentation functions. We call the ⊥ -integrated fragmentation functions D(z) (unpolarised), LD(z) (longitudinally polarised) and LT D(z) (transversely polarised). For the ⊥ -dependent functions we use Mulders’ terminology. Occasionally, other notation will be introduced, for the sake of clarity, or to maintain contact with the literature on the subject. In particular, we shall follow these rules: • the subscripts 0; L; T in the distribution and fragmentation functions denote the polarisation

state of the quark (0 indicates unpolarised, and the subscript L is actually omitted in the familiar helicity distribution and fragmentation functions); • the superscripts 0; L; T denote the polarisation state of the parent hadron.

8

V. Barone et al. / Physics Reports 359 (2002) 1–168 Table 1 Notation for the distribution and the fragmentation functions (JJM denotes the Ja,e– Ji–Mulders classi?cation Distribution functions

Fragmentation functions

JJM

This paper

JJM

This paper

f1 g1 h1 g1T

f; q Lf; Lq LT f; LT q g1T

D1 G1 H1 G1T

D LD LT D G1T

h⊥ 1L

h⊥ 1L

⊥ H1L

⊥ H1L

h⊥ 1T

h⊥ 1T

⊥ H1T

⊥ H1T

f1T⊥

f1T⊥ ; LT0 f

⊥ D1T

⊥ D1T

h⊥ 1

0 h⊥ 1 ; LT f

H1⊥

H1⊥ ; L0T D

Thus, for instance, LLT f represents the distribution function of transversely polarised quarks in a longitudinally polarised hadron (it is related to Mulders’ h⊥ 1L ). The Ja,e–Ji–Mulders terminology is compared to ours in Table 1. The correspondence with other notation encountered in the literature [26,27] is LN fq=N ↑ ≡ LT0 f; LN fq↑ =N ≡ L0T f; LN Dh=q↑ ≡ 2L0T Dh=q : Finally, we recall that the name transversity, as a synonym for transverse polarisation, was proposed by Ja,e and Ji [19]. In [28,29] it was noted that “transversity” is a pre-existing term in spin physics, with a di,erent meaning, and that its use therefore in a di,erent context might cause confusion. In this report we shall ignore this problem, and use both terms, “transverse polarisation distributions” and “transversity distributions” with the same meaning. 1.3. Conventions We now list some further conventions adopted throughout the paper. The metric tensor is g = g = diag(+1; −1; −1; −1): The totally antisymmetric tensor 0123

=−

0123 =

A generic four-vector

!

(1.3.1) is normalised so that

+ 1: A

(1.3.2)

is written, in Cartesian contravariant components, as

A = (A0 ; A1 ; A2 ; A3 ) = (A0 ; A):

(1.3.3)


The light-cone components of A are de?ned as 1 A± = √ (A0 ± A3 ); 2 and in these components A is written as

9

(1.3.4)

A = (A+ ; A− ; A⊥ ):

(1.3.5)

The norm of A is given by A2 = (A0 )2 − A2 = 2A+ A− − A2⊥ and the scalar product of two four-vectors

(1.3.6) A

and

B

is

A · B = A0 B0 − A · B = A+ B− + A− B+ − A⊥ · B⊥ :

(1.3.7)

Our fermionic states are normalised as p|p = (2)3 2E3 (p − p ) = (2)3 2p+ (p+ − p+ )(p⊥ − p⊥ );

(1.3.8)

u(p; s) u(p; s ) = 2p ss

(1.3.9)

with E = (p2 +m2 )1=2 . The creation and annihilation operators satisfy the anticommutator relations {b(p; s); b† (p ; s )} = {d(p; s); d† (p ; s )} = (2)3 2Ess 3 (p − p ):

(1.3.10)

2. Longitudinal and transverse polarisation The representations of the PoincarVe group are labelled by the eigenvalues of two Casimir operators, P 2 and W 2 (see e.g., [30]). P is the energy-momentum operator, W is the Pauli– Lubanski operator, constructed from P and the angular-momentum operator J W = −

1 ! J! P : 2 eigenvalues of P 2 and

(2.0.1)

W 2 are m2 and −m2 s(s + 1) respectively, where m is the mass of The the particle and s its spin. The states of a Dirac particle (s = 1=2) are eigenvectors of P and of the polarisation operator , ≡ −W · s=m P |p; s = p |p; s; W ·s 1 |p; s = ± |p; s; m 2 where s is the spin (or polarisation) vector of the particle, with the properties −

s2 = − 1; In general,

s · p = 0:

(2.0.2) (2.0.3) (2.0.4)

s

may be written as p·n (p · n)p ; s = ;n + m m(m + p0 ) where n is a unit vector identifying a generic space direction.

(2.0.5)

10


The polarisation operator , can be re-expressed as 1 ,= 5 s=p= 2m and if we write the plane-wave solutions of the free Dirac equation in the form e−ip·x u(p) (positive energy); (x) = e+ip·x v(p) (negative energy);

(2.0.6)

(2.0.7)

with the condition p0 ¿0, , becomes , = + 12 5 s=

(positive-energy states);

(2.0.8a)

when acting on positive-energy states, (p= − m)u(p) = 0, and , = − 12 5 s=

(negative-energy states);

(2.0.8b)

when acting on negative-energy states, (p= + m)v(p) = 0. Thus the eigenvalue equations for the polarisation operator read (/ = 1; 2) ,u(/) = + 12 5 s=u(/) = ± 12 u(/)

(positive energy);

,v(/) = − 12 5 s=v(/) = ± 12 v(/)

(negative energy):

(2.0.9)

Let us consider now particles which are at rest in a given frame. The spin s is then (set p = 0 in Eq. (2.0.5)) s = (0; n)

(2.0.10)

and in the Dirac representation we have the operator 0 ·n 1 5 s= = 2 0 − · n acting on u(/) =

’(/) 0

;

v(/) =

0 1(/)

(2.0.11)

:

(2.0.12)

Hence the spinors u(1) and v(1) represent particles with spin 12 · n = + 12 in their rest frame whereas the spinors u(2) and v(2) represent particles with spin 12 ·n = − 12 in their rest frame. Note that the polarisation operator in the form (2.0.8a), (2.0.8b), is also well de?ned for massless particles. 2.1. Longitudinal polarisation For a longitudinally polarised particle (n = p= |p|), the spin vector reads |p| p0 p s = ; m m |p|

(2.1.1)


11

and the polarisation operator becomes the helicity operator ,=

·p ; 2|p|

(2.1.2)

with = 5 0 . Consistently with Eq. (2.0.7), the helicity states satisfy the equations ·p u± (p) = ± u± (p); |p| ·p v± (p) = ∓ v± (p): |p|

(2.1.3)

Here the subscript + indicates positive helicity, that is spin parallel to the momentum ( · p¿0 for positive-energy states, ·p¡0 for negative-energy states); the subscript − indicates negative helicity, that is spin antiparallel to the momentum (·p¡0 for positive-energy states, ·p¿0 for negative-energy states). The correspondence with the spinors u(/) and v(/) previously introduced is: u+ = u(1) , u− = u(2) , v+ = v(2) , v− = v(1) . In the case of massless particles one has ,=

·p 1 = 5 : 2|p| 2

(2.1.4)

Denoting again by u± ; v± the helicity eigenstates, Eqs. (2.1.3) become for zero-mass particles 5 u± (p) = ± u± (p); 5 v± (p) = ∓ v± (p):

(2.1.5)

Thus helicity coincides with chirality for positive-energy states, while it is opposite to chirality for negative-energy states. The helicity projectors for massless particles are then 1 (1 ± 5 ) positive-energy states; (2.1.6) P± = 21 2 (1 ∓ 5 ) negative-energy states: 2.2. Transverse polarisation Let us come now to the case of transversely polarised particles. With n · p = 0 and assuming that the particle moves along the z direction, the spin vector (2.0.5) becomes, in Cartesian components s = s⊥ = (0; n⊥ ; 0);

where n⊥ is a transverse two-vector. The polarisation operator takes the form 1 − 2 5 ⊥ · n⊥ = 12 0 ⊥ · n⊥ (positive-energy states); ,= 1 1 (negative-energy states) 2 5 ⊥ · n⊥ = − 2 0 ⊥ · n⊥

(2.2.1)

(2.2.2)

12


and its eigenvalue equations are 1 =⊥ u↑↓ = 2 5 s

± 12 u↑↓ ;

1 =⊥ v↑↓ = 2 5 s

∓ 12 v↑↓ :

(2.2.3)

The transverse polarisation projectors along the directions x and y are (x)

P↑↓ = 12 (1 ± 1 5 ); (y)

P↑↓ = 12 (1 ± 2 5 )

(2.2.4)

for positive-energy states, and (x)

P↑↓ = 12 (1 ∓ 1 5 ); (y)

P↑↓ = 12 (1 ∓ 2 5 )

for negative-energy states. The relations between transverse wave functions)   u(x) = (1= √2)(u+ + u− ); ↑  u(x) = (1= √2)(u+ − u− ); ↓

(2.2.5) polarisation states and helicity states are (for positive-energy   u(y) = (1= √2)(u+ + iu− ); ↑  u(y) = (1= √2)(u+ − iu− ): ↓

(2.2.6)

2.3. Spin density matrix The spinor u(p; s) of a particle with polarisation vector s satis?es u(p; s)u(p; s) = (p= + m) 12 (1 + 5 s=): If the particle is at rest then s = (0; s) = (0; s⊥ ; 5) and (2.3.1) gives 1

1 2 (1 + · s) 0 : u(p; s)u(p; s) = 2m 0 0

(2.3.1)

(2.3.2)

Here one recognises the spin density matrix for a spin-half particle ! = 12 (1 + · s):

(2.3.3)

This matrix provides a general description of the spin structure of a system, that is also valid when the system is not in a pure state. The polarisation vector s = (5; s⊥ ) is, in general, such that s2 6 1: in particular it is s2 = 1 for pure states, s2 ¡1 for mixtures. Explicitly, ! reads 1 + 5 sx − isy 1 != : (2.3.4) 2 sx + isy 1 − 5


13

The entries of the spin density matrix have an obvious probabilistic interpretation. If we call Pm (nˆ) the probability that the spin component in the nˆ direction is m, we can write 5 = P1=2 (z) ˆ − P−1=2 (z); ˆ sx = P1=2 (x) ˆ − P−1=2 (x); ˆ sy = P1=2 (y) ˆ − P−1=2 (y): ˆ In the high-energy limit the polarisation vector is p ; s = 5 + s⊥ m where 5 is (twice) the helicity of the particle. Thus we have

(2.3.5) (2.3.6)

(1 + 5 s=)(p= + m) = (1 + 5 5 + 5 s=⊥ )(p= + m); (1 + 5 s=)(m − p=) = (1 − 5 5 + 5 s=⊥ )(m − p=)

(2.3.7)

and the projector (2.3.1) becomes (with m → 0) u(p; s)u(p; s) = 12 p=(1 − 5 5 + 5 s=⊥ ):

(2.3.8)

If u5 (p) are helicity spinors, calling !55 the elements of the spin density matrix, one has 1 =(1 2p

− 5 5 + 5 s=⊥ ) = !55 u5 (p)u 5 (p);

(2.3.9)

where the r.h.s. is a trace in helicity space. 3. Quark distributions in DIS Although the transverse polarisation distributions cannot be probed in fully inclusive DIS for the reasons mentioned in the Introduction, it is convenient to start from this process to illustrate the ?eld-theoretical de?nitions of quark (and antiquark) distribution functions. In this manner, we shall see why the transversity distributions LT f decouple from DIS even when quark masses are taken into account (which would in principle allow chirality-Pip distributions). We start by reviewing some well-known features of DIS (for an exhaustive treatment of the subject see e.g., [31]). 3.1. Deeply inelastic scattering Consider the inclusive lepton–nucleon scattering (see Fig. 2, where the dominance of onephoton exchange is assumed) l(‘) + N (P) → l (‘ ) + X (PX );

(3.1.1)

where X is an undetected hadronic system (in brackets we put the four-momenta of the particles). Our notation is as follows: M is the nucleon mass, m‘ the lepton mass, s‘ (s‘ ) the spin

14


Fig. 2. Deeply inelastic scattering.

four-vector of the incoming (outgoing) lepton, S the spin four-vector of the nucleon, ‘ = (E; ‘), and ‘ = (E ; ‘ ) the lepton four-momenta. Two kinematic variables (besides the centre-of mass energy s = (‘ + P)2 , or, alternatively, the lepton beam energy E) are needed to describe reaction (3.1.1). They can be chosen among the following invariants (unless otherwise stated, we neglect lepton masses): q2 = (‘ − ‘ )2 = − 2EE (1 − cos #); =

P·q ; M

Q2 Q2 = (the Bjorken x); 2P · q 2M P·q y= ; P·‘ where # is the scattering angle. The photon momentum q is a spacelike four-vector and one usually introduces the quantity Q2 ≡ −q2 , which is positive. Both the Bjorken variable x and the inelasticity y take on values between 0 and 1. They are related to Q2 by xy = Q2 =(s − M 2 ). The DIS cross-section is x=

d =

d 3 ‘ 1 e4 L W 2 ; 4‘ · P Q4 (2)3 2E

(3.1.2)

where the leptonic tensor L is de?ned as (lepton masses are retained here) L = [ul (‘ ; sl ) ul (‘; sl )]∗ [u l (‘ ; sl ) ul (‘; sl )] sl

= Tr[(‘= + ml ) 12 (1 + 5 s=l ) (‘= + ml ) ]

(3.1.3)


and the hadronic tensor W is d 3 PX 1 W = (2)4 4 (P + q − PX )PS |J (0)|X X |J (0)|PS : 2 X (2)3 2EX Using translational invariance this can be also written as 1 W = d 4 eiq· PS |J ()J (0)|PS : 2

15

(3.1.4)

(3.1.5)

It is important to recall that the matrix elements in (3.1.5) are connected. Therefore, vacuum transitions of the form 0|J ()J (0)|0PS |PS are excluded. Note that in (3.1.3) and (3.1.4) we summed over the ?nal lepton spin but did not average over the initial lepton spin, nor sum over the hadron spin. Thus we are describing, in general, the scattering of polarised leptons on a polarised target, with no measurement of the outgoing lepton polarisation (for comprehensive reviews on polarised DIS see [29,32,33]). In the target rest frame, where ‘ · P = ME, (3.1.2) reads 2 E d /em = L W ; dE d: 2MQ4 E

(3.1.6)

where d: = d cos # d’. The leptonic tensor L can be decomposed into a symmetric and an antisymmetric part under ↔ interchange (A) L = L(S) (‘; ‘ ) + iL (‘; sl ; ‘ );

(3.1.7)

and, computing the trace in (3.1.3), we obtain L(S) = 2(‘ ‘ + ‘ ‘ − g ‘ · ‘ );

L(A) = 2ml

! ! s‘ (‘

− ‘ ) :

(3.1.8a) (3.1.8b)

If the incoming lepton is longitudinally polarised, its spin vector is sl =

5l ‘ ; ml

5l = ± 1

(3.1.9)

and (3.1.8b) becomes L(A) = 25l

! ‘

!

q :

(3.1.10)

Note that the lepton mass ml appearing in (3.1.8b) has been cancelled by the denominator of (3.1.9). In contrast, if the lepton is transversely polarised, that is sl = sl⊥ , no such cancellation occurs and the process is suppressed by a factor ml =E. In what follows we shall consider only unpolarised or longitudinally polarised lepton beams. The hadronic tensor W can be split as W = W(S) (q; P) + iW(A) (q; P; S);

(3.1.11)

16


where the symmetric and the antisymmetric parts are expressed in terms of two pairs of structure functions, F1 ; F2 and G1 ; G2 , as q q 1 (S) W = −g + 2 W1 (P · q; q2 ) 2M q 1 P·q P·q P − 2 q P − 2 q W2 (P · q; q2 ); (3.1.12a) + 2 M q q 1 1 (3.1.12b) W(A) = ! q! MS G1 (P · q; q2 ) + [P · qS − S · qP ] G2 (P · q; q2 ) : 2M M Eqs. (3:1:12a; b) are the most general expressions compatible with the requirement of gauge invariance, which implies q W = 0 = q W :

(3.1.13)

Using (3.1.7), (3.1.11) the cross-section (3.1.6) becomes 2 /em E (S) (S) d (A) − L(A) ]: = [L W W dE d: 2MQ4 E

(3.1.14)

The unpolarised cross-section is then obtained by averaging over the spins of the incoming lepton (sl ) and of the nucleon (S) and reads 2 E (S) (S) 1 1 d /em dunp : (3.1.15) = = L W dE d: 2 s 2 dE d: 2MQ4 E l

S

Inserting Eqs. (3.1.8a) and (3.1.12a) into (3.1.15) one obtains the well-known expression 2 E 2 4/em dunp 2 # 2 # 2W1 sin : (3.1.16) = + W2 cos dE d: Q4 2 2 Di,erences of cross-sections with opposite target spin probe the antisymmetric part of the leptonic and hadronic tensors 2 d(+S) d(−S) /em E (A) (A) − − : = 2L W dE d: dE d: 2MQ4 E

(3.1.17)

In the target rest frame the spin of the nucleon can be parametrised as (assuming |S| = 1) S = (0; S ) = (0; sin / cos , ignoring terms O(M 2 =Q2 ), is 2Mx 1 − y cos = sin >; cos / = cos > + Q 2Mx sin / = sin > − 1 − y cos = cos > (3.1.27) Q


19

and hence we obtain

2 d(−S) 4/em 4Mx d(+S) − 1 − y(g1 + g2 ) sin > cos = ; = − 2 (2 − y)g1 cos > + d x dy d’ d x dy d’ Q Q (3.1.28a)

which demonstrates that when the target spin is perpendicular to the photon momentum (>==2) DIS probes the combination g1 + g2 ; and d(+S⊥ ) d(−S⊥ ) 4/2 4Mx − 1 − y(g + g ) cos =: (3.1.28b) = − em 1 2 d x dy d’ d x dy d’ Q2 Q This result can be obtained in another, more direct, manner. Splitting the spin vector of the nucleon into a longitudinal and transverse part (with respect to the photon axis): S = S|| + S⊥ ;

(3.1.29)

where 5N = ± 1 is (twice) the helicity of the nucleon, the antisymmetric part of the hadronic tensor becomes W(A) =

2M ! q! (g1 + g2 )]: [S|| g1 + S⊥ P·q

(3.1.30)

Thus, if the nucleon is longitudinally polarised the DIS cross-section depends only on g1 ; if it is transversely polarised (with respect to the photon axis) what is measured is the sum of g1 and g2 . We shall use expression (3.1.30) when studying the quark content of structure functions in the parton model, to which we now turn. 3.2. The parton model In the parton model the virtual photon is assumed to scatter incoherently o, the constituents of the nucleon (quarks and antiquarks). Currents are treated as in free ?eld theory and any interaction between the struck quark and the target remnant is ignored. The hadronic tensor W is then represented by the handbag diagram shown in Fig. 4 and reads (to simplify the presentation, for the moment we consider only quarks, the extension to antiquarks being rather straightforward) 1 2 d 3 PX d4 k d4 W = ea (2 ) 3 4 4 (2) a (2) 2E (2) (2) X X ×[u() =(k; P; S)]∗ [u() =(k; P; S)](2)4 4 (P − k − PX )(2)4 4 (k + q − );

(3.2.1)

where a is a sum over the Pavours, ea is the quark charge in units of e, and we have introduced the matrix elements of the quark ?eld between the nucleon and its remnant

=i (k; P; S) = X | i (0)|PS :

(3.2.2)

20


Fig. 4. The so-called handbag diagram. Fig. 5. The ∗ N collinear frame. Note our convention for the axes.

We de?ne the quark–quark correlation matrix Aij (k; P; S) as d 3 PX (2)4 4 (P − k − PX )PS | j (0)|X X | i (0)|PS : Aij (k; P; S) = 3 2E (2) X X

(3.2.3)

Using translational invariance and the completeness of the |X states this matrix can be reexpressed in the more synthetic form Aij (k; P; S) = d 4 eik· PS | N j (0) i ()|PS : (3.2.4) With the de?nition (3.2.3) the hadronic tensor becomes d4 k d4 ea2 (2 )(2)4 4 (k + q − ) Tr[A = ] W = 4 4 (2) (2) a d4 k = ea2 ((k + q)2 ) Tr[A (k= + q=) ]: 4 (2) a

(3.2.5)

In order to calculate W , it is convenient to use a Sudakov parametrisation of the fourmomenta at hand (the Sudakov decomposition of vectors is described in Appendix A). We introduce the null vectors p and n , satisfying p2 = 0 = n2 ;

p · n = 1;

n+ = 0 = p−

(3.2.6)

and we work in a frame where the virtual photon and the proton are collinear. As is customary, the proton is taken to be directed along the positive z direction (see Fig. 5). In terms of p and n the proton momentum can be parametrised as M2 (3.2.7) n p : 2 Note that, neglecting the mass M , P coincides with the Sudakov vector p . The momentum q of the virtual photon can be written as P = p +

q P · qn − xp ;

(3.2.8)


21

where we are implicitly ignoring terms O(M 2 =Q2 ). Finally, the Sudakov decomposition of the quark momentum is 2) (k 2 + k⊥ : (3.2.9) n + k⊥ 2/ In the parton model one assumes that the handbag-diagram contribution to the hadronic tensor 2 . This means that we can write k approximately as is dominated by small values of k 2 and k⊥

k = /p +

k /p :

(3.2.10)

The on-shell condition of the outgoing quark then implies 1 ((k + q)2 ) (−Q2 + 2/P · q) = (/ − x); 2P · q

(3.2.11)

that is, k xP . Thus the Bjorken variable x ≡ Q2 =(2P · q) is the fraction of the longitudinal momentum of the nucleon carried by the struck quark: x = k + =P + . (In the following we shall also consider the possibility of retaining the quark transverse momentum; in this case (3.2.9) will be approximated as k xP + k⊥ .) Returning to the hadronic tensor (3.2.5), the identity ! = [g! g + g g! − g g! − i

!

] 5

(3.2.12)

allows us to split W into symmetric (S) and antisymmetric (A) parts under ↔ interchange. Let us ?rst consider W(S) (i.e., unpolarised DIS): 1 2 d4 k k+ (S) W = e x − + [(k + q ) Tr(A ) + (k + q ) Tr(A ) 2P · q a a (2)4 P −g (k ! + q! )Tr(A ! )]:

(3.2.13)

From (3.2.8) and (3.2.9) we have k + q (P · q)n and (3.2.13) becomes 1 2 d4 k k+ (S) W = e x − + [n Tr(A ) + n Tr(A ) − g n! Tr(A ! )]: (3.2.14) 2 a a (2)4 P Introducing the notation d4 k k+ D ≡ x − + Tr(DA) (2)4 P d− ixP+ − PS | N (0)D (0; − ; 0⊥ )|PS = P+ e 2 dE iEx = e PS | N (0)D (En)|PS ; 2 where D is a Dirac matrix, W(S) is written as 1 2 e [n + n − g n! ! ]: W(S) = 2 a a

(3.2.15)

(3.2.16)

22


We have now to parametrise , which is a vector quantity containing information on the quark dynamics. At leading twist, i.e., considering contributions O(P + ) in the in?nite momentum frame, the only vector at our disposal is p P (recall that n = O(1=P + ) and k xP ). Thus we can write d4 k k+ ≡ x − + Tr( A) (2)4 P dE iEx = (3.2.17) e PS | N (0) (En)|PS = 2f(x)P ; 2 where the coeKcient of P , which we called f(x), is the quark number density, as will become clear later on (see Sections 4.2 and 4.3). From (3.2.17) we obtain the following expression for f(x): d− ixP+ − f(x) = PS | N (0) + (0; − ; 0⊥ )|PS : (3.2.18) e 4 Inserting (3.2.17) into (3.2.16) yields W(S) = ea2 (n P + n P − g )fa (x):

(3.2.19)

a

The structure functions F1 and F2 can be extracted from W by means of two projectors (terms of relative order 1=Q2 are neglected) 1 4x2 F1 = P1 W = P P − g W ; (3.2.20a) 4 Q2 x 12x2 F2 = P2 W = P P −g W : (3.2.20b) 2 Q2 Since (P P =Q2 )W = O(M 2 =Q2 ) we ?nd that F1 and F2 are proportional to each other (the so-called Callan–Gross relation) and are given by x F2 (x) = 2xF1 (x) = − g W(S) = ea2 xfa (x); (3.2.21) 2 a which is the well-known parton model expression for the unpolarised structure functions, restricted to quarks. To obtain the full expressions for F1 and F2 , one must simply add to (3.2.20b) the antiquark distributions fNa , which were left aside in the above discussion. They read (the rôle of and N is interchanged with respect to the quark distributions: see Section 4.2 for a detailed discussion) d− ixP+ − N f(x) = PS |Tr[ + (0) N (0; − ; 0⊥ )]|PS (3.2.22) e 4 and the structure functions F1 and F2 are F2 (x) = 2xF1 (x) = ea2 x[fa (x) + fNa (x)]: a

(3.2.23)


23

3.3. Polarised DIS in the parton model Let us turn now to polarised DIS. The parton-model expression of the antisymmetric part of the hadronic tensor is 1 2 d4 k k+ (A) W = e x − + ! (k + q)! Tr( 5 A): (3.3.1) 2P · q a a (2)4 P With k = xP this becomes, using the notation (3.2.15) e2 a 5 : W(A) = ! n! 2 a

(3.3.2)

= O(1)) At leading twist the only pseudovector at hand is S|| (recall that S|| = O(P + ) and S⊥ and 5 is parametrised as (a factor M is inserted for dimensional reasons)

5 = 2M Lf(x)S|| = 25N Lf(x)P :

(3.3.3)

Here Lf(x), given explicitly by d− ixP+ − Lf(x) = PS | N (0) + 5 (0; − ; 0⊥ )|PS ; (3.3.4) e 4 is the longitudinal polarisation (i.e., helicity) distribution of quarks. In fact, inserting (3.3.3) in (3.3.2), we ?nd e2 a W(A) = 25N ! n! p (3.3.5) Lfa (x): 2 a Comparing with the longitudinal part of the hadronic tensor (3.1.30), which can be rewritten as (A) W; long = 25N

! n

!

p g1 ;

we obtain the usual parton model expression for the polarised structure function g1 1 2 g1 (x) = e Lfa (x): 2 a a

(3.3.6)

(3.3.7)

Again, antiquark distributions LfqN should be added to (3.3.7) to obtain the full parton model expression for g1 1 2 g1 (x) = e [Lfa (x) + LfNa (x)]: (3.3.8) 2 a a The important lesson we learned is that, at leading twist, only longitudinal polarisation contributes to DIS. 3.4. Transversely polarised targets Since S⊥ is suppressed by a power of P + with respect to its longitudinal counterpart S|| , transverse polarisation e,ects in DIS manifest themselves at twist-three level. Including subdominant

24


contributions, Eq. (3.3.3) becomes 5 = 2M Lf(x)S|| + 2MgT (x)S⊥ ;

(3.4.1)

where we have introduced a new, twist-three, distribution function gT , de?ned as (we take the nucleon spin to be directed along the x-axis) P+ i P+ d− ixP+ − gT (x) = 5 = PS | N (0) i 5 (0; − ; 0⊥ )|PS : (3.4.2) e 2M M 4 As we are working at twist-three (that is, with quantities suppressed by 1=P + ) we take into account the transverse components of the quark momentum, k xp + k⊥ . Moreover, quark mass terms cannot be ignored. Reinstating these terms in the hadronic tensor, we have 1 2 d4 k k+ 1 ! ! W(A) = ea x − (k + q) Tr[ A] − Tr[i A] : m ! 5 q 5 2P · q a (2)4 P+ 2

(3.4.3)

Notice that now we cannot simply set k ! + q! P · qn , as we did in the case of longitudinal polarisation. Let us rewrite Eq. (3.4.3) as 1 ! W(A) = ea2 5 + LW(A) ; (3.4.4) ! q 2P · q a where LW(A) =

1 2P · q

! a

ea2

1 ! i 5 @ − mq i 5 : 2

!

(3.4.5)

If we could neglect the term LW(A) then, for a transversely polarised target, we should have, using Eq. (3.4.1) 2 2M ! q! S⊥ ea a W(A) = (3.4.6) g (x): P·q 2 T a Comparing with Eq. (3.1.30) yields the parton-model expression for the polarised structure function combination g1 + g2 : 1 2 a g1 (x) + g2 (x) = e g (x): (3.4.7) 2 a a T This result has been obtained by ignoring the term LW(A) in the hadronic tensor, rather a strong assumption, which seems lacking in justi?cation. Surprisingly enough, however, Eq. (3.4.5) turns out to be correct. The reason is that at twist-three one has to add an extra term W(A)g into (3.4.4), arising from non-handbag diagrams with gluon exchange (see Fig. 6) and which exactly cancel out LW(A) . Referring the reader to the original papers [34] (see also [35]) for a detailed proof, we limit ourselves to presenting the main steps.


25

Fig. 6. Higher-twist contribution to DIS involving quark–gluon correlation.

For the sum LW(A) + W(A)g one obtains 1 2 1 (A) (A)g ! ! e LW + W = mq i 5 ! i 5 D (En) − 4P · q a a 2 − D (En) − D (En) + · · · ;

(3.4.8)

where D = @ − igA and the ellipsis denotes terms with the covariant derivative acting to the left and the gluon ?eld evaluated at the space–time point 0. We now resort to the identity E 1 2 E 5 ! = g!

− g! − i

! 5 :

(3.4.9)

Multiplying by D! and using the equations of motion (iD= − mq ) = 0 we ultimately obtain E 1 2 mq E 5 = D

− D +

! ! i 5 D ;

(3.4.10)

which implies the vanishing of (3.4.8). Concluding, DIS with transversely polarised nucleon (where transverse refers to the photon axis) probes a twist-three distribution function, gT (x), which, as we shall see, has no probabilistic meaning and is not related in a simple manner to transverse quark polarisation. 3.5. Transverse polarisation distributions of quarks in DIS Let us focus now on the quark mass term appearing in the antisymmetric hadronic tensor— see Eqs. (3.4.5) and (3.4.8). We have shown that actually it cancels out and does not contribute to DIS. Its structure, however, is quite interesting. It contains, in fact, the transverse polarisation distribution of quarks, LT f, which is the main subject of this report. The decoupling of the quark mass term thus entails the absence of LT f from DIS, even at higher-twist level. The matrix element i! 5 admits a unique leading-twist parametrisation in terms of a tensor structure containing the transverse spin vector of the target S⊥ and the dominant Sudakov vector p ! i! 5 = 2(p S⊥ − p! S⊥ )LT f(x):

(3.5.1)

26


The coeKcient LT f(x) is indeed the transverse polarisation distribution. It can be singled out by contracting (3.5.1) with n! , which gives (for de?niteness, we take the spin vector directed along x) d− ixP+ − 1! 1 LT f(x) = 2 in! 5 = PS | N (0)i1+ 5 (0; − ; 0⊥ )|PS : (3.5.2) e 4 Eq. (3.4.10) can be put in the form of a constraint between LT f(x) and other twist-three distributions embodied in D and i 5 D! . Let us consider the partonic content of the last two quantities. The gluonic (non-handbag) contribution W(A)g to the hadronic tensor involves traces of a quark–gluon–quark correlation matrix. We introduce the following two quantities: dE1 dE2 iE1 x2 iE2 (x1 −x2 ) D (E2 n) ≡ PS | N (0) D (E2 n) (E1 n)|PS ; (3.5.3a) e e 2 2 dE1 dE2 iE1 x2 iE2 (x1 −x2 ) i 5 D (E2 n) ≡ e PS | N (0)i 5 D (E2 n) (E1 n)|PS : e 2 2 (3.5.3b) These matrix elements are related to those appearing in (3.4.8) by d x2 D (E2 n) = D (E2 n); d x2 i 5 D (E1 n) = i 5 D (E1 n):

(3.5.4a) (3.5.4b)

At leading order (which for the quark–gluon–quark correlation functions means twist-three) the possible Lorentzian structures of D and i 5 D are D = 2MGD (x1 ; x2 )p

/
p/ n< S⊥! ;

(3.5.5a)

i 5 D = 2M G˜ D (x1 ; x2 )p S⊥ + 2M G˜ D (x1 ; x2 )p S⊥ :

(3.5.5b)

Here three multiparton distributions, GD (x1 ; x2 ), G˜ D (x1 ; x2 ) and G˜ D (x1 ; x2 ), have been introduced. One of them, G˜ (x1 ; x2 ), is only apparently a new quantity. Multiplying Eq. (3.5.5b) by n and exploiting the gauge choice A+ = 0, it is not diKcult to derive a simple connection between G˜ (x1 ; x2 ) and the twist-three distribution function gT (x2 ) [34] G˜ D (x1 ; x2 ) = x2 (x1 − x2 )gT (x2 ):

(3.5.6)

G˜ D (x1 ; x2 )

can be eliminated in favour of the more familiar gT (x2 ). We are now in Hence the position to translate Eq. (3.4.10) into a relation between quark and multiparton distribution functions. Using (3.5.1) and (3:5:4a)–(3:5:6) in (3.4.10) we ?nd mq dy[GD (x; y) + G˜ D (x; y)] = xgT (x) − (3.5.7) LT f(x): M By virtue of this constraint, the transverse polarisation distributions of quarks, that one could naYZvely expect to be probed by DIS at a subleading level, turn out to be completely absent from this process.


27

Fig. 7. The quark–quark correlation matrix A.

4. Systematics of quark distribution functions In this section we present in detail the systematics of quark and antiquark distribution functions. Our focus will be on leading-twist distributions. For the sake of completeness, however, we shall also sketch some information on the higher-twist distributions. 4.1. The quark–quark correlation matrix Let us consider the quark–quark correlation matrix introduced in Section 3.2 and represented in Fig. 7, Aij (k; P; S) = d 4 eik· PS | N j (0) i ()|PS : (4.1.1) Here, we recall, i and j are Dirac indices and a summation over colour is implicit. The quark distribution functions are essentially integrals over k of traces of the form Tr(DA) = d 4 eik· PS | N (0)D ()|PS ; (4.1.2) where D is a Dirac matrix structure. In Section 3.2, A was de?ned within the naYZve parton model. In QCD, in order to make A gauge invariant, a path-dependent link operator

L(0; ) = P exp −ig ds A (s) ; (4.1.3) 0

where P denotes path-ordering, must be inserted between the quark ?elds. It turns out that the distribution functions involve separations of the form [0; − ; 0⊥ ], or [0; − ; ⊥ ]. Thus, by working in the axial gauge A+ = 0 and choosing an appropriate path, L can be reduced to unity. Hereafter we shall simply assume that the link operator is unity, and just omit it. The A matrix satis?es certain relations arising from hermiticity, parity invariance and timereversal invariance [15]: A† (k; P; S) = 0 A(k; P; S) 0 ˜ P; ˜ 0 ˜ −S) A(k; P; S) = 0 A(k;

(hermiticity); (parity);

˜ P; ˜ † 5 ˜ S)C A∗ (k; P; S) = 5 CA(k;

(time-reversal);

(4.1.4a) (4.1.4b) (4.1.4c)

28


where C = i 2 0 and the tilde four-vectors are de?ned as k˜ = (k 0 ; −k). As we shall see, the time-reversal condition (4.1.4c) plays an important rôle in the phenomenology of transverse polarisation distributions. It is derived in a straightforward manner by using T () T † = ˜ and T |PS = (−1)S−Sz |P; ˜ S˜ , where T is the time-reversal operator. The crucial − i 5 C (−) point, to be kept in mind, is the transformation of the nucleon state, which is a free particle state. Under T , this goes into the same state with reversed P and S . The most general decomposition of A in a basis of Dirac matrices,

D = {5; ; 5 ; i 5 ; i 5 }; 1 2

is (we introduce a factor

(4.1.5)

for later convenience)

A(k; P; S) = 12 {S5 + V + A 5 + iP5 5 + 12 iT 5 }:

(4.1.6)

The quantities S, V , A , P5 and T are constructed with the vectors k , P and the pseudovector S . Imposing the constraints (4:1:4a), (4.1.4c) we have, in general, S = 12 Tr(A) = C1 ;

(4.1.7a)

V = 12 Tr( A) = C2 P + C3 k ;

(4.1.7b)

A = 12 Tr( 5 A) = C4 S + C5 k · SP + C6 k · Sk ;

(4.1.7c)

1 (4.1.7d) Tr( 5 A) = 0; 2i 1 T = Tr( 5 A) = C7 P [ S ] + C8 k [ S ] + C9 k · SP [ k ] ; (4.1.7e) 2i where the coeKcients Ci = Ci (k 2 ; k · P) are real functions, owing to hermiticity. If we relax the constraint (4.1.4c) of time-reversal invariance (for the physical relevance of this, see Section 4.8 below) three more terms appear: P5 =

V = · · · + C10

!

S P! k ;

(4.1.7d )

P5 = C11 k · S; T = · · · + C12

(4.1.7b )

!

P! k :

(4.1.7e )

4.2. Leading-twist distribution functions We are mainly interested in the leading-twist contributions, that is the terms in Eqs. (4:1:7a)– (4.1.7e) that are of order O(P + ) in the in?nite momentum frame. The vectors at our disposal are P , k xP , and S 5N P =M + S⊥ , where the approximate + equality signs indicate that we are neglecting terms suppressed by (P )−2 . Remember that the transverse spin vector S⊥ is of order (P + )0 . For the time being we ignore quark transverse momentum k⊥ (which in DIS is integrated over). We shall see later on how the situation becomes more complicated when k⊥ enters the game.


29

At leading order in P + only the vector, axial, and tensor terms in (4.1.6) survive and Eqs. (4:1:7b, c, e) become 1 V = d 4 eik· PS | N (0) ()|PS = A1 P ; (4.2.1a) 2 1 A = d 4 eik· PS | N (0) 5 ()|PS = 5N A2 P ; (4.2.1b) 2 1 ] T = d 4 eik· PS | N (0) 5 ()|PS = A3 P [ S⊥ ; (4.2.1c) 2i where we have introduced new real functions Ai (k 2 ; k · P). The leading-twist quark correlation ] matrix (4.1.6) is then (we use P [ S⊥ = 2iP= S=⊥ ) A(k; P; S) = 12 {A1 P= + A2 5N 5 P= + A3 P= 5 S=⊥ }:

(4.2.2)

From (4:2:1a)–(4.2.1c) we obtain A1 =

1 Tr( + A); 2P +

(4.2.3a)

5N A2 =

1 Tr( + 5 A); 2P +

(4.2.3b)

i S⊥ A3 =

1 1 Tr(ii+ 5 A) = + Tr( + i 5 A): + 2P 2P

(4.2.3c)

The leading-twist distribution functions f(x), Lf(x) and LT f(x) are obtained by integrating A1 , A2 and A3 , respectively, over k, with the constraint x = k + =P + , that is,     2   f(x)   d4 k   A1 (k ; k · P)   k+ 2 Lf(x) = (4.2.4) A2 (k ; k · P) x − + ;   (2)4  P  L f(x)     2 A3 (k ; k · P) T i = (1; 0) that is, using (4:2:3a)–(4.2.3c) and setting for de?niteness 5N = + 1 and S⊥ 1 d4 k f(x) = Tr( + A)(k + − xP + ); 2 (2)4 1 d4 k Lf(x) = Tr( + 5 A)(k + − xP + ); 2 (2)4 1 d4 k LT f(x) = Tr( + 1 5 A)(k + − xP + ): 2 (2)4

(4.2.5a) (4.2.5b) (4.2.5c)

Finally, inserting the de?nition (4.1.1) of A in (4:2:5a)–(4.2.5c), we obtain the three leading-twist distribution functions as light-cone Fourier transforms of expectation values of quark-?eld

30


bilinears [36]: d− ixP+ − f(x) = PS | N (0) + (0; − ; 0⊥ )|PS ; e 4 d− ixP+ − e Lf(x) = PS | N (0) + 5 (0; − ; 0⊥ )|PS ; 4 d− ixP+ − PS | N (0) + 1 5 (0; − ; 0⊥ )|PS : LT f(x) = e 4

(4.2.6a) (4.2.6b) (4.2.6c)

The quark–quark correlation matrix A integrated over k with the constraint x = k + =P + d4 k Aij (x) = Aij (k; P; S)(x − k + =P + ) (2)4 dE iEx = (4.2.7) e PS | N j (0) i (En)|PS ; 2 in terms of the three leading-twist distribution functions, reads A(x) = 12 {f(x)P= + 5N Lf(x) 5 P= + LT f(x)P= 5 S=⊥ }:

(4.2.8)

Let us now complete the discussion introducing the antiquarks. Their distribution functions are obtained from the correlation matrix N Aij (k; P; S) = d 4 eik· PS | i (0) N j ()|PS : (4.2.9) Tracing AN with the Dirac matrices D gives Tr(DA) = d 4 eik· PS |Tr[D (0) N ()]|PS :

(4.2.10)

N Lf, N LT fN care is needed with the signs. By charge conjuIn deriving the expressions for f, gation, the ?eld bilinears in (4.1.4a) transform as N (0)D () → ±Tr[D (0) N ()]; (4.2.11) where the + sign is for D = , i 5 and the − sign for D = 5 . We thus obtain the antiquark density number: 1 d4 k + N N f(x) = Tr( + A)(k − xP + ) 2 (2)4 d− ixP+ − = PS |Tr[ + (0) N (0; − ; 0⊥ )]|PS ; (4.2.12) e 4 the antiquark helicity distribution 1 d4 k + N N Lf(x) = − Tr( + 5 A)(k − xP + ) 2 (2)4 d− ixP+ − = PS |Tr[ + 5 (0) N (0; − ; 0⊥ )]|PS (4.2.13) e 4


and the antiquark transversity distribution d4 k 1 + N N LT f(x) Tr( + 1 5 A)(k − xP + ) = 2 (2)4 d− ixP+ − = PS |Tr[ + 1 5 (0) N (0; − ; 0⊥ )]|PS : e 4

31

(4.2.14)

Note the minus sign in the de?nition of the antiquark helicity distribution. If we adhere to the de?nitions of quark and antiquark distributions, Eqs. (4:2:6a)–(4.2.6c) and (4:2:12)–(4:2:14), the variable x ≡ k + =P + is not a priori constrained to be positive and to range from 0 to 1 (we shall see in Section 4.3 how the correct support for x comes out, hence justifying its identi?cation with the Bjorken variable). It turns out that there is a set of symmetry relations connecting quark and antiquark distribution functions, which are obtained by continuing x to negative values. Using anticommutation relations for the fermion ?elds in the connected matrix elements PS | N () (0)|PS c = − PS | (0) N ()|PS c ;

(4.2.15)

one easily obtains the following relations for the three distribution functions: N f(x) = − f(−x);

(4.2.16a)

N Lf(x) = Lf(−x);

(4.2.16b)

N = − LT f(−x): LT f(x)

(4.2.16c)

Therefore, antiquark distributions are given by the continuation of the corresponding quark distributions into the negative x region. 4.3. Probabilistic interpretation of distribution functions Distribution functions are essentially the probability densities for ?nding partons with a given momentum fraction and a given polarisation inside a hadron. We shall now see how this interpretation comes about from the ?eld-theoretical de?nitions of quark (and antiquark) distribution functions presented above. Let us ?rst of all decompose the quark ?elds into “good” and “bad” components: =

(+)

+

(−) ;

(4.3.1)

where 1 ∓ ± (±) = 2

:

(4.3.2)

The usefulness of this procedure lies in the fact that “bad” components are not dynamically independent: using the equations of motion, they can be eliminated in favour of “good” components and terms containing quark masses and gluon ?elds. Since in the P + → ∞ limit (+) dominates over (−) , the presence of “bad” components in a parton distribution function signals

32


Fig. 8. (a) A connected matrix element with the insertion of a complete set of intermediate states and (b) a semiconnected matrix element.

higher twists. Using the relations √ N + = 2 † (+) ; (+) √ N + 5 = 2

† (+) 5 (+) ;

√ N ii+ 5 = 2

† i (+) 5 (+)

the leading-twist distributions (4:2:6a)–(4.2.6c) can be re-expressed as [36] d− ixP+ − † √ e f(x) = PS | (+) (0) + (+) (0; − ; 0⊥ )|PS ; 2 2 d− ixP+ − † √ e Lf(x) = PS | (+) (0) 5 (+) (0; − ; 0⊥ )|PS ; 2 2 d− ixP+ − † √ e LT f(x) = PS | (+) (0) 1 5 (+) (0; − ; 0⊥ )|PS : 2 2

(4.3.3a) (4.3.3b) (4.3.3c)

(4.3.4a) (4.3.4b) (4.3.4c)

Note that, as anticipated, only “good” components appear. It is the peculiar structure of the quark-?eld bilinears in Eqs. (4:3:4a)–(4.3.4c) that allows us to put the distributions in a form that renders their probabilistic nature transparent. A remark on the support of the distribution functions is now in order. We already mentioned that, according to the de?nitions of the quark distributions, nothing constrains the ratio x ≡ k + =P + to take on values between 0 and 1. The correct support of the distributions emerges, along with their probabilistic content, if one inserts into (4:3:4a)–(4.3.4c) a complete set of intermediate states {|n} [37] (see Fig. 8). Considering, for instance, the unpolarised distribution we obtain from (4.3.4a) 1 f(x) = √ ((1 − x)P + − Pn+ )|PS | (+) (0)|n|2 ; (4.3.5) 2 n where n incorporates the integration over the phase space of the intermediate states. Eq. (4.3.5) clearly gives the probability of ?nding inside the nucleon a quark with longitudinal momentum k + =P + , irrespective of its polarisation. Since the states |n are physical we must have Pn+ ¿ 0, that is En ¿ |Pn |, and therefore x 6 1. Moreover, if we exclude semiconnected diagrams like


33

that in Fig. 8b, which correspond to x¡0, we are left with the connected diagram of Fig. 8a and with the correct support 0 6 x 6 1. A similar reasoning applies to antiquarks. Let us turn now to the polarised distributions. Using the Pauli–Lubanski projectors P± = 1 1 1 2 (1 ± 5 ) (for helicity) and P↑↓ = 2 (1 ± 5 ) (for transverse polarisation), we obtain 1 ((1 − x)P + − Pn+ ){|PS |P+ (+) (0)|n|2 − |PS |P− (+) (0)|n|2 }; Lf(x) = √ 2 n (4.3.6a) 1 LT f(x) = √ ((1 − x)P + − Pn+ ){|PS |P↑ 2 n

2 (+) (0)|n|

− |PS |P↓

2 (+) (0)|n| }:

(4.3.6b)

These expressions exhibit the probabilistic content of the leading-twist polarised distributions Lf(x) and LT f(x): Lf(x) is the number density of quarks with helicity + minus the number density of quarks with helicity − (assuming the parent nucleon to have helicity +); LT f(x) is the number density of quarks with transverse polarisation ↑ minus the number density of quarks with transverse polarisation ↓ (assuming the parent nucleon to have transverse polarisation ↑). It is important to notice that LT f admits an interpretation in terms of probability densities only in the transverse polarisation basis. The three leading-twist quark distribution functions are contained in the entries of the spin density matrix of quarks in the nucleon (5(x) is the quark helicity density, s⊥ (x) is the quark transverse spin density): !++ !+− 1 + 5(x) sx (x) − isy (x) 1 !55 = = : (4.3.7) 2 sx (x) + isy (x) !−+ !− − 1 − 5(x) Recalling the probabilistic interpretation of the spin density matrix elements discussed in Section 2.3, one ?nds that the spin components s⊥ ; 5 of the quark appearing in (4.3.7) are related to the spin components S⊥ ; 5N of the parent nucleon by 5q (x)f(x) = 5N Lf(x);

(4.3.8a)

s⊥ (x)f(x) = S⊥ LT f(x):

(4.3.8b)

4.4. Vector, axial and tensor charges If we integrate the correlation matrix A(k; P; S) over k, or equivalently A(x) over x, we obtain a local matrix element (which we call A, with no arguments) Aij = d 4 k Aij (k; P; S) = d x Aij (x) = PS | N j (0) i (0)|PS ; (4.4.1) which, in view of (4.2.2), can be parametrised as A = 12 [gV P= + gA 5N 5 P= + gT P= 5 S=⊥ ]:

(4.4.2)

34


Here gV ; gA and gT are the vector, axial and tensor charge, respectively. They are given by the following matrix elements, recall (4:2:1a)–(4.2.1c): PS | N (0) i (0)|PS = 2gV P ;

(4.4.3a)

PS | N (0) 5 i (0)|PS = 2gA MS ;

(4.4.3b)

PS | N (0)i 5 i (0)|PS = 2gT (S P − S P ):

(4.4.3c)

Warning: the tensor charge gT should not be confused with the twist-three distribution function gT (x) encountered in Section 3.4. Integrating Eqs. (4:2:6a)–(4.2.6c) and using the symmetry relations (4:2:16a)–(4.2.16c) yields +1 1 N d x f(x) = d x[f(x) − f(x)] (4.4.4a) = gV ; −1 0 +1 1 N d x Lf(x) = d x[Lf(x) + Lf(x)] = gA ; (4.4.4b) −1 0 +1 1 N d x LT f(x) = d x[LT f(x) − LT f(x)] (4.4.4c) = gT : −1

0

Note that gV is simply the valence number. As a consequence of the charge conjugation properties of the ?eld bilinears N , N 5 and N i 5 , the vector and tensor charges are the ?rst moments of Pavour non-singlet combinations (quarks minus antiquarks) whereas the axial charge is the ?rst moment of a Pavour singlet combination (quarks plus antiquarks). 4.5. Quark–nucleon helicity amplitudes The DIS hadronic tensor is related to forward virtual Compton scattering amplitudes. Thus, leading-twist quark distribution functions can be expressed in terms of quark–nucleon forward amplitudes. In the helicity basis these amplitudes have the form A5; 5 , where 5; 5 (; ) are quark (nucleon) helicities. There are in general 16 amplitudes. Imposing helicity conservation, − 5 = − 5;

i:e:; + 5 = + 5 ;

(4.5.1)

only 6 amplitudes survive: A++; ++ ;

A− −; − − ;

A+−; +− ;

A−+; −+ ;

A+−; −+ ;

A−+; +− :

(4.5.2)

Parity invariance implies A5; 5 = A−−5; − −5

(4.5.3)

and gives the following 3 constraints on the amplitudes: A++; ++ = A− −; − − ; A++; − − = A− −; ++ ; A+−; −+ = A−+; +− :

(4.5.4)


35

Fig. 9. The three quark–nucleon helicity amplitudes.

Time-reversal invariance, A5; 5 = A 5 ; 5 ;

(4.5.5)

adds no further constraints. Hence, we are left with three independent amplitudes (see Fig. 9) A++; ++ ;

A+−; +− ;

A+−; −+ :

(4.5.6)

Two of the amplitudes in (4.5.6), A++; ++ and A+−; +− , are diagonal in the helicity basis (the quark does not Pip its helicity: 5 = 5 ), the third, A+−; −+ , is o,-diagonal (helicity Pip: 5 = − 5 ). Using the optical theorem we can relate these quark–nucleon helicity amplitudes to the three leading-twist quark distribution functions, according to the scheme f(x) = f+ (x) + f− (x)∼Im(A++; ++ + A+−; +− );

(4.5.7a)

Lf(x) = f+ (x) − f− (x)∼Im(A++; ++ − A+−; +− );

(4.5.7b)

LT f(x) = f↑ (x) − f↓ (x)∼Im A+−; −+ :

(4.5.7c)

In a transversity basis (with ↑ directed along y) | ↑ = √12 [|+ + i|−]; | ↓ = √12 [|+ − i|−];

(4.5.8)

the transverse polarisation distributions LT f is related to a diagonal amplitude LT f(x) = f↑ (x) − f↓ (x)∼Im(A↑↑; ↑↑ − A↑↓; ↑↓ ):

(4.5.9)

Reasoning in terms of parton–nucleon forward helicity amplitudes, it is easy to understand why there is no such thing as leading-twist transverse polarisation of gluons. A hypothetical LT g would imply an helicity Pip gluon–nucleon amplitude, which cannot exist owing to helicity conservation. In fact, gluons have helicity ±1 but the nucleon cannot undergo an helicity change L = ± 2. Targets with higher spin may have an helicity-Pip gluon distribution. If transverse momenta of quarks are not neglected, the situation becomes more complicated and the number of independent helicity amplitudes increases. These amplitudes combine to form six k⊥ -dependent distribution functions (three of which reduce to f(x), Lf(x) and LT f(x) when integrated over k⊥ ).

36


4.6. The So>er inequality From the de?nitions of f, Lf and LT f, that is, Lf(x) = f+ (x) − f− (x), LT f(x) = f↑ (x) − f↓ (x) and f(x) = f+ (x) + f− (x) = f↑ (x) + f↓ (x), two bounds on Lf and LT f immediately follow: |Lf(x)| 6 f(x);

(4.6.1a)

|LT f(x)| 6 f(x):

(4.6.1b)

Similar inequalities are satis?ed by the antiquark distributions. Another, more subtle, bound, simultaneously involving f, Lf and LT f, was discovered by So,er [38]. It can be derived from the expressions (4:5:7a)–(4.5.7c) of the distribution functions in terms of quark–nucleon forward amplitudes. Let us introduce the quark–nucleon vertices a5 :

and rewrite Eqs. (4:5:7a)–(4.5.7c) in the form f(x)∼Im(A++; ++ + A+−; +− )∼ (a∗++ a++ + a∗+− a+− ); X Lf(x)∼Im(A++; ++ − A+−; +− )∼ (a∗++ a++ − a∗+− a+− ); X ∗ LT f(x)∼Im A+−; −+ ∼ a− − a++ :

(4.6.2a) (4.6.2b) (4.6.2c)

X

From

|a++ ± a− − |2 ¿ 0;

(4.6.3)

using parity invariance, we obtain a∗++ a++ ± a∗− − a++ ¿ 0;

(4.6.4)

X

X

X

that is f+ (x) ¿ |LT f(x)|;

(4.6.5)

which is equivalent to f(x) + Lf(x) ¿ 2|LT f(x)|:

(4.6.6)

An analogous relation holds for the antiquark distributions. Eq. (4.6.6) is known as the So,er inequality. It is an important bound, which must be satis?ed by the leading-twist distribution functions. The reason it escaped attention until a relatively late discovery in [38] is that it involves three quantities that are not diagonal in the same basis. Thus, to be derived, So,er’s


37

Fig. 10. The So,er bound on the leading-twist distributions [38] (note that there LT q(x) was called q(x)).

inequality requires consideration of probability amplitudes, not of probabilities themselves. The constraint (4.6.6) is represented in Fig. 10. We shall see in Section 5.5 that the So,er bound, like the other two—more obvious— inequalities (4:6:1a), (4.6.1b), is preserved by QCD evolution, as it should be. 4.7. Transverse motion of quarks Let us now account for the transverse motion of quarks. This is necessary in semi-inclusive DIS, when one wants to study the Ph⊥ distribution of the produced hadron. Therefore, in this section we shall prepare the ?eld for later applications. The quark momentum is now given by k xP + k⊥ ;

(4.7.1) , k⊥

which is zeroth order in P + and thus suppressed by one power of where we have retained + P with respect to the longitudinal momentum. At leading twist, again, only the vector, axial and tensor terms in (4.1.6) appear and Eqs. (4:1:7b), (4.1.7c), (4.1.7e) become V = A1 P ;

1 ˜ A 1 k⊥ · S ⊥ P ; M 5N ˜ [ ] 1 ] ] T = A3 P [ S⊥ + ; A2 P k⊥ + 2 A˜ 3 k⊥ · S⊥ P [ k⊥ M M A = 5N A2 P +

(4.7.2a) (4.7.2b) (4.7.2c)

38


where we have de?ned new real functions A˜ i (k 2 ; k · P) (the tilde signals the presence of k⊥ ) and introduced powers of M , so that all coeKcients have the same dimension. The quark–quark correlation matrix (4.1.6) then reads 1 1 A(k; P; S) = A1 P= + A2 5N 5 P= + A3 P= 5 S=⊥ + A˜ 1 k⊥ · S⊥ 5 P= 2 M 1 ˜ 5N ˜ (4.7.3) + A2 P= 5 k=⊥ + 2 A3 k⊥ · S⊥ P= 5 k=⊥ : M M We can project out the Ai ’s and A˜ i ’s, as we did in Section 4.2 1 Tr( + A) = A1 ; 2P + 1 1 Tr( + 5 A) = 5N A2 + k⊥ · S⊥ A˜ 1 ; + 2P M 1 5N i ˜ 1 i i ˜ Tr(ii+ 5 A) = S⊥ A3 + A3 : k⊥ A2 + 2 k⊥ · S⊥ k⊥ + 2P M M Let us rearrange the r.h.s. of the last expression in the following manner: 2 k⊥ 1 i i ˜ i ˜ S⊥ A3 + 2 k⊥ · S⊥ k⊥ A3 = S⊥ A3 + A3 M 2M 2 1 2 ij 1 i j k⊥ + k⊥ g⊥ S⊥j A˜ 3 : − 2 k⊥ M 2

(4.7.4a) (4.7.4b) (4.7.4c)

(4.7.5)

If we integrate Eqs. (4:7:4a)–(4:7:4c) over k with the constraint x = k + =P + , the terms proportional to A˜ 1 , A˜ 2 and A˜ 3 in (4:7:4b)–(4:7:5) vanish. We are left with the three terms proportional 2 =2M 2 )A ˜ 3 , which give, upon integration, the three to A1 , A2 and to the combination A3 + (k⊥ distribution functions f(x), Lf(x) and LT f(x), respectively. The only di,erence from the pre2 =2M 2 ) A ˜3 vious case of no quark transverse momentum is that LT f(x) is now related to A3 +(k⊥ and not to A3 alone 2 k⊥ d4 k ˜ A3 + (4.7.6) A3 (x − k + =P + ): LT f(x) ≡ (2)4 2M 2 Three of If we do not integrate over k⊥ , we obtain six k⊥ -dependent distribution functions. 2 ), Lf(x; k2 ) and L f(x; k2 ), are such that f(x) = d 2 k f(x; k2 ), them, which we call f(x; k⊥ ⊥ T ⊥ ⊥ ⊥ etc. The other three are completely new and are related to the terms of the correlation matrix containing the A˜ i ’s. We shall adopt Mulders’ terminology for them [15,16]. Introducing the notation 1 d k+ d k− [D] A ≡ Tr(DA)(k + − xP + ) 2 (2)4 d− d 2 ⊥ i(xP+ − −k⊥ ·⊥ ) = e PS | N (0)D (0; − ; ⊥ )|PS ; (4.7.7) 2(2)3


39

we have +

2 A[ ] = Pq=N (x; k⊥ ) = f(x; k⊥ );

A[

+

A[i

5 ]

i+

(4.7.8a)

2 = Pq=N (x; k⊥ )5(x; k⊥ ) = 5N Lf(x; k⊥ )+

5 ]

k⊥ · S ⊥

i = Pq=N (x; k⊥ )s⊥ (x; k⊥ ) i 2 LT f(x; k⊥ ) = S⊥

5N i ⊥ 1 2 + )− 2 k h (x; k⊥ M ⊥ 1L M

M

2 ); g1T (x; k⊥

i j k⊥ k⊥

(4.7.8b)

1 2 ij 2 + k⊥ g⊥ S⊥j h⊥ 1T (x; k⊥ ); 2 (4.7.8c)

where Pq=N (x; k⊥ ) is the probability of ?nding a quark with longitudinal momentum fraction x and transverse momentum k⊥ , and 5(x; k⊥ ), s⊥ (x; k⊥ ) are the quark helicity and transverse spin densities, respectively. The spin density matrix of quarks now reads 1 + 5(x; k⊥ ) sx (x; k⊥ ) − isy (x; k⊥ ) 1 !55 = (4.7.9) 2 sx (x; k⊥ ) + isy (x; k⊥ ) 1 − 5(x; k⊥ ) and its entries incorporate the six distributions listed above, according to Eqs. (4:7:8a) and (4:7:8b). Let us now try to understand the partonic content of the k⊥ -dependent distributions. If the 2 ), which coincides with target nucleon is unpolarised, the only measurable quantity is f(x; k⊥ Pq (x; k⊥ ), the number density of quarks with longitudinal momentum fraction x and transverse 2. momentum squared k⊥ If the target nucleon is transversely polarised, there is some probability of ?nding the quarks transversely polarised along the same direction as the nucleon, along a di,erent direction, or longitudinally polarised. This variety of situations is allowed by the presence of k⊥ . Integrating over k⊥ , the transverse polarisation asymmetry of quarks along a di,erent direction with respect to the nucleon polarisation, and the longitudinal polarisation asymmetry of quarks in a transversely polarised nucleon disappear: only the case s⊥ ||S⊥ survives. Referring to Fig. 11 for the geometry in the azimuthal plane and using the following parametrisations for the vectors at hand (we assume full polarisation of the nucleon): k⊥ = (|k⊥ | cos =k ; −|k⊥ | sin =k );

(4.7.10)

S⊥ = (cos =S ; −sin =S );

(4.7.11)

s⊥ = (|s⊥ | cos =s ; −|s⊥ | sin =s );

(4.7.12)

we ?nd for the k⊥ -dependent transverse polarisation distributions of quarks in a transversely polarised nucleon (± denote, as usual, longitudinal polarisation whereas ↑↓ denote transverse polarisation) 2 Pq↑=N ↑ (x; k⊥ ) − Pq↓=N ↑ (x; k⊥ ) = cos(=S − =s )LT f(x; k⊥ )

+

2 k⊥

2M 2

2 cos(2=k − =S − =s )h⊥ 1T (x; k⊥ )

(4.7.13a)

40


Fig. 11. Our de?nition of the azimuthal angles in the plane orthogonal to the ∗ N axis. The photon momentum, which is directed along the positive z axis, points inwards. For our choice of the axes see Fig. 5.

and for the longitudinal polarisation distribution of quarks in a transversely polarised nucleon |k⊥ | 2 Pq+=N ↑ (x; k⊥ ) − Pq−=N ↑ (x; k⊥ ) = ): (4.7.13b) cos(=S − =k )g1T (x; k⊥ M Due to transverse motion, quarks can also be transversely polarised in a longitudinally polarised nucleon. Their polarisation asymmetry is |k⊥ | 2 Pq↑=N + (x; k⊥ ) − Pq↓=N + (x; k⊥ ) = (4.7.13c) cos(=k − =s )h⊥ 1L (x; k⊥ ): M As we shall see in Section 6.5, the k⊥ -dependent distribution function h⊥ 1L plays a role in the azimuthal asymmetries of semi-inclusive leptoproduction. 4.8. T -odd distributions Relaxing the time-reversal invariance condition (4:1:4c)—we postpone the discussion of the physical relevance of this until the end of this subsection—two additional terms in the vector and tensor components of A arise 1 V = · · · + A1 ! P k⊥! S⊥ ; (4.1.7b ) M 1 T = · · · + A2 ! P! k⊥ ; (4.1.7e ) M ⊥ and h⊥ which give rise to two k⊥ -dependent T -odd distribution functions, f1T 1 [ + ]

ij ⊥ k⊥i S⊥j

⊥ 2 (x; k⊥ ); f1T M ij [ii+ 5 ] 2 ⊥ k⊥j ⊥ A =··· − ): h (x; k⊥ M 1

A

=··· −

(4.8.1a) (4.8.1b)


41

⊥ , is Let us see the partonic interpretation of the new distributions. The ?rst of them, f1T related to the number density of unpolarised quarks in a transversely polarised nucleon. More precisely, it is given by

Pq=N ↑ (x; k⊥ ) − Pq=N ↓ (x; k⊥ ) = Pq=N ↑ (x; k⊥ ) − Pq=N ↑ (x; −k⊥ )

= −2

|k⊥ |

M

⊥ 2 (x; k⊥ ): sin(=k − =S )f1T

(4.8.2a)

The other T -odd distribution, h⊥ 1 , measures quark transverse polarisation in an unpolarised hadron [16] and is de?ned via |k⊥ |

2 (4.8.2b) sin(=k − =s )h⊥ 1 (x; k⊥ ): M We shall encounter again these distributions in the analysis of hadron production (Section ⊥ and h⊥ , 7.4). For later convenience we de?ne two quantities LT0 f and L0T f, related to f1T 1 respectively, by (for the notation see Section 1.2)

Pq↑=N (x; k⊥ ) − Pq↓=N (x; k⊥ ) = −

2 LT0 f(x; k⊥ ) ≡ −2 2 )≡− L0T f(x; k⊥

|k⊥ |

M

⊥ 2 (x; k⊥ ); f1T

|k⊥ |

(4.8.3a)

2 ): (4.8.3b) h⊥ (x; k⊥ M 1 It is now time to comment on the physical meaning of the quantities we have introduced in ⊥ and this section. One may legitimately wonder whether T -odd quark distributions, such as f1T ⊥ h1 that violate the time-reversal condition (4.1.4c) make any sense at all. In order to justify the existence of T -odd distribution functions, their proponents [39] advocate initial-state e,ects, which prevent implementation of time-reversal invariance by naYZvely imposing the condition (4.1.4c). The idea, similar to that which leads to admitting T -odd fragmentation functions as a result of ?nal-state e,ects (see Section 6.4), is that the colliding particles interact strongly with non-trivial relative phases. As a consequence, time reversal no longer implies the constraint ⊥ (4.1.4c). 4 If hadronic interactions in the initial state are crucial to explain the existence of f1T ⊥ and h1 , these distributions should only be observable in reactions involving two initial hadrons (Drell–Yan processes, hadron production in proton–proton collisions, etc.). This mechanism is known as the Sivers e,ect [40,41]. Clearly, it should be absent in leptoproduction. A di,erent way of looking at the T -odd distributions has been proposed in [42– 44]. By relying on a general argument using time reversal, originally due to Wigner and recently revisited by ⊥ and Weinberg [45], the authors of [44] show that time reversal does not necessarily forbid f1T ⊥ h1 . In particular, an explicit realisation of Weinberg’s mechanism, based on chiral Lagrangians, ⊥ and h⊥ may emerge from the time-reversal preserving chiral dynamics of quarks shows that f1T 1 in the nucleon, with no need for initial-state interaction e,ects. If this idea is correct, the T -odd distributions should also be observable in semi-inclusive leptoproduction. A conclusive statement on the matter will only be made by experiments.

4

Thus “T -odd” means that condition (4.1.4c) is not satis?ed, not that time reversal is violated.

42


4.9. Twist-three distributions At twist-three the quark–quark correlation matrix, integrated over k, has the structure [14] M 5N e(x) + gT (x) 5 S=⊥ + (4.9.1) hL (x) 5 [p=; n=] ; A(x) = · · · + 2 2 where the dots represent the twist-two contribution, Eq. (4.2.8), and p; n are the Sudakov vectors (see Appendix A). Three more distributions appear in (4.9.1): e(x), gT (x) and hL (x). We already encountered gT (x), which is the twist-three partner of Lf(x): dE iEx + twist-4 terms: e PS | N (0) 5 (En)|PS = 25N Lf(x)p + 2MgT (x)S⊥ 2 (4.9.2a) Analogously, hL (x) is the twist-three partner of LT f(x) and appears in the tensor term of the quark–quark correlation matrix: dE iEx ] + 2MhL (x)p[ n] + twist-4 terms: e PS | N (0)i 5 (En)|PS = 2LT f(x)p[ S⊥ 2 (4.9.2b) The third distribution, e(x), has no counterpart at leading twist. It appears in the expansion of the scalar ?eld bilinear: dE iEx (4.9.2c) e PS | N (0) (En)|PS = 2Me(x) + twist-4 terms: 2 The higher-twist distributions do not admit any probabilistic interpretation. To see this, consider for instance gT (x). Upon separation of into good and bad components, it turns out to be P+ d− ixP+ − † gT (x) = PS | (+) (0) 0 1 5 (−) (0; − ; 0⊥ ) e M 4 −

† 0 1 − (−) (0) 5 (+) (0; ; 0⊥ )|PS :

(4.9.3)

This distribution cannot be put into a form such as (4.3.6a), (4.3.6b). Thus gT cannot be regarded as a probability density. Like all higher-twist distributions, it involves quark–quark– gluon correlations and hence has no simple partonic meaning. It is precisely this fact that makes gT (x) and the structure function that contains gT (x), i.e., g2 (x; Q2 ), quite subtle and diKcult to handle within the framework of parton model. It should be borne in mind that the twist-three distributions in (4.9.1) are, in a sense, “e,ective” quantities, which incorporate various kinematical and dynamical e,ects that contribute to higher twist: quark masses, intrinsic transverse motion and gluon interactions. It can be shown [15] that e(x), hL (x) and gT (x) admit the decomposition e(x) =

mq f(x) + e(x); ˜ M x

(4.9.4a)


43

Fig. 12. The quark–quark–gluon correlation matrix AA .

hL (x) =

mq Lf(x) 2 ⊥(1) − h1L (x) + h˜L (x); M x x

mq LT f(x) 1 (1) + g1T (x) + g˜T (x); M x x where we have introduced the weighted distributions k2 2 h⊥(1) (x) ≡ d 2 k⊥ ⊥2 h⊥ (x; k⊥ ); 1L 2M 1L gT (x) =

(1) g1T (x) ≡

d 2 k⊥

2 k⊥

2M 2

2 g1T (x; k⊥ ):

(4.9.4b) (4.9.4c)

(4.9.5a) (4.9.5b)

The three tilde functions e(x), ˜ h˜L (x) and g˜T (x) are the genuine interaction-dependent twist-three parts of the subleading distributions, arising from non-handbag diagrams like that of Fig. 6. To understand the origin of such quantities, let us de?ne the quark correlation matrix with a gluon insertion (see Fig. 12) ˜ ˜ 4 ˜ P; S) = d d 4 z eik· AAij (k; k; ei(k−k)·z PS |=N j (0)gA (z)=i ()|PS : (4.9.6) Note that in the diagram of Fig. 12 the momenta of the quarks on the left and on the right of the unitarity cut are di,erent. We call x and y the two momentum fractions, i.e., k = xP; k˜ = yP (4.9.7) and integrate (4.9.6) over k and k˜ with the constraints (4.9.7) + d4 k d 4 k˜ ˜ AAij (x; y) = AA (k; k; P; S)(x − k + =P + )(y − k˜ =P + ) 4 4 (2) (2) dE dJ iEy iJ(x−y) = PS |=N j (0)gA (Jn)=i (En)|PS ; (4.9.8) e e 2 2 where in the last equality we set E = P + − and J = P + z − , and n is the usual Sudakov vector. If a further integration over y is performed, one obtains a quark–quark–gluon correlation matrix where one of the quark ?elds and the gluon ?eld are evaluated at the same space–time

44

V. Barone et al. / Physics Reports 359 (2002) 1–168 Table 2 The quark distributions at twist 2 and 3a Quark distributions S

0

L

T

Twist 2

f(x)

Lf(x)

LT f(x)

Twist 3 (∗)

e(x) h(x)

hL (x) eL (x)

gT (x) fT (x)

a

S denotes the polarisation state of the parent hadron (0 indicates unpolarised). The asterisk indicates T -odd quantities.

point: AAij (x) =

dE iEx e PS |=N j (0)gA (En)=i (En)|PS : 2

(4.9.9)

The matrix AA (x; y) makes its appearance in the calculation of the hadronic tensor at the twist-three level. It contains four multiparton distributions GA , G˜ A , HA and EA ; and has the following structure: M AA (x; y) = {iGA (x; y) ⊥ S⊥ P= + G˜ A (x; y)S⊥ 5 P= 2 +HA (x; y)5N 5 ⊥ P= + EA (x; y) ⊥ P= }:

(4.9.10)

Time-reversal invariance implies that GA , G˜ A , HA and EA are real functions. By hermiticity G˜ A and HA are symmetric whereas GA and EA are antisymmetric under interchange of x and y. ˜ are indeed related to GA , G˜ A , HA and EA , in particular It turns out that g˜T (x); h˜L (x) and e(x) to the integrals over y of these functions. One ?nds, in fact, [46] xg˜T (x) = dy [GA (x; y) + G˜ A (x; y)]; (4.9.11a) ˜ (4.9.11b) xhL (x) = 2 dy HA (x; y); (4.9.11c) xe(x) ˜ = 2 dy EA (x; y): For future reference we give in conclusion the T -odd twist-three quark–quark correlation matrix, which contains three more distribution functions [16,47] M i A(x)|T -odd = fT (x) ⊥ S⊥ − i5N eL (x) 5 + h(x)[p=; n=] : (4.9.12) 2 2 We shall ?nd these distributions again in Section 7.3.1. The quark (and antiquark) distribution functions at leading twist and twist-three are collected in Table 2.


45

4.10. Sum rules for the transversity distributions A noteworthy relation between the twist-three distribution hL and LT f, arising from Lorentz covariance, is [48] d ⊥(1) hL (x) = LT f(x) − (x); (4.10.1) h d x 1L where h⊥(1) 1L (x) has been de?ned in (4.9.5a). Combining (4.10.1) with (4.9.4b) and solving for we obtain [14] (quark mass terms are neglected) h⊥(1) 1L 1 1 dy dy ˜ ˜ LT f(y) + hL (x) − 2x (4.10.2) h (y): hL (x) = 2x 2 2 L y x x y On the other hand, solving for hL leads to

h⊥(1) 3 d 1L = xLT f(x) − xh˜L (x): (4.10.3) x dx x2 If the twist 3 interaction dependent distribution h˜L (x) is set to zero one obtains from (4.10.2) 1 dy LT f(y) (4.10.4) hL (x) = 2x 2 x y and from (4.10.3) and (4.10.1) 1 1 dy ⊥(1) 2 LT f(y): (4.10.5) h1L (x) = − xhL (x) = − x 2 2 x y Eq. (4.10.4) is the transversity analogue of the Wandzura–Wilczek (WW) sum rule [49] for the g1 and g2 structure functions, which reads 1 dy WW 2 2 (4.10.6) g2 (x; Q ) = − g1 (x; Q ) + g1 (y; Q2 ); x y where g2WW is the twist-two part of g2 . In partonic terms, in fact, the WW sum rule can be rewritten as (see (3.4.7)) 1 dy gT (x) = Lf(y); (4.10.7) x y which is analogous to (4.10.5) and can be derived from (4.9.4a) and from a relation for gT (x) similar to (4.10.1). 5. Transversity distributions in quantum chromodynamics As well-known, the principal e,ect of QCD on the naYZve parton model is to induce, via renormalisation, a (logarithmic) dependence on Q2 [50 –52], the energy scale at which the distributions are de?ned or (in other words) the resolution with which they are measured. The two techniques with which we shall exemplify the following discussions of this dependence and of the general calculational framework are the renormalisation group equations [53,54] (RGE)

46


applied to the operator–product expansion [55] (OPE) (providing a solid formal basis) and the ladder-diagram summation approach [56,57] (providing a physically more intuitive picture). The variation of the distributions as a function of energy scale may be expressed in the form of the standard DGLAP so-called evolution equations. Further consequences of higher-order QCD are mixing and, beyond the leading logarithmic (LL) approximation, eventual scheme ambiguity in the de?nitions of the various quark and gluon distributions; i.e., the densities lose their precise meaning in terms of real physical probability and require further conventional de?nition. In this section we shall examine the Q2 evolution of the transversity distribution at LO and NLO. In particular, we shall compare its evolution with that of both the unpolarised and helicity-dependent distributions. Such a comparison is especially relevant to the so-called So,er inequality [38], which involves all three types of distribution. The section closes with a detailed examination of the question of parton density positivity and the generalised so-called positivity bounds (of which So,er’s is then just one example). 5.1. The renormalisation group equations In order to establish our conventions for the de?nition of operators and their renormalisation, etc., it will be useful to briePy recall the RGE as applied to the OPE in QCD. Before doing so let us make two remarks related to the problem of scheme dependence. Firstly, in order to lighten the notation, where applicable and unless otherwise stated, all expressions will be understood to refer to the so-called minimal modi?ed subtraction (MS) scheme. A further complication that arises in the case of polarisation at NLO is the extension of 5 to d = 4 dimensions [58– 60]. An in-depth discussion of this problem is beyond the scope of the present review and the interested reader is referred to [61], where it is also considered in the context of transverse polarisation. For a generic composite operator O, the scale independent so-called bare (OB ) and renormalised (O(2 )) operators are related via a renormalisation constant Z(2 ), where is then the renormalisation scale: O(2 ) = Z −1 (2 )OB :

(5.1.1)

The scale dependence of O(2 ) is obtained by solving the RGE, which expressed in terms of the QCD coupling constant, /s = g2 =4, is 2

@O(2 ) + (/s (2 ))O(2 ) = 0; @2

(5.1.2)

where (/s (2 )), the anomalous dimension for the operator O(2 ), is de?ned by (/s (2 )) = 2

@ ln Z(2 ): @2

The formal solution is simply 2 2 O(Q ) = O( ) exp −

/s (Q2 )

/s (2 )

(5.1.3) (/s ) d/s ; eld manifolds. In the non-perturbative supersymmetry approach at hand, we are not dealing with such types of :eld manifold. This is not an academic point, as is shown by counterexamples: there exist cases, namely two-dimensional systems in class D or DIII, where the naive application of the Mermin–Wagner–Coleman theorem leads to the erroneous prediction of a vanishing DoS. (The non-linear sigma model for weakly disordered 2d systems in these classes predicts a zero-energy DoS which diverges in the thermodynamic limit [47,49].) However, for the systems in the classes AIII, CI, and C studied in the present paper, the inclusion of the Goldstone modes indeed leads to a vanishing DoS. It is very instructive to explore the phenomenon in the zero-dimensional case (i.e. the case of ergodic systems, where spatial Juctuations of the Goldstone modes are frozen out). The reason why d = 0 is a good case to study is that the zero-dimensional non-linear sigma model integral can be performed explicitly (c.f., e.g., Ref. [84]). It turns out that the results, including the vanishing of the DoS at zero energy, agree with the phenomenological random-matrix theory approach to systems of class C, CI, and AIII [34,63,64]. Moreover, the simplicity of the 0d-case makes it particularly straightforward to deduce what is missing in the diagrammatic approach. After all, it cannot be that perturbative diagrammatic approaches are completely oblivious to the existence of the Goldstone modes mentioned above. In fact, it has been shown in Ref. [34] that relevant classes of diagrams are missed within the SCBA approach to computing the Green function. “Relevant” here means that the diagrams in question diverge as the energy approaches zero. This behaviour is indicative of the fact that these diagram classes represent the perturbative implementation of the Goldstone modes discussed above (very much like the standard diFusion mode is the :rst-order perturbative contribution to the Goldstone modes induced by the spontaneous breaking of the symmetry between retarded and advanced Green functions for disordered metals). More recent diagrammatic approaches [86] have included these modes into the perturbative theory of the disordered d-wave superconductor. In intermediate energy regimes, where the IR singularity of the Goldstone modes has not yet become virulent, the extended diagrammatic formulation represents an alternative to the :eld-theoretical approach. Summarizing, we :nd that (i) the inclusion of Goldstone modes is crucial for a correct description of the quasi-particles of a d-wave superconductor (in particular its DoS), (ii) that within diagrammatics these modes are represented by singular diagram classes which (iii) can be brought under control in intermediate, but not in the lowest energy regimes. Finally, we comment on work where a non-vanishing zero-energy DoS has been obtained in a non-perturbative manner. In particular, Lee [2] considered the DoS on the background

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A. Altland et al. / Physics Reports 359 (2002) 283–354

of an ensemble of impurities at the unitarity limit. The strength of these impurities makes a comparison with our analysis diScult. (The construction of the :eld theory crucially relies on the existence of a parametric separation between the energetic extension of the nodal region and the much smaller width of the disorder distribution. See however, Note added in proof and Ref. [96].) Another type of non-perturbative approach has been put forward by Ziegler et al. [4]. In their work, a non-vanishing DoS was obtained on the basis of rigorous estimates applied to the single-particle Green function. Upon close inspection, however, their line of reasoning can be dismissed by its lack of relevance for superconductors. Indeed, the Hamiltonian they consider is extremely special and non-generic in class CI. It is formulated on a lattice and, in the notation of Ref. [4], reads d

H = − (∇2 + ) 3 + "ˆ 1 ; d where the kinetic energy −∇2 and the non-local d-wave order parameter "ˆ are taken to act by

(∇2 )(r) = (r + 2e1 ) + (r − 2e1 ) + (r + 2e2 ) + (r − 2e2 ) ; d

("ˆ

)(r) = "( (r + e1 ) + (r − e1 ) − (r + e2 ) − (r − e2 ))

and is a random chemical potential. The choice for ∇2 has the arti:cial feature that it allows hopping only between third-nearest neighbours, which leads to a conservation law alien to superconductors: H commutes with the operator D3 , where Dr; r = (−1)x1 +x2 *r; r multiplies by plus one on the sites of an A sublattice (x1 + x2 even) and by minus one on the sites of the B sublattice (x1 + x2 odd). Since D3 has two eigenvalues ±1, the Hilbert space decomposes into two sectors not coupled by H . The :rst one consists of all particle states on the A sublattice and hole states on the B sublattice, while the second sector contains all the complementary states. Without loss of generality we may restrict the Hamiltonian to one sector. The particle–hole degree of freedom then becomes redundant—we can make a particle–hole transformation on the B sites, so that the :rst sector becomes all particles and the second sector all holes—and the Hamiltonian reduces to d H˜ = − (∇2 + )|A + (∇2 + )|B + "ˆ ;

where—(∇2 + )|A denotes the restriction of—(∇2 + ) to the A sublattice. H˜ is a generalized discrete Laplacian augmented by a random potential. It belongs to the symmetry class AI, the Wigner–Dyson class with < = 1, which is to say that the Hamiltonian matrix is real symmetric, and the large-scale behaviour of its two-particle Green’s function is controlled by the cooperon and diFusion modes well known from the theory of disordered metals. There exist no quantum interference modes aFecting the single-particle Green’s function. Thus the quasi-particle Hamiltonian of Ziegler, Hettler and Hirschfeld models a time-reversal invariant normal metal rather than a superconductor! As one would expect, it can be proved [4] that such a Hamiltonian has a non-zero density of states at E = 0 for a wide class of distributions of the random potential . However, this result tells us nothing about a disordered d-wave superconductor. The built-in D3 conservation law eliminates all the modes of quantum interference that are characteristic of the superconductor and act to suppress the density of states at zero energy. As a corollary, we


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conclude that the ZHH model provides no test of the accuracy of the self-consistent T -matrix approximation widely used for superconductors. To preclude any misunderstanding, let us emphasize that the ZHH model is built on a superJuid condensate, which may of course exhibit the Meissner eFect and other phenomena associated with superconductivity. However, the issue at hand is not the nature of charge transport by the superconducting condensate. As was emphasized in Section 1, we are not studying the condensate, but rather its quasi-particle excitations, their spectral statistics and their transport properties, which can be probed experimentally via spin or thermal transport. From this perspective the ZHH Hamiltonian, albeit built on a superconducting ground state, models a normal metal. (More precisely, it gives the behaviour of a thermal insulator, since quantum interference eFects ultimately drive the model to strong localization in two dimensions.)

6. Numerical analyses of the quasi-particle spectrum Much of the early work on quasi-particles in disordered d-wave superconductors was analytical. More recently, a number of numerical investigations exploring the eFects of disorder scattering appeared (see, e.g., Refs. [20,21,85]). Taking the soft and hard scattering regimes in turn, the present section reviews elements of these works, and relates them to the results discussed in previous sections. 6.1. Hard scattering A comprehensive analysis of quasi-particle spectra in time-reversal invariant d-wave superconductors with point-like scatterers (symmetry class CI) appeared in Refs. [20]. Going beyond the mere diagonalization of the lattice Hamiltonian (3), these papers determined the order parameter self-consistently. Moreover, the role of a nesting symmetry of particle-hole type, which is present in the case of a half-:lled band and refers to momentum transfers q = (± =a; ± =a), was explored. Without going into quantitative detail, the main results of these papers can be summarized as follows: • The self-consistent T -matrix approximation fails to correctly describe the DoS below a certain

energy scale, the value of which increases with disorder.

• At zero energy the DoS vanishes in all cases but the extreme one of scatterers at the unitarity

limit. In that particular case, spectral weight accumulates at the band center, reJecting the creation of impurity bound states. For the special limit of zero chemical potential, corresponding to fully realized particle-hole nesting symmetry, the low-energy DoS diverges logarithmically in accord with the analysis of Lee and PVepin [17]. • Away from zero energy (and for generic scatterers), a regime of linearly increasing DoS, tentatively identi:ed as the DoS pro:le predicted by Senthil and Fisher [13], is observed. We must caution, however, that this identi:cation does not convince us for weakly disordered systems: as discussed earlier, the linear suppression appears in an insulating phase of separated localization volumes. Given the large size of the localization length in weakly disordered

336


Fig. 4. Density of states for " = 1, correlation length 0 = 2, and disorder strengths W = 0; 1; : : : ; 8; 10 (bottom to top at E = 0). All energies are measured in units of the hopping matrix element. The system size is L = 33 and the level broadening (introduced so as to suppress oscillations on the scale of the mean level spacing) = 0:05. The inset shows the same data on a smaller scale with = 0:0005. The :nite DoS at E = 0 is due to the :nite level broadening . Fig. 5. Double logarithmic plot of the density of states of Fig. 4. Disorder ranges from W = 1–10. Dots (•) represent data and lines power law :ts in the respective intervals. Inset: density of states for W = 2 and L = 15 (dotted), 25 (short-dashed), 35 (long-dashed), and 45 (solid). Note that the numerical uncertainties are considerably smaller than the amplitude of the Juctuations.

two-dimensional systems, it is not clear whether the separation of characteristic length scales can be realized on lattice sizes accessible to numerical computation. • Self-consistency leads to a further suppression of the DoS, in particular in systems with binary-alloy type scatterers close to the unitarity limit. • As was explained in Ref. [86], the nesting symmetry is of little relevance except for the case of unitary scatterers. 6.2. Soft scattering The quasi-particle spectrum of time-reversal invariant d-wave superconductors with soft scatterers (class AIII) has been investigated numerically in Ref. [85]. We will now review the results of that work in some detail. The starting point was the usual lattice Hamiltonian de:ned in Eq. (2), but with the assumption of long-range correlated disorder so as to stay within the soft-scattering regime. Speci:cally, the on-site disorder potential was de:ned by |ri − rj |2 W i = √ fj exp − ; 02 j where = j exp(−2|rj2 |=02 ), and fj were independent random variables drawn from a uniform distribution on [ − 12 ; 12 ]. The results are shown in Fig. 4.


337

Fig. 6. Exponents extracted from the :tted curves in Fig. 5 as a function of the disorder strength W for " = 1. The solid curve is the result of NTW, Eq. (33). Fig. 7. Density of states for values of the order parameter " = 0:1–1.0 (top to bottom) and disorder strength W = 3. Each curve is shifted by a factor of 1.2 for clarity.

The suppression of inter-node scattering by smoothing the potential is very strong: for a potential correlated over just two lattice spacings (0 = 2), the inter-node scattering matrix elements are reduced by a factor of about 10−8 as compared to the intra-node matrix elements. Although such matrix elements will become relevant at very large scales, it is expected that potentials with 0 ¿ 2 place small systems of size up to 100 × 100 :rmly inside the pure symmetry class AIII. In practice, this means that each of the low-energy sectors associated with the four nodes in the dispersion relation is described by Dirac fermions subject to pure gauge disorder. Fig. 4 shows the DoS of the system for various disorder strengths. The quantitative analysis of the spectral data shows that three diFerent regimes can be distinguished: (i) for low energies, ¡ Emin , the structure of the spectrum is dominated by :nite size eFects. (For the lattice analysed in Ref. [85] Emin E0 =10 where E0 denotes the total width of spectrum.) The most apparent of these eFects is a disorder and system size dependent bump in the DoS. At very low energies, the DoS vanishes, as is seen in the high resolution inset of Fig. 4. (ii) For high energies, ¿ Emax ≈ E0 =2, the structure of the spectrum depends on non-universal lattice eFects. (iii) Most interesting is the intermediate regime, Emin ¡ ¡ Emax . In this region, the energy-dependent DoS exhibits power law behaviour (cf. Fig. 5). Of course, an energy window of width Emax =Emin ∼ 5 provides a rather poor statistical basis for establishing power law behaviour. Nevertheless, the procedure seems justi:ed as it is not just one power law with a single exponent but rather a two-parameter family of exponents (W; ) that is analysed. Here W measures the strength of the disorder while = t=" is the anistropy parameter. In Fig. 6, the exponents obtained by :tting the DoS determined numerically are compared with the NTW prediction (33) for the isotropic case = 1. Speci:cally, for the present system, L(E) ∼ |E | ;

=

1 − 2g= ; 1 + 2g=

g=

W2 : 32[t

338


Fig. 8. Exponents for W = 3 as a function of the order parameter ". The solid curve is the result of NTW, Eq. (33).

Similarly, Fig. 7 displays exponents obtained for :xed disorder strength but diFerent anisotropy parameters. Notice that the comparison between the numerical data and the family of analytical exponents does not involve undetermined :t parameters. The applicability of the NTW scaling law is limited to small disorder strengths, g ¡ 1. In Ref. [87] it has been argued that for larger values of g, the DoS becomes energy-independent. This prediction is supported by the numerical data (Fig. 8). To summarize, numerical analysis of the quasi-particle spectra in d-wave superconductors reveals the need to distinguish between three diFerent types of disorder: scatterers at the unitarity limit, non-unitary point-like impurities, and soft scattering potentials. Although the comparison with analytical predictions is impeded by :nite size eFects, there exists reasonable agreement for each of these types. 7. Discussion This concludes our survey of the inJuence of disorder on the quasi-particle properties of disordered d-wave superconductors. Since we included the majority of the discussion in Section 1, we will limit our remarks to some key points: broadly speaking, the analysis above emphasized that the low-energy transport properties of the model d-wave system depend sensitively on the nature of the impurity potential. In two dimensions, a potential which is short-ranged in space places the system in the spin insulator phase, where all quasi-particle states are localized. On the other hand, a potential which contains only forward-scattering components leads to a marginally perturbed WZW theory in which the zero-energy quasi-particle states are critical, and the density of states vanishes as a power law. The low-energy theory in this case belongs to a one-parameter family of :xed points, each identi:ed through a diFerent value of the disorder-coupling strength. This is reJected in a low-energy density of states which varies with energy as a power law with an exponent that depends on the strength of disorder.


339

Experimentally, the relevant scattering phenomenology is likely to be somewhere in between the cases considered above: at intermediate energy scales, signatures of the critical theory may well be visible in thermal transport measurements, although the behaviour at very low energies must, ultimately, be that of the spin insulator. The collapse of the critical theory in the presence of strong disorder was attributed to a cancellation of the multi-valued WZW terms arising from the diFerent nodal sectors of the theory. In fact, such cancellations are guaranteed by symmetry to occur in lattice models which exhibit a Dirac-node structure at the Fermi level (such as the random -Jux model). In such lattice models, the nodes arise in pairs related by parity. Under the same parity transformation, the WZW term is mapped onto a partner with opposite sign. Therefore, when the :elds belonging to each nodal sector are locked by strong disorder, the diFerent WZW terms add and cancel pairwise. Finally, the formalism developed and investigated above is not entirely speci:c to d-wave superconductors. The global structures of the theory relied only on the existence of a Dirac-like spectrum of the clean system. We believe that the general scheme outlined above could be applied in the investigation of other model systems with a gapless linear density of states such as gapless semiconductors, and superJuids. Acknowledgements We would like to acknowledge useful discussions with Patrick Lee, Catherine PVepin, and Alexei Tsvelik. Furthermore, we are particularly grateful to Bodo Huckestein for providing access to the numerical data presented in Section 6. Note added in proof After the completion of this manuscript an extension of the :eld theory to the case of pointlike distributed disorder was derived [96]. While the new formalism agrees with the central result of PVepin and Lee [17]—a zero energy singularity of the DoS—it also shows that the formation of that singularity is a peculiarity of the half :lled system. Away from half :lling, the generic phenomenology of a d-wave superconductor of class CI system is observed. Appendix A. Gradient expansion and the chiral anomaly To complement our derivation of the eFective action for the soft-scattering limit from non-Abelian bosonization, we include here a derivation of the same action from the gradient expansion. The motivation is largely of pedagogical nature: to facilitate comparison with existing works in the literature on weakly disordered fermion systems, it is useful to present more than a single route to the construction of the critical theory. We also wish to point at some unexpected diSculties that are encountered in the standard approach to derive the low-energy eFective action of the d-wave superconductor (or for that matter any disordered relativistic fermion system). To be speci:c we will formulate the gradient expansion for a soft-scattering system that is time-reversal invariant (class AIII). The inclusion of perturbations driving the model to any

340


of the other three classes is straightforward. However, since this appendix mainly serves a pedagogical purpose, we will limit ourselves to the discussion of just one class. As discussed in Section 2, the independence of the nodes entails a decoupling of the low-energy theory into two X and (2; 2). X We discuss one speci:c sector, say (1; 1), X anticipating pairs of nodal sectors (1; 1) that the full theory can later be obtained by straightforward combination of both sub-sectors. X sector derived in Section 3 and given Our starting point is the soft-mode action for the (1; 1) by Eq. (22). Rearranging matrix blocks, the action can be brought into the simpler form %M −1 9 S[M ] = STr ln ; (A.1) 9X %M where the block decomposition is in PH-space and we have omitted the superscript (1) on the derivative operator for notational simplicity. To further simplify the notation, we have set the two characteristic velocities vi (i = 1; 2) temporarily to unity (i.e. v = 1, and = 1). (In fact, some authors attempt to get rid of these scales altogether by means of a coordinate rescaling xi → vi xi . However, as we are going to discuss below, this seemingly innocuous manipulation may lead to inconsistencies once the nodes are coupled. Moreover, the inJuence of such a rescaling on the unspeci:ed source components of the action must be treated with caution. We will therefore re-instate the scales vi towards the end of this section.) The bold-face notation STr means that we are taking the (super)trace over both superspace and Hilbert space. To compute a low-energy action from the above expression, we can follow one of at least three diFerent routes: • The most direct approach would be to introduce coordinates on the :eld manifold, say by M = eX , to expand around unity: M = 5 + X + · · ·, and then to derive a low-energy action for the X ’s via a straightforward gradient expansion. Owing to the overall GL(2|2) invariance

of the model, such an approach determines the low-energy action not just in the vicinity of unity but rather on the entire manifold. • Alternatively, as in the main body of the text, one may resort to an entirely symmetry-oriented approach and obtain the structure of the low-energy action by means of current algebra and non-Abelian bosonization. This was the route taken by NTW [5]. • Finally, a third option is not to introduce coordinates on the :eld manifold but to attempt a gradient expansion directly based on the original degrees of freedom M . While such schemes are standard in applications of non-linear sigma models to disordered metallic systems, we here run into diSculties, caused by the appearance of ill-de:ned momentum integrals. The way to overcome this problem is :rst to subject (A.1) to a UV-regularization scheme and only then to expand in the spatial Juctuations of the :elds. This will be our method of choice in this section. Its main advantages are that it is computationally eScient and better exposes the global structures of the theory than a coordinate-based approach does. A.1. Chiral anomaly Before subjecting the action functional to a gradient expansion, let us :rst make some pedagogical remarks. For the time being, let M be a :eld taking values in some matrix group G,


341

and let us consider the functional determinant Z[M ] = Det DM = eTr ln DM = e−S[M ] ; %M −1 9

DM =

9X

%M

:

(When G is a group of supermatrices, Det has to be replaced by SDet−1 .) Our goal is to expand ln(1= Z) in gradients to produce a low-energy eFective action for M . Now, if G is a group of unitary matrices M −1 = M † , the determinant is not real: −1† %M −9 X = Det Z −9X %M † %M −1 9X Z; = Det = 9 %M so ln Z has an imaginary part. On general :eld-theoretic grounds, we expect this imaginary part to be a multi-valued functional of WZW type. How can we compute Im ln Z? A natural idea is to “take the square root”: M ≡ T2 T1−1 , and manipulate the determinant as follows: %T1 T2−1 9 Det 9X %T2 T1−1 % T1−1 9T1 T1 0 T2−1 0 = Det : 0 T2 0 T1−1 T2−1 9XT2 % One might now be tempted to assume the multiplicativity of Det, which would lead to Z being equal to % T1−1 9T1 Z = Det : T2−1 9XT2 % Dropping the diagonal factors on the left and right seems especially innocuous when T1 and T2 are unitary. The motivation for trying to pass from Z to Z is that the latter can be computed exactly by a standard procedure (see, e.g., Ref. [88] and the next subsection) in the limit of small %. The result, lim ln Z [T1 ; T2 ] = + 2W [T1 T2−1 ]

%→0

is expressed by the celebrated WZW functional: 1 i[M ] W [M ] = d 2 r Tr 9 M −1 9 M + ; 16 24 [M ] = d 3 r G Tr M −1 9 MM −1 9 MM −1 9G M :

342


Note, however, the inequality Re W [M ] ¿ 0 for unitary M . Thus, if Z were equal to Z , the constant :elds M (r) = M0 would minimize rather than maximize the Boltzmann weight Z[M ]. We would then be forced to conclude that the :eld theory with action S[M ] = − ln Z[M ] is unstable with respect to spatial Juctuations and does not exist. By extending the argument to the supersymmetric setting, we would :nd the theory with action (A.1) to be sick. On the other hand, we know (e.g. from non-Abelian bosonization) that this is not the case, so there must be something wrong with the present argument. Where is the error? The answer is that the manipulation taking Z into Z disregards the existence of the notorious chiral anomaly and is correct only for T1 = T2 , the case of a pure gauge transformation. In other words, for a gauge transformation with an axial component (T1 = T2 ) the passage from Z to Z is accompanied by a Jacobian diFerent from unity. Indeed, for G = U(1); M = ei’ , straightforward application of the method of Abelian bosonization [89] gives 1 i’ −ln Z[e ] = d 2 r(9 ’)2 = ln Z [ei’ ] : 8 By analogy, we expect that also in the non-Abelian case, correct evaluation of Z[M ] yields a stable theory with the proper sign of the coupling. A safe way of computing the gradient expansion is to :rst UV regularize the Dirac operator DM after which axial gauge transformations can readily be performed. Returning to our original problem, we notice that a technically convenient way of regularizing in the ultraviolet is to add to the action (A.1) a term M −1 9 −STr ln ; 9X M which vanishes by supersymmetry when is taken to be a positive in:nitesimal ( → 0+). The resulting expression, −1 %M −1 9 M −1 9 S[M ] = STr ln 9X %M 9X M is indeed manifestly well-behaved in the ultraviolet. (The diFerence between M and %M becomes negligible for large eigenvalues of the Dirac operator, in which case the two matrix factors cancel each other and the action approaches zero.) Setting M = T2 T1−1 and using the cyclic invariance of the trace, we rewrite the action functional as % T1−1 9T1 T1−1 9T1 S = STr ln − STr ln : T2−1 9XT2 % T2−1 9XT2 Because % now acts as a mass, the low-energy limit of the theory is captured entirely by the second term. The :rst contribution becomes appreciable only for momenta larger than %, where it cancels the second term. Thus the role of the :rst contribution has been relegated to that of a UV regulator. We are, of course, at liberty to replace it by some other UV regularization scheme. Doing so and expanding the second term in gradients, we safely arrive at the WZW action, now with the correct overall sign. This will be demonstrated in more detail in the next subsection.


343

A.2. Heat kernel regularization For our purposes, it is convenient to use Schwinger’s proper-time regularization (see, e.g., Ref. [88] where the same procedure was applied to the non-Abelian Schwinger model, i.e. massless 1 + 1 dimensional Dirac fermions coupled to an SU(Nc ) gauge :eld). Without loss, we equate T ≡ T1 = T2−1 and rewrite the action in the form T −1 9T S = −STr ln ; T 9XT −1 = −STr ln(2 − T 9XT −2 9T ) ; where a UV cutoF at the momentum scale % is implied. The proper-time regularization scheme for an elliptic operator H is implemented by ∞ ds −Tr ln H = Tr e−sH : s 1= Notice that the lower integration bound 1= cuts oF the contributions to ln Det H from eigenvalues of H greater than and thus regularizes in the ultraviolet. Applying this scheme to our action (with cutoF = %2 ), we obtain ∞ 2 ds X −2 S= STr e−s( −T 9T 9T ) : 1=%2 s This expression is both UV and IR :nite and could in principle serve as the starting point for a gradient expansion. Much easier than the direct evaluation of S, however, is the evaluation of its variation *S. We will therefore proceed by varying S with respect to some parameter, ˙ and :nally we will reconstruct S by integrating S˙ with say t; then we will compute *S ≡ S, respect to t. Thus we consider some one-parameter family of :elds T (r; t) with T (r; 0) = 5 and T (r; 1) = T (r), and we diFerentiate with respect to t. This results in ∞ 2 X −2 ˙ S= ds STr(T˙ 9XT −2 9T + T 9XT −2 9T˙ − T 9X(T −1 T˙ T −2 + T −2 T˙ T −1 )9T )e−s( −T 9T 9T ) : 1=%2

We next use the cyclic invariance of the supertrace to convert the integrand into a total derivative with respect to the integration variable s: ∞ d 2 −1 2 X −1 X −2 S˙ = ds e− s STr(T −1 T˙ + T˙ T −1 ) (esT 9T 9T − esT 9T 9T ) : ds 1=%2 Performing the integral over s, setting the in:nitesimal to zero, and making the integration over real space ( d 2 r) explicit, we obtain the expression −2 −2 ˙ S = d 2 r STr(T −1 T˙ + T˙ T −1 )(r)r|e−% H1 − e−% H2 |r ;

344


where H1 = − T −1 9 ◦ T 2 9X ◦ T −1 ; H2 = − T 9X ◦ T −2 9 ◦ T and the symbol ◦ means composition of operators. Our next task is to compute the diagonal parts of the heat kernels r|e−sHi |r (i = 1; 2), for small values of the dimensionful parameter s = 1=%2 . This is a standard exercise in semiclassical analysis, and its solution can be found in textbooks [90]. Re-expressing H1 in the form H1 = − 14 (9 − iA )2 + B ; where the non-Abelian gauge potential A and :eld strength B are functions of T and its derivatives (which for brevity we do not specify here), we have the standard short-time expansion B(r) 1 r|e−sH1 |r = − + O(s) : s The same can be done for H2 instead of H1 . By taking the diFerence of the two expansions, we obtain 1 −2 −2 r|e−% H1 − e−% H2 |r = − ([T −1 9T; 9XTT −1 ] + 9(9XTT −1 ) + 9X(T −1 9T ))(r) + O(1=%2 ) : On dimensional grounds, the term O(1=%2 ) must involve four derivatives and therefore becomes negligible for wavelengths much larger than the short-distance cutoF 1=%. ˙ We then arrive at The above expansion is now substituted into the expression for S. 1 S˙ = − d 2 r STr(T −1 T˙ + T˙ T −1 )(9X(T −1 9T ) + 9(9XTT −1 ) + T −1 9T 9XTT −1 − 9XTT −2 9T ) : It is not hard to verify that, on making the identi:cation M = T −2 , this expression coincides with i 1 d 2 −1 ˙ S=− d r STr(9 M 9 M ) + d 2 r STr(9t MM −1 9 MM −1 9 MM −1 ) : 8 dt 4 Integrating over time and noticing that S = 01 S˙ dt, we obtain the WZW functional given in (30), with g = 0. Finally, we undo the rescaling made at the beginning of the calculation, and arrive at the anisotropic eFective action i 1 S[M; ] = d 2 r STr(−1 91 M −1 91 M + −1 92 M −1 92 M ) : [M ] − 12 8 X is obtained by exchanging coordinates The corresponding result for the other pair of nodes (2; 2) x1 ↔ x2 . Notice that this result has the peculiar feature of being independent of the disorder. In the isotropic case = 1, the model becomes completely universal in the sense that its coupling constants assume :xed values. By the arguments reviewed in Appendix B.2, this WZW model is equivalent to free relativistic fermions, i.e. our original model in the absence of disorder. But this raises the question where the information on the presence of impurities was lost. After all


345

it is hard to conceive that the disorder strength g should be reJected only in the value of the UV momentum cutoF % ∼ e−4t"=g . Our analysis using non-Abelian bosonization described in the main text reveals the fact that the action above should be supplemented by a current–current interaction with coupling given by g. This begs the question how the existence of this term got lost in the standard scheme of saddle-point approximation plus gradient expansion. The key to the answer of this question lies in the presence of massive modes in the model, a fact we have ignored thus far. Indeed, we had immediately reduced the Hubbard–Stratonovich :eld Q to its Goldstone-mode content, Q → i%M . In fact, however, the :eld Q also contains massive modes, i.e. modes P which are not compatible with the chiral symmetry of the Hamiltonian and, therefore, Juctuate at a :nite energy cost. One way of handling the situation would be to put Q = i%PM , where the P’s are not just set to unity but integrated out. This procedure results in the appearance of an additional current–current interaction STr(M −1 9XM ) STr(M 9M −1 ) : Unfortunately, it turns out to be impossible to reliably determine the value of the coupling constant of this perturbation within the standard scheme: to obtain the additional term, one has to integrate out massive Juctuations, and yet the mass gap characterizing the modes P does not suSce to justify a Gaussian approximation to this integral. The reason for the last fact is that the disorder-generated perturbation of the :eld theory is strictly RG marginal, which means that there is no dynamically generated mass scale in the problem. Thus, no intrinsic mechanism stabilizing the Gaussian approximation exists, Juctuations are important, and to determine the coupling constant, the P-integral must be performed exactly. It goes without saying that this is diScult to do in practice. Ultimately, it is this de:ciency of the standard scheme which forces us to build our theory on the less standard approach of non-Abelian bosonization. Appendix B. Dirac fermions in a random vector potential As was reviewed in the main text, the low-energy quasi-particles of a dirty d-wave superconductor behave, in the single-node approximation, as Dirac fermions in a random vector potential. We have argued that non-Abelian bosonization takes that theory into a super-symmetric WZW model, from which we rederived the critical exponent for the density of states. In the present appendix, we elaborate on this issue and supply more of the technical details. B.1. WZW model of type A|A Let us begin by establishing the general context with a few remarks. The characteristic feature of a WZW model is the multi-valued term [M ]. Multi-valued functionals of this type were studied in the context of Hamiltonian mechanics by Novikov [91], and became :rmly established in :eld theory through Witten’s celebrated paper [38] on non-Abelian bosonization. Ever since, WZW models de:ned on compact groups have been an object of intense study. There exists a vast amount of literature on them, and they have been solved in great detail. To a much lesser extent, :eld theorists have also studied WZW models of the non-compact type.

346


The target spaces of these models are not groups but are non-compact symmetric spaces, the simplest example being SL(2; C)=SU(2). One of the rare occurrences of such a model is found in [92]. The target space of the WZW model with which we will be concerned with, transcends the classical setting in that it is a superspace. It turns out that the proper mathematical construction of a WZW model with superspace target is a delicate matter. Indeed, to de:ne a functional integral on Euclidean space-time, one needs a target space with a Riemannian structure, providing for an action functional that is bounded from below. Sadly, the invariant geometry on supergroups, such as GLC (n|n), U(n|n), GLR (n|n), OSpR (2n|2n) etc., or even on symmetric quotients such as GLC (n|n)=U(n|n), is never Riemannian, but always of inde:nite signature. Therefore, WZW models, and non-linear sigma models in general, do not exist on supergroups, at least not in the literal sense (i.e. without some procedure of analytic continuation of the :elds). One can easily appreciate this point by looking, for example, at the complex Lie supergroup GLC (1|1), with the standard (bi-)invariant metric given by % = − STr dg−1 dg, g ∈ GLC (1|1). To understand the properties of this metric tensor, it suSces to examine it on the tangent space at the group unit, which is the complex Lie algebra glC (1|1). Elements of this Lie algebra are written as a X= ;

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