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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 The´orique, Ecole Normale Superieure, 24 rue Lhomond, 75231 Paris Cedex, France. E-mail: [email protected] G.E. BROWN (Nuclear physics), Institute for Theoretical Physics, State University of New York at Stony Brook, Stony Brook, NY 11974, USA. E-mail: [email protected] D.K. CAMPBELL (Non-linear dynamics), Dean, College of Engineering, Boston University, 44 Cummington Street, Boston, MA 02215, USA. E-mail: [email protected] G. COMSA (Surfaces and thin ﬁlms), Institut fur . Physikalische und Theoretische Chemie, Universit.at Bonn, Wegelerstrasse 12, D-53115 Bonn, Germany. E-mail: [email protected] J. EICHLER (Atomic and molecular physics), Hahn-Meitner-Institut Berlin, Abteilung Theoretische Physik, Glienicker Strasse 100, 14109 Berlin, Germany. E-mail: [email protected] M.P. KAMIONKOWSKI (Astrophysics), Theoretical Astrophysics 130-33, California Institute of Technology, 1200 East California Blvd., Pasadena, CA 91125, USA. E-mail: [email protected]

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Instructions to authors

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

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

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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.

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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.

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

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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 rˆole 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

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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.

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

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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)

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

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)

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

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)

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

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)

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

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

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

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

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

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)

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

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)

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

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 <; sin / sin <; cos /):

(3.1.18)

Taking the direction of the incoming lepton to be the z-axis, we have ‘ = E(1; 0; 0; 1); ‘ = E (1; sin # cos ’; sin # sin ’; cos #):

(3.1.19)

Inserting (3.1.8b) and (3.1.12b) into Eq. (3.1.17), with the above parametrisations for the spin and the momentum four-vectors, for the cross-section asymmetry in the target rest frame we

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

17

Fig. 3. Lepton and spin planes. The lepton plane is taken here to coincide with the xz plane, i.e., ’ = 0.

now obtain 2 E d(+S) d(−S) 4/em − − {[E cos / + E (sin # sin / cos = + cos # cos /)]MG1 = dE d: dE d: Q2 E

+ 2EE [sin # sin / cos = + cos # cos / − cos /]G2 };

(3.1.20)

where = = < − ’ is the azimuthal angle between the lepton plane and the (‘ˆ; Sˆ ) plane. (See Fig. 3.) In particular, when the target nucleon is longitudinally polarised (that is, polarised along the incoming lepton direction), one has / = 0 and the spin asymmetry becomes 2 E d⇒ d⇐ 4/em − − = [(E + E cos #)MG1 − Q2 G2 ]: dE d: dE d: Q2 E

(3.1.21a)

When the target nucleon is transversely polarised (that is, polarised orthogonally to the incoming lepton direction), one has / = =2 and the spin asymmetry is 2 E 2 d⇑ d⇓ 4/em − − = sin #[MG1 + 2EG2 ]: dE d: dE d: Q2 E

(3.1.21b)

A remark on the terminology is in order here. The terms “longitudinal” and “transverse” are somewhat ambiguous, insofar as a reference axis is not speci?ed. From an experimental point of view, the “longitudinal” and “transverse” polarisations of the nucleon are in reference to the lepton beam axis. Thus “longitudinal” (“transverse”) indicates parallel (orthogonal) to this axis. We use the large arrows ⇒ and ⇑ to denote these two cases, respectively. From a theoretical point of view, it is simpler to refer to the direction of motion of the virtual photon. One then speaks of the “longitudinal” (S|| ) and “transverse” (S⊥ ) spin of the nucleon, meaning by this spin parallel and perpendicular, respectively, to the photon axis. When the target is “longitudinally” or “transversely” polarised in this sense, we shall make explicit reference to S|| and S⊥ in the cross-section. Later, it will be shown how to pass from d⇒ − d⇐ and d⇑ − d⇓ to d(+S|| ) − d(−S|| ) and d(+S⊥ ) − d(−S⊥ ). Note that, in general, d⇒ is a combination of d(S|| ) and d(S⊥ ). We shall return to this point in Section 9.2.

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

It is customary to introduce the dimensionless structure functions F1 (x; Q2 ) ≡ MW1 (; Q2 ); F2 (x; Q2 ) ≡ W2 (; Q2 ); g1 (x; Q2 ) ≡ M 2 G1 (; Q2 ); g2 (x; Q2 ) ≡ M2 G2 (; Q2 ):

(3.1.22)

In the Bjorken limit Q2 ?xed; (3.1.23) 2M F1 , F2 , g1 and g2 are expected to scale approximately, that is, to depend only on x. In terms of F1 , F2 , g1 and g2 , the hadronic tensor reads q q 2 P·q P·q (S) 2 P − 2 q P − 2 q F2 (x; Q2 ); W = 2 −g + 2 F1 (x; Q ) + q P·q q q (3.1.24a) 2M ! q! S ·q (A) 2 2 W = S g1 (x; Q ) + S − (3.1.24b) P g2 (x; Q ) : P·q P·q ; Q2 → ∞;

x=

The unpolarised cross-section then becomes (as a function of x and y) 2 s xym2N d 4/em 2 2 2 xy F1 (x; Q ) + 1 − y − F2 (x; Q ) ; = d x dy Q4 s

(3.1.25)

whereas the spin asymmetry (3.1.20), in terms of g1 and g2 , takes on the form d(+S) d(−S) − d x dy d’ d x dy d’ 2 2Mx 4/em 2 2 2 1 − y[yg1 (x; Q ) + 2g2 (x; Q )] sin / cos = ; = 2 (2 − y)g1 (x; Q ) cos / + Q Q (3.1.26) where we have neglected contributions of order M 2 =Q2 . Note that the term containing g2 is suppressed by one power of Q. This makes the measurement of g2 quite a diKcult task. It is useful at this point to re-express the cross-section asymmetry (3.1.26) in terms of the angle > between the spin of the nucleon S and the photon momentum q = l − l . The relation between /, = and >, 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

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

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

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

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)

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

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)

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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 rˆole 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)

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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

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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.

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

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)

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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.

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

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)

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

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 rˆole 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.

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

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

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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

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

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

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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

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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)

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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)

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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⊥ ).

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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

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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)

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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

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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)

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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)

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⊥ , 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.

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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)

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

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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

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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.

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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)

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

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 ; <(/s )

(5.1.4)

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47

where <(/s ) is the RGE function governing the renormalisation scale dependence of the e,ective QCD coupling constant /s (2 ): <(/s ) = 2

@/s (2 ) : @2

(5.1.5)

At NLO the anomalous dimension (2 ) and the QCD <-function <(/s ) can then be expanded perturbatively as /s (0) /s 2 (1) (/s ) = + + O(/s3 ); (5.1.6) 2 2 / 2 /s s <1 + O(/s3 ) : (5.1.7) <(/s ) = − /s <0 + 4 4 4 2 2 2 The ?rst two coeKcients of the <-function are: <0 = 11 3 CG − 3 TF = 11 − 3 Nf and <1 = 3 (17CG − 2 10CG TF − 6CF TF ) = 102 − 38 3 Nf , where CG = Nc and CF = (Nc − 1)=2Nc are the usual Casimirs related, respectively, to the gluon and the fermion representations of the colour symmetry group, SU(Nc ), and TF = 12 Nf , for active quark Pavour number Nf . This leads to the following NLO expression for the QCD coupling constant: /s (Q2 ) 1 <1 ln ln Q2 =2 1− 2 ; (5.1.8) 4 <0 ln Q2 =2 <0 ln Q2 =2

where is the QCD scale parameter. A generic observable derived from the operator O may be de?ned by f(Q2 )∼PS |O(Q2 )|PS . Inserting the above expansions into (5.1.4), the NLO evolution equation for f(Q2 ) = O is then obtained (note that the equations apply directly to O if it already represents a physical observable): −2 (0) =<0 −(4 (1) =<1 −2 (0) =<0 ) f(Q2 ) /s (Q2 ) <0 + <1 /s (Q2 )=4 ; = f(2 ) /s (2 ) <0 + <1 /s (2 )=4

(5.1.9)

which, to NLO accuracy, may be expanded thus

/s (Q2 ) /s (2 )

−2 (0) =<0

/s (2 ) − /s (Q2 ) (1) <1 (0) 1+ − : <0 2<0

(5.1.9 )

In order to obtain physical hadronic cross-sections at NLO level, the NLO contribution to f has then to be combined with the NLO contribution to the relevant hard partonic cross-section; indeed, it is only this combination that is fully scheme independent. It turns out that the quantities typically measured experimentally (i.e., cross-sections, DIS structure functions or, more simply, quark distributions) are related via Mellin-moment transforms to expectation values of composite quark and gluon ?eld operators. The de?nition we adopt for the Mellin transform of structure functions, anomalous dimension, etc. is as

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

follows: 5 ; 6 f(n) =

0

1

d x xn−1 f(x):

(5.1.10)

We may also de?ne the Altarelli–Parisi [12] (AP) splitting function, P(x), as precisely the function of which the Mellin moments, Eq. (5.1.10), are just the anomalous dimensions, (n). Note that P(x) may be expanded in powers of the QCD coupling constant in a manner analogous to and therefore also depends on Q2 . In x-space the evolution equations may be written in the following schematic form: d f(x; Q2 ) = P(x; Q2 ) ⊗ f(x; Q2 ); (5.1.11) d ln Q2 where the symbol ⊗ stands for a convolution in x, 1 dy x g(x) ⊗ f(x) = f(y); g y y x

(5.1.12)

which becomes a simple product in Mellin-moment space. With these expressions it is then possible to perform numerical evolution either via direct integration of (5.1.11) using suitable parametric forms to ?t data, or in the form of (5:1:9 ) via inversion of the Mellin moments. The operators governing the twist-two 7 evolution of moments of the f1 ; g1 and h1 structure functions (in this section we shall use f1 , g1 and h1 to generically indicated unpolarised, helicity and transversity weighted parton densities respectively) are 8 Of1 (n) = S N 1 iD2 : : : iDn ;

(5.1.13a)

Og1 (n) = S N 5 1 iD2 : : : iDn ;

(5.1.13b)

Oh1 (n) = S N 5 1 2 iD3 : : : iDn :

(5.1.13c)

where the symbol S conventionally indicates symmetrisation over the indices 1 ; 2 ; : : : ; n while the symbol S indicates simultaneous antisymmetrisation over the indices 1 and 2 and symmetrisation over the indices 2 ; 3 ; : : : ; n . 5.2. QCD evolution at leading order The Q2 evolution coeKcients for f1 and g1 at LO have long been known while the LO Q2 evolution for h1 was ?rst speci?cally presented in [18]. However, the ?rst calculations of the one-loop anomalous dimensions related to the operators governing the evolution of LT q(x; Q2 ) date back, in fact, to early (though incomplete) work on the evolution of the transverse-spin DIS 5

The de?nition with n − 1 replaced by n is also found in the literature. We choose to write n as an argument to avoid confusion with the label indicating perturbation order. 7 There are, of course, possible higher-twist contributions too, but we shall ignore these here. 8 All composite operators appearing herein are to be considered implicitly traceless. 6

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

49

Fig. 13. Example one-loop diagrams contributing to the O(/s ) anomalous dimensions of LT q.

structure function g2 [62], which, albeit in an indirect manner, involves the operators of interest here. Mention (though again incomplete) may also be found in [63]. Following this, and with various approaches, the complete derivation of the complex system of evolution equations for the twist-three operators governing g2 was presented [35,64,65]. Among the operators mixing with the leading contributions one ?nds the following: O(n) = S m N 5 1 2 iD3 : : : iDn ;

(5.2.1)

where m is the (current) quark mass. It is immediately apparent that this is none other than the twist-two operator responsible for LT q(x), multiplied here however by a quark mass and thereby rendered twist three—its evolution is, of course, identical to the twist-two version. For reasons already mentioned, see for example Eq. (4.5.9) and the discussion following, calculation of the anomalous dimensions governing LT q(x) turns out to be surprisingly simpler than for the other twist-two structure functions, owing to its peculiar chiral-odd properties. Indeed, as we have seen, the gluon ?eld cannot contribute at LO in the case of spin-half hadrons as it would require helicity Pip of two units in the corresponding hadron–parton amplitude. Thus, in the case of baryons the evolution is of a purely non-singlet type. Note that this is no longer the case for targets of spin greater than one half and, as pointed out in [18], a separate contribution due to linear gluon polarisation is possible; we shall also consider the situation for spin-1 mesons and=or indeed photons in what follows. In dimensional regularisation (d = 4 − j dimensions) calculation of the anomalous dimensions requires evaluation of the 1= j poles in the diagrams depicted in Fig. 13 (recall that at this order there is no scheme dependence). Although not present in baryon scattering, the linearly polarised gluon distribution, LT g, can contribute to scattering involving polarised spin-1 mesons. Thus, to complete the discussion of leading-order evolution, we include here too the anomalous dimensions for this density. For the four cases that are diagonal in parton type (spin-averaged and helicity-weighted [12]; transversity and linear gluon polarisation [18]) one ?nds in Mellin-moment space:   n 3 1 1 qq (n) = CF  + −2 ; (5.2.2a) 2 n(n + 1) j j=1

L qq (n) = qq (n);

(5.2.2b)

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



3 LT qq (n) = CF  − 2 2 

LT gg (n) = CG 

n 1 j=1

11 −2 6n

j

  = qq (n) − CF

n 1 j=1

j

1 ; n(n + 1)

(5.2.2c)



 − 2 TF :

3

(5.2.2d)

The equality expressed in Eq. (5.2.2b) is a direct consequence of fermion-helicity conservation by purely vector interactions in the limit of negligible fermion mass. The ?rst point to stress is the commonly growing negative value, for increasing n, indicative of the tendency of all the x-space distributions to migrate towards x = 0 with increasing Q2 . In other words, evolution has a degrading e,ect on the densities. Secondly, in contrast to the behaviour of both q and Lq, the anomalous dimensions governing LT q do not vanish for n = 0 and hence there is no sum rule associated with the tensor charge [14]. Moreover, comparison to Lq reveals that LT qq (n)¡L qq (n) for all n. This implies that for (hypothetically) identical starting distributions (i.e., LT q(x; Q02 ) = Lq(x; Q02 )), LT q(x; Q2 ) everywhere in x will fall more rapidly than Lq(x; Q2 ) with increasing Q2 . We shall return in more detail to this point in Section 5.5. For completeness, we also present the AP splitting functions (i.e., the x-space version of the 1 anomalous dimensions) (n) = 0 d x xn−1 P(x), for the pure fermion sector: 1 + x2 (0) = CF ; (5.2.3a) Pqq 1−x + (0) (0) = Pqq ; LPqq 1 + x2 (0) (0) − 1 + x = Pqq (z) − CF (1 − z); LT Pqq = CF 1−x +

(5.2.3b) (5.2.3c)

where the “plus” regularisation prescription is de?ned in Appendix C in Eq. (C.1) and we also have made use of the identity given in Eq. (C.2). Naturally, the plus prescription is to be ignored (0) (0) ¡LPqq is when multiplying functions that vanish at x = 1. Once again, the inequality LT Pqq manifest for all x¡1, indeed, one has (0) 1 2 [Pqq (x)

(0) (0) + LPqq (x)] − LT Pqq (x) = CF (1 − x) ¿ 0:

(5.2.4)

The non-mixing of the transversity distributions for quarks, LT q, and gluons, LT g, is a,orded a physical demonstration via the ladder-diagram summation technique. In Fig. 14 the general leading-order one-particle irreducible (1PI) kernels are displayed. If the four external lines are all quarks (i.e., a gluon rung, see Fig. 14a), the kernel is clearly diagonal (in parton type) and therefore contributes to the evolution of LT q. For the case in which one pair of external lines are quarks and the other gluons (i.e., a quark rung, see Figs. 14b and c), helicity conservation along the quark line in the chiral limit implies a vanishing contribution to transversity evolution. Likewise, the known properties of four-body amplitudes, namely t-channel helicity conservation, preclude any contribution that might mix the evolution of LT q and LT g.

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

51

Fig. 14. The 1PI kernels contributing to the O(/s ) evolution of LT q in the axial gauge.

The same reasoning clearly holds at higher orders since the only manner for gluon and quark ladders to mix is via diagrams in which an incoming quark line connects to its Hermitianconjugate partner. Thus, quark-helicity conservation in the chiral limit will always protect against such contributions. Before continuing to NLO, a comment is in order here on the recent debate carried out in the literature [66 – 68] regarding the calculation of the anomalous dimensions for h1 and the validity of certain approaches. The authors of [66] attempt to calculate the anomalous dimensions relevant to h1 exploiting a method based on [69]. The motivation is that use of so-called time-ordered or old-fashioned perturbation theory in the WeizsYacker–Williams approximation [70,71] (as adopted in, e.g., [18]) encounters a serious diKculty: it is only applicable to the region x¡1 while the end-point (x = 1) contribution cannot be evaluated directly. Where there is some conservation law (e.g., quark number), then appeal to the resulting sum rule allows indirect extraction of this point (the usual -function contribution). In the case of transversity no such conserved quantum number exists and one might doubt the validity of such calculations. Indeed, the claim in [66] was that a direct calculation, based on a dispersion relation approach, yields a di,erent result to that reported in Eq. (5.2.3c) above. A priori, from a purely theoretical point of view, this apparent discrepancy is hard to credit: were it real, then it would imply precisely the type of ambiguity to which the singularity structure of the theory on the light-cone is supposedly immune. In [66] the anomalous dimensions are calculated via the one-loop corrections to the classic handbag diagram (see, e.g., Fig. 4) with, however, on-shell external quark states and no lower hadronic blob. In order to mimic the required chiral structure, one of the upper vertices is taken to be 5 and the other or 5 for g1 or h1 respectively. The results for g1 are in agreement with other approaches while the anomalous dimensions for h1 di,er in the coeKcient of the -function contribution. As the authors themselves point out, a careful examination of the calculation then reveals that, in the Feynman gauge they adopt, the single contribution leading to the di,erence arises owing to the scalar vertex (i.e., spin-Pip) correction. What casts doubt on such ?ndings is the fact that in a physical gauge, as used for example in the ladder-diagram summation approach [56,57] mentioned earlier, precisely all such vertex corrections are in fact absent (to this order in QCD). Moreover, it is just this property that gives rise, in that approach, to the universal short-distance behaviour, independent of the particular nature of the vertices involved. Various cross checks of these potentially disturbing ?ndings have been performed [67,68] with the conclusion that the original calculations are after all correct. In particular, BlYumlein

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

[68] has produced a very thorough appraisal of the situation. Moreover, he has uncovered a fatal conceptual oversight: the scalar current is not conserved. 9 To appreciate the relevance of this observation let us briePy recall the salient points of the RGE approach (for more detail the reader is referred to [68]). A product of two currents (as in the peculiar Compton amplitude under consideration) may be expanded as j1 (z)j2 (0) = C(n; z)O(n; 0); n

where typically then ji = jV; A (i.e., vector or axial vector currents) but here a scalar current jS must be introduced. The RGE for the Wilson coeKcients C(n; z) is D + j1 (g) + j2 (g) − O (n; g)]C(n; z) = 0;

where ji (g) and O (n; g) are the anomalous dimensions of the currents ji and the composite operators O(n) respectively, and (neglecting quark masses) the RG operator is de?ned as @ @ D = 2 2 + <(g) : @ @g Thus, the LL corrections to the Compton amplitude have coeKcients C (n; g) = j1 (g) + j2 (g) − O (n; g): The point then is that while in the better-known spin-averaged and helicity-weighted cases ji = 0 for both currents (axial and=or vector), the scalar current necessary for the transversity case is not conserved and jS = 0. Thus, in contrast to the former, C and − O do not coincide in the calculation for h1 . The discrepancy in [66] is due precisely to the neglect of jS . 5.3. QCD evolution at next-to-leading order Reliable QCD analysis of the sort of data samples we have come to expect in modern experiments requires full NLO accuracy. For this it is necessary to calculate both the anomalous dimensions to two-loop level and the constant terms (i.e., the part independent of ln Q2 ) of the so-called coeKcient function (or hard-scattering process) at the one-loop level, together, of course, with the two-loop <-function. The two-loop anomalous-dimension calculation for h1 has now been presented in three papers: [72,73] using the MS scheme in the Feynman gauge and [74] using the MS scheme in the light-cone gauge. These complement the earlier two-loop calculations for the better-known twist-two structure functions: f1 [75 –81] and g1 [82–84]. Such knowledge has been exploited in the past for the phenomenological parametrisation of f1 [85 –87] and g1 [88–90] in order to perform global analyses of the experimental data; and will certainly be of value when the time comes to analyse data on transversity. The situation at NLO is still relatively simple, as compared to the unpolarised or helicityweighted cases. Examples of the relevant two-loop diagrams are shown in Fig. 15. It remains impossible for the gluon to contribute, for the reasons already given. The only complication is the usual mixing, possible at this level, between quark and antiquark distributions, for which 9

We are particularly grateful to Johannes BlYumlein for illuminating discussion on this point

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

53

Fig. 15. Example two-loop diagrams contributing at O(/s2 ) to the anomalous dimensions of LT q.

quark helicity conservation poses no restriction since the quark and antiquark lines do not connect directly to one another, see Fig. 15d. It is convenient to introduce the following combinations of quark transversity distributions (the ± subscript is not to be confused with helicity): LT q± (n) = LT q(n) ± LT q(n); N

(5.3.1a)

(n); LT q˜+ (n) = LT q+ (n) − LT q+

(5.3.1b)

LT M(n) =

LT q+ (n);

(5.3.1c)

q

where q and q represent quarks of di,ering Pavours. The speci?c evolution equations may then be written as (e.g., see [91]) d LT q− (n; Q2 ) = LT qq; − (n; /s (Q2 ))LT q− (n; Q2 ); d ln Q2

(5.3.2a)

d LT q˜+ (n; Q2 ) = LT qq; + (n; /s (Q2 ))LT q˜+ (n; Q2 ); d ln Q2

(5.3.2b)

d LT M(n; Q2 ) = LT MM (n; /s (Q2 ))LT M(n; Q2 ): d ln Q2

(5.3.2c)

Note that the ?rst moment (n = 1) in Eq. (5.3.2a) corresponds to evolution of the nucleon’s tensor charge [14,19,92]. The splitting functions LT qq; ± and LT MM have expansions in powers

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

of the coupling constant that take the following form: / / 2 s s LT ii (n; /s ) = LT (0) LT (1) (5.3.3) qq (n) + ii (n) + · · · ; 2 2 where {ii} = {qq; ±}; {MM} and we have taken into account the fact that LT qq; + , LT qq; − and LT MM are all equal at LO. It is convenient then to introduce [91] (1) (1) LT (1) qq; ± (n) = LT qq (n) ± LT qqN (n);

(5.3.4a)

(1) (1) LT (1) MM (n) = LT qq; + (n) + LT qq; PS (n):

(5.3.4b)

Since it turns out that LT (1) qq; PS (n) = 0, the two evolution equations (5.3.2b), (5.3.2c) may be replaced by a single equation: d LT q+ (n; Q2 ) = LT qq; + (n; /s (Q2 ))LT q+ (n; Q2 ): (5.3.2b ) d ln Q2 The formal solution to Eqs. (5.3.2a) and (5.3.2b ) is well-known (e.g., see [93]) and reads /s (Q02 ) − /s (Q2 ) <1 (1) 2 (0) LT qq; ± (n) − LT qq (n) LT q± (n; Q ) = 1 + <0 2<0 ×

(0)

(Q2 )

/s /s (Q02 )

−2LT< qq (n) 0

LT q± (n; Q02 );

(5.3.5)

with the input distributions LT q± (n; Q02 ) given at the input scale Q0 . Of course, the corresponding LO expressions may be recovered from the above expressions by setting the NLO quantities, LT (1) ij; ± and <1 , to zero. In the MS scheme the (1) (n) relevant to h1 are as follows: 10 2 3 LT (1) qq; J (n) = CF { 8 +

2 n(n+1) J−

˜ − 3S2 (n) − 4S1 (n)[S2 (n) − S2 (n=2)] − 8S(n) + S3 (n=2)}

134 22 + 12 CF Nc { 17 12 − (2=m(n + 1))J− − 9 S1 (n)+ 3 S2 (n)+4S1 (n)[2S2 (n) − S2 (n=2)]

˜ − S3 (n=2)} + 23 CF TF {− 14 + + 8S(n)

10 3 S1 (n)

− 2S2 (n)};

where J = ± and the S functions are de?ned by n j −k ; Sk (n) = j=1

Sk ( n2 ) = 2k ˜ = S(n)

n

n

j −k ;

(5.3.7a) (5.3.7b)

j=2;even

(−1)j S1 (j)j −2 :

j=1 10

(5.3.6)

Note that 23 S1 (n) in the second line of (A:8) in [77] should read 23 S3 (n).

(5.3.7c)

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Fig. 16. Comparison between f1 and h1 of the variation with n of (a) the two-loop anomalous dimensions (1) n for Nf = 3 (circles) and 5 (triangles), and (b) the combination (1) (n)=2<1 − (0) (n)=2<0 for Nf = 3 and 5; from [72].

Fig. 17. The LO and NLO Q2 -evolution of (a) the tensor charge and (b) the second moments of h1 (x; Q2 ) and f1 (x; Q2 ) (both are normalised at Q2 = 1 GeV2 ), from [72].

In Fig. 16 we show the n dependence of the two-loop anomalous dimensions (as presented in [72] 11 ). From the ?gure, one clearly sees that for n small, LT (1) (n) is signi?cantly larger than (1) (n) but, with growing n, very quickly approaches (1) (n) while maintaining the inequality LT (1) (n)¿ (1) (n). For the speci?c moments n = 1 (corresponding to the tensor charge) and n = 2, we display the Q2 variation in Fig. 17. To express the corresponding results in x space it is convenient to introduce the following de?nitions: 12 2x ; (1 − x)+ 1+x 1 dz 1−z 1 = − 2Li2 (−x) − 2 ln x ln(1 + x) + ln2 x − 2 ; S2 (x) = x ln z z 2 6 LT R(0) (x) =

(5.3.8) (5.3.9)

1+x

11

According to the convention adopted for the moments in [72], n = 0 there corresponds to n = 1 in the present report. 12 In order to avoid confusion with the tensor-charge anomalous dimensions, the notation adopted here corresponding to LT R(0) (x) is di,erent than that commonly adopted.

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where Li2 (x) is the usual dilogarithm function. In the MS scheme, de?ning (1) (1) (1) LT Pqq; ± (x) = LT Pqq (x) ± LT PqqN (x);

(5.3.10)

cf. Eq. (5.3.4a), one then has (1) LT Pqq (x) = CF2 {1 − x − [ 32 + 2 ln(1 − x)] ln xLT R(0) (x) + [ 38 − 12 2 + 6O(3)](1 − x)}

+ 12 CF CG {−(1 − x) + [ 67 9 + + [ 17 12 +

11 2 9

11 3

ln x + ln2 x − 13 2 ]LT R(0) (x)

− 6O(3)](1 − x)}

+ 23 CF TF {[ − ln x − 53 ]LT R(0) (x) − [ 14 + 13 2 ](1 − x)}; (0) 1 LT Pq(1) qN (x) = CF [CF − 2 CG ]{−(1 − x) + 2S2 (x)LT R (−x)};

(5.3.11a) (5.3.11b)

where O(3) ≈ 1:202057 is the usual Riemann Zeta function. Note that the plus prescription is to be ignored in LT R(0) (−x). To complete this section we also report on the corresponding NLO calculation for linear (transverse) gluon polarisation [94]. As already noted, LT g is precluded in the case of spin-half hadrons—it may, however, be present in objects of spin one, such as the deuteron or indeed even the photon [95]. " 2 (1) (0) 1 1 2 LT Pgg (x) = CG2 ( 67 18 + 2 ln x − 2 ln x ln(1 − x) − 6 )LT R (x) 1 − x3 (0) 8 + + S2 (x)LT R (−x) + ( 3 + 3O(3))(1 − x) 6x 1 − x3 4 (0) 10 + CG TF − 9 LT R (x) + − 3 (1 − x) 3x 2(1 − x3 ) − CF TF + (1 − x) : 3x

(5.3.12)

The corresponding expression in Mellin-moment space for the anomalous dimensions is 13 1 1 n 67 (1) 2 8 n ˜ LT gg (n) = CG + + S1 (n)(2S2 ( ) − ) + S3 ( ) − 4S(n) 3 2(n − 1)(n + 2) 2 9 2 2 1 4 20 n(n + 1) − CF TF + CG TF − + S1 (n) + : (5.3.13) 3 9 (n − 1)(n + 2) (n − 1)(n + 2) As noted in [94] the result for the part ∼CF TF in (5.3.12) was presented in [95] for the region x¡1 (corresponding to the two-loop splitting function for linearly polarised gluons into photons). However, the two calculations appear to be at variance: the results of [95] imply a small-x behaviour of O(1=x2 ) for the relevant splitting function, which would then be more singular than the unpolarised case. 13

We are very grateful to Werner Vogelsang for providing us with the exact expression.

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There are two aspects of the splitting function (5.3.12) that warrant particular comment. Firstly, the small-x behaviour changes signi?cantly on going from LO to NLO. At LO, the splitting function is O(x) for x → 0 whereas at NLO there are O(1=x) terms (as in the unpolarised case): we have (1) LT Pgg (x) ≈

1 (N 2 + 2Nc TF − 4CF TF ) + O(x) 6x c

(x → 0):

(5.3.14)

Notice that all logarithmic terms ∼x ln2 x cancel in this limiting region. The second comment regards the so-called supersymmetric limit: namely CF = Nc = 2TF [65], which was investigated for the unpolarised and longitudinally polarised NLO splitting functions in [82–84,96], for the time-like case in [97] and for the case of transversity in [94]. In the supersymmetric limit the LO splitting functions for quark transversity and for linearly polarised gluons are equal [18,95]: 3 2x (0) (0) LT Pqq (x) = Nc + (1 − x) = LPgg (x): (5.3.15) (1 − x)+ 2 Hence, we may consider linear polarisation of the gluon as the supersymmetric partner to transversity (see also [65]). Indeed, as was natural, we have already applied the terminology without distinction to spin-half and spin-one. In [94] the check was performed that the supersymmetric relation still holds at NLO. To do so it is necessary to transform to a regularisation scheme that respects supersymmetry, namely dimensional reduction. As noted in [94] the transformation is rendered essentially trivial owing to the absence of O(j) terms in the d-dimensional LO splitting functions for transversity or linearly polarised gluons at x¡1; such terms are always absent in dimensional reduction but may be present in dimensional regularisation. Thus, at NLO the results for the splitting functions for quark transversity—see Eqs. (5.3.11a), (5.3.11b)—and for linearly polarised gluons, Eq. (5.3.12), automatically coincide for x¡1 with their respective MS expressions in dimensional reduction. These expressions may therefore be immediately compared in the supersymmetric limit and indeed for CF = Nc = 2TF (1) (1) LT Pqq; + (x) ≡ LT Pgg (x)

(x¡1):

(5.3.16)

Note, in addition, that the supersymmetric relation is trivially satis?ed for x = 1; see [83,84], where the appropriate factorisation-scheme transformation to dimensional reduction for x = 1 is given. 5.4. Evolution of the transversity distributions The interest in the e,ects of evolution in the case of transversity is two-fold: ?rst, there is the obvious question of the relative magnitude of the distributions at high energies given some low-energy starting point (e.g., a non-perturbative model calculation, for a detailed discussion and examples see Section 8) and second is the problem raised by the So,er inequality. It is to the ?rst that we now address our attention while we shall deal with latter shortly.

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Let us for the moment simply pose the question of the e,ect of QCD evolution [72,74,94, 98–112] on the overall magnitude of the transversity densities that might be constructed at some low-energy scale. As already noted above, there is no conservation rule associated with the tensor charge of the nucleon (cf. the vector and axial-vector charges) and, indeed, the sign of the anomalous dimensions at both LO and NLO is such that the ?rst moment of h1 falls with increasing Q2 . Thus, one immediately deduces that the tensor charge will eventually disappear in comparison to the vector and axial charges. Such behaviour could have a dramatic impact on the feasibility of high-energy measurement of h1 and thus requires careful study. A analytic functional form for the LO anomalous dimensions governing the evolution of h1 is LT (0) (n) = 43 { 32 − 2[ (n + 1) + E ]};

(5.4.1)

where (z) = d ln D(z)=d z is the digamma function and E = 0:5772157 is the Euler–Mascheroni constant. Since LT (0) (1) =− 23 , the ?rst moment of h1 and the tensor charges, q = 01 d x(LT q− LT q), N decrease with Q2 as

/s (Q2 ) q(Q ) = /s (Q02 ) 2

−2LT (0) qq (1)=<0

q(Q02 ) =

/s (Q02 ) /s (Q2 )

−4=27

q(Q02 );

(5.4.2)

where, to obtain the second equality, we have set Nf = 3. Despite the smallness of the exponent,

−4=27, we shall see that the evolution of LT q(x; Q2 ) is rather di,erent from that of the helicity

distributions Lq(x; Q2 ), especially for small x. At NLO this becomes

−2LT (0) qq (1)=<0 /s (Q2 ) q(Q ) = /s (Q02 ) /s (Q02 ) − /s (Q2 ) <1 (1) (0) × 1+ LT qq; − (1) − LT qq (1) q(Q02 ) <0 2<0 2

/s (Q2 ) = /s (Q02 )

4 27

337 2 2 1− (/s (Q0 ) − /s (Q )) ; 486

(5.4.3)

where in the second equality we have used LT (1) qq; − (1) =

19 2 257 13 181 13 CF − CF Nc + CF TF = − + Nf ; 8 72 18 18 27

(5.4.4)

and we have once again set Nf = 3. Recall that the ?rst moments of the q → qg polarised and unpolarised splitting functions, vanish to all orders in perturbation theory and that the g → qqN polarised anomalous dimension L qg (1) is zero at LO; thus, Lq(Q2 ) is constant. This can be seen analytically by the following argument [99] based on the double-log approximation. The leading behaviour of the parton distributions for small x is governed by the rightmost singularity of their anomalous dimensions in Mellin-moment space. From Eq. (5.4.1) we see that this singularity is located at n = − 1 for

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Fig. 18. Evolution of the helicity and transversity distributions for the u Pavour [99]. The dashed curve is the input LT u = Lu at Q02 = 0:23 GeV2 taken from the GRV [114] parametrisation. The solid (dotted) curve is LT u (Lu) at Q2 = 25 GeV2 . The dot-dashed curve is the result of the evolution of LT u at Q2 = 25 GeV2 driven by Pqq , i.e., with the term LT P in Ph turned o,.

LT q at LO. Expanding LT (n) around this point gives 8 LT (n)∼ + O(1): 3(n + 1)

(5.4.5)

Equivalently, in x-space, expanding the splitting function LT P in powers of x yields 8 LT P(x)∼ x + O(x2 ): (5.4.6) 3 In contrast, the rightmost singularity for Lq in moment space is located at n = 0 and the splitting functions LPqq and LPqg behave as constants as x → 0. Therefore, owing to QCD evolution, LT q acquires and an extra suppression factor of one power of x with respect to Lq at small x. We note too that at NLO the rightmost singularity for LT q is located at n = 0, so that NLO evolution renders the DGLAP asymptotics for x → 0 in the case of transversity compatible with Regge theory [113]. As mentioned earlier, this problem may be investigated numerically by integrating the DGLAP equations (5.1.11) with suitable starting input for h1 and g1 . As a reasonable trial model one may assume the various LT q and Lq to be equal at some small scale Q02 and then allow the two types of distributions to evolve separately, each according to its own evolution equations. The input hypothesis LT q(x; Q02 ) = Lq(x; Q02 ) is suggested by various quark-model calculations of LT q and Lq [14,98] (see also Section 8 here), in which these two distributions are found to be very similar at a scale Q02 . 0:5 GeV2 . For Lq(x; Q02 ), we then use the leading-order GlYuck–Reya–Vogelsang (GRV) parametrisation [114], whose input scale is Q02 = 0:23 GeV2 . The result for the u-quark distributions is shown in Fig. 18 (the situation is similar for the other Pavours). The dashed line is the input, the solid line and the dotted line are the results of the evolution of LT u and Lu, respectively, at Q2 = 25 GeV2 . For completeness, the evolution of LT u when driven only by Pqq —i.e., with the LT P term turned o,, see Eq. (5:2:3c)—is also shown (dot-dashed line). The large di,erence in the evolution of LT u (solid curve) and Lu

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Fig. 19. Comparison of the Q2 -evolution of LT u(x; Q2 ) and Lu(x; Q2 ) at (a) LO and (b) NLO, from [72].

N Q2 ) and Lu(x; Fig. 20. Comparison of the NLO Q2 -evolution of LT u(x; N Q2 ), LT d(x; N Q2 ), from [72].

(dotted curve) at small x is evident. Note also the di,erence between the correct evolution of LT u and the evolution driven purely by Pqq (dot-dashed curve). As a further comparison of the behaviour of h1 and g1 , in Figs. 19(a) and (b) we display the LO and NLO Q2 -evolution of LT u and Lu, starting, respectively, from the LO and NLO input function for Lu given in [88] for Q2 = 0:23 and 0:34 GeV2 . Although LO evolution leads to a signi?cant divergence between LT u and Lu at Q2 = 20 GeV2 , 14 this tendency is strengthened by the NLO evolution, in particular, in the small-x region. Although the evolution of Lu shown in Fig. 19 is a,ected by mixing with the gluon distribution, the non-singlet quark distributions also show the same trend. In Fig. 20, we compare the NLO Q2 -evolution of LT u, N LT dN and Lu, N starting from the same input distribution function (the NLO input function for the sea-quark distribution to g1 given in 14

Such a di,erence was also pointed out in [100].

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61

[88]). The di,erence between LT uN and LuN is again signi?cant. Although the input sea-quark distribution is taken to be Pavour symmetric (LT uN = LT dN at Q2 = 0:34 GeV2 ), NLO evolution violates this symmetry owing to the appearance of LT Pq(1) qN —see (5.3.11b). However, this e,ect is very small, as is evident from Fig. 20 and discussed in [105]. 5.5. Evolution of the So>er inequality and general positivity constraints Particular interest in the e,ects of evolution arises in connection with the So,er inequality [38], Eq. (4.6.6). It has been argued [115] that this inequality, which was derived within a parton model framework, may be spoilt by radiative corrections, much as the Callan–Gross relation. Such an analogy, however, is somewhat misleading, since the So,er inequality is actually very similar to the more familiar positivity bound |Lq(x)| 6 q(x). The LO evolution of the inequality is governed by Eq. (5.2.4) and hence it is not endangered, as pointed out in [99]. At NLO the situation is complicated by the well-known problems of scheme dependence, etc. Indeed, it is perhaps worth remarking that the entire question of positivity is ill-de?ned beyond LO, inasmuch as the parton distributions themselves as physical quantities become ill-de?ned: a priori there is no guarantee in a given scheme that any form of positivity will survive higher-order corrections. This observation may, of course, be turned on its head and used to impose conditions on the scheme choice such that positivity will be guaranteed [116]. At any rate, if this is possible then at the hadronic level any natural positivity bounds should be respected, independently of the regularisation scheme applied. An instructive and rather general manner to examine the problem is to recast the system of evolution equations into a form analogous to the Boltzmann equation [101]. First of all, let us rewrite Eq. (5.1.11) in a slightly more suggestive form for the non-singlet case: 1 dq(x; t) dy x (5.5.1) = P ; t q(y; t); dt y x y where t = ln Q2 . One may thus interpret the equation as describing the time, t, evolution of densities, f(x; t), in a one-dimensional x space. The Pow is constrained to run from large to small x owing to the ordering x¡y under the integral. Such an interpretation facilitates dealing with the infrared (IR) singularities present in the expressions for P(x). Indeed, a key element is provided by consideration of precisely the IR singularities [117,118]. Let us now rewrite the plus regularisation in the following form: 1 dy P+ (x; t) = P(x; t) − (1 − x) P(y; t); (5.5.2) 0 y which then permits the evolution equations to be rewritten as 1 1 dy x dq(x; t) dy P(y; t): = q(y; t)P ; t − q(x; t) dt y x y 0

(5.5.3)

Reading the second term as describing the Pow of partons at the point x [117], the kinetic interpretation is immediate. It is useful to render the analogy more direct by the change of variables

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y → y=x in the second term, leading to the following more symmetric form: x 1 y dq(x; t) dy x dy = q(y; t)P ;t − q(x; t)P ;t : dt y x x y 0 x

(5.5.4)

In this fashion the equation has been translated into a form analogous to the Boltzmann equation: namely, 1 dq(x; t) dy[(y → x; t)q(y; t) − (x → y; t)q(x; t)]; (5.5.5) = dt 0 where the one-dimensional analogue of the Boltzmann “scattering probability” may be de?ned as 1 x (y → x; t) = Q(y − x) P (5.5.6) ;t : y y Cancellation of the IR divergencies between contributions involving real and virtual gluons is therefore seen to occur as a consequence of the continuity condition on “particle number”; i.e., the equality of Pow in and out in the neighbourhood of y = x in both terms of Eq. (5.5.5). In the spin-averaged case the particle density (at some initialisation point) is positive by definition. Now, the negative second term in Eq. (5.5.5) cannot change the sign of the distribution because it is “diagonal” in x, i.e., it is proportional to the function at the same point x. When the distribution is suKciently close to zero, it stops decreasing. This is true for both “plus” and (1 − x) terms, for any value of their coeKcients (if positive, it only reinforces positivity of the distribution). Turning next to the spin-dependent case, for simplicity we consider ?rst the Pavour non-singlet and allow the spin-dependent and spin-independent kernels to be di,erent, as they indeed are at NLO. Rather than the usual helicity sums and di,erences, it turns out to be convenient to cast the equations in terms of de?nite parton helicities. Although such a form mixes contributions of di,erent helicities, the positivity properties emerge more clearly. We thus have dq+ (x; t) = P++ (x; t) ⊗ q+ (x; t) + P+− (x; t) ⊗ q− (x; t); dt

(5.5.7a)

dq− (x; t) = P+− (x; t) ⊗ q+ (x; t) + P++ (x; t) ⊗ q− (x; t); dt

(5.5.7b)

where P+± (z; t) = 12 [P(z; t) ± LP(z; t)] are the evolution kernels for helicity non-Pip and Pip, respectively. For x¡y, positivity of the initial distributions, q± (x; t0 ) ¿ 0 or |Lq(x; t0 )| 6 q(x; t0 ), is preserved if both kernels P+± are positive, which is true if |LP(z; t)| 6 P(z; t)

(z¡1):

(5.5.8)

Terms that are singular at z = 1 cannot alter positivity as they only appear in the diagonal (in helicity) kernel, P++ ; non-forward scattering is completely IR safe. Once again in the kinetic interpretation, the distributions q+ and q− stop decreasing on approaching zero.

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To extend the proof to include the case in which there is quark–gluon mixing is trivial—we need the full expressions for the evolutions of quark and gluon distributions of each helicity: dq+ (x; t) qq qq (x; t) ⊗ q+ (x; t) + P+− (x; t) ⊗ q− (x; t) = P++ dt qg qg + P++ (x; t) ⊗ g+ (x; t) + P+− (x; t) ⊗ g− (x; t);

(5.5.9a)

dq− (x; t) qg qg (x; t) ⊗ q+ (x; t) + P++ (x; t) ⊗ q− (x; t) = P+− dt qg qg + P+− (x; t) ⊗ g+ (x; t) + P++ (x; t) ⊗ g− (x; t);

(5.5.9b)

dg+ (x; t) gq gq (x; t) ⊗ q+ (x; t) + P+− (x; t) ⊗ q− (x; t) = P++ dt gg gg (x; t) ⊗ g+ (x; t) + P+− (x; t) ⊗ g− (x; t); + P++

(5.5.9c)

dg− (x; t) gq gq (x; t) ⊗ q+ (x; t) + P++ (x; t) ⊗ q− (x; t) = P+− dt gg gg (x; t) ⊗ g+ (x; t) + P++ (x; t) ⊗ g− (x; t): + P+−

(5.5.9d)

Since inequality (5.5.8) is clearly valid separately for each type of parton [116], |LPij (z; t)| 6 Pij (z; t)

(z¡1; i; j = q; g);

(5.5.10)

all the kernels appearing on the r.h.s. of this system, are positive. With regard to the singular terms, they are again diagonal (in parton type here) and hence cannot a,ect positivity. The validity of the equations at LO is guaranteed via their derivation, just as the (positive) helicity-dependent kernels were in fact ?rst calculated in [12]. At NLO, the situation is more complex [116]. To conclude, the maintenance of positivity under Q2 evolution has two sources: (a) inequalities (5.5.10), leading to the increase of distributions and (b) the kinetic interpretation of the decreasing terms. For the latter, it is crucial that they are diagonal in x, helicity and also parton type, which is a prerequisite for their IR nature. We now ?nally return to the So,er inequality: in analogy with the previous analysis, it is convenient to de?ne the following “super” distributions: Q+ (x) = q+ (x) + LT q(x);

(5.5.11a)

Q− (x) = q+ (x) − LT q(x):

(5.5.11b)

According to the So,er inequality, both distributions are positive at some scale (say Q02 ) and the evolution equations for the non-singlet case take the form (henceforth the argument t will be suppressed) dQ+ (x) Q Q (x) ⊗ Q+ (x) + P+− (x) ⊗ Q− (x); (5.5.12a) = P++ dt dQ− (x) Q Q (x) ⊗ Q+ (x) + P++ (x) ⊗ Q− (x); (5.5.12b) = P+− dt

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where the “super” kernels at LO are just Q (0) (z) = 12 [Pqq (z) P++

+

Ph(0) (z)] =

1 (1 + z)2 + 3(1 − z) ; CF 2 (1 − z)+

Q (0) P+− (z) = 12 [Pqq (z) − Ph(0) (z)] = 12 CF (1 − z):

(5.5.13a) (5.5.13b)

One can easily see, that the inequalities analogous to (5.5.10) are satis?ed, so that both Q Q (z) and P+− (z) are positive for z¡1, while the singular term appears only in the diagonal P++ kernel. Thus, both requirements are ful?lled and the So,er inequality is maintained under LO evolution. The extension to the singlet case is trivial owing to the exclusion of gluon mixing. Therefore, only evolution of quarks is a,ected, leading to the presence of the same extra terms on the r.h.s., as in Eq. (5.5.9a): dQ+ (x) qG qG Q Q (x) ⊗ Q+ (x) + P+− (x) ⊗ Q− (x) + P+− (x) ⊗ G+ (x) + P++ (x) ⊗ G− (x); = P++ dt (5.5.14a) dQ− (x) qG qG Q Q (x) ⊗ Q+ (x) + P++ (x) ⊗ Q− (x) + P+− (x) ⊗ G+ (x) + P++ (x) ⊗ G− (x); = P+− dt (5.5.14b) which are all positive and singularity free; this concludes the demonstration that positivity is indeed preserved. 6. Transversity in semi-inclusive leptoproduction While it is usual to adopt DIS as the de?ning process and point of reference when discussing distribution functions, as repeatedly noted and explicitly shown in Section 3, the case of transversity is somewhat special in that it does not appear in DIS. However, owing to the topology of the contributing Feynman diagrams, transversity does play a rˆole in semi-inclusive DIS, owing to the presence of two hadrons: one in the initial state and the other in the ?nal state [17,18,20,119 –122]. This process is the subject of the present section. 6.1. DeFnitions and kinematics Semi-inclusive—or, to be more precise, single-particle inclusive—leptoproduction (see Fig. 21) is a DIS reaction in which a hadron h, produced in the current fragmentation region, is detected in the ?nal state (for the general formalism see [15,123]) l(‘) + N (P) → l (‘ ) + h(Ph ) + X (PX ):

(6.1.1)

With a transversely polarised target, one can measure quark transverse polarisation at leading twist either by looking at a possible asymmetry in the Ph⊥ distribution of the produced hadron (the so-called Collins e,ect [15 –17,124]), or by polarimetry of a transversely polarised ?nal hadron (for instance, a 0 hyperon) [15,18,125,126]. Transversity distributions also appear in the Ph⊥ -integrated cross-section at higher twist [15,120].

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Fig. 21. Semi-inclusive deeply inelastic scattering.

Fig. 22. Lepton and hadron planes in semi-inclusive leptoproduction.

We de?ne the invariants x=

Q2 ; 2P · q

y=

P·q ; P·‘

z=

P · Ph : P·q

(6.1.2)

We shall be interested in the limit where Q2 ≡ −q2 ; P · q; Ph · q and Ph · P become large while x and z remain ?nite. The geometry of the process is shown in Fig. 22. The lepton scattering plane is identi?ed by l and l . The virtual photon is taken to move along the z-axis. The three-momenta of the virtual photon q and of the produced hadron Ph de?ne a second plane, which we call the hadron plane. The spin S of the nucleon and the spin Sh of the produced hadron satisfy S 2 = Sh2 = − 1 and S · P = Sh · Ph = 0. The cross-section for the reaction (6:1:1) is 1 d 3 PX d = 4‘ · P s X (2)3 2EX l

×(2)4 4 (P + ‘ − PX − Ph − ‘ )|M|2

d 3 l d 3 Ph ; (2)3 2E (2)3 2Eh

(6.1.3)

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where we have summed over the spin sl of the outgoing lepton. The squared matrix element in (6:1:3) is |M|2 =

e4 [u (‘ ; sl ) ul (‘; sl )]∗ [ul (‘ ; sl ) ul (‘; sl )] q4 l ×X; Ph Sh |J (0)|PS ∗ X; Ph Sh |J (0)|PS :

(6.1.4)

Introducing the leptonic tensor [ul (‘ ; sl ) ul (‘; sl )]∗ [ul (‘ ; sl ) ul (‘; sl )] L = sl

= 2(‘ ‘ + ‘ ‘ − g ‘ · ‘ ) + 2i5l

! ‘

!

q

(6.1.5)

and the hadronic tensor d 3 PX 1 W = (2)4 4 (P + q − PX − Ph ) (2)4 X (2)3 2EX ×PS |J (0)|X; Ph Sh X; Ph Sh |J (0)|PS

(6.1.6)

the cross-section becomes d =

1 e4 d 3 l d 3 Ph 4 L W (2) : 4‘ · P Q4 (2)3 2E (2)3 2Eh

(6.1.7)

In the target rest frame (‘ · P = ME) one has 2Eh

2 d /em E = L W : d 3 Ph dE d\ 2MQ4 E

(6.1.8)

In terms of the invariants x, y and z, Eq. (6.1.8) reads 2Eh

2 y /em d L W : = d 3 Ph d x dy Q4

(6.1.9)

If we decompose the momentum Ph of the produced hadron into a longitudinal (Ph|| ) and a transverse (Ph⊥ ) component with respect to the ∗ N axis and if |Ph⊥ | is small compared to the energy Eh , then we can write approximately 1 d 3 Ph = d z d 2 Ph⊥ ; 2Eh 2z

(6.1.10)

and re-express Eq. (6.1.9) as 2 y d /em = L W : d x dy d z d 2 Ph⊥ 2Q4 z

(6.1.11)

Instead of working in a ∗ N collinear frame, it is often convenient to work in a frame where the target nucleon and the produced hadron move collinearly (the hN collinear frame, see Appendix B.2). In this frame the virtual photon has a transverse momentum qT , which is related to Ph⊥ , up to 1=Q2 corrections, by qT −Ph⊥ =z. Thus Eq. (6.1.11) can be

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

67

written as 2 d /em = yzL W : d x dy d z d 2 qT 2Q4

(6.1.12)

Let us now evaluate the leptonic tensor. In the ∗ N collinear frame the lepton momenta can be parametrised in terms of the Sudakov vectors p and n as x Q2 ‘ = (1 − y)p + ; n + ‘⊥ y 2xy

(6.1.13a)

x Q2 (1 − y) ‘ = p + n + ‘⊥ y 2xy

(6.1.13b)

2 = ((1 − y)=y 2 ) Q 2 . The symmetric part of the leptonic tensor then becomes with l⊥

L(S) = −

Q2 4(1 − y) [1 + (1 − y)2 ]g⊥ + t t y2 y2

4Q2 (1 − y) 2(2 − y) + )+ (t ‘⊥ + t ‘⊥ y y2

‘⊥ ‘⊥ 1 + g⊥ ; 2 2 l⊥

where t = 2xp + q ; the antisymmetric part reads 2 Q (2 − y) ! Q2 ! ! (A) = 5l ! L p n + ‘ n − 2x‘⊥ p : y x ⊥

(6.1.14a)

(6.1.14b)

At leading-twist level, only semi-inclusive DIS processes with an unpolarised lepton beam probe the transverse polarisation distributions of quarks [15]. Therefore, in what follows, we shall focus on this case and take only the target nucleon (and, possibly, the outgoing hadron) to be polarised. At twist-three there are also semi-inclusive DIS reactions with polarised leptons, which allow extracting LT f. For these higher-twist processes we shall limit ourselves to presenting the cross-sections without derivation. 6.2. The parton model In the parton model the virtual photon strikes a quark (or antiquark), which later fragments into a hadron h. The process is depicted in Fig. 23. The relevant diagram is the handbag diagram with an upper blob representing the fragmentation process. Referring to Fig. 23 for the notation, the hadronic tensor is given by (for simplicity we consider only the quark contribution) 1 2 d 3 PX d4 k d4 W = e (2)4 a a X (2)3 2EX (2)4 (2)4 ×(2)4 4 (P − k − PX )(2)4 4 (k + q − )(2)4 4 ( − Ph − PX ) ×[1(; N Ph ; Sh ) =(k; P; S)]∗ [1(; N Ph ; Sh ) =(k; P; S)];

(6.2.1)

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Fig. 23. Diagram contributing to semi-inclusive DIS at LO.

where =(k; P; S) and 1(; Ph ; Sh ) are matrix elements of the quark ?eld

, de?ned as

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

(6.2.2)

1(; Ph ; Sh ) = 0| (0)|Ph Sh ; X :

(6.2.3)

We now introduce the quark–quark correlation matrices d 3 PX (2)4 4 (PX + k − P)=i (k; P; S)=N j (k; P; S) Aij (k; P; S) = 3 2E (2) X X = d 4 eik· PS | N j (0) i ()|PS and Rij (; Ph ; Sh ) = =

X

d 3 PX (2)4 4 (Ph + PX − )1i (; Ph ; Sh )1Nj (; Ph ; Sh ) (2)3 2EX

X

d 3 PX (2)3 2EX

d 4 ei· 0| i ()|Ph Sh ; X Ph Sh ; X | N j (0)|0:

(6.2.4)

(6.2.5)

Here A is the matrix already encountered in inclusive DIS, see Sections 3.2 and 4.1, which incorporates the quark distribution functions. R is a new quark–quark correlation matrix (sometimes called decay matrix), which contains the fragmentation functions of quarks into a hadron h. An average over colours is included in R. Inserting Eqs. (6:2:4); (6:2:5) into (6:2:1) yields d4 k d4 W = ea2 4 (k + q − ) Tr[A R ]: (6.2.6) 4 4 (2) (2) a

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69

It is an assumption of the parton model that k 2 , k · P, 2 and · Ph are much smaller than Stated di,erently: when these quantities become large, A and R are strongly suppressed. Let us work in the hN collinear frame (see Appendix B.2), the photon momentum is 1 1 q −xP + Ph + qT = −xP + ; Ph− ; qT : (6.2.7) z z

Q2 .

We recall that Ph⊥ −z qT . The quark momenta are k /P + kT = (P + ; 0; 0T );

(6.2.8a)

(6.2.8b)

Thus the delta function in (6.2.6) can be decomposed as 4 (k + q − ) = (k + + q+ − + )(k − + q− − − )2 (kT + qT − T ) (k + − xP + )(k − − Ph− =z)2 (kT + qT − T ):

(6.2.9)

which implies / = x and < = 1=z, that is k xP + kT ;

(6.2.10a)

1 Ph + T : z The hadronic tensor (6.2.6) then becomes d k + d k − d 2 kT d+ d− d 2 T W = ea2 (2)4 (2)4 a

(6.2.10b)

×(k + − xP + )(k − − Ph− =z)2 (kT + qT − T ) Tr[A R ]:

(6.2.11)

Exploiting the delta functions in the longitudinal momenta, we obtain d k − d 2 kT d+ d 2 T W = ea2 (2)4 (2)4 a ×2 (kT + qT − T ) Tr[A R ]k + =xP+ ; − =P− =z : h

(6.2.12)

To obtain the ?nal form of W , we must insert the explicit expressions for A and R into (6.2.12). The former has been already discussed in Section 4.1. In the following we shall concentrate on the structure of R. 6.3. Systematics of fragmentation functions The fragmentation functions are contained in the decay matrix R, which we rewrite here for convenience (from now on X incorporates the integration over PX ) Rij (; Ph ; Sh ) = d 4 ei· 0| i ()|Ph Sh ; X Ph Sh ; X | N j (0)|0: (6.3.1) X

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We have omitted the path-ordered exponential L = P exp(−ig ds A (s)), needed to make (6.3.1) gauge invariant, since in the A+ = 0 gauge a proper path may be chosen such that L = 1. Hereafter the formalism will be similar to that developed in Section 4.1 for A and, therefore, much detail will be suppressed. The quark fragmentation functions are related to traces of the form Tr[DR] = d 4 ei· Tr 0| i ()|Ph Sh ; X Ph Sh ; X | N j (0)D|0; (6.3.2) X

where D is a Dirac matrix. R can be decomposed over a Dirac matrix basis as 1 i R(; Ph ; Sh ) = S5 + V + A 5 + iP5 5 + T 5 ; 2 2

(6.3.3)

where the quantities S, V , A , P5 , T , constructed with the momentum of the fragmenting quark , the momentum of the produced hadron Ph and its spin Sh , have the general form 15 [15,121,127–129] S=

1 Tr(R) = C1 ; 2

(6.3.4a)

V =

1 Tr( R) = C2 Ph + C3 + C10 2

Sh Ph! ;

(6.3.4b)

A =

1 Tr( 5 R) = C4 Sh + C5 · Sh Ph + C6 · Sh ; 2

(6.3.4c)

P5 =

1 Tr( 5 R) = C11 · Sh ; 2i

(6.3.4d)

T =

!

1 Tr( 5 R) = C7 Ph[ Sh] + C8 [ Sh] + C9 · Sh Ph[ ] + C12 2i

!

Ph! :

(6.3.4e)

The quantities Ci = Ci (2 ; · Ph ) are real functions of their arguments, owing to the hermiticity property of R. The presence of the terms with coeKcients C10 , C11 and C12 , which were forbidden in the expansion of the A matrix by time-reversal invariance, is justi?ed by the fact that in the fragmentation case we cannot naYZvely impose a condition similar to (4.1.4c), that is R∗ (; Ph ; Sh ) = 5 CR(; ˜ P˜ h ; S˜ h )C † 5 :

(6.3.5)

In the derivation of (4.1.4c) the simple transformation property of the nucleon state |PS under T is crucial. However, R contains the states |Ph Sh ; X which are out-states with possible ?nal-state interactions between the hadron and the remnants. Under time reversal they do not simply invert their momenta and spin but transform into in-states T |Ph Sh ; X ; out ˙ |P˜ h S˜ h ; X˜ ; in: 15

(6.3.6)

We consider here spin- 12 (or spin-0) hadrons. For the production of spin-1 hadrons see below, Section 6:9.

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71

Fig. 24. A hypothetical mechanism giving rise to a T -odd fragmentation function.

These may di,er non-trivially from |P˜ h S˜ h ; X˜ ; out owing to ?nal-state interactions, which can generate relative phases between the various channels open in the |in → |out transition. Thus, the terms containing C10 , C11 and C12 are allowed in principle. The fragmentation functions related to these terms are called T -odd fragmentation functions [15,128,129]. One of them, called H1⊥ , gives rise to the so-called Collins e,ect [15,17,129]. A generic mechanism giving rise to T -odd fragmentation functions is shown diagrammatically in Fig. 24. What is needed, in order to produce such fragmentation functions, is an interference diagram in which the ?nal-state interaction (represented in ?gure by the dark blob) between the produced hadron and the residual fragments cannot be reabsorbed into the quark–hadron vertex [130]. It has been argued in [131] that the relative phases between the hadron and the X system might actually cancel in the sum over X . This would cause the T -odd distributions to disappear. Only experiments will settle the question. Working in a hN collinear frame, the vectors (or pseudovectors) appearing in (6:3:4a)– (6.3.4e) are Ph ;

1 Ph + T z

and

Sh

5h P + ShT ; Mh h

(6.3.7)

where we have to remember that the transverse components are suppressed by a factor 1=Ph− (that is, 1=Q) compared to the longitudinal ones. To start with, consider the case of collinear kinematics. If we ignore T , at leading twist (that is at order O(Ph− )) the terms contributing to (6.3.3) are V =

1 Tr( R) = B1 Ph ; 2

(6.3.8a)

A =

1 Tr( 5 R) = 5h B2 Ph ; 2

(6.3.8b)

T =

1 ] ; Tr( 5 R) = B3 Ph[ ShT 2i

(6.3.8c)

where we introduced the functions Bi (2 ; · Ph ). The decay matrix then reads R(; Ph ; Sh ) = 12 {B1 P=h + 5h B2 5 P=h + B3 P=h 5 S=hT }:

(6.3.9)

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Recalling that Ph has only a Ph− component, Eqs. (6:3:8a)–(6:3:8c) become 1 − − Tr( R) = B1 ; 2Ph

(6.3.10a)

1 − − Tr( 5 R) = 5h B2 ; 2Ph

(6.3.10b)

1 i− i − Tr(i 5 R) = ShT B3 : 2Ph

(6.3.10c)

The three leading-twist fragmentation functions: the unpolarised fragmentation function Dq (x), the longitudinally polarised fragmentation function LDq (x), and the transversely polarised fragmentation function LT Dq (x), are obtained by integrating B1 , B2 and B3 , respectively, over , with the constraint 1=z = − =Ph− . For instance z d4 D(z) = B1 (2 ; · Ph )(1=z − − Ph− ) 2 (2)4 z d4 = Tr( − R)(− − Ph− =z) 4 (2)4 z d+ iPh− + =z 0| (+ ; 0; 0⊥ )|Ph Sh ; X Ph Sh ; X | N (0) − |0: (6.3.11) = e 4 X 2 The normalisation of D(z) is such that d z z D(z) = 1; h

(6.3.12)

Sh

where h is a sum over all produced hadrons. Hence, D(z) is the number density of hadrons of type h with longitudinal momentum fraction z in the fragmenting quark. Analogously, we have for LD(z) (with 5h = 1) z d4 LD(z) = B2 (2 ; · Ph )(1=z − − =Ph− ) 2 (2)4 z d+ iPh− + =z 0| (+ ; 0; 0⊥ )|Ph Sh ; X Ph Sh ; X | N (0) − 5 |0 (6.3.13) = e 4 X 2 i = (1; 0) for de?niteness) and for LT D(z) (with ShT z d4 LT D(z) = B3 (2 ; · Ph )(1=z − − =Ph− ) 2 (2)4 z d+ iPh− + =z 0| (+ ; 0; 0⊥ )|Ph Sh ; X Ph Sh ; X | N (0)i1− 5 |0: = e 4 X 2

(6.3.14)

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Fig. 25. Kinematics in (a) the ∗ N frame and (b) the hN frame.

Note that LT D(z) is the fragmentation function analogous to the transverse polarisation distribution function LT f(x). In the literature LT D is often called H1 (z) [15]. Introducing the -integrated matrix z d4 R(z) = R(; Ph ; Sh )(1=z − − =Ph− ); (6.3.15) 2 (2)4 the leading-twist structure of the fragmentation process is summarised in the expression of R(z), which is R(z) = 12 {D(z)P=h + 5h LD(z) 5 P=h + LT D(z)P=h 5 S=hT }:

(6.3.16)

The probabilistic interpretation of D(z), LD(z) and LT D(z) is analogous to that of the corresponding distribution functions (see Section 4.3). If we denote by Nh=q (z) the probability of ?nding a hadron with longitudinal momentum fraction z inside a quark q, then we have (using ± to label longitudinal polarisation states and ↑↓ to label transverse polarisation states) D(z) = Nh=q (z);

(6.3.17a)

LD(z) = Nh=q+ (z) − Nh=q− (z);

(6.3.17b)

LT D(z) = Nh=q↑ (z) − Nh=q↓ (z):

(6.3.17c)

6.4. T -dependent fragmentation functions In the collinear case (kT = T = 0) the produced hadron is constrained to have zero transverse momentum (Ph⊥ = − z qT = 0). Therefore, in order to investigate its Ph⊥ distribution within the parton model, one has to account for the transverse motion of quarks (in QCD transverse momenta of quarks emerge at NLO owing to gluon emission). The kinematics in the ∗ N and hN frames is depicted in Fig. 25 (for simplicity the case of no transverse motion of quarks inside the target is illustrated).

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Reintroducing T , we have at leading twist [15,121,127–129] V =

1 1 B Tr( R) = B1 Ph + 2 Mh 1

Ph T! ShT ;

(6.4.1a)

A =

1 1 ˜ B1 T · ShT Ph ; Tr( 5 R) = 5h B2 Ph + 2 Mh

(6.4.1b)

T =

!

1 5 1 ] + h B˜ 2 Ph[ T] + 2 B˜ 3 T · ShT Ph[ T] Tr( 5 R) = B3 Ph[ ShT 2i Mh Mh +

1 B Mh 2

!

Ph! T ;

(6.4.1c)

where we have introduced new functions B˜ i (2 ; · Ph ) (the tilde signals the presence of T ), Bi (2 ; · Ph ) (the prime labels the T -odd terms) and inserted powers of Mh so that all coeKcients have the same dimension. Multiplying Eqs. (6:4:1a)–(6:4:1c) by P yields 1 ij 1 − − Tr( R) = B1 + M B1 T Ti ShTj ; 2Ph h 1 1 Tr( − 5 R) = 5h B2 + T · ShT B˜ 1 ; 2P − Mh 5 1 2T ˜ i− i B3 ShT Tr(i 5 R) = B3 + + h B˜ 2 Ti 2 − 2P Mh 2Mh 1 1 ij 1 j ij − 2 B˜ 3 Ti T + 2T g⊥ ShTj + B Tj : 2 Mh 2 T Mh

(6.4.2a) (6.4.2b)

(6.4.2c)

The eight T -dependent fragmentation functions are obtained from the B coeKcients as follows: 1 d+ d− ) = B1 (2 ; · Ph )(1=z − − =Ph− ); (6.4.3) D(z; 2 T 2z (2)4 etc., where T ≡ −z T is the transverse momentum of the hadron h with respect to the fragmenting quark, see Eq. (6.2.10b). If the transverse motion of quarks inside the target is ignored, then T coincides with Ph⊥ . De?ning the integrated trace 1 d+ d− [D] R ≡ Tr(DR)(− − Ph− =z) 4z (2)4 1 d+ d 2 T i(Ph− + =z−T ·T ) e Tr 0| (+ ; 0; ⊥ )|Ph Sh ; X Ph Sh ; X | N (0)D|0; = 4z X (2)3 (6.4.4)

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75

we obtain from (6.4.2a)–(6.4.2c) −

R[ ] = Nh=q (z; T ) = D(z; 2 T)+ R[

−

5]

1 Mh

ij ⊥ 2 T Ti ShTj D1T (z; T );

= Nh=q (z; T )5 (z; T ) = 5h LD(z; 2 T)+

R[i

i−

(6.4.5a)

5 ]

1 T · ShT G1T (z; 2 T ); Mh

(6.4.5b)

= Nh=q (z; T )s i⊥ (z; T ) 5h i ⊥ H (z; 2 T) Mh T 1L 1 2 ij 1 1 i j ⊥ − 2 T T + T g⊥ ShTj H1T (z; 2 T)+ 2 M Mh h

i = ShT LT D(z; 2 T)+

ij ⊥ 2 T Tj H1 (z; T );

(6.4.5c)

; 5 ) is the spin of the quark and N (z; ) is the probability of ?nding a hadron where s = (s⊥ h=q T with longitudinal momentum fraction z and transverse momentum T = − z T , with respect to the quark momentum, inside a quark q. In (6:4:5a)–(6:4:5c) we have adopted a more traditional notation for the three fragmentation functions, D, LD and LT D, that survive upon integration over T whereas we have resorted ⊥, G , to Mulders’ terminology [15] for the other, less familiar, fragmentation functions, D1T 1T ⊥ ⊥ ⊥ H1L , H1T and H1 (note that in Mulders’ scheme D, LD and LT D are called D1 , G1L and H1T , respectively, and D1 , G1 and H1 , once integrated over T ). The integrated fragmentation 2 functions D(z) and LD(z), are obtained from D(z; 2 T ) and LD(z; T ), via (6.4.6a) D(z) = d 2 T D(z; 2 T );

LD(z) =

d 2 T LD(z; 2 T );

(6.4.6b)

whereas LT D(z) is given by 2 2 2 ⊥ 2 T LT D(z) = d T LT D(z; T ) + H (z; T ) : 2Mh2 1T

(6.4.6c)

Among the unintegrated fragmentation functions, the T -odd quantity H1⊥ (z; 2 T ) plays an important rˆole in the phenomenology of transversity as it is related to the Collins e,ect, i.e., the observation of azimuthal asymmetries in single-inclusive production of unpolarised hadrons at leading twist. In partonic terms, H1⊥ is de?ned—see Eq. (4.8.2b) for the corresponding distribution function h⊥ 1 —via Nh=q↑ (z; T ) − Nh=q↓ (z; T ) =

|T |

Mh

sin(= − =s )H1⊥ (z; 2 T );

(6.4.7)

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Fig. 26. Toy model for fragmentation.

where = and =s are the azimuthal angles of the quark momentum and polarisation, respectively, de?ned in a plane perpendicular to Ph . The angular factor in (6.4.7), that is (recall that Ph is directed along −z) sin(= − =s ) =

( ∧ Ph ) · s ; | ∧ Ph | |s |

(6.4.8)

is related to the so-called Collins angle (see Section 6.5), as we now show. First of all, note that on neglecting O(1=Q) e,ects, azimuthal angles in the plane perpendicular to the hN axis coincide with the azimuthal angles de?ned in the plane perpendicular to the ∗ N axis. Then, if we ignore the intrinsic motion of quarks inside the target, we have T = − Ph⊥ =z and = = =h − :

(6.4.9)

The angle in (6.4.8) is therefore = − =s = =h − =s − = − AC − ;

(6.4.10)

so that sin(= − =s ) = sin AC ;

(6.4.11)

where AC , the azimuthal angle between the spin vector of the fragmenting quark and the momentum of the produced hadron, is what is known as the Collins angle [17]. Just to show how the T -odd fragmentation function H1⊥ may arise from non-trivial ?nal-state interactions, as discussed in Section 6.3, let us consider a toy model [130] (see Fig. 26) that provides a simple example of the mechanism symbolically presented in Fig. 24. Thus, we assume that the quark, with momentum and mass m, fragments into an unpolarised hadron, leaving a remnant which is a point-like scalar diquark. The fragmentation function H1⊥ is contained in the tensor component of the matrix R 1 i R(; Ph ) = · · · + T 5 ; (6.4.12) 2 2 where—see Eq. (6.4.1c), T = · · · +

1 B Mh 2

!

! Ph ;

(6.4.13)

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

so that, using 5 = 12 i /< /< , 1 1 R(; Ph ) = ··· + B Ph : 2 Mh 2

77

(6.4.14)

If we describe the hadron h by a plane wave iPh ·x h (x)∼u(Ph )e

(6.4.15)

it is easy to show that the fragmentation matrix R is i −i R(; Ph ) ∼ u(Ph )u(Ph ) =−m =−m =+m =+m (P=h + Mh ) 2 ; (6.4.16) 2 2 −m − m2 where we have omitted inessential factors. We cannot extract a factor proportional to Ph (hence, producing H1⊥ ) from (6.4.16). Let us now suppose that a residual interaction of h with the intermediate state generates a phase in the hadron wave function. If, for instance, in (6.4.16) we make the replacement (assuming only two fragmentation channels) ∼

u(Ph ) → u(Ph ) + ei1 =u(Ph )

(6.4.17)

by a little algebra one can show that a term of the type (6.4.14) emerges in R, with M B2 ∼ 2 h 2 sin 1: (6.4.18) −m Therefore, if the interference between the fragmentation channels produces a non-zero phase 1, T -odd contributions may appear. The proliferation of channels, however, might lead, as suggested in [131], to the vanishing of such phases and of the resulting T -odd fragmentation functions. Another microscopic mechanism that may give rise to a T -odd fragmentation function has been recently investigated in [132]. Using a simple pseudoscalar coupling between pions and quarks to model the fragmentation process, these authors show that the inclusion of one-loop self-energy and vertex corrections generates a non-vanishing H1⊥ . 6.5. Cross-sections and asymmetries in semi-inclusive leptoproduction We shall now calculate the trace in (6.2.12). At leading twist, as already mentioned, transverse polarisation distributions are probed by unpolarised lepton beams. In this case, the leptonic tensor is symmetric and couples to the symmetric part of W , that is 1 2 d k − d 2 kT d+ d 2 T W (S) = ea 2 a (2)4 (2)4 ×2 (kT + qT − T ) Tr[A { R } ]k + =xP+ ; − =P− =z : h

(6.5.1)

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Using the Fierz identity we can decompose the trace in (6.5.1) as Tr[A { R } ] = 12 {Tr[A]Tr[R] + Tr[iA 5 ]Tr[iR 5 ] − Tr[A / ]Tr[R / ] − Tr[A / 5 ]Tr[R / 5 ] + 12 Tr[iA/< 5 ]Tr[iR/< 5 ]}g

+ 12 Tr[A { ]Tr[R } ] + 12 Tr[A { 5 ]Tr[R } 5 ] }

+ 12 Tr[iA/{ 5 ]Tr[iR/ 5 ]:

(6.5.2)

If we insert Eqs. (4:7:2a)–(4.7.2c) and (6:4:1a)–(6.4.1c) into (6.5.1) and integrate over k − and + making use of Eqs. (4:7:8a)–(4.7.8c) and (6:4:5a)–(6.4.5c), after some algebra we obtain [15,121,127,128] (S) 2 2 W =2 ea z d T d 2 kT 2 (kT + qT − T ) a

×

−gT

f(x; kT2 )D(z; 2 T)

1 + Mh

! 2 ⊥ 2 T T! ShT f(x; kT ) D1T (z; T )

{ }

−[ST ShT + ST · ShT gT ]LT f(x; kT2 )LT D(z; 2 T) { }

[ST T + ST · T gT ]T · ShT ⊥ LT f(x; kT2 ) H1T (z; 2 T) Mh2 # { }! { }! ST T T! + T T ST! − LT f(x; kT2 ) H1⊥ (z; 2 T) + ··· : 2Mh

−

(6.5.3)

In (6.5.3) we have considered only unpolarised and transversely polarised terms, and we have omitted the kT -dependent contributions (in the following we shall assume that transverse motion of quarks inside the target can be neglected). Neglecting higher-twist (i.e., O(1=Q)) contributions, the transverse (T ) vectors and tensors appearing in (6.5.3) coincide with the corresponding perpendicular (⊥) vectors and tensors. The (S) is performed by means of the identities [15] contraction of W (S) with the leptonic tensor L 2Q2 [1 + (1 − y)2 ]; y2 4Q2 (1 − y) { } (S) [a⊥ b⊥ + a⊥ · b⊥ g⊥ ]L = |a⊥ ||b⊥ | cos(=a + =b ); y2 4Q2 (1 − y) 1 { }! { }! (S) =− |a⊥ ||b⊥ |sin(=a + =b ); [a⊥ ⊥ b⊥! + b⊥ ⊥ a⊥! ]L 2 y2 (S) g⊥ L = −

(6.5.4a) (6.5.4b) (6.5.4c)

where =a and =b are the azimuthal angles in the plane perpendicular to the photon–nucleon axis. Combining Eq. (6.5.3) with Eqs. (6:5:4a)–(6.5.4b) leads quite straightforwardly to the parton-model formul^ for the cross-sections. To obtain the leading-order QCD expressions, one must simply insert the Q2 dependence into the distribution and fragmentation functions.

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6.5.1. Integrated cross-sections Consider, ?rst of all, the cross-sections integrated over Ph⊥ . In this case, the kT and T integrals decouple and can be performed, yielding the integrated distribution and fragmentation functions. Hence we obtain 2 s d 1 4/em 2 ea x = [1 + (1 − y)2 ]fa (x)Da (z) 4 d x dy d z Q 2 a −(1 − y)|S⊥ | |Sh⊥ | cos(=S + =Sh )LT fa (x)LT Da (z) : (6.5.5) As one can see, at leading twist, the transversity distributions are probed only when both the target and the produced hadron are transversely polarised. From (6.5.5) we can extract the transverse polarisation Ph of the detected hadron, de?ned so that (‘unp’ = unpolarised) d = dunp (1 + Ph · Sh ):

(6.5.6)

↑ If we denote by Phy the transverse polarisation of h along y, when the target nucleon is → the transverse polarisation of h along x, when the target polarised along y (↑), and by Phx nucleon is polarised along x (→), we ?nd 2 ea LT fa (x)LT Da (z) 2(1 − y) ↑ → a Phy = − Phx = : (6.5.7) 2 2 1 + (1 − y) a ea fa (x)Da (z)

If the hadron h is not transversely polarised, or—a fortiori—is spinless, the leading-twist

Ph⊥ -integrated cross-section does not contain LT f. In this case, in order to probe the transversity distributions, one has to observe the Ph⊥ distributions, or consider higher-twist contributions

(Section 6.6). In the next section we shall discuss the former possibility.

6.5.2. Azimuthal asymmetries We now study the (leading-twist) Ph⊥ distributions in semi-inclusive DIS and the resulting azimuthal asymmetries. We shall assume that the detected hadron is spinless, or that its polarisation is not observed. For simplicity, we also neglect (at the beginning, at least) the transverse motion of quarks inside the target. Thus (6.5.3) simpli?es as follows (recall that only the unpolarised and the transversely polarised terms are considered) W (S) = 2 ea2 z d 2 T 2 (T + Ph⊥ =z) a

×

−g⊥ f(x)D(z; 2 T)

−

{ }! T T!

(ST

{ }! T ST! )

+ T 2Mh

#

LT f(x)H1⊥ (z; 2 T) + ··· : (6.5.8)

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Contracting W (S) with the leptonic tensor (6.1.14a) and inserting the result into (6.1.11) gives the cross-section 2 s 4/em d 1 2 2 = ea x ) [1 + (1 − y)2 ]fa (x)Da (z; Ph⊥ 2 4 d x dy d z d Ph⊥ Q 2 a |Ph⊥ | ⊥ 2 + (1 − y) |S |sin(=S + =h )LT fa (x)H1a (z; Ph⊥ ) : (6.5.9) zMh ⊥ From this we obtain the transverse single-spin asymmetry d(S⊥ ) − d(−S⊥ ) d(S⊥ ) + d(−S⊥ ) 2 0 2 2(1 − y) a ea LT fa (x)LT Da (z; Ph⊥ ) |S⊥ | sin(=S + =h ): = 2 2 1 + (1 − y)2 a ea fa (x)Da (z; Ph⊥ )

AhT ≡

(6.5.10)

2 ) as—see (6.4.7) Here we have de?ned the T -odd fragmentation function L0T D(z; Ph⊥ 2 L0T D(z; Ph⊥ )=

|Ph⊥ |

zMh

2 H1⊥ (z; Ph⊥ ):

(6.5.11)

Note that our L0T D is related to LN D of [26] by L0T D = LN D=2 (our notation is explained in Section 1.2). The existence of an azimuthal asymmetry in transversely polarised leptoproduction of spinless hadrons at leading twist, which depends on the T -odd fragmentation function H1⊥ and arises from ?nal-state interaction e,ects, was predicted by Collins [17] and is now known as the Collins e,ect. The Collins angle AC was originally de?ned in [17] as the angle between the transverse spin vector of the fragmenting quark and the transverse momentum of the outgoing hadron, i.e., AC = =s − =h :

(6.5.12)

Thus, one has sin AC =

(q ∧ P h ) · s : |q ∧ Ph | |s |

(6.5.13)

Since, as dictated by QED (see Section 6.7), the directions of the ?nal and initial quark spins are related to each other by (see Fig. 27) =s = − =s ;

(6.5.14)

(6.5.13) becomes AC = − =s − =h . Ignoring the transverse motion of quarks in the target, the initial quark spin is parallel to the target spin (i.e., =s = =S ) and AC can ?nally be expressed in terms of measurable angles as AC = − =S − =h :

(6.5.15)

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Fig. 27. The transverse spin vectors and the transverse momentum of the outgoing hadron in the plane perpendicular to the ∗ N axis. AC is the Collins angle.

If the transverse motion of quarks in the target is taken into account the cross-sections become more complicated. We limit ourselves to a brief overview of them. Let us start from the unpolarised cross-section, which reads 2 s e2 dunp 4/em a = (6.5.16) x[1 + (1 − y)2 ]I [fa Da ]; d x dy d z d 2 Ph⊥ Q4 2 a where we have introduced the integral I , de?ned as [15] I [f D](x; z) ≡ d 2 kT d 2 T 2 (kT + qT − T ) f(x; kT2 ) D(z; 2 T)

=

d 2 kT f(x; kT2 )D(z; |Ph⊥ − z kT |2 ):

The cross-section for a transversely polarised target takes the form 2 s d(S⊥ ) 4/em = |S | ea2 x(1 − y) ⊥ Q4 d x dy d z d 2 Ph⊥ a hˆ · ⊥ ⊥ ×I LT fa H1a sin(=S + =h ) + · · · ; Mh

(6.5.17)

(6.5.18)

where hˆ ≡ Ph⊥ = |Ph⊥ | and a term giving rise to a sin(3=h − =S ) asymmetry, but not involving LT f, has been omitted. As we shall see in Section 9.2.2 there are presently some data on semi-inclusive DIS o, nucleons polarised along the scattering axis, that are of a certain interest

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for the study of transversity. It is therefore convenient, in view of the phenomenological analysis of those measurements, to give also the unintegrated cross-section for a longitudinally polarised target, which, although not containing LT f, depends on the Collins fragmentation function H1⊥ , a crucial ingredient in the phenomenology of transversity. One ?nds 2 s d(5N ) 4/em =− 5N ea2 x(1 − y) 2 4 d x dy d z d Ph⊥ Q a 2(hˆ · ⊥ )(hˆ · k⊥ ) − ⊥ · k⊥ ⊥ ⊥ ×I h1La H1a sin(2=h ): (6.5.19) MMh Note the characteristic sin(2=h ) dependence of (6.5.18) and the appearance of the k⊥ -dependent distribution function h⊥ 1L . One can factorise the x and z dependence in the above expressions by properly weighting the cross-sections with some function that depends on the azimuthal angles [16,133]. This procedure also singles out the di,erent contributions to the cross-section for a given spin con?guration of the target (and of the incoming lepton). To see how it works let us consider the case of a transversely polarised target. We rede?ne the azimuthal angles so that the orientation of the lepton plane is given by a generic angle =‘ in the transverse space. Eq. (6.5.18) then becomes 2 s d(S⊥ ) hˆ · ⊥ 2/em 2 ⊥ = |S⊥ | ea x(1 − y)I LT fa H1a sin(=S + =h − 2=‘ ) d x dy d z d=‘ d 2 Ph⊥ Q4 M h a (6.5.20) + sin(3=h − =S − 2=‘ ) term The weighted cross-section that projects the ?rst term of (6.5.20) out, and leads to a factorised expression in x and z, is d(S⊥ ) |P | d=‘ d 2 Ph⊥ h⊥ sin(=S + =h − 2=‘ ) Mh z d x dy d z d=‘ d 2 Ph⊥ =

2 s 4/em ⊥(1) |S | ea2 x(1 − y)LT fa (x)H1a (z); ⊥ Q4 a

(6.5.21)

where the weighted fragmentation function H1⊥(1) is de?ned as 2 T ⊥(1) 2 2 H1 (z) = z d T H1⊥ (z; z 2 2T ): (6.5.22) 2Mh2 For a more complete discussion of transversely polarised semi-inclusive leptoproduction, with or without intrinsic quark motion, we refer the reader to the vast literature on the subject [15,16,48,121,126,127,133–138]. In Section 9.2 we shall present some predictions and some preliminary experimental results on AhT . 6.6. Semi-inclusive leptoproduction at twist-three Let us now see how transversity distributions appear at the higher-twist level. We shall consider only twist-three contributions and limit ourselves to quoting the main results without derivation (which may be found in [15]).

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83

If the lepton beam is unpolarised, the cross-section for leptoproduction of unpolarised (or spinless) hadrons with a transversely polarised target is 2 s d(S⊥ ) M 2 4/em |S | e 2(2 − y) 1−y = ⊥ a d x dy d z Q4 Q a Mh H˜ a (z) × sin =S xLT fa (x) M z # a ˜ M (z) H h L − 5h cos =S x2 gTa (x)LDa (z) + ; (6.6.1) xLT fa (x) M z where the factor M=Q signals that (6.6.1) is a twist-three quantity. Adding (6.6.1) to the transverse component of (6.5.5) gives the complete Ph⊥ -integrated cross-section of semi-inclusive DIS o, a transversely polarised target up to twist-three. Note that in (6.6.1) the leading-twist transversity distributions LT f(x) are coupled to the twist-three fragmentation functions H˜ (z) and H˜ L (z), while the leading-twist helicity fragmentation function LD(z) is coupled to the twist-three distribution gT (x). H˜ (z) is a T -odd fragmentation function. At twist-three, the transversity distributions also contribute to the scattering of a longitudinally polarised lepton beam. The corresponding cross-section is 2 s d(5l ; S⊥ ) 4/em M 2 5 |S | e 2y 1 − y =− ⊥ l d x dy d z Q4 Q a a ˜ M (z) E a h × cos =S x2 gTa (x)Da (z) + xLT fa (x) M z # a E˜ L (z) Mh + 5h sin =S : (6.6.2) xLT f(x) M z Here, again, the leading-twist transversity distributions LT f(x) are coupled to the twist-three ˜ fragmentation functions E(z) and E˜ L (z), and the leading-twist unpolarised fragmentation function D(z) is coupled to the twist-three distribution gT (x). E˜ L (z) is a T -odd fragmentation function. Up to order O(1=Q), there are no other observables in semi-inclusive leptoproduction involving transversity distributions. The twist-two and twist-three contributions to semi-inclusive leptoproduction involving the transversity distributions LT f are collected in Tables 3 and 4. 6.7. Factorisation in semi-inclusive leptoproduction It is instructive to use a di,erent approach, based on QCD factorisation, to rederive the results on semi-inclusive DIS presented in Section 6.5. We start by considering the collinear case, that is ignoring the transverse motion of quarks both in the target and in the produced hadron. In this case a factorisation theorem is known to hold. This theorem was originally demonstrated for the production of unpolarised particles [139 –143] and then also shown to apply if the detected

84

V. Barone et al. / Physics Reports 359 (2002) 1–168 Table 3 The contributions to the Ph⊥ -integrated cross-section involving the transversity distributionsa Cross-section integrated over Ph⊥ ‘

N

h

observable

Twist-two

0

T

T

LT f(x)LT D(z)

Twist-three

0 0 L L

T T T T

0 L 0 L

LT f(x)H˜ (z)(∗) LT f(x)H˜ L (z) ˜ LT f(x)E(z) LT f(x)E˜ L (z)(∗)

a

T; L and 0 denote transverse, longitudinal and no polarisation, respectively. The asterisk labels T -odd observables. Table 4 Contributions to the Ph⊥ distributions involving transversity. The produced hadron h is taken to be unpolariseda Ph⊥ distribution (h unpolarised) ‘

N

observable

Twist-two

0

T

LT f ⊗ H1⊥ (∗)

Twist-three

0 0 L

T T T

LT f ⊗ H˜ (∗) LT f ⊗ H1⊥ (∗) LT f ⊗ E˜

a

The notation is as in Table 3.

particles are polarised [144]. In contrast, when the transverse motion of quarks is taken into account, factorisation is not proven and can only be regarded as a reasonable assumption. 6.7.1. Collinear case The QCD factorisation theorem states that the cross-section for semi-inclusive DIS can be written, to all orders of perturbation theory, as J J d = d dO fa (; )!5 5 d ˆ 55 JJ (x=; Q=; /s ())Dh=b (O; ); (6.7.1) ab 55 JJ

where ab is a sum over initial (a) and ?nal (b) partons, !55 is the spin density matrix of parton a in the nucleon, and d ˆ is the perturbatively calculable cross-section of the hard subprocesses that contribute to the reaction. In (6.7.1) is the fraction of the proton momentum carried by the parton a; O is the fraction of the momentum of parton b carried by the produced hadron, and is the factorisation scale. Lastly, Dh=b (z) is the fragmentation matrix of parton b

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85

Fig. 28. Lepton–quark (antiquark) scattering.

into the hadron h ++ Dh=b Dh=b = −+ Dh=b

+− Dh=b −− Dh=b

:

(6.7.2)

It is de?ned in such a manner that 1 JJ 1 ++ −− D (z) = [Dh=b (z) + Dh=b (z)] = Dh=b (z); 2 J h=b 2

(6.7.3)

where Dh=b (z) is the usual unpolarised fragmentation function, that is the probability of ?nding a hadron h with longitudinal momentum fraction z inside a parton b. The di,erence of the diagonal elements of Dh=b (z) gives the longitudinal polarisation fragmentation function ++ 1 2 [Dh=b (z)

−− − Dh=b (z)] = 5h LDh=b (z);

(6.7.4a)

whereas the o,-diagonal elements are related to transverse polarisation +− 1 2 [Dh=b (z)

−+ + Dh=b (z)] = Shx LT Dh=b (z);

(6.7.4b)

+− 1 2 [Dh=b (z)

−+ − Dh=b (z)] = Shy LT Dh=b (z):

(6.7.4c)

Note that Dh=b is normalised such that for an unpolarised hadron it reduces to the unit matrix. At lowest order the only N → lq(q) N (see elementary process contributing to d ˆ is lq(q) Fig. 28). Thus, the sum runs only over quarks and antiquarks, and a = b. Eq. (6.7.1) ab then becomes (omitting energy scales) dO d d ˆ J J E Eh 3 3 = d 2 fa ()!5 5 E E Dh=a (O): (6.7.5) 3 3 d l d Ph O d l d 55 JJ a 55 JJ

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The elementary cross-section in (6.7.5) is (sˆ = xs is the centre-of-mass energy squared of the partonic scattering, with the hat labelling quantities de?ned at the subprocess level) d ˆ 1 1 E E = M5/J< M5∗ /J < 4 (‘ + k − ‘ − ) 3 3 d l d 55 JJ 322 sˆ 2 = where

d ˆ dy

55 JJ

=

sy ˆ 2

/<

d ˆ dy

55 JJ

4 (‘ + k − ‘ − );

1 1 M5/J< M5∗ /J < ; 16sˆ 2

(6.7.6)

(6.7.7)

/<

with the sum being performed over the helicities of the incoming and outgoing leptons. Working in the hN collinear frame, where the photon momentum is q −xP + z −1 Ph + qT , that is, in light-cone components, q (−xP + ; z −1 Ph− ; qT ), the energy–momentum conservation delta function may be written as 4 (‘ + k − ‘ − ) = 4 (q + k − ) (q+ + k + )(q− − − )2 (qT ) =

2xz ( − x)(O − z) 2 (qT ): Q2

(6.7.8)

The integrations over and O in (6.7.5) can now be performed and the cross-section for semi-inclusive DIS (expressed in terms of the invariants x; y; z and of the transverse momentum of the outgoing hadron Ph⊥ ) reads d ˆ d J J = fa ()!5 5 Dh=a (z)2 (Ph⊥ ): (6.7.9) 2 d x dy d z d Ph⊥ dy 55 JJ a 55 JJ

Note the 2 (Ph⊥ ) factor coming from the kinematics of the hard subprocess at lowest order. Integrating the cross-section over the hadron transverse momentum we obtain d d ˆ JJ fa ()!55 Dh=a (z): (6.7.10) = d x dy d z dy 55 JJ a 55 JJ

Let us now look at the helicity structure of the lq scattering process. By helicity conservation, the only non-vanishing scattering amplitudes are (y = − tˆ= sˆ = 12 (1 − cos Q)) M++++ = M− − − − = 4i e2 ea M+−+− = M−+−+ = 2i e2 ea

1 1 = 2i e2 ea ; cos Q y

1 + cos Q 1−y = 2i e2 ea ; 1 − cos Q y

(6.7.11a) (6.7.11b)

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where Q is the scattering angle in the lq centre-of-mass frame. The elementary cross-sections contributing to (6.7.10) are % 1 1$ d ˆ d ˆ |M++++ |2 + |M+−+− |2 = = dy ++++ dy − − − − 16sˆ 2 =

d ˆ dy

d ˆ = dy +−+−

−+−+

=

' 4/2 xs ea2 & 1 + (1 − y)2 ; 4 Q 2

1 4/2 xs 2 ∗ = e (1 − y) Re M++++ M+−+− 16sˆ Q4 a

and the cross-section (6.7.10) then reads d ˆ d −− ++ fa (x) (!++ Dh=a + !− − Dh=a ) = d x dy d z dy ++++ a d ˆ +− −+ + !+− Dh=a + !−+ Dh=a : dy +−+−

(6.7.12a) (6.7.12b)

(6.7.13)

Inserting (6:7:12a), (6.7.12b) into (6.7.13) and using (4:3:7)–(4:3:8b) and (6:7:3)–(6:7:4c), we obtain 2 s d 4/em ea2 x = d x dy d z Q4 a 1 × [1 + (1 − y)2 ][fa (x)Dh=a (z) + 5N 5h Lfa (x)LDh=a (z)] 2 + (1 − y)|S⊥ ||Sh⊥ | cos(=S + =Sh )LT fa (x)LT Dh=a (z) (6.7.14) which coincides with the result already obtained in Section 6.5. In the light of the present derivation of (6.7.14), we understand the origin of the y-dependent factor in (6.5.7) and (6.5.10). This factor, aˆT ≡

2(1 − y) d ˆ +−+− = d ˆ ++++ 1 + (1 − y)2

(6.7.15)

is a spin transfer coeKcient, i.e., the transverse polarisation of the ?nal quark generated by an initial transversely polarised quark in the lq → lq process. To see this, let us call HJJ the quantity d ˆ HJJ ≡ !5 5 (6.7.16) dy 55 JJ and introduce the spin density matrix of the ?nal quark, de?ned via HJJ = Hunp !JJ ;

(6.7.17)

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where Hunp = H++ + H− − = (d ) ˆ ++++ : We ?nd explicitly !++ ! = aˆT !+−

aˆT !−+ !− −

(6.7.18)

;

(6.7.19)

and, recalling that the ?nal quark travels along −z, we ?nally obtain for its spin vector s sx = − aˆT sx ;

sy = aˆT sy :

(6.7.20)

Thus, the initial and ?nal quark spin directions are specular with respect to the y axis. The factor aˆT is also known as the depolarisation factor. It decreases with y, being unity at y = 0 and zero at y = 1. 6.7.2. Non-collinear case If quarks are allowed to have transverse momenta, QCD factorisation is no longer a proven property, but only an assumption. In this case we write, in analogy with (6.7.5), d dO 2 E Eh 3 3 = d d k d 2 T Pa (; kT )!5 5 T 2 d l d Ph O a 55 JJ

×E E

d ˆ 3 d l d 3

55 JJ

JJ Dh=a (O; T );

(6.7.21)

where Pa (; kT ) is the probability of ?nding a quark a with momentum fraction x and transverse momentum kT inside the target nucleon, and Dh=a (O; T ) is the fragmentation matrix of quark a into the hadron h, having transverse momentum T ≡ = − z T with respect to the quark momentum. Evaluating Eq. (6.7.21), as we did with Eq. (6.7.5), we obtain the cross-section in 2 terms of the invariants x; y; z and of Ph⊥ d 2 = d kT d 2 T Pa (; kT )!5 5 d x dy d z d 2 Ph⊥ a 55 JJ

d ˆ × dy

55 JJ

JJ Dh=a (z; T )2 (z kT − T − Ph⊥ ):

(6.7.22)

Inserting the elementary cross-sections (6:7:12a), (6.7.12b) in (6.7.22) and writing explicitly the sum over the helicities, we obtain 2 s 4/em d 2 2 = ea x d kT d 2 T Pa (x; kT ) d x dy d z d 2 Ph⊥ Q4 a 1 −− ++ × + !− − Dh=a ] [1 + (1 − y)2 ][!++ Dh=a 2

V. Barone et al. / Physics Reports 359 (2002) 1–168 +− −+ + (1 − y)[!+− Dh=a + !−+ Dh=a ]

89

×2 (z kT − T − Ph⊥ ):

(6.7.23)

Let us suppose now that the hadron h is unpolarised. Using the correspondence (4.3.7) between the spin density matrix elements and the spin of the initial quark, and the analogous relations for the fragmentation matrix obtained from (6:4:5a)–(6.4.5c), that is 1 ++ 1 ij −− ⊥ ) = D(z; 2 Ti ShTj D1T (z; 2 (6.7.24a) (D + Dh=a T)+ T ); 2 h=a Mh T 1 1 ++ −− ) = 5h LD(z; 2 T · ShT G1T (z; 2 (6.7.24b) (Dh=a − Dh=a T)+ T ); 2 Mh 1 +− 5h 1 ⊥ −+ 1 ) = ShT LT D(z; 2 H (z; 2 (Dh=a + Dh=a T)+ T) 2 Mh T 1L 1 1 1j 1 j 1j ⊥ − 2 T1 T + 2T g⊥ ShTj H1T (z; 2 Tj H1⊥ (z; 2 T)+ T ); 2 Mh T Mh (6.7.24c) −

1 +− 5h 2 ⊥ −+ 2 ) = ShT LT D(z; 2 H (z; 2 (D − Dh=a T)+ T) 2i h=a Mh T 1L 1 2 2j 1 1 1 j ⊥ − 2 T T + T g⊥ ShTj H1T (z; 2 T)+ 2 Mh Mh

2j ⊥ 2 T Tj H1 (z; T );

(6.7.24d)

the transverse polarisation contribution to the cross-section turns out to be 2 s d(S⊥ ) 4/em 2 2 = − e x(1 − y) d k d 2 T Pa (x; kT ) T a d x dy d z d 2 Ph⊥ Q4 a 1 ⊥ 2 (sx Ty + sy Tx )H1a (z; 2 (6.7.25) T ) (z kT − T − Ph⊥ ): Mh If, for simplicity, we neglect the transverse momentum of the quarks inside the target, then s⊥ Pa (x) = S⊥ LT fa (x). The integration over T can be performed giving the constraint T ≡ −z T = Ph⊥ , and Eq. (6.7.25) becomes, with our convention for the axes and azimuthal angles 2 s d(S⊥ ) 4/em |P | ⊥ 2 = |S | ea2 x(1 − y)LT fa (x) h⊥ H1a (z; Ph⊥ ) sin(= + =S ): ⊥ 2 4 d x dy d z d Ph⊥ Q zM h a ×

(6.7.26)

Since T = − Ph⊥ =z, we have = + =S = =h − + =S = AC − and (6.7.26) reduces to the transverse polarisation term of (6.5.9).

(6.7.27)

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Fig. 29. Two-particle leptoproduction.

6.8. Two-hadron leptoproduction Another partially inclusive DIS reaction that may provide important information about transversity is two-particle leptoproduction (see Fig. 29): l(‘) + N (P) → l (‘ ) + h1 (P1 ) + h2 (P2 ) + X (PX )

(6.8.1)

with the target transversely polarised. In this reaction two hadrons (for instance, two pions) are detected in the ?nal state. Two-hadron leptoproduction has been proposed and studied by various authors [124,130,131, 145] as a process that can probe the transverse polarisation distributions of the nucleon, coupled to some interference fragmentation functions. The idea is to look at angular correlations of the form (P1 ∧ P2 ) · s , where P1 and P2 are the momenta of the two produced hadrons is the transverse spin vector of the fragmenting quark. These correlations are not forand s⊥ bidden by time-reversal invariance owing to ?nal-state interactions between the two hadrons. To our knowledge, the ?rst authors who suggested resonance interference as a way to produce non-diagonal fragmentation matrices of quarks were Cea et al. [146] in their attempt to explain the observed transverse polarisation of hyperons produced in pN interactions [3]. Hereafter we shall consider an unpolarised lepton beam and unpolarised hadrons in the ?nal state. The cross-section for the reaction (6.8.1) reads, cf. Eq. (6.1.7) 1 e4 d 3 l d 3 P1 d 3 P2 4 d = L W (2) ; (6.8.2) 4‘ · P Q4 (2)3 2E (2)3 2E1 (2)3 2E2 where L is the usual leptonic tensor, Eq. (6.1.5), and W is the hadronic tensor 1 d 3 PX (2)4 4 (P + q − PX − P1 − P2 ) W = 4 (2) X (2)3 2EX ×PS |J (0)|X; P1 P2 X; P1 P2 |J (0)|PS :

(6.8.3)

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91

Fig. 30. Diagram contributing to two-hadron leptoproduction at lowest order.

Following [130], we introduce the combinations Ph ≡ P1 + P2 ; and the invariants P · P1 z1 = ; P·q

R ≡ 12 (P1 − P2 )

(6.8.4)

P · P2 ; P·q

(6.8.5)

z2 =

P · Ph z2 z1 P · P1 =1 − ; = = P·q z P · Ph z in terms of which the cross-section becomes 2 /em d y = L W : 2 2 3 4 d x dy d z d d Ph⊥ d R⊥ 4(2) Q (1 − )z Using z = z1 + z2 =

2 d 2 R⊥ = 12 d R⊥ d=R = 12 (1 − ) dMh2 d=R ;

(6.8.6)

(6.8.7)

(6.8.8)

where Mh2 = Ph2 = (P1 + P2 )2 is the invariant-mass squared of the two hadrons and =R is the azimuthal angle of R in the plane perpendicular to the ∗ N axis, the cross-section can then be re-expressed as 2 y d /em = (6.8.9) L W : d x dy d z d d 2 Ph⊥ dMh2 d=R 2(2)3 Q4 z In the parton model (see Fig. 30) the hadronic tensor has a form similar to that of the single-particle case d k − d 2 kT d+ d 2 T W = eq2 (2)4 (2)4 q ×2 (kT + qT − T ) Tr[A > ]k + =xP+ ; − =P− =z ; h

(6.8.10)

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except that there appears now a decay matrix for the production of a pair of hadrons d 4 O ei·O 0| i (O)|P1 P2 ; X P1 P2 ; X | N j (0)|0: (6.8.11) >ij (; P1 ; P2 ) = X

Working in a frame where P and Ph are collinear (transverse vectors in this frame are denoted, as usual, by a T subscript), the matrix (6.8.11) can be decomposed as was (6.3.3). At leading twist the contributing terms are (remember that hadrons h1 and h2 are unpolarised) V = 12 Tr( >) = B1 Ph ;

1 2

A = Tr( 5 >) = T =

1 B M1 M2 1

(6.8.12a) !

Ph R! T ;

1 1 [B Tr( 5 >) = 2i M1 + M2 2

!

Ph! T + B3

(6.8.12b) !

Ph! R ];

(6.8.12c)

where M1 and M2 are the masses of h1 and h2 , respectively. In (6.8.12c) Bi and Bi are functions of the invariants constructed with , P, Ph and R. The prime labels the so-called T -odd terms (but one should bear in mind that T -invariance is not actually broken). Multiplying Eqs. (6:8:12a)–(6.8.12c) by P results in 1 Tr( − >) = B1 ; 2Ph−

(6.8.13a)

1 1 − ij − Tr( 5 >) = M M B1 T RTi Tj ; 2Ph 1 2

(6.8.13b)

1 1 i− ij ij − Tr(i 5 >) = M + M [B2 T Tj + B3 T RTj ]: 2Ph 1 2

(6.8.13c)

Introducing the integrated trace 1 d+ d− 1 − [D] − Tr(D>) − Ph > = 4z (2)4 z 1 dO+ d 2 T i(Ph− O+ =z−T ·T ) = e 4z X (2)3 ×Tr 0| (O+ ; 0; 0⊥ )|P1 P2 ; X P1 P2 ; X | N (0)D|0;

(6.8.14)

we can rewrite Eqs. (6:8:13a)–(6.8.13c) as [130] −

2 >[ ] = Nh1 h2 =q (z; ; T ; RT ) = D(z; ; 2 T ; RT ; T · RT );

>[

−

5]

= Nh1 h2 =q (z; ; T ; RT )5q =

1 M1 M2

ij ⊥ 2 2 T RTi Tj G1 (z; ; T ; RT ; T

(6.8.15a) · RT );

(6.8.15b)

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>[i

i−

5 ]

93

i = Nh1 h2 =q (z; ; T ; RT )s⊥

=

1 2 [ ij Tj H1⊥ (z; ; 2 T ; RT ; T · RT ) M1 + M2 T

ij 2 2 ˜⊥ (6.8.15c) T RTj H 1 (z; ; T ; RT ; T · RT )]; where Nh1 h2 =q (z; ; T ; RT ) is the probability for a quark q to produce two hadrons h1 ; h2 . ⊥ In Eq. (6.8.15c), D, G1⊥ , H1⊥ and H˜ 1 are interference fragmentation functions of quarks ⊥ into a pair of unpolarised hadrons. In particular, H1⊥ and H˜ 1 are related to quark transverse polarisation in the target. H1⊥ has an analogue in the case of single-hadron production (where it ⊥ has been denoted by the same symbol), while H˜ 1 is a genuinely new function. It is important ⊥ to notice that H˜ 1 is the only fragmentation function, besides D, that survives when the quark

+

transverse momentum is integrated over. The symmetric part W (S) of the hadronic tensor, the component contributing to the crosssection when the lepton beam is unpolarised (as in our case), is given by (with the same notation as in Section 6.5 and retaining only the unpolarised and the transverse polarisation terms) W (S) = 2 ea2 z d 2 T d 2 kT 2 (kT + qT − T ) a

×

−gT f(x; kT2 )D(z; 2 T)

−

{ }! T T!

ST

{ }!

+ T T ST! 2(M1 + M2 )

2 ×LT f(x; kT2 )H1⊥ (z; ; 2 T ; RT ; T · RT ) −

{ }! T RT!

ST

# ×

⊥ 2 LT f(x; kT2 )H˜ 1 (z; ; 2 T ; RT ; T

{ }!

+ RT T ST! 2(M1 + M2 )

· RT ) + · · · :

(6.8.16)

Let us now neglect the intrinsic motion of quarks inside the target. This implies that (S)

T =−Ph⊥ =z. Contracting W (S) with the leptonic tensor L by means of the relations (6.5.4a)– (6.5.4c) and integrating over Ph⊥ , we obtain the cross-section (limited to the unpolarised and

transverse polarisation contributions) d d x dy d z d dMh2 d=R 2 s 1 4/em 2 = ea x [1 + (1 − y)2 ]fa (x)Da (z; ; Mh2 ) 3 4 (2) Q a 2 + (1 − y)

|S⊥ ||R⊥ |

M1 + M2

⊥ 2 ˜ sin(=S + =R )LT fa (x)H 1a (z; ; R⊥ ) :

2 . The fragmentation functions appearing here are integrated over Ph⊥

(6.8.17)

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We de?ne now the interference fragmentation function LT I (z; ; Mh2 ) as |R⊥ | ˜ ⊥ H (z; ; Mh2 ) ˙ Nh1 h2 =q↑ (z; ; R⊥ ) − Nh1 h2 =q↓ (z; ; R⊥ ); LT I (z; ; Mh2 ) = M1 + M2 1 where, we recall, 2 R⊥ = (1 − )Mh2 − (1 − )M12 − M22 :

Integrating (6.8.17) over , we ?nally obtain 2 s d 1 4/em 2 = ea x [1 + (1 − y)2 ]fa (x)Da (z; Mh2 ) 2 3 4 2 d x dy d z dMh d=R (2) Q a 2 + (1 − y)|S⊥ | sin(=S + =R )LT fa (x)LT Ia (z; Mh ) : From (6.8.20) we obtain the transverse single-spin asymmetry d(S⊥ ) − d(−S⊥ ) AhT1 h2 ≡ d(S⊥ ) + d(−S⊥ ) 2 2 2(1 − y) a ea LT fa (x)LT Ia (z; Mh ) |S⊥ | sin(=S + =R ); = 2 2 1 + (1 − y)2 a ea fa (x)Da (z; Mh )

(6.8.18)

(6.8.19)

(6.8.20)

(6.8.21)

which probes the transversity distributions along with the interference fragmentation function LT I . We can introduce, into two-hadron leptoproduction, the analogue of the Collins angle AC of single-hadron leptoproduction, which we call AC . We de?ne AC as the angle between the ?nal and R , i.e., quark transverse spin s⊥ ⊥ AC ≡ =s − =R :

(6.8.22)

We have sin AC ≡

(P h ∧ R ) · s ( P 2 ∧ P 1 ) · s = : |Ph ∧ R||s | |P2 ∧ P1 ||s |

(6.8.23)

Since =s = − =s , where =s is the azimuthal angle of the initial quark transverse spin, we can also write AC = − =s − =R :

(6.8.24)

If the initial quark has no transverse momentum with respect to the nucleon, then =s = =S and AC is given, in terms of measurable angles, by AC = − =S − =R :

(6.8.24 )

In the language of QCD factorisation the cross-section for two-hadron leptoproduction is written as d D(z; Mh2 ; =R ) d d = fa (x)!5 5 : (6.8.25) dy 55 JJ d x dy d z dMh2 d=R dMh2 d=R J J a 55 JJ

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95

Fig. 31. Leptoproduction of two hadrons h1 and h2 via resonance (h; h ) formation. Fig. 32. The factor sin 0 sin 1 sin(0 − 1 ) obtained from phase shifts (?gure from [131]).

What we have found above is that the fragmentation matrix d D=dMh2 d=R factorises into z- and Mh2 -dependent fragmentation functions and certain angular coeKcients. For the case at hand, the angular dependence is given by the factor sin(=S + =R ) in (6.8.20). An explicit mechanism giving rise to an interference fragmentation function like LT I has been suggested by Ja,e et al. [131,147] (a similar mechanism was considered earlier in a di,erent but related context [146]). The process considered in [131,147] and shown diagrammatically in Fig. 31 is the production of a + − pair, via formation of a (I = 0, L = 0) and ! (I = 1, L = 1) resonance. The single-spin asymmetry then arises from interference between the s- and p-wave of the pion system. Similar processes are K production near to the K ∗ resonance, and K KN production near to the =. In all these cases two mesons, h1 and h2 , are generated from the decay of two resonances h (L = 0) and h (L = 1). The ?nal state can be written as a superposition of two resonant states with di,erent relative phases |h1 h2 ; X = ei0 |h; X + ei1 |h ; X :

(6.8.26)

The interference between the two resonances is proportional to sin(0 − 1 ). The values of 0 and 1 depend on the invariant mass Mh of the two-meson system. It turns out that the interference fragmentation function LT I has the following structure LT I (z; Mh2 )∼ sin 0 sin 1 sin(0 − 1 )LT Iˆ(z; Mh2 );

(6.8.27)

2 where the phase factor √ sin 0 sin 1 sin(0 − 1 ) depends on Mh . The maximal value that this factor can attain is 3 3=8.

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The two-hadron spin-averaged fragmentation function Dh1 h2 (z; Mh2 ) is the superposition of the unpolarised fragmentation functions of the two resonances weighted by their phases Dh1 h2 (z; Mh2 ) = sin 2 0 Dh (z) + sin 2 1 Dh (z):

(6.8.28)

The resulting single-spin asymmetry is then d(S⊥ ) − d(−S⊥ ) AhT1 h2 ≡ d(S⊥ ) + d(−S⊥ ) ˙

2(1 − y) |S | sin 0 sin 1 sin(0 − 1 )sin(=S + =R ) 1 + (1 − y)2 ⊥ 2 2 ˆ a ea LT fa (x)LT I a (z; Mh ) : × 2 2 f (x)[sin 2 D ˆ ˆ =a (z)] D e (z) + sin a 0 1 h=a h a a

(6.8.29)

We remark that the angle = de?ned in [131] corresponds to our =S − =R − =2. In the case of two-pion production, 0 and 1 can be obtained from the data on phase shifts [148]. The factor sin 0 sin 1 sin(0 − 1 ) is shown in Fig. 32. It is interesting to observe that the experimental value of this quantity reaches 75% of its theoretical maximum. 6.9. Leptoproduction of spin-1 hadrons As ?rst suggested by Ji [149] (see also [150]) the transversity distribution can be also probed in leptoproduction of vector mesons (e.g., !; K ∗ ; =). The fragmentation process into spin-1 hadrons has been fully analysed, from a formal viewpoint, in [151,152]. The polarisation state of a spin-1 particle is described by a spin vector S and by a rank-2 spin tensor T ij . The latter x ; S y ; S xy ; S xx [151]. The transversity distribution contains ?ve parameters, usually called SLL ; SLT LT TT TT LT f emerges when an unpolarised beam strikes a transversely polarised target. The cross-section in this case is [151] (we retain only the terms containing LT f and we use the notation of Section 6.5.2) 2 s 4/em d(S⊥ ) = |S | ea2 x(1 − y) ⊥ d x dy d z d 2 Ph⊥ Q4 a a × |SLT | sin (=LT + =S )I [LT fa H1LT ]

+ |STT | sin (2=TT + =S − =h )I

+ SLL sin(=S + =h )I

hˆ · ⊥

Mh

Mh

a LT fa H1TT

⊥a LT fa H1LL

+ |SLT | sin(=LT

hˆ · ⊥

2(hˆ · ⊥ )2 − 2⊥ ⊥a − =S − 2=h )I LT fa H1LT 2Mh2

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

− |STT | sin(2=TT − =S − 3=h ) # (hˆ · ⊥ )[2(hˆ · ⊥ )2 − 32⊥ =2] ⊥a ×I LT fa H1TT + ···: 3

Mh

97

(6.9.1)

For simplicity, we omitted the subscript h in the tensor spin parameters (which are understood to pertain to the produced hadron). The azimuthal angles =LT and =TT are de?ned by xy Sy STT tan =LT = LT ; tan = = (6.9.2) TT x xx SLT STT and ( ( x )2 + (S y )2 ; xx )2 + (S xy )2 : |SLT | = (SLT | S | = (STT (6.9.3) TT LT TT ⊥ ; Note that LT f couples in (6.9.1) to ?ve di,erent fragmentation functions: H1LT ; H1TT ; H1LL All these functions are T -odd. If we integrate the cross-section over Ph⊥ only one term survives, namely 2 s d(S⊥ ) 4/em a |S || S | sin(= + = ) ea2 x(1 − y)LT fa (x)H1LT (z): (6.9.4) = LT LT S ⊥ d x dy d z Q4 a ⊥ ; H⊥ . H1LT 1TT

The fragmentation function appearing here, H1LT , is called hˆ1N by Ji [149]. It is a T -odd and chirally odd function which can be measured at leading twist and without considering intrinsic transverse momenta. Probing the transversity by (6.9.4) requires polarimetry on the produced meson. For a self-analysing particle this can be done by studying the angular distribution of its decay products (e.g., !0 → + − ). Thus, the vector-meson fragmentation function H1LT represents a speci?c contribution to two-particle production near the vector-meson mass. 6.10. Transversity in exclusive leptoproduction processes Let us now consider the possibility of observing the transversity distributions in exclusive leptoproduction processes. Collins et al. [153] remarked that the exclusive production of a transversely polarised vector meson in DIS, that is lp → lVp, involves the chirally odd o,-diagonal parton distributions in the proton. These distributions (also called “skewed” or “o,-forward” distributions) depend on two variables x and x since the incoming and outgoing proton states have di,erent momenta P and P , with (P − P)2 = t (the reader may consult [154 –156] on skewed distributions). For instance, the o,-forward transversity distribution (represented in Fig. 33a) contains a matrix element of the form PS | N (0) + ⊥ 5 (− )|P S . At low x the difference between x and x is small and the o,-diagonal distributions are completely determined by the corresponding diagonal ones. In [157] it was shown that the chirally odd contribution to vector-meson production (see Fig. 33b) is actually zero at LO in /s . This result was later extended in [158], where it was observed that the vanishing of the chirally odd contribution is due to angular momentum and chirality conservation in the hard scattering and hence holds at leading twist to all orders in the strong coupling. Thus, the (o,-diagonal) transversity distributions cannot be probed in exclusive vector-meson leptoproduction.

98

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Fig. 33. The o,-diagonal transversity distribution (a) and its contribution to exclusive vector-meson production (b).

7. Transversity in hadronic reactions The second class of reactions probing quark transversity is hadron scattering with at least one of the two colliding particles in a transverse polarisation state. We shall ?rst consider the case where both initial hadrons are transversely polarised. In particular, Drell–Yan production with two transversely polarised hadrons turns out to be the most favourable reaction for studying the transversity distributions. Indeed, the pioneering work of Ralston and Soper [9] and Pire and Ralston [159] concentrated precisely on this process. We shall then see how transversity may emerge even when only one of the colliding hadrons is transversely polarised. This possibility, however, is more uncertain. 7.1. Double-spin transverse asymmetries When both hadrons are transversely polarised, the typical observables are double-spin transverse asymmetries of the form d(ST ; ST ) − d(ST ; −ST ) ATT = : (7.1.1) d(ST ; ST ) + d(ST ; −ST ) Since there is no gluon transversity distribution for spin-half hadrons, transversely polarised pp reactions which are dominated at the partonic level by qg or gg scattering are expected to yield a very small ATT [21,160]. Thus, direct-photon production (with lowest-order subprocesses N and two-gluon-jet gq → q and qqN → g), heavy-quark production (qqN → QQN and gg → QQ), production (gg → gg and qqN → gg) do not seem to be promising reactions to detect quark transverse polarisation. The only good candidate process for measuring transversity in doubly polarised pp (or pp) N DY collisions is Drell–Yan lepton pair production [9,14,28]. We shall see that at lowest order ATT contains combinations of the products N B ): LT f(xA )LT f(x The advantage of studying quark transverse polarisation via Drell–Yan is twofold: (i) transversity distributions appear at leading-twist level; (ii) the cross-section contains no unknown quantities, besides the transversity distributions themselves. This makes theoretical predictions relatively easier, with respect to other reactions.

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99

7.2. The Drell–Yan process Drell–Yan lepton-pair production is the process A(P1 ) + B(P2 ) → l+ (‘) + l− (‘ ) + X;

(7.2.1)

where A and B are protons or antiprotons and X is the undetected hadronic system. The centre-of-mass energy squared of this reaction is s = (P1 + P2 )2 2P1 · P2 (in the following, the hadron masses M1 and M2 will be systematically neglected, unless otherwise stated). The lepton pair originates from a virtual photon (or from a Z 0 ) with four-momentum q = ‘ + ‘ . Note that, in contrast to DIS, q is a time-like vector: Q2 = q2 ¿0. This is also the invariant mass of the lepton pair. We shall consider the deeply inelastic limit where Q2 ; s → ∞, while the ratio E = Q2 =s is ?xed and ?nite. The Drell–Yan (DY) cross-section is d =

2 82 /em d 3 ‘ d3 ‘ 4 2L W (2) ; sQ4 (2)3 2E (2)3 2E

(7.2.2)

where the leptonic tensor, neglecting lepton masses and ignoring their polarisation, is given by Q2 L = 2 ‘ ‘ + ‘ ‘ − (7.2.3) g 2 and the hadronic tensor is de?ned as 1 d 3 PX (2)4 4 (P1 + P2 − q − PX ) W = (2)4 X (2)3 2EX ×P1 S1 ; P2 S2 |J (0)|X X |J (0)|P1 S1 ; P2 S2 1 = d 4 eiq· P1 S1 ; P2 S2 |J (z)J (0)|P1 S1 ; P2 S2 : 4

(2)

(7.2.4)

The phase space in (7.2.2) can be rewritten as d3 ‘ d: d 4 q d 3 ‘ = ; (2)3 2E (2)3 2E 8(2)6

(7.2.5)

where : is the solid angle identifying the direction of the leptons in their rest frame. Using (7.2.5) the DY cross-section takes the form 2 d /em L W : = d 4 q d: 2sQ4

(7.2.6)

We de?ne now the two invariants x1 =

Q2 ; 2P1 · q

x2 =

Q2 : 2P2 · q

(7.2.7)

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In the parton model x1 and x2 will be interpreted as the fractions of the longitudinal momenta of the hadrons A and B carried by the quark and the antiquark which annihilate into the virtual photon. In a frame where the two colliding hadrons are collinear (A is taken to move in the positive z direction), the photon momentum can be parametrised as q =

Q2 Q2 P + P + qT : x2 s 1 x1 s 2

(7.2.8)

Neglecting terms of order O(1=Q2 ), one ?nds Q2 =x1 x2 s 1, that is E = Q2 =s = x1 x2 , and therefore q = x1 P1 + x2 P2 + qT = (x1 P1+ ; x2 P2− ; qT ):

(7.2.9)

Note that x1

P2 · q ; P1 · P2

x2

P1 · q : P1 · P2

(7.2.10)

In terms of x1 , x2 and qT the DY cross-section reads 2 d /em L W : = d x1 d x2 d 2 qT d: 4Q4

(7.2.11)

It is customary [9,161] to introduce three vectors Z , X and Y de?ned as P1 · q P2 · q P1 − P ; Z = P1 · P2 P1 · P2 2 1 [P2 · Z P˜ 1 − P1 · Z P˜ 2 ]; X = − P1 · P2 1 ! P1 P2! q : Y = P1 · P2

(7.2.12a) (7.2.12b) (7.2.12c)

where P˜ 1; 2 = P1; 2 − (P1; 2 · q=q2 )q . These vectors are mutually orthogonal and orthogonal to q , and satisfy Z 2 −Q2 ;

X 2 Y 2 −qT2 :

(7.2.13)

Thus, they form a set of spacelike axes and have only spatial components in the dilepton rest frame. Using (7.2.9), Z , X and Y can be expressed as Z = x1 P1 − x2 P2 ;

X = qT ;

Y =

T qT ;

(7.2.14)

where T

≡

1 P1 · P2

!

P1 P2! :

In terms of the unit vectors X Y ; yˆ = √ ; xˆ = √ −X 2 −Y 2

(7.2.15) Z zˆ = √ ; −Z 2

q qˆ = ; Q2

(7.2.16)

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101

Fig. 34. Drell–Yan dilepton production.

Fig. 35. The geometry of Drell–Yan production in the rest frame of the lepton pair.

the lepton momenta can be expanded as ‘ = 12 q + 12 Q(sin Q cos = xˆ + sin Q sin = yˆ + cos Q zˆ );

(7.2.17a)

‘ = 12 q − 12 Q(sin Q cos = xˆ + sin Q sin = yˆ + cos Q zˆ ):

(7.2.17b)

The geometry of the process in the dilepton rest frame is shown in Figs. 34 and 35. The leptonic tensor then reads L = − 1 Q2 [(1 + cos2 Q)g − 2 sin2 Q zˆ Oˆ 2

⊥

+ 2 sin2 Q cos 2=(xˆ xˆ + 12 g⊥ ) + sin2 Q sin 2= xˆ{ yˆ }

+ sin 2Q cos = zˆ{ xˆ} + sin 2Q sin = zˆ{ yˆ } ];

(7.2.18)

where = g − qˆ qˆ + zˆ zˆ : g⊥

(7.2.19)

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Fig. 36. The parton-model diagram for the Drell–Yan hadronic tensor.

In the parton model, calling k and k the momenta of the quark (or antiquark) coming from hadron A and B, respectively, the hadronic tensor is (see Fig. 36) 1 2 d4 k d4 k 4 W = ea (k + k − q)Tr[A1 AN 2 ]: (7.2.20) 3 a (2)4 (2)4 Here A1 is the quark correlation matrix for hadron A, Eq. (4.1.1), AN 2 is the antiquark correlation matrix for hadron B, Eq. (4.2.9), and the factor 1=3 has been added since in A1 and AN 2 summations over colours are implicit. It is understood that, in order to obtain the complete expression of the hadronic tensor, one must add to (7.2.20) a term with A1 replaced by A2 and AN 2 replaced by AN 1 , which accounts for the case where a quark is extracted from B and an antiquark is extracted from A. In the following formul^ we shall denote this term symbolically by [1 ↔ 2]. Hereafter the quark transverse motion (which is discussed at length in [161]) will be ignored and only the ordinary collinear con?guration will be considered. We now evaluate the hadronic tensor in a frame where A and B move collinearly, with large longitudinal momentum. Setting k 1 P1 , k 2 P2 and using (7.2.9), the delta function in (7.2.20) gives 4 (k + k − ) = (k + + k + − q+ )(k − + k − − q− )2 (qT ) (1 P1+ − x1 P1+ )(2 P2− − x2 P2− )2 (qT )

=

1

(1 P1+ P2−

− x1 )(2 − x2 )2 (qT ):

(7.2.21)

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103

The hadronic tensor then becomes 1 2 d k − d 2 kT d k + d 2 kT 2 W = ea (q T ) 3 a (2)4 (2)4 ×Tr[A1 AN 2 ]k + =x1 P+ ;k − =x2 P− + [1 ↔ 2]: 1

(7.2.22)

2

Since the leptonic tensor is symmetric, only the symmetric part of W contributes to the cross-section. For Tr[A1 { AN 2 } ] we resort to the Fierz decomposition (6.5.2), with the replacements A → A1 , R → AN 2 . Using (4:2:5a)–(4.2.5c) and (4:2:12)–(4:2:14), we then obtain (the spins of the two hadrons are S1 = (S1T ; 51 ) and S2 = (S2T ; 52 )) 1 2 d 2 qT W = e {−gT [fa (x1 )fNa (x2 ) + 51 52 Lfa (x1 )LfNa (x2 )] 3 a a − [S1T S2T + S1T · S2T gT ]LT fa (x1 )LT fNa (x2 )} + [1 ↔ 2]: { }

(7.2.23)

In contracting the leptonic and the hadronic tensors, it is convenient to pass from the AB collinear frame to the ∗ A collinear frame. We recall that, at leading twist, the transverse (T ) vectors approximately coincide with the vectors perpendicular to the photon direction (denoted by a subscript ⊥): S1T S1⊥ , S2T S2⊥ , gT g⊥ . Therefore, the contraction L W can be performed by means of the following identities: −g⊥ L = Q2 (1 + cos2 Q); {

(7.2.24a)

}

[S1⊥ S2⊥ + S1⊥ · S2⊥ g⊥ ]L = − Q2 sin2 Q|S1⊥ ||S2⊥ |cos(2= − =S1 − =S2 );

(7.2.24b)

where Q is the polar angle of the lepton pair in the dilepton rest frame and =S1 (=S2 ) is the azimuthal angle of S1⊥ (S2⊥ ), measured with respect to the lepton plane. For the Drell–Yan cross-section we ?nally obtain /2 ea2 d = em2 {[fa (x1 )fNa (x2 ) + 51 52 Lfa (x1 )LfNa (x2 )](1 + cos2 Q) d: d x1 d x2 4Q a 3 + |S1⊥ ||S2⊥ |cos(2= − =S1 − =S2 )LT fa (x1 )LT fNa (x2 ) sin2 Q} + [1 ↔ 2]:

(7.2.25)

From this we derive the parton-model expression for the double transverse asymmetry: ADY TT =

d(S1⊥ ; S2⊥ ) − d(S1⊥ ; −S2⊥ ) d(S1⊥ ; S2⊥ ) + d(S1⊥ ; −S2⊥ )

sin2 Q cos(2= − =S1 − =S2 ) = |S1⊥ | |S2⊥ | 1 + cos2 Q

2 N a ea LT fa (x1 )LT fa (x2 ) + [1 2 N a ea fa (x1 )fa (x2 ) + [1 ↔

↔ 2]

2]

; (7.2.26)

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and we see that a measurement of ADY TT directly provides the product of quark and antiquark transverse polarisation distributions LT f(x1 )LT f(x2 ), with no mixing with other unknown quantities. Thus, the Drell–Yan process seems to be, at least in principle, a very good reaction to probe transversity. Note that in leading-order QCD, Eq. (7.2.26) is still valid, with Q2 dependent distribution functions, namely ADY TT = |S1⊥ | |S2⊥ | ×

sin2 Q cos(2= − =S1 − =S2 ) 1 + cos2 Q

ea2 LT fa (x1 ; Q2 )LT fNa (x2 ; Q2 ) + [1 ↔ 2] : 2 2 N 2 a ea fa (x1 ; Q )fa (x2 ; Q ) + [1 ↔ 2]

a

(7.2.27)

Here LT f(x; Q2 ) are the transversity distributions evolved at LO. In Section 9.1 we shall see some predictions for ADY TT . 7.2.1. Z 0 -mediated Drell–Yan process If Drell–Yan dilepton production is mediated by the exchange of a Z 0 boson, the vertex ei , where ei is the electric charge of particle i (quark or lepton), is replaced by (Vi + Ai 5 ) , where the vector and axial-vector couplings are Vi = T3i − 2ei sin2 #W ; Ai = T3i :

(7.2.28)

The weak isospin T3i is + 12 for i = u and − 12 for i = l− ; d; s. The resulting double transverse asymmetry has a form similar to (7.2.26), with the necessary changes in the couplings. Omitting the interference contributions, it reads Z ADY; = |S1⊥ | |S2⊥ | TT

×

2 a (Va

sin2 Q cos(2= − =S1 − =S2 ) 1 + cos2 Q

− A2a )LT fa (x1 )LT fNa (x2 ) + [1 ↔ 2]

2 a (Va

+ A2a )fa (x1 )fNa (x2 ) + [1 ↔ 2]

:

(7.2.29)

7.3. Factorisation in Drell–Yan processes With a view to extending the results previously obtained to NLO in QCD, we now rederive them in the framework of QCD factorisation [144]. The Drell–Yan cross-section is written in a factorised form as (hereafter we omit the exchanged term [1 ↔ 2]) 2 (2) N 2 d1 d2 !(1) ˆ 2 =2 ; /s (2 ))]// << ; (7.3.1) d = / / fa (1 ; )!< < fa (2 ; )[d (Q a // <<

where 1 and 2 are the momentum fractions of the quark (from hadron A) and antiquark (from B), !(1) and !(2) are the quark and antiquark spin density matrices, and (d ) ˆ // << is the cross-section matrix (in the quark and antiquark spin space) of the elementary subprocesses. As usual, denotes the factorisation scale.

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Fig. 37. The qqN → l+ l− process contributing to Drell–Yan production at LO.

At LO d ˆ incorporates a delta function of energy–momentum conservation, namely 4 (k + k − q), which sets 1 = x1 , 2 = x2 and qT = 0. Thus Eq. (7.3.1) becomes (omitting the scales) (1) (2) d ˆ d = !/ / !< < fa (1 )fNa (2 ); (7.3.2) d: d x1 d x2 d: << // a // <<

where the only subprocess is qqN → l+ l− (see Fig. 37) and its cross-section is 1 ∗ d ˆ = M/< M/ < : d: // << 642 sˆ

(7.3.3)

The contributing scattering amplitudes are M++++ = M− − − − ;

M++− − = M− −++

and the cross-section (7.3.2) reduces to (1) (2) d ˆ d (1) (2) = (!++ !++ + !− − !− − ) d: d x1 d x2 d: ++++ a d ˆ (1) (2) (1) (2) + (!+− !+− + !−+ !−+ ) fa (x1 )fNa (x2 ): d: +−+−

(7.3.4)

(7.3.5)

Here the spin density matrix elements are, for the quark 1 !(1) ++ = 2 (1 + 5); 1 !(1) +− = 2 (sx − isy );

1 !(1) − − = 2 (1 − 5); 1 !(1) −+ = 2 (sx + isy )

(7.3.6a)

and for the antiquark 1 N !(2) ++ = 2 (1 + 5); 1 !(2) +− = 2 (sNx − isNy );

1 N !(2) − − = 2 (1 − 5); 1 !(2) −+ = 2 (sNx + isNy );

(7.3.6b)

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so that (7.3.5) becomes d 1 d ˆ d ˆ N = (1 + 55) + (sx sNx − sy sNy ) fa (x1 )fNa (x2 ): d: d x1 d x2 2 a d: ++++ d: +−+−

(7.3.7)

At LO, the scattering amplitudes for the qqN annihilation process are M++++ = M− − − − = e2 ea (1 − cos Q);

(7.3.8a)

M++− − = M− −++ = e2 ea (1 + cos Q)

(7.3.8b)

and the elementary cross-sections then read d ˆ d ˆ 1 = = (|M++++ |2 + |M++− − |2 ) d: ++++ d: − − − − 642 sˆ 2 e2 1 /em a (1 + cos2 Q); 2sˆ 3 d ˆ d ˆ 1 /2 e 2 1 ∗ = = M++− − ) = em a sin2 Q: 2 Re(M++++ 2 d: +−+− d: −+−+ 64 sˆ 2sˆ 3

(7.3.9a)

=

(7.3.9b)

Inserting (7:3:9a, b) into (7.3.7) we obtain d /2 ea2 N = em2 {(1 + 55)(1 + cos2 Q) + (sx sNx − sy sNy )sin2 Q}fa (x1 )fNa (x2 ): d: d x1 d x2 4Q a 3

(7.3.10)

Using 5fa (x1 ) = 51 Lfa (x1 );

s⊥ fa (x1 ) = S1⊥ LT fa (x1 );

(7.3.11a)

5NfNa (x2 ) = 52 LfNa (x2 );

sN⊥ fNa (x2 ) = S2⊥ LT fNa (x2 )

(7.3.11b)

we obtain /2 ea2 d = em2 {[fa (x1 ) fNa (x2 ) + 51 52 Lfa (x1 )LfNa (x2 )](1 + cos2 Q) d: d x1 d x2 4Q a 3 + |S1⊥ | |S2⊥ |cos(2= − =S1 − =S2 )LT fa (x1 )LT fNa (x2 )sin2 Q};

(7.3.12)

which is what we obtained in Section 7.2 in a di,erent manner (see Eq. (7.2.25)). Note that the angular factor appearing in ADY TT —Eq. (7.2.26)—is the elementary double-spin transverse asymmetry of the qqN scattering process, namely d ( ˆ s⊥ ; sN⊥ ) − d ( ˆ s⊥ ; −sN⊥ ) aˆTT ≡ d ( ˆ s⊥ ; sN⊥ ) + d ( ˆ s⊥ ; −sN⊥ ) =

sin2 Q d ˆ +−+− cos(2= − =s − =sN): (sx sNx − sy sNy ) = d ˆ ++++ 1 + cos2 Q

(7.3.13)

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The Drell–Yan cross-section is most often expressed as a function of the rapidity of the virtual photon y de?ned as y≡

1 q + 1 x1 = ln : ln 2 q− 2 x2

In the lepton c.m. frame y = 12 (1 + cos Q). From x1 x2 ≡ E = Q2 =s, we obtain √ √ x1 = Eey ; x2 = Ee−y

(7.3.14)

(7.3.15)

and (7.3.12) becomes (dy dQ2 = s d x1 d x2 ) 2 e2 d /em a = {[fa (x1 )fNa (x2 ) + 51 52 Lfa (x1 )LfNa (x2 )](1 + cos2 Q) d: dy dQ2 4Q2 s a 3 + |S1⊥ | |S2⊥ |cos(2= − =S1 − =S2 )LT fa (x1 )LT fNa (x2 )sin2 Q}:

(7.3.16)

If we integrate over cos Q, we obtain 2 d 2/em e2 {[fa (x1 )fNa (x2 ) + 51 52 Lfa (x1 )LfNa (x2 )] = dy dQ2 d= 9Q2 s a a + 12 |S1⊥ | |S2⊥ |cos(2= − =S1 − =S2 )LT fa (x1 )LT fNa (x2 )}:

(7.3.17)

Let us now extend the previous results to NLO. Here we are interested in the transverse polarisation contribution to the Drell–Yan cross-section, which can be written as (reintroducing the scales) dT d ˆ T (Q2 =2 ; /s ) 2 e d d (7.3.18) = LT fa (1 ; 2 )LT fa (2 ; 2 ): 1 2 a 2 d= dy dQ2 d= dy dQ a We have seen that at LO, i.e., O(/s0 ), the elementary cross-section is LO :

d ˆ T /2 = em2 cos(2= − =s − =sN)(1 − x1 )(2 − x2 ); 2 dy dQ d= 9Q s

(7.3.19)

where =s (=sN) is the azimuthal angle of the quark (antiquark) spin, with respect to the lepton plane. Integrating over y we obtain LO :

2 d ˆ T /em = cos(2= − =s − =sN) (1 − z) dQ2 d= 9Q2 sˆ

(7.3.20)

with z ≡ E=1 2 = Q2 =1 2 s. At NLO, i.e., O(/s ), the subprocesses contributing to Drell–Yan production are those shown in Fig. 38: virtual-gluon corrections and real-gluon emission. The NLO cross-section d ˆ T =dy dQ2 d= exhibits ultraviolet singularities (arising from loop integrations), infrared singularities (due to soft gluons), and collinear singularities (when the gluon is emitted parallel to the quark or the antiquark). Summing virtual and real diagrams, only the collinear divergences survive. Working in d = 4 − 2j they are of the type 1= j. These singularities are subtracted and absorbed in the de?nition of the parton distributions.

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Fig. 38. Elementary processes contributing to the transverse Drell–Yan cross-section at NLO: (a, b) virtual-gluon corrections and (c) real-gluon emission.

The NLO elementary cross-sections have been computed by several authors with di,erent methods [61,74,108,162]. 16 Vogelsang and Weber [108] were the ?rst to perform this calculation using massive gluons to regularise the divergences. Soon after the authors of [162] presented a calculation based on dimensional reduction. The result was then translated into dimensional regularisation in [61]. As a check of the expression given in [61], Vogelsang [74] has shown how to exploit the earlier result obtained in [108]. From the detailed structure of the collinear singularities for both dimensional and o,-shell regularisation, it is straightforward to translate results from one scheme to another. The expression for d ˆ T =dy dQ2 d= is rather cumbersome and we do not repeat it here (instead, we refer the reader to the original papers). The y-integrated cross-section is more legible and N scheme [74] reads, in the MS 2 d ˆ T /em NLO : = cos(2= − =s − =sN) dQ2 d= 9Q2 sˆ ln(1 − z) 4z ln z 6z ln2 z ×CF 8z − − 1−z + 1−z 1−z # 2 2 + 4(1 − z) + − 8 (1 − z) : (7.3.21) 3 (1) The quantity in curly brackets is the NLO Wilson coeKcient LT CDY for Drell–Yan. If we call ˜ LT C the quantity to be added to the Wilson coeKcient in order to pass from the scheme of [61] to MS, the result (7.3.21) coincides with that of [61] for the choice LT C˜ = − (1 − z). On the other hand, the expression for LT C˜ claimed in [61] as providing the translation to the MS scheme (in dimensional regularisation) is LT C˜ = − (1 − z)+2(1 − z). In [74] it is noted that the reason for this di,erence lies in the discrepancy between the calculation presented there and that of [61] for the (4 − 2j)-dimensional LO splitting function, where extra O(j) terms were found. The correctness of the result in [74] for this quantity is supported by the observation that the d-dimensional 2 → 3 squared matrix element for the process qqN → + − g (with transversely polarised incoming (anti)quarks) given there factorises into the product of the d-dimensional 16

We recall that NLO corrections to unpolarised Drell–Yan were presented in [163,164], and to longitudinally polarised Drell–Yan in [165].

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109

(0) 2 → 2 squared matrix element for qqN → + − multiplied by the splitting function LT Pqq in d = 4 − 2j dimensions, when the collinear limit of the gluon aligning parallel to one of the incoming quarks is correctly performed. It is thus claimed that the result of [61] for the NLO transversely polarised Drell–Yan cross-section (in dimensional regularisation) corresponds to a di,erent (non-MS) factorisation scheme. Finally, a ?rst step towards a next-to-next-to-leading order (NNLO) calculation of the transversely polarised Drell–Yan cross-section was taken in [166].

7.3.1. Twist-three contributions to the Drell–Yan process At twist-three, transversity distributions are also probed in Drell–Yan processes when one of the two hadrons is transversely polarised and the other is longitudinally polarised. In this case, ignoring subtleties related to quark masses and transverse motion (so that hL (x) = h˜L (x) and gT (x) = g˜T (x), see Section 4.9), the cross-section is [14,46,61] (the transversely polarised hadron is A) 2 e2 d /em a · · · + |S1⊥ |52 sin 2Q cos(= − =S1 ) = d: d x1 d x2 4Q2 a 3 # 2M 2M1 a a 2 × (7.3.22) x1 gT (x1 )LfNa (x2 ) + x2 LT fa (x1 )hNL (x2 ) ; Q Q where the dots denote the leading-twist contributions presented in Eq. (7.2.25). The transversity distribution of quarks in hadron A is coupled to the twist-three antiquark distribution hNL . The longitudinal–transverse asymmetry resulting from (7.3.2) is (we assume the masses of the two hadrons to be equal, i.e., M1 = M2 ≡ M ) ADY LT = |S1⊥ |52 ×

a

2 sin 2Q cos(= − =S1 ) M 1 + cos2 Q Q a

ea2 [x1 gTa (x1 )LfNa (x2 ) + x2 LT fa (x1 )hNL (x2 )] : 2 N a ea fa (x1 )fa (x2 )

(7.3.23)

Let us now consider the case where one of the two hadrons is unpolarised while the other is transversely polarised. Time-reversal invariance implies the absence of single-spin asymmetries, even at twist-three. Such asymmetries might arise as a result of initial-state interactions that generate T -odd distribution functions. If such a mechanism occurs, relaxing the naive time-reversal invariance condition (see Section 4.8), the Drell–Yan cross-section acquires extra terms and (7.3.22) becomes 2 e2 d /em a · · · + |S1⊥ | sin 2Q sin(= − =S1 ) = d: d x1 d x2 4Q2 a 3 # 2M 2M1 ˜ a 2 × (7.3.24) x1 fT (x1 )fNa (x2 ) + x2 LT fa (x1 )h˜Na (x2 ) : Q Q

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Fig. 39. One of the diagrams contributing to the Drell–Yan cross-section at twist-three.

˜N Here f˜T (x) and h(x) are the twist-three T -odd distribution functions introduced in Section 4.8. From (7.3.24) we obtain the single-spin asymmetry ADY T = |S1⊥ | ×

2 sin 2Q sin(= − =S1 ) M 1 + cos2 Q Q

a

ea2 [x1 fTa (x1 )fNa (x2 ) + x2 LT fa (x1 )hNa (x2 )] : 2 N a ea fa (x1 )fa (x2 )

(7.3.25)

The existence of T -odd distribution functions has also been advocated by Boer [47] to explain, at leading-twist level, an anomalously large cos 2= term in the unpolarised Drell–Yan cross-section [167–169], which cannot be accounted for by LO or NLO QCD [170] (it can however be attributed to higher-twist e,ects [171–173]). Boer has shown that, on introducing initial-state T -odd e,ects, the unpolarised Drell–Yan cross-section indeed acquires a cos 2= con2 2 N⊥ tribution involving the product h⊥ 1 (x1 ; k⊥ )h1 (x2 ; k⊥ ). If hadron A is transversely polarised, the 2 )hN⊥ (x ; k 2 ). same mechanism generates a sin(= + =S1 ) term, which depends on LT f(x1 ; k⊥ 2 ⊥ 1 It must be stressed once again that the mechanism based on initial-state interactions is highly hypothetical, if not at all unlikely. However, it was shown in [46,174] that single-spin asymmetries might arise in Drell–Yan processes owing to the so-called gluonic poles in twist-three multiparton correlation functions [175 –177]. Let us briePy address this issue (for a general discussion of higher twists in hadron scattering see [34,35,178–180]). At twist-three the Drell–Yan process is governed by diagrams such as that in Fig. 39. The hadronic tensor is then (we drop the subscripts 1 and 2 from the quark correlation matrices for simplicity) 1 d4 k d4 k 2 N ] W = ea (k + k − q)Tr[A A 3 a (2)4 (2)4

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

−

d4 k (2)4

× Tr

d4 k (2)4

111

k=˜ − q= (k˜ − q)2 + ij

d 4 k˜ (k + k − q) (2)4 # N AA A + · · · ;

(7.3.26)

where we have retained only one of the twist-three contributions, and AA is the quark–quark– gluon correlation matrix de?ned in (4.9.6). Neglecting 1=Q2 terms, the quark propagator in (7.3.26) gives (k˜ = y1 P1 ) − x1 − y1 k=˜ − q= →− : (7.3.27) − 2x2 P2 x1 − y1 + ij (k˜ − q)2 + ij Let us now introduce another quark–quark–gluon correlator dE dJ iEy iJ(x−y) AFij (x; y) = PS |=N j (0)F + (Jn)=i (En)|PS ; e e 2 2

(7.3.28)

which can be parametrised as, see the analogous decomposition of AA (x; y) Eq. (4.9.10), M AF (x; y) = {iGF (x; y) ⊥ S⊥ P= + G˜ F (x; y)S⊥ 5 P= 2 + HF (x; y)5N 5 ⊥ P= + EF (x; y) ⊥ P= }:

In the A+ = 0 gauge one has F + = @+ A⊥ and relation between AA (x; y) and AF (x; y) (x − y)AA (x; y) = − iAF (x; y): Thus, if some projection of AF (x; x) is non-zero,

(7.3.29)

by partial integration one ?nds the following (7.3.30) AA (x; x)

the corresponding projection of must have a pole (the “gluonic pole”). From (7.3.27) and (7.3.30), we see that the trace in the twist-three term of (7.3.26) contains the quantity (P.V. stands for principal value) x1 − y1 −i AA (x1 ; y1 ) = A (x1 ; y1 ) x1 − y1 + ij x1 − y1 + ij F = P:V:

−i AF (x; y) − (x1 − y1 )AF (x1 ; x1 ): x−y

(7.3.31)

Keeping the real term in (7.3.31) one ultimately ?nds that the Drell–Yan cross-section with one transversely polarised hadron and one unpolarised hadron involves, at twist-three, the multiparton distributions GF (x1 ; x1 ) and EF (x1 ; x1 ) (the former is proportional to the distribution T (x1 ; x1 ) introduced in [174,176]). The single-spin asymmetry is then expressed as 2 sin 2Q sin(= − =S1 ) M ADY T ˙ |S1⊥ | 1 + cos2 Q Q 2 a e [G (x1 ; x1 )fNa (x2 ) + LT fa (x1 )EFa (x2 ; x2 )] : (7.3.32) × a a F 2 N a ea fa (x1 )fa (x2 )

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V. Barone et al. / Physics Reports 359 (2002) 1–168 Table 5 Contributions to the Drell–Yan cross-section involving transversitya Drell–Yan cross-section A

B

Observable

Twist-two

T

T

N 2) LT f(x1 )LT f(x

Twist-three

T

0

T

L

N 2 )(∗) LT f(x1 )h(x LT f(x1 )hNL (x2 )

a

The asterisk denotes T -odd terms.

To establish a connection between (7.3.32) and (7.3.25), let us now invert F + = @+ A⊥ , hence obtaining 1 −i AA (x; y) = (x − y)[AA(∞) (x) + AA(−∞) (x)] + P:V: (7.3.33) AF (x; y): 2 x−y If we impose antisymmetric boundary conditions [46], i.e., AA(∞) (x) = − AA(−∞) (x);

then (7.3.33) reduces to −i AA (x; y) = P:V: A (x; y): x−y F

(7.3.34)

(7.3.35)

and (7.3.31) becomes (“e, ” stands for “e,ective”) x1 − y1 AAe, ≡ A (x1 ; y1 ) x1 − y1 + ij A = AA (x1 ; y1 ) − (x1 − y1 )AF (x1 ; x1 ):

(7.3.36)

Now, the important observation is that AA and AF have opposite behaviour with respect to time reversal and hence AAe, has no de?nite behaviour under this transformation. Consequently, the T -even functions of AF can be identi?ed with T -odd functions in the e,ective correlation matrix e, e, AAe, . This mechanism gives rise to two e,ective T -odd distributions f˜T (x) and h˜ (x), which are related to the multiparton distribution functions by [46] (omitting some factors) e, f˜ T (x)∼ dy Im GAe, (x; y)∼GF (x; x); (7.3.37a) ˜he, (x)∼ dy Im EAe, (x; y)∼EF (x; x): (7.3.37b) In the light of this correspondence one can see that Eq. (7.3.25), based on T -odd distributions, and Eq. (7.3.32), based on multiparton distributions, translate into each other. Thus, at least in the case in which the T -odd functions appear at twist-three, there is an explanation for them in terms of quark–gluon interactions, with no need for initial-state e,ects. In conclusion, let us summarise the various contributions of transversity to the Drell–Yan cross-section in Table 5.

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Fig. 40. Hadron–hadron scattering with inclusive production of a particle h.

7.4. Single-spin transverse asymmetries Consider now inclusive hadron production, A + B → h + X . If only one of the initial-state hadrons is transversely polarised and the ?nal-state hadron is spinless (or its polarisation is unobserved), what is measured is the single-spin asymmetry d(ST ) − d(−ST ) AhT = : (7.4.1) d(ST ) + d(−ST ) As we shall see, single-spin asymmetries are expected to vanish in leading-twist in perturbative QCD (this observation is originally due to Kane et al. [6]). They can arise, however, as a consequence of intrinsic e,ects of quark transverse motion [17,40,124] and=or higher-twist contributions [7,8,175,176]. In the former case, one probes the following quantities related to transversity: • Distribution functions: LT f(x) (transversely polarised quarks in a transversely polarised ⊥ (x; k2 ) (unpolarised quarks in a transversely polarised hadron), h⊥ (x; k2 ) (transhadron), f1T T T 1

versely polarised quarks in an unpolarised hadron).

• Fragmentation functions: H1⊥ (z; 2T ) (transversely polarised quarks fragmenting into an un⊥ (z; 2 ) (unpolarised quarks fragmenting into a transversely polarised polarised hadron), D1T T

hadron).

The twist-three single-spin asymmetries involving the transversity distributions contain, besides the familiar unpolarised quantities, the quark–gluon correlation function EF (x; y) of the incoming unpolarised hadron and a twist-three fragmentation function of the outgoing hadron (see below, Section 7.4.2). Let us now enter into some detail. We consider the following reaction (see Fig. 40): A↑ (PA ) + B(PB ) → h(Ph ) + X;

(7.4.2)

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where A is transversely polarised and the hadron h is produced with a large transverse momentum PhT , so that perturbative QCD is applicable. In typical experiments A and B are protons while h is a pion. 2 and of the Feynman The cross-section for (7.4.2) is usually expressed as a function of PhT variable t−u 2P xF ≡ √hL = ; (7.4.3) s s where PhL is the longitudinal momentum of h, and s; t; u are the Mandelstam variables s = (PA + PB )2 ;

t = (PA − Ph )2 ;

u = (PB − Ph )2 :

(7.4.4)

The elementary processes at lowest order in QCD are two-body partonic processes a(ka ) + b(kb ) → c(kc ) + d(kd ):

(7.4.5)

In the collinear case we set 1 kc = Ph : z and the partonic Mandelstam invariants are ka = xa PA ;

kb = xb PB ;

sˆ = (ka + kb )2 xa xb s;

(7.4.6) (7.4.7a)

xa t ; z xu uˆ = (kb − kc )2 b : z Thus the condition sˆ + tˆ + uˆ = 0 implies xa t − xb u z=− : xa xb s tˆ = (ka − kc )2

(7.4.7b) (7.4.7c)

(7.4.8)

According to the QCD factorisation theorem the di,erential cross-section for the reaction (7.4.2) can formally be written as d = !a/ / fa (xa ) ⊗ fb (xb ) ⊗ d ˆ // ⊗ Dh=c (z): (7.4.9) abc //

Here fa (fb ) is the distribution of parton a (b) inside the hadron A (B), !a// is the spin density matrix of parton a, Dh=c is the fragmentation matrix of parton c into hadron h, and d =d ˆ tˆ is the elementary cross-section: 1 1 d ˆ = M/< M/∗ < ; (7.4.10) d tˆ // 16sˆ2 2 <

where M/< is the scattering amplitude for the elementary partonic process (see Fig. 41). If the produced hadron is unpolarised, or spinless, as will always be the case hereafter, only the diagonal elements of Dh=c are non-vanishing, i.e., Dh=c = Dh=c , where Dh=c is the

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

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Fig. 41. Elementary processes contributing to hadron–hadron scattering.

unpolarised fragmentation function. Together with helicity conservation in the partonic subprocess, this implies / = / . Therefore, in (7.4.10) there is no dependence on the spin of hadron A and all single-spin asymmetries are zero. To escape such a conclusion we must consider either the intrinsic transverse motion of quarks, or higher-twist e,ects. 7.4.1. Transverse motion of quarks and single-spin asymmetries Let us ?rst of all see how the transverse motion of quarks can generate single-spin asymmetries. This can happen in three di,erent ways:

(1) Intrinsic T in hadron h implies that Dh=c is not necessarily diagonal (owing to T -odd e,ects at level of fragmentation functions). (2) Intrinsic kT in hadron A implies that fa (xa ) in (7.4.9) should be replaced by the probability density Pa (xa ; kT ), which may depend on the spin of hadron A (again, owing to T -odd e,ects but at the level of distribution functions). (3) Intrinsic kT in hadron B implies that fb (xb ) in (7.4.9) should be replaced by Pb (xb ; kT ). The transverse spin of parton b inside the unpolarised hadron B may then couple to the transverse spin of parton a inside A (this too is a T -odd e,ect at the level of distribution functions).

E,ect 1 is the Collins e,ect [17], e,ect 2 is the Sivers e,ect [40], and e,ect 3 is the e,ect studied by Boer [47] in the context of Drell–Yan processes (Section 7.3.1). We stress that all these intrinsic T ; kT , or kT e,ects are T -odd. When the intrinsic transverse motion of quarks is taken into account, the QCD factorisation theorem is not proven. We assume, however, its validity and write a factorisation formula similar to (7.4.9), that is explicitly d 1 2 2 Eh 3 = d xa d xb d kT d kT d 2 T d Ph z abc // <<

×Pa (xa ; kT )!a/ / Pb (xb ; kT )!b< <

d ˆ d tˆ

// <<

Dh=c (z; T );

(7.4.11)

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where

d ˆ d tˆ

// <<

=

1 M/< M/∗ < : 2 16sˆ <

(7.4.12)

To start with, let us consider the Collins mechanism for single-spin asymmetries [17,26]. We take into account the intrinsic transverse motion of quarks inside the produced hadron h (which is responsible for the e,ect), and we neglect the transverse momenta of all other quarks. Thus Eq. (7.4.11) becomes d 1 Eh 3 = d xa d xb d 2 T d Ph z abc //

×fa (xa )!a/ / fb (xb )

d ˆ d tˆ

//

Dh=c (z; T );

(7.4.13)

and the elementary cross-sections are given by (7.4.10), with T retained. We are interested in transverse spin asymmetries d(ST ) − d(−ST ). Therefore, since we are neglecting the intrinsic kT motion inside A, the spin density matrix elements of our concern are !a+− and !a−+ , and the contributing elementary cross-sections are d ˆ +−+− = d ˆ −+−+ and d ˆ +−−+ = d ˆ −++− . Using Eqs. (4:3:7) and (6:7:24a–d) we ?nd, with our choices of axes Eh

d(ST ) d(−ST ) − Eh 3 d Ph d 3 Ph 1 = − 2|ST | d xa d xb d 2 T LT fa (xa )fb (xb ) z abc d ˆ d ˆ × sin(= + =S ) + sin(= − =S ) L0T Dh=c (z; 2T ); d tˆ +−+− d tˆ +−−+

(7.4.14)

where = and =S are the azimuthal angles of T and ST , respectively, and L0T Dh=c is the T -odd fragmentation function related to H1⊥ , see (6.5.11). In particular, if the spin of hadron A is directed along y, Eq. (7.4.14) takes the form d(ST ) d(−ST ) d xb Eh 3 − Eh = −2|ST | d xa d 2 T LT fa (xa )fb (xb ) d Ph d 3 Ph z abc

×LTT (x ˆ a ; xb ; T )L0T Dh=c (z; 2T );

where the elementary double-spin asymmetry LTT ˆ is given by d ˆ d ˆ ˆ ↑ b → c↓ d) d (a ˆ ↑ b → c↑ d) d (a : LTT ˆ = − = − d tˆ +−+− d tˆ +−−+ d tˆ d tˆ

(7.4.15)

(7.4.16)

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Eq. (7.4.15) gives the single-spin asymmetry under the hypothesis that only the Collins mechanism (based on the existence of the T -odd fragmentation functions L0T Dh=c or H1⊥ ) is at work. Another source of single-spin asymmetries in hadron-hadron scattering is the Sivers e,ect [26,39,40,81], which relies on T -odd distribution functions. This mechanism predicts a single-spin asymmetry of the form d(ST ) d(−ST ) d xb Eh 3 − Eh = |S | d x d 2 kT LT0 fa (xa ; kT2 )fb (xb ) T a d Ph d 3 Ph z abc

×

d ˆ (xa ; xb ; kT )Dh=c (z); d tˆ

(7.4.17)

⊥ ) is the T -odd distribution de?ned in (4.8.3a). where LT0 f, (related to f1T Finally, the e,ect studied by Boer in [47] gives rise to an asymmetry involving the other T -odd distribution, L0T f (or h⊥ 1 ), de?ned in (4.8.3b). This asymmetry reads d(ST ) d(−ST ) d xb − Eh = −2|ST | d xa d 2 kT LT fa (xa )L0T fb (xb ; kT2 ) Eh 3 d Ph d 3 Ph z abc

×LTT ˆ (xa ; xb ; kT )Dh=c (z);

(7.4.18)

where the elementary asymmetry is LTT ˆ =

d (a ˆ ↑ b↑ → cd) d (a ˆ ↑ b↓ → cd) : − d tˆ d tˆ

(7.4.19)

The caveat of Section 4.8 with regard to initial-state interaction e,ects, which are assumed to generate the T -odd distributions, clearly applies here and makes both the Sivers and the Boer mechanisms highly conjectural. In the next section we shall see how single-spin asymmetries emerge at higher twist. 7.4.2. Single-spin asymmetries at twist three In the 1980s Efremov and Teryaev [7,8] pointed out that non-vanishing single-spin asymmetries can be obtained in perturbative QCD if one resorts to higher twist. These asymmetries were later evaluated in the context of QCD factorisation by Qiu and Sterman, who studied direct photon production [175,176] and, more recently, hadron production [177]. This program has been extended to cover the chirally odd contributions by Kanazawa and Koike [182,183]. Here we limit ourselves to quoting the main general results of these works (for a phenomenological analysis see Section 9.1). At “twist-three” the cross-section for the reaction (7.4.2) can be formally written as d = {GFa (xa ; ya ) ⊗ fb (xb ) ⊗ d ˆ ⊗ Dh=c (z) abc

+ LT fa (xa ) ⊗ EFb (xb ; yb ) ⊗ d ˆ ⊗ Dh=c (z) (3) + LT fa (xa ) ⊗ fb (xb ) ⊗ d ˆ ⊗ Dh=c (z)};

(7.4.20)

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where GF (xa ; xb ) and EF (xa ; xb ) are the quark–gluon correlation functions introduced in Section (3) is a twist-three fragmentation function (that we do not specify further), and d , ˆ d ˆ 7.3.1, Dh=c and d ˆ are cross-sections of hard partonic subprocesses. The ?rst line in (7.4.20), which does not contain the transversity distributions, corresponds to the chirally-even mechanism studied in [177]. The second term in (7.4.20) is the chirally-odd contribution analysed in [182]. The elementary cross-sections can be found in the original papers. In Section 9.1 we shall see how the predictions based on Eq. (7.4.20) compare with the available data on single-spin asymmetries in hadron production. In practice, it turns out that the transversity-dependent term is negligible [183]. 8. Model calculations of transverse polarisation distributions As we have no experimental information on the transversity distributions yet, model calculations are presently the only way to acquire knowledge of these quantities. This section is devoted to such calculations. We shall see how the transverse polarisation distributions have been computed using various models of the nucleon and other non-perturbative tools. In particular, three classes of models will be discussed in detail: (1) relativistic bag-like models, such as the MIT bag model and the colour dielectric model, which are dominated by the valence component; (2) chiral soliton models, in which the sea plays a more important rˆole and contributes significantly to various observables; (3) light-cone models, based on the Melosh rotation. Results obtained in other models, not included in the above list (for instance, diquark spectator models), via QCD sum rules and from lattice calculations will also be reported. Quite obviously, the presentation of all models will be rather sketchy, our interest being essentially in their predictions for the quark and antiquark transversity distributions. 17 What models provide is the nucleon state (i.e., the wave functions and energy spectrum), which appears in the ?eld-theoretical expressions (4:2:5a)–(4.2.5c) of quark distributions. In general, however, it is impossible to solve the equations of motion exactly for any realistic model. Hence, one must resort to various approximations, which clearly a,ect the results of the calculation. In order to test the validity of the approximations (and of the models) one may check that the computed distributions ful?ll the valence-number sum rules 1 1 N d x[u(x) − u(x)] = 2; d x[d(x) − d(x)] =1 (8.0.1) 0

0

and that they satisfy other theoretical constraints, such as the So,er inequality (see Section 4.6) q(x) + Lq(x) ¿ 2|LT q(x)|: 17

(8.0.2)

Throughout this section the transversity distributions will be denoted by LT q and the tensor charges will be called q.

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119

As seen in Section 5, the renormalisation of the operators in the matrix elements of Eqs. (4:2:5a)–(4.2.5c) introduces a scale dependence into the parton distributions. In contrast, when computed in any model, the matrix elements of (4:2:5a)–(4.2.5c) are just numbers, with no scale dependence. The problem thus arises as to how to reconcile QCD perturbation theory with quark models. Since the early days of QCD various authors [184 –186] have proposed the answer to this question: the twist-two matrix elements computed in quark models should be interpreted as representing the nucleon at some Fxed, low scale Q02 (we shall call it the model scale). In other terms, quark models provide the initial condition for QCD evolution. The experience accumulated with the radiative generation models [187–189] has taught us that, in order to obtain a picture of the nucleon at large Q2 in agreement with experiment (at least in the unpolarised case), the nucleon must contain a relatively large fraction of sea and glue, even at low momentum scales. Purely valence models are usually unable to ?t the data at all well. Although the model scale has the same order of magnitude in all models (Q0 ∼ 0:3–0:8 GeV), its precise value depends on the details of the model and on the procedure adopted to determine it. The smallness of Q02 clearly raises another problem, namely to what extent one can apply perturbative evolution to extrapolate the quark distributions from such low scales to large Q2 . This problem is still unresolved (for an attempt to model a non-perturbative evolution mechanism in an e,ective theory see [190,191]) although the success of ?ts based on radiatively generated parton distributions [187,192–194] inspires some con?dence that the realm of perturbative QCD may extend to fairly small scales. 8.1. Bag-like models In bag-like models (the MIT bag model [195 –197] and the colour dielectric model [198– 200] (CDM)) con?nement is implemented by situating the quarks in a region characterised by a value of the colour dielectric constant of order unity. The value of the dielectric is zero outside the nucleon, that is, in the true vacuum, from which the coloured degrees of freedom are expelled. A certain amount of energy is associated with the excitation of non-perturbative gluonic degrees of freedom. This energy is described by the so-called vacuum pressure in the MIT bag model and by excitations of a phenomenological scalar ?eld in the CDM. The two models (MIT bag and CDM) di,er in the following points. In the MIT bag model the interior of the bag is supposedly described by perturbative QCD and quarks have current masses; con?nement is imposed by special boundary conditions at the surface of the bag and the bag itself has no associated dynamics. In contrast, in the CDM chiral symmetry is broken both inside and outside the nucleon, in a manner somewhat similar to the -model [201,202]. Quark con?nement is due to interaction with the phenomenological scalar ?eld which describes the non-perturbative gluonic degrees of freedom. There is no rigid separation between “inside” and “outside”, and con?nement is implemented dynamically. 8.1.1. Centre-of-mass motion A problem arising in many model calculations is the removal of spurious contributions to physical observables due to the centre-of-mass motion (for a comprehensive discussion of this matter, see the book by Ring and Schuck [203]). The origin of the problem lies in the fact

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that the solution of classical equations of motion (i.e., the mean-?eld approximation) breaks translational invariance. A way to restore this invariance is to ?rst de?ne the quantum state of the nucleon at rest, and then minimise the normally ordered Hamiltonian in this state. In the evaluation of speci?c operators, boosted states of the nucleon are also required. The diKculty of boosting the states makes the procedure hard to fully implement. A (non-relativistic) method frequently used to construct momentum eigenstates is the so-called Peierls–Yoccoz projection [203]. Writing |R; 3q to denote a three-quark state centred at R: |R; 3q = b†1 (R)b†2 (R)b†3 (R)|R; 0q;

(8.1.1)

where |R; 0q is the quantum state of the empty bag, a generic nucleon eigenstate of momentum P may be written as 1 |P = d 3 ReiP·R |R; 3q (8.1.2) N3 (P 0 ; P ) with normalisation 1 0 2 |N3 (P ; P )| = 0 d 3 R eiP·R R; 3q|0; 3q: (8.1.3) 2P The expectation value of the normally ordered Hamiltonian in the projected zero-momentum eigenstate is P = 0| : Hˆ : |P = 0 E= : (8.1.4) P = 0|P = 0 Minimisation of this quantity amounts to solving a set of integro-di,erential equations, for which a variational approach is generally adopted. In the literature this procedure is known as “variation after projection” (VAP), to be distinguished from the simpler “variation before projection” (VBP), which consists of minimising the unprojected Hamiltonian ?rst and then using the solutions in the Peierls–Yoccoz projection (8.1.2). For a detailed discussion of these techniques see [204,205]. 8.1.2. The quark distributions in bag models In the (projected) mean-?eld approximation, the matrix elements de?ning the distribution functions can be rewritten in terms of single-particle (quark or antiquark) wave functions, after inserting a complete set of states between the two fermionic ?elds and N [206,207]. The intermediate states that contribute are 2q and 3q1qN states for the quark distributions, and 4q states for the antiquark distributions (see Fig. 42). The leading-twist quark distribution functions read (f is the Pavour) qf (x) = P(f; /; m)F/ (x); (8.1.5a) /

Lqf (x) =

m

/

LT qf (x) =

m

/

m

3

P(f; /; m)(−1)(m+ 2 +i/ ) G/ (x); 3

P(f; /; m)(−1)(m+ 2 +i/ ) H/ (x);

(8.1.5b) (8.1.5c)

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121

Fig. 42. Intermediate states in the parton model: (a) 2q, (b) 3q1qN and (c) 4q.

where

 F/ (x)   d 3 p/ G/ (x) = A/ (p/ )((1 − x)P + − p/+ )  (2)3 (2p/0 )  H/ (x)  p/z 2  u (p ) + 2u(p )v(p ) + v2 (p/ );  / / /   |p/ |     z z 2  p p / 1 u2 (p/ ) + 2u(p/ )v(p/ ) / + v2 (p/ ) 2 −1 ; × |p | |p | / /  2     p/⊥ 2 p/z  2 2  + v (p/ ) 1 − :  u (p/ ) + 2u(p/ ) v(p/ ) |p/ | |p/ |

(8.1.6)

In (8.1.6) u and v are, respectively, the upper and lower components of the single-quark wave functions, m is the projection of the quark spin along the direction of the nucleon spin, and P(f; /; m) is the probability of extracting a quark (or inserting an antiquark) of Pavour f and spin m, leaving a state generically labelled by the quantum number /. The index i/ takes the value 0 when a quark is extracted and 1 when an antiquark is inserted. The overlap function A/ (p/ ) contains the details of the intermediate states. The antiquark distributions are obtained in a similar manner (the index i/ is 1 for the 4q states). When a quark (or an antiquark) is inserted, it can give rise to an in?nite number of states. Among all four-particle intermediate states, only that corresponding to a quark or an antiquark inserted into the ground state is usually considered because excited intermediate states have larger masses and, as will be clear in the following, give a negligible contribution to the distribution functions. Concerning the antiquark distributions, we recall from Section 4.2 that the following formal expressions hold: q(x) N = − q(−x);

(8.1.7a)

Lq(x) N = Lq(−x);

(8.1.7b)

LT q(x) N = − LT q(−x):

(8.1.7c)

Although some authors use these relations to calculate the antiquark distributions by extending to negative x the quark distributions, it should be recalled that this is an incorrect procedure. The reason, explained in Section 4.3, is that for x ¡ 0 there are semiconnected diagrams that

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contribute to the distributions whereas in computing the quark distribution functions in the physical region one considers only connected diagrams (as stressed by Ja,e [37], this indeed deFnes the parton model). It is important to note that, in the non-relativistic limit, where the lower components of the quark wave functions are neglected, the three currents in Eq. (8.1.6) coincide. This implies, in the light of (8:1:5b), (8.1.5c), that ignoring relativistic e,ects the helicity distributions are equal to the transversity distributions. This is obviously only valid at the model scale (i.e., at very low Q2 ) since, as shown in Section 5.4, QCD evolution discriminates between the two distributions, in particular at small x. In Eqs. (8.1.6) energy–momentum conservation is enforced by the delta function. This is also responsible for the correct support of the distributions, which vanish for x ¿ 1. In fact, rewriting the integral in (8.1.6) as ∞ 3 + + d p/ ((1 − x)P − p/ ) = 2 dp/ p/ ; (8.1.8) p/ ¡

where (m/ is the mass of the intermediate state) ) 2 ) ) M (1 − x)2 − m2/ ) ); p/¡ = )) 2M (1 − x) )

(8.1.9)

one sees that for x → 1 the lower limit of integration p/¡ tends to in?nity and (8.1.8) gives zero. The distributions are centred at the point xN = 1 − (m/ =M ), which is positive for the 2q term and negative for the 4q and 3q1qN terms. In the latter case only the tails of the distributions (which are centred in the non-physical region x ¡ 0) contribute to the physical region 0 6 x 6 1. More massive intermediate states would lead to distributions shifted to more negative x values, and hence are negligible. Let us now address the problem of the saturation of the So,er inequality. First of all, note that the three quantities F/ ; G/ and H/ de?ned in (8.1.6) satisfy F/ (x) + G/ (x) = 2H/ (x):

(8.1.10)

This has led to the erroneous conclusion [38] that the inequality is saturated for a relativistic quark model, such as the MIT bag model. It is clear from Eqs. (8:1:5a)–(8.1.5c) that the spin-Pavour structure of the proton, which results in the appearance of the probabilities P(f; /; m), spoils this argument and prevents in general the saturation of the inequality. So,er’s inequality is only saturated in very speci?c (and quite unrealistic) cases. For instance, it is saturated when P(f; /; − 12 ) = 0, which happens if the proton is modelled as a bound state of a quark and scalar diquark. It is interesting to note that in SU(6) the hyperon is indeed a bound state of a scalar–isoscalar ud diquark and an s quark: thus the transversity distribution of the latter attains the maximal value compatible with the inequality. Another instance of saturation occurs when F/ = G/ = H/ and P(f; /; − 12 ) = 2 P(f; /; 21 ). It is easy to verify that this happens for the d quark distribution in a non-relativistic model of the proton with an SU(6) wavefunction. Apart from these two special cases, So,er’s inequality should not generally be expected to be saturated.

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8.1.3. Transversity distributions in the MIT bag model Calculations of structure functions within the MIT bag model have been performed by various authors, using di,erent versions of the model [206 –210]. The transversity distributions, however, have been evaluated only in the simplest non-chiral version of the MIT bag. In the ?rst calculation of LT q [14], the distributions were estimated using the formalism developed in [208], with no attempt to restore translational invariance. The single-quark wave functions of the non-translationally invariant bag are directly used in the evaluation of the matrix elements. The single-quark contribution to the transversity distributions, corresponding to H/ (x) in (8.1.6), is then 2 ∞ !n MR ymin 2 2 ymin H (x) = dyy t0 + 2t0 t1 ; (8.1.11) + t1 y y 2(!n − 1)j02 (!n ) |ymin| where !n is the nth root of the equation tan !n = − !n =(!n − 1) and ymin = xRM − !n . R and M are the bag radius and nucleon mass, respectively, and RM = 4 !n . The functions t0 and t1 are de?ned as 1 tl (!n ; y) = du u2 jl (u !n )jl (uy); (8.1.12) 0

where jl is the lth order spherical Bessel function. For completeness, we also give the unpolarised and helicity contributions ∞ ymin !n MR 2 2 F(x) = dyy t0 + 2t0 t1 (8.1.13a) + t1 ; y 2(!n − 1)j02 (!n ) |ymin | G(x) =

!n MR 2(!n − 1)j02 (!n )

∞ ymin ymin 2 2 2 dyy t0 + 2t0 t1 −1 : × + t1 2 y y |ymin|

(8.1.13b)

To obtain the quark distributions one must insert F(x); G(x) and H (x) into Eqs. (8:1:5a)– (8.1.5c), along with the probabilities P(f; /; n). In [14,208] only valence quarks were assumed to contribute to the distributions, hence the intermediate states |/ reduce to the diquark state alone. Therefore, the quark distributions are just proportional to F(x); G(x) and H (x). In particular, with an SU(6) spin-Pavour wave function one has simply 4 Lu(x) = G(x); 3

1 Ld(x) = − G(x) 3

(8.1.14)

and analogous relations for LT u and LT d with G(x) replaced by H (x). The quantities F(x); G(x) and H (x) are plotted in Fig. 43. As one can see, the transversity distributions are not so di,erent from those for helicity. Since translational invariance is lost, the distributions do not have the correct support. Thus, the integral of F(x) over x between 0 1 and 1 is not unity, as it should be. In particular, one has 0 F(x) d x = 0:90. This normalisation

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Fig. 43. Single-quark contributions to the distribution functions in the MIT bag model of [14]. Table 6 Axial and tensor charges in various models. Tensor charges evolved in LO QCD from the intrinsic scale of the model (Q02 ) to Q2 = 10 GeV2 are also shown. See the text for details Model [Ref.]

Lu

Ld

LM

u

d

|u=d|

Q0 [GeV]

u(Q2 )

d(Q2 )

NRQM ? MIT [14] ♦ CDM [98] ⊕ CQSM1 [235] × CQSM2 [238] + CQM [243] ⊗ LC [92] ◦ Spect. [264] ∗ Latt. [270,275] .

1.33 0.87 1.08 0.90 0.88 0.65 1.00 1.10 0.64

−0:33 −0:22 −0:29 −0:48 −0:53 −0:22 −0:25 −0:18 −0:35

1 0.65 0.79 0.37 0.35 0.43 0.75 0.92 0.29

1.33 1.09 1.22 1.12 0.89 0.80 1.17 1.22 0.84

−0:33 −0:27 −0:31 −0:42 −0:33 −0:15 −0:29 −0:25 −0:23

4.03 4.04 3.94 2.67 2.70 5.33 4.03 4.88 3.65

0.28 0.87 0.40 0.60 0.60 0.80 0.28 0.25 1.40

0.97 0.99 0.99 0.97 0.77 0.72 0.85 0.83 0.80

−0:24 −0:25 −0:25 −0:37 −0:29 −0:13 −0:21 −0:17 −0:22

problem can ∞ be overcome (although in a very ad hoc manner) by integrating between −∞ and ∞, since −∞ F(x) d x = 1. Proceeding in this manner, for the tensor charges one obtains 4 ∞ 1 ∞ H (x) d x = 1:09; d ≡ − H (x) d x = − 0:27 (8.1.15) u ≡ 3 −∞ 3 −∞ to be compared with the axial charges obtained similarly: Lu = 0:87 and Ld = − 0:22 (see Table 6). Stratmann [211] recomputed q(x); Lq(x) and LT q(x) in the MIT bag model, introducing a Peierls–Yoccoz projection to partially restore the translational invariance. However, he did not perform a VAP calculation. Since the masses of the intermediate states were not computed within the model, the number sum rules turned out to be violated. Another problem of the approach of [211] is that the antiquark distribution functions were evaluated using Eqs. (8:1:7a)–(8.1.7c) (we have already commented on the inconsistency of such a procedure in Section 8.1.2).

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Fig. 44. The transversity distribution h1 (x; Q02 ) (continuous line) and the spin distribution function g1 (x; Q02 ) (dashed line) for the MIT bag model at the initial scale Q02 . The corresponding evolved distributions h1 (x; Q2 ) (dotted line) and g1 (x; Q2 ) (dot-dashed line) are obtained starting from (a) Q02 = 0:079 GeV2 and (b) Q02 = 0:75 GeV2 . From [212].

The MIT bag model was also used to compute the transversity distributions in [212]. The technique adopted in this work is rather di,erent from that discussed above and is based on a non-relativistic reduction of the relativistic wave function (for a discussion of this non-covariant approach see also [213]). The results of [212] for h1 = 12 f ef2 LT qf and g1 = 12 f ef2 Lqf are shown in Fig. 44. 8.1.4. Transversity distributions in the CDM The transversity distributions were calculated in the colour dielectric model in [98]. In particular, the chiral version of the CDM was used, in which the splitting between the masses of the nucleon and the delta resonance, or between the scalar and the vector diquark, is due to the exchange of pions, instead of perturbative gluons. Although this model su,ers a number of drawbacks, its main technical advantage with respect to the MIT bag model is that it allows a full VAP procedure to be performed, since con?nement is implemented by a dynamical ?eld, not by a static bag surface. For the same reason, the valence-number sum rules turn out to be ful?lled (to within a few percent) if the masses of the intermediate states are consistently computed within the model. As we shall see, the So,er inequalities are also satis?ed, for both quarks and antiquarks. The Lagrangian of the chiral CDM is g L = i N @ + N ( + i 5 · ) 1 + 12 (@ 1)2 − 12 M 2 12 + 12 (@ )2 + 12 (@ )2 − U (; );

(8.1.16)

where U (; ) is the usual Mexican-hat potential, see e.g., [214]. L describes a system of interacting quarks, pions, sigma and a scalar–isoscalar chiral singlet ?eld 1. The chromodielectric ?eld 1 incorporates non-perturbative gluonic degrees of freedom. Through their interaction with

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Fig. 45. The transversity distributions xLT q(x) in the colour dielectric model (from [98]). Left: the quark distriN The distributions are shown at the butions xLT u and xLT d. Right: the antiquark distributions xLT uN and xLT d. model scale Q02 = 0:16 GeV2 and at Q2 = 25 GeV2 .

the 1 ?eld, the quarks acquire a mass that increases strongly at the boundary of the bag, hence leading to absolute con?nement. The parameters of the model are: the chiral meson masses m = 0:14 GeV, m = 1:2 GeV (the precise value of this parameter is actually irrelevant), the pion decay constant f = 93 MeV, the quark–meson coupling constant g, and the mass M of the 1 ?eld. The parameters g and M , which are the only free parameters of the model, can be uniquely ?xed by reproducing the average nucleon-delta mass and the isoscalar radius of the proton. The chiral CDM Lagrangian (8.1.16) contains a single-minimum potential for the chromodielectric ?eld 1: V (1) = 12 M 2 12 . A double-minimum version of the CDM is also widely studied and used (see for instance [215]). The structure functions computed in the two versions of the chiral CDM do not di,er sensibly. 18 The solution of the ?eld equations for the chiral CDM proceeds through the introduction of the so-called hedgehog ansatz, which corresponds to a mean-?eld approximation [217]. The technique used to compute the physical nucleon state is based on a double projection of the mean-?eld solution onto linear- and angular-momentum eigenstates. This technique has also been used to compute the static properties of the nucleon [214], the unpolarised and the longitudinally polarised distribution functions [215], and the nucleon electromagnetic form factors [218]. We refer the reader to these papers for more detail. A di,erent technique to obtain states of de?nite angular momentum and isospin, based on the quantisation of the collective degrees of freedom associated with the rotation of the hedgehog state, will be mentioned in Section 8.2.1. Let us simply remark that in the chiral CDM the chiral ?eld cannot develop a non-zero winding number and its value is always very small. Thus, the choice of a speci?c technique to obtain physical states from the hedgehog is less critical in the chiral CDM than in other models. In Fig. 45 we show the results of the calculation in [98]. One of the features of the distributions computed in the CDM is their rapid fallo, and vanishing for x ¿ 0:6. This is due 18

The single-minimum CDM seems to be preferable in the light of quark matter calculations [216].

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to the soft con?nement of the quarks, which do not carry large momenta. It should also be stressed that the Peierls–Yoccoz procedure, which is a non-relativistic approximation, becomes unreliable at large x. Note also that the sea contribution is rather small. As for the model scale Q02 , in [98,215,219] it was determined by matching the value of the momentum fraction carried by the valence, as computed in the model, with that obtained by evolving backward the value experimentally determined at large Q2 . The result is Q02 = 0:16 GeV2 . Proceeding in a similar manner, Stratmann found Q02 = 0:08 GeV2 [211] in the MIT bag. The CDM distributions evolved from Q02 = 0:16 GeV2 to Q2 = 25 GeV2 are also shown in Fig. 45. Needless to say, perturbative evolution from such low Q02 values should be taken with some caution. The tensor charges computed in the CDM are (at Q02 = 0:16 GeV2 ) u = 1:22;

d = − 0:31;

(8.1.17)

whereas the axial charges are Lu = 1:08 and Ld = − 0:29 (see Table 6). 8.2. Chiral models In chiral models the qqN excitations are described in terms of e,ective degrees of freedom represented by chiral ?elds. There is now a huge variety of models of this type and, as already seen in Section 8.1, bag-like models also admit chiral versions. In this section we shall focus on two models: the chiral quark soliton model (CQSM) [220,221], which can be also derived from the Nambu–Jona–Lasinio model by imposing non-linear constraints on the chiral ?elds [222–228], and the chiral quark model (CQM) [229]. The main di,erence between these two models is that in the CQSM chiral symmetry is dynamically broken within the model itself, while in the CQM quarks have large dynamical masses arising from a process of spontaneous chiral symmetry breaking which is not described by the model. Another important di,erence, rePected in the name of the two models, is that in the CQSM a non-trivial topology is introduced, which is crucial for stabilising the soliton whereas in the CQM chiral ?elds are treated as a perturbation. Finally, while the CQSM is a non-con?ning model, con?nement may be introduced into the CQM, starting from a non-chiral con?ning model and dressing the quarks with chiral ?elds. As for the spin structure of the nucleon, chiral models are characterised by a depolarisation of valence quarks, due to a transfer of total angular momentum of the nucleon into the orbital angular momentum of the sea, described by the chiral ?elds. This feature has made the chiral models quite popular for the study of the spin structure of the nucleon. 8.2.1. Chiral quark soliton model The basic idea of the chiral quark soliton model is to describe the low-energy behaviour of QCD by two e,ective degrees of freedom, Nambu–Goldstone pions and quarks with a dynamical mass. The model is described by the following vacuum functional [220,221] 4 N 5 N exp{iSe, [(x)]} = D D exp i d x (i @ − mU ) (8.2.1)

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with U = exp[ia (x)E a ];

(8.2.2a)

U 5 = exp[ia (x)E a 5 ] = 12 (1 + 5 )U + 12 (1 − 5 )U † :

(8.2.2b)

Here is the quark ?eld, m is the e,ective quark mass arising from the spontaneous breakdown of chiral symmetry, and U is the SU(2) chiral pion ?eld. A possible derivation of the e,ective action (8.2.1) is based on the instanton model of QCD vacuum [220]. The CQSM describes the nucleon as a state of Nc valence quarks bound by a self-consistent hedgehog-like pion ?eld whose energy, in fact, coincides with the aggregate energy of quarks from the negative-energy Dirac continuum. This model di,ers from the -model [201,202] in that no kinetic energy is associated to the chiral ?elds, which are e,ective degrees of freedom, totally equivalent to the qqN excitations of the Dirac sea (the problem of double-counting does not arise). The CQSM ?eld equations are solved as follows. For a given time-independent pion ?eld U = exp(ia (x)E a ), one ?nds the spectrum of the Dirac Hamiltonian: HAn = En An ;

(8.2.3)

which contains the upper and lower Dirac continua (distorted by the presence of the external pion ?eld), and may also contain discrete bound-state levels, if the pion ?eld is strong enough. If the pion ?eld has winding number 1, there is exactly one bound-state level which travels all the way from the upper to the lower Dirac continuum as one increases the spatial size of the pion ?eld from zero to in?nity. This level must be ?lled to obtain a non-zero baryon-number state. Since the pion ?eld is colour blind, in the discrete level one may place Nc quarks in a state that is antisymmetric in colour. Calling Elev (with −M 6 Elev 6 M ) the energy of the discrete level, the nucleon mass is obtained by adding Nc Elev to the energy of the pion ?eld (which coincides exactly with the overall energy of the lower Dirac continuum) and subtracting the free continuum. The self-consistent pion ?eld is thus found by minimising the functional # M = min Nc Elev [U ] + (En [U ] − En(0) ) : (8.2.4) U

En ¡0

From symmetry considerations one looks for the minimum in a hedgehog con?guration x Uc (x) = exp[ia (x)E a ] = exp[i(n · )P(r)]; r = |x|; n = ; (8.2.5) r where P(r) is the pro?le of the soliton. The latter is then obtained using a variational procedure. At lowest order in 1=Nc , the CQSM essentially corresponds to a mean-?eld picture. Some observables, however, vanish at zeroth order in 1=Nc (this is the case, as we shall see, of unpolarised isovector and polarised isoscalar distribution functions) and for these quantities a calculation at ?rst order in 1=Nc is clearly needed.

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Within the CQSM several calculations of distribution functions have been performed [230 – 234]. In particular, the transversity distributions were computed in [235 –238]. The calculations mainly di,er in the order of 1=Nc expansion considered. We shall see that this expansion is related to the expansion in the collective angular velocity : of the hedgehog solution, and hence to the collective quantisation of the hedgehog solitons. The ?rst application of the CQSM to transversity is contained in [235], where only the tensor charges were computed. In that paper the CQSM was extended from two to three Pavours with a chiral SU (3)R ⊗ SU (3)L symmetry (for a review see [239]). Corrections of order 1=Nc were taken into account in building the quantised soliton. The procedure adopted in [235] for constructing states with de?nite spin and Pavour out of the hedgehog is the so-called cranking procedure [203,240]. The parameters of the model are the constituent mass of the u and d quarks, the explicit SU(3) symmetry breaking term for the mass of the s quark, and a cuto, needed to render the theory ?nite. These parameters are ?xed from hadronic spectroscopy. In particular the constituent mass of the quarks is m = 420 MeV and the symmetry breaking term corresponds to an extra mass of 180 MeV for the s quark. The values of the tensor charges obtained are (the model scale is taken to be Q02 = 0:36 GeV2 , see Table 6) u = 1:12;

d = − 0:42;

s = − 0:01:

(8.2.6)

These quantities are not much a,ected by the value of the constituent mass and of the SU(3) symmetry breaking term. We stress the importance of the 1=Nc corrections. Without these corrections the tensor (and also the axial) charge of the u quark would be equal and opposite to that of the d quark. It is also important to notice that, while the axial singlet charge is substantially reduced owing to the presence of the chiral ?elds, the tensor charges are close to those obtained in other models where the chiral ?elds are absent, or play a minor rˆole. A ?rst attempt to compute the x dependence of the transversity distributions in the CQSM was made in [236]. In this calculation, however, no 1=Nc corrections were taken into account, and hence they obtained LT u + LT d = 0, which is a spurious—and unrealistic—consequence of the zeroth-order approximation adopted. The two most sophisticated calculations in the CQSM are those of [237] and [238]. In [237], both centre-of-mass motion corrections and 1=Nc contributions were taken into account. The correct support for the distributions is obtained by using a procedure that transforms the distributions computed in the rest frame into the distributions in the in?nite momentum frame. This procedure essentially amounts to using the relation >(1 − x) (8.2.7) fRF (−ln(1 − x)): 1−x An important limitation of this work is that only the valence contribution to the distribution functions is considered. The transversity distributions at the scale of the model as computed in [237] are shown in Fig. 46. Wakamatsu and Kubota [238] went beyond the valence quark approximation of [237] and included vacuum-polarisation e,ects. Thus they were also able to compute the antiquark distributions, which had only been previously evaluated in [98]. A further improvement is the treatment of the temporal non-locality of the bilinear operators which appear in the distribution functions. fIMF (x) =

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Fig. 46. Longitudinal and transverse polarisation distributions in the valence quark approximation of [237].

The generic operator A† (0)Oa A(− ) is expanded as [241] A† (0)Oa A(− ) O˜ a + i− 21 {:; O˜ a };

(8.2.8)

˙ Eq. (8.2.8) implies that the non-locality of where O˜ a ≡ A† (0)Oa A(0) and : = − iA† (t)A(t). † − the operator A (0)Oa A( ) causes a rotational correction proportional to the collective angular velocity :. In contrast to the calculation of [237], in [238] no centre-of-mass motion corrections are performed and therefore the distributions do not have the correct support. The antiquark distributions are obtained using the relations (8:1:7a)–(8.1.7c), and the caveat concerning such a procedure thus applies. In Figs. 47 and 48 the quark and antiquark helicity and transversity distributions computed in [238] are shown. While the quark distributions are not too di,erent from those computed in the CDM (see Fig. 45), the T uN distribution has a di,erent sign. This is a consequence of the di,erent technique used in the calculation of antiquark distributions in [98,238] (explicit evaluation of LT qN in [98] vs. use of (8:1:7a)–(8.1.7c) in [238]). The tensor charges obtained in [238] are (again at Q02 = 0:36 GeV2 , see Table 6) u = 0:89;

d = − 0:33:

(8.2.9)

Very recently the technique adopted in [238] was criticised in [242]. In this work, however, the isoscalar and isovector distributions are computed at a di,erent order in 1=Nc and cannot therefore be combined to give the single-Pavour distributions. 8.2.2. Chiral quark model In the chiral quark model of Manohar and Georgi [229] the relevant degrees of freedom at a scale below 1 GeV are constituent quarks and Goldstone bosons. This model was used in [243] to compute the quark and antiquark distribution functions. The CQM is particularly interesting

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Fig. 47. Longitudinal and transverse polarisation distributions of quarks, at the scale of the model and after the perturbative evolution. From [238].

Fig. 48. Longitudinal and transverse polarisation distributions of antiquarks, at the scale of the model and after the perturbative evolution. From [238].

for the study of the spin structure of the nucleon, as it predicts a depolarisation of constituent quarks due to the emission of Goldstone bosons into P-wave states. In the CQM model, the u; d and s quarks are assumed to develop large dynamical masses as a consequence of a mechanism of spontaneous chiral symmetry breaking, which lies outside the

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Fig. 49. Diagrams contributing to the constituent quark structure: (a) the Goldstone boson spectator process, (b) the process probing the structure of the Goldstone boson. The solid lines represent quarks and dashed lines represent Goldstone bosons. From [243].

model itself. We denote these “bare” massive states by |u0 ; |d0 , etc. Once they are dressed by Goldstone bosons, the constituent u and d quark states are √ aJ a |u = Z |u0 + a |d+ + |u0 + aK |sK + + |uJ; (8.2.10a) 2 6 √ aJ a |d = Z |d0 + a |u− + |d0 + aK |sK 0 + |dJ; (8.2.10b) 6 2 where Z is the renormalisation constant for a “bare” constituent quark (it turns out to be about 0:7) and |ai |2 are the probabilities of ?nding the Goldstone bosons in the dressed constituent quark states. These probabilities are related to each other by the underlying SU (3)R ⊗ SU (3)L symmetry of the model. There is a single free parameter which may be ?xed by computing the axial coupling gA . Thus the CQM is a perturbative e,ective theory in the Goldstone boson expansion. There are three types of contributions to the quark distribution functions. The ?rst corresponds to the probability of ?nding a bare quark f0 inside a dressed quark f, and is the bare quark distribution renormalised by the Z factor. The other two contributions correspond to diagrams (a) and (b) of Fig. 49. The spin-independent term corresponding to diagram (a) is given by 1 dy x qj (x) = : (8.2.11) Pj/=i (y)qi y x y Here the splitting function P(y)j/=i is the probability of ?nding a quark j carrying a momentum fraction y and a (spectator) Goldstone boson / (/ = ; K; J) inside a constituent quark i. Diagram (b) of Fig. 49 corresponds to probing the internal structure of the Goldstone bosons. This process gives the following contribution: dy1 dy2 x y1 qk (x) = Vk=/ P/j=i qi (y2 ); (8.2.12) y1 y2 y1 y2 where Vk=/ (x) is the distribution function of quarks of Pavour k inside the Goldstone boson /. In analogy with (8.2.11), (8.2.12), the longitudinal and transverse polarisation distributions contain the splitting functions LP(y) and LT P(y), respectively.

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Fig. 50. The u-quark distribution functions: (a) u(x), (b) Lu(x) and (c) LT u(x), respectively. In each ?gure, the result with dressed constituent quarks is shown by the solid curve and that without dressing by the dashed curve. The latter corresponds to the spectator model calculation of [245] (see Section 8.4). From [243].

In [243] the splitting functions are computed within the CQM, the bare quark distributions are obtained from a covariant quark–diquark model [18,244,245], and the quark distribution functions in the Goldstone bosons are taken from phenomenological parametrisations [193] for pions and from models [246 –248] for kaons. The following relation is found: P(y) + LP(y) = 2LT P(y);

(8.2.13)

which implies saturation of the So,er inequality. The dominant contribution to the dressing of the constituent quarks is due to pion emission. The pion cloud a,ects the u sector reducing both the helicity and transversity distributions in a similar manner, as can be seen in Fig. 50. The situation is quite di,erent in the d sector. In fact, while the renormalisation and meson cloud corrections approximately cancel each other in the helicity distribution Ld, for the transversity distribution LT d these corrections are both positive. Thus, with respect to the bare distributions, Ld is almost unmodi?ed whereas LT d is drastically reduced. The di,erence between Ld(x) and LT d(x) is an important and peculiar feature of the model of [243]; the results for the d distributions are shown in Fig. 51. One can see that the meson cloud suppresses [T d much more than its helicity counterpart. In terms of tensor charges, d is reduced by about 40% by the pion depolarisation e,ect while the corresponding axial charge is almost unchanged. The tensor and axial charges are collected in Table 6. Notice that depolarisation due to Goldstone boson emission is a signi?cant e,ect, although not suKcient to reproduce the small value of LM observed experimentally. 8.3. Light-cone models This section is devoted to models using the so-called front-form dynamics to describe the nucleon in the in?nite momentum frame, and the Melosh rotation to transform rest-frame quark states into in?nite momentum states. We start by recalling the general idea of front-form dynamics, originally due to Dirac [249], and then present the calculations of the transversity distributions based on this approach.

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Fig. 51. The d-quark distribution functions: (a) Ld(x) and (b) LT d(x), respectively. The notation is the same as in Fig. 50. From [243].

8.3.1. Forms of dynamics and Melosh rotation As shown by Dirac [249] (see also [250]), we have in general a certain freedom in describing the dynamics of a system. Various choices of variables de?ning the initial conditions and of operators generating the evolution of the system are possible. We shall refer to each of these choices as a “form” of dynamics. The state of a system is de?ned on a hypersurface M in Minkowski space that does not contain time-like directions. To characterise the state unambiguously, M must intersect every world-line once and only once. The most familiar example of such a surface is, of course, the time instant x0 = 0. Among the ten generators of the PoincarVe algebra, there are some that map M into itself, not a,ecting the time evolution, and others that drive the evolution of the system and contain the entire dynamics. The latter generators are called Dirac “Hamiltonians”. Hereafter we shall only be interested in two forms of dynamics: the instant-form and the front-form (for a more general discussion we refer the reader to [250]). In the usual form of dynamics, the instant-form, the initial conditions are set at some instant of time and the hypersurfaces M are Pat three-dimensional surfaces only containing directions that lie outside the light-cone. The generators of rotations and space translations leave the instant invariant and do not a,ect the dynamics. The remaining four generators (boosts and time translations) are the “Hamiltonians”. In the front-form dynamics one considers instead three-dimensional surfaces in space–time formed by a plane-wave front advancing at the velocity of light, e.g., the surface x+ = 0. The quantities P 1 ; P 2 ; P + ; M 12 ; M −+ ; M 1+ and M 2+ are associated with transformations that leave this front invariant. The remaining PoincarVe generators, namely P − , M 1− and M 2− are the “Hamiltonians”. The advantage of using front-form dynamics is that the number of PoincarVe generators a,ecting the dynamics of the system is reduced and there is one more PoincarVe generator that transforms the states without evolving them. Working within front-form dynamics, there is an important transformation, namely the ↑↓ Melosh–Wigner (MW) rotation [251,252], which relates the spin wave functions qRF in the

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↑↓ rest frame (RF) to the spin wave functions qIMF in the in?nite momentum frame (IMF). 19 The Melosh–Wigner rotation is ↑ ↑ ↓ qIMF = w[(k + + m)qRF + (k 1 + ik 2 )qRF ];

(8.3.1a)

↓ ↑ ↓ qIMF = w[ − (k 1 − ik 2 )qRF + (k + + m)qRF ];

(8.3.1b)

2 ]−1=2 and k + = k 0 + k 3 . where w = [(k + + m)2 + k⊥ The reason that the MW rotation is relevant in DIS is that this process probes quark dynamics on the light-cone rather than the constituent quarks in the rest frame [253,254]. As for the spin structure, in front-form dynamics the spin of the proton is not simply the sum of the spins of the individual quarks, but is the sum of the MW-rotated spins of the light-cone quarks [255]. Calculations of distribution functions using MW have recently appeared in the literature, see e.g., [256,257].

8.3.2. Transversity distributions in light-cone models B.-Q. Ma has reconsidered the problem of the spin of the nucleon in the light of the e,ects of the MW rotation [258]. Applying the MW rotation, the quark contribution to the spin of the nucleon is reduced. In particular, from Eqs. (8.3.1a), (8.3.1b) one can show that the observed (i.e., IMF) axial charge LqIMF is related to the constituent quark axial charge LqRF as follows: LqIMF = Mq LqRF ;

(8.3.2)

where Mq =

2 (k + + m)2 − k⊥ 2 (k + + m)2 + k⊥

and Mq is its expectation value in the three-quark state Mq = d 3 kMq |](k)|2 ;

(8.3.3)

(8.3.4)

where ](k) is the (normalised) momentum wave function of the three-quark state. By choosing two di,erent reasonable wave functions (harmonic oscillator and power-law fall o,) the calculation in [259] gives Mq = 0:75 (both for u and d if we assume mu = md ), which leads to a reduction of LM from 1 (the constituent quark model value) down to 0:75. The e,ect of the MW rotation on the tensor charges was discussed in [92]. One ?nds that the IMF tensor charge is related to the constituent quark tensor charge by qIMF = M˜ q qRF ;

(8.3.5)

where M˜ q = 19

(k + + m)2 ; 2 (k + + m)2 + k⊥

(8.3.6)

Note that in literature the rest-frame wave functions are also called “instant-form” wave functions, and the in?nite-momentum-frame wave functions are also called “light-cone” wave functions.

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Fig. 52. The quark spin distributions xLqRF (x) (solid curves), xLT q(x) (dashed curves), and xLq(x) (dotted curves) in the light-cone SU(6) quark-spectator model, for (a) u quarks and (b) d quarks. From [260].

and, again, M˜ q is the expectation value of M˜ q . From (8.3.3) and (8.3.6) one ?nds an important connection between the MW rotation for the longitudinal and the transverse polarisation, namely 1 + Mq = 2M˜ q :

(8.3.7)

With the value Mq = 0:75, this implies M˜ q = 0:88 and, using the SU(6) values RF = 43 and dRF = − 13 , one obtains [92] (we omit the “IMF” subscript) u = 1:17;

d = − 0:29:

(8.3.8)

The transverse polarisation distributions were computed using the MW rotation in [260]. In this paper a simple relation connecting the spin distributions of quarks in the rest frame LqRF (x), the quark helicity distributions Lq(x) and the quark transversity distributions [T q(x) was derived. It reads LqRF (x) + Lq(x) = 2LT q(x):

(8.3.9)

Adopting a diquark spectator model [261] to compute the rest-frame distributions, the authors of [260] obtain the curves shown in Fig. 52. From (8.3.9) and the measured values of the quantities Dp; n ≡ d xg1p; n ; gA =gV and Ls, it is possible to obtain predictions for the tensor charges, as shown by Ma and Schmidt [262]. Taking gA =gV = 6(Dp − Dn ) these authors ?nd (the ranges are determined by the experimental and theoretical uncertainties) u = 0:84–1:09;

d = − (0:23–0:51):

(8.3.10)

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Using the value gA =gV = 1:2573 from neutron < decay (denoted case 2), they ?nd instead u = 0:89–1:11;

d = − (0:29–0:53):

(8.3.10 )

Another calculation of the transversity distributions based on the MW rotation is presented in [263]. These authors use a three-quark wave function obtained by solving the SchrYodinger equation with a hypercentral phenomenological potential (for details see [256]). The e,ect of the MW rotation is to introduce a signi?cant di,erence between longitudinal and transverse polarisation already at the model scale. 8.4. Spectator models As we have seen, the main ingredients in model calculations of quark distributions are the nucleon–quark vertices and the masses of the intermediate states. In the spectator model the set of intermediate states is reduced to only the diquark states and the vertex is parametrised in some manner, for instance assuming an SU(6) spin-Pavour structure. This model was used in [244] to estimate unpolarised and longitudinally polarised distributions and, in [245,264] to compute the transversity distributions. In [243,245] it was used as the starting point for the perturbative dressing of quarks by chiral ?elds. As already mentioned, the model contains the diquark masses as free parameters. Typical values of these masses are in the range 600 –800 MeV, with a splitting of the scalar and vector diquark masses of the order of 100 –200 MeV. The parameters entering the vertices are their Dirac structure and form factors. The calculations in [264] and in [245] di,er mainly in the choice of the parameters and in a more-or-less simple form for the vertex. The results of [245] for the distribution functions are those already presented in Section 8.2.2 in Figs. 50 and 51 (they correspond to the undressed contributions, i.e., to the dashed curves). Similar results were obtained in [264], where, ?xing the parameters so as to obtain the experimental value of the axial coupling, the tensor charges were found to be u = 1:22;

d = − 0:25:

(8.4.1)

In [244] the scale of this model was estimated to be Q02 = 0:063 GeV2 . 8.5. Non-perturbative QCD calculations 8.5.1. QCD sum rules In the QCD sum-rule approach (see for instance [265]) one considers correlation functions of the form 2 ,(q ) = i d 4 eiq· 0|T (j()j(0))|0; (8.5.1) where j(x) = q(x)Dq(x) N is a quark current (all indices are omitted for simplicity). The vacuum polarisation (8.5.1) is computed in two di,erent ways. On one hand, it is modelled by a dispersion relation, expressing its imaginary part in terms of resonances exchanged in the s-channel. On the other hand, in the limit of small light-cone separations, i.e., large Q2 ≡ −q2 , one can

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make an operator–product expansion (OPE) of T (j()j(0)), thus relating ,(q2 ) to quark condensates qq N . The two theoretical expressions of ,(q2 ) are then equated, after performing a Borel transformation, which allows one to pick up only the lowest-lying resonances in a particular channel. The result is an expression for the matrix elements of certain quark currents in a hadron state, in terms of quark and resonance parameters, condensates and the Borel mass MB . The generalisation of this procedure to three-point correlators is straightforward. A QCD sum rule calculation of the tensor charges was reported by He and Ji in [266]. Following the method presented in [267], they consider the three-point correlation function 2 2 , (q ) = i d 4 d 4 O eiq· 0|T (j (O)J()J(0)) N |0; (8.5.2) where j is the quark tensor current N q(O) j (O) = q(O)

(8.5.3)

and J() is the nucleon interpolating ?eld, i.e., J() =

T abc [ua ()C ub ()] 5 dc ():

(8.5.4)

(Here a, b and c are colour indices, the superscript T indicates transpose and C is the charge conjugation matrix.) The interpolating current J is related to the nucleon spinor U (P) by 0|J(0)|P = 5U (P);

(8.5.5)

where 5 is a coupling strength. Computing (8.5.2), using OPE on one hand and resonance saturation on the other, for the tensor charges He and Ji obtained (at a scale 2 = M 2 ) u = 1:0 ± 0:5;

d = 0:0 ± 0:5:

(8.5.6)

The uncertainty corresponds to a variation of the Borel mass MB2 from M 2 to 2M 2 . In a subsequent paper [268] He and Ji presented a more re?ned QCD sum rule calculation of q taking into account operators of higher orders. Instead of the three-point function approach adopted in [266], they used the external-?eld approach. Their starting point is the two-point correlation function in the presence of an external constant tensor ?eld Z , (Z ; q2 ) = i d 4 eiq· 0|T (J()J(0)) N |0Z : (8.5.7) The coupling between quarks and Z is described by the additional term N qZ LL = gq q

(8.5.8)

in the QCD Lagrangian. Referring to the original paper for the details of the calculation, we give here the results for u and d at the scale 2 = M 2 u = 1:33 ± 0:53;

d = 0:04 ± 0:02;

where the error is an estimated theoretical one.

(8.5.9)

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A similar study of tensor charges in the QCD sum rule framework was carried out by Jin and Tang [269], who discussed in detail various sources of uncertainty, and in particular the dependence of the results on the vacuum tensor susceptibility induced by the external ?eld. Finally, we recall that QCD sum rules have also been used to compute the transversity distributions LT q(x). This was done in [69] by considering a four-point correlator. The ranges of validity of the various approximations adopted and the sensitivity of the results on the Borel mass considerably restrict the interval of x over which a reliable calculation can be performed. The result of [69] is LT u 0:5 for 0:3 . x . 0:5, with no apparent variation in this range (the Q2 scale is estimated to be Q2 ≈ 5–10 GeV2 ). 8.5.2. Lattice Lattice evaluations of the tensor charges have been presented by various groups [270 –274]. The lattice approach is based on a hypercubic discretisation of the Euclidean path integral for QCD and a Monte Carlo computation of the resulting partition function. The lattice size must be large enough so that ?nite size e,ects are small. An important parameter is the lattice spacing a. The continuum limit corresponds to a → 0. Usually, di,erent values of quark masses are used and a linear extrapolation is made to the chiral limit of massless quarks. Aoki et al. [270] performed a simulation on a 163 × 20 lattice with spacing a 0:14 fm, in the quenched approximation (which corresponds to setting the fermion determinant in the partition function equal to one). They obtained u = 0:839(60);

d = − 0:231(55)

(8.5.10)

at a scale = a−1 1:4 GeV. For comparison, the axial charges computed with the same lattice con?guration are [275] Lu = 0:638(54);

Ld = − 0:347(46):

(8.5.11)

Similar results were reported in [272]. The continuum limit was investigated by Capitani et al. [273], who found for the di,erence u − d the value u − d = 1:21(4):

(8.5.12)

Finally, a lattice calculation of the tensor charges in full QCD (that is, with no quenching assumption) has been recently carried out by Dolgov et al. [274]. Using 163 × 32 lattices, these authors obtain u = 0:963(59);

d = − 0:202(36):

(8.5.13)

8.6. Tensor charges: summary of results In Table 6 we compare the results for tensor charges computed in the models discussed above. We also show the value of the axial charges. 20 20

The only model where the polarised strange quark distribution has been computed is the CQSM1 of [235]. They ?nd Ls = − 0:05 and s = − 0:01.

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Fig. 53. The tensor charges computed in various models and evolved to Q2 = 10 GeV2 . For the symbols see Table 6. The MIT and CDM points are slightly displaced for clarity.

To allow a homogeneous comparison, we evolved the tensor charges from the model scales Q02 to Q2 = 10 GeV2 in LO QCD. Given the very low input scales, the result of this evolution should be taken with caution (it serves to give a qualitative idea of the trend). Unfortunately, Q0 has not been evaluated in the same manner in all models. As discussed earlier, a possible way to estimate the model scale is to ?x it in such a manner that, starting from the computed value of the momentum fraction carried by the valence, and evolving it to larger Q2 , one ?ts the experimentally observed value. This procedure, with slight di,erences, has been adopted to ?nd the intrinsic scale in the non-relativistic quark model (NRQM) [212], in the MIT bag model [186,211], in the CDM [98,215,219] and in the spectator model [244]. For the light-cone (LC) model of [92] we have taken the same scale as in the NRQM, since the starting point of that calculation is the rest-frame spin distributions of quarks. In other calculations, in particular in chiral models, the authors have chosen Q0 as the scale up to which the model is expected to incorporate the relevant degrees of freedom. The variety of procedures adopted to determine Q0 adds a further element of uncertainty in the results for q(Q2 ) presented in Table 6. The evolved tensor charges are collected in Fig. 53. As one can see, they span the ranges u = 0:7 − 1;

d = − (0:1 − 0:4)

at Q2 = 10 GeV2 :

(8.6.1)

It is important to notice that, since the evolution of the tensor charges is multiplicative, the ratio u=d does not depend on Q2 . As one can see from Table 6, most of the calculations give for |u=d| a value of the order of 4, or larger. Chiral soliton models CQSM1 and CQSM2, in contrast, point to a considerably smaller value, of the order of 2.7. An experimental measurement of the tensor charges may then represent an important test of these models. Note also that the introduction of chiral ?elds in a perturbative manner, as in the CQM, actually has the e,ect of increasing |u=d| owing to a strong reduction in d. A possible way to extract the u=d

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transversity ratio is to make a precision measurement of the ratio of azimuthal asymmetries in + =− leptoproduction. This could be done in the not so distant future (see Section 10). 9. Phenomenology of transversity We now review some calculations of physical observables (typically, double-spin and singlespin asymmetries) related to transversity. 21 Due to the current lack of knowledge about LT q and the related fragmentation functions, the available predictions are quite model-dependent and must be taken with a grain of salt. They essentially provide an indication of the order of magnitude of some phenomenological quantities. We also discuss two recent results on azimuthal asymmetries in pion leptoproduction that may ?nd an explanation in the coupling of transversity to a T -odd fragmentation function arising from ?nal-state interactions. 9.1. Transverse polarisation in hadron–hadron collisions 9.1.1. Transverse double-spin asymmetries in Drell–Yan processes The Drell–Yan transverse double-spin asymmetries were calculated at LO in [98,100] and at NLO in [104,105] (see also [276]). Earlier estimates [21,277] su,ered a serious problem (they assumed the same QCD evolution for Lq and LT q), which led to too optimistic values for ADY TT . In [100] the equality LT q(x; Q02 ) = Lq(x; Q02 ) was assumed to hold at a very low scale (the input Q02 = 0:23 GeV2 of the GRV distributions [114]), as suggested by various non-perturbative and con?nement model calculations (see Section 8). The transversity distributions were then evolved according to their own Altarelli–Parisi equation at LO. The resulting asymmetry (divided by the partonic asymmetry) is shown in Fig. 54. Its value is just a few percent, which makes the planned Relativistic Heavy Ion Collider (RHIC) measurement of ADY TT rather diKcult. The asymmetry for the Z 0 -mediated Drell–Yan process is plotted in Fig. 55 and has the same order of magnitude as the electromagnetic one. The authors of [105] use a di,erent procedure to estimate the transversity distributions. They set |LT q| = 2(q + Lq) at the GRV scale, thus imposing the saturation of So,er’s inequality. This yields the maximal value for ADY TT . The transversity distributions are evolved at NLO. The NLO corrections are found to be relatively small, although non-negligible. The predicted curves for ADY TT are shown in Fig. 56. Summarising the results of the calculations of ADY TT , we can say that at the typical energies of √ the RHIC experiments [278,279] ( s ¿ 100 GeV) one expects for the double-spin asymmetry, integrated over the invariant mass Q2 of the dileptons, a value ADY TT ∼ (1–2)%; 21

at most:

In this section, the transversity distributions will be denoted by LT q.

(9.1.1)

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Fig. 54. Drell–Yan longitudinal and transverse double-spin asymmetries normalised to the partonic asymmetry, as a function of xa − xb (i.e., x1 − x2 ) for two values of the dilepton invariant mass (left), and as a function of the invariant mass of the dilepton pair M 2 for two values of the c.m. energy (right). From [100].

Fig. 55. Longitudinal and transverse double-spin asymmetries (normalised to the partonic asymmetry) for the Z 0 -mediated Drell–Yan process. From [100].

√ It is interesting to note that as s falls the asymmetry tends to increase, as it was ?rst pointed √ out in [100]. Thus, at s = 40 GeV, which would correspond to the c.m. energy of the proposed ˜ experiment [280], ADY (but later cancelled) HERA-N TT could reach a value of ∼ (3–4)%. DY Model calculations of ATT are reported in [98,263]. The longitudinal-transverse Drell–Yan asymmetry ADY LT (see Section 7.3.1) was estimated in [281,282] and found to be ?ve to ten times smaller than the double-transverse asymmetry. Polarised proton–deuteron Drell–Yan processes were investigated in [283–286].

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Fig. 56. The Drell–Yan transverse double-spin asymmetry as a function √ of the virtual photon rapidity y and of the ˜ dilepton √ invariant mass M , for two values of the c.m. energy: (a) s = 40 GeV (corresponding to HERA-N ) and (b) s = 200 GeV (corresponding to RHIC). The error bars represent the estimated statistical uncertainties of the two experiments. From [105].

9.1.2. Transverse single-spin asymmetries In the early 1970s data on single-spin asymmetries in inclusive pion hadroproduction [287– 289] provoked a certain theoretical interest as it was widely held that large e,ects could not be reproduced within the framework of perturbative QCD [6]. In 1991 the E704 experiment at Fermilab extended the results on large single-spin asymmetries in inclusive pion hadroproduction with a transversely polarised proton [290,291] to higher pT . These surprising results have prompted intensive theoretical work on the subject. As a matter of fact, one year before the recent Fermilab measurements, Sivers had suggested that single-spin asymmetries could originate, at leading twist, from the intrinsic motion of quarks in the colliding protons [40,41]. This idea was pursued by the authors of [39,181], who pointed out that the Sivers e,ect is not forbidden by time-reversal invariance [17] provided one takes into account soft interactions in the initial state. In so doing, T -odd distribution functions are introduced (see Sections 4.8 and 7.4.1). A di,erent mechanism was proposed by Collins [17]. It relies on the hypothesis of Fnal-state interactions, which would allow polarised quarks with non-zero transverse momentum to fragment into an unpolarised hadron (the Collins e,ect already discussed in Sections 6.5 and 7.4.1). Finally, as seen in Section 7.4.1, another way to produce single-spin asymmetries is to assume the existence of a T -odd transverse polarisation distribution of quarks in the unpolarised initial-state hadron. All the above e,ects manifest themselves at leading twist. We shall concentrate on the Collins mechanism, which appears, among the three hypothetical sources of single-spin asymmetries just mentioned, the likeliest one (as repeatedly stressed, initial-state interactions are de?nitely harder to unravel than Fnal-state interactions).

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The Collins e,ect was investigated phenomenologically in [26], under the hypothesis that it is the only mechanism contributing to single-spin asymmetries. The authors of [26] propose a simple parametrisation for the Collins fragmentation function L0T D=q (z; ⊥ ) (see Section 6.5 and note that in [26] a function LN D=q↑ is de?ned, which is related to our L0T D=q by LN D=q↑ = 2L0T D=q ): (z) / L0T D=q (z; ⊥ ) = N ⊥ (9.1.2) z (1 − z)< ; M where M = 1 GeV and it is assumed that L0T D=q is peaked around the average value ⊥ ≡ 2⊥ 1=2 . The z dependence of ⊥ (z) is obtained from a ?t to LEP measurements of the transverse momentum of charged pions inside jets [292] (remember that ⊥ −Ph⊥ =z neglecting the intrinsic motion of quarks inside the target). Isospin and charge conjugation invariance allows one to reconstruct the entire Pavour structure of quark fragmentation into pions, giving the relations L0T D+ =u = L0T D− =d = L0T D+ =dN = L0T D− =u = 2L0T D0 =u = 2L0T D0 =d = 2L0T D0 =dN = 2L0T D0 =u = L0T D=q :

(9.1.3)

Only valence quarks in the incoming protons are considered in [26]. Their transverse polarisation distributions are taken to be proportional to the unpolarised distributions, according to LT u(x) = PTu=p u(x);

LT d(x) = PTd=p d(x);

(9.1.4)

where the transverse polarisation PTu=p of the u quark is set equal to 2=3, as in the SU(6) model, whereas the transverse polarisation of the d quark is left as a free parameter. The result of the ?t of the single-spin asymmetry data is shown in Fig. 57. Good agreement is obtained if L0T D=q saturates at large z the positivity constraint |L0T D=q | 6 D=q , otherwise the value of the single-spin asymmetry AT is too small at large xF . It also turns out that the resulting transversity distribution of the d quark violates the So,er bound 2|LT d| 6 d + Ld. Boglione and Leader pointed out [293] that, since Ld is negative in most parametrisations, the So,er constraint for the d distributions is a rather strict one. A ?t to the AT data that satis?es the So,er inequality was performed in [293], with good results provided one allows Ld to become positive at large x. In this case too, the positivity constraint on L0T D=q has to be saturated at large z. The inferred transversity distributions are shown in Fig. 58. Another calculation of the single-spin asymmetry in pion hadroproduction, based on the Collins e,ect, is presented in [294]. These authors generate the T -odd fragmentation function by the Lund string mechanism and obtain fair agreement with the E704 data by assuming the following behaviour for the transversity distributions: LT u(x) LT d(x) − → 1; as x → 1: (9.1.5) u(x) d(x) A comment on the applicability of perturbative QCD to the analysis of the E704 measurements is in order. First of all, we have already pointed out that factorisation with intrinsic transverse momenta of quarks is not a proven property but only a (plausible) hypothesis. Second, and more important, the E704 data span a range of |P⊥ | that ranges up to 4 GeV for 0 in the

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Fig. 57. Fit of the data on AT for the process p↑ p → X [290,291] assuming that only the Collins e,ect is active; the upper, middle, and lower sets of data and curves refer to + , 0 and − , respectively. From [26].

Fig. 58. The transversity distributions obtained in [293] from a ?t to the E704 data. The curves correspond to di,erent parametrisations of the helicity distributions (see [293] for details). The ?gure is taken from [48].

central region, where the asymmetry is small, and up to only 1:5 GeV for ± ; 0 in the forward region, where the asymmetry is large. At such low values of transverse momenta perturbative QCD is not expected to be completely reliable, since cross-sections tend to rise very steeply

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Fig. 59. The ?t of the single-spin asymmetry data (here AT is called AN ) performed in [177].

as |P⊥ | → 0. What allows some con?dence that a perturbative QCD treatment is nevertheless meaningful is the fact that both intrinsic ⊥ e,ects and higher twists (see below) regularise the cross-sections at P⊥ = 0. A phenomenological analysis of the E704 results, based on the Sivers e,ect as the only source of single-spin asymmetries, was carried out in [39,181]. For other (model) calculations of AT see [295,296]. As shown in Section 7.4.2, single-spin asymmetries may also arise as a result of twist-three effects [175,177,182,183]. Qiu and Sterman have used the ?rst, chirally even, term of Eq. (7.4.20) to ?t the E704 data on AT , setting GF (x; x) = Kq(x);

(9.1.6)

where K is a constant parameter. Their ?t is shown in Fig. 59. Another twist-three contribution, the second term in Eq. (7.4.20), involves the transversity distributions. This term has been evaluated by Kanazawa and Koike [182,183] with an assumption similar to (9.1.6) for the multiparton distribution EF , i.e., EF (x; x) = K LT q(x):

(9.1.7)

They found that, owing to the smallness of the hard partonic cross-sections, this chirally odd contribution to single-spin asymmetries turns out to be negligible. Clearly, in order to discriminate between leading-twist intrinsic ⊥ e,ects and higher-twist mechanisms a precise measurement of the P⊥ dependence of the asymmetry is needed, in particular at large P⊥ . Given the current experimental information on AT it is just impossible to draw de?nite conclusions as to the dynamical source of single-spin transverse asymmetries.

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9.2. Transverse polarisation in lepton–nucleon collisions Let us turn now to semi-inclusive DIS on a transversely polarised proton. As discussed at length in Section 6, there are three candidate reactions for determining LT q at leading twist: (1) semi-inclusive leptoproduction of a transversely polarised hadron with a transversely polarised target; (2) semi-inclusive leptoproduction of an unpolarised hadron with a transversely polarised target; (3) semi-inclusive leptoproduction of two hadrons with a transversely polarised target. We shall review some calculations concerning the ?rst two reactions. Two-hadron production is more diKcult to predict, as it involves interference fragmentation functions for which we have at present no independent information from other processes (a model calculation is presented in [145]). 9.2.1. hyperon polarimetry We have seen in Section 6.5 that detecting a transversely polarised hadron h↑ in the ?nal state of a semi-inclusive DIS process with a transversely polarised target, lp↑ → l h↑ X , probes the product LT q(x) LT Dq (z) at leading twist. The relevant observable is the polarisation of h↑ , which at lowest order reads (we take the y axis as the polarisation axis) 2 2 2 ↑ q; qN eq LT q(x; Q )LT Dh=q (z; Q ) h Py = aˆT (t) ; (9.2.1) 2 2 2 q; qN eq q(x; Q )Dh=q (z; Q ) where aˆT (y) = 2(1 − y)=[1 + (1 − y)2 ] is the elementary transverse asymmetry (the QED depolarisation factor). In this class of reactions, the most promising is production. The polarisation is, in fact, easily measured by studying the angular distribution of the → p decay. The transverse polarisation of ’s produced in hard processes was studied a long time ago in [297,298] and more recently in [126]. From the phenomenological viewpoint, the main problem is that, in order to compute the quantity (9.2.1), one needs to know the fragmentation functions LT Dh=q (z; Q2 ) besides the transversity distributions. ↑ A prediction for Py has been recently presented by Anselmino et al. [299]. These authors assume, at some starting scale Q02 , the relations D=u = D=d = D=s = D=u = D=dN = D=sN ≡ D=q ;

(9.2.2a)

LT D=u = LT D=d = LT D=u = LT D=dN = N LT D=s = N LD=s ;

(9.2.2b)

where N is a free parameter. For D=q and LD=s they use the parametrisation of [300] at Q02 = 0:23 GeV2 . As for the transversity distributions, saturation of the So,er bound is assumed and sea densities are neglected. Leading-order QCD evolution is applied. In Fig. 60 we show the ↑ results of [299] for Py , with three di,erent choices of N and /: the ?rst scenario corresponds to the SU(6) non-relativistic quark model (the entire spin of the is carried by the strange quark, i.e., N = 0); the second scenario corresponds to a negative N ; and the third scenario corresponds to all light quarks contributing equally to the spin (i.e., N = 1). Other predictions for the polarisation are o,ered by Ma et al. [318].

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Fig. 60. The polarisation of hyperons produced in semi-inclusive DIS, as predicted in [299]. Fig. 61. The target spin and the lepton and photon momenta. Note that q is directed along the negative z axis.

9.2.2. Azimuthal asymmetries in pion leptoproduction A potentially relevant reaction for the study of transversity is leptoproduction of unpolarised hadrons (typically pions) with a transversely polarised target, lp↑ → l hX . In this case, as seen in Section 6.5, LT q may be probed as a consequence of the Collins e,ect (a T -odd contribution to quark fragmentation arising from ?nal-state interactions). In this case one essentially measures 2 ). LT q(x)H1⊥q (z; Ph⊥ Preliminary results on single-spin transverse asymmetries in pion leptoproduction have been recently reported by the SMC [24] and the HERMES collaboration [23]. Before presenting them, we return to a kinematical problem already addressed in Section 3.1: the de?nition of the target polarisation. From the experimental point of view, the DIS target is “longitudinally” (“transversely”) polarised when its spin S is parallel (perpendicular) to the initial lepton momentum ‘. If we parametrise S as (see Fig. 61) S = |S|(sin #S cos =S ; −sin #S sin =S ; −cos #S )

(9.2.3)

and ‘ as ‘ = E(sin # ; 0; −cos # )

(9.2.4)

the angle / between S and ‘ is given by cos / = sin # sin #S cos =S + cos # cos #S :

(9.2.5)

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Thus, we have / = 0 ⇒ “longitudinal” polarisation; / = ⇒ “transverse” polarisation: 2 We use quotation marks when adopting the experimental terminology. From the theoretical point of view, it is more convenient to focus on the target and ignore the leptons. Thus the target is said to be longitudinal (transverse) polarised when its spin is parallel (perpendicular) to the photon momentum, i.e., #S = 0 ⇒ longitudinal polarisation; #S = ⇒ transverse polarisation: 2 The absence of quotation marks signals the theoretical terminology. DIS kinematics gives 2Mx sin # = 1 − y + O(M 3 =Q3 ); Q cos # = 1 −

2M 2 x2 (1 − y) + O(M 4 =Q4 ) Q2

and inverting (9.2.5) we obtain 2Mx 1 − y sin / cos =S ; cos #S cos / − Q 2Mx sin #S sin / + 1 − y cos / cos =S ; Q

(9.2.6a) (9.2.6b)

(9.2.7a) (9.2.7b)

where we have neglected O(M 2 =Q2 ) terms. If the target is “longitudinally” polarised (i.e., / = 0), one has cos #S 1; sin #S

2Mx 1 − y cos =S ; Q

so that (setting =S = 0 since the lepton momenta lie in the xz plane) 2Mx |S⊥ | 1 − y|S|: Q

(9.2.8a) (9.2.8b)

(9.2.9)

Therefore, when the target is “longitudinally” polarised, its spin has a non-zero transverse component, suppressed by a factor 1=Q. This means that there is a transverse single-spin asymmetry given by (see (6.5.10)) 2 2 ) e LT fa (x)L0T Da (z; Ph⊥ 2Mx AT aˆT (y) aa 2 |S| 1 − y sin(= ): (9.2.10) 2 Q a ea fa (x)Da (z; Ph⊥ )

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Fig. 62. The SMC data [24] on the transverse single-spin asymmetry in pion leptoproduction, as a function of the Collins angle.

We stress that the 1=Q factor in (9.2.10), which mimics a twist-three contribution, has a purely kinematical origin. This is the situation explored by HERMES [23]. If the target is “transversely” polarised (i.e., / = =2), one has 2Mx cos #S − 1 − y cos =S ; (9.2.11a) Q sin #S 1; (9.2.11b) and |S⊥ | |S|:

(9.2.12)

In this case, neglecting 1=Q2 kinematical e,ects, the target is also transversely polarised; the measured transverse single-spin asymmetry is unsuppressed and is given by Eq. (6.5.10). This is the situation of the SMC experiment [24]. Let us now come to the data. The SMC [24] presented a preliminary measurement of AT for pion production in DIS of unpolarised muons o, a transversely polarised proton target at s = 188:5 GeV2 and x 0:08;

y 0:33;

z 0:45;

Q2 5 GeV2 :

(9.2.13)

Two data sets, with P⊥ = 0:5 GeV and P⊥ = 0:8 GeV, are selected. The result for the total amount of events is (note that SMC use a di,erent choice of axes and, moreover, their Collins angle has the opposite sign with respect to ours) +

(9.2.14a)

−

(9.2.14b)

AT = aˆT = (0:11 ± 0:06) sin(= + =S ); AT = aˆT = − (0:02 ± 0:06) sin(= + =S ):

The SMC data are shown in Fig. 62. Evaluating the depolarisation factor aˆT at y 0:33, Eqs. (9:2:14a), (9.2.14b) imply +

(9.2.15a)

−

(9.2.15b)

AT = (0:10 ± 0:06) sin(= + =S ); AT = − (0:02 ± 0:06) sin(= + =S ):

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Fig. 63. The HERMES data [23] on the transverse single-spin asymmetry in pion leptoproduction, as a function of x (left) and |P⊥ | (right).

The HERMES experiment at HERA [23] has reported results on AT for positive and negative pions produced in DIS of unpolarised positrons o, a “longitudinally” polarised proton target at s = 52:6 GeV2 and in the kinematical ranges: 0:023 6 x 6 0:4;

0:1 6 y 6 0:85;

0:2 6 z 6 0:7;

Q2 ¿ 1 GeV2 :

(9.2.16)

The transverse momentum of the produced pions is |P⊥ | . 1 GeV. The HERMES result is (see Fig. 63) +

(9.2.17a)

−

(9.2.17b)

AT = + [0:022 ± 0:004(stat:) ± 0:004(syst:)]sin = ; AT = − [0:001 ± 0:005(stat:) ± 0:004(syst:)]sin = :

The conventions for the axes and the Collins angle used by HERMES are the same as ours. There appears to be a sign di,erence between the SMC and HERMES results. Unfortunately, the proliferation of conventions does not help to settle sign problems. According to the discussion above, the HERMES data, which are obtained with a “longitudinally” polarised target, gives a transverse single-spin asymmetry suppressed by 1=Q. Thus, higher-twist longitudinal e,ects might be as relevant as the leading-twist Collins e,ect. The result (9:2:17a), (9:2:17b) should be taken with this caveat in mind. Another, even deeper, reason to be very cautious in interpreting the SMC and HERMES results is the low values of |P⊥ | covered by the two experiments. This makes any perturbative QCD analysis rather problematic. Anselmino and Murgia [301] have recently analysed the SMC and HERMES data (for another analysis see [319]) and extracted bounds on the Collins fragmentation function L0T D=q . They simplify the expression of the single-spin transverse asymmetry (6.5.10) by assuming that the transversity of the sea is negligible, i.e., LT qN 0, using Eq. (9.1.3) for the fragmentation functions into pions and similar relations for D=q , and ignoring the non-valence quark contributions

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in pions. Thus, the single-spin transverse asymmetries become 4LT u(x) L0T D=q (z; P⊥ ) + AT ∼ ; N D=q (z; P⊥ ) 4u(x) + d(x) LT d(x) L0T D=q (z; P⊥ ) − ; AT ∼ d(x) + 4u(x) D=q (z; P⊥ ) L0T D=q (z; P⊥ ) 4LT u(x) + LT d(x) 0 : AT ∼ N D=q (z; P⊥ ) 4u(x) + d(x) + 4u(x) + d(x)

(9.2.18a) (9.2.18b) (9.2.18c)

Saturating the So,er inequality, the authors of [301] derive a lower bound for the quark analysing power L0T D=q =D=q from the data on AT . From the SMC result they ?nd |L0T D=q | & 0:24 ± 0:15; z 0:45; P⊥ 0:65 GeV (9.2.19a) D=q and from the HERMES data |L0T D=q | & 0:20 ± 0:04(stat:) ± 0:04(syst:); D=q

z ¿ 0:2:

(9.2.19b)

These results, if con?rmed, would indicate a large value of the Collins fragmentation function and would therefore also point to a relevant contribution of the Collins e,ect in other processes. More data at higher P⊥ would clearly make a perturbative QCD study de?nitely safer. For another determination of the Collins analysing power see below, Section 9.3. As already recalled, the interpretation of the HERMES result is made diKcult by the fact that the target is “longitudinally” polarised (that is |S⊥ | ∼ (M=Q) |S| and |S|| | ∼ |S|). Thus, focusing on the dominant 1=Q e,ects, there are in principle two types of contributions to the cross-section: (1) leading-twist contributions for a transversely polarised target; (2) twist-three contributions for a longitudinally polarised target. Type 1 is O(1=Q) owing to the kinematical relation (9.2.9); type 2 is O(1=Q) owing to dynamical twist-three e,ects. The sin = asymmetry measured by HERMES receives the following contributions [15,48,121,133,302–305] (we omit the factors in front of each term): M M ˜ Asin = ∼ |S⊥ |LT q ⊗ H1⊥(1) + |S|| |hL ⊗ H1⊥(1) + h |S|| |h⊥(1) 1L ⊗ H Q Q ∼

M M M ⊥(1) ˜ |S|LT q ⊗ H1 + |S|hL ⊗ H1⊥(1) + h |S|h⊥(1) 1L ⊗ H ; Q Q Q

(9.2.20)

where h⊥(1) and H1⊥(1) are de?ned in (4.9.5a) and (6.5.22), respectively. The ?rst term in 1L (9.2.20) is the type 1 term described above and corresponds to the Collins e,ect studied in [301]. The other two terms (type 2) were phenomenologically investigated in [48,302–305]. In order to analyse the data by means of (9.2.20), extra input is needed, given the number of unknown quantities involved. As we have seen in Section 6.5, when the target is longitudinally polarised there is also a sin 2= asymmetry, which appears at leading twist and has the form ⊥(1) Asin 2= ∼ |S|| |h⊥(1) : 1L ⊗ H1

(9.2.21)

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Fig. 64. The single-spin azimuthal asymmetry in pion leptoproduction as computed in [303], compared to the HERMES data [23]. The solid line corresponds to LT q = Lq. The dashed line corresponds to saturation of the So,er inequality.

The smallness of Asin 2= , as measured by HERMES (see Fig. 63), seems to be an indication in favour of h⊥(1) 0. If we make this assumption [303], recalling (4.10.1) and (4.10.2), we 1L obtain h˜L (x) = hL (x) = LT q(x) and (9.2.20) reduces to a single term of the type LT q ⊗ H1⊥(1) . Using a simple parametrisation [17] for the Collins fragmentation function, namely H1⊥ (x; 2⊥ ) M MC = 2h 2 2 D(x; ⊥ ) MC + ⊥

(9.2.22)

with MC as a free parameter, the authors of [303] ?t the HERMES data fairly well (see Fig. 64). In [48] it was pointed out that the HERMES data on the sin 2= asymmetry do not necessarily ˜ imply h⊥(1) 1L = 0. If one assumes the interaction-dependent distribution hL (x) to be vanishing, so that from (4:95a) and (4.10.2) one has (neglecting quark mass terms) 1 1 dy ⊥(1) 2 h1L (x) = − xhL (x) = − x LT q(y) (9.2.23) 2 2 x y then it is still possible to obtain a sin = asymmetry of the order of few percent (as found by HERMES), with the sin 2= asymmetry suppressed by a factor 2. The approximation (9.2.23) was also adopted in [306], where an analysis of the HERMES and SMC data in the framework of the chiral quark soliton model is presented. In conclusion, we can say that the interpretation of the HERMES and SMC measurements is far from clear. The experimental results seem to indicate that transversity plays some rˆole but the present scarcity of data, their errors, our ignorance of most of the quantities involved in the

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Fig. 65. Kinematics of two-hadron production in e+ e− annihilation.

process, and, last but not least, uncertainty in the theoretical procedures make the entire matter still rather vague. More, and more precise, data will be of great help in settling the question. 9.3. Transverse polarisation in e+ e− collisions. An independent source of information on the Collins fragmentation function H1⊥ is inclusive two-hadron production in electron–positron collisions (see Fig. 65): e+ e− → h1 h2 X:

(9.3.1)

This process was studied in [307–309]. It turns out that the cross-section has the following angular dependence (we assume that two alike hadrons are produced, omit the Pavour indices and refer to Fig. 65 for the kinematical variables) d N 2 ) + C sin2 Q2 cos(2=1 )H1⊥(1) (z1 )HN ⊥(1) ˙ (1 + cos2 Q2 )D(z1 )D(z (z2 ); 1 d cos Q2 d=1

(9.3.2)

where C is a constant containing the electroweak couplings. Thus, the analysis of cos(2=1 ) asymmetries in the process (9.3.1) can shed light on the ratio between unpolarised and Collins fragmentation functions. Efremov and collaborators [310 –312] have carried out such a study using the DELPHI data on Z 0 hadronic decays. Under the assumption that all produced particles are pions and that fragmentation functions have a Gaussian T dependence, they ?nd * ⊥+ HN 1 = (6:3 ± 1:7)%; (9.3.3) D where the average is over Pavours and the kinematical range covered by data. The result (9.3.3) is an indication of a non-zero fragmentation function of transversely polarised quarks into unpolarised hadrons. The authors of [310 –312] argue that a more careful study of the Q2 dependence of the experimentally measured cross-section could increase the value (9.3.3) up to ∼ 10%. An analysing power of this order of magnitude would make the possibility of observing the Collins e,ect in the future experiments rather tangible.

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10. Experimental perspectives In this section, which completes the bulk of our report, we outline the present experimental situation and the future prospects. The study of transversity distributions is a more-or-less important fraction of the physics program of many ongoing and forthcoming experiments in various laboratories (DESY, CERN and Brookhaven). An overview of the experimental state of the ?eld can be found in the Proceedings of the RIKEN-BNL Workshop on “Future Transversity Measurements” [313]. 10.1. ‘N experiments 10.1.1. HERMES The HERMES experiment uses the HERA 27:5 GeV positron (or electron) beam incident on a longitudinally polarised H or D gas-jet target. The hydrogen polarisation is approximately 85%. Running since 1995, HERMES has already provided a large amount of data on polarised inclusive and semi-inclusive DIS. We have discussed (see Section 9.2.2) their (preliminary) result of main concern in this report, that is the observation of a relatively large azimuthal spin asymmetry in semi-inclusive DIS on a longitudinally polarised proton target, which may involve the transversity distributions via the Collins e,ect (as we have noted, similar ?ndings have been reported by the SMC at CERN). HERMES plans to continue data taking in the period 2001–2006, after the HERA luminosity upgrade (which should increase the average luminosity by a factor 3). Two of the foreseen ?ve years of running should be dedicated to a transversely polarised target with an expected statistics of 7 × 106 reconstructed DIS events. The foreseen target polarisation is ∼ 75%. The transverse polarisation program at HERMES includes [314] (besides the extraction of the spin structure function g2 ): (i) a measurement of the twist-three azimuthal asymmetry in semi-inclusive pion production with a longitudinally polarised lepton beam; (ii) the study of the Collins e,ect in the scattering of an unpolarised lepton beam o, a transversely polarised target; and (iii) a measurement of the transverse asymmetry in leptoproduction of two correlated mesons. According to the estimates presented in [314,315], HERMES should be able to determine both the transversity distributions and the Collins fragmentation function (at least for the dominant u Pavour) with good statistical accuracy. 10.1.2. COMPASS COMPASS is a new ?xed target experiment at CERN [25], with two main programs: the “muon program” and the “hadron program”. The former (which upgrades SMC) aims to study the spin structure of the nucleon with a high-energy muon beam. COMPASS will use polarised muons of 100 –200 GeV, scattering o, polarised proton and deuteron targets. Expected polarisations are 90% and 50% for the proton and the deuteron, respectively. The transverse polarisation physics program is similar to that of HERMES, but covers di,erent kinematical regions. In particular, single-spin asymmetries in hadron leptoproduction will be measured. These will provide the transversity distributions via the Collins e,ect. According to an estimate presented in [25] LT q should be determined with a ∼ 10% accuracy in the intermediate-x region. Data taking by COMPASS has started in 2001.

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10.1.3. ELFE Electron Laboratory for Europe (ELFE) is a continuous electron-beam facility, which has been discussed since the early 1990s. The latest proposal is for construction at CERN by exploiting the cavities and other components of LEP not required for LHC [316]. The maximum energy of the electron beam would be 25 GeV. The very high luminosity (about three orders of magnitude higher than HERMES and COMPASS) would allow accurate measurements of semi-inclusive asymmetries with transversely polarised targets. In particular, polarimetry in the ?nal state should reach a good degree of precision (a month of running time, with a luminosity of 1034 cm−2 s−1 , allows the accumulation of about 106 ’s with transverse momentum greater than 1 GeV=c. 10.1.4. TESLA-N The TESLA-N project [317] is based on the idea of using one of the arms of the e+ e− collider TESLA to produce collisions of longitudinally polarised electrons on a ?xed proton or deuteron target, which may be either longitudinally or transversely polarised. The basic parameters are: electron beam energy 250 GeV, an integrated luminosity of 100 fb−1 per year, a target polarisation ∼ 80% for protons, ∼ 30% for deuterium. The transversity program includes the measurement of single-spin azimuthal asymmetries and two-pion correlations [314]. The proposers of the project have estimated the statistical accuracy in the extraction of the transversity distributions via the Collins e,ect and found values comparable to the existing determinations of the helicity distributions. They have also shown that the expected statistical accuracy in the measurement of two-meson correlations is encouraging if the interference fragmentation function is not much smaller that its upper bound. 10.2. pp experiments 10.2.1. RHIC The Relativistic Heavy Ion Collider (RHIC, Fig. 66) at the Brookhaven National Laboratory operates with gold ions and protons. With the addition of Siberian snakes and spin rotators, there will be the possibility of accelerating intense polarised proton beams up to energies of 250 GeV per beam. The spin-physics program at RHIC will study reactions involving two polarised proton beams with both longitudinal and transverse spin orientations, at an average centre-of-mass energy of 500 GeV (for an overview of spin physics at RHIC see [22]). The expected luminosity is up to ∼ 2 × 1032 cm−2 s−2 , with 70% beam polarisation. Two detectors will be in operation: STAR (see, e.g., [279]) and PHENIX (see, e.g., [278]). The former is a general purpose detector with a large solid angle; the latter is a dedicated detector mainly for leptons and photons. Data taking with polarised protons will start in 2001. The most interesting process involving transversity distributions to be studied at RHIC is Drell–Yan lepton pair production mediated by ∗ or Z 0 . As seen in Section 9.1, the expected double-spin asymmetry ADY TT is just few percent but may be visible experimentally, provided the transversity distributions are not too small. Single-spin Drell–Yan measurements could be a good testing ground for the existence of transversity in unpolarised hadrons arising from T -odd initial-state interaction e,ects [47].

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Fig. 66. An overview of RHIC.

11. Conclusions The transverse polarisation of quarks represents an important piece of information on the internal structure and dynamics of hadrons. In the previous sections we have tried to substantiate this statement reviewing the current state of knowledge. In conclusion, let us try to summarise what we have learned so far about transversity. • The transverse polarisation (or transversity) distributions LT q are chirally odd leading-twist

quantities that do not appear in fully inclusive DIS, but do appear in semi-inclusive DIS processes and in various hadron-initiated reactions. • The QCD evolution of LT (x; Q2 ) is known up to NLO, and turns out to be di,erent from the evolution of the helicity counterpart. • Many models (and other non-perturbative tools) have been used to calculate the transversity distributions in the nucleon. These computations show that, at least for the dominant u sector, at low momentum scales LT q is not so di,erent from Lq.

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• The phenomenology of transversity is very rich. It includes transversely polarised Drell–Yan

processes, leptoproduction of polarised baryons and mesons, correlated meson production and, via the Collins e,ect, lepto- and hadro-production of pions. • Only two preliminary results, that may have something to do with transversity, are currently available. The HERMES and SMC collaborations have found non-vanishing azimuthal asymmetries in pion leptoproduction, which may be explained in terms of transverse polarisation distributions coupling to a T -odd fragmentation function (Collins e,ect). However, no de?nite conclusion about the physical explanation of these ?ndings is possible as yet.

The intense theoretical e,ort developed over the last decade must now be put to fruition by a vigorous experimental study of transversity. Many collaborations around the world (at Brookhaven, DESY and CERN) aim at measuring quark transverse polarisation in the nucleon. This is certainly a complex task since the foreseen values of some of the relevant observables are close to the sensitivity limits of the experiments. Nevertheless, the variety and accuracy of the measurements planned for the coming years permit a certain con?dence that the veil of ignorance surrounding quark transversity will at last begin to dissolve. Acknowledgements We are grateful to Mauro Anselmino, Alessandro Bacchetta, Elena Boglione, Umberto D’Alesio, Bo-Qiang Ma, Francesco Murgia, Sergio Scopetta, Oleg Teryaev and Fabian Zomer for various discussions on the subject of this report. Appendix A. Sudakov decomposition of vectors We introduce two light-like vectors (the Sudakov vectors) 1 p = √ (; 0; 0; ); 2 1 n = √ (−1 ; 0; 0; −−1 ); 2

(A.1a) (A.1b)

where is arbitrary. These vectors satisfy p2 = 0 = n2 ;

p · n = 1;

n+ = 0 = p− :

(A.2)

In light-cone components they read p = (; 0; 0⊥ );

(A.3a)

n = (0; −1 ; 0⊥ ):

(A.3b)

A generic vector A can be parametrised as (a Sudakov decomposition) A = /p +
(A.4)

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159

with A⊥ = (0; A⊥ ; 0). The modulus squared of A is 2 : A2 = 2/< − A⊥

(A.5)

Appendix B. Reference frames B.1. The ∗ N collinear frames In DIS processes, we call the frames where the virtual photon and the target nucleon move collinearly “ ∗ N collinear frames”. If the motion takes place along the z-axis, we can represent the nucleon momentum P and the photon momentum q in terms of the Sudakov vectors p and n as P = p + 12 M 2 n p ;

(B.1)

q P · qn − xp = Mn − xp ;

(B.2)

where the approximate equality sign indicates that we are neglecting M 2 with respect to large scales such as Q2 , or (P + )2 in the in?nite momentum frame. Conventionally we always take the nucleon to be directed in the positive z direction. With the identi?cation (B.1) the parameter appearing in the de?nition of the Sudakov vectors (A.1a), (A.1b) coincides with P + and ?xes the speci?c frame. In particular: • in the target rest frame (TRF) one has

P = (M; 0; 0; 0);

(B.3)

q = (; 0; 0; − 2 + Q2 ) (B.4) √ √ and ≡ P +√= M= 2. The Bjorken limit in this frame corresponds to q− = 2 → ∞ with q+ = − Mx= 2 ?xed. • in the inFnite momentum frame (IMF) the momenta are

1 P √ (P + ; 0; 0; P + ); 2 1 M M + + q √ − xP ; 0; 0; − + − xP : 2 P+ P

(B.5) (B.6)

Here we have P − → 0 and ≡ P + → ∞. In this frame the vector n is suppressed by a factor of (1=P + )2 with respect to p . By means of the Sudakov vectors we can construct the perpendicular metric tensor g⊥ which 2 2 projects onto the plane perpendicular to p and n, and to P and q (modulo M =Q terms) = g − (p n + p n ): g⊥

(B.7)

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Transverse vectors in the ∗ N frame (or “perpendicular” vectors) will be denoted by a ⊥ subscript. Another projector onto the transverse plane is ! p! n : ⊥ =

Consider now the spin vector of the nucleon. It may be written as M2 5N 5N S = p − ; n + S⊥ p + S⊥ M 2 M

(B.8)

(B.9)

where 5N2 + S2⊥ 6 1 (the equality sign applies to pure states). The transverse spin vector S⊥ is of order O(1), thus it is suppressed by one power of P + with respect to longitudinal spin S = 5N p =M . Finally, in semi-inclusive DIS the momentum Ph of the produced hadron h may be parametrised in the ∗ N collinear frame as ; Ph zq + xzP + Ph⊥

(B.10)

where z=

P · Ph 2x = 2 P · Ph : P·q Q

(B.11)

B.2. The hN collinear frames In polarised semi-inclusive DIS it is often convenient to work in a frame where the target nucleon N and the produced hadron h are collinear (a “hN collinear frame”). In the family of such frames the momenta of N and h are parametrised, in terms of two Sudakov vectors p and n , as M 2 n p ; 2 M 2x Q2 z Q2 z Ph h2 p + n n : Q z 2x 2x

P = p +

(B.12) (B.13)

The projectors onto the transverse plane (vectors lying in this plane will be denoted by the subscript T ) are gT = g − (p n + p n ); ! p! n : T =

(B.14) (B.15)

In hN collinear frames the photon acquires a transverse momentum Q2 (B.16) n + qT : 2x Comparing this to the Sudakov decomposition (B.10) of the momentum of the produced hadron we obtain q = − xp +

−zqT : Ph⊥

(B.17)

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The spin vector of h is 5 Sh = h Ph + ShT : (B.18) Mh The relation between transverse vectors in the ∗ N frame (⊥-vectors) and transverse vectors in the hN frame (T -vectors) is 2x a⊥ = aT − 2 [a · qT P + a · PqT ]: (B.19) Q Therefore, if we neglect order 1=Q corrections, that is if we ignore higher-twist e,ects, we can identify transverse vectors in ∗ N collinear frames with transverse vectors in hN collinear frames (in other terms, we have g⊥ gT and ⊥ T ). Appendix C. Mellin moment identities We ?rst recall here the de?nition of the so-called plus regularisation, necessary for the IR singularities present in the AP splitting kernels: 1 1 f(x) [f(x) − f(1)] dx = dx : (C.1) (1 − x)+ (1 − x) 0 0 A convenient identity regarding the above plus symbol is:   n−1 1 1 n−1   n−1 x x 1 dx f(x) = dx − (1 − x) f(x):  (1 − x) (1 − x) j + 0 0 +

(C.2)

j=1

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Physics Reports 359 (2002) 169–282

Higgs physics at LEP-1 Andr!e Sopczak ∗ Lancaster University; Lancaster LA1 4YW; UK Received June 2001; editor : J:A: Bagger

Contents 1. Introduction 2. One-doublet standard model 2.1. Theoretical framework 2.2. Search in the low-mass range 2.3. Search in the high-mass range 2.4. Combination of the search results from all LEP experiments 3. One-doublet and one-singlet model 3.1. Theoretical framework 3.2. Search for invisibly decaying Higgs bosons 4. Two-doublet model 4.1. Theoretical framework 4.2. Limits on sin( − ) from searches for Higgs boson bremsstrahlung 4.3. General Z lineshape limits 4.4. Limits on cos( − ) and neutral Higgs boson masses from the total Z width

171 172 172 185 187 202 207 207 209 211 211 221 221 226

4.5. Search for neutral Higgs boson pair-production 4.6. Search for charged Higgs boson pair-production 4.7. Search for the reactions Z → : → :h; :A 5. Minimal Supersymmetric Standard Model 5.1. Theoretical framework 5.2. Basic radiative corrections and mass limits 5.3. Full one-loop radiative corrections and mass limits 6. Conclusions Acknowledgements Appendix A. Details of the search methods A.1. The reaction Z → Z? H A.2. The reaction Z → hA A.3. The reaction Z → H+ H− References

∗

Corresponding author. EP Division, CERN, CH 1211 Geneva 23, Switzerland. E-mail address: [email protected] (A. Sopczak). c 2002 Elsevier Science B.V. All rights reserved. 0370-1573/02/$ - see front matter PII: S 0 3 7 0 - 1 5 7 3 ( 0 1 ) 0 0 0 6 2 - X

226 229 230 235 235 242 245 253 254 254 254 260 269 275

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Abstract In e+ e− annihilations at the Z pole near 91 GeV center-of-mass energy, the complete results from the general searches for the Higgs boson of the Minimal Standard Model (MSM) and for neutral and charged Higgs bosons of non-minimal Higgs models are reviewed for the four LEP experiments at CERN: ALEPH, DELPHI, L3 and OPAL. The search for Higgs bosons has been made possible by the analysis of their large data sets consisting of a total of about 18 million hadronic Z decays recorded from 1989 to 1995. No indication of a signal was observed in any search channel. Interpretations of the experimental results are given in the MSM and in extensions with an extra Higgs boson singlet and c 2002 doublet, as well as in the Minimal Supersymmetric extension of the Standard Model (MSSM). Elsevier Science B.V. All rights reserved. PACS: 12.60.−i; 12.60.Fr; 14.80.−j; 14.80.Bn; 14.80.Cp

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171

1. Introduction The Grst Z boson was registered on August 13, 1989 at the Large Electron Positron collider (LEP-1) at CERN, and during six years of data-taking at the Z resonance, large data sets were recorded each year with increasing luminosity. In 1994, the LEP-1 design luminosity was exceeded and in 1995 the data taking at the Z resonance was completed by the LEP experiments ALEPH, DELPHI, L3 and OPAL. Details of these experiments can be found in Refs. [1– 4]. A total of about 4.5 million hadronic Z boson decays were recorded by each experiment, and to this day new results from these data are being released. This report reviews the results from the LEP-1 data for the pursuit of one of the most challenging quests of experimental particle physics: the search for Higgs particles [5–9]. The interest in the Higgs boson search is closely related to the understanding of the mechanism which is responsible for generating masses. The vacuum, deGned as the lowest energy state of a dynamical system, can have a structure. The importance of understanding the vacuum structure for the progress of modern physics has been pointed out in Ref. [10]. Historically, the quantum theory of electromagnetism, quantum electrodynamics (QED), predicts a vacuum structure in which virtual electron–positron pairs can be generated, and this prediction has been conGrmed to a high level of precision. In the Higgs model the picture is more subtle. The vacuum is Glled with Higgs Gelds and ordinary particles couple to the ground state of these Gelds. In 1964, Peter Higgs and others pointed out that non-zero vacuum expectation values of these Gelds are important in gauge theories and change the nature of the particles described by these theories [5 –9] in that the particles obtain an e:ective mass. Glashow, Weinberg and Salam applied Higgs’ idea to the SU(2)L × U(1)Y gauge symmetry with left-handed doublets and right-handed singlets, thus establishing the basis for the present Standard Model of particle physics [11–14]. The renormalizability of the theory, which is necessary to obtain sensible predictions and is only possible when a Higgs boson is involved, was proven in 1971 by ’t Hooft and Veltman, who received the Nobel Prize in 1999 [15 –18]. The relation of the gauge boson masses as a function of the gauge couplings are predicted with high precision after radiative corrections. Today, all existing experimental results are in very good agreement with the Standard Model predictions. After the discovery of the top quark in 1994 [19 –21], for which indirect evidence already existed before [22], only the Higgs boson remains undiscovered to complete the SM. The discovery of a light Higgs boson would support Supersymmetric extensions of the Standard Model, which predicts a light Higgs boson below about 130 GeV. The non-observation would exclude Supersymmetry with all its beautiful implications. In the so-called Minimal Standard Model (MSM) of particle physics one Higgs boson is predicted as will be explained in Section 2. The most important production mechanism at LEP is Higgs boson bremsstrahlung o: a Z boson in analogy to the radiation of a photon o: an accelerated electron. The search for this MSM Higgs boson is determined only by the (unknown) Higgs boson mass, since the decay properties of the Higgs boson depend only on its mass. For each Higgs boson decay channel, the background from several well-known Z boson decay modes are studied. In extensions of the MSM additional Higgs boson production mechanisms are possible and the Higgs boson decay properties depend on further parameters. This leads to a variety of di:erent

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Table 1.1 Overview of di:erent Higgs boson sectors in Standard and Supersymmetric models. Supersymmetric models require at least two Higgs boson doublets. The Minimal Standard Model (MSM), its Minimal Supersymmetric extension (MSSM) and the Non-minimal MSSM (NMSSM) contain a distinctive Higgs boson structure Standard Model

Supersymmetric Model

One-doublet (MSM) One-doublet and one-singlet (simplest Majoron model) Two-doublet Two-doublet and one-singlet

— — Two-doublet (MSSM) Two-doublet and one-singlet (NMSSM)

possible signatures in the detectors. Table 1.1 gives an overview of Higgs boson sectors with di:erent numbers of Higgs boson doublets and singlets. The experimental search for the MSM Higgs boson is reviewed in Section 2 and details of the event selection are given in Appendix A. Section 2 also summarizes the particle spectrum of the Standard Model of particle physics and the properties of the minimal Higgs sector. Interpretations and combined mass limits are given in the one-doublet Higgs boson model of the MSM. The next possible extension of the Higgs boson sector is the addition of an extra Higgs boson singlet. In Section 3 experimental results are interpreted in the simplest Majoron model with one Higgs boson doublet and one Higgs boson singlet. The general two-doublet Higgs model is discussed in Section 4, and in addition to the various search channels, direct and indirect constraints from experimental analyses are summarized. Details of the experimental searches for non-minimal Higgs bosons are reviewed in Appendix A and in particular the important b-quark tagging with microvertex detectors and -lepton identiGcation in hadronic events are addressed. In Section 5 the idea of Supersymmetry is introduced. Supersymmetric models strongly motivate the two-doublet Higgs boson scenario. The important implications of Supersymmetry for the Higgs boson phenomenology are summarized. Higher order corrections in the Minimal Supersymmetric extension of the Standard Model (MSSM) modify signiGcantly the tree level predictions and are very important in the interpretation of the experimental results. The search techniques and methods of interpretation of the LEP-1 data which are reviewed in this report can be largely applied to the Higgs boson searches in the future. Since 1996, LEP-2 operated with a center-of-mass energy above the W+ W− threshold, thus raising the kinematic reach for a Higgs boson discovery. The ongoing analyses of LEP-2 data are not reviewed in this article and the reader is referred to Ref. [23] for the latest results. The hunt for Higgs bosons will also be undertaken at the Tevatron at Fermilab, the Large Hadron Collider (LHC) at CERN and a planned linear collider of at least 500 GeV center-of-mass energy (LC500). 2. One-doublet standard model 2.1. Theoretical framework This section describes the main constituents of the Standard Model of particle physics. All matter consists of leptons, quarks and gauge bosons. Leptons exist in three generations

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Table 2.1 Fermion properties in the Standard Model. The Qavor, electric charge and mass for the quarks and leptons of the three generations are compared. The observation of small neutrino masses was recently reported [26 –28] Generation

Flavor

Electric charge

1

u d

−1=3

e e

−1

2

c s

3

2=3

0

2=3

−1=3

0

−1

t b

−1=3

2=3

−1

0

Mass (GeV) 0.002– 0.007 0.003– 0.09 ≈0 0:511 × 10−3

1.1–1.4 0.06 – 0.17 ≈0 0.106

169 –179 4.1– 4.4 ≈0 1.78

(families): electrons, muons and taus with their corresponding neutrinos. The six quarks are also ordered in three families distinguished by their Qavors (see Table 2.1): up (u) and down (d), charm (c) and strange (s), top (t) and bottom (b). Some of the basic properties of fermions are summarized in Table 2.1 [24]. A comprehensive review of the Standard Model is given in Ref. [25]. Each lepton and quark has one anti-particle with opposite charge. Quarks exist in three colors. Thus, matter consists of 12 leptons and 36 quarks. Normal matter is built up of upand down-type quarks and electrons. Other fermions can be detected in high-energy physics experiments. It is remarkable that no reason is known for the existence of the particles in the second and third families. All fundamental forces, electromagnetic, weak, strong, and gravitational, are transmitted by mediators. The gravitational force is many orders of magnitude weaker than the other forces, and it is not yet known how to incorporate it in the Standard Model. In this work, the ‘Standard Model’ includes (a) the electroweak theory, which is the uniGcation of electromagnetic (QED) and weak interactions and (b) QCD, which describes strong interactions between quarks and gluons. The mediators (gauge bosons with spin = 1) of the three fundamental forces of the Standard Model are listed in Table 2.2. In the Standard Model, all fermion and boson masses are generated by the so-called Higgs mechanism described below. Cosmological models suggest that mass generation occurred during a very early epoch of the Universe, and that prior to the spontaneous breakdown of a higher symmetry, only a plasma of massless particles existed. The change of the vacuum structure to a new vacuum with a stable minimum in the Higgs potential determines the Higgs particle

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Table 2.2 Gauge boson properties in the Standard Model. The theoretical prediction of the mass agrees with the measurements Force

Gauge bosons

Electro-magnetic

Neutral weak Charged weak

Z

Strong

gi

W±

Electric charge

Mass (GeV) prediction [25]

Mass experiment [24]

0

0

6 2 × 10−16 eV

0

91:0 ± 0:8 79:7 ± 0:9

91:1882 ± 0:0022 GeV 80:419 ± 0:056 GeV

0

0

¡ a few MeV

±1

(i = 1; : : : ; 8)

spectrum and their masses. The discovery of one or more Higgs bosons would prove the concept of the Higgs mechanism. The importance of spontaneous symmetry breaking was Grst realized in condensed matter physics. In the Landau–Ginzburg theory of superconductivity [29], developed in the early 1950s, the MeiTner–Ochenfeld e:ect, which describes the screening of magnetic Qux in a superconductor, is explained by the generation of an e:ective mass for photons. The Landau– Ginzburg potential is of order 4 ( is the wave function): VLG () = m2 2 + 4 , where is the self-coupling constant, m2 = a(T − Tc ), Tc is the critical temperature and a is a positive constant. For T ¿ Tc , the free energy has a minimum at the vacuum expectation || = 0. If T ¡ Tc , the free energy has a minimum at ||2 = − m2 =2 ¿ 0. The conserved electric current, ˜ ∗ ) − e||2 A, where e is the electron charge and A is the photon Geld, ˜ − ∇ j = − (i=2)(∗ ∇ corresponds to the invariance of the electromagnetic Lagrangian under a phase transformation. The second term leads to an e:ective photon mass, which explains why the magnetic Geld cannot penetrate the superconductor. This idea was then also applied to particle physics to explain mass generation. The Higgs sector of the Standard Model consists of at least one complex Higgs doublet Geld to generate masses for all three heavy gauge bosons of the weak interaction. The Standard Model with one Higgs doublet is called Minimal Standard Model (MSM). In this section, the theoretical properties of this Higgs sector are discussed, with emphasis on implications for experimental Higgs boson searches. The Higgs potential is shown in Fig. 2.1 and has the form V () = 2 † + († )2 ; where and are real constants, and is the complex Higgs doublet Geld: + 1 + i2 = : = 3 + i4 0

(2.1)

(2.2)

In analogy with the Landau–Ginzburg theory, for 2 ¿ 0 the parabolic potential has only one minimum at zero. For values of 2 ¡ 0, the potential has an inGnite number of degenerate

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Fig. 2.1. Illustration of the Higgs potential with the form of a ‘Mexican hat’. The ground state is degenerate and the choice of one speciGc value spontaneously breaks the symmetry.

ground states (Fig. 2.1) and the symmetry of the ground state is spontaneously broken: A particular ground state does not display the symmetry of the Lagrangian. The leading term in the Higgs potential has to be of order 4 , since odd powers of would cause an unstable vacuum, i.e. the potential has no global minimum. The 2 term leads to a Higgs boson mass and terms i (i ¿ 6) reduce to an e:ective 4 interaction in four space–time dimensions, as shown in non-perturbative calculations [30]. 2.1.1. Gauge boson masses Without the Higgs mechanism, the Z and W± gauge bosons of the SU(2)L × U(1)Y theory would be massless. However, the weak interactions are known to be of short range from neutron decay and thus are mediated by massive bosons. This is di:erent from the electromagnetic interactions which are long range and mediated by massless photons. Masses for the gauge bosons are generated explicitly through spontaneous symmetry breaking via the Higgs mechanism. The Higgs potential Eq. (2.1) for 2 ¡ 0 and ¿ 0 has the shape of a ‘Mexican hat’ (Fig. 2.1). The potential V () develops a minimum at Gnite values of ||: ||2 = † = 12 (12 + 22 + 32 + 42 ) = −

2 : 2

(2.3)

This manifold of points where V () is minimal is invariant under SU(2)L transformations. In order to expand (x) around a particular minimum, one can choose 1 = 2 = 4 = 0, 3 = − 2 =, thereby explicitly breaking the SU(2)L symmetry. This corresponds to breaking of the rotational symmetry of the ground state depicted in Fig. 2.1. The Higgs Geld (x) can be

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expanded around that particular minimum 0 1 ; (x) = 2 v + h(x)

(2.4)

where h(x) is the Higgs Geld expanded around the vacuum expectation value (VEV), v. The Higgs doublet has four degrees of freedom Eq. (2.2), three of which are transformed to give masses to the W± and the Z bosons. One neutral physical Higgs Geld remains. The Higgs boson mass is given by √ 92 V 2 mH = 2 = v2 = 2 (2.5) 9 =v as the curvature of the physical Higgs Geld at the minimum of the potential. The Higgs boson mass is not predicted by the theory, since the Higgs self-coupling is a free parameter. A mass term for the charged vector bosons is generated: mW = 12 vg. The neutral gauge boson Gelds mix and after diagonalization their physical masses are obtained: mZ = 12 v g2 + g2 ; m = 0 : (2.6) The ratio of the coupling constants for the SU(2)L and U(1)Y group deGnes the weak mixing angle W , tan W ≡ g =g. The Higgs mechanism leads to the important mass relation: mW =mZ = cos W

(2.7)

(details can be found in standard text books [31,32]). The -parameter deGnes the deviation from the above mass relation in terms of physical parameters: mW ≡ : (2.8) mZ cos W By deGnition, = 1 in the MSM. The experimental world average [24] = 0:9998 ± 0:0008

(2.9)

is in complete agreement with the theoretical prediction. This represents a stringent constraint on extensions of the MSM. 2.1.2. Fermion masses This subsection outlines the generation of fermion masses and their couplings to the Higgs boson in MSM. The Higgs Geld generates masses for the fermions and it can couple to the fermion Gelds while preserving the SU(3)C × SU(2)L × U(1)Y gauge invariance. The Lagrangian shows the interaction of the electron with the Higgs boson explicitly: + e 0 Le = − ge (Ue e) UL eR + eU R (− U ) : (2.10) 0 e L

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Analogous terms exist for the other fermions. One arbitrary Yukawa coupling gf which is deGned as the coupling constant between one boson and a fermion pair, exists for each fermion. The substitution of the Higgs Geld by its expansion around a particular vacuum state with non-zero VEV, Eq. (2.4), breaks the symmetry spontaneously, and a fermion mass term is generated. From the expansion of the Lagrangian, a relationship between the Yukawa coupling and the fermion mass mf is obtained: gv (2.11) mf = √f : 2 The fermion masses are not predicted as gf is unknown, although the value of the VEV v is known from -decay: v = 2−1=4 GF−1=2 = 246 GeV. This relation of Fermi’s constant GF to the VEV is based on the fact that the Fermi theory of -decay must be recovered in the Standard Model for low momentum transfer. Eq. (2.11) has a major implication for the Higgs boson search: The Yukawa coupling gf is proportional to the fermion mass, so the decay rate is proportional to the fermion mass squared, i.e. the Higgs boson prefers to decay into the most massive kinematically accessible fermion pair until the decay into the heavy gauge bosons is allowed for mH ¿ 2mW . 2.1.3. One-loop potential In this subsection, the implications of one-loop radiative corrections and results of nonperturbative calculations for the Higgs boson search are summarized. A renormalization procedure is applied in order to remove inGnite terms, which arise when, in addition to the tree level, higher order radiative corrections, represented by loop graphs, are included in the calculations. InGnities introduced by Feynman diagrams have di:erent degrees of divergence. They can be absorbed by a redeGnition of 2 ; and the Higgs Geld. The conventional renormalization conditions [33] are given for the renormalized Higgs boson mass R , for the renormalized coupling constant R , and for the wave function renormalization at a given physical energy scale M by d 4 V d 2 V 92 ! 2 2 R ≡ −! (pi = 0) = ; R ≡ ; ≡1; (2.12) d 2 =0 d 4 =0 9p2 p2 =M 2 where ! is the inverse Higgs propagator for a particle with four-momentum p. In the framework of the MSM, the Higgs potential acquires one-loop contributions from Higgs boson self-interactions (scalar), Higgs–gauge boson interactions (vector) and Higgs–fermion interactions: Ve: () = Vtree + Vradiative ;

Vradiative = Vscalar + Vvector + Vfermion :

(2.13)

The physical energy scale M is arbitrary and is Gxed by the choice of the renormalization scheme: changing it redeGnes the renormalized Higgs boson masses and renormalized coupling constants. 2.1.4. Higgs boson mass limits from vacuum stability The relationship between the top quark and Higgs boson masses, the Coleman–Weinberg lower Higgs boson mass limit and implications from possible phase transitions at Gnite temperature are described in the following sections.

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Fig. 2.2. Higgs boson and top mass relation required for vacuum stability. The large top mass requires a large Higgs boson mass.

Top quark and Higgs boson mass relation. The large top-quark mass mtop = 175 GeV enhances the importance of radiative corrections for Vfermion , since the fermion masses are proportional to the Yukawa coupling constants gf , see Eq. (2.11). Therefore, the most massive fermions contribute most to the loop diagrams. The Higgs boson search at LEP is kinematically restricted to about mH ¡ 115 GeV. Thus, the main contributions to radiative corrections are due to the vector boson and fermions [33]: Vradiative () Vvector + Vfermion = −4

B = 1:75 × 10

1−

mtop 76:2 GeV

1 4 4 B ln ; 64#2 M4

4 :

(2.14)

The large top-quark mass yields B 6 0. If B is suVciently negative, the e:ective potential Ve: is unbounded and the stable minima of the ‘Mexican hat’ potential disappear. In this case, if the Higgs Geld increases, it would gain more energy out of the vacuum to rise to even higher values. All the energy of the Universe could be absorbed; this is an unacceptable scenario. The turnover point will occur for B + 12 = 0, for some large value of . The self-coupling, , determines directly the Higgs boson mass, since the VEV v is known. The possible fermionic destabilization of the vacuum was realized in Ref. [34]. A summary of the relation between mt and mH , required by vacuum stability, is given in Ref. [33], from which Fig. 2.2 is taken. At the beginning of data-taking at LEP in 1989, the conclusion drawn for the Higgs boson search at LEP in the framework of the MSM was that vacuum stability requires mH ¿ 250 GeV for a top mass of about 160 GeV. This Higgs boson mass is too large to be produced at LEP energies.

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Coleman–Weinberg mass generation. In 1972, S. Coleman and E. Weinberg Grst introduced the appealing scenario that spontaneous symmetry breaking is exclusively produced by radiative corrections [35]. By setting the Higgs tree-level mass parameter in Eq. (2.1) to zero, a lower limit on the Higgs boson mass can be obtained. As before, the Higgs boson self-interaction can be neglected in the e:ective potential. The e:ective one-loop potential possesses a global minimum at a non-zero value of . Thus, as a result of radiative corrections, the symmetry can be spontaneously broken and masses are generated. The induced Higgs boson mass is given by m2CW = 8Bv2 ;

(2.15)

where the physical energy scale M is chosen to be at the VEV: M = v = 246 GeV. The Higgs boson mass requirement is given by mH ¿ mCW :

(2.16)

Owing to the large top-quark mass, m2CW is negative and the lower Higgs boson mass limit presents no constraint. In more complicated Higgs models, similar expressions for the Coleman– Weinberg mass have been derived and are outlined in the discussion of the general two-doublet model in Section 4.1. Aside from Gts to the Z lineshape, no other indirect measurements are sensitive to the Higgs boson mass. At present, no theoretical developments predict the Higgs boson mass, and no promising perspectives are in sight. Finite temperature and phase transition. A new dimension can be added to the theoretical discussion via cosmological considerations described in this section. The Higgs potential has far-reaching implications in cosmology. The hope is that the interplay between particle physics and cosmology will ultimately reveal the structure of the Higgs vacuum, thus giving insight into the generation of matter at the very beginning of evolution. Experiments at large accelerators are currently able to study the physics at energies that were present during the early evolution of the Universe. The understanding of the interplay between physics of the smallest and largest dimensions has improved greatly over the last years but is not conclusive. The following paragraph is therefore more speculative than factual, and has little inQuence on the experimental search regions. This is one of many possibilities proposed by theorists and is outlined here to illustrate the type of thinking going on today. In the framework of the cosmological Standard Model (Big Bang theory), energies around 100 GeV, which are investigated at LEP, correspond to the Universe at an age of about 10−10 s. By drawing the analogy with the MeiTner–Ochsenfeld e:ect, Kirzhits and Linde [36,37] considered the possibility that at the high temperatures present in the early stages of the Universe, symmetries were restored which are spontaneously broken today. Consequently, during the evolution of the Universe, there was a phase transition. In Gnite temperature calculations, new terms are added to the Higgs potential as for the one-loop corrections. These Geld-theoretical calculations at Gnite temperatures can be done in a similar way to calculations on the lattice, since the Green’s functions obey periodic boundary conditions in both cases. The additional terms have the form of the free energy of an ideal massive Fermi and Bose gas, summed over all contributing fermions and bosons. The predictions of the modiGed potential cannot solve the main question of whether the required phase transition

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can reveal the Higgs sector and the Higgs boson mass. The underlying reason is cosmological in nature. In the case of a Grst-order phase transition at temperature T ≈ mZ , energy is released and a perturbation, a so-called bubble, is introduced in the homogeneous space. These bubbles would contribute to cluster and galaxy formation, having observable e:ects. Unfortunately, the initially created bubbles are so small that the inhomogeneities are washed out in the further development of the Universe because of the high temperatures present in that epoch. In our present understanding, all information about the origin of Higgs bosons is lost. Conclusive information about the Higgs bosons will not be discovered until more aspects of the interplay between cosmology and particle physics are known. It is theoretically not excluded that our current vacuum state is not the absolute minimum and thus might change into a lower one if energy increases above a certain threshold. An immense release of energy might be the consequence. 2.1.5. Triviality upper Higgs boson mass bound This subsection outlines a very general theoretical upper limit on the MSM Higgs boson mass. It is called ‘triviality bound’. Non-perturbative calculations in Geld theory, which address the signiGcance of the Higgs potential in a very general context, are not only an intellectual challenge, but they also give a stringent upper limit on the Higgs boson mass. In the following section, Grst the one-loop approximation, then a non-perturbative triviality Higgs boson mass limit in the one-doublet model are discussed. Some upper limits on Higgs boson masses in multi-doublet models are derived from triviality arguments (Section 4.1). The upper Higgs boson mass limit is due to the renormalization behavior of the Higgs self-interaction coupling constant and its proportionality to the Higgs boson mass. The renormalization group equations (RGE) can result in the Higgs self-coupling going to inGnity as the energy cut-o: %, deGned as the energy scale up to which the theory is valid, increases. In this case the Higgs mechanism will not work: For each (non-trivial) self-coupling this divergence occurs. The energy scale where cut-o: energy and Higgs boson mass are equal deGnes the upper Higgs boson mass limit. One-loop analysis. The RGE govern the scale dependence of the Higgs self-coupling: d = d t = (t), where t ≡ ln M at the energy scale M . The -function is given in the Grst-order loop calculation by [38] (t) =

6 2 (t) : #2

(2.17)

By focusing on large Higgs boson masses with a strong self-coupling, the contributions from gauge bosons and fermions to one-loop calculations can be neglected. 1 The RGE relate the coupling constant (M ) to (%): % 1 1 6 = + 2 ln : (M ) (%) # M

1

Complementary assumptions were made in the discussion in Section 2.1.3.

(2.18)

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Fig. 2.3. Theoretical upper and lower bounds on the Higgs boson mass. The upper triviality bound is given for the energy scale % where physics beyond the SM is required. The lower bound results from the requirement of vacuum stability. For the measured top mass of 175 GeV, the allowed Higgs mass range is about 130 –180 GeV (thick vertical line) for % = 1016 GeV.

By removing the cut-o: (%=M → ∞), the one-loop calculation results in the triviality of the theory ( → 0). For a Gxed Gnite cut-o:, the largest value (M ) for a given M is obtained in the limit of inGnite bare coupling ((%) → ∞): (M ) 6

#2 8v2 #2 ⇒ m2H 6 : 6 ln %=M 6 ln %=M

(2.19)

The physical mass scale of the Standard Model is of order M = mW . The choice of %=mW = 10, which contributes only logarithmically, leads to a Higgs boson mass limit: mH 6 600 GeV. This value is consistent with Fig. 2.3, which is described below. Non-perturbative triviality bound. The justiGcation of the one-loop Higgs boson mass upper limit is given by Monte Carlo lattice calculations [39]. As for the one-loop calculation, the gauge couplings and the Yukawa coupling are switched o:. The Monte Carlo study supports the existence of a trivial Gaussian Gxed point at R = 0. Simulations show that for di:erent values of the bare coupling , the renormalized coupling goes to zero as the bare mass m goes to its critical value mc (). The simulated Qow of R can be Gtted by R = const ln(||)−* ;

= 1 − m2 =m2c () ;

(2.20)

where the logarithmic dependence is due to the RGE with the marginal critical exponent (* = 1) at the trivial Gaussian Gxed point of the 4 potential. The computer simulation leads to a triviality upper mass bound, which conGrms the perturbative conclusion. On the lattice, the dimensionless Higgs correlation length + = 1=amH , where a = #=% is the lattice spacing, relates

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the Higgs boson mass to the cut-o:: % = #+ : mH

(2.21)

On approaching the critical line (inGnite correlation length) which separates the area of broken and unbroken symmetry, the existence of the trivial Gaussian Gxed point implies that R → 0 for any Gxed bare , manifesting the fact that after renormalization macroscopic physics is independent of the chosen microscopic conGguration. This is the underlying idea of universality in the renormalization theory. The critical area is deGned by having small cut-o: artifacts. The boundary of the critical area near the critical line is determined by Monte Carlo simulations. The result is a relatively small value, +min = 2, on the boundary. The corresponding upper Higgs boson mass limit is determined to be mH 6 2:6v = 640 GeV. In conclusion, one can speculate that the qualitative agreement between the one-loop calculation and the lattice simulation extends to the other results drawn from the perturbative calculation. The uniqueness of the 4 potential lies in the logarithmic evolution of the renormalized coupling. In perturbation theory, this corresponds to a renormalizable theory. The ultimate success of lattice calculations would be to Gnd a non-trivial Gxed point which predicts the Higgs boson mass. An update with higher order calculations of the vacuum stability and triviality limits is given in Fig. 2.3 (from Ref. [40]). The lower limit from the vacuum stability requirement is weaker compared to the limit given in Fig. 2.3. For a top mass of 175 GeV the limit is 130 GeV compared to the previous limit of about 270 GeV. The upper limit from the triviality requirement that the SM remains valid up to an energy scale % is given for the range % = 103 – 1016 GeV. For % = 1016 GeV an upper limit on the Higgs boson mass of about 180 GeV is obtained. 2.1.6. Indirect limits from the Z lineshape, top-quark and W masses The Z lineshape depends on the Higgs boson and top mass through radiative corrections. Thus, the Z-lineshape measurement predicted successfully the top-quark mass before it was observed directly. After the direct top-mass determination the Z-lineshape measurement gave bounds on the Higgs boson mass [41– 43]. The diVculty in determining the Higgs boson mass indirectly is the large top mass dependence compared to the small Higgs boson mass dependence due to quadratic and logarithmic virtual contributions, respectively. The progress made at the Tevatron and LEP colliders with respect to precise measurements of electroweak observables, mainly top-quark and W-boson masses, has led to improved indirect constraints on the Higgs boson mass. The ,2 sensitivity is shown in Fig. 2.4 (from Ref. [43]). Fig. 2.5 (from Ref. [43]) shows a comparison of top and W mass measurements and theoretical predictions for di:erent MSM Higgs boson masses. Light Higgs boson masses are favored and an upper mass limit of 165 GeV at 95% CL is derived [43,44]. The current precision does not allow a distinction to be made between the MSM and an extension. Further measurements of top and W masses will improve the indirect Higgs boson mass bounds.

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Fig. 2.4. ,2 -Gt on the Higgs boson mass from Z-lineshape measurements. The central value is in the mass region already excluded at LEP-2 (shaded area). The Y,2 curve can be interpreted as an indirect upper Higgs boson mass limit. Fig. 2.5. Measured top and W masses with error ellipses. In the MSM (grey region) light Higgs boson masses are favored.

2.1.7. Implications for the Higgs boson search at LEP-1 This section summarizes the MSM Higgs boson production and decay at LEP-1 energies around the Z pole. In the Higgs boson bremsstrahlung process [45 – 47], called Higgsstrahlung, the Z decays into a Higgs boson and an o:-mass-shell Z: e+ e− → Z → HZ? , as illustrated in Fig. 2.6. The di:erential rate for this process, normalized to the Z → f fU decay rate, is given at the tree level as a function of the Higgs boson mass by [48] 1 d !(Z → Hf fU) = 2 + − !(Z → ) dx 4# sin W cos2 W ×

(1 − x + x2 =12 + 2r 2 =3)(x2 − 4r 2 )1=2 ; (x − r 2 )2 + (!Z =mZ )2

(2.22)

where is the Gne structure constant, W the Weinberg angle, x = 2EH =mZ ; EH the energy of the Higgs boson and r ≡ mH =mZ . The total production rate is obtained by integration over the kinematic range 2r 6 x 6 1 + r 2 . Fig. 2.7 shows the number of expected Higgs events, normalized to one million hadronic Z decays for (a) the neutrino channel where Z? decays into neutrino pairs, and (b) the muon channel where Z? decays into a muon pair. Radiative corrections have been taken into account by the Improved Born Approximation [49] and by top triangle graph contributions [50,51]. The Higgs boson decay mode determines the Higgs signature in the detectors. Fig. 2.8 (from Ref. [52]) presents the possible Higgs boson decays and partial widths in the Higgs boson mass

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Fig. 2.6. Feynman graph of Higgsstrahlung production. Fig. 2.7. Higgs boson production rate as a function of the Higgs boson mass per one million hadronic Z decays.

range between 10 MeV and 100 GeV. For the experimental analysis, a low and a high Higgs boson mass range can be distinguished, since in the Grst case, many di:erent decay modes are important and in the second case the bbU decay mode is dominant. (a) Low-mass Higgs boson range. In the low-mass range (mH ¡ 15 GeV) the Higgs boson can decay into many di:erent quark and lepton channels, if the decay is kinematically allowed. An uncertainty of the Higgs boson decay branching ratios is due to the uncertainty of the bare quark masses, which determine the coupling strength for the quarks to the Higgs boson. For Fig. 2.8 the following bare quark masses were taken: mu = 5 MeV; md = 10 MeV; ms = 150 MeV; mc = 1:5 GeV; mb = 5 GeV. However, the experimental thresholds for Higgs boson decay into quark pairs are determined by the mass of the corresponding meson pair. The lightest possible Higgs boson decay into quarks is h → ## for mH ¿ 270 MeV. In the range between 2 m# and 2 GeV, the gluon pair-production via a triangle loop-diagram involving all fermions can be the dominant channel as pointed out in Refs. [53–55]. In this region non-perturbative fragmentation e:ects are important and the expected branching ratio into muons, pions and kaons is uncertain. QCD corrections [56] decrease the partial width into quarks, thus enhancing the + − branching ratio. Very light Higgs bosons decay into a + − or e+ e− pair. As outlined in Section 2.2, Higgs bosons could also decay with secondary vertices in the case of very small masses owing to their long lifetime. Finally, it should be pointed out that even for heavy Higgs bosons which would be accessible in the mass reach of LEP, the Higgs boson decay width is negligibly small. Thus, in all searches for the Higgs boson at LEP, the resolution of the mass reconstruction is determined only by the detector resolution. (b) High-mass Higgs boson range. In the high-mass range (mH ¿ 15 GeV) the dominant Higgs boson decay is H → bbU . Fig. 2.9 shows the H and Z decay rates. The Higgs boson production in association with Z? → qqU is the dominant channel, but it is not used because of the very similar event topology and large production rate of the hadronic background. Therefore, the search is performed in the neutral and charged lepton channels.

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Fig. 2.8. Higgs boson decay modes and their partial widths. A heavy Higgs boson decays predominantly into a bbU pair. For smaller masses the Higgs decays into charm or strange quarks, and for masses below the quark threshold it decays into a pair of leptons. Fig. 2.9. Higgs and Z boson decay fractions for a Higgs boson mass of 60 GeV. The search channels are determined by the particles from the decays of Higgs and Z bosons.

2.2. Search in the low-mass range In this section the mass region below the bbU threshold is addressed for the Z → Z? H process. A search for Higgs bosons in this mass region was important, since the Higgs boson signature depends strongly on its mass. Thus, the Higgs boson could have been discovered in various low-mass search channels. Furthermore, the Higgs boson could have a lower production rate than predicted in the MSM, in which case these low-mass signatures are also important, although the MSM Higgs boson mass limit is already at a higher mass. The search for Higgs bosons with low mass has become less important in the last few years, since very large production rates are expected in the MSM and even with large suppression of the production cross section, this mass region has been ruled out during the Grst years of data-taking. There are three classes of signatures (Figs. 2.10 –2.12) in the low-mass Higgs boson range: mono-jet type events, Higgs boson decays into various light hadronic Gnal states, and events with secondary vertices. Mono-jets. Mono-jets are expected in this mass region between about 4 and 15 GeV. Fig. 2.10 shows an example of a mono-jet recoiling to an e+ e− or + − pair, and a mono-jet signature recoiling to a U pair. Such mono-jets have not been observed and the mass region is excluded at 99% CL for the MSM Higgs boson [57– 67]. Various @nal states. Various Gnal states are expected as illustrated in Fig. 2.11. No indication of a Higgs signal in any channel has been found, and the mass region below 4 GeV is excluded at 99% CL [57– 67]. As an example, Table 2.3 summarizes the He+ e− Higgs decay channels investigated by L3 and gives the corresponding Higgs boson detection eVciencies.

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Fig. 2.10. Diagrammatic view of a mono-jet Higgs boson signal. The mono-jet results from a small invariant mass between two quarks of the decay of a light Higgs boson.

Fig. 2.11. Diagrammatic view of a light Higgs boson signal with various Gnal states. In this case the Higgs boson cannot decay into a pair of heavy quarks and a light meson or lepton pair is expected. Fig. 2.12. Diagrammatic view of a Higgs boson signal with a secondary vertex. In this case the Higgs boson lifetime is so large that the Higgs boson travels in the detector before it decays. This is expected for a Higgs boson mass below the decay threshold into a muon pair. For an even lighter Higgs boson mass the Higgs boson can travel through the detector without leaving a trace and only the decay products of the associated Z decay can be detected.

Table 2.3 L3 selection eVciencies (in %) for a low-mass Higgs boson in the He+ e− channel, for the Higgs decaying into charged particles mH (GeV)

0.01

0.1

0.22

0.3

1.0

3.6

H → e + e− H → H → #+ #− H → K+ K−

8.2 — — —

7.4 — — —

— 22.0 — —

— — 9.4 —

13.6 28.0 17.0 13.0

— 24.0 15.0 16.0

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Fig. 2.13. Higgs boson decay length (a) below and (b) above the H → + − threshold. Fig. 2.14. DELPHI number of expected very low-mass Higgs boson events. The result of an analysis of a Higgs boson decaying outside the detector without leaving a direct signature is combined with an analysis where the Higgs boson travels in the detector before its decay, leaving a V-shape signature.

Secondary vertices. A displaced vertex is expected for very low Higgs boson masses as illustrated in Fig. 2.12. For mH ¡ 2m the Higgs boson has a decay length such that it does not decay at the primary interaction point. Two signatures can be distinguished: (a) the Higgs decays outside the detector, and (b) the Higgs decays inside the detector material, leaving a ‘V’ signature. Fig. 2.13 (from Ref. [68]) shows the decay length. Searches for these signatures have been performed by all LEP experiments, and no indication of a signal has been observed [57,69 –72]. An example of the number of expected Higgs boson events from an earlier analysis is given in Fig. 2.14 (from Ref. [69]). 2.3. Search in the high-mass range This section explains the search strategy, compares the numbers of signal and background events and discusses systematic errors contributing to setting a mass limit. The LEP-1 results reported by ALEPH, DELPHI, L3 and OPAL are reviewed in detail. The integrated luminosities delivered to each LEP experiment are shown in Table 2.4 and in Fig. 2.15. Table 2.5 gives an overview of the covered mass ranges. The production process is shown in Fig. 2.6 and the branching into the di:erent possible search channels are shown in Fig. 2.9. The HqqU channel with the largest branching fraction is not investigated due to the large hadronic background. Therefore, only results for the H and Hll channels are reviewed. Fig. 2.16 shows a diagrammatic view of a high-mass Higgs boson signal for the neutrino, electron and muon, and tau search channels.

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Table 2.4 Integrated luminosities delivered to each LEP experiment between 1989 and 1995 at the Z resonance. In 1995 the LEP energy was increased above the Z resonance Year

1989

1990

1991

1992

1993

1994

1995

L (pb−1 )

1.2

7.6

17.3

28.6

40.0

64.5

40.0

Table 2.5 Mass ranges of Higgs boson searches at LEP-1. The LEP experiments have analyzed the neutrino channel and charged lepton channels over a large mass range Experiment

Channel

Mass range (GeV)

Reference

ALEPH

HU He+ e− H + −

50 –70 50 –70 50 –70

[57,58] [57,58] [57,58]

DELPHI

HU He+ e− H + −

35 –70 35 –70 35 –70

[59,60,73] [59,60,73] [59,60,73]

L3

HU He+ e− H + −

0 –15, 15 –50, 50 –70 0 – 4, 4 –30, 30 –70 2–15, 15 –70

[61,62] [61,62] [61,62]

OPAL

HU He+ e− H + −

0 –70 30 – 66 30 – 66

[63– 67] [63– 67] [63– 67]

Fig. 2.15. Integrated luminosities delivered to each LEP experiment at the Z resonance. The luminosity increased signiGcantly each year. Owing to the transition to higher center-of-mass energies, the 1995 luminosity is lower than in the previous year.

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Fig. 2.16. Diagrammatic view of a high-mass Higgs boson signal for the neutrino, electron and muon, and tau search channels.

2.3.1. H results The decay of the Higgs boson leads to two acoplanar jets and the neutrinos from the Z decay to large missing energy. The dominant physics background arises from hadronic events where some energy is not detected. The analysis strategy is similar for all experiments. First, an event preselection is applied in order to use the di:erence in the basic event topology between signal and background. The measured visible energy is required to be low, since much energy escapes undetected in signal events because of the neutrino production, while in the background from hadronic events almost all energy is deposited in the detector. This background is further reduced by the fact that the jets are mostly back-to-back, while for the signal a large acoplanarity (in the plane perpendicular to the beam axis) between the two jets is expected. After the preselection, further variables are deGned which separate signal and background. These involve for example how well the missing energy direction is separated from the jets. For the background a smaller value is expected. Typically 15 selection variables are deGned and Gne tuned with signal and background simulation. Then they are applied to the data. This procedure ensures that the analysis is unbiased by the data. As an example, details from the ALEPH event selection are given in Appendix A.1.1. In the following, for each experiment the number of observed events and the expected background are given. Furthermore, systematic errors are discussed for each experiment. A quantitative summary of all systematic errors is given at the end of this section. All systematic errors are included in the derivation of the Higgs boson mass limits. Individual mass limits are given for each experiment after the presentation of the results in the Hll channel. ALEPH. ALEPH has observed no candidate event in this channel [57,58]. In the mass range 50 –70 GeV a background of 0:62 ± 0:09 events is expected from four-fermion processes (0.42 from UqqU and 0.20 from + − qqU) and 0:50 ± 0:47 from e+ e− → qqU. The four-fermion processes are illustrated in Fig. 2.17. The detection eVciency is 38.3% for a 60 GeV and 29.8% for 65 GeV Higgs boson. The detection eVciencies and the number of expected background events are determined by Monte Carlo simulation and can therefore be a:ected by systematic e:ects of the detector simulation. In order to study these systematic e:ects, a data sample of multi-hadronic events with a hard photon is selected. These events result from the radiative process e+ e− → qqU or from the decay of an energetic #0 in one of the jets. The photon must have more than 20 GeV and a polar angle such that |cos | ¡ 0:95. Also, the photon is required not to be in the vicinity

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Fig. 2.17. Feynman graphs of four-fermion background. The reactions for the annihilation, two-photon, bremsstrahlung and conversion process are shown.

of a charged particle track. In order to have the best possible measurement of the photon energy, it is required not to be near a non-instrumented inter-module gap of the electromagnetic calorimeter. The photon is then removed from the event and the remaining hadronic part of the event is passed through the preselection. 5857 events are selected. There are two cases. In one case, an event is a genuine qqU event, then the photon is typically isolated from the jets. This event is similar to the mainly two-jet HU signal. On the other hand, an event with the #0 decay resembles the three-jet background with the energy of at least one jet mismeasured. Thus, both simulated signal eVciency and background rejection can be checked using such a data sample. In order to check the Monte Carlo simulation, the simulated hadronic events with an energetic photon are selected, passed through the preselection and then compared with the data sample using an absolute normalization. As an example, Fig. 2.18 (from Ref. [58]) shows the distribution of the isolation angle A for data and simulated events. The good agreement in the region where genuine qq are dominant gives conGdence in the determination of the detection eVciency at the level of 1%. Also, the distribution agrees well between simulation and data in the region of hadronic events with an

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◦

Fig. 2.18. ALEPH HU isolation angle A for simulated qqU , and qqU events with a #0 . The cut A ¿ 31 is determined by an optimization procedure and it separates well the genuine qqU events. ◦

energetic #0 . Fig. 2.18 also shows that the cut values of the isolation angle A ¿ 31 determined by the minimization process (Appendix A.1.1) are well set to reject the three-jet background. The systematic error of 1% is included in the determination of the mass limit, which is given together with the lepton channel later in the section. DELPHI. The DELPHI H analysis [59,60] is based on a luminosity of 34:6 pb−1 which corresponds to 1.0 million hadronic Z decays. 2 No candidate is selected. DELPHI combined a neural network and probabilistic analysis in this channel. For the neural network analysis the background expectation is below 1.4 events at 95% CL since no event is selected in the sample of simulated background events. For the probabilistic analysis 1:1 ± 0:3 background events are expected. The combined neural network and probabilistic analysis results in a selection eVciency of 34.5% for a 60 GeV Higgs boson. Sources of uncertainties of the simulated eVciencies are the reconstructed energies, momenta and tracks, as well as the description of insensitive detector regions. Individual contributions to the systematic error of 1.6% are determined by varying several parameters: • the charged particle resolutions are smeared according to the experimental momentum reso• • • • • 2

lutions, leading to 0.83% error; the electromagnetic energy is varied by 5%, leading to 0.59% error; the hadronic energy is varied by 5%, leading to 0.68% error; the track eVciency is decreased by 5%, leading to 0.72% error; the overall calorimeter eVciency is decreased by 5%, leading to 0.58% error; the insensitive detector region is enlarged by approximately 40%, leading to 0.39% error.

For the combined result an update with 1.6 million hadron events is included [73].

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Fig. 2.19. L3 HU detection eVciency as a function of expected background events. The indicated cut is placed where 0.25 background events are expected. Fig. 2.20. L3 study of qqU data to check the simulated Higgs mass resolution. After removing the hard photon from the qqU data sample, the di:erence between the invariant mass of the hadronic system and the mass of the hadronic system computed only from the photon energy is compared. The mass resolution of the hadronic system is 15%.

In conclusion, systematic errors have little inQuence on the experimental sensitivity to detect a Higgs boson. The mass limit is calculated in combination with the lepton channel later in this section. L3. L3 detects no candidate event. The expected background is 0.02 events from + − bbU , 0.01 events from UqqU, and 0.25 events from qqU Gnal states. The signal eVciency for a 60 GeV Higgs is 26.6%. The dependence of its detection eVciency is shown in Fig. 2.19 (from Ref. [62]) as a function of the number of expected background events. The uncertainty in the simulated detection eVciency has been studied. The most important uncertainties are due to the energy calibration and the angle resolution. These errors amount to 0.35% on the detection eVciency for a 60 GeV signal. The statistical error due to the limited number of simulated signal events is 2.4%. Adding the errors on the cross section and the branching fraction of 1% each, the error on the number of expected events to be observed is 3.1%. The mass limit from this search alone is 58:1 GeV. The precision of the Monte Carlo simulations in reconstructing mass resolutions was also studied with qqU events [74]. In order to compare the predicted mass resolution of hadronically decaying Higgs bosons with data events, a sample of qqU events with a hard photon is selected. After removing the photon from the reconstruction, their topology is similar to Z → HU events. For data, the quantity Ymqq is deGned as the di:erence between the invariant mass of the hadronic system and the mass of the hadronic system computed only from the photon energy. For the simulated Higgs boson signal, Ymqq is the reconstructed Higgs boson mass. Fig. 2.20 (from Ref. [74]) shows the result of the comparison. The mass resolution of the hadronic system is 15%. The energies of the hard photon in the qqU data sample allow the

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investigation of a hadronic invariant mass range of about 25 –65 GeV. Data and Monte Carlo simulations agree within the statistical errors. OPAL. OPAL Gnds two candidate events, consistent with the expected background of 2:30 ± 0:43 events. Compared to ALEPH and L3, OPAL chooses a search mass region which extends to lower Higgs boson masses. Therefore, a larger number of background events is expected. The background consists of 0:54 ± 0:24 qqU; 0:35 ± 0:35 + − , and 1:41 ± 0:04 four-fermion events. In the Higgs boson mass region larger than 50 GeV, fewer than 0.6 background events are expected. One candidate event is a mono-jet, and the other a di-jet event. The mono-jet event has a visible mass of 6:3 ± 0:8 GeV and a recoiling mass of 78:5 ± 1:3 GeV, corresponding to the Higgs boson mass. The di-jet event has a visible mass of 24:8 ± 3:0 GeV and a recoiling mass of 34:9 ± 7:7 GeV. The detection eVciency for the MSM Higgs boson is about 50% at a Higgs boson mass of 40 GeV. For higher masses it falls to about 15% at 65 GeV due to the fact that heavy Higgs bosons have a more collinear jet signature which is more similar to the qqU background. On the other hand the selection eVciency falls for lighter Higgs boson masses to about 30% at 12 GeV. This is due to the small transverse momentum which makes these events more similar to two-photon background. The uncertainty on the number of expected Higgs boson events due to luminosity and cross section is 0.5% and 1.0%, respectively. The integrated luminosity is determined by counting multi-hadron events. OPAL has taken into account several e:ects in the determination of the uncertainty of the detection eVciency: the fragmentation uncertainty, the uncertainty on S , a variation of the selection cuts by about one standard deviation of the experimental resolutions and the statistical error on the simulated signal sample. The last error is dominant and varies between 1.8% and 5.0%. The systematic and statistical errors are summed in quadrature and the selection eVciencies are reduced by one standard deviation. The result of this channel is expressed as limits on the production rate for a large mass range, normalized to the MSM prediction. OPAL gives explicitly the formula used [66]: 1(e+ e− → h)BR (h → qqU) 1(e+ e− → HMSM ) =

BR (Z → ) U

N95 (mh ) ; + − Ei [3(Ei ; mh )1(e e → HMSM )(Ei ; mH )L(Ei )]

(2.23)

where the 1’s are the production cross sections, 3 is the selection eVciency, and L is the integrated luminosity at energy Ei . The function N95 (mh ) is the number of expected events needed for a 95% CL using Poisson statistics. The number of background events and their mass resolution is taken into account. The resulting exclusion curve, based on the OPAL analysis of the HU channel, is given in Fig. 2.21 (from Ref. [63– 67]). From this search alone a MSM Higgs boson is excluded up to 60:6 GeV. 2.3.2. Hll results The reaction e+ e− → Z → HZ? (H → hadrons)(Z? → ‘+ ‘− ), where ‘ = e or , leads to two jets from the Higgs decay and a pair of leptons. The identiGcation of the leptons and jets allows

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Fig. 2.21. OPAL general limit on a scalar production in association with a Z normalized to the Z → HZ? cross section.

the dominant background of double semileptonic decays Z → bbU → e+ e− X and the four-fermion process e+ e− → e+ e− qqU to be reduced. In the event preselection, events with a pair of leptons and jets are identiGed; thus, purely leptonic or purely hadronic background events are rejected. In the next step, emphasis is put on the isolation between leptons and jets, since in signal events the leptons are mostly well separated from the direction of the jets. As an example of an event selection, details from L3 [61,62] are given in Appendix A.1.2. First, the number of observed events and the expected background are given for each experiment. Then, systematic errors are discussed and individual mass limits are presented in combination with the previously discussed results from the HU channel. ALEPH. ALEPH Gnds three data events, while 2.7 events are expected from the four-fermion background. The quark Qavor is determined from the simulation to be 1.1 ‘+ ‘− bbU and 1.6 non-b events (supposedly, mainly ‘+ ‘− ccU events, owing to the longer c-meson lifetime compared to the light-quark mesons). All three data events are in the H + − channel with the masses 49:7 ± 0:5 GeV, 51:5 ± 0:5 GeV, and 66:9 ± 0:3 GeV. The detection eVciencies for a 60 GeV Higgs boson before the b-tagging cut are 50.0% for the He+ e− and 61.0% for the H + − search, and after the cut 39.4% and 48.1%, respectively. The errors on the detection eVciencies after b-tagging are 0.5%. Systematic e:ects on the detection eVciency are due to small di:erences in the lepton identiGcation of simulation and data. These e:ects are studies with e+ e− → ‘+ ‘− events [75]. The correction factors are (−1:7 ± 0:3)% for He+ e− and (0:8 ± 0:3)% for H + − . The b-tagging eVciency has been cross-checked with qqU events [76]. A similar selection as for the H‘+ ‘− events is applied, where instead of the lepton selection, a photon selection is applied. The photon selection is similar to the one used to check the HU channel. The events containing energetic photons from #0 decays are rejected by the isolation requirement. An additional rejection for events with #0 ’s is applied by ALEPH to reduce this contamination by a factor two using the fact that the electromagnetic shower proGle is broader for #0 ’s than

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195

Fig. 2.22. ALEPH b-tagging probability for qqU data and simulated events, used to determine the systematic error of the b-tagging eVciency.

for photons. A data sample of 2729 events is obtained and 2643 events are expected from the q qU simulation. In detail, 297 bbU , 867 ccU , 333 ssU , 311 d dU and 835 uuU events are selected. The distribution of Puds is now compared between the qqU data sample and its simulation. Fig. 2.22 (from Ref. [58]) shows good agreement between data and simulation. The requirement Puds ¡ 0:05 results in a rejection of (76:7 ± 0:8)% of the data and (75:9 ± 0:8)% of the simulated events. Thus, data events are slightly more rejected than simulated events, so that the simulation slightly overestimates the number of events which pass this requirement. The eVciency of the simulated Higgs signal has to be slightly reduced. The Puds distribution for bbU Gnal states is Qatter than for lighter quarks, as seen in Fig. 2.22, and the e:ect is reduced. Conservatively, ALEPH assumes that the same di:erence between data and simulation of (+1:1 ± 1:4)% is identical for all quark Qavors. From this value and the fact that the b-tagging requirement has an eVciency of 78.8% on the H‘+ ‘− simulation, the correction factor for the detection eVciency is (−0:3 ± 0:5)%. The event selection for the Hll channel also gives a small eVciency for the decay channels involving ’s, since the decay of pairs leads to missing energy and in some decay channels to pairs of electrons and muons. There are two possible channels with a pair in the Gnal state (either the Higgs boson or the Z boson can decay into a pair): (H → hadrons)(Z? → + − ) and

(H → + − )(Z? → hadrons) :

(2.24)

The Higgs boson is expected to decay into a pair at a rate of about 7%, while the Z decay rate into a pair is about 3%. In the following the derivation of the mass limit from the H and Hll is described. The production cross section as a function of the Higgs boson mass is well known. The numbers of

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Table 2.6 ALEPH number of expected signal events to be observed in the H, U He+ e− , and H + − channels mH (GeV)

HU

H‘+ ‘−

Total

50 55 60 65

25.17 12.23 5.12 1.73

8.45 4.24 1.87 0.69

33.62 16.47 6.99 2.42

expected events determined in the HU and H‘+ ‘− searches are given after branching fractions and detection eVciencies have been taken into account. The numbers of expected signal events obtained by ALEPH are given in Table 2.6. Systematic errors on the numbers of expected events are taken into account. The sum of the systematic errors is less than 2%. Several sources contribute: • The determination of the number of multi-hadronic events leads to an uncertainty of 0.2%

[77].

• The error on the top-quark mass results in an uncertainty in the Higgs boson production cross section of 0.1% for a 64 GeV Higgs boson. A top mass of 175 ± 9 GeV as the combined

value from the CDF and D0 collaborations is taken. • The uncertainties in the selection procedure, i.e. the lepton identiGcation and the b-tagging, lead to an uncertainty on the detection eVciency of the order of ±1%. • An uncertainty of 1% on the H → bbU partial decay width is due to the uncertainty on the b-quark pole mass. It results in an uncertainty of ±0:7% on the number of expected Higgs events, since mostly bbU are selected. The statistical error is only 0.2% because of the large number of simulated signal events. The sum of the systematic errors is less than 2%. The number of expected events has been reduced by this value in order to derive a lower limit on the Higgs boson mass. The three candidate events have been taken into account according to their masses and mass resolutions using the method described in Refs. [57,58]. Fig. 2.23 (from Ref. [58]) shows the number of expected events for the HU channel and the combination of HU and H‘+ ‘− channels, as well as the line for the 95% CL. The mass regions of the candidates are not near the region where the limit is set, therefore about three expected events suVce to set the 95% CL. The resulting mass limit is 63:9 GeV. DELPHI. Four H‘+ ‘− candidates are selected by DELPHI in the event sample corresponding to 1 million hadronic Z decays, taken in 1991 and 1992 [59,60]. The number is in agreement with 2.5 expected events from hadronic reactions and 2.7 events from four-fermion processes. Table 2.7 lists the candidate events and their reconstructed mass from the 1991 and 1992 data set, as well as a candidate from the 1990 data sample. Only the 1990 candidate has a reconstructed mass above 30 GeV, and the reconstructed masses vary over a large range. None of the candidates has a high probability of b-tagging as expected from a Higgs signal. Details of the important b-tagging are given in Appendix A.2.1.

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197

Fig. 2.23. ALEPH number of expected e+ e− → HZ events. The intersection of the line of expected events with the line of 95% CL marks the observed mass limit. Two candidate events increase the 95% CL line according to the measured mass resolution. Table 2.7 DELPHI H‘+ ‘− candidate events and their reconstructed masses. No accumulation at one mass is observed and no candidate is selected above 35 GeV Year

Channel

Reconstructed mass

1990 1992 1992 1992 1992

e+ e− e+ e− e+ e− e+ e−

35:4 ± 5:0 15:4+3:8 −3:2 19:2+3:7 −2:3 18:9+4:8 −1:9 27:8 ± 1:9

+ −

The systematic errors are estimated by varying the selection criteria. For a 60 GeV Higgs boson in the He+ e− channel, the resulting change of the selection eVciency is between −3:0% and +1:3%. For a 70 GeV Higgs boson the systematic error reduces the selection eVciency by −7:5%. In the H + − channel, the variation of the selection criteria results in a change of the selection eVciency between −0:6% and +0:7% for a 60 GeV Higgs boson and up to −2:7% for a 70 GeV Higgs boson. The statistical error is about 1.5% for the electron and muon channel. The Higgs boson mass limit is derived from the combination of the neutrino and both charged lepton channels. Fig. 2.24 shows the selection eVciencies for the three search channels. The number of expected events is reduced by one standard deviation of the systematic error and in addition by 2% because of the uncertainty in the Higgs boson production and decay, and 0.5%

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Fig. 2.24. DELPHI eVciency and number of expected e+ e− → HZ events based on the event sample of 1.0 million hadron Z decays. The individual results from the neutrino, electron and muon channels are shown as well. In the absence of high mass candidate events the 95% CL line is at 3.0 and the intersection with the number of expected events gives the mass limit of 55:7 GeV.

because of the normalization uncertainty of the number of hadronic Z decays. Fig. 2.24 (from Ref. [60]) shows the number of expected events in each channel separately and the combined number of expected events. In the absence of candidate events in the 50 GeV mass region, the limit is set where the number of expected events intersects the 3.0 event line for a 95% CL. The mass limit is 55:7 GeV [59,60]. An update with 1.6 million hadronic events increased this limit to 58:3 GeV [73]. L3. L3 Gnds two candidate events in the He+ e− selection. The Grst event has a recoil mass of 31:4 ± 1:5 GeV, recorded in 1991, and the second has a recoil mass of 67:6 ± 0:7 GeV, recorded

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Table 2.8 L3 H‘+ ‘− detection eVciency. The decrease at 70 GeV in the H + − channel is due to an invariant mass cut mH (GeV)

He+ e− eVciency (%)

H + − eVciency (%)

50 60 65 70

46.6 42.2 39.9 35.8

36.1 32.3 26.2 3.3

in 1992. The two events are consistent with the expected background for a recoiling mass larger than 30 GeV. From fully simulated events using the FERMISV generator [78], 3:0 ± 0:7 background events are determined, and less than one event is expected from the hadronic event simulation. Other background sources are negligible. For comparison, in the mass range 50 –70 GeV; 1:3 ± 0:3 background events are expected. The detection eVciencies are given in Table 2.8. No H + − candidate event is observed in the data. The background from the four-fermion process + − qqU is 0:65 ± 0:11 events for recoiling masses larger than 30 GeV. Other background sources are negligible. The systematic errors on the number of expected Higgs boson events to be observed in the detector are due to • the uncertainty in the normalization with the number of hadronic events leading to an error

of less than 1%,

• the theoretical prediction of the Higgs boson production rate of less than 1%, • the uncertainty in the Higgs boson decay branching fractions resulting in an uncertainty of

0.7%.

In the L3 He+ e− channel an error of less than 3% is due to the electron isolation criterion and of 1.5% is due to the limited statistics of the signal simulation. The total He+ e− error on the expected number of Higgs boson events is 3.7%. In the H + − channel the uncertainty of the detection eVciency is 6.5%, and the total error is 7.0%. L3 derives a combined mass limit of the HU and H‘+ ‘− searches. The candidates in the He+ e− channel are not compatible with a Higgs boson mass in the vicinity of 60 GeV and no other candidates are observed. Therefore, the 95% limit is set where the number of expected events to be observed in the detector is 3. The detection eVciencies are reduced by one standard deviation of the total error. The numbers of expected events are given in Table 2.9, where the H → + − branching fraction is Gxed to 9%. These numbers of expected events are plotted in Fig. 2.25 (from Ref. [62]) together with the 95% CL line. The resulting lower mass limit for the MSM Higgs boson is 60:2 GeV. OPAL. OPAL observes no candidate in the He+ e− search and one candidate in the H + − channel with recoil mass of 61:2 ± 1:0 GeV. A total of 0.8 background events is expected from four-fermion events. The expected background from hadronic Z decays having a genuine e+ e− or + − pair has been determined using the high statistics data sample of Z → bbU events. The application of the selection shows that less than 0.11 such background events are expected in the data set of 4.4 million hadron Z decays. The simulation of the 4 million hadronic Z decays

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Table 2.9 L3 number of expected e+ e− → HZ events. The rate in the neutrino channel is much larger because of the larger Z → U branching compared to Z → e+ e− and Z → + − mH (GeV)

50

55

60

65

70

HU channel He+ e− channel H + − channel

11.30 2.50 1.88

5.30 1.27 0.92

2.17 0.57 0.42

0.553 0.228 0.145

0.113 0.072 0.006

Fig. 2.25. L3 number of expected e+ e− → HZ events. The mass limit of 60:2 GeV is set where the line of expected events intersects the 95% CL line. The candidate at 67:6 ± 0:7 GeV increases the 95% CL line from 3.0 to 4.7 with the given mass resolution.

with full detector simulation was used to determine the number of fake lepton pairs. In general, fake leptons can occur in rare cases in the following scenarios. • Electrons could be faked from an overlap of a charged hadronic track with a photon or #0 .

Also, photons could convert in the detector material, leading to an electron pair.

• Fake muons could be detected if punch-through occurs. In this case, charged particles pass

the calorimeters and enter the muon chambers.

A subsample of the simulated hadronic Z decays is chosen where the lepton identiGcation is not applied and the lepton isolation is loosened. Then, the lepton identiGcation is applied, and the resulting rejection factor is 7 × 10−4 . Now, the full data sample is taken and all cuts except the lepton identiGcation are applied. The resulting rejection factor is 2 × 10−6 . The resulting total rejection factor is 1:4 × 10−9 and less than 0.01 hadronic Z events are expected in the ‘+ ‘− qqU data sample.

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201

Table 2.10 OPAL eVciencies and number of expected e+ e− → HZ events + −

mH (GeV)

HU (%)

HU Nexp

He+ e− (%)

H + − (%)

H‘ Nexp

30 40 50 54 56 58 60 62 66

47.7 51.6 38.6 33.1 30.5 28.7 25.7 21.4 13.0

203.8 84.0 20.4 10.8 7.4 5.2 3.4 2.1 0.6

9.8 22.1 24.2 25.3 24.5 23.2 21.5 20.8 17.1

8.6 23.1 33.6 29.4 29.9 28.0 30.8 23.8 21.6

13.1 12.0 4.8 2.8 2.0 1.4 1.0 0.6 0.2

‘

total Nexp

216.9 96.0 25.2 13.6 9.2 6.6 4.5 2.7 0.8

The detection eVciencies for H‘+ ‘− are shown as a function of the Higgs boson mass in Table 2.10. The systematic errors on the detection eVciencies are: • 5% from signal Monte Carlo statistics, • 1% from the total integrated luminosity, • 1% from the uncertainty in the production cross section.

About 5% uncertainty is added in quadrature for the limited statistics of the simulation. For the derivation of a lower limit on the Higgs boson mass, the eVciencies are reduced by the total errors (7% for the charged lepton channels and between 1.8% and 5% for the neutral lepton channel). OPAL derives a combined mass limit from the HU and H‘+ ‘− channels. The mass limit from the HU channel alone is 60:6 GeV and no HU candidate event is observed in its vicinity. Adding the number of expected events from the H‘+ ‘− channel would increase this limit to 62:3 GeV. However, the H + − candidate with mass 61:2 ± 1:0 GeV is close to this mass value. Having one candidate, the 95% CL line is at 4.7 instead of 3.0 events. The resulting mass limit reduces to 59:6 GeV. Fig. 2.26 (from Ref. [67]) shows the number of expected Higgs boson events and the 95% CL line as well as the location of the candidate event. OPAL has tested other methods to determine the 95% CL line taking into account the mass resolution of the candidate event. These methods give rise to mass limits within the uncertainties of the above value. The fact that the mass limit decreases when the H‘+ ‘− is included raises the question of whether the H‘+ ‘− contributes at all to the sensitivity in the Higgs search near 60 GeV. In order to answer this question, OPAL compares the luminosities needed to exclude a 60:6 GeV signal with and without the H‘+ ‘− channel taking into account the numbers of expected signal and background events. At this mass, 3.0 events are expected from the HU and 0.9 events from the H‘+ ‘− channel. The background above 50 GeV Higgs boson mass is 0.6 events in the HU and 0.4 in the H‘+ ‘− channel. Using Poisson statistics 3.9 expected signal events with 1.0 expected background event gives higher sensitivity than 3.0 expected signal with 0.6 background events. This corresponds to a smaller luminosity needed if the H‘+ ‘− is included. Therefore, the OPAL Higgs boson mass limit is 59:6 GeV.

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Fig. 2.26. OPAL number of expected e+ e− → HZ events. The mass limit from the H‘+ ‘− alone is 60:6 GeV. This limit is reduced to 59:6 GeV when combined with the H channel because of a candidate event in the region where the limit is set.

The detection eVciencies and the number of expected Higgs boson events in the mass range 50 –70 GeV are summarized in Table 2.11. Table 2.12 summarizes the systematic and statistical errors. 2.4. Combination of the search results from all LEP experiments In order to derive a combined mass limit, Grst the Higgs boson candidates from the LEP experiments are reviewed. One of the three ALEPH H + − candidate events described in the previous section is shown in Fig. 2.27 (from Ref. [58]). The recoiling mass of the + − pairs is 49:7 GeV. Fig. 2.28 (from Ref. [62]) shows a He+ e− candidate from the L3 selection. Table 2.13 lists the Higgs boson candidates [57– 67] in the mass range from 30 to 70 GeV. The most precise measurement of the mass corresponding to the Higgs mass is calculated from the e+ e− and + − pairs (recoiling mass). Note that the OPAL low-mass candidates, described in the previous section, are not listed. Table 2.14 summarizes the Higgs boson mass limits in the MSM given by the LEP experiments [57– 67,73]. The number of expected events is given by each LEP experiment [57– 67,73], and shown in Fig. 2.29 for combined data corresponding to a total of 14 million hadronic Z decays.

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203

Table 2.11 Overview of detection eVciencies for a 50 – 70 GeV Higgs boson. The eVciencies in brackets are determined by interpolation from the nearest Higgs boson masses used in the publication Experiment

mH = 50 GeV ALEPH DELPHI L3 OPAL

mH = 55 GeV ALEPH DELPHI L3 OPAL

mH = 60 GeV ALEPH DELPHI L3 OPAL

mH = 65 GeV ALEPH DELPHI L3 OPAL

mH = 70 GeV ALEPH DELPHI L3 OPAL

EVciency (%)

Expected events

HU

He+ e−

H + −

HU

He+ e−

(46.2) 50.0 34.8 (38.6)

(46.1) 35.6 46.6 (24.2)

(46.1) 52.8 36.1 (30.8)

25.2 8.0 11.3 20.4

0.96 2.5 (2.0)

(41.7) 45.6 (30.1) (31.7)

(51.2) 36.6 (54.3) (24.9)

(51.2) 54.5 (38.4) (29.7)

12.2 4.0 5.3 8.7

0.56 1.3 (1.0)

38.3 34.5 28.6 25.7

39.4 32.4 42.2 21.5

48.1 54.0 32.3 30.8

5.12 1.6 2.17 3.4

1.27 0.26 0.57 0.45

29.8 22.0 16.0 15.1

(34.7) 29.8 39.9 (18.0)

(34.7) 48.2 26.2 (22.1)

1.73 0.40 0.55 0.8

0.07 0.23 (0.15)

(26.7) 10.6 9.2 13.0

(27.7) 17.1 35.8 17.1

(27.7) 37.7 3.3 21.6

(0.52) 0.06 0.11 (0.24)

0.02 0.07 (0.58)

8.45

4.2

0.69

(0.16)

H + −

Sum

1.8 1.88 (2.8)

33.6 10.8 15.7 25.2

0.67 0.92 (1.0)

16.5 5.3 7.5 10.7

0.92 0.38 0.42 0.65

7.0 2.3 3.2 4.5

0.16 0.16 (0.15)

2.42 0.63 0.93 1.1

0.04 0.01 (0.007)

(0.68) 0.12 0.19 (0.31)

To a good approximation, a combined Higgs mass limit can be set by the summation of the number of expected Higgs events. The calculation of the 95% CL limit takes the background events into account and corrects for a reduction of up to 25% due to tighter selection cuts with increasing statistics. This reduction is introduced since a larger luminosity, corresponding to the combined data set of the four LEP experiments, gives also a higher background. Consequently, experiments would tighten their selection cuts to optimize the sensitivity. An example of how the detection eVciency decreases with increasing luminosity is given in Table A.1. The combined mass limit for LEP-1 is mH ¿ 65:6 GeV :

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Table 2.12 Overview of systematic and statistical errors in per cent. Details are given in the text Experiment

Channel

Production

ALEPH

HU He+ e− H + −

0.7 0.7 0.7

DELPHI

HU He+ e− H + −

L3

OPAL

Selection

Statistics

Total

1.0 1.0 1.0

0.2 0.2 0.2

¡2 ¡2 ¡2

2.0 2.0 2.0

¡ 3:7 ¡ 7:5 ¡ 2:7

1.6 1.5 1.6

¡ 4:5 ¡ 7:9 ¡ 3:7

HU He+ e− H + −

1.7 1.7 1.7

0.35 ¡3 6.5

2.4 1.5 2.4

3.1 3.8 6.9

HU He+ e− H + −

1.3 1.4 1.4

¡1 5 5

3.2 5 5

3.5 7 7

Fig. 2.27. ALEPH 49:7 GeV H + − candidate. The muons are pointing to the upper left corner, opposite the hadronic activity.

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205

Fig. 2.28. L3 67:6 GeV He+ e− Higgs boson candidate shown in the plane perpendicular to the beam line. The lines in the TEC represent the reconstructed charged tracks. The size of the symbols indicating individual calorimetric hits (towers in the BGO electromagnetic calorimeter and boxes in the hadron calorimeter) corresponds to the energy deposition in that hit. The towers which appear in the TEC region in this projection belong to the BGO endcaps.

Table 2.13 Overview of high-mass Higgs boson candidates in the range 30 to 70 GeV. The candidates are distributed over a large mass range Experiment

Event type

Year

Mass (GeV)

ALEPH

+ − qqU + − qqU + − qqU

1993 1994 1995

51:4 ± 0:5 49:7 ± 0:5 66:9 ± 0:3

DELPHI

e+ e− q qU

1990

35:4 ± 5:0

L3

e+ e− q qU e+ e− q qU

1991 1992

31:4 ± 1:5 67:6 ± 0:7

OPAL

+ − qqU

1993

61:2 ± 1:0

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Table 2.14 Overview of individual Higgs boson mass limits at 95% CL from the LEP-1 results. Similar mass limits are observed by all LEP experiments, although the size of the analyzed data sample varies between them, since the Higgs boson production cross section decreases quickly for heavy Higgs bosons

Data sample Hadronic Z decays (106 ) Mass limit (GeV)

ALEPH

DELPHI

L3

OPAL

1989 –1995 4.5 63.9

1990 –1993 1.6 58.3

1990 –1994 3.1 60.1

1990 –1995 4.4 59.6

Fig. 2.29. Combined Higgs boson mass limit. The numbers of expected events to be observed in individual experiments and their summation. A reduction factor is applied as explained in the text. The lower solid line gives the 95% CL limit, taking into account the observed candidate events, their mass resolution and the background expectation.

The Gnal combined mass limit varies less than about 1 GeV from that calculated by other methods [79 –82]. The evolution of the combined mass limits, using the method described before, is shown in Table 2.15. The evolution of the Higgs mass limits for each experiment is shown in Fig. 2.30. The sensitivity is compared assuming 50% eVciency in the H + − , He+ e− and HU channels. Fig. 2.29 shows that, with larger statistics, the reduction of four-fermion background is crucial for high sensitivity, since the e:ect of the background events is clearly seen in the 95% CL line. This has been achieved with enhanced microvertex application for b-quark tagging.

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207

Table 2.15 Evolution of LEP-1 Higgs boson lower mass limits at 95% CL. The combined mass limits can be compared directly since the same method of combination was used as described in the text. The Gnal LEP-1 mass limit was almost reached with the inclusion of the 1994 data. SigniGcantly higher mass sensitivity required an increase of the center-of-mass energy beyond the scope of LEP-1 Including data of year

1991

1993

1994

1995

Hadronic Z decays (106 ) Combined limit (GeV)

2.0 59.3 [83]

6.0 63.5 [84]

12 65.1 [85]

14 65.6 [86]

Fig. 2.30. Evolution of Higgs boson mass limits. The solid line shows the expected sensitivity taking 50% detection eVciency in the search channels. With increasing luminosity the mass limit lies below this line since the selection cuts have to be tightened to cope with the increasing background in order to obtain roughly zero background.

3. One-doublet and one-singlet model 3.1. Theoretical framework A straightforward class of extensions of the MSM which are qualitatively di:erent from the MSM is the Majoron-type models [87–91]. In addition to the Higgs doublet or triplet Gelds, the characteristic feature of the Majoron models is the presence of a complex SU(2)×U(1) singlet

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scalar. The spontaneous breaking of the U(1) symmetry leads to a Goldstone boson (called the Majoron, J) which, in the fermion and gauge sector, couples only to right-handed neutrinos owing to the singlet nature of the Geld. In the Higgs sector, the coupling of the Majoron to the Higgs bosons might be large and the Higgs bosons could therefore predominantly decay into an invisible Majoron pair: H → JJ :

The simplest Majoron model based on one Higgs doublet and one singlet Geld * is considered. The complex doublet and singlet Gelds have six degrees of freedom. Three degrees of freedom are absorbed to give masses to the Z and W bosons, thus in addition to the Majoron, two Higgs bosons are predicted. The mixing of the real parts of and * leads to two massive Higgs bosons: H = R cos − *R sin ; S = R sin + *R cos ; ◦

where is the mixing angle, which may be constrained to be in the range 0 –45 without any loss of generality. The imaginary part of the singlet Geld is identiGed with the Majoron. The free parameters of this model are the masses of the two Higgs bosons H and S, the mixing angle and the ratio of the vacuum expectation values of the and * Gelds (tan ≡ v =v* ). 3 The production rate of the H and S bosons is reduced with respect to the MSM Higgs boson by a factor of cos2 and sin2 , respectively. The decay widths of the H and S into the heaviest possible fermion–antifermion pair are reduced by the same factor. Their decay widths into a Majoron pair are proportional to the complementary factors (cos2 for S and sin2 for H). Explicitly, if the H → bbU decay is kinematically allowed, then [92] √ 3 2GF U !(H → bb) = (3.1) mH m2b (1 − 4m2b =m2H )3=2 cos2 ; 8# √ 2GF 3 (3.2) !(H → JJ) = m tan2 sin2 : 32# H The relative branching ratios may then be expressed as 3=2 m2b mb 2 2 2 U rH ≡ !(H → bb)=!(H → JJ) = 12 cot cot 1 − 4 2 ; mH mH

mb rS ≡ !(S → bbU )=!(S → JJ) = 12 mS 3

2

m2 tan cot 1 − 4 2b mS 2

2

(3.3)

3=2

:

Note that tan is deGned similarly in the two-doublet model, as described in the next chapter.

(3.4)

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209

Fig. 3.1. Limits on the production cross sections of invisibly decaying Higgs bosons. The region above the line is excluded at 95% CL.

Making use of the above equations, the numbers of expected H and S events are

rH 1 2 U NH (mH ) = L1MSM (mH ) cos 3(H → bb) + 3(H → JJ) ; 1 + rH 1 + rH

rS 1 NS (mS ) = L1MSM (mS ) sin2 3(S → bbU ) + 3(S → JJ) ; 1 + rS 1 + rS

(3.5) (3.6)

where L is the integrated luminosity, 1MSM (m) is the cross section for the production of the MSM Higgs boson of mass m, and 3 is the eVciency for detecting the relevant decay modes. The combination of the results from the searches for the MSM Higgs boson and the Higgs boson decaying into the invisible Majorons allows the setting of limits on Higgs boson masses, independent of other parameters. At the Z pole, the Higgs boson is produced via the Higgsstrahlung process, Z → Z? H. The Higgs boson escapes detection in its invisible decay mode. The dominant signature is then an unbalanced hadronic event in which the Z? decays via the qqU mode. As this mode (Z? → qqU) accounts for about 70% of all Z decays, the sensitivity of detecting it is enhanced, compared to the search for the MSM Higgs bosons via the neutrino channel Z → Z? H → UH. 3.2. Search for invisibly decaying Higgs bosons Model-independent limits on the production cross section are given in Fig. 3.1 based on data from L3 [61,62] as a function of the invisible Higgs boson mass at 95% CL. In the model

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Fig. 3.2. Limits on the masses of invisibly decaying Higgs bosons as a function of the mixing angle sin2 , at 95% CL. S and H are the Higgs bosons in the Majoron model with expected production rates for rS = rH = 0, which implies a 100% branching ratio into invisible Majorons.

Fig. 3.3. ALEPH He+ e− invisible Higgs boson candidate. The mass recoiling against the e+ e− pair, i.e. the mass of the invisible Higgs boson, is 61.3 GeV.

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with one doublet and one singlet Higgs Geld, a large region in the plane of Higgs boson mass versus mixing angle is excluded, as shown in Fig. 3.2 for the special case in which the invisible Higgs boson decay is dominant, i.e. large tan ; r ≈ 0. In addition to the invisible Higgs boson decays expected in the one-doublet and one-singlet model, other models also predict invisible Higgs boson decays. For example, Supersymmetric models with broken R-parity or possible h → ,˜0 ,˜0 decays 4 predict invisible Higgs boson decays. In the ALEPH analysis one candidate event is observed, which is consistent with a 61.3 GeV Higgs boson decaying invisibly. This candidate is shown in Fig. 3.3 (from Ref. [93]). It is compatible with the expected rate from the four-fermion background. 4. Two-doublet model 4.1. Theoretical framework A possible extension, beyond the MSM with one Higgs boson doublet, is a model with two doublets. Of particular interest (Section 5) are Supersymmetric extensions of the Standard Model. The two-doublet Higgs model implies the existence of a larger number of Higgs particles and includes the one-doublet Higgs boson signatures with a reduced production rate. It is therefore important to search for Higgs bosons in the whole mass range when new data become available. Higgs boson signatures predicted in the one-doublet model, which are already experimentally excluded for a certain Higgs boson mass, can still be found when the experiment is sensitive to a lower Higgs boson production rate. Such a discovery would give strong support to the two-doublet Higgs theory. This section outlines the implications of the two-doublet Higgs model for an experimental search. 4.1.1. General constraints Extensions of the MSM Higgs sector must satisfy three general constraints arising from (a) the -parameter measurement, (b) the absence of Qavor-changing neutral currents, and (c) the unitarity requirement. (a) -parameter. An extension of the minimal Higgs sector must not spoil the successful predictions of the Standard Model, among which are the correct prediction of the Z and W± mass relation. A general Higgs scenario with an arbitrary number of singlets and doublets is compatible with = 1, Eq. (2.8). Higgs representations beyond two doublets, where is given as a function of the total SU(2)L isospin T, the U(1)Y hypercharge Y and the VEV v(T; Y) of each multiplet, require complicated parameter arrangements to match the measured value. The largest contribution to the error on , as given in Eq. (2.9), is due to the uncertainty of the W mass (Table 2.2). Therefore, the increase of the LEP energy above the WW production threshold in 1996 has reduced the error on mW and as a consequence the error on . Also, the CDF and D0 experiments have reduced the error on mW with increasing data statistics. A precise determination of is important, since any deviation from = 1 requires ‘New Physics’ 4

,˜0 is the lightest Supersymmetric particle, named neutralino.

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beyond the MSM. Unless a deviation from = 1 is found, Higgs models consisting of only singlets and doublets are favored. (b) Flavor-changing neutral currents. The absence of Qavor-changing neutral currents (FCNC) [24,94] strongly constrains extensions of the MSM. In the MSM, FCNC are forbidden at the tree-level. In a more general Higgs model, this is no longer the case. In order to avoid large FCNC, which contradict the low measured rate of KL → + − , one of the two following requirements must be satisGed: the Higgs boson masses are of order 1 TeV which will suppress FCNC, or all fermions of a given electric charge must couple to the same Higgs doublet. In the latter case, a theorem by Glashow and Weinberg [95] ensures that FCNC, mediated by gauge bosons, are absent. This possibility is favored over an unnaturally high Higgs boson mass. Glashow and Weinberg’s theorem does not determine the fermion couplings uniquely. In a general two-doublet model, there are two possible scenarios for the couplings of the Higgs bosons to fermions: (1) One doublet couples to up-type fermions and the other doublet couples to down-type fermions. This structure is required in the Minimal Supersymmetric extension of the Standard Model (MSSM). (2) One doublet couples to up-type and down-type fermions, while the other doublet does not couple to fermions at all. (c) Unitarity. An extended Higgs model has to fulGll the unitarity bound, which prevents scattering amplitudes from growing without bounds as a function of energy. This is achieved by a renormalizable gauge theory with a non-trivial cancellation among Feynman diagrams that involve Higgs Gelds. In the MSM, the unitarity requirement yields the tree-level relation gHZZ = gmZ , where gHZZ is the coupling between the MSM Higgs and a Z pair and g is the gauge coupling. In a more elaborate Higgs sector, the unitarity problems are cured by a combination of Feynman diagrams involving additional Higgs Gelds. Thus, unitarity relates the MSM Higgs couplings to the Higgs couplings of an extended Higgs sector. A Higgs sector consisting of doublets and singlets Hi , which couple to the Z with the coupling gHi ZZ and to fermions with a coupling gHi ff , must obey the following sum rules: i

(gHi ZZ )2 = (gHZZ )2 ;

(gHi ZZ )(gHi ff ) = (gHZZ )(gH: );

(4.1)

i

where gHZZ and gH: are the corresponding couplings in the MSM. An obvious consequence of these sum rules is that the production rate of a Higgs particle in an extended Higgs sector must always be smaller than or equal to the production rate of the Higgs boson in the MSM. In summary, within the above three constraints, the Higgs sector of the Standard Model may have a variety of structures that are consistent with the observed experimental data. The simplest extension of the minimal Higgs model is the two-doublet model. In this extension, the absence of FCNC and = 1 are guaranteed without unnaturally Gne-tuning any parameter (see Section 5.1 for details). The couplings between Higgs bosons and gauge bosons as well as between Higgs bosons and fermions are smaller than those in the MSM.

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4.1.2. Higgs boson mass spectrum The spectrum of Higgs particles in the two-doublet model is richer than that of the MSM. Two complex doublets of Higgs Gelds have eight degrees of freedom: + + 1 2 1 = ; 2 = : (4.2) 0 1 20 The most general gauge-invariant Higgs potential must respect the discrete symmetry i ↔ −i , i = 1; 2, in order to avoid FCNC. It has the form V (1 ; 2 ) = 1 (1† 1 − v12 )2 + 2 (2† 2 − v22 )2 + 3 [(1† 1 − v12 )(2† 2 − v22 )]2 + 4 [(1† 1 )(2† 2 ) − (1† 2 )(2† 1 )] + 5 [Re(1† 2 ) − v1 v2 cos +]2 + 6 [Im(1† 2 ) − v1 v2 sin +]2 : The VEVs

v1 = 1 =

0 v1

and

v2 = 2 =

0 v2 ei+

(4.3)

(4.4)

minimize the potential for arbitrary positive parameters i (i = 1; : : : ; 6) and arbitrary phase +. This potential with spontaneously broken symmetry is analogous to the MSM potential, given in Eq. (2.1). For sin + = 0, the CP symmetry of the Lagrangian is broken because of the phase +. This leads to large CP violation, in contradiction to measurements; thus, + is set to zero. (In the MSM, CP violation can be incorporated by introducing a CP-violating phase in the CKM matrix.) The Higgs spectrum is obtained by expanding the Higgs Gelds around their minima. Three Goldstone bosons are identiGed by their derivative couplings to the gauge Gelds. Performing the expansion of the gauge-invariant terms in the Lagrangian L = |D 1 |2 + |D 2 |2 + · · · with the covariant derivative D = (9 − (i=2)g · W − (i=2)g X ), the gauge boson masses and an orthogonal basis of the neutral gauge boson mass eigenstates are obtained. The resulting gauge boson masses are given by mZ = (v12 + v22 )

g2 ; 2 cos W

m = 0;

mW = (v12 + v22 )

g2 : 2

(4.5)

Thus, the quadratically summed VEVs must be equal to the VEVs of the MSM. The ratio of the VEVs deGnes a key parameter: tan ≡ v2 =v1 :

(4.6)

The mass eigenstates of the neutral Higgs bosons are derived from their mass-mixing matrix: 2 4v1 (1 + 3 ) + v22 5 4(3 + 5 )v1 v2 M= : (4.7) 4(3 + 5 )v1 v2 4v22 (2 + 3 ) + v12 5

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Diagonalization introduces a second key parameter: ; the neutral mixing angle :

(4.8)

Physical Higgs boson masses for two charged Higgs bosons H± and three neutral Higgs bosons h, H, and A are obtained: mH± = 4 (v12 + v22 );

mA = 6 (v12 + v22 ) ; 2 1 2] : mH; h = 2 [M11 + M22 ± (M11 − M22 )2 + 4M12

(4.9)

The convention mH ¿ mh is used throughout this review. Thus, the mass spectrum, which is derived from the gauge invariant CP-conserving Higgs potential with spontaneously broken symmetry, consists of Gve physical Higgs bosons. Eight initial degrees of freedom (six from the -parameters and two from the VEVs) can be expressed as four di:erent Higgs boson mass parameters mH± , mA , mH , mh and tan , while the remaining three degrees of freedom are absorbed giving masses to the gauge bosons. 5 In summary, the Higgs spectrum of the two-doublet model consists of • one neutral pseudo-scalar A, • two neutral scalars H and h, and • two charged scalars H± .

4.1.3. Production Charge conjugation C, parity P, and total angular momentum J , quantum numbers of the Higgs bosons allow one to identify the possible Higgs production mechanisms. 6 The J CP quantum numbers are 1− − for the photon, 1− − for the Z, and J P = 1− for the W± . These multiplicative quantum numbers must be conserved during the Higgs production process. Applying the parity and charge conjugation operators to the Higgs Gelds, the following quantum numbers are assigned: J CP (A) = 0+− ;

J CP (H) = 0++ ;

J CP (h) = 0++ ;

J P (H± ) = 0+ :

(4.10)

The CP-odd nature of the A boson forbids its bremsstrahlung emission o: the Z or the W± . Furthermore, the interactions Z → hh and Z → AA are forbidden (the Z wave function is anti-symmetric, while the wave functions for the pairs of hh and AA are symmetric). The only remaining interactions for the Higgs production, near the Z resonance, are (a) (b) (c) (d) 5 6

bremsstrahlung process: Z → Z∗ h, Z → Z∗ H, neutral pair-production: Z → hA, Z → HA, charged pair-production: Z → H+ H− , bremsstrahlung o: b or :

In the following chapter on Supersymmetry the same parameters are used. For charged particles no C quantum number is deGned.

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Fig. 4.1. Feynman graphs of Higgs boson production in the two-doublet Higgs model.

U , Z → bbU → bbh U U , Z → bb → bbA + − Z → → + − h, Z → + − → + − A. The production graphs are shown in Fig. 4.1. It is important to point out that the Higgs bremsstrahlung coupling ghZZ and neutral Higgs pair-production coupling gZhA are complementary functions of the mixing angle and tan : ghZZ ˙ sin( − );

gZhA ˙ cos( − ) :

(4.11)

For process (a), the production width for the h of the two-doublet model and the HMSM boson of the MSM are related by sin2 ( − ) =

!(Z → hZ? ) : !(Z → HMSM Z? )

(4.12)

Thus, for sin2 ( − ) ≈ 1 similar event rates are expected as for the MSM Higgs boson. For process (b), the Z partial width for the Higgs boson pair production is proportional to cos2 ( − ): 2 2 mh mA 1 × cos2 ( − ) ; U 3=2 ; (4.13) !(Z → hA) = !(Z → ) 2 m2Z m2Z where (a; b) = (1 − a − b)2 − 4ab. The expected event rate can be very large up to about 10% of the total number of produced Z events. For process (c), the partial width of the Z decay into a charged Higgs pair at the tree-level depends only on the mass of the charged Higgs boson [49,96]: 7 2 3 4m2H± m G 1 F 3 − sin2 W H ; = 1 − : (4.14) !(Z → H+ H− ) = √ Z ± ± H m2Z 6 2# 2 7

Radiative corrections are calculated. First results [97] were corrected and now two independent groups agree [98,99].

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Fig. 4.2. Charged Higgs boson pair-production rates per one million Z events. The rate reduces quickly near the kinematic threshold of mZ =2.

The number of expected events is shown in Fig. 4.2 for one million hadronic Z decays. Even near the kinematic limit a large number of charged Higgs bosons are expected, for example more than 1000 events of a 40 GeV charged Higgs boson. For process (d), the production cross section is proportional to sin2 = cos2 for the scalar and tan2 for the pseudoscalar Higgs boson. Three experimentally relevant conclusions can be drawn. • First, it follows that the same Higgs boson production and decay process for a certain mass

that has been excluded in the MSM could still have escaped detection. A discovery of Higgs bosons with low production rate and with the kinematics of the Bjorken process would strongly suggest that a non-minimal Higgs structure is realized in nature. • Second, in the two-doublet model, the Higgs boson production via the Bjorken process and the neutral Higgs boson pair-production are complementary. However, the Higgs bremsstrahlung production and neutral Higgs boson pair-production are not explicitly predicted, since and tan are unknown parameters of the theory. • For large tan process (d) can dominate the pseudoscalar Higgs boson production. 4.1.4. Decay The Yukawa interaction determines the Higgs boson decay branching ratios into fermion pairs. This also depends on and tan . The Higgs boson coupling has the general form of g: = const mf f(; tan ) ;

(4.15)

where = h; H; A and f(; tan ) depends on the choice of one of the two possible Higgs couplings to the fermions. Table 4.1 summarizes the dependence of the function f on and tan for the case when up-type quarks and leptons couple to Higgs doublet 1 and down-type quarks couple to Higgs doublet 2 . The factors are identical for the second and third families. The ratios described by f satisfy the second unitarity sum rule, Eq. (4.1), for all and tan .

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Table 4.1 Higgs–fermion coupling in the two-doublet Higgs model for the case when one doublet couples to up-type fermions and the other couples to down-type fermions Decay

h → + −

→ ccU

→ bbU

A → + −

→ ccU

→ bbU

f(; tan ) =

sin cos

cos sin

sin cos

tan

cot

tan

The Yukawa interaction of the Higgs with the fermions implies that the Higgs boson decays into the heaviest kinematically accessible fermion pair. This general feature guides the search for the neutral Higgs bosons. In a two-doublet model, many Higgs decay channels are possible for a given Higgs boson mass owing to the two arbitrary parameters and tan . Independently of the choice of and tan , the following general features are found (if the decays are kinematically allowed): • Higgs boson decays into + − pairs are dominant over decays into e+ e− or + − , • decay rates into c-quarks are larger than decay rates into u-quarks (where the t-quark is

kinematically not accessible), and

• decays into b-quarks are the leading down-type decay modes.

The ratio of Higgs boson decays into leptons and hadrons depends on and tan , since only the massive (charged) leptons couple to the Higgs boson. It is important to note that because of the trigonometric functions, the Higgs branching ratio into a + − or bbU pair can vanish simultaneously. In this case, the ccU branching fraction dominates. The existence of charged Higgs bosons would result in new phenomena. A summary of the predictions on charged Higgs boson decays is given. When only charge conservation is considered, the following decay channels are open: H+ → e+ ; + ; + ; udU; cdU; t dU; usU; csU; t sU; ubU ; cbU ; t bU :

(4.16)

The decay channels involving the top quark will be ignored in this work, since the top mass is higher than the available LEP energy. Thus, the Higgs boson decays into the three heaviest quark pairs H± → ubU ; csU; cbU are the dominant hadronic decay channels. Furthermore, the + channel is dominant over other leptonic channels. Taking the experimental and theoretical knowledge of the Cabibbo–Kobayashi–Maskawa (CKM) matrix elements into account [24], Higgs decays are further constrained. The CKM matrix has the form   Vud Vus Vub    Vij =  (4.17)  Vcd Vcs Vcb  : Vtd Vts Vtb The charged Higgs decay coupling to the quark pair ij is proportional to the factor Vij . The diagonal elements of the matrix Vij are of order 1 while the o:-diagonal elements are largely

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Fig. 4.3. Charged Higgs boson mass limits from b → s constraints based on 1992 CLEO data with LO calculations.

suppressed. Therefore, H+ → csU is the dominant hadronic decay channel. In summary, the most important decay channels are H+ → + and H+ → csU. The relative rates of the resulting Gnal states can be parametrized in terms of the leptonic branching ratio Rl ≡ BR (H+ → + ). The relative decay fractions are given by !(Z → H+ H− → cs) ˙ 2Rl (1 − Rl ) ; !(Z → H+ H− → csUcs U ) ˙ (1 − Rl )(1 − Rl ) ; !(Z → H+ H− → + − ) U ˙ R2l :

(4.18)

b → s limit. In general, the charged Higgs boson can contribute indirectly to various pro0 cesses which are mediated by W bosons. These processes are, for example, B0 − BU mixing, b → s and leptonic -decays. Most important is the b → s reaction which is addressed here. A strong limit on the charged Higgs boson mass can result from the CLEO measurement of BR (b → s ) ¡ 5:4 × 10−4 [100]. Using calculations of the charged Higgs contribution to this decay rate [101,102], already in 1993, a charged Higgs boson with mH± ¡ 260 GeV was excluded independently of the tan value, as shown in Fig. 4.3. The latest rate from the CLEO Collaboration [103] is BR (b → s ) = (3:15 ± 0:35 ± 0:32 ± 0:26) × 10−4 :

(4.19)

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Fig. 4.4. Charged Higgs boson mass limits from b → s constraints with NLO calculations. The lines show constant predicted rates in the two-doublet Higgs model. Experimental results give an upper limit and thus exclude indirectly light charged Higgs boson masses.

This value combined with the ALEPH measurement [104] of BR (b → s ) = (3:11 ± 0:80 ± 0:72) × 10−4

(4.20)

yields as average BR (b → s ) = (3:14 ± 0:48) × 10−4 :

(4.21)

This average gives an upper limit on the branching fraction of 3:93 × 10−4 at 95% CL. The limit on the charged Higgs boson mass resulting from next to leading order (NLO) calculations is shown in Fig. 4.4 (from Ref. [105]) as a function of tan . Compared to the charged Higgs boson mass limit shown before in Fig. 4.3 the new mass limit is only slightly weaker. However, these limits can vanish completely [106] if Supersymmetric particles cancel the charged Higgs contribution to the branching fraction. Therefore, the direct searches cannot be substituted by these indirect bounds. 4.1.5. One-loop potential Much progress has been made in theoretical investigations of the e:ects of radiative corrections in the framework of the two-doublet model. Four main aspects were already discussed in Section 2.1 for the MSM (vacuum stability, Coleman–Weinberg limit, phase transitions at Gnite temperature and the triviality Higgs boson mass bound). They also apply in the two-doublet Higgs model as outlined in the following paragraphs. Top quark and Higgs boson mass relation. The existence of a second Higgs doublet destabilizes the vacuum further in the presence of a heavy top quark. For a given top mass (mt ˙ gt v2 ), the Higgs–fermion coupling gt has to be larger in the two-doublet model than in the MSM, since the VEVs v1 and v2 of the doublets are smaller, while the quadratic sum of the VEVs is identical in both models, Eq. (4.5). A study of the vacuum destabilization based on one-loop calculations [107] shows that in the top quark and Higgs boson mass parameter space, a region similar to that in the one-doublet case is excluded by vacuum instability. However, no experimentally relevant conclusions for the two-doublet Higgs search can be drawn.

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Coleman–Weinberg lower limit. The lower Higgs boson mass limit in the Coleman–Weinberg case, where the tree-level Higgs boson mass is set to zero, can be obtained in complete analogy to the one-doublet Higgs model. In the two-doublet model, the B factor of Eq. (2.14) includes the contributions of all Gve Higgs bosons. The new bosonic contribution increases the Coleman– Weinberg mass mCW . It can become positive while mt is large. In the neutral Higgs sector of the one-doublet model, relation (2.16) is replaced by [33] m2H cos2 ( − ) + m2h sin2 ( − ) ¿ m2CW :

(4.22)

As for the one-doublet case, the large top mass requires m2CW to be negative and thus no limit exists. Finite temperature and phase transition. The possibility of a phase transition in the framework of the two-doublet model at the electroweak energy scale (≈ 100 GeV) has been investigated [108–110]. An upper Higgs boson mass limit can be obtained, based on the idea that the matter–anti-matter asymmetry, which is present in the Universe, is generated at the electroweak phase transition. The experimentally determined lower Higgs boson mass limit excludes, in the one-doublet Higgs model (Section 2), this cosmological theory. In the two-doublet model, the upper Higgs boson mass limit from phase transition vanishes when heavy Higgs bosons are also present. Therefore, the above cosmology argument favors a Higgs scenario beyond the MSM. Triviality upper Higgs boson mass bound. An upper bound for the Higgs boson mass in the doublet version of the Standard Model was derived [111]. Encouraged by the fact that in the one-doublet case, the one-loop calculation and the correct lattice calculation give a very similar upper Higgs boson mass limit, the one-loop investigation is also pursued in the two-doublet case. The upper Higgs boson mass limit increases to 1 TeV in the two-doublet model for most choices of and tan . In summary, one-loop radiative correction calculations for the two-doublet model do not yet lead to conclusive predictions for experimental Higgs search. 4.1.6. Implications for the Higgs boson search at LEP-1 A summary of the theoretical implications for the Higgs search at LEP in the framework of the general two-doublet Higgs model is in order. Extensions of the Standard Model Higgs sector must be consistent with two important experimental results on neutral currents: Grst, the -parameter is very nearly equal to one; and second, there are stringent limits on Qavor-changing neutral currents. Models that contain only Higgs doublets automatically satisfy the Grst constraint, and can satisfy the second without unnatural Gne-tuning of the parameters. A model with two Higgs doublets illustrates new phenomena of more general models: • The rate of Higgs bremsstrahlung Z → Z? h is suppressed compared to the MSM prediction by a factor of sin2 ( − ), where and are free parameters of the two-doublet model. • Independent limits on cos2 ( − ) were obtained from constraints on hA contributions to the Z width. The limits on sin2 ( − ) and cos2 ( − ) were combined to exclude a region in

the (mh ; mA ) plane. A mass pair (mh ; mA ) is excluded if the corresponding upper limit on

sin2 ( − ) from the bremsstrahlung process is lower than the lower limit coming from the

pair-production process. Details are given in the next section.

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• The neutral pair-production mechanism leads to multi-jet and=or multi-lepton Gnal states,

such as

U bU ; + − bbU ; + − + − ; Z → hA → bbb U bb U bU : Z → hA → AAA → bbb

(4.23)

Although Higgs bosons tend to decay into the most massive kinematically accessible fermion pair, no unique prediction of the branching ratios can be made because of the unknown values of the parameters and tan . Therefore, most generally, negative search results for each channel can be given as limits on the production branching ratio: !(Z → hA)BR (hA → X)=!(Z → qqU) ;

(4.24)

as a function of the (mh ; mA ) mass combination, where X stands for the visible Gnal states as deGned in processes (4.23). The searches for these signatures are reported in Section A.2. • In the charged Higgs sector of the two-doublet model, the predicted production rate depends only on the charged Higgs boson mass. The decays of the pair-produced charged Higgs bosons can be constrained to three dominant channels and the relative branching fractions are parametrized by one parameter. The searches for the signatures of the charged Higgs bosons in the three dominant processes Z → H+ H− → csUcs U ; cs; + −

(4.25)

are reported in Section A.3. Negative search results are reported in each channel as limits on the charged Higgs boson mass and the leptonic Higgs decay branching ratio. Production rates for Higgs boson bremsstrahlung and neutral Higgs boson pair-production are complementary. Therefore, a Higgs boson cannot escape detection if it is kinematically accessible. The search for Higgs bremsstrahlung in the MSM Higgs decay channels with reduced production rates is particularly important. For some parameter combinations, the Yukawa process e+ e− → bbU x, where x stands for h or A, could be the Grst discovery channel. In the absence of a signal, the experimental results set limits on the parameters of the general two-doublet Higgs model. 4.2. Limits on sin( − ) from searches for Higgs boson bremsstrahlung In the two-doublet Higgs model, the combined LEP limits from Higgs boson bremsstrahlung searches of Fig. 2.29 can be interpreted as a limit on the parameter sin2 ( − ) based on Eq. (4.12). The limit is shown in Fig. 4.5 for the assumption that the branching modes are identical to those of the MSM. 4.3. General Z lineshape limits As an example, details of a study of general limits from Z lineshape measurements are reviewed [112]. The high statistics of the combined LEP lineshape data are used to derive constraints on hypothetical extensions of the MSM. Limits for simple tests on models which predict

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Fig. 4.5. Combined limits on sin2 ( − ) from LEP-1 data. The region above the line is excluded at 95% CL.

Fig. 4.6. Illustration of possible e:ects in extensions of the MSM which can be constrained by a comparison of measured Z parameters with MSM predictions.

additional Z boson decays or modiGed Z-couplings are given and applied in the two-doublet Higgs model. The data considered comprise about eight million visible Z decays, recorded between 1989 and 1993 [22,113]. The Z parameters were obtained by Gtting the lineshape of the Z decay into charged leptons and hadrons. All measurements are in agreement with the MSM predictions. Details of the experimental analysis and similar interpretations can be found in the corresponding publications of the four LEP experiments [22]. Severe limits on ‘New Physics’ beyond the MSM were obtained from precision measurements of the Z parameters. Any hypothetical Z decay into new particles Z → X (Fig. 4.6a), radiative contributions from non-MSM virtual particles (Fig. 4.6b), or modiGcations to the MSM Z-couplings (Fig. 4.6c) are constrained by measurements of the total Z width !Z , the invisible Z width !Zinv , the leptonic widths !Zee ; !Z ; !Z , or the ratio of the hadronic to leptonic Z decay width R. Thus, constraints on physics beyond the MSM can be expressed as limits on deviations from the MSM Z decay width predictions. In particular, such limits can be used to constrain the

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Table 4.2 Measured Z parameters, MSM predictions, and their lower and upper limits for one-sided 95% CLs. All decay widths are given in MeV Parameter

!Z !Zinv !Zee !Z !Z R

Measurements

Theory (MSM)

Mean value

Lower bound

Upper bound

Mean value

Lower bound

Upper bound

2497:4 ± 3:8 499:8 ± 3:5 83:85 ± 0:21 83:95 ± 0:30 84:26 ± 0:34 20:795 ± 0:040

2491.2 494.1 83.51 83.46 83.70 20.729

2503.6 505.5 84.19 84.44 84.82 20.861

2496.5 501.6 83.95 83.95 83.75 20.767

2480.6 499.7 83.56 83.56 83.37 20.692

2512.3 503.4 84.33 84.33 84.13 20.842

existence of Higgs bosons in models with more than one Higgs doublet; charginos, neutralinos and light gluinos in Supersymmetric models (Section 5) with or without R-parity conservation; additional heavy charged or neutral leptons; or anomalous gauge boson couplings. 4.3.1. Measurement and theory Table 4.2 summarizes the measured values of !Z ; !Zinv ; !Zee ; !Z ; !Z , and R, as well as their MSM upper and lower bounds for one-sided 95% CLs. One-sided CLs are used because a new decay would always increase the Z width; they are derived assuming Gaussian errors by extending the 11 error to 1:641 [114]. The measured values are averages from the four LEP experiments taking into account common systematic errors [113]. Theoretical upper and lower bounds are obtained with an analytical program (ZFITTER version 4.6 [115 –118]) by varying the strong coupling constant s , the top-quark mass mt , and the MSM Higgs mass mh , independently within their one-sided 95% CL limits. The uncertainty in these values constitutes the dominant error on the MSM predictions. The world average s (mZ ) = 0:125 ± 0:005 [113] was used. 8 This average is based on data from N experiments, ppU colliders, the SLD measurement of the left–right asymmetry and the LEP experiments. The top-quark mass mt = (174±10+13 −12 ) GeV [119] was taken when the analysis was performed, then new top mass values were released [19 –21]. For mh a previously combined lower mass limit [84], resulting from the data of the four LEP experiments (Section 2.4) and a theoretical upper mass bound following from consistency arguments in the MSM [25] were used. Thus, the ranges used for s ; mt and mh were 0:117 ¡ s (mZ ) ¡ 0:133;

(148 ¡ mt ¡ 201) GeV;

(63:5 ¡ mh ¡ 1000) GeV :

The central values of the MSM predictions are the arithmetic means of the upper and lower bounds.

8

The latest value of s is given in Ref. [43].

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Table 4.3 Allowed changes of !Z ; !Zinv ; !Zee ; !Z ; !Z , and R due to non-MSM contributions, using twice one-sided 95% CL limits. ‘max’ indicates the maximum experimental value minus the minimum theoretical value, and ‘min’ indicates the minimum experimental value minus the maximum theoretical value. The interval and di:erence are deGned in the text. All decay widths are given in MeV Parameter

Interval

Di:erence

Sum

(>!Z )min (>!Z )max (>!Zinv )min (>!Zinv )max (>!Zee )min (>!Zee )max (>!Z )min (>!Z )max (>!Z )min (>!Z )max (>R)min (>R)max

−21:1

−0:9

−22:0

23.0 −9:3 5.8 −0:82 0.63 −0:87 0.88 −0:43 1.45 −0:113 0.169

0 0 1.8 0 0.10 0 0 −0:51 0 −0:028 0

23.0

−9:3

7.6

−0:82

0.73

−0:87

0.88

−0:94

1:45

−0:141

0.169

In order to obtain a conservative limit on non-MSM e:ects from !Z , one considers the intervals: th (!Z )exp min − (!Z )max

and

th (!Z )exp max − (!Z )min ;

where both the experimental and theoretical limits are taken at the one-sided 95% CL. Similar intervals are deGned for the other parameters listed in Table 4.2. If the value of the predicted mean value minus the measured mean value is negative (positive), it is added to the lower (upper) limit. This conservative approach avoids the setting of tighter constraints than allowed by the agreement between theory and measurement. Otherwise, for example a measurement of the central value of the total Z decay width signiGcantly below the MSM expectation would naively lead to too strong a bound on physics processes beyond the MSM. Table 4.3 summarizes the intervals and di:erences obtained. Considering a new decay channel Z → X, the decay ratios of X are deGned as xj ≡ !(X → j)=!(X → anything); and j = h; l; i for hadrons, leptons, and invisible particles, respectively. In this deGnition, xh + xl + xi = 1. The hadronic and leptonic branching ratios of the Z are bh and bl . In the deGnition of R, the hadronic Z decays are summed over all Gve quark types produced at LEP, while the leptonic Z decay width is given for a massless charged lepton pair assuming lepton universality. Letting !ZX ≡ !(Z → X), then (1) the limit on !ZX from !Z is given by !ZX 6 (>!Z )max = 23:0 MeV ;

(4.26)

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Table 4.4 One-sided 95% CL lower and upper limits on Y!(Z) for Z decaying into any, invisible, charged leptonic, and hadronic channels. The corresponding branching ratio upper limits on YBr (Z) are also given Origin

Decay mode

Y!(Z)

(MeV)

YBr (Z) (in %)

!Z !Zinv !Zee !Z !Z R

Z → anything Z → invisible Z → e + e− Z → + − Z → + − Z → hadrons

−22:0 −9:3 −0:82 −0:87 −0:94 −12

23.0 7.6 0.73 0.88 1.45 14

0.92 0.30 0.029 0.035 0.058 0.56

(2) the limit on !ZX from !Zinv is given by xi !ZX 6 (>!Zinv )max = 7:6 MeV ; (3) a contribution from Z → X decays would change the ratio R = bh =bl by bh xl !Z bh + !ZX xh !ZX xh >R = − ≈R − ; bl !Z bh 3bl !Z bl + 13 !ZX xl

(4.27)

(4.28)

which is an approximation for !ZX !Z . For xh =1 and xl =0; (>R)max leads to !ZX 6 14 MeV. For xh = 0 and xl = 1; (>R)min results in !ZX 6 1:7 MeV; this limit is however weaker than those from !Zee ; !Z ; !Z . Radiative contributions from non-MSM virtual particles or modiGcations to the MSM Z-couplings are constrained by the upper and lower limits given in Table 4.3. The most stringent limits on deviations from the non-MSM e:ects on the Z decay widths are summarized in Table 4.4. Both upper and lower limits are given at one-sided 95% CL. As a consequence, modiGed MSM Z-couplings or amplitudes of non-MSM radiative corrections are constrained to the interval at 90% CL. The limits on new decay modes obtained from !Z are independent of the decay branching fractions, while the limits from !Zinv constrain only invisible Z decay modes. The limits from !Zee ; !Z ; !Z , and R constrain the corresponding leptonic and hadronic Z-couplings, respectively. The limits for unspeciGed and invisible decay modes are of most general use. The limits on !Zee are tighter, since the Zee-coupling contributes both to Z production and decay. One should note that the charged leptonic and hadronic limits are not able to constrain Z decays if the resulting new particles subsequently decay; dedicated searches are necessary for such speciGc Gnal states. This is due to the precise selection criteria applied for leptonic and hadronic Z decay event topologies. If a model predicts the invisible, charged leptonic and hadronic branching fractions of Z decays, a ,2 -method allows the setting of tighter constraints. The results have been updated over several years. First results were derived from 1990 and 1991 LEP data. Only slightly tighter limits were obtained by including the 1992 data as they were entirely taken on the Z pole. (A scanning of the Z lineshape requires data-taking

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at di:erent center-of-mass energies.) By including the 1993 data which contain both o:- and on-peak results, limits are signiGcantly improved; the experimental errors are reduced by about a factor of two compared to previous measurements. In this regard, little improvement was expected from the 1994 data as they were again only taken on the Z-pole. For unspeciGed Z decays, the 1993 improvement of Y!(Z) is mainly due to the increased predicted MSM lower bound on !Z following from the top mass measured at the Fermilab experiments. As an example, an application of the constraints on the Z width is given in the next section. 9 4.4. Limits on cos( − ) and neutral Higgs boson masses from the total Z width A limit on cos2 ( − ) in the two-doublet Higgs model is derived. The value cos2 ( − ) can be constrained by the limits from precision Z-lineshape measurements owing to the large production rate of Z → hA. The Z decay width into neutral Higgs boson pairs in the two-doublet Higgs model is given by Eq. (4.13). !(Z → ) U is derived from a combined Z-lineshape Gt: !(Z → ) U = 166:6 ± 1:2 MeV [113]. Without any assumption on the Higgs decay modes and based on Eq. (4.13), the constraint Y!(Z) 6 23:0 MeV sets a limit on cos2 ( − ) as a function of mh and mA : 2 2 2!ZX 2 −3=2 mh mA cosmax ( − ) = ; : (4.29) !(Z → ) U m2Z m2Z Fig. 4.7 shows the excluded cos2 ( − ) range at 95% CL as a function of mh for mA = 20 GeV. Further constraints on cos2 ( − ) can result from an analysis of the one-loop vertex corrections to the Zbb-coupling involving additional neutral and charged Higgs bosons. Such corrections could decrease !(Z → bbU ) and thus the hadronic decay width, depending on the unknown parameters of the two-doublet Higgs model [121,122]. In this case the limit Y!(Z → hadrons) ¿ −12 MeV applies. 4.5. Search for neutral Higgs boson pair-production In conjunction with a constraint on sin2 ( − ), derived from the search for the MSM Higgs boson (Fig. 4.5), the cos2 ( −) limit, given in Fig. 4.7, leads to an exclusion of a large (mh ; mA ) parameter range. Such a limit was already presented in Ref. [84] by excluding (mh ; mA ) values where sin2 ( − ) + cos2 ( − ) ¿ 1. The excluded region from the combination of sine and cosine limits is presented in Fig. 4.8. General limits on hA production are set by DELPHI [123] based on the four-jet selection described in Appendix A.2.2. The number of observed data events from selection (1) is 1956, while 1956 ± 38 ± 140 background events are expected. The dominant QCD background process is illustrated in Fig. 4.9. A slightly larger number of background compared to data events would give a stronger limit than equal numbers. Therefore, this di:erence is set conservatively to zero to derive the above constraint. Statistical and systematic errors are added in quadrature and the quoted eVciencies are reduced by the combined error to derive the limit. 9

Now, the analysis of all LEP-1 data and the direct precision measurements of the mW at LEP-2 result in Y!(Z) ¡ 6:3 MeV at 95% CL [120].

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Fig. 4.7. General limits on cos2 ( − ) in the two-doublet Higgs model as a function of mh for mA = 20 GeV. The limit is based on the constraint Y!(Z → anything) 6 23:0 MeV, set by the precision lineshape measurements. No assumptions on the decay branching ratios of the Higgs bosons are made. Fig. 4.8. Combined limits on (mh ; mA ). The region below the line is excluded at 95% CL from the combination of the sin2 ( − ) and cos2 ( − ) limit as described in the text.

Fig. 4.9. Feynman graphs of QCD four-jet production. The production can be via the emission of a hard (energetic) gluon and a subsequent splitting into two gluons; the emission of two hard gluons; or the gluon emission with a splitting into two quarks. Only the last of these processes leads in about 0.02% of the total Z decays into the production of four b-quarks, which is irreducible background for the search of neutral Higgs boson pair-production.

The following limit on a Higgs boson signal is set at 95% CL: U cU) !(Z → hA → bbc ¡ 2:5 × 10−3 : !Z

(4.30)

Then, DELPHI applies more speciGc criteria in order to select events with two bbU pairs. Selection (2) described in Appendix A.2.2 gives 105 data events, which is consistent with

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Fig. 4.10. DELPHI bbU bbU data event with four b-jets tagged using a microvertex detector. The central beam position (primary vertex) and secondary vertices are indicated by the error ellipses.

the expectation of 97 ± 9 ± 10 background events. The following limit on a Higgs boson signal is set at 95% CL if both Higgs masses are above 35 GeV: U bU ) !(Z → hA → bbb ¡ 3:5 × 10−4 !Z

(4.31)

and it decreases to the following value for a mass of 15 GeV: U bU ) !(Z → hA → bbb ¡ 5:5 × 10−4 : !Z

(4.32)

Only one of the selected data events has four jets identiGed as b-jets. Fig. 4.10 (from Ref. [123]) shows this event on the scale of the microvertex detector. The error ellipses indicate the reconstructed primary and secondary vertices. In comparison with the DELPHI results, general limits on the hA production are also given by L3 [70,71]. They are shown in Fig. 4.11 (from Ref. [85]). Note that DELPHI normalized this limit to !Z , Eqs. (4.31) and (4.32), while L3 normalized their limit to !(Z → qqU).

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Fig. 4.11. L3 limits on !(Z → hA → bbU bbU )=!(Z → qqU) as a function of mh and mA at 95% CL. The regions for mh and mA values below about 20 GeV are not covered, since the jet-Gnding algorithm required a minimum invariant mass. Fig. 4.12. L3 limits on !(Z → hA → AAA → bbU bbU bbU )=!(Z → qqU) as a function of mh and mA at 95% CL. The kinematic requirement 2mA ¡ mh constrains the search region to the shown area.

The h → AA decay can be dominant if it is kinematically allowed. No indication of a Higgs boson has been observed and limits on the production rate are set, for example, on six ’s or U bU in six b’s of about 10−3 [70,71,93,123,124]. In particular, the DELPHI selections for the bbb U bU because of the more eVcient b-tagging and the U bb sample (2) give higher eVciencies for bbb larger sphericity of the six b-quark events. The selection of sample (2) gives about twice the U bb U bU compared to the bbb U bU channel. Branching ratio eVciency for e+ e− → hA → AAA → bbb limits are given from L3 [70,71] in Fig. 4.12 (from Ref. [85]). In the absence of a signal in the Z → hA → + − bbU and + − + − as outlined in Appendix A.2.5, branching ratio limits are given from L3 [70,71] in Figs. 4.13 and 4.14 (from Ref. [85]). 4.6. Search for charged Higgs boson pair-production The charged Higgs boson production process is illustrated in Fig. 4.1. Before LEP began operation the mass limit was about 20 GeV [125]. In this section the mass limits from the three decay channels (leptonic + − , U semi-leptonic csU− , U and hadronic csU cs U ) are shown individually and also the combined limit is presented. The limits from OPAL [126], given in Fig. 4.15, are based on their analysis reviewed in Section A.3. The two events in the + − U and the one event in the csU− U channel are considered candidate events in the derivation of the limits. The combined, exclusion line is the envelope of the three individual searches. A more detailed combination based on combining exclusion conGdence levels does not give signiGcantly stronger limits. This is due to the sharply falling production cross section near the kinematic

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Fig. 4.13. L3 limits on !(Z → hA → + − bbU )=!(Z → qqU) as a function of mh and mA at 95% CL. The axial asymmetry is due to the sensitivity decrease for a very low energy + − pair in a very energetic hadronic environment from a heavy Higgs boson decay into a bbU pair. Fig. 4.14. L3 limits on !(Z → hA → + − + − )=!(Z → qqU) as a function of mh and mA at 95% CL. Most sensitivity was obtained for light masses, since this analysis was tuned for complementarity with the searches involving a Higgs boson decay into a bbU pair.

limit. The mass limit is 44:1 GeV independent of the branching fraction. Similar results near the kinematic limit were given by DELPHI [127], L3 [70,71] and ALEPH [128]. While much attention was given to the search for charged Higgs bosons by all LEP experiments, which are predicted in the framework of the general two-doublet Higgs model, extensions of the two-doublet model could predict doubly charged Higgs bosons [129 –131]. These models received support by the recent observation of small neutrino masses [26 –28]. A search for pair-produced doubly charged Higgs bosons (H±± ) from Z decays was performed by OPAL [132]. Under the strong assumption that this Higgs boson decays dominantly to like-sign charged leptons, a distinct four-fermion signature for short-lived H±± is expected. For long-lived H±± they search for charged tracks with ionization d E= d x signiGcantly larger than that for a singly-charged track. No indication of a signal has been found. Fig. 4.16 shows the excluded regions in the gll –mH±± plane, where gll is the coupling strength of the Higgs boson to lepton pairs. The third component of the isospin IL3 of H±± is not predicted and the excluded regions are shown for IL3 = 0; ±1. 4.7. Search for the reactions Z → : → :h; :A U bU , bbU + − and + − + − as in In this search channel (Fig. 4.1d), the same Gnal states of bbb the hA decay modes are present. However, the precise event structure is di:erent. In order to determine the selection eVciencies these radiative Higgs events have to be simulated. Results from ALEPH [133], DELPHI [134] and L3 [70,71,135] are reviewed. The detection eVciencies

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Fig. 4.15. OPAL charged Higgs mass limits at 95% CL. The exclusion contours of the leptonic, semileptonic, and hadronic channels, as well as their combination, are shown as a function of the charged Higgs boson mass and its leptonic decay branching fraction. plane for the third component of the H±± isospin, IL3 = 0; ±1. Fig. 4.16. OPAL excluded regions in the gll –m±± H Regions A and B are excluded by the search for short-lived and for long-lived H±± , respectively. The area on the left of the dashed line is excluded by the Z-lineshape measurement. The hatched area is not excluded.

from L3 have been determined using the event generator [136] and keeping the event selection cuts for the hA searches unchanged. A later analysis including a larger data sample is presented from ALEPH using the generator [137] and tuning the selection cuts for low-mass Higgs bosons. Both event generators are based on the matrix element calculation [138] for the processes U bU , • e+ e− → bbU x → bbb + − U • e e → bbx → bbU + − , • e+ e− → + − x → + − bbU , • e+ e− → + − x → + − + − , where x stands for the h or A Higgs boson. U bU channel due to the large Higgs boson branching fraction and Most important are the bbb the channels with a + − and bbU pair due to the small expected background. In the L3 analysis of the bbU + − channel, a distribution of the -isolation angle is shown in Fig. 4.17 (from Refs. [70,71,135]) for the simulated Higgs boson mass of 30 GeV in the reaction e+ e− → + − h → + − bbU . The cut separates well signal and background and the data agree with the background simulation. After additional selection cuts, no data event survives in the 1991–1993 data set of about 1.6 million hadronic Z decays. The sensitivity region of this analysis requires mh ¿ 2m . For lower Higgs boson masses, the decay into lighter particles becomes an important analogue to the low-mass MSM Higgs boson searches described in Appendix 2:2.

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Fig. 4.17. L3 bbU h -isolation angle for data, simulated background and a 30 GeV Higgs boson in the reaction e+ e− → bbU x → bbU + − . The indicated cut separates well expected signal and background.

In the ALEPH analysis [133], the Higgs boson mass region from 100 MeV up to 40 GeV is covered. In the low mass region the search concentrates on a low charged multiplicity system, possibly resulting from a Higgs boson decay, and an energetic + − pair. The signature is named llV . The background generally increases because of the e+ e− → ‘+ ‘− ‘+ ‘− four-fermion background for smaller invariant masses as shown in Fig. 4.18. ALEPH has searched in four categories: A For Higgs bosons below about 5 GeV, the detection eVciency is about 30% for e+ e− → + − A, if mA ¿ 2m . For lower Higgs boson masses the dominant decay mode is an e+ e− pair. In this channel, photon conversion in detector material gives an important background, and selection sensitivities are much lower. The general llV search leads to 530 data events while 533:5 ± 10:8 are expected for the 4.5 million hadronic Z decays in the 1989 to 1995 data sample. B For Higgs boson masses above 5 GeV an invariant mass cut of 1:25 GeV is introduced to reduce the background. The selection eVciency reduces to typically 20%, while 32 data and 30:0 ± 0:8 background events are expected. U bU channel becomes important. In C For Higgs boson masses above the bbU threshold, the bbb the data sample up to 1994, 644 data events are selected and 594 ± 45 background events from the processes shown in Fig. 4.9 are selected. D In the same mass region above the bbU threshold, the + − bbU channel is also important. Six events are selected which are counted as candidates in deriving limits due to the larger uncertainty of the expected events in this channel. Typical detection eVciencies in searches C and D are 10%. Thus, owing to the much lower background rate, this channel gives U bU channel. higher sensitivity than in the bbb

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U invariant mass of e+ e− and + − pairs for data and simulated background in the search Fig. 4.18. ALEPH bbA of light Higgs bosons where h → e+ e− or + − .

Interpretation. In the two-doublet model, the Yukawa process e+ e− → f + f − h or f + f − A could be dominant. This would be the case if sin( − ) is small such that the bremsstrahlung process (Fig. 4.1a) is too suppressed to be detected. On the other hand, mA is beyond the kinematic U reach, such that no hA events could be produced. In this scenario, the process e+ e− → bbh + − or h could be the discovery channel. The situation is analogous with large sin( − ) and U or + − A could be discovered Grst, as pointed out already in mh . In this case e+ e− → bbA Ref. [139]. Experimentally, the Yukawa process was not investigated in the Grst few years of LEP-1 operation, since the number of expected Higgs boson events was insuVcient for sensitivity. Those experimental estimates were based on the available integrated luminosity, together with an estimate of the expected detection eVciency. The production cross sections for the Yukawa process decrease quickly with the Higgs boson mass. For f + f − h and f + f − A, the tan dependence of the cross section is characteristic 1(e+ e− → f + f − x) ˙ tan2 ;

(4.33)

where x stands for the h or A Higgs boson. The Higgs decay branching ratios are given in Section 2. In the framework of the two-doublet model, excluded regions are presented as a function of the Higgs boson mass and tan . Pre-LEP results [140] and the L3 result combining

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Fig. 4.19. L3 bbU h excluded region at 95% CL from searches for bbU bbU , bbU + − and + − + − . In comparison with a previous result from g − 2 measurements [140] (upper curve) the excluded region is largely increased. Fig. 4.20. ALEPH bbU A excluded region at 95% CL from searches A to D as described in the text.

U bU , bbU + − and + − + − channels are shown in Fig. 4.19 (from Refs. [70,71,135]) for bbb = in the mh versus tan plane. The L3 exclusion contour begins above the + − threshold. U bU threshold, since the h → + − branching ratio reduces, The limit becomes weaker at the bbb U bU sensitivity only starts at about 20 GeV Higgs boson mass because of the four-jet and the bbb

clustering with a rather large ycut parameter. A signiGcant increase of sensitivity is achieved compared to the previous results shown. Fig. 4.20 (from Ref. [133]) shows the excluded regions in the mA versus tan plane from ALEPH. The results from the four di:erent searches A to D are presented. Note that the sharp peak at the + − threshold is due to the fact that much more sensitivity is obtained for large charge multiplicity. Below the + − threshold the Higgs boson decays dominantly into neutral particles. The analysis B for the llV signature has the largest sensitivity. Above the bbU threshold U bU search D. the + − qqU analysis C gives much larger sensitivity than the bbb In the 1999 analysis from DELPHI, the important possibility of a light Higgs boson which has escaped detection is also addressed. The production reactions with both scalar and pseudoscalar Higgs bosons are studied. The search resulted in improved limits based on all LEP-1 data in comparison with the previous results. Fig. 4.21 (from Ref. [134]) shows the limits for the U bU Gnal states. Note that the limits are about equal for scalar and pseudoscalar + − + − and bbb U bU Gnal states is at a Higgs production. The transition of sensitivity between + − + − and bbb boson mass of 10 GeV.

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Fig. 4.21. DELPHI bbU h and bbU A excluded region at 95% CL from searches in the + − + − and bbU bbU channels. The Yukawa coupling enhancement factor corresponds to tan for bbU A and sin = cos for bbU h.

5. Minimal Supersymmetric Standard Model 5.1. Theoretical framework Supersymmetry models [141–146] have gained much attention over the last twenty-Gve years. They are promising extensions of the Standard Model owing to some very attractive theoretical features. First, their theoretical motivations are outlined. The two-doublet Higgs model, described in the previous sections, is the underlying Higgs boson structure of the MSSM. This model implies important Higgs boson mass relations. The implications at the tree-level for the Higgs boson phenomenology at LEP are reviewed. Radiative corrections to the MSSM change dramatically the predictions for the neutral Higgs sector, as outlined in Section 5.2. 5.1.1. Motivation of Supersymmetry In the Standard Model, radiative corrections to the Higgs boson mass mH depend on the cut-o: scale %, and are of the form m2H = m20 + cg2 %2 ;

(5.1)

where m0 is the bare Higgs boson mass, g is the Higgs coupling constant and c is a constant of order one [33]. An analogous cut-o: dependence has been known for a long time for QED. The corrections to the fermion masses diverge proportionally to ln(%=mf ). The signiGcant difference is that the divergence is logarithmic and not quadratic as in the Higgs case. Even when the QED cut-o: is at the Planck scale (≈ 1019 GeV), the corrections are still of the order of the physical fermion masses. Assuming that the Standard Model describes the physics up to the uniGcation energy scale of the electromagnetic, weak and strong force (≈ 1015 GeV), and taking the upper Higgs boson mass limit into account (Section 2.1), a Gne-tuning of the bare mass to at least one part in 1024 is required [147], which follows directly from Eq. (5.1).

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The Gne tuning is a major theoretical Qaw of the MSM Higgs sector. The Standard Model becomes an unnatural theory in the sense of ’t Hooft’s deGnition of natural [148]: “At any energy scale , a physical parameter or set of physical parameters i ( ) is allowed to be very small only if the replacement i ( ) = 0 would increase the symmetry of the system.” Supersymmetry is exactly this larger symmetry and the MSM becomes an e:ective low-energy theory. The Standard Model is recovered in the limit when Supersymmetry particles decouple (their masses are set much heavier than the SM particles). Many other examples of an e:ective theory exist where the deeper underlying structure escaped detection for a long time. In Supersymmetry each fermion has a bosonic partner. For each boson which contributes to a Feynman loop diagram, there exists a second loop involving a fermionic loop. The fermion loops contribute with a factor of (−1) compared to the bosonic graphs. Thus, all loop graphs cancel for exact Supersymmetry. In exact Supersymmetry, the masses of the Supersymmetric partners are equal to the masses of the Standard Model particles. Consequently, the cancellation is to all orders of perturbation theory (renormalization theorem). Evidently, Supersymmetry cannot be an exact symmetry, since experiments have shown that the Supersymmetric partners must have a much larger mass than their counterparts. However, the mass di:erences between the Standard Model and Supersymmetric particles cannot be larger than the cut-o: scale, %2 ≈ |m2SM − m2SUSY |. The Gne-tuning problem appears when % is of the order of 1 TeV. At this energy scale the corrections of the Higgs boson mass become as large as the bare mass, Eq. (5.1). Masses of the Supersymmetric particles are expected to be below this energy scale. This explains the intense search at LEP for these particles. Since Supersymmetry is not exact, Supersymmetric models improve the renormalization behavior but do not cancel all loop diagrams completely. Therefore, radiative corrections in the MSSM have to be considered. They have a strong impact on the Higgs sector. Another important theoretical argument for considering Supersymmetry as an extension of the Standard Model is based on the Supersymmetric group algebra. In Supersymmetric models, an operator Q exists, which transforms a bosonic Geld into a fermionic Geld, QB = F. Let P be a four-momentum, then Supersymmetry can be deGned by the algebra j

{Qi ; QU } = 2( ) P >ij ;

[Qi ; P ] = 0;

[P ; P ] = 0 ;

(5.2)

where Latin indices label the di:erent generators and Greek subscripts are Dirac indices. Unlike the Standard Model algebra, the Supersymmetry algebra is related to space–time translations. This feature has far-reaching consequences, since General Relativity also arises from local space–time translations. Thus, Supersymmetric models have a chance to be the framework for the uniGcation of electromagnetic, weak and strong forces of the Standard Model with the gravitational force. Owing to the renormalization theorem, theories based on Supersymmetry promise better renormalization behavior than quantum gravity models. With regard to the two-doublet Higgs model, Supersymmetry constrains the Higgs sector, which results in precise experimental predictions. The Higgs sector is important for the veriGcation or exclusion of the MSSM.

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Fig. 5.1. Evolution of the three coupling constants in the MSM. A single uniGcation point is excluded by more than seven standard deviations.

5.1.2. Experimental status of Supersymmetry No direct experimental evidence for Supersymmetry has been found. Many searches for Supersymmetric particles have recently been performed by the LEP experiments, resulting in improvements on existing lower mass limits. The negative results of the searches for Supersymmetric particles at LEP near the Z resonance gave mass limits very close to the kinematic reach of 45 GeV. For LEP-2, mass limits are generally close to the kinematic limit for pair-produced particles. The latest update of the current searches can be found in Ref. [149]. For a review see Ref. [150]. Experimental indications that Supersymmetry may be the correct extension of the Standard Model are obtained in the framework of Grand UniGcation Theories (GUT) [151]. These theories predict the uniGcation of electromagnetic, weak and strong coupling constants at an energy scale of about 1015 eV. The LEP experiments have contributed to measuring the slope of the three running coupling constants, leading to the conclusion that the uniGcation of the forces is not possible in the Standard Model, while it is possible in Supersymmetry models. Figs. 5.1 and 5.2 (from Ref. [151]) illustrate this important argument for the MSSM. 5.1.3. Higgs boson mass relations As in the MSM, the fermions obtain masses via a Yukawa coupling to the Higgs bosons. Supersymmetry not only constrains the two-doublet model, but also requires the existence of a second Higgs doublet 10 because of the following two arguments. First, the most general SU(2)L ×U(1)Y invariant superpotential requires two Higgs superGelds ˆ H 1 = (H1 ; H˜ 1 ) and Hˆ 2 = (H2 ; H˜ 2 ), where H˜ 1; 2 are the fermionic superpartners (Higgsinos) of the 10

Any even number of Higgs doublets is supported by Supersymmetry. In this work, only the simplest version with two Higgs doublets is considered.

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Fig. 5.2. Evolution of the three coupling constants in the MSSM. A single uniGcation point is Gtted at the grand uniGcation scale of 106 GeV.

Table 5.1 SU(2)L × U(1)Y superGeld quantum numbers. Under SU(3)C transformations, all Gelds are singlets. Mass terms for charged leptons and up- and down-type quarks can be constructed without destroying Supersymmetry Hˆ 1

Hˆ 2

Qˆ

Lˆ

Uˆ

Dˆ

Rˆ

21=2

2−1=2

21=6

2−1=2

11=3

1−1=3

1−1

Higgs boson Gelds. It has the form i i i WF = 3ij [gl Hˆ 1 LUˆj Rˆ + gd Hˆ 1 QUˆj Dˆ + gu Hˆ 2 QUˆj Uˆ ] :

(5.3)

ˆ Lˆ are SU(2)L quark and lepton superGelds, Uˆ ; Dˆ are the SU(2)L quark singlet superHere Q; Gelds, Rˆ is the SU(2)L charged lepton singlet superGeld, i; j are the SU(2)L indices, and gl; d; u are the Yukawa coupling constants for leptons, down- and up-type quarks, respectively. The second Higgs boson doublet with opposite hypercharge is needed in order to give masses to down quarks while conserving the SU(2)L × U(1)Y invariance. This is the Supersymmetric analog of Eq. (2.10). The SU(2)L and U(1)Y quantum numbers of the superGelds are summarized in Table 5.1. In the Supersymmetry Model, it is not possible to generate a second Higgs doublet from the Grst doublet with opposite hypercharge, as in the MSM case, since Supersymmetry would be explicitly broken. While in the MSM the Higgs doublet Geld and its complex conjugate Geld give masses to up- and down-type quarks respectively, see (2.10), the complex conjugate Supersymmetric Higgs Geld would not be invariant under Supersymmetry transformations.

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The second argument for the necessity of two Higgs doublets arises because the sum of triangle graphs involving fermions must vanish to ensure the absence of anomalies which destroy the renormalizability of the theory. An analogous argument in the Standard Model exists: fermionic contributions to triangle graphs require the existence of the top quark to cancel the anomalies. Anomalies in the Higgs sector of the Standard Model do not appear, since only a bosonic Higgs exists. In the following study, only the bosonic part of the Higgs sector in the Supersymmetry model will be discussed, since the fermionic parts mix with other Supersymmetric particles and their discussion is outside the scope of this review. The bosonic Higgs Gelds are analogous to the two-doublet Higgs Gelds of Section 4, Eq. (4.2), and have the form 0 + 2 1 H1 = ; H = ; (5.4) 2 20 1− with hypercharge − 12 and + 12 , respectively. The Supersymmetry-breaking terms in the Lagrangian are constrained in order not to destroy the main motivation for Supersymmetry, the cancellation of the quadratic cut-o: contributions to the Higgs boson mass. Soft Supersymmetry-breaking terms, deGned as being associated with a dimensionful parameter and spontaneous Supersymmetry-breaking terms, both of dimension four, break Supersymmetry in such a way as not to regenerate quadratic divergences. The introduction of complex SU(2)L × U(1)Y singlet Gelds N , in addition to the two Higgs doublets, is consistent with the Supersymmetry requirement. Although these singlets arise naturally in Superstring theories, they could contradict the naturalness motivation of Supersymmetry. The introduction of singlet Gelds goes beyond the MSSM and decreases the predictive power of the theory. These theories are called non-minimal supersymmetric extensions of the standard model (NMSSM). In the absence of singlet Gelds, the most general superpotential that respects the global symmetry of baryon and lepton conservation has the form W = 3ij H1i H2j + WF ;

(5.5)

where has the dimension of mass and WF is given by Eq. (5.3). The scalar Geld potential V , which describes the bosonic Higgs sector, is derived from the superpotential W . Note that the gauge couplings g and g are introduced by a Supersymmetric gauge term in the Lagrangian [32]. After adding soft Supersymmetry-breaking mass terms mi and rearranging the terms to recover the form of the potential of the general two-doublet model, Eq. (4.3), the bosonic Higgs potential is given by V = (m21 + | |2 )H1i∗ H1i + (m22 + | |2 )H2i∗ H2i − m212 (3ij H1i H2j + h:c:) + 18 (g2 + g2 )[H1i∗ H1i + H2i∗ H2i ] + 12 g2 |H1i∗ H2i |2 :

(5.6)

Hence, the same Gve physical Higgs Gelds, which are identiGed in the general two-doublet model (Section 4), are expanded around their VEV. The comparison of the above Supersymmetric

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Higgs potential with the potential of the general two-doublet model, given in Eq. (4.3), leads to constraints on the six degrees of freedom of the general two-doublet model: 1 = 2 ;

(5.7)

3 = 18 (g2 + g2 ) − 1 ;

(5.8)

4 = 21 − 12 g2 ;

(5.9)

5 = 6 = 21 − 12 (g2 + g2 ) :

(5.10)

The last relation assures CP conservation, since the complex phase + of the general two-doublet model can be absorbed by a Geld redeGnition. Relations for the mi parameters are also found by comparison with the two-doublet potential: m21 = | |2 + 21 v22 − 12 m2Z ;

(5.11)

m22 = | |2 + 21 v12 − 12 m2Z ;

(5.12)

m212 = − 12 v1 v2 (g2 + g2 − 41 ) :

(5.13)

In the general two-doublet model, the Higgs boson masses are expressed in terms of the VEV v1 ; v2 and the parameters i ; i = 1; : : : ; 6 in Eqs. (4.7) and (4.9). From Eqs. (5.9) and (5.10), the mass of the neutral pseudoscalar A, and the mass of the charged Higgs bosons H± , are related to the mass of the charged gauge boson W: m2A = m2H± − m2W :

(5.14)

The neutral scalar masses are related to mZ , mA , and tan by 2 2 2 1 mH; h = 2 [mA + mZ ± (m2A + m2Z )2 − 4m2Z m2A cos2 ] :

(5.15)

These relations have crucial implications for the Higgs boson search at LEP: mH± ¿ mW ;

mH ¿ mZ ;

mh 6 mZ |cos 2| 6 mZ :

mA ¿ mh ; (5.16)

The last relation is very important, although radiative corrections increase the upper limit up to about 40 GeV (Section 5.2). This exciting prediction gives a much higher probability for discoveries of the neutral Higgs boson arising in the MSSM than that of the Higgs boson in the MSM. As shown in Section 2.1, the theoretical upper mass bound for the MSM Higgs is much higher than mZ . The Grst and second mass in Eq. (5.16) implies that the search for the charged and the second neutral Higgs bosons at LEP-1 energies is not expected in the framework of the MSSM.

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In addition, Supersymmetry is a promising framework for the Higgs boson search at LEP, since the mixing angle (4.8) and tan (4.6) are functions of the Higgs boson masses based on the relations (5.7) – (5.10): 2 2 mH + m2h mA − m2Z sin 2 = − sin 2 ; cos 2 = − cos 2 : (5.17) mH − m2h mH − m2h Therefore, the expected production cross-section, which is dependent on two arbitrary parameters in the general two-doublet model, can be expressed as a function of the two neutral Higgs boson masses. For the experimental search, it is convenient to choose (mh ; mA ) as free parameters. The third parameter tan is determined by Eq. (5.15). If Supersymmetry is the deeper underlying structure of the Standard Model, then, at the limit where Supersymmetry e:ects decouple, the Higgs structure of the MSM has to be recovered. By removing the pseudoscalar Higgs boson (mA → ∞ for arbitrary Gxed tan ), Eq. (5.15) requires mH → ∞, then Eq. (5.17) implies sin 2 = − sin 2 and cos 2 = − cos 2. Consequently, the hZZ coupling suppression factor sin( − ) → 1 and the h production in the Supersymmetric model becomes indistinguishable from the Higgs boson production in the MSM. The H± , H and A bosons are removed simultaneously and the Standard Model scenario is recovered. In this particular case, the important remnant of the MSSM is the mass relation mh 6 mZ . This relation will be modiGed when radiative corrections are included. 5.1.4. Implications for the Higgs boson search at LEP This section outlines the implication of these theoretical considerations for the searches for Higgs bosons at LEP in the framework of the MSSM at the tree-level. The MSSM is a special case of the general two-doublet model discussed before. Owing to the mass relations in (5.16), only the search for the neutral scalar h and the neutral pseudoscalar A at the LEP-1 energy is possible. The lightest neutral scalar h can be produced via the Higgsstrahlung process Z → hZ?

(5.18)

or via the Higgs pair-production process Z → hA :

(5.19)

The production rates of these processes are complementary and they are only a function of the neutral Higgs boson masses (mh ; mA ). The partial width of the Z decay via process (5.18) is proportional to sin2 ( − ) and in process (5.19) it is proportional to cos2 ( − ). This term is given by cos2 ( − ) =

m2h (m2Z − m2h ) : m2A (m2Z + m2A − 2m2h )

(5.20)

The pair production is dominant when mh ≈ mA , i.e. tan 1. Several thousands of pairproduced Higgs bosons are expected per one million collected hadronic Z events within the kinematically accessible parameter space (mh ; mA ), where mh + mA ¡ mZ . Fig. 5.3 shows the regions of the (mh ; mA ) plane where the number of pair-produced neutral Higgs boson events is larger than 250, 1250 and 12 500. These high event rates would lead to a clear signal.

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Fig. 5.3. Expected Z → hA events in tree-level calculation in the MSSM per one million hadronic Z decays in the (mh ; mA ) parameter space. In region (a) more than 250, in region (b) more than 1250, and in region (c) more than 12 500 Higgs boson events are expected. Table 5.2 Branching fractions into fermions (in %) of the neutral Higgs bosons h and A for mh = mA = 42 GeV at the tree-level calculation. The h decay products are listed on the horizontal axis and the A decay products on the vertical axis. All other modes have negligible branching fractions h\A

+ −

ssU

bbU

+ − ssU bbU

0 0 4

0 0 1

4 1 89

The MSSM also predicts the branching ratios of the various allowed Higgs boson decay channels as a function of (mh ; mA ). The Higgs boson decays into up-type fermions (up-type quarks and charged leptons) are suppressed in the region of dominant Higgs boson pair production, where tan 1 (Table 4.1). Quantitatively, an example of the relative Higgs boson decay branching ratios is given in Table 5.2. The e:ects of possible Higgs boson decays into Supersymmetric counterparts of Standard Model particles are expected to be small, since low-mass Supersymmetric particles are excluded. The implication of recent studies on radiative corrections in the MSSM alters the excluded mass regions which were derived from tree-level calculations. In order to interpret the experimental results, reviewed in Appendix A, in the framework of the MSSM, a detailed study of the e:ects of radiative corrections is crucial. First interpretations by the LEP experiments until 1991 did not take radiative corrections into account. 5.2. Basic radiative corrections and mass limits The importance of radiative corrections for the Higgs boson mass spectrum in the MSSM was pointed out by Li and Sher in 1984 [152] and then by several other authors [153–155].

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Owing to these corrections, the neutral Higgs boson masses can increase by several tens of GeV and the Higgs couplings can change substantially. In particular, the mass relations in (5.16) are no longer valid. Early studies of radiative corrections in the MSSM did not take into account the possibility that the top-quark mass and its supersymmetric partner, the stop quark, could be very massive [152,156 –159]. Much theoretical progress has been made since [160 –164]. The consequences for the experimental Higgs search have also been worked out [161,165,166]. Theoretical research has improved the precision of the predictions by including the contribution of additional particles to the one-loop graphs. Full one-loop diagrammatic calculations were performed [167–172] and interpretations of data given [173]. Later, further progress was made in giving good approximate formulae [174,175]. In 1998, two-loop diagrammatic calculations for the leading scalar quark sector became available [176]. This section outlines the implications of radiative corrections for the Higgs boson search within the framework of the MSSM. First, a simple case is reviewed to explain the e:ects of radiative corrections on the Higgs boson production rates. Details of a comprehensive study with full one-loop corrections are then given. 5.2.1. Primary assumptions The main implications of the MSSM for the Higgs search can be extracted by making the following assumptions [161,165,166]. When top and stop quarks have large masses, they contribute most to the corrections of the tree-level calculations. The radiative corrections to the Higgs potential include terms proportional to mt ; m2t and m4t ; only one term depends on m4t . Owing to the large top mass, this term gives the leading e:ect if the scalar top mass is large. Since only the m4t term a:ects the neutral Higgs sector, the e:ects of radiative corrections to the charged Higgs sector are small in this approximation. It is further assumed that the value of tan is not too large (tan 6 mt =mb ≈ 50); otherwise, bottom and sbottom couplings to the Higgs are large and the bottom–sbottom loops can no longer be neglected. In addition, it is assumed that all Supersymmetric partners are degenerate in mass and do not mix. Their common mass mq˜ is related to the Supersymmetry-breaking mass scale m, by mq˜2 = m2t + m2 . In Section 5.3 all of these assumptions are relaxed. 5.2.2. Higgs boson production and decay Under the above assumptions, the e:ects of radiative corrections to tree-level calculations can be simply summarized with a single parameter for a given mt and mt˜ when no mixing in the scalar top sector is assumed: m2t˜ 3W m4t 3≡ ln ; (5.21) 2# m2W m2Z m2t where W = EM = sin2 W . Radiative corrections alter the mass relations and the mass mixing angle. The correction 3= sin is added to the squared mass mixing matrix for the neutral scalars. The diagonalization of the radiative corrected mass matrix gives the physical Higgs boson masses:

1 2 3 2 2 mH; h ; = ±F ; m + mZ 1 + (5.22) 2 A sin

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where

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 2 2 F =  mA + mZ 1 +

3 sin

2 2

− 4m2A m2Z cos2 2

1=2 3 3 −4 m2 m2 sin − 4 m4 cos2  : sin A Z sin Z

(5.23)

Eq. (4.9) is recovered when the radiative corrections are set to zero (3 = 0). The weak mixing angle and tan are related by −(m2A + m2Z ) sin 2 sin 2 = : (5.24) F This relation is analogous to the tree-level relation (5.17) where (mh ; mA ) are chosen as free parameters. The ( − ) value has changed after radiative corrections, and so has the expected number of pair-produced Higgs boson events compared to the tree-level prediction. 5.2.3. Top and stop quark masses The mass of the top quark and of its supersymmetric partner, the stop boson, determine the size of the radiative corrections. The stop mass is unknown, only an experimentally lower mass limit exists which depends on the left–right stop mixing angle. The natural upper mass limit is the upper energy scale of low-energy Supersymmetry which is around 1 TeV. The top mass was determined Grst from precision Z-lineshape measurements [22] and discovered at CDF and D0 [19 –21]. The current top mass is 174:3 ± 5:1 GeV [24]. In order to illustrate the e:ects of radiative corrections on the expected Higgs signal, the following top and stop masses were chosen: mt = 175 GeV;

mt ¡ mt˜ ¡ 1000 GeV :

(5.25)

In particular, the assumption mt ¡ mt˜ has to be relaxed in a general analysis. This mass range corresponds to an 3 range of 0 ¡ 3 ¡ 1. 5.2.4. Implications for the Higgs boson search at LEP Limits obtained in the general two-doublet model can be reinterpreted in the MSSM as excluded regions in the mass parameter space. The theoretically preferred parametrization (mA ; tan ) of the available phase space is transformed into the experimentally relevant (mh ; mA ) plane. Without radiative corrections, there is a one-to-one correspondence. However, with radiative corrections, one or two (mh ; mA ) pairs can correspond to a single (mA ; tan ) pair, when tan is constrained to the range 1:0 ¡ tan 6 50. This ambiguity exists only in a small (mh ; mA ) region. For the prediction of the number of expected events, the tan value which corresponds to the smaller Higgs production cross section is chosen, for the most conservative calculation. The e:ect of radiative corrections in the (mh ; mA ) plane is shown in Fig. 5.4. The case 3 = 0 corresponds to the tree-level (Fig. 5.3). The region where more than 250 Z → hA events per one million hadronic Z decays are expected shifts to the region below the diagonal for larger values of 3 when radiative corrections become large. In this case, the situation is analogous to

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Fig. 5.4. Expected Z → hA events per one million hadronic Z decays in the (mh ; mA ) parameter space including radiative corrections in the MSSM. The regions where more than 250 events are expected are shown with radiative corrections parametrized by 3 for (a) 3 = 0:00, (b) 3 = 0:01, (c) 3 = 0:10 and (d) 3 = 1:00. Fig. 5.5. Comparison of excluded (mh ; mA ) regions in the MSSM with the assumptions given in the text. The dark region is excluded, the hatched region allowed, and the light region not allowed by the theory.

the tree-level prediction with mh and mA exchanged. Thus, all the parameter space is relevant for the experimental search in the MSSM. The general unitarity constraint, Eq. (4.1), requires that bremsstrahlung Higgs boson production and Higgs boson pair-production be complementary. Consequently, even in the most involved radiative correction scenario, the Higgs cannot escape detection if it is kinematically accessible and all (mh ; mA ) mass combinations are included in the search for the Higgs bosons. 5.2.5. Neutral Higgs boson mass limits The LEP experiments have interpreted their results mainly as a function of top and stop masses only. Fig. 5.5 (from Refs. [70,71,93,123,124]) shows the MSSM results of the four LEP experiments for independent variations over top and stop masses (except DELPHI, which has Gxed top and stop masses). A more general analysis with a larger theoretical precision [173] has revealed a new unexcluded mass region as reviewed in the next section. 5.3. Full one-loop radiative corrections and mass limits At the tree-level, the Higgs sector of the MSSM can be e:ectively parametrized in terms of two free variables, for example, the masses of two Higgs bosons. Then, as reviewed before, the stop mass dependence is important. After including one-loop corrections which involve all

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Supersymmetric particles, Higgs boson masses and couplings depend on additional unknown parameters of the MSSM. These unknown parameters are referred to as SUSY parameters. Several approaches have been developed to compute radiative corrections to the tree-level approximation: • the E:ective Potential Approach (EPA) [161,165,166], • the Renormalization Group Equations (RGE) approach [177–179], and • the Full one-loop Diagrammatic Calculations (FDC) in the on-shell renormalization scheme

[167–172]. 11

The interpretations in the MSSM of the searches for Higgs bosons at the Z resonance, which are performed by the LEP experiments [70,71,93,123,124], were based on common assumptions. 1. Radiative corrections to the MSSM Higgs boson masses, production and decay rates are considered in the EPA approximation. Only the leading part of the contribution arising from top and stop quarks is considered, as given in Eq. (5.21). 2. In the most simpliGed version of the EPA, Eq. (5.21), only one free SUSY parameter mt˜ is taken into account. 3. In the EPA interpretations, mt˜ ¿ mt is assumed. 5.3.1. Parameter space of the MSSM Experimental searches should rely as little as possible on theoretical assumptions. Hence, a less constrained parameter space compared to previous studies is considered. One can explore the dependence of the masses and couplings of the Higgs bosons on all soft-breaking parameters in the MSSM Lagrangian. Some relations and simpliGcations are applied in order to decrease the number of free parameters, after checking that the results in the Higgs sector are insensitive to those assumptions. The results presented here are based on the measurements from the L3 collaboration. The data set corresponds to about 408 000 hadronic Z decays [70,71,180]. The experimental results from this data set were taken as input for the detailed study of excluded mass regions. The most general version of the MSSM Lagrangian contains a large number of free parameters. Most of the SUSY parameters have a small impact on the Higgs sector. Numerical simulations were used to identify important parameters for the Higgs boson phenomenology [173]. These parameters have been varied independently: • (mh ; mA ) or (mH ; mA )—the investigated Higgs boson mass combinations. • msq —the common mass parameter for all squarks. The assumption of the same mass param-

eters for the three squark generations has a small e:ect. Results depend mostly on the stop mass parameter and only minimally on the masses of other sfermions. • mg —the gaugino mass. The commonly used GUT relation for the SU(2) and U(1) gaugino masses were assumed: mU(1) = 53 tan2 W mg . This assumption also has little impact on the results. • —the mixing parameter of the Higgs doublets in the superpotential. 11

Recently, two-loop corrections were calculated [176].

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Table 5.3 Ranges of SUSY parameters used for independent variation in the study of the MSSM neutral Higgs boson searches Parameter

msq (GeV)

mg (GeV)

(GeV)

A

Range

200 –1000

200 –1000

−500 to +500

−1 to +1

• A—the mixing parameter in the sfermion sector. As for msq only one universal mixing parameter was considered for all squark generations. The mixing term is deGned as Amsq − = tan .

For this analysis the top-quark mass is Gxed to mt = 175 GeV. In order to study the e:ect of the variation of the SUSY parameters described above, they were scanned in the ranges given in Table 5.3. Possible sources of uncertainties are connected with the increase of the range of the variation over the MSSM parameter space. The change of the limits shown in Table 5.3 has only a small e:ect on the results discussed in the next sections. A decrease of the lower limit for the sfermion and gaugino masses causes in most cases at least one SUSY particle to be light and observable. Such parameter combinations are rejected. An increase of the upper limits of the sfermion and gaugino masses inQuences the upper limit on mh . This e:ect is rather weak as it depends only logarithmically on the squark masses. Increasing the mixing parameter A can be more signiGcant. 12 Such possibilities are, however, unlikely both from the theoretical and experimental point of view. They are diVcult to generate in Grand UniGed Theories (GUT) and can have signiGcant e:ects in many low-energy processes. Large A values lead to a large splitting of the sfermion masses and usually to the existence of a light or negative-mass sfermion. Non-diagonal soft-breaking couplings are the source of sizeable Qavor-changing neutral currents (and, if complex, CP breaking e:ects [184]), which are ruled out by the data. Therefore, the choice of the bounds shown in Table 5.3 is well motivated and results are found to be stable against small variations. The parameters shown in Table 5.3 are the input parameters for the calculations of the physical sfermion, chargino, and neutralino masses. Some parameter combinations can be unphysical (e.g. negative squark masses) or experimentally excluded. Such cases are removed by imposing the following constraints: stop and chargino are required to be heavier than mZ =2, and the neutralino to be heavy (or weakly coupled) in agreement with the bound on contributions to the Z width beyond the MSM: Y!Zmax ¡ 31 MeV [112]. An additional constraint is applied to tan , deGned as the ratio of the vacuum expectation values of the Higgs doublets. Although at the one-loop level tan dependents on the renormalization scheme, this dependence is rather weak [167–170]. Tree-level experimental bounds on tan are assumed to hold approximately, and its value is constrained to 0:5 6 tan 6 50. The lower bound is based on Ref. [185]. A variation of the upper bound has no signiGcant e:ect on the results. In this approach tan is a function of (mh ; mA ) and the SUSY parameters listed in Table 5.3. 12

Now, −2 ¡ A ¡ 2 is considered [181–183].

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Fig. 5.6. Upper h mass bound for various values of SUSY parameters. For the same set of heavy SUSY parameters, given in Table 5.4, the FDC and the EPA are compared; the maximum FDC value is about 5 GeV lower. For the FDC the change of the maximum Higgs boson mass is shown when only the squark mass parameter msq is reduced to 200 GeV, only the gaugino parameter mg reduced to 200 GeV, and only the scalar top mixing parameter A is set to 1. The theoretical upper bound is largely reduced for light squarks, increased by 2 GeV for light gauginos, and increased by 8 GeV for large mixing. Table 5.4 Fixed heavy SUSY parameters used for comparison with the EPA results Parameter

msq (GeV)

mg (GeV)

(GeV)

A

Value

1000

1000

100

0

5.3.2. Uncertainties in the predictions Several possible sources of uncertainties exist in the interpretation of the experimental results in the framework of the MSSM. Two of them have been addressed by taking into account a larger set of the MSSM parameters, and by using FDC to calculate the one-loop radiative corrections. For Gxed values of mA and tan and for an equivalent set of SUSY parameters, the FDC gives typically mh values reduced by 5 –7 GeV in comparison with EPA. The cross sections e+ e− → Zh; Ah di:er typically by 15 –30%; the di:erence can be much larger for some sets of model parameters. As an example, Fig. 5.6 (from Ref. [186]) illustrates the dependence on the SUSY parameters of the upper limits on mh as a function of mA . The solid line is plotted assuming all SUSY particles to be heavy (msq = mg = 1 TeV) and no mixing in the sfermion sector (A = 0). The values of these parameters are listed in Table 5.4. The dotted line shows the case of light squarks

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(msq = 200 GeV) (the contribution of sleptons is negligible). The top and stop masses have the largest impact on the upper mh bound. However, from an experimental point of view, the dependence on gaugino masses and on mixing in the squark sector can also be very important: for instance changing mg from 1 TeV to 200 GeV shifts mh by 2 GeV. The shift can be positive or negative, depending on the other parameters. A change of A from 0 to 1 shifts mh by 8 GeV. It is noted that the lower bound on tan a:ects the theoretically allowed regions in the (mh ; mA ) plane. The change of tan ¿ 1 to tan ¿ 0:5 extends the theoretically allowed region from about mh = 60–80 GeV for small mA . For comparison of the EPA and the FDC, the dashed line shows the upper bound on mh obtained in the EPA for the same set of SUSY parameters (Table 5.4) as for the solid lines. The EPA upper bound is higher by about 5 GeV. This di:erence is particularly important in the study of the kinematically accessible mass region at LEP-2. Two-loop calculations reduce the upper bounds; in this sense the resulting limits are conservative. Some estimates were given of the two-loop corrections for the MSSM Higgs boson masses [187–189]. A detailed study [190] shows that, with the correct deGnition of the physical top-quark mass as the pole of the propagator (consistent with the FDC method), two-loop corrections to the h mass are negative and small. Later, higher order corrections were calculated [176] and are included in the LEP-2 results [181–183]. 5.3.3. Neutral Higgs boson mass limits In order to derive bounds on h and A masses, the limits on the Higgs boson production rates from the L3 collaboration [70,71,180] were used. In the mass plane (mh ; mA ), each point with a step size of 1 GeV up to Higgs boson masses of 120 GeV was analyzed separately. For each mass combination, the production cross sections of the reactions 13 e+ e− → hZ? → hfUf , e+ e− → hA, and the branching ratios for h and A decays were computed as a function of the parameters given in Table 5.3. Then, the number of expected Higgs boson events for each investigated Gnal state and each mass bin was calculated. The following channels were taken into account. • Single h production processes: e+ e− → hZ? → he+ e− , h + − , h. U • hA pair-production processes: U bb U , bb U , e+ e− → hA → + − + − , + − bb + − U U U e e → hA → AAA → bbbbbb. (for mh ¿ 2mA )

In addition, the combined LEP-1 limit on non-standard Z decays Y!Zmax ¡ 31 MeV was applied [112].

The possibility of Higgs boson production via bremsstrahlung o: a b-quark e+ e− → bbU → bbU h is not discussed. This channel could be signiGcant for large values of tan [191] (Section 4.7). 13

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√

Fig. 5.7. Excluded mass regions in the (mh ; mA ) plane for s = mZ from di:erent search channels: Z lineshape (thick-solid), Z → Z? h (thin-solid), + − bbU (dotted), + − + − (dotted–dashed), bbU bbU (thin-dashed), 6b (thick-dashed). √

Fig. 5.8. Excluded regions at s = mZ in the (mh ; mA ) plane. Dotted line: EPA (epsilon approximation). Thin solid line: FDC result for heavy SUSY parameters. Thick solid line: FDC results with a full scan over the SUSY parameters. Very thick line: new unexcluded region in FDC.

A given (mh ; mA ) combination is excluded if, for all SUSY parameter sets (from the ranges deGned in Table 5.3 and for Gxed mt = 175 GeV), the expected number of events in at least one of the channels is excluded at 95% CL. Fig. 5.7 shows regions in the (mh ; mA ) plane which are excluded by the individual channels listed above. A comparison of the excluded regions of Fig. 5.7 with the combined excluded region of Fig. 5.8 shows that the sum of the partial exclusion regions is smaller than the combined one. This is due to the scanning over the SUSY parameters. For Gxed (mh ; mA ), one can Gnd the parameter combinations for which the cross section for a given channel is particularly low. It is unlikely that the cross sections are very low in all channels simultaneously owing to the well-known complementarity of the cross sections of e+ e− → hZ and e+ e− → hA reactions. This complementarity holds approximately even after the inclusion of non-leading vertex corrections. Figs. 5.8 and 5.9 show the regions in the (mh ; mA ) and (mA ; tan ) planes that were excluded by the simultaneous analysis of all channels. Three regions are distinguished. (i) Excluded regions after performing a full scan over the SUSY parameter space and using the FDC method in cross section and branching ratio calculations.

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√

Fig. 5.9. Excluded regions at s = mZ in the (mA ; tan ) plane. Dotted line: EPA (epsilon approximation). Thin solid line: FDC result for heavy SUSY parameters. Thick solid line: FDC results with a full scan over the SUSY parameters.

(ii) As above, but varying only msq and assuming that the other SUSY parameters are constrained to the values shown in Table 5.4. This allows comparison with the EPA approximation for Gxed msq . (iii) Excluded regions with radiative corrections calculated in the simpliGed EPA [70,71] (epsilon approximation), Eq. (5.21), where only the leading corrections from the top and stop loops are taken into account. In this case results depend on msq only. The range 175 GeV 6 msq 6 1000 GeV was used. Fig. 5.8 reveals an interesting result for the excluded regions in the (mh ; mA ) plane. The full scan over the SUSY parameter space (thick solid line) gives, in comparison with the epsilon approximation (iii) (dotted line), a substantial additional triangle-shaped unexcluded mass range for 45 GeV ¡ mA ¡ 80 GeV and 25 GeV ¡ mh ¡ 50 GeV, which is marked with a bold solid line. The existence of this region can be understood in the following way: In the range mh + mA ¡ mZ the reaction e+ e− → hA is allowed kinematically and both main discovery channels, e+ e− → hZ? and e+ e− → hA, contribute. If radiative corrections reduce the cross section of one of them below the experimental sensitivity, the complementary cross section (Eqs. (4.12) and (4.13)) will be large enough to exclude this mass combination. The unexcluded triangle begins just above the mh + mA = mZ limit. In this range the bremsstrahlung cross section e+ e− → hZ? can be small for some SUSY parameters. We identify points where it is suppressed by a factor of 25 compared with the MSM prediction for e+ e− → HMSM Z? . The complementary process is already forbidden kinematically. An example for an unexcluded mass point is given in Table 5.5. The cross sections for Higgs boson bremsstrahlung and pair-production obtained in FDC are listed for a chosen set of

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Table 5.5 Example of an unexcluded mass point. Cross sections for Higgs boson bremsstrahlung and pair-production for a chosen set of SUSY parameters. All mass parameters are given in GeV, and cross sections in pb mh

mA

mt

msq

mg

A

tan

mt˜1

mt˜2

1hA

1hU

32

64

175

200

200

−500

−1

3.2

81

362

0.0

0.25

SUSY parameters. The simple version of the EPA used by the LEP experiments does not allow √ mt1˜ mt2˜ ¡ mt . Therefore, no corresponding cross section exists for the given FDC example. Low unexcluded mh values are obtained for low physical stop masses of the order of 50 –200 GeV and large mixing in the sfermion sector (A = ± 1, large ). In such cases the splitting between the left and right stop masses is large. The other SUSY parameters have smaller inQuence on the shape of the unexcluded region. With increasing mA , the cross section for the e+ e− → hZ? reaction becomes less sensitive to the SUSY parameters and similar to the e+ e− → HMSM Z? cross section (calculated at mh = mHMSM ) because of the known decoupling e:ect [179,192]. The di:erence between cross sections calculated in the MSSM and MSM decreases as 1=m4A . Above mA ≈ 100 GeV, the bremsstrahlung production of h is suVcient to establish, independent of the SUSY parameters, a Higgs boson mass bound of about 55 GeV. Fig. 5.8 shows that the regions obtained in approaches (ii) (thin solid line) and (iii) (dotted line) are similar. The excluded area in (ii) is only slightly larger than in (iii). The few GeV distance between the lines reQects the di:erence in the mh values calculated in the EPA and the FDC. This shows that the EPA result can be approximately recovered for a speciGc set of SUSY parameters given in Table 5.4. Fig. 5.9 shows the results in the (mA ; tan ) plane. No signiGcant di:erences in the shape of the excluded regions obtained in the EPA (dotted line) and the full scan FDC for heavy SUSY parameters (thick solid line) is visible. The newly unexcluded region is located between the thick and thin solid lines. However, the (mh ; mA ) variables are more useful and natural from the experimental point of view and in this parametrization the signiGcant di:erence between the standard LEP-1 analyses and the scan is quantitatively more visible. In summary, experimental results, based on the data given in Refs. [70,71], give mh ¿ 45 GeV and mA ¿ 45 GeV for almost the entire Supersymmetric parameter space at 95% CL. Important exceptions exist for Higgs boson masses as low as 25 GeV for some combinations of SUSY parameters. The unexcluded mass region arises because of the parameter scan, not the di:erent calculations of EPA and FDC. 5.3.4. Beyond the two-doublet Higgs model Additional Higgs singlets can mix with the two Higgs doublets. Extensions of the MSSM result in Non-Minimal Supersymmetric Standard Models [193,194] (NMSSM). In these models, the mass relations between the Higgs bosons of the MSSM are weaker. The particle spectrum is enlarged while cross sections for individual processes are reduced, owing to a generalized sum rule. The signatures are more complex but have many similarities with the MSSM scenario. Including one-loop corrections, the mass of the lightest scalar Higgs boson is restrained to about 156 GeV [194]. This prediction is some 20 GeV larger than with the MSSM.

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A larger, even number of doublets in the Supersymmetric extension of the Standard Model results in a wider particle spectrum, less predictive power and similar general features. Therefore, the simple MSSM with only two Higgs doublets is the usually considered extension of the Standard Model. Models with Higgs boson triplets, which respect the = 1 constraint, are a further possible extension of the MSM. An experimental search for signals of Higgs boson triplets is diVcult because of the large number of free parameters involved in the prediction of production and decay rates. Theoretical investigations to explain a large top mass have led to speculations that a ttU condensate can substitute the Higgs boson [195]. At present, the expected signatures of the ttU condensate are experimentally indistinguishable from the Standard Model Higgs signatures. In composite models, excited leptons and quarks are expected. The search by the four LEP experiments [149] has excluded the existence of these excited states up to the kinematically accessible threshold. In addition, the precision measurements of the Z parameters disfavor composite models [196]. Recently, the case of models with a large number of Higgs boson singlets and doublets has been considered [197]. A ‘no-lose theorem’ is derived, i.e. even in the most complex case, a Higgs boson signature will be found. 6. Conclusions The sensitivity for the Higgs boson searches at LEP-1 has exceeded expectation, although no signal has been observed. The pre-LEP expectations for the sensitivity of the MSM Higgs boson at LEP-1 was about 30 GeV [32], while a Higgs mass larger than 65:6 GeV was excluded at 95% CL from the combined data of the four LEP experiments. The latest results from Higgs boson searches at LEP-2 are given in Ref. [23]. A variety of search channels with very di:erent event topologies have been analyzed in models beyond the MSM. The di:erent models described in this review, which are organized according to their Higgs boson scenarios, have been compared with the data. In the two-doublet Higgs model, searches for neutral and charged Higgs bosons have led to various limits on their production rates. Charged Higgs bosons are excluded independently of the decay mode up to the kinematic reach of LEP-1 of about 45 GeV. For the neutral Higgs bosons, mass limits and limits on sin2 ( − ) and cos2 ( − ) are obtained. In the framework of the MSSM, interpretations are given including the e:ects of radiative corrections. Mass limits were obtained by all LEP experiments close to the kinematic reach for a Gxed MSSM benchmark parameter set. It has been shown that a general scan over the MSSM parameters reduces signiGcantly the benchmark mass limits, since for some parameter combinations both hZ and hA cross sections can be small. Many search techniques have been developed over the 6 years of LEP-1 operation. This knowledge was applied and extended for the analysis of LEP-2 data at center-of-mass energies up to about 209 GeV. In particular the b-quark tagging, the search in the four-quark channel (no LEP-1 sensitivity), and statistical methods for separating signal and background have been improved. Most search channels, event shape variables, methods to identify leptons with high

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precision and the addressed theoretical models will continue to be important for the Higgs boson searches at future colliders like the Tevatron, the Large Hadron Collider (LHC) or a TeV electron–positron collider. Acknowledgements I would like to thank my friends, colleagues, fellow Higgs hunters, in particular from the LEP Higgs working group and from the theory for many fruitful discussions. I also wish to express my gratitude to the readers of the draft for their comments: Wim de Boer and Giovanni Crosetti from DELPHI; Jean-Paul Martin and Hanna Nowak from L3; Ron Settles from ALEPH; Peter Sherwood from OPAL; Wolfgang Hollik and Kurt Riesselmann (theory) and Edith Borie. Appendix A. Details of the search methods This appendix gives details about the Higgs boson and background reactions and the methods to distinguish them. Three classes of searches for Higgs bosons are distinguished: (a) searches for Higgs boson bremsstrahlung, (b) neutral Higgs boson pair-production, (c) charged Higgs boson pair-production, and the production graphs are shown in Fig. 4.1. Examples of the selection variables are given for the Higgs boson bremsstrahlung in the neutrino channel from ALEPH and in the charged lepton channel from L3. For searches beyond the MSM, details of the searches for neutral Higgs boson pair production are discussed from DELPHI and L3, and for the charged Higgs boson pair production from OPAL. A.1. The reaction Z → Z? h In the high-mass Higgs boson range (mH ¿ 15 GeV), most Higgs bosons are expected to have the decay mode H → bbU . The decay of the Z boson determines the search channels for the Z → Z? H reaction. The neutrino (Z → ), U electron (Z → e+ e− ), muon (Z → + − ), and + − tau (Z → ) channels are most important owing to their distinct signatures and relatively large branching fractions. Typical Higgs boson signatures are illustrated in Fig. 2.15. All LEP experiments exploit the same search channels. Owing to the large background in the H+ − channel, this decay channel was analyzed only in the Grst two years of LEP data-taking, up to about 1991. With increasing data statistics, this channel no longer added sensitivity. 14 However, since about 17% of the ’s decay into electrons or muons, this channel also leads in about 5.8% of the + − production to a clean pair of electrons or muons. These decay modes are covered by the dedicated searches of the He+ e− and H + − channels. Already at the beginning of the Higgs analysis at LEP, it was realized that the HqqU channel (not shown in Fig. 2.15) is not feasible for the Higgs boson search owing to the very large irreducible 14

In order to determine the detection eVciency, a statistical method was used based on counting the number of signal and background events.

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hadronic background. This channel was not used at all at LEP-1. However, in the searches at LEP-2, with a much better signal-to-background ratio, the HqqU channel has been exploited and contributes most sensitivity. A.1.1. UqqU channel The searches are performed for a signature of acoplanar jets accompanied by large missing energy. No hadronic activity is allowed to be observed recoiling against the jets. For a heavy Higgs boson its detection is more diVcult, since its signature is more similar to the bbU and qqU

background events. The LEP detectors were simulated in detail in order to describe precisely the data. Simulated background and its relevance for the UqqU channel is given. • The most important background is due to multi-jet events from e+ e− → q qU in which at

least one jet momentum is mismeasured. OPAL [63– 67], for example, simulated 7.5 million

e+ e− → q qU events with JETSET version 7.4 [198,199] and identiGed three reasons why jet

momenta could be mismeasured:

◦ much energy of a heavy quark is taken away by an energetic neutrino, ◦ other production mechanisms of neutrinos are present in the event, ◦ the production of KL0 ’s whose energy is only poorly measured in the hadron calorimeter.

In order to study these events an additional event sample is simulated where these events are enhanced by a preselection on the generator level. This data sample corresponds to 33 million hadronic Z decays. • The reaction e+ e− → + − could lead to acoplanar hadronic jets in the hadronic decay

modes. Missing energy always arises because of the neutrinos in the decay.

• For very light Higgs bosons, much energy is taken away by the neutrinos of the Z decay.

In this case the two-photon reaction becomes an important background. In this reaction with four fermions in the Gnal state, the initial electrons are likely to escape in the beam pipe. • The four-fermion process, in which at least one neutrino is produced, is very similar to that of the signal. On the other hand, the expected production cross section is very small. Details of the event selections in the H channel are given for Gnal LEP-1 results from ALEPH [57,58]. The large number of data events is reduced by a preselection. The same cut variables are Gne-tuned in the Gnal event selection. They use the following event preselection [57].

The visible invariant mass of an event must be less than 70 GeV. At least eight charged particle tracks with |cos tr | ¡ 0:95 must be observed. The tracks must originate near the nominal collision point (20 cm in z and 2 cm coaxial). Events with energy loss around the beam direction are rejected by demanding that the fraction ◦ ◦ of energy beyond 30 must be larger than 60% and the energy within 12 of the beam axis has to be smaller than 3 GeV. • In order to reject most events with back-to-back jets from Z decays, events are divided into two hemispheres by a plane perpendicular to the thrust axis. The angle of the total momenta measured in the two hemispheres deGnes the acollinearity angle. It is required to be ◦ less than 165 . • • • •

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• Events from two-photon collisions are rejected by requiring the visible energy to be larger than 25 GeV if the total momentum transverse to the beam axis is less than 10% of the

center-of-mass energy. • A few remaining Z → + − and + − q qU four-fermion events with very small q qU-invariant mass are rejected by requiring the invariant masses in both hemispheres to be larger than 2:5 GeV. • Events with a hard initial state photon along the beam axis are removed. These events could have a large acollinearity angle and thus be rejected by requiring that the missing momentum direction does not point along the beam axis, i.e. is at an angle to the beam axis of more ◦ than 21:8 . In addition, the angle of the projection of the momentum directions of the two hemispheres onto the plane perpendicular to the beam axis (acoplanarity angle) must be ◦ smaller than 175 .

Three variables are deGned to further reduce the background. The cuts on these variables are set by the optimization procedure. U events with two semileptonic decays. The event is forced into • Background from e+ e− → bbg three jets. For the background the jets are planar and thus the sum S of the three jet-jet ◦ angles is close to 360 . U events in which the energy of one jet is largely taken away • Background from e+ e− → bbg by a neutrino from a semileptonic decay. An isolation angle A is deGned as the largest cone around the total missing momentum direction containing less than 1 GeV. For the background events this angle tends to be small, since the energy deposit of the charged lepton is near the direction of the escaping neutrino. • Background from e+ e− → q qg U events accompanied by a hard initial state photon. The acoplanarity angle *, as deGned before, is applied. The simulated distribution and the data events are shown in Fig. A.1 for S, while all other cuts have been applied. The method of setting the selection cuts is discussed. In the ALEPH analysis [57,58] the location of the most critical selection cuts is determined with an optimization procedure [200]. The value of the 95% CL upper limit on the signal production cross section 195 = N95 (x)=3(x)L

(A.1)

is minimized, where N95 is a function of the cut position x and the expected background, 3 is the simulated signal acceptance. In the absence of any signal ∞ b2 (x) b3 (x) −b(x) N95 = kn Pb (n) = e 3:00 + 4:74b(x) + 6:30 (A.2) + 7:75 + ··· ; 2! 3! n=0

where kn is the 95% CL Poisson upper limit for n observed events, Pb (n) is the Poisson distribution for observing n events with a background of b events. In the special case that no background is expected, the formula reduces to 3.0 events to be observed in the data and none seen to obtain a 95% CL lower limit. ALEPH determines the dependence of the background on the cut position b(x) by smoothing and extrapolating the simulated background distribution of the variable x after all other cuts have been applied. If the cuts are unchanged and the data set increases with larger recorded

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◦

Fig. A.1. ALEPH HU sum of jet angles S for data and background simulation. The indicated cut at 342 is determined by an optimization procedure, leading to zero candidates.

Table A.1 ALEPH HU Gnal selection cuts determined by an optimization procedure and their evolution with increasing data statistics Including data from

1992

1993

1994

1995

106 hadronic Z Cut on S Cut on A Cut on * EVciency (mH = 60 GeV)

1.2 345.0 25.8 164 43.0

1.9 343.8 28.0 162 41.3

3.7 342.5 30.5 160 39.2

4.5 342.0 31.0 159 38.3

luminosity, the number of background events increases. As a result N95 increases. A new minimization of 195 is performed by tightening the cuts. Table A.1 shows the optimized cut values as well as their evolution with increasing data statistics. A.1.2. ‘+ ‘− qqU channel The signature in this channel is a pair of energetic, isolated leptons and a hadronic system. It results from the reaction e+ e− → Z → HZ? (H → hadrons)(Z? → ‘+ ‘− ) where ‘ = e or . In these channels the lepton identiGcation is most important. The dominant background processes are double semileptonic decays Z → bbU → e+ e− X and the four-fermion process e+ e− → e+ e− qqU. Owing to the four-fermion processes shown in Fig. 2.17, more background events are expected for the He+ e− than for the H + − channel.

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Details for the event selection are reviewed from L3 [61,62]. First, a preselection is applied: • The background from e+ e− → e+ e− , + − or + − is removed by requiring at least 16

clusters in the electromagnetic calorimeter.

• The background from e+ e− → q qU is reduced by requiring that the two most energetic clusters have energy larger than 3 GeV, their sum of energies is larger than 15 GeV, and their angle ◦

is larger than 40 .

The He+ e− and H + − channels are analyzed separately. An electron can be well identiGed by its shower shape pattern in the electromagnetic calorimeter. Two matrices of 3 × 3 and 5 × 5 crystals are deGned around the most energetic crystal. An important selection criterion is the ratio of the energy deposited in the inner and outer matrix. • These electron candidates are subjected to an isolation criterion. The hadronic and electro◦

◦

magnetic energy is summed in a cone of 5 and 15 around the BGO cluster direction. Good isolation is achieved by requiring that almost the entire energy is deposited in the inner cone. • The three most energetic electron candidates are selected and all pairs of two jets are considered. The event is accepted if the following conditions are fulGlled: ◦ A pair of electron candidates is found where the most energetic cluster matches in azimuthal angle with exactly one charged track and the second most energetic cluster with at least one track. ◦ The pair of electron candidates must have opposite charges if the less energetic one has an energy lower than 18 GeV. For higher energy electrons the measured charge has a larger uncertainty. The background in the He+ e− search is further reduced by the following Gnal cuts: • In order to address the double semileptonic b-quark decays Z → bbU → e+ e− X, the transverse

momenta of the two electrons with respect to the nearest jet direction are calculated. The sum of the transverse momenta is required to be larger than 10 GeV and the smaller one larger than 1 GeV. • Background from the four-fermion process e+ e− q qU (Fig. 2.17) is reduced by requiring that 2mee + mrecoil be larger than 80 GeV, where mrecoil is the recoiling mass against the electron pair. The e:ect of this cut is shown in Fig. A.2 (from Ref. [61]) for the data sample recorded from 1991 to 1993. Fig. A.3 (from Ref. [62]) shows the agreement between data and simulated background events in the recoil mass spectrum. Two events are observed while 3:0 ± 0:7 are expected from the background. In the H + − search, the signature is a pair of energetic, isolated muons and a hadronic system from the Higgs boson decay. The most important background processes are double semileptonic decays Z → bbU → + − X and the four-fermion process e+ e− → + − qqU. Initially a preselection is applied to select di-muon events which could be produced in association with a Higgs boson decay. • Events with two muon tracks are selected.

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Fig. A.2. L3 H‘+ ‘− missing versus recoil mass distribution for data, and simulated signal events. The indicated cut removes two 1992 candidates.

Fig. A.3. L3 He+ e− Gnal recoil mass spectrum. The two observed background events are consistent with the 3:0 ± 0:7 expected background events from four-fermion processes.

• Background from low multiplicity processes, such as + − e+ e− and + − #+ #− , is

rejected by requiring more than four charged tracks in the central tracking chamber and at least eight clusters in the electromagnetic and hadronic calorimeters.

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Table A.2 L3 H + − Gnal selection cuts. The selection variables are described in the text Variable

Cut

D1; 2 D 1 · D2 min(E1BGO ; E2BGO ) h E1; 2 E1h · E2h

¡ 2:5 GeV ¡ 0:25 GeV2 ¡ 0:2 GeV ¡ 4 GeV ¡ 1 GeV2

• Background from hadronic Z decays and four-fermion processes are reduced by requiring that

the event thrust value be smaller than 0.92.

• In order to reduce this background further, the muon momenta p1; 2 must be larger than 3:4 GeV each and their sum must be larger than 20 GeV.

In addition to the very precise information from the L3 muon detector, the muon pair must fulGll the following cut: • The muon tracks are extrapolated to the vertex region. It is required that their distance of

closest approach to the interaction point is smaller than 3.5 standard deviations along the beam axis and in the plane perpendicular to it.

The Gnal selection in the H + − channel is based on an isolation requirement of the muons. For each muon three selection variables are deGned: • D ≡ Ejet − p , where Ejet is the energy of the jet closest to the muon, and p is the muon

momentum. ◦ • Two cones are considered around the muon direction. Their half-opening angles are 15 and ◦ 3 around the direction of the muon. The electromagnetic energies in these cones E3BGO and ◦ BGO BGO BGO BGO ≡ E15◦ − E3◦ . E15◦ are used to deGne E h − Eh . • Similarly, the hadronic energies in these cones are used to deGne Eh ≡ E15 ◦ ◦ 3

Single- and two-dimensional cuts are applied using the three variables. Their optimal cut positions are summarized in Table A.2. No candidate event passes the selection in the data and the detection eVciency for a 60 GeV signal is 32.3%. A.2. The reaction Z → hA The search for neutral Higgs boson pair-production concentrates mainly on the reactions • • • •

U bU , Z → hA → bbb U bb U bU , Z → hA → AAA → bbb + − U Z → hA → bb, Z → hA → + − + − .

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Fig. A.4. L3 simulation of Higgs boson masses (mh ; mA ). The simulated Higgs boson masses mh = 22 GeV and mA = 52 GeV are well separated with a mass resolution of 3:5 GeV and 5:4 GeV, respectively.

Fig. A.5. L3 mass-,2 for data, qqU and signal simulations. The mass reconstruction of a signal hypothesis reduces strongly the large QCD background.

In the Grst years of the search, the analyses in the hadronic channels are mostly based on kinematic cuts using the event topology. Higgs boson masses were reconstructed from the possible pairings of the jets as shown in Figs. A.4 and A.5 (from Refs. [70,71,135,180]). The b-content was enhanced by the selection of semileptonic b-decays as shown in Fig. A.6

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Fig. A.6. L3 b-jet tagging with semileptonic b-decays. Good b-jet purity is achieved for events with large transverse lepton momentum.

(from Ref. [201]). With increasing statistics the number of background events from QCD processes increased. The Feynman graphs for the reactions leading to the four-jet events are shown in Fig. 4.9. The semileptonic branching fraction into electrons and muons is only about 10% for each. Therefore, most sensitivity for h and A Higgs bosons was reached in the channels involving a decay although Higgs bosons tend to decay into the heaviest kinematically accessible quark pair. The branching fraction of the Higgs bosons into pairs is typically less than 10%. It was therefore most important to exploit the pure hadronic channel further by reducing the large QCD background. Later all experiments were equipped with microvertex detectors. These detectors allow the tagging of b-Qavored jets with secondary vertices using the fact that B mesons are formed. B mesons have a long lifetime (B = 1:5 ps), which gives a larger number of detectable secondary vertices. The search for non-minimal Higgs bosons proGted much from the new techniques of b-quark tagging to reduce the QCD background, since about • 70% of the Z decays are hadronic, • 15% of the Z decays are bbU pairs, and only • 0.02% of the Z decays lead to two bbU pairs.

Mostly, we expect four b-quarks in the Gnal state of the hA production. With ideal b-quark tagging we could reduce the hadronic background by a factor of Gve if we require a bbU pair, and the background could be reduced by a factor of about 3300 if only the four b-quark events are selected. The b-tagging eVciency and purity will be a most important factor in reaching maximal sensitivity. In the following the DELPHI analysis [123] is outlined as an example of the b-tagging U bU . The analysis comprises about one million application in the search for e+ e− → hA → bbb hadronic Z decays recorded during the years 1991 and 1992. Owing to the large expected event rates and the unique signatures one obtains limits close to the kinematic limit. An initial selection

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for hadronic Z decays gives 96:0 ± 0:5% eVciency. A simulated background sample compatible with the number of data events is used. This rather small sample leads to statistical errors similar to those from the data, which has to be taken into account when data and background distributions are compared. Three mass combinations are simulated with the full detector simulation. These mass combinations are for equal and very di:erent h and A masses, and for the case that h decays into a ccU pair. The mass combinations are (40,40), (60,15) and (20,60). In order to obtain a precise determination of the selection eVciencies for the whole mass plane of h and A masses, a simpliGed detector simulation is used which has the advantage that many mass combinations and thus a Gne grid of mass points can be simulated in a reasonable time period. The precision of the simpliGed simulation is checked by comparing the selection eVciencies for the same mass points, using the full and simpliGed detector simulation. A.2.1. b-quark tagging Owing to the longer lifetime of B mesons the underlying idea is the identiGcation and counting of secondary vertices. In order to Gnd secondary vertices, Grst, the primary vertex has to be determined. This vertex position is determined from all charged tracks in the event and uses about 100 events of the same machine Gll. Tracks not matching with a primary vertex are removed and the Gt is repeated until an acceptable , is found. DELPHI proceeds to remove tracks until the probability of all remaining tracks coming from a common vertex is larger than 1%. The precision is about 60 m in the plane perpendicular to the beam pipe. The longitudinal interaction position was determined with a precision of 40 m. This error is dominated by the size of the beam spot and it also takes into account the possible displacements of the beam during the run. Now, the impact parameter of a charged particle can be deGned as the minimal distance between its track in the plane perpendicular to the beam axis and the reconstructed primary vertex. Particular attention has to be given to the sign of the impact parameter. It is taken as positive if the projected track intersects the axis of the closest jet after the point of closest approach; otherwise, it is taken as negative. In the DELPHI analysis the simulated impact parameter resolution is increased by 10% to match the data. Fig. A.7 (from Ref. [123]) shows the impact parameter distributions after the correction is applied. Quality cuts are applied to the tracks, rejecting those which do not have associated hits in two microvertex layers. Tracks with an impact parameter larger than 2 mm are also rejected, since they likely result from KS0 decays or photon conversions. The number of remaining tracks with an impact parameter larger than 2:51 of its measurement error is determined and shown in Fig. A.8 (from Ref. [123]). These tracks are called tracks with lifetime information. The agreement between simulation and data is good. U bU and bbb U bb U bU channels A.2.2. bbb First, the spherical character of these events is used to separate them from the QCD background. This can be either achieved by requiring small thrust value [180] as applied for example in the L3 preselection [70,71], or using Fox–Wolfram moments [202]. DELPHI [123] requires the Fox–Wolfram moments H2 and H4 to be less than 0.6. After the cut on the Fox–Wolfram

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Fig. A.7. DELPHI bbU bbU impact parameter distribution. Data (with error bars) and simulation (histograms) are in good agreement. Fig. A.8. DELPHI bbU bbU tracks with lifetime information. Data (with error bars) and simulation (histograms) are in good agreement.

moments, 9.4% data and 9.2% simulated hadronic background remains. The remaining signal eVciencies are 92% for (40,40) and 77% for (15,60) mass combinations. Subsequently, the JADE algorithm [203] is used to form jets from the energy clusters in the event. DELPHI sets the parameter ycut = 0:01 which corresponds to a minimal invariant mass between the jets of 9:1 GeV. Events with less than four jets are rejected and events with more than four jets are forced to have exactly four jets by increasing the ycut parameter. The remaining background consists mostly of four-jet QCD events. Fig. 4.9 shows the corresponding Feynman diagrams. After this rather simple preselection, b-quark tagging is applied. The method which had already been applied in 1990 in the MARK-II collaboration [204] is used. The underlying idea to identify events with b-quarks is the requirement of a certain number of tracks with secondary vertices as deGned before. This method of b-quark tagging has proved to be rather insensitive to background from wrong association of microvertex hits to the charged track, strange particle decay and photon conversions. The number of these tracks can either be counted in the whole event or per jet. The b-tagging selection cuts are chosen to optimize the sensitivity. The sensitivity is proportional to 3hA = Nbackground , since with the large number of background events Gaussian statistics can be applied. Based on the preselected four-jet sample, two di:erent selections are applied: • The Grst selection (sample 1) focuses on events with at least one bbU pair by requiring more than four tracks with lifetime information in the whole event. This analysis also includes the case in which only one of the Higgs bosons decays into a bbU pair. • In the second selection (sample 2), more speciGc criteria are applied in order to select events with two bbU pairs. The simplest selection would be to require at least two tracks with lifetime information. However, in this case, the selection eVciency would be as low as 20% because of the limited solid angle covered by the microvertex detector. Therefore, the following looser selection is applied: two jets must each have at least two tracks with lifetime information. In addition, the remaining jets must have two tracks with lifetime information. U bU Gnal state with a production rate of about 0.03%, bbc U cU Gnal states Compared to the bbb occur at a much higher rate of about 0.2% in hadronic Z decays. An even larger background

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Table A.3 DELPHI bbU bbU background expectation contained in sample 1 and 2. In both samples most background results from double gluon emission as illustrated in Fig. 4.9 Final state

Sample 1 (in %)

bbU gg bbU ccU bbU bbU No bbU

≈5 ≈5

86

4

Sample 2 (in %) ≈ 69

14 13 4

U rate arises from bbU events with two hard gluons. These bbgg events are in fact the dominant background. Based on the simulation and the available parton information, it is possible to determine the origin of the background remaining in samples (1) and (2) as shown in Table A.3. DELPHI determines that systematic errors are mainly due to the b-tagging eVciency and the four-jet simulation. • In the simulation 10% more events are selected than in the data with more than four tracks

with lifetime information. This discrepancy is caused by a better track–microvertex association eVciency in the simulation than in the data, and also a slightly better impact parameter resolution. A correction factor of 0.9 is consequently applied to the background simulation. In addition, a systematic uncertainty of 5% is assigned. • The four-jet rate after the event preselection is 10% higher in the data than in the simulation. Since this increase is independent of the b-tagging cuts, it should a:ect signal and background simulation in the same way. The event rate is corrected and half of the correction factor, 5%, is taken as systematic error. • The JETSET prediction for the rate of four b-quark Gnal states has a theoretical uncertainty. The four-quark rate in sample (2) is (13 ± 4 ± 5)%, where the Grst error is statistical and the second arises from the uncertainty in the production rate. DELPHI has checked this systematic error by comparing Matrix Element and Parton Shower simulation with JETSET. The di:erences are well within the quoted errors. The resulting systematic errors on the background expectation are ±7 for sample (1) and ±10 for sample (2). The signal eVciencies are determined for selections (1) and (2) using the 2000 simulated signal events for the di:erent mass combinations. Selection (1) gives (8:5 ± 0:5)% U cU signal and only a small variation of the eVciency is found for di:erent eVciency for a bbc mass combinations in the range deGned by mh ≈ 40–60 GeV and mA ≈ 20 GeV. The number of data events passing all cuts is 1899. This is consistent with the number of expected background events of 1956 ± 38 ± 140. U bU signal if both Higgs boson masses Selection (2) gives an eVciency of (7 ± 1)% for a bbb are above 25 GeV and (8 ± 1)% if both masses are above 35 GeV. The selection eVciency drops to (5 ± 0:5)% if one of the Higgs boson masses is 15 GeV. The number of data events is 105, which is consistent with the expectation of 97 ± 9 ± 10 background events.

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◦

Fig. A.9. Illustration of the -isolation angle, . The inner cone has a half-angle of 10 around a charged track, while the outer cone has the half-opening angle as deGned in the text.

A.2.3. -lepton tagging The main topic of this analysis is the isolation and identiGcation of -leptons in events where the hadronic activity from bbU decays is also present. Particular attention has been given to the scenario of a light + − pair recoiling against a heavy bbU pair which has much hadronic activity. In this case -isolation becomes diVcult, as the angle between the + and the − is quite small. Furthermore, bbU fragmentation products may penetrate opposite hemispheres, thus + − and bbU hemispheres can no longer be distinguished. Details from L3 [70,71] are given for the search channels involving -leptons. Background mainly arises from Z → + − and hadronic Z → qqU decays. In order to distinguish signal and ◦ background events, an inner cone with opening angle 10 around a -candidate and an outer cone with isolation angle is considered. Tau candidates are deGned as charged tracks which are ◦ associated to an electron or a muon or satisfy all of the following requirements: 1 6 Ntr10 6 3, ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ 1 6 Ncl10 6 8, E 10 ¿ 1 GeV; E 10 −30 ¡ E 10 ; Ncl10 −30 6 5, where Ntr10 ; Ncl10 and E 10 denote the number of charged tracks, the number of calorimetric clusters and the calorimetric energy ◦ within a cone of half-angle 10 around a charged track. The isolation angle is deGned as the opening angle of the outer cone for which the ◦ energy deposition between inner and outer cone is 10% of the energy in the inner cone E 10 . The isolation angle provides a good basis for a Qexible -isolation cut. Fig. A.9 illustrates the deGnition. This selection variable can be tuned depending on the expected -isolation of the signal signature. A.2.4. Invariant mass of -pairs The reconstruction of the invariant mass of the -pair is based on the fact that, in spite of the presence of (one or two) neutrinos among the -decay products, the momentum of

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Fig. A.10. Illustration of the reconstruction of the -momenta p1; 2 = pj1; 2 + p1; 2 using the direction of the -jets pj1; 2 and the missing momentum vector P.

each of the two ’s can be fully reconstructed using the measured energy and the direction of the visible -decay products and the missing momentum vector. If the masses of the particles from the -decay are small compared to their momenta, the direction of the ’s is, to a good approximation, the same as that of its observable decay products (either a -jet or an electron or a muon). If the two ’s are not back to back (in which case the reconstruction of the individual momenta of the two ’s is not possible), the missing momentum vector can be used to determine the momentum carried by the neutrinos from the decay of each of the two ’s, using the following relations: P cos j1 P = p1 + p2 cos j1 j2 ; P sin j1 P = p2 sin j1 j2 ;

(A.3)

where P is the magnitude of the measured missing momentum vector P, j1 P the angle between the missing momentum vector and one of the two -jets and j1 j2 the angle between the two -jets (Fig. A.10). The -momenta p1; 2 = pj1; 2 + p1; 2 can be reconstructed using the direction of the -jets pj1; 2 and P. The magnitudes p1 and p2 of the momenta carried by the neutrinos from the decay of the two ’s can be calculated from Eq. (A.3), provided the two -jets are not collinear (i.e. ◦ if j1 j2 = 180 ). The -pair invariant mass is given by m = [2p1 p2 (1 − cos j1 j2 )]1=2

(A.4)

in terms of the reconstructed -momenta p1; 2 = pj1; 2 + p1; 2 : Assuming that the two ’s come from the decay of the h, the h mass resolution can be improved by constraining the energy of the -pair E = p1 + p2 to the energy of the h from Z → hA decay m2A − m2h 1 Eh = mZ 1 − : (A.5) 2 m2Z

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Fig. A.11. L3 + − bbU distribution of reconstructed invariant + − masses with mh = mA = 22 GeV. The indicated cuts are placed around the signal region. The data are in good agreement with the background simulation.

Fig. A.12. L3 + − bbU acollinearity angle. The indicated cut separates well the simulated signal and background. No indication of a signal is visible in the data.

For any (mh ; mA ) mass pair, the measured momenta of the two ’s multiplied by a factor Eh =E are used to calculate the invariant mass of the -pair. The resulting signal mass distribution is compared to the data and the background prediction in Fig. A.11 (from Ref. [180]). A.2.5. + − bbU and + − + − channels In the search channel involving a -pair recoiling to a jet system, the invariant mass of the -pair can be reconstructed using kinematic constraints. Fig. A.12 (from Ref. [70,71]) shows the acollinearity for a simulated + − bbU Higgs signal in comparison with data and background simulation. The + − + − signature has been searched for and no signal has been observed. The most important background originates from Z → + − events. This background can be largely

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Fig. A.13. L3 + − + − track multiplicity. In the event hemisphere with the lower number of tracks, typically two tracks are expected from the signal while from Z → + − background mostly a single track is expected. The data agree well with the background.

suppressed by requiring exactly two tracks in one hemisphere as expected from one-prong h → + − decays, shown in Fig. A.13 (from Ref. [70,71]). No indication of a signal has been observed. A.3. The reaction Z → H+ H− The searches for the signatures of the charged Higgs bosons are performed by all LEP experiments in the three dominant channels csUcs U ; cs and + − . The theoretical motivation was given in Section 4. Compared to the neutral Higgs boson searches, the kinematic constraint mH+ = mH− is used to increase the sensitivity. The harder kinematic constraint gives a mass resolution better than 1 GeV for a charged Higgs boson. Also in comparison with the neutral Higgs boson pair-production, it should be noted that the expected irreducible background in the U bU channel, since no b-tagging can be purely hadronic channel is much larger than for the bbb applied. Details of the search for charged Higgs bosons are given from the OPAL analysis [126]. The analysis is based on data collected between 1989 and 1994. The total integrated luminosity is 110 pb−1 , corresponding to about three million hadronic Z decays. About 80% of the data is taken at the peak and about 10% each at ±2 GeV. Owing to the large production rate, the data taken above the central value of the Z peak have contributed signiGcantly to the mass reach. A.3.1. csUcs U channel In this four-jet channel the main background results from higher order QCD reactions Z → q qgg U and qqq U qU as illustrated in Fig. 4.9. As a preselection only events with Evis ¿40 GeV are considered. In addition, the event has to be longitudinally balanced, | (Ei cos i )|= Ei ¡ 0:65,

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Fig. A.14. OPAL csUcs U number of jets for data, simulated background and signal. Four jets are expected for the signal, while QCD background gives mostly two jets. The data agree with the background. U sphericity distribution for data, simulated background and signal. The sphericity of the signal Fig. A.15. OPAL csUcs events is large compared to the background and the observed data.

where Ei and i are the energy and polar angles of a cluster i. These cuts reject two-photon and Z → qqU events. In the next step, jets are formed. OPAL uses the Durham algorithm [205] with the YCLUS jet Gnder [206]. Recall √that the ycut parameter is deGned such that the minimal invariant mass between the jets is ycut s. The jet resolution parameter ycut is set to 0.01. Events with at least four jets are selected. For the events with more than four jets, the jet Gnder is reapplied with a larger ycut value until exactly four jets are constructed. The event sphericity is calculated and is required to be larger than 0.3. Figs. A.14 and A.15 show, respectively, the number of jets and the sphericity distribution for data, 3.5 million simulated Z → hadrons events and a 42 GeV simulated H+ H− signal. These cuts reduce the data sample by a factor of about 20 compared to its original size. The detection eVciencies vary between 55% and 61% in the mass range 36 –45 GeV. In order to improve the mass resolution of the jets, a kinematic Gt is applied to the remaining events assuming the H+ H− → csUcs U hypothesis. The four-momenta of the four jets and their errors are considered. Energy and momentum conservation and the two equal invariant masses of the jet pairs (mH+ = mH− ) are required in the Gt. The Gt is applied to three possible mass combinations and the one with the smallest ,2 is chosen. Events which do not match the H+ H− → csUcs U hypothesis are rejected by requiring that ,2 ¡ 45 for the Gve degrees of freedom. Fig. A.16 shows the invariant mass distribution of the jet pairs. OPAL gives an explanation for the small

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Fig. A.16. OPAL csUcs U invariant mass distribution for data, simulated signal and background. No peak structure is observed in the data.

di:erence in shape between the data and the simulated Z → hadrons distributions, attributing it to the known inaccuracies in modeling the phase-space distribution of the events with more than three partons [207]. In addition, uncertainties in the simulation of the fragmentation contribute to the discrepancy. The invariant mass distribution of the simulated H+ H− signal has a narrow peak close to the nominal Higgs boson mass and a broad tail towards lower masses owing to incorrect jet associations. The searches of the other LEP experiments are also based on Gnding a narrow peak in the invariant mass distribution. OPAL parametrizes the data and signal mass distributions shown in Fig. A.16. The data are well described by a third order polynominal and the signal is described by a third order polynominal and a Gaussian with the width 1 centered around the peak position m0 . The signal area contained in the Gaussian part is denoted by f. The parameters m0 , 1, and f are determined as a function of the Higgs boson mass by linear interpolation between a set of six simulated Higgs boson masses. The data mass region m0 ± 1:641 is investigated, where 1:641 corresponds to a one-sided 95% CL interval. The data are scanned for a peak structure in a step size of 0:25 GeV, where its interval is smaller than the expected mass resolution in order to ensure no artiGcial eVciency reduction during the scan. In order to determine the background, a third order polynomial is Gtted to the data in the mass range 30 –45 GeV excluding the interval m0 ± 1:641. The Gtted function is subtracted from the entire data distribution and a Gaussian with parameters m0 and 1 is Gtted to the distribution of the di:erence. The area of the Gaussian is determined. The 95% CL upper limit is set if the number of expected Higgs boson events exceeds Nmax = + 1:641 ; (A.6) 3f 3f where 3 is the detection eVciency and 1(=3) is the error determined from the propagation of errors of the detection eVciency; and f are determined from the Gt.

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The main error on the detection eVciency is due to the uncertainty in the modeling of the hadronization and fragmentation process. From the comparison of the data and simulated Z → hadrons distributions, the systematic error is estimated to be below 5%. The statistical error from the limited Monte Carlo signal statistics is 3%, leading to a total error of 6%. Compared to this error, the systematic error resulting from the determination of is small. For the branching fraction BR (H± → ) = 0, OPAL has set the lower mass limit of 44:2 GeV at 95% CL. The limit is derived from the Nmax value and the number of expected Higgs boson events as a function of the Higgs boson mass. The shift between data and simulation is −100 MeV. Therefore, the Higgs mass limit for this analysis is 44:3 GeV. A.3.2. cs channel The signal is characterized by two hadronic jets, a , and missing energy. In order to reduce the large background from Z → hadrons, only the -decays with one charged track are considered. Two-photon events are reduced by rejecting events with large energy Qow in the forward calorimeters or in the endcap electromagnetic calorimeters. Owing to the expected missing energy in the signal events, the total event energy is required to be between 30 and 70 GeV. All of the cuts reduce the data sample to about 5% of its original size. The event is divided into two hemispheres by a plane with the thrust axis as normal vector. In each hemisphere a summed momentum vector is calculated, and from their axis acollinearity and acoplanarity angles are calculated. The requirements cos acol ¡ 0:95 and cos acop ¡ 0:99 are applied by OPAL to further reduce the background from hadronic Z decays. In addition, the missing momentum vector must not point towards the beam axis, |miss | ¡ 0:94, to reduce two-photon events and qqU background. These cuts reduce the data sample by a factor of about 25. In the next step of the selection, an isolated, charged track from a one-prong -decay is searched for. The energy of the charged track must lie between 3 and 15 GeV and the energy ◦ of track and associated cluster must be larger than 7:5 GeV in a cone of 13 . The isolation ◦ requirement is that no other charged track must be found in a cone of 30 half-opening angle and the energy between the inner and outer cone must not exceed 0:4 GeV. These requirements reduce the data further by a factor of about 20. The tracks and clusters outside the wider isolation cone are assigned to the H → cs decay. Their invariant mass is constrained to mcs ¡ 55 GeV. Heavy Higgs bosons will be produced with low kinematic energy; therefore, the two hadron jets are typically in opposite hemispheres and their energy is about mH =2. The cs-system is divided into two jets using the plane perpendicular to its thrust axis. Both jets are required to have energies between 10 and 30 GeV. Finally, a cut is applied in the plane of mcs versus cos cs as shown in Fig. A.17, where cos cs is the acollinearity angle of the two jet momenta. Near the kinematic production threshold of the H+ H− pair, cos cs ≈ 1. Fig. A.17 shows the cut by which the number of data events is reduced from Gve to one. The remaining candidate event has mcs ≈ 34 GeV and cos cs ≈ 0:2. The application of the same selection to a Monte Carlo sample corresponding to the data of 3.5 million hadronic events gives four candidates before and exactly one after the Gnal cut. The detection eVciency varies between 6% and 10% in the mass range of 35 –46 GeV. The statistical error is less than 10% and systematic errors result mainly from modeling the

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Fig. A.17. OPAL cs invariant mass versus acollinearity angle. The correlation between the opening angle of the cs-jets and their reconstructed invariant mass is shown for data and simulated charged Higgs boson signals of 36, ◦ 40 and 45 GeV. For the highest simulated mass the opening angle is about 180 . From Gve selected data events only the 34 GeV candidate event is located in the Gnal selection region between the two solid lines. The expected QCD background is consistent with the observation of one candidate event.

fragmentation. The variation of the fragmentation parameters results in an estimate of 5% systematic error. For BR (H →) = 0:5, the lower Higgs boson mass limit is 45:0 GeV at 95% CL. The limit is derived from the number of expected Higgs events, reduced by 11% systematic error, and the 95% CL level at 4.74 events due to the one candidate [114]. A.3.3. + − U channel In this channel a pair of low-multiplicity narrow jets and missing energy are expected. The focus is put on the identiGcation of the ’s and separation from Z → + − background. The -decay products are strongly collimated and a cone algorithm is deGned. The cone algorithm starts with the particle having the highest energy, where both tracks or ◦ clusters are considered. In a cone of 20 half-opening angle the particle with the next highest energy is searched for. If such a particle is found its momentum is added to deGne the axis of a new cone and the procedure is repeated until no further particles can be added to that cone. Of the remaining particles, the particle with the highest energy is taken as the starting point of a new cone. The above procedure is repeated until all particles are assigned to a cone. The energy of the cone is deGned as the sum of the particle energies. In order to avoid double counting, the momentum sum of a track is subtracted if it points towards a given cluster. This is also applied in the calculation of the total event energy Evis . Events with exactly two cones and with at least one charged particle per cone are selected. The energy of a cone has to be larger than 2 GeV and the axis of the cone must not point near the beam-pipe, |cos | ¡ 0:7. More than 99% of the beam energy must be contained in the two

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Fig. A.18. OPAL + − U visible energy versus acoplanarity angle for (a) data and (b) simulated Higgs boson signal. The data sample is largely reduced to three events passing the cut indicated by the vertical and horizontal lines, while most simulated signal events remain. The data events are consistent with the expectation from e+ e− → + −

background.

cones. In addition, no charged track must be outside the cones, and no cluster must point in the endcap electromagnetic calorimeter. ◦ The requirement that the acoplanarity angle is larger than 20 is most important to reject background from Z → ‘+ ‘− . The acoplanarity is deGned by the angle of the axes of the two cones projected to the plane perpendicular to the beam axis. The distribution of the visible energy versus the acoplanarity angle is shown in Fig. A.18 for the data and for a simulated + − + − 44 GeV √ H H → U signal. Most of the two-photon events are removed by requiring Evis = s ¿ 0:15. The acoplanarity distribution after this cut is shown in the insert. Events with large acoplanarity angle are due to + − decays involving a high momentum neutrino. After the two-dimensional cut indicated in Fig. A.18, three data events survive. One of the OPAL candidate events has a cone with an electron pair from a photon conversion. Also, the event has unreconstructed hits from a low momentum positive charged track. The event is consistent with an e+ e− → + − hypothesis and it is removed by a dedicated algorithm applied to all events to identify conversions. The other two events are also consistent with e+ e− → + − background, where the photon is included in one of the cones, thus giving rise to a large acoplanarity angle. The two data events are consistent with the e+ e− → + −

background expectation of 1:0 ± 0:5 events. Conservatively, both data events are considered candidates when deriving the limit. The resulting detection eVciency is about 30% for mH± ¿ 35 GeV. For BR (H+ → + ) = 1, the lower Higgs mass limit is 45:5 GeV at 95% CL calculated from the number of expected Higgs events and using 6.3 events as the 95% CL limit with two candidate events [114].

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Physics Reports 359 (2002) 283–354

Theories of low-energy quasi-particle states in disordered d-wave superconductors Alexander Altlanda , B.D. Simonsb; ∗ , M.R. Zirnbauerc a

Theoretische Physik III, Ruhr-Universitat-Bochum, 44780 Bochum, Germany b Cavendish Laboratory, Madingley Road, Cambridge CB3 0HE, UK c Institut fur Theoretische Physik, Universitat zu Koln, Zulpicher Strasse 77, 50937 Koln, Germany Received June 2001; editor: C:W:J: Beenakker Contents 1. Introduction 1.1. Intrinsic and extrinsic symmetries 1.2. Symmetry class AIII 1.3. Symmetry class CI 1.4. Symmetry class C 1.5. Symmetry class A 1.6. Self-consistent theories 2. Dirty d-wave (basics) 2.1. The Hamiltonian 2.2. Generic symmetries 2.3. Field-integral formulation 2.4. Symmetries of the Gaussian :eld theory 2.5. Disorder average 2.6. Renormalization group 3. How to proceed? 4. Non-Abelian bosonization 4.1. Hard scattering, T-invariance (class CI) 4.2. Soft scattering, T-invariance (class AIII) 4.3. Realistic scenario? 4.4. Soft scattering, broken T-invariance (class A)

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4.5. Hard scattering, broken T-invariance (class C) 5. Comparison with other approaches 6. Numerical analyses of the quasi-particle spectrum 6.1. Hard scattering 6.2. Soft scattering 7. Discussion Acknowledgements Note added in proof Appendix A. Gradient expansion and the chiral anomaly A.1. Chiral anomaly A.2. Heat kernel regularization Appendix B. Dirac fermions in a random vector potential B.1. WZW model of type A|A B.2. Functional integral solution Appendix C. WZW theory for systems with boundary References

∗

Corresponding author. E-mail address: [email protected] (B.D. Simons). c 2002 Elsevier Science B.V. All rights reserved. 0370-1573/02/$ - see front matter PII: S 0 3 7 0 - 1 5 7 3 ( 0 1 ) 0 0 0 6 5 - 5

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Abstract The physics of low-energy quasi-particle excitations in disordered d-wave superconductors is a subject of ongoing intensive research. Over the last decade, a variety of conceptually and methodologically diFerent approaches to the problem have been developed. Unfortunately, many of these theories contradict each other, and the current literature displays a lack of consensus on even the most basic physical observables. Adopting a symmetry-oriented approach, the present paper attempts to identify the origin of the disagreement between various previous approaches, and to develop a coherent theoretical description of the diFerent low-energy regimes realized in weakly disordered d-wave superconductors. We show that, depending on the presence or absence of time-reversal invariance and the microscopic nature of the impurities, the system falls into one of four diFerent symmetry classes. By employing a :eld-theoretical formalism, we derive eFective descriptions of these universal regimes as descendants of a common parent :eld theory of Wess–Zumino–Novikov–Witten type. As well as describing the properties of each universal regime, we analyse a number of physically relevant crossover scenarios, and discuss reasons for the disagreement between previous results. We also touch upon other aspects of the phenomenology of the d-wave superconductor such as quasi-particle localization properties, the spin c 2002 Elsevier quantum Hall eFect, and the quasi-particle physics of the disordered vortex lattice. Science B.V. All rights reserved. PACS: 74.20.−z; 74.25.Fy; 71.23.−k; 71.23.An; 72.15.Rn

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1. Introduction In recent years it has become clear that the superconducting phase of the hole-doped cuprate superconductors is of unconventional, d-wave, symmetry. Motivated by this observation, the properties of quasi-particle states in d-wave superconductors have come under intense scrutiny [1–20]. A particular subject that turned out to be not only experimentally relevant but also theoretically intricate is the inJuence of static disorder on the large-scale features of the d-wave phase. Leaving aside the pair-breaking eFect of the disorder which strongly inJuences the high-temperature properties, attempts to resolve the impact of impurity scattering on low temperature spectral and transport properties of a disordered d-wave superconductor have ignited controversy in the recent literature. The key feature which distinguishes d-wave from the more conventional s-wave superconductors, and makes the disorder problem harder to solve, is the presence of four isolated “nodes” on the Fermi surface in the vicinity of which low-lying Dirac-type [22] quasi-particle excitations exist. Beginning with the work of Gorkov and Kalugin [1], and later Lee [2], the earliest considerations, based on approximate self-consistent treatments, concluded that an arbitrarily weak impurity potential induces a :nite density of states (DoS) at the Fermi surface, and leads to weak localization of all quasi-particle states in the two-dimensional (2d) system. Although the :rst result found support, at least super:cially, in an analysis by Ziegler et al. [4], these conclusions were found to be in contradiction with those of other authors. In particular, following the early considerations of Refs. [1,2], Nersesyan et al. [5] (NTW) developed a complementary approach in which the quasi-particle Green functions of a disordered d-wave superconductor were mapped on a conformally invariant fermion-replica :eld theory. Based on this mapping, NTW proposed that the 2d system is characterized by critical properties of the quasi-particle states at the Fermi level. Their model predicted that the DoS vanishes as (E) ∼ |E | with some disorder-dependent exponent . To further add to this controversy, Senthil et al. [6] (SFBN) proposed a description by an alternative fermion-replica :eld theory, namely the principal chiral non-linear sigma model over the symplectic group Sp(2r) (with r = 0). Void of any signatures of criticality, the large-scale behaviour of the two-dimensional model was argued to describe a phase of localized states, the “spin insulator”, thereby :nding qualitative agreement with the early considerations and contradicting the :ndings of NTW. In contradiction to both NTW and the early works, the quasi-particle DoS at the Fermi level was found to vanish with a universal exponent of unity. Since this work, further analysis of the :eld-theoretic scheme generated fresh, and seemingly contradictory proposals. Fendley and Konik [19] have argued that, when the scattering between pairs of opposite nodes is neglected, the low-energy properties of the weakly disordered system are described by a critical :eld theory that has Sp(2r) (with r = 0) for its target space and includes a Wess–Zumino term. The same theory, now based on a peculiar implementation of a time-reversal symmetry breaking lattice operator, had been suggested by Fukui [18]. Albeit belonging to a diFerent universality class, both proposals [18,19] share key aspects with the theory of NTW, including the existence of extended quasi-particle states. How, if indeed at all, can the largely contradictory approaches be reconciled with each other? And is it possible to unify elements of these approaches into one coherent theoretical formulation? This is the :rst complex of questions we are going to address in this paper.

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A second and more pragmatic point we are going to address concerns the microscopic nature of the impurity scattering. It will turn out that much of the controversy in the :eld-theoretical literature described above can be attributed to the sensitivity of the d-wave quasi-particle physics to the range of the scattering potential. In a number of recent publications [23–25], it has been argued that both experimental and theoretical evidence hints at a strong enhancement of forward scattering in d-wave superconductors. Without going into details, we here merely state the essence of the argument, viz. that a signi:cant amount of inter-node scattering would have a strong pair-breaking eFect, thereby being in conJict with the very formation of stable d-wave order. Actually :guring out the meaning of the attribute “signi:cant” is a delicate issue we will not even attempt to address. Notwithstanding this uncertainty, one may ask the principal question of what kind of low-energy theory governs the behaviour of quasi-particle Hamiltonians that are strongly anisotropic (albeit certainly not exclusively forward scattering) on the bare microscopic level. In particular, the question arises whether, as in disordered metallic systems, the impurity form factor merely leads to a renormalization of the scattering time or whether it might have a more substantial eFect. As we are going to argue below, the latter is the case here: the extreme cases of pure forward scattering and isotropic scattering fall into distinct universality classes with qualitatively diFerent properties. And, although the isotropic limit is ultimately attractive (in the renormalization group sense), one might speculate that under realistic conditions (:nite temperatures, experimental resolutions, etc.) the forward-scattering limit could well be the relevant one. A second aspect that makes the limit of weak forward scattering an interesting object of study concerns the physics of the mixed state: as with conventional type-II superconductors, the application of a magnetic :eld of intermediate strength (Hc1 ¡ H ¡ Hc2 ) forces vortices into the d-wave superconductor. The impact of vortex formation on the low-energy density of quasi-particle states of pure d-wave superconductors has been the subject of an ongoing debate [26 –31]. Recently, signi:cant progress was made by Franz and Tesanovic [28], who exploited a novel singular gauge transformation, mapping the problem again on the problem of Dirac fermions. This transformation, which is analogous to that applied in the theory of the fractional quantum Hall eFect, is applicable even in the presence of randomness. As a result, one obtains an eFective low-energy model, essentially Dirac fermions with random scalar and vector potential, that can be applied to explore the quasi-particle DoS of the disordered vortex lattice. We will return to this subject below. Although, for a non-expert, the detailed :eld-theoretical analysis contained in this paper is often somewhat involved, we believe that the main conclusions of our survey of the dirty “d-wave system”—for lack of a better terminology we will often refer to the quasi-particles of a disordered d-wave superconductor by that word—should be widely accessible. We have, therefore, chosen to summarize the complete phase diagram here. At the same time, this allows us to organize and place into context many of the existing :eld-theoretical works on the subject. 1.1. Intrinsic and extrinsic symmetries While, at :rst sight, the various points brought up above seem unrelated, the microscopic analysis below will show that they do, in fact, all :nd their origin in a common microscopic mechanism. Just as for the clean d-wave Gorkov Hamiltonian [32,33], the Hilbert space for

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Fig. 1. Symmetry classes realized in dirty d-wave superconductors with conserved spin. The two types of perturbation causing crossover between classes are indicated.

the low-energy Hamiltonian of the pure forward-scattering system foliates into four sectors not connected by the disorder potential. These nodal sectors can be grouped into pairs related by parity. The low-energy physics of the individual sectors turns out to be more intricate than that of the complete system, which is comprised of four Dirac nodes coupled by impurity scattering. In general it will, of course, be the behaviour of the full system that matters. There are, however, a number of situations in which signatures of the decoupled system remain visible. Exploring these scenarios, which are accompanied by a drastic change in the phenomenology of the system, is one of the major issues to be addressed in this paper. The origin of the relative non-triviality of the decoupled system can be understood qualitatively by noticing that the structure of any theory describing the low-lying quasiparticle excitations of a disordered system is essentially :xed by two types of symmetries: “intrinsic” symmetries, such as the behaviour of the Hamiltonian under spin-rotation, timereversal, particle–hole transformations, etc., and “extrinsic” symmetries such as translational or rotational invariance or, for that matter, parity. Speci:cally, the unperturbed d-wave superconductor is a particle–hole symmetric, spin-rotation and time-reversal invariant system, thereby falling into the symmetry class CI in the classi:cation of Ref. [34]. The implementation of these intrinsic symmetries into an eFective low-energy theory of a superconductor system was formulated some time ago by Oppermann [35], and was recently rederived in the context of d-wave superconductivity by SFBN [6]. The behaviour of the d-wave system under extrinsic symmetry operations is more remarkable. In fact, as we will see, it is disregard for these symmetries that has driven astray some of the :eld theories based on intrinsic symmetry: not all terms allowed by the intrinsic symmetries respect the extrinsic symmetries. The point is that individual nodes are non-invariant under “parity”, by which we mean any operation that reverses the orientation of two-dimensional space [36]. Of course, the full four-node system emerging from a manifestly invariant Gorkov Hamiltonian is parity invariant, but individual nodes are not. Remarkably, this parity non-invariance and the intrinsic symmetries conspire to let the system of isolated nodes fall into a symmetry class that diFers from the Wigner–Dyson classes and the classes commonly attributed to bulk superconductors. The complete phase diagram of the d-wave system in fact separates into a total of four distinct regions (see Fig. 1) distinguished by the presence=absence of time-reversal invariance and by the correlation radius of the disorder potential. Taking each region in turn, we here summarize the phenomenology and the basic properties of the respective low-energy theories [37].

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1.2. Symmetry class AIII So, beginning with the time-reversal invariant system with only forward scattering, one :nds that the corresponding symmetry class is not CI, but rather AIII, i.e. a symmetry class that is typically realized in disordered relativistic (chiral) fermion systems. Indeed, it is straightforward to verify that the low-energy physics of individual nodes is described by a model of Dirac fermions in the presence of some quenched random gauge :eld. The implications of this correspondence for the disordered system were :rst noticed and analysed in the seminal work of NTW [5], who pointed out that the low-energy physics of the individual nodes is critical. NTW argued that a general mechanism uncovered long ago by Witten [38] applies in particular to the d-wave system: in a Lagrangian formulation based on non-Abelian bosonization, non-trivial transformation behaviour with respect to parity is reJected by the presence of a Wess–Zumino– Novikov–Witten (WZW) term in the action functional. Speci:cally, NTW mapped the d-wave system on a :eld theory with U(1) × SU(r) × SU(N ) symmetry (r being the number of replicas). In the limit of isolated nodes (N = 1), the theory has the :eld manifold U(1) × SU(r), with the SU(r) sector being governed by a so-called level-1 WZW action. The latter theory enjoys the existence of an in:nite-dimensional (current algebra) symmetry, which makes it solvable to a large extent. Several of its features, including the existence of algebraically decaying correlations (absence of localization!), and the scaling of the DoS in the vicinity of the Fermi energy can be derived rigorously. Within the framework of the supersymmetric formulation, the properties of the forwardscattering system are found to be described by two copies of a WZW model of level 1 and type A|A, i.e. a WZW model whose :elds take values in a supermanifold that, for reasons discussed in body of the paper, is labelled by A|A. Each copy derives from a pair of nodes on the Fermi surface related by parity or inversion symmetry. Formally, the total zero-energy eFective action is given by S = W [M1 ; ] + W [M2−1 ; −1 ] + g d 2 r O00 where the dimensionless parameter (de:ned microscopically later) speci:es the degree of anisotropy of the Dirac cones, and i W [M; ] ≡ [M ] + g d 2 r O00 12 1 − d 2 r STr(−1 91 M −1 91 M + 92 M −1 92 M ) 8 is the anisotropic WZW action. The :elds M1 and M2 take values in (a maximal Riemannian subspace of) the supergroup GL(2|2), [M ] represents the WZW term, and O00 ; O00 are low-angle scattering operators whose speci:c structure will be described in Section 4. Note that, since the WZW term is odd under M → M −1 (i.e. [M −1 ] = − [M ]), the WZW terms for the two sectors of nodes carry opposite sign, which reJects the fact that they derive from Dirac operators carrying opposite orientations. Further, the fact that GL(2|2) and not GL(1|1)—the supersymmetric analog of a fermion replica theory with group manifold U(r)—appears as the relevant degree of freedom, has to do with the behaviour of the system under time reversal. This aspect, unimportant for the physics of the system of isolated nodes, becomes vitally relevant once inter-node scattering is switched on.

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Perhaps the most characteristic feature of models of Dirac fermions subject to Abelian gauge randomness is their stability under renormalization. Indeed, the low-energy theory de:ned by the action W is a :xed point of the renormalization group for each value of the dimensionless disorder strength g, as long as g does not exceed some critical value. The absence of running couplings bears consequences for various physical observables. One :nds that, for low energies, the DoS vanishes as (E) ˙ |E | , with the g-dependent family of exponents 1 − 2g= = ; 1 + 2g= in accord with the analysis of NTW. We also recapitulate that the eigenstates of the random gauge Dirac system are extended [39]. While exhibiting critical spectral and transport properties, the eFective action for d-wave superconductors in class AIII is unstable against the inJuence of short-range isotropic scattering, a matter not correctly handled by NTW. Impurity scattering involving large momentum transfer couples the nodes and leads to a “locking” of the :elds at the diFerent nodes. This process bears two consequences for the structure of the low-energy theory: :rst, and in contradiction to the naive expectation, the :eld con:gurations surviving the locking of the formerly independent GL(2|2) :elds take values in a diFerent manifold, viz. OSp(2|2). Second, the locking M1 = M2 implies that in the full action the WZW terms cancel, which makes the restoration of parity invariance in the coupled system explicit. (In parentheses, we note that the absence of a WZW term is a suScient but not a necessary condition for parity invariance of physical observables: for a single WZW theory, the sign change in [M ] caused by a parity operation can be compensated by a :eld transformation M = eX → M −1 = e−X , i.e. the :eld X is a pseudo-scalar. However, in a system comprised of two coupled sub-theories carrying opposite orientations, this compensating transformation is inhibited on either sub-theory. Moreover, there exists no room for a WZW term in the locked full theory, as the sum of coupled Dirac theories for the d-wave system is anomaly-free.) Since it was the WZW term that rendered the NTW theory critical, it is evident that the low-energy behaviour of the full theory will be of diFerent type. 1.3. Symmetry class CI A fermion-replica theory of the full system, a non-linear sigma model on the :eld manifold Sp(2r), was :rst derived and analysed by SFBN. Essentially, this model is a d-wave version of the Sp(2r) non-linear sigma model formulation of time-reversal invariant super-conducting glasses developed earlier by Oppermann [35]. Void of any elements supporting critical behaviour, the predictions of that theory diFer greatly from NTW: all quasi-particle states are localized, a fact that led SFBN to coin the term “spin insulator”; the DoS still vanishes at zero energy but in a manner altogether diFerent from the scaling obtained by NTW. All in all, the low-energy theory of the composite four-node system seems to have little overlap with the critical theory of the decoupled forward-scattering system. Once the :elds belonging to the diFerent nodal sectors become locked, the total eFective action from above takes the form S[M ] = W [M; ] + W [M −1 ; −1 ], i.e. + −1 d 2 r STr(9 M −1 9 M ) ; S[M ] = − 8

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where the :eld M now belongs to the group manifold OSp(2|2). Technically, this is a principal chiral non-linear sigma model of type D|C. It represents a supersymmetric extension of the action derived by SFBN [6], and leads to the prediction of quasi-particle localization of all states in two dimensions—the spin insulator phase—and a DoS that vanishes at zero energy. It seems unlikely that symmetry class AIII is realized in the physical d-wave system. As mentioned earlier, the bare microscopic theory will probably contain a large forward-scattering amplitude, plus a certain amount of inter-node scattering rendering it a member of symmetry class CI. This hard-scattering admixture is a relevant perturbation which means that the theory will ultimately Jow towards the strong-coupling :xed point in class CI. On the other hand, realistic experimental conditions are likely to prevent the system from exploring all limiting regions of the phase diagram. This motivates us to ask how fast the Jow towards the strong-coupling regime takes place and what properties the crossover theories, situated somewhere in between the independent-node and the strongly coupled theory, possess. We will return to this issue below. 1.4. Symmetry class C An alternative mechanism of allowing elements of the decoupled theory to resurface is by breaking parity explicitly. At least theoretically, this is realized in the so-called spin quantumHall (SQH) system [12,40], where the nodal degeneracy is lifted via the addition of an idxy component to the order parameter. The resulting low-energy phase shares much of its phenomenology—the formation of edge states, quantized values of the transverse component of the conductivity tensor, etc.—with the integer quantum Hall eFect and has, therefore, been christened the spin quantum Hall state. (The pre:x “spin” indicates that in the superconductor problem, the behaviour of spin currents or thermal currents is considered. Because of the non-conservation of quasi-particle charge, the study of charge transport coeScients becomes meaningless.) Originally, it was suggested [40] that the formation of a secondary id-order parameter component giving the d + id state might be quasi-spontaneously driven by a weak external magnetic :eld. Recent work [41] has cast some doubt on the applicability of this scenario and, therefore, on the experimental relevance of the spin quantum Hall eFect. Theoretically, however, the d + id system remains an interesting object of study: :rst, it represents a rare example of a system with an exactly solvable quantum-Hall phase transition driven by disorder [42]. Second, unlike with d-wave superconductors in the previously discussed symmetry class CI, aspects of the physics of individual nodes survive the thermodynamic limit. In fact, the id-component does more than lift the node degeneracy, implying that it does not lead to a resurrection of the WZW model: the inclusion of an idxy component not only breaks parity but also time-reversal invariance, and hence reduces the symmetry of the system from CI down to C. Within a fermion-replica approach, the class C system is described by a :eld theory taking values on the manifold Sp(2r)=U(r). This :eld theory does not support a WZW term. It does, however, admit for the existence of a closely related parity-breaking term in the Lagrangian, viz. a topological theta term. Based on phenomenological considerations, it has in fact been argued [12] that a non-linear sigma model over Sp(2r)=U(r) with a theta term should be the relevant theory of the disordered d + id system.

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Exploring how such a theory may emerge as a descendant of the class CI theory of the non-degenerate node system will be a further topic to be addressed below. The explicit form of the resulting supersymmetric soft-mode action for the system of class C reads as 1 0 0 − d 2 r STr(11 9 Q9 Q + 12 Q9 Q9 Q) ; (1) 8 where the WZW term has transformed into a topological theta term. In contrast to the conventional integer quantum Hall eFect exhibited by normal systems, the :elds of the saddle-point manifold Q belong to the coset space OSp(2|2)=GL(1|1), with the relevant Riemannian symmetric superspace inside that manifold being of type DIII|CI. Unlike its normal relative, the critical theory for the spin quantum Hall eFect for class C is readily amenable to theoretical investigation. Indeed the properties of this phase have been explored in a number of recent numerical and analytical works. Surprisingly, a mapping [12] of the class C theory onto a network model [43] shows the transition to be in the same universality class as classical percolation [42]. This correspondence, which applies only for the statistics of various low-order moments, relies on a remarkable cancellation of interference channels, and allows several of the critical exponents to be determined exactly [42,44]. Very recently, a proposal for the general critical theory of the spin quantum Hall transition has been made by Bernard and LeClair [45], although its validity still remains a matter of some controversy. 1.5. Symmetry class A To exhaust the number of diFerent low-energy scenarios realized in the d-wave system, one may consider a situation where a perturbation breaking time-reversal symmetry is superimposed on an impurity potential causing only forward scattering. Below we will identify the resulting system as a member of symmetry class A, i.e. the standard Wigner–Dyson class of unitary symmetry. We :nd that, as with its class C counterpart, the class A system also supports a (spin) quantum Hall transition. Somewhat surprisingly, however, that transition turns out to fall into the universality class of the standard integer quantum Hall transition. Formally, the low-energy theory is again described by an action functional including a theta term, similar to the one above. The main diFerences with the previously discussed case are that (a) one obtains two copies S[Q; ] of the action, one for each nodal sector, and (b) that the :elds Qi (i = 1; 2) belong to the coset space GL(2|2)=(GL(1|1) × GL(1|1)). This means that the :eld theory is of type AIII|AIII which is the supersymmetric extension of Pruisken’s theory for the conventional integer quantum Hall eFect. Finally, we remark on a second and, arguably, more physically motivated scenario in which the classes A and C are engaged—the mixed phase of a type-II superconductor. On symmetry grounds alone, one would expect the Hamiltonian of the vortex phase of the dirty d-wave system to belong to one of these classes. Although the nature of the quasi-particle states in the vortex phase of the clean d-wave system has been addressed in a number of (seemingly controversial) publications [26 –31,46], a :eld-theoretic analysis of this disordered system has not been given in the literature. This motivates us to focus our discussion of the symmetry classes C and A on the mixed state, not on the spin quantum Hall system.

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Furthermore, for brevity, we have chosen not to comment upon the inJuence of the breaking of spin rotational invariance either by magnetic impurities or spin-orbit scatterers. Such systems, which lie outside the scope of present work, have been addressed in the recent literature and we refer to Refs. [47– 49] for a discussion of the phenomenology of these systems. Altogether, the key characteristics of the :eld theories for the d-wave system are summarized in the following table: Class T

Hard scattering

Symmetry group

Saddle-point manifold

NLM

Fermion-replica analog

AIII A

+ −

− −

GL(2|2) × GL(2|2) GL(2|2)

A|A AIII|AIII

CI C

−

+

+ +

OSp(2|2) × OSp(2|2) OSp(2|2)

GL(2|2) GL(2|2)=(GL(1|1) ×GL(1|1)) OSp(2|2) OSp(2|2)=GL(1|1)

U(2r) U(2r)=(U(r) ×U(r)) Sp(2r) Sp(2r)=U(r)

D|C DIII|CI

Note that the contents of this table are in complete agreement with Ref. [50], where each of the ten symmetry classes, as enumerated by the large families of Cartan’s list of irreducible symmetric spaces, was put in correspondence with the saddle-point manifold of a non-linear sigma model. This coincidence is by no means accidental. Although Ref. [50] was formulated in the random-matrix setting, the translation scheme employed there is of universal validity and unerringly identi:es the internal (:eld-theoretical) global symmetries for each symmetry class. The present treatment has the virtue of being well adapted to the speci:c case of d-wave superconductors, and shows quite explicitly how the hierarchy of Goldstone modes emerges in this context. 1.6. Self-consistent theories Before leaving this introductory section, let us return to the controversy between the predictions made by the various :eld-theoretical approaches, and the self-consistent diagrammatic schemes developed in the literature. Readers who :nd the following, purely qualitative, remarks cryptic are referred to Section 5 for a more substantial discussion. The last few years have seen a build-up of confusion concerning the behaviour of the density of states at very small energies: straightforward diagrammatic analysis predicts a :nite DoS at zero energy, whereas, in the :eld theory approaches, unbounded Juctuations of Goldstone modes lead to a vanishing DoS. An intuitive argument in favour of a :nite DoS is that any average over random potential Juctuations should introduce some imaginary self-energy in the Green function of the Dirac operator. This self-energy would lead to a smearing of spectral structures over some energy window of width given by and would, therefore, be in conJict with the formation of a cusp in the energy-dependent DoS. This argument, however, is overly simple; in particular it does not account for the full complexity of the scattering problem in random Dirac (or gapless superconductor) systems. Let us try to discuss this point on a somewhat more technical level: The various classes of diagrams appearing in perturbative approaches can be grouped, roughly speaking, into two categories: diagrams of the self-energy type (commonly evaluated in SCBA

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or similar approximations), and various types of ladder diagrams (alias diFusion modes). While the former are usually associated with some smearing of average single-particle quantities by disorder, the latter describe mesoscopic Juctuations which become operative at long range. More precisely, ladder diagrams describe the long-range quantum interference between retarded and advanced single-particle Green functions. For normal-conducting systems, the distinction between these diagrammatic elements is perfectly canonical: single-particle Green functions are renormalized by self-energy diagrams whereas higher-order Green functions with poles on both sides of the real energy axis may be inJuenced by diFusion modes. For superconductors however, and for gapless superconductors in particular, this distinction breaks down. In previous diagrammatic approaches, the DoS of dirty d-wave superconductors was computed with (self-consistent) account for self-energy diagrams. This led to some smearing and, therefore, to a :nite DoS. However, as was shown in Ref. [34] for the prototypical case of superconductor=normal quantum dots, the single-particle DoS of a gapless superconductor may be aFected by diFusion-like modes. This phenomenon can be understood on various levels of sophistication. It can be explained by qualitative semi-classical reasoning [34] or, more technically, by the fact that unlike in normal systems the single-particle Green function of a superconductor possesses poles on both sides of the real energy axis. While for a bulk s-wave superconductor, all poles are shifted far into the complex plane by the bulk order parameter (a manifestation of Anderson’s Theorem [51]), the situation in gapless systems is diFerent. Here, multiple impurity scattering suppresses the self-energy. Technically, the suppression is caused by ladder diagrams that sum up to yield diFusion modes. These modes enter already at the single-particle level [34], and are missing from previous diagrammatic works on d-wave superconductivity. In the limit of zero energy, the diFusion modes become massless (or of in:nite range) implying that a purely perturbative approach is met with a problem. (Technically, in two dimensions, the contribution of each diFusion mode scales as g−1 ln(E=E0 ), where E0 is some cutoF energy and g a measure of the dimensionless (spin) conductance of the system. For E suSciently small, the expansion becomes uncontrolled.) At this point it is good to remember that all the concepts discussed above have a :eld-theoretical analog, with an optional non-perturbative extension. In particular, the diFusion modes of the diagrammatic approach have the signi:cance of perturbative excitations of the Goldstone modes discussed above—very much like a spin wave can be interpreted as a perturbative manifestation of the Goldstone mode related to the intrinsic spin-rotation symmetry of a ferromagnet. The Juctuations of these modes, restricted only by :nite energies, become stronger and stronger as the energy approaches zero. For suf:ciently small energies, (a) perturbative (diagrammatic) schemes for treating these modes are ruled out, and (b) the non-perturbative analyses discussed below produce a vanishing DoS. (In passing we note that such mechanisms are not unfamiliar in mesoscopic physics. For example, for energy diFerences ! = E1 − E2 larger than the mean level spacing ", spectral correlations of generic normal disordered systems can be treated by diFusion-mode diagrammatic approaches. For !=" ¡ 1, perturbation theory breaks down, a phenomenon that manifests itself in the appearance of unphysical divergences. In this regime, a non-perturbative integration over the appropriate Goldstone mode [52] produces the correct result.) Summarizing, we :nd that (i) the issue of the DoS is crucially related to long-range modes of quantum interference operating on the single-particle level, (ii) to obtain the correct, vanishing, behaviour of the DoS,

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a non-perturbative treatment of these modes is inevitable but that (iii) signatures for the incompleteness of previous perturbative analyses can be found within the diagrammatic framework. Now, it is important to realize that the arguments above by no means amount to a categoric prediction of a vanishing zero-energy DoS in bulk d-wave superconductors. The point is that the weak disorder :eld-theoretical approach is based on certain model assumptions which leave room for complementary scenarios. Therefore, to complete our introductory discussion, let us brieJy mention a number of competing theories of the disordered d-wave system which lie outside the :eld-theoretic scheme outlined above. Typically, such theories rely on peculiarities of the d-wave system. For example, Balatsky and Salkola [9] have proposed a mechanism whereby the weak coupling of “marginally-bound” impurity states at the Fermi level leads to the absence of quasi-particle localization. When subject to a single impurity in the unitarity limit, a quasi-particle state is created at zero energy [53,54]. In a recent work, PVepin and Lee [17] have proposed that, when subjected to many such impurities, these quasi-particle states broaden into a narrow delocalized band around zero energy. On this basis, these authors have proposed that the properties of the dirty d-wave system at very low energies are characterized by extended quasi-particle states. Based on the RG analysis contained in this paper, it is our belief that, even for scatterers in the unitarity limit, there must ultimately be a cross-over at low-energy scales to the physics advocated by SFBN. Finally, some authors [20] have inquired into the importance of maintaining self-consistency of the d-wave order parameter in a disordered background. As others before, in this work we will take a pragmatic view and will not attempt a synthesis of self-consistent aspects of the theory. A second and probably more severe limitation regards the role of interactions. As with more conventional disordered normal and superconductor systems, the low-energy physics of quasi-particles in d-wave superconductors, too, will be aFected by various mechanisms of inter-particle interactions: zero-bias anomalies, interaction contributions to transport coeScients, interaction induced renormalization of the propagation in the Cooper channel, and more. Beyond that, one should expect that collective excitations, both of purely electronic and of phononic type, contribute to (thermal) transport coeScients, and further modify the low-energy behaviour of quasi-particle states. None of these eFects will be considered in the present paper. (For a discussion of interaction eFects in d-wave superconductors, see Ref. [14].) Needless to say, this may impose severe limitations on the relevance of the present work for the physics of real d-wave superconductors. To what extent do the predictions of the weakly disordered theory bear comparison with experiment? As mentioned above, superconductors of unconventional d-wave symmetry are realized in the high-temperature cuprate superconductors. Here the inJuence of both magnetic and non-magnetic impurities has been investigated intensively. In d-wave systems, both types of disorder are pair-breaking and lead to a rapid depression of Tc with doping. However, at very low doping concentrations, Tc remains a signi:cant fraction of its optimal value. In this case the inJuence of disorder on the low-energy quasi-particle properties has been studied extensively. In particular, recent angle resolved photoemission experiments [55] con:rm both the integrity of the large-scale nodal structure of the spectrum and con:rm the existence of long-lived quasi-particle states near the Fermi level. The same measurements show a large anisotropy of the Dirac nodes with, for example, Bi2 Sr 2 CaCu2 O8 exhibiting a ratio of t=" ≡ = 20.

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At very low temperatures, phase coherence properties of the quasi-particle states should be accessible via spin transport. The latter have been investigated indirectly through measurements of the thermal conductivity at very low temperatures: relating the thermal conductance %(T ) to the quasi-particle spin conductivity s through the Einstein relation, one obtains the Wiedemann– Franz ratio %(T )=Ts = (42 =3)kB2 . Taking the bare value for the spin conductance from theory, one obtains [6,15] k2 %(T 1=() 2 = 4 kB2 s → B ( + −1 ) : T 3 3 Remarkably, this result is found to be in close agreement with experiment [57]. In BiSCCO the extrapolated value of the longitudinal part of the thermal conductivity remains constant over a wide range of impurity concentrations. This result has been triumphed (see, e.g., Ref. [58]) as experimental support for the self-consistent theory of Lee [2]. However, it should be noted that this result does not sit comfortably with the class CI theory, which predicts that the spin conductance should exhibit weak localization corrections which renormalize s down from its universal bare value. Whether the seeming coincidence of the measured conductance and the bare value signi:es strongly enhanced forward scattering, or whether, as seems more likely, it can be explained in terms of a short phase coherence length, is as yet unclear. On this point, it is interesting to note that low-temperature measurements of the quasi-particle conductivity in the underdoped cuprate YBa2 Cu4 O8 [59] indicate that the quasi-particle states are localized. Although it would be useful to explore the thermal Hall transport [60], experiments have been so far unable to access the same very low temperature regime. The paper is organized as follows: in Section 2 we will formulate some basic concepts required for the construction of a low-energy theory of the d-wave system. This will include a symmetry analysis of the disordered Hamiltonian, its representation in terms of a (supersymmetric) generating functional, and a renormalization group analysis of the relevancy of the various channels of disorder. In Section 3 we ask how an eFective low-energy theory can be distilled from the microscopic functional integral. We will :nd that standard schemes, such as a straightforward gradient expansion, are met with severe diSculties. In Section 4, non-Abelian bosonization will be applied as an alternative method to derive the low-energy theory. In a number of subsections the predictions of this theory for the four symmetry variants introduced above will be discussed. In Section 5 we compare these results with the phenomenology predicted by other approaches, such as self-consistent T -matrix approaches. To complement the theoretical analysis of the low-energy properties of the d-wave system, in Section 6 we present a limited review of the progress made in the numerical investigation of the density of states and show how these results relate to the diFerent regimes discussed in the text. Finally, we close with a brief discussion in Section 7. 2. Dirty d -wave (basics) In this section we discuss the essential concepts needed for the construction of a low-energy eFective theory describing the quasi-particles of a disordered spin-singlet superconductor with

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dx2 −y2 symmetry. We start out by reviewing some elementary features of the Gorkov Hamiltonian for that system. Much attention will be paid to the generic symmetries, which turn out to be quite subtle. We then express the Green functions of the Hamiltonian as a :eld integral. The result is a supersymmetric generating functional, which will serve as the basic platform for the low-energy eFective :eld theories developed in later sections. Similar considerations for a class of related two-dimensional systems, having a low-energy limit modelled by a disordered Dirac equation, were :rst made by Fradkin [22]. 2.1. The Hamiltonian Basic to the theory of quasi-particle transport in spin-singlet superconductors are the Gorkov equations [32]. They follow from a quasi-particle Hamiltonian which in second quantized language, and on a lattice of sites i, is written as † cj↑ tij − *ij "ij H= (ci↑ ; ci↓ ) ; (2) † "ij −tij + *ij cj↓ ij

where tij are the hopping matrix elements, and "ij is the lattice d-wave order parameter. Time-reversal and spin-rotation invariance constrain these matrices to be real symmetric: † tij = tji = tXij and "ij = "ji = "X ij . Their precise form will be speci:ed shortly. The operators ci create spin- 12 fermions, is the chemical potential, and the sum ij extends over nearest neighbours on a two-dimensional square lattice with spacing a. By making a particle–hole transformation on down spins, d↑ ≡ c↑ ;

d↓ ≡ c↓† ;

the Hamiltonian is cast in the concise form † H= di ((tij − *ij )3 + "ij 1 )dj ; ij

where 1 ; 2 ; 3 are the Pauli matrices. By way of warning, we remark that this transformation makes it seem as though charge were conserved and spin was not. In reality, the opposite is true: quasi-particle charge is not conserved, whereas spin is. To facilitate future manipulations, we perform a =2-rotation d → exp(i1 =4)d, to transform to † H= di ((tij − *ij )2 + "ij 1 )dj : (3) ij

The matrix elements tij and "ij consist of clean and dirty parts. The clean part of the Hamiltonian, H0 , which we focus on :rst, assumes its simplest form in momentum space: † H0 = dk [(t(k) − )2 + "(k)1 ]dk : k

The kinetic energy and the d-wave order parameter are taken to be t(k) = t(cos(kx a) + cos(ky a)) ; "(k) = "(cos(kx a) − cos(ky a)) :

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Fig. 2. The quasi-particle spectrum of a clean d-wave superconductor showing the existence of four Dirac nodes.

It is the momentum dependence of the order parameter "(k) that distinguishes d-wave superconductors from their relatives with s-wave pairing. The dispersion relation of H0 , (k) = ± (t(k) − )2 + "(k)2 is displayed in Fig. 2 for the special value = 0 (the half-:lled band). The most important feature of the function (k) is that it vanishes (for = 0) at the four points (kx ; ky ) = a−1 (±=2; ±=2) : When the chemical potential is shifted away from half :lling, these nodal points move to diFerent locations in the Brillouin zone. No qualitative change is caused by that, and we shall therefore stick to the case = 0. After the introduction of small momentum oFsets by setting (kx ; ky ) = (±=(2a) + *kx ; ±=(2a) + *ky ), the dispersion relation in the vicinity of the nodal points is approximated as

(k + *k) ±a t 2 k22 + "2 k12 ; where k12 ≡ *kx ∓ *ky . Thus the dispersion relation near the nodal points is linear in k, and the low-energy physics will therefore be of Dirac type. Anticipating that we will be primarily interested in the nodal regions, we introduce four species of fermion operators, dn*k ≡ dk n +*k ; X 2; 2X} labels the nodes, with the assignments to node momenta k n being where n ∈ {1; 1; 1 → a−1 (−=2; =2) ; 2 → a−1 (=2; =2) ; 2X → a−1 (−=2; −=2) ; 1X → a−1 (=2; −=2) :

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If we restrict attention to the low-energy sector, the Hamiltonian of the clean system decomposes into four nodal sub-Hamiltonians: n† H0 = dk H0n (k)dnk : n; k

With the de:nition of the characteristic speed v and the anisotropy parameter by t v = ta; = " the continuum real space representations of the operators H0a are X

H01 = − H01 = − iv(−1 1 91 + 2 92 ) ; X

H02 = − H02 = − iv(−1 1 92 + 2 91 ) : Here 9i ≡ 9xi (i = 1; 2) and x12 ≡ x ∓ y. The structure of H0 makes the connection between clean d-wave superconductors and the two-dimensional Dirac Hamiltonian manifest. An equivalent but more convenient representation of H0 reads H012 H0 = ; H021 where the four nodal sub-Hamiltonians H0a have been assembled into a single block H012 , and H021 = H012† . The explicit form of H012 in the four-component nodal space is   −i9(1) 1   (1)   i9   1X  H012 = v  (4) (2)    2 −i9X   (2) 2X X i9 with 9(1) = −1 91 − i92 ; (2)

9X

= −1 92 − i91 :

(2) The overbar over 9X does not denote complex conjugation, but is motivated by the observa(2) tion that, for = 1; 9X vanishes on holomorphic functions f(x1 + ix2 ), while 9(1) annihilates antiholomorphic functions f(x1 − ix2 ). We now introduce disorder. In general, both the normal part of the Hamiltonian and the order parameter will contain a random component, which we denote by Vimp and "imp , respectively. Speci:cally, we assume that

• Vimp and "imp are uncorrelated Gaussian distributed variables with • zero mean: Vimp = "imp = 0,

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• and the same variance:

g 2

Ximp (r)Ximp (r ) = f(|r − r |=0) ;

where X = V or X = ". The parameter 0 is a correlation length which we assume to be much larger than the lattice constant a. (This assumption is motivated by the fact that impurity scattering in real d-wave superconductors seems to be predominantly forward [23–25]. To reproduce that feature, we have to take Vimp and "imp to be slowly varying.) The strength of the is measured by the constant g, and f is a correlation function normalized to unity: disorder d 2 r f(|r|=0) = 1. Later, we will argue that taking Vimp and "imp from the same distribution is simply a matter of technical convenience. Allowing the distribution of disorder in the two channels to be diFerent will not change the universality class, and therefore not alter the qualitative behaviour. The stochastic part of the quasi-particle Hamiltonian then has the form   12 Himp  ; Himp ≡ Vimp 2 + "imp 1 =  12† Himp 12 = " where Himp imp − iVimp . Projection gives  w00 w− w−0 w w0−  − w00 12 Himp =  w0 w0 w00 w0− w−0 w−−

of this operator on the space of four low-energy sectors  w0 w0    w  w00

1 1X ; 2 2X

(5)

where the matrix elements are random functions of position given by wpq = ei(px+qy) ("imp − iVimp ) : They describe stochastic scattering between nodes that diFer by a momentum (p; q) (c.f. Fig. 3). Writing gpq ≡ d 2 r wX pq (0)wpq (r) for the coupling between the nodes, and assuming the Gaussian disorder speci:ed above, we have gpq = g−p−q = g d x dy ei(px+qy) f(|r|=0) : Note two consequences which now follow from the inequality 0a : (i) the variances gpq for large momentum transfer (p; q) are suppressed as compared to the variance g in the soft channel, and (ii) the various scattering channels are approximately statistically independent.

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Fig. 3. Impurity scattering between the nodes of a d-wave superconductor. The ellipsoidal lobes indicate the moX 2; 2. X By way of example, the four diFerent scattering channels mentum support of the four low-energy sectors 1; 1; −1 wpq with mean momentum transfer (p; q) = a [(0; 0); (; 0); (0; −); (; −)] acting on node 1 are indicated.

The sum of the pure (4) and disordered (5) operators constitutes the full Gorkov Hamiltonian: H 12 12 H= ; H 12 ≡ H012 + Himp ; (6) H 12† which forms the basis for our analysis below. Before turning to the generic symmetries of this Hamiltonian, we introduce a perturbation which is both interesting and physically relevant. In the present paper we do not touch upon the subject of perturbations involving the electron spin, but we will repeatedly comment on the role of time-reversal invariance and the eFects of breaking it. Time-reversal symmetry is broken in the mixed state of type-II superconductors, where a magnetic :eld penetrates in the form of vortices. Exactly, how vortices inJuence the low-energy density of quasi-particle states of a pure d-wave superconductor has been the subject of an ongoing debate [26 –31]. Progress was recently made by Franz and Tesanovic (FT) [28], who suggested to perform a singular gauge transformation similar to what is done in the Chern–Simons gauge theory of the fractional quantum Hall eFect. Since each vortex carries half a magnetic Jux quantum, a unit cell consisting of two vortices is used. Dividing the vortex lattice into A and B sublattices, FT make a singular gauge transformation (centered around vortices) in the particle sector for the A sublattice, and in the hole sector for the B sublattice. The purpose of the transformation is to cancel the magnetic :eld on average, thereby allowing to describe the quasi-particle states as Bloch waves. Although FT focus on the case of a perfectly regular vortex lattice, the same transformation can be applied when the vortex positions are given a random component. After linearization and projection on a single node

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(and after rotation by ei1 =4 ), the gauge-transformed Hamiltonian for, say, node 1 is mv A H = H01 + ((v2 + v2B )0 + 2 (v2A − v2B ) + −1 1 (v1A − v1B )) : 2 Here vA; B are the supercurrent velocities for the A and B sublattices. We see that the diFerences of the velocities add to Vimp and "imp , while their sum enters as a scalar potential (proportional to the unit matrix 0 = 12 in particle–hole space). The latter fact was :rst understood by Volovik [61], and is the message to carry away from the above discussion: the main consequence of time-reversal symmetry breaking by vortices is the addition of a scalar potential to the single-node Dirac Hamiltonian. Anticipating the discussion of the next subsection, we remark that the diFerent treatment of the particle and hole sectors by FT breaks the symplectic symmetry (7) of the Gorkov Hamiltonian. For our purposes, this causes no problems. The only agent we will need is the scalar potential proportional to v2A + v2B , and this can be produced also from a gauge transformation that treats particles and holes on an equal footing. 2.2. Generic symmetries An essential pre-requisite to the construction of the low-energy eFective theory for dirty d-wave superconductors, just as for disordered metals, is a solid understanding of the fundamental symmetries of the quasi-particle Hamiltonian. In recent years it has become clear that, because of quantum interference channels that owe their existence to the particle-hole degree of freedom of the Gorkov equations, the low-energy quasi-particles of disordered superconductors transcend the established framework of the “threefold way” [62], namely the Wigner–Dyson classi:cation scheme by unitary, orthogonal, or symplectic symmetry. Dirty superconductors generically fall into one of four non-standard symmetry classes [34]. Borrowing the notation from Cartan’s table of symmetric spaces, these have been termed C; CI; D, and DIII. Speci:cally, spin-singlet superconductors (with conserved quasi-particle spin) belong to the classes C or CI, depending on whether time-reversal symmetry is broken or not. While these are the generic symmetry classes of dirty superconductors, other classes appear, and complicate the symmetry pattern, if system-speci:c conservation laws of exact or approximate nature are present. As we have reviewed, the low-energy quasi-particles of d-wave superconductors organize into four nodal sectors. In the physically relevant case of “soft” disorder, the scattering between nodes is suppressed as compared to the scattering within nodes. Of course, on very large time scales any small amount of scattering couples the nodes, leading to generic low-energy behaviour. On intermediate time scales, however, the nodes can be regarded as isolated. What symmetry class the disordered quasi-particles con:ned to a single node belong to, is not immediate from general principles. By inspection, we will :nd that the class is AIII if time reversal invariance is present, and A if not. While the latter is just the Cartan label for the “unitary” Wigner–Dyson class, an example for the former are Dirac fermions in a random U(N ) gauge :eld (see e.g. [63,64]). Having sketched the general picture, we now turn to the details. We begin by recalling that the Gorkov Hamiltonian for a disordered spin-singlet superconductor satis:es the relation C: H = − 2 H T 2

(7)

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which we refer to as “particle–hole” (PH) symmetry. Its physical origin is conservation of spin in conjunction with Fermi statistics, and it is violated if and only if spin ceases to be a constant of the motion. For systems where time-reversal invariance holds, which implies the absence of external :elds and complex order parameter components such as idxy , the Hamiltonian matrix can be chosen to be real symmetric: H = HX = H T . Because of the rotation by ei1 =4 made in going to Eq. (3), time-reversal symmetry here does not take its canonical form but is expressed by T: H = 1 H T 1 :

(8)

When both C and T are valid symmetries, the quasi-particles of the superconductor are said to be in class CI. When T is broken but C is still present, the symmetry class changes to C. The presence of both C and T constrains H to be of the form Z H= ; ZT = Z : (9) ZX Comparing this with the previous section we notice that the Hamiltonian (6) is indeed oFdiagonal but H 12 , unlike Z, is not symmetric. Looking even further back we see that, before linearization and projection on the four nodes, the oF-diagonal block Hij12 = "ij − itij = Hji12 still obeyed the constraint of being a symmetric matrix. What is the origin of this apparent discrepancy? The answer is this. In writing dn*k ≡ dk n +*k and making the assignment to nodes, we used di<erent conventions in the “particle” (d↑ ) and “hole” (d↓ ) sectors, so as to arrange for H 12 in (4) to have entries only on the diagonal. If we had taken the conventions to be the same, we would have obtained H 12 = Z0 + Zimp with   0 i9(1)   (1) 0  −i9   Z0 = v  (10) (2)   0 i9X   (2) −i9X 0 and



w−  w  00 Zimp =   w0− w0

w00 w− w−0 w0

w0− w−0 w−− w00

 w0 w0    ; w00  w

(11)

both of which are symmetric, in agreement with (9). As follows from the expression for Z0 , here one has to imagine that the pure Dirac Hamiltonian Jips the particle-hole spinor between X This interpretation, the particle state of one node, say 1, and the hole state of its conjugate, 1. while perfectly valid and compatible with momentum conservation, is not the one commonly adopted in the literature. There, one follows the convention of interchanging the two hole states, so that the Dirac Hamiltonian simply Jips between the particle and hole states of one and the

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same node. To facilitate comparison with the literature, we tacitly made this change of basis in Section 2.1. The transformation between the two bases is eFected by a matrix we denote by   1 0 1 1 0  1X   (1 =  : (12)   0 1 2 1 0 2X By multiplying the particle–hole spinor with this permutation matrix in the hole sector (but not in the particle sector), the symmetry relations (7) and (8) are transformed into C : H = − (2 ⊗ (1 )H T (2 ⊗ (1 ) ; T : H = (1 ⊗ (1 )H T (1 ⊗ (1 ) :

(13)

As is easily veri:ed, these relations are obeyed by the Gorkov Hamiltonian of Eq. (6). In summary, we distinguish between two choices of basis, and hence between two ways of writing the Hamiltonian. Both have their respective advantages and disadvantages, and depending on the given context we will use one or the other, whichever is better suited. We write He = UHU † for the Hamiltonian in the canonical representation (9). There now exist two diFerent scenarios. If the inter-node scattering is assumed to be strong, a low-energy quasi-particle prepared in a given initial con:guration will quickly attain a state where its wavefunction is uniformly spread over the space of four nodes. In that case, the presence of the nodal exchange operator (1 in (13) is of no consequence, and we are back to the generic situation where the symmetry class is C or CI depending on whether time-reversal invariance is broken or not. On the other hand, if the inter-node scattering in Himp is negligibly weak, the four low-energy sectors decouple, and then the two individual operations in (13), both of which relate nodes to their conjugates, become ineFective. What remains eFective is an operation relating each isolated node to itself. By applying the symmetries C and T in sequence, C

T

H → − (2 ⊗ (1 )H T (2 ⊗ (1 )→ − 3 H3 ; we see two facts: (i) the product operation CT does not mix the low-energy sectors and (ii) if both C and T are good symmetries, the nodal sub-Hamiltonians are odd under conjugation by 3 (the latter follows more directly from the basic equation (3)). Given the presence of disorder, no further symmetries are expected. Thus, the quasi-particle Hamiltonian for an isolated node is constrained only by H = − 3 H3 : This relation is the de:ning equation of the “chiral” symmetry class AIII. Any perturbation that breaks time-reversal invariance destroys T and hence CT. Under such conditions, the nodal sub-Hamiltonians do not have any symmetry other than Hermiticity.

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We say that Hamiltonians of the last type belong to class A, the standard Wigner–Dyson class with unitary symmetry. This concludes our analysis of symmetries. We have seen that, depending on the hard or soft nature of the scattering potential and the presence or absence of time-reversal invariance, the Gorkov Hamiltonian for dirty d-wave superconductors belongs to one of four symmetry classes: CI, C, AIII, or A. These symmetries have physical consequences (such as singular vertex corrections to the density of states, the thermal conductivity, etc.), which are best evaluated in a :eld-theoretical formalism. 2.3. Field-integral formulation Our goal is to compute the disorder average of the Green function, G(E) = (E − H )−1 , and for this purpose we employ the machinery of supersymmetry [52]. In that method, Green functions are generated from a Gaussian functional integral, T T T D8D ei 8 (E−H ) + (8 j+k ) ; Z[j; k] = where Im E ¿ 0 is assumed and H , de:ned in (6), is the Gorkov Hamiltonian projected on the four nodal regions of the d-wave superconductor. S SX ≡ ; 8≡ ; ; % are :elds that have 2 × 2 × 4 components each, where S; SX (;; %) denote complex (Grassmann) :elds with 2 × 4 components in the tensor product of PH and node space. While convergence of the integral requires the commuting components of the :elds 8 and to be related by complex conjugation, the anticommuting components of the two :elds are independent. As usual, Green function matrix elements are obtained by diFerentiating twice with respect to the sources j and k. A noteworthy feature due to the symmetry C in (13) is that retarded (Im E ¿ 0) and advanced (Im E ¡ 0) Green functions are related by G(E) = − (2 ⊗ (1 )G T (−E)(2 ⊗ (1 ) : This has the consequence that the functional Z[j; k] generating a single Green function can also be used to compute the disorder average of the two-particle Green function G(0+ )G(0− ) at zero energy (E = 0± ≡ 0 ± i*). As the emphasis here will be on the basic structure of the theory (rather than on speci:c observables), we temporarily suppress the source content and focus on the functional Z[0]. In the next step, we are going to adapt the functional integral to the particular symmetries of the Gorkov Hamiltonian. In normal conductors, time-reversal symmetry is known to give rise, via the so-called Cooperon mode, to quantum interference corrections to diFusion that are infrared singular in dimension d 6 2. Similar modes appear in the present case [34], as a result of the discrete symmetries C and T. In the impurity diagram technique, these modes

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emerge as divergent series of ladder graphs, or maximally crossed diagrams. For a semiclassical interpretation of these modes we refer to [34]. The :eld-theoretic approach elevates their status to those of Goldstone modes due to the breaking of certain global continuous symmetries, which in turn derive from C and T. In view of this, we now exercise special care to translate the fundamental symmetries of the Hamiltonian into symmetries of the functional integral. For that purpose we :nd it convenient to transform to the representation UHU † = He = 1 Re Z − 2 Im Z introduced in the previous subsection. Using it, we write T T X ; Z[0] = exp(iE(<˜ +
˜ Z =˜ + i=s Z=; X LD = i =

s

˜ =: LE = − 2iE =

(15)

A node-resolved representation of the Lagrangian is obtained by making the decompositions    X  =1 =1  =X   X   1 =  = ≡  X ; =˜ ≡  1X  :  =2   =2  =X =X X 2

2

The overbar here does not mean complex conjugation. Following the conventions of conformal :eld theory, we use it to denote :elds whose correlation functions are antiholomorphic in the limit of zero disorder. In the :nal step, we substitute the nodal decomposition into (15), which

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gives LD ≡ L1D + L2D + L12 D ; s

(1)

X 1 (v9(1) + iw00 )=X X + 2=sX(v9X L1D = 2= 1 1 s

+ iwX 00 )=1

s

+ iw− =X 1 =X 1 + iw− =X 1X=X 1X + iwX − =1s =1 + iwX − =1sX=1X ; (2)

L2D = 2=2s (v9X

s

+ iw00 )=2X + 2=X 2X(v9(2) + iwX 00 )=X 2

s s + iw−− =2s =2 + iw =2sX=2X + iwX −− =X 2 =X 2 + iwX =X 2X=X 2X ; s

s

s

X X X L12 D = 2i(w0− =1 =2 + w0 =1 =2X + w−0 =1X =2

s +w0 =X 1X=2X + wX 0− =1s =X 2 + wX 0 =1s =X 2X + wX −0 =1sX=X 2 + wX 0 =1sX=X 2X) ;

(16)

X X where L1D ; L2D , and L12 D govern the pairs of nodes (1; 1); (2; 2), and the coupling between them, respectively. The factors of two in these expressions arise from combining terms: =s A= ± =s A= = 2=s A= for a symmetric (or antisymmetric) operator A. Eqs. (15) and (16) cast the dirty d-wave problem into the :eld-theoretical form that all subsequent analysis will be based on. 2.4. Symmetries of the Gaussian >eld theory We next explore the symmetries of the Lagrangian (16). The outcome will pre-determine the structure of the low-energy eFective :eld theories that derive from it. To prepare the stage, we recall that the word “symmetries” has two aspects to it. Firstly, there exists a symmetry group, which acts by transformations that leave the Lagrangian invariant. This is the proper meaning of the word “symmetry”. Secondly, the degrees of freedom of the theory, the :elds, take values in a target manifold. In current speak the latter, too, is sometimes referred to as the “symmetry” of the theory, although it is of course diFerent from the symmetry group. The remark we wish to make is the trivial and yet important statement that these two meanings must be kept apart. In the present section we are going to discuss the :rst aspect, namely the symmetry group of the action functional. The structure of the target manifold will be discussed in Section 3, after the introduction of the relevant low-energy degrees of freedom. We begin by considering the case of highest symmetry where all of the disorder except for w00 is switched oF. Symmetry breaking caused by hard scattering wpq and the perturbations that break time-reversal invariance, will be discussed afterwards. Class AIII (forward scattering, T-invariance): on setting E = 0 and wpq = 0 (for pq = 00), the Lagrangian reduces to four independent terms: s

(1)

X 1 (v9(1) + iw00 )=X X + 2=sX(v9X L = 2= 1 1 (2)

+ 2=2s (v9X

+ iw00 )=2X +

+ iwX 00 )=1

s 2=X 2X(v9(2)

+ iwX 00 )=X 2

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X which are distinguished by the dichotomy of holomorphic (=) versus antiholomorphic (=) X and (2; 2). X This Lagrangian describes :elds, and by the grouping into pairs of nodes: (1; 1) (anisotropic) Dirac fermions in a random Abelian vector potential A = w00 ; AX = wX 00 . Since the four terms are identical in structure, but involve diFerent partial derivatives 9(1) = 9(2) for non-vanishing anisotropy = 1, the symmetry group will be a Cartesian product of four copies of the same group, which is readily identi:ed as the supergroup GL(2|2). Indeed, looking at, say, the :rst two terms, and noting that w00 (r) and wX 00 (r) are just complex numbers, we see that a local transformation =1sX(r) → =1sX(r)T (z1 )−1 ;

=1 (r) → T (z1 )=1 (r) ;

s s =X 1 (r) → =X 1 (r)TX (zX1 )−1 ;

=X 1X(r) → TX (zX1 )=X 1X(r) ;

(17)

where z1 ≡ x1 + ix2 and zX1 ≡ x1 − ix2 , leaves the action functional invariant. Because =1 ; =1X comprise two commuting and two anticommuting components, the invertible supermatrix T is of size (2 + 2) × (2 + 2). Hence the symmetry group of the present case is GL(2|2) or, rather, four independent copies thereof. As we shall see, the local nature of the symmetry group makes the present theory exactly solvable. Note that here, and throughout this section, the word “symmetry group” means the complexi>ed [65] symmetry group, which ignores the relation that exists between the bosonic :eld components and their complex conjugates. We mention in passing that the symmetry group of the totally clean limit (w00 = wX 00 = 0) described by free :elds, is four copies of the orthosymplectic supergroup OSp(4|4). This will come to play a role when we turn to the method of non-Abelian bosonization. Class A (soft scattering, no T-invariance): as we saw in Section 2.1, the Hamiltonian He ceases to be oF-diagonal in PH space (while remaining diagonal in CC and node space) when time-reversal symmetry is broken. Hence the breaking of T introduces terms into the Lagrangian that mix the two sectors of holomorphic and antiholomorphic :elds. Consequently, invariance of the Lagrangian requires the actions of GL(2|2) in these sectors to be related to each other. This “locking” of group actions is the only eFect T-breaking has on the symmetries, and therefore the net result is that the four copies of GL(2|2) get reduced to two copies, corresponding to X and (2; 2). X the two pairs of nodes (1; 1) Let us look at the locking of GL(2|2) group actions in more detail. As we recalled in Section 2.1, the supercurrent Jow in the mixed state acts as a scalar potential (vA + vB )1 ⊗ (3 . After transformation to the canonical representation, He = UHU † , this reads (vA + vB )3 ⊗ (3 . Consider therefore the T-breaking perturbation T s T (3 ⊗ (3 )8 = <˜ (3 −
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is de:ned by (TA)s = As T s . From (14) we read oF that As is obtained from A by taking the ordinary transpose, and then exchanging the two commuting (anticommuting) components by multiplication with 1 (i2 ). The equation gT 1 g = 1 de:nes an orthogonal group O(2; C), and gT 2 g = 2 the symplectic group Sp(2; C). Thus the condition T s T = 1 determines an orthosymplectic subgroup OSp(2|2) of GL(2|2). The full symmetry group consists of two copies X and antiholomorphic :elds of (2; 2) X of OSp(2|2), since the holomorphic :elds of the pair (1; 1) can be transformed independently of the remaining :elds. The following remark may be helpful. Given the physical distinction between the BB and FF sectors, there are two versions of OSp to consider. The :rst one acts by symplectic transformations in the former sector and by orthogonal transformations in the latter. (We refer to this as “symplectic” bosons and “orthogonal” fermions for short.) The OSp(4|4) symmetry group of the free-:eld problem is of this kind. In the second version of OSp, which is the relevant one here, the roles of the orthogonal and symplectic groups are interchanged (“orthogonal” bosons and “symplectic” fermions). As another aside, we mention that en route from soft to hard scattering one could imagine a scenario where the scattering couples each node only with its conjugate [5]. Such a scenario would lead to a diFerent symmetry group and eventually to a diFerent target manifold [19]. However, we see no physical room for this theoretical possibility and will not pursue it here. Class C (hard scattering, no T-invariance): the symmetry group for this :nal case can be approached from class A by including hard scattering, or from class CI by breaking T. In either way, one :nds that the remaining symmetry is just a single copy of the group OSp(2|2) with orthogonal bosons and symplectic fermions. This is the orthosymplectic supersymmetry omnipresent in class C [50,66,67,45]. For future reference, the symmetry groups for the four diFerent cases are summarized in the following table, where the notation ×n G means n independent copies of the group G. T

Hard scattering

Symmetry group

+ + – –

– + – +

×4 GL(2|2) ×2 OSp(2|2) ×2 GL(2|2) OSp(2|2)

2.5. Disorder average We now carry out the disorder average. By integrating the Gaussian generating functional over the distribution of the matrix elements wpq , which are subject to the correlation laws stated in Section 2.1, we obtain X e− (LD +LE ) d2 r ; Zav [0] ≡ Z[0] = D=D=

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where LD = 2vL0 + Ldis ; s

s

(1)

(2)

X 1 9(1) =X X + =sX9X =1 + =X 2X9(2) =X 2 + =2s 9X = X ; L0 = = 1 2 1 Ldis = gO00 + g O00 + g0 O0 + g O

(18)

and all perturbations Opq are quartic in the :elds: X s1 =X X=1s = X + 4=2s = X=X s2 =X X ; O00 = 4= 1

s

1

2

2

s

X 1 =X X=X 2 =X X + 4=1s = X=2s = X ; O00 = 4= 1 2 1 2 s

s

s

s

X 1 =2 =1s =X 2 + 4=X 1X= X=sX=X X + 4=X 1X=2 =sX=X 2 + 4=X 1 = X=1s =X X ; O0 = 4= 2 1 2 2 2 1 s

s

s

s

X 1X=X X=sX= X + =X 1 =X 1 =1s =1 + =sX= X=X 2X=X X + =2s =2 =X 2 =X 2 : O = = 1 1 1 2 2 2 In the initial stage of the calculation, disorder averaging produces expressions of the form 2 d r d 2 r f(|r − r |=0)(=s =)(r)(=s =)(r ) ; where f is the correlation function of the disorder. To arrive at the above expressions for Opq , we omitted the :nite spread of this function due to a non-zero value of 0. This approximation is justi:ed by the fact that all corrections from expansion around the local limit r − r = 0 carry at least two derivatives, which renders them irrelevant in the renormalization group (RG) sense. Note that all of the perturbations Opq are marginal by power counting at the free-fermion point. Eq. (18) de:nes the disorder-averaged theory. Before going further, we will inquire into the nature of the RG Jow caused by the perturbations Opq . 2.6. Renormalization group In this section we perform a one-loop renormalization group analysis to explore the relevance of the :ve couplings gpq . There exists a standard formula [68,69] to use for that purpose, which refers to the operator product expansion (OPE) of the perturbating operators. The general statement is that, if the short-distance expansion for a set of marginal perturbations O(i) has the form *‘ ij (k) O(i) (r)O( j) (0) d 2 r = 2 ck O (0) + · · · ‘ ‘¡|r|¡‘+*‘ k

the corresponding couplings gk renormalize according to the equation dgk =
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To apply this formula to the problem at hand, we need the OPE for the fundamental :elds. Apart from an overall multiplicative constant, which can be removed by a conformal rescaling of the :elds, these are =X 1;X A (x1 ; x2 )=X 1; B (0; 0) = =1;X A (x1 ; x2 )=1; B (0; 0) = =2;X A (x1 ; x2 )=2; B (0; 0) = =X 2;X A (x1 ; x2 )=X 2; B (0; 0) =

*AB ; x2 − i −1 x1 −*AB

x2 + i −1 x1

;

*AB ; x1 − i −1 x2 −*AB

x1 + i −1 x2

:

√ Here = = t=" is the (square root of the) anisotropy parameter, and A; B is a composite index built from the BF and CC indices of the :elds. As usual in this context, the above relations have to be understood as identities that hold under the functional integral sign. By using the free-:eld expansions in conjunction with Wick’s theorem, we can now work out the OPE for the set of composite operators Opq . By a straightforward if tedious calculation, this yields the RG Jow equations

g˙ = 0 ; 2 2 + g0 ; g˙ = 12 g

g˙0 = gg0 + 14 g g0 ; 2 ); g˙ = 2(gg + g0

(19)

where g˙i stands for dgi =d ln ‘ (times some unimportant constant which we do not specify). Notice the following features: • The beta functions contain no terms of linear order in the couplings, which expresses the fact

that we are dealing with a set of marginal perturbations.

• In the physical regime of positive couplings, the beta functions are never negative, so none

of the couplings decreases under the RG Jow.

• None of the couplings supports its own Jow. In particular, for the model with only forward

scattering (gpq = 0) the coupling constant g is truly marginal.

• For generic initial data, the nature of the Jow is marginally relevant, i.e. the couplings (save

for the constant g ) increase under the Jow, although the rate of increase vanishes in the limit of weak disorder gi 0. • The anisotropy parameter does not enter the one-loop RG equations. The fact that the anisotropy has disappeared from the RG equations can be understood heuristically as follows. The OPE scheme for the set {Opq } encodes the one-loop renormalization of the

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four-fermion vertices of the theory. These vertices are related to the fermion self-energy through a Ward identity. To explore the eFects of the anisotropy, one may therefore directly analyse the self-energy diagrams. The latter have been shown [5] to be unaFected by the anisotropy, to leading (or one-loop) order. The physical reason is that the :rst-order self-energy diagram measures the Born scattering rate between the four low-energy sectors. This rate is not aFected by the ellipsoidal shape (expressing the anisotropy) of the low-energy lobes in momentum space; it only depends on the total phase volume available. In higher orders of perturbation theory, the situation changes and the self-energy begins to be aFected by the anisotropy in momentum space. In particular, higher-order scattering between neighbouring nodes is suppressed [5]. Accordingly, we expect that the beta functions at higher loop orders do depend on the anisotropy of the model. Let us stress two messages that emerge from the current section: (i) owing to the quadratic dependence of the beta functions on their arguments, the RG Jow is marginal (or very slow) in the limit of small couplings (or weak disorder), and (ii) for generic values of the disorder-generated couplings, the renormalization group Jow drives the system away from the free-fermion theory. In combination with the phenomenological input that the inter-node couplings gpq are small as compared to the intra-node coupling g, the :rst message leads us to expect a prolonged crossover intervening between a ballistic regime at short scales, and the diFusive regime governed by a non-linear sigma model at large scales.

3. How to proceed? Anticipating that a direct perturbative analysis of the theory (18) will not capture all of the important physics, one is tempted to employ the “standard” scheme for dealing with weakly disordered non-interacting particles: Hubbard–Stratonovich transformation followed by a saddle-point analysis and gradient expansion. It turns out, however, that this approach is beset with a number of problems, none of which appears in systems with a non-relativistic kinetic energy. These dif:culties will eventually force us to adopt a strategy where elements of the standard scheme are supplemented, and even superseded, by an alternative scheme tailored to relativistic fermions: non-Abelian bosonization. For pedagogical reasons, we will here proceed in a conservative fashion and continue somewhat further along the much-trodden path (Hubbard–Stratonovich transformation and so on), introducing along the way a few concepts of general validity. In particular, we will identify the degrees of freedom of the hierarchy of low-energy eFective theories, and the way they are associated with the global symmetries discussed in Section 2.4. As a by-product, we will be able to make the connection to previous self-consistent approaches. To begin with, we set the couplings gpq to zero and focus on the functional integral Zfs [0] ≡ Zav [0]|gpq =0 ; containing only the largest perturbation, gO00 + g O00 . The :rst step of the standard approach is to make a Gaussian transformation, called the “Hubbard–Stratonovich transformation”. Introducing two auxiliary :elds P and Q, and exploiting the equality of the unrenormalized coupling

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constants g = g , it is straightforward to show that 2 2 ˜ Zfs [0] = D=D= DQDP e−1=g STr(Q +P )

˜s

s

˜

×e− (LE +2vL0 += (iQ−P)=+= (1 (iQ+P)(1 =) 2 2 = DQDPe−1=g STr(Q +P )

E ×exp − STr ln + i 1 2

iQ − P

iZ0

iZX 0

(1 (iQ + P)(1

;

where the values of P and Q are supermatrices of linear size 4 × 2 × 2 acting in nodal space and internal space (the latter being the tensor product of charge conjugation and boson-fermion space). For simplicity we here ignore the issue of convergence of integrals, deferring a more strict treatment until Section 4. We mention in passing that it can be understood at this stage why the restriction to equally distributed disorder in the normal and order parameter channels makes no essential diFerence: one can relax this assumption by introducing two more Hubbard–Stratonovich :elds, R and S, coupling to the PH matrices 1 and 2 , respectively. On varying the action with respect to these :elds, one :nds that they vanish on the saddle-point level. Since a non-vanishing saddle point is a necessary condition for the formation of Goldstone modes, which control the infrared behaviour, we conclude that these :elds and hence the choice of (un)equally distributed disorder, are of no importance. With the aim of subjecting the functional integral to a stationary-phase analysis, we now vary the action with respect to the :elds Q and P to generate a set of saddle-point equations. We :rst look for solutions (P0 ; Q0 ) that are diagonal in both nodal and internal space. For vanishing energy E, the solutions are P0 = 0 ; Q0 = − i% ; where the real parameter % is de:ned implicitly through d2 k 1 2 : = 2 2 g (2) % + (k)2

(20)

This equation, which can be regarded as the Dirac analog of the SCBA equation for disordered normal-conducting systems (with −Q0 representing the self-energy), has been discussed in the literature (see, e.g., Refs. [2,5]). Its solution is %2 = 2 e−8[t=g ; where represents a UV cutoF that regularizes the integral over momenta, which would otherwise diverge logarithmically. The parameter % can be regarded as a self-consistently determined self-energy. This interpretation is strengthened by re-instating sources j; k into Zfs [0], and then

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using this generating functional to compute matrix elements of the physical Green function. Doing so, one :nds that these are given by the inverse of the operator % − iE iZ0 (21) iZX 0 % − iE and the information content of the above saddle-point approximation coincides with that of the diagrammatic SCBA approach. However, as we know from experience with disordered normal-conducting systems, there exists much physics that does not unfold at the saddle-point level, but resides in the Juctuations of Q. The same is true here. The saddle point (Q0 ; P0 ) = (−i%; 0) breaks the symmetry ×4 GL(2|2) for class AIII as recorded in the table at the end of Section 2.4. By the Goldstone mechanism, this leads to the appearance of massless modes, and one expects that it is the Juctuations in these “soft” modes, rather than any single saddle point, that determines the behaviour of the system at large scales or low energies. Let us therefore ask what happens when the functional integral Zfs [0] is subjected to a global symmetry transformation taken from ×4 GL(2|2). (As we saw, the symmetry is actually local, but we should hesitate to draw any conclusions from that, as the axial part of the symmetry is anomalous; see Appendix A.1). Since these transformations do not mix the nodal sectors X and (2; 2), X we can concentrate on one of them, say (1; 1). X Consider the action functional (1; 1) evaluated at the saddle point, % −iv(2 9(1) 1 ; STr ln (1) 2 iv(2 9X % where (2 denotes the second Pauli matrix acting in nodal space, and the energy E has temporarily been set to zero. Then recall the transformation (17) with the matrices (T; TX ) ∈ GL(2|2) × GL(2|2), which are now taken to be constant in space. By transferring this transformation to the argument of the logarithm, we obtain the “rotated” action   %M s 0 0 −v9(1)    0 %M −1 v9(1)  0   1   ; (22) STr ln (1)  2 %M 0 v9X  0    (1) s −1 X −v9 0 0 %(M ) s where M = T TX . This expression states that not only the diagonal matrix i%, but in fact any GL(2|2) con:guration i% × diag(M s ; M −1 ; M; (M s )−1 ) is a solution of the saddle-point equation. X is isomorphic to GL(2|2). Put diFerently, the saddle-point manifold for the nodal sector (1; 1) X On including an identical factor for the other sector, (2; 2), the total saddle-point manifold of the model with only forward scattering becomes GL(2|2) × GL(2|2). (Note that a complete description of the saddle-point approximation would have to specify the real submanifold to be integrated over. The present discussion gives only the complex saddle-point manifold for simplicity.) We note in passing that what we are encountering here is reminiscent of the phenomenon of chiral symmetry breaking in quantum chromodynamics. The appearance of a chiral symmetry

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group GL × GR = GL(2|2) × GL(2|2) (still maintaining the focus on a single nodal sector), with independent left and right factors, is directly connected with the oddness of the Hamiltonian under conjugation by 3 in PH space: 3 H3 = − H . The latter symmetry is broken by the saddle point Q0 = − i%, which is even under conjugation by 3 . In four space-time dimensions, the role of i3 = 1 2 is played by i5 = 0 1 2 3 , for massless fermions one has 5 H5 = − H , and M becomes the pion :eld. In QCD, as in the present case, the saddle-point manifold is given by a diagonal subgroup GL = GR (the “axial” symmetry transformations) of the chiral symmetry group. The main diFerence turns out to be that chiral symmetry is not truly broken (although its axial part is anomalous) in the vacuum of our 2d theory. For the case of forward scattering with T-invariance (class AIII) the above discussion answers the question, raised at the beginning of Section 2.4, concerning the relationship between the global symmetries of the :eld theory and its degrees of freedom. The former are represented by s T and TX , and the latter by M = T TX . The next step of the standard scheme would be to allow M to vary slowly and carry out a gradient expansion of the action functional. However, for reasons that are spelled out below, we do not pursue this approach here but will switch to an altogether diFerent strategy in the next section. Before doing so, we extend the construction of saddle-point manifolds to the other symmetry classes: CI, A, and C. The procedure is always the same: we let the global symmetry group act on the diagonal saddle-point Q0 = − i%, and the saddle-point manifold is then simply the result of this group action. Class CI (hard scattering, T-invariance): Recall from Section 2.4 that the couplings gpq , which mix the nodes and lead from class AIII to CI, reduce the symmetry group from four copies of GL(2|2) to two copies of OSp(2|2). Because there are no other changes, the above form of s the symmetry-transformed action functional continues to hold, with M still given by M = T TX . s −1 The orthosymplectic group property, T s = T −1 and TX = TX , entails M s = M −1 . As T and TX vary over OSp(2|2), so does M , and hence the (complex) saddle-point manifold is isomorphic to that group. The symmetry group is still chiral, acting on the :eld M (r) independently on the s left and right by M (r) → TM (r)TX . X Class A (soft scattering, no T-invariance): again, we focus without loss on the pair (1; 1). According to the discussion in Section 2.4, breaking of T-invariance by a supercurrent Jow s locks the left and right actions of GL(2|2) to each other by TX = @T −1 @ with @ = 3CC ⊗ (3 . s Since M and T are node-diagonal, this equation simpli:es to TX = 3CC T −1 3CC . The expression s for M = T TX thus becomes M = T3CC T −1 3CC , which is invariant under translations T → Th by elements h that leave 3CC :xed: h3CC h−1 = 3CC . These are easily seen to form a subgroup GL(1|1) × GL(1|1) of GL(2|2). Hence the complex saddle-point manifold is isomorphic to GL(2|2)=(GL(1|1) × GL(1|1)) : This coset space is very familiar [52] as the target manifold of the non-linear sigma model for systems in the Wigner–Dyson class with unitary symmetry (alias class A). The standard symbol X for the :eld in this case is Q = M3CC . Identical statements apply to the pair (2; 2). s CC Class C (hard scattering, no T-invariance): As before, the constraint TX = 3 T −1 3CC from T-breaking leads to Q = T3CC T −1 . What is diFerent now is that the :eld is completely locked in node space, and T ∈ OSp(2|2) obeys the condition T s = T −1 . Again, the :eld Q does not change when T is multiplied on the right by an element h that :xes 3CC under conjugation.

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Such elements turn out to form a subgroup GL(1|1), and Q therefore parametrizes the coset space OSp(2|2)=GL(1|1). This concludes our discussion of saddle-point manifolds. A summary of the essential results was tabulated in the introductory section. Let us mention that major elements of that table have appeared in the literature, although in the framework of the fermion-replica trick: • Prior to the celebrated discovery that the order parameter of high-temperature superconductors

has dx2 −y2 symmetry, Oppermann [35] had studied a system he called the “superconducting Ising glass”, and derived for it a non-linear sigma model over the compact group Sp(2r), with r = 0 owing to the use of the replica trick. Oppermann’s superconducting Ising glass is a time-reversal invariant system with local superconducting order and conserved spin. Thus it has all the prerogatives of class CI, and the target manifold Sp(2r) is just what one expects on the basis of the general classi:cation [50]. For the important case of d-wave superconductivity, the Sp(2r) non-linear sigma model was recently rediscovered by Senthil et al. [6]. • According to Nersesyan et al. [5], the eFect of impurity scattering on an isolated node of a d-wave superconductor is described by a non-linear sigma model (more precisely, by a WZW model) on U(r). We will discuss a supersymmetric variant of that model in some detail later. • The non-linear sigma model with target manifold Sp(2r)=U(r) was identi:ed as the lowenergy eFective theory for dirty d-wave superconductors in class C by Senthil et al. [6]. The supersymmetric extension of that theory had appeared in [70] (to describe the physics of low-energy quasi-particle excitations in disordered superconductor=normal metal junctions) and [66] (as a description of quasi-particles in the core of a disordered vortex). • The appearance of the Wigner–Dyson class of unitary symmetry (class A) for d-wave superconductors in the mixed state is implicit to several recent papers [61,46,28].

The merit of the present work is that it assembles the various symmetry classes and :eld theories into a single coherent scheme. After this extensive tour of symmetries and their :eld-theoretical realization, we are now ready to attack the concrete goal of giving a bona :de construction of the low-energy eFective theories. Equipped with a strong background in the supersymmetric theory of disordered metals, one would deem the strategy to follow quite obvious: one should decompose the Hubbard– Stratonovich :elds P; Q into Goldstone (or massless) modes M and a complementary set of massive modes. One would then integrate over the latter in Gaussian approximation and, :nally, expand to lowest order in the gradients ∇M , measuring the energy cost due to spatial variations in the massless :elds. It came as a surprise to us that this program is corrupted by two sources of severe diSculty: • In the course of carrying out the gradient expansion we are confronted with a two-dimensional

version of the chiral anomaly (see Appendix A.1).

• For the physically interesting limit of systems with T-invariance and forward scattering only

(class AIII), we are unable to isolate and eliminate the massive :elds in a controlled manner.

The presence of an anomaly leads to disastrous results when the gradient expansion is carried out in the most straightforward way. This, however, is a surmountable problem; once the nature

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of the anomaly has been understood, a less naive and properly regularized expansion scheme can be set up to yield the correct answer. The second problem is more serious. Its origin is that Dirac fermions are perturbed by a random Abelian vector potential (which is the concrete form taken by the disorder in the present realization of class AIII) in a truly marginal way. What this means is that the system retains the conformal invariance of the free-fermion theory, with no mass scale being generated. As a result, any attempt to integrate out the “massive” modes perturbatively is doomed to fail. In principle, one could try to go beyond the Gaussian saddle-point approximation for the massive modes but, in practice, this seems unfeasible. In this situation it comes as a relief that an alternative approach to the problem exists: the method of non-Abelian bosonization due to Witten [38].

4. Non-Abelian bosonization The utility of non-Abelian bosonization as a tool to construct the low-energy eFective action for the quasi-particles of a d-wave superconductor was :rst recognized by Nersesyan et al. [5]. These authors used the fermion-replica trick to deal with Dirac fermions perturbed by various types of disorder. By bosonizing the disorder-averaged fermion-replica theory, they arrived at a WZW model. This model had the attractive feature of exact solvability, which enabled NTW to predict some exact values for the scaling exponent of the low-energy density of states. As discussed in Section 1, however, the original approach of NTW has a number of weak spots, one of which, namely the neglect of the fact that the multi-valued term in the WZW action changes sign when the orientation of position space is reversed, bears drastic consequences. Following NTW’s original strategy, it is the goal of this section to introduce a bosonization scheme in which (i) all symmetries of the system are fully included, (ii) the dependence of the WZW action on orientation is accounted for, and (iii) the physics of the crossover regimes in between the limiting cases of hard and soft scattering is included. We will begin by reviewing a few elements of the general bosonization approach to supersymmetric theories, and then discuss the application of these concepts to the case at hand. Originally, non-Abelian bosonization was introduced to construct bosonic representations of relativistic two-dimensional fermion models with continuous internal symmetries [38]. More recently, the approach has been extended [49] to the supersymmetric case of relativistic fermions supplemented by a bosonic ghost system, which is the setup needed to treat problems with disorder. Below we will review some elements of this approach, in a form tailored to the d-wave application. Readers who are familiar with bosonization will :nd that there are no structural diFerences to the original, fermionic version. Readers who are not may wish to consult :rst an introduction to standard non-Abelian bosonization, e.g. Witten’s original and highly pedagogical article [38]. A detailed account of the principal supersymmetric extension of the approach can be found in Ref. [49]. Before turning to the problem of the d-wave superconductor, let us begin with some preliminary considerations of a generic model: consider the functional integral X Z ≡ D D X e−S0 [ ; ] (: : :) ; X ;

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where and X are :elds with 2m bosonic and 2m fermionic components and the ellipses (: : :) ; X stand for certain operators constructed from these :elds. The action functional S0 denotes the free supersymmetric action t X S0 [ ; ] = d 2 r ( X 9 X + t 9X ) ; (23) where → t is an “orthosymplectic” transpose which di<ers from the s-operation introduced earlier in that it de:nes a skew symmetric scalar product: t

=−

t

:

(Although skew symmetry is the only property that matters for our purposes, it may be helpful to recall the concrete realization given in [49] for n = 1: t = (<; ; b; −c) and = (; <; c; b)T , where Greek and Roman letters denote fermionic and bosonic components, respectively.) It is implicitly assumed that the existence of the functional integral is ensured by the presence of some convergence generating term in the action. (For an explicit example of such a term, see below.) Furthermore, let M be a :eld taking values in the (anomalous) symmetry group of S0 , which is the complex supergroup OSp(2m|2m) of matrices ful:lling M t M = 1, with the matrix operation M → M t de:ned by (M )t = t M t . The basic statement of non-Abelian bosonization then is that the functional integral Z de:ned above is equivalent to Z ≡ DM e−W [M ] (: : :)M ; C|D

where W [M ] ≡ −

1 16

d 2 r STr(9 M −1 9 M ) +

i [M ] ; 24

[M ] is the WZW functional, and the ellipses stand for a representation of (: : :) ; X in terms of the matrix :eld M to be speci:ed momentarily. The phrase “equivalent” here means that, with the transcription (: : :) ; X ↔ (: : :)M understood, the two functional integrals Z and Z produce identical results. Following Witten, an explicit expression for can be written down by choosing some extension of the :eld M to a one-parameter family of :elds M˜ (t) with the property M˜ (0) = 1 and M˜ (1) = M . Then 1 −1 −1 −1 [M ] = d 2 r dt G STr(M˜ 9G M˜ M˜ 9 M˜ M˜ 9 M˜ ) ; 0

where (G; ; ) represents the coordinate triple (t; x1 ; x2 ), and G is the fully antisymmetric tensor. The bosonization dictionary needed for the transcription (: : :) ; X ↔ (: : :)M is given as follows: t

↔ J ≡ (2)−1 M 9M −1 ;

X X t ↔ JX ≡ (2)−1 M −1 9XM ;

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X t ↔ ‘−1 M ; X

t

↔ ‘−1 M −1 ;

(24)

where ‘ is some length scale serving to UV-regularize the Dirac theory. Finally, the subscript “C |D” in the de:nition of Z indicates that the functional integration over M does not really extend over OSp(2m|2m). The reason is that the invariant metric STr(M −1 dM )2 on this group, as on any complex group, is not Riemannian. This in turn implies that a functional integral controlled by the metric term STr(9 M −1 9 M ) cannot be de:ned by an unrestricted integration over OSp(2m|2m). (This term has inde:nite sign, which spoils the convergence of the integral.) For ordinary groups the obvious remedy is to restrict the complex group to a compact real subgroup; one passes, for example, from O(2m; C) to O(2m). No such remedy exists for supergroups such as OSp(2m|2m), as these do not possess subgroups on which the invariant metric becomes of de:nite sign. To rescue the functional integral, one has to abandon the group structure and restrict OSp(2m|2m) to what is called a Riemannian symmetric superspace of type C |D in the terminology of Ref. [50]. Referring for a detailed discussion to Ref. [49], we here merely mention that the BB sector of this manifold is given by the non-compact coset space Sp(2m; C)=Sp(2m), while the FF sector is the compact group O(2m). We next apply 2this general apparatus to the d-wave system. Firstly, it is necessary to cast the free action d r L0 (we here set v = 1) of the system into the form of the action S0 above. To this end we de:ne XX = t s s 1 X ≡ (=X 1 ; −=X 1X); X ≡ ; 1 1 =X 1 = 1 t s s ; 1 ≡ 1 ≡ (=1X ; =1 ); −=1X X X t ≡ (=X s2X; −=X s2 ); X ≡ =2 ; 2 2 =X 2X =2X t s s : (25) 2 ≡ 2 ≡ (=2 ; =2X ); −=2 t t t Note X 1 X 2 = − X 2 X 1 and 1t 2 = − 2t 1 (skew symmetry), but X 1 that the free part of the Lagrangian can be represented as t (n) L0 = ( X 9(n) X + nt 9X n ) ; n=1;2

LE = − 2iE

n

n=1;2

t 2 =+ 2

X . One then veri:es 1

n

Xt

n n

:

To make this Lagrangian amenable to a direct application of the bosonization rules, we temporarily remove the anisotropy inherent to the operators 9(n) . This can be done by rescaling the

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X according to x1 → −1 x1 ; x2 → x2 and the coordinates coordinates for the sector of nodes (1; 1) X according to x1 → x1 ; x2 → −1 x2 . The derivative operators for the complementary sector (2; 2) (2) then assume the isotropic form, 9(1) → 91 − i92 ; 9X → 92 − i91 . (It must be kept in mind, however, that in the two sectors diFerent scaling operations were performed. This type of scaling is only meaningful for those elements of the theory which do not involve a coupling between neighbouring nodes. We will therefore undo the scaling immediately after the individual sectors have been bosonized.) The bosonization of the free Lagrangian L0 then leads to the sum W [M1 ] + W [M2 ] of two WZW actions, where W [Mn ] represents the sector (n; n) X and Mn (n = 1; 2) are two independent :elds taking values in the Riemannian symmetric superspace of type C |D associated with the supergroup OSp(4|4). From the symmetry of the orthosymplectic currents, Jn = n nt = − (2 Jns (2 t s and JX n = X n X n = − (2 JX n (2 , the :eld Mn inherits the property Mn−1 = (2 Mns (2 : As before, (k (k = 1; 2; 3) denotes the Pauli matrices acting in node space (presently, the two-component space underlying the de:nition (25)). To prepare the bosonization of the remaining operators, including those that couple the nodes, we now undo the above scaling operation. This is achieved by making the replacements W [M1 ] → W [M1 ; ];

W [M2 ] → W [M2 ; −1 ] ;

where i 1 W [M; ] ≡ [M ] − 24 16

d x1 d x2

×STr(−1 91 M −1 91 M + 92 M −1 92 M )

is the anisotropic analog of the WZW action above. Notice that, owing to its topological character, the WZW term [M ] is not aFected by the scaling operation. We now bosonize the remaining content of the theory, by making use of the dictionary. We obtain t O00 = ( X (3 X )( nt (3 n ) n

n=1;2

↔ O00

n

STr((3 JX n ) STr((3 Jn ) ;

n=1;2 t ( X 1 (3

X )( X t (3 X ) + ( 1t (3 1 )( 2t (3 2 ) 2 1 2 ↔ STr((3 JX 1 ) STr((3 JX 2 ) + STr((3 J1 ) STr((3 J2 ) ; =

O =

1 Xt X ( n (k n )( nt (k 2

n)

n=1;2 k=1;2

↔

1 STr((k JX n ) STr((k Jn ) ; 2 n=1;2 k=1;2

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1 t O0 = X 1 2

2

Xt

2 1

+

X t (k 1

k=1;2;3

↔ ‘−2 STr M1 M2 + ‘−2 LE = − 2iE

n=1;2

2

X t (k 2

1

STr M1 (k M2 (k ;

k=1;2;3

X t nt n

↔ −2i

E STr Mn : ‘

The sum S ≡ W [M1 ; ] + W [M2 ; −1 ] +

n=1;2

d 2 r Lpert ;

Lpert = gO00 + g O00 + g O + g0 O0 + LE

(26) (27)

represents our :nal result for the low-energy eFective action of the disordered d-wave superconductor: two WZW actions, which are coupled by a number of marginally relevant operators due to inter-node and intra-node scattering of the quasi-particles. Note that the rotational symmetry of Euclidean space (or Lorentz invariance of Minkowski space) is broken by the term O00 , which is a relict of the order parameter symmetry dx2 −y2 of the superconductor. In the following subsections we are going to discuss some physical consequences of the :eld theory (27). 4.1. Hard scattering, T-invariance (class CI) We :rst discuss the case where inter-node scattering is present, and is strong. By construction of the bosonized theory, the RG Jow equations for its couplings coincide with those for the perturbed Dirac theory. From (19) we then know that all couplings, including g0 , increase under renormalization. While O00 , O00 , and O are current–current perturbations, the operator O0 is seen to act as a “potential”. We expect such a perturbation to make some of the :elds massive and remove them from the low-energy theory. To elucidate this eFect, we parametrize the :elds Mn as follows: + An + A− Bn n (n = 1; 2) : Mn = exp − Cn −A+ n + An s The requirement Mn−1 = (2 Mns (2 ∈ OSp(4|4) is satis:ed by imposing the conditions (A± n) = ± ± s s An , Bn = Bn , and Cn = Cn for n = 1; 2. In this parametrization, the expansion of O0 around unity reads − − 2 2 + 2 ‘2 O0 = STr((A+ 1 ) + (A2 ) + (A1 + A2 ) + B1 C1 + B2 C2 ) + · · · :

It is now extremely important that, by construction [49] of the Riemannian symmetric superspace of type C |D, this potential energy is bounded from below by zero. The low-energy con:gurations of the :elds M1 ; M2 are those that minimize O0 , which implies − − + A+ 1 = A2 = A1 + A2 = Bn = Cn = 0

(n = 1; 2) :

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By inserting the solution of this equation into the parametrization for Mn , we see that the low-energy :elds are scalar in node space: M 0 M1 = = M2−1 ; 0 M where M , a supermatrix of size (2+2)2 , is subject to the condition M −1 = M s . The last equation de:nes the orthosymplectic supergroup OSp(2|2) with “orthogonal” bosons and “symplectic” fermions. (Technically speaking, M takes values in a Riemannian symmetric superspace of type D|C inside OSp(2|2).) Inserting the constrained form of M1 and M2 into the operators O00 , O00 , and O , we see that all of these vanish. The eFective action thus reduces to SeF = 2W [M ; ] + 2W [M −1 ; −1 ] + −1 =− d 2 r STr 9 M −1 9 M : (28) 8 This is isotropic and independent of the disorder strength, and its FF part coincides with the replica :eld theory written down by Senthil et al. [6]. The WZW terms have cancelled because [M1 ] + [M2 ] = 0 for M1 = M2−1 . In view of recent statements to the contrary [18], we emphasize that the cancellation of WZW terms is a robust feature that does not depend on any speci:c model assumptions made (as long as the disorder is generic, placing the model in class CI). The basic mechanism behind the cancellation is the dependence of the WZW term on parity or, equivalently, the choice of orientation of two-dimensional space: its sign gets reversed by the transformation x1 ↔ x2 . Although this dependence by itself does not forbid the presence of a WZW term for parity-invariant systems (as the sign change can be absorbed by a target space isometry M → M −1 ), it does so for a d-wave superconductor in zero magnetic :eld. The crucial fact here is that the pure Dirac X correspond to opposite orientations (they map Hamiltonians for the two pairs of nodes (1; 1) onto each other by the transformation x1 ↔ x2 ). Therefore, in the bosonized theory they are represented by WZW actions with topological coupling constants that still carry opposite signs, provided that uniform conventions for assigning Dirac bilinears to WZW :elds are in force. (In the above treatment, we arranged for the signs to be identical by choosing our conventions to be diFerent for the two nodal sectors. This was done for notational convenience.) When any kind of scattering (consistent with the generic symmetries of the system) between neighbouring nodes is turned on, the two WZW :elds M1 and M2 become locked in the low-energy theory, now by M1 = M2 , and the topological couplings with diFering signs inevitably add up to zero. The model (28), with the WZW term being absent, is called the principal chiral non-linear sigma model on OSp(2|2) (more precisely, on a Riemannian symmetric superspace of type D|C, which is a supermanifold based on the direct product of the non-compact space R+ = SO(2; C)=SO(2) with the compact group Sp(2)). Referring for a more detailed discussion of its replica analog to Ref. [6], we here review only one salient feature of this theory. According to Friedan [71], the one-loop beta function of any 2d non-linear model is determined by the Ricci curvature of the target manifold. For symmetric superspaces, just like for ordinary symmetric spaces, the Ricci curvature is proportional to the metric tensor, and in the present case is easily shown to be positive [72]. This means that the beta function is positive, and the coupling t = 8=( + −1 ) therefore increases under renormalization. By plausible extrapolation to strong

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coupling (t → ∞), one then expects the theory to be in an insulating phase with vanishing (spin) conductance s = 2=(t) → 0. This phase was named the “spin insulator” in Ref. [6]. The local density of states of the spin insulator is predicted to vanish linearly, (E) ∼ |E |, at ultra-low energies, by a scaling argument due to Senthil and Fisher [13]. On the basis of the supersymmetric :eld theory (28), we can phrase the argument as follows. The growth of the coupling t under renormalization means that, while there is asymptotic freedom (accompanied only by small :eld Juctuations) at short scales, the Juctuations of M grow strong for large wavelengths. The scale for crossover from weak to strong coupling is set by the localization −1 length, 0 ∼ e+ [6]. Let us therefore partition the system into blocks of linear size 0. To mimic the crossover between short-range order and long-range disorder, we take the :elds on diFerent blocks to be independent, and on each individual block to be spatially constant. By doing the :eld integral in this simple approximation we :nd (E) = 0 f(E0 02 ) ; where 0 is the SCBA density of states, which depends only weakly on energy, and the scaling function f(x) (with asymptotic limit f(∞) = 1) has the small-x expansion f(x) = x=4 + O(x2 ). Thus the local density of states goes to zero linearly at E = 0, with the characteristic energy scale being given by (0 02 )−1 , the level spacing for one localization volume. 4.2. Soft scattering, T-invariance (class AIII) We now set the inter-node couplings g ; g0 to zero. What then remains is the free-fermion theory, as represented by the sum of WZW actions W [M1 ] + W [M2 ], marginally perturbed by the operator gO00 + g O00 . Two observations simplify the analysis of this theory. Firstly, we may omit the term O00 when calculating the density of states. The physical reason is that in the X and take the absence of inter-node coupling we can project on one pair of nodes, say (1; 1), other one into account by multiplying the :nal answer with a factor of 2. Thus, we may set X 2 = 2 = 0 and drop the :eld M2 . Alternatively, the reduction can be seen by reasoning within the :eld-theoretical formulation, where it comes about because the functional integral giving X has an invariance under BRST the contribution to the density of states from the pair (1; 1) X Secondly, we may rescale the coordinates to make transformations acting in the sector (2; 2). the :eld theory isotropic. We are then facing an isotropic WZW model with a current–current perturbation: W [M ] + g d 2 r STr((3 J ) STr((3 JX ) : (29) From our earlier computation of the RG beta functions we know that the coupling constant g does not Jow. This can be veri:ed directly from the WZW model by observing that the OPE of the Abelian current STr (3 J with itself does not generate any UV singularities. Thus the above action is a :xed point for the renormalization group, and the theory is conformally invariant. What are its properties? A quick approach to the answer is to recall that the physical problem we are dealing with here is the much studied [5,73,39] problem of Dirac fermions in a random Abelian vector potential. This problem can doubtless be analysed via the above action based on OSp(4|4). Unfortunately, that formulation does not seem to be the most eScient one to use

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and, in any case, it is not the one used in related previous work. What forced us to introduce the OSp(4|4) target space was the fact that we do not know of any other way of bosonizing the perturbations that couple the neighbouring nodes. In the present problem, however, where the coupling between the nodes is absent, the symmetry group consists of two copies of GL(2|2), the subgroup of elements h of OSp(4|4) with the property h(3 h−1 = (3 . The excess of :eld degrees of freedom in OSp(4|4), as compared to GL(2|2), means that there is some redundancy. An eScient method of solution will try to avoid this redundancy, and can avoid it as follows. In standard non-Abelian bosonization (without bosonic ghosts), two schemes are distinguished: bosonization takes the free-fermion theory into a level-one WZW model either over the orthogonal group O(2n), or over the unitary group U(n). While the two bosonization schemes are equivalent in the free limit, the O(2n) version has the advantage of exposing the full set of symmetries of the free theory of n Dirac fermions (or 2n Majorana fermions), whereas the U(n) version is better adapted to the treatment of certain perturbations. The latter scheme was developed particularly by A\eck [74] in his treatment of the 1d Hubbard and Heisenberg models. By straightforward transcription of Ref. [49], we can generalize the U(n) bosonization scheme to include a bosonic ghost system. The resulting theory is again a supersymmetric WZW model, with an action functional of the same form as before, but the :eld now takes values in the supergroup GL(2|2) (or, rather, in a Riemannian symmetric superspace of type A|A inside GL(2|2)). Deferring the details of this generalization to Appendix B, we here take a short cut. To arrive at the bosonized theory in the desired form, we start from the action functional (29) and simply insert for M1 the reduced expression M 0 M1 = ; 0 (M s )−1 where M ∈ GL(2|2) is parametrized, for example, by XBB XBF † † ; XBB = XBB ; XFF = − XFF : M = exp XFB XFF The action functional for M then becomes i[M ] 1 S[M ] = − d 2 r STr 9 M −1 9 M + 8 12 g + 2 d 2 r STr(M −1 9M ) STr(M 9XM −1 ) :

(30)

The same result is obtained by direct application of the GL(2 |2) bosonization rules to the supersymmetric Dirac theory with the marginal perturbation g O00 d 2 r. The numerical (or bosonic) part of this action has the important property of being bounded from below (for g not in excess of a certain critical value), and hence leads to a well-de:ned functional integral. (Positivity of the action is ensured by the construction of the :eld manifold as a Riemannian symmetric superspace.) Past solutions of the physical problem of Dirac fermions in a random Abelian vector potential have used the equivalent representation by a replica sine-Gordon model [5], or diagonalization of the quantum Hamiltonian by a Bogoliubov transformation [73], or the study of Kac–Moody

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current algebras with U(1|1) × U(1|1) symmetry [39]. A closely related model has been investigated via sigma model techniques in [75]. Here we are going to take a diFerent approach: we will solve the problem by direct manipulation of the functional integral. Following Knizhnik and Zamolodchikov [76], this is possible by exploiting the local (in:nite-dimensional) symmetry GL × GR (with G = GL(2|2)). For better readability, the technical details of this computation have been relegated to Appendix B, and we now just give a summary of the most important points. The basic idea is to study the response of the functional integral to :eld variations of the form *X M (r) ≡ −X (r)M (r) ; where X (r) takes values in the Lie algebra gl(2|2). As usual, variation of the action yields the Noether current: 1 *X S = − d 2 r J9X X ; 2g JX = − STr(XM 9M −1 ) + STr(X ) STr(M 9M −1 ) : From the stationarity condition *X S = 0, one infers that J is a holomorphic conserved current, i.e. 9XJA = 0 for spatially constant A and on solutions of the equations of motion. In addition, the theory has a conserved current JX which is antiholomorphic: 9JX A = 0. The expression for it is determined by the change of S in response to variations *Y M (r) = M (r)Y (r). In the following we concentrate on the holomorphic sector. By exploiting the invariance of the functional expectation value : : : JB (0) (where the ellipses indicate additional operator insertions) w.r.t. the variable change M → e−X M = M + *X M + · · · with *X M = − XM , one deduces the OPE for the currents: f(A; B) J[A; B] (0) + JA (z)JB (0) = + ··· ; z2 z 2g f(A; B) = − STr(AB) + STr(A) STr(B) : (31) Like any operator product expansion, it is to be understood as an identity that holds under the functional integral sign. The dots indicate terms which remain :nite in the limit z → 0. The behaviour of correlation functions under scale changes or, more generally, under conformal transformations z → (z), is determined by the OPE of the :elds with the holomorphic component of the stress–energy–momentum tensor, T (z). To obtain the latter, one demands that the OPE between T (z) and the currents starts as T (z)JA (0) = JA (0)=z 2 + · · ·, which expresses the fact that JA (z) has conformal dimension (1; 0). Assuming T (z) to be of the Sugawara form and using Eq. (31), one then :nds %ij 1 − 2g= T (z) = Ji (z)Jj (z) + (32) Je (z)Je (z) : 2 2 The notation here means the following: choosing some basis {ei } of the Lie superalgebra gl(2; 2), we set Ji = Jei , and %ij = − STr ei ej , and we raise the indices of the metric tensor by %ij %jk = *ik . The current Je , whose square gives the second term in the formula for T (z), represents the generator e = id (unit matrix).

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The observable we are interested in is the local density of quasi-particle states at low energy. Recall from (15) that a :nite energy E is accounted for in the :eld theory by a perturbation X s1 =1 + =X s1X= X). This bosonizes to −2i(E=‘) d 2 r STr(M + M −1 ). Moreover, the −2iE d 2 r(= 1 local density of states at r = 0 is given by the expectation value of the bosonic (or fermionic) s s part of =X 1 (0)=1 (0) + =X 1X(0)=1X(0), which bosonizes to (2‘)−1 Tr(M (0) + M (0)−1 ). Thus, the density of states is determined by the expectation value of the fundamental :eld M and its inverse M −1 . To work out its scaling behaviour, we need the conformal dimension of M , which is the number "M multiplying the leading singularity in the OPE of T (z) with M (0): T (z)M (0) = "M M (0)=z 2 + · · · : Given the Sugawara form (32) of the stress–energy–momentum tensor, the value of "M follows from the OPE JA (z)M (0) = − AM (0)=z + · · · ; which reJects the fact that, by construction, JA represents the generator of left translations M → e−A M . Using that the quadratic Casimir in the fundamental representation of GL(2|2) vanishes (%ij ei ej = 0), one :nds "M = 12 − g=. The total scaling dimension of M is the sum (from holomorphic and antiholomorphic sectors) " = "M + "X M = 1 − 2g= : To calculate the local density of states, (E), we add −2iE d 2 r STr(M + M −1 ) + G Tr(M (0) + M (0)−1 ) to the action functional and diFerentiate with respect to the source G at G = 0. If the short-distance cutoF is raised by a renormalization group transformation r → br, the energy scales as E → b2−" E and the local source as G → b−" G. Consequently, (E) obeys the homogeneity relation (E) = b−" (b2−" E) : The solution is a power law " 1 − 2g= (E) ˙ |E | ; = = : (33) 2 − " 1 + 2g= Thus the density of states is expected to vary algebraically with a non-universal exponent that depends on the disorder strength g. This is in agreement with a result :rst found by Nersesyan et al. [5]. Let us mention in passing that the critical line for Dirac fermions in a random Abelian vector potential has been argued [39] to be unstable with respect to an in:nite number of relevant perturbations. In our opinion, this instability is not a true eFect but disappears when the theory is formulated on a target manifold with Riemannian structure (see Appendix B.1). 4.3. Realistic scenario? Having reviewed the two extreme cases of strongly coupled and completely decoupled nodes, it is pertinent to address what kind of scenario will prevail under realistic conditions. By the

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very nature of the question, no clear-cut answer exists. Yet assuming that the basic modelling by the microscopic Hamiltonian (3) has some :nite overlap with experimental reality, we believe that two basic scenarios are conceivable: Systems with a signi>cant amount of large-momentum transfer scattering on the microscopic level. Here the adjective “signi:cant” refers to a regime in which the strength of the inter-node coupling is in excess of the other characteristic energy scales of the problem (such as temperature, experimental resolution, etc., which couple to the action via the energy parameter E). In that case, relative Juctuations of the two :elds M1 and M2 are negligible, and it is only the isotropic content of the theory, described by the action (28), that matters. A more complicated situation arises in systems whose microscopic Hamiltonian contains only soft scattering: in systems where the inter-node scattering is a small perturbation, we run into a crossover scenario. The bare theory is given by two WZW actions, weakly perturbed by inter-node scattering. As we saw, renormalization makes the inter-node couplings grow. Therefore, the theory Jows to strong coupling on asymptotically large scales and the :elds for the diFerent nodes will again be locked. However, that :xed-point regime may be preceded by a large intermediate region in which the critical properties of the WZW theory prevail. In that case we expect for the DoS a power law which is both non-trivial and non-universal, as predicted by (33). Summarizing, the d-wave system renormalizes to a universal limit described by the rotationally and parity invariant action (28). There may, however, be an extended crossover region in which the system of decoupled single-node actions dominates. The question of which type of scenario prevails in real and numerical experiments is system speci:c and not for the present approach to decide. 4.4. Soft scattering, broken T-invariance (class A) We have argued in Section 2.1 that perturbations breaking time-reversal invariance reduce the symmetry of the d-wave superconductor without inter-node scattering from class AIII to A. In the mixed state, as we recall from Section 2.1, T-breaking is due primarily to the supercurrent Jow, which gives rise to a scalar potential in the quasi-particle Hamiltonian projected on a single node. This issue is resumed in the present subsection, where we address the perturbation X by of the class AIII Dirac Lagrangian (for the pair of nodes 1; 1) imv s vs (=X 1 3CC =1 + =1sX3CC =X 1X) ; 2 where vs = v2A + v2B is the second component of the supercurrent velocity. According to the bosonization rules derived in Appendix B, this perturbation transforms into s =X 1 3CC =1 + =1sX3CC =X 1X → ‘−1 STr 3CC (M + M −1 ) :

As before, M takes values in (the Riemannian symmetric superspace of type A|A inside) GL(2|2). Thus the complete action functional is imv S[M ] + d 2 r vs STr 3CC (M + M −1 ) ; 2‘

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where S[M ] was de:ned in Eq. (30). The perturbation has positive scaling dimension 2 − " = 1 + 2g= and is therefore relevant. We expect such a perturbation to cause a reduction of the :eld manifold. Unlike the symmetrybreaking terms we considered earlier, the present one is imaginary-valued, giving rise to oscillations in the functional integral. The argument for :eld reduction therefore cannot be the rigorous kind of potential-energy argument that applied to the crossover from class AIII to CI. What we must rely on now is stationary phase, i.e. the fact that contributions from :eld con:gurations with M -dependent phases tend to cancel out. Starting from the identity element M = 1 and conjugating by a symmetry transformation for class A, M → TM3CC T −1 3CC with T ∈ GL(2|2), we get a stationary-phase manifold parametrized by M3CC = 3CC M −1 = T3CC T −1 ≡ Q : By construction, the above T-breaking perturbation vanishes on this manifold, STr 3CC (M + M −1 ) → 0 : As for the other terms in the Lagrangian, it is straightforward to show that they reduce to STr 9 M −1 9 M → STr 9 Q 9 Q ; STr(M 9M −1 ) STr(M −1 9XM ) → 0 : The fate of the WZW term [M ] under the :eld reduction is less trivial. Doing the same steps as in Section 7 of Ref. [49], we :nd that [M ] reduces to a topological theta term on the manifold of Q-matrices: i[M ] → Stop [Q] ≡ iJ d 2 r Ltop ; 12 Ltop = STr Q 9 Q 9 Q 16i with J = ± . The ambiguity in the sign of J results from the multivaluedness of the WZW term: to calculate [M ] one must extend the :eld M to a 3-space bounded by the two-dimensional position space, and there is no unique way of doing that. However, if the position space is closed (as is implied by its being a boundary) the ambiguity in the sign of J has no consequence. In that case it can be shown that the topological term is a winding number or, in other words, the integral of the topological density Ltop over any closed surface takes quantized values in 2iZ, so that a shift of J=2 by any integer n is unobservable in the functional integral. For the pair X the situation is similar. of nodes (2; 2) To summarize, we :nd that the low-energy eFective action for the pair of nodes (n; n) X of a d-wave superconductor in symmetry class A is 1 S (n; n)X [Q] = ± Stop [Q] − d 2 r STr(−sn 91 Q91 Q + sn 92 Q92 Q) ; 8 where s1 = 1 and s2 = − 1. In the isotropic limit = 1, this action functional is familiar from the integer quantum Hall eFect, where it is known as Pruisken’s action [78] with longitudinal electrical conductivity xx = 1= and Hall electrical conductivity xy = J=2 = ± 1=2. The physics associated with this action is (i) a :nite and smooth density of states at E = 0, and (ii) critical behaviour in the transport coeScients.

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At :rst sight, one might be suspicious of our non-zero result for the coupling xy = J=2. What is puzzling about it is this. Since the Hall conductivity transforms as a pseudoscalar, i.e. has its sign reversed by the exchange of Cartesian coordinates x ↔ y, a non-vanishing Hall response breaks parity. On the other hand, the massless Dirac theory is invariant under parity (if the transformation x ↔ y is accompanied by the exchange of the left-moving and right-moving :elds) and so is the WZW model (if x ↔ y is combined with M ↔ M −1 ). It goes without saying that a parity-invariant theory cannot generate pseudoscalar observables. On these grounds, recent papers have stated [79,46,80] that the thermal Hall conductivity of a d-wave superconductor in the mixed phase vanishes in the linear (or Dirac) approximation. Does it follow from this symmetry argument that the reduction to the non-linear sigma model and=or our non-zero 2result for the coupling xy must be false? It does not. Indeed, recall the invariance of exp iJ d r Ltop under shifts J → J ± 2, when the position space has no boundary. This invariance means that xy = 12 and xy = − 12 give equivalent :eld theories for the case of a closed system. Thus, although xy is a pseudoscalar, the non-linear sigma model at xy = ± 12 enjoys the property of being parity-invariant, and there does not exist any conJict with the parity invariance of the massless Dirac theory (or the WZW model). At the same time, it would be a preposterous proposition to claim that Pruisken’s non-linear sigma model for the integer quantum Hall eFect (IQHE) at xy = ± 12 describes a parity-invariant physical system with vanishing Hall response! How do we then reconcile parity invariance at xy = ± 12 with IQHE phenomenology? The discrepancy is resolved by observing that the Hall conductivity is not a Fermi-edge quantity, which is to say that it is not determined by the Green’s functions at a single energy E = EF only, but is given by a sum over energies. This feature of the Hall conductivity, which distinguishes it from the longitudinal conductivity xx , poses a diSculty to the non-linear sigma model formulation: since the :eld theory is derived for a :xed energy, all the information needed to reconstruct the Hall response in the bulk cannot be contained in it. In other words, the topological coupling xy of the non-linear sigma model does not coincide with the total Hall conductivity of the bulk, and hence the parity invariance of the :eld theory at xy = ± 12 does not imply that the Hall response of the quantum Hall system vanishes. Nonetheless, as was pointed out by Pruisken [78], it is possible to monitor the Hall response :eld-theoretically by studying systems that have a boundary. There exist chiral boundary currents at the edges of quantum Hall systems, and the topological term of the non-linear sigma model (for a :xed energy E = EF ) does capture the physics of these boundary currents. In fact, when a boundary is present, the invariance under shifts xy → xy ± 1 no longer holds, parity symmetry is broken for all non-zero values of xy including xy = ± 12 , and the topological term gives rise to long-range current–current correlations at the edges of the system. The message from this is that, in order to get an unambiguous :eld-theoretic view of thermal and spin Hall transport in a d-wave superconductor, we are well advised to consider geometries with a boundary. Having understood that, we run into another diSculty: A consistent extension of a single WZW model to a system with boundaries is not possible! Instead, a geometric construction detailed in Appendix C will show the following: • The modelling of a system

and M2 .

with boundary 9

requires, at least, two WZW :elds M1

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• These :elds are coupled at the boundary through a condition (M1 )−1 |9 = M2 |9 (plus addi-

tional conditions on the derivatives of the :elds to ensure current conservation.) This mechanism reJects the fact that specular reJection at the boundaries of a :nite systems couples quasi-particle species of diFerent parity. (Specular reJection inverts one of the momentum components while leaving the other invariant: a parity operation.) • As far as the Hall conductivity of the coupled system is concerned, everything depends on the masses m1; 2 of the two :eld species: (i) if both :elds are massless, they enter the theory with equal weight and the Hall conductivity vanishes by symmetry. (ii) In highly anisotropic cases, e.g. m1 m2 , the :eld M1 decouples from the low energy sector of the theory. Under these conditions, xy = ± 12 depending on which :eld has become heavy, and the sign of m1; 2 . Finally, (iii) if the masses |m1 | |m2 | are approximately equal, it is the signs of m1; 2 that determine the Hall conductivity. If the signs are the same, we expect to get xy = ± 1; if they diFer, we expect xy = 0. What can we infer from these observations about the problem at hand? Recall that the two X and (2; 2) X are represented by the WZW :elds M1 and M2 , pairs of conjugate nodes (1; 1) respectively, with our conventions being such that the total WZW term is [M1 ] + [M2 ]. Given these low-energy eFective :elds, we impose the boundary condition (M1 )−1 |9 = M2 |9 . (Alternatively, the boundary conditions could mix in high-energy :elds corresponding to excited quasi-particle bands of the superconductor.) From what we said before, this boundary condition is forced by the requirement that the WZW functional be mathematically well-de:ned. If the parity invariance of the microscopic Hamiltonian is broken by the presence of a magnetic :eld in the form of vortices and by the supercurrent Jow, there exists no fundamental symmetry principle that would protect the :elds from acquiring (small) masses m1; 2 in the low-energy eFective theory. More detailed analysis would then be required to decide which scenario (i–iii) applies in the case of a realistic d-wave superconductor (for the most complete investigation to date, see [80]). However, as far as theory is concerned, let us re-iterate that the WZW model and the original Dirac fermion model represent equivalent descriptions of the system. In view of the consistency problems outlined above, and their resolution by an extended theory, one expects that a faithful description of Hall transport in the Dirac fermion language, too, has to involve all four nodal fermion species simultaneously. Therefore, symmetry-based statements that the Hall conductivity of a single Dirac fermion system vanishes [46] should be taken with a grain of salt. To avoid confusion, we elaborate on one other subtle point that may seem paradoxical. Recall that what we set out to calculate was the single-particle Green function, giving the density of states. Since the :eld theory we have derived for it contains massless (Goldstone) :elds Q, one might be misled into thinking that our theory predicts singular corrections to the density of states. In the present instance, this is not so. The reason why we obtained massless :elds is that, in the course of the derivation, we chose to carry out a doubling of :eld variables (introducing charge conjugation space, so as to accommodate the symplectic symmetry of the Gorkov Hamiltonian, which becomes important as soon as the coupling between nodes is turned on). This means that we are eFectively working in the two-particle sector. Put diFerently, the X although the basic quasi-particle present theory involves all of the :elds for the nodes (1; 1),

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Hamiltonian connects only half of them. As a result, when we compute the density of states the source :elds couple only to a subset of the Grassmann :elds. Thus there is a global fermionic symmetry, which causes the :eld integral to collapse by the usual BRST mechanism and yield a vanishing correction to the density of states. The :eld theory stays alive only for the two-particle Green function, viz. transport. 4.5. Hard scattering, broken T-invariance (class C) The :nal case to consider is the simultaneous presence of inter-node scattering and breaking of time-reversal invariance. Having switched to the GL(2|2) bosonization scheme for convenience of presentation and solution in the previous two subsections, we must now return to the more general OSp(4|4) scheme in order to accommodate the inter-node couplings. The T-breaking perturbation by a supercurrent Jow is rewritten in terms of n ; X n as s i=˜ (3CC ⊗ (3 )= = i STr (3CC ⊗ (3 )(

1

X t + X 2t ) : 1 2

By the OSp(4|4) bosonization rules, this bosonizes to i‘−1 STr (3CC ⊗ (3 )(M1 + M2−1 ) :

(34)

Now recall that the eFect of inter-node scattering is to lock the :elds (M1 = M2−1 ) and to reduce M1 to a scalar in node space, M 0 ; M1 = 0 M with M s = M −1 . Inserting this expression for M1 and using Tr (3 = 0, we :nd that the perturbation disappears. Thus, in the linear approximation used here, the breaking of time-reversal symmetry by the supercurrent Jow ceases to be eFective in the node-locked limit. While this may seem surprising at :rst, it is not hard to comprehend in physical terms. The perturbation we are considering represents the Doppler shift due to the motion of quasi-particles X 2) X are located relative to the supercurrent Jow vs . Because the node 1(2) and its conjugate 1( at opposite momenta, the corresponding quasi-particle states experience opposite Doppler shifts. Therefore, scattering between opposite notes causes the Doppler eFect to alternate randomly in time. It is clear that the node-locked limit will be attained over scales large enough in order for a quasi-particle to undergo many scattering events between the nodes. On such scales, and in modes of long-range quantum interference that are stable with respect to disorder averaging, the Doppler shift averages to zero. However, this does not mean that the presence of the supercurrent Jow is of no consequence. At the free-fermion point, the :eld M has scaling dimension one, which makes the perturbation (34) more relevant in the RG sense than the marginal perturbations that represent inter-node scattering. (Precisely speaking, this is true only for wavelengths smaller than the spacing between vortices. On large scales, the spatial oscillations of the supercurrent velocity cause the perturbation to average to zero.) We should therefore evaluate the consequences of the perturbation (34) before locking the nodes. Doing so, and appealing to the same argument as in the previous subsection, we expect the perturbation (34) to reduce the :eld Mn (n = 1; 2)

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to stationary-phase manifolds which now are given by M1 = T1 (3CC ⊗ (3 )T1−1 (3CC ⊗ (3 ) ; M2 = (3CC ⊗ (3 )T2 (3CC ⊗ (3 )T2−1 with Tn ∈ OSp(4|4), the diagonal subgroup of the chiral symmetry group of the C |D WZW model. Next, we take the eFects of inter-node scattering into account (with the plausible assumption that reduction of the :eld manifold does not change the relevant nature of the perturbations from inter-node scattering). By the mechanism described in Section 4.1, the :elds will be locked: T 0 T1 = T2 = ; 0 T where T now belongs to the other variant of OSp(2|2) (orthogonal bosons, symplectic fermions). By inserting the locked :elds into the WZW action, we arrive at a non-linear sigma model with action 8 1 S[Q] = − d 2 r STr 9 Q9 Q ; t = ; (35) t + −1 where Q = T3CC T −1 parametrizes a Riemannian symmetric superspace of type D|C. Note that in the linear order of approximation used, there is no topological theta term in this non-linear sigma model. Precisely, the same :eld theory (without a de:nite value for the coupling t) was obtained in [67] from a more phenomenological approach. The known physics of this system is a Zeeman-:eld dependent weak-localization correction to the thermal conductivity [67], and a density of states that vanishes as E 2 at ultra-low energies [13]. Rather than elaborating on these properties, we turn to another way of breaking time-reversal symmetry, which is to make the order parameter of the superconductor complex by including an imaginary component. Motivated by recent developments that were reviewed in the Section 1, we consider an idxy component with momentum dependence iG sin(kx a) sin(ky a). In the Gorkov Hamiltonian, such a term enters as the Pauli matrix 2 acting in particle-hole space. If we again rotate by ei1 =4 and linearize around the nodes, the term becomes proportional to 3 and acts as X and a Dirac mass term, with the mass parameter being positive for one pair of nodes, say (1; 1), negative for the other pair. In the :eld theory, upon bosonization, the perturbation is represented by the operator iG (36) STr 3CC (M1 − M2−1 ) ‘ which is relevant at the free-fermion point. The observation we made earlier for the case of a supercurrent Jow applies here as well: if we assume the limit of locked :elds (M1 = M2−1 ), the perturbation vanishes. This is again plausible, since the Dirac masses resulting from sin(kx a) sin(ky a) carry opposite signs for neighbouring nodes, which, as we recall, transform into each other by kx → −kx or ky → −ky . Hence the Dirac mass of a quasi-particle subject to inter-node scattering is zero on average. As before, however, the correct procedure is :rst to deal with the relevant mass perturbation, and only then to impose :eld locking. Proceeding in this way, we would again be led to the action (35) for the low-energy eFective theory.

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We should, however, be suspicious of the last result, as it does not conform with what is expected on symmetry grounds. The point is that the action of the non-linear sigma model (35) is invariant under a parity transformation x1 → x1 and x2 → −x2 , whereas the perturbed WZW theory we are starting from is not. Indeed, if we combine the parity transformation with :eld inversion Mi → Mi−1 , so as to preserve the WZW action, the sign of the perturbation gets reversed: STr 3CC M1−1 = STr 3CC M1t = − STr M1 3CC ;

where the last equality follows from STr(AB) = STr(AB)t = STr(Bt At ) and the fact that 3CC is odd under the orthosymplectic transpose de:ned by (3CC )t = t (3CC )t . Thus, parity is broken in the initial theory, and it is reasonable to expect that it will still be broken in the :nal theory. As was :rst pointed out by Pruisken et al. [77,78] in the related context of the integer quantum Hall eFect, the way to break parity invariance in the non-linear sigma model is to add a theta term with topological angle J ∈ Z to the action. From what has been said, we expect such a term in the low-energy eFective action when a secondary idxy component of the order parameter is present. To compute the angle J, we abandon non-Abelian bosonization and revert to the standard method: Hubbard–Stratonovich transformation and saddle-point approximation, followed by a gradient expansion. (Note that, although non-Abelian bosonization is perfectly suited to describe the Jow of the :eld theory away from the free-fermion point, it is a less reliable tool for understanding how the theory arrives at and Jows into the non-linear sigma model.) The last step is readily performed using the method of heat kernel regularization reviewed in Appendix A.2. As a result we obtain the action presented in Eq. (1), with couplings 2G + −1 0 0 11 = = ; 12 : % A summary of the physics of this model, and references to the relevant literature, were given in Section 1. 5. Comparison with other approaches In this section, we comment on the second controversy mentioned in Section 1: the incompatibility of :eld theory and self-consistent approaches. Several arguments have been put forward to explain the existence of conJicting results for observables as basic as the mean DoS (:nite at zero energy in the self-consistent approaches, vanishing in :eld theory). It has, for example, been argued that the vanishing of the DoS in :eld theory might be due to an ill-de:ned continuum limit [82]. On the other hand, it has been pointed out [83] that the notorious appearance of logarithmic divergences in the diagrammatic results hints at an insuScient UV-regularization. However, in hindsight it looks like these, UV related aspects of the problem carry minor if any signi:cance. The origin of the discrepancy between :eld theory and early self-diagrammatic approaches to the problem lies somewhere else: First of all, it is important to realize that the vanishing of the DoS obtained in the :eldtheoretical description is due to a mechanism which can simply not be included into diagrammatic approaches. Indeed, we have seen in Section 3 that the breaking of the chiral symmetry of the Hamiltonian on the saddle-point level leads to the appearance of Goldstone modes.

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In early perturbative approaches to the problem [1–3], these modes had not been included at all. In the :eld theory, however, they are responsible for much of the physical behaviour of the system. In particular, it is the strong Juctuation of these modes in low-dimensional systems that leads to a vanishing of the DoS at E = 0. It is tempting to explain this vanishing by the Mermin–Wagner–Coleman theorem as was done by NTW. Indeed, a non-vanishing DoS would have meant that a continuous symmetry of the :eld theory was spontaneously broken. (As with normal disordered systems, the DoS plays the role of an order parameter in the :eld theory.) The Mermin–Wagner–Coleman theorem essentially states that, as a consequence of unbounded Goldstone mode Juctuations, spontaneous breaking of a continuous symmetry does not occur in dimensions d 6 2. There is, however, a caveat with that argument, which is that the Mermin–Wagner–Coleman theorem explicitly refers to compact >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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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

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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.

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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

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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.

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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

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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,

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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 :

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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.

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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 ;

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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

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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.

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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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Juctuations in ’1 . Of course one could rescue the stability by putting the theory on a lattice with, say, nearest-neighbour coupling for the U(1|1) :eld. However, the state of lowest energy of such a lattice theory would be ferromagnetic in ’2 , but antiferromagnetic in the ’1 . As a result the ground state would not be invariant under global supersymmetry transformations. Even worse, the partition function of the theory (in the absence of sources) would be identically zero and hence not normalizable to unity. This follows essentially from the fact [93] that U(1|1) has vanishing volume w.r.t. its invariant Berezin–Haar measure. All these considerations make it rather unlikely that one could ever make sense, for our purposes, of a Euclidean space-time nonlinear sigma model or WZW model with target space U(1|1) or more generally U(n|n). The construction that solves the diSculty was described in a general mathematical setting in Ref. [50]. Its :eld-theoretic implementation was discussed in some detail in Section 6 of Ref. [94] and Section 7:1 of Ref. [49]. The basic idea is easily understood. To make the numerical part of the metric tensor da2 − db2 positive de:nite, we should take a ≡ x from the real numbers, and b ≡ iy from the imaginary numbers. By returning to group level via exponentiation, we then get manifolds MF = U(1) and MB = GL(1; C)=U(1). In the general case of n Green functions, we obtain MF = U(n) and MB = GL(n; C)=U(n). These are Riemannian symmetric spaces of compact and non-compact type, respectively. On incorporating the fermions into the exponential mapping, we are led to some sort of superspace. An equivalent procedure is to start from the complex supergroup GLC (n|n) and restrict the bosonic degrees of freedom to MB ×MF . (This construction does not let us impose any reality constraints on the fermions [84]. Fortunately, there exists no fundamental principle that would force us to do so.) Let Xn denote the resulting space. Xn is not a group, but belongs to the category of Riemannian symmetric superspaces [50]. It is called type A|A, which tells us that both the BB and the FF sector are symmetric spaces belonging to the A (or “unitary”) series. Its main property is that the metric tensor % = − STr dg−1 dg restricts to a Riemannian structure on MB × MF . Thus, the :eld theory to be studied has the target space Xn . We denote the :eld by M , and set n = 2 (although nothing essential depends on that choice). By slight generalization of the action functional (30), we consider kG S[M ] = kW [M ] + 2 d 2 r STr(M 9M −1 ) STr(M −1 9XM ) with W [M ] being the WZW functional for GL(2|2): i[M ] 1 W [M ] = − d 2 r STr 9 M −1 9 M + : 8 12 This is the theory we shall now analyse. As a by-product we will get (by setting G = 0) a justi:cation of the bosonization rules that leads to (30). B.2. Functional integral solution The main tool for dealing with the WZW functional is a relation due to Polyakov and Wiegmann [95]: 1 d 2 r STr g−1 9Xg 9h h−1 : W [gh] = W [g] + W [h] +

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Using it, one sees that W [M ] is invariant under local transformations M (z; z) X → g(z)M (z; z)h( X z) X −1 ; where g(z) and h(z) X take values in GLC (2|2). We will see that a deformation of this symmetry survives in the presence of a non-vanishing coupling G. Consider now the expectation value F[M ] where F[M ] is some functional of the :eld M concentrated at a set of points r1 ; r2 ; : : : ; rN . Exploiting the invariance of the functional integration measure under left translations, we make a change of integration variables M (r) → e−X (r) M (r), and denote the :rst variation of the :eld by *X M (r) = − X (r)M (r). The resulting variation of the action de:nes the current J : 1 *X S = − d 2 r J9X X ; 2kG JX = − k STr(XM 9M −1 ) + STr(X )STr(M 9M −1 ) : To derive this expression for the current, we made use of the Polyakov–Wiegmann formula, and we used the identity 9 STr(M −1 9XM ) = 9X STr(M −1 9M )

in conjunction with two partial integrations to cancel some terms. (The boundary terms produced by these partial integrations at the points r1 ; : : : ; rN cancel each other.) The invariance of the expectation value F[M ] w.r.t. the variation *X M implies 1 2 *X F[M ] + F[M ] =0 : d r J9X X From this relation it immediately follows that 9XJA = 0, called the equation of motion, holds under the functional integral sign (away from the points of support of F[M ]) for any spatially constant A ∈ glC (2|2). As an additional consequence one has 1 d z JX (z) · F[M ] = *X F[M ] ; 2i where is any closed contour that circles once around the points of support of F[M ]. By specializing to F[M ] = M (0)F1 [M ], where F1 is some other functional, one gets the operator product expansion between the current and the :eld: AM (0) JA (z)M (0) = − + ··· z and by taking F[M ] = JB (0)F1 [M ] one gets the OPE for the currents themselves: kf(A; B) J[A; B] (0) JA (z)JB (0) = + + ··· ; z2 z 2G f(A; B) = − STr(AB) + STr(A)STr(B) : The dots indicate terms that remain :nite in the limit z → 0.

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Identical considerations can be made starting from :eld variations *Y M (r) = M (r)Y (r), which generate right translations M (r) → M (r)eY (r ). They lead to a conserved current 2kG JX Y = − k STr(YM −1 9XM ) + STr(Y )STr(M −1 9XM ) ; which is antiholomorphic, i.e. satis:es 9JX A = 0 for spatially constant A. The behaviour of correlation functions under conformal transformations is determined by the stress–energy–momentum tensor, written in components as T dx dx = Tzz dz 2 + TzXzXdzX2 : Conformal invariance implies TzzX = 0 = TzzX , and 9XTzz = 0 = 9TzXzX (on solutions of the equations of motion). We focus on the holomorphic sector and set T (z) = 2Tzz (z). Classical considerations based on the general formula T = (9 ’i )9L= 9(9 ’i ) − * L suggest kG k Tcl = STr(M 9M −1 )2 − STr 2 (M 9M −1 ) ; 2 but this, it turns out, is modi:ed by quantum Juctuations. The correct expression for T (z) is obtained by demanding that the leading singularity in the operator product T (z)JA (0) be z −2 JA (0), as results from JA being a holomorphic conserved current. The explicit form of the stress–energy–momentum tensor is obtained by the Sugawara construction, which represents T (z) as a quadratic form in the currents. Let {ei } be some basis of the Lie superalgebra gl(2; 2) with metric %ij = − STr ei ej . Indices are raised by %ij %jk = *ik . We write Ji ≡ Jei for short, and Je = Jid for the current corresponding to the unit matrix. Setting 1 T˜ (z) = %ij : Ji (z)Jj (z) : 2k where the colons mean normal ordering (i.e. subtraction of the short-distance singularities of the operator product at coinciding points), and using the OPE for the currents and the associativity of the operator product algebra, we get 2G %ij lim z 2 T˜ (z)JA (0) = JA (0) − STr(A)Je (0) + J[e ;[e ; A]] (0) : z→0 2k i j The last term is a quantum correction, which vanishes in the limit k → ∞. For gl(2; 2) one veri:es the relation %ij [ei ; [ej ; A]] = 2e STr A. This allows to rewrite the right-hand side of the preceding relation as 1 2G JA (0) + − STr(A)Je (0) : k To obtain the required form of the operator product T (z)JA (0) = JA (0)=z 2 + · · ·, we must cancel the second term, which is achieved by de:ning 1 − 2Gk= T (z) = T˜ (z) + : Je (z)Je (z) : : 2k 2 On setting k = 1 and renaming G to g, we arrive at the expression for T (z) claimed in Eq. (32). Note that the symbol for normal ordering can be dropped, as the expression is already :nite as it stands [39].

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Our :nal step is the computation of the OPE of T (z) with the fundamental :eld M (0). This is readily done from the formula for T (z) and the OPE between the currents and M (0). Using the fact that the quadratic Casimir invariant evaluated in the fundamental representation of gl(2; 2) vanishes (%ij ei ej = 0), we :nd "M M (0) + ··· ; T (z)M (0) = z2 where "M = (1 − 2Gk=)=(2k 2 ) is the (holomorphic) scaling dimension of the :eld M . The total dimension is 1 − 2Gk= "M + "X M = : k2 Note that the dimension becomes negative for disorder strengths G greater than =2k. If we substitute a M = exp
X 1 9= X + =sX9X=1 : L== 1 1 Since the OPEs involving the currents match, too, we expect that the two theories are equivalent. The bosonization rules are the same as in (24) but for factors of two: =1s =1sX ↔ −1 M 9M −1 ; s =X 1X=X 1 ↔ −1 M −1 9XM ; s =1 =X 1 ↔ ‘−1 M ;

=X 1X=1sX ↔ ‘−1 M −1 :

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Appendix C. WZW theory for systems with boundary In the main text we had said that a single WZW theory represents too narrow a framework to consistently describe systems with boundaries: the WZW functional exp i[M ]=12 for maps M from the position space to the target space G (the Riemannian symmetric superspace of type A|A in GL(2|2)) makes sense only if is closed (i.e. has no boundaries). The reason is that, to de:ne [M ] as an integral of the 3-form STr(M −1 dM )∧3 , one needs to extend M to a map M˜ : B → G where B is some 3-space. But in order for the WZW functional to be independent of the choice of extension M˜ , the boundary of B must coincide with (9B = ), which is possible only if is closed (9 = 99B = 0). Thus the WZW functional is ill-de:ned in the presence of a boundary. To the extent that the WZW model is a faithful representation of the Dirac theory, we expect the implementation of a boundary to be problematic in the latter, too. How can we cure this disease and arrive at a meaningful functional integral for systems with a boundary? There exists only one solution to the problem [81], and it requires having a minimum of two WZW theories. 1 Let us assume there exists a pair of WZW :elds M and M , with the sum of WZW terms being [M ] + [M ]. The boundary condition we impose is M −1 |9 = M |9 . (To ensure current conservation, we also need to impose a condition on the normal derivatives of the :elds at the boundary.) This boundary condition guarantees that the two maps M : → G and M : → G combine to a single map ∪ (− ) → G, where ∪ (− ) consists of two copies of the position space, glued together at the boundary 9 . Since the double covering ∪ (− ) is a closed manifold, we can regard it as the boundary of a 3-space B, so the independence of the choice of extension is restored and the total WZW functional is again well-de:ned. Having understood this, the question arises: where do we take the second WZW theory from? Depending on the physics of the problem at hand, a number of options are conceivable: If the original problem mapped onto an even number of WZW species anyway—this is the case for our d-wave system—it is natural to glue these :elds pairwise together at the boundaries. This procedure is discussed in the text. However, one may also contemplate situations where the low-energy sector of the original problem contains only a single WZW theory. (In the language of fermions, this amounts to considering a single Dirac fermion species.) In this case, a second WZW :eld or, equivalently, a second Dirac species, changes the low-energy content of the theory. To repair this, we must give a mass to the second species, so as to prevent it from contributing to the long-range behaviour of the correlation functions. Note the important fact that the mass of the second species breaks parity, so we can no longer conclude on symmetry grounds that pseudoscalar observables such as the Hall conductivity vanish! If we again carry out the low-energy reduction to the non-linear sigma model, we will get the same answer as in Section 4.4, as the auxiliary second species drops out in the low-energy limit. Moreover,

1

A consistent formulation of a single WZW model with boundary becomes possible [56] if one imposes boundary conditions that break the (left–right) symmetry. However, in the present context such boundary conditions are ruled out by the fact that the d-wave system (in class CI) has a global OSpL (2|2) × OSpR (2|2) symmetry, and this exact symmetry cannot be lost on passing to the eFective :eld theory description.

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we expect the sign ambiguity xy = ± 12 to be resolved: it is the sign of the mass of the second species that will determine the sign of xy . The second scenario is not new but is known from the work of Ludwig et al. [73] who studied a disordered electron model on a square lattice with a magnetic Jux of per plaquette. The low-energy limit of that model consists of two species of Dirac fermions, one with a light mass and the other with a heavy mass. The Hall conductivity of the model does not vanish when the light species becomes massless (and hence the low-energy sector parity-invariant), and is determined by the sign of the Dirac mass of the heavy species. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]

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[33] Central to this paper are the Green functions of quasi-particles subject to an eFective, spatially varying pairing :eld. These objects, and the underlying Hamiltonian, have :rst been formulated and investigated by Gorkov. For this reason, we use the terminology “Gorkov equations” and “Gorkov Hamiltonian” (deviating from the commonly used attribute “Bogoliubov–deGennes”). [34] A. Altland, M.R. Zirnbauer, Phys. Rev. B 55 (1997) 1142. [35] R. Oppermann, Physica A 167 (1990) 301; V.E. Kravtsov, R. Oppermann, Phys. Rev. B 43 (1991) 10 865. [36] In its standard de:nition, the parity operation denotes a point reJection at the origin, (x; y) → (−x; −y). Within the context of anomalous superconductivity, however, “parity” mostly stands for a planar reJection, (x; y) → (x; −y), or any other discrete transformation that reverses the orientation of 2d space. [37] While the classi:cation scheme is complete (in the sense that it does provide closure under renormalization), it is important to recognize that it does not uniquely determine the physical behaviour of a system. For example, inside the symmetry classes D and DIII in two dimensions, a number of diFerent physical behaviours (localized, critical, or metallic) is possible. [38] E. Witten, Comm. Math. Phys. 92 (1984) 455. [39] C. Mudry, C. Chamon, X.-G. Wen, Phys. Rev. B 53 (1996) 7638; Nucl. Phys. B 466 (1996) 383. [40] R.B. Laughlin, Phys. Rev. Lett. 80 (1998) 5188. [41] M.R. Li, P.J. Hirschfeld, P. W^olJe, Phys. Rev. B 63 (2001) 05 4504. [42] I. Gruzberg, A.W.W. Ludwig, N. Read, Phys. Rev. Lett. 82 (1999) 4524. [43] V. Kagalovsky, B. Horovitz, Y. Avishai, J.T. Chalker, Phys. Rev. Lett. 82 (1999) 3516. [44] J.L. Cardy, Phys. Rev. Lett. 84 (2000) 3507. [45] D. Bernard, A. LeClair, cond-mat=0003075. [46] J. Ye, Phys. Rev. Lett. 86 (2001) 316. [47] T. Senthil, M.P.A. Fisher, Phys. Rev. B 61 (2000) 9690. [48] N. Read, D. Green, Phys. Rev. B 61 (2000) 10 267. [49] M. Bocquet, D. Serban, M.R. Zirnbauer, Nucl. Phys. B 578 (2000) 628. [50] M.R. Zirnbauer, J. Math. Phys. 37 (1996) 4986. [51] P.W. Anderson, Phys. Rev. Lett. 3 (1959) 325. [52] K.B. Efetov, Adv. Phys. 32 (1983) 53. [53] N. Nagaosa, T.K. Ng, Phys. Rev. B 51 (1995) 15 588. [54] S. Krivenko, G. Khaliullin, JETP Lett. 62 (1995) 723; G. Khaliullin, et al., Phys. Rev. B 56 (1997) 11 882. [55] A. Kaminski, et al., Phys. Rev. Lett. 84 (2000) 1788. [56] K. Gawedzki, hep-th=0108044; K. Gawedzki, I. Todorov, P. Tran-Ngoc-Bich, hep-th=0101170. [57] M. Chiao, et al., Phys. Rev. B 62 (2000) 3554. [58] J. Orenstein, A. Millis, Science 288 (2000) 468. [59] N.E. Hussey, et al., Phys. Rev. Lett. 85 (2000) 4140. [60] Y. Zhang, et al., Phys. Rev. Lett. 84 (2000) 2219. [61] G.E. Volovik, JETP Lett. 58 (1993) 469. [62] F.J. Dyson, J. Math. Phys. 3 (1962) 1199. [63] J.J.M. Verbaarschot, I. Zahed, Phys. Rev. Lett. 70 (1993) 3852. [64] J.J.M. Verbaarschot, Phys. Rev. Lett. 72 (1994) 2531. [65] Examples of a group and its complex extension are U(N ) and GL(N; C). Another example is O(N ) and O(N; C), the latter being the orthogonal group over the complex number :eld. More generally, we call a manifold MC the complex extension of a manifold M if M is a submanifold of MC with half the dimension and the complexi:cation Tp M + iTp M of the tangent space of M in every point p ∈ M coincides with the tangent space of MC in p. [66] R. Bundschuh, C. Cassanello, D. Serban, M.R. Zirnbauer, Nucl. Phys. B 532 (1998) 689. [67] R. Bundschuh, C. Cassanello, D. Serban, M.R. Zirnbauer, Phys. Rev. B 59 (1999) 4382. [68] A.B. Zamolodchikov, Sov. J. Nucl. Phys. 46 (1987) 1090. [69] J.L. Cardy, Scaling and Renormalization in Statistical Physics, Cambridge University Press, Cambridge, 1996. [70] A. Altland, B.D. Simons, D. Taras-Semchuk, Adv. Phys. 49 (2000) 321. [71] D.H. Friedan, Ann. Phys. 163 (1985) 318. [72] S. Helgason, DiFerential Gometry, Lie Groups and Symmetric Spaces, Academic Press, New York, 1978.

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Physics Reports 359 (2002) 355–528

The quark–gluon plasma: collective dynamics and hard thermal loops Jean-Paul Blaizot ∗; 1 , Edmond Iancu1 Service de Physique Theorique2 , CE-Saclay 91191 Gif-sur-Yvette, France Received June 2001; editor: G:E: Brown Contents 1. Introduction 1.1. Scales and degrees of freedom in ultrarelativistic plasmas 1.2. One-loop polarization tensor from kinetic theory 1.3. Kinetic equations for quantum particles 1.4. QCD kinetic equations and hard thermal loops 1.5. E8ect of collisions 1.6. E8ective theory for soft and ultrasoft excitations 1.7. Outline of the paper 2. Quantum :elds near thermal equilibrium 2.1. Equilibrium thermal :eld theory 2.2. Nonequilibrium evolution of the quantum :elds 2.3. Mean :eld and kinetic equations 3. Kinetic theory for hot QCD plasmas 3.1. NonAbelian versus nonlinear e8ects 3.2. Mean :elds and induced sources

357 358 363 365 367 370 371 373 374 375 387 397 411 411 415

3.3. Approximation scheme 3.4. The nonAbelian Vlasov equations 3.5. Kinetic equations for the fermionic excitations 3.6. Summary of the kinetic equations 4. The dynamics of the soft excitations 4.1. Solving the kinetic equations 4.2. Equations of motion for the soft :elds 4.3. Collective modes, screening and Landau damping 4.4. Hamiltonian theory for the HTLs 5. Hard thermal loops 5.1. Irreducible amplitudes from induced sources 5.2. The HTL e8ective action 5.3. Hard thermal loops 5.4. HTL resummation and beyond 6. The lifetime of the quasiparticles 6.1. The fermion damping rate in the Born approximation 6.2. Higher-order corrections

∗

422 430 433 437 438 438 443 444 449 456 456 458 460 465 470 471 473

Corresponding author. Tel.: +33-1-6908-7386; fax: +33-1-6908-8120. E-mail addresses: [email protected] (J.-P. Blaizot), [email protected] (E. Iancu). 1 Member of CNRS. 2 Laboratoire de la Direction des Sciences de la MatiDere du Commissariat aD l’Energie Atomique and UnitGe de Recherche AssociGee au CNRS (URA2306). c 2002 Elsevier Science B.V. All rights reserved. 0370-1573/02/$ - see front matter PII: S 0 3 7 0 - 1 5 7 3 ( 0 1 ) 0 0 0 6 1 - 8

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6.3. The Bloch–Nordsieck approximation 6.4. Large-time behaviour 6.5. Discussion 6.6. Damping rates in QCD 7. The Boltzmann equation for colour excitations 7.1. The collision term 7.2. Coloured and colourless excitations 7.3. Ultrasoft amplitudes 7.4. The Boltzmann–Langevin equation: noise and correlations

475 478 480 482 483 484 487 490

8. Conclusions 499 Acknowledgements 500 Appendix A. Notation and conventions 501 Appendix B. Diagrammatic calculations of hard thermal loops 504 B.1. The soft gluon polarization tensor 508 B.2. The soft fermion propagator 518 References 523

495

Abstract We present a uni:ed description of the high temperature phase of QCD, the so-called quark–gluon plasma, in a regime where the e8ective gauge coupling g is suKciently small to allow for weak coupling calculations. The main focus is the construction of the e8ective theory for the collective excitations which develop at a typical scale gT , well separated from the typical energy of single particle excitations which is the temperature T . We show that the short wavelength thermal Luctuations, i.e., the plasma particles, provide a source for long wavelength oscillations of average :elds which carry the quantum numbers of the plasma constituents, the quarks and the gluons. To leading order in g, the plasma particles obey simple gauge-covariant kinetic equations, whose derivation from the general Dyson–Schwinger equations is outlined. By solving these equations, we e8ectively integrate out the hard degrees of freedom, and are left with an e8ective theory for the soft collective excitations. As a by-product, the “hard thermal loops” emerge naturally in a physically transparent framework. We show that the collective excitations can be described in terms of classical :elds, and develop for these a Hamiltonian formalism. This can be used to estimate the e8ect of the soft thermal Luctuations on the correlation functions. The e8ect of collisions among the hard particles is also studied. In particular we discuss how the collisions a8ect the lifetimes of quasiparticle excitations in a regime where the mean free path is comparable with the range of the relevant interactions. Collisions play also a decisive role in the construction of the e8ective theory for ultrasoft excitations, with momenta ∼ g2 T , a topic which is brieLy addressed at the end of this paper. c 2002 Elsevier Science B.V. All rights reserved. PACS: 11.10.Wx; 12.38.Mh; 11.15.Kc; 12.38.Cy; 05.20.Dd; 05.60.+w

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1. Introduction It is currently believed that matter at high density (several times ordinary nuclear matter density) or high temperature (beyond a few hundred MeV) becomes simple: all known hadrons are expected to dissolve into a plasma of their elementary constituents, the quarks and the gluons, forming a new state of matter: the quark–gluon plasma [1,2]. The transition from the quark–gluon plasma to hadronic matter is one of several transitions occurring in the early universe [3]. It is supposed to take place during the :rst few microseconds after the big bang, when the temperature is of the order of 200 MeV. At a higher temperature, of the order of 250 GeV, another transition takes place, the electroweak transition above which all particles become massless and form another ultrarelativistic plasma. The study of this phase transition and of the corresponding plasma is an interesting and active :eld of research (see e.g. [4,5]). The electroweak plasma has many features in common with the quark–gluon plasma, and we shall allude to some of them in the course of this paper. However we shall concentrate here mainly on the quark–gluon plasma. Indeed, much of the present interest in the quark–gluon plasma is coming from the hope to observe it in laboratory experiments, by colliding heavy nuclei at high energies. An important experimental program is underway, both in the USA (RHIC at Brookhaven), and in Europe at CERN. (For general references on the :eld, see [2,6,7].) It is therefore of the utmost importance to try and specify theoretically the expected properties of such a plasma. Part of our motivations in writing this report is to contribute to this e8ort. The existence of weakly interacting quark matter was anticipated on the basis of asymptotic freedom of QCD [8]. But the most compelling theoretical evidences for the existence of the quark–gluon plasma are coming from lattice gauge calculations (for recent reviews see e.g. [9 –11]). These are at present the unique tools allowing a detailed study of the transition region where various interesting phenomena are taking place, such as colour decon:nement or chiral symmetry restoration. In this report, we shall not consider this transition region, but focus rather on the high temperature phase, exploiting the fact that at suKciently high temperature the e8ective gauge coupling constant becomes small enough to allow for weak coupling calculations [12–15]. The picture of the quark–gluon plasma which emerges from these weak coupling calculations is a simple one, and in many respect the quark–gluon plasma is very much like an ordinary electromagnetic plasma in the ultrarelativistic regime [16 –18], with however speci:c e8ects related to the nonAbelian gauge symmetry [19 –21]. To zeroth order in an expansion in powers of the coupling g, the quark–gluon plasma is a gas of noninteracting quarks and gluons. The interactions appear to alter only slightly this simple picture: they turn those plasma particles which have momenta of the order of the temperature into massive quasiparticles, and generate collective modes at small momenta which can be described accurately in terms of classical :elds. One thus sees a hierarchy of scales and degrees of freedom emerging that invites us to construct e8ective theories for these various degrees of freedom. Weak coupling techniques can be used to this aim [19,20,22–26]; once the e8ective theories are known they can be used to also describe non perturbative phenomena [27,28]. It is indeed important to keep in mind that weak coupling approximations are not to be identi:ed with strictly perturbative calculations. A celebrated counter example is that of the

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presently much discussed phenomenon of color superconductivity [29]. Staying in the realm of high temperature QCD, we note that weak coupling expansions generate terms which are odd in g, and these can only be obtained through in:nite resummations. Such resummations appear naturally in the construction of e8ective theories alluded to earlier. The possibility to identify and perform such resummations o8ers a chance to extrapolate weak coupling results down to temperature where the coupling is not really small (recall that the dependence of the coupling on the temperature is only logarithmic, and it is only for T Tc , where Tc is the decon:nement temperature, that the coupling is truly small). Recent works indicate that this strategy may indeed be successful [30 –32]. As well known, severe infrared divergences occur in high order perturbative calculations. These divergences, usually associated with those of an e8ective three dimensional theory, are not easily overcome by analytic tools. Lattice calculations indicate that the strong longwavelength Luctuations responsible for such divergences survive at high temperature and give signi:cant contributions to the parameters characterizing the long distance behaviour of the correlation functions (e.g. the so-called screening masses [33,34]). While those results may suggest the existence of new, nonperturbative, degrees of freedom, there is no evidence that these degrees of freedom contribute signi:cantly to thermodynamical quantities. On the contrary, both recent lattice results [35], and the analytical resummations mentioned above, support the conclusion that this contribution is small. A :nal motivation for pushing these analytical techniques is the possibility they o8er to study dynamical quantities. These are diKcult to obtain on the lattice, but are essential in any attempt to study real phenomena. Indeed much of this report will be devoted to dynamical features of the quark gluon plasma, emphasizing in particular its kinetic and transport properties. In fact, as we shall discover, kinetic theory appears to be a powerful tool for integrating out degrees of freedom when constructing e8ective theories. Finally, it may be added that dynamical information, in particular that on the plasma quasiparticles and its collective modes, can be relevant also for the calculation of thermodynamical quantities [30 –32]. The goal of this review is twofold. On the one hand, we wish to o8er a consistent description of the quark–gluon plasma in the weak coupling regime, summarizing recent progress and pointing out some open problems. On the other hand, we shall give a pedagogical introduction to some of the techniques that we have found useful in dealing with this problem. We emphasize that most of the discussion will concern a plasma in equilibrium or close to equilibrium, and the present work is but a little step towards the ultimate goal of treating more realistic situations such as met in nuclear collisions for instance. We hope nevertheless that some of the techniques introduced here can be extended to treat these more complex situations, and indeed some have already been used to this aim. A more precise view of the content of this paper is detailed in the rest of this section, where we shall introduce, in an elementary fashion, most of the important concepts to be used. An explicit outline is given in Section 1.7. 1.1. Scales and degrees of freedom in ultrarelativistic plasmas We consider the decon:ned phase of QCD, the quark–gluon plasma, in thermal equilibrium at a temperature T . This a nonAbelian gauge theory with “colour” gauge group SU(N ) and Nf

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Lavours of quarks, whose dynamics is described by the following Lagrangian: a a LQCD = − 14 F F + S i (iD = )ij

j

:

(1.1)

The notations above are explained in detail in Appendix A, but Eq. (1.1) can be easily read by referring to QED: LQCD di8ers indeed from the corresponding QED Lagrangian, LQED , a = 9 Aa − 9 Aa − gf abc Ab Ac only by the nonlinear terms in the colour :eld strength tensor F (showing that gluons, unlike photons, do couple among themselves) and by the presence of colour indices: for the adjoint representation (a; b; : : : ; with a running from 1 to N 2 − 1) in the case of the gluons, and for the fundamental representation (i; j; : : : ; with i running from 1 to N ) in the case of the fermions (quarks and antiquarks). Note that, for simplicity, the fermions are taken as massless. That is, we consider an ultrarelativistic plasma where the masses of the dynamical quarks satisfy mq T and thus can be ignored. As in QED, the masslessness of the gluons follows from the requirement of gauge symmetry, which also :xes the form of the interactions in Eq. (1.1). In thermal equilibrium, typical interactions in the system involve :elds with momenta of order T , so that the relevant coupling “constant” is the QCD running coupling g() evaluated at a scale ∼ T . If the temperature is high enough, T QCD , the system is weakly coupled, g1, and in a :rst approximation is described by the ideal gas (g → 0). In this limit, the plasma particles are distributed in momentum space according to the Bose–Einstein or Fermi–Dirac distributions: 1 1 Nk = jk ; nk = jk ; (1.2) e −1 e +1 where jk = k ≡ |k|, ≡ 1=T , and chemical potentials are assumed to vanish. In such an ultrarelativistic system, the particle density n is not an independent parameter, but is determined by the temperature: n ˙ T 3 . Accordingly, the mean interparticle distance n−1=3 ∼ 1=T is of the same order as the thermal wavelength T = 1=k of a typical particle in the thermal bath for which k ∼ T . Thus the particles of an ultrarelativistic plasma are quantum degrees of freedom for which in particular the Pauli principle can never be ignored. In the weak coupling regime (g1), the interactions do not alter signi:cantly the picture. The hard degrees of freedom, i.e. the plasma particles with momenta k ∼ T , remain the dominant degrees of freedom and since the coupling to gauge :elds occurs typically through covariant derivatives, Dx = 9x + igA(x), the e8ect of interactions on particle motion is a small perturbation unless the :elds are very large, i.e., unless A ∼ T=g, where g is the gauge coupling: only then do we have 9x ∼ T ∼ gA, where 9x ∼ k is a hard space–time gradient. We should note here that often in this report we shall rely on considerations, such as the one just presented, which are based on the magnitude of the gauge :elds. Obviously, such considerations depend on the choice of a gauge. What we mean is that there exists a large class of gauge choices for which they are valid. And we shall verify a posteriori that within such a class, our :nal results are gauge invariant. Note however that thermal Luctuations could make it diKcult to give a gauge independent meaning to colour inhomogeneities on scales much larger than 1=g2 T [36]. Considering now more generally the e8ects of the interactions, we note that these depend both on the strength of the gauge :elds and on the wavelength of the modes under study. A measure of the strength of the gauge :elds in typical situations is obtained from the magnitude

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of their thermal Luctuations, that is AS ≡ A2 (t; x). In equilibrium A2 (t; x) is independent of t and x and given by A2 = G(t = 0; x = 0) where G(t; x) is the gauge :eld propagator. In the noninteracting case we have (with jk = k): d3 k 1 A2 = (1 + 2Nk ) : (1.3) (2)3 2jk

Here we shall use this formula also in the interacting case, assuming that the e8ects of the interactions can be accounted for simply by a change of jk (a more complete calculation is presented in Appendix B). We shall also ignore the (divergent) contribution of the vacuum Luctuations (the term independent of the temperature in Eq. (1.3)). For the plasma particles jk = k ∼ T and A2 T ∼ T 2 . The associated electric (or magnetic) :eld Luctuations are E 2 T ∼ (9A)2 T ∼ k 2 A2 T ∼ T 4 and give a dominant contribution to the plasma energy density. As already mentioned, these short wavelength, or hard, gauge :eld Luctuations produce a small perturbation on the motion of a plasma particle. However, this is not so for an excitation at the momentum scale k ∼ gT , since then the two terms in the covariant derivative 9x and gAS T become comparable. That is, the properties of an excitation with momentum gT are expected to be nonperturbatively renormalized by the hard thermal Luctuations. And indeed, the scale gT is that at which collective phenomena develop, the study of which is one of the main topic of this report. The emergence of the Debye screening mass mD ∼ gT is one of the simplest examples of such phenomena. Let us now consider the thermal Luctuations at this scale gT T , to be referred to as the soft scale. We shall see that these Luctuations can be accurately described by classical :elds. In fact, since jk ∼ gT T , one can replace Nk by T= jk in Eq. (1.3); thus, the associated occupation numbers are large, Nk 1. Introducing an upper cut-o8 gT in the momentum integral, one then gets gT T 2 A gT ∼ d 3 k 2 ∼ gT 2 : (1.4) k √ Thus AS gT ∼ gT so that gAS gT ∼ g3=2 T is still of higher order than the kinetic term 9x ∼ gT . In that sense the soft modes with k ∼ gT are still perturbative, i.e. their self-interactions can be ignored in a :rst approximation. Note however that they generate contributions to physical observables which are not analytic in g2 , as shown by the example of the order g3 contribution to the energy density of the plasma: !pl 1 T 3 (3) ∼ d 3 k !pl !pl =T ∼ !pl !pl ∼ g3 T 4 ; (1.5) ! e − 1 pl 0 where !pl ∼ gT is the typical frequency of a collective mode. Moving down to lower momenta, one meets the contribution of the unscreened magnetic Luctuations which play a dominant role for k ∼ g2 T . At that scale, to be referred to as the ultrasoft scale, it becomes necessary to distinguish the electric and the magnetic sectors (which provide comparable contributions at the scale gT ). The electric Luctuations are damped by the Debye screening mass (Nk = jk T=(k 2 + m2D ) ≈ T=m2D when k ∼ g2 T ) and their contribution, of order g4 T 2 , is negligible in comparison with that of the magnetic Luctuations. Indeed, because

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Fig. 1. A typical n-loop contribution to the pressure which is infrared divergent.

of the absence of static screening in the magnetic sector, we have there jk ∼ k and g2 T 1 2 2 A g T ∼ T d 3 k 2 ∼ g2 T 2 ; (1.6) k 0 so that gAS g2 T ∼ g2 T is now of the same order as the ultrasoft derivative 9x ∼ g2 T : the Luctuations are no longer perturbative. This is the origin of the breakdown of perturbation theory in high temperature QCD. To appreciate the diKculty from another perspective, let us :rst observe that the dominant contribution to the Luctuations at scale g2 T comes from the zero Matsubara frequency g2 T g2 T 1 1 2 3 A g2 T = T d k 2 ∼T d3 k 2 : (1.7) 2 ! + k k 0 0 n n Thus the Luctuations that we are discussing are those of a three dimensional theory of static :elds. Following Linde [37,38] consider then the higher order corrections to the pressure in hot Yang–Mills theory. Because of the strong static Luctuations most of the diagrams of perturbation theory are infrared (IR) divergent. By power counting, the strongest IR divergences arise from ladder diagrams, like the one depicted in Fig. 1, in which all the propagators are static, and the loop integrations are three-dimensional. Such n-loop diagrams can be estimated as ( is an IR cuto8): n k 2(n−1) 2(n−1) 3 g T d k ; (1.8) (k 2 + 2 )3(n−1) which is of the order g6 T 4 ln(T=) if n = 4 and of the order g6 T 4 (g2 T=)n−4 if n ¿ 4. (The various factors in Eq. (1.8) arise, respectively, from the 2(n − 1) three-gluon vertices, the n loop integrations, and the 3(n − 1) propagators.) According to this equation, if ∼ g2 T , all the diagrams with n ¿ 4 loops contribute to the same order, namely to O(g6 ). In other words, the correction of O(g6 ) to the pressure cannot be computed in perturbation theory. Having identi:ed the main scales and degrees of freedom, our task will be to construct appropriate e8ective theories at the various scales, obtained by eliminating the degrees of freedom at higher scales. This will be done in steps. In fact the main part of this work will be devoted to the construction of the e8ective theory at the scale gT obtained by eliminating the hard degrees of freedom with momenta k ∼ T . We shall consider some aspects of the e8ective theory at the scale g2 T only in Section 7.

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The soft excitations at the scale gT can be described in terms of average 4elds. Such average :elds develop for example when the system is exposed to an external perturbation, such as an external electromagnetic current. Staying with QED, we can summarize the e8ective theory for the soft modes by the equations of motion 9 F = jind + jext

(1.9)

, and that is, Maxwell equations with a source term composed of the external perturbation jext an extra contribution jind which we shall refer to as the induced current. The induced current is generated by the collective motion of the charged particles, i.e. the hard fermions. In the absence of the external current, Eq. (1.9) describes the longwavelength collective modes which carry the quantum numbers of the photon, i.e., the soft plasma waves. Similarly, we shall see that the Dirac equation with an appropriate induced source "ind (x) describes collective longwavelength excitations with fermionic quantum numbers [18]:

iD = #(x) = "ind (x) :

(1.10)

The induced sources jind and "ind may be regarded as a functionals of the average gauge :elds A (x) and fermion :eld #(x). Once these functionals are known, the equations above constitute a closed system of equations for the soft :elds. The main problem is to calculate the induced sources jind and "ind . This is done by considering the dynamics of the hard particles in the background of the soft :elds A and #. Let us restrict ourselves here to the induced current. This can be obtained using linear response theory. To be more speci:c, consider as an example a system of charged particles on which is acting a perturbation of the form d x j (x)A (x), where j (x) is the current operator and A (x) some applied gauge potential. Linear response theory leads to the following relation for the induced current: ind R R j = d 4 y% (x − y)A (y); % (x − y) = − i&(x0 − y0 )[j (x); j (y)]eq ; (1.11) R (x − y) is also referred to as the polarization operwhere the (retarded) response function % ator. 3 Note that in Eq. (1.11), the expectation value is taken in the equilibrium state. Thus, within linear response, the task of calculating the basic ingredients of the e8ective theory for soft modes reduces to that of calculating appropriate equilibrium correlation functions. This can be done by a variety of techniques which will be reviewed in Section 2. In fact we shall need the response function only in the weak coupling regime, and for particular kinematical conditions which allow for important simpli:cations. In leading order in weak coupling, the polarization tensor is given by the one-loop approximation. In the kinematical regime of interest, where the incoming momentum is soft while the loop momentum is hard, we can write %(!; p) = g2 T 2 f(!=p; p=T ) with f a dimensionless function, and in leading order in p=T ∼ g, % is of the form g2 T 2 f(!=p). This particular contribution of the one-loop polarization tensor is an example of what has been called a “hard thermal loop” (HTL) [39 – 42,19,20]; for photons 3 Eq. (1.11) holds as written if A is an external potential. However, the polarization tensor is usually taken to be the one-particle irreducible piece of the response function (Section 5.1). With that de:nition, formula (1.11) still holds with A including also the induced current.

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in QED, this is the only one. It turns out that this hard thermal loop can be obtained from simple kinetic theory, and the corresponding calculation is done in the next subsection. In nonAbelian theory, linear response is not suKcient: constraints due to gauge symmetry force us to take into account speci:c nonlinear e8ects and a more complicated formalism needs to be worked out. Still, simple kinetic equations can be obtained in this case also, but in contrast to QED, the resulting induced current is a non linear functional of the gauge :elds. As a result, it generates an in:nite number of “hard thermal loops”. Actually, we shall see that even in QED, gauge invariance forces the fermionic induced source "ind to depend non linearly upon the gauge :elds, which entails the occurrence of an in:nite number of hard thermal loops with two external fermion lines and an arbitrary number of photon external lines. 1.2. One-loop polarization tensor from kinetic theory As indicated in the previous subsection, in the kinematical regime considered, the one loop polarization tensor can be obtained using elementary kinetic theory. Since this approach will be at the heart of the forthcoming developments in this paper, we present here this elementary calculation. We consider an electromagnetic plasma and momentarily assume that we can describe its charged particles in terms of classical distribution functions fq (p; x) giving the density of particles of charge q (q = ± e) and momentum p at the space–time point x = (t; r) [43]. We consider then the case where collisions among the charged particles can be neglected and where the only relevant interactions are those of particles with average electric (E) and magnetic (B) :elds. Then the distribution functions obey the following simple kinetic equation, known as the Vlasov equation [44,43]: 9fq 9fq 9fq +v· +F· =0 ; 9t 9r 9p

(1.12)

where v = d jp =dp is the velocity of a particle with momentum p and energy jp (for massless particles v = pˆ ), and F = q(E + v ∧ B) is the Lorentz force. The average :elds E and B depend themselves on the distribution functions fq . Indeed, the induced current d3 p v (f+ (p; x) − f− (p; x)) ; (1.13) jind (x) = e (2)3 where v ≡ (1; v), is the source term in the Maxwell equations (1.9) for the mean :elds. When the plasma is in equilibrium, the distribution functions, denoted as fq0 (p) ≡ f0 (jp ), are isotropic in momentum space and independent of the space–time coordinates; the induced current vanishes, and so do the average :elds E and B. When the plasma is weakly perturbed, the distribution functions deviate slightly from their equilibrium values, and we can write: fq (p; x) = f0 (jp ) + *fq (p; x). In the linear approximation, *f obeys (v · 9x )*fq (p; x) = − qv · E

df0 ; d jp

(1.14)

where v · 9x ≡ 9t + v · ∇. The magnetic :eld does not contribute because of the isotropy of the equilibrium distribution function.

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It is convenient here to set df0 *fq (p; x) ≡ −qW (x; v) ; d jp

(1.15)

thereby introducing a notation which will be used in various forms throughout this report. Since df0 fq (p; x) = f0 (jp ) − qW (x; v)

f0 (jp − qW (x; v)) ; (1.16) d jp W (x; v) may be viewed as a local distortion of the momentum distribution of the plasma particles. The equation for W is simply (v · 9x )W (x; v) = v · E(x) : Contrary to Eq. (1.12), the linearized equations (1.14) or of f with respect to p, and can be solved by the method derivative of *f(p; x) along the characteristic de:ned by perturbation is introduced adiabatically so that the :elds (" → 0+ ) when t0 → −∞, we obtain the retarded solution t dt e−"(t−t ) v · E(x − v(t − t ); t ) ; W (x; v) =

(1.17) (1.17) do not involve the derivative of characteristics: v · 9x is the time dx=dt = v. Assuming then that the and the Luctuations vanish as e"t0

−∞

and the corresponding induced current: d 3p df0 ∞ 2 jind (x) = − 2e v d, e−", v · E(x − v,) : (2)3 d jp 0

(1.18)

(1.19)

Since E = − ∇A0 − 9A= 9t, the induced current is a linear functional of A . At this point we assume explicitly that the particles are massless. In this case, v is a unit vector, and the angular integral over the direction of v factorizes in Eq. (1.19). Then, using Eq. (1.11) as de:nition for the polarization tensor % (x − y), and the fact that the Fourier transform of 0∞ d, e−", f(x − v,) is if(Q)=(v · Q + i"), with Q = (!; q) and f(Q) the Fourier transform of f(x), one gets, after a simple calculation [16]: v v d. 2 % (!; q) = mD −*0 * 0 + ! ; (1.20) 4 ! − v · q + i" where the angular integral d. runs over all the orientations of v, and mD is the Debye screening mass: 2e2 ∞ df0 2 mD = − 2 dpp2 : (1.21) 0 d jp As we shall see, Eq. (1.20) is the dominant contribution at high temperature to the one-loop polarization tensor in QED [17], i.e., the QED hard thermal loop, provided one substitutes for f0 the actual quantum equilibrium distribution function, that is, f0 (jp ) = np , with np given in Eq. (1.2). After insertion in Eq. (1.21), this yields m2D = e2 T 2 =3. In the next subsection, we shall address the question of how simple kinetic equations emerge in the description of systems of quantum particles, and under which conditions such systems

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can be described by seemingly classical distribution functions where both positions and momenta are simultaneously speci:ed. We shall later :nd that the expression obtained for the polarization tensor using simple kinetic theory generalizes to the nonAbelian case. This is so in particular because the kinematical regime remains that of the linear Vlasov equation, with straight line characteristics. 1.3. Kinetic equations for quantum particles In order to discuss in a simple setting how kinetic equations emerge in the description of collective motions of quantum particles, we consider in this subsection a system of non relativistic fermions coupled to classical gauge :elds. Since we are dealing with a system of independent particles in an external :eld, all the information on the quantum many-body state is encoded in the one-body density matrix [45 – 47]: /(r; r ; t) ≡

†

(r ; t) (r; t) ;

(1.22)

where and † are the annihilation and creation operators, and the average is over the initial equilibrium state. It is on this object that we shall later implement the relevant kinematical approximations. To this aim, we introduce the Wigner transform of /(r; r ; t) [48,49]: s s

(1.23) f(p; R; t) = d 3 s e−ip · s / R + ; R − ; t : 2 2 The Wigner function has many properties that one expects of a classical phase space distribution function as may be seen by calculating the expectation values of simple one-body observables. For instance the average density of particles n(R) is given by d3 p n(R; t) = /(R; R; t) = f(p; R; t) : (1.24) (2)3 Similarly, the current operator: (1=2mi)( j(R; t) =

†∇

− (∇

1 (∇y − ∇x )/(y; x; t)||y−x|→0 = 2mi

†)

) has for expectation value:

d3 p p f(p; R; t) : (2)3 m

(1.25)

These results are indeed those one would obtain in a classical description with f(p; R; t) the probability density to :nd a particle with momentum p at point R and time t. Note however that while f is real, due to the hermiticity of /, it is not always positive as a truly classical distribution function would be. Of course f contains the same quantum information as /, and it does not make sense to specify quantum mechanically both the position and the momentum. However, f behaves as a classical distribution function in the calculation of one-body observables for which the typical momenta p that are involved in the integration are large in comparison with the scale 1= characterizing the range of spatial variations of f, i.e. p1. By using the equations of motion for the :eld operators, i ˙ (r; t) = [H; ], where H is the single particle Hamiltonian, one obtains easily the following equation of motion for the density matrix: i9t / = [H; /] :

(1.26)

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In fact we shall need the Wigner transform of this equation in cases where the gradients with respect to R are small compared to the typical values of p. Under such conditions, the equation of motion reduces to 9 (1.27) f + ∇p H · ∇R f − ∇R H · ∇p f = 0 : 9t where we have kept only the leading terms in an expansion in ∇R . For particles interacting with gauge potentials A (X ), the Wigner transform of the single particle Hamiltonian in Eq. (1.27) takes the form p2 e2 e H (R; p; t) = − A · p + A2 (R; t) + eA0 (R; t) : (1.28) 2m m m Assuming that the :eld is weak and neglecting the term in A2 , one can write Eq. (1.27) in the form e 9t f + v · ∇R f + e(E + v ∧ B) · ∇p f + (pj 9j Ai )∇ip f = 0 ; (1.29) m where we have set v = (p − eA)=m. This equation is almost the Vlasov equation (1.12): it di8ers from it by the last term which is not gauge invariant. The presence of such a term, and the related gauge dependence of the Wigner function, obscure the physical interpretation. It is then convenient to de:ne a gauge invariant density matrix: /(r; G r ; t) ≡ where (s = r − r )

†

(r ; t) (r; t)U (r; r ; t) ;

U (r; r ) = exp −ie

r

r

(1.30)

dz · A(z; t) ≈ exp(−ies · A(R))

(1.31)

and the integral is along an arbitrary path going from r to r. Actually, in the last step we have used an approximation which amounts to chose for this path the straight line between r to r; furthermore, we have assumed that the gauge potential does not vary signi:cantly between r to r. (Typically, /(r; r ) is peaked at s = 0 and drops to zero when s & T where T is the thermal wavelength of the particles. What we assume is that over a distance of order T the gauge potential remains approximately constant.) Note that in the calculation of the current (1.25), only the limit s → 0 is required, and that is given correctly by Eq. (1.31) (see also Eq. (1.33) below). With the approximate expression (1.31) the Wigner transform of Eq. (1.30) is simply G G f(R; k) = f(R; k + eA). By making the substitution f(R; p) = f(R; p − eA) in Eq. (1.29), one veri:es that the noncovariant term cancels out and that the covariant Wigner function fG obeys indeed Vlasov’s equation. In the presence of a gauge :eld, the previous de:nition (1.25) of the current su8ers from the lack of gauge covariance. It is however easy to construct a gauge invariant expression for the current operator, 1 1 † 1 † j= ∇ − eA − ∇ + eA ; (1.32) 2m i i whose expectation value in terms of the Wigner transforms reads d 3 p p − eA d3 k k G f(R; k; t) : (1.33) f(R; p; t) = j(R; t) = (2)3 m (2)3 m

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The last expression involving the covariant Wigner function makes it clear that j(R; t) is gauge invariant, as it should. The momentum variable of the gauge covariant Wigner transform is often referred to as the kinetic momentum. It is directly related to the velocity of the particles: k = mv = p − eA. As for p, the argument of the noncovariant Wigner function, it is related to the gradient operator and is often referred to as the canonical momentum. In order to understand the structure of the equations that we shall obtain for the QCD plasma, it is :nally instructive to consider the case where the particles possess internal degrees of freedom (such as spin, isospin, or colour). The density matrix is then a matrix in internal space. As a speci:c example, consider a system of spin 1=2 fermions. The Wigner distribution reads [50] f(p; R) = f0 (p; R) + fa (p; R)5a ;

(1.34)

where the 5a are the Pauli matrices, and the fa are three independent distributions which describe the excitations of the system in the various spin channels; together they form a vector that we can interpret as a local spin density, f = (1=2)Tr(f). When the system is in a magnetic :eld with Hamiltonian H = − 0 · B, the time derivative of f acquires a new contribution −20 B ∧ f ;

(1.35)

which accounts for the spin precession in the magnetic :eld. Thus the time dependence of the distribution function along the characteristics is determined by the following operator: d 9 = + v · ∇ − 20 B ∧ : dt 9t

(1.36)

It is important to realize that all the di8erential operators above and in the Vlasov equation apply to the arguments of the distribution functions, and not to the coordinates of the actual particles. Note however that equations similar to the ones presented here can be obtained for classical spinning particles. When the angular momentum of such particles is large, it can indeed be treated as a classical degree of freedom, and the corresponding equations of motion have been obtained by Wong [51]. After replacing spin by colour, these equations have been used by Heinz [52,53] in order to write down transport equations for classical coloured particles. By implementing the relevant kinematical approximations one then recovers [54] the nonAbelian Vlasov equations to be derived below, i.e., Eqs. (1.37) and (1.38). (See also Refs. [55 –59] for related work.) We shall not pursue this line of reasoning however, since we do not :nd it technically useful (it does not bring any simpli:cations) and it is physically misleading (see however Refs. [60]). Besides, the kinetic equation describing soft fermionic excitations (like Eq. (1.39) below) are not readily obtained in this way. 1.4. QCD kinetic equations and hard thermal loops We are now ready to present the equations that we shall obtain for the QCD plasma. These equations are for generalized one-body density matrices describing the long wavelength collective motions of the hard particles. They look formally as Vlasov equations, the main

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ones being [23,18] dnk ; dk

(1.37)

dNk ; dk

(1.38)

[v · Dx ; *n± (k; x)] = ∓ gv · E(x) [v · Dx ; *N (k; x)] = − gv · E(x)

(v · Dx )=(k; x) = − igCf (Nk + nk )v=#(x) ;

(1.39)

In these equations, v = (1; v), v = k=k, Aa (x) and #(x) are average gauge and fermionic :elds, and *n± , *N and = are gauge-covariant Wigner functions for the hard particles. The :rst two Wigner functions are density matrices describing the colour oscillations of the quarks and the gluons, respectively: *n± = *na± t a and *N = *Na T a . The last one (=) is that of a more exotic density matrix which mixes bosons and fermions degrees of freedom, = ∼ A; it determines the induced fermionic source "ind in Eq. (1.10). The right hand sides of the equations specify the quantum numbers of the excitations that they are describing: soft gluon for the :rst two, and soft quark for the last one. One of the major di8erence between the QCD equations above and the linear Vlasov equation for QED is the presence of covariant derivatives in the left hand sides of Eqs. (1.37) – (1.39). These play a role similar to that of the magnetic :eld in Eq. (1.35) for the distribution functions of particles with spin. (Note that the equation for = holds also in QED, with a covariant derivative there as well.) Eqs. (1.37) – (1.39) have a number of interesting properties which will be discussed in Section 3. In particular, they are covariant under local gauge transformations of the classical :elds, and independent of the gauge-:xing in the underlying quantum theory. By solving these equations, one can express the induced sources as functionals of the background :elds. To be speci:c, consider the induced colour current: d3 k a jind (x) ≡ 2g v Tr (T a *N (k; x)) ; (1.40) (2)3 where *N is the gluon density matrix (the quark contribution reads similarly). Quite generally, the induced colour current may be expanded in powers of A , thus generating the one-particle irreducible amplitudes of the soft gauge :elds [23] ab abc / jind a = % Ab + 12 7 / Ab Ac + · · · :

(1.41)

ab = *ab % Here, % is the polarization tensor, and the other terms represent vertex corrections. These amplitudes are the hard thermal loops for soft gluons [42,19,20,22]. There are similar HTLs for the soft fermions; they are generated by expanding "ind . Diagrammatically, the HTLs are obtained by isolating the leading order contributions to one-loop diagrams with soft external lines (see Appendix B for some explicit such calculations). It is worth noticing that the kinetic equations isolate directly these hard thermal loops, in a gauge invariant manner, without further approximations.

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The gluon density matrix can be parameterized as in Eq. (1.15): *N (k; x) = − gW (x; v)(dNk =d k) ;

(1.42)

where Nk ≡ 1=(ek − 1) is the Bose–Einstein thermal distribution, and W (x; v) ≡ Wa (x; v)T a is a colour matrix in the adjoint representation which depends upon the velocity v = k=k (a unit vector), but not upon the magnitude k = |k| of the momentum. A similar representation holds for the quark density matrices *n± (k; x). Then the colour current takes the form d. a a (x) = m2D jind (1.43) v W (x; v) 4 with m2D ∼ g2 T 2 . The kinetic equations for *N and *n± can then be written as an equation for Wa (x; v): (v · Dx )ab Wb (x; v) = v · Ea (x) :

(1.44)

This di8ers from the corresponding Abelian equation (1.17) merely by the replacement of the ordinary derivative 9x ∼ gT by the covariant one Dx = 9x + igA. Accordingly, the soft gluon polarization tensor derived from Eqs. (1.43) – (1.44), i.e., the “hard thermal loop” % , is formally identical to the photon polarization tensor obtained from Eq. (1.17) and given by Eq. (1.20) [39,40]. The reason for the existence of an in:nite number of hard thermal loops in QCD is the presence of the covariant derivative in the left hand side of Eq. (1.44). A similar observation can be made by writing the induced electromagnetic current in the form d. 1 2 4 jind (x) = mD d y x| (1.45) v |y v · E(y) = d 4 y 5j (x; y)E j (y) : 4 v·9 This expression, which is easily obtained from the expression (1.19) of *f, de:nes the conductivity tensor 5 . The generalization of this expression to QCD amounts essentially to replacing the ordinary derivative by a covariant one. Even though obtained via one-loop calculations, the HTLs are nonperturbative in the sense that, at soft momenta .gT , they are as large as the corresponding tree-level amplitudes. For instance, the expression (1.20) for the photon (or gluon) polarization tensor shows that % ∼ m2D ∼ g2 T 2 , which at soft momenta p ∼ gT is as large as the tree-level inverse propagator −1 (p) ≡ g p2 − p p ∼ (as obtained from the left hand side of the Maxwell equation (1.9)) G0 g2 T 2 . In Section 5.3, we shall similarly verify that the 3-gluon and 4-gluon HTL vertex “corrections” generated by expanding the induced current as in Eq. (1.41) are actually as large as the corresponding tree-level vertices. Thus, the HTL cannot be perturbatively expanded out at soft momenta, but rather must be treated as a part of the leading order e;ective theory at soft momenta. This is summarized by the equations of motion for the soft :elds with the induced sources in the HTL approximation. For instance, for soft gluons a (D F )a (x) = jind (x) ; a as given by Eq. (1.43). with jind

(1.46)

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1.5. E;ect of collisions Until now, we have been discussing independent particles moving in average self-consistent :elds. It can be argued that in weak coupling and for long wavelength excitations, this is the dominant picture. There are situations however where collisions among the plasma particles cannot be ignored. We shall consider in this report two such cases. One concerns the lifetime of the single particle excitations to be discussed in Section 6. The other refers to the study of ultrasoft excitations at the scale g2 T which will be presented in Section 7. The determination of the lifetimes of single particle excitations played an essential role in the development of the subject and led in particular to the identi:cation of the hard thermal loops [42,61,62]. Physically, the lifetime of a quasiparticle excitation is limited by its collisions with the other particles in the plasma. The collision rate can be estimated directly in the form 8 = n5v, where n ∼ T 3 is the density of plasma particles, 5 the collision cross section, and v the velocity equalto the speed of light. Restricting ourselves :rst to the Coulomb interaction, we can write 5 = dq2 (d5=dq2 ), with d5=dq2 ∼ g4 =q4 . Thus, 1 4 3 8∼g T dq2 4 ; (1.47) q which is badly infrared divergent. From our previous calculation of the polarization tensor, one knows, however, that in the plasma the Coulomb interaction is screened, so that the e8ective electric photon propagator is not 1=q2 but 1=(q2 + m2D ), where mD ∼ gT is the Debye screening mass. With this e8ect taken into account, the collision rate becomes 1 8 ∼ g4 T 3 2 ∼ g2 T ; (1.48) mD which is now :nite, and of order g2 T . However, screening corrections at the scale gT [63], as encoded in the hard thermal loops, are not suKcient to eliminate all the divergences due to the magnetic interactions [42,55,64 – 66]; they leave an estimate for the lifetime mD dq 2 8∼g T (1.49) q which is logarithmically divergent [42]. This infrared problem occurs both in Abelian and nonAbelian gauge theories. In QCD, it is commonly bypassed by advocating the infrared cut-o8 provided by a “magnetic mass” ∼ g2 T , so that 8 ∼ g2 T ln(1=g). But such a solution cannot apply for QED where one does not expect any magnetic screening [17,67]. In Section 6, we shall analyse the origin of these infrared divergences and show that the dominant ones can be resummed in closed form for the retarded propagator of the quasiparticle excitation. This will be achieved by considering as an intermediate step the propagation of a test particle in a background of random ultrasoft (and mostly static) thermal Luctuations. The retarded propagator is obtained by averaging over these Luctuations. Remarkably, the resulting damping is nonexponential, the retarded propagator being of the form SR (t) ∼ exp(−g2 Tt ln(t mD )) [68]. We shall see that such a particular behaviour also emerges in

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a treatment of the collisions using a generalized Boltzmann equation in a regime where the mean free path is comparable with the range of the relevant interactions [69]. The second case where the collisions become important is in the study of ultrasoft perturbations at the scale g2 T or smaller. To give a crude estimate of these collisional e8ects, one may use the relaxation time approximation, and write the kinetic equation as (v · Dx )ab Wb (x; v) = v · Ea (x) −

W a (x; v) ; ,col

(1.50)

where ,col is a typical relaxation time. It is important here to distinguish between colour and colourless excitations. The relaxation of colour is dominated by the singular forward scattering which yields ,col ∼ 1=(g2 T ln(1=g)) [55,65,25]. Then, Eq. (1.50) shows that the e8ect of the collisions become a leading order e8ect for inhomogeneities at the scale 9x ∼ g2 T , or less. Colourless Luctuations, such as Luctuations in the momentum or the electric charge distributions, involve a colour independent Luctuation W . The corresponding kinetic equation reduces to a simple drift term v · 9x in the left hand side (no colour mean :eld) and a collision term in the right hand side. This collision term involves now large angle scatterings, and the resulting relaxation time is much larger, ,el ∼ 1=(g4 T ln(1=g)) [63,70,71]. In that case, collisions become important only for space–time inhomogeneities at scale ∼1=g4 T . Of course, the relaxation time approximation is only a crude approximation. (For coloured excitations, this is not even a gauge-invariant approximation [72].) A complete Boltzmann equation [26] will be derived in the last section of this report, by extending the techniques used to derive the collisionless kinetic equations in Section 3. In the same way as the induced current calculated from the solution of the Vlasov equation (1.44) generates directly the hard thermal loops, we shall see that the induced current calculated with the solution of the Boltzmann equation isolates the leading-order contributions to an in:nite set of multi-loop diagrams where the external momenta are ultrasoft [72]. These amplitudes share many properties with the hard thermal loops, although they correspond typically to multiloop diagrams. These amplitudes are logarithmically infrared divergent, so are best understood as ingredients of the e8ective theory for ultrasoft excitations at a scale gT , with playing in their calculation the role of an IR cuto8 [25]. 1.6. E;ective theory for soft and ultrasoft excitations We have concentrated so far on the dynamics of hard degrees of freedom in external background :elds, possibly taking into account the e8ect of collisions when considering very long wavelength excitations. But it is also of interest to consider the e8ective theory for the soft degrees of freedom obtained by “eliminating” the hard ones. As mentioned earlier, for soft photons in QED this e8ective theory reduces to the Maxwell equations with an induced current, and the same holds for gluons in QCD, with the Maxwell equations replaced by the Yang– Mills equations and with the colour current (1.40). Similarly, the soft fermionic excitations are described, in both QED and QCD, by the Dirac equation (1.10) with the induced source "ind built out of =(k; x), cf. Eq. (1.39). If we want to study for instance the collective excitations of the plasma these equations of motion are all what is needed.

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There are cases however where one needs to take into account the e8ect of such collective modes on correlation functions (an example is actually provided by the calculation of the damping rate of quasiparticle excitations). To do so, one needs to go one step further and determine the Boltzmann weight associated with such modes. The problem is made easier by the fact that soft bosonic excitations can be described by classical :elds [73,74] which may be identi:ed with the average :elds introduced before. For excitations at the scale gT , one can construct a Hamiltonian description of the dynamics of these classical :elds. In terms of the :elds W a introduced earlier, the Hamiltonian is remarkably simple [24,75,76]: d. a 1 H= d 3 x Ea · Ea + Ba · Ba + m2D (1.51) W (x; v) W a (x; v) : 2 4 As we shall see in Section 4, when appropriate Poisson brakets are introduced, the Hamiltonian (1.51) generates indeed the correct dynamics [24,77]. It will also be shown in Section 4 that this Hamiltonian provides the correct Boltzmann weight to integrate over soft Luctuations [77]. The calculation of real time correlation functions reduces then to the calculation of a functional integral where the integration variables are the gauge :elds and the auxiliary :elds W , and the functional integration amounts to an average over the initial conditions for the classical :eld equations of motion. This allows in particular for numerical calculations of the real time correlation functions on a three-dimensional lattice. An important application, which has received much attention in recent years [4,78,28,27] is the evaluation of the anomalous baryon number violation rate at high temperature. This is de:ned as [4] 2 2 ∞ g 7≡ dt d 3 x [Eai Bai (t; x)][Eai Bai (0; 0)] ; (1.52) 2 8 −∞ and receives contributions typically from the nonperturbative magnetic modes with momenta k ∼ g2 T and energies ! . g4 T [78]. Recently, this has been computed via lattice simulations of the classical e8ective theory with Hamiltonian (1.51) [27]. (See also Refs. [57,58] for a di8erent lattice implementation of the HTL e8ects, and Refs. [79 –82] for numerical calculations within purely Yang–Mills classical theory, without HTLs.) The e8ective theory that we have just outlined leads to ultraviolet divergences. However, it is de:ned with an ultraviolet cuto8 gT T . The coeKcients of the e8ective theory, which are the hard thermal loops, must also be calculated with an infrared cuto8 , so that the cuto8 dependence of the parameters in the e8ective theory (here the Debye mass) cancels against the cuto8 dependence of the classical thermal loops. Without such a matching, which turns out to be diKcult to implement in QCD, the calculation of correlation functions within the classical e8ective theory remains linearly sensitive to the ultraviolet cuto8 [74,78,77,83–85]. This is clearly exhibited by the numerical results for 7, Eq. (1.52), obtained in [82]. In order to reduce the sensitivity to the scale it has been suggested to go one step further and eliminate also the soft degrees of freedom down to a scale g2 T gT . This can be done starting from the classical e8ective theory for soft :eld and integrating explicitly over the soft degrees of freedom. This is the approach followed by BXodeker, who showed that the resulting theory at the scale g2 T takes the form of a Boltzmann–Langevin equation [25]. Results of numerical simulations based on (a simpli:ed version of) this equation have been given in [28]

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(cf. Section 7 below). The collision term obtained by BXodeker is identical to that appearing in the Boltzmann equation that we have obtained following a completely di8erent route [26]. The reason for this will be detailed in Section 7, where we also show that the noise term in the Langevin equation is simply related to the collision integral through the Luctuation–dissipation theorem. The building blocks of the new e8ective theory are the ultrasoft amplitudes mentioned above. As already mentioned, these amplitudes depend logarithmically on the separation scale , but this dependence will eventually cancel against the cuto8 dependence of the loop corrections in the e8ective theory. 1.7. Outline of the paper We now present the outline of this paper. Section 2 is a pedagogical introduction to most of the techniques that we shall be using. This includes a short review of equilibrium thermal :eld theory in the imaginary time formalism, a description of near equilibrium longwavelength excitations, the use of Wigner transform to obtain kinetic equations. To keep things as simple as possible, the formalism is developed for the case of a real scalar :eld. In Section 3, we begin to implement these techniques in the case of QCD. In particular, we present the main steps in the derivation of the kinetic equations for the hard particles. These kinetic equations are solved explicitly in Section 4. This leads to e8ective equations of motion for the soft modes of the plasma. These soft modes could be excitations of the plasma driven by external disturbances. They also appear as long-wavelength Luctuations in the plasma in equilibrium. The issue of calculating the e8ect of such Luctuations on real time correlation functions is addressed. We show that this can be formulated conveniently in terms of an e8ective theory for classical :elds. The construction of this e8ective theory is explicitly given. The induced current which provides the source for the soft mode propagation is a non linear functional of the gauge :elds. It may be viewed as a generating functional for an in:nite set of one loop amplitudes, the so-called hard thermal loops. Some of these hard thermal loops are explicitly constructed in Section 5 and their properties analysed. A few applications are mentioned. Section 6 addresses the issue of the damping of the plasma excitations. This is a problem which has triggered much of the work on the hard thermal loops, but whose general solution requires going beyond the hard thermal loop approximation. It provides an interesting illustration of the e8ects of collisions in a regime where the range of the relevant interactions is comparable with the mean free path of the particles. In Section 7 we consider some of the physics taking place at the scale g2 T . For modes with such momenta, collision terms in the Boltzmann equation become relevant. We show that there exists an in:nite set of amplitudes, which we called ultrasoft amplitudes, which become of the same order of magnitude as the hard thermal loops, and which are generated by the Boltzmann equation. This equation is an essential ingredient in the e8ective theory for ultrasoft excitations which is brieLy presented. Finally Section 8 summarizes the conclusions.

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Appendix A contains a summary of the notation used throughout. Appendix B presents detailed calculations, in the hard thermal loop approximation, of one loop diagrams that are referred to in the main text. 2. Quantum #elds near thermal equilibrium In most of this paper, we shall study generically how a system initially in thermal equilibrium responds to a weak and slowly varying disturbance. This section summarizes the main tools that will be needed in such a study. It starts with a short review of equilibrium thermal :eld theory using the imaginary time formalism. Then we turn to o8-equilibrium situations and derive the equations of motion for the appropriate Green’s functions. The last subsection is devoted to the implementation of the longwavelength approximation using gradient expansions. This allows us to transform the general equations of motion into simpler kinetic equations. Much of the material of this section is fairly standard, and many results will be mentioned without proof. More complete presentations can be found for instance in Refs. [144,86,87,12–14] for equilibrium situations, and in Refs. [43,88–93] for the nonequilibrium ones. In order to bring out the essential aspects of the formalism while avoiding the complications speci:c to gauge theories, we shall consider in this section only a scalar :eld theory, with Lagrangian 1 2

L = 9 ;9 ; −

m2 2 ; − V (;) 2

1 1 m2 2 = (90 ;)2 − (∇;)2 − ; − V (;) ; 2 2 2

(2.1)

where V (;) is a local potential. The initial equilibrium state is described by the canonical density operator: D=

e−H ; Z

(2.2)

where H is the hamiltonian of the system and Z the partition function. For the scalar :eld, m2 2 1 2 1 3 2 (2.3) H= d x + (∇;) + ; + V (;) ; 2 2 2 where (x) is the :eld canonically conjugate to ;(x). We may express D in terms of the eigenstates |n of H (H |n = En |n) and probabilities pn (pn = e−En =Z): D= |npn n| : (2.4) n

We consider a time-dependent perturbation of the form Hj (t) − H = d 3 x j(t; x);(x) :

(2.5)

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Under the action of such a perturbation, the system evolves away from the equilibrium state. The density operator at time t is given by the equation of motion iD˙ j = [Hj (t); D(t)] ;

(2.6)

where D˙ j ≡ 9t Dj . It can be written as Dj (t) = |n; t pn n; t |

(2.7)

n

with time-independent pn ’s (the same as in equilibrium); the state |n; t is the solution of the SchrXodinger equation which coincides initially with the eigenstate |n. Note

that the evolution described by Eq. (2.6) conserves the entropy S = − kB Tr D ln D = − kB n pn ln pn . All the approximations that we shall consider here ful:l this property. 2.1. Equilibrium thermal 4eld theory Before embarking into the discussion of nonequilibrium dynamics, it is useful to review brieLy the formalism of thermal :eld theory in equilibrium. We shall in particular recall how perturbation theory can be used to calculate the partition function: Z ≡ Tr exp{−H } = exp{−En } ; (2.8) n

from which all the thermodynamical functions can be obtained. The simplest formulation of the perturbation theory for thermodynamical quantities is based on the formal analogy between the partition function (2.8) and the evolution operator U (t; t0 ) = exp{−i(t − t0 )H }, where the time variable t is allowed to take complex values. Speci:cally, we can write Z = Tr U (t0 − i; t0 ), with arbitrary (real) t0 . More generally, we shall de:ne an operator U (,) ≡ exp(−,H ), where , is real, but often referred to as the imaginary time (, = i(t − t0 ) with t − t0 purely imaginary). The evaluation of the partition function (2.8) by a perturbative expansion involves the splitting of the hamiltonian into H = H0 + H1 , with H1 H0 . For instance, for the scalar :eld theory in Eq. (2.1), it is convenient to take 1 m2 2 3 1 2 2 H0 = d x + (∇;) + H1 = d 3 xV (;) : (2.9) ; ; 2 2 2 We then set U (,) = exp(−,H ) = exp(−,H0 ) exp(,H0 ) exp(−,H ) = U0 (,)UI (,) ;

(2.10)

where U0 (,) ≡ exp(−H0 ,). The operator UI (,) is called the interaction representation of U . We also de:ne the interaction representation of the perturbation H1 : H1 (,) = e,H0 H1 e−,H0

(2.11)

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and similarly for other operators. It is easy to verify that UI (,) satis:es the following di8erential equation: d (2.12) UI (,) + H1 (,)UI (,) = 0 d, with the boundary condition UI (0) = 1 :

(2.13)

The solution of the above di8erential equation, with the boundary condition (2.13), can be written formally in terms of the time ordered exponential: e−H = e−H0 T, exp −

0

d,H1 (,)

:

(2.14)

The symbol T, implies an ordering of the operators on its right, from left to right in decreasing order of their imaginary time arguments: T, exp −

=1 −

0

0

=1 −

0

d,H1 (,)

1 d,H1 (,) + 2 d,H1 (,) +

0

0

d,1 d,2 T[ H1 (,1 )H1 (,2 )] + · · ·

d,1

,1

0

d,2 H1 (,1 )H1 (,2 ) + · · · :

(2.15)

Using Eq. (2.14), one can rewrite Z in the form Z = Z0

T exp −

0

d,H1 (,)

; 0

where, for any operator O, −H0 e O0 ≡ Tr O : Z0

(2.17)

Alternatively, one may write the partition function as the following path integral: Z =N D(;) exp − d, d 3 x LE (x) ; ;(0)=;()

(2.16)

0

(2.18)

where N is a normalization constant which cancels out in the calculation of expectation values, but which needs to be treated with care in the evaluation of thermodynamical functions. In Eq. (2.18), ;(,; x) ≡ ;(t = t0 − i,; x) and 1 1 m2 2 LE = (9, ;)2 + (∇;)2 + (2.19) ; + V (;) : 2 2 2 The functional integration runs over :eld con:gurations which are periodic in imaginary time, ;(, = 0) = ;(, = ).

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In the rest of this report, we shall refer to both the path integral and the operator formalisms, the choice of either one depending on which is the most convenient for the question under study. In both formalisms, imaginary time Green’s functions or propagators appear. These have special properties which are recalled in the next subsections. 2.1.1. The imaginary time Green’s functions By adding to LE in Eq. (2.18) a source term −j(x);(x), where j(x) is an arbitrary external current, one transforms Z into the generating functional Z[j] of the imaginary-time Green’s functions: T, ;(x1 );(x2 ) : : : ;(x n ) = Tr {D T, ;(x1 );(x2 ) : : : ;(x n )} *n Z[j] 1 = : Z *j(x1 )*j(x2 ) : : : *j(x n ) j=0

(2.20)

In this formula, ;(x) (with x = (t; x), t = t0 − i,; 0 6 , 6 ) is a :eld operator in the Heisenberg representation, ;(t; x) = eiH (t−t0 ) ;(x)e−iH (t−t0 ) = eH, ;(x) e−H, :

(2.21)

The connected Green’s functions, for which we reserve throughout the notation G (n) (x1 ; x2 ; : : : ; x n ), are given by *n ln Z[j] (n) G (x1 ; x2 ; : : : ; x n ) = : (2.22) *j(x1 )*j(x2 ) : : : *j(x n ) j=0 For space–time translational invariant systems, they depend e8ectively only on n − 1 relative coordinates. In particular, for the 2-point function, we shall write G (2) (x1 ; x2 ) = G (2) (x1 − x2 ) ≡ G(x). The imaginary time Green’s functions obey periodicity conditions. For instance, for the 2-point function, we have G(, − ) = G(,)

for 0 6 , 6 ;

G(, + ) = G(,)

for − 6 , 6 0 ;

(2.23)

where , ≡ ,1 − ,2 . (Here, and often below, when focusing on temporal properties we do not mention the spatial coordinates for simplicity.) To prove these relations, note that, for 0 6 , 6 , G(,) = Tr {D ;(,);(0)} =

1 Tr {e−H (−,) ; e−,H ;} ; Z

(2.24)

where Eq. (2.21) has been used. On the other hand, − 6 , − 6 0, so that G(, − ) = Tr {D ;(0);(, − )} =

1 Tr {e−H ; e−H (−,) ; e(−,)H } ; Z

(2.25)

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which coincides indeed with G(,), Eq. (2.24), because of the cyclic invariance of the trace. The periodicity conditions (2.23) allow us to represent G(,) by a Fourier series: 1 −i!n , G(,) = e G(i!n ) ; (2.26) n where the frequencies !n = 2nT , with integer n, are called Matsubara frequencies. The free propagator G0 (x − y) is de:ned as (see Eq. (2.17)): G0 (x − y) ≡ T, ;I (x);I (y)0 ;

(2.27)

where ;I (x) is the interaction representation of the :eld operator (cf. Eq. (2.11)): ;I (t; x) = eH0 , ;(x) e−H0 , :

(2.28)

It satis:es the equation of motion: (−92, − ∇2x + m2 )G0 (,; x) = *(,) *(3) (x) ;

(2.29)

with the periodic boundary conditions (2.23). This equation is easily solved using the Fourier representation (2.26). One gets G0 (i!n ; k) =

1

j2k + !n2

;

(2.30)

√ where jk = k2 + m2 . The imaginary-time propagator G0 (,) can be recovered from its Fourier coeKcients (2.30) by performing the frequency sum in Eq. (2.26). After a simple calculation (see Appendix B), one obtains the following expression for G0 (,; k), valid for − 6 , 6 : +∞ d k0 −k0 , G0 (,; k) = /0 (k)(&(,) + N (k0 )) ; (2.31) e −∞ 2

where N (k0 ) = 1=(ek0 − 1) is the Bose–Einstein occupation factor, and /0 (k) = 2(k0 )*(k 2 − m2 ) = (*(k0 − jk ) − *(k0 + jk )) ; jk

(2.32)

(with (k0 ) = k0 = |k0 |) is the spectral function of a free relativistic scalar particle of mass m. In the imaginary-time formalism, the thermal perturbation theory has the same structure as the perturbation theory in the vacuum, the only di8erence being that the integrals over the loop energies are replaced by sums over the Matsubara frequencies [12,14]. Appendix B provides examples of explicit computations in this formalism. Note that because the heat bath provides a preferred frame, explicit Lorentz invariance is lost, which makes some calculations more complicated than the equivalent ones at zero temperature. 2.1.2. Analyticity properties and real-time propagators The imaginary-time propagator G(,) may be written quite generally as G(,) = &(,) G ¿ (,) + &(−,) G ¡ (,) ;

(2.33)

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where the functions G ¿ and G ¡ are de:ned by G ¿ (x; y) ≡ Tr {D;(x);(y)} ; G ¡ (x; y) ≡ Tr {D;(y);(x)} = G ¿ (y; x) ;

(2.34)

with the :elds ;(x) in the Heisenberg representation (2.21). In these equations, all the time variables are complex variables of the form t = t0 − i, to start with. However, as we shall see, the functions G ¿ and G ¡ are analytic functions of their time arguments, with certain domains of analyticity to be speci:ed below. They can be used to construct real-time Green’s functions, such as the time-ordered, or Feynman, propagator: G(x; y) = &(x0 − y0 )G ¿ (x; y) + &(y0 − x0 )G ¡ (x; y) ;

(2.35)

as well as retarded and advanced propagators: GR (x; y) ≡ i&(x0 − y0 )[G ¿ (x; y) − G ¡ (x; y)] ; GA (x; y) ≡ −i&(y0 − x0 )[G ¿ (x; y) − G ¡ (x; y)] ;

(2.36)

where x0 and y0 are both real time variables. These functions enter the description of the response of the system to small external perturbations (cf. Section 2.2.1). To see the origin of the analyticity, note that, by de:nition, 1 G ¿ (t) = Tr {D;(t);(0)} = Tr {e−H (−it) ;e−iHt ;} : (2.37) Z To evaluate the trace in Eq. (2.37), we may introduce a complete set |n of energy eigenstates, H |n = En |n, and thus obtain 1 −En e |n|;|m|2 eit(En −Em ) : (2.38) G ¿ (t) = Z m; n If we assume that the exponentials control the convergence of this sum, we expect the trace to be convergent as long as − ¡ Im t ¡ 0. Similarly, we expect G ¡ (t) to exist for all t in the region 0 ¡ Im t ¡ . In these respective domains, G ¿ (t) and G ¡ (t) are both analytic functions. They also exist, as generalized functions, for t approaching the boundaries of their respective analyticity domains, and, in particular, for real values of t [94,88,87]. For complex time variables, the periodicity conditions (2.23) translate into the following condition on the analytic functions G ¿ and G ¡ (0 6 Im t 6 ): G ¡ (t) = G ¿ (t − i) ;

(2.39)

also known as the Kubo–Martin–Schwinger (KMS) condition [95,96]. The real-time functions G ¿ and G ¡ satisfy hermiticity properties: (G ¿ (y; x))∗ = G ¿ (x; y) and (G ¡ (y; x))∗ = G ¡ (x; y). This is easily veri:ed. For instance (G ¿ (y; x))∗ = (Tr {D;(y);(x)})∗ = Tr {(D;(y);(x))† } = Tr {;(x);(y)D} = G ¿ (x; y) ;

(2.40)

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where in writing the last two equalities we have used the hermiticity of ;(x) (x0 real) and of D, and the cyclic invariance of the trace. The hermiticity property (2.40), together with de:nitions (2.36), imply (GR (x; y))∗ = GA (y; x). For the real scalar :eld we have the additional symmetry condition G ¿ (x; y) = G ¡ (y; x) (cf. Eq. (2.34)), which ensures that GR (x; y) and GA (x; y) are real functions, with GA (x; y) = GR (y; x). The analytic functions G0¿ and G0¡ corresponding to the free scalar :eld can be read o8 Eq. (2.31): 1 G0¿ (,; k) = {(1 + Nk )e−jk , + Nk ejk , } ; 2jk G0¡ (,; k) =

1 {N e−jk , + (1 + Nk )ejk , } : 2jk k

(2.41)

where Nk ≡ N (jk ). By replacing , → it (with real t) in these equations, and using de:nitions (2.35) and (2.36), we get 1 G0 (t; k) = {&(t)e−ijk t + &(−t)eijk t + 2Nk cos jk t } ; (2.42) 2jk sin jk t sin jk t ; G0A (t; k) = − &(−t) : (2.43) G0R (t; k) = &(t) jk jk 2.1.3. Spectral representations for the propagator and the self-energy The analytic properties of the Green’s functions, considered as functions of complex times, entail corresponding properties of their Fourier transforms, which we shall now summarize. Let G ¿ (k) (with k = (k 0 ; k); k0 real) be the Fourier transform of the real-time function G ¿ (x): G ¿ (k) = d 4 xeik·x G ¿ (x) ; (2.44) and similarly for G ¡ (k). The hermiticity of G ¿ (x) (cf. Eq. (2.40)) implies that G ¿ (k) is a real function, and similarly for G ¡ (k). Furthermore, the KMS condition (2.39) implies the following relation: G ¿ (k0 ; k) = ek0 G ¡ (k0 ; k) :

(2.45)

Consider then the spectral density /(k). This is related to the functions G ¿ (k) and G ¡ (k) by /(k) ≡ G ¿ (k) − G ¡ (k) = d 4 x eik·x [;(x); ;(0)] =

2 −En e |n|;|m|2 (*(k0 + En − Em ) − *(k0 − En + Em )) : Z m; n

(2.46)

In writing the second line, all reference to the spatial momenta has been omitted, for simplicity. For the noninteracting system, Eq. (2.46) reduces to the free spectral density (2.32).

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By rotational symmetry, /(k) is a function of k0 and |k|. The dependence on k0 it is such that /(−k0 ) = − /(k0 ) and k0 /(k0 ) ¿ 0, as can be deduced from Eq. (2.46). Furthermore, the equal-time commutation relation [i9t ;(t; x); ;(0; x )]t=0 = *(x − x ) can be used to obtain the sum rule: d k0 (2.47) k0 /(k) = d 3 x e−ik·x [i9t ;(x); ;(0)]t=0 = 1 : 2 By combining Eqs. (2.45) and (2.46), we obtain G ¿ (k) = /(k)[1 + N (k0 )];

G ¡ (k) = /(k)N (k0 ) :

(2.48)

Formulae (2.46) and (2.48) show that the functions G ¿ (k) and G ¡ (k) are positive de:nite, and suggest the following interpretation for them [88]: For positive k0 ; G ¡ (k) is proportional to the average density of particles with momentum k and energy k0 , while G ¿ (k) measures the density of states available for the addition of an extra particle with four-momentum k . A similar interpretation may be given for negative k0 , by exchanging the roles of G ¿ and G ¡ (recall the identity 1 + N (−k0 ) = − N (k0 ), so that G ¿ (−k0 ) = G ¡ (k0 )). By inverting Eq. (2.44), and using Eq. (2.48), one obtains, for − 6 Im x0 6 0, d 4 k −ik·x G ¿ (x) = e /(k)[1 + N (k0 )] ; (2.49) (2)4 which generalizes Eq. (2.31). This expression, when continued to imaginary time t → −i,; 0 6 , 6 , gives the function G ¿ (,; k) and, by inversion of Eq. (2.26), the Matsubara propagator G(i!n ; k) = d, ei!n , G ¿ (,; k) 0

=

∞

−∞

d k0 /(k0 ; k) : 2 k0 − i!n

(2.50)

In going from the :rst to the second line, we used ei!n = 1 and [1 + N (k0 )](e−k0 − 1) = − 1. According to Eq. (2.50), the Fourier coeKcient G(i!n ; k) can be regarded as the value of the function ∞ d k0 /(k0 ; k) G(!; k) = ; (2.51) −∞ 2 k0 − ! for ! = i!n . This function is often referred to as the analytic propagator. It is the unique continuation of the Matsubara propagator G(i!n ; k) which is analytic o8 the real axis and does not grow as fast as an exponential as |!| → ∞ [94]. Note that Eq. (2.51) relates the spectral density to the discontinuity of G(!) across the real axis i/(k0 ; k) = G(k0 + i"; k) − G(k0 − i"; k) ;

(2.52)

with " → 0+ . The causal Green’s functions are also simply related to the analytic propagator. For instance, the Fourier transform of the retarded 2-point function (2.36): GR (k) = d 4 x eik·x−"x0 GR (x) ; (2.53)

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may be obtained as the limit of the analytic propagator G(!; k), Eq. (2.51), as ! approaches the real energy axis from above: ∞ d k0 /(k0 ; k) GR (k0 ; k) = ; (2.54) −∞ 2 k0 − k0 − i" that is, GR (k0 ; k) = G(! = k0 + i"; k) :

(2.55)

Similarly, for the advanced 2-point function (see Eq. (2.36)) we have GA (k0 ; k) = G(! = k0 − i"; k) = GR∗ (k0 ; k) :

(2.56)

By using the spectral representation (2.54), we can extend the de:nition of the retarded propagator to any complex energy ! such that Im ! ¿ 0: it then follows that GR (!) is an analytic function in the upper half-plane, where it coincides with the analytic propagator (2.51). In the lower half-plane, on the other hand, GR (!) is de:ned by continuation across the real axis, and it may have there singularities. Similarly, the advanced propagator GA (!) can be de:ned as an analytic function in the lower half plane. The analyticity properties that we have discussed have an important consequence, known as the Buctuation–dissipation theorem, which relates the dissipation properties of a system to its various correlations. To exhibit such a relation, let us :rst observe that by combining Eqs. (2.54), (2.55) and (2.52), we can write /(k) = 2 Im GR (k) :

(2.57)

Thus the spectral function /(k) may be obtained from the imaginary part of the retarded propagator which describes the dissipation of the single particle excitations (see the end of this subsection). But once the spectral density is known, the various correlations can be calculated according to Eqs. (2.48). The previous discussion can be readily extended to the self-energy ?, de:ned by the Dyson– Schwinger equation: G −1 = G0−1 + ? :

(2.58)

Up to a possible singular part at , = 0 (see Eqs. (2.115) – (2.116) below for an example), we can write ?(,) = &(,)?¿ (,) + &(−,)?¡ (,) ;

(2.59)

where the self-energies ?¿ and ?¡ share the analytic properties of the 2-point functions G ¿ and G ¡ , respectively. After continuation to complex values of time, they satisfy the KMS condition ?¡ (t) = ?¿ (t − i) (for 0 6 Im t 6 ), and can be given the following representations in momentum space: ?¿ (k) = − 7(k)[1 + N (k0 )];

?¡ (k) = − 7(k)N (k0 ) ;

(2.60)

where: −7(k) ≡ ?¿ (k) − ?¡ (k) :

(2.61)

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One can also de:ne an analytic self-energy (analytic continuation of ?(i!n )) with the spectral representation (up to the possible subtraction of a singular part): ∞ d k0 −7(k0 ; k) ?(!; k) = (2.62) k0 − ! −∞ 2 with 7(k0 ; k) de:ned in Eq. (2.61). The Dyson–Schwinger equation can be used to relate the retarded propagator to the retarded self-energy: GR (k0 ; k) =

(k0 +

i")2

−1 ; − j2k − ?R (k0 ; k)

(2.63)

where ?R (k0 ; k) ≡ ?(k0 +i"; k). Note that, with the present conventions, 7(k0 ; k) =−2Im ?R (k0 ; k). By using Eq. (2.52), one obtains the spectral density as /(k0 ; k) =

(k02

−

j2k

7(k0 ; k) : − Re ?R (k0 ; k))2 + (7(k0 ; k)=2)2

(2.64)

The sign properties of /(k0 ; k), discussed after Eq. (2.46), require Re ?R (k0 ) to be even and 7(k0 ) to be odd functions of k0 , with k0 7(k0 ) ¿ 0. In particular, ?¿ (k) and ?¡ (k) are negative de:nite in our present conventions (see Eqs. (2.60)). For a free particle, ? = 0, and the spectral function is a sum of delta functions (see Eq. (2.32)). When 7 is small and not too strongly dependent on k0 , the spectral density (2.64) does not di8er too much from the free particle one. In such cases, the associated single-particle excitations are often referred to as quasiparticles. To be more speci:c, let Ek be the positive-energy solution (whenever it exists) of the equation k02 = j2k + Re ?R (k0 ; k). If 7 is a slowly varying function of k0 in the vicinity of Ek , then, for k0 close to Ek , the spectral density (2.64) has a Lorentzian shape: /(k0 Ek ; k)

zk 28k ; 2Ek (k0 − Ek )2 + 82k

(2.65)

while the retarded propagator (2.63) develops a simple pole at k0 = Ek − i8k : GR (k0 Ek ; k)

zk −1 : 2Ek k0 − Ek + i8k

(2.66)

In writing these equations, we have denoted: 1 9Re ?R −1 zk ≡ 1 − ; 2Ek 9k0 k0 =Ek 8k ≡

zk 7(k0 = Ek ; k) ; 4Ek

(2.67)

and we have assumed that 8k Ek . Eqs. (2.65) and (2.66) describe quasiparticles with energy Ek and width 8k . For negative energy, there is another pole in GR , at k0 = − Ek − i8k . Note that both poles lie in the lower half-plane, in agreement with the analytic structure of the retarded propagator discussed before.

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The quantity 8k controls the lifetime of the corresponding quasiparticle excitation, as measured by the behaviour of the retarded propagator at large times. The retarded propagator is given by (cf. Eq. (2.54)): ∞ ∞ d k0 −ik0 t d k0 −ik0 t GR (t; k) = GR (k0 ; k) = i&(t) /(k0 ; k) : (2.68) e e −∞ 2 −∞ 2 Whenever Eqs. (2.65) and (2.66) are valid, GR (t; k) ∼ e−iEk t e−8k t at large times (for a positive energy state), so that |GR (t; k)|2 ˙ exp{−28k t }. We shall refer to the quantity ,(k) = 1=28k as the lifetime of the excitation, and to 8k as the quasiparticle damping rate. Note that, even if it is generic, the exponential decay is by no means universal. A more complicated behaviour can occur whenever some of the aforementioned assumptions are not satis:ed. In Section 6, we shall encounter an example of such a nontrivial evolution in time [68,97–99]. 2.1.4. Classical 4eld approximation and dimensional reduction In the high temperature limit, → 0, the imaginary-time dependence of the :elds frequently becomes unimportant and can be ignored in a :rst approximation. The integration over imaginary time becomes then trivial and the partition function (2.18) reduces to 3 Z ≈ N D(;) exp − d xH(x) ; (2.69) where ; ≡ ;(x) is now a three-dimensional :eld, and 1 m2 2 H = (∇;)2 + (2.70) ; + V (;) : 2 2 The functional integral in Eq. (2.69) is recognized as the partition function for static three dimensional :eld con:gurations with energy d 3 xH(x). We shall refer to this limit as the classical 4eld approximation. Ignoring the time dependence of the :elds is equivalent to retaining only the zero Matsubara frequency in their Fourier decomposition. Then the Fourier transform of the free propagator is simply T G0 (k) = 2 : (2.71) jk This may be obtained directly from Eq. (2.26) keeping only the term with !n = 0, or from Eq. (2.31) by ignoring the time dependence and using the approximation T 1 N (jk ) = jk ≈ : (2.72) e − 1 jk Both approximations make sense only for jk T , implying N (jk )1. In this limit, the energy density per mode jk N (jk ) ≈ T is as expected from the classical equipartition theorem. Also, because N (jk ) ≈ 1 + N (jk ) ≈ T= jk , the two propagators G0¿ and G0¡ in Eq. (2.41) become equal, and the analytic properties discussed in Section 2.1.2 are lost. That G ¿ ≈ G ¡ in the classical limit is in agreement with the fact that the :eld operator ;(x) becomes a commuting c-number in this limit. We shall discuss later, in Section 2.2.4, how to construct real time propagators in the classical :eld approximation.

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385

The classical :eld approximation may be viewed as the leading term in a systematic expansion. To see that, let us expand the :eld variables in the path integral (2.18) in terms of their Fourier components: 1 −i!n , ;(,) = e ;(i!n ) ; (2.73) n where the !n ’s are the Matsubara frequencies. This takes care automatically of the periodic boundary conditions. The path integral (2.18) can then be written as Z = N1 D(;0 ) exp{−S[;0 ]} ; (2.74) where ;0 ≡ ;(!n = 0) and exp{−S[;0 ]} = N2 D(;n =0 ) exp −

0

d,

d 3 xLE (x)

:

(2.75)

The quantity S[;0 ] may be called the e8ective action for the “zero mode” ;0 . Aside from the direct classical :eld contribution that we have already considered, this e8ective action receives also contributions from integrating out the nonvanishing Matsubara frequencies. Diagrammatically, S[;0 ] is the sum of all the connected diagrams with external lines associated to ;0 , and in which the internal lines are the propagators of the nonstatic modes ;n =0 . Thus, a priori, S[;0 ] contains operators of arbitrarily high order in ;0 , which are also nonlocal. In practice, however, one wishes to expand S[;0 ] in terms of local operators, i.e., operators with the schematic structure am; n ∇m ;n0 with coeKcients am; n to be computed in perturbation theory. To implement this strategy, it is useful to introduce an intermediate scale (T ) which separates hard (k & ) and soft (k . ) momenta. All the nonstatic modes, as well as the static ones with k & are hard (since K 2 ≡ !n2 + k 2 & 2 for these modes), while the static (!n = 0) modes with k . are soft. Thus, strictly speaking, in the construction of the e8ective theory along the lines indicated above, one has to integrate out also the static modes with k & . The bene:ts of this separation of scales are that (a) the resulting e8ective action for the soft :elds can be made local (since the initially nonlocal amplitudes can be expanded out in powers of p=K, where p is a typical external momentum, and K & is a hard momentum on an internal line), and (b) the e8ective theory is now used exclusively at soft momenta, where classical approximations such as (2.72) are expected to be valid. This strategy, which consists in integrating out the non-static modes in perturbation theory in order to obtain an e8ective three-dimensional theory for the soft static modes (with !n = 0 and k ≡ |k| . ), is generally referred to as “dimensional reduction” [100 –105]. It is especially useful in view of nonperturbative lattice calculations, which are easier to perform in lower dimensions [105,106] (see also Section 5.4.3 below). As an illustration let us consider a massless scalar theory with quartic interactions; that is, m = 0 and V (;) = (g2 =4!);4 in Eq. (2.1). The ensuing e8ective action for the soft :elds (which we shall still denote as ;0 ) reads g32 () 4 h() 6 1 2 1 3 2 2 S[;0 ] = F() + d x (∇;0 ) + M ();0 + ;0 + ; + ZL ; 2 2 4! 6! 0 (2.76)

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Fig. 2. One-loop tadpole diagram for the self-energy of the scalar :eld.

where F() is the contribution of the hard modes to the free-energy, and ZL contains all the other local operators which are invariant under rotations and under the symmetry ; → −;, i.e., all the local operators which are consistent with the √symmetries of the original Lagrangian. We have changed the normalization of the :eld (;0 → T ;0 ) with respect to Eqs. (2.69) – (2.70), so as to absorb the factor in front of the e8ective action. The e8ective “coupling constants” in Eq. (2.76), i.e. M 2 (); g32 (); h() and the in:nitely many parameters in ZL, are computed in perturbation theory, and depend upon the separation scale , the temperature T and the original coupling g2 . To lowest order in g; g32 ≈ g2 T; h ≈ 0 (the :rst contribution to h arises at order g6 , via one-loop diagrams), and M ∼ gT , as we shall see shortly. Note that Eq. (2.76) involves in general nonrenormalizable operators, via ZL. This is not a diKculty, however, since this is only an e8ective theory, in which the scale acts as an explicit ultraviolet (UV) cuto8 for the loop integrals. Since the scale is arbitrary, the dependence on coming from such soft loops must cancel against the dependence on of the parameters in the e8ective action. Let us verify this cancellation explicitly in the case of the thermal mass M of the scalar :eld, and to lowest order in perturbation theory. To this order, the scalar self-energy is given by the tadpole diagram in Fig. 2. The mass parameter M 2 () in the e8ective action is obtained by integrating over hard momenta within the loop in Fig. 2 (cf. Eq. (B.19)): g2 M () = T 2 n 2

g2 = 2

d 3 k (1 − *n0 ) + &(k − )*n0 (2)3 !n2 + k 2

d3 k (2)3

T N (k) 1 − &( − k) 2 + k 2k k

;

(2.77)

where the &-function in the second line has been generated by writing &(k − ) = 1 − &( − k). The :rst term, involving the thermal distribution, gives the contribution g2 2 Mˆ ≡ 2

d 3 k N (k) g2 2 = T : (2)3 k 24

(2.78)

As it will turn out, this is the leading-order (LO) scalar thermal mass, and also the simplest example of what will be called “hard thermal loops” (HTL). The second term, involving 1=2k, in Eq. (2.77) is quadratically UV divergent, but independent of the temperature; the standardrenormalization procedure at T = 0 amounts to simply removing this term (see Section 2.3.3).

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The third term, involving the &-function, is easily evaluated. One :nally gets: g2 g2 T 2 6 2 2 ˆ 1− 2 : M () = M − 2 T ≡ 4 24 T

387

(2.79)

The -dependent term above is subleading, by a factor =T 1. The one-loop correction to the thermal mass within the e8ective theory is given by the same diagram in Fig. 2, but where the internal :eld is static and soft, with the massive propagator 1=(k 2 + M 2 ()), and coupling constant g32 ≈ g2 T . Since the typical momenta in the integral will be k & M , and M ∼ Mˆ ∼ gT , we choose gT . We then obtain g2 T d3 k 2 D( − k) 2 *M () = 3 2 (2) k + M 2 () M g2 T g2 ˆ g2 T M

1 − − = arctan MT ; (2.80) 2 4 2 42 8 where the terms neglected in the last step are of higher order in Mˆ = or =T . As anticipated, the -dependent terms cancel in the sum M 2 ≡ M 2 () + *M 2 (), which then provides the physical thermal mass within the present accuracy: M 2 = M 2 () + *M 2 () =

g2 T 2 g2 ˆ − MT : 24 8

(2.81)

The LO term, of order g2 T 2 , is the HTL Mˆ . The next-to-leading order (NLO) term, which involves the resummation of the thermal mass M () in the soft propagator, is of order g2 Mˆ T ∼ g3 T 2 , and therefore nonanalytic in g2 . This nonanalyticity is related to the fact that the integrand in Eq. (2.80) cannot be expanded in powers of M 2 =k 2 without running into infrared divergences. In Section 2.2.4, we shall see how e8ective theories based on a classical :eld approximation can be used to compute time-dependent correlations. Then, in Section 4.4 we shall extend this strategy to gauge theories. In that case however, the problem of matching the coeKcients of the e8ective theory with those of the original one can be a delicate one. 2.2. Nonequilibrium evolution of the quantum 4elds We consider now situations where the system, initially in equilibrium, is perturbed by an external source which starts acting at some time t0 . We take the external source to be a current j(x) linearly coupled to the scalar :eld. The evolution of the system is then described by the Hamiltonian Hj (t) of Eq. (2.5). The density operator at time t is given by (cf. Eq. (2.6)): Dj (t) = Uj (t; t0 )DUj−1 (t; t0 ) ;

(2.82)

where D is the density operator at time t0 and Uj (t; t0 ), the evolution operator, satis:es i9t Uj (t; t0 ) = Hj (t)Uj (t; t0 );

Uj (t0 ; t0 ) = 1 :

(2.83)

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An operator Uj (t2 ; t1 ) can be de:ned similarly for arbitrary t1 . Such operators, which may be viewed as “time translation operators”, satisfy the group property: Uj (t2 ; t1 ) = Uj (t2 ; t3 )Uj (t3 ; t1 ) :

(2.84)

In particular since for any t1 ; Uj (t1 ; t1 ) = 1, we have Uj−1 (t2 ; t1 ) = Uj (t1 ; t2 ). Eq. (2.83) can be formally integrated to yield the following expression for Uj (t2 ; t1 ): t2 ˜ Uj (t2 ; t1 ) = T exp −i Hj (t) dt ; (2.85) t1

where the symbol T˜ orders the operators from right to left, in increasing or decreasing order of their time arguments depending, respectively, on whether t2 ¿ t1 or t2 ¡ t1 (i.e., we use the same symbol for what are usually distinguished as chronological or antichronological ordering operators; the reason for this will become more evident when we discuss contour propagators). In other words, T˜ [Hj (t) Hj (t )] = Hj (t )Hj (t) if, in going from t1 to t2 along the time axis, one :rst meets t and then t ; in the opposite case, T˜ [Hj (t) Hj (t )] = Hj (t) Hj (t ). 2.2.1. Retarded response functions The expectation value of any operator O at time t can be calculated from the density operator solution of the equation of motion (2.83). We assume here that O does not depend explicitly on time. Then, Tr Dj (t)O = Tr D Oj (t) = Oj (t) ;

(2.86)

Oj (t) ≡ Uj−1 (t; t0 )OUj (t; t0 ) = Uj (t0 ; t)OUj (t; t0 )

(2.87)

where and Uj (t; t0 ) is the evolution operator de:ned in the previous subsection. If j = 0; Tr DO = O is time-independent and corresponds to the equilibrium expectation value. The di8erence *Oj (t) ≡ Oj (t) − O is a measure of the response of the system to the external perturbation. If the departure from equilibrium is small, we may attempt to calculate Oj (t) as an expansion in powers of j. To do so, the following identities are useful (t; t ¿ t0 ): *Uj (t; t0 ) *Uj (t0 ; t) i i (2.88) = &(t − t )Uj (t; t0 );j (t ); = − &(t − t ) ;j (t )Uj (t0 ; t) : *j(t ) *j(t ) (The second identity follows from the :rst one by noticing that Uj (t0 ; t) = Uj−1 (t; t0 ), and that *Uj−1 = − Uj−1 *Uj Uj−1 :) From these identities, we get easily * Oj (t) = &(t − t )[Oj (t); ;j (t )] (2.89) i *j(t ) and, more generally ∞ 0 (−i)n d 4 y1 d 4 y2 : : : d 4 yn &(t − y10 )&(y10 − y20 ) : : : &(yn−1 − yn0 ) *Oj (t) = n=1

×[ : : : [[O(t); ;(y1 )]; ;(y2 )] : : : ;(yn )] j(y1 )j(y2 ) : : : j(yn ) :

(2.90)

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In this equation, the symbol [ : : : [[; ]] : : : ] denotes nested commutators, and ;(t) and O(t) are operators in the Heisenberg representation without the source, that is O(t) = eiH (t−t0 ) Oe−iH (t−t0 )

(2.91)

and similarly for ;(t). The expectation values in the r.h.s. of Eq. (2.90) are equilibrium expectation values, computed in the absence of external sources. In particular, the average :eld induced in the system by the external current can be expanded as ∞ 1 E(x) = − d 4 y1 d 4 y2 : : : d 4 yn GR(1+n) (x; y1 ; y2 ; : : : ; yn )j(y1 )j(y2 ) : : : j(yn ) ; (2.92) n! n=1

where GR(1+n) (x; y1 ; y2 ; : : : ; yn ) ≡ (−i)n+2

P

0 &(t − y10 )&(y10 − y20 ) · · · &(yn−1 − yn0 )

×[ : : : [[;(x); ;(y1 )]; ;(y2 )] : : : ;(yn )]

(2.93)

is a retarded Green’s function with n + 1 external legs. The sum in Eq. (2.93) runs over all the n! permutations of the labels y1 ; y2 ; : : : ; yn , so that the function GR(1+n) is symmetric with respect to its y-arguments. On the other hand, this is a causal function with respect to x, since it vanishes for x0 ¡ yi0 (i = 1; 2; : : : ; n), that is, prior to the action of the perturbation. As already noted the statistical averages in the formulae above are taken over the initial equilibrium thermodynamical ensemble, with the canonical density operator D = Dj (t0 ) = e−H =Z. Thus, in principle, it is possible to study the response of the system to external perturbations by computing only equilibrium Green’s functions. This is especially convenient for weak perturbations, when the expansion in Eqs. (2.90) and (2.92) can be limited to its :rst term: this is the linear response approximation. In this case, Eq. (2.92) reduces to E(x) = − d 4 yGR (x − y)j(y) ; (2.94) where GR (x − y) = i&(x0 − y0 )[;(x); ;(y)] is the retarded propagator (2.36), studied in the previous section. If one could limit oneself to the study of linear response, the imaginary time formalism presented in the previous section could therefore be suKcient. However, as we shall see later, in nonAbelian gauge theories, Green’s functions with di8erent numbers of external legs are related by Ward identities. In other words, non Abelian gauge symmetry forces us to go beyond linear response, even when studying the response to weak external perturbations. This means that we shall need to consider n-point functions such as (2.93), whose calculation is generally diKcult. At this stage, some extra formalism is needed, and this will be developed in the next section. 2.2.2. Contour Green’s functions The main technical feature of the formalism to be described now, and which allows one to exploit the full power of :eld theoretical techniques in the calculation of nonequilibrium n-point functions, is the use of a complex time path surrounding the real-time axis. This has

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been originally introduced by Schwinger [107] and Keldysh [108] (see also Ref. [109]; for a recent presentation of this formalism see [110,87,14]). We shall also refer to the formalism of Kadano8 and Baym [88], which exploits the analytic properties of the Green’s functions in order to derive real time equations of motion for the Green’s functions from the corresponding equations in imaginary-time. Consider then the time-ordered 2-point function in the presence of j: G(t1 ; t2 ) = T;j (t1 );j (t2 ) ≡ Tr(DT;j (t1 );j (t2 )) ≡ &(t1 − t2 )G ¿ (t1 ; t2 ) + &(t2 − t1 )G ¡ (t1 ; t2 ) ;

(2.95)

where ;j (t) is the :eld operator in the Heisenberg representation in the presence of the sources, given by Eq. (2.87). By making explicit the various evolution operators in Eq. (2.95), we can write, e.g., 1 (2.96) Tr {DUj (t0 ; t1 );Uj (t1 ; t2 );Uj (t2 ; t0 )} ; Z where we have used the group property (2.84). Imagine now writing all the evolution operators in terms of ordered exponentials, as in Eq. (2.85), thus generating a chain of time dependent Hamiltonians. These operators follow, along the chain, di8erent ordering prescriptions, depending from which Uj they originate. Assume, for instance, that t0 ¡ t2 ¡ t1 , in which case the 2-point function G ¿ (t1 ; t2 ) coincides with the time-ordered propagator in Eq. (2.95); then, the evolution operators are chronologically ordered from t0 to t2 and from t2 to t1 , and anti-chronologically ordered from t1 to t0 . This is a source of complications which, however, can be bypassed by allowing all time variables to run on an appropriate contour in the complex time plane. We then extend the de:nition of the evolution operator to complex time variables, i.e., we de:ne Uj (z2 ; z1 ) as the solution of Eq. (2.83) with t replaced by a complex variable z. The evolution operator becomes then a translation operator in the complex time plane and the equation can be formally integrated along any given contour C. Such a contour can be speci:ed by a function z(u), where the real parameter u is continuously increasing along C. The contour evolution operator can then be written as (compare with Eq. (2.85)): Uj (C) = TC exp −i Hj (z) d z ; (2.97) G ¿ (t1 ; t2 ) =

C

where the operator TC orders the operators Hj (zi ) from right to left in increasing order of the parameters ui (zi = z(ui )). Note that Eq. (2.97) involves (2.98) Hj (z) ≡ H + d 3 xj(z; x);(x) ; this requires the extension of the external source j to complex time arguments, which we leave arbitrary at this stage. We de:ne a contour &-function &C : &C (z1 ; z2 ) = 1 if z1 is further than z2 along the contour (we write then z1 z2 ), while &C (z1 ; z2 ) = 0 if the opposite situation holds (z1 ≺ z2 ). In terms of the coordinate u along the contour, &C (z1 ; z2 ) = &(u1 − u2 ), with z1 = z(u1 ) and z2 = z(u2 ).

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Fig. 3. Complex-time contour for the evaluation of the thermal expectation values: C = C+ ∪ C− ∪ C0 .

We shall need also a contour delta function, which we de:ne by −1 9z *C (z1 ; z2 ) ≡ *(u1 − u2 ) : 9u

(2.99)

Consider now the speci:c contour depicted in Fig. 3. This may be seen as the juxtaposition of three pieces: C = C+ ∪ C− ∪ C0 . On C+ , z = t takes all the real values between t0 to tf , with tf larger than all the times of interest. On C− , we set z = t − i" (" → 0+ ) and t runs backward from tf to t0 . Finally, on C0 , z = t0 − i,, with 0 ¡ , 6 . This particular contour allows us to replace the various orderings of operators that we have met by a single ordering along the contour. Thus, the ordering along the contour coincides with the chronological ordering on C+ , with antichronological ordering on C− , and with ordering according to the imaginary time on C0 . We then generalize the Heisenberg representation (2.87) to :elds de:ned on the contour: ;j (z) = Uj−1 (z; t0 );Uj (z; t0 )

(2.100)

and de:ne the ordered product of two such operators by TC (;j (z1 );j (z2 )) = &C (z1 ; z2 );j (z1 );j (z2 ) + &C (z2 ; z1 );j (z2 );j (z1 ) :

(2.101)

This allows us to extend de:nition (2.95) of the o8-equilibrium propagator as follows: G(z1 ; z2 ) = Tr {DTC ;j (z1 );j (z2 )} ≡ &C (z1 ; z2 )G ¿ (z1 ; z2 ) + &C (z2 ; z1 )G ¡ (z1 ; z2 ) :

(2.102)

The physical nonequilibrium Green’s functions in real-time (cf. Eq. (2.95)) are obtained from the corresponding contour functions by choosing appropriately the time arguments on C+ and C− and identifying the external source j(z; x) with the physical perturbation in Eq. (2.5), i.e., j(z = t) = j(z = t − i") ≡ j(t). Thus, one easily veri:es that for the choice z1 = t1 − i" ∈ C− and z2 = t2 ∈ C+ , the contour two-point function G ¿ (z1 ; z2 ) obtained from Eq. (2.102) reduces to the physical Green’s function G ¿ (t1 ; t2 ) in Eq. (2.96). Similarly, by choosing z1 = t1 ∈ C+ and z2 = t2 − i" ∈ C− in Eq. (2.102), one obtains G ¡ (t1 ; t2 ), while for both z1 and z2 on C+ one gets the time-ordered, or Feynman, propagator G(t1 ; t2 ) of Eq. (2.95).

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So far we have not used the part C0 of the contour. It becomes useful whenever the initial density operator is the canonical density operator (2.2). Indeed, as already noticed in Section 2.1, this can be represented as an evolution operator along the contour C0 . Such a representation allows us to treat the statistical average over the initial state on the same footing as the time evolution, and in particular to perform approximations on the initial state which are consistent with those made on the evolution equation. With this in mind, we then de:ne the following generating functional: Z[j] ≡ Tr Uj (t0 − i; t0 ) ;

(2.103)

where j(z) is an arbitrary function along the contour to start with. This generating functional may be written as the following path integral: −H 4 Z[j] = Tr e TC exp −i d xj(x);(x)

=N

;(t0 )=;(t0 −i)

C

D; exp i

C

4

d x(L(x) − j(x);(x))

;

(2.104)

where d 4 x = d z d 3 x, and the integral C d z runs from t0 to t0 − i along the contour; the periodicity conditions at t0 and t0 − i are the same as in Eq. (2.18). General (connected) n-point contour Green’s functions are obtained as in *n ln Z[j] G (n) (z1 ; z2 ; : : : ; zn ) = : (2.105) *j(z1 )*j(z2 ) : : : *j(zn )

All the manipulations which lead to the perturbative expansion can now be simply extended to complex time arguments lying on the contour. To do perturbative calculations, we need the free contour propagator: G0 (z1 ; z2 ) ≡ &C (z1 ; z2 )G0¿ (z1 ; z2 ) + &C (z2 ; z1 )G0¡ (z1 ; z2 ) d 4 k −i(k0 z−k·x) = e /0 (k)[&C (z) + N (k0 )] ; (2)4

(2.106)

where z = z1 − z2 , x = x1 − x2 , /0 (k) is the free spectral density, Eq. (2.32), and the second line follows from Eq. (2.49) for G ¿ together with a similar equation for G ¡ (cf. Eq. (2.48)). A similar representation, but with /0 (k) replaced by /(k) (the full spectral density), holds also for the exact contour propagator in thermal equilibrium. Note :nally that we can exploit the analytic properties of the thermal Green’s functions to deform the contour C in the complex time plane. This is clear at least in thermal equilibrium, where the analyticity properties of the functions G ¿ (z) and G ¡ (z) discussed in Section 2.1.2 imply that the contour 2-point function (i.e., Eq. (2.102) with j = 0) is well de:ned for any contour C such that Im z is nonincreasing along the contour. The choice of a speci:c contour is a matter of convenience, and di8erent contours may lead to slightly di8erent formalisms (see, e.g., [87,111,14]). In fact, any such contour may be viewed as a particular deformation of the Matsubara contour used in Section 2.1. For any contour C, the Green’s functions satisfy boundary conditions which generalize the KMS conditions in equilibrium (cf. Eq. (2.39)).

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For instance: G(t0 ; z) = G(t0 − i; z) G (3) (t0 ; z ; z ) = G (3) (t0 − i; z ; z );

etc :

(2.107)

Under suitable conditions these analytic properties may hold also for the nonequilibrium (j = 0) Green’s functions [88], although no rigorous proof can be given in general. 2.2.3. Equations of motion for Green’s functions After all these preparations, we are now ready to write down the equations of motion satis:ed by the contour Green’s functions. The mean :eld equation is most easily obtained by taking the expectation value with the statistical operator of the equations of motion obeyed by the :eld operators in the Heisenberg representation: dV 2 2 (−9 − m )E(x) − (2.108) (x) = j(x) ; d; where E(x) ≡ ;(x), 92 = 92z − ∇2 , and the angular brackets denote expectation values. Eq. (2.108) is conveniently rewritten as (−92 − m2 )E(x) = j(x) + j ind (x) ; where ind

j (x) ≡

dV (x) d;

(2.109)

(2.110)

will be called the induced current because it plays, in the r.h.s. of Eq. (2.108), the same role as the external current, namely the role of a source for the average :eld E. For a ;4 -theory, where V (;) = (g2 =4!);4 , we have explicitly j ind (x) =

g2 3 g2 g2 ; (x) = (E3 (x) + G (3) (x; x; x)) + E(x)G(x; x) : 3! 3! 2!

(2.111)

By di8erentiating Eq. (2.109) with respect to j(y), and using i(*E(x)=*j(y)) = G(x; y), one obtains an equation for G(x; y): 2 2 (−9x − m )G(x; y) − i d 4 z?(x; z)G(z; y) = i*C (x; y) ; (2.112) C

where *C (x0 ; y0 ) is the contour delta function, Eq. (2.99), and the self-energy ? is given by: * *j ind (x) dV =−i (x) : (2.113) ?(x; y) ≡ −i *E(y) *E(y) d; In writing Eq. (2.112), the following chain of identities has been used: ind *j ind (x) 4 *j (x) *E(z) i =i d z = i d 4 z?(x; z)G(z; y) : *j(y) *E(z) *j(y) C C

(2.114)

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For a free theory (? = 0), the solution to Eq. (2.112) with the KMS boundary conditions (i.e., the free contour propagator) is given by Eq. (2.106). The self-energy (2.113) admits the following decomposition, similar to that of G, Eq. (2.35): ?(x; y) = − i?* (x)*C (x; y) + &C (x0 ; y0 )?¿ (x; y) + &C (y0 ; x0 )?¡ (x; y) :

(2.115)

We have isolated here a possible singular piece ?* . For instance, for the ;4 theory, g2 2 (E (x) + G(x; x)) : 2 The nonsingular components obey the KMS condition: ?* (x) =

(2.116)

?¡ (t0 ; z) = ?¿ (t0 − i; z) ;

(2.117)

a consequence of de:nition (2.113), and conditions (2.107) which are satis:ed by the n-point Green’s functions. The equations of motion in real-time for the mean :eld and the 2-point functions are obtained by choosing the contour in Fig. 3 and letting the external time variables x0 and y0 take values on the real-time pieces of this contour. For x0 ∈ C+ , Eq. (2.109) goes formally unmodi:ed: −(92 + m2 )E(x) = j(x) + j ind (x) :

(2.118)

Consider now Eq. (2.112): by choosing x0 ∈ C+ and y0 ∈ C− , and by using the decompositions (2.35) and (2.115), we obtain an equation for G ¡ (x; y): x0 2 2 * ¡ (9x + m + ? (x))G (x; y) = −i d 4 z[?¿ (x; z) − ?¡ (x; z)]G ¡ (z; y) t0

+i

t0

−i

y0

d 4 z?¡ (x; z)[G ¿ (z; y) − G ¡ (z; y)]

t0 −i

t0

d 4 z?¡ (x; z)G ¿ (z; y) ;

(2.119)

where the :rst two integrals in the r.h.s. run along the real axis and we have isolated in the third integral the contribution from the imaginary-time piece of the contour. A similar equation for G ¿ follows similarly if, starting from Eq. (2.112), one chooses x0 ∈ C− and y0 ∈ C+ . It is instructive to consider the restriction of the equation above to the case of equilibrium, and in particular to verify that it is then independent of the initial time t0 , as it should. In equilibrium, the various propagators and self-energies are expected to be functions only of time ¿ (x − z) and, similarly di8erences and to admit spectral representations like Eq. (2.49) for Geq (cf. Eq. (2.60)), d 4 k −ik·(x−z) ¿ ?eq (x − z) = − e 7(k)[1 + N (k0 )] : (2.120) (2)4 It is easy to verify that such representations are indeed consistent with Eq. (2.119). In fact, by inserting these representations in Eq. (2.119), and using properties like e−k0 [1+N (k0 )] = N (k0 ), one :nds that, in thermal equilibrium, this equation is independent of the initial time t0 , and it

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395

can be rewritten in momentum space as (−k 2 + m2 + ?* )G ¡ (k) = − ?R (k)G ¡ (k) − ?¡ (k)GA (k) ;

(2.121)

where we have used de:nitions (2.36) for the retarded and advanced Green’s functions, together with similar de:nitions for ?R and ?A . This equation is solved indeed by G ¡ (k) = /(k)N (k0 ) with /(k) given by Eq. (2.64). Eq. (2.119) and the corresponding one for G ¿ could have been obtained also by analytic continuation of the imaginary-time Dyson–Schwinger equations [88]. Speci:cally, one may start with Eq. (2.112) written along the Matsubara contour, that is (with x0 = t0 − i,x and y0 = t0 − i,y ): d,z d 3 z?(x; z)G(z; y) (−92,x − ∇2x + m2 + ?* (x))G(x; y) + 0

(3)

= *(,x − ,y )* (x − y) ;

(2.122)

and then deform the contour in the complex time plane, by exploiting the analytic properties of the nonequilibrium Green’s functions (see previous subsection). This simple technique will be used in connection with gauge theories, in Sections 3 and 7 below [18,23,26]. In what follows it will often be convenient to let t0 → −∞ and to assume that the external sources are switched o8 adiabatically in the remote past. Then, for :xed values of the real-time arguments x0 and y0 , and for any z0 on the vertical piece of the contour, the real parts of the time di8erences x0 − z0 and y0 − z0 go to in:nity. In this limit, the 2-point correlations G ¿ (z; y) and G ¿ (z; x) are expected to die away suKciently fast, for the contributions of the imaginary-time integrals in Eq. (2.119) to become negligible [88]. In fact, for the kind of nonequilibrium situations to be considered below, and which involve only longwavelength perturbations, the correlation function G ¿ (z; y) is dominated by hard degrees of freedom (k ∼ T ), and decays over a characteristic range |x0 − y0 | ∼ 1=T (cf. Eq. (B.28)). Thus, neglecting the imaginary-time integrals in Eq. (2.119) is justi:ed as soon as x0 1=T or y0 1=T . We are thus led to the following set of equations: (92x + m2 + ?* (x))G ¡ (x; y) = − (92x + m2 + ?* (x))G ¿ (x; y) = −

∞

−∞

∞

−∞

d 4 z[?R (x; z)G ¡ (z; y) + ?¡ (x; z)GA (z; y)] ; d 4 z[?R (x; z)G ¿ (z; y) + ?¿ (x; z)GA (z; y)] ;

(2.123)

where we have extended the de:nitions of the retarded and advanced Green’s functions and self-energies to nonequilibrium situations. The presence of these functions has allowed us to extend the upper bound of the z0 integral to +∞. (In equilibrium, the Fourier transform of the :rst equation (2.123) coincides with Eq. (2.121), as it should.) By taking the di8erence of the two equations above, one obtains an equation satis:ed by the retarded propagator GR (x; y) (cf. Eq. (2.36)): ∞ (92x + m2 + ?* (x))GR (x; y) + d 4 z?R (x; z)GR (z; y) = *(4) (x − y) : (2.124) −∞

Note that, while the correlation functions G ¿ and G ¡ and the corresponding self-energies are coupled by Eqs. (2.123), the retarded Green’s function GR is determined by the retarded self-energy ?R alone. A similar observation applies to the advanced functions GA and ?A .

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Eqs. (2.123) and (2.124) are the equations obtained by Kadano8 and Baym [88], in the framework of nonrelativistic many-body theory. In these equations, any explicit reference to the initial conditions has disappeared. Thus the KMS conditions only enter as boundary conditions to be satis:ed by the various Green’s functions in the remote past. The same set of equations has been shown by Keldysh [108] to describe the general nonequilibrium evolution of a quantum system, with the density matrix of the initial state determining the appropriate boundary conditions (see also [43,92]). To make progress, the above equations must be supplemented with some approximation allowing us to express the self-energy ? in terms of the propagator G. This generally results in complicated, nonlinear and integro-di8erential, equations for G. Moreover, in o8-equilibrium situations, we generally loose translational invariance, so we cannot analyze these equations with the help of Fourier transforms. However, for slowly varying (or soft) o8-equilibrium perturbations, these equations can be transformed into kinetic equations [88,92,112–114], as will be explained in Section 2.3 below. 2.2.4. Correlation functions in the classical 4eld approximation There are situations where one wishes to evaluate the real time correlation functions in the classical :eld approximation (cf. Section 2.1.4). Although the techniques developed above could in principle be used, it is more eKcient to proceed di8erently. Consider for instance the calculation of the correlation function Gcl (x; y) ≡ Ecl (x)Ecl (y) ;

(2.125)

where the brackets denote the classical thermal averaging (cf. Eq. (2.129) below), and Ecl (t; x) and %cl (t; x) are classical :elds, whose time dependence is obtained by solving the classical equations of motion, 9H 9H 90 % = − 90 E = ; =% (2.126) 9E 9% with % the :eld canonically conjugate to E, and H the classical Hamiltonian M2 2 1 2 1 3 2 (2.127) H= d x % + (∇E) + E + V (E) ≡ d 3 x H(x) : 2 2 2 Note that V may contain a perturbation which drives the system out of equilibrium. Initially however the perturbation vanishes and the system is in thermal equilibrium. To be speci:c we shall denote by Heq the corresponding Hamiltonian. The initial conditions are Ecl (t0 ; x) = E(x);

%cl (t0 ; x) = %(x) ;

(2.128)

and the classical :eld con:gurations E(x) and %(x) are statistically distributed according to the Boltzmann weight e−Heq . The correlator (2.125) is then obtained by averaging over the initial conditions according to −1 D%(x)DE(x)Ecl (t; x)Ecl (t ; y)e−Heq (%; E) (2.129) Gcl (t; x; t ; y) = Zcl where Zcl is the classical partition function: Zcl = D%(x)DE(x)e−Heq (%; E) :

(2.130)

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As an application of Eq. (2.129), let us compute Gcl (x; y) for a free scalar :eld (V = 0). The solution Ecl (x) to the free equations of motion, (920 − ∇2 + M 2 )E(x) = 0 ;

(2.131)

with the initial conditions (2.128) reads

Ecl (t; x) =

d 3 k ik·x e (2)3

sin jk t E(k)cosjk t + %(k) jk

;

(2.132)

√ with jk = k 2 + M 2 , and %(k) the Fourier transform of %(x), etc. In this case, the functional integral in Eq. (2.129) can be exactly computed since Gaussian, and yields:

Gcl0 (x; y)

≡T

d 3 k ik·(x−y) cos jk (x0 − y0 ) e = (2)3 j2k

d 4 k −ik·(x−y) T e /0 (k) ; (2)4 k0

(2.133)

where /0 (k) is the free spectral density (2.32). One recognizes in Eq. (2.133) the classical limit of the correlators in Eq. (2.48). Indeed, at soft momenta, N (k0 ) T=k0 1, and therefore G0¿ (k) G0¡ (k)

T /0 (k) = Gcl0 (k) : k0

(2.134)

Note that in the classical :eld approximation the spectral function is still related to the imaginary part of the retarded propagator as in Eq. (2.57), but is no longer given by the di8erence of the functions G ¿ (k) and G ¡ (k) (see Eq. (2.46)). In the presence of interactions, the averaging over initial conditions using the functional integral (2.129) will develop ultraviolet divergences, so these make sense only if supplemented with an UV cuto8 . This situation is quite similar to that discussed in Section 2.1.4 for the static case, and will not be discussed further here (see Refs. [74,115] for more details).

2.3. Mean 4eld and kinetic equations We are now ready to implement in the scalar case the general approximation scheme that will be used in the rest of this paper for gauge theories. Starting from the general equations for the Green’s functions that we have derived in the previous subsection, we specialize to longwavelength perturbations and use a gradient expansion to reduce the general equations of motion to simpler kinetic equations. The self-energies in these equations are obtained through weak coupling expansion combined with mean :eld approximations. Assuming furthermore weak deviations from the equilibrium, we then arrive at a closed system describing the dynamics of the longwavelength excitations of the system in the high temperature limit. Then, we analyse the role of the collisions in the damping of single particle excitations. In the last subsection

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we reconsider the kinetic equations in the time representation; this will be useful later when dealing with problems where coherence e8ects are important. 2.3.1. Wigner functions In thermal equilibrium, the system is homogeneous, the average :eld vanishes (we do not consider here the possibility of spontaneous symmetry breaking), and the two-point functions depend only on the relative coordinates s = x − y . In the high temperature limit T m, the thermal particles have typical momenta k ∼ T and typical energies jk ∼ T ; the 2-point functions are peaked around s = 0, their range of variation being determined by the thermal wavelength T = 1=k ∼ 1=T (cf. Eq. (B.28)). In what follows, we are interested in o8-equilibrium deviations which are slowly varying in space and time. That is, we assume that the system acquires space–time inhomogeneities over a typical scale T . The :eld ; develops then a nonvanishing average value E(x), and the 2-point functions depend on both coordinates x and y. It is then convenient to introduce relative and central coordinates: x + y s ≡ x − y ; X ≡ (2.135) 2 and use the Wigner transforms of the 2-point functions. These are de:ned as Fourier transforms with respect to the relative coordinates s . For instance, the Wigner transform of G ¡ (x; y) is s

s ¡ (2.136) G (k; X ) ≡ d 4 seik·s G ¡ X + ; X − 2 2 with similar de:nitions for the other 2-point functions like G ¿ , GR , GA and the various self-energies. Note that in order to avoid the proliferation of symbols, we use the same symbols for the 2-point functions and their Wigner transforms, considering that the di8erent functions can be recognized from their arguments. The hermiticity properties of the 2-point functions, as discussed in Section 2.1.2 (cf. Eqs. (2.40) and (2.36)), imply similar properties for the corresponding Wigner functions. For instance, from (G ¿ (x; y))∗ = G ¿ (y; x) we deduce that G ¡ (k; X ) is a real function, (G ¡ (k; X ))∗ = G ¡ (k; X ), as in thermal equilibrium, and similarly for G ¿ (k; X ). Also, (GA (k; X ))∗ = GR (k; X ). Similar properties hold for the various self-energies. Moreover, for a real scalar :eld, we have the additional relations G ¿ (k; X ) = G ¡ (−k; X ) and GA (k; X ) = GR (−k; X ), which follow since G ¿ (x; y) = G ¡ (y; x) and GA (x; y) = GR (y; x) (cf. Eqs. (2.34) and (2.36)). For slowly varying disturbances, taking place over a scale T , we expect the s dependence of the 2-point functions to be close to that in equilibrium. Thus, typically, k ∼ 9s ∼ T , while 9X ∼ 1=T . The general equations of motion written down in Section 2.2.3 can then be simpli:ed with the help of a gradient expansion, using k and X as most convenient variables. 2.3.2. Kinetic equations We shall construct below the equation satis:ed by G ¡ (k; X ) to leading order in the gradient expansion. The starting point is Eq. (2.123) for G ¡ (x; y), namely, 2 2 * ¡ (2.137) (9x + m + ? (x))G (x; y) = − d 4 z[?R (x; z)G ¡ (z; y) + ?¡ (x; z)GA (z; y)] ;

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together with an analogous equation where the di8erential operator is acting on y: 2 2 * ¡ (9y + m + ? (y))G (x; y) = − d 4 z[G ¡ (x; z)?A (z; y) + GR (x; z)?¡ (z; y)] :

399

(2.138)

(To obtain Eq. (2.138), start with the second Eq. (2.123) for G ¿ (x; y), interchange the space– time variables x and y , and use symmetry properties like G ¿ (y; x) = G ¡ (x; y), GA (z; x) = GR (x; z), etc.) When the system is inhomogeneous, the 2-point functions like G ¡ (x; y) depend separately on the two arguments x and y, so that the two equations (2.137) and (2.138) are independent. In order to carry out the gradient expansion, we consider the di8erence of Eqs. (2.137) and (2.138), to be brieLy referred to as the di;erence equation in what follows. After replacing x and y by the coordinates s and X (see Eq. (2.135)), we rewrite the derivatives as 9x = 9s + 12 9X ;

9y = − 9s + 12 *X ;

92x − 92y = 29s · 9X ;

(2.139)

and perform an expansion in powers of 9X , keeping only the terms involving at most one soft derivative 9X . For instance, s

s

?* (x) − ?* (y) = ?* X + − ?* X −

(s · 9X )?* (X ) : (2.140) 2 2 We then perform a Fourier transform s → k and get an equation involving Wigner functions. By Fourier transform, (s · *X )?* (X ) → −i(9X ?* )9k :

(2.141)

Furthermore, it is easily veri:ed that the convolutions in the r.h.s. of Eqs. (2.137) – (2.138) transform as i d 4 zA(x; z)B(z; y) → A(k; X )B(k; X ) + {A; B}P:B: + · · · ; (2.142) 2 where {A; B}P:B: denotes a Poisson bracket: {A; B}P:B: ≡ 9k A · 9X B − 9X A · 9k B

(2.143)

and the dots stand for terms which involve at least two powers of the soft derivative. Thus, the di8erence equation involves, for instance, d 4 z[?R (x; z)G ¡ (z; y) − G ¡ (x; z)?A (z; y)] i (2.144) 2 where all the functions in the r.h.s. are Wigner transforms (i.e., they are functions of k and X ). At this stage, it is convenient to introduce the following Wigner functions: → (?R − ?A )G ¡ + {?R + ?A ; G ¡ }P:B: ;

/(k; X ) ≡ G ¿ (k; X ) − G ¡ (k; X ) ; 7(k; X ) ≡ ?¡ (k; X ) − ?¿ (k; X ) ;

(2.145)

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which provide nonequilibrium generalizations of the spectral densities /(k), Eq. (2.46), and 7(k), Eq. (2.61). In these terms, the Wigner transforms GR (k; X ) and ?R (k; X ) admit the following representations: ∞ ∞ d k0 /(k0 ; k; X ) d k0 7(k0 ; k; X ) GR (k; X ) = (k; X ) = − ; ? : (2.146) R −∞ 2 k0 − k0 − i" −∞ 2 k0 − k0 − i" Similar relations (with −i" → i") hold for the corresponding advanced functions. Note also the relations: GR (k; X ) − GA (k; X ) = i/(k; X ) ; GR (k; X ) + GA (k; X ) = 2 Re GR (k; X ) :

(2.147)

Similar relations hold for the self-energies ?R , ?A , and 7. By using these relations, and the manipulations indicated above, the di8erence equation reduces to the following equation for G ¡ (k; X ): 2(k · 9X )G ¡ + (9X ?* )9k G ¡ = −7G ¡ − /?¡ + {?¡ ; Re GR }P:B: + {Re ?R ; G ¡ }P:B: :

(2.148)

By using de:nition (2.143) of the Poisson bracket, together with the identity 7G ¡ + /?¡ = G ¿ ?¡ − ?¿ G ¡ (cf. Eq. (2.145)), one :nally rewrites this equation as 4 9Re ? 9G ¡ 9Re ? 9G ¡ 2k − + − {?¡ ; Re GR }P:B: = − (G ¿ ?¡ − ?¿ G ¡ ) ; (2.149) 9k 9X 9X 9k where Re ? ≡ Re ?R + ?* . Eq. (2.149) holds to leading order in the gradient expansion (that is, up to terms involving at least two powers of the soft derivative), and to all orders in the interaction strength. In equilibrium, both sides of Eq. (2.149) are identically zero. This is obvious for the terms in the l.h.s., which involve the soft derivative 9X , and can be easily veri:ed for the terms in the r.h.s. by using the KMS conditions for G and ? (cf. Eq. (2.45)). Thus Eq. (2.148) describes the o8-equilibrium inhomogeneity in G ¡ (k; X ), and can be seen as a quantum generalization of the Boltzmann equation (see below). The Wigner function G ¡ (k; X ) plays here the role of the phase-space distribution function f(k; X ). The drift term on the l.h.s. of Eq. (2.149) generalizes the usual kinetic drift term 9t + v · ∇X by including self-energy corrections: The real part of the self-energy acts as an e8ective potential whose space–time derivative provides a “force” term (9X Re ?)(9k G ¡ ). The momentum dependence of ? modi:es the “velocity” of the particles: v → v − (1=2k0 )9k Re ?. The terms on the r.h.s. describe collisions. We shall see below that, for on-shell excitations, these collision terms acquire the standard Boltzmann form. Finally, the Poisson bracket {?¡ ; Re GR }P:B: has a less transparent physical interpretation, which should be clari:ed, however, by the following discussion of the spectral density. The :rst two terms in the l.h.s. of Eq. (2.149) can be recognized as the Poisson bracket −{Re GR−1 ; G ¡ }P:B: , where GR−1 (k; X ) ≡ −k 2 + m2 + ?R (k; X ); see Eq. (2.150) below. 4

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401

The o8-equilibrium spectral density /(k; X ) is most easily obtained from the retarded propagator GR (k; X ) (cf. Eq. (2.146)), for which we can get kinetic equations. These are derived in the same way as above, by performing a gradient expansion in Eq. (2.124) and an analogous equation involving 92y . Unlike the previous calculation, however, the gradient expansion is performed here on the sum of the two equations for GR (k; X ). One gets then (k 2 − m2 − ?* (X ) − ?R (k; X ))GR (k; X ) = − 1 :

(2.150)

This equation contains no soft derivative 9X (the :rst corrections involve at least two powers of the soft gradients). Accordingly, the retarded o8-equilibrium Green’s function GR (k; X ) is related to the corresponding self-energy ?R (k; X ) in the same way as the respective functions in equilibrium (recall Eq. (2.63)). In particular, the associated spectral density is the straightforward generalization of Eq. (2.64), namely: 7(k; X ) /(k; X ) = 2 : (2.151) 2 * (k − m − ? (X ) − Re ?R (k; X ))2 + (7(k; X )=2)2 The o8-equilibrium inhomogeneity enters Eqs. (2.150) and (2.151) only via their parametric dependence on X [88,92]. It is also useful to note that /(k; X ) = G ¿ (k; X ) − G ¡ (k; X ) satis:es a kinetic equation which follows from Eq. (2.148) for G ¡ (k; X ) together with a corresponding equation for G ¿ (k; X ). This reads: 9Re ? 9/ 9Re ? 9/ 2k − + = − {7; Re GR }P:B: : (2.152) 9k 9X 9X 9k (It is straightforward to verify that Eq. (2.151) satis:es indeed this kinetic equation.) Remarkably, the collision terms have mutually cancelled in the di8erence of the two equations for G ¡ and G ¿ . The terms in the l.h.s. of Eq. (2.152) describe drift and mean :eld e8ects, as discussed in connection with Eq. (2.149). The Poisson bracket in the r.h.s. (the di8erence of the corresponding PBs in the equations for G ¡ and G ¿ ) accounts for the o8-equilibrium inhomogeneity in the width 7(k; X ). In fact, if this term is neglected, then the corresponding solution of Eq. (2.152) is simply /(k; X ) = 2(k0 )*(k 2 − m2 − Re ?(k; X )) :

(2.153)

This de:nes a quasiparticle approximation, to be further discussed in Sections 2.3.3 and 2.3.4 below. Conversely, whenever one needs to go beyond such an approximation and include :nite width e8ects, one has to also take into account the Poisson brackets, for consistency [116,117]. For instance, the role of the PBs for insuring conservation laws in systems with broad resonances is discussed in Ref. [118]. 2.3.3. Mean 4eld approximation A mean :eld approximation is obtained if we neglect the interactions among the particles beyond their interactions with the average :elds, that is, in particular, if we neglect the collision terms. From the point of view of the Dyson–Schwinger equations, this corresponds to a truncation of the hierarchy at the level of the 2-point functions: all the connected n-point functions with n ¿ 3 are set to zero. Thus, in the kinetic equations (2.148) – (2.149), we shall neglect all self-energy terms except for the tadpole ?* , Eq. (2.116).

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We thus get the following closed set of equations for the mean :eld E and the Wigner function G ¡ (k; X ): −(92X + m2 )E(X ) = j(X ) + j ind (X ) ; 1 * k k · 9X + (9X ? )9 G ¡ (k; X ) = 0 ; 2

(2.154)

(2.155)

where the induced current is (cf. Eq. (2.111)): j ind (X ) =

g2 E(X )(E2 (X ) + 3G ¡ (X; X )) : 6

(2.156)

In this equation, G ¡ (X; X ) denotes the function G ¡ (x; y) for x = y = X , that is, the integral over k of the Wigner transform G ¡ (k; X ). The spectral density of the hard quasiparticles (k ∼ T ) in the mean :eld approximation follows from Eq. (2.150) with ?R (k; X ) = 0. We obtain (cf. Eq. (2.151)): /(k; X ) = 2(k0 )*(k02 − Ek2 (X )) ;

(2.157)

with Ek2 (X ) ≡ k2 + M 2 (X ) and M 2 (X ) ≡ m2 + ?* (X ) = m2 +

g2 2 (E (X ) + G ¡ (X; X )) : 2

(2.158)

Thus, the mean :eld approximation automatically leads to a quasiparticle approximation. The spectral density (2.157) satis:es (cf. Eq. (2.152)): [k · 9X + 12 (9X ?* )9k ]/(k; X ) = 0 :

(2.159)

By using this equation, it is easy to see that the solution to Eq. (2.155) can be written as G ¡ (k; X ) = /(k; X )N (k; X ) = 2*(k02 − Ek2 (X )){&(k0 )N (k; X ) + &(−k0 )[N (−k; X ) + 1]} ;

(2.160)

where we have separated, in the second line, the positive and negative energy components of the on-shell Wigner function N (k; X ). The structure of the second line follows by using G ¿ (k; X ) = G ¡ (k; X ) + /(k; X ), together with the symmetry property G ¿ (k; X ) = G ¡ (−k; X ). The density matrix N (k; X ) satis:es a kinetic equation analogous to the Vlasov equation (9t + vk · ∇x − ∇x Ek · ∇k )N (k; t; x) = 0

(2.161)

and can be interpreted as a phase-space distribution function for quasiparticles with momentum k and energy Ek (X ). In Eq. (2.161), vk (X ) = k=Ek (X ) = ∇k Ek (X ) is the quasiparticle velocity, and the spatial gradient of the quasiparticle energy Ek (X ) acts like a force on the quasiparticle. Since, for a given :eld con:guration E(X ), the quasiparticle mass squared M 2 (X ) depends on the distribution functions, via Eq. (2.158), the kinetic equations should in principle be solved

J.-P. Blaizot, E. Iancu / Physics Reports 359 (2002) 355–528

simultaneously with the “gap equation”: g2 2 g2 d4 k ¡ 2 2 M (X ) = m + E (X ) + G (k; X ) 2 2 (2)4 d 3 k 2N (k; X ) + 1 g2 g2 = m2 + E2 (X ) + : 2 2 (2)3 2Ek (X )

403

(2.162)

That would correspond to a self-consistent one-loop approximation, which is similar to the large-N limit for the O(N ) scalar model [119]. However, we shall not pursue here the analysis of these equations in full generality, but rather restrict ourselves to the case of small 4eld oscillations, E → 0. In this case, the mean :eld and the kinetic equations decouple since, to leading order in E; j ind (X ) M 2 E(X ), where M 2 is now a constant, solution of the gap equation: d 3 k 2N (Ek ) + 1 g2 2 2 M =m + ; (2.163) 2 (2)3 2Ek and Ek2 = k 2 + M 2 . The linearized mean :eld equation reads then −(92X + M 2 )E(X ) = 0 :

(2.164)

Thus, in this weak :eld approximation, the same mass M characterizes the longwavelength oscillations of the mean :eld E, which we can regard as collective excitations of the system, and the short wavelength excitations associated rather to single particle excitations. Eq. (2.164) shows that this mass M sets the scale of the soft space–time variations: −1 ∼ 9X ∼ M . Furthermore, from Eq. (2.156) one deduces that “small :elds” means gEM : the contribution to the mass (or to the induced current) is then dominated by the short wavelength, or hard, thermal Luctuations. In order to compute M , one should :rst eliminate the UV divergences from the gap equation (2.163). Although such questions will play a minor role in our discussions, it is nevertheless instructive to see how this can be achieved in this simple example. Divergences occur in the following integral, which we compute with an upper cut-o8 : 1 M2 M2 d3 k 1 2 = I () − M I () + ln + ··· ; (2.165) 1 2 2 (2)3 2Ek 2(4)2 2 where is an arbitrary subtraction scale, I1 () ≡

2 ; 2(4)2

I2 () ≡

2 1 ln ; 2(4)2 2

(2.166)

and the dots stand for terms which vanish as → ∞. If we de:ne the renormalized mass and coupling constant via m2r m2 ≡ + I1 (); gr2 g2

1 1 ≡ 2 + I2 () ; gr2 g

(2.167)

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then we obtain a gap equation which is free of UV divergences: d 3 k N (Ek ) g2 M 2 M 2 g2 : M 2 = m2r + r 2 ln 2 + r 2(4) 2 (2)3 Ek

(2.168)

It is easy to verify that the above renormalization procedure renders :nite also the inhomogeneous gap Eq. (2.162). Furthermore, relations (2.167) between the bare and renormalized parameters do not involve the temperature, that is, they are the same as in the vacuum. In particular, Eq. (2.167) implies that the renormalized coupling constant satis:es dgr2 g4 = r 2 ; d ln (4)

(2.169)

which ensures that the solution M 2 of Eq. (2.168) is independent of . The gap equation (2.168) can be solved numerically; the corresponding result is discussed, e.g., in Refs. [120,32]. Alternatively, in the high temperature limit T m, and in the weak coupling regime g2 1, we have T M as well, and the solution to Eq. (2.168) can be obtained in a high temperature expansion: g2 2 g2 (2.170) T − MT + O(g2 M 2 ln(T=)) : 24 8 (Here and below, we denote the renormalized parameters simply as m2 and g2 .) The leading contribution of the thermal Luctuations (the “hard thermal loop”) is of the order of g2 T 2 , and comes from hard momenta k ∼ T M within the integral of Eq. (2.168), for which one can neglect M as compared to k: g2 d 3 k N (k) g2 2 2 (2.171) = T = Mˆ : 3 2 (2) k 24 M 2 (T ) = m2 +

The subleading thermal e8ect in Eq. (2.170) comes from the contribution of soft momenta (k ∼ M T ) to the integral of Eq. (2.168), for which one can approximate N (Ek ) ≈ T=Ek : g2 d3 k N (Ek ) N (k) g2 T T 2 ≈ 2 dk k − − 2 (2)3 Ek k 4 Ek2 k 2 g2 M 2 T dk g2 = = − MT : (2.172) 42 k2 + M 2 8 In particular, for m = 0 and to lowest order in g one can replace M by Mˆ in Eq. (2.172), and thus recover the NLO result for the thermal mass given in Eq. (2.81). 2.3.4. Damping rates from kinetic equations The approximations developed in the previous subsection are suKcient to give a consistent description of the dynamics of the longwavelength excitations of an ultrarelativistic plasma of scalar particles. In the case of gauge theories we shall show explicitly, in Sections 3, 4 and 5, that this approximation scheme isolates the dominant contributions in a systematic expansion in powers of the gauge coupling. Now, there are many interesting physical phenomena whose description requires going beyond this mean :eld approximation. This is the case in particular

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405

of transport phenomena, or of the damping of various excitations. In both cases, the collisions play an essential role. We shall then consider the kinetic equation with the collision terms included, that is, the quantum Boltzmann equation (2.149). For a weakly interacting system, which has long-lived single-particle excitations, it is a good approximation to work in a quasiparticle approximation (see Section 2.3.2). We shall then look for solutions of the form: G ¡ (k; X ) = /(k; X )N (k; X );

G ¿ (k; X ) = /(k; X )[1 + N (k; X )] ;

(2.173)

where /(k; X ) = 2(k0 )*(k02 − Ek2 (X )) is the spectral density in the mean-:eld approximation, Eq. (2.157). With this Ansatz for /, the l.h.s. of Eq. (2.159) vanishes, which suggests to neglect, for consistency, the Poisson brackets in the r.h.s. of Eq. (2.152) and those in the r.h.s. of Eq. (2.148) [88,92]. With these approximations, Eq. (2.149) becomes (k · 9X )G ¡ + 12 (9X ?* )9k G ¡ = − 12 (G ¿ ?¡ − ?¿ G ¡ ) :

(2.174)

This equation has to be complemented with approximations for ?¿ and ?¡ consistent with the previous approximations. Then, as we shall see, it becomes the Boltzmann equation. As in the previous subsection, we can decompose the Wigner functions into positive and negative energy components (cf. Eq. (2.160)). Then, by isolating the positive-energy component of Eq. (2.174), we obtain the following equation for the distribution function N (k; X ): (9t + vk · ∇x − ∇x Ek · ∇k )N (k; X ) =−

1 {[1 + N (k; X )]?¡ (k; X ) − N (k; X )?¿ (k; X )} ; 2Ek

(2.175)

where ?(k; X ) ≡ ?(k0 = Ek ; k; X ) is the on-shell self-energy, and the other notations are as in Eq. (2.161). As an application, let us now consider the single particle excitation which is obtained by adding, at t0 = 0, a particle with momentum p (with p ∼ T ) to a system initially in equilibrium. We want to compute the relaxation rate for this elementary excitation. Since, for a large system, this is a small perturbation, we can neglect all mean :eld e8ects (so that, e.g., ∇x Ep = 0), and assume N (p; t) to be only a function of time. From Eq. (2.175) we get 2Ep

9 N (p; t) = − [1 + N (p; t)]?¡ (p; t) + N (p; t)?¿ (p; t) : 9t

(2.176)

2 2 Here, Ep = (p2 + Mˆ )1=2 , with Mˆ = g2 T 2 =24 (the zero-temperature mass m is set to zero). Since the self-energies in the r.h.s. depend a priori on N (p; t) itself, this equation is generally nonlinear. However, for momenta k = p, the distribution function does not change appreciably from the equilibrium value N (Ek ), so that, to leading order in the perturbation, we can use the equilibrium self-energies (2.60). These read ?¿ (p) = − 7(p)[1 + N (Ep )] and ?¡ (p) = − 7(p)N (Ep ), where 7(p) ≡ 7(p0 = Ep ; p) is the discontinuity of the (equilibrium) self-energy on the mass shell. With this approximation, we get a linear equation:

2Ep

9 N (p; t) = − [N (p; t) − N (Ep )]7(p) ; 9t

(2.177)

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Fig. 4. Leading-order contribution to the collisional self-energy in ;4 theory. Fig. 5. Elementary processes leading to the damping of the single-particle excitation of momentum p (cf. Eq. (2.184)).

whose solution is of the form (*N (p; t) ≡ N (p; t) − N (Ep )): *N (p; t) = *N (p; 0)e−28(p)t ;

(2.178)

with 8(p) ≡ 7(p)=4Ep . We thus recover the relation between the lifetime of the single particle excitation and the imaginary part of the self-energy on the mass shell, already mentioned at the end of Section 2.1.3. The leading order contribution to 7 comes from the two loop diagram in Fig. 4. Thus, 7 ∼ g4 T 2 , and therefore 8 ∼ (7=Ep ) ∼ g4 T for a hard excitation (Ep ∼ T ), while 8 ∼ g3 T for a soft one (Ep ∼ M ∼ gT ). In both cases we have , ∼ 1=81=Ep , which corresponds to long-lived excitations, as required for the validity of the quasiparticle approximation. Although the latter is not a self-consistent approximation (the collision term generates a width which is not included in the spectral densities which are used to estimate it), the neglected terms are of higher order than those we have kept. To compute 7, one can directly evaluate the on-shell imaginary part of the self-energy in Fig. 4, using equilibrium perturbation theory [121–123]. Alternatively, one can :rst construct the collision term associated to this self-energy, and then extract 7 as the coeKcient of *N (p; t) in Eq. (2.177). Since the resulting collision term is interesting for other applications [124] than the one discussed here, and since it clari:es the physical interpretation of the damping in terms of collisions, this is the method we shall follow here. The self-energy in Fig. 4 can be easily evaluated in the x representation: ?(x; y) = −

g4 (G(x; y))3 ; 6

(2.179)

where the time variables x0 and y0 take values along the contour of Fig. 3. By taking x0 and y0 real, with x0 later (respectively, earlier) than y0 , we get ?¿ (x; y) = −

g4 ¿ (G (x; y))3 ; 6

?¡ (x; y) = −

g4 ¡ (G (x; y))3 ; 6

or, after a Wigner transform, g4 ¿ d[k1 ; k2 ; k3 ]G ¿ (k1 ; X )G ¿ (k2 ; X )G ¿ (k3 ; X ) ? (p; X ) = − 6

(2.180)

(2.181)

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407

with a similar expression for ?¡ (p; X ). Here, we have set d[k1 ; k2 ; k3 ] ≡

d 4 k1 d 4 k2 d 4 k3 (2)4 *(4) (k1 + k2 + k3 − p) : (2)4 (2)4 (2)4

(2.182)

In the quasiparticle approximation (2.173), the associated collision term reads g4 ¡ ¿ ¿ ¡ (G ? − G ? )(p; X ) = − d[k1 ; k2 ; k3 ]/(p; X )/(k1 ; X )/(k2 ; X )/(k3 ; X ) 6 ×{N (p; X )[1 + N (k1 ; X )][1 + N (k2 ; X )][1 + N (k3 ; X )] −[1 + N (p; X )]N (k1 ; X )N (k2 ; X )N (k3 ; X )} :

(2.183)

This collision term has the standard Boltzmann structure, with a gain term and a loss term: it involves the matrix element squared for binary collisions (which here is simply |Mpk1 →k2 k3 |2 = g4 =6), together with statistical factors for the on-shell external particles. We consider now again the particular case of a single particle excitation with momentum p. Then, as already discussed, N (p; t) ≡ N (Ep ) + *N (p; t), while all the other particles are in equilibrium: N (ki ; X ) = N (ki0 ). The collision term (2.183) then takes the form −g4 *N (p; t) d[k1 ; k2 ; k3 ]/0 (k1 )/0 (k2 )/0 (k3 ){[1 + N1 ][1 + N2 ][1 + N3 ] − N1 N2 N3 } 6 ≡ −*N (p; t)7(p) ;

(2.184)

where p0 = Ep ; /0 (k) = 2(k0 )*(k02 − Ek2 ) and Ni ≡ N (ki0 ). It can be easily veri:ed that 7(p) de:ned as above coincides indeed with the on-shell discontinuity of the two-loop self-energy in Fig. 4. By inspection of Eq. (2.184), one can identify the physical processes responsible for the damping. There are two elementary processes: the three-particle decay of the incoming :eld (see Fig. 5a), and the binary collision with a particle from the thermal bath (see Fig. 5b). The statistical factors corresponding to the 3-body decay are: [1 + N (E1 )][1 + N (E2 )][1 + N (E3 )] − N (E1 )N (E2 )N (E3 ) ;

(2.185)

with the :rst term describing the direct (decay) process, and the second one representing the inverse (recombination) process. Because of the 3-particle threshold at p0∗ = 3M , this decay process is not e8ective on the mass-shell p0 = Ep , so it does not contribute to the damping of the single-particle excitation in Eq. (2.178). On the other hand, the binary collisions, which are accompanied by statistical factors of the type: N (E1 )[1 + N (E2 )][1 + N (E3 )] − [1 + N (E1 )]N (E2 )N (E3 ) ;

(2.186)

have no kinematical threshold, and contribute indeed to the on-shell damping rate. The :nal result for 8 follows after performing the phase-space integral in Eq. (2.184). For an arbitrary external momentum p, this is quite complicated, and 8(p) can be obtained only numerically [122,123]. However, in the zero momentum limit, an analytical calculation has been given, with

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the result [121]: 8(p = 0) =

g2 Mˆ g3 T √ : = 64 64 24

(2.187)

More generally, the Boltzmann equation (2.175) with the collision term (2.183) describes a variety of nonequilibrium phenomena in the weakly coupled scalar :eld theory and allows one to compute transport coeKcients [125 –127,50,124]. Its solution accomplishes a nontrivial resummation of the ordinary perturbation theory, including in particular that of ladder diagrams [128,122,124,26,72,129]. The kinetic approach is not only technically simpler and physically more transparent than the diagrammatic approach, but also allows for relatively straightforward extensions to gauge theories, as it will be discussed in Section 7. 2.3.5. Time representation and Fermi’s golden rule When one applies the techniques discussed in the previous subsection to the calculation of the damping of charged excitations in hot gauge theories, one is confronted with the diKculty that the on-shell imaginary part of the self-energy is infrared divergent, as mentioned in Section 1.5. The physical origin of this problem is the fact that the mean free path is about the same as the range of the relevant interactions, so that the particles cannot be considered as freely moving (i.e., as on-shell excitations) between successive collisions. This invalidates a simple description in terms of the Boltzmann equation [130], or of the standard perturbation theory in the energy representation [42,131]. In preparation for the detailed discussion of this problem in Section 6, here we reformulate the calculation of the decay of a single particle excitation using kinetic theory by using the time representation. (See also Refs. [69,132] for a similar construction.) As in the previous subsection, we consider an excitation which is obtained by adding, at t = t0 , a particle with momentum p and energy Ep = (p2 + Mˆ 2 )1=2 to a system initially in equilibrium. For t ¿ t0 , the 2-point functions have the generic structure in Eq. (2.173), where we assume that *N (k; t) = 0 for any k = p, while for p (the momentum of the added particle), G ¡ (p; X ) = G0¡ (p) + *G(p; X );

G ¿ (p; X ) = G0¿ (p) + *G(p; X ) ;

(2.188)

where G0¡ (p) and G0¿ (p) are the free equilibrium two-point functions and *G(p; X ) ≡ *G(p; p0 ; t) = 2*(p0 − Ep )

1 *N (p; t) ; 2Ep

(2.189)

with *N (p; t0 ) describing the initial perturbation. The only di8erence with respect to the previous discussion in Section 2.3.4 is that here we shall not take the limit t0 → −∞, but rather keep a :nite (although relatively large: t − t0 1=T ) time interval t − t0 . This will prevent us from taking the on-shell limit when evaluating the collision terms. In order to derive the kinetic equation satis:ed by N (p; t), it is useful to observe, from Eqs. (2.188) – (2.189), that dp0 ¡ *N (p; t) = 2Ep (2.190) *G (p; p0 ; t) = 2Ep *G ¡ (p; x0 = y0 = t) : 2

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We thus need the equation satis:ed by *G ¡ (p; x0 ; y0 ) in the equal time limit. The starting point is Eq. (2.119) for G ¡ (p; x0 ; y0 ), which for x0 = y0 = t simpli:es to t 2 2 2 ¡ d z0 {?¿ (p; t; z0 )G ¡ (p; z0 ; t) (9x0 + p + M )|x0 =y0 =t G (p; x0 ; y0 ) = −i t0

− ?¡ (p; t; z0 )G ¿ (p; z0 ; t)} :

(2.191)

We have neglected here the vertical piece of the contour (i.e., the third integral in the r.h.s. of Eq. (2.119)) since this becomes irrelevant for suKciently large times t − t0 1=T . To perform the gradient expansion in time, we replace x0 , y0 by s ≡ x0 − y0 and t ≡ (x0 + y0 )=2, and proceed as in Section 2.3.2, but without introducing the Wigner transform in time. For instance (with the momentum variable p left implicit) ¿ ¡ d z0 ? (x0 ; z0 )G (z0 ; y0 ) ≡ d z0 ?¿ (x0 − z0 ; (x0 + z0 )=2)G ¡ (z0 − y0 ; (z0 + y0 )=2)

d z0 ?¿ (x0 − z0 ; t)G ¡ (z0 − y0 ; t) ;

(2.192)

where in the :rst line we have rewritten the two-time functions as functions of the relative and central time variables, and in the second line we have used the fact that the o8-equilibrium propagators are peaked at small values of the relative time (i.e., |x0 − z0 | . 1=T ) to write (x0 + z0 )=2 (z0 + y0 )=2 t to leading order in the gradient expansion. In the equal-time limit x0 = y0 = t, the last expression becomes (with z0 changed into t ) t t−t0 ¿ ¡ dt ? (t − t ; t)G (t − t; t) ≡ ds ?¿ (s; t)G ¡ (−s; t) : (2.193) 0

t0

After considering similarly the equation where the temporal derivative acts on y0 , and taking the di8erence of the two equations, we get t−t0 29s 9t G ¡ (p; s; t)|s=0 = −i ds{?¿ (s; t)G ¡ (−s; t) − ?¡ (s; t)G ¿ (−s; t) 0

¡

+ G (s; t)?¿ (−s; t) − G ¿ (s; t)?¡ (−s; t)} ;

(2.194)

where the p-dependence of the functions in the r.h.s. is implicit. Eq. (2.194) is the :nite-time generalization of the Boltzmann equation (2.174). By using the symmetry properties G ¡ (p; s; t) = G ¿ (−p; −s; t);

?¡ (p; s; t) = ?¿ (−p; −s; t) ;

(2.195)

¡ (p; s; t) = ?¡ (p; s), with p ≡ |p|), together with the isotropy of the equilibrium state (e.g., ?eq one can easily check that the r.h.s. of this equation vanishes in thermal equilibrium, as it should. Moreover, for the single-particle excitation of interest, the self-energies can be taken as in thermal equilibrium (cf. the discussion after Eq. (2.176)), so that Eq. (2.194) reduces to t−t0 29s 9t *G(p; s; t)|s=0 = − i ds[?¿ (p; s) − ?¡ (p; s)]*G(p; −s; t) ; (2.196) −(t−t0 )

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or, equivalently (cf. Eq. (2.189)), t−t0 9 ds eiEp s 7(p; s)*N (p; t) ; 2Ep *N (p; t) = − 9t −(t−t0 ) with the de:nition (cf. Eq. (2.60)): ¿

¡

7(p; s) ≡ −[? (p; s) − ? (p; s)] =

dp0 −ip0 s 7(p0 ; p) : e 2

(2.197)

(2.198)

We are interested in relatively large time intervals t − t0 , of the order of the mean free time between successive collisions in the plasma. For systems with short range interactions, like the scalar theory discussed throughout this section, the leading behaviour at large time is obtained by letting t → ∞ in the integration limits in Eq. (2.197). Then, the unrestricted integral over s simply reconstructs the on-shell Fourier component of 7, ∞ ds eiEp s 7(p; s) = 7(p0 = Ep ; p) ; (2.199) −∞

so that Eq. (2.197) reduces to the usual Boltzmann equation (2.177) describing the relaxation of single-particle excitations. Since the integration limits in Eq. (2.197) involve only the relative time t − t0 , it is clear that this on-shell limiting behaviour is also obtained by letting t0 → −∞, which is indeed how Eq. (2.177) has been derived in Section 2.3.4. This explains our emphasis on keeping t0 :nite in this subsection. This allows us to treat also systems with long-range interactions, like gauge theories, for which the on-shell limit of the self-energy is ill-de:ned. Speci:cally, we shall see in Section 6.5 that for gauge theories, 7(p; s) is only slowly decreasing with s (like 1=s), so that the unrestricted integral in Eq. (2.199) is logarithmically divergent. But even in that case, the :nite-time equation (2.197) is still well de:ned, and correctly describes the behaviour at large times. By also using Eq. (2.198), this equation is :nally rewritten as (with t0 = 0 from now on): sin(p0 − Ep )t 9 dp0 Ep *N (p; t) = − *N (p; t) : (2.200) 7(p0 ; p) 9t p0 − Ep 2 For a :xed large time, the function R(t; p0 − E) ≡

sin(p0 − E)t ; p0 − E

(2.201)

is strongly peaked around p0 = E, with a width ∼ 1=t. In the limit t → ∞, R(t; p0 − E) → *(p0 − E), which enforces energy conservation: This limit is known as Fermi’s “golden rule”. In the absence of infrared complications, one can use this limit to obtain the large time behaviour of Eq. (2.200), and thus obtain Eq. (2.177). For gauge theories, however, this naX\ve large-time limit leads to singularities, so that the time dependence of R(t; p0 − E) must be kept. The correct kinetic equation rather reads then 9 *N (p; t) = − 28(p; t)*N (p; t) ; 9t

(2.202)

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with a time-dependent damping rate sin(p0 − Ep )t 1 dp0 : 8(p; t) ≡ 7(p0 ; p) 2Ep 2 p0 − Ep

411

(2.203)

As we shall see in Section 6, this quantity is well de:ned even in gauge theories, because 1=t acts e8ectively as an infrared cut-o8. But this also entails that, in such cases, 8(p; t) remains explicitly time dependent even for asymptotically large times, so that the decay is nonexponential in time. 3. Kinetic theory for hot QCD plasmas With this section, we begin the study of collective excitations of the quark–gluon plasma. We assume that the temperature T is high enough for the condition g ≡ g(T )1 to be satis:ed, and proceed with a weak coupling expansion. As discussed in the introduction, a convenient way to study the collective phenomena is to investigate the response of the plasma to “soft” external perturbations with typical Fourier components P ∼ gT . We shall consider here external sources which produce excitations with the same quantum numbers as the plasma constituents. As we shall see, fermions and bosons play symmetrical roles in the ultrarelativistic plasmas, and the soft fermionic excitations have a collective nature, similar to that of the more familiar plasma waves. The plasma particles act collectively as induced sources for longwavelength average S respectively. The 4elds, either gluonic or fermionic, which will be denoted as Aa , # and #, induced sources can be expressed in terms of 2-point Green’s functions, and their determination is the main purpose of this section. 3.1. NonAbelian versus nonlinear e;ects As emphasized in the Introduction, if we were to study Abelian plasmas, the formalism of the linear response theory would be suKcient for our purpose. In a nonAbelian theory, Ward identities, to be discussed in Section 5.3.3, force us to go beyond this simple approximation. To see how it comes about, we examine more closely here the distinction between Abelian and nonAbelian plasmas. Consider :rst the response of the QED plasma to a soft electromagnetic background :eld, with gauge potentials A . The response function is the induced current jind (x) ≡ j (x), where j (x) = e S (x)8 (x) is the current operator. In the linear response approximation, (3.1) jind (x) = d 4 y % (x − y)A (y) where % (x − y) is the polarization tensor:

(3.2)

Eq. (3.1) is consistent with the Abelian gauge symmetry because the polarization tensor is transverse. Indeed, the condition 9 % = 0 guarantees that the induced current is conserved, 9 jind (x) = 0, and that expression (3.1) is gauge invariant (the contribution of a pure gauge potential A = 9 & cancels out).

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In fact, one can make the gauge invariance explicit by using the transversality of % to reexpress jind in terms of the physical electric :eld; going over to momentum space, with P = (!; p) and E j (P) = i(!Aj (P) − pj A0 (P)), we can write (with j ≡ jind ): j (P) = 5j (P)E j (P);

5j (P) ≡ (i=!)%j (P) ;

(3.3)

where the conductivity tensor 5j (P) satis:es P 5j = 0 and %0 = − ipj 5j . The kinetic theory in Section 1.3 provides us with an explicit expression for 5i (see Eqs. (1:20)–(1:121)): d. v vi i 2 5 (!; p) ≡ imD ; (3.4) 4 ! − v · p + i" which turns out to be the correct result to leading order in e when P ∼ eT . The linear relation between the induced current and the applied gauge potential exhibited in Eq. (3.1) stops to be valid when the :rst nonlinear corrections become comparable to the linear term. Since % ˙ m2D ∼ e2 T 2 , the linear term in jind is of order e2 T 2 A. The :rst nonlinear correction involves the photon four-point vertex function and is of order e4 A3 . Thus, the linear approximation in QED holds as long as AT=e, or E ∼ eTAT 2 . Consider now QCD. For suKciently weak gauge :elds Aa , the linear approximation is valid here as well. Then, the induced colour current takes the form a jind (x) = d 4 y %ab (x; y)Ab (y) ; (3.5) where the polarization tensor receives contributions from all the coloured particles (quarks, gluons, and also ghosts in gauges with unphysical degrees of freedom), and is diagonal in colour, %ab (x; y) = *ab % (x; y). For inhomogeneities at the scale gT (9x A ∼ gTA), and to leading order in g, % has the same expression as in QED, Eq. (1.20), but with m2D ∼ g2 T 2 . Thus, the eight components of the colour current are decoupled and are individually conserved: 9 ja = 0. However, the linear approximation holds in QCD only for :elds much weaker than in QED. This can be seen in various ways. For instance, consider the equations of motion for the soft mean :elds, that is, the Yang–Mills equations with the induced current ja as a source in the r.h.s.: a [D ; F ]a (x) = jind (x) :

(3.6) jind

Since [D ; [D ; F ]] = 0, this equation requires to be covariantly conserved, i.e., to satisfy [D ; j ] = 0, a condition which is generally not consistent with the linear approximation (3.5). In fact, the linearized conservation law 9 j = 0 becomes a good approximation to the correct law [D ; j ] = 0 only for :elds which are so weak that the mean :eld term gA can be neglected within the soft covariant derivative D : gA 9x . For 9x ∼ gT , this requires AT . But in this limit, all the other nonlinear e8ects in Eq. (3.6) can be neglected as well, so this equation reduces to a set of uncoupled Maxwell equations, one for each colour. In other terms, the linear response approximation for the induced current is valid only for :elds which are so weak that they are e8ectively Abelian. a This conclusion is corroborated by an analysis of the current jind . Under gauge transformations of the background :elds, this must transform as a colour vector in the adjoint representation

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413

(so as to insure the covariance of Eq. (3.6)). That is, under the in:nitesimal gauge transformation h(x) = 1 + i&a (x)T a , 1 A (x) → A (x) − 9 &(x) − i[A (x); &(x)] ; g

(3.7)

the current j ≡ ja T a should transform as j → j + *j , with *j (x) = − i[j (x); &(x)] :

(3.8)

Now, under the same transformation, the variation of the linearized current (3.5) is instead *j (x) = − i d 4 y % (x; y)[A (y); &(y)] : (3.9) For a nonlocal response function % (x; y) this is di8erent from the correct transformation law (3.8). This suggests that in the presence of a nonAbelian gauge symmetry there is an interplay between nonlinear and nonlocal e8ects. This can be made more visible by rewriting the induced colour current in terms of a “conductivity” 5i , as in Eq. (3.3): a i jind (x) = d 4 y 5ab (x; y)Ebi (y) : (3.10) Under a gauge transformation h(x) = exp(i& a (x)T a ), the electric :eld transforms as a colour vector: Eai (x) → hab (x)Ebi (x). In order for the induced current to transform similarly, the conductivity must transform as i (x; y) → haaS(x)5aiSbS(x; y)h†bb 5ab S (y) :

(3.11)

Since 5 is generally nonlocal (see, e.g., Eq. (3.4)), this is satis:ed only if the conductivity is itself a functional of the gauge :elds, i.e., relation (3.10) is nonlinear. The particular solution that we shall obtain as the outcome of our approximation scheme satis:es the above requirement in a simple way: the dependence of the conductivity on the gauge :elds is simply given by a parallel transporter. We have i 5ab (x; y|A) = 5i (x − y)Uab (x; y|A) ;

(3.12)

where 5i (x − y) is independent of colour (actually, it coincides with the Abelian conductivity (3.4)), and x Uab (x; y|A) = P exp −ig d z A (z) (3.13) y

is the parallel transporter along the straight line joining y and x. Under a gauge transformation, the parallel transporter becomes Uab (x; y|A) → Uab (x; y|Ah ) = haaS(x)UaSbS(x; y|A)h−1 S (y) ; bb which insures the correct transformation law (3.11) for the conductivity tensor.

(3.14)

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By expanding the exponential in the Wilson line (3.13), it is possible to express the current (3.10) as a formal series in powers of the gauge potentials: 1 abc / a ab jind = % Ab + 7 / Ab Ac + · · · : (3.15) 2 The coeKcients in this series are the one-particle irreducible amplitudes of the soft :elds in thermal equilibrium (cf. Section 5.1): % ∼ g2 T 2 is the polarization tensor, while the other terms (of the generic form 7(n+1) An ) are corrections to the (n + 1)-gluon vertices. The magnitude of the latter can be estimated as follows: since these terms arise solely from expanding the Wilson line (3.13), they scale like 7(n+1) An ∼ %A(glA)n−1 , where l ∼ |x − y| is the typical range of the nonlocality of the response, as controlled by 5i (x − y) (cf. Eq. (3.12)). Now, |x − y| ∼ 1= 9x ∼ 1=gT (cf. Eq. (3.4)), so that the nonlinear e8ects become important when A ∼ T , or E ∼ 9A ∼ gT 2 . For such :elds, glA ∼ 1, and all the terms in expansion (3.15) are of the same order, namely of order g2 T 3 . Repeating the argument in momentum space, one :nds that the n-gluon vertex correction scales like 7(n) (P) ∼ g2 T 2 (g=P)n−2 , where P is a typical external momentum. For P ∼ gT , such vertices are as large as the corresponding tree level vertices, whenever the latter exist. For instance, % ∼ g2 T 2 ∼ D0−1 (with D0−1 ∼ P 2 the tree-level inverse propagator of the gluon), and similarly 7(3) ∼ g3 (T 2 =P) ∼ g2 T ∼ 70(3) (with 70(3) ∼ gP the tree-level three-gluon vertex). Within the approximations that we assume here implicitly, and which will be detailed in Section 3.3, the soft amplitudes in the r.h.s. of Eq. (3.15) are the gluon “hard thermal loops” (HTL). We conclude these general remarks with a few words on the strategy that we shall follow a below. Consider the induced colour current jind as an example: This is a nonequilibrium response function, but all the coeKcients in expansion (3.15) are equilibrium amplitudes. This suggests two possible strategies for computing this current: (i) Within the equilibrium formalism in Section 2.1, one could evaluate the coeKcients in Eq. (3.15) one by one, by computing loop diagrams at :nite temperature. (ii) Alternatively, one could use the nonequilibrium techniques developed in Sections 2.2 and 2.3 to compute directly the induced current in terms of the soft mean :elds, by deriving, and then solving, appropriate equations of motion. The :rst strategy has been adopted in the original derivation of the hard thermal loops from :nite-temperature Feynman graphs [39 – 41,19,20]. In this framework, the HTLs emerge as the dominant contributions to one-loop amplitudes with soft external lines (p ∼ gT ) and hard loop momenta (k ∼ T ), and are obtained as the leading order in an expansion in powers of p=k ∼ g. Many of the remarkable properties of the HTLs have been identi:ed, and studied, within the equilibrium formalism [19,20,22,133–140]. Here, however, we shall rather follow the second strategy, which exploits the nonequilibrium formalism to construct directly the induced current (or other response functions) [18,23,75,76, 26,72]. Aside from the fact that it generates all the soft amplitudes at once, this approach has also the advantage that the kinematical approximations leading to HTLs, and which exploit the separation of scales in the problem (gT T ), are more naturally and more economically implemented at the level of the equations of motion, rather than on Feynman diagrams. As in the scalar theory discussed in Section 2, these approximations will lead from the general Kadano8–Baym equations for QCD to relatively simple kinetic equations.

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With respect to the scalar case of Section 2.3, the main new ingredient here, which is also the main source of technical complications, is, of course, gauge symmetry. We shall see below that it is possible to derive kinetic theory for QCD in a gauge invariant way. To this aim, it will be convenient to work with gauge mean :elds as strong as Aa ∼ T , for which all the nonlinear e8ects associated with gauge symmetry are manifest. In this case, gA ∼ gT ∼ 9x , so that not only the interactions, but also the soft inhomogeneities, and the nonlinear mean :eld e8ects are controlled by powers of the coupling constant. It is then possible to maintain gauge symmetry explicitly via a systematic expansion in powers of g. In particular, by computing the colour current induced by such :elds to leading order in g, we include all the nonlinear e8ects displayed in Eq. (3.15), and therefore all the gluon HTLs. 3.2. Mean 4elds and induced sources At this point, it is convenient to introduce some more formalism: the so-called “background :eld gauge” [141–143], which will allow us to preserve explicit gauge covariance with respect S at all intermediate steps. We stress however that the to the background :elds Aa , # and #, choice of this particular gauge is only a convenience: the :nal kinetic equations to be obtained are independent of the gauge choice. In fact, these equations have been originally constructed in covariant gauges, and shown to be independent of the parameter in the gauge :xing term (9 Aa )2 = [18,23]. 3.2.1. The background 4eld gauge The generating functional Z[j] of a nonAbelian gauge theory may be expressed as the following functional integral, which we write in imaginary time: a 1 a *G 1 a 2 4 2 a Z[j] = DA det exp − d x ; (3.16) (F ) + (G [A]) + j Aa 4 2 *& b where G a [A] is the gauge :xing term (for example, G a = 9 Aa for the so-called covariant gauges, and G a = 9i Aai for Coulomb gauges), is a free parameter (to be referred to as the gauge :xing parameter) and *G a =*& b is the functional derivative of G a [A] with respect to the parameter & a (x) of the in:nitesimal gauge transformations: 1 1 *Aa = − 9 & a + fabc Ab & c = − [D ; &]a : (3.17) g g Since the gauge-:xed Lagrangian in Eq. (3.16) (including the Faddeev–Popov determinant) is not gauge-invariant, the equations of motion derived from it have no simple transformation properties under the gauge transformations of the external sources or of the average :elds. It is however possible to develop a formalism which guarantees these simple properties. This is the method of the background :eld gauge [141,142]. In this method, one splits the gauge :eld into a classical background :eld Aa , to be later identi:ed with the average :eld, and a Luctuating quantum :eld aa , and one de:nes a new generating functional a *G˜ 1 a 1 ˜a 4 2 2 b ˜ exp − d x ; Z[j; A] = Da det (F [A + a]) + (G [a]) + j ab 4 2 *& b (3.18)

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a where the new gauge-:xing term G˜ is chosen so as to be covariant under the gauge transformations of the background :elds. Speci:cally, consider the following gauge transformations of the various :elds and sources:

A → hA h† − (i=g)h9 h† ; a → ha h† ;

j → hj h† ;

L → hLh† :

(3.19)

(Note the homogeneous transformations of the quantum gauge :elds a and of the ghost :elds L; LS to be introduced shortly.) Then, the following, Coulomb-type, gauge-:xing term a G˜ ≡ [Di [A]; ai ]a = 9i aai − gfabc Abi aic ;

(3.20)

a b is manifestly covariant under transformations (3.19): G˜ → hab G˜ . (A gauge-:xing term of the a covariant type can be similarly de:ned with G˜ ≡ [D [A]; a ]a .) a The Faddeev–Popov determinant in Eq. (3.18) involves the variation of G˜ in the following gauge transformation: 1 1 aa → aa − 9 & a + fabc (Ab + ab )& c = − [D [A + a]; &]a : (3.21) g g

This determinant is written as a functional integral over a set of anticommuting “ghost” :elds a in the adjoint representation, La and LS : a *G˜ 4 Sa i b S = DLDL exp − d xL (Di [A]D [A + a])ab L : (3.22) det *& b We thus obtain ˜ A] = Z[j;

4 b S S DaDLDL exp −SFP [a; L; L; A] − d xj ab ;

with the Faddeev–Popov action 1 1 a 4 a 2 i 2 i b S A] = d x SFP [a; L; L; ; (F [A + a]) + (Di [A]a ) + LS (Di [A]D [A + a])ab L 4 2

(3.23)

(3.24)

where D [A + a] = 9 + ig(A + a ) is the covariant derivative for the total :eld A + a , and a [A + a] is the corresponding :eld strength tensor. F The essential property of the complete action in Eq. (3.23), including the sources, is to be invariant with respect to the gauge transformations (3.19). Because of this symmetry, the gen˜ A] is invariant under the normal gauge transformations of its arguments, erating functional Z[j; given by the :rst line of Eq. (3.19). To see this, it is suKcient to accompany the gauge transformations of j and A by a change in the integration variables a ; L and LS of the form indicated in the second line of Eq. (3.19): The combined transformations do not modify the full action, ˜ A] guarantees neither the functional measure (since they are unitary). This symmetry of Z[j; the covariance of the Green’s functions under the gauge transformations (3.19) of the external :eld and current, which is the property we were after.

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˜ A] with respect to the external Speci:cally, by functionally di8erentiating W˜ [j; A] ≡ −ln Z[j; current ja , one generates the connected Green’s functions of the :elds aa . They depend on the background :eld Aa through the gauge :xing procedure. Consider, for instance, the average :eld *W˜ [j; A] ; (3.25) ab (x) = *jb (x) We wish to show that, under the gauge transformation (3.19), a = ab T b transforms as follows: a → a = ha h−1 :

(3.26)

We have a =

*W˜ [j ; A ] *jb (x)

−1 = Z˜ [j ; A ]

= Z˜

−1

[j; A]

Da DLS DL a exp{−SFP [a ; L ; LS ; j ; A ]}

S j; A]} DaDLSDL (ha h−1 ) exp{−SFP [a; L; L;

= ha h−1 :

(3.27)

(Note that the action SFP has been temporarily rede:ned so as to include the coupling to the external current.) In going from the second to the third line, we have changed the integration variables according to Eq. (3.19), and used the invariance of the functional measure and of the full action under the transformations (3.19). Similarly, it is easy to verify that the 2-point function *2 W˜ [j; A] ab G : (3.28) (x; y) ≡ Taa (x)ab (y) = *ja (x)*jb (y) transforms covariantly (x; y) → haaS(x)Ga (x; y)h†bb Gab S (y) : SbS

The ghost propagator,

b

Mab (x; y) ≡ TLa (x)LS (y) ;

(3.29) (3.30)

has the same transformation property. Similar covariance properties hold for the higher point Green’s functions, and for the various self-energies. At this point, we require the background :eld Aa to be precisely the average :eld in the system. First, note that, for an arbitrary external current ja , the total average :eld is Ab + ab . To see this, perform a shift of the integration variable a in the functional integral (3.18) of the form ab → ab − Ab , to get b ˜ A] = Z[j; A] exp Z[j; j Ab ; (3.31)

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where Z[j; A] is the usual generating functional, Eq. (3.16), but evaluated in an unconventional gauge which depends on the background :eld Aa . Eq. (3.31) implies W˜ [j; A] = W [j; A] − (j; A), so that the total average :eld in the system is *W [j; A] *W˜ [j; A] = + Ab (x) = ab (x) + Ab (x) ; *jb (x) *jb (x)

(3.32)

as anticipated. This becomes equal to Ab if ab (x) = 0 :

(3.33)

This condition, which has a gauge-invariant meaning since the average values of the quantum :elds transform homogeneously (see Eq. (3.26)), implies a functional relation between the external current and the background :eld, which we write as j[A]. The functional 7[A] ≡ W˜ [j[A]; A] ;

(3.34)

is the e;ective action, whose functional derivatives are the one-particle-irreducible amplitudes in the background :eld gauge. By construction, 7[A] is invariant with respect to the gauge transformations of its argument, but, in general, it depends on the gauge-:xing parameter . In what follows, we shall not construct the e8ective action (3.34) (see however Section 5.2), but we shall instead write directly the equations of motion for the average :elds and the 2-point functions, and we shall impose on these equations the consistency condition (3.33). The resulting equations will be then covariant with respect to the gauge transformations of the classical mean :elds. Let us now add fermionic :elds and sources. The full generating functional, that we shall use in the rest of this section, reads then b S S S S ˜ Z[j; "; "; S A; #; #] = DaDLDLD D exp −SFP − (j ab + "S + ") ; (3.35) where the Faddeev–Popov action SFP depends on both the quantum and the background :elds 1 a SFP ≡ Scl (A + a; # + ; #S + S ) + (3.36) (Di [A]ai )2 + LS (Di [A]Di [A + a])ab Lb : 2 In the above equation, Scl is the usual QCD action in imaginary time, Eq. (A.11), but evaluated for the shifted :elds A + a; # + , and #S + S . As in Eq. (3.18), the external sources ja ; " and "S are coupled only to the quantum :elds. Some attention should be paid to the boundary conditions in the functional integral (3.35). The gluonic :elds to be integrated over are periodic in imaginary time, with period : a (, = 0) = a (, = ). The fermionic :elds ; S satisfy antiperiodic boundary conditions (e.g., (, = 0) = − (, = )) (see, e.g., [46,14]). Finally, the ghost :elds L and LS are periodic in spite of their Grassmannian nature: this is because the Faddeev–Popov determinant is de:ned on the space of periodic gauge :elds [144,145].

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The partition function (3.35) is invariant under the gauge transformations of the background :elds and of the external sources, that is, transformations (3.19) together with S −1 ; # → h#; #S → #h " → h";

"S → "h S −1 :

(3.37)

Accordingly, the associated Green’s functions are covariant under the same transformations. Finally, the classical :elds A; # and #S are identi:ed with the respective average :elds by requiring that (cf. Eq. (3.33)): a = = S = 0 : (3.38) In constructing the kinetic theory below, it will be convenient to use the (strict) Coulomb gauge, which o8ers the most direct description of the physical degrees of freedom. This gauge is de:ned either by Eq. (3.36) with → 0, or, which is operationally simpler, by imposing the transversality constraint Di [A]ai = 0 ;

(3.39)

within the functional integral (3.35). In this gauge, the gluons Green’s functions are (covariantly) transverse, that is Dxi [A]Gi (x; y) = 0

(3.40)

and similarly for the higher point functions. At tree-level and with A = 0, the only nontrivial components of the retarded propagator are *ij − kˆi kˆj 1 (0) G00 (k) = − 2 ; Gij(0) (k) = − : (3.41) k (k0 + i")2 − k2 That is, the electric gluon is static, and the same is also true for the Coulomb ghost: M(0) (k) = 1=k2 . Accordingly, G ¿(0) (k) = (*ij − kˆi kˆj )G0¿ (k) ; (3.42) G ¡(0) (k) = (*ij − kˆi kˆj )G0¡ (k); ij

ij

[where, e.g., G0¡ (k) = /0 (k)N (k0 ), and /0 (k) = 2(k0 )*(k 2 ); cf. Eq. (2.48)], while all the other ¡(0) components (like G00 ) just vanish. That is, in this gauge, only the physical transverse gluons

are part of the thermal bath at tree-level. This is convenient since ghosts or electric gluons do not contribute to the polarization e8ects to be considered below. (In gauges with propagating unphysical degrees of freedom, the contributions from ghost and longitudinal gluons cancel each other in the :nal results, thus leaving only the contributions of the transverse gluons [19,23]; alternatively, the former can be kept unthermalized using the formalism of Refs. [146].) Keeping this in mind, we shall completely ignore the ghosts in what follows. 3.2.2. Equations of motion for the mean 4elds The mean :eld equations in the background :eld method are easily derived from the generating functional (3.35), and read b Dab [A + a]F [A + a] − g(#S + S )8 t a (# + ) = ja (x) ; (3.43) iD = [A + a](# + ) = "(x) ;

(3.44)

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S Here, D† [A] = 9 − igAa T a , and the derivatogether with the Hermitian conjugate equation for #. ← tive 9 acts on the function on its left. The physically interesting equations are obtained after imposing conditions (3.38), and can be written compactly as a a ind a S [D ; F (x)]a − g#(x)8 (x) ; (3.45) t #(x) = j (x) + j iD = #(x) = "(x) + "ind (x) :

(3.46)

Here and in what follows, D or F denote the covariant derivative or the :eld strength tensor associated to the background :eld Aa (x). The left hand sides of the above equations are the same as at tree-level. All the quantum and medium e8ects are included in the induced sources jind a and "ind in the right hand sides. The induced colour current jind a may be written as jaind (x) = jfa (x) + jga (x)

with the two terms representing, respectively, the quark and gluon contributions: j a (x) = g S (x)8 t a (x) ; f

jga (x) = gfabc 7/ ab (D a/ )c + g2 fabc fcde ab ad a e ;

(3.47) (3.48) (3.49)

where 7/ ≡ 2g/ g − g g/ − g g/ : Finally, the induced fermionic source reads: "ind (x) = g8 t a aa (x) (x) :

(3.50)

In equilibrium, both the mean :elds and the induced sources vanish. This follows from symmetry: in equilibrium, the expectation values involve thermal averages over colour singlet states, and elementary group theory can then be used to prove that, in this case, all terms on the r.h.s. of Eqs. (3.47) – (3.49) indeed vanish. Similarly, "ind is nonvanishing only in the presence of fermionic mean :elds. We shall compute later in this section the induced sources as functionals of the average :elds. S and gauge (Aa ) mean :elds, it is convenient to Since we consider both fermionic (# and #) separate the corresponding induced e8ects by writing: jind a ≡ jAa + j a :

(3.51)

The :rst piece, jA ≡ jind [A ; # = #S = 0], is the colour current which is induced by gauge :elds alone. The second piece, j , denotes the contribution of the fermionic mean :elds; in general, this is also dependent on the gauge :elds Aa . Similarly, we identify quark and gluon contributions by writing jf = jfA + jf ;

jg = jgA + jg ;

(3.52)

for the two pieces of the induced current in Eq. (3.47). 3.2.3. Induced sources and 2-point functions By inspecting Eqs. (3.47) – (3.50), one sees that the induced sources are entirely expressed in terms of 2-point functions. (The only exception is the induced current jga which also contains

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the 3-point function ab ad a e . However, the leading contribution to this 3-point function contains at least two powers of g more than the other terms, so that it can be ignored at leading order.) We now introduce speci:c notations for the various 2-point functions which will appear in the forthcoming developments. Aside from the usual quark and gluon propagators, Sij (x; y) ≡ T i (x) S j (y) = − ab (x; y) G

≡

Taa (x)ab (y) =

* i (x) ; *"j (y)

*aa (x) − ; *jb (y)

(3.53)

we shall also need the following “abnormal” propagators: Ki b (x; y) ≡ T i (x)ab (y) = −

* i (x) *ab (y) − = ; *jb (y) *"Si (x)

H ib (x; y) ≡ Tab (x) S i (y) = −

* S i (y) *ab (x) = ; *jb (x) *"i (y)

(3.54)

which vanish in equilibrium, and which mix fermionic and bosonic degrees of freedom. The time ordered propagators are further separated into components which are analytic functions of their time arguments (see Sections 2.1.2 and 2.2.3). For example, the fermion propagator is written as (with colour indices omitted): S(x; y) = &(,x − ,y ) (x) S (y) − &(,y − ,x ) S (y) (x) ≡ &(,x − ,y )S ¿ (x; y) − &(,y − ,x )S ¡ (x; y) ;

(3.55)

where the minus sign appears because of the anticommutation property of the fermionic :elds. In particular, for a free massless fermion, we have (cf. Eq. (B.14)) S0¿ (k) = k=/0 (k)[1 − n(k0 )];

S0¡ (k) = k=/0 (k)n(k0 ) :

(3.56)

Similar de:nitions hold for the other 2-point functions G; M; K and H , but without the minus sign. For instance, ab ¿ab ¡ab G (x; y) = &(,x − ,y )G (x; y) + &(,y − ,x )G (x; y) :

(3.57)

After continuation to real time, the functions above have hermiticity properties which generalize Eq. (2.40): e.g., (S ¿ )† (x; y) = 80 S ¿ (x; y)80 and (G ¿ )† (x; y) = G ¿ (x; y), or, more explicitly, ¿ab (x; y))∗ = G ¿ba (y; x). Note also the following symmetry property, which will be useful (G later: ¿ab ¡ba G (x; y) = G (y; x) :

(3.58)

The induced sources in Eqs. (3.48) – (3.50) involve products of :elds with equal time arguments ,x = ,y . They may be expressed in terms of the analytic components of the above

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propagators by taking the limit ,y − ,x = " → 0+ : jfa (x) = g Tr(8 t a S ¡ (x; x)) ; ¡ jga (x) = ig 7/ Tr T a Dx G/ (x; y)|y→x+ ; ¡ "ind (x) = g8 t a Ka (x; x) :

(3.59)

Here, the traces involve both spin and colour indices, and y → x+ stands for ,y − ,x → 0+ (or for y0 − x0 → i0+ after continuation to real time). Because the induced sources involve products of :elds at the same point, one could expect to encounter ultraviolet divergences when calculating them. However, this will not be the case in our leading order calculation. Indeed, as we shall verify later, the dominant contribution to the induced sources arises entirely from the thermal particles; this contribution is ultraviolet :nite, owing to the presence of the thermal occupation factors. At this point, it is easy to verify the gauge transformation properties of the induced sources. We have already emphasized that the Green’s functions are covariant under the transformations: A → hA h−1 − (i=g)h9 h−1 ;

# → h#;

S −1 : #S → #h

(3.60)

For instance, the gluon 2-point function transforms according to Eq. (3.29), and, similarly: Sij (x; y) → hik (x)Skl (x; y)h−1 lj (y) ; Kia (x; y) → hij (x)Kjb (x; y)h˜ab (y) :

(3.61)

˜ (We have denoted by h(x) and h(x) the elements of the gauge group in the fundamental and the adjoint representations, respectively.) It is then easy to see that the induced currents in Eqs. (3.59) transform as colour vectors in the adjoint representation, while "ind i (x) transforms like #i (x), i.e., as a colour vector in the fundamental representation. Accordingly, the mean :eld equations (3.45) and (3.46) are gauge covariant. 3.3. Approximation scheme In this section we develop the approximations that allow us to construct kinetic equations for the o8-equilibrium 2-point functions in Eq. (3.59). These approximations are intended to retain the terms of leading order in g in the induced sources, given that the coupling constant enters not only the interaction vertices, but also the space–time inhomogeneities of the plasma (since 9x ∼ gT ), and, for the reasons explained in Section 3.1, also the amplitudes of the mean :elds: S ∼ gT 3 . The constraint on ## S is introduced for consistency gA ∼ gT (or F ∼ gT 2 ), and ## with the Yang–Mills equation (3.45), together with the previous constraint on A : this insures, S t a # ∼ g2 T 3 , which is of the same order as the terms involving Aa (like jA ∼ e.g., that g#8 % A ∼ g2 T 2 A) within the same equation. The starting point is provided by the imaginary-time Dyson–Schwinger equations for the 2-point functions, as obtained by di8erentiating the mean :eld equations with respect to the

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Fig. 6. The one-loop quark self-energy.

Fig. 7. O8-equilibrium e8ects to lowest order: (a) a mean :eld insertion in the fermion propagator; (b) the corresponding contribution to the induced current.

external sources. To this aim, we consider the mean :eld equations (3.43) – (3.44) for arbitrary values of the average quantum :elds ab , etc., and use identities like the one listed in the r.h.s.’s of Eqs. (3.53) – (3.54). After di8erentiation, we set the average values of the quantum :elds to zero (recall Eq. (3.38)). Then, the equations thus obtained are continued towards real time, by exploiting the analytic properties of the various Green’s functions and self-energy. The :nal outcome of this procedure are generalizations of Kadano8–Baym equations for QCD (cf. Sections 2.2.3 and 7.1). 3.3.1. Mean 4eld approximation The :rst approximation to be performed is a mean 4eld approximation (cf. Section 2.3.3), which is equivalent to the one-loop approximation of the diagrammatic approach. To justify this approximation, consider the equation for the quark propagator S(x; y), obtained by di8erentiating Eq. (3.44) with respect to "(y): a a −iD = x S(x; y) − g8 t #(x)H (x; y) + d 4 z?(x; z)S(z; y) = *(x − y) : (3.62) Here, ?(x; y) is the quark self-energy, de:ned as (compare to Eq. (2.113)) *"ind (x) d 4 z?(x; z)S(z; y) ≡ − = gTa=(x) (x) S (y)c : *"(y)

(3.63)

In thermal equilibrium, the :rst contribution to ? arises at one-loop order (see Fig. 6), and is ?eq ∼ g2 T . The induced current jf , Eq. (3.59), involves the o8-equilibrium deviation of the propagator, *S ≡ S − Seq , which can be obtained from perturbation theory. The diagrams contributing to *S contain at least one mean :eld insertion. Let us consider insertions of the gauge :eld Aa , for de:niteness: the lowest order contribution *S (0) is shown in Fig. 7a; when the hard line with momentum k ∼ T is closed on itself, this generates the one-loop contribution to

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Fig. 8. One-loop corrections to the single :eld insertion in Fig. 7a.

Fig. 9. One-loop corrections to the induced current in Fig. 7b: (a) self-energy correction; (b) vertex correction.

the induced current displayed in Fig. 7b. The :rst “radiative” corrections to Fig. 7a come from the self-energy term in Eq. (3.62) and are displayed in Fig. 8. The diagram (Fig. 8a) is obtained by inserting the equilibrium self-energy, Fig. 6, in any of the external lines in Fig. 7a. The diagram (Fig. 8b) involves the (lowest-order) o;-equilibrium self-energy *?(0) , and is obtained by replacing the internal fermion propagator in Fig. 6 by *S (0) . By closing the external lines in Fig. 8 on themselves, one obtains the two-loop corrections to the induced current shown in Figs. 9a and b. Power counting suggests that these two-loop corrections in Fig. 9 can be neglected in leading order, since they are suppressed by a factor of g2 with respect to the one-loop contribution in Fig. 7b. But when both the external and the internal gluon lines in Fig. 9 are soft, naX\ve power counting can be altered by infrared e8ects. For, in that case, the internal fermion propagators are nearly on-shell and read 1=(k · P), with k ∼ T (the hard momentum running along the quark loop) and P . gT (any of the soft momenta carried by the gluons, or a linear combination of them); the smallness of P gives rise to an enhancement over the naX\ve estimates. A careful analysis shows that the relative magnitude of the two-loop corrections depends upon the external momentum P: (a) if P ∼ gT , then the two-loop diagrams are indeed suppressed, but only by one power of g [19]; (b) if P . g2 T , then the one- and two-loop contributions become equally important [25,26,129], as are also higher loop diagrams to be presented in Section 7. The same conclusion can be reached by analysing directly the equations of motion. Consider for instance the gluon propagator G(x; y) in a soft colour background :eld Aa (x). Its Wigner transform G(k; X ) obeys a kinetic equation similar to Eq. (2.174) for the scalar :eld, which involves three types of terms: a drift term (k · 9X )G ¡ , a mean :eld term, and a collision term C(k; X ) = − (G ¿ ?¡ − ?¿ G ¡ ). The relative magnitude of the collision term is determined by comparing it with the drift term. In order to compare (k · 9X )G ¡ with C(k; X ), one should :rst recall that both vanish in equilibrium (cf. Section 2.3.2). Let us then set G ¡ (k; X )

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¡ (k)+*G ¡ (k; X ) and ?¡ (k; X ) ≡ ?¡ (k)+*?¡ (k; X ). The drift term becomes (k ·9 )*G ¡ , = Geq X eq while: ¡ ¡ ¡ ¿ C(k; X ) = − (?eq *G ¿ − ?eq *G ¡ ) + (*?¿ Geq − *?¡ Geq ) + ··· ;

(3.64)

where the dots stand for terms which are quadratic in the o8-equilibrium deviations. Since ?eq (k) ∼ g2 T 2 is :xed by the physics in equilibrium, the importance of the self-energy corrections in the kinetic equation depends upon the scale 9X of the inhomogeneity: If 9X ∼ gT , then the collision terms are suppressed by one power of g and can be neglected to leading order. If 9X ∼ g2 T or less, the collision terms are as important as the drift term. (See however Section 7, where “accidental” cancellations will be discussed which alter slightly this argument.) To summarize, when studying the collective dynamics at the scale gT and to leading order in g, we can restrict ourselves to a mean :eld approximation where the hard particles interact only with the soft mean :elds. The relevant equations for the 2-point functions read then (in Coulomb’s gauge, cf. Eq. (3.39)): D = x S ¡ (x; y) = ig8 t a #(x)H ¡a (x; y) ;

(3.65)

¡ab D = x K ¡b (x; y) = − igt a 8 #(x)G (x; y) ;

(3.66)

¡ ¡ a (g D2 − D D + 2igF )ab y Kb (x; y) = − gS (x; y)8 t #(y) ;

(3.67)

¡cb a ¡b ¡b a S (g/ D2 − D D/ + 2igF/ )ac x G/ (x; y) = g#(x)8 t K (x; y) + gH (y; x)8 t #(x) :

(3.68)

They must be supplemented with the gauge-:xing conditions (cf. Eq. (3.40)): Dxi Gi ¡ (x; y) = 0; Dxi Hi¡a (x; y) = 0;

¡ Gj (x; y)Dyj† = 0 ;

Dyi Ki¡a (x; y) = 0 ;

(3.69)

and the initial conditions chosen such that, in the absence of the external sources, the system is in equilibrium: the mean :elds vanish, and the 2-point functions reduce to the corresponding functions in equilibrium. To the order of interest, the latter are the corresponding free functions (cf. Eqs. (3.42) and (3.56)): ¡ Geq (x; y) G0¡ (x − y);

¡ Seq (x; y) S0¡ (x − y);

¡ Keq (x; y) = 0 :

(3.70)

Since Eqs. (3.65) – (3.68) involve only the “smaller” components, like G ¡ , we shall often omit the upper indices “¡” in what follows. Given the transformation laws in Eqs. (3.29) and (3.61), it is easily seen that Eqs. (3.65) – (3.68) are covariant under the gauge transformations (3.60) of the average :elds. By solving these equations without further approximations, one would obtain the induced sources to one-loop order (see Figs. 10 and 11 for some corresponding diagrams). However, by itself, the mean :eld approximation is not a consistent approximation: additional powers of g are hidden in the soft o8-equilibrium inhomogeneities, and these will be isolated with the help of the gradient expansion.

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Fig. 10. The quark current jfA induced by a colour :eld A , in the one-loop approximation. The blobs represent gauge :eld insertions.

Fig. 11. Some typical contributions to the induced fermionic source "ind in the one-loop approximation. The blobs represent either gauge :eld, or fermionic :eld insertions.

3.3.2. Gauge-covariant Wigner functions As in the scalar theory in Section 2, the equations of motion for the 2-point functions are :rst rewritten in terms of Wigner functions, in order to facilitate the gradient expansion. If Gab (x; y) is a generic 2-point function, its Wigner transform reads (cf. Eq. (2.136)): s s

Gab (k; X ) ≡ d 4 seik·s Gab X + ; X − : (3.71) 2 2 (From now on, we use calligraphic letters to denote the Wigner functions.) Unlike Gab (x; y), which is separately gauge-covariant at x and y (see, e.g., Eq. (3.29)), its Wigner transform Gab (k; X )—which mixes the two points x and y in its de:nition (3:71)—is not covariant. However, it is possible to construct a gauge covariant Wigner function. Consider :rst the following function: s

s s

s

GG ab (s; X ) ≡ UaaS X; X + GaSbS X + ; X − Ubb X − ;X ; (3.72) S 2 2 2 2

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where U (x; y) is the nonAbelian parallel transporter, also referred to as a Wilson line: U (x; y) = P exp −ig d z A (z) : (3.73) 8

In Eq. (3.73), Aa = Aa T a ; 8 is an arbitrary path going from y to x, and the symbol P denotes the path-ordering of the colour matrices in the exponential. Under the gauge transformations of A , the Wilson line (3.73) transforms as (in matrix notations): U (x; y) → h(x)U (x; y)h† (y) ;

(3.74)

so that function (3.72) is gauge-covariant at X for any given s: G X ) → h(X )G(s; G X )h† (X ) : G(s; Correspondingly, its Wigner transform GG ab (k; X ) ≡ d 4 s eik·s GG ab (s; X )

(3.75) (3.76)

transforms covariantly as well: For any given k; GG (k; X ) → h(X )GG (k; X )h† (X ). In principle, any of the two Wigner functions (3.71) and (3.76) could be used to compute the induced sources (3.59). However, only the new Wigner function in Eq. (3.76) will satisfy a gauge-covariant equation of motion, which makes its physical interpretation more transparent. From now on, we shall use systematically gauge-covariant Wigner functions, denoted as G (k; X ). For instance, G (k; X ), or GG (k; X ); K S

s s ˜ s

G ia (k; X ) ≡ d 4 s eik·s Uij X; X + s Kjb K X + ;X − (3.77) U ba X − ; X ; 2 2 2 2 where U (U˜ ) is the parallel transporter in the fundamental (adjoint) representation. Under the gauge transformation (3.60), a

b

G i (k; X ) → hij (X )K G j (k; X )h˜ab (X ) : K

(3.78)

By using Eq. (3.59), one can express the induced sources in terms of these Wigner functions: d4 k G (k; X )) ; Tr (8 t a S (3.79) jfa (X ) = g (2)4 i d4 k a / a X jg (X ) = g7 Tr T k GG / (k; X ) + [D ; GG / (k; X )] ; (3.80) (2)4 2 d4 k a G a ind " (X ) = g 8 t K (k; X ) : (3.81) (2)4 At this stage, the path 8 in the Wilson line (3.73) is still arbitrary. In particular, if 8 is chosen as the straight line joining x and y, the transition from non-covariant to gauge-covariant Wigner functions, e.g., from Eqs. (3.71) to (3.76), can be interpreted as the replacement of the canonical momentum kˆ = i9s by the kinetic momentum pˆ = kˆ − gA (X ) [52,147,148]. In fact, most of our results will be independent of the exact form of 8. This is so because we need U (x; y) only in situations where x is close to y, as we argue now.

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G X ) remains close to its For soft and relatively weak background :elds, the function G(s; value in equilibrium, so it is peaked at s = 0, and vanishes when s & 1=T . Over such a short scale, the mean :eld A does not vary signi:cantly, and we can write, for any path 5 8 joining x and y, g d z A (z) ≈ g(s · A(X )) ; (3.82) 8

up to terms which involve, at least, one soft derivative 9X A ∼ gTA (and which do depend upon the path). Furthermore, for s ∼ 1=T; gs · A ∼ g (since gA ∼ gT ), so we can expand the exponential in Eq. (3.73) in powers of g and get, to leading nontrivial order, Uab (x; y) *ab − ig(s · Aab (X )) :

(3.83)

The present use of the Wilson line should be contrasted with that in Section 3.1, where the parallel transporter in Eq. (3.12) covers a relatively large space–time separation |x − y| ∼ 1=gT determined by the inhomogeneity in the system. In that case, the parallel transporter cannot be expanded as in Eq. (3.83), for the reasons explained in Section 3.1. However, the corresponding path 8 is then :xed by the dynamics of the hard particles (cf. Section 4.1.1). The constraint on the amplitudes of the mean :elds entails a similar constraint on the o8-equilibrium deviation *G ≡ G − Geq : as we shall see later, *G ∼ (gA=T )Geq ∼ gGeq . Thus, by writing G ≡ Geq + *G;

GG ≡ Geq + *GG ;

(3.84)

ab = *ab G , Geq eq

we can easily obtain the following relation in Eq. (3.72), and recalling that between *GG and *G, valid to leading order in g: G X ) *G(x; y) + ig(s · A(X ))Geq (s) ; *G(s;

(3.85)

or, after a Wigner transform, *GG (k; X ) *G(k; X ) + g(A(X ) · 9k )Geq (k) :

(3.86)

For an abnormal Wigner function, the equilibrium contribution vanishes, so that ordinary and G K . Similar simgauge-covariant Wigner functions coincide to leading order in g: e.g., K pli:cations can be performed on Eq. (3.80) for the gluonic current, to get d4 k jga (X ) = g Tr T a {−k *GG (k; X ) + *GG (k; X )k } ; (3.87) 4 (2) where the following property has been used: G X ) Dx G(x; y) = 9s G(s; : y=x

s=0

(3.88)

Note :nally that, within the same approximations, the gauge-:xing conditions (3.69) imply that the gauge-covariant gluon Wigner function is (spatially) transverse, as at tree-level, k i *GG i = 0, 5

Strictly speaking, Eq. (3.82) is a good approximation provided 8 never goes too far away from x and y, that is, provided |z − x| = O(1=T ) for any point z on 8.

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G i = kiH G i = 0. We can thus write and similarly k i K *GG ij (k; X ) ≡ (*ij − kˆi kˆj )*GG (k; X ) :

(3.89)

As we shall verify in Section 3.4.1, the spatial components above are the only ones to contribute to the induced current to leading order in g (this is speci:c to Coulomb’s gauge [23,26]). Thus, :nally, d4 k a jg (X ) = 2g k Tr {T a *GG (k; X )} ; (3.90) (2)4 where the overall factor of 2 comes from the sum over transverse polarization states. 3.3.3. Gauge-covariant gradient expansion In this section, we show how to extract the terms of leading order in g in Eqs. (3.65) – (3.68). This involves approximations similar to those already performed in the previous subsection (in relation with Eqs. (3.83), (3.86) and (3.87)), and which take into account the dependence on g associated with the soft inhomogeneities (9X ∼ gT ), the amplitudes of the mean :elds (A ∼ T or F ∼ gT 2 ), and the magnitude of the o8-equilibrium deviations *GG ∼ gG0 . Since these approximations are related by gauge symmetry, we shall refer to them globally as the gauge-covariant gradient expansion. ¡ab (x; y) in the presence of a soft background Consider then the gluon 2-point function G :eld Aa , but without fermionic :elds (# = #S = 0). Like in the scalar theory in Section 2.3.2, we start with the following two Kadano8–Baym equations for G ≡ G ¡ (here, in the mean :eld approximation; cf. Eq. (3.68)): (g/ D2 − D D/ + 2igF/ )x G/ (x; y) = 0 ; G/ (x; y)(g/ (D† )2 − D/† D † + 2igF/ )y = 0 ;

(3.91)

and take their di8erence. This involves, in particular, O(x; y) ≡ Dx2 G(x; y) − G(x; y)(Dy† )2 ;

(3.92)

where Dx2 = 92x + 2igA · 9x + ig(9 · A) − g2 A2 , and Minkowski indices are omitted to simplify the writing (they will be reestablished when needed). In this subsection we shall illustrate our approximations by focusing on O. After replacing the coordinates x and y by s and X (cf. Eqs. (2.135) and (2.139)), we have, typically, s ∼ 1=T; 9s ∼ T and 9X ∼ gT . We then perform an expansion in powers of 9X and keep only terms which involve, at most, one soft derivative 9X . For instance, A (X + s=2) ≈ A (X ) + (1=2)(s · 9X )A (X ) : Proceeding as in Section 2.3.2, and paying attention to the colour algebra, we obtain O(s; X ) = 29s · 9X G + 2ig[A (X ); 9s G] + ig{A (X ); 9X G } + ig{(s · 9X )A ; 9s G } + ig{(9X · A); G } − g2 [A2 (X ); G] −

g2 {(s · 9X )A2 ; G } + · · · ; 2

(3.93)

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where the right parentheses (the braces) denote commutators (anticommutators) of colour matrices, and the dots stand for terms which involve at least two soft derivatives 9X . At this point, we use the fact that A ∼ gT and *G ≡ G − Geq ∼ gGeq (as it will be veri:ed a posteriori), with Geq ≈ G0 in the present approximation (cf. Eq. (3.70)). By keeping only terms of leading order in g, one simpli:es Eq. (3.93) to O(s; X ) ≈ 2(9s · 9X )*G + 2ig[A ; 9s *G] + 2ig(s · 9X )A (9s G0 ) + 2ig(9X · A)G0 ;

(3.94)

where all the terms in the r.h.s. are of order g2 T 2 G0 . After a Fourier transform with respect to s, O(s; X ) becomes −iO(k; X ) with O(k; X ) ≈ 2[k · DX ; *G(k; X )] + 2gk (9 X A (X ))9 G0 (k) :

(3.95)

Here, *G(k; X ) is the ordinary Wigner transform of *G(x; y), Eq. (3.71), but it can be expressed in terms of the gauge-covariant Wigner function *GG (k; X ) with the help of Eq. (3.86). This :nally yields (0) O (k; X ) ≈ 2[k · DX ; *GG (k; X )] − 2gk P FP (X )9 G (k) ;

(3.96)

where the Minkowski indices have been reintroduced. We recognize here the familiar structure of the Vlasov equation, generalized to a nonAbelian plasma: Eq. (3.96) involves a (gauge-covariant) drift term (k · DX )*GG , together with a “force term” proportional to the background :eld strength tensor. In fact, this “force term” involves the equilibrium distribution function G0 ≡ G0¡ , so, in this respect, it is closer to the linearized version of the Vlasov equation, Eq. (1.14). However, unlike Eq. (1.14), its nonAbelian counterpart in Eq. (3.96) is still nonlinear, because of the presence of the covariant drift operator (k · DX ), and because the nonAbelian :eld strength tensor is itself nonlinear. 3.4. The nonAbelian Vlasov equations In this section, we construct the kinetic equations which determine the colour current induced by a soft gauge :eld Aa . (The fermionic mean :elds # and #S are set to zero in what follows.) According to Eqs. (3.79) and (3.90), we need the equations satis:ed by the quark and gluon G and *GG , in the presence of the background :eld Aa . From the discussion Wigner functions, *S in the previous subsection, we anticipate that these equations are nonAbelian generalizations of the (linearized) Vlasov equation. 3.4.1. Vlasov equation for gluons Since we expect the transverse components *GG ij to be the dominant ones, we focus on the spatial components ( = i and = j) of Eqs. (3.91): Dx2 Gij − Dix D0x G0j + 2igFi/ (x)G/j = 0 ; † † Djy + 2igGi/ F / (y) = 0 : Gij (Dy† )2 − Gi0 D0y

(3.97)

(We have also used Eq. (3.69) to simplify some terms in these equations.) We now take their di8erence, to be succinctly referred to as the di;erence equation (cf. Section 2.3.2). In doing

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so, we :rst meet (cf. Eqs. (3.92) and (3.96)): Dx2 Gij − Gij (Dy† )2 → 2[k · DX ; *GG ij ] − 2gk P FP (X )9 Gij(0) (k) :

(3.98)

Note the following identity, which will be useful later: k P FP 9 Gij(0) (k) ≡ k P FP 9 [(*ij − kˆi kˆj )G0 (k)]

ki *jl + kj *il − 2kˆi kˆj kl G0 : (3.99) k2 This shows that the r.h.s. of Eq. (3.98) involves also nontransverse components. These will be canceled by the other terms in Eqs. (3.97), as we explain now. The components Gi0 and G0j vanish in equilibrium, and remain small out of equilibrium (see below), but nevertheless their contribution to the di8erence equation is nonnegligible. This is so since the hard derivatives 9s0 9si ∼ T 2 multiplying these components do not cancel in the di8erence equation, in contrast to what happens in the terms involving *Gij (cf. Eq. (3.98), where we remember that 92x − 92y = 29s · 9X ∼ gT 2 ). One can evaluate these components from Eqs. (3.91) with = 0 and = j. This gives [26] = (*ij − kˆi kˆj )k P FP 9 G0 − k P FPl

*lj − kˆl kˆj ab ba G0 (k); Gi0 (k; X ) = G0i (−k; X ) ; k2 which provides the following contribution to the di8erence equation: G0j (k; X ) ≈ 2igF0l

(3.100)

ki *jl + kj *il − 2kˆi kˆj kl G0 (k) : (3.101) k2 Finally, there is a contribution from the terms involving the :eld strength tensor in Eqs. (3.97). To the order of interest, it reads † † −(Dix D0x G0j − Gi0 D0y Djy ) → −2gk 0 F0l (X )

−2ig(Fil (X )Glj (s) − Gil (s)Flj (X )) → −2gG0 (k)(Fil kˆl kˆj + Fjl kˆl kˆi ) : (0)

(0)

(3.102)

Together, the contributions in Eqs. (3.101) and (3.102) precisely cancel the nontransverse piece in the r.h.s. of Eq. (3.98), as anticipated. To conclude, *GG ij is transverse indeed, as required by the gauge condition (3.89), and can be written as *GG ij = (*ij − kˆi kˆj )*GG , with *GG (k; X ) satisfying [k · DX ; *GG (k; X )] = gk P FP (X )9 G0 (k) :

(3.103)

Since DX ∼ gT and gFP ∼ (DX )2 ∼ g2 T 2 , it follows that *GG ∼ (DX =T )G0 ∼ gG0 , as anticipated. (By comparison, G0j ∼ (gF0i =T 2 )G0 ∼ g2 G0 is of higher order in g.) Since G0 ≡ G0¡ (k) = 2(k0 )*(k 2 )N (k0 ), and therefore k P FP 9k G0 (k) = 2*(k 2 )k i Fi0 (dN=d k0 ) ;

(3.104)

it follows that *GG (k; X ) has support only on the tree-level mass-shell k 2 = 0. By also using the symmetry property (3.58), we can write *GG ab (k; X ) ≡ 2*(k 2 ){&(k0 )*Nab (k; X ) + &(−k0 )*Nba (−k; X )} ;

(3.105)

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where *Nab (k; X ) is a density matrix satisfying the following, Vlasov-type, equation [23] (with v = (1; k=k) and E i = F i0 = Eai T a ): [v · DX ; *N (k; X )] = − gv · E(X )

dN : dk

(3.106)

This equation implies that *N has the same colour structure as the electric :eld E(x), that is, *N ≡ *N a T a , with the components *N a (k; x) transforming as a colour vector in the adjoint representation. In terms of this density matrix, the induced current (3.90) can be :nally written as 6 d3 k d3 k Aa a jg (X ) = 2g v Tr (T *N (k; X )) = 2gN v *N a (k; X ) : (3.107) (2)3 (2)3 Aside from its covariance under the gauge transformations of the background :eld, Eq. (3.106) is also independent of the gauge-:xing for the quantum :elds, as proven in Ref. [23]. 3.4.2. Vlasov equation for quarks We now brieLy consider the corresponding equation for the quark 2-point function S ≡ S ¡ . Starting with Eq. (3.65), namely D = x S(x; y) = 0;

S(x; y)D = †y = 0

and using (with 5 ≡ (i=2)[8 ; 8 ]) g D = xD = x = Dx2 + 5 F (x) ; 2 one obtains the following di8erence equation: g Dx2 S(x; y) − S(x; y)(Dy† )2 + (5 F (x)S(x; y) − S(x; y)5 F (y)) = 0 : 2

(3.108)

(3.109)

(3.110)

Then one proceeds with a gauge-covariant gradient expansion, as in Section 3.3.3, which :nally G (k; X ) (valid to leading order yields the following equation for the covariant Wigner function *S in g): G (k; X )] = gk · F(X ) · 9k S0¡ − i g F (X )[5 ; S0 ] : [k · DX ; *S (3.111) 4 ˜ ≡ k=/0 (k)n(k 0 ) (cf. Eq. (3.56)), and calculating the Dirac By also using S0 ≡ S0¡ (k) = k=M(k) commutator [5 ; k=], we :nally get the simple equation: ˜ G (k; X )] = gk=k · F(X ) · 9k M(k) [k · DX ; *S ;

(3.112)

G is a colour matrix of the form *S G = *S G a t a , with the components which implies that *S G a transforming as a colour vector. Eq. (3.112) also shows that *S G has the same spin and *S 6

The upper script A is to recall that this is only the contribution of the soft gauge :elds A to the current; cf. Section 3.2.2.

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mass-shell structure as the free 2-point function S0¡ : G (k; X ) = k=2*(k 2 ){&(k0 )*n+ (k; X ) + &(−k0 )*n− (−k; X )} : *S

(3.113)

The density matrices *n± (k; X ) ≡ *na± (k; X )t a satisfy the following kinetic equation: dn [v · DX ; *n± (k; X )] = ∓ gv · E(X ) k ; (3.114) dk which is the nonAbelian version of the Vlasov equation for quarks. The induced current jfAa reads :nally (cf. Eq. (3.79)): d3 k jfAa (X ) = gNf v (*na+ (k; X ) − *na− (k; X )) : (3.115) (2)3 3.5. Kinetic equations for the fermionic excitations A noteworthy feature of the ultrarelativistic plasmas is the existence of collective modes with fermionic quantum numbers. The associated collective motion involves both quarks and gluons, whose mutual transformations, over a long space–time range, can be described as a propagating fermionic :eld #(x) [18,23]. We shall now establish the kinetic equations which determine the corresponding induced sources "ind and j . 3.5.1. Equation for "ind The fermionic source "ind (x) is given by Eq. (3.81), where, to the order of interest, we G a (k; X ) by the ordinary Wigner transform can replace the gauge covariant Wigner function K Ka (k; X ) (cf. after Eq. (3.85)). The kinetic equation for Ka (k; X ) is obtained from the equations of motion (3.66) and (3.67) for Ka (x; y), that is (0) D = x Ka (x; y) = − igt a 8 #(x)G (x; y) ;

(3.116)

2 a (g D˜ − D˜ D˜ + 2igF˜ )ab y Kb (x; y) = − gS0 (x; y)8 t #(y) ;

(3.117)

˜ is the covariant derivative in the fundamental (adjoint) representation. (To simplify where D(D) notations, the colour indices for the fundamental representation are not shown explicitly; see, e.g., Eq. (3.54).) These equations describe a process where a hard gluon scatters o8 a soft fermionic :eld # and gets “turned into” a hard quark, or a hard antiquark annihilates against # to generate a hard gluon. In the right hand sides of the above equations, we have replaced the full gluon and fermion (0) propagators G and S by their free counterparts G and S0 . This is correct in leading order since the o8-equilibrium deviations are suppressed by one power of g: e.g., *S ∼ gS0 . (For S ∼ gT 3 ; see the deviations induced by the fermionic :elds, this relies on the constraint ## Section 3.5.2 below.) Because of that, these equations are linear in #, although nonlinear in Aa . As for the gluon Wigner function in Section 3.4.1, here too there is a hierarchy among the components of Ka (k; X ): The spatial components Kai are the large ones, and are transverse: k i Ki = 0. The temporal component Ka0 is smaller, K0 ∼ gKi , so it does not contribute to the induced source "ind directly; its only role is to remove the longitudinal component of the r.h.s.

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of Eq. (3.116). (Thus, K0 plays here the same role as G0j in Section 3.4.1.) Then Eq. (3.116) with = i reduces to 2

(D˜ y )ab Kib (x; y) = − gSij(0) (x − y)8 j t a #(y) ; where Sij(0) (s)

≡

d 4 k −ik·s e (*ij − kˆi kˆj )S0 (k) : (2)4

(3.118)

(3.119)

To the same order, Eq. (3.116) simpli:es to Dx2 Kia (x; y) ≈ −igt a 8 8 j #(x)9x Gji(0) (x − y) ;

(3.120)

where the r.h.s. is transverse indeed (cf. Eq. (3.42)). We now subtract Eq. (3.118) from Eq. (3.120) and transform the di8erence equation as in the previous subsections. One then :nds, for instance, ab 2 (Dx2 *ab − (D˜ y )ab )Kib (x; y) ≈ 2((9X + igA(X ))*ab + igA˜ (X )) · 9s Kib (s; X ) ;

(3.121)

which, in contrast to Eq. (3.94), does not involve the soft derivative 9X A of the background gauge :eld (this is so since Ka vanishes in equilibrium). Thus, there will be no “Lorentz force” in the kinetic equation for Kai (k; X ), which :nally reads ab

k · [(9X + igA(X ))*ab + igA˜ (X )]Kbi (k; X ) g = − i /0 (k)(N (k0 ) + n(k0 ))k=8 j (*ij − kˆi kˆj )t a #(X ) : (3.122) 2 The di8erential operator in the l.h.s. is a covariant derivative acting on vectors in the direct product of the fundamental and the adjoint representation. Thus, Eq. (3.122) is consistent with the transformation law (3.78) for Kai (k; X ). In a pictorial representation of the solution Kai (k; X ), the fundamental gauge :eld A = Aa t a (respectively, the adjoint :eld A˜ = Aa T a ) in the l.h.s. of Eq. (3.122) is responsible for gauge :eld insertions along the external quark leg (respectively, gluon leg) of Kai (cf. Fig. 12). The induced source (3.81) involves the quantity K = (k; X ) ≡ t a 8 Ka ≈ t a 8i Kai . To get the equation satis:ed by this quantity, multiply Eq. (3.122) by t a from the left, and use the identities t a t c + ifabc t b = t c t a and t a t a = (N 2 − 1)=2N ≡ Cf to obtain d−2 (3.123) Cf /0 (k)(N (k0 ) + n(k0 ))k=#(X ) : 2 The factor (d − 2) = 2—which arises via 8i k=8 j (*ij − kˆi kˆj ) = (d − 2)k=—indicates that only the transverse gluons are involved in this collective motion. This is a consequence of the fact that Eq. (3.123) is gauge-:xing independent, as actually proven in Refs. [18,23]. ab Note that the adjoint gauge :eld A˜ (X ) has disappeared in going from Eqs. (3.122) to (3.123): the latter involves only the covariant derivative in the fundamental representation, which is consistent with the fact that K = (k; X ) must transform in the same way as #(X ) under a gauge rotation of the background :elds. (k · DX )K = (k; X ) = − ig

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˜ denotes insertions of the gauge mean Fig. 12. Pictorial representation of the solution Kai (k; X ) to Eq. (3.122): A(A) :eld in the fundamental (adjoint) representation.

Since the r.h.s. of Eq. (3.123) is proportional to /(k), it follows that K = (k; X ) = 2*(k 2 ){&(k0 )=(k; X ) + &(−k0 )=(−k; X )}

(3.124)

with the density matrix =(k; X ) satisfying (with d = 4, and v = (1; k=k)): (v · DX )=(k; X ) = − igCf (Nk + nk )v=#(X ) : Finally, the fermionic source "ind (X ) is obtained as d3 k 1 =(k; X ) : "ind (X ) = g (2)3 k

(3.125)

(3.126)

Eq. (3.125) is the analog of the Vlasov equation, the fermionic :eld playing in the former the same role as the colour electric :eld in the r.h.s. of the latter. There is, however, a noticeable di8erence: the equilibrium distributions Nk and nk enter the r.h.s. of Eq. (3.125), while their variations, dN=d k and dn=d k, enter the r.h.s.’s of the Vlasov equations (3.106) and (3.114), respectively. This reLects the di8erence in the mechanism by which the hard particles react to the propagation of a colour :eld, or of a fermionic :eld. In the :rst case, the :eld slightly changes the momentum of the hard particles causing a modi:cation of their distribution functions. In the second case, the basic mechanism at work is a conversion of hard gluons into hard fermions, or vice-versa, the soft fermionic :eld bringing the necessary quantum numbers, but no momentum. The same mechanism acts also in QED, where the long-range fermionic excitations are described by Eq. (3.125) with gCf → e, and the nonAbelian covariant derivative D = 9 + igt a Aa replaced by the Abelian one, D = 9 + ieA [18]. 3.5.2. Equations for j Since the background fermionic :elds carry also colour, they induce a colour current j a when acting on the hard particles. In the general equations (3.65) and (3.68) for S and G, the e8ects of the fermionic :elds # and #S are mixed with those of the gauge :elds A . It is convenient

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to separate these e8ects, by writing A

G (k; X ) = S0 (k) + *S G (k; X ) + *S (k; X ) ; S

(3.127)

G A , and similarly *GG A , denote the together with a similar decomposition for GG . Here, *S o8-equilibrium deviations induced by the gauge :elds when # = #S = 0, and obey the kinetic equations established in Section 3.4. The other pieces, *S and *G , denote the additional deviations which emerge in the presence of soft fermionic :elds, and which are responsible for the piece j of the induced current: d4 k a {J a (k; X ) + Jga (k; X )} : (3.128) j (X ) = g (2)4 f We have introduced here the following phase-space current densities (cf. Eqs. (3.79) and (3.90)): Jf a (k; X ) ≡ Tr 8 t a *S (k; X );

Jga (k; X ) ≡ 2k Tr T a *G (k; X ) ;

(3.129)

in terms of which the kinetic equations pertinent to j are most conveniently written as [18,23] (with K = a ≡ 8 Ka , etc.) S )t a K [k · DX ; J (k; X )]a = igk {#(X = (k; X ) − H = (k; X )t a #(X )} f

g 2

S )K − i Nk {#(X = a (k; X ) − H = a (k; X )#(X )} ;

(3.130)

g S )K = a (k; X ) − H = a (k; X )#(X )} : (3.131) Nk {#(X 2 These equations are gauge-:xing independent, and covariant with respect to the gauge transS formations of the background :elds. The expressions in their r.h.s.’s are bilinear in # and #, as necessary for the conservation of the fermionic quantum number. (The general equations satis:ed by the o8-equilibrium propagators *S and *G can be found in [23], where it is S ∼ gT 3 , these deviations are perturbatively small: *S ∼ gS0 and also veri:ed that, for ## *G ∼ gG0 .) Quite remarkably, the genuinely nonAbelian terms, proportional to N , in the right hand sides of these two equations cancel in their sum, that is, in the equation satis:ed by the total current J ≡ Jf + Jg , which reads simply S )t a K = (k; X ) − H = (k; X )t a #(X )} : (3.132) [k · DX ; J (k; X )] = igk t a {#(X [k · DX ; Jg (k; X )]a = i

Thus, the total current J is e8ectively determined by the Abelian-like piece of the quark current Jf alone (i.e., the :rst term in the r.h.s. of Eq. (3.130)). Not surprisingly then, Eq. (3.132) has a direct analog in QED, which reads [18] S )K (k · 9X )J (k; X ) = igk {#(X = (k; X ) − H = (k; X )#(X )} : (3.133) Since the r.h.s. of Eq. (3.132) (or (3.133)) has support only at k 2 = 0, the current can be expressed in terms of on-shell density matrices, as follows: J (k; X ) = 2k 2*(k 2 ){&(k 0 )*n+ (k; X ) + &(−k 0 )*n− (−k; X )} :

(3.134)

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Note that the net contribution to the induced current J is due to the hard fermions only, because of the cancellations alluded to before. The colour density matrices *n± = *n±a t a satisfy the following kinetic equation (with S= ≡ =† 80 ): g S S=(k; X )t a #(X )} : [v · DX ; *n± (k; X )]a = ± i {#(X )t a =(k; X ) − (3.135) 2k In terms of them, the induced current j is :nally written as d3 k j a (X ) = 2g v Tr t a (*n+ (k; X ) − *n− (k; X )) : (2)3

(3.136)

3.6. Summary of the kinetic equations Let us summarize now the kinetic equations which represent the main result of this section. These are written here for the various density matrices, and read (cf. Eqs. (3.114), (3.106), (3.125), and (3.135)): dnk ; dk

(3.137)

dNk ; dk

(3.138)

[v · Dx ; *n± (k; x)] = ∓ gv · E(x) [v · Dx ; *N (k; x)] = − gv · E(x)

(v · Dx )=(k; x) = − igCf (Nk + nk )v=#(x) ; [v · Dx ; *n± (k; x)]a = ± i

g S S=(k; x)t a #(x)} : {#(x)t a =(k; x) − 2k

(3.139) (3.140)

In these equations, v = (1; v), with v = k=k a unit vector which represents the velocity of the hard, and massless, thermal particles. In writing the equations above, we have used the lower case letter x to denote the space–time variable (rather than the upper case variable X which was introduced earlier for the Wigner transform). This notation, which will be used systematically from now on, should not give rise to confusion, as there is no other space–time variable left. As repeatedly emphasized, Eqs. (3.137) – (3.140) are gauge-:xing independent, although they have been derived here by working in the background-:eld Coulomb gauge. This is so since they describe the collective motion of the hard particles, which, to the order of interest, are the same as the physical degrees of freedom of an ideal plasma (quarks, antiquarks, and transverse gluons). These equations are also covariant under the gauge transformations of the soft background S and nonlinear in the :elds A which enter the covariant drift term v · Dx . :elds A , # and #, The fermionic density matrix = is a colour vector in the fundamental representation, while all the other density matrices, which determine the induced colour current, are adjoint colour vectors. Thus, in the Abelian limit (hot QED), the equation satis:ed by = remains nonlinear

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(with the Abelian covariant derivative D = 9 + ieA ), while all the other equations involve the ordinary drift operator v · 9x . These equations have an eikonal structure: in the presence of soft background :elds, the hard particles follow on the average straight-line characteristics, although they may exchange momentum with the soft gauge :elds. The covariant derivative within the drift term induces a colour precession of the various density matrices, which will become manifest in the solutions to Eqs. (3.137) – (3.140), to be presented in the next section. 4. The dynamics of the soft excitations By solving the kinetic equations, which we shall do in this section, one can express the induced sources in terms of the soft mean :elds. Then, the Yang–Mills and Dirac equations including these sources form a closed system of equations which describe the dynamics of the soft excitations of the plasma. This de:nes a classical e8ective theory for the soft :elds, which, as we shall see in the last part of this section, can be given a Hamiltonian formulation. By using this formulation, the calculation of correlation functions at large space-time distances can be reduced to averaging over the initial conditions products of :elds obeying the classical equations of motion. This averaging involves functional integrals which can be calculated using lattice techniques, which is especially useful in applications to nonperturbative problems such as those mentioned at the end of this section. 4.1. Solving the kinetic equations The kinetic equations (3.137) – (3.140) are all :rst-order di8erential equations which can be solved, at least formally, once the initial conditions are speci:ed. We shall consider retarded boundary conditions and assume that, as t → −∞, both the average :elds and the induced sources vanish adiabatically, leaving the system initially in equilibrium. The kinetic equations involve, in their left hand sides, the covariant line derivative v · Dx which makes them nonlinear with respect to Aa . If we were to solve these equations for a :xed v ≡ (1; v), we could get rid of the nonlinear terms by choosing the particular (light-cone) axial gauge v Aa (x) = 0. In this gauge, (v · D)ab = *ab v · 9 and v · Ea (x) = − v · (90 Aa + ∇Aa0 ), as for Abelian :elds. However, in calculating induced sources like (3.107) or (3.126), we have to integrate over all the directions of v. It is therefore necessary to solve the kinetic equations in an arbitrary gauge. 4.1.1. Green’s functions for v · Dx In order to proceed systematically, we start by de:ning a retarded Green’s function by −i(v · Dx )GR (x; y; v) = *(4) (x − y) ;

(4.1)

where D = 9 + igA is the covariant derivative in either the fundamental or the adjoint representation, and GR is a colour matrix in the corresponding representation. A unit matrix in the appropriate representation is implicit in the right hand side.

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In the absence of gauge :elds (A = 0), Eq. (4.1) is easily solved by Fourier analysis. We thus get (with " → 0+ ): d 4 p −ip·(x−y) −1 e GR (x; y; v) = (2)4 v · p + i" = i &(x0 − y0 )*(3) (x − y − v(x0 − y0 )) :

(4.2)

This expression is readily extended to nonvanishing gauge :elds. The corresponding solution to Eq. (4.1) can be written as GR (x; y; v) = i &(x0 − y0 )*(3) (x − y − v(x0 − y0 ))U (x; y) ∞ d, *(4) (x − y − v,)U (x; x − v,) : =i 0

(4.3)

For later use, we note here also the corresponding advanced Green’s function: GA (x; y; v) = −i &(y0 − x0 )*(3) (y − x + v(x0 − y0 ))U (x; y) ∞ d, *(4) (x − y + v,)U (x; x + v,) : = −i 0

(4.4)

In these equations, U (x; y) is the parallel transporter (3.73) along the straight line 8 joining x and y. In particular, , U (x; x − v,) = P exp −ig dt v · A(x − v(, − t)) ; (4.5) 0

where the path joining x − v, to x is parameterized by (t; x(t)) with x(t) = x − v, + vt. In order to verify, for instance, that (4.3) is the correct solution to Eq. (4.1) we may use the following formula for the line-derivative of the parallel transporter: (v · 9x )U (x; y)|y=x−v, = − ig v · A(x)U (x; x − v,) :

(4.6)

Under a gauge transformation A → hA h−1 − (i=g)h9 h−1 , the above Green’s functions transform as GR; A (x; y; v) → h(x)GR; A (x; y; v)h−1 (y), a property which may also be veri:ed directly on Eqs. (4.1). Thus, solutions (4.3) – (4.4) are related to solutions (4.2) by the gauge transformation which connects the axial gauge v · A = 0 to an arbitrary gauge. 4.1.2. The induced colour current In order to solve Eqs. (3.137) and (3.138), it is convenient to express :rst the quark and gluon density matrices *n± and *N in terms of new functions, Wa (x; v), solutions of [v · Dx ; W (x; v)] = F (x)v :

(4.7)

Here we use matrix notations, with W ≡ Wa t a for quarks, and W ≡ Wa T a for gluons. It follows from Eq. (4.7) that the quantities Wa (x; v) satisfy v Wa (x; v) = 0 ;

(4.8)

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so that Wa0 = vi Wai . From the equations above, and Eqs. (3.137) – (3.138) we have *na± (k; x) = ∓ gWa0 (x; v)

dn ; dk

*N a (k; x) = − gWa0 (x; v)

dN : dk

(4.9)

As already mentioned in Section 1.2, the functions Wa0 measure the local distortions of the momentum distributions (see also Section 4.1.4). Eq. (4.7) is easily solved with the help of the Green’s functions introduced above. Using the retarded Green’s function (4.3) one gets a b W (x; v) = −i d 4 yGRab (x; y; v)F (y)v

=

∞

0

b d,Uab (x; x − v,)F (x − v,)v ;

or, in matrix notations, ∞ d,U (x; x − v,)F (x − v,)v U (x − v,; x) : W (x; v) = 0

(4.10)

(4.11)

Once the solution of the kinetic equation is known, one can calculate the induced current A in closed form. By inserting Eqs. (4.9) in the expressions (3.115) and (3.107), jA ≡ jfA + jb and performing the integration over k = |k|, one obtains d. jAa (x) = m2D (4.12) v W0a (x; v) : 4 Here, the integral d. runs over all the directions of the unit vector v, and mD is the Debye screening mass (cf. Section 4.3.2 below), ∞ dNk g2 dnk 2 2 mD ≡ − 2 d kk 2N + 2Nf 2 0 dk dk = (2N + Nf )

g2 T 2 : 6

(4.13)

The induced current (4.12) is covariantly conserved, [D ; jA (x)] = 0 ; as is most easily seen using Eqs. (4.12) and (4.7): d. d. A 0 [D ; j (x)] ˙ [v · Dx ; W (x; v)] = v · E(x) = 0 : 4 4 According to Eqs. (4.10) and (4.12), the induced current can also be written as ∞ d. jAa (x) = m2D d,Uab (x; x − v,)v · Eb (x − v,) : v 4 0

(4.14)

(4.15)

(4.16)

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It transforms as a colour vector in the adjoint representation. The parallel transporter in Eq. (4.16), which ensures this property, also makes it a nonlinear functional of the gauge :elds. 4.1.3. The fermionic induced sources The retarded solution to Eq. (3.139) for =(k; x) reads =(k; x) = − gCf (Nk + nk )v= d 4 y GR (x; y; v)#(y) ;

(4.17)

where GR is now in the fundamental representation. When this is inserted in Eq. (3.126), we obtain the fermionic source with retarded conditions ∞ d. "ind (x) = − i!02 d, U (x; x − v,)#(x − v,) : (4.18) v= 4 0 Here, !02 =

g2 Cf 82

∞

0

d k k(nk + Nk ) =

g2 Cf 2 T ; 8

(4.19)

is the plasma frequency for fermions [39,41]. In the Abelian case, "ind looks formally the same as in Eq. (4.18), but with !02 = e2 T 2 =8. That is, both in QCD and in QED the fermionic source "ind is linear in the fermionic :eld #, but nonlinear in the gauge :elds A . As explained in Section 3.1, the nonlinearity is a consequence of the gauge symmetry together with the nonlocality of the response functions. The presence of the parallel transporter in Eq. (4.18) ensures that "ind transforms in the same way as # under gauge rotations. After similarly solving Eq. (3.140), one obtains the colour current j = j a t a as j (x) = g!02 t a

d. v 4

0

∞

dt

0

∞

S − vt)v=U (x − vt; x)t a U (x; x − vs)#(x − vs) : ds #(x (4.20)

It satis:es the following continuity equation ("Sind ≡ "ind† 80 ): S a "ind − "Sind t a #) : [D ; j ] = igt a (#t In QED, the corresponding current j reads ∞ ∞ d. 2 S − vt)v=U (x − vt; x − vs)#(x − vs) : j (x) = e!0 dt ds #(x v 4 0 0

(4.21)

(4.22)

4.1.4. More on the structure of the kinetic equations The equations of motion for a classical particle of mass m and charge e, moving in an electromagnetic background :eld, may be given two equivalent forms. In terms of the kinetic

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momentum k , related to the velocity of the particle (k = k 0 v ), we have d k = eF v : dt

(4.23)

In terms of the canonical momentum p = k + eA , the equation reads dp = ev 9 A : dt

(4.24)

While the kinetic momentum k is independent of the choice of the gauge, and Eq. (4.23) is manifestly gauge covariant, the canonical momentum p depends on the gauge, as obvious from Eq. (4.24). We show now that the kinetic equations for colour (or charge) oscillations can be also written in two di8erent forms, whose interpretation is similar to that of Eqs. (4.23) and (4.24). We consider :rst a QED plasma. Then Eq. (4.7) reduces to (v · 9x )W (x; v) = F (x)v ;

(4.25)

where, we recall, the velocity v is a constant unit vector. This may be rewritten as d W (t; x(t); v) = F (t; x(t))v ; dt

(4.26)

where d=dt is the total time derivative along the characteristic x(t) = x0 + vt. This is the same as Eq. (4.23) for constant velocity in its r.h.s. Thus, eW (x; v) may be interpreted as the kinetic 4-momentum acquired by a charged particle following, at constant velocity v, a straight line trajectory which goes through x at time t. (For W 0 (x; v), this interpretation has been already given in Section 1.2.) Then, condition (4.8) simply reLects the fact that the energy transferred by the :eld, eW 0 , coincides with the mechanical work done by the Lorentz force, ev · Zk = evi W i . In the nonAbelian case, the Luctuations *n± and *N are matrices in colour space, that is, W = Wa T a . The colour vector of components Wa precesses in the background gauge :eld. This precession is induced by the covariant derivative in Eq. (4.7). Viewing this precession along the characteristic as an additional source of time-dependence for the colour vector Wa , one can write d W (t; x(t); v) ≡ [(9t + v · ∇)*ac − gfabc (v · Ab )]Wc ; dt a

(4.27)

so that Eq. (4.7) may be given a form similar to Eq. (4.26). A di8erent form of the kinetic equation, which corresponds to Eq. (4.24) for the canonical momentum, is obtained by de:ning a (x; v) ≡ A (x) + W (x; v) :

(4.28)

F v = 9 (v · A) − [v · D; A ] ;

(4.29)

Since

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Eqs. (4.7) and (4.8) give immediately [v · D; a ] = 9 (v · A) :

(4.30)

In the Abelian case, Eq. (4.30) can be rewritten in a form analogous to Eq. (4.24) for p : d (4.31) a (t; x(t); v) = (v · 9x )a = 9 (v · A) ; dt showing that ea is the change in the canonical momentum p ≡ k + eA of a charged particle following the trajectory x(t) = x0 + vt. A similar interpretation holds for QCD, with aa a colour vector subject to the precession described by Eq. (4.27). The relevance of these new functions follows from the fact that a (x; v) is a gauge potential of zero :eld strength [76], i.e., 9 a − 9 a + ig[a ; a ] = 0 :

(4.32)

A particular projection of Eq. (4.32) has been obtained by Taylor and Wong [22] by enforcing the gauge invariance of the e8ective action 7A , Eq. (5.17). The zero :eld strength condition is at the origin of interesting formal developments relating the e8ective action of the HTLs to the eikonal of a Chern–Simon theory [134,137,24]. Finally, by combining Eq. (4.12) with W 0 = − A0 + a0 together with Eq. (4.30) for a0 , one obtains the following expression for the induced current: ∞ d. Aa 2 a 2 j (x) = − *0 mD A0 + mD d, Uab (x; x − v,)(v · A˙ b (x − v,)) ; (4.33) v 4 0 where A˙ ≡ 90 A . This expression will be useful later. 4.2. Equations of motion for the soft 4elds By solving the kinetic equations we have expressed the induced sources in terms of the soft average :elds. The equations for the mean :elds become then a closed system of equations describing the dynamics of long wavelength excitations ( ∼ 1=gT ) of the plasma. These equations, which generalize the usual Dirac and Yang–Mills equations, read ∞ d. 2 iD = #(x) = − i!0 d, U (x; x − v,)#(x − v,) ; (4.34) v= 4 0 and a S [D ; F (x)]a − g#(x)8 t #(x) ∞ d. a 2 d, Uab (x; x − v,)v · Eb (x − v,) : = j (x) + mD v 4 0

(4.35)

The colour current j a , not written explicitly here, can be found in Eq. (4.20). These equations have a number of noteworthy properties: (a) They are gauge-covariant; this follows immediately from the covariance of the various induced currents. They are also gauge-:xing independent, i.e., they are independent of the choice of the gauge in the quantum generating functional (3.16).

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(b) The induced sources in the r.h.s. of these equations are nonlocal, and nonlinear. As already mentioned, the nonlinearity is a consequence of the gauge symmetry and the nonlocality. (c) The induced sources in the r.h.s. are of the same order in g as the tree-level terms in the l.h.s. of the equations. That is, the propagation of the soft modes is non-perturbatively renormalized by their interactions with the hard particles. For mean :elds as strong as S ∼ gT 3 , all the nonlinear terms in Eqs. (4.34) – (4.35) are allowed, i.e. F ∼ gT 2 and ## of the same order. (e) The linearized versions of the above equations read, in momentum space, (p2 g − p p + % (p))A (p) = 0 ; (−p= + ?(p))#(p) = 0 ;

(4.36)

where % (p) and ?(p) are the self-energies for soft gluons and, respectively, soft fermions in the HTL approximation (the retardation prescription on these self-energies is implicit here). They are given by Eq. (1.20) (with m2D from Eq. (4.13)) in the case of % , and, respectively, by Eq. (4.44) below in the case of ?. Eqs. (4.36) de:ne the excitation energies of the collective modes which carry the quantum numbers of the elementary constituents, gluons or quarks. These modes have been :rst studied in Refs. [39 – 41], and will be discussed in the next subsection (see also Refs. [149,150,21,14] for more details). 4.3. Collective modes, screening and Landau damping The collective behaviour at the scale gT results in plasma waves, as well as screening and dissipative phenomena, which are encoded in the mean :eld equations (4.34) – (4.35), or their linear version (4.36). This section is devoted to a brief presentation of these collective phenomena. 4.3.1. Collective modes The collective plasma waves are propagating solutions to Eqs. (4.34) – (4.35). In the weak :eld, or Abelian limit, to which we shall restrict ourselves in this subsection, these are solutions to the linearized equations (4.36). That is, they are eigenvectors, with zero eigenvalues, of the matrices in the left hand sides of these equations. Note that some of the solutions to the :rst equation correspond to spurious excitations coming from the lack of gauge :xing. In order to proceed systematically and identify the physical degrees of freedom, we recognize that the matrices in the l.h.s. of Eqs. (4.36) are nothing but the inverse propagators in the HTL approximation. Such propagators—to be generally referred to as the “HLT-resummed propagators”, and denoted by ∗G for gluons, and ∗ S for fermions—are constructed in detail in Appendix B. We shall mostly use the gluon propagator in Coulomb’s gauge, where ∗G has the following nontrivial components (compare with Eq. (3.41)), corresponding to longitudinal (or electric) and transverse (or magnetic) degrees of freedom: G00 (!; p) ≡∗ML (!; p);

∗

Gij (!; p) ≡ (*ij − pˆ i pˆ j )∗MT (!; p) ;

(4.37)

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Fig. 13. Dispersion relation for soft excitations in the linear regime: (a) soft fermions; the upper branch is the “normal” fermion, with dispersion relation !+ (p), while the lower branch, with the characteristic plasmino minimum, is the “abnormal” mode, with energy !− (p); (b) soft gluons (or linear plasma waves), with the upper (lower) branch corresponding to transverse (longitudinal) polarization.

where (with ! → ! + i" for retarded boundary conditions): −1 −1 ∗ ∗ ML (!; p) = 2 MT (!; p) = 2 ; ; 2 p + %L (!; p) ! − p − %T (!; p)

(4.38)

and the electric (%L ) and magnetic (%T ) polarization functions are de:ned as %L (!; p) ≡ −%00 (!; p);

%T (!; p) ≡ 12 (*ij − pˆ i pˆ j )%ij (!; p) :

(4.39)

Explicit expressions for these functions can be found in Eqs. (B.63). The dispersion relations for the modes are obtained from the poles of the propagators, that is, p2 + %L (!L ; p) = 0;

!T2 = p2 + %T (!T ; p) ;

(4.40)

for longitudinal and transverse excitations, respectively. The solutions to these equations, !L (p) and !T (p), are displayed in Fig. 13b. The longitudinal mode is the analog of the familiar plasma oscillation. It corresponds to a collective oscillation of the hard particles, and disappears when pgT . Both dispersion relations are time-like (!L; T (p) ¿ p), and show a gap at zero momentum (the same for transverse and longitudinal modes since, when p → 0, we recover isotropy). As we shall see in Section 4.3.3, there is no Landau damping for the soft modes in the HTL approximation. Rather, these modes get attenuated via collisions in the plasma, a mechanism which matters at higher order in g and which will be discussed in Section 6. For small pmD , the dispersion relations read: 2 2 !T2 = !pl + 35 p2 + · · · ; !L2 = !pl + 65 p2 + · · · ; (4.41) √ where !pl ≡ mD = 3 is the plasma frequency (the gap in Fig. 13b). For large momenta, pmD , one has

!T2 p2 + m2D =2;

!L2 p2 (1 + 4xL ) ;

(4.42)

where xL ≡ exp{−(2p2 =m2D ) − 2}. Thus, with increasing momentum,√the transverse branch becomes that of a relativistic particle with an e8ective mass m∞ ≡ mD = 2 (commonly referred

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to as the “asymptotic mass”). Although, strictly speaking, the HTL approximation does not apply at hard momenta, the above dispersion relation !T (p) remains nevertheless correct for p ∼ T where it coincides with the light-cone limit of the full one-loop result [151]: m2D : (4.43) 2 The longitudinal branch, on the other hand, approaches the light cone exponentially, but, as already mentioned, it disappears from the spectrum, as its residue is exponentially suppressed [149] (see Appendix B). We turn now to soft fermionic excitations. The corresponding HTL is easily obtained from Eq. (4.18) with A → 0, and reads (cf. Section 5.3.1): d. v= ?(!; p) = !02 : (4.44) 4 ! − v · p + i" m2∞ ≡ %T1−loop (!2 = p2 ) =

Let us :rst consider in more detail what happens in the long wavelength limit, p → 0. From Eq. (4.44) one gets then !02 80 : ! The spectrum at p = 0 is thus obtained from the poles of 80 ∗ : S(!) = −! + !02 =! ?(!; p = 0) =

(4.45) (4.46)

For each eigenstate of 80 , corresponding to the eigenvalues ±1, there are two poles, at ! = ± !0 . Thus, the pole at ! = !0 is double degenerate, and similarly for the pole at ! = − !0 . This degeneracy is removed by a small (zero-temperature) mass, or, as we shall see, by a :nite momentum p, which leads to the split spectrum shown in Fig. 13a. Consider then the HTL-resummed propagator ∗ S at :nite momentum. This has the following structure (cf. Section B.2.3): ∗

ˆ +∗M− (!; p)− (p) ˆ ; S(!; p)80 = ∗M+ (!; p)+ (p)

(4.47)

ˆ are projectors. with the functions ∗M± given by Eq. (B.104), and the matrices ± ≡ (1 ± 80 8 · p)=2 ∗ ∗ ∗ Charge conjugation exchanges the poles of M+ and M− . M+ has two poles, one at positive !, with energy !+ (p), and another one at negative !, with energy −!− (p); these go over to ±!0 as p → 0. Correspondingly, ∗M− has poles at !− and −!+ . In the limit p!0 , the branches ±!+ (p) describe the “normal” (anti)fermion, with a dispersion relation 2 2 !+ (p) p2 + M∞ ;

2 M∞ ≡ 2!02 ;

√

(4.48)

describing the propagation of a massive particle with the “asymptotic” mass M∞ = 2!0 . In the same limit, the “abnormal” branches ±!− (p) disappear from the spectrum since their residues are exponentially suppressed [150]. Incidentally, Eq. (4.48) is also correct for p ∼ T , where it coincides with the full one-loop result [151]. For generic soft momenta, on the other hand, all the four branches are present in the spectrum, and describe collective excitations in which the hard quarks get converted into hard gluons, or

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vice versa, giving rise to longwavelength oscillations in the number density of the hard fermions. In particular, for p!0 , p p !+ (p) !0 + + · · · ; !− (p) !0 − + · · · ; (4.49) 3 3 so that the “abnormal” (or “plasmino”) branch is actually decreasing at small p, down to a minimum at p = pmin ≈ 0:408!0 , and then it increases and approaches ! = p, as shown in Fig. 13a. The origin of this peculiar behaviour is a collective phenomenon whereby the single particle strength at small momentum p is split by the coupling of the soft modes to the hard particles which form a continuum of states with energy |!| ¡ p. Due to this coupling, a fraction of the anti-fermion strength, initially at energy ! = − p, is pushed up to the positive energy ! = !− (p), producing the abnormal branch. For small p the behaviour of !− (p) is therefore that of a negative energy state: it decreases as p increases. This physical interpretation is made explicit by the construction of the plasmino state at zero temperature and large chemical potential in Ref. [152]. We note :nally that particular solutions of the nonlinear equations (4.35) have also been found, in Refs. [153,154,21]. These solutions describe nonlinear plane-waves propagating through the plasma, and represent truly nonAbelian collective excitations. 4.3.2. Debye screening The screening of a static chromoelectric :eld by the plasma constituents is the natural nonAbelian generalization of the Debye screening, a familiar phenomenon in classical plasma physics [43]. In coordinate space, screening means that the range of the gauge interactions is reduced as compared to the vacuum. In momentum space, this corresponds to a softening of the infrared behaviour of the various n-point functions. Consider a static colour :eld, that is, a :eld con:guration which, at least in some particular gauge, can be represented by time-independent gauge potentials Aa (x). For such :elds, expression (4.33) of the colour current reduces to its :rst, local, term (a static colour charge density): jA a (x) = − *0 m2D Aa0 (x) :

(4.50)

The equations of motion (4.35) simplify accordingly: [Di ; E i (x)] + m2D A0 (x) = 0 ; ijk [Dj ; Bk ] = ig[A0 ; E i ] :

(4.51)

They di8er from the corresponding equations in the vacuum only by the presence of a “mass” term m2D A20 for the electrostatic :elds. To appreciate the role of this mass, consider :rst the Abelian case, where the equations above are linear (−M + m2D )A0 (x) = /(x) ; ZAi − ∇i (∇ · A) = 0 :

(4.52)

The :rst equation, in which we have added an external source with charge density /(x) = Q *(3) (x), is easily solved by Fourier transform and yields the familiar screened Coulomb

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potential:

d3 p eip·r e−mD r = Q : (4.53) (2)3 p2 + m2D 4r The second equation (4.52) shows that the static magnetic :elds are not screened, which can be related to the fact that plasma particles carry no magnetic charges. The same conclusion can be reached by an analysis of the e8ective photon (or gluon) propagators (4.38) in the static limit ! → 0. Eqs. (B.63) imply A0 (r) = Q

%L (0; p) = m2D ; and therefore: ∗

ML (0; p) =

%T (0; p) = 0 ;

−1

∗

m2D

MT (0; p) =

1 ; p2

(4.55) + which clearly shows that the Debye mass acts as an infrared cut-o8 ∼ gT in the electric sector, while there is no such cut-o8 in the magnetic sector. In nonAbelian plasmas, Debye screening may be accompanied by interesting nonlinear e8ects. A particular solution to Eqs. (4.51) is [140] e−mD r 1 Aa0 = Arˆa ; Aai = aij rˆj ; (4.56) r r where A is a constant and rˆi = xi =r. This solution is a superposition of the Wu-Yang magnetic monopole [155] and a screened electrostatic :eld. More generally, it has been shown in Ref. [140] that all the (SU (2)-radially symmetric) solutions which are regular at in:nity approach at the origin the monopole solution (4.56). That is, all such solutions show electric screening, but there is no sign of magnetic screening, in spite of the nonAbelian coupling between electric and magnetic :elds in Eqs. (4.51). Moreover, there are no :nite energy solutions (no static solitons), in complete analogy to what happens in the vacuum (i.e., for mD = 0) [156]. p2

;

(4.54)

4.3.3. Landau damping For time-dependent :elds, there exists a di8erent screening mechanism associated to the energy transfer to the plasma constituents. In Abelian plasmas, this mechanism is known as Landau damping [43]. For simplicity, let us start with this Abelian case, and compute the mechanical work done by a longwavelength electromagnetic :eld acting on the charged particles. The rate of energy transfer has the familiar expression [43] d EW (t) (4.57) = d 3 x E(t; x) · j(t; x) ; dt where j i (p) = %Ri (p)A (p) is the induced current. Consider, for instance, a periodic electric :eld of the form E(t; p) = E(p) cos !0 t. From Eq. (4.57), one can compute the average energy loss over one period T0 = 2=!0 , with the following result: 1 dEW d3 p i = E (−p)(−Im %Rij (!0 ; p))E j (p) 3 dt 2!0 (2) m2D d. = (4.58) *(! − v · p)|v · E(p)|2 ; 2 4

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where we have used the following expression for the imaginary part of the retarded polarization tensor (cf. Eq. (1.20)) d. Im %R (!; p) = − m2D ! (4.59) v v *(! − v · p) : 4 The expression in Eq. (4.58) is nonnegative: on the average, the energy is transferred from the electromagnetic :eld to the particles. The *-function in Eq. (4.58) shows that the particles which absorb energy are those moving in phase with the :eld (i.e., the particles whose velocity component along p is equal to the :eld phase velocity: v · pˆ = !=p). Since in ultrarelativistic plasmas v is a unit vector, only space-like (|!| ¡ p) :elds are damped in this way. To see how this mechanism leads to screening, consider the e8ective photon (or gluon) propagator in the hard thermal loop approximation (cf. Eq. (4.38)), and focus on the magnetic propagator. For small but nonvanishing frequencies the corresponding polarization function %T (!; p) is dominated by its imaginary part, which vanishes linearly as ! → 0 (see Eq. (4.59)), in contrast to the real part which vanishes quadratically. Speci:cally, the second equation (B.63) yields ! %T (!p) = − i m2D + O(!2 =p2 ) ; (4.60) 4 p and therefore ∗

MT (!p)

p2

1 : − i(!=4p) m2D

(4.61)

In the computation of scattering cross sections, the relevant matrix element squared is proportional to (see, e.g., Eq. (6.3)) 1 |∗MT (!; p)|2 4 ; (4.62) p + (m2D !=4p)2 which shows that Im %T (p) acts as a frequency-dependent IR cuto8 at momenta p ∼ (!m2D )1=3 . That is, as long as the frequency ! is di8erent from zero, the soft momenta are dynamically screened by Landau damping [63]. Dynamical screening occurs also for the longitudinal interactions, but in this case it is less important, since Debye screening dominates at small frequency. Furthermore, in the case of QCD, the study of Landau damping is complicated by nonlinear e8ects. The nonAbelian expression for the rate of mechanical work (see Eq. (4.67) below) involves the nonlinear colour current (4.16); accordingly, all the n-point HTL amplitudes (self-energy and vertices) develop imaginary parts (see Section 5). Moreover, the Landau damping is also operative for soft fermions, both in QCD and in QED; this is described, e.g., by the imaginary part of the fermion self-energy, Eq. (4.44). 4.4. Hamiltonian theory for the HTLs There exists a concise and elegant formulation of the e8ective theory for the soft :elds dynamics as a Hamiltonian theory [24,75,77]. At :rst sight, this may be surprising since the

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corresponding equation of motion, namely Eq. (4.35), (the fermionic :elds are set to zero in this section): ∞ d. a 2 [D ; F (x)] = mD d, Uab (x; x − v,)v · Eb (x − v,) ; (4.63) v 4 0 is nonlocal in space and time, and also dissipative: Because of Landau damping, energy is transferred between the particles and the background :elds. However, at the expense of keeping the :eld W0a (x; v) as a soft degree of freedom (summarizing the e8ects of the plasma particles), one can rewrite Eq. (4.63) as to the following system of local equations (cf. Eqs. (4.7) and (4.12)): d. a 2 [D ; F (x)] = mD v W0a (x; v) ; 4 [v · Dx ; W0 (x; v)]a = v · Ea (x) : (4.64) The :elds Aa (x) and W0a (x; v) will be regarded as independent degrees of freedom in the following. 4.4.1. The energy of the colour 4elds In order to obtain the Hamilton function for these degrees of freedom, we start by computing the energy E carried by the longwavelength colour excitations [138,139,24,75]. We can write E = EYM (t) + EW (t) ≡ d 3 xE(t; x) ; (4.65) where EYM (t) is the energy stored in the colour :elds at time t, 1 EYM (t) = d 3 x (Ea (t; x) · Ea (t; x) + Ba (t; x) · Ba (t; x)) ; (4.66) 2 while EW (t) is the polarization energy, that is, the energy transferred by the colour :eld to the plasma constituents, as mechanical work. Energy conservation dE=dt = 0, together with the :rst equation (4.64), imply dEW (t) (4.67) = d 3 x Ea (t; x) · ja (t; x) ; dt where ja is the induced current (4.12): d. 2 ja (t; x) = mD (4.68) v W0a (x; v) : 4 We recognize in Eq. (4.67) the nonAbelian generalization of Eq. (4.58). By using the equation of motion for W0a (x; v) (i.e. the second equation (4.64)), we can write t EW (t) = dt d 3 x Ea (t ; x ) · ja (t ; x ) −∞

= m2D =

m2D 2

d. 4 d. 4

t

−∞

dt

t

−∞

dt

d 3 x W0a (t ; x ; v)[v · Dx ; W0 (t ; x ; v)]a d 3 x (v · 9x )(W0a W0a ) :

(4.69)

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The integral over t can now be done (we assume the :elds to vanish at spatial in:nity and at t → −∞), and yields d. 0 m2D 3 d x (4.70) EW (t) = W (x; v) Wa0 (x; v) : 2 4 a Together, Eqs. (4.65), (4.66) and (4.70) express the energy associated with the propagation of a longwavelength colour wave in a hot QCD plasma [75]. Clearly, this quantity is positive de:nite, which reLects the stability of the plasma with respect to longwavelength colour oscillations [24]. The energy Lux density, or Poynting vector, of the propagating colour waves can be computed similarly, with the result [75] m2D d. a a S(x) = E (x) × B (x) + (4.71) v Wa0 (x; v) Wa0 (x; v) : 2 4 Then, the energy conservation can be also written in local form, as 90 E(x) + 9i S i (x) = 0 ;

(4.72)

where E(x) is the energy density in Eq. (4.65). Note, however, that the above expressions are local only when expressed in terms of both the gauge :elds Aa (x) and the auxiliary :elds Wa0 (x; v). 4.4.2. Hamiltonian analysis In the temporal gauge Aa0 = 0, Eqs. (4.64) become (with W0a (x; v) ≡ W a (x; v) in what follows) 90 Aai = − Eia ; −90 Eia

a

+ ijk (Dj Bk )

= m2D

d. vi W a (x; v) ; 4

(4.73)

(90 + v · D)ab Wb = v · Ea ; together with Gauss’ law (the = 0 component of the :rst equation (4.64)): d. a a a 2 G (x) ≡ (D · E) + mD W (x; v) = 0 : 4

(4.74)

This last equation contains no time derivative and should therefore be regarded as a constraint. We show now that Eqs. (4.73) can be given a Hamiltonian structure. To this aim, consider the conserved energy in Eq. (4.65) which we denote here by H : 1 d. a H= d 3 x Ea · Ea + Ba · Ba + m2D (4.75) W (x; v) W a (x; v) : 2 4 This expression is independent of the choice of gauge. However, in the gauge Aa0 = 0, we can make it act as a Hamiltonian, that is, as the generator of the time evolution. As independent degrees of freedom, we choose the vector potentials Aia (x), the electric :elds Eai (x), and the density matrices W a (x; v). (Note that the latter are :elds on the extended con:guration space E 3 × S 2 , with E 3 being the ordinary three-dimensional coordinate space and S 2 the unit sphere

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spanned by v.) Then, following Nair [24], we organize this as a Hamiltonian system by introducing the following Poisson brackets: {Eia (x); Abj (y)} = − *ab *ij *(3) (x − y) ; {Eia (x); W b (y; v)} = vi *ab *(3) (x − y) ;

(4.76)

m2D {W a (x; v); W b (y; v )} = (gfabc W c + (v · Dx )ab )*(3) (x − y)*(v; v ) : Here, *(v; v ) is the delta function on the unit sphere, normalized such that d. (4.77) *(v; v ) f(v) = f(v ) ; 4 and all other brackets are assumed to vanish. We also assume the standard properties for the Poisson brackets, namely antisymmetry, bilinearity and Leibniz identity: {AB; C } = A{B; C } + {A; C }B. It is then straightforward to verify that (a) the Poisson brackets (4.76) satisfy the Jacobi identity (a necessary consistency condition) and (b) the equations of motion (4.73) follow as canonical equations for the Hamiltonian (4.75). For instance, 90 W a = {H; W a }, and similarly for Eia and Aai . The Hamiltonian in Eq. (4.75) is remarkably simple: it is quadratic in the auxiliary :elds W0a . Up to the colour indices, this piece would be the same in QED. Thus, all the nonAbelian complications are encoded in the Yang–Mills piece of H and in the nontrivial Poisson brackets (4.76). 4.4.3. E;ective classical thermal 4eld theory We shall now use the above Hamiltonian formulation of the HTL e8ective theory to write down a generating functional for the thermal correlations of the soft :elds, in the classical approximation. As emphasized in Section 2.2.5, the classical approximation is correct only at soft momenta, so we shall introduce an ultraviolet cuto8 , with gT T , to eliminate the hard (k & T ) Luctuations from the classical theory. Correspondingly, will act as an infrared cuto8 for the hard, quantum, modes (see below). We denote by Eia (x); Aai (x) and Wa (x; v) the initial conditions for the HTL equations of motion (4.73). The partition function reads as follows (compare with Eq. (2.130)): Zcl = DEai DAai DWa *(Ga ) e−H ; (4.78) where Ga and H are expressed in terms of the initial :elds as in Eqs. (4.74) and (4.75). Eq. (4.78) can be rewritten as a a 3 a a a a 2 a a Zcl = DA0 DAi exp − d x(Bi Bi + (Di A0 ) (Di A0 ) + mD A0 A0 ) ; (4.79) 2 where the temporal components Aa0 of the gauge :elds have been reintroduced as Lagrange multipliers to enforce Gauss’ law, and the Gaussian functional integrals over Eia and Wa have been explicitly performed. In particular, the integral over the W’s has generated the screening mass for the electrostatic :elds, as expected. We recognize in Eq. (4.79) the static limit (5.19) of the HTL action.

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More generally, time-dependent correlations of the soft :elds are obtained by averaging products of :elds Aia (t; x) obeying Eqs. (4.73). These correlations can be generated from a a a a a 4 a a Zcl [Ji ] = DEi DAi DW *(G ) exp −H + d xJi (x)Ai (x) ; (4.80) where Aia (t; x) is the solution to Eqs. (4.73) (in particular, Aai (t0 ; x) = Aai (x)), and the external current Jia is introduced as a device to generate the correlations of interest via functional differentiations, but does not enter the equations of motion for the :elds. It can be veri:ed [77] that the phase-space measure DEai DAai DWa is invariant under the time evolution described by Eqs. (4.73), so that Zcl [J ] is independent of the initial time t0 , as it should. Since the dynamics is also gauge-invariant, it is suKcient to enforce Gauss’ law at t = t0 , as we did in Eq. (4.80). As a simple, but still nontrivial, check of Eq. (4.80), let us consider the Abelian limit, where the equations of motion can be solved analytically, and the functional integral in Eq. (4.80) can be computed exactly, since Gaussian [77]. We know already the result that we want to obtain: This should read (compare with Eq. (2.133)): 1 4 4 i cl j Zcl [Ji ] = Zcl [0] exp − d x d y J (x) Gij (x − y)J (y) ; (4.81) 2 d 4 k −ik·(x−y) ∗ cl e /ij (k)Ncl (k0 ) ; (4.82) Gij (x − y) ≡ (2)4 where Ncl (k0 ) = T=k0 , and ∗/ij (p) is the photon spectral density in the HTL approximation and in the temporal gauge (cf. Eqs. (B.68) – (B.71)): ki kj ∗ /L (!; k) : (4.83) !2 The 2-point function (4.82) is the classical limit of the corresponding quantum correlator, which reads d 4 k −ik·(x−y) ∗ ∗ G (x; y) = e / (k)[&(x0 − y0 ) + N (k0 )] : (4.84) (2)4 /ij (!; k) = (*ij − kˆi kˆj )∗/T (!; k) +

∗

In the classical limit N (k0 ) ≈ T=k0 1, so that the 2-point functions G ¿ and G ¡ reduce to the unique classical correlator G cl (see Sections 2.1.4 and 2.2.5). For instance, in the transverse sector, T ∗ /T (!; k) = GTcl (!; k) : (4.85) ! It is our purpose here to verify that Eqs. (4.81) – (4.83) emerge indeed from the computation of the Abelian version of the functional integral (4.80). To this aim, we have to solve :rst the linearized equations of motion (4.73): d. i 2 2 ij i j j 2 [(90 − ∇ )* + 9 9 ]A (x) = mD v W (x; v) ; 4 ∗ ¿ GT (!; k)

∗ GT¡ (!; k)

(90 + v · ∇)W (x; v) = v · E(x) ;

(4.86)

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with the initial conditions Ai (t0 ; x) = Ai (x);

i A˙ (t0 ; x) = − Ei (x);

W (t0 ; x; v) = W(x; v) :

(4.87)

To simplify the presentation we shall limit ourselves here to the transverse sector, and consider the transverse projection of the :rst equation (4.86), d. i (920 + k2 )AiT = m2D (4.88) v W (t; k; v) ; 4 T i j

with k · AT = 0 and vTi = (*ij − kˆ kˆ )v j . We choose as initial conditions AiT = 0 and ETi = 0, but let W(x; v) be arbitrary. That is, for the gauge :elds, we choose as initial values the corresponding average values in thermal equilibrium. All the Luctuations are generated by the randomly chosen initial conditions W (t0 ; x; v), that is, by the long wavelength initial Luctuations in the charge density of the hard fermions. These Luctuations will generate time-dependent gauge :elds, via the equations of motion (4.86). Consider then the solution W (x; v) to the Vlasov equation (i.e., the second equation (4.86)) which we write as W = Wind + WL ;

(4.89)

where Wind is the solution to the Vlasov equation with zero initial condition: Wind (t0 ; x; v) = 0, and WL is the Luctuating piece, solution of the homogeneous equation (90 + v · ∇)WL = 0

(4.90)

with the initial condition WL (t0 ; x; v) = W(x; v). It follows that (for x0 ¿ t0 ): Wind (x; v) = − i d 4 y &(y0 − t0 )GR (x; y|v)v · E(y) ; WL (x; v) = W(x − v(x0 − t0 ); v) ;

(4.91)

where GR (x; y|v) is the retarded Green’s function in Eq. (4.2). Eq. (4.89) implies a similar i + Si , with decomposition for the current: j i = jind i (x) = − d 4 y &(y0 − t0 )%Rij (x − y)Aj (y) ; jind i

S

(x) = m2D

d. i v W(x − v(x0 − t0 ); v) ; 4

where %Rij is the (retarded) HTL polarization tensor given in Eq. (1.20). The Maxwell equation (4.88) now becomes ∞ 2 2 i (90 + k )AT + dy0 %T (x0 − y0 ; k)AiT (y0 ) = SiT ; t0

(4.92)

(4.93)

which should be compared with the equation that we had before, namely the :rst equation (4.36): the crucial di8erence is the presence of the Luctuating current Si (x) in the right hand

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side, which is independent of the gauge :elds, and acts as a “noise term” to induce the thermal correlations of the classical electromagnetic :elds. Speci:cally, Eqs. (4.78) and (4.75) imply that the initial conditions W are Gaussian random variables with zero expectation value and the following, local, 2-point correlation: W(x; v)W(y; v ) = (T=m2D )*(3) (x − y)*(v; v ) :

This immediately implies: d. i j (3) i j 2 S (x)S (y) = mD T v v * (x − y − v(x0 − y0 )) ; 4

(4.94)

(4.95)

which is nonlocal; that is, the Luctuating current Si (x) is not a “white noise”. For t0 → −∞, Eq. (4.93) can be easily solved by Fourier transform to yield (recall that AiT = ETi = 0): AiT (k) = ∗MT (k)SiT (k) ;

(4.96)

where ∗MT (k) is the retarded magnetic propagator in the HTL approximation, cf. Eq. (4.38). The gauge :eld correlation induced by the “noise term” Si (x) is :nally obtained as i j

cl

AiT (k)AT (p) = (2)4 *(4) (p + k)(*ij − kˆ kˆ )G˜ T (k) ; j∗

with cl G˜ T (!; k) ≡ m2D T |∗MT (k)|2

(4.97)

d. 2 v 2*(! − v · k) 4 T

= −2(T=!)|∗MT (k)|2 Im %T (!; k) ;

(4.98)

where in writing the second line we have recognized Im %T from Eq. (4.59). This can be rewritten as cl G˜ T (!; k) = (T=!)T (!; k) ;

(4.99)

where T is the o8-shell (or Landau damping) piece of the transverse photon spectral density in the HTL approximation (cf. Eqs. (B.71) and (B.76) in the appendix). Thus, by averaging over the initial conditions W alone, one has generated the Landau damping piece of the magnetic propagator. Similarly, by averaging over the initial :elds AiT and pole ETi , one generates also the pole piece, 7 ∗/T ˙ *(!2 − !T2 (k)) [77]. Altogether, this gives the classical transverse 2-point function in the expected form (cf. Eq. (4.85)): GTcl (!; k) = (T=!)∗/T (!; k) :

(4.100)

An entirely similar result holds for the longitudinal 2-point function as well [77], which completes the veri:cation of the result announced in Eqs. (4.81) – (4.83). One thus sees, on the 7

This identi:cation of the Landau damping spectral density T with the average over W, and of the pole spectral with the average over Ai and Ei , holds only in the limit t0 → −∞ [77]. density ∗/pole T

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example of the 2-point function in QED, that the physics of HTLs is correctly reproduced by the classical theory. We note that in this calculation the intermediate scale did not play any role. In QCD, however, because the equations of motion for the soft modes are nonlinear, the average over the initial conditions generate soft thermal loops which are linearly ultraviolet divergent (as for the scalar theory in Section 2.1.4; recall, especially, Eq. (2.80) there). In this case, the intermediate scale , with gT T , is necessary. As compared with the classical theory in Section 2.2.5, the new complication here is that the cuto8 procedure must be implemented in a way consistent with gauge symmetry. Furthermore, in numerical solutions of the equations of motion using the lattice techniques, an additional complication arises from the fact that the lattice regularization breaks rotational and dilation symmetries. As a consequence, the ultraviolet divergences of the lattice theory cannot be all absorbed into just one parameter, a “lattice Debye mass”. In spite of many e8orts [74,83,77,84], no complete solution to such problems has been found. Nevertheless, the HTL e8ective theory has been already implemented on a lattice [27] (see also Refs. [57,28,157]), and applied to the calculation of the anomalous baryon number violation rate in a high-temperature Yang–Mills theory (cf. Section 1.6). Remarkably, the results obtained in this way, even without any matching, appear to be rather insensitive to lattice artifacts, and are moreover consistent with some previous numerical calculations [57] (where the HTLs are simulated via classical coloured “test particles” [54]), and also with the theoretical predictions in Refs. [78,25]. 5. Hard thermal loops In the previous section we have obtained explicit expressions for the induced sources as functionals of the mean :elds. These functionals may be used to obtain, by successive di8erentiations with respect to the :elds, the e8ective propagators and vertices for the soft :elds. The resulting expressions are the so-called “hard thermal loops” (HTL) [42,19,20,22], i.e., the leading order thermal corrections to the one-particle-irreducible (1P-I) amplitudes with soft external lines. These amplitudes may be used as building blocks to improve perturbative calculations through various resummation schemes. These will be discussed in the last section of this chapter, which contains also digressions on the limitation of weak coupling calculations and how these can be overcome using lattice calculations. 5.1. Irreducible amplitudes from induced sources S with respect to its :eld arguments, By taking the derivatives of the e8ective action 7[A; #; #] we obtain the equations of motion for the mean :elds (all the subsequent formulae hold for time arguments along a complex contour of the type discussed in Section 2): j = −

*7 ; *A

"= −

*7 ; *#S

"S =

*7 ; *#

(5.1)

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where j ; "; "S are external sources. By writing 7 = Scl + 7ind , where Scl is the classical action (A.11), we get *7 a S ja (x) = [D ; F (x)]a − g#(x)8 t #(x) − a ind ; *A (x) *7ind : (5.2) S *#(x) Then, a comparison with Eqs. (3.45) – (3.46) allows us to identify the induced sources as the :rst derivatives of 7ind : *7 *7ind *7ind ; "Sind (x) = − jinda (x) = ind ; "ind (x) = : (5.3) S *#(x) *Aa (x) *#(x) Accordingly, all the n-point 1P-I Green’s functions with n ¿ 2 can be obtained by di8erentiating the induced sources with respect to the mean :elds. Unless otherwise speci:ed, we set the :elds to zero after di8erentiation. That is, we compute the equilibrium 1P-I Green’s functions. For instance, the gluon 1P-I 2-point function (which coincides with the gluon inverse propagator) is obtained as *2 7 ab (D−1 )ab = (D0−1 )ab (5.4) (x; y) = (x; y) + % (x; y) ; *Aa (x)*A b (y) where "(x) = iD = #(x) −

ab 2 −1 (D0−1 )ab (5.5) (x; y) = * (−g 9 + (1 − )9 9 )*C (x − y) is the corresponding free propagator written here in a covariant gauge with gauge :xing term (9 · Aa )2 =2, and *jind a (x) ab % (x; y) = (5.6) *A b (y) is the gluon polarization tensor. For fermions we write similarly *2 7 = S0−1 + ? ; (5.7) S −1 (x; y) = S *#(y)*#(x)

with the free propagator S0−1 (x; y) = − i9=x *(x − y) and the self-energy *"ind (x) ?(x; y) = : (5.8) *#(y) More di8erentiations yield the irreducible (or proper) vertices. For instance, the quark–gluon vertex is *3 7 = g8 t a *C (x − y)*C (y − z) + g7a (x; y; z) ; (5.9) a S *#(z)*#(y)*A (x) whose induced part is obtained either from the induced color current jind , or from the fermionic source "ind , according to *2 jinda (x) *2 "ind (y) : (5.10) = a g7a (x; y; z) = S *A (x)*#(z) *#(z)*#(y)

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Similarly, the proper three-gluon vertex is obtained as *2 jind a (x) abc g7 / (x; y; z) = : *Ab (y)*A/c (z)

(5.11)

The induced piece 7ind of the e8ective action depends in general on the speci:c form of the gauge :xing term G a [A] in the generating functional (3.16), and, within a given class of gauges (i.e. for a given G a [A]), on the gauge parameter . Moreover, as a functional of the S 7ind is generally not invariant under the gauge transformations of classical :elds Aa , # and #, its arguments. However, in the HTL approximation, the induced sources are both independent of the quantum gauge :xing, and also covariant under the gauge transformations of the classical :elds. Besides, the induced current is covariantly conserved, [D ; jind ] = 0, cf. Eqs. (4.14) and (4.21). Together, these conditions guarantee that the HTL e8ective action (to be denoted as 7HTL ) is both gauge-:xing independent, and invariant under the gauge transformations of its :eld arguments. The latter property can be easily veri:ed as follows: Under the in:nitesimal gauge transformation (we omit the fermionic :elds, for simplicity) 1 A → A + *A ; *A = − [D ; &] ; (5.12) g the induced action changes as (cf. Eq. (5.3)) 1 4 ind *7ind = d xj a *Aa = d 4 x[D ; j ind ]a &a ; (5.13) g C C where the second equality follows after an integration by parts (the surface term has been assumed to vanish). Clearly, the gauge invariance of 7ind requires that j ind is covariantly conserved, a condition satis:ed indeed at the HTL level, i.e. for 7ind = 7HTL . 5.2. The HTL e;ective action S it Since the induced sources are known explicitly in terms of the classical :elds Aa , # and #, is tempting to try and use Eq. (5.3) to construct also the e8ective action in explicit form. Note, however, that the induced sources are known only for real-time arguments, and for retarded (or advanced) boundary conditions (cf. Section 4.1). Thus, the contour action cannot be derived by simply integrating Eqs. (5.3) with the induced sources in Section 4.1. Still, if we temporarily ignore the boundary conditions, it is possible to write down a functional which generates these induced sources, and summarizes in a compact form many of the remarkable features of the HTLs. Consider QED :rst, and let us construct the e8ective action which generates the electromag netic induced current jind = % A . Since this is linear in A , the :rst equation (5.3) can be trivially integrated to give (with 7A denoting the purely photonic piece of 7HTL ): 1 d4 p 7A = A (−p)% (p)A (p) 2 (2)4 v v d. d4 p 1 2 = mD F ( − p) F (p) ; (5.14) 4 4 (2)4 (v · p)2

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where the second line follows by using Eq. (1.20) for % and some simple algebraic manipulations. Expression (5.14) is well de:ned only for :elds F which carry time-like momenta, |!| ¿ p, for which the denominator (v · p)2 is nonvanishing. As discussed in Section 4.3.3, these are the :elds which propagate without dissipation. The polarization tensor obtained by di8erentiating 7ind twice (cf. Eq. (5.6)) comes out necessarily symmetric: % (x − y) =

*2 7ind = % (y − x) ; *A (x)*A (y)

(5.15)

or equivalently: % (p) = % (−p). This symmetry property is satis:ed by the contour C (x; y), but it is inconsistent with the retarded prescription (one rather has self-energy % %R (y − x) = %A (x − y)). However, under the conditions for which 7A is well de:ned, the boundary conditions play no role. Eq. (5.14) admits a straightforward generalization to QCD. Speci:cally, the induced colour current in Eq. (4.12) can be formally rewritten as v d. Aa 2 j (x) = mD d 4 y x; a (5.16) y; b v · Eb (y) : 4 v·D This can be generated, via Eq. (5.3), by the following “action” (see Ref. [23] for an explicit proof): v v 1 2 d. 4 4 7A ≡ mD d x d y Tr F (x) x y F (y) : (5.17) 2 4 −(v · D)2 Formally, this functional is obtained from the Abelian action (5.14) by simply replacing the ordinary derivative (v · 9)2 with the covariant one (v · D)2 [135]. For time-independent :elds Aa (x), Eq. (5.17) reduces to a (screening) mass term for the electrostatic potentials (cf. Eq. (4.50)): 7Astatic = 12 m2D Aa0 (x)Aa0 (x) :

(5.18)

Within the imaginary time formalism, this provides an e8ective three-dimensional action for soft (k ∼ gT ) and static (!n = 0) Matsubara modes: 1 ij a 1 i a 2 1 2 a a 7static = d 3 x (5.19) Fa Fij + (D A0 ) + mD A0 A0 ; 4 2 2 This coincides, as expected, with the leading-order result of the dimensional reduction (cf. Section 2.1.4) in QCD (see Refs. [105,104], and references therein). Let us :nally add the fermionic :elds. The HTL e8ective action is written as 7HTL = 7A +7 , with 7 satisfying S *7 =*#(x) = "ind (x);

*7 =*Aa (x) = j a (x) :

After rewriting "ind in Eq. (4.18) as follows: v= d. ind 2 4 y #(y) ; " (x) = !0 d y x 4 i(v · D)

(5.20)

(5.21)

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it becomes clear that the :rst equation (5.20) is satis:ed by v= d. 2 4 4 S y #(y) : 7 = !0 d x d y#(x) x 4 i(v · D)

(5.22)

It is then simply to verify that the above 7 provides also the correct current j (x) of Eq. (4.20). The above construction of the HTL e8ective action from kinetic theory follows closely Refs. [23]. Originally, this action has been derived by Taylor and Wong [22] (although in a form di8erent from Eq. (5.17)), by exploiting the properties of the hard thermal loops, in particular, their gauge symmetry (cf. Section 5.3. below). The manifestly gauge invariant action in Eq. (5.17) has been :rst presented in Refs. [135,136]. (See also [158].) 5.3. Hard thermal loops By di8erentiating the expressions for the induced sources obtained in Section 4.1, it is straightforward to construct the 1P-I amplitudes of the soft :elds. Given the boundary conditions that we have chosen in solving the kinetic equations, this procedure naturally generates the corresponding retarded amplitudes. 5.3.1. Amplitudes with one pair of external fermion lines From Eq. (4.18), the soft quark self-energy in a background gauge :eld A is obtained as *"ind (x) d. ?(x; y) = (5.23) = − !02 v= GR (x; y; v) : *#(y) 4 For A = 0, this reduces to d. v= 2 ?(p) = !0 ; 4 v · p + i"

(5.24)

where the small imaginary part i" implements the retarded conditions. The angular integral in Eq. (5.24) is performed in Appendix B. Since "ind is linear in #, there is no polarization amplitude with more than one pair of soft external fermions. On the other hand, Eq. (4.18) is nonlinear in the gauge mean :elds (through the parallel transporter), and it generates an in:nite series of vertex functions between a quark pair and any number of soft gluons (or photons). To be speci:c, we de:ne the correction to the amplitude between a quark pair and n soft gluons by *n n gn 7a11:::a ?(y1 ; y2 ) ; (5.25) n :::n (x1 ; : : : ; x n ; y1 ; y2 ) = *Aan (x n ) : : : *Aa11 (x1 ) with ?(y1 ; y2 ) given by Eq. (5.23). In doing these di8erentiations, we use the formula *GR (x1 ; x2 ; v) = − gv GR (x1 ; y; v)t a GR (y; x2 ; v) ; (5.26) *Aa (y) which follows from Eqs. (4.3) and (4.5). The normalization we choose for the amplitudes (5.25) is such that 7(n) depends on g only through !02 . In all the amplitudes (5.25), y10 is the largest

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time, while y20 is the smallest one. The relative chronological ordering of the n gluon lines is arbitrary, and, in fact, the amplitudes are totally symmetric under their permutations. Up to minor changes due to the color algebra, all the amplitudes obtained in this way coincide with the corresponding abelian amplitudes [18]. We give now the explicit expressions for the amplitudes involving a quark pair and one or two gluons. After one di8erentiation with respect to A , Eq. (5.23) yields the quark–gluon vertex correction: d. a 2 7 (x; y1 ; y2 ) = !0 8 (5.27) v v GR (y1 ; x; v)t a GR (x; y2 ; v) : 4 The time arguments above satisfy y10 ¿ x0 ¿ y20 . For A = 0, we de:ne the Fourier transform of 7a by (2)4 *(4) (p + k1 + k2 )7a (p; k1 ; k2 ) ≡ d 4 x d 4 y1 d 4 y2 exp{i(p · x + k1 · y1 + k2 · y2 )}7a (x; y1 ; y2 ) ; and we get 7a (p; k1 ; k2 ) =

−t

a

!02 8

v v d. ≡ t a 7 (p; k1 ; k2 ) : 4 (v · k1 + i")(v · k2 − i")

(5.28)

(5.29)

Since all the external momenta are of the order gT , g7 ∼ g!02 =k 2 ∼ g is of the same order as the bare vertex g8 . Thus, the complete quark–gluon vertex at leading order in g is gt a∗7 , where ∗7 ≡ 8 + 7 . Consider now the vertex between a quark pair and two gluons. This vertex does not exist at tree level, and in leading order it arises entirely from the hard thermal loop. We have: v v v/ d. ab 7 (p1 ; p2 ; k1 ; k2 ) = −!02 8/ 4 (v · k1 + i")(v · k2 − i") tatb tbta : (5.30) × + v · (k1 + p1 ) + i" v · (k1 + p2 ) + i" Alternatively, we can derive the amplitudes (5.25) from expression (4.20) for the induced current j . The resulting amplitudes will obey di8erent boundary conditions since the time argument of j (x) is now the largest one. It is convenient to rewrite Eq. (4.20) as d. a 2 j (x) = gt !0 v d 4 y1 d 4 y2 4 S 1 )v=GA (y1 ; x; v)t a GR (x; y2 ; v)#(y2 ) ; #(y

(5.31)

where de:nitions (4.3) and (4.4) have been used. Then, the correction to the quark–gluon vertex is (recall the :rst equality in Eq. (5.10)) d. a 2 (5.32) 7 (x; y1 ; y2 ) = !0 8 v v GA (y1 ; x; v)t a GR (x; y2 ; v) ; 4

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where now the time arguments satisfy x0 ¿ max(y10 ; y20 ), the chronological order of y1 and y2 being arbitrary (compare, in this respect, with Eq. (5.27) above). For A = 0, we have v v d. a a 2 7 (p; k1 ; k2 ) = − t !0 8 ; (5.33) 4 (v · k1 − i")(v · k2 − i") which di8ers from (5.29) solely by the i"’s in the denominators reLecting the respective boundary conditions. If we further di8erentiate Eq. (5.32) with respect to A , we generate amplitudes of the type (5.25), in which x10 is the largest time. 5.3.2. Amplitudes with only gluonic external lines The amplitudes involving only soft gluons may be derived from the induced current jA given in Eqs. (4.16) or (4.33). A :rst di8erentiation in Eq. (4.33) yields (cf. (5.6)) d. v v ab % (p) = m2D *ab −*0 *0 + p0 : (5.34) 4 v · p + i" This coincides with the electromagnetic polarization tensor (1.20) derived in Section 1.3. A second di8erentiation of Eq. (4.33) with respect to A yields the three-gluon vertex (recall Eq. (5.11)). With the Fourier transform de:ned as in Eq. (5.28), we obtain p30 p20 d. v v v/ abc abc 2 7 / (p1 ; p2 ; p3 ) = if mD − ; (5.35) 4 v · p1 + i" v · p3 − i" v · p2 − i" where the imaginary parts in the denominators correspond to the time orderings x10 ¿ x20 ¿ x30 for the :rst term inside the parentheses, and x10 ¿ x30 ¿ x20 for the second term. We can rewrite abc ≡ if abc 7 this more symmetrically as 7 / / with p10 − p20 m2 d. 7 / (p1 ; p2 ; p3 ) = D v v v/ 3 4 (v · p1 + i")(v · p2 − i") p20 − p30 p30 − p10 + : (5.36) + (v · p2 − i")(v · p3 − i") (v · p3 − i")(v · p1 + i") This vanishes for zero external frequencies (static external gluons), in agreement with Eq. (5.18). For pi ∼ gT , g7 / ∼ g2 T ∼ gpi is of the same order as the corresponding tree-level vertex. 5.3.3. Properties of the HTLs Originally, the “hard thermal loops” have been identi:ed in one-loop diagrams in thermal equilibrium. The self-energy corrections (5.24) and (5.34) have been obtained :rst by Klimov [39] and by Weldon [40,41]. These early works have been put in a new perspective by Braaten and Pisarski [42,19,133], and by Frenkel et al. [20,22], who recognized that, in hot gauge theories, both the propagators and the vertex functions receive thermal contributions of order T 2 in the limit of high temperature and soft external momenta. It is worth emphasizing that a HTL is just a part of the corresponding one-loop correction, namely that part which arises by integration over hard loop momenta (this is the origin of the name “hard thermal loop”), and after performing kinematical approximations allowed by the smallness of the external momenta pi . gT with respect to the hard loop momentum k ∼ T .

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In fact, if we consider that all the pi ’s are precisely of the order gT , then the HTL is the leading order piece in the expansion in powers of g which takes into account the assumed g-dependence of the external four-momenta [19]. Alternatively, since all the hard thermal loops are proportional to T 2 , they can be also obtained as the leading order terms in the high-temperature expansion of the one-loop amplitudes [20,39 – 41]. When compared to the general one-loop corrections in vacuum, the hard thermal loops have remarkably simple features which we examine now. (i) Independence with respect to gauge 4xing. The hard thermal loops are gauge-:xing independent for arbitrary values of their external momenta. For the self-energies, this has been noticed already in Refs. [39 – 41]. For higher-point vertex functions, it has been veri:ed either by explicit calculations in various gauges [20,19], or by induction [19]. An explicit proof for all the HTLs has been given within kinetic theory, in Ref. [23]. As emphasized in Section 3, the gauge-:xing independence reLects the fact that only the physical, on-shell, excitations of the ideal quark–gluon plasma contribute to the collective motions to the order of interest. (ii)Ward identities. The hard thermal loops are connected by simple Ward identities, similar to those satis:ed by the tree-level propagators and vertices, or by the QED amplitudes. The :rst such identities can be easily read o8 the equations in the previous subsection. For instance, Eqs. (5.24), (5.29) and (5.30) imply p 7 (p; k1 ; k2 ) = ?(k1 ) − ?(k1 + p) ab (p1 ; p2 ; k1 ; k2 ) = ifabc 7 c (p1 + p2 ; k1 ; k2 ) p1 7

+7 b (p2 ; k1 ; k2 + p1 )t a − t a 7 b (p2 ; k1 + p1 ; k2 ) ;

(5.37)

while Eqs. (5.34) and (5.36) yield p % (p) = 0 ; p1 7 / (p1 ; p2 ; p3 ) = % / (p3 ) − % / (p2 ) :

(5.38)

These identities follow directly from the conservation laws for the induced colour current, and ultimately express the fact that the HTL e8ective action is invariant under the gauge transformations of its :eld arguments. For instance, by successively di8erentiating Eq. (4.14) for jA with respect to A one obtains Ward identities relating HTLs with gluonic external lines, like those in Eq. (5.38). Similarly, identities like those in Eq. (5.37) can be obtained by di8erentiating Eq. (4.21) for j . (iii) Nonlocal structure. The speci:c nonlocality of the HTLs, in 1=(v · p) (where v is the velocity of the hard particle around the loop, and p a linear combination of the external momenta) :nds its origin in the (covariant) drift term in the kinetic equations, and reLects the eikonal propagation of the hard particles in the soft background :elds. In particular, the limit v · p → 0 may lead to singularities in the HTLs. We distinguish two types of such singularities: (a) infrared divergences when the external momenta tend to zero (for instance, note the singular behaviour of the transverse gluon self-energy %T (!; p) in Eq. (4.60) as !; p → 0 with !p); (b) collinear divergences for light-like (P 2 ≡ !2 − p2 = 0) external momenta, in which case the angular integration over v (like, e.g., in Eqs. (5.24), (5.34) or (5.36)) leads to logarithmic

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Fig. 14. One-loop corrections to the quark–gluon vertex, with the external gluon attached to the internal gluon line (a), respectively to the internal fermion line (b).

singularities (note the logarithmic branching points at ! = ± p in the gluon self-energies in Eqs. (B.63) – (B.64), or in the fermion self-energies in Eqs. (B.97) – (B.98)). (iv) Cancellations. In the diagrammatic calculation of the HTLs, one has observed some “accidental” compensations with interesting consequences: • For instance, in QED, the only photon HTL is the polarization tensor: while HTL-like contributions show up also in individual diagrams with more external photons (n ¿ 3), these

contributions appear to cancel each other when all the diagrams contributing to a given vertex function in the HTL approximation are added together [135]. • Also, in QCD, interesting cancellations occur when computing HTLs with quark and gluon external lines [22]. The simplest example is provided by the quark–gluon vertex 7a . To one loop order, the vertex correction is obtained from the two diagrams in Fig. 14. Both these diagrams contain hard thermal loops, namely, 2 2 v v= d. a ag T N 7(a) (p; k1 ; k2 ) = t ; 8 2 4 (v · k1 )(v · k2 ) 2 2 v v= N d. a ag T Cf + : (5.39) 7(b) (p; k1 ; k2 ) = − t 8 2 4 (v · k1 )(v · k2 ) When combining the two contributions, the terms proportional to N=2 cancel, so that we are left only with the term proportional to Cf , which originates from Fig. 14b. Similar cancellations occur also for the diagrams with more external gluons, so that the corresponding hard thermal loops are all proportional to Cf [22] (as one can verify from Eqs. (5.29) or (5.30)). If, at a :rst glance, these cancellations may seem accidental, note however that they are essential to ful:l the Ward identities (5.37). All these compensations :nd a clear interpretation at the level of the kinetic equations. They reLect the fact that the only nonlinear e8ects which persist in the present approximations are those required by gauge symmetry: • In QED, the electromagnetic current jA is linear in the gauge :elds, and also gauge-invariant

(cf. Section 3.1); thus, there is no room for purely multi-photon HTL vertices.

• In QCD, the cancellation of HTL-like contributions associated with soft gluon insertions in

diagrams with fermionic legs corresponds to the disappearance of the adjoint background

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˜ ) in going from Eq. (3.122) for Kai (k; X ) to Eq. (3.123) for K :eld k · A(X = (k; X ), and also to the compensation of the terms proportional to N between the two components (fermionic and gluonic) of the induced current j (cf. Eqs. (3.130) –(3.132)). Such compensations are imposed by gauge symmetry; they ensure, e.g., that Kai (k; X ) transforms as a fundamental colour vector (i.e., like #(X )) under a gauge rotation of the background :elds. (v) Nonperturbative character. For external momenta of order gT , the hard thermal loops are of the same order in g as the corresponding tree-level amplitudes, whenever the latter exist. That is, the e8ects induced by the collective motion at the scale gT are leading order e8ects, and not just perturbative corrections. This observation is the basis of the resummation programme proposed by Braaten and Pisarski [42,19], to be discussed in the next subsection. 5.4. HTL resummation and beyond In high-temperature gauge theories, the naX\ve perturbation theory breaks down at the soft scale gT , because of the large collective e8ects. This crucial observation, due to Pisarski [159,42] led subsequently Braaten and Pisarski [42,19,133] to propose a reorganization of the perturbative expansion where the hard thermal loops are included at the tree level. In some respects, the resummation of hard thermal loops can be seen as a generalization of the resummation of ring diagrams in the computation of the correlation energy for a high-density electron gas, by Gell-Mann and Brueckner [160]. Many other examples can be found in the literature, both in nonrelativistic many-body physics [161,45 – 47], and in relativistic plasmas [162–172]. 5.4.1. The Braaten–Pisarski resummation scheme At a formal level, the resummed theory is de:ned by the e8ective action 7e8 = Scl + 7HTL where Scl is the classical action for QCD, Eq. (A.11), and 7HTL is the generating functional of HTLs described in Sections 5.1 and 5.2 (cf. Eqs. (5.17) and (5.22)). In practice, this means that the Feynman rules to be used for the soft :elds are de:ned so as to include the HTL self-energies and vertices. On the other hand, the bare Feynman rules are to be applied for the hard :elds [42,19]. Indeed, the leading corrections to the self-energy of a hard :eld are O(g2 ), while the corrections to a vertex in which any leg is hard are, at most, O(g) [19]; these are truly perturbative corrections, and do not call for resummation. Thus, when computing a Feynman graph, one is instructed to use the bare (thermal) propagators for all the internal lines which carry hard momenta, and the bare vertices for all the interaction vertices which involve, at least, one pair of hard :elds. But for the soft internal lines, and the vertices with only soft external legs, one must use e;ective propagators and vertices. The e8ective quark and gluon propagators ∗ S and ∗ G are obtained by inverting ∗

−1 −1 G (p) = G0 (p) + % (p);

∗ −1

S

(p) = S0−1 (p) + ?(p) ;

(5.40)

and are given explicitly in Appendix B (cf. Sections B:1:3 and B:2:3). Consider now the e8ective vertices connecting (soft) quarks and gluons. The three-particle vertex reads: ∗

7a (p; k1 ; k2 ) = t a 8 + 7a (p; k1 ; k2 ) ≡ t a∗7 (p; k1 ; k2 ) ;

(5.41)

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with 7a given by Eq. (5.29). It satis:es the following Ward identity: p∗ 7 (p; k1 ; k2 ) =∗ S −1 (k1 ) −∗ S −1 (k1 + p) ;

(5.42)

which follows from Eqs. (5.37) and (5.40). At tree-level, 70 ≡ 8 is the only quark–gluon vertex. In the e8ective theory, on the other hand, we have an in:nite series of new vertices, which are related through Ward identities, and which connect a quark-antiquark pair to any number of gluons. For instance, the corresponding four-particle (2 quarks–2 gluons) vertex ab (p ; p ; k ; k ) = 7ab (p ; p ; k ; k ) (cf. Eq. (5.30)), and is related to the three-particle reads ∗ 7 1 2 1 2 1 2 1 2 vertex in Eq. (5.41) by the second Ward identity (5.37). A similar discussion applies to the e8ective vertices with only gluon legs. When performing perturbative calculations, one has to :x the gauge. This only a8ects the form of the gluon propagator (recall that the HTLs are gauge-:xing independent). Also, in gauges with ghosts, one must use bare Feynman rules for the ghost propagator and vertices. Indeed, it can be veri:ed that there are no HTL corrections for the amplitudes with ghost external lines [19,20,23], a property which reLects the gauge-invariance of the HTL e8ective action. Consider now the systematics of the resummed perturbation theory. Since the HTLs are now included at the tree-level of the e8ective theory, one must be careful to avoid overcounting. A standard procedure consists in adding and subtracting the action 7HTL to the bare action Scl , by writing Scl ≡ (Scl + 7HTL ) − 7HTL = 7e8 + *S :

(5.43)

In the e8ective expansion, the tree-level amplitudes are generated by 7e8 ≡ Scl + 7HTL , while the reminder *S ≡ −7HTL is treated perturbatively as a counterterm (i.e., a quantity which is formally of one-loop order) to ensure that the HTLs are not double counted. In practice, this requires a systematic separation of soft and hard momenta. As an example, consider the :rst correction to the soft gluon self-energy beyond the HTL of Eq. (5.34). By power counting, this is of order g3 T 2 , and involves three types of contributions [19]: (a) one-loop diagrams with soft loop momentum (Fig. 15); (b) one-loop diagrams with hard internal momentum and with the HTL subtracted; (c) two-loop diagrams with only hard internal momenta. In case (a), all the momenta (internal and external) are soft, so one has to use e8ective propagators and vertices (cf. Fig. 15). In case (b), the subtraction of the HTL is ensured by the corresponding counterterm in −7HTL . This calculation has been done in Refs. [173,174] where

Fig. 15. E8ective one-loop diagrams contributing to the soft gluon self-energy to next-to-leading order. All the lines in these diagrams are soft, so all the propagators and vertices are de:ned to include the corresponding HTLs.

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in particular the next-to-leading order correction to the plasma frequency in QCD has been √ 2 = "g N !2 , where !2 = g2 NT 2 =9 is the leading order result (we consider here a obtained: *!pl pl pl purely Yang–Mills plasma), and the coeKcient " ≈ −0:18 is found to be gauge-:xing independent, as expected from general arguments [175]. (See also Ref. [121] for a similar calculation in the scalar theory with quartic self-interactions.) 5.4.2. Some applications of HTL-resummed perturbation theory We shall brieLy mention here some other applications of the HTL-resummed perturbation theory. More can be found in the textbook by LeBellac [14], and also in the review paper by Thoma [176]. The simplest version of the HTL resummation applies to the calculation of static quantities (like the thermodynamical functions, or the time-independent correlations) in the imaginary-time formalism. Then, only the internal lines with zero Matsubara frequency (!n = 0) can be soft, and require resummation [177,178]. Since in the static limit the HTLs collapse to the Debye mass term (cf. Eq. (5.18)), the resummation then reduces to including m2D in the electric propagator. By using this technique, the free energy has been computed up to order g5 for massless scalar ;4 theory [179,180], Abelian gauge theories [181,182], and QCD [178,183]. These results have been reobtained by using the dimensionally reduced e8ective theory in Refs. [184,185]. The whole machinery of the HTL resummation comes into play when considering dynamical quantities, like time-dependent correlators. The most celebrated example in this respect is the calculation of the quasiparticle damping rate 8 (see Section 6 below). The original attempts to compute 8 have met with various conceptual problems which have been a major stimulus for several interesting progress in hot gauge theories. It was this problem, dubbed for some time the “plasmon puzzle”, which triggered the discovery of the hard thermal loops, and the study of their remarkable properties. An historical account of this subject, together with references to previous work, can be found in Ref. [62]. The HTL-resummed calculation of 8 for excitations with zero momentum (gluons or fermions) is presented in [19,61,186,187]. Quite generally, one expects the predictions of the e8ective theory to be di8erent from those of the bare theory for all the quantities which are sensitive to soft momenta. In particular, most of the logarithmic infrared divergences of the bare expansion are eliminated by the resummation of the HTLs. An important example in this sense, which is also the earliest one to have shown the role of dynamical screening in removing IR divergences, is the calculation of the viscosity in hot QCD, by Baym et al. [63]. This example is quite generic for the transport phenomena based on momentum-relaxation processes (other examples are the charge and quark di8usivities) [188,189,70]. By contrast, the transport coeKcients for colour remain IR sensitive even after the inclusion of the HTLs [55,65,25], as it will be explained in Section 7 below. Here are some more examples of applications of the HTL perturbation theory to the calculation of dynamical quantities: the calculation of the collisional energy-loss of charged or colored partons [190 –193], the Primako8 production of axions from a QED plasma [194 –196], and the photon production by a quark–gluon plasma, for both hard [197–199], or soft [200,201] photons. In the particular case of the photon production rate, it is the Landau damping of a soft fermion which provides infrared :niteness [197,198] (in the bare perturbation theory, there is a logarithmic divergence associated with the exchange of a massless quark). Another example

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where the resummation enters in a decisive way is the calculation of the production rate of soft dileptons in a hot quark–gluon plasma [202]. Note, however, that collinear divergences identi:ed in higher orders [200,204] raise doubts about the consistency of the original calculations in [202,197,198]. In spite of signi:cant recent work and progress [200,203–206], the complete calculation of the production rates for photons and dileptons in the quark–gluon plasma remains an open problem. 5.4.3. Other resummations and lattice calculations If the resolution of the “plasmon puzzle” was one of the :rst, and most remarkable, successes of the HTL perturbation theory, it is still in relation with the damping rate calculation that the limits of the HTL resummation :rst emerged. Simultaneously with the :rst successful calculation of the damping rates for excitations with zero momentum, it was found that for :nite-momentum excitations, infrared divergences remain even after including the HTLs [42,131], [207–215]. As we shall see later, in Section 6, this diKculty is related to the fact that the HTLs do not play any role in the static magnetic sector. The nonperturbative contributions of the magnetic Luctuations, which occur at O(g6 ) in the free energy (cf. Section 1.1), occur already at O(g4 ) in the static magnetic self-energy %T (0; p). Various theoretical arguments [86,216 –220] predict that (static) magnetic screening should be nonperturbatively generated at the scale g2 T , and this is indeed con:rmed by lattice calculations [221–223]. For practical purposes, this may be represented as a simple magnetic mass %T (0; p) ≈ m2mag ∼ (g2 T )2 , although the precise nature of the screening mechanism is not yet fully understood. By contrast, in Abelian gauge theories it can be proven that, to all orders in the coupling constant, there is no static magnetic screening [67]. Similarly, the next-to-leading order contribution to the Debye mass in QCD, of O(g3 ), is logarithmically IR divergent, but the coeKcient in front of the logarithm can be computed perturbatively [64], from the one-loop e8ective diagram in Fig. 16. This yields the positive correction *m2D 2PNTmD ln(1=g), where P = g2 =4, mD is the LO Debye mass, Eq. (4.13), and the logarithm has been generated as ln(mD =mmag ) ln(1=g) to logarithmic accuracy (see also Ref. [224]). In fact, as shown in Ref. [66], a similar problem occurs in the Abelian context

Fig. 16. E8ective one-loop contribution, of O(g3 ), to the electric polarization function %L (0; p) in QCD. All the lines in this diagram are static. The external lines, as well as the internal line marked by a blob, are electric and dressed by the Debye mass. The other internal one is magnetic and therefore massless.

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of scalar QED, where no magnetic mass is expected. There, the IR divergence of perturbation theory has been cured via an all-order resummation of soft photon e8ects in the vicinity of the mass-shell [66]. A systematic framework for the nonperturbative calculation of the thermodynamical quantities and of the static correlations is provided by :nite-temperature lattice QCD [9]. By using four-dimensional lattice simulations, the QCD pressure has been computed for a pure SU(3) Yang–Mills theory in Refs. [225,226], and, more recently, also for QCD with two and three light quarks [227]. The electric and magnetic screening masses have been similarly computed in Refs. [228,223]. Since lattice calculations are easier to perform in lower dimensions, their eKciency can be increased by using dimensional reduction. The corresponding e8ective theory for QCD (or the electroweak theory) is a three-dimensional SU(N ) gauge theory coupled to a massive adjoint “Higgs” (the electrostatic :eld A0a of the original theory in D = 4) with all the interactions permitted by the symmetries in the problem [105,104,184]. The parameters in this e8ective theory (the mass of the scalar :eld and the strengths of the various e8ective interactions) can be obtained by matching the soft correlation functions calculated in the original theory and the e8ective theory, to the order of interest [104,229,184,106]. To lowest order, this yields the e8ective theory in Eq. (5.19). The combination of dimensional reduction and three-dimensional lattice calculations has allowed for systematic studies of the phase transition in the electroweak theory [105,106,230,5,231] and its minimal supersymmetric extension [232], and of the static long-range correlations in high-temperature QCD [221–223,33,34]. In QCD, estimates have been obtained in this way for the nonperturbative O(g6 )-contribution to the free energy [221], for the magnetic screening mass [222,223], and for the Debye mass [33,34]. (See also [233] for a nonperturbative de:nition of the Debye mass, which has been used for the numerical calculations in [34].) Whenever lattice calculations in both D = 3 (with dimensional reduction) and D = 4 are available, the results agree reasonably well (see Refs. [222,234,223] for explicit comparisons). The Debye mass found on the lattice is very well :tted by the following formula [34]: mD lattice mD = mD + PNT ln 2 + 7:0 + O(g3 T ) : (5.44) gT It di8ers signi:cantly from the lowest order perturbative prediction (the HTL value mD ) up to temperatures as high as T ∼ 107 Tc . This shows that the Debye mass, as well as other long-range correlators, receive at most temperatures of interest important nonperturbative contributions from the longwavelength Luctuations in the plasma. Returning to the thermodynamical functions which are dominated by the hard degrees of freedom, one may expect such nonperturbative contributions to be quantitatively small. And indeed the lattice data [225 –227] show a (slow) approach of the ideal-gas result from below with deviations of not more than some 10 –15% for temperatures a few times the decon:nement temperature. Recent lattice calculations using dimensional reduction provide further evidence that the total, nonperturbative, contribution of the soft modes to the free energy is rather small [35]. This being said, it is worth reminding that naX\ve perturbation theory is inadequate to describe the thermodynamics of the quark–gluon plasma. Already the next-to-leading order perturbative correction, the so-called plasmon e8ect which is of order g3 [169], signals the inadequacy of

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the conventional thermal perturbation theory because, in contrast to the leading-order correction of O(g2 ), it leads to a free energy in excess of the ideal-gas value. In fact, for the O(g3 ) e8ect to be less important that the O(g2 ) negative correction, the QCD coupling constant must be as low as P . 0:05, which would correspond to temperatures as high as & 105 Tc . This suggests a further reorganization of perturbation theory where more information on the plasma quasiparticles is included already at tree-level, and this is the place where the HTL come back into the game. A possible strategy is the so-called “screened perturbation theory” [235] where the HTL-resummed Lagrangian 7e8 in Eq. (5.43) is now used at all momenta, soft and hard. The eKciency of this method in improving the convergence of perturbation theory has been demonstrated in the context of scalar :eld theories, via calculations up to two-loop [235,236] and three-loop [237] order. Recently, this scheme has been extended to QCD [30], where, however, only one-loop calculations have been presented so far. A problem with this approach is that, at any :nite loop order, the UV structure of the theory is modi:ed: new (eventually temperature-dependent) divergences occur and must be subtracted, thus introducing a new source of renormalization scheme dependence [237,238]. In gauge theories, this is further complicated by the nonlocality of the HTLs [30]. An alternative approach has been worked out in Refs. [31,32] and uses the HTLs only in the kinematical regimes where they are accurate. This approach is based on a self-consistent (“E-derivable” [239]) two-loop approximation to the thermodynamic potential, but focuses on the entropy which has the simple form (given here for a scalar :eld): d 4 k 9n S=− {Im ln G −1 − Im % Re G } : (5.45) (2)4 9T This e8ectively one-loop expression is correct up to terms of loop-order 3 (i.e., of O(g4 ) or higher) provided G and % are the self-consistent one-loop propagator and self-energy [239]. Thus, any explicit two-loop interaction contribution to the entropy has been absorbed into the spectral properties of quasiparticles. Remarkably, this holds equally true for fermionic [240] and gluonic [31,32] interactions. Expression (5.45) is manifestly UV :nite, the statistical factors providing an ultraviolet cut-o8. Based on formula (5.45), approximately self-consistent calculations have been proposed [31,32] where the self-energy % is determined in HTL-resummed perturbation theory (with manifestly gauge-invariant results), but the entropy is evaluated exactly, by numerically integrating Eq. (5.45) with this approximate self-energy. The results compare very well with the lattice data for all temperatures above T 2:5Tc (with Tc the critical temperature for the decon:nement phase transition). This method has been applied successfully also to plasmas with non vanishing baryonic density (i.e., with a nonzero chemical potential) [31,32], for which lattice calculations are not yet available. 6. The lifetime of the quasiparticles Because of their interactions with the particles in the thermal bath, all the excitations of a plasma have :nite lifetimes. In the weak coupling regime, one expects these lifetimes to be long. This was indeed veri:ed in Section 2.3.4 for a scalar theory with a quartic interaction.

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Fig. 17. Fermion–fermion elastic scattering in the resummed Born approximation. As usual the blob on the photon propagator represents the screening correction, in the hard thermal loop approximation.

However, in gauge theories, the perturbative calculation of the lifetimes is plagued with infrared divergences, both in Abelian and nonAbelian plasmas [42,131] [207–215]. Although screening corrections contribute to cure much of the problem, these are not enough. In this section we shall identify the physical origin of the problem and show how it can be solved, at least in the case of QED, by an all order resummation of soft photon e8ects. This resummation is reminiscent of the Bloch–Nordsieck calculation at zero temperature [241,242], and calls upon kinematical approximations which have been met several times along this review. The calculation that we shall present is also interesting from the point of view of kinetic theory, as it provides an example where coherence e8ects between successive scatterings need to be taken into account, thus preventing a simple description via a Boltzmann equation. 6.1. The fermion damping rate in the Born approximation In this subsection, we compute the damping rate of a hard electron (p ∼ T ) in a hot QED plasma, to leading order in e. The damping is caused by collisions involving a photon exchange with the electrons of the heat bath. The relevant Feynman graph is depicted in Fig. 17. The collision rate is obtained by integrating the corresponding matrix element squared |M|2 over the thermal phase space for the scattering partners. At the order of interest, we can treat the (hard) external fermion lines as free massless Dirac particles. On the other hand, since, as we shall verify later, the scattering rate is dominated by soft momentum transfers, q . eT , it is essential to include the screening corrections on the photon line. We are thus led to the following expression for the damping rate: 1 8p = d p˜ 1 d p˜ 2 d p˜ 3 (2)4 *(4) (p + p1 − p2 − p3 ) 4j ×{n1 (1 − n2 )(1 − n3 ) + (1 − n1 )n2 n3 }|M|2 ; |M|2

(6.1) ∗ G (q)

is computed with the e8ective photon propagator where the matrix element squared given by Eq. (4.37). The other notations in Eq. (6.1) are as follows: all the external particles are on their mass-shell (i.e., j = p and ji = pi for i = 1; 2; 3), and d p˜ i ≡ d 3 pi =((2)3 2ji ). The statistical factors

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ni ≡ n(ji ) take care of the Pauli principle for the two processes (direct and inverse) associated with the diagram of Fig. 17. Note that, for fermions, the rates of these two processes have to be added together to give the depopulation of the state with momentum p [88]. Except for this change of sign, expression (6.1) of the damping rate has the same structure as that for a scalar particle obtained from Eqs. (2.184) and (2.186). In the regime where q . eT pi , the matrix element simpli:es to [97] ˆ · (v1 × q) ˆ ∗MT (q0 ; q)|2 ; |M|2 64e4 p2 p12 |∗ML (q0 ; q) + (v × q)

(6.2) ˆ v1 ≡ pˆ1 , and where v ≡ p, are, respectively, the electric (l) and the magnetic (t) photon propagators in the HTL approximation, as de:ned in Eq. (4.38). By using Eq. (6.2), and performing some of the momentum integrals in Eq. (6.1), we can rewrite 8p as a double integral over the energy q0 and the magnitude q = |q| of the momentum of the virtual photon: q 2 2 q 1 e4 T 3 ∞ dq0 ∗ 8

| ML (q0 ; q)|2 + dq 1 − 02 |∗MT (q0 ; q)|2 : (6.3) 12 2 q −q 2 ∗M L; T (q0 ; q)

Note that, as a result of our kinematical approximations, the damping rate has become independent of p. The integration limits on q0 , namely |q0 | 6 q, arise from the kinematics: the exchanged photon is necessarily space-like. Finally, the momentum integral is infrared divergent, which is why we have introduced the lower cuto8 . If we were to use a bare photon propagator in (6.3), i.e. |ML (q0 ; q)|2 = 1=q4 and |MT (q0 ; q)|2 = 1=(q02 − q2 )2 , one would :nd that the resulting q-integral is quadratically divergent: e4 T 3 ∞ dq e4 T 3 8

˙ : (6.4) 8 q3 2 This divergence is softened by screening e8ects which are di8erent in the longitudinal (electric) and in the transverse (magnetic) channel. In the electric sector, the Debye screening provides a natural IR cuto8, namely the electric mass mD ∼ eT . Accordingly, the electric contribution to 8 is :nite, and of the order 8L ∼ e4 T 3 =m2D ∼ e2 T . In the magnetic sector, the dynamical screening due to Landau damping is not suKcient to completely remove the IR singularity in 8T . A logarithmic divergence remains, which we now analyze. The leading IR contribution to 8T comes from small photon momenta, q . mD , where we can use the approximate expression (4.61) to write q 1 e4 T 3 ∞ dq0 8T

dq : (6.5) 2 4 2 24 −q 2 q + (mD q0 =4q) In general, Eq. (4.61) holds only for suKciently low frequencies q0 q; but it can nevertheless be used to study the IR divergence of 8T since, in the limit of small momenta qmD , |∗MT (q0 ; q)|2 is strongly peaked at q0 = 0, with a width Mq0 ∼ q3 =m2D q. In fact, when q → 0, 1 4 |∗MT (q0 ; q)|2 4 → *(q0 ) ; (6.6) 2 2 q + (mD q0 =4q) qm2D and this limiting behaviour is suKcient to extract the IR-divergent piece of Eq. (6.5) which reads e2 T mD dq e2 T mD 8T

= ln : (6.7) 4 q 4

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We have introduced the upper cuto8 mD ∼ eT to approximately account for the correct UV behaviour of the integrand in Eq. (6.5): namely, as qmD , the integrand is decreasing like m2D =q3 , so that the q-integral is indeed cuto8 at q ∼ mD . (Incidentally, the :nal result in Eq. (6.7) is the same as the exact result for 8 = 8T + 8L obtained by evaluating the integrals in Eq. (6.3) with a sharp IR momentum cuto8 equal to [98].) Thus the logarithmic divergence is due to collisions involving the exchange of very soft, quasistatic (q0 → 0), magnetic photons, which are not screened by plasma e8ects. This situation is quite generic: in both QED and QCD, the IR complications which remain after the resummation of the HTLs are generated by very soft magnetic photons, or gluons, with momenta qgT and frequencies q0 . q3 =m2D q (see also the discussion at the end of Section B.1.4 in the appendix, and Fig. 32 there). Note also that, if we ignore temporarily this IR problem, both the electric and the magnetic damping rates are of order e2 T , rather than e4 T as one would naively expect by looking at the diagram in Fig. 17. This situation, sometimes referred to as anomalous damping [131], is a consequence of the strong sensitivity of the cross section to the IR behaviour of the photon propagator. By comparison, the other processes contributing to the damping of the fermion, namely the Compton scattering and the annihilation process, are less IR singular because they involve the exchange of a virtual fermion; as a result, these contributions are indeed of order e4 T . Note :nally that there is no IR problem in the calculation of the damping rate at zero temperature and large chemical potential [244,245]. In that case too, the dominant contribution to 8 comes from the exchange of soft magnetic photons (or gluons in QCD). In the vicinity of the Fermi surface, 8 is proportional to |E − |, where E is the fermion energy and the chemical potential. (The electric photons alone would give a contribution proportional to (E − )2 , a behaviour familiar in nonrelativistic Fermi liquids [246].) 6.2. Higher-order corrections While the above calculation of the interaction rate in the (resummed) Born approximation is physically transparent, for the analysis of the higher order corrections it is more convenient to obtain 8 from the imaginary part of the retarded self-energy. To lowest order, one can write 8p = −

1 tr(p= Im ∗ ?R (p0 + i"; p))|p0 =p ; 4p

(6.8)

with ∗ ?(p) given by the (resummed) one-loop diagram in Fig. 18. This diagram is evaluated in Appendix B, where we also verify that the resulting expression for 8 (Eq. (6.8)) coincides with the interaction rate obtained above, in Eq. (6.3). Let us turn now to higher order contributions to ?, and focus on those diagrams which can be obtained by dressing the fermion propagator by an arbitrary number of soft photon lines. An example of such a diagram is given in Fig. 19. We refer to this class of diagrams as to the “quenched approximation” (no fermion loops are included except for the hard thermal loops dressing the soft photons lines). One can verify [68,97] that the leading infrared contribution to 8p comes from these diagrams where the internal fermion lines are hard and nearly on-shell.

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Fig. 18. The resummed one-loop self-energy of a hard fermion.

Fig. 19. A generic n-loop diagram (here, n = 6) which is responsible for infrared divergences in perturbation theory. All the photon lines are soft and dressed by the hard thermal loop. The fermion line is hard and nearly on-shell.

The individual contributions of these diagrams to the damping rate contain power-like infrared divergences. An explicit calculation to two-loop order can be found in Appendix C of Ref. [97]. As we shall see, it is possible to resume all these leading IR contributions and obtain a :nite result. This is most conveniently done by formulating the perturbation theory in the time (rather than the energy) representation. As we shall see, the inverse of the time acts e8ectively as an IR cuto8, needed to account for coherence e8ects between successive scatterings. In the energy representation, one essentially assumes that the particles return on their mass shell after each scattering, and this assumption is not satis:ed in the present case where the typical mean free path is comparable to the range of the relevant interactions. We come back to this in the discussion later. Because of the aforementioned coherence e8ects, it is furthermore convenient to consider approximations for the propagator rather than for the corresponding self-energy. Consider then the contour propagator −iS(x − y) ≡ TC (x) S (y) = &C (x0 ; y0 )S ¿ (x − y) − &C (y0 ; x0 )S ¡ (x − y) ;

(6.9)

where the time variables x0 and y0 lie on a contour C in the complex time plane, as explained in Section 2. In the “quenched” approximation (in the sense of Fig. 19), S(x − y) is given by the following functional integral: −1 S(x − y) = Z DAG(x; y|A)eiSC [A] ; (6.10)

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where G(x; y|A) is the tree-level propagator in the presence of a background electromagnetic :eld, that is, the solution of the Dirac equation: −iD=x G(x; y|A) = *C (x; y)

(6.11)

with antiperiodic boundary conditions G(t0 ; y0 |A) = − G(t0 − i; y0 |A) ;

(6.12)

and similarly for y0 . Furthermore, SC [A] is the e8ective action for soft photons in the HTL approximation (which we write here in a covariant gauge): 1 1 1 SC [A] = d 4 x − F F − (9 · A)2 + d 4 x d 4 y A (x)% (x; y)A (y) 4 2L 2 C C C 1 −1 ≡ d 4 x d 4 y A (x)∗ G (x − y)A (y) : (6.13) 2 C C The gauge :elds to be integrated over in Eq. (6.10) satisfy the periodicity condition A (t0 ; x) = A (t0 − i; x). Correspondingly, the photon propagator satis:es the KMS condition (cf. Eq. (2.39)): ∗

G (t0 − y0 ) =∗ G (t0 − y0 − i) ;

(6.14)

and can be given the following spectral representation (cf. Eq. (2.106)): d 4 q −iq·(x−y) ∗ ∗ G (x − y) = − i e / (q)[&C (x0 ; y0 ) + N (q0 )] ; (6.15) (2)4 where ∗/ (q) is the photon spectral density in the HTL approximation, Eqs. (B.68) – (B.71), and N (q0 ) = 1=(eq0 − 1). The resulting fermion propagator, given by Eq. (6.10), satis:es the KMS condition: S(t0 − y0 ) = − S(t0 − y0 − i) ; and can be given the following spectral representation: d 4 p −ip·(x−y) S(x − y) = i e /(p)[& G C (x0 ; y0 ) − n(p0 )] ; (2)4

(6.16) (6.17)

where /(p) G is the fermion spectral density in the present approximation, and n(p0 ) = 1=(ep0 +1). To illustrate the previous equations, we display in Fig. 20 a typical diagram contributing to G(x; y|A) in perturbation theory. This diagram involves n photon insertions and contributes to order en . By integrating over the :elds A (x) in Eq. (6.10) one e8ectively closes the external photon lines in Fig. 20 into e8ective photon propagators. In this way, one generates all the Feynman graphs such as those illustrated in Fig. 19, that is, all the diagrams of the quenched approximation. 6.3. The Bloch–Nordsieck approximation As suggested by the previous analysis, a quasiparticle decays mainly through collisions involving the exchange of soft (q . eT ) virtual photons with the hard fermions of the heat bath.

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Fig. 20. A typical diagram contributing to G(x; y|A) to order en in perturbation theory (here, n = 4). This diagram involve n = 4 photon :eld insertions, and n + 1 = 5 bare fermion propagators S0 (including the external lines).

When the quasiparticle is hard (p ∼ T ), we can perform kinematical approximations similar to those widely used in relation with soft photon e8ects (see, e.g., Refs. [242,243]). The main outcome of these approximations is the replacement of the Dirac equation (6.11) by the following equation, known as the Bloch–Nordsieck (BN) equation [241,242]: −i(v · Dx )G(x; y|A) = *C (x; y) ;

(6.18)

where v = (1; v) and v is a :xed parameter, to be identi:ed with the velocity of the hard quasiparticle (here v is a unit vector). To get some justi:cation for this approximation one may analyse the perturbative solution of (6.11) in the relevant kinematical domain. Consider then, in the energy-momentum representation, a hard (p ∼ T ) electron, nearly on-shell (p0 p), and propagating through a soft (q . eT ) electromagnetic background :eld. A typical Feynman diagram contributing to the Dirac propagator, solution of equation (6.11), is displayed in Fig. 20. In such a diagram, the free propagator, when expanded near the mass shell, takes the form (p0 + q0 )80 − (p + q) · S0 (p0 + q0 ; p + q) = − (p0 + q0 )2 − (p + q)2 80 − v · −1

≡ h+ (v)G0 (p0 + q0 ; p + q) ; (6.19) 2 p0 + q0 − v · (p + q) where q = (q0 ; q) is a linear combination of the external photon momenta, v = p=p, v = (1; v). The matrices h+ (v) coming from the fermion propagators combine with the various photon– fermion vertices to give a global contribution: h+ (v)81 h+ (v)82 : : : h+ (v)8n h+ (v) = v1 v2 : : : vn h+ (v)

(6.20)

for n external photon lines. Note the factorization of the matrix h+ (v) which is independent of the photon momenta and plays no dynamical role. Thus, within the present kinematical approximations, the diagram in Fig. 20 could as well have been evaluated with −1 ; (6.21) G0 (p + q) = (p0 + q0 ) − v · (p + q) as the fermion propagator and 7 = v as the photon–fermion vertex. We recognize here the Feynman rules generated by the BN equation (6.18), provided we identify in the latter the vector v with the velocity of the hard particle.

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The Bloch–Nordsieck equation (6.18) de:nes a Green’s function of the covariant derivative v · D, and in this sense it is formally identical to Eq. (4.1) in Section 4.1. However, the solutions of Eqs. (6.18) and (4.1) di8er because of the respective boundary conditions. In Section 4.1, Eq. (4.1) is solved for retarded (or advanced) boundary conditions. In principle, the thermal BN equation (6.18) is to be solved with antiperiodic boundary conditions (cf. Eq. (6.12)). However, as discussed in Refs. [97,98] these antiperiodic boundary conditions, which greatly complicate the solution of the BN equation, are not needed to obtain the dominant behaviour at large time of the fermion propagator: this is indeed identical to that of a test particle, by which we mean a particle which is distinguishable from the plasma particles, and is therefore not part of the thermal bath. The propagator of a test particle has only one analytic component, namely S ¿ (S ¡ vanishes since the thermal bath acts like the vacuum for the :eld operators of the test particle). Therefore, for real times x0 ; y0 ∈ C+ , the contour propagator of a test particle coincides with the retarded propagator (cf. Eqs. (2.36) and (6.9)): S(x − y) = i&(x0 − y0 )S ¿ (x − y) = SR (x − y) :

(6.22)

The resulting propagator S ≡ SR is still given by Eq. (6.10), but now G ≡ GR obeys retarded conditions and is therefore given explicitly by Eq. (4.3): it depends on the background :eld only through the parallel transporter (4.5). Accordingly, the functional integration in (6.10) is straightforward and yields SR (t; p) = i&(t)e−it(v·p) M(t) ; where the quantity M(t) ≡ Z −1 DAU (x; x − vt)eiSC [A] 2 t t e ∗ = exp − ds1 ds2 v G (v(s1 − s2 ))v 2 0 0

(6.23)

(6.24)

contains all the nontrivial time dependence. The s1 and s2 integrations in Eq. (6.24) can be performed by using the spectral representation (6.25). We then obtain (omitting an irrelevant phase factor): 1 − cos t(v · q) d4 q 2 M(t) = exp −e /(q)N ˜ (q0 ) ; (6.25) (2)4 (v · q)2 where /(q) ˜ ≡ v ∗/ (q)v :

(6.26)

It is interesting to observe that this result can be also obtained in the framework of the classical :eld theory of Section 4.4.3. Indeed, the integral over q in Eqs. (6.24) and (6.25) being dominated by soft momenta, one can replace there N (q0 ) by T=q0 , so that, when written in the temporal gauge A0 = 0, the Feynman propagator ∗ Gij in these equations reduces to the classical correlator Gijcl in Eq. (4.82). Then, Eq. (6.24) is e8ectively the same as Eq. (4.81) with the eikonal current density t i i J (z) = ev ds *(4) (z − x + v(t − s)) : (6.27) 0

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Thus, result (6.24) can be seen as the result of the classical averaging over the initial conditions for the HTL e8ective theory, that is, −1 M(t) = Zcl DEi DAi DW*(Ga )U (x; x − vt |Acl )e−H ; (6.28) where

U (x; x − vt |Acl ) = exp i

4

d zJ

i

(z)Aicl (z)

;

(6.29)

and Aicl (x) is the solution to the classical equations of motion (4.86) with the initial conditions {Ei ; Ai ; W}. 6.4. Large-time behaviour We are now in a position to study the large-time behaviour of the fermion propagator, as described by the function M(t). Let us :rst observe that for a :xed time t, the function: 1 − cos t(v · q) ; (6.30) f(t; v · q) ≡ (v · q)2 in Eq. (6.25) is strongly peaked around v · q ≡ q0 − v · q = 0, with a width ∼ 1=t. In the limit t → ∞, f(t; v · q) → t*(v · q). In the absence of infrared complications, we could use this limit in Eq. (6.25) to obtain M(t → ∞) ∼ e−8t , with d4 q 2 /(q)N ˜ (q0 )*(v · q) : (6.31) 8 ≡ e (2)4 We recognize in Eq. (6.31) the one-loop damping rate (see Eq. (B.107)). We know, however, that 8 is infrared divergent (cf. Eq. (6.5)), so a di8erent strategy must be used to extract the large time behaviour of M(t). In the Coulomb gauge, the photon spectral density reads (cf. Eqs. (B.68) – (B.76)): ˆ 2 ∗/T (q0 ; q) : /(q ˜ 0 ; q) =∗/L (q0 ; q) + 1 − (v · q) (6.32) The infrared problems come from the magnetic sector and, more precisely, from the IR limit, q → 0, where we can use the approximation: ∗

/T (q0 ; q)N (q0 )

2 m2D qT → T 2 *(q0 ) 2 q6 + (m2D q0 =4)2 q

as q → 0 :

(6.33)

(Since ∗/T (q0 ; q) = 2 Im ∗MT (q0 + i"; q), the above equation is, of course, equivalent to Eq. (6.6) for the magnetic propagator.) In the computation of M(t), it is convenient to isolate the singular behaviour in Eq. (6.33) by writing (with N (q0 ) T=q0 ): T 1 1 ∗ /T (q0 ; q)N (q0 ) ≡ 2T*(q0 ) 2 − 2 + T (q0 ; q) : (6.34) 2 q q0 q + mD A contribution ˙ 1=(q2 + m2D ) has been subtracted from the singular piece—and implicitly included in T (q0 ; q)—to avoid spurious ultraviolet divergences: written as they stand, both

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terms in the r.h.s. of Eq. (6.34) give UV-:nite contributions. In fact, it turns out [98] that with this particular choice of a regulator, the nonsingular contribution to M(t), i.e., that coming from ˆ 2 ∗/T + T ; /˜ non-sing ≡ ∗ /L − (v · q)

(6.35)

is precisely zero, so that the net contribution arises entirely from the piece proportional to *(q0 ) in Eq. (6.34). The latter is easily evaluated as d3 q 1 1 1 − cos t(v · q) 2 ln M(t) = −e T − (2)3 q2 q2 + m2D (v · q)2 = −PTt (ln(mD t) + (8E − 1) + O(e; 1=mD t)) ; (6.36) where P = e2 =4 and 8E = 0:5772157 is Euler’s constant. Thus, at very large times, the decay of the retarded propagator is not exponential. In fact, M(t) is decreasing faster than an exponential. It follows that the Fourier transform ∞ ∞ −i!t SR (!; p) = dt e SR (t; p) = i dt eit(!−v·p+i") M(t) ; (6.37) −∞

0

exists for any complex (and :nite) !. In contrast to what one would expect from perturbation theory, the quasiparticle propagator has no pole, nor any other kind of singularity, at the mass-shell. However, the associated spectral density ∞ /(!; G p) = 2 Im SR (!; p) = 2 dt cos t(v · p)M(t) (6.38) 0

(with v · p = ! − v · p) retains the shape of a resonance strongly peaked around the perturbative mass-shell ! ∼ v · p, with a typical width of order ∼ e2 T ln(1=e). (See Fig. 21, where we also represent, for comparison, the Lorentzian spectral function /L (j) = 28=(j2 + 82 ), with j = v · p and 8 = PT ln(1=e).)

Fig. 21. The spectral density /( G j) (full line) and the lorentzian /L (j) (dashed line) for e = 0:08. All the quantities are made adimensional by multiplication with appropriate powers of mD (e.g., j=mD is represented on the abscissa axis, and mD / on the vertical axis).

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As the previous analysis shows, the leading logarithmic behaviour at large times—i.e., the term ln(mD t) in Eq. (6.36), which is the counterpart of the IR divergence ln(mD =) in the energy representation—is due to scattering involving the exchange of quasistatic magnetic photons. Thus, in the path integrals in Eqs. (6.24) and (6.28), the dominant contribution comes from integration over static magnetic :eld con:gurations. To isolate this contribution in Eq. (6.28), it is enough to replace there the classical solution Aicl (t; x) by its initial condition Ai (x); this gives (the integrals over Ei and W contribute only to the normalization factor): −1 3 2 M(t) ≈ Z DAi U (x; x − vt |Ai ) exp − d z Fij : (6.39) 4 The same integral would have been obtained by restricting the integration in the quantum path-integral in Eq. (6.24) to the static Matsubara modes of the transverse :elds. It can be easily veri:ed that integral (6.39) yields indeed the asymptotic behaviour exhibited in Eq. (6.36): M(mD t 1) exp{−PTt ln(mD t)}. (The scale mD ∼ eT enters this calculation as an ad hoc upper momentum cuto8, which is necessary since the integral in Eq. (6.39) has a spurious, logarithmic, ultraviolet divergence, due to the reduction to the static modes [68]. In the full calculation including also the nonstatic modes and leading to Eq. (6.36), this cuto8 has been provided automatically by the screening e8ects at the scale eT .) We conclude this subsection with two comments: First, we notice that the previous analysis has been extended to massive test particles [98], and also to soft quasiparticles, that is, to collective fermionic excitations with typical momenta p ∼ eT [97]. It has also been shown that the result presented here is gauge invariant [98]. Second, we note that a similar problem has been investigated in a completely di8erent context, that of the propagation of electromagnetic waves in random media [247] (see also [248]). 6.5. Discussion At this point, a few comments on the results that we have obtained are called for. In particular we have earlier alluded to the fact that the inverse of the time acts e8ectively as an infrared cuto8. We wish to see now more explicitly how this occurs, both in the calculation of the propagator, and in that of the damping rate from kinetic theory. Consider the one-loop correction to the retarded propagator SR (t; p); for t ¿ 0, this is given by t1 t (2) *SR (t; p) = − dt1 dt2 S0 (t − t1 ; p)?R(2) (t1 − t2 ; p)S0 (t2 ; p) ; (6.40) 0

0

where S0 (t; p) is the free retarded propagator and ?R(2) (t; p) is the retarded one-loop selfenergy. Since, in the BN approximation, S0 (t; p) = i&(t)e−it(v·p) , we immediately obtain *SR(2) (t) = − S0 (t) *M(t), with t dt (t − t )eit (v·p) ?R(2) (t ; p) *M(t; p) ≡ i 0 t

it dt eipt ?R(2) (t ; p) ; (6.41) 0

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481

where the last, approximate, equality holds in the large time limit. The above expression is well de:ned although the limit t → ∞ of the integral over t (which is precisely the on-shell self-energy ?R(2) (! = p)) does not exist. In fact [98] e−ipt 1 ?R(2) (t; p) −iPT (6.42) for t t mD does not decrease fast enough with time to have a Fourier transform. However, t dt = − PTt ln(mD t) (6.43) *M(t; p) PTt 1=mD t is :nite and, as shown in Refs. [97,98], this second order correction to the retarded propagator exponentiates in an all-order calculation: 1 SR (t; p) ˙ exp(−PTt ln mD t) for t : (6.44) mD In other terms, the full BN result in Eq. (6.25) is nothing but the exponential of the one-loop correction to the propagator in the time representation: M(t) = exp{−*M(t)}. We shall recover this mechanism of exponentiation from a di8erent point of view, that of kinetic theory. As shown at the end of Section 2, the single-particle excitation with momentum p can be described as an o8-equilibrium deviation *N (p; t) ≡ N (p; t) − N (p) in the distribution function, which obeys Eq. (2.202). Here, we compute the time-dependent damping rate 8(p; t) for an electron, to leading order in perturbation theory. According to Eq. (2.203) we need the discontinuity 7(p0 ; p) = − tr(p= Im ∗ ?R (p0 + i"; p)) ; (6.45) which can be extracted from Eqs. (B.88) and (B.89) in Appendix B. For large enough times t 1=mD , we need this quantity only in the vicinity of the mass-shell (|p0 − p|mD ), where it reads (compare to Eq. (6.31)): 1 d4 q /(q)N ˜ (q0 )*(p0 − p − q0 + v · q) 7(p0 ; p) = 2e2 2p (2)4 e2 T mD ; (6.46) ln 2 |p0 − p| up to terms which vanish as p0 → p. When inserted into Eq. (2.203), this yields sin(v · q)t d4 q 2 8(t) = e /(q)N ˜ (q0 ) ; (6.47) 4 (2) v·q which is independent of p. The naive in:nite-time limit of this expression, using sin(v · q)t= (v · q) → *(v · q), coincides with the usual one-loop damping rate in Eq. (6.31), and is IR divergent. But for any :nite t, the expression of 8(t) given by Eq. (6.47) is well de:ned and can be used in Eq. (2.202) to get =

t

*N (p; t) = e−2 0 dt 8(t ) *N (p; 0) ; with (cf. Eq. (6.47)) t 1 − cos t(v · q) d4 q 2 dt 8(t ) = e /(q)N ˜ (q0 ) : 4 (2) (v · q)2 0

(6.48) (6.49)

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A short comparison with Eq. (6.25) reveals that t M(t) = exp − dt 8(t ) : 0

(6.50)

Note that, in both approaches, the :nal result emerges as the exponential of a one-loop correction in the time representation. In this one-loop correction the inverse of the time plays the role of an infrared cut-o8, and perturbation theory in the time representation can be applied for suKciently small times (here, t . 1=gT ). Another approach leading to a similar exponentiation of the one-loop result is the so-called “dynamical renormalization group” developed in Refs. [99,69,132]. 6.6. Damping rates in QCD Let us now turn to QCD. As already mentioned, the one-loop calculations of damping rates are a_icted by the same IR problem as in QED. However, in QCD we may expect these divergences to be screened at the scale g2 T by the self-interactions of the magnetic gluons. That is, we expect the quasiparticle (quark or gluon) propagator to decay exponentially according to mD g2 T M(t → ∞) exp −Cr + O(1) ; (6.51) t ln 4 mmag where Cr is the Casimir factor of the appropriate colour representation (i.e., Cf = (N 2 − 1)=2N for a quark, and Cg = N for a gluon), and mmag ∼ g2 T is the “magnetic mass”. (We consider here a hard quasiparticle, with momentum p & T . Except for the magnetic mass, the leading term displayed in Eq. (6.51) is determined by the one-loop calculation.) Note however that the magnetic mass matters only at times t & 1=g2 T . For intermediate times, 1=gT t 1=g2 T , relying on the analogy with the Abelian problem, we may expect a non exponential decay law: g2 T 2 (6.52) M(1=gT t 1=g T ) exp −Cr t(ln(mD t) + O(1)) : 4 To verify the behaviour in Eqs. (6.51) – (6.52), a nonperturbative analysis is necessary. As shown in Ref. [98], the BN approximation leads to the following functional representation of the quasiparticle propagator (compare with Eq. (6.24)): M(t) ≡ Z −1 DA Tr(U (x; x − vt))eiSC [A] ; (6.53) where U (x; x − vt) is the nonAbelian parallel transporter, Eq. (4.5), and SC [A] is the contour e8ective action for soft gluons in the HTL approximation. Now, in QCD the action is not quadratic and the functional integral (6.53) cannot be computed analytically. A possible continuation would be to treat the soft gluons in the classical approximation, and thus replace Eq. (6.53) by the nonAbelian version of Eq. (6.28), to be eventually computed on a classical lattice (cf. Section 4.4.3). A more economical proposal [98] is to use “dimensional reduction”, as in Eq. (6.39). This should be enough to generate the nonperturbative magnetic screening, and therefore the leading logarithm ln(mD =mmag ) ln(1=g) of the asymptotic behaviour in Eq. (6.51) (but not also the constant term of O(1) under the logarithm which is sensitive to the nonstatic modes).

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Fig. 22. One-loop diagrams for the soft fermion self-energy in the e8ective expansion.

To conclude, let us recall that for quasiparticles with zero momentum the damping rates are :nite and of order g2 T . The IR problems are absent in this case since the magnetic interactions do not contribute. The corresponding damping rates have been computed for both gluons [61] and fermions [186,187], to one-loop order in the e8ective theory. Since all the external and internal lines are soft, the corresponding diagrams involve resummed propagators and vertices (see, e.g., Fig. 22 for the case of a soft external fermion). The resulting damping rates were shown to be gauge-:xing independent, a property which relies strongly on the Ward identities satis:ed by the HTLs [19,61]. 7. The Boltzmann equation for colour excitations As we have already argued, as long as we are interested in the collective excitations with wavelength ∼ 1=gT we can ignore, in leading order in g, collisions among the plasma particles. However collisions become a dominant e8ect for colour excitations with wavelength ∼ 1=g2 T , and colorless excitations with wavelength ∼ 1=g4 T . In this section we focus on excitations with wavelength ∼ 1=g2 T to which we shall refer as ultrasoft excitations. We shall then reconsider brieLy the approximations which led us in Section 3 to kinetic equations, and study the dynamics of hard particles (k ∼ T ) in the background of ultrasoft :elds Aa (X ) such that 9X ∼ gA ∼ g2 T . As we shall see, in leading order, the role of the soft degrees of freedom (k ∼ gT ) is merely to mediate collisions between the plasma particles. The resulting kinetic equation is a Boltzmann equation, whose solution implicitly resums an in:nite number of diagrams of perturbation theory. These diagrams generalize the HTLs to the case where the external lines are ultrasoft and are called “ultrasoft amplitudes”. In order to specify the separation between soft and ultrasoft :elds, we shall introduce an intermediate scale such that g2 T gT . The ultrasoft amplitudes depend logarithmically on this scale which plays the role of an infrared cuto8 in their calculation. An alternative description of the ultrasoft dynamics relies on the fact that it is essentially that of classical :elds. Already the soft modes are classical, and to leading order their dynamics is entirely contained in the classical equations of motion given in Section 4. Furthermore, in order to calculate correlation functions in real time, one can use the hamiltonian formulation

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of Section 4.4 in order to perform the necessary averages over the initial conditions. Then, the non-perturbative dynamics can be studied for instance via classical lattice simulations. The e8ective theory presented in Section 4.4.3 turns out to have a relatively strong (linear) dependence upon its ultraviolet cuto8, which may lead to lattice artifacts. However, as suggested by BXodeker [25], by integrating out the soft modes in classical perturbation theory, one obtains an e8ective theory for the ultrasoft :elds which is only logarithmically sensitive to the scale introduced above, and could therefore be better suited for numerical calculations. This e8ective theory involves the Boltzmann equation alluded to before supplemented by a noise term which, as we shall see, is related to the collision term by the Luctuation–dissipation theorem. 7.1. The collision term For simplicity, throughout this section we shall restrict ourselves to a Yang–Mills plasma without quarks, and use the background :eld Coulomb gauge, as de:ned in Eq. (3.39). The ¡ (x; y) read (compare to Eqs. Kadano8–Baym equations for the gluon 2-point function G (2.123)): ¡ (g/ D2 − D D/ + 2igF/ )x G/ (x; y) ¡ ¡ = d 4 z {g ?R/ (x; z)G/ (z; y) + ?/ (x; z)GA/ (z; y)g } ;

(7.1)

G¡/ (x; y)(g/ (D† )2 − D/† D † + 2igF/ )y ¡ ¡ = d 4 z {g GR/ (x; z)?/ (z; y) + G/ (x; z)?A/ (z; y)g } :

(7.2)

and

The subsequent analysis of Eqs. (7.1) and (7.2) proceeds as in Sections 3.3–3.4: we construct the di8erence of the two equations, introduce gauge-covariant Wigner functions, and then perform a gauge-covariant gradient expansion which is controlled by powers of g2 (since DX ∼ g2 T ). The new feature is the emergence of the collision term, coming from the terms involving self-energies in Eqs. (7.1) and (7.2). To leading order accuracy, we can restrict ourselves to a quasiparticle approximation, in the sense of Section 2.3.4; that is, we can ignore the o8-shell e8ects for the hard particles (here, the transverse gluons), together with the Poisson brackets generated by the gradient expansion of the self-energy terms (cf. Section 2.3.2). This means that, to leading nontrivial order, the (gauge-covariant) Wigner functions conserve the same structure as in the mean :eld approximation (cf. Section 3.4.1), namely: ¡ GG ij (k; X ) = (*ij − kˆi kˆj )[G0¡ (k) + *GG (k; X )] (7.3) with (compare with Eq. (3.105)) dN : *GG ab (k; X ) = − /0 (k)Wab (k; X ) d k0 ¿ A similar equation holds for GG ij with G0¡ replaced by G0¿ .

(7.4)

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The :nal kinetic equation is conveniently written as an equation for *GG (k; X ), and reads (in matrix notations): 2[k · DX ; *GG (k; X )] − 2gk F (X )9 k G0¡ (k) = C(k; X )

(7.5)

with the collision term C(k; X ) (a colour matrix with elements Cab (k; X )) to be constructed now. To this aim, consider a typical convolution term in the r.h.s. of Eq. (7.1) or (7.2); to leading order in the gradient expansion, this yields (with Minkowski indices omitted, for simplicity): d 4 zG(x; z)?(z; y) → G(k; X )?(k; X ) + · · · ; (7.6) where as compared to Eq. (2.142) we have neglected the Poisson bracket term. By collecting all the terms coming from the r.h.s. of Eqs. (7.1) and (7.2), and paying attention to the ordering of the colour matrices, we obtain C(k; X ) = i(?R G¡ − G¡ ?A + ?¡ GA − GR ?¡ ) 1 = − ({G¿ ; ?¡ } − {?¿ ; G¡ }) − i[Re ?R ; G¡ ] + i[Re GR ; ?¡ ] ; 2

(7.7)

where [; ] and {; } stand here for colour commutators and anticommutators, respectively. In writing the second line above, we have also used relations (2.145) and (2.146). We now proceed with further approximations. Since A ∼ gT , gF ∼ g9X A ∼ g4 T 2 , and ¡ . Similarly, writing ?¡ = ?¡ +*?¡ , one :nds *?¡ ∼ g2 ?¡ . Eq. (7.5) implies that *G¡ ∼ g2 Geq eq eq Thus, we can linearize C(k; X ) in Eq. (7.7) with respect to the o8-equilibrium Luctuations. ab = *ab G ), the two Since the equilibrium two-point functions are diagonal in colour (e.g., Geq eq commutators in Eq. (7.7) vanish after linearization, while the anticommutators yield: ¿ ¡ ¡ ¿ C(k; X ) −(Geq *?¡ + *G¿ ?eq ) + (*?¿ Geq + ?eq *G¡ ) :

(7.8)

It is straightforward to rewrite this in a manifestly gauge-covariant way. To this aim, it is enough to replace the noncovariant Luctuations *G and *? by the corresponding gauge-covariant expressions *GG and *?G (cf. Eq. (3.86)): *G(k; X ) = *GG (k; X ) − g(A(X ) · 9k )Geq (k) ;

(7.9)

and similarly for *?(k; X ). One then gets ¡

¿

¿

¡

¿ G ¡ ¡ ¿ G C(k; X ) = − (Geq *? + *GG ?eq ) + (*?G Geq + ?eq *G ) :

(7.10)

This turns out to be the same expression as above, Eq. (7.8), except for the replacement of ordinary Wigner functions by gauge-covariant ones: The corrective terms in Eq. (7.9) cancel out in C(k; X ) since they contribute a term proportional to the collision term in equilibrium, which is zero: ¿ ¡ ¡ ¿ g(A(X ) · 9k )(Geq ?eq − Geq ?eq ) = 0 :

(7.11)

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Fig. 23. Self-energy describing collisions in the (resummed) Born approximation. All the lines represent o8-equilibrium propagators. The continuous lines refer to the hard colliding particles in Fig. 17 (here, hard gluons). The wavy lines with a blob denote soft gluon propagators dressed by the hard thermal loops. ¿ G¿, Actually, Eq. (7.10) can be simpli:ed even further: to the order of interest, Geq 0 ¡ ¿ ¡ G ¡ and Eq. (7.3) implies that, *G G G G

* G = * G . It follows that Geq 0 ¿ ¡ (7.12) C(k; X ) −7(k)*GG (k; X ) + (*?G (k; X )G0¡ (k) − *?G (k; X )G0¿ (k)) ¡ − ?¿ . with 7(k) = ?eq eq The structure of the collision term is independent of the speci:c form of the collisional self-energies *?G ¿ and *?G ¡ , and is solely a consequence of the (gauge-covariant) gradient expansion. However, for consistency with the gradient expansion, these self-energies have to be computed to leading order in g2 . As we shall verify, these are obtained from the two-loop diagram in Fig. 23. After linearization the collision term may be represented by the four processes displayed in Fig. 24, where each diagram involves a Luctuation *G denoted by a cross while all other propagators are equilibrium propagators. The collision term associated to the diagrams in Fig. 24 is constructed in detail in Ref. [26] and can be written as follows: Cab (k; X ) = − d T|Mpk→p k |2 N (k0 )N (p0 )[1 + N (k0 )][1 + N (p0 )]

×{N (NWab (k; X ) − (T a T b )cd Wcd (k ; X )) + (T a T b )ccS (T c T cS)ddS (Wdd S (p; X ) − WddS (p ; X ))} :

(7.13)

In this equation, |Mpk→p k |2 ˙ g4 is the matrix element squared corresponding to the one-gluon exchange depicted in Fig. 17, and d T is a compact notation for the measure of the phase-space integral: d4 p d4 q dT ≡ /0 (k)/0 (p)/0 (p + q)/0 (k − q) : (7.14) 4 (2) (2)4 The four terms within the braces in Eq. (7.13) are in one to one correspondence with the diagrams in Figs. 24a–d. The collision term (7.13) is formally of order g4 , since proportional to |M|2 . However, because of the sensitivity of the phase space integrals to soft momenta, it may be enhanced and

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Fig. 24. Pictorial representation of the linearized collision term. Each one of the four diagrams correspond to o8-equilibrium Luctuations in one of the colliding :elds (the one which is marked with a cross). All the unmarked propagators are in equilibrium.

become of order g2 . The mechanism here is similar to that leading to the anomalous damping rate discussed in Section 6, and in fact the :rst term in Eq. (7.13) (the one involving Wab (k; x)) is the same as −7(k)*GG (k; x) (cf. Eq. (7.12)), that is, it is proportional to the damping rate of a hard excitation. Whether the collision integral is of order g2 or g4 depends however on subtle cancellations which are studied in the next subsection. 7.2. Coloured and colourless excitations The construction of the collision term Cab (k; x) in Eq. (7.13) involves two kinds of gradient expansions [26]: one in powers of Dx =k ∼ g2 , and another in powers of 9x =q, where q is the momentum exchanged in the collision. The latter assumes that the range of the interactions (as measured by 1=q) is much shorter than the range of the inhomogeneities ∼ 1= 9x . It is this approximation that makes the collision term local in x. As we shall argue now, this is a good approximation for colourless Luctuations, but is only marginally correct for coloured ones. G and Wab = *ab W . The various colour traces in For colourless Luctuations, *GG ab = *ab *G, a b Eq. (7.13) are then elementary (e.g., (T T )cc = N*ab ), and yield Cab = *ab C, with C(k; x) = −N 2 d T|Mpk→p k |2 N0 (k0 )N0 (p0 )[1 + N0 (k0 )][1 + N0 (p0 )] ×{W (k; x) − W (k ; x) + W (p; x) − W (p ; x)} :

(7.15)

What is remarkable about Eq. (7.15) is that the phase-space integral is dominated by relatively hard momentum transfers gT . q . T , even though each of the four individual terms in the r.h.s. is actually dominated by soft momenta. This is a consequence of the cancellation of the leading infrared contributions among the various terms [26]. For instance, for soft q, W (k ; x) ≡ W (k − q; x) ≈ W (k; x), so that the IR contributions to the :rst two terms in Eq. (7.15) cancel each other. A similar cancellation occurs between the last two terms in

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Eq. (7.15), namely W (p; x) and W (p ; x). Thus, in order to get the leading IR (qT ) behaviour of the full integrand in Eq. (7.13) one needs to expand W (k ; x) and W (p ; x) in powers of q. By doing so, one generates, in leading order, an extra factor of q2 in the integrand which removes the most severe IR divergences in the collision integral. (This is the familiar factor 1 − cos & ≈ q2 =2, with & the scattering angle, which characterizes the transport cross sections.) As a result, the integral in Eq. (7.15) leads to relaxation rates typically of order g4 T ln(1=g), where the logarithm originates from screening e8ects at the scale gT . Such rates control the transport coeKcients like the shear viscosity [63,70] or the electric conductivity [71]. (See also Refs. [131,151,249] where similar cancellations are identi:ed via diagrammatic calculations in Abelian gauge theories.) Under such conditions, the e8ects of the collisions become important for inhomogeneities at the scale g4 T ; the inequality 9x q is then very well satis:ed because q is here relatively hard, gT . q . T . However, the fact that the relaxation rates are not saturated by small angle scattering implies that to calculate them, even to leading order in g, one has to consider all the collisions with one particle exchange (including, e.g., Compton scattering). In terms of self-energy diagrams for the collision term, this means that one has to include all the two-loop diagrams contributing to ?¿ and ?¡ , and not only the diagram in Fig. 23. The situation is di8erent for colour relaxation. The longwavelength colour excitations are described by a density matrix W (k; x) in the adjoint representation: W (k; x) ≡ Wa (k; x)T a . The colour algebra in Eq. (7.13) can then be performed by using the following identities: 2 N abe N Tr(T a T b T c ) = ifabc ; = if (T a T b )ccS (T c T cS)ddS (T e )dd : (7.16) S 2 4 The resulting collision term is C ≡ Ca T a , with 2 Ca (k; x) = −N d T|Mpk→p k |2 N0 (k0 )N0 (p0 )[1 + N0 (k0 )][1 + N0 (p0 )] 1 1 × Wa (k; x) − Wa (k ; x) − (Wa (p; x) + Wa (p ; x)) : (7.17) 2 4 The cancellation of the leading infrared contributions no longer takes place and we can simply set k = k and p = p in Eq. (7.17) which then simpli:es to [250,26]: N2 dN dN Ca (k; x) − d T|Mpk→p k |2 {Wa (k; x) − Wa (p; x)} : (7.18) 2 d k0 dp0

For soft q, the matrix element |M|2 has been already evaluated in Eq. (6.2): |M|2 = 16g4 k2 p2 | ∗Ml (q) + (qˆ × v) · (qˆ × v ) ∗Mt (q)|2 ;

(7.19)

ˆ The phase-space measure (7.14) can be similarly simpli:ed. This where v ≡ kˆ and v ≡ p. eventually yields a simpler equation for Wa (k; x) which, remarkably, is consistent with Wa (k; x) being independent of the magnitude |k| of the hard momentum, as in the HTL approximation (cf. Eq. (4.9)). We thus write Wa (k; x) = g{&(k0 )Wa (x; v) − &(−k0 )Wa (x; −v)} ;

(7.20)

where a factor of g is introduced to keep in line with the normalization in Eq. (4.9). In particular, the induced colour current preserves the structure in Eq. (4.12).

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Finally, the Boltzmann equation reads [26] (v · Dx )ab Wb (x; v) = v · Ea (x) − m2D

g2 NT 2

d. E(v · v ){W a (x; v) − W a (x; v )} : 4

489

(7.21)

The angular integral above runs over all the directions of the unit vector v , and mD is the Debye mass, m2D = g2 T 2 N=3 for the pure Yang–Mills theory. Furthermore, d4 q 2 *(q0 − q · v)*(q0 − q · v )| ∗Ml (q) + (qˆ × v) · (qˆ × v ) ∗Mt (q)|2 ; E(v · v ) ≡ (2) (2)4 (7.22) with the two delta functions expressing the energy conservation at the two vertices of the scattering process in Fig. 17. The collision term in Eq. (7.21) is local in x, but nonlocal in v. As it stands, it is infrared divergent. At this point, one should recall that we are eventually interested in the e8ective theory for the ultrasoft :elds which can be separated from the soft degrees of freedom that we are “eliminating” by an intermediate scale such that g2 T gT . This scale acts as an IR cuto8 for the collision term, which is therefore :nite, but logarithmically dependent on . For instance, the damping rate of a hard gluon, given by the :rst term (local in v) of the collision integral is 7(k0 = k) g2 NT d. m2D E(v · v ) = =8 ; (7.23) 2 4 4k (Up to a colour factor, this is the same equation as Eq. (6.3). Note also that a cancellation has taken place making the contribution of the damping rate to the collision integral only half of what it would normally give; compare, in this respect, Eqs. (7.21) and (7.23) above to Eq. (2.177) in Section 2.3.4.) The integral over v can be analytically computed, with the simple result [98,72] (with P = g2 =4) mD 8 = PNT ln : (7.24) Note that for colour excitations at the scale g2 T , the inequality 9x q is only marginally satis:ed since the collision term is logarithmically sensitive to all momenta . q . gT . We have no such a simple exact result for the full quantity E(v · v ), but it is nevertheless straightforward to extract its dependence from Eq. (7.22): this is obtained by retaining only the singular piece of the matrix element for magnetic scattering, namely using Eq. (6.6) to get E(v · v )

mD 2 (v · v )2 ln ; 2 2 2 mD 1 − (v · v )

(7.25)

By using Eq. (7.23), the Boltzmann equation for colour relaxation is :nally written as (v · Dx )ab Wb (x; v) = v · Ea (x) − 8{W a (x; v) − W a (x; v)} ;

(7.26)

where we have introduced the following compact notation: for an arbitrary function of v, say F(v), we denote by F(v) its angular average with weight function E(v · v ) (that is, its average

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with respect to the scattering cross section): (d. =4)E(v · v )F(v ) F(v) ≡ ; (d. =4)E(v · v )

(7.27)

which is still a function of v. From Eq. (7.26), it is clear that the quasiparticle damping rate 8 sets also the time scale for colour relaxation: ,col ∼ 1=8 ∼ 1=(g2 T ln(1=g)) [55]. We conclude this subsection by noticing that Eq. (7.26) is invariant under the gauge transformations of the background :eld, and also with respect to the choice of a gauge for the Luctuations with momenta k & . In Ref. [26], Eq. (7.21) was derived in Coulomb gauge, but we expect it to be gauge-:xing independent since it involves only the o8-equilibrium Luctuations of the (hard) transverse gluons, together with the (gauge-independent) matrix element squared (7.19). Finally we note that the Boltzmann equation (7.21) (with the collision term approximated as in Eq. (7.25)) has been also obtained in Ref. [59] by using a classical transport theory of colour particles [51,52,54]. 7.3. Ultrasoft amplitudes By solving the Boltzmann equation, one can obtain Wa (x; v), and thus the induced current ja (x) as a functional of the :elds Aa (x), which can be expanded in the form 1 abc / ab ja = % Ab + 7 / Ab Ac + · · · (7.28) 2 ab ; 7abc , etc., in this expansion are one-particle-irreducible amplitudes for the The coeKcients % / ultrasoft :elds in thermal equilibrium (cf. Section 5.1), and will be referred to as the ultrasoft amplitudes (USA) in what follows. These are the generalizations of the HTLs to the case where the external legs carry momenta of order g2 T or less. Speci:cally, these are the leading contributions of the hard and soft degrees of freedom to the amplitudes of the ultrasoft :elds. 7.3.1. General properties The ultrasoft amplitudes share many of the remarkable properties of the HTLs: (i) They are gauge-:xing independent (like the Boltzmann equation itself), (ii) satisfy the simple Ward identities shown in Eq. (5.38) (these follow from the conservation law D j = 0 for the current), and (iii) reduce to the usual Debye mass m2D (cf. Eq. (4.50)) in the static limit ! → 0. To verify this last point, it is convenient to use the decomposition (4.28) for W a (x; v): W a (x; v) ≡ −Aa0 (x) + Aa (x; v) : The :rst term does not contribute to the collision term since left with the following equation for Aa (x; v):

(7.29) A0a (x)

(v · Dx )ab Ab (x; v) = 90 (v · Aa ) − 8{Aa (x; v) − Aa (x; v)} :

−

A0a (x) = 0.

One is then (7.30)

Thus, for time-independent :elds Aa (x), the homogeneous Eq. (7.30) with retarded boundary conditions admits the trivial solution Aa (x; v) = 0, and therefore ja (x) = − *0 m2D Aa0 (x) ;

(7.31)

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as in the HTL approximation (cf. Eq. (4.50)): all the ultrasoft vertices with n ¿ 3 external lines vanish, while % (! = 0; p) = − *0 * 0 m2D . Thus, to this order, the physics of the static Debye screening is not a8ected by the collisions among the hard particles. This is consistent with the results in [64,224] according to which the :rst correction to m2D , of order g3 T 2 ln(1=g), is due to soft and ultrasoft loops (cf. Fig. 16). Consider now time-dependent :elds. In order to analyse the solutions of the Boltzmann equation (7.26), it is convenient to write the collision term as an operator acting on W a (x; v): Ca (x; v) = −8{Wa (x; v) − Wa (x; v)} d. 4 C (x; x ; v; v )W b (x ; v ) ≡ −(CW )a (x; v) ≡− d x 4 ab with the following kernel, which is nonlocal but symmetric in v and v : *C a (x; v) Cab (x; x ; v; v ) = − *W b (x ; v ) = *ab *(4) (x − x )8{*(2) (v; v ) − *(2) (v; v )} : Then, the solution to the Boltzmann equation (7.26) can be formally written as d. 1 4 W (x; v) = d x x; v x ; v v · E(x ) : 4 v · D + C

(7.32)

(7.33)

(7.34)

This expression exhibits in particular the role of the collisions in smearing out the divergences of the HTLs at v · D → 0 (cf. Section 5.3.3). 7.3.2. The ultrasoft polarization tensor and its diagrammatic interpretation Solution (7.34) can be used to derive an expression for the ultrasoft polarization tensor % . To this aim, one needs the induced current only to linear order in A , so one can replace v · D by v · 9 in Eq. (7.34), and use the momentum representation. It is also convenient to use the decomposition (7.29), so as to obtain the tensor % in a manifestly symmetric form (compare with Eq. (5.34)): d. d. 1 ab 2 ab % (P) = mD * −*0 * 0 + ! v v v v : (7.35) 4 4 v · P + iC A diagrammatic interpretation of this formula is obtained by formally expanding out the (0) (1) (0) + % + : : : ; where % is the HTL given by collision term. One thus obtains % = % Eq. (5.34), and v d. v v (1) 2 % (P) = −i8!mD − 4 v · P v · P v·P 2 d. d. v v 4 g NT v = −i!mD − ; (7.36) E(v · v ) v · P v · P v · P 2 4 4 where in the second line we have used de:nition (7.27) of the angular averaging together with (0) is a real quantity while the :rst-order iteration in Eq. (7.23) for 8. For time-like momenta % Eq. (7.39) is purely imaginary, reLecting the dissipative role of the collisions.

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(0) Fig. 25. A generic one-loop diagram contributing to the HTL % . The internal continuous lines denote hard transverse gluons; the external wavy line is an ultrasoft gluon.

(1) Fig. 26. The diagrams contributing to the :rst iteration % (P) of the polarization tensor, Eq. (7.36); the continuous lines are hard transverse gluons; the internal wavy lines are soft gluons with momenta g2 T . q . gT (with the blob denoting HTL resummation).

(N ) Higher order iterations % , proportional to 8N , can be written down similarly. Note how2 ever that, for P ∼ g T , the contribution in Eq. (7.36) is of the same order in g as the HTL (5.34). Thus, the iterative expansion is only formal. It is only used here for the comparison with perturbative calculations of % in terms of Feynman diagrams, and to identify the nature of the resummations achieved by the Boltzmann equation [26,72] (see also Refs. [128,122,124]). (0) Thus the zeroth order iteration % is the HTL, which is the leading order contribution in an (1) expansion in powers of P=k of the one loop diagram of Fig. 25. The :rst order iteration % is obtained via a similar expansion from the three diagrams displayed in Figs. 26 [129]. The internal wavy lines in these diagrams are soft gluons dressed with the HTL. In the language of the Boltzmann equation, these are the soft quanta exchanged in the collisions between the hard particles (the latter being represented by the continuous lines in Figs. 26). As shown in [26,72], these are precisely the diagrams generated by the :rst iteration of the collision term in Eq. (7.13). (N ) The higher order iterations % with N ¿ 2 can be similarly given a diagrammatic interpretation, by iterating the diagrams for the collision term in Fig. 24 [26]. A typical diagram contributing to % which is obtained in this way is shown in Fig. 27. The continuous lines

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Fig. 27. A generic ladder diagram contributing to the ultrasoft polarization tensor, as obtained from the Boltzmann equation.

Fig. 28. (a) A ladder diagram generated by iterations of the :rst two pieces, Wa (k; x) and (−1=2)Wa (k ; x), of the collision term (7.17); the smooth lines are eikonal propagators with a damping rate 28. (b) The sum of all the ladders in (a); the thick line is an eikonal propagator with a damping rate 8.

with a blob represent the following dressed eikonal propagator (compare with Eq. (4.2)): −1 ; (7.37) v · P + 2i8 as obtained after resumming the self-energy corrections to the hard propagators, i.e., by iterating the self-energy insertion in Fig. 26a, or, equivalently, the :rst piece Wa (k; x) of the collision term (7.17). The continuous lines without a blob in Fig. 27 are thermal correlation functions like G0¿ and G0¡ , or derivatives of them. The vertex corrections (the wavy lines, or ladders) inside any of the hard loops in Fig. 27 are generated by iterating the second piece, (−1=2)Wa (k ; x), of the collision term (7.17). The net e8ects of these vertex corrections is to replace 28 by 8 in the eikonal propagator (7.37). This relies on the approximation Wa (k; x) − (1=2)Wa (k ; x) ≈ (1=2)Wa (k; x) (which has been used in going from Eq. (7.17) to (7.18)), and is illustrated in Fig. 28, where the thick internal line denotes the following eikonal propagator: −1 : (7.38) v · P + i8 Finally the wavy lines relating di;erent hard loops in Fig. 27 are generated by iterating the diagrams in Figs. 24c and d, or, equivalently, the last two pieces (−1=4)(Wa (p; x)+Wa (p ; x)) ≈ (−1=2)Wa (p; x) of the collision term (7.17). A similar diagrammatic interpretation holds for the n-point ultrasoft vertices (see Ref. [72] for more details). 7.3.3. Leading log approximation and colour conductivities A simple approximation where the polarization tensor can be calculated in closed form is the “leading-logarithmic approximation” [25,250,129], which relies on the following observation: for colour inhomogeneities at the scale g2 T , the collision term, which is of order 8 ∼ g2 T ln(1=g), wins over the drift term v · Dx ∼ g2 T by a “large” logarithm ln(1=g). Thus, to leading logarithmic

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accuracy (LLA), one can neglect the drift term in the l.h.s. of Eq. (7.26) and, for consistency, use approximation (7.25) in the collision term. Then the Boltzmann equation (7.26) reduces to v · Ea (x) = 8{W a (x; v) − W a (x; v)} :

(7.39)

The electric :eld in this equation is assumed to be transverse. Neglecting the drift term is in general not allowed for longitudinal :elds; in particular we have seen that the collision term vanishes for static longitudinal :elds. Eq. (7.39) has the following solution: W a (v) =

v · Ea : 8

(7.40)

This is easily veri:ed: the above W a (v) is an odd function of v, while the approximate E(v · v ) in Eq. (7.25) is even, so that W a (v) ≈ 0 to LLA. (Note that this would not hold with the whole collisional cross-section in Eq. (7.22).) After insertion in Eq. (4.12), the approximation (7.40) for W a (v) generates the following, local, colour current: ja = 5Ea

with 5 ≡

1 m2D 4T = : 38 9 ln(mD =)

(7.41)

Although 5 is not really a physically measurable quantity, at the level of approximation at which we are working it behaves as such. One could therefore expect it to be independent of the arbitrary scale separating soft and ultrasoft degrees of freedom. For that to happen, however, one needs to perform a complete calculation at the scale g2 T , that is, one needs to compute the contributions of the ultrasoft :elds themselves. These are given by loop diagrams of the ultrasoft e8ective theory, with acting then as an ultraviolet (UV) cuto8. Because the UV divergences in the e8ective theory are only logarithmic (this can be veri:ed by power counting), the loop corrections in the e8ective theory will lead to terms proportional to ln . Without doing any calculation, one expects these terms to cancel the -dependence in the ultrasoft amplitudes, leaving in place of the natural scale of the e8ective theory, that is, g2 T . To LLA, the constant term under ln(1=g) can be neglected. Thus, in LLA, the color conductivity is obtained by simply replacing ln(mD =) ≈ ln(1=g) in Eq. (7.41) [55,65,25,250]. We should notice here an important di8erence with the HTLs. Recall that the HTLs, obtained after integrating out the hard (p ∼ T ) modes, are leading order amplitudes at the scale gT in a strict expansion in powers of g. Moreover, to this order, they are independent of the scale separating T from gT . This is so since, if computed with an infrared cuto8 , the HTLs would depend linearly of this scale (see, e.g., Eq. (2.80)), and the corresponding dependence would be suppressed by a factor =T as compared to the leading order contribution, of order g2 T 2 . Because of that, in writing the HTLs we have generally omitted their explicit dependence on the separation scale . The ultrasoft amplitudes, on the other hand, depend logarithmically on the separation scale (g2 T gT ), so has to be kept explicitly as an IR cuto8 when computing the USA’s. This logarithmic dependence also implies that the contributions of the ultrasoft :elds to the respective amplitudes are of the same order in g as the USA’s themselves; thus, the latter are not dominant quantities, but only part of the full amplitudes at the scale g2 T . Now, the

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remaining contributions, due to the interactions of the ultrasoft :elds, are fully nonperturbative, and can in general be obtained only from a numerical calculation using for instance the lattice techniques. Thus, in order to compute ultrasoft physical correlations already to leading order, one has to perform a numerical calculation within the e8ective theory. A possible way to do that will be discussed in the next subsection. 7.4. The Boltzmann–Langevin equation: noise and correlations In order to compute thermal correlations of the ultrasoft :elds in real time, like, e.g.,

Ai (x)Aj (y), one could rely on the Luctuation-dissipation theorem (see Section 2.1.3) in order

to construct the thermal correlators from the retarded response functions; these can in turn be obtained from the solution of the Boltzmann equation with retarded boundary conditions (cf. Eq. (7.28)). For instance, the two-point function is obtained as (in the classical approximation where G ¿ ≈ G ¡ ≈ Gcl ): d 4 k −ik·(x−y) T Gcl (x; y) ≡ A (x)A (y) = e / (k) ; (7.42) (2)4 k0

where / = 2 Im GR , and GR−1 = G0−1 + %R , with %R (formally) constructed in Section 7.3.2. Similar representations can be written for the thermal self-energy: d 4 k −ik·(x−y) T % (x; y) = j (x)j (y) = e (−2 Im %R (k)) ; (7.43) 4 (2) k0 and also for the higher order n-point correlators. In practice, however, this strategy is not useful because the Boltzmann equation in general can only be solved numerically, and it is not easy to extract the spectral function; besides, the correlators one needs to evaluate do not necessarily have the simple form of a product of gauge :elds (see, e.g., Eq. (1.52)). An alternative procedure relies on the fact that the ultrasoft dynamics is that of classical :elds which obey the equations of motion discussed in Section 4. Correlation functions can then be obtained by averaging over the initial conditions appropriate products of the :elds which solve the classical equations of motion (cf. Section 4.4). Following BXodeker [25], let us split the classical :elds into soft and ultrasoft components (A → A + a; W → W + w, etc.), where capitals (lower cases) denotes ultrasoft (soft) components. The equations for the ultrasoft :elds take the form d. D F = m2D v W (x; v) ; 4 (v · Dx )ab Wb = v · Ea + gfabc (v · ab wc ) ; (7.44) where (ab wc ) means that only the ultrasoft components (with k . ) have to be kept in the product of :elds. The soft :elds a and w obey themselves equations of motion which relates them to the ultrasoft background :elds. In leading order the self-interactions of the soft :elds

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can be neglected and their equations of motion read simply d. a 2 9 f (x) = mD v wa (x; v) ; 4 v · 9x wa (x; v) = v · ea (x) + ha (x; v) ;

(7.45)

where f = 9 a − 9 a and ha describes the coupling between soft and ultrasoft :elds: ha (x; v) ≡ gfabc [v · Ab (x)wc (x; v) + v · ab (x)W c (x; v)] :

(7.46)

Both the soft and the ultrasoft :elds are thermal Luctuations whose typical amplitudes have been estimated in Section 1.2 as a ∼ g1=2 T and A ∼ gT . The corresponding estimates for the :elds w and W follow from Eq. (4.94), which gives W ∼ (Tk 3 =m2D )1=2 where k is a typical spatial gradient: thus for k ∼ gT; w ∼ g1=2 T and for k ∼ g2 T; W ∼ g2 T . Since e ∼ g3=2 T 2 , while h ∼ gAw ∼ g5=2 T 2 at most, Eqs. (7.45) can be solved perturbatively in h for arbitrary initial conditions for the soft :elds a and w. By inserting the corresponding solutions into Eqs. (7.44) and performing the average over the initial conditions for the soft :elds, one recovers the Boltzmann equation from the second equation (7.44) [25]. Thus, it does not make any di8erence whether one eliminates the soft :elds in the quantum theory or in the classical one. However, as noticed by BXodeker, the equation of motion for W contains a source term independent of the ultrasoft :elds: the term (aw) where a and w are solutions of their equations of motion in the absence of ultrasoft background, i.e. the solution of Eq. (7.45) with h = 0. The average over the soft initial conditions of this terms vanishes, but it is present for arbitrary conditions, and it has a nonvanishing correlator. Such a term plays the role of a noise term and can be used in a Boltzmann–Langevin equation to e8ectively perform the averaging over the ultrasoft initial conditions. Note that this averaging is not a trivial task here: While the e8ective theory at the scale gT could be put in a Hamiltonian form, thus providing the Boltzmann weight for the initial conditions, no such a simple structure exists in the e8ective theory at the scale g2 T . Having identi:ed the strategy that we wish to follow, we can go back to general principles and use the Luctuation–dissipation theorem together with the known structure of the collision term in the Boltzmann equation (7.26) in order to deduce the statistics of the noise term in the Boltzmann–Langevin equation. Consider then such an equation: (v · Dx )ab Wb (x; v) + 8{W a (x; v) − W a (x; v)} = v · Ea (x) + a (x; v) :

(7.47)

where a (x; v) is the noise term, to be constrained by the collision term in Eq. (7.32). The latter has the following properties: It is (i) linear in the colour distribution W a (x; v), (ii) local in space and time, (iii) diagonal in colour, (iv) nonlocal in the velocity v, and (v) independent of the gauge mean :elds Aa . Correspondingly, the noise can be chosen as Gaussian, “white” (i.e., local in x = (t; x)), and colourless (i.e., diagonal in colour), but nonlocal in v. That is, its only nontrivial correlator is the two-point function a (x; v) b (x ; v ), which is of the form a (x; v) b (x ; v ) = J(v; v ) *ab *(4) (x − x ) J(v; v )

(7.48)

with independent of the gauge :elds, and therefore also independent of x (since the background :eld is the only source of inhomogeneity in the problem). The following steps are quite similar to the discussion in Section 4.6.3. According to Eq. (7.47), there are two sources for colour excitations Wa (x; v) at the scale g2 T : the mean

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:eld Ea (x) and the noise term a (x; v). The full solution can thus be written as a W a (x; v) = Wind (x; v) + Wa (x; v) ;

(7.49)

where Waind (x; v) is the solution in the absence of noise (i.e., the solution to the Boltzmann equation (7.26)), and Wa (x; v) is a Luctuating piece satisfying (v · Dx )ab Wb (x; v) + 8{Wa (x; v) − Wa (x; v)} = a (x; v) :

(7.50)

Thus, Wa (x; v) is proportional to , and generally also dependent upon the mean :eld A . The colour current is correspondingly decomposed as d. a 2 j (x) = mD (7.51) v (Waind (x; v) + Wa (x; v)) ≡ jind a (x) + Sa (x) ; 4 with Sa (x) denoting the Luctuating current, which acts as a noise term in the Yang–Mills equation: 8 (D F )a (x) = jind a (x) + Sa (x) ;

(7.52)

and generates the thermal correlators A (x)A (y) : : : A/ (z) of the ultrasoft :elds. The correlators of S can be obtained from the original two-point function in Eq. (7.48) by solving the equations of motion. A priori, this is complicated by the nonlinear structure of the equations. However, since the unknown function J(v; v ) is independent of the background :eld Aa , it can be obtained already in the weak :eld limit, that is, by an analysis of the linearized equations of motion. In this limit, jind = %R A , with %R as constructed in Section 7.3.2, while the Luctuation–dissipation theorem (7:43) implies T ∗ 4 (4) Sa (P)Sb (P ) = (2) * (P + P )*ab −2 Im %R (!; p) : (7.53) ! Furthermore, W satis:es the linearized version of Eq. (7.50) (in momentum space, and with colour indices omitted): (v · P)W(P; v) + i8{W(P; v) − W(P; v)} = i (P; v) ; which is formally solved by d.1 i v1 (P; v1 ) ; W(P; v) = v 4 v · P + iC

(7.54)

(7.55)

where (cf. Eq. (7.33)) C(v; v ) = 8{*(2) (v; v ) − *(2) (v; v )} :

8

Note that current conservation becomes more subtle in the presence of the noise [25].

(7.56)

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This, together with Eqs. (7.48) and (7.51), implies d. d.1 d. d.2 ∗ 4 (4) 4 S (P)S (P ) = (2) * (P + P )mD 4 4 4 4 −i i v2 ×v v v v1 J(v1 ; v2 ) v v · P + iC v · P − iC

(7.57)

which is to be compared with Eq. (7.53) and the known expression for the imaginary part of the retarded polarization tensor (cf. Eq. (7.35)): d. d.1 d. d.2 Im %R (!; p) = −!m2D v v 4 4 4 4 1 1 C × v v (v ; v ) v (7.58) 1 1 2 v · P − iC v2 : v · P + iC Clearly, the above equations are consistent with each other provided J(v; v ) =

2T C(v; v ) ; m2D

(7.59)

X which is the result obtained by Bodeker. Incidentally, the above derivation together with Eq. (7.33) show that the noise correlator (7.48) admits the following representation: a (x; v) b (x ; v ) = −

2T *C a (x; v) ; m2D *W b (x ; v )

(7.60)

which is consistent with some general properties discussed in Ref. [251]. The inclusion of thermal Luctuations via a local noise term, like in Eq. (7.47), is very convenient for numerical simulations. In order to compute the thermal correlators of the ultrasoft :elds, it is now suKcient to solve the coupled system of Boltzmann–Langevin and Yang–Mills equations only once, i.e., for a single set of (arbitrary) initial conditions, but for large enough times. Thus, in order to compute, e.g., the 2-point function Ai (t; x)Aj (t ; x ), it is enough the take the product of the solution Ai (t; x) with itself at two space–time points (t1 ; x1 ) and (t2 ; x2 ) such that t1 − t2 = t − t , x1 − x2 = x − x , and t1 and t2 are large enough. In practice, the only numerical calculations within this e8ective theory [28] have been performed until now in the leading logarithmic approximation, where the theory drastically simpli:es [25]: it then reduces to a local stochastic equation for the magnetic :elds, of the form (below, the cross product stands for both the vector product, and the colour commutator): D × B = 5E + ; j

Sia (x)Sb (y) = 2T5*ab *ij *(4) (x − y)

(7.61)

2 =8 and 8 = PNT ln(1=g). Note with the colour conductivity in the LLA (cf. Eq. (7.41)): 5 = !pl 0 0 i that, in this approximation, the noise term Sa (x) in the Yang–Mills equations is both white and Gaussian. This is to be contrasted with the general noise Sa (x) in Eqs. (7.50) – (7.51)

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which in general is a nonlinear functional of the gauge :elds, and has an in:nite series of nonlocal n-point correlators (see, e.g., the two-point function in Eq. (7.57)). Besides being local, the e8ective theory in Eq. (7.61) is also ultraviolet :nite [25,252], thus allowing for numerical simulations which are insensitive to lattice artifacts [28]. It turns out, however, that this LLA is not very accurate when applied to the computation of the hot baryon number violation rate: the numerical result in [28] is only about 20% of the corresponding result obtained in lattice simulations of the full HTL e8ective theory [58,27]. Recently, it has been argued [253,254] that the local form (7.61) of the ultrasoft theory remains valid also to “next-to-leading logarithmic accuracy” (NLLA), i.e., at the next order in an expansion in powers of the inverse logarithm ln−1 ≡ 1=ln(1=g). The only modi:cation refers to the value of the parameter 5, which now must be computed to NLLA. This in turns involves the matching between two calculations: the expansion of the solution to the Boltzmann equation to NLLA [72,254], and a perturbative calculation within the “high energy” sector of the ultrasoft theory [254] (by which we mean loop diagrams with internal momenta of order g2 T ln(1=g) which must be computed with USA-resummed propagators and vertices). The complete result for 5 to NLLA has been obtained in Ref. [254] (see also [255]). Quite remarkably, by using this result within the simpli:ed e8ective theory (7.61), one obtains an estimate for the hot sphaleron rate which is rather close (within 20%) to the HTL result in Refs. [58,27]. 8. Conclusions This report has been mostly concerned with the longwavelength collective excitations of ultrarelativistic plasmas, with emphasis on the high temperature, decon:ned phase of QCD, where the coupling “constant” is small, g(T )1, and the basic degrees of freedom are those which are manifest in the Lagrangian, i.e., the quarks and the gluons. In this regime, the dominant degrees of freedom, the plasma particles, have typical momenta of order T . Other important degrees of freedom are soft, collective excitations, with typical momenta ∼ gT . In this work we have constructed a theory for those collective excitations which carry the quantum numbers of the elementary constituents. The e8ective theory for the collective excitations takes the familiar form of coupled Yang– Mills and Vlasov equations. The separation of scales between hard and soft degrees of freedom leads to kinematical simpli:cations which allowed us to reduce the Dyson Schwinger equations for the Green’s functions to kinetic equations for the plasma particles, by performing a gradient expansion compatible with gauge symmetry. The resulting theory is gauge invariant. In this construction, the gauge coupling plays an essential role which was not recognized in previous works aiming at developing a kinetic theory for the quark–gluon plasma. Aside from the fact that it measures the strength of the coupling, g also characterizes the typical momentum of the collective modes (∼ gT ), and also the amplitude of the :eld oscillations. The solution of the kinetic equations provides the source for Yang–Mills :elds, the so-called induced current, which can be regarded as the generating functional for the hard thermal loops. These are the dominant contributions of the one loop amplitudes at high temperature and soft external momenta, and they need to be resummed on soft internal lines when doing perturbative calculations. Such resummations may be taken into account by a reorganization of perturbation

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theory, sometimes called “HTL perturbation theory”, and this has led to numerous applications. Note however that since the HTL e8ective action is non local, high order calculations within that scheme may become rapidly diKcult. Recent progress indicates that HTL may also be useful in thermodynamical calculations. In the most naive approach, the HTL e8ective action reduces, for the calculation of static quantities, to the three dimensional e8ective action for QCD. This however is of limited use for analytical calculations because of the infrared divergences of the three dimensional theory. However other schemes exist which allow to take into account the full spectral information on the quasiparticles which is correctly coded in the HTL. In particular it was shown recently that self-consistent calculations of the entropy of a quark–gluon plasma are able to reproduce accurately the lattice data for T & 2:5Tc . This suggests that the quasiparticle picture of the quark–gluon plasma remains valid even in regimes where the coupling is not small. Further support of this picture is provided by the calculation of the quasiparticle damping rate. This calculation played an important role in the development of the subject and led in particular to the identi:cation of the hard thermal loops. However HTL are not suKcient to obtain the full answer. We have seen that further resummations are necessary in order to eliminate the IR divergences left over by the HTL resummations. Theses divergences signal the necessity to take into account coherence e8ects related to the fact that particles never come quite on shell between collisions. We have presented various ways to do this. But in spite of the fact that the damping of single particle excitations is unconventional, it remains small in weak coupling. The calculation of the damping rate provides one example of a calculation where one needs to take into account the e8ect of soft thermal Luctuations. Because these can be treated as classical :elds, a possible way to proceed is to use the classical theory developed in this paper. Then the integration of soft Luctuations amounts to averaging over the initial conditions for the classical :elds, with a Boltzmann weight which was explicitly given. In some applications, it is necessary to go beyond and to consider explicitly the e8ective theory for ultrasoft Luctuations, obtained after integrating not only the hard degrees of freedom, but also the soft ones. In this case, no Hamiltonian could be found. Rather the kinetic theory for the hard particles is a Boltzmann equation, and the averaging over the ultrasoft initial conditions is done via a noise term in a Boltzmann Langevin equation.

Acknowledgements During the preparation of this work, we have bene:ted from useful discussions and correspondence with many people, whom we would like to thank: P. Arnold, G. Baym, D. BXodeker, D. Boyanovsky, E. Braaten, P. Danielewicz, H. de Vega, F. Guerin, U. Heinz, H. Heiselberg, K. Kajantie, F. Karsch, M. Laine, M. LeBellac, L. McLerran, G. Moore, S. MrGowczyGnski, B. MXuller, J.Y. Ollitrault, R. Parwani, R. Pisarski, A. Rebhan, K. Rummukainen, D. Schi8, M. Shaposhnikov, D. Schi8, A. Smilga, A. Weldon, and L. Ya8e. We address special thanks to Tony Rebhan for his careful reading of the manuscript and his numerous remarks. Part of this work was done while one of us (E.I.) was a fellow in the Theory Division at CERN, which we thank for hospitality and support.

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Appendix A. Notation and conventions In order to facilitate the reading of this report, we summarize here our notation and conventions, and list some of the most important symbols indicating for each of them the section where it is :rst introduced. We shall always use the Minkowski metric (even within the imaginary time formalism), g = g = diag(1; −1; −1; −1)

(A.1)

and natural units, ˝ = c = 1. We consider a SU(N ) gauge theory with Nf Lavours of massless quarks, whose Lagrangian is: 1 a a S L = − F F + i (iD = )ij 4

j

:

(A.2)

The corresponding action reads S = d 4 x L. In this equation, the colour indices for the adjoint representation (a; b; : : :) run from 1 to N 2 − 1, while those for the fundamental representation (i; j; : : :) run from 1 to N . The sum over the quark Lavours is implicit, and so will be also that over colours whenever this cannot lead to confusion. The generators of the gauge group in di8erent representations are taken to be Hermitian and traceless. They are denoted by t a and T a , respectively, for the fundamental and the adjoint representations, and are normalized thus:

Tr(t a t b ) = 12 *ab ;

Tr(T a T b ) = N*ab :

(A.3)

It follows that (T a )bc = − ifabc ;

Tr(T a T b T c ) = ifabc

N ; 2

T a T a = N;

t a t a = Cf ≡

N2 − 1 ; 2N

(A.4)

where fabc are the structure constants of the group [t a ; t b ] = ifabc t c :

(A.5)

We use, without distinction, upper and lower positions for the color indices. Furthermore, in Eq. (A.2), D = ≡ 8 D , with the usual Dirac matrices 8 satisfying {8 ; 8 } = 2g , and D ≡ 9 + igAa t a the covariant derivative in the fundamental representation. More generally, we shall use the symbol D to denote the covariant derivative in any of the group representations, i.e. D = 9 + igA , where A is a colour matrix, i.e., A ≡ Aa t a in the fundamental representation and A ≡ Aa T a in the adjoint representation. For any matrix O(x) acting in a representation of the colour group, we write [D ; O(x)] ≡ 9 O(x) + ig[A (x); O(x)] :

(A.6)

a is de:ned as The gauge :eld strength tensor F a a F ≡ [D ; D ]=(ig) = F t

a with F = 9 Aa − 9 Aa − gfabc Ab Ac :

(A.7)

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The electric and magnetic :elds are Eai = Fai0 ;

Bai = − (1=2)ijk Fajk :

(A.8) h(x) = exp(i& a (x)t a ),

and read The gauge transformations are implemented by unitary matrices i A → hA h−1 − h9 h−1 ; → h ; D → hD : (A.9) g Whenever we need to distinguish between various representations of the colour group, we use a tilde to denote quantities in the adjoint representation. For instance, we write D˜ = 9 + igAa T a , ˜ = exp(i& a (x)T a ), so that, in the gauge transformation (A.9), and h(x) Eai (x) → h˜ab (x)Ebi (x); Bai (x) → h˜ab (x)Bbi (x): (A.10) If, in the same transformation, the matrix O(x) transforms covariantly, O(x) → h(x)O(x)h−1 (x), then the same holds for its covariant derivative [D ; O(x)] → h(x)[D ; O(x)]h−1 (x). When working in the imaginary-time formalism, we write x = (x0 ; x) = (t0 − i,; x), with t0 real and arbitrary, and 0 6 ,4 6 ; therefore, 90 = i9, and d x0 = − id,. We de:ne the Euclidean action by writing eiS ≡ ei d xL ≡ e−SE , with 1 a a S 3 SE = d,d x =) : (A.11) F F + (−iD 4 0 Unless it may induce confusion, we generally omit the subscript E on the imaginary-time quantities. The thermal occupation numbers for gluons, N (k0 ), and quarks, n(k0 ), are written as 1 1 N (k0 ) = k0 ; n(k0 ) = (k0 −) ; (A.12) e −1 e +1 where ≡ 1=T , and is a chemical potential. In fact, we consider here mostly a plasma with = 0, but many of the results are easy to generalize to arbitrary . We now pursue with an enumeration of the symbols used in the text, presented here in alphabetical order. In parentheses we indicate the sections where they are introduced. Aa (x): the gauge vector potential, usually identi:ed with the soft classical background :eld, i.e., the gauge mean 4eld (1:1; 3:1; 3:2:1). aa (x): the (typically hard) quantum Luctuations of the gauge :eld (3:2:1). = 1=T : the inverse temperature. Ca (x; v), Cab (x; x ; v; v ): the collision term for the colour Boltzmann equation and its kernel (7:3:1). D, Dj : the statistical density operator, in or out of thermal equilibrium (2; 2:2:1). D [A] = 9 + igA : the covariant derivative with the gauge :eld Aa . "ind (x): the induced fermionic source in QED or QCD (3:2:2). Eia , Aai , Wa ; Ga , H: initial conditions for the classical equations of motion in the HTL e8ective theory; the associated Gauss’ operator and Hamiltonian (4:4:3). *f(k; X ), *n(k; X ), *N (k; X ): o8-equilibrium density matrices for hard electrons (1:2), quarks (3:4:2) and gluons (3:4:1). a = 9 Aa − 9 Aa − gf abc Ab Ac : the gauge :eld strength tensor. F a G a , G˜ : gauge-:xing terms in the quantum path integral for a gauge theory (3:2:1).

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G, G0 , G (n) , G : time-ordered Green’s functions (2-point, free, n-point), in real- or complextime, for scalars (2:1:1; 2:2:2) and gluons (3:2:1). GR , GA : (i) retarded and advanced 2-point functions, in or out of thermal equilibrium, for scalars (2:1:2; 2:2:2) or gluons (7:1); (ii) retarded and advanced Green’s functions for the drift operator in the kinetic equations (4:1:1). Gcl : the classical thermal 2-point correlation, for scalars (2:2:4) or gluons (4:4:3; 7:4). G ¡ , G ¿ : analytic 2-point functions, for scalars (2:1:2; 2:2:2) and gluons (3:2:3). G ¡ (k; X ), G ¿ (k; X ), GR (k; X ), GA (k; X ): various Wigner functions for scalars (2:3:1). G¡ (k; X ), G¿ (k; X ), GR (k; X ), GA (k; X ): various Wigner functions for gluons (3:3:2; 7:1). ∗ G , ∗ M , ∗ M : the HTL-resummed propagator for soft gluons, and its longitudinal and L T transverse components (4:3:1; B:1:3). 7: the discontinuity of the self-energy (2:1:3; 2:3:2). 8, 8(p; t): the quasiparticle damping rate (2:1:3; 2:3:4; 2:3:5; 6:1). 7, 7ind , 7HTL , 7A , 7# : the e8ective action, its induced piece, the HTL e8ective action and its various components (5:5:1; 5:5:2). H , H0 , H1 , Hj (t): (i) the Hamiltonian for a generic :eld theory (total, free, interacting, in the presence of an external source) (2; 2:1; 2:2:1; 2:2:3); (ii) the Hamiltonian for the HTL e8ective theory (4:4:3). h(x) = exp(i&(x)), with & ≡ & a T a or & ≡ & a t a : a SU(N ) gauge transformation. I(v; v ): noise–noise correlator for the colour Boltzmann–Langevin equation (7.4). j (x): the external source driving the system out of equilibrium (2; 2:2), and also the argument of the generating functional Z[j] (2:2:2 and 3:2:1). j ind (x): the induced source in the scalar :eld theory (2:2:3). jind (x), jinda (x): the induced electromagnetic (1:2) and colour (3:2:2) currents. jA , j jf , jg : the colour current induced by the gauge (A) or fermionic ( ) mean :elds acting on hard quarks (f) or gluons (g) (3:2:2). K a , H a , Ka , Ha : the “abnormal” propagators and their Wigner transforms (3:2:3; 3:3:2). : separation scale between hard and soft momenta (2:1:4; 4:4:3). =(k; X ): “abnormal” density matrix (3:5:1). mD , !pl , !0 , m∞ , M∞ : Debye mass, plasma frequencies (for gluons and fermions), asymptotic masses (for gluons and fermions) (4:1:2; 4:1:3; 4:3:1). mmag : the magnetic screening mass (5:4:3). : (i) infrared cuto8 (1:1; 6:1); (ii) separation scale between soft and ultrasoft momenta (7). Nk , nk : thermal occupation factors for single-particle bosonic, or fermionic, states (1.1). a , Sa : colour “noise terms”, for the Boltzmann–Langevin (7.4) and, respectively, Yang–Mills (4:4:3; 7:4) equations. ;, ;0 , E, Ecl : scalar :elds: the quantum :eld (2), its static Matsubara mode (2:1:4), the average :eld (2:2:1; 2:2:3), the classical :eld (2:2:4). E(v · v ): collisional cross-section for hard particles with velocities v and v (7.2). ab : the photon (1.2) or gluon (4:2; 5:3:2; 7:3:2; B:1) polarization tensor. % , % , S : the (typically hard) quantum fermionic :elds (3:2:1). S the soft fermionic mean :elds (3:2:1). #, #: /0 , /: the (free) spectral density, in or out of thermal equilibrium (2:1:1; 2:1:3; 2:3:2).

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∗/

∗ ∗ , /L , /T : the gluon spectral density in the ∗ S, S: the fermion propagator: in general (3:2:3)

HTL approximation (4:4:3; B:1:3). and in the HTL approximation (4:3:1; B:2:3). ?, ?¿ , ?¡ , ?R , ?A : self-energies for scalars (2:1:3; 2:2:3), quarks (3:3:1), or gluons (7.1); the associated Wigner functions are denoted as ?¡ (k; X ), etc. (2:3:2; 7:1). 5i , 5: electromagnetic (1:4; 3:1) and colour (7:3:3) conductivities. ˜ TC : symbols for operator ordering, in real- or complex-time (2:1; 2:2; 2:2:2). T, T, , T, U (x; y|A), U (x; y): Parallel transporters, or Wilson lines (1:3; 3:1; 3:3:2; 4:1:1). U (t; t0 ), Uj (t2 ; t1 ), Uj (z; t0 ): evolution operators, in real- or complex-time (2:1; 2:2:1; 2:2:2). W (x; v), Wa (x; v), Wa (x; v): reduced density matrices for charge (1.2) or colour (4.1.2, 7.2) oscillations of the hard particles. !n = 2nT , or !n = (2n + 1)T with integer n: Matsubara frequencies for bosons or fermions (2.1.1, Appendix B). Z, Zcl : the thermal partition function: quantum (2.1) and classical (2:1:4; 2:2:4). Z[j], Zcl [J ]: the generating functional of thermal correlations: quantum (2:2:2; 3:2:1) and classical (4:4:3). ˜ A]: the generating functional in the background :eld gauge (3:2:1). Z[j; a La and LS : the anticommuting “ghost” :elds (3:2:1). Appendix B. Diagrammatic calculations of hard thermal loops In this appendix, we present a few explicit calculations of Feynman diagrams in the imaginary time formalism. In particular, we obtain in this fashion the gluon and fermion self-energies in the hard thermal loop (HTL) approximation. Unless otherwise stated, all calculations are for massless QCD at zero chemical potential. Ultraviolet divergences are regulated by dimensional continuation (4 → d = 4 − 2), but we keep the fermions as four-component objects, Tr(8 8 ) = 4g . We denote a generic four-momentum as k = (k0 ; k), k0 = i!n = inT , with n even (odd) for bosonic (fermionic) :elds. The scalar product is de:ned with the Minkowski metric, so that, for instance, k 2 = k02 − k2 = − !n2 − k2 . The measure of loop integrals is denoted by the following condensed notation: {d k } ≡ T [d k] ≡ T (dk); (dk) ; (B.1) n;even

where

n;odd

d d−1 k : (2)d−1 For a free scalar particle with mass m, the Matsubara propagator is given by Eq. (2.30), namely (in this appendix, we prefer to denote this propagator as M, rather than G0 ): 1 −1 M(k) = 2 = ; k0 = i!n = i2nT ; (B.2) !n + k2 + m2 k 2 − m2 while for a massive fermion we have −1 ˜ ˜ S0 (k) = (k= + m)M(k); M(k) ≡ 2 ; k0 = i!n = i(2n + 1)T ; (B.3) k − m2 (dk) ≡

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with k= ≡ 8 k = i!n 80 − k · 8. Note that the only di8erence between the functions M(k) and ˜ M(k) lies in the odd or even character of the corresponding Matsubara frequencies. The gluon propagator in the covariant gauge, with a gauge-:xing term (9 Aa )2 =2, is 0 G (k) = − g M(k) + ( − 1)k k M2 (k) ;

(B.4)

with M(k) given by Eq. (B.2) with m = 0. The Coulomb gauge propagator will be also used. When computing Feynman graphs, we have to calculate sums over the internal Matsubara frequencies. These may be done by appropriate contour integration, or by introducing a mixed Fourier representation of the propagators, as we shall explain shortly. As an example of the use of contour integration, let us compute the imaginary-time propagator M(,) by performing the following frequency sum (cf. Eq. (2.26)): M(,; k) = T e−i!n , M(i!n ; k) ; (B.5) n

where !n = 2nT . For 0 ¡ , ¡ , we have M(,) = M¿ (,), and we can replace the sum in Eq. (B.5) by d! e−!, ¿ M (,; k) = − M(!; k) ; (B.6) 2i e−! − 1 √ where M(!; k) = (j2k − !2 )−1 , jk = k2 + m2 , and the integration contour is indicated in Fig. 29. Note that the choice of the function is such that one can close the contour without getting contribution from grand circles at in:nity. The integration is then trivial and yields: 1 M¿ (,; k) = {(1 + Nk )e−jk , + Nk ejk , } ; (B.7) 2jk where Nk ≡ N (jk ) = 1=(ejk − 1). For , ¡ 0 one could use instead d! e−!, ¡ M (,) = M(!) ; 2i e! − 1 which gives 1 {N e−jk , + (1 + Nk )ejk , } : M¡ (,; k) = 2jk k

(B.8)

(B.9)

By putting together the previous results, we derive an expression for M(,; k) valid for − 6 , 6 : +∞ d k0 −k0 , M(,; k) = /0 (k)(&(,) + N (k0 )) ; (B.10) e −∞ 2 where /0 (k) is the spectral density for a free particle, Eq. (2.32). For fermions, we obtain similarly +∞ d k0 −k0 , ˜ M(,; k) = /0 (k)(&(,) − n(k0 )) : (B.11) e −∞ 2 where n(k0 ) = 1=(ek0 + 1). By using the simple identities ek0 N (k0 ) = 1 + N (k0 );

ek0 n(k0 ) = 1 − n(k0 ) ;

(B.12)

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Fig. 29. Integration contour for Eqs. (B.5) – (B.6). Dashed line: original contour used to reproduce the Matsubara sum. Solid line: deformed contour used to calculate the integral. The dots on the imaginary axis correspond to Matsubara frequencies !n , the crosses on the real axis to ±jk .

˜ one can easily verify that the functions M(,) and M(,) obey the expected (anti)periodicity conditions for − 6 , 6 . For instance, for 0 ¡ , 6 : ˜ − ; k) = − M(,; ˜ k) : M(, − ; k) = M(,; k); M(, (B.13) The corresponding representations for the quark and for the gluon propagators are then (cf. Eqs. (B.3) and (B.4)): +∞ d k0 −k0 , S0 (,; k) = /0 (k)(k= + m)(&(,) − n(k0 )) ; e −∞ 2 +∞ d k0 −k0 , 0 / (k)(&(,) + N (k0 )) ; (B.14) G (,; k) = e −∞ 2 where / (k) = /F (k) + (1 − )/ (k) ;

(B.15)

and /F (k) = − g 2(k0 )*(k 2 ) is the gluon spectral density in Feynman gauge ( = 1), while / (k) ≡ −2(k0 )k k * (k 2 )[* (k 2 ) is the derivative of *(k 2 ) with respect to k 2 ]. In the Feynman 0 (,; k) reduces to −g M(,; k). gauge, G

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The mixed Fourier representation of the propagators given by Eqs. (B.10) – (B.14) can be used to facilitate the evaluation of Matsubara sums in Feynman diagrams. To this aim, we write, for instance, M(i!n ; k) = d, ei!n , M¿ (,; k) ; (B.16) 0

and perform the sums over Matsubara frequencies by using (with integer l): ei!n , = (±)l *(, − l) ; T n

(B.17)

l

where the plus (minus) sign corresponds to even (odd) Matsubara frequencies. Explicit examples of the procedure will be given below. As a trivial application, we use Eqs. (B.16), (B.17) and (B.7) to obtain (henceforth, we set m = 0, and therefore jk = k): 1 + 2Nk 1 T = M¿ (, = 0; k) = ; (B.18) 2 2 !n + k 2k n so that

[d k]M(k) =

=

d d−1 k 1 + 2Nk 2k (2)d−1 d 3 k Nk T 2 = : (2)3 k 12

(B.19)

In going from the :rst to the second line of Eq. (B.19), we have used the fact that, under dimensional regularization, d d−1 k 1 =0 (B.20) (2)d−1 2k and we have set d = 4 in the evaluation of the temperature-dependent piece, which is UV :nite. (If some other UV regularization scheme is being used—e.g., an upper cut-o8 —then the zero-temperature piece in Eq. (B.19) gives a nonvanishing contribution, which is quadratically divergent as → ∞. This divergence can be removed by renormalization at T = 0, and the :nal result is the same as above. See also the discussion in Section 2.3.3, after Eq. (2.165).) The integral (B.19) appears, for instance, in the calculation of the tadpoles in Fig. 2 and in Fig. 30c below, or in the evaluation of the gauge :eld Luctuations in Section B.1.4. In an entirely similar way, we obtain d d−1 k 1 − 2nk d 3 k nk T2 ˜ {d k }M(k) − − = = = : (B.21) (2)3 k 24 (2)d−1 2k Finally, let us consider the Fourier transform of the 2-point function M¡ (t; k) obtained from Eq. (B.9) by analytic continuation to real time. Let us denote by M¡ T (t; k) the :nite temperature contribution. We have Nk M¡ cos(kt) ; (B.22) T (t; k) = k

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where we have taken jk = k. The Fourier transform of this expression is easily obtained in the form 1 x−t x+t ¡ MT (t; x) = 2 + : (B.23) 4 x (n)2 + (x + t)2 (n)2 + (x − t)2 n¿1

By using the summation formula: 1 y 1 = + ; coth n2 2 + y2 2y2 2y

(B.24)

n¿0

one gets T (B.25) (h(Tx+ ) + h(Tx− )) ; 8x with x± ≡ x ± t, and: 1 h(u) ≡ coth(u) − : (B.26) u For u1, h(u) ≈ u=3, so Eq. (B.25) yields T2 M¡ (t = 0; x = 0) = ; (B.27) T 12 in agreement with Eq. (B.19). For u1; h(u) = 1 − 1=u + O(e−u ), so at large x the equal-time (t = 0) 2-point function decreases as M¡ 3 1 T (t = 0; x)

(B.28) for x1=T : ¡ MT (t = 0; x = 0) xT M¡ T (t; x) =

This slow decrease (∼ 1=x) is due to the static (!n = 0) Matsubara mode. To clearly see this, let us rederive Eq. (B.28) starting with Eq. (2.26) (with , = 0) where the contributions of the various Matsubara modes are explicitly separated. We have d3 k T −|!n |x eik·x M(0; x) = T = e (2)3 !n2 + k2 4x n n T T −!n x = 1+2 e = coth(Tx) ; (B.29) 4x 4x n¿1

where the :rst term within the braces in the second line, ∼ 1=x, is the contribution of the static mode. (Note that M(0; x) = M¡ T (0; x) + MT =0 (0; x) where the vacuum contribution is MT =0 (0; x) = 1=42 x2 :) B.1. The soft gluon polarization tensor ab (p) is given by the four diagrams in To one-loop order, the gluon polarization tensor % Fig. 30, which we shall evaluate in the hard thermal loop approximation, valid when the external gluon line carries energy and momentum of order gT . Since the colour structure of this tensor ab = *ab % , we shall omit colour indices in what follows. For complementarity with is trivial, %

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Fig. 30. The one-loop gluon polarization tensor.

the analysis in Section 3, where the Vlasov equations which determine the HTL % have been constructed in the Coulomb gauge, here we shall rather work in the Feynman gauge (i.e., the covariant gauge with = 1; cf. Eq. (B.4)). As already emphasized in the main text, the :nal result for the HTL will be gauge-independent. B.1.1. The quark loop (a) Consider :rst the contribution of the quark loop in Fig. 30a, which we denote by % . A straightforward application of the Feynman rules gives, for Nf quark Lavours, g2 Nf (a) {d k } Tr {8 S0 (i!r ; k)8 S0 (i!r − i!n ; k − p)} ; % (i!n ; p) = (B.30) 2 with p0 = i!n = i2nT and k0 = i!r = i(2r+1)T , with integers n and r. The trace in Eq. (B.30) refers to spin variables only (the colour trace has been already evaluated and it is responsible (a) (a) for the factor 1=2). It is easily veri:ed that % (p) is transverse, p % (p) = 0, so that it has only two independent components (see below). In the calculation, the external energy p0 = i!n is purely imaginary and discrete to start with. To implement the property that p0 is soft, an analytic continuation to real energy is necessary. This becomes possible after performing the sum over the internal Matsubara frequency i!r . To do so, we use the mixed Fourier representations of the two quark propagators in Eq. (B.30) together with the identity (B.17), and obtain g2 Nf (a) % (i!n ; p) = − (dk) d, e−i!n , Tr {8 S0 ( − ,; k)8 S0 (,; k − p)} : (B.31) 2 0 We then use the integral representation (B.14) for the two propagators in Eq. (B.31), and denote by k0 and q0 the respective energy variables. The ,-integral is then easily evaluated e−(i!n +q0 ) − e−k0 e−q0 − e−k0 d, e−i!n , e−k0 (−,) e−q0 , = = ; (B.32) k0 − q0 − i!n k0 − q0 − i!n 0

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where we used the fact that exp(−i!n ) = 1. By also using the identity (1 − n(k0 ))(1 − n(q0 ))(e−q0 − e−k0 ) = n(q0 ) − n(k0 ) ; we :nally get (a) % (i!n ; p) =

g2 Nf 2

(dk)

×Tr (8 k=8 q=)

∞

−∞

d k0 2

∞

−∞

(B.33)

dq0 /0 (k)/0 (q) 2

n(k0 ) − n(q0 ) ; k0 − q0 − i!n

(B.34)

with the notations k = (k0 ; k), q = (q0 ; q) and q ≡ k − p. Expression (B.34) can now be continued in the complex energy plane by simply replacing i!n → p0 with p0 o8 the real axis: ∞ d k0 ∞ dq0 (a) 2 % (p0 ; p) = 2g Nf (dk) /0 (k)/0 (q) −∞ 2 −∞ 2 ×[k q + q k − g (k · q)]

n(k0 ) − n(q0 ) : k0 − q0 − p0

(B.35)

In going from Eq. (B.34) to Eq. (B.35), we have also performed the spin trace. Let us now focus on the spatial components of the polarization tensor (B.35). After integration over k0 and q0 , one obtains g2 Nf 1 (a) %ij (p0 ; p) = (dk) 2 jk jq n(jk ) − n(jq ) n(jk ) − n(jq ) × (ki qj + qi kj + *ij (jk jq − k · q)) + jk − jq + p0 jk − jq − p0 + (ki qj + qi kj − *ij (jk jq + k · q)) 1 − n(jk ) − n(jq ) 1 − n(jk ) − n(jq ) × + : jk + jq + p0 jk + jq − p0

(B.36)

As T → 0, the statistical factors vanish, and k i qj + qi k j − *ij (jk jq + k · q) g2 Nf 1 1 ij %T=0 (p) = (dk) + : 2 jk j q jk + jq + p0 jk + jq − p0 (B.37) In four space–time dimensions, this expression develops ultraviolet divergences, which are eliminated by the gluon wave-function renormalization [256]. The thermal contribution %T ≡ % − %T=0 has no ultraviolet divergences since the statistical factors n(jk ) are exponentially decreasing for k T . Thus, in evaluating %T , we can set d = 4. To isolate the HTL in Eq. (B.36), we set p0 → ! + i", with real ! and " → 0+ (retarded boundary conditions), and assume that both ! and p ≡ |p| are of the order gT . By de:nition, the hard thermal loop is the leading piece in the expansion of % (p) in powers of g, including the assumed g-dependence of the external four-momentum [19]. It can be veri:ed that the HTL in

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Eq. (B.36) arises entirely from the integration over hard loop momenta k ∼ T . The contribution of the soft momenta k . gT is suppressed (here, by a factor g2 ) because of the smallness of the associated phase space. Furthermore, the contribution of the very high momenta, k T , is exponentially suppressed by the thermal occupation numbers. This last argument does not apply to the vacuum piece, Eq. (B.37); however, after renormalization, the :nite contribution of %T=0 (p) is of the order g2 p2 , which for p ∼ gT is down by a factor of g2 as compared to the HTL (we anticipate that the latter is O(g2 T 2 )). Let us evaluate now the contribution of the hard loop momenta k ∼ T . Since p ∼ gT k, we can write jq ≡ |k − p| k − v · p

(B.38)

which allows us to simplify the energy denominators in Eq. (B.36) as follows: jk + jq ± ! 2k;

jk − j q ± ! v · p ± ! :

(B.39)

In these equations, v ≡ k=k denotes the velocity of the hard particle, |v| = 1. Note that we have two types of energy denominators: Those involving the di;erence of the two internal quark energies, which are soft (jk − jq v · p ∼ gT ), and those involving the sum of the respective energies, which are hard (jk + jq 2k ∼ T ). The soft denominators, whose form is reminiscent of the Bloch–Nordsieck approximation described in Section 6.3, are associated with the scattering of the soft gluon on the hard thermal quarks. Because of the Pauli principle, such processes occur with the following statistical weight: dn ; (B.40) dk that is, they are suppressed by one power of g at soft momenta p ∼ gT . Because of this suppression, they contribute to the same order as the terms involving hard denominators, associated to vacuum-like processes where the soft gluon :eld turns into a virtual quark-antiquark pair. These are accompanied by statistical factors like n(jk )[1 − n(jq )] − [1 − n(jk )]n(jq ) = n(jk ) − n(jq ) v · p

[1 − n(jk )][1 − n(jq )] − n(jk )n(jq ) = 1 − n(jk ) − n(jq ) 1 − 2n(k) :

(B.41)

Note however that, to the order of interest, the hard denominators are independent of the external energy and momentum, so that they give only a constant contribution to % . After performing the following simpli:cations: ki qj + qi kj + *ij (jk jq − k · q) 2ki kj ; ki qj + qi kj − *ij (jk jq + k · q) 2(ki kj − *ij k 2 ) ; we obtain %ij(a) (!; p) ≈ −2g2 Nf g2 T 2 Nf = 6

ki kj dn v · p ki kj − *ij k 2 n(k) (dk) + k2 d k ! − v · p k2 k d. !vi vj ; 4 ! − v · p

(B.42)

(B.43)

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where the second line follows from the :rst one by using: ∞ 1 ∞ dn 2 T 2 d k kn(k) = − d k k2 = : (B.44) 2 0 dk 12 0 The angular integral d. runs over all the directions of the unit vector v. (a) The other space–time components of % can be computed in a similar fashion. The complete result to leading order in g reads v v g2 T 2 Nf d. (a) 0 0 % (!; p) ≈ −* * + ! : (B.45) 6 4 ! − v · p + i" This coincides with the corresponding result of the kinetic theory (namely, the quark piece of Eq. (5.34)). B.1.2. The ghost and gluon loops Consider now the other pieces of the one-loop polarization tensor, as given by the diagrams with one gluon or ghost loop in Figs. 30b–d. The tadpole diagram (Fig. 30c) is easily evaluated as (in Feynman’s gauge): (c) 2 % = − g g (d − 1)N [d k]M(k) : (B.46) It is momentum-independent. The factor N (number of colours) arises from the colour trace, while the factor (d − 1) originates from the four-gluon vertex. The contribution of the gluon loop in Fig. 30b is g2 N (b) 0 0 % (p) = [d k]75 (−p + k; p; −k)G0/ (k)7/ " (−k; p; −p + k)G0"5 (p − k) ; 2 (B.47) 0 where 7 / is the bare three-gluon vertex, 0 7 / (p; q; k) = g (p − q)/ + g / (q − k) + g/ (k − p) :

(B.48)

Instead of performing the Matsubara sum directly in Eq. (B.47), which would be complicated by the energy dependence of the vertices, we follow Ref. [19] and make :rst some of the simpli:cations which are allowed as long as we are interested only in the hard thermal loop. Since, by assumption, the external momentum p is soft, while the integral over k is dominated by hard internal momenta, the terms linear in p in the three-gluon vertex can be neglected next to those linear in k: 0 0 75 (−p + k; p; −k) 75 (k; 0; −k) = 7P5 k P ;

(B.49)

with the notation 7 / ≡ 2g g/ − g/ g − g g / :

(B.50)

(The reader might doubt of the validity of Eq. (B.49) at this stage since, strictly speaking, the external energy is generally hard to start with, p0 = i2nT . Note, however, that p0 is

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a dummy variable in the summation over the internal Matsubara frequencies k0 . After the analytic continuation, p0 → ! becomes soft, with ! ∼ gT , while k0 remains hard, k0 ∼ T .) Since 7P5 7 /" g/ g"5 = 4(d − 2)gP g + 2(gP g + gP g ) ;

(B.51)

0 (k) = − g M(k) in Feynman’s gauge, we deduce that and G g2 N (b) % (p) ≈ − [d k]((4d − 6)k k + 2g k 2 )M(k)M(p − k) : (B.52) 2 After performing similar manipulations on the ghost loop in Fig. 30d, we obtain (in covariant gauges, the ghost propagator coincides with the scalar propagator M(k)): (d) 2 % (p) ≈ g N [d k]k k M(k)M(p − k) : (B.53)

By adding together Eqs. (B.46), (B.52) and (B.53), we end up with the following expression for the one-gluon-loop polarization tensor: (g) % (p) ≈ −(d − 2)g2 N [d k]{2 k k M(k)M(p − k) + g M(k)} ; (B.54) where the upperscript g stays for “gluons”. Although this has been derived here in Feynman’s gauge, the :nal result (B.54) is actually gauge-:xing independent [39,40,19], as also shown by the kinetic theory [23]. The factor d − 2 = 2 in Eq. (B.54) shows that only the two physical, transverse, degrees of freedom of the hard gluons are involved in the hard thermal loop. The contributions of the unphysical degrees of freedom (longitudinal gluons and ghosts) mutually cancel in the sum of Eqs. (B.46), (B.52) and (B.53). At this point, it is interesting to consider also the contribution of the quark loop in Fig. 30a that would have been obtained if, before summing over the Matsubara frequencies, we had implemented the same kinematical approximations as above. For the quark loop in Eq. (B.30), this amounts to the replacement Tr(8 k=8 (k= − p=)) 4(2k k − g k 2 ) ; ˜ and one gets (we recall that S(k) = k=M(k)) (a) ˜ M(k ˜ − p) + g M(k) ˜ }: (p) ≈ 2g2 Nf {d k }{2 k k M(k) %

(B.55) (B.56)

The similitude between Eqs. (B.56) and (B.54) suggests that the HTL content of Eq. (B.54) can be obtained by following the same steps as for the quark loop in Section B.1.1. The resulting expression is identical to Eq. (B.45), with the factor g2 T 2 Nf =6 replaced by g2 T 2 N=3. Thus, the complete hard thermal loop for the soft gluon self-energy reads [39,40] v v d. 2 0 0 % (!; p) = mD −* * + ! (B.57) 4 ! − v · p + i" with the Debye mass in Eq. (4.13). Eq. (B.57) coincides with expression (5.34) derived from kinetic theory. The small imaginary part i" in the denominator of Eq. (B.57) implements the retarded boundary conditions. This is relevant only for space-like (!2 ¡ p2 ) external momenta, since it is only

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Fig. 31. Illustration of the Landau damping mechanism: a space-like photon (or gluon) can be absorbed or emitted by an on-shell thermal fermion.

for such momenta that the energy denominator ! − v · p can vanish. In that case, the polarization tensor (B.57) develops an imaginary part, d. 2 Im % (!; p) = − mD ! (B.58) v v *(! − v · p) ; 4 which describes the absorption or the emission of a space-like gluon, with four-momentum p = (!; p), by a hard particle (quark or gluon) from the thermal bath (see, Fig. 31 for an example). According to Eq. (B.58), Im % (!; p) vanishes linearly in the static limit, ! → 0. This may be easily understood by inspection of the thermal phase space available for the processes in Fig. 31. This is proportional to n1 (1 − n2 ) − (1 − n1 )n2 = n1 − n2 ;

(B.59)

where n1 and n2 are the statistical factors for the two thermal fermions, with energies 1 and 2 = 1 + !, respectively. As ! → 0, we may write n1 − n2 −!(dn=d1 ), which vanishes linearly with !. B.1.3. The HTL gluon propagator In order to construct the propagator of the soft gluon in the HTL approximation, we have to invert the equation ∗

−1 −1 G ≡ G0 + % ;

(B.60)

where % is the polarization tensor of Eq. (B.57), which is transverse: p % (p) = 0 :

(B.61)

Actually, this property holds for the whole one-loop contribution to % [86,40], but, in contrast to what happens at zero temperature, it is generally not satis:ed (except in ghost-free gauges, like axial gauges) beyond the one-loop approximation [171,174]. In order to invert Eq. (B.60), we need to :x the gauge. The physical interpretation is more transparent in the Coulomb gauge, where the only nontrivial components of ∗ G are the electric (or longitudinal) one, ∗ G00 (p0 ; p) ≡ ∗ ML (p0 ; p), and the magnetic (or transverse) one, ∗ G (p ; p) = (* − p ˆ i pˆ j ) ∗ MT (p0 ; p). At tree-level, we have ML (p0 ; p) = − 1=p2 , correspondij ij 0 ing to the instantaneous Coulomb interaction, and MT (p0 ; p) = − 1=(p02 − p2 ), whose poles at p0 = ± p are associated with the massless transverse gluon excitations.

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Being transverse, the polarization tensor (B.57) is determined by only two scalar functions, which we choose to be its longitudinal (L) and transverse (T) components with respect to p. Speci:cally, we write p0 pi %00 (p0 ; p) = − %L (p0 ; p); %0i (p0 ; p) = − 2 %L (p0 ; p) ; p %ij (p0 ; p) = (*ij − pˆ i pˆ j )%T (p0 ; p) − pˆ i pˆ j

p02 %L (p0 ; p) ; p2

(B.62)

where p = |p| and pˆ i = pi =p. These de:nitions are appropriate for the Coulomb gauge. The explicit forms of the scalar functions %L; T (p0 ; p) follow easily from Eqs. (B.57) and (B.62): %L (p0 ; p) = m2D (1 − Q(p0 =p)) ; p2 − p2 m2 p 2 %T (p0 ; p) = D 02 1 − 0 2 Q(p0 =p) 2 p p0

(B.63)

with Q(x) ≡

x x+1 ln : 2 x−1

(B.64)

For real energy p0 = ! + i" and space-like momenta (|!| ¡ p), the function Q(!=p) has a nonvanishing imaginary part: ! Im Q(!=p) = − (B.65) &(p − |!|) : 2p Correspondingly, the polarization functions %L and %T acquire imaginary parts which describe the Landau damping of soft space-like gluons. At high temperature, and in the hard thermal loop approximation, we have then ∗

ML (p0 ; p) =

p2

−1

+ %L (p0 ; p)

;

∗

MT (p0 ; p) =

p02

−

p2

−1 : − %T (p0 ; p)

These functions can be given the following spectral representations ∞ dp0 ∗/T (p0 ; p) ∗ MT (!; p) = ; p0 − ! −∞ 2 ∞ 1 dp0 ∗/L (p0 ; p) ∗ ML (!; p) = − 2 + ; p p0 − ! −∞ 2

(B.66)

(B.67)

where ∗/L and ∗/T are the corresponding spectral densities, ∗

/L; T (p0 ; p) = 2 Im ∗ ML; T (p0 + i"; p) :

(B.68)

Note the subtraction performed in the spectral representation of ∗ ML (!; p): this is necessary since ∗ ML (!; p) → −1=p2 as |!| → ∞. By taking ! → 0 in Eqs. (B.67), and using

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∗ M (0; p) = L

− 1=(p2 + m2D ) and

∗ M (0; p) = 1=p2 , T

one obtains the following “sum rules”:

dp0 ∗ 1 1 /L (p0 ; p) = 2 − 2 ; 2p0 p p + m2D 1 dp0 ∗ /T (p0 ; p) = 2 : 2p0 p

(B.69)

Besides, the transverse density ∗/T satis:es the usual sum-rule (2.47): dp0 ∗ p0 /T (p0 ; p) = 1 : 2

(B.70)

The spectral functions ∗/L and ∗/T have the following structure (with s = L or T): ∗

/s (p0 ; p) ≡ 2 Im ∗ Ms (p0 + i"; p) = 2(p0 )zs (p)*(p02 − !s2 (p)) + s (p0 ; p)&(p2 − p02 ) ;

(B.71)

where, in the second line, the *-function corresponds to the (time-like) poles of the resumed propagator ∗ Ms (p0 ; p), with energy p0 = ± !s (p) and residue zs (p) (cf. Section 4.3.1), while the function s (p0 ; p), with support at space-like momenta, corresponds to Landau damping (cf. Section 4.3.3). Speci:cally, the mass-shell residues are de:ned by −zs (p) ∗ for p02 ≈ !s2 (p) ; Ms (p0 ; p) ≈ 2 (B.72) 2 p0 − !s (p) which, together with the pole condition ∗ M−1 s (p0 = ± !s ; p) = 0 and Eqs. (B.63) for %T and %L , implies the following expressions for zT and zL : zT−1 (p) =

2!T2 (!T2 − p2 ) ; m2D !T2 − (!T2 − p2 )2

zL−1 (p) =

2!L2 (!L2 − p2 ) : p2 (m2D + p2 − !L2 )

(B.73)

It can be veri:ed from the above formulae that the residues zs (p) are positive √ functions [75], with the following limiting behaviour: At small p; p!pl (with !pl ≡ mD = 3 the frequency of the longwavelength plasma oscillations; cf. Section 4.3.1), zT (p) ≈ 1 −

p2 ; 2 5!pl

zL (p) ≈

2 !pl ; p2

while for large momenta, p!pl , 2 3!pl 8p2 ln −3 ; zT (p) ≈ 1 − 2 4p2 3!pl

(B.74) 8p2 2p2 zL (p) ≈ exp − 2 − 2 : 2 3!pl 3!pl

(B.75)

Thus, as mentioned in Section 4.3.1, the longitudinal mode is exponentially suppressed at pgT . The functions L and T are explicitly given by (cf. Eq. (B.63)): p0 L (p0 ; p) = m2D | ∗ ML (p0 ; p)|2 ; p T (p0 ; p) = m2D

p0 (p2 − p02 ) ∗ | MT (p0 ; p)|2 ; 2p3

(B.76)

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and satisfy the following sum-rules, which follow by inserting the decomposition (B.71) in Eqs. (B.69) and (B.70): 1 dp0 1 zL (p) L (p0 ; p) = 2 − 2 − ; 2p0 p p + m2D !L2 1 dp0 zT (p) T (p0 ; p) = 2 − ; 2p0 p !T2 dp0 (B.77) p0 T (p0 ; p) = 1 − zT (p) : 2 B.1.4. The gauge 4eld Buctuations We can use the spectral functions that we have just obtained to calculate the magnitude of the gauge :eld Luctuations. These are given by d4 k 2 A = N (k0 )/(k) ; (B.78) (2)4 where /(k) is one of the spectral functions (B.68). The dominant Luctuations in the plasma have momenta k ∼ T , and can be estimated using the free spectral density /0 (k) = 2(k0 )*(k 2 ). One then :nds (ignoring the vacuum contribution, see Eq. (B.19): d 3 k N (jk ) 2 A T = ∼ T2 : (B.79) (2)3 jk The longwavelength Luctuations can be evaluated using the classical :eld approximation whereby N (k0 ) ≈ T=k0 . One then obtains d k0 2 3 A soft ≈ T d k /(k) ; (B.80) 2k0 where the momentum integral is to be limited to soft momenta k T . The k0 integral can be calculated with the help of the “sum rules” (B.69). In the case of longitudinal (electric) :elds with typical momenta k ∼ gT , one obtains d3 k 2 2 A gT ≈ TmD ∼ TmD ∼ gT 2 : (B.81) k 2 (k 2 + m2D ) Transverse (magnetic) :elds are not screened at small frequencies in the HTL approximation. Therefore 1 d k0 ∗ /T (k0 ; k) = 2 : (B.82) 2k0 k Then the typical Luctuations at the scale k ∼ are 3 d k 2 A ≈ T ∼ T : k2

(B.83)

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Inparticular, for ∼ g2 T , the magnetic Luctuations A2 g2 T ∼ g2 T 2 become nonperturbative: g A2 ∼ for ∼ g2 T . Fluctuations with longer wavelengths 1=g2 T are expected to be damped by the nonperturbative magnetic screening at the scale g2 T . Note that the ultrasoft magnetic Luctuations A2 g2 T carry very soft frequencies k0 . g4 T . Indeed, for k ∼ g2 T , the contribution of the magnetic modes to the integral in Eq. (B.78) is saturated by the Landau damping piece T of the spectral density ∗/T , as it can be easily veri:ed on the second equation (B.77). This gives (cf. Eqs. (B.76) and (4.62)): 1 m2D k T (k0 k)

: k0 2 k 6 + (m2D k0 =4)2

(B.84)

For k ∼ g2 T , this is strongly peaked at small frequencies k0 . k 3 =m2D g4 T , as illustrated in Fig. 32. B.2. The soft fermion propagator The one-loop fermion self-energy is displayed in Fig. 33, and is evaluated as 2 ?(p) = − g Cf [dq]8 S0 (p − q)8 G (q) ;

(B.85)

where p0 = i!n = i(2n + 1)T , q0 = i!m = i2mT , with integers n and m, and the gluon propagator G is here taken, for convenience, in the Coulomb gauge. In view of further applications, we shall compute Eq. (B.85) with a resummed gluon propagator of the general form (B.67). B.2.1. The one-loop self-energy After performing the Matsubara sum and the continuation to complex values of the external energy p0 , we obtain the analytic one-loop self-energy in the form ?(p) = ?C (p) + ?L (p) + ?T (p) ;

(B.86)

where ?C denotes the contribution of the instantaneous Coulomb interaction, as arising from the term −1=p2 in the second line of Eq. (B.67): +∞ 1 d k0 2 ?C (p) = g Cf (dk) (B.87) /0 (k)(1 − n(k0 ))80 k=80 ; (p − k)2 −∞ 2 while ?L (p) and ?T (p) are, respectively, the contributions of the longitudinal and transverse gluons: +∞ d k0 +∞ dq0 1 + N (q0 ) − n(k0 ) 2 ?L (p) = − g Cf (dq) (B.88) /0 (k)/L (q)80 k=80 2 −∞ 2 k0 + q0 − p0 −∞

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Fig. 32. The functions T (q0 ; q) and T (q0 ; q)=q0 in terms of q0 =!pl for q = 0:5!pl . All the quantities are made adimensional by multiplying them by appropriate powers of !pl .

Fig. 33. The one-loop quark self-energy.

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and ?T (p) = −g2 Cf

(dq)

+∞

−∞

×(*ij − qˆi qˆj )8i k=8 j

d k0 2

+∞

−∞

dq0 /0 (k)/T (q) 2

1 + N (q0 ) − n(k0 ) : k0 + q0 − p0

(B.89)

In these equations, k = (k0 ; k) (with k = p − q) is the four-momentum carried by the internal quark line. Consider now the strict one-loop calculation, i.e., the one involving the tree-level gluon propagator. For a bare gluon, /L = 0, and therefore ?L = 0 as well. Furthermore, /T = /0 , so that Eq. (B.89) becomes (after performing the Dirac trace): +∞ d k0 +∞ dq0 2 ?T (p0 ; p) = −2g Cf (dq) /0 (k)/T (q) 2 −∞ 2 −∞ ˆ ˆ ×(k0 80 − (S · q)(k · q))

1 + N (q0 ) − n(k0 ) : k0 + q0 − p0

The energy integrations can be easily performed with the result 1 ?T (p0 ; p) = −2g2 Cf (dq) 2jk 2jq 1 − n(jk ) + N (jq ) n(jk ) + N (jq ) ˆ ˆ × (jk 80 − (S · q)(k · q)) + jk + jq − p0 jk − jq − p0 n(jk ) + N (jq ) 1 − n(jk ) + N (jq ) ˆ ˆ · q)) + ; − (jk 80 + (S · q)(k jk − jq + p0 jk + jq + p0

(B.90)

(B.91)

where jq ≡ |q| and jk ≡ |k| = |p − q|. The zero-temperature piece of this expression contains ultraviolet divergences which can be removed by renormalization [256]. The :nite-temperature piece is UV :nite, because of the thermal factors, and we can set d = 4 in its evaluation. B.2.2. The hard thermal loop To isolate the HTL in the previous expressions, we consider soft external energy and momentum, ! ∼ p ∼ gT T , and keep only the leading order contribution. Once again, this contribution arises by integration over hard loop momenta, q ∼ T , and it can be isolated by performing the same kinematical approximations as in the previous subsections. It can then be veri:ed that the Coulomb piece (B.87) contains no HTL (i.e., no contribution proportional to T 2 ). Thus, only the physical, transverse, gluons contribute to the HTL, in agreement with the results from kinetic theory (cf. Section 3.5.1). Consider Eq. (B.91) for p0 = ! + i", with ! real. For p ∼ gT q ∼ T , we can write: ˆ and, to leading order in g, we can even replace k −q and jk q jk q − v · p with v ≡ q, everywhere except in the energy denominators. For instance, we shall write ˆ ˆ q(80 − v · S) = qv= ; jk 80 + (S · q)(k · q)

(B.92)

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where v ≡ (1; v). As in the previous discussion of the gluon HTL, we encounter both soft and hard energy denominators, jk − jq ± ! v · p ± !;

jk + jq ± ! 2k ;

(B.93)

but in the present case, only those terms involving soft denominators survive to leading order. Indeed, the corresponding numerators in Eq. (B.91) involve the sum of statistical factors n(jk )+ N (jq ) ≈ n(q) + N (q) which, unlike the di8erence n(jk ) − n(jq ) in Eq. (B.40), is not suppressed when the external momentum p is soft. We thus get d 3 q n(q) + N (q) 80 − v · S 80 + v · S 2 ?(!; p) ≈ 2g Cf + (2)3 2q2q !−v·p !+v·p ∞ 2 g Cf v= = dq q(n(q) + N (q)) d. ; (B.94) 3 (2) 0 !−v·p up to terms of higher order in g. The radial integral is evaluated as ∞ 2 T 2 2 T 2 2 T 2 dq q(n(q) + N (q)) = + = ; 12 6 4 0 so that, :nally [39,41], v= d. 2 ?(!; p) ≈ !0 ; 4 ! − v · p + i"

(B.95)

(B.96)

where !0 is the fermionic plasma frequency given in Eq. (4.19). We thus recover the quark HTL (5.24) derived from kinetic theory. Eq. (B.96) has been obtained here in the Coulomb gauge, but it is gauge-:xing independent, as shown in Refs. [39,41,19,23]. After performing the angular integration in Eq. (B.96), the :nal result for the quark HTL may be rewritten as ?(!; p) = a(!; p)80 + b(!; p)pˆ · S ;

(B.97)

where a(!; p) ≡

!02 Q(!=p); !

b(!; p) ≡ −

!02 [Q(!=p) − 1] ; p

(B.98)

and the function Q(x) is de:ned in Eq. (B.64). For real energy (p0 = ! + i") and space-like momenta (|!| ¡ p), the functions in Eq. (B.98) develop imaginary parts which describe the Landau damping of the soft fermion :eld. B.2.3. The propagator of the soft quark The propagator of the soft quark in the hard thermal loop approximation is obtained by inverting the Dyson–Schwinger equation ∗ −1

S

(p) = S0−1 (p) + ?(p) :

(B.99)

To this aim, it is useful to observe that both the tree-level propagator S0 (p) and the self-energy ?(p) are chirally symmetric (e.g., {85 ; ?(p)} = 0), so that they can be decomposed into

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simultaneous eigenstates of chirality (85 ) and helicity ( · p). Speci:cally, we write !80 − p · S −1 −1 ˆ + ˆ ; S0 (!; p) = − = h+ (p) h− (p) !2 − p 2 !−p !+p

(B.100)

ˆ = (80 ∓ pˆ · S)=2 and pˆ ≡ p=p, and similarly where h± (p) ˆ + (!; p) − h+ (p)? ˆ − (!; p) ; ?(!; p) = h− (p)?

(B.101)

with (cf. Eq. (B.97)) ˆ ?± (!; p) ≡ ± 12 tr(h± (p)?(!; p)) = ± a(!; p) + b(!; p) :

(B.102)

ˆ ≡ h± (p)8 ˆ 0 project onto spinors whose chirality is equal (h+ ) or Note that the matrices ± (p) opposite (h− ) to the helicity. (To see this, it is useful to recall the identity 5i = 85 80 8i , where 5i ≡ −ijk 8 j 8k =2i, so that 85 ± = (85 ± · p)=2.) ˆ By using the above representations, one can easily invert Eq. (B.99) to get the full propagator ∗

ˆ + ∗ M− (!; p)h− (p) ˆ ; S(!; p) = ∗ M+ (!; p)h+ (p)

∗

M± (!; p) =

(B.103)

where −1

: (B.104) ! ∓ (p + ?± (!; p)) The poles of the e8ective propagator ∗ S(!; p) determine the mass-shell condition for soft fermionic excitations, to leading order in g (cf. Section 4.3.1). B.2.4. The one-loop damping rate To leading order in g, the (hard) fermion damping rate can be readily extracted from the previous formulae in this section. As explained in Section 6.3, this requires the resummation of the internal gluon line, which is soft in the kinematical regime of interest. By using Eqs. (B.88) and (B.89) with the gluon spectral densities in the HTL approximation (cf. Eqs. (B.68) – (B.76)), we obtain 1 8≡− tr(p= Im ∗ ?R (p0 + i"; p))|p0 =p 4p +∞ d k0 +∞ dq0 = g2 Cf (dq) *(k0 + q0 − p)[1 + N (q0 ) − n(k0 )] 2 −∞ 2 −∞ ˆ ˆ ∗/T (q) + [k0 + (v · k)] ∗/L (q)} ; ×/0 (k){2[k0 − (v · q)(k · q)]

(B.105)

ˆ After performing the integral over k0 , and using the where k = (k0 ; k), k = p − q and v = p. inequality qp to do a few kinematical approximations, e.g. p−q p − v · q = p − q cos & ; 1 + N (q0 ) − n(p−q ) N (q0 ) T=q0 ; we can bring Eq. (B.105) into the form d4 q T 2 8 g Cf *(q0 − q cos &)( ∗/L (q) + (1 − cos2 &) ∗/T (q)) : (2)4 q0

(B.106) (B.107)

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523

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CONTENTS VOLUME 359 V. Barone, A. Drago, P.G. Ratcliﬀe. Transverse polarisation of quarks in hadrons

1

A. Sopczak. Higgs physics at LEP-1

169

A. Altland, B.D. Simons, M.R. Zirnbauer. Theories of quasi-particle states in disordered d-wave superconductors

283

J.-P. Blaizot, E. Iancu. The quark–gluon plasma: collective dynamics and hard thermal loops

355

Contents of volume 359

529

Forthcoming issues

530

PII: S 0 3 7 0 - 1 5 7 3 ( 0 1 ) 0 0 1 0 5 - 3

530

FORTHCOMING ISSUES* C.M. Varma, Z. Nussinov, W. van Saarloos. Singular Fermi liquids P. Tabeling. Two-dimensional turbulence: a physicist approach J.A. Krommes. Fundamental descriptions of plasma turbulence in magnetic ﬁelds J.D. Vergados. The neutrinoless double beta decay from a modern perspective C.-I. Um, K.-H. Yeon, T.F. George. The quantum damped harmonic oscillator P. Reimann. Brownian motors: noisy transport far from equilibrium T. Yamazaki, N. Morita, R. Hayano, E. Widmann, J. Eades. Antiprotonic helium J.K. Basu, M.K. Sanyal. Ordering and growth of Langmuir–Blodgett ﬁlms: X-ray scattering studies G.E. Brown, M. Rho. On the manifestation of chiral symmetry in nuclei and dense nuclear matter S. Nussinov, M.A. Lampert. QCD inequalities C.A.A. de Carvalho, H.M. Nussenzveig. Time delay L.S. Ferreira, G. Cattapan. The role of the D in nuclear physics C. Chandre, H.R. Jauslin. Renormalization-group analysis for the transition to chaos in Hamiltonian systems S. Stenholm. Heuristic ﬁeld theory of Bose–Einstein condensates A. Konechny, A. Schwarz. Introduction to M(atrix) theory and noncommutative geometry

*The full text of articles in press is available from ScienceDirect at http://www.sciencedirect.com. PII: S 0 3 7 0 - 1 5 7 3 ( 0 1 ) 0 0 1 0 6 - 5

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