Transverse Spin Physics

TRANSVERSE

SPIN PHYSICS

This page is intentionally left blank

Vlncenzo Barone University ofPiemonte Orientals Italy

Philip G Ratclifife University oflnsubria, Italy

TRANSVERSE

SPIN PHYSICS vP World Scientific

New Jersey • London • Singapore • Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: Suite 202, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

Wassily Kandinsky "Circles in the circle" The Philadelphia Museum of Art

TRANSVERSE SPIN PHYSICS Copyright © 2003 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 981-238-101-5

This book is printed on acid-free paper. Printed in Singapore by Mainland Press

A Gianna, Armando e Daniele (VB) To Sandra, Alberto and Alice (PGR)

This page is intentionally left blank

Preface

This book is devoted to the theory and phenomenology of transverse-spin effects in high-energy hadronic physics. Contrary to common past belief, it has now become rather clear that such effects are far from irrelevant. A decade or so of intense theoretical work has shed much light on the subject and brought to surface an entire class of possible new phenomena, which now await thorough experimental investigation. Over the next few years a number of experiments worldwide (at BNL, CERN, DESY and JLAB) are due to run with transversely polarised beams and targets, providing data that will enrich our knowledge of the transverse-spin structure of hadrons. We therefore feel that now is the right moment to assess the state of the art and this will be the principal aim of the volume. The outline is as follows. After a few introductory remarks in Chap. 1, attention is directed in Chap. 2 to polarised deeply-inelastic scattering (DIS), in particular to DIS on transversely polarised targets, which probes the transverse-spin structure function 52- The existing data are reviewed and discussed (for completeness and comparison, a brief presentation of longitudinally polarised DIS and of the helicity structure of the proton is also provided). In Chap. 3 we illustrate the transverse-spin structure of the proton in some detail: the leading-twist and twist-three distribution functions, including A T / (the transversity distribution) and gx (the parton distribution contributing to 52), are introduced. Model calculations of these quantities are also presented. In Chap. 4, after some general considerations on the renormalisation-group equation and the operator-product expansion, we discuss the evolution of transversity at leading and next-toleading order in quantum chromodynamics (QCD). In Chap. 5 we study the vii

Vlll

Preface

#2 structure function and its related sum rules, within the framework of perturbative QCD. The last three chapters are devoted to the phenomenology of transversity, in the context of Drell-Yan processes (Chap. 6), inclusive leptoproduction (Chap. 7) and inclusive hadroproduction (Chap. 8). The interpretation of some recent single-spin asymmetry data is discussed and finally the prospects for future measurements are also reviewed. This book is partially based on an earlier review paper written in collaboration with Alessandro Drago, to whom we are extremely grateful for a long and fruitful collaboration on the subject of transversity. It is a pleasure to thank all the friends and colleagues who have shared with us their knowledge of high-energy spin phenomena in the course of the years. In particular, we should like to mention: Mauro Anselmino, Johannes Blumlein, Daniel Boer, Sigfrido Boffi, Elena Boglione, Alessandro Bacchetta, Umberto D'Alesio, Stefano Forte, Paolo Gambino, Bob Jaffe, Xiangdong Ji, Elliot Leader, Bo-Qiang Ma, Piet Mulders, Francesco Murgia, Enrico Predazzi, Marco Radici, Giovanni Ridolfi, Jacques Soffer, Oleg Teryaev, Werner Vogelsang, Fabian Zomer.

Torino and Como, July 2002

Vincenzo Barone

Philip G. Ratcliffe

Contents

Preface

vii

Chapter 1 Introduction 1.1 The transverse-spin structure function and the transversity distributions 1.2 A first look at gi 1.3 A prelude to transversity 1.4 Notation and terminology 1.5 Conventions

2 3 5 9 11

Chapter 2 Polarised deeply-inelastic scattering 2.1 Basics of DIS 2.2 The unpolarised cross-section 2.3 Polarised cross-sections 2.4 Target polarisation 2.5 Forward virtual Compton scattering 2.6 Spin asymmetries 2.7 The partonic content of structure functions 2.7.1 Unpolarised structure functions 2.7.2 The longitudinal spin structure function 2.7.3 The transverse-spin structure function 2.8 Mellin moments of polarised structure functions 2.8.1 The first moment of g\ 2.8.2 The Bjorken sum rule 2.8.3 The Wandzura-Wilczek relation

13 14 17 19 22 24 26 28 31 33 34 39 40 42 43

ix

1

x

Contents

2.8.4 The Burkhardt-Cottingham sum rule 2.8.5 The Efremov-Leader-Teryaev sum rule 2.9 Experimental results on polarised structure functions 2.10 Transverse spin in electroweak DIS

44 46 46 53

Chapter 3 The transverse-spin structure of the proton 3.1 The quark-quark correlation matrix 3.2 Leading-twist distribution functions 3.3 Probabilistic interpretation of distribution functions 3.4 Vector, axial and tensor charges 3.5 Quark-nucleon helicity amplitudes 3.6 The Soffer inequality 3.7 Transverse motion of quarks 3.8 Twist-three distributions 3.9 Sum rules for Arf and gr 3.10 T-odd distributions 3.11 Model calculations 3.11.1 Models for the transversity distributions 3.11.2 Calculations of the tensor charges 3.11.3 Models for g2

57 57 59 62 65 66 68 70 75 81 83 87 88 96 98

Chapter 4 The QCD evolution of transversity 103 4.1 The renormalisation-group equation 104 4.2 QCD evolution at leading order 109 4.3 QCD evolution at next-to-leading order 115 4.4 Fragmentation functions at next-to-leading order 123 4.5 Evolution of the transversity distributions 126 4.6 Evolution of the Soffer inequality and positivity constraints . . 131 4.7 The low-z behaviour of h\ 135 Chapter 5 The g2 structure function in Q C D 5.1 The operator-product expansion—non-singlet 5.2 Ladder-diagram summation 5.3 Singlet g2 in LO 5.4 Non-singlet and singlet coefficients g2 in NLO 5.5 Sum rules for g2 in QCD 5.5.1 The Burkhardt-Cottingham sum rule in QCD 5.5.2 The Wandzura-Wilczek relation in QCD

137 139 143 146 146 147 147 149

Contents

5.6 5.7 5.8

5.5.3 The Efremov-Leader-Teryaev sum rule in QCD Low-a: behaviour of g2 Twist-three evolution equations in the large-iVc limit Evolution of the g? fragmentation function

xi

....

150 150 151 154

Chapter 6 Transversity in Drell—Yan production 157 6.1 Double transverse-spin asymmetries 157 6.2 The Drell-Yan process 158 6.2.1 Z°-mediated Drell-Yan processes 165 6.3 Factorisation in Drell-Yan processes 165 6.4 Twist-three contributions to the Drell-Yan process 171 6.5 Predictions for Drell-Yan double transverse-spin asymmetries . 176 6.6 Transversity at RHIC 179 Chapter 7 Transversity in inclusive leptoproduction 7.1 Single-particle leptoproduction: definitions and kinematics . . . 7.2 The partonic description of semi-inclusive DIS 7.3 The fragmentation matrix 7.4 Time reversal and transverse polarisation 7.5 Leading-twist fragmentation functions 7.5.1 Ky-dependent fragmentation functions 7.6 The Collins fragmentation function 7.7 Cross-sections and asymmetries of inclusive leptoproduction . . 7.7.1 Integrated cross-sections 7.7.2 Single-spin azimuthal asymmetries 7.8 Factorisation in semi-inclusive DIS 7.8.1 The collinear case 7.8.2 The non-collinear case 7.8.3 Sudakov form factors 7.9 Inclusive leptoproduction at twist three 7.10 Two-particle leptoproduction 7.11 Leptoproduction of spin-one hadrons 7.12 Transversity in exclusive leptoproduction processes 7.13 Phenomenological analyses and experimental results 7.13.1 A0 hyperon polarimetry 7.13.2 Azimuthal asymmetries in pion leptoproduction . . . . 7.13.3 Transverse polarisation in e+e~ collisions 7.14 Experimental perspectives

183 183 187 190 192 194 197 200 203 205 205 211 211 216 218 219 220 229 231 232 232 234 241 242

xii

Contents

7.14.1 7.14.2 7.14.3 7.14.4

HERMES COMPASS ELFE TESLA-N

242 243 244 244

Chapter 8 Transversity in inclusive hadroproduction 245 8.1 Inclusive hadroproduction with a transversely polarised target 245 8.2 Transverse motion of quarks and single-spin asymmetries . . . 248 8.3 Single-spin asymmetries at twist three 251 8.3.1 Experimental results and phenomenology 252 Appendix A Polarisation of a Dirac particle A.l The polarisation operator and spin vector A.2 Longitudinal polarisation A.3 Transverse polarisation A.4 The spin density matrix

257 257 259 260 261

Appendix B

Sudakov decomposition of yectors

263

Appendix C

Projectors for structure functions

265

Appendix D Reference frames D.l The 7*AT collinear frames D.2 The hN collinear frames Appendix E

Appendix F

267 267 268

Dimensional regularisation and minimal subtraction

271

Mellin-moment identities

275

Bibliography

277

Index

291

Chapter 1

Introduction

The idea that transverse-spin effects should be suppressed at high energies has been a tenacious prejudice in hadronic physics over the last thirty years or so. 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 first point to bear in mind is the distinction between transverse polarisation itself and its measurable effects. As well-known, for example, even the ultra-relativistic electrons and positrons of the LEP storage ring are significantly polarised in the transverse plane (Knudsen et al., 1991) owing to the Sokolov-Ternov effect (Sokolov and Ternov, 1964). The real problem is to identify processes sensitive to such polarisation; while this is not always easy, it is certainly not impossible. The spin (or polarisation) vector sM of a high-energy particle may be decomposed into a longitudinal component stf, parallel to the particle momentum p M , and a transverse component s^_, perpendicular to p^:

sM = < + «i = - P " + *£ • ii

C1-1)

m

Here A is twice the helicity of the particle and m is its mass. It is clear that, since s2 = A2 + s x = — 1, in a frame where the particle is moving very fast the transverse spin components are suppressed with respect to the longitudinal component by a factor m/E (E being the particle energy). This explains, for example, why the effects of the transverse polarisation of the lepton beam in deeply-inelastic scattering (DIS) are unobservable. For the same reason, the contribution of the transverse-spin structure function g2 to the DIS cross-section is of order xM/Q (where M is the target mass 1

2

Introduction

and Q2 is the four-momentum transfer squared). The above argument should not, however, lead to the false impression that all transverse-polarisation phenomena are subdominant. In fact, some of them - those related to the so-called "transversity" of quarks - are neither kinematically nor dynamically suppressed and represent the leading contribution to certain hadronic processes. Historically, the first extensive discussion of transverse-spin effects in high-energy hadronic physics followed the discovery in 1976 that A0 hyperons produced in pN interactions, even at relatively high px, exhibit an anomalously large transverse polarisation (Bunce et al., 1976). Such a result requires a non-zero imaginary part in the off-diagonal elements of the fragmentation matrix of quarks into A0 hyperons. It was soon pointed out that this is forbidden at leading-twist in QCD and may arise only as an 0{\IPT) effect (Kane, Pumplin and Repko, 1978; Efremov and Teryaev, 1982,1985). It thus took a while to fully realise that transverse-polarisation phenomena are indeed sometimes unsuppressed. This was shown in the pioneering paper of Ralston and Soper (1979) on longitudinally and transversely polarised Drell-Yan processes, but the idea remained almost unnoticed for a decade. An issue related to hadronic transverse spin, and investigated theoretically in the same period, is the g^ spin structure function (Hey and Mandula, 1972; Heimann, 1974). After some early attempts by Feynman (1972) to incorporate g^ into the parton model, it was realised that g