The general question, crucial to an understanding of the internal structure of the nucleon, of how to split the total angular momentum of a photon or gluon into spin and orbital contributions is one of the most important and interesting challenges faced by gauge theories like Quantum Electrodynamics and Quantum Chromodynamics. This is particularly challenging since all QED textbooks state that such an splitting cannot be done for a photon (and a fortiori for a gluon) in a gauge-invariant way, yet experimentalists around the world are engaged in measuring what they believe is the gluon spin! This question has been a subject of intense debate and controversy, ever since, in 2008, it was claimed that such a gauge-invariant split was, in fact, possible. We explain in what sense this claim is true and how it turns out that one of the main problems is that such a decomposition is not unique and therefore raises the question of what is the most natural or physical choice. The essential requirement of measurability does not solve the ambiguities and leads us to the conclusion that the choice of a particular decomposition is essentially a matter of taste and convenience. In this review, we provide a pedagogical introduction to the question of angular momentum decomposition in a gauge theory, present the main relevant decompositions and discuss in detail several aspects of the controversies regarding the question of gauge invariance, frame dependence, uniqueness and measurability. We stress the physical implications of the recent developments and collect into a separate section all the sum rules and relations which we think experimentally relevant . We hope that such a review will make the matter amenable to a broader community and will help to clarify the present situation.C.Lorce@ulg.ac.be 30 1. The Stueckelberg symmetry 30 2. Towards a more refined classification 31 3. Origin and geometrical interpretation of the Stueckelberg symmetry 33 4. Measurability and the controversy about Stueckelberg symmetry 35 D. The Lorentz transformation properties 38 1. The standard approach 38 2. Critique of the standard approach 39 3. Lorentz transformation law of the pure-gauge and physical fields 40 V. The proton spin decomposition 41 A. The QCD energy-momentum and covariant angular momentum tensors 42 B. Decompositions of the proton momentum and the proton spin 44 1. The canonical decompositions 44 2. The kinetic decompositions 45 3. The master decomposition 46 C. Non-abelian Stueckelberg and Lorentz transformations 48 Bel,z 74 2. Lattice calculation of J q Bel,T 76 3. Evaluation of L q Ji,z in a longitudinally polarized nucleon, from GPDs 77 4. Evaluation of L q Ji,z in a longitudinally polarized nucleon, from GTMDs 78 B. Expressions for the canonical version of L 79 C. The orbital angular momentum in quark models 79 D. The phase-space distribution of angular momentum 82 VIII. Qualitative summary and experimental implications 84 A. Gauge invariance and measurability 84 B. Two kinds of decompositions 85 C. Sum rules vs. relations 86 1...

Comprehensive review paper on the theory and phenomenology of polarized deep inelastic scattering, to appear in Physics ReportsComment: 113 pages, latex, 40 figures not included (hard copies available via mail upon request to anselmino@to.infn.it

We present a new NLO QCD analysis of the world data on inclusive polarized deep inelastic scattering. Comparing to our previous analysis: i) the values of g A and a 8 = 3F −D are updated ii)the MRST'99 instead of the MRST'98 parametrization for the input unpolarized parton densities is used and iii) the recent SLAC E155 proton data on the spin asymmetry A1 are included in the analysis. A new set of polarized parton densities is extracted from the data and the sensitivity of the results to different positivity constraints is discussed.

Motivated by recent dramatic developments in the field, this book provides a thorough introduction to spin and its role in elementary particle physics. Starting with a simple pedagogical introduction to spin and its relativistic generalisation, the author successfully avoids the obscurity and impenetrability of traditional treatments of the subject. The book surveys the main theoretical and experimental developments, as well as discussing exciting plans for the future. Emphasis is placed on the importance of spin-dependent measurements in testing QCD and the Standard Model. This book will be of value to graduate students and researchers working in all areas of quantum physics and particularly in elementary particle and high energy physics. It is suitable as a supplementary text for graduate courses in theoretical and experimental particle physics.

The high energy and large p(T) inclusive polarized process, (A,SA) + (B,SB) -> C + X, is considered under the assumption of a generalized QCD factorization scheme. For the first time all transverse motions, of partons in hadrons and of hadrons in fragmenting partons, are explicitly taken into account; the elementary interactions are computed at leading order with noncollinear exact kinematics, which introduces many phases in the expressions of their helicity amplitudes. Several new spin and k(perp)-dependent soft functions appear and contribute to the cross sections and to spin asymmetries; we put emphasis on their partonic interpretation, in terms of quark and gluon polarizations inside polarized hadrons. Connections with other notations and further information are given in some Appendixes. The formal expressions for single and double spin asymmetries are derived. The transverse single spin asymmetry A(N), for p(uparrow) p -> pi X processes is considered in more detail, and all contributions are evaluated numerically by saturating unknown functions with their upper positivity bounds. It is shown that the integration of the phases arising from the noncollinear kinematics strongly suppresses most contributions to the single spin asymmetry, leaving at work predominantly the Sivers effect and, to a lesser extent, the Collins mechanism

The single transverse spin asymmetry in D meson production at RHIC can provide a clean measure of the gluon Sivers distribution function. At intermediate rapidity, D production is largely dominated by the elementary gg → cc channel, where there cannot be any transverse spin transfer. Therefore, any transverse single spin asymmetry observed for D's produced in p ↑ p interactions can only originate from the Sivers effect in the gluon distribution functions. A sizeable transverse single spin asymmetry measured by PHENIX or STAR experiments would then give direct information on the size of the gluon Sivers distribution function.Parton distribution and fragmentation functions are phenomenological quantities which have to be obtained from experimental observation and cannot be theoretically predicted. When parton intrinsic transverse momenta are taken into account, a large number of pdf's and ff's arise: the main difficulties in gathering experimental information on these spin and k ⊥ dependent functions is that most often more of them contribute to the same physical observable, making it extremely hard to estimate each single one separately. The Sivers function ∆ N f (x, k ⊥ ) 1 , which describes the probability density of finding unpolarized partons inside a transversely polarized proton, is one of these functions. It plays a crucial role since it can explain single spin asymmetries in terms of parton dynamics 2 . We suggest here an experiment to be conducted at RHIC which can isolate the gluon Sivers effect, making it possible to reach direct independent 1

This work presents, in two volumes, a comprehensive and unified treatment of modern theoretical and experimental particle physics at a level accessible to beginning research students. The emphasis throughout is on presenting underlying physical principles in a simple and intuitive way, and the more sophisticated methods demanded by present day research interests are introduced in a very gradual and gentle fashion. Volume 1 covers electroweak interactions, the discovery and properties of the 'new' particles, the discovery of partons and the construction and predictions of the simple parton model. Volume 2 deals at some length with CP-violation, but is mainly devoted to QCD and its application to 'hard' processes. A brief coverage of 'soft' hadronic physics is included. This work will provide a comprehensive reference and textbook for all graduate students and researchers interested in modern particle physics.

Motivated by the need for an absolute polarimeter to determine the beam polarization for the forthcoming RHIC spin program, we study the spin dependence of the proton-proton elastic scattering amplitudes at high energy and small momentum transfer. In particular, we examine experimental evidence for the existence of an asymptotic part of the helicity-flip amplitude φ 5 which is not negligible relative to the largely imaginary average non-flip amplitude φ + = 1 2 (φ 1 + φ 3 ). We discuss theoretical estimates of r 5 = m φ 5 / √ −t Im φ + based upon several approaches: extrapolation of low and medium energy Regge phenomenological results to high energies, models based on a hybrid of perturbative QCD and non-relativistic quark models, and models based on eikonalization techniques. We also apply the rigorous, model-independent methods of analyticity and unitarity. We find the preponderence of evidence at currently available energy indicates that r 5 is small, probably less than 10%. The best available experimental limit comes from Fermilab E704: combined with rather weak theoretical assumptions those data indicate that |r 5 | < 15%. These bounds are important because rigorous methods allow much larger values. Furthermore, in contradiction to a widelyheld prejudice that r 5 decreases with energy, general principles allow it to grow as fast as ln s asymptotically, and some of the models we consider show an even faster growth in the RHIC range. One needs a more precise measurement of r 5 or to bound it to be smaller than 5% in order to use the classical Coulomb-nuclear interference technique for RHIC polarimetry. Our results show how important the measurements of spin dependence at RHIC will be to our understanding of proton structure and scattering dynamics. As part of this study, we demonstrate the surprising result that protonproton elastic scattering is self-analysing, in the sense that all the helicity amplitudes can, in principle, be determined experimentally at small momentum transfer without a knowledge of the magnitude of the beam and target polarization.