On the Origin of Model Relations among Transverse-Momentum Dependent Parton Distributions

Transverse-momentum dependent parton distributions (TMDs) are studied in the framework of quark models. In particular, quark-model relations among TMDs are reviewed, elucidating their physical origin in terms of the quark-spin structure in the nucleon. The formal aspects of the derivation of these relations are complemented with explicit examples, emphasizing how and to which extent the conditions which lead to relations among TMDs are implemented in different classes of quark models.Comment: 38 pages, 3 figures, 3 table

The angular momentum controversy: What's it all about and does it matter?

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.Comment: 96 pages, 11 figures, 5 tables, review prepared for Physics Report

Modeling the transverse momentum dependent parton distributions

We review quark model calculations of the transverse momentum dependent parton distributions (TMDs). For the T-even TMDs, we discuss the physical origin of model relations which hold in a large class of quark models. For the T-odd TMDs we review results in a light-cone constituent quark model (CQM) with the final state interaction effects generated via single-gluon exchange mechanism. As phenomenological application, we show the good agreement between results in the light-cone CQM and available experimental data for the Collins asymmetry.Comment: 8 pages, 2 figures, 2 tables; invited talk at "The 4th Workshop on Exclusive Reactions at High Momentum Transfer", Jefferson Lab., May 18-21, 201

Quark Wigner Distributions and Orbital Angular Momentum

We study the Wigner functions of the nucleon which provide multidimensional images of the quark distributions in phase space. These functions can be obtained through a Fourier transform in the transverse space of the generalized transverse-momentum dependent parton distributions. They depend on both the transverse position and the three-momentum of the quark relative to the nucleon, and therefore combine in a single picture all the information contained in the generalized parton distributions and the transverse-momentum dependent parton distributions. We focus the discussion on the distributions of unpolarized/longitudinally polarized quark in an unpolarized/longitudinally polarized nucleon. In this way, we can study the role of the orbital angular momentum of the quark in shaping the nucleon and its correlations with the quark and nucleon polarizations. The quark orbital angular momentum is also calculated from its phase-space average weighted with the Wigner distribution of unpolarized quarks in a longitudinally polarized nucleon. The corresponding results obtained within different light-cone quark models are compared with alternative definitions of the quark orbital angular momentum, as given in terms of generalized parton distributions and transverse-momentum dependent parton distributions.Comment: 29 pages, 10 figures, 3 table

Models for TMDs and numerical methods

We study the connection between the quark orbital angular momentum and the pretzelosity transverse-momentum dependent parton distribution function. We discuss the origin of this relation in quark models, identifying as key ingredient for its validity the assumption of spherical symmetry for the nucleon in its rest frame. Finally we show that the individual quark contributions to the orbital angular momentum obtained from this relation can not be interpreted as the intrinsic contributions, but include the contribution from the transverse centre of momentum which cancels out only in the total orbital angular momentum.Comment: 43 pages, 8 figures; proceedings of International School of Physics "Enrico Fermi", Course CLXXX - "Three-dimensional Partonic Structure of the Nucleon", 28 June - 8 July 2011, Varenna (Italy

The Parton Orbital Angular Momentum: Status and Prospects

Theoretical progress on the formulation and classification of the quark and gluon orbital angular momenta (OAM) is reviewed. Their relation to parton distributions and open questions and puzzles are discussed. We give a status report on the lattice calculation of the parton kinetic and canonical OAM and point out several strategies to calculate the quark and gluon canonical OAM on the lattice.Comment: 16 pages, contribution to the EPJA speical issue on "3D Structure of the Nucleon

Pretzelosity TMD and Quark Orbital Angular Momentum

We study the connection between the quark orbital angular momentum and the pretzelosity transverse-momentum dependent parton distribution function. We discuss the origin of this relation in quark models, identifying as key ingredient for its validity the assumption of spherical symmetry for the nucleon in its rest frame. Finally we show that the individual quark contributions to the orbital angular momentum obtained from this relation can not be interpreted as the intrinsic contributions, but include the contribution from the transverse centre of momentum which cancels out only in the total orbital angular momentum.Comment: 8 pages; references added, version to appear in Phys. Lett.