International Journal of Modern Physics: Conference Series

The introduction talk given at the beginning of QCD Evolution workshop held in Thomas Jefferson National Accelerator Facility (Jefferson Lab) on May 14 -17, 2012.

The study of the transverse spin and transverse momentum structure of the nucleon is an important part of the scientific program of COMPASS, a fixed target experiment taking data at the CERN SPS since 2002. In these ten years COMPASS has produced a number of interesting results by measuring the forward going hadrons produced in deep inelastic scattering of a 160 GeV muon beam off polarized deuteron and proton targets. The COMPASS contribution to the understanding of the transverse structure of the nucleon, and the possible future contributions, are briefly reviewed here.

The investigation of the partonic degrees of freedom beyond collinear approximation (3D description) has been gained increasing interest in the last decade. At the HERMES experiment, azimuthal asymmetries in hard exclusive reactions and in semi-inclusive deep-inelastic scattering of electrons and positrons off a (polarized) hydrogen and deuterium target have been measured. Such asymmetries provide new insights on crucial aspects of the parton dynamics. By measuring various hadron types in the initial and final states, flavor sensitivity is achieved. Non zero signals are reported for azimuthal asymmetries with respect the transverse target polarization in real-photon exclusive-electroproduction, which are related (still in a model dependent way) to the elusive quark orbital motion. Evidence is reported of the poorly known transversity function and of naive-T-odd transverse-momentum-dependent parton distribution functions related to spin-orbit effects. Evidence of spin-orbit effects in quark fragmentation is also observed, which are opposite in sign for favored and disfavored processes.

We calculate the double spin asymmetry (DSA) A_LT in proton-proton (pp) collisions for the production of photons, hadrons, and jets in the collinear twist-3 framework. These processes allows us for the first time to access a complete set of collinear twist-3 functions. We also give numerical estimates for the asymmetries of which photon production shows the most promise of a measurable effect. Furthermore, we comment on how these asymmetries would allow us to probe the gluon helicity down to as of yet unexplored x values, remark on the importance of these effects in determining the evolution of the Efremov-Teryaev-Qiu-Sterman (ETQS) function T_F(x, x), and discuss their usefulness in resolving the true mechanism behind the large transverse single spin asymmetries (SSAs) measured in pp collisions.

This contribution summarizes recent results from the RHIC spin program and the measurement of light quark fragmentation functions at Belle.

Transverse Spin and Transverse Momemtum Dependent (TMD) distribution study has been one of the main focuses of hadron physics in recent years. The initial exploratory Semi-Incluisve Deep-Inelastic-Scattering (SIDIS) experiments with transversely polarized proton and deuteron from HERMES and COMPASS attracted great attention and lead to very active efforts in both experiments and theory. QCD factorization has been carefully studied. A SIDIS experiment on the neutron with a polarized ³He target was performed at JLab. Recently published results will be shown. Precision TMD experiments are planned at JLab after the 12 GeV energy upgrade. The approved experiments with a new SoLID spectrometer on both the proton and neutron will be presented. Proper QCD evolution treatments beyond collinear cases become crucial for the precision study of the TMDs. Experimentally, Q² evolution and higher-twist effects are often closely related. The experience of study higher-twist effects in the cases of moments of the spin structure functions will be discussed.

One of the most important nonperturbative methods for solving QCD is quantization at fixed light-front time τ = t + z/c – Dirac's "Front Form". The eigenvalues of the light-front QCD Hamiltonian predict the hadron spectrum and the eigensolutions provide the light-front wavefunctions which describe hadron structure. More generally, we show that the valence Fock-state wavefunctions of the light-front QCD Hamiltonian satisfy a single-variable relativistic equation of motion, analogous to the nonrelativistic radial Schrödinger equation, with an effective confining potential U which systematically incorporates the effects of higher quark and gluon Fock states. We outline a method for computing the required potential from first principles in QCD. The holographic mapping of gravity in AdS space to QCD, quantized at fixed light-front time, yields the same light front Schrödinger equation; in fact, the soft-wall AdS/QCD approach provides a model for the light-front potential which is color-confining and reproduces well the light-hadron spectrum. One also derives via light-front holography a precise relation between the bound-state amplitudes in the fifth dimension of AdS space and the boost-invariant light-front wavefunctions describing the internal structure of hadrons in physical space-time. The elastic and transition form factors of the pion and the nucleons are found to be well described in this framework. The light-front AdS/QCD holographic approach thus gives a frame-independent first approximation of the color-confining dynamics, spectroscopy, and excitation spectra of relativistic light-quark bound states in QCD.

Azimuthal asymmetries in high-energy processes, most pronounced showing up in combination with single or double (transverse) spin asymmetries, can be understood with the help of transverse momentum dependent (TMD) parton distribution and fragmentation functions. These appear in correlators containing expectation values of quark and gluon operators. TMDs allow access to new operators as compared to collinear (transverse momentum integrated) correlators. These operators include nontrivial process dependent Wilson lines breaking universality for TMDs. Making an angular decomposition in the azimuthal angle, we define a set of universal TMDs of definite rank, which appear with process dependent gluonic pole factors in a way similar to the sign of T-odd parton distribution functions in deep inelastic scattering or the Drell-Yan process. In particular, we show that for a spin 1/2 quark target there are three pretzelocity functions.

For transversely polarized nucleons the distribution of quarks in the transverse plane is transversely shifted and that shift can be described in terms of Generalized Parton Distributions (GPDs). This observation provides a 'partonic' derivation of the Ji-relation for the quark angular momentum in terms of GPDs. Wigner distributions are used to show that the difference between the Jaffe-Manohar definiton of quark orbital angular momentum and that of Ji is equal to the change of orbital angular momentum due to the final state interactions as the struck quark leaves the target in a DIS experiment.

We discuss the quark phase-space or Wigner distributions of the nucleon which combine in a single picture all the information contained in the generalized parton distributions and the transverse-momentum dependent parton distributions. In particular, we present results for the distribution of unpolarized quarks in a longitudinally polarized nucleon obtained in a light-front constituent quark model. We show how the quark orbital angular momentum can be extracted from the Wigner distributions and compare it with alternative definitions.

We consider the definition of unpolarized transverse-momentum-dependent parton distribution functions while staying on-the-light-cone. By imposing a requirement of identical treatment of two collinear sectors, our approach, compatible with a generic factorization theorem with the soft function included, is valid for all non-ultra-violet regulators (as it should), an issue which causes much confusion in the whole field. We explain how large logarithms can be resummed in a way which can be considered as an alternative to the use of Collins-Soper evolution equation. The evolution properties are also discussed and the gauge-invariance, in both classes of gauges, regular and singular, is emphasized.

We address a connection between the energy evolution of the polygonal light-like Wilson exponentials and the geometry of the loop space with the gauge invariant Wilson loops of a variety of shapes being the fundamental degrees of freedom. The renormalization properties and the differential area evolution of these Wilson polygons are studied by making use of the universal Schwinger quantum dynamical approach. We discuss the appropriateness of the dynamical differential equations in the loop space to the study of the energy evolution of the collinear and transverse-momentum dependent parton distribution functions.

In this talk, we examine the existing calculations of QCD evolution kernels for the scale dependence of two sets of twist-3 quark-gluon correlation functions, T_q,F(x, x) and , which are the first transverse-momentum-moment of the naive-time-reversal-odd Sivers and Boer-Mulders function, respectively. The evolution kernels at the leading order in strong coupling constant α_s were derived by several groups with apparent differences. We identify the sources of discrepancies and are able to reconcile the results from various groups.

We review the relation between the QCD perturbative resummation and evolution. Particularly we emphasize briefly the idea on how to use the evolution equation to resum the important double logarithms appeared in the perturbative QCD expansion when the process involves two distinctive scales Q₁ ≫ Q₂. We present the resummation formalism for the single transverse spin asymmetry for both semi-inclusive deep inelastic scattering and Drell-Yan production processes.

Transverse single spin asymmetries in inclusive deep-inelastic lepton-nucleon scattering can be generated through multi-photon exchange between the leptonic and the hadronic part of the process. Here we consider two-photon exchange, and mainly focus on the transverse target spin asymmetry. In particular, we investigate the case where two photons couple to different quarks. Such a contribution involves a quark-photon-quark correlator in the nucleon, which has a (model-dependent) relation to the Efremov-Teryaev-Qiu-Sterman quark-gluon-quark correlator T_F. Using different parameterizations for T_F we compute the transverse target spin asymmetry for both a proton and a neutron target, and compare the results to recent experimental data. Potential implications on our general understanding of single spin asymmetries in hard scattering processes are discussed as well.

Recently, theoretical developments have led to the QCD evolution equations for the unpolarized Transverse Momentum Dependent (TMD) distribution functions and for the Sivers function (TMD-evolution). We tested whether the proposed TMD-evolution can already be observed in the SIDIS data on the Sivers asymmetry. Although very preliminary, our analysis shows that data are compatible with such an evolution with a clear indication of evolution in the x-dependent data subsets.

Starting from a definition of transverse momentum-dependent parton distributions for semi-inclusive deep inelastic scattering and the Drell-Yan process, given in terms of matrix elements of a quark bilocal operator containing a staple-shaped Wilson connection, a scheme to determine such observables in lattice QCD is developed and explored. Parametrizing the aforementioned matrix elements in terms of invariant amplitudes permits a simple transformation of the problem to a Lorentz frame suited for the lattice calculation. Results for the Sivers and Boer-Mulders transverse momentum shifts are presented, focusing in particular on their dependence on the staple extent and the Collins-Soper evolution parameter.

A recent investigation of the single spin asymmetry (SSA) in low virtuality electroproduction/photoproduction of J/ψ in color evaporation model is presented. It is shown that the asymmetry is sizable and can be used as a probe for the still unknown gluon Sivers function.

We consider the cross section for semi-inclusive deep inelastic scattering in Fourier space, conjugate to the outgoing hadron's transverse momentum, where convolutions of transverse momentum dependent parton distribution functions and fragmentation functions become simple products. Individual asymmetric terms in the cross section can be projected out by means of a generalized set of weights involving Bessel functions. Advantages of employing these Bessel weights are that they suppress (divergent) contributions from high transverse momentum and that soft factors cancel in (Bessel-) weighted asymmetries. Also, the resulting compact expressions immediately connect to previous work on evolution equations for transverse momentum dependent parton distribution and fragmentation functions and to quantities accessible in lattice QCD. Bessel-weighted asymmetries are thus model independent observables that augment the description and our understanding of correlations of spin and momentum in nucleon structure.

We present the first steps in an effort to incorporate the physics of transverse spin asymmetries into the saturation formalism of high energy QCD. We consider a simple model in which a transversely polarized quark scatters on a proton or nuclear target. Using the light-cone perturbation theory the hadron production cross section can be written as a convolution of the light-cone wave function squared and the interaction with the target. To generate the single transverse spin asymmetry (STSA) either the wave function squared or the interaction with the target has to be T-odd. In this work we use the lowest-order q → q G wave function squared, which is T-even, generating the STSA from the T-odd interaction with the target mediated by an odderon exchange. We study the properties of the obtained STSA, some of which are in qualitative agreement with experiment: STSA increases with increasing projectile x_F and is a non-monotonic function of the transverse momentum k_T. Our mechanism predicts that the quark STSA in polarized proton–nucleus collisions should be much smaller than in polarized proton–proton collisions. We also observe that the STSA for prompt photons due to our mechanism is zero within the accuracy of the approximation.

The photon impact factor for the BFKL pomeron is calculated in the next-to-leading order (NLO) approximation using the operator expansion in Wilson lines. The result is represented as a NLO k_T-factorization formula for the structure functions of small-x deep inelastic scattering.

The high-energy QCD factorization for Deep Inelastic Scattering and for proton-nucleus collisions using Wilson line formalism and factorization in rapidity is discussed. We show that in DIS the factorization in rapidity reduces to the k_T-factorization when the 2-gluon approximation is applied, provided that the composite Wilson line operator is used in the high-energy Operator Product Expansion. We then show that the inclusive forward cross-section in proton-nucleus collisions factorizes in parton distribution functions, fragmentation functions and dipole gluon distribution function at one-loop level.

Recently, by performing the complete next-to-leading order calculation, we have demonstrated the one-loop factorization for inclusive hadron productions in pA collisions in the saturation formalism. The differential cross section is written into a factorization form in the coordinate space at the next-to-leading order, while the naive form of the convolution in the transverse momentum space does not hold. The rapidity divergence with small-x dipole gluon distribution of the nucleus is factorized into the energy evolution of the dipole gluon distribution function, which is known as the Balitsky-Kovchegov equation. Furthermore, the collinear divergences associated with the incoming parton distribution of the nucleon and the outgoing fragmentation function of the final state hadron are factorized into the splittings of the associated parton distribution and fragmentation functions, which allows us to reproduce the well-known DGLAP equation. The hard coefficient function, which is finite and free of divergence of any kind, is evaluated at one-loop order. This result is important, not only for the phenomenological applications to the inclusive hadron production in p-A collisions at RHIC and future LHC experiment, but also for theoretically promoting the rigorous developments towards a complete QCD factorization in small-x physics.

Forward rapidity di-hadron azimuthal angular correlations in high energy proton-nucleus and proton-proton collisions are sensitive to quadrupoles; traceless correlator of 4 Wilson lines whereas single inclusive particle production iNVOLVES only dipoles, traceless correlator of 2 Wilson lines. We discuss the progress made in understanding the energy (rapidity) evolution of the quadrupole as well as its various limits.

Our original suggestion to investigate exclusive π⁰ electroproduction as a method for extracting from data the tensor charge, transversity, and other quantities related to chiral odd generalized parton distributions is further examined. We now explain the details of the process: i) the connection between the helicity description and the cartesian basis; ii) the dependence on the momentum transfer squared, Q², and iii) the angular momentum, parity, and charge conjugation constraints (J^PC quantum numbers).

We discuss the information on the spin structure of the proton that one can access by studying transverse momentum distributions, generalized parton distributions and their connection through generalized transverse momentum distributions. A "flexible" parametrization is proposed to extract this information from deeply virtual exclusive experiments.

A brief review of the theory of generalized parton distributions (GPDs) is given. We discuss the basic concepts of the GPD theory and relationship between GPDs and simpler phenomenological functions, viz. form factors, parton densities and distribution amplitudes. A special emphasis is given to the formulation of GPDs in terms of double distributions.

We discuss recent developments in building models for GPDs that are based on the formalism of double distributions (DDs). A special attention is given to a careful analysis of the singularity structure of DDs. The DD formalism is applied to construction of a model GPDs with a singular Regge behavior. Within the developed DD-based approach, we discuss the structure of GPD sum rules. It is shown that separation of DDs into the so-called "plus" part and the D-term part may be treated as a renormalization procedure for the GPD sum rules. This approach is compared with an alternative prescription based on analytic regularization.

We present recent developments on the role of the running coupling constant at the intersection of perturbative and nonperturbative QCD. A number of experiments show a smooth transition from small to large scales given by the four-momentum transfer in the reactions. This is at variance with perturbative QCD where the running coupling constant becomes infinite when the scale equals Λ_QCD. Approaches using an effective coupling constant could help interpret the opposite trend of data as compared to standard perturbative QCD predictions. We give an overview of the role of the coupling constant in the procedure to match nonperturbative hadronic model to perturbative QCD and we propose an extraction of an effective coupling constant from inclusive electron proton scattering data at large Bjorken x.

We summarize a new analysis of the distribution of charged leptons produced in decays of Z and γ* bosons in the Collins-Soper-Sterman (CSS) formalism for transverse momentum resummation. By comparing the distribution measured at the Tevatron with the resummed CSS cross section with approximate Wilson coefficients, we constrain the magnitude of the nonperturbative Gaussian smearing factor and analyze its uncertainty caused by variations in scale parameters. We find excellent agreement between the data and our theoretical prediction, provided by the RESBOS resummation program. The nonperturbative factor that we obtained can be used to update resummed QCD predictions for precision measurements in inclusive W and Z production and for comparisons to various models of nonperturbative dynamics.