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Register a new userEvolution of transverse-distance dependent parton densities at large-$x_B$ and geometry of the loop space
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We discuss possible applications of the equations of motion in the generalized Wilson loop space to the phenomenology of the three-dimensional parton distribution functions in the large-$x_B$ approximation. This regime is relevant for future experimental programs to be launched at the (approved) Jefferson Lab 12 GeV upgrade and the (planned) Electron-Ion Collider. We show that the geometrical evolution of the Wilson loops corresponds to the combined rapidity and renormalization-group equation of the transverse-distance dependent parton densities in the large-$x_B$ factorization scheme.
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QCD factorization approach in the field-theoretic description of the semi-inclusive hadronic processes in the large Bjorken $x$ approximation implies extraction of the three-dimensional parton distribution functions as a convolution of a collinear jet function and soft transverse-distance dependent (TDD) function defined as a vacuum average of a partially light-like Wilson loop. The soft function can be interpreted, therefore, as an element of generalized loop space. A class of classically conformal-invariant transformations of the elements of this space is generated by the non-local area derivative operator which corresponds to a diffeomorphism in the loop space and determines equations of motion, the latter being associated with the rapidity evolution of the soft TDD functions. We propose a large-$x$ TDD factorization framework and discuss practical applications of this approach to the phenomenology of the TDDs accessible in future experimental programs at the Jefferson Lab 12 GeV and the Electron-Ion Collider.
There is considerable controversy about the size and importance of nonperturbative contributions to the evolution of transverse-momentum-dependent (TMD) parton distribution functions. Standard fits to relatively high-energy Drell-Yan data give evolution that when taken to lower Q is too rapid to be consistent with recent data in semi-inclusive deeply inelastic scattering. Some authors provide very different forms for TMD evolution, even arguing that nonperturbative contributions at large transverse distance b_T are not needed or are irrelevant. Here, we systematically analyze the issues, both perturbative and nonperturbative. We make a motivated proposal for the parameterization of the nonperturbative part of the TMD evolution kernel that could give consistency: with the variety of apparently conflicting data, with theoretical perturbative calculations where they are applicable, and with general theoretical nonperturbative constraints on correlation functions at large distances. We propose and use a scheme- and scale-independent function A(b_T) that gives a tool to compare and diagnose different proposals for TMD evolution. We also advocate for phenomenological studies of A(b_T) as a probe of TMD evolution. The results are important generally for applications of TMD factorization. In particular, they are important to making predictions for proposed polarized Drell-Yan experiments to measure the Sivers function.
I review some open questions relating to the large transverse momentum divergences in transverse moments of transverse momentum dependent (TMD) parton correlation func- tions. I also explain, in an abbreviated and summarized form, recent work that shows that the resulting violations of a commonly used integral relation are not perturbatively suppressed. I argue that this implies a need for more precise definitions for the correlation functions used to describe transverse moments.
Parton distribution functions (PDFs) are nonperturbative objects defined by nonlocal light-cone correlations. They cannot be computed directly from Quantum Chromodynamics (QCD). Using a standard lattice QCD approach, it is possible to compute moments of PDFs, which are matrix elements of local operators. Recently, an alternative approach has been proposed, based on the introduction of quasi-parton distribution functions (quasi-PDFs), which are matrix elements of equal-time spatial correlations and hence calculable on lattice. Quasi-PDFs approach standard PDFs in the limit of very large longitudinal proton momenta $P^z$. This limit is not attainable in lattice simulations, and quasi-PDFs fail to reproduce PDFs at high fractional longitudinal momenta. In this paper, we propose a method to improve the reconstruction of PDFs by combining information from quasi-PDFs and from the Mellin moments of regular PDFs. We test our method using the diquark spectator model for up and down valence distributions of both unpolarized and helicity PDFs. In the future, the method can be used to produce PDFs entirely based on lattice QCD results.
Equations of motion for the light-like QCD Wilson loops are studied in the generalized loop space (GLS) setting. To this end, the classically conformal-invariant non-local variations of the cusped Wilson exponentials lying (partially) on the light-cone are formulated in terms of the Frechet derivative. The rapidity and renormalization-group behaviour of the gauge-invariant quantum correlation functions (in particular, the three-dimensional parton densities) are demonstrated to be connected to certain geometrical properties of the Wilson loops defined in the GLS.
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