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Register a new userUnderstanding the large-distance behavior of transverse-momentum-dependent parton densities and the Collins-Soper evolution kernel
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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.
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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.
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.
The rapidity anomalous dimension (RAD), or Collins-Soper kernel, defines the scaling properties of transverse momentum dependent distributions and can be extracted from the experimental data. I derive a self-contained nonperturbative definition that represents RAD without reference to a particular process. This definition makes possible exploration of the properties of RAD by theoretical methods on one side, and the properties of QCD vacuum with collider measurements on another side. To demonstrate these possibilities, I compute the power correction to RAD, its large-b asymptotic, and compare these estimations with recent phenomenological extractions.
The Collins-Soper kernel relates transverse momentum-dependent parton distribution functions (TMDPDFs) at different energy scales. For small parton transverse momentum $q_Tsim Lambda_text{QCD}$, this kernel is non-perturbative and can only be determined with controlled uncertainties through experiment or first-principles calculations. This work presents the first exploratory determination of the Collins-Soper kernel using the lattice formulation of Quantum Chromodynamics. In a quenched calculation, the $N_f=0$ kernel is determined at scales in the range 250 MeV $< q_T < 2$ GeV, and an analysis of the remaining systematic uncertainties is undertaken.
At small transverse momentum $q_T$, transverse-momentum dependent parton distribution functions (TMDPDFs) arise as genuinely nonperturbative objects that describe Drell-Yan like processes in hadron collisions as well as semi-inclusive deep-inelastic scattering. TMDPDFs naturally depend on the hadron momentum, and the associated evolution is determined by the Collins-Soper equation. For $q_T sim Lambda_mathrm{QCD}$ the corresponding evolution kernel (or anomalous dimension) is nonperturbative and must be determined as an independent ingredient in order to relate TMDPDFs at different scales. We propose a method to extract this kernel using lattice QCD and the Large-Momentum Effective Theory, where the physical TMD correlation involving light-like paths is approximated by a quasi TMDPDF, defined using equal-time correlation functions with a large-momentum hadron state. The kernel is determined from a ratio of quasi TMDPDFs extracted at different hadron momenta.
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