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Determining the Nonperturbative Collins-Soper Kernel From Lattice QCD

Markus A. Ebert, Iain W. Stewart, Yong Zhao

TL;DR

The paper presents a lattice QCD strategy to determine the nonperturbative Collins-Soper kernel $oldsymbol{\\gamma_\\zeta^q(\mu,b_T)}$ that governs TMDPDF evolution in the nonperturbative region. By employing LaMET to relate equal-time, boosted-quark matrix elements (quasi-TMDPDFs) to lightcone TMDPDFs and forming ratios at different momenta, soft contributions cancel and one can extract $oldsymbol{\\gamma_\\zeta^q}$ through a known short-distance coefficient. The framework includes explicit one-loop relations for the quasi-TMDPDF and a nonperturbative soft mismatch $g^S_q(b_T,\mu)$, enabling a controlled lattice-to-continuum matching. A concrete one-loop illustration confirms the method reproduces the correct Collins-Soper kernel, highlighting its potential for first-principles inputs to TMD phenomenology. The approach promises improved control over nonperturbative evolution in QCD transverse-momentum dynamics and offers clear paths for future refinements and extensions to gluons and alternative boosted states.

Abstract

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 $q_T \sim Λ_\mathrm{QCD}$ 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.

Determining the Nonperturbative Collins-Soper Kernel From Lattice QCD

TL;DR

The paper presents a lattice QCD strategy to determine the nonperturbative Collins-Soper kernel that governs TMDPDF evolution in the nonperturbative region. By employing LaMET to relate equal-time, boosted-quark matrix elements (quasi-TMDPDFs) to lightcone TMDPDFs and forming ratios at different momenta, soft contributions cancel and one can extract through a known short-distance coefficient. The framework includes explicit one-loop relations for the quasi-TMDPDF and a nonperturbative soft mismatch , enabling a controlled lattice-to-continuum matching. A concrete one-loop illustration confirms the method reproduces the correct Collins-Soper kernel, highlighting its potential for first-principles inputs to TMD phenomenology. The approach promises improved control over nonperturbative evolution in QCD transverse-momentum dynamics and offers clear paths for future refinements and extensions to gluons and alternative boosted states.

Abstract

At small transverse momentum , 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 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.

Paper Structure

This paper contains 9 sections, 32 equations, 2 figures.

Table of Contents

  1. Introduction
  2. TMD Factorization
  3. TMDPDFs on the Lattice
  4. Relating quasi-TMDPDF and TMDPDF
  5. Explicit relation at one loop
  6. Extracting gammazeta on Lattice
  7. Illustration at one loop
  8. Conclusions and Outlook
  9. (Quasi) TMDPDF at NLO

Figures (2)

  • Figure 1: Illustration of the Wilson line structure of (a) the $n$-collinear beam function $B_{q}$ and (b) the soft function $S^q$, defined in Eqs. \ref{['eq:beam']} and \ref{['eq:soft']}. The Wilson lines (solid) extend to infinity in the directions indicated and are joined there by transverse Wilson lines. The $\tau$ dependence that regulates singularities from these Wilson lines is not shown. Adapted from Ref. Li:2016axz.
  • Figure 2: (a): Illustration of the Wilson line structure of the quasi beam function $\tilde{B}_q$ in Eq. \ref{['eq:qbeam']}. (b): Behavior of a longitudinal separation $b^z$ (blue solid) under a Lorentz boost along the $z$ direction (orange dotted), and its approximate limit $-\gamma b^z {\bar{n}}$.