Towards Quasi-Transverse Momentum Dependent PDFs Computable on the Lattice

TL;DR

This work tackles the challenge of computing nonperturbative transverse-momentum dependent parton distributions (TMDPDFs) from lattice QCD by formulating quasi-TMDPDFs built from equal-time, finite-length Wilson lines via the LaMET framework. A key contribution is the infrared-consistency test showing that naive constructions of quasi beam and soft functions fail to reproduce the TMDPDF infrared structure at one loop, prompting the introduction of a bent quasi soft function that restores the correct $b_T$-dependent logarithms and enables a perturbative matching. The authors provide explicit one-loop results for the lightlike TMDPDF, naive quasi-beam/soft, and the bent-soft construction, demonstrating a viable matching kernel at this order and highlighting the potential of using ratios of quasi-TMD PDFs to access Collins-Soper evolution nonperturbatively. Together, these results lay out a concrete path toward lattice-based determinations of TMDPDFs (and the Collins-Soper kernel) or, at minimum, robust ratio-based constraints, while also outlining open questions for higher-order validation.

Abstract

Transverse momentum dependent parton distributions (TMDPDFs) which appear in factorized cross sections involve infinite Wilson lines with edges on or close to the light-cone. Since these TMDPDFs are not directly calculable with a Euclidean path integral in lattice QCD, we study the construction of quasi-TMDPDFs with finite-length spacelike Wilson lines that are amenable to such calculations. We define an infrared consistency test to determine which quasi-TMDPDF definitions are related to the TMDPDF, by carrying out a one-loop study of infrared logarithms of transverse position $b_T\sim Λ_{\rm QCD}^{-1}$, which must agree between them. This agreement is a necessary condition for the two quantities to be related by perturbative matching. TMDPDFs necessarily involve combining a hadron matrix element, which nominally depends on a single light-cone direction, with soft matrix elements that necessarily depend on two light-cone directions. We show at one loop that the simplest definitions of the quasi hadron matrix element, the quasi soft matrix element, and the resulting quasi-TMDPDF all fail the infrared consistency test. Ratios of impact parameter quasi-TMDPDFs still provide nontrivial information about the TMDPDFs, and are more robust since the soft matrix elements cancel. We show at one loop that such quasi ratios can be matched to ratios of the corresponding TMDPDFs. We also introduce a modified "bent" quasi soft matrix element which yields a quasi-TMDPDF that passes the consistency test with the TMDPDF at one loop, and discuss potential issues at higher orders.

Figures (13)

• Figure 1: Graphs of the Wilson line structure of the $n$-collinear beam function $B_{q}$ (a) and the soft function $S^q$ (b), defined in Eqs. \ref{['eq:beamfunc']} and \ref{['eq:softfunc']}. The Wilson lines (solid) extend to infinity in the directions indicated. Adapted from Ref. Li:2016axz.
• Figure 2: Illustration of the collinear (orange) and soft (green) modes contribution to the $\vec{q}_T$ measurement. The hard modes (blue) describe offshell modes producing the energetic final state. The dashed lines indicate that the degeneracy in $p^+ p^-$ has to be resolved to properly define separate collinear and soft functions.
• Figure 3: Illustration of the Wilson line structure of the $n$-collinear beam function $B$ (a) and the soft function $S$ (b), with Wilson lines truncated at some finite length $L$. The corresponding Wilson line paths with infinite-long Wilson lines is shown in Fig. \ref{['fig:wilsonlines']}.
• Figure 4: One loop diagrams for the TMD soft function with finite-length Wilson lines in Feynman gauge, up to mirror diagrams. The labels indicate the Wilson line paths in position space.
• Figure 5: Illustration of the Wilson line structure of the quasi beam function (a), and the behavior of the longitudinal separation under a Lorentz boost along the $z$ direction (b).