Practical quasi parton distribution functions

TL;DR

The paper develops a practical framework to extract parton distribution functions from lattice QCD using quasi-PDFs, addressing the fundamental power divergences introduced by Wilson lines via a nonperturbative subtraction informed by a static quark–antiquark potential. It then formulates and executes a coordinate-space one-loop perturbative matching between continuum and lattice, exploring both 3D and 2D UV-cutoff schemes, and demonstrates the approach with naive fermion actions, including mean-field improvement and link-smearing to reduce lattice artifacts. The work provides explicit expressions and numerical results for the one-loop matching coefficients, clarifying how to isolate the physical quasi distributions from lattice-regulated counterparts and outlining the path toward nonperturbative matching and extension to more realistic lattice actions and to TMDs/GPDs.

Abstract

A completely new strategy to calculate parton distribution functions on the lattice has recently been proposed. In this method, lattice calculable observables, called quasi distributions, are related to normal distributions. The quasi distributions are known to contain power-law UV divergences arise from a Wilson line in the non-local operator, while the normal distributions only have logatithmic UV divergences. We propose possible method to subtract the power divegence to make the matching of the quasi with the normal distributions well-defined. We also demonstrate the matching of the quasi quark distribution between continuum and lattice implementing the power divergence subtraction. The matching calculations are carried out by one-loop perturbation.

Figures (7)

• Figure 1: Diagrammatic expression of Feynman rules for non-local quark bilinear operator relevant to one-loop perturbative calculation. For $O_{\delta z}^{(2)\mu\nu, AB}(p,q,k)$ (right), two gluon lines have a common momentum $k$ with opposite sign, where this setting is sufficient for the perturbative calculation at one-loop level.
• Figure 2: One-loop diagrams for $\delta\Gamma$s.
• Figure 3: Sunset diagram for quark wave-function renormalization.
• Figure 4: Comparison of functions in table \ref{['TAB:correspondence_cutoff']} between two dimensional and three dimensional cutoff scheme.
• Figure 5: One-loop matching coefficients for each individual diagrams: quark self-energy, vertex-type, sail-type and tadpole-type, as well as their total contribution. The linear divergence is subtracted and the MF improvement is used. Three cases of gluon link smearing are considered for a Wilson line in the non-local operator: unsmear (left), HYP1 Hasenfratz:2001hp (center) and HYP2 DellaMorte:2005yc (right). Both three dimensional (circle symbols) and two dimensional (star symbols) UV cutoff cases are shown.