Quark Pseudo-Distributions at Short Distances

TL;DR

This paper provides a one-loop analysis of the small-distance behavior of the Ioffe-time distribution, clarifying UV and short-distance effects, and offers formulas useful for extracting parton distribution functions from lattice QCD.

Contribution

It presents the first explicit one-loop calculation of the Ioffe-time distribution's short-distance behavior, separating UV divergences from perturbative evolution effects.

Findings

UV divergences vanish at zero $z_3^2$ with finite cut-off.

The ratio of ITDs removes UV singularities but retains short-distance logarithms.

Explicit one-loop expression for reduced ITD aids PDF extraction from lattice QCD.

Abstract

We perform a one-loop study of the small-$z_3^2$ behavior of the Ioffe-time distribution (ITD) ${\cal M} (\nu, z_3^2)$, the basic function that may be converted into parton pseudo- and quasi-distributions. We calculate the corrections at the operator level, so that our results may be later used for pseudo-distribution amplitudes and generalized parton pseudo-distributions. We separate two sources of the $z_3^2$-dependence at small $z_3^2$. One is related to the ultraviolet (UV) singularities generated by the gauge link, and another to short-distance logarithms generating perturbative evolution of parton densities. Our calculation explicitly shows that, for a finite UV cut-off, the UV-singular terms vanish when $z_3^2=0$. The UV divergences are absent in the ratio ${\cal M} (\nu, z_3^2)/{\cal M} (0, z_3^2)$ ("reduced" ITD). Still, it has a non-trivial short-distance behavior due to $\ln z_3^2 \Lambda^2$ terms generating perturbative evolution of the parton densities. We give an explicit expression, up to constant terms, for the reduced ITD at one loop. It may be used in extraction of PDFs from the lattice QCD simulations. We also use our results to get new insights concerning the structure of parton quasi-distributions at one-loop level.