Threshold resummation for computing large-$x$ parton distribution through large-momentum effective theory

TL;DR

This paper develops a threshold resummation framework for large-$x$ parton distribution functions using lattice QCD, enabling more accurate calculations by resumming large logarithms and connecting to known perturbative results.

Contribution

It introduces a full renormalization group resummation of threshold logarithms for large-$x$ PDFs in lattice QCD, consistent with NNLO calculations, and explores related universal functions.

Findings

Derived a factorization formula with jet and Sudakov functions.

Performed NNLO calculations of the heavy-light Sudakov form factor.

Established the relation between space-like and time-like jet functions.

Abstract

Parton distribution functions (PDFs) at large $x$ are poorly constrained by high-energy experimental data, but extremely important for probing physics beyond standard model at colliders. We study the calculation of PDFs at large-$x$ through large-momentum $P^z$ expansion of the lattice quasi PDFs. Similar to deep-inelastic scattering, there are two distinct perturbative scales in the threshold limit where the matching coefficient can be factorized into a space-like jet function at scale $P^z|1-y|$ and a pair of heavy-light Sudakov form factors at scale $P^z$. The matching formula allows us to derive a full renormalization group resummation of large threshold logarithms, and the result is consistent with the known calculation to the next-to-next to leading order (NNLO). This paves the way for direct large-$x$ PDFs calculations in lattice QCD. As by-products, we find that the space-like jet function is related to a time-like version calculated previously through analytic continuation, and the heavy-light Sudakov form factor, calculated here to NNLO, is a universal object appearing as well in the large momentum expansion of quasi transverse-momentum-dependent PDFs and quasi wave-function amplitudes.