Renormalization and Matching for the Collins-Soper Kernel from Lattice QCD

TL;DR

This paper advances the calculation of the Collins-Soper kernel, essential for TMDPDF evolution, by providing one-loop renormalization in RI'/MOM scheme, conversion to MS scheme, and a position space method for lattice QCD determination.

Contribution

It introduces a one-loop renormalization and conversion procedure for quasi-TMDPDF operators and proposes a position space approach to simplify lattice calculations of the Collins-Soper kernel.

Findings

Renormalization of staple-shaped Wilson line operators in RI'/MOM scheme.

Conversion factor from RI'/MOM to MS scheme.

A new position space method for kernel calculation.

Abstract

The Collins-Soper kernel, which governs the energy evolution of transverse-momentum dependent parton distribution functions (TMDPDFs), is required to accurately predict Drell-Yan like processes at small transverse momentum, and is a key ingredient for extracting TMDPDFs from experiment. Earlier we proposed a method to calculate this kernel from ratios of the so-called quasi-TMDPDFs determined with lattice QCD, which are defined as hadronic matrix elements of staple-shaped Euclidean Wilson line operators. Here we provide the one-loop renormalization of these operators in a regularization-independent momentum subtraction (RI$^\prime$/MOM) scheme, as well as the conversion factor from the RI$^\prime$/MOM-renormalized quasi-TMDPDF to the $\overline{\rm MS}$ scheme. We also propose a procedure for calculating the Collins-Soper kernel directly from position space correlators, which simplifies the lattice determination.