Perturbative Corrections to Quark TMDPDFs in the Background-Field Method: Gauge Invariance, Equations of Motion, and Multiple Interactions

TL;DR

This paper computes perturbative corrections to quark TMDPDFs using a background-field method, emphasizing gauge invariance and the role of equations of motion, and explores multiple interactions at all orders of strong coupling.

Contribution

It extends the background-field approach to include all-order multiple interactions and clarifies the treatment of EoM-related singularities in quark TMDPDFs.

Findings

Proper identification and removal of EoM-related singularities.

Gauge-invariant TMDPDF operator with transverse gauge link.

Analysis of multiple interactions at all orders of strong coupling.

Abstract

We calculate the perturbative corrections in the strong coupling to the unpolarized quark transverse-momentum dependent parton distribution function (TMDPDF) operator within a background-field framework, extending the approach of Ref. [1]. We focus on ensuring gauge invariance, identifying two key components needed: a gauge-invariant TMDPDF operator with a transverse gauge link at spatial infinity, and accounting of the equations of motion (EoM) of the background fields. We go beyond next-to-leading order in strong coupling expansion, considering multiple interactions with the background field at all orders of strong coupling. By examining the interplay between quark and gluon contributions, we show that spurious singularities, proportional to EoM, can be misinterpreted as genuine divergences in QCD factorization unless properly identified and removed.