Structure of divergences in Drell-Yan process with small transverse momentum

TL;DR

This paper analyzes the divergence structure in the Drell-Yan process at small transverse momentum, proposing a method within soft-collinear effective theory to disentangle and handle various divergences for a consistent factorization.

Contribution

It introduces a systematic approach to separate and cancel divergences in the Drell-Yan process, clarifying the interplay of soft and collinear parts and their renormalization group behavior.

Findings

Rapidity divergence is managed with the δ regulator in Wilson lines.

Soft function cancels the mixing of UV and IR divergences.

The combined collinear and soft parts exhibit consistent RG behavior at one loop.

Abstract

We consider the structure of divergences in Drell-Yan process with small transverse momentum. The factorization proof is not trivial because various kinds of divergences are intertwined in the collinear and soft parts at high orders. We prescribe a method to disentangle the divergences in the framework of the soft-collinear effective theory. The rapidity divergence is handled by introducing the $\delta$ regulator in the collinear Wilson lines. The collinear part, which consists of the transverse-momentum-dependent parton distribution function (TMDPDF), is free of the rapidity divergence after the soft zero-bin subtraction. There still remains the problem of mixing between the ultraviolet and infrared divergences, which forbids the renormalization group description. We show that the mixing is cancelled by the soft function. This suggests that the collinear and soft parts should be treated as a whole in constructing a consistent factorization theorem. The renormalization group behavior of the combined collinear and soft parts is presented explicitly at one loop. We also show that the integrated PDF can be obtained by integrating the TMDPDF over the transverse momentum.