Evolution of unpolarized transverse momentum dependent parton distributions of twist-3

TL;DR

This paper derives evolution equations for four unpolarized twist-3 TMD PDFs using diagram expansion and exponential regulator, providing results crucial for high-order perturbative calculations.

Contribution

It presents a direct derivation of twist-3 TMD PDF evolution equations with an exponential regulator, differing from previous background field methods.

Findings

Eight homogeneous evolution equations for twist-3 TMD PDFs.

Agreement with some existing results, but sign differences with others.

Facilitates future high-order perturbative calculations of twist-3 TMD PDFs.

Abstract

We revisit the calculation of the evolution equations for four unpolarized twist-3 transverse momentum dependent (TMD) parton distribution functions (PDFs) at $\mathcal {O}(\alpha_s)$. Unlike the existing calculations in the literature, which are based on the background field method, we derive the evolution equations directly through diagram expansion. Additionally, instead of using the $\delta$ regulator, we employ the exponential regulator to regularize the rapidity divergences. We find that the evolutions of the four twist-3 TMD PDFs are governed by eight homogeneous equations. Our evolution kernels agree with one set of results in the literature, but differ from another by a sign. Considering the advantages of the exponential regulator in high-order perturbative calculations, our results are crucial for computing the twist-3 TMD PDFs at $\mathcal {O}(\alpha_s^2)$.