# Twist-2 matching of transverse momentum dependent distributions

## TL;DR

This paper performs a systematic NLO analysis of the large-$q_T$ matching of TMD distributions to twist-2 distributions, identifying which distributions match and providing explicit matching coefficients.

## Contribution

It provides the complete set of twist-2 TMD distributions that match at NLO, including the first calculation of their matching coefficients and insights into the pretzelosity distribution.

## Key findings

- Matching coefficients for unpolarized, helicity, transversity, and linearly polarized gluon distributions are calculated.
- Pretzelosity matching coefficient is zero at this order, explaining its small phenomenological value.
- Rapidity divergence cancellation imposes constraints on TMD operator structure and $ ext{gamma}_5$-definition.

## Abstract

We systematically study the large-$q_T$ (or small-$b$) matching of transverse momentum dependent (TMD) distributions to the twist-2 integrated parton distributions. Performing operator product expansion for a generic TMD operator at the next-to-leading order (NLO) we found the complete set of TMD distributions that match twist-2. These are unpolarized, helicity, transversity, pretzelosity and linearly polarized gluon distributions. The NLO matching coefficients for these distributions are presented. The pretzelosity matching coefficient is zero at the presented order, however, it is evident that it is non-zero in the following orders. This result offers a natural explanation of the small value of pretzelosity found in phenomenological fits. We also demonstrate that the cancellation of rapidity divergences by the leading order soft factor imposes the necessary requirement on the Lorentz structure of TMD operators, which is supported only by the TMD distributions of leading dynamical twist. Additionally, this requirement puts restrictions on the $\gamma_5$-definition in the dimensional regularization.