# One-loop renormalization of staple-shaped operators in continuum and lattice regularizations

## TL;DR

This paper provides one-loop renormalization results for staple-shaped nonlocal quark operators in continuum and lattice QCD, including divergence analysis, operator mixing, and conversion schemes, aiding lattice TMD studies.

## Contribution

It introduces the first comprehensive one-loop renormalization and mixing analysis for staple-shaped operators in both continuum and lattice regularizations, with practical conversion formulas.

## Key findings

- Identified linear and logarithmic divergences including cusp divergences.
- Calculated mixing coefficients among nonlocal operators on the lattice.
- Provided conversion factors to translate lattice results to MS-bar scheme.

## Abstract

In this paper we present one-loop results for the renormalization of nonlocal quark bilinear operators, containing a staple-shaped Wilson line, in both continuum and lattice regularizations. The continuum calculations were performed in dimensional regularization, and the lattice calculations for the Wilson/clover fermion action and for a variety of Symanzik-improved gauge actions. We extract the strength of the one-loop linear and logarithmic divergences (including cusp divergences), which appear in such nonlocal operators; we identify the mixing pairs which occur among some of these operators on the lattice, and we calculate the corresponding mixing coefficients. We also provide the appropriate RI'-like scheme, which disentangles this mixing nonperturbatively from lattice simulation data, as well as the one-loop expressions of the conversion factors, which turn the lattice data to the MS-bar scheme. Our results can be immediately used for improving recent nonperturbative investigations of transverse momentum-dependent distribution functions (TMDs) on the lattice. Finally, extending our perturbative study to general Wilson-line lattice operators with n cusps, we present results for their renormalization factors, including identification of mixing and determination of the corresponding mixing coefficients, based on our results for the staple operators.