Transverse-momentum-dependent wave functions and Soft functions at one-loop in Large Momentum Effective Theory

TL;DR

This paper provides a one-loop proof of TMD factorization in LaMET, calculating perturbative corrections to TMD wave functions and soft functions, aiding future first-principle extractions from lattice data.

Contribution

It offers the first one-loop validation of TMD factorization in LaMET and detailed calculations of perturbative corrections using modern techniques.

Findings

One-loop hard functions are extracted.

Perturbative corrections depend on the operator used.

Corrections are less sensitive to transverse separation.

Abstract

In large-momentum effective theory (LaMET), the transverse-momentum-dependent (TMD) light-front wave functions and soft functions can be extracted from the simulation of a four-quark form factor and equal-time correlation functions. In this work, using expansion by regions we provide a one-loop proof of TMD factorization of the form factor. For the one-loop validation, we also present a detailed calculation of ${\cal O}(\alpha_s)$ perturbative corrections to these quantities, in which we adopt a modern technique for the calculation of TMD form factor based the integration by part and differential equation. The one-loop hard functions are then extracted. Using lattice data from Lattice Parton Collaboration on quasi-TMDWFs, we estimate the effects from the one-loop matching kernel and find that the perturbative corrections depend on the operator to define the form factor, but are less sensitive to the transverse separation. These results will be helpful to precisely extract the soft functions and TMD wave functions from the first-principle in future.