A unified formulation of one-loop tensor integrals for finite volume effects

TL;DR

This paper introduces a unified method for calculating one-loop tensor integrals in finite volume settings, facilitating automated and precise finite volume correction computations relevant for lattice QCD analyses.

Contribution

It presents a new unified formulation for one-loop tensor integrals in finite volume, including a generic numerical formula and analysis of reduction techniques.

Findings

Feasible tensor decomposition with a unit space-like vector

Derived a generic formula for tensor coefficient computation

Analyzed Passarino-Veltmann reduction in finite volume

Abstract

A unified formulation of one-loop tensor integrals is proposed for systematical calculations of finite volume corrections. It is shown that decomposition of the one-loop tensor integrals into a series of tensors accompanied by tensor coefficients is feasible, if a unit space-like four vector $n^\mu$, originating from the discretization effects at finite volume, is introduced. A generic formula has been derived for numerical computations of all the involved tensor coefficients. For the vanishing external three-momenta, we also investigate the feasibility of the conventional Passarino-Veltmann reduction of the tensor integrals in a finite volume. Our formulation can be easily used to realize the automation of the calculations of finite volume corrections to any interesting quantities at one-loop level. Besides, it provides finite volume result in a unique and concise form, which is suited for, e.g., carrying out precision determination of physical observable from modern lattice QCD data.