Numerical calculation of one-loop integration with hypergeometric functions

TL;DR

This paper derives analytical expressions for one-loop scalar functions using Lauricella's hypergeometric functions and develops a numerical library for their stable computation, validated against existing tools.

Contribution

It provides exact analytical formulas for one-loop scalar functions in terms of hypergeometric functions and introduces a numerical library for their stable evaluation.

Findings

Derived explicit formulas for two-, three-, and four-point functions.

Developed a numerical library for hypergeometric function evaluation.

Validated results against the golem95 package.

Abstract

One-loop two-, three- and four-point scalar functions are analytically integrated directly such that they are expressed in terms of Lauricella's hypergeometric function $F_D$. For two- and three-point functions, exact expressions are obtained with arbitrary combination of kinematic and mass parameters in arbitrary space-time dimension. Four-point function is expressed in terms of $F_D$ up to the finite part in the expansion around 4-dimensional space-time with arbitrary combination of kinematic and mass parameters. Since the location of the possible singularities of $F_D$ is known, information about the stabilities in the numerical calculation is obtained. We have developed a numerical library calculating $F_D$ around 4-dimensional space-time. The numerical values for IR divergent cases of four-point functions in massless QCD are calculated and agreed with golem95 package.