One loop integration with hypergeometric series by using recursion relations

TL;DR

This paper introduces a novel method for evaluating one-loop integrals in quantum field theory using recursion relations and hypergeometric functions, enabling more efficient calculations of multi-point functions.

Contribution

It develops a recursion relation based on Bernstein theorem that expresses one-loop integrals via a new hypergeometric function, providing explicit power series coefficients.

Findings

Derived recursion relations connecting n-point to (n+1)-point functions.

Expressed one-loop integrals using a new hypergeometric function related to Aomoto-Gelfand functions.

Validated numerical results against LoopTools for two- and three-point functions.

Abstract

General one-loop integrals with arbitrary mass and kinematical parameters in $d$-dimensional space-time are studied. By using Bernstein theorem, a recursion relation is obtained which connects $(n+1)$-point to $n$-point functions. In solving this recursion relation, we have shown that one-loop integrals are expressed by a newly defined hypergeometric function, which is a special case of Aomoto-Gelfand hypergeometric functions. We have also obtained coefficients of power series expansion around 4-dimensional space-time for two-, three- and four-point functions. The numerical results are compared with LoopTools for the case of two- and three-point functions as examples.