Scalar one-loop vertex integrals as meromorphic functions of space-time dimension d

TL;DR

This paper derives representations for scalar one-loop vertex Feynman integrals as meromorphic functions of space-time dimension d, using hypergeometric functions, facilitating calculations at special points and for higher-loop amplitudes.

Contribution

It introduces new representations of scalar one-loop vertex integrals as meromorphic functions of d using hypergeometric functions, enabling easier evaluation at singular points and for higher-point functions.

Findings

Derived explicit hypergeometric function representations of integrals

Provided methods for evaluating integrals at asymptotic and exceptional points

Established recursion relations for higher n-point functions

Abstract

Representations are derived for the basic scalar one-loop vertex Feynman integrals as meromorphic functions of the space-time dimension $d$ in terms of (generalized) hypergeometric functions $_2F_1$ and $F_1$. Values at asymptotic or exceptional kinematic points as well as expansions around the singular points at $d=4+2n$, $n$ non-negative integers, may be derived from the representations easily. The Feynman integrals studied here may be used as building blocks for the calculation of one-loop and higher-loop scalar and tensor amplitudes. From the recursion relation presented, higher n-point functions may be obtained in a straightforward manner.