A New Hypergeometric Representation of One-Loop Scalar Integrals in $d$ Dimensions

TL;DR

The paper addresses the computation of scalar one-loop integrals with arbitrary external momenta and masses in any space-time dimension by introducing a dimension-difference equation approach. It expresses $n$-point results as hypergeometric series with Gram determinant ratio expansion variables and a boundary term, and provides explicit $2$-, $3$-, and $4$-point representations in terms of $_2F_1$, $F_1$, and $F_S$, plus simple $d=4-2\varepsilon$ integral forms for ε-expansion. The method is illustrated with Bhabha-scattering diagrams, and the boundary term is determined via large-$d$ asymptotics or a differential-equation approach, enabling practical higher-order perturbative calculations.

Abstract

A difference equation w.r.t. space-time dimension $d$ for $n$-point one-loop integrals with arbitrary momenta and masses is introduced and a solution presented. The result can in general be written as multiple hypergeometric series with ratios of different Gram determinants as expansion variables. Detailed considerations for $2-,3-$ and $4-$point functions are given. For the $2-$ point function we reproduce a known result in terms of the Gauss hypergeometric function $_2F_1$. For the $3-$point function an expression in terms of $_2F_1$ and the Appell hypergeometric function $F_1$ is given. For the $4-$point function a new representation in terms of $_2F_1$, $F_1$ and the Lauricella-Saran functions $F_S$ is obtained. For arbitrary $d=4-2ε$, momenta and masses the $2-,3-$ and $4-$point functions admit a simple one-fold integral representation. This representation will be useful for the calculation of contributions from the $ε-$ expansion needed in higher orders of perturbation theory. Physically interesting examples of $3-$ and $4-$point functions occurring in Bhabha scattering are investigated.