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

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.