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=4−2 ε, 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.