Laurent series expansion of a class of massive scalar one-loop integrals to ${\cal O}(\ep^2)

Abstract

We use dimensional regularization to calculate the ${\cal O}(\ep^2)$ expansion of all scalar one-loop one-, two-, three- and four-point integrals that are needed in the calculation of hadronic heavy quark production. The Laurent series up to ${\cal O}(\ep^2)$ is needed as input to that part of the NNLO corrections to heavy flavor production at hadron colliders where the one-loop integrals appear in the loop-by-loop contributions. The four-point integrals are the most complicated. The ${\cal O}(\ep^2)$ expansion of the three- and four-point integrals contains in general polylogarithms up to ${\rm Li}_4$ and functions related to multiple polylogarithms of maximal weight and depth four.