Angular integrals in d dimensions

TL;DR

This work provides a general, all-order framework for evaluating $d$-dimensional angular integrals with arbitrary denominators in perturbative field theory by expressing them in terms of the multivariable $H$-function via Mellin–Barnes representations. It unifies and extends known results by handling massless and massive momenta, and by keeping the denominator exponents symbolic, yielding explicit hypergeometric forms (Gauss ${}_2F_1$ and Appell $F_1$) in concrete cases. The authors demonstrate the method through detailed examples up to three denominators, discuss epsilon expansions, and outline computational strategies (MB deformation, MB.m) for higher-order terms, highlighting relevance to NNLO subtraction schemes. The approach provides a versatile tool for phase-space integrals in higher-order perturbative calculations and suggests potential for differential-reduction techniques in $d$ dimensions.

Abstract

We discuss the evaluation of certain d dimensional angular integrals which arise in perturbative field theory calculations. We find that the angular integral with n denominators can be computed in terms of a certain special function, the so-called H-function of several variables. We also present several illustrative examples of the general result and briefly consider some applications.