Massless on-shell box integral with arbitrary powers of propagators

TL;DR

This paper derives a general expression for the massless one-loop box integral with arbitrary propagator powers in any space-time dimension, using hypergeometric functions and Gr"obner basis techniques.

Contribution

It introduces a novel method to reduce the integral to hypergeometric functions via a third order differential equation solved with Gr"obner basis methods.

Findings

Integral expressed as a sum over three hypergeometric functions

Recurrence relations from Gr"obner basis provided

First terms of epsilon expansion calculated

Abstract

The massless one-loop box integral with arbitrary indices in arbitrary space-time dimension $d$ is shown to reduce to a sum over three generalised hypergeometric functions. This result follows from the solution to the third order differential equation of hypergeometric type. To derive the differential equation, the Gr\"obner basis technique for integrals with noninteger powers of propagators was used. A complete set of recurrence relations from the Gr\"obner basis is presented. The first several terms in the $\varepsilon =(4-d)/2$ expansion of the result are given.