Massless on-shell box integral with arbitrary powers of propagators

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.