One-loop pentagon integral with one offshell leg in $d$ dimensions from differential equations in $\epsilon$-form

TL;DR

This paper develops a method using differential equations in epsilon-form to compute a one-loop pentagon integral with one offshell leg, providing a simple integral representation valid in any dimension.

Contribution

It introduces a reduction to epsilon-form for the differential equations, enabling an exact, simplified integral representation of the integral in arbitrary dimensions.

Findings

Derived a one-fold integral representation valid in any dimension

Performed epsilon expansion and analytical continuation

Simplified the calculation of the one-loop pentagon integral

Abstract

We apply differential equations technique to the calculation of the one-loop massless diagram with one offshell legs. Using reduction to $\epsilon$-form, we managed to obtain a simple one-fold integral representation exact in space-time dimensionality. Expansion of the obtained result in $\epsilon$ and analytical continuation to physical region are discussed.