Analytical Result for Dimensionally Regularized Massless On-shell Double Box

TL;DR

This work analytically evaluates the dimensionally regulated, massless on-shell (unit propagator powers) double box Feynman integral for general $s$ and $t$. It develops a Mellin-Barnes representation from the alpha-parametrization, resolves the resulting $\epsilon$-poles via a careful contour-residue analysis, and computes the last MB integral to obtain a closed-form expression in terms of polylogarithms ${\rm Li}_a(-t/s)$ (up to $a=4$) and generalized polylogarithms $S_{a,b}(-t/s)$ (with $a=1,2$ and $b=2$), with an equivalent form in $s/t$. The paper also provides the backward-scattering value $t=-s$ and demonstrates that the method extends to massless on-shell box integrals with any integer propagator powers, offering asymptotic expansions in the limits $t/s\to 0$ and $s/t\to 0$. This MB-based approach thus yields both exact analytic results and practical expansions for important two-loop amplitudes such as Bhabha scattering.

Abstract

The dimensionally regularized massless on-shell double box Feynman diagram with powers of propagators equal to one is analytically evaluated for general values of the Mandelstam variables s and t. An explicit result is expressed either in terms of polylogarithms Li_a(-t/s), up to a=4, and generalized polylogarithms S_{a,b}(-t/s), with a=1,2 and b=2, or in terms of these functions depending on the inverse ratio, s/t.

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