Analytical Evaluation of Dimensionally Regularized Massive On-Shell Double Boxes

TL;DR

This work extends Mellin–Barnes techniques within dimensional regularization to massive on-shell two-loop, four-point Feynman integrals relevant for Bhabha scattering at NNLO. It derives sixfold MB representations for planar double boxes of both types and an eightfold MB representation for the non-planar topology, then resolves singularities in the regulator $\psilon$ and provides analytic results expressed in terms of polylogarithms and related functions, with some coefficients given in closed form (e.g., $c^1_2(x)=\frac{3}{2}\ln^2\frac{1-x}{1+x}$) and others as higher-weight polylogarithms. The master integrals are cross-validated numerically via finite MB integrals and sector decomposition at Euclidean points, and the results highlight the potential need for generalized two-dimensional harmonic polylogarithms (2dHPL) or GHPLs for full analytic compactness. A notable finding is the successful extraction of double-pole and finite parts for planar topologies, while the non-planar case remains challenging numerically, indicating directions for methodological development in sector decomposition to handle massive, sign-varying kinematics. Overall, the paper provides concrete MB-based analytic access to key NNLO contributions to Bhabha scattering and clarifies current limitations in the massive non-planar sector.

Abstract

The method of Mellin-Barnes representation is used to calculate dimensionally regularized massive on-shell double box Feynman diagrams contributing to Bhabha scattering at two loops.

Figures (1)

• Figure 1: Massive on-shell double boxes: (a) planar double box of the first type, (b) planar double box of the second type and (c) non-planar double box. The solid lines denote massive, the dotted lines massless particles.