Analytic results for two-loop planar master integrals for Bhabha scattering

TL;DR

This paper analytically computes two-loop planar master integrals for Bhabha scattering in QED, using differential equations, and introduces results involving multiple polylogarithms and elliptic polylogarithms, advancing precision in theoretical calculations.

Contribution

It provides the first complete analytic evaluation of these integrals, including elliptic cases, using canonical differential equations and polylogarithmic functions.

Findings

All master integrals expressed in terms of multiple polylogarithms except one

Derived a compact elliptic multiple polylogarithm result for a specific integral

Enhanced analytical understanding of two-loop Bhabha scattering calculations

Abstract

We analytically evaluate the master integrals for the second type of planar contributions to the massive two-loop Bhabha scattering in QED using differential equa- tions with canonical bases. We obtain results in terms of multiple polylogarithms for all the master integrals but one, for which we derive a compact result in terms of elliptic mul- tiple polylogarithms. As a byproduct, we also provide a compact analytic result in terms of elliptic multiple polylogarithms for an integral belonging to the first family of planar Bhabha integrals, whose computation in terms of polylogarithms was addressed previously in the literature.