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

TL;DR

This paper analytically evaluates two-loop master integrals for Bhabha scattering in QED using a novel differential equations approach with a pure function basis, providing explicit results up to weight four.

Contribution

It introduces a new basis of master integrals that simplifies differential equations, enabling straightforward integration and boundary condition determination for two-loop Bhabha scattering integrals.

Findings

Explicit analytical expressions for master integrals up to weight four.

Identification of relevant function class: Goncharov polylogarithms and Chen iterated integrals.

Simplified canonical form of differential equations for these integrals.

Abstract

We evaluate analytically the master integrals for one of two types of planar families contributing to massive two-loop Bhabha scattering in QED. As in our previous paper, we apply a recently suggested new strategy to solve differential equations for master integrals for families of Feynman integrals. The crucial point of this strategy is to use a new basis of the master integrals where all master integrals are pure functions of uniform weight. This allows to cast the differential equations into a simple canonical form, which can straightforwardly be integrated order by order in epsilon in dimensional regularization. The boundary conditions are also particularly transparent in this setup. We identify the class of functions relevant to this problem to all orders in epsilon. We present the results up to weight four for all except one integrals in terms of a subset of Goncharov polylogarithms, which one may call two-dimensional harmonic polylogarithms. For one integral, and more generally at higher weight, the solution is written in terms of Chen iterated integrals.