Master integrals for massive two-loop Bhabha scattering in QED

TL;DR

The paper delivers a comprehensive catalog and analysis of scalar master integrals needed for massive two-loop Bhabha scattering in QED, addressing one- and multi-flavor configurations. It employs state-of-the-art techniques including IBP/LI reduction, differential equations, sector decomposition, and Mellin-Barnes methods to obtain analytic and semi-analytic ε-expansions, primarily in terms of harmonic polylogarithms up to weight 4. By providing explicit results for several challenging MIs (notably in the 3- and 4-point sectors) and documenting the divergent structures, it lays the groundwork for a complete two-loop prediction when combined with real-emission and electroweak corrections. The published Mathematica files MastersBhabha.m and HPL4.m make the results accessible for practical use in precision luminosity calculations at future colliders.

Abstract

We present a set of scalar master integrals (MIs) needed for a complete treatment of massive two-loop corrections to Bhabha scattering in QED, including integrals with arbitrary fermionic loops. The status of analytical solutions for the MIs is reviewed and examples of some methods to solve MIs analytically are worked out in more detail. Analytical results for the pole terms in epsilon of so far unknown box MIs with five internal lines are given.

Figures (12)

• Figure 1: The two-loop self-energy diagrams for massive Bhabha scattering. SE3 to SE5 are needed for the renormalization of external fermion lines.
• Figure 2: The two-loop vertex diagrams for massive Bhabha scattering.
• Figure 3: The two-loop box diagrams for massive Bhabha scattering.
• Figure 4: The five one-loop MIs. External solid (dashed) lines describe on- (off-)shell momenta.
• Figure 5: The six two-loop 2-point MIs. External solid (dashed) lines describe on (off) -shell momenta.