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

TL;DR

The paper delivers a complete analytic evaluation of the master integrals for a planar two-loop Bhabha scattering family in QED using a canonical differential equations approach. By choosing a basis of pure, uniform-weight integrals, the authors cast the system into d f = ε d à f, enabling straightforward ε-by-epsilon integration and transparent boundary conditions. They identify the relevant function space—Chen iterated integrals and a two-dimensional generalization of harmonic polylogarithms (2dHPLs / Goncharov polylogarithms)—and provide explicit results up to weight four, with all higher-order terms accessible from the same framework. The results are validated against existing literature and numerical methods, and the authors discuss representations, numerical strategies, and potential extensions to single-valued function bases and alternative variable mappings. This work advances precise NNLO predictions for Bhabha scattering and clarifies the function space needed for massive two-loop 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.

Figures (4)

• Figure 1: One-loop and two-loop family of integrals (2a) for Bhabha scattering. Solid lines indicate massive propagators, and dashed lines massless ones.
• Figure 2: Master integrals for the integral family of one-loop Bhabha integrals.
• Figure 3: Two integration paths are shown connecting the base point (black dot) and the argument of the function (circle). The Euclidean region for real $x,y$ with $0<x<1, 0<y<1$ is delimited by dashed lines. Note that this picture is a $\mathbb{R}^2$ slice of $\mathbb{C}^2$. Chen iterated integrals are homotopy invariant (on the space where the singular points are removed.)
• Figure 4: Master integrals for integral family (2a) of Bhabha integrals. Dots denote doubled propagators. An asterisk indicates numerators not shown in the picture, and a dagger indicates that the master integral is a linear combinations of integrals (possibly with dots), that are not shown.