Planar box diagram for the (N_F = 1) 2-loop QED virtual corrections to Bhabha scattering

TL;DR

The paper delivers a complete analytic determination of the Master Integrals for planar 2-loop box diagrams with one electron loop ($N_F=1$) contributing to Bhabha scattering in QED, without neglecting the electron mass. By reducing all relevant scalar integrals to 14 Master Integrals via IBP, LI, and symmetry identities and solving the resulting differential equations in Mandelstam variables, the authors obtain closed-form, $D-4$ Laurent expansions expressed in terms of 1d and 2d harmonic polylogarithms up to weight 3. The results include explicit expressions for the two nontrivial 5-denominator MIs, their asymptotic expansions in key kinematic regions, and the full Laurent expansion of the 6-denominator scalar integral in terms of lower-denominator MIs. These analytic results enhance the precision of next-to-next-to-leading order QED corrections to Bhabha scattering and provide essential building blocks for the full two-loop amplitude. The work also supplies appendices with loop propagator definitions, a 1-loop box result, and the harmonic polylogarithm formalism, facilitating practical implementation and cross-checks.

Abstract

In this paper we present the master integrals necessary for the analytic calculation of the box diagrams with one electron loop (N_{F}=1) entering in the 2-loop (α^3) QED virtual corrections to the Bhabha scattering amplitude of the electron. We consider on-shell electrons and positrons of finite mass m, arbitrary squared c.m. energy s, and momentum transfer t; both UV and soft IR divergences are regulated within the continuous D-dimensional regularization scheme. After a brief overview of the method employed in the calculation, we give the results, for s and t in the Euclidean region, in terms of 1- and 2-dimensional harmonic polylogarithms, of maximum weight 3. The corresponding results in the physical region can be recovered by analytical continuation. For completeness, we also provide the analytic expression of the 1-loop scalar box diagram including the first order in (D-4).

Figures (8)

• Figure 1: 2-loop box diagram relevant for the purpose of the paper; the momenta $p_1$ and $p_2$ are incoming, $p_3$ and $p_4$ outgoing.
• Figure 2: The 6-denominator topology.
• Figure 3: The set of four 5-denominator topologies. The last one is the product of a 1-loop box and a tadpole.
• Figure 4: The set of six independent 4-denominator topologies.
• Figure 5: The set of six independent 3-denominator topologies.