Two-Loop N_F=1 QED Bhabha Scattering Differential Cross Section

TL;DR

The authors deliver a complete, UV-renormalized two-loop ($\alpha^4$) QED calculation for Bhabha scattering with $N_F=1$, preserving the full mass $m$ and kinematic dependence ($s,t$). They reduce all loop integrals to $14$ Master Integrals via the Laporta algorithm and solve them with differential equations, obtaining results in $1$- and $2$-dimensional harmonic polylogarithms up to weight $3$. UV renormalization is performed in the on-shell scheme, and IR poles remain only to be canceled by real soft-photon emission, yielding a finite physical cross section when combined with real radiation. The work provides a non-approximate, mass-dependent differential cross section suitable for precision luminosity determinations at $e^+e^-$ colliders and advances explicit two-loop QED predictions using modern analytic techniques.

Abstract

We calculate the two-loop virtual, UV renormalized corrections at order α^4 (N_F=1) in QED to the Bhabha scattering differential cross section, for arbitrary values of the squared c.m. energy s and momentum transfer t, and on-shell electrons and positrons of finite mass m. The calculation is carried out within the dimensional regularization scheme; the remaining IR divergences appear as polar singularities in (D-4). The result is presented in terms of 1- and 2-dimensional harmonic polylogarithms, of maximum weight 3.

Figures (4)

• Figure 1: Tree-level diagrams.
• Figure 2: One-loop diagrams.
• Figure 3: Two-loop $N_F=1$ diagrams.
• Figure 4: Counterterm diagrams at one and two loops.