Two-Loop N_F =1 QED Bhabha Scattering: Soft Emission and Numerical Evaluation of the Differential Cross-section

TL;DR

The paper delivers a UV- and IR-finite differential cross-section for $e^+e^- \to e^+e^-$ Bhabha scattering in pure QED at two-loop accuracy with $N_F=1$, keeping finite electron mass $m$ and all kinematic dependence on $s$ and $t$. It combines previously computed two-loop virtual corrections with soft-photon real-emission contributions at orders $\mathcal{O}(\alpha^3)$ and $\mathcal{O}(\alpha^4(N_F=1))$, performing a soft-photon phase-space integration with cutoff $\omega$ to demonstrate IR cancellation and produce a numerically evaluable result. The work provides explicit real-emission calculations via integrals $I_{ij}$ and factors $J_{ij}$, shows the cancellation of IR poles among virtual and real pieces (including reducible and box diagrams), and reports small but positive higher-order corrections relative to the dominant negative $\mathcal{O}(\alpha^3)$ terms. Numerical implementations in Mathematica and Fortran77 confirm the expected behavior across a wide range of beam energies $E$ and scattering angles, validating the mass-dependent expansion and offering publicly available codes for precise luminosity determinations at current and future $e^+e^-$ colliders.

Abstract

Recently, we evaluated the virtual cross-section for Bhabha scattering in pure QED, up to corrections of order alpha^4 (N_F =1). This calculation is valid for arbitrary values of the squared center of mass energy s and momentum transfer t; the electron and positron mass m was considered a finite, non vanishing quantity. In the present work, we supplement the previous calculation by considering the contribution of the soft photon emission diagrams to the differential cross-section, up to and including terms of order alpha^4 (N_F=1). Adding the contribution of the real corrections to the renormalized virtual ones, we obtain an UV and IR finite differential cross-section; we evaluate this quantity numerically for a significant set of values of the squared center of mass energy s.

Figures (15)

• Figure 1: Diagrams contributing to the real corrections at order $\alpha^3$.
• Figure 2: Cancellation of the IR divergencies in the one-loop vertex diagrams.
• Figure 3: Cancellation of the IR divergencies in the one-loop box diagrams.
• Figure 4: Diagrams contributing to the real corrections at order $\alpha^4(N_F = 1)$.
• Figure 5: Cancellation of the IR divergencies of the two-loop $N_F=1$ reducible diagrams.