Two-Loop Bhabha Scattering in QED

TL;DR

This work delivers non-approximate analytic photonic corrections to Bhabha scattering at order α^4 in QED, incorporating irreducible and reducible two-loop vertex effects and one-loop interferences with full electron mass dependence. It employs dimensional regularization and the Laporta-Remiddi reduction to master integrals solved by differential equations, cross-checked against small-mass limits and Penin's results, and pairs virtual corrections with soft-photon emission to ensure IR finiteness. The analysis provides detailed, gauge-consistent expressions and outlines the status of two-loop photonic box contributions, with Appendix B offering the m^2/s → 0 expansion to enable indirect cross-comparisons. The results underpin precise phenomenology for luminosity measurements in e+e− colliders and validate existing high-precision QED calculations while highlighting remaining theoretical gaps for the full two-loop box contributions with finite mass.

Abstract

In the context of pure QED, we obtain analytic expressions for the contributions to the Bhabha scattering differential cross section at order alpha^4 which originate from the interference of two-loop photonic vertices with tree-level diagrams and from the interference of one-loop photonic diagrams amongst themselves. The ultraviolet renormalization is carried out. The IR-divergent soft-photon emission corrections are evaluated and added to the virtual cross section. The cross section obtained in this manner is valid for on-shell electrons and positrons of finite mass, and for arbitrary values of the center of mass energy and momentum transfer. We provide the expansion of our results in powers of the electron mass, and we compare them with the corresponding expansion of the complete order alpha^4 photonic cross section, recently obtained in hep-ph/0501120. As a by-product, we obtain the contribution to the Bhabha scattering differential cross section of the interference of the two-loop photonic boxes with the tree-level diagrams, up to terms suppressed by positive powers of the electron mass. We evaluate numerically the various contributions to the cross section, paying particular attention to the comparison between exact and expanded results.

Figures (18)

• Figure 1: Two-loop irreducible diagrams contributing to the photonic vertex corrections of the electron current in the $t$-channel. The two diagrams, analogous to Figs (c) and (d), which have an insertion of a one-loop vertex and self-energy correction on the outgoing electron line, are not shown.
• Figure 2: Two-loop photonic vertex diagrams contributing to Bhabha scattering.
• Figure 3: Two-loop reducible vertex diagrams contributing to Bhabha scattering.
• Figure 4: One-loop vertex diagrams contributing to Bhabha scattering.
• Figure 5: One-loop box diagrams contributing to Bhabha scattering.