Two-Loop Fermionic Corrections to Massive Bhabha Scattering

TL;DR

This paper provides a complete computation of two-loop fermionic corrections to Bhabha scattering in QED, retaining exact heavy-fermion mass dependence and performing a high-energy expansion under $m_e^2 \ll m_f^2 \ll s,t,u$. The authors reduce all relevant diagrams to a small set of Master Integrals and evaluate them with differential equations, harmonic polylogarithms, and Mellin-Barnes techniques, distinguishing electron and heavier-fermion contributions. They combine virtual corrections with soft-photon emission to obtain IR-finite NNLO predictions, and present extensive analytic results and numerical tables illustrating the impact of heavy-fermion loops. The findings show that heavier fermions contribute at a level comparable to electron loops in certain kinematics, and that these corrections are essential for precision luminosity determinations at future colliders like the ILC, while remaining manageable relative to photonic two-loop effects. The work also provides a framework and data for integrating these corrections into phenomenological tools and for future extensions to include additional fermion species and higher-order effects.

Abstract

We evaluate the two-loop corrections to Bhabha scattering from fermion loops in the context of pure Quantum Electrodynamics. The differential cross section is expressed by a small number of Master Integrals with exact dependence on the fermion masses me, mf and the Mandelstam invariants s,t,u. We determine the limit of fixed scattering angle and high energy, assuming the hierarchy of scales me^2 << mf^2 << s,t,u. The numerical result is combined with the available non-fermionic contributions. As a by-product, we provide an independent check of the known electron-loop contributions.

Figures (5)

• Figure 1: Classes of Bhabha-scattering one-loop diagrams. A thin fermion line represents an electron, a thick one can be any fermion. The full set of graphs can be obtained through proper permutations. We refer to webPage:2006xx for the reproduction of the full set of graphs.
• Figure 2: Classes of Bhabha-scattering two-loop diagrams containing at least one fermion loop. We use the conventions of Figure \ref{['1loop']}. Note that class 2a contains three topologically different subclasses. We refer to webPage:2006xx for the reproduction of the full set of graphs.
• Figure 3: Counterterm-dependent Feynman rules relevant for Bhabha scattering for $i=1$ (one loop) and $i=2$ (two loops). Note that in the on-mass-shell scheme $e^2=4\pi\alpha$ at all orders in perturbation theory.
• Figure 4: Ratio of the fermionic NNLO corrections to the differential cross section respect to the tree-level result for $\sqrt{s}=10$ GeV and $\sqrt{s}=500$ GeV. A solid line represents the electron-loop contributions, a dotted one the sum of electron- and muon-loop ones, and a dashed one includes also $\tau$ leptons.
• Figure 5: Same as Figure \ref{['2plots']}, including the photonic contributions of Arbuzov:1998duGlover:2001evPenin:2005eh (dash-dotted lines).