Two-Loop Heavy-Flavor Contribution to Bhabha Scattering

TL;DR

The study addresses the precision calculation of Bhabha scattering by evaluating the two-loop QED corrections from heavy-flavor vacuum polarization with arbitrary mass $m_f$ in the regime $s,t,u \gg m_e^2$. The authors implement a small-$m_e$ expansion, infrared/collinear factorization, and a reduction of two-loop four-point integrals via the Laporta algorithm and differential equations, expressing results in Harmonic Polylogarithms and providing a closed form for the core function $f(\rho,x)$ with $\rho=m_f^2/s$. They present numerical estimates for representative collider energies, showing muon loops dominate at low energy (about $0.45$ per mille) and leptonic heavy-flavor corrections approach $1.3\%$ at $\sqrt{s}=500$ GeV, while the top-quark contribution remains small. Overall, the work completes the QED part of the two-loop Bhabha corrections and offers detailed guidance for luminosity determinations at facilities such as KLOE and the ILC, with emphasis on the regime $s,t,u \gg m_e^2$ and the importance of exact $m_f^2/s$ dependence.

Abstract

We evaluate the two-loop QED corrections to the Bhabha scattering cross section which involve the vacuum polarization by heavy fermions of arbitrary mass m_f >> m_e. The results are valid for generic values of the Mandelstam invariants s,t,u >> m_e^2.

Figures (6)

• Figure 1: The two-loop diagrams associated with the logarithmic dependence of the corrections to the Bhabha scattering amplitude on $m_e$. Actually the diagram $(a)$ is free of electron mass logarithms. The bold arrow circle corresponds to the heavy-flavor vacuum polarization.
• Figure 2: Two-loop corrections to the Bhabha scattering differential cross section at $\sqrt{s}=1$ GeV due to a closed loop of $\tau$-lepton (dotted line), $c$-quark (dashed line) and $b$-quark (solid line) for $m_c=1.25$ GeV and $m_b=4.7$ GeV.
• Figure 3: Two-loop corrections to the Bhabha scattering differential cross section at $\sqrt{s}=1$ GeV due to a closed loop of muon (dashed line). The solid line represents the sum of the contributions of the muon, $\tau$-lepton, $c$-quark and $b$-quark.
• Figure 4: Two-loop leptonic corrections to the Bhabha scattering differential cross section at $\sqrt{s}=500$ GeV. The dash-dotted line represents the electron contribution including the soft-pair radiation. The dashed and dotted lines represent the contributions of muon and $\tau$-lepton, respectively. The solid line is the sum of the three contributions.
• Figure 5: Two-loop corrections to the Bhabha scattering differential cross section at $\sqrt{s}=500$ GeV due to a closed loop of top quark for $m_t=170.9$ GeV.