Three-loop helicity amplitudes of four-lepton scattering in QED
Giulio Crisanti, Thomas Dave, Pierpaolo Mastrolia, Jonathan Ronca, Sid Smith, William J. Torres Bobadilla
TL;DR
The work tackles the problem of computing complete analytic three-loop virtual corrections to four-fermion scattering in massless QED, focusing on $e^{+}e^{-}\to\mu^{+}\mu^{-}$, $e^{+}e^{-}\to e^{+}e^{-}$, and $e^{+}\mu^{-}\to\mu^{+}e^{-}$. It adopts an optimized IBP reduction within a canonical differential-equation framework, reducing the three-loop integrals to a set of $533$ master integrals and solving for the master integrals across all kinematic regions through analytic continuation, with boundary conditions fixed numerically against known constants. The finite remainders of the amplitudes are expressed in terms of generalised polylogarithms up to transcendental weight $w=6$ in the dimensionless ratio $x=-t/s$, after UV renormalisation in the $\overline{\text{MS}}$ scheme and infrared subtraction using SCET-inspired operators. The results include explicit helicity amplitudes for all channels and are accompanied by a public repository containing the UV-renormalised and IR-finite form factors and amplitudes, providing a crucial benchmark for high-precision, loop-level QED calculations and cross-checks against QCD in the Abelian limit.
Abstract
We present the analytic expressions of the three-loop virtual corrections to the helicity amplitudes of 2 -> 2 four-fermion scattering processes in massless QED. The contributing Feynman diagrams are grouped into integrand families characterised by independent Symanzik polynomials and decomposed in terms of master integrals using an optimised integration-by-parts strategy. Upon the renormalisation of the ultraviolet divergences and the extraction of the universal infrared pole structure, the finite results are expressed in terms of generalised polylogarithms up to transcendental weight six. Amplitudes for dimuon production in electron-positron annihilations, electron-muon scattering, and Bhabha scattering are explicitly derived.