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Lepton-pair scattering with an off-shell and an on-shell photon at two loops in massless QED

Simon Badger, Jakub Kryś, Ryan Moodie, Simone Zoia

TL;DR

This work addresses the need for precise two-loop QED corrections to electron–muon scattering at the MUonE experiment by computing massless-lepton helicity amplitudes for $0\to \ell\bar{\ell}\gamma\gamma^*$ and expressing the four-point master integrals in a basis of algebraically independent multiple polylogarithms. The authors employ finite-field techniques and a canonical differential-equation framework to reconstruct analytic amplitudes from numerical samples, yielding a compact MPL representation up to weight four for all relevant integral families with a single off-shell leg. They organize the amplitude into gauge-invariant subamplitudes, project onto a four-vector basis, and handle all permutations of integral topologies with an optimized IBP reduction workflow and a momentum-twistor parametrisation for kinematics, including appropriate UV renormalisation and IR factorisation. The resulting analytic expressions and master-integral basis enable efficient, stable two-loop computations that feed N$^3$LO predictions for $e\mu\to e\mu$ and related ISR/RSV observables, contributing directly to MUonE's precision program.

Abstract

We compute the two-loop QED helicity amplitudes for the scattering of a lepton pair with an off-shell and an on-shell photon, $0\to\ell\bar\ellγγ^*$, using the approximation of massless leptons. We express all master integrals relevant for the scattering of four massless particles with a single external off-shell leg up to two loops in a basis of algebraically independent multiple polylogarithms, which guarantees an efficient numerical evaluation and compact analytic representations of the amplitudes. Analytic forms of the amplitudes are reconstructed from numerical evaluations over finite fields. Our results complete the amplitude-level ingredients contributing to the N$^3$LO predictions of electron-muon scattering $eμ\to eμ$, which are required to meet the precision goal of the future MUonE experiment.

Lepton-pair scattering with an off-shell and an on-shell photon at two loops in massless QED

TL;DR

This work addresses the need for precise two-loop QED corrections to electron–muon scattering at the MUonE experiment by computing massless-lepton helicity amplitudes for and expressing the four-point master integrals in a basis of algebraically independent multiple polylogarithms. The authors employ finite-field techniques and a canonical differential-equation framework to reconstruct analytic amplitudes from numerical samples, yielding a compact MPL representation up to weight four for all relevant integral families with a single off-shell leg. They organize the amplitude into gauge-invariant subamplitudes, project onto a four-vector basis, and handle all permutations of integral topologies with an optimized IBP reduction workflow and a momentum-twistor parametrisation for kinematics, including appropriate UV renormalisation and IR factorisation. The resulting analytic expressions and master-integral basis enable efficient, stable two-loop computations that feed NLO predictions for and related ISR/RSV observables, contributing directly to MUonE's precision program.

Abstract

We compute the two-loop QED helicity amplitudes for the scattering of a lepton pair with an off-shell and an on-shell photon, , using the approximation of massless leptons. We express all master integrals relevant for the scattering of four massless particles with a single external off-shell leg up to two loops in a basis of algebraically independent multiple polylogarithms, which guarantees an efficient numerical evaluation and compact analytic representations of the amplitudes. Analytic forms of the amplitudes are reconstructed from numerical evaluations over finite fields. Our results complete the amplitude-level ingredients contributing to the NLO predictions of electron-muon scattering , which are required to meet the precision goal of the future MUonE experiment.
Paper Structure (16 sections, 59 equations, 2 figures, 2 tables)

This paper contains 16 sections, 59 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Structure of the amplitude
  3. Setup of the calculation
  4. Computation of the MI
  5. Construction of the special function basis
  6. Expression in terms of MPL
  7. Performance and validation
  8. Conclusions
  9. Definition of the Feynman integral families
  10. Optimised IBP reduction procedure for amplitudes with many permuted integral families
  11. Rational parametrisation of the kinematics
  12. Renormalisation and IR structure
  13. Analytic continuation
  14. Electron-line corrections to $e^- \mu^- \to e^- \mu^- \gamma$.
  15. Corrections to $e^- e^+ \to \gamma \gamma^*$.
  16. ...and 1 more sections

Figures (2)

  • Figure 1: Representative Feynman diagrams for the subamplitudes defined in \ref{['eq:nldecomp']}. The off-shell external leg is indicated by a bold line.
  • Figure 2: The two-loop amplitudes include six ordered integral families: three penta-triangles, a double-box, and two crossed double-boxes. All are planar except for the crossed double-boxes. The off-shell external leg is indicated by a bold line. External legs have outgoing momenta.