Locally finite two-loop amplitudes for off-shell multi-photon production in electron-positron annihilation

TL;DR

This paper develops a local subtraction scheme for two-loop QED amplitudes involving off-shell photons, enabling their numerical evaluation in four dimensions by removing divergences point-by-point in the integrand.

Contribution

It introduces a novel local IR and UV divergence subtraction method for two-loop amplitudes, facilitating numerical computation of finite remainders in four dimensions.

Findings

Counterterms remove divergences point-by-point in the integrand.

Remainders are integrable in four dimensions and suitable for numerical methods.

Uncovered new symmetrization techniques for IR factorization in loop diagrams.

Abstract

We study the singularity structure of two-loop QED amplitudes for the production of multiple off-shell photons in massless electron-positron annihilation and develop counterterms that remove their infrared and ultraviolet divergences point by point in the loop integrand. The remainders of the subtraction are integrable in four dimensions and can be computed in the future with numerical integration. The counterterms capture the divergences of the amplitudes and factorize in terms of the Born amplitude and the finite remainder of the one-loop amplitude. They consist of simple one- and two-loop integrals with at most three external momenta and can be integrated analytically in a simple manner with established methods. We uncover novel aspects of fully local IR factorization, where vertex and self-energy subdiagrams must be modified by new symmetrizations over loop momenta, in order to expose their tree-like tensor structures and hence factorization of IR singularities prior to loop integration. This work is a first step towards isolating locally the hard contributions of generic gauge theory amplitudes and rendering them integrable in exactly four dimensions with numerical methods.