The integrand form of infrared singularities of two-loop QCD scattering amplitudes

TL;DR

The paper tackles the universal infrared (IR) singularities of two-loop massless QCD amplitudes by representing the singular part in terms of Feynman integrals that mirror the bare amplitude topologies, and by adopting a pure UV/IR scheme that yields a locally finite finite part. It introduces an integrand-level reconstruction where UV and IR operators are realized as linear combinations of amplitude-compatible integrands with $\epsilon$-independent coefficients, matched to integrated Catani-Becher-Neubert expressions, and uses a minimal-basis Master Integral framework to ensure completeness. A concrete reconstruction for two-loop massless QCD yields explicit integrand forms for the pure UV and IR operators, including a new two-loop IR tripole piece, and demonstrates how the finite part can be rendered locally finite for the process $q\bar{q}\to gg$ by subtracting the singular part and performing an IBP reduction to locally finite Master Integrals. The resulting locally finite representation improves numerical stability and provides a path to 4D amplitude formulations, with potential extensions to higher loops, massive particles, and integrand-level strategies that bypass intermediate integrated forms.

Abstract

In this work, we express the singular part of a scattering amplitude in terms of Feynman integrals compatible with topologies appearing in the bare amplitude, and we choose a basis of locally finite Master Integrals. In two-loop massless QCD, we find such a representation of the amplitude singularities using a systematic ansatz reconstruction of the integrand from a predicted integrated form. As an example application, we write the finite part of an amplitude for the digluon production in quark annihilation for some helicity configurations as manifestly locally finite.