Multi-loop Amplitudes and Resummation

TL;DR

This work develops a factorization framework for QCD hard-scattering amplitudes that links fixed-order multi-loop results to all-orders resummation. By decomposing amplitudes into jet, soft, and hard components and employing the soft anomalous dimension, it explains the universal exponentiation of infrared poles and connects Catani’s ε-pole structure to universal soft radiation. The approach enables predictions of pole structures to high orders using lower-order form factors and anomalous dimensions, and is illustrated with explicit two-loop quark processes, showing consistency with direct calculations. The results have implications for threshold resummation and may guide cross-section calculations at NNLO and beyond, including potential extensions to inelastic processes.

Abstract

We explore the relation between resummation and explicit multi-loop calculations for QCD hard-scattering amplitudes. We describe how the factorization properties of amplitudes lead to the exponentiation of double and single poles at each order of perturbation theory. For these amplitudes, previously-observed relations between single and double poles in different 2 to 2 processes can now be interpreted in terms of universal functions associated with external partons and process-dependent anomalous dimensions that describe coherent soft radiation. Catani's proposal for multiple poles in dimensionally-continued amplitudes emerges naturally.