A Numerical Unitarity Formalism for One-Loop Amplitudes

TL;DR

The paper presents a semi-numerical four-dimensional unitarity framework (EGK) to efficiently compute the cut-constructible part of one-loop amplitudes for multi-leg processes. By expanding the integrand in a reduced set of propagator terms and using a Neerven-Vermaseren basis to parameterize loop momenta, it extracts box, triangle, and bubble coefficients from quadruple and lower cuts via products of tree amplitudes. Numerical results for 4-, 5-, and 6-gluon amplitudes show favorable computational timing compared with traditional integration-by-parts approaches, and the method is extended to $D$-dimensions to recover the full amplitude including the rational part. The work provides a practical, scalable algorithm for automated NLO calculations in complex QCD processes and lays groundwork for broader applications in collider phenomenology.

Abstract

The unitarity method for calculating one-loop amplitudes provides algorithms of polynomial complexity. This is primarily beneficial for the computation of multi-leg one loop amplitudes and it is therefore of great interest to develop a numerical implementation of the unitarity method. We describe a recently-developed, efficient, semi-numerical unitarity method for the computation of the cut-constructible part of one-loop amplitudes.

Figures (1)

• Figure 1: The generic $N$-point loop amplitude.