Unitarity cuts and reduction to master integrals in d dimensions for one-loop amplitudes

TL;DR

The paper develops a two-step, d-dimensional unitarity reduction for one-loop amplitudes that reads master-integral coefficients directly from the integrand. By combining a pure four-dimensional cut integration with a mass parameter and dimensional shift identities, it separates the 4D phase space from the (-2ε)-dimensional piece, enabling algebraic extraction of coefficients. The authors derive explicit recursion/reduction relations for bubbles, triangles, boxes, and pentagons and validate the approach with five- and four-gluon all-plus amplitudes, showing exact agreement with known results after mapping to dimensionally shifted bases. This framework offers a scalable, dimensionally robust method for computing one-loop amplitudes in QCD, reducing reliance on traditional tensor reductions and improving numerical stability. The methods are poised to simplify high-multiplicity one-loop calculations and provide systematic, algebraic access to complete amplitudes including rational pieces in d dimensions.

Abstract

We present an alternative reduction to master integrals for one-loop amplitudes using a unitarity cut method in arbitrary dimensions. We carry out the reduction in two steps. The first step is a pure four-dimensional cut-integration of tree amplitudes with a mass parameter, and the second step is applying dimensional shift identities to master integrals. This reduction is performed at the integrand level, so that coefficients can be read out algebraically.