Unitarity Cuts with Massive Propagators and Algebraic Expressions for Coefficients

TL;DR

This work generalizes the d-dimensional unitarity-cut framework to cases with massive propagators, delivering explicit reduction formulas that yield coefficients for pentagon, box, triangle, and massive bubble master integrals. By refining spinor-based phase-space integration in $(4-2\,\epsilon)$ dimensions and introducing a canonical splitting strategy, the authors provide algebraic, automatable expressions for all coefficients, including handling of mass effects, tadpoles, and massless bubbles. They unify box, triangle, and bubble reductions and discuss pentagon contributions, tying the approach to the Ossola–Papadopoulos–Pittau (OPP) method while emphasizing residue-based, symbolically computable results. The methodology enables efficient, programmable one-loop computations with massive propagators, with clear guidance on input tree amplitudes and on using quadruple cuts for localization.

Abstract

In the first part of this paper, we extend the d-dimensional unitarity cut method of hep-ph/0609191 to cases with massive propagators. We present formulas for integral reduction with which one can obtain coefficients of all pentagon, box, triangle and massive bubble integrals. In the second part of this paper, we present a detailed study of the phase space integration for unitarity cuts. We carry out spinor integration in generality and give algebraic expressions for coefficients, intended for automated evaluation.