One-loop Integral Coefficients from Generalized Unitarity

TL;DR

This work extends generalized unitarity to compute all four-dimensional one-loop scalar-integral coefficients with arbitrary masses. By systematically applying quadruple, triple, double, and single cuts, the method isolates box, triangle, bubble, and tadpole contributions, using carefully chosen spinor-based parametrizations and infinity-behavior analyses to separate residues from asymptotic terms. The approach relies on a unitarity bootstrap, combining four-dimensional cuts with on-shell recursion to recover rational terms, enabling complete one-loop amplitudes in QCD, electroweak, and beyond. The framework promises automated, mass-general one-loop calculations across the Standard Model and supersymmetric theories, leveraging compact tree-level inputs from helicity methods.

Abstract

I describe a method for determining the coefficients of scalar integrals for one-loop amplitudes in quantum field theory. The method is based upon generalized unitarity and the behavior of amplitudes when the free parameters of the cut momenta approach infinity. The method works for arbitrary masses of both external and internal legs of the amplitudes. It therefore applies not only to QCD but also to the Electroweak theory and to quantum field theory in general.

Figures (5)

• Figure 1: a quadruple cut.
• Figure 2: Tree level amplitudes contributing to a particular box topology.
• Figure 3: a triple cut.
• Figure 4: a double cut.
• Figure 5: a single cut.