Double-Cut of Scattering Amplitudes and Stokes' Theorem

TL;DR

The paper develops an analytic, Stokes' theorem–based method to compute double-cuts of one-loop amplitudes by transforming the phase-space integration into a two-variable rational problem in $z$ and $\\bar{z}$, enabling direct extraction of two-point function coefficients via Hermite Polynomial Reduction; a formal Generalised Cauchy Formula governs the z–\\bar{z} integration and residue structure, with explicit results for the $I_2$ double-cut and a pathway to systematic coefficient extraction in the master integral decomposition.

Abstract

We show how Stokes' Theorem, in the fashion of the Generalised Cauchy Formula, can be applied for computing double-cut integrals of one-loop amplitudes analytically. It implies the evaluation of phase-space integrals of rational functions in two complex-conjugated variables, which are simply computed by an indefinite integration in a single variable, followed by Cauchy's Residue integration in the conjugated one. The method is suitable for the cut-construction of the coefficients of 2-point functions entering the decomposition of one-loop amplitudes in terms of scalar master integrals.

Figures (1)

• Figure 1: Double-cut of one-loop amplitude in the $K^2$-channel.