Cross Section Evaluation by Spinor Integration I: The massless case in 4D

TL;DR

This paper introduces a novel Lorentz-invariant method for calculating total cross sections in massless 4D particle interactions by reducing complex phase space integrations to simpler one-dimensional forms, inspired by unitarity cut techniques.

Contribution

It presents a new integration method that simplifies phase space calculations for massless particles in four dimensions, avoiding complex region carving and using spinor integration techniques.

Findings

Reduces three-dimensional momentum integrations to one dimension.

Maintains manifest Lorentz invariance throughout the calculation.

Applicable to massless particles in 4D with simple integration regions.

Abstract

To get the total cross section of one interaction from its amplitude ${\cal M}$, one needs to integrate $|{\cal M}|^2$ over phase spaces of all out-going particles. Starting from this paper, we will propose a new method to perform such integrations, which is inspired by the reduced phase space integration of one-loop unitarity cut developed in the last few years. The new method reduces one constrained three-dimension momentum space integration to an one-dimensional integration, plus one possible Feynman parameter integration. There is no need to specify a reference framework in our calculation, since every step is manifestly Lorentz invariant by the new method. The current paper is the first paper of a series for the new method. Here we have exclusively focused on massless particles in 4D. There is no need to carve out a complicated integration region in the phase space for this particular simple case because the integration region is always simply $[0,1]$.