$\Lambda$ Scattering Equations

TL;DR

This paper introduces a novel double-cover formulation of the CHY scattering equations, enabling a new computational approach called the Lambda algorithm for evaluating scattering amplitudes more efficiently.

Contribution

It develops a double-cover version of the CHY scattering equations with a parameter Lambda, enhancing redundancy and leading to a new algorithm for amplitude calculation.

Findings

The Lambda algorithm simplifies the evaluation of CHY integrals.

The new representation breaks amplitudes into sums of smaller components.

Enhanced redundancy allows fixing puncture locations more flexibly.

Abstract

The CHY representation of scattering amplitudes is based on integrals over the moduli space of a punctured sphere. We replace the punctured sphere by a double-cover version. The resulting scattering equations depend on a parameter $\Lambda$ controlling the opening of a branch cut. The new representation of scattering amplitudes possesses an enhanced redundancy which can be used to fix, modulo branches, the location of four punctures while promoting $\Lambda$ to a variable. Via residue theorems we show how CHY formulas break up into sums of products of smaller (off-shell) ones times a propagator. This leads to a powerful way of evaluating CHY integrals of generic rational functions, which we call the $\Lambda$ algorithm.