Cross-ratio Identities and Higher-order Poles of CHY-integrand

TL;DR

This paper introduces a systematic decomposition algorithm using cross-ratio identities to evaluate complex CHY-integrands with higher-order poles efficiently, simplifying their computation to integrals over simple poles.

Contribution

It presents a novel, analytic, and easy-to-implement method combining cross-ratio identities with the $ ext{ extLambda}$-algorithm for evaluating complex CHY-integrands.

Findings

Decomposes higher-order pole integrands into simple pole integrands.

Enhances efficiency for large particle numbers and complex pole structures.

Provides a method applicable where the $ ext{ extLambda}$-algorithm alone is insufficient.

Abstract

The evaluation of generic Cachazo-He-Yuan(CHY)-integrands is a big challenge and efficient computational methods are in demand for practical evaluation. In this paper, we propose a systematic decomposition algorithm by using cross-ratio identities, which provides an analytic and easy to implement method for the evaluation of any CHY-integrand. This algorithm aims to decompose a given CHY-integrand containing higher-order poles as a linear combination of CHY-integrands with only simple poles in a finite number of steps, which ultimately can be trivially evaluated by integration rules of simple poles. To make the method even more efficient for CHY-integrands with large number of particles and complicated higher-order pole structures, we combine the $\Lambda$-algorithm and the cross-ratio identities, and as a by-product it provides us a way to deal with CHY-integrands where the $\Lambda$-algorithm was not applicable in its original formulation.