CHY Loop Integrands from Holomorphic Forms

TL;DR

This paper develops a method to construct two-loop CHY integrands for $$ theory using holomorphic forms on Riemann surfaces, extending the CHY approach beyond tree level and verifying results for up to seven particles.

Contribution

It introduces a novel construction of two-loop CHY integrands from holomorphic forms and extends the $$-algorithm for analytical verification.

Findings

Constructed two-loop CHY integrands from holomorphic forms.

Extended the $$-algorithm to two loops for analytical checks.

Verified integrands for up to seven external particles.

Abstract

Recently, the Cachazo-He-Yuan (CHY) approach for calculating scattering amplitudes has been extended beyond tree level. In this paper, we introduce a way of constructing CHY integrands for $\Phi^3$ theory up to two loops from holomorphic forms on Riemann surfaces. We give simple rules for translating Feynman diagrams into the corresponding CHY integrands. As a complementary result, we extend the $\Lambda$-algorithm, originally introduced in arXiv:1604.05373, to two loops. Using this approach, we are able to analytically verify our prescription for the CHY integrands up to seven external particles at two loops. In addition, it gives a natural way of extending to higher-loop orders.