The Role of M\"obius Constants and Scattering Functions in CHY Scalar Amplitudes

TL;DR

This paper systematically analyzes how M"obius constants and scattering functions influence the integrations in CHY scalar amplitudes, providing a step-by-step method and illustrative examples for computing these amplitudes.

Contribution

It introduces a systematic approach to perform the integrations in CHY scalar amplitudes, clarifying the role of M"obius constants and scattering functions, with detailed examples.

Findings

The choice of M"obius constants affects intermediate calculations but not the final amplitude.

A step-by-step integration method exposes individual propagators in Feynman diagrams.

Illustrative examples demonstrate the practical application of the method for five- and nine-point amplitudes.

Abstract

The integrations leading to the Cachazo-He-Yuan (CHY) double-color $n$-point massless scalar amplitude are carried out one integral at a time. M\"obius invariance dictates the final amplitude to be independent of the three M\"obius constants $\sigma_r, \sigma_s, \sigma_t$, but their choice affects integrations and the intermediate results. The effect of the M\"obius constants, the two colors, and the scattering functions on each integration is investigated. A systematic way to carry out the $n-3$ integrations is explained, each exposing one of the $n-3$ propagators of the Feynman diagrams. Two detailed examples are shown to illustrate the procedure, one a five-point amplitude, and the other a nine-point amplitude.