The Role of Möbius Constants and Scattering Functions in CHY Scalar Amplitudes

TL;DR

The paper develops a systematic, one-integration-at-a-time approach to CHY double-color scalar amplitudes, clarifying how Möbius constants $σ_r, σ_s, σ_t$ and scattering functions influence intermediate steps while leaving the final amplitude Möbius-invariant. It introduces crystals and defects to identify integration units, showing that each integration exposes a propagator and produces factorization into partial amplitudes $J_{ar S}$ and $J_{ar T}$ connected by $1/k^2$, reproducing the full amplitude as a sum over compatible pairings (pairing diagrams) that map to Feynman diagrams. Through a five-point diagonal-color example and a nine-point non-diagonal-color example, the method demonstrates how the correct propagator structure emerges and fictitious poles cancel, reinforcing the pairing-diagram interpretation. The work also connects partial/off-shell/shifted amplitudes to BCFW-like factorization, ensuring a gauge-independent, diagrammatic understanding of CHY amplitudes and their relation to conventional quantum-field-theory structures.

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. Möbius invariance dictates the final amplitude to be independent of the three Möbius constants $σ_r, σ_s, σ_t$, but their choice affects integrations and the intermediate results. The effect of the Möbius 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.