CHY representations for gauge theory and gravity amplitudes with up to three massive particles

TL;DR

This work extends Cachazo–He–Yuan (CHY) representations to tree-level gauge and gravity amplitudes containing up to three massive particles. The authors show that these amplitudes factor into sums over the $(n-3)!$ scattering-equation solutions and, when recast in suitable momentum invariants, are identical to their massless counterparts, effectively achieving mass independence. The key approach is dimensional reduction from massless amplitudes in higher dimensions, implemented via careful choices of polarization and a CHY formalism built from a propagator matrix, double-partial amplitudes, and Pfaffian structures. The results unify a wide class of mixed massive–massless amplitudes and reproduce known massless limits and several nontrivial massive cases, while clarifying why four or more massive particles fall outside this CHY representation. This provides a systematic pathway toward CHY descriptions of more general massive theories and hints at extensions to fermions and other sectors.

Abstract

We show that a wide class of tree-level scattering amplitudes involving scalars, gauge bosons, and gravitons, up to three of which may be massive, can be expressed in terms of a Cachazo-He-Yuan representation as a sum over solutions of the scattering equations. These amplitudes, when expressed in terms of the appropriate kinematic invariants, are independent of the masses and therefore identical to the corresponding massless amplitudes.

Figures (1)

• Figure 1: A cubic diagram in which thin lines denote massless propagators, while the thick lines denote massive propagators (with possibly different masses)