CHY formula and MHV amplitudes

TL;DR

This paper demonstrates that a specific rational solution of the scattering equations uniquely supports MHV amplitudes in Yang-Mills and gravity, linking the CHY formula to known amplitude formulas.

Contribution

It proves the unique support of MHV amplitudes by a special solution of scattering equations and develops covariant techniques for explicit computations.

Findings

The special solution reproduces Parke-Taylor and Hodges formulas.

Other solutions do not support MHV amplitudes.

The special solution supports anti-MHV amplitudes.

Abstract

In this paper, we study the relation between the Cachazo-He-Yuan (CHY) formula and the maximal-helicity-violating (MHV) amplitudes of Yang-Mills and gravity in four dimensions. We prove that only one special rational solution of the scattering equations found by Weinzierl support the MHV amplitudes. Namely, localized at this solution, the integrated CHY formula reproduces the Parke-Taylor formula for Yang-Mills amplitudes as well as the Hodges formula for gravitational amplitudes. This is achieved by developing techniques, in a manifestly M\"obius covariant formalism, to explicitly compute relevant reduced Pfaffians/determinants. We observe and prove two interesting properties (or identities), which facilitate the computations. We also check that all the other $(n-3)!-1$ solutions to the scattering equations do not support the MHV amplitudes, and prove analytically that this is indeed true for the other special rational solution proposed by Weinzierl, that actually supports the anti-MHV amplitudes.