New Formulas for Amplitudes from Higher-Dimensional Operators

TL;DR

This work addresses tree-level amplitudes from higher-dimensional operators in gauge and gravity theories within the CHY framework. It introduces a new gauge-invariant object ${\\cal P}_n$ that generalizes the reduced Pfaffian and yields compact CHY expressions for $F^3$, $R^2$, and $R^3$ amplitudes, connected by KLT/double-copy relations such as ${\\cal I}^{F^3} = {\\cal C}_n {\\cal P}_n$, ${\\cal I}^{R^3} = {\\cal P}_n {\\cal P}_n$, and ${\\cal I}^{R^2} = {\\cal P}_n Pf' \\Psi_n$. In four dimensions, ${\\cal P}_n$ is orthogonal to Pf' and vanishes on the same sector, explaining the vanishing of $R^2$ amplitudes and enabling a self-dual and anti-self-dual decomposition of $F^3$ and $R^3$, with a Parke-Taylor-like formula for the self-dual $F^3_+$ case. These results illuminate the 4d structure of higher-dimensional operator amplitudes, connect to string corrections and double-copy structure, and suggest new directions for loop integrands and worldsheet models.

Abstract

In this paper we study tree-level amplitudes from higher-dimensional operators, including $F^3$ operator of gauge theory, and $R^2$, $R^3$ operators of gravity, in the Cachazo-He-Yuan formulation. As a generalization of the reduced Pfaffian in Yang-Mills theory, we find a new, gauge-invariant object that leads to gluon amplitudes with a single insertion of $F^3$, and gravity amplitudes by Kawai-Lewellen-Tye relations. When reduced to four dimensions for given helicities, the new object vanishes for any solution of scattering equations on which the reduced Pfaffian is non-vanishing. This intriguing behavior in four dimensions explains the vanishing of graviton helicity amplitudes produced by the Gauss-Bonnet $R^2$ term, and provides a scattering-equation origin of the decomposition into self-dual and anti-self-dual parts for $F^3$ and $R^3$ amplitudes.