The S Matrix of 6D Super Yang-Mills and Maximal Supergravity from Rational Maps

TL;DR

This paper introduces new formulas for six-dimensional super Yang-Mills and supergravity scattering amplitudes using rational maps and scattering equations, revealing novel structures and redundancies, and extends these results to lower dimensions and massive cases.

Contribution

The paper provides the first explicit formulas for 6D super Yang-Mills and supergravity amplitudes based on rational maps and scattering equations, including novel treatments for even and odd particle numbers.

Findings

Formulas for 6D $ ext{N}=(1,1)$ SYM amplitudes

Formulas for 6D $ ext{N}=(2,2)$ SUGRA amplitudes

Extensions to 5D and 4D massive amplitudes

Abstract

We present new formulas for $n$-particle tree-level scattering amplitudes of six-dimensional $\mathcal{N}=(1,1)$ super Yang-Mills (SYM) and $\mathcal{N}=(2,2)$ supergravity (SUGRA). They are written as integrals over the moduli space of certain rational maps localized on the $(n-3)!$ solutions of the scattering equations. Due to the properties of spinor-helicity variables in six dimensions, the even-$n$ and odd-$n$ formulas are quite different and have to be treated separately. We first propose a manifestly supersymmetric expression for the even-$n$ amplitudes of $\mathcal{N}=(1,1)$ SYM theory and perform various consistency checks. By considering soft-gluon limits of the even-$n$ amplitudes, we deduce the form of the rational maps and the integrand for $n$ odd. The odd-$n$ formulas obtained in this way have a new redundancy that is intertwined with the usual $\text{SL}(2, \mathbb{C})$ invariance on the Riemann sphere. We also propose an alternative form of the formulas, analogous to the Witten-RSV formulation, and explore its relationship with the symplectic (or Lagrangian) Grassmannian. Since the amplitudes are formulated in a way that manifests double-copy properties, formulas for the six-dimensional $\mathcal{N}=(2,2)$ SUGRA amplitudes follow. These six-dimensional results allow us to deduce new formulas for five-dimensional SYM and SUGRA amplitudes, as well as massive amplitudes of four-dimensional $\mathcal{N}=4$ SYM on the Coulomb branch.