Lie Polynomials and a Twistorial Correspondence for Amplitudes

TL;DR

This paper explores the mathematical structure of scattering amplitudes in gauge and gravity theories using Lie polynomials, twistorial geometry, and a new correspondence that unifies various amplitude formulas.

Contribution

It introduces a twistorial framework linking moduli space geometry, Lie polynomials, and scattering amplitudes, providing a natural interpretation of CHY and ABHY formalisms.

Findings

Establishes a twistorial correspondence between moduli space and momentum invariants.

Provides a geometric interpretation of CHY half-integrands and scattering forms.

Generalizes the associahedral $n-3$-planes in the space of kinematic invariants.

Abstract

We review Lie polynomials as a mathematical framework that underpins the structure of the so-called double copy relationship between gauge and gravity theories (and a network of other theories besides). We explain how Lie polynomials naturally arise in the geometry and cohomology of $\mathcal{M}_{0,n}$, the moduli space of $n$ points on the Riemann sphere up to Mobi\"us transformation. We introduce a twistorial correspondence between the cotangent bundle $T^*_D\mathcal{M}_{0,n}$, the bundle of forms with logarithmic singularities on the divisor $D$ as the twistor space, and $\mathcal{K}_n$ the space of momentum invariants of $n$ massless particles subject to momentum conservation as the analogue of space-time. This gives a natural framework for Cachazo He and Yuan (CHY) and ambitwistor-string formulae for scattering amplitudes of gauge and gravity theories as being the corresponding Penrose transform. In particular we show that it gives a natural correspondence between CHY half-integrands and scattering forms, certain $n-3$-forms on $\mathcal{K}_n$, introduced by Arkani-Hamed, Bai, He and Yan (ABHY). We also give a generalization and more invariant description of the associahedral $n-3$-planes in $\mathcal{K}_n$ introduced by ABHY.}