Positive Geometries, Corolla Polynomial and Gauge Theory Amplitudes

TL;DR

This paper links the geometry of associahedra and Corolla polynomials to gauge theory amplitudes, providing new geometric and combinatorial methods to compute Yang-Mills amplitudes at tree and one-loop levels.

Contribution

It introduces a novel geometric framework connecting associahedron structures and Corolla polynomials to Yang-Mills amplitudes, extending parametric representations to loop levels.

Findings

Identifies S-matrix of Yang-Mills with a scalar via associahedron form contraction.

Extends Corolla polynomial methods to obtain one-loop planar integrands.

Provides explicit constructions for tree-level and one-loop gluon amplitudes.

Abstract

Arkani-Hamed, Bai, He, and Yan (ABHY) discovered a convex realisation of the associahedron whose combinatorial and geometric structure generates tree-level amplitudes in bi-adjoint scalar theory. In this paper, we identify S-matrix of Yang-Mills theory with a scalar obtained by contracting the canonical form of ABHY associahedron with a multi-vector field (MVF) in the kinematic space. Components of this MVF are determined by the combinatorial structures that underlie the associahedron and Corolla polynomial that was introduced by Kreimer, Sars, and van Suijlekom (KSVS) in [2]. KSVS used the Corolla polynomial to obtain (at all orders in the loop expansion) the parametric representation of gauge theory Feynman integral from the corresponding Feynman integral in $\phi^{3}$ theory. Using the full power of Corolla polynomial, we then extend these results to obtain Yang-Mills one loop planar integrand by contracting the Corolla generated MVF with the canonical form defined by $\hat{D}_{n}$ polytope discovered by Arkani-Hamed, Frost, Plamondon, Salvatori, Thomas. We also demonstrate that KSVS representation of Corolla graph differential in the parametric space can be readily extended to "spin up" the curve integral formulae for $\textrm{Tr}\phi^{3}$ amplitude discovered in [3,4] and give an explicit construction of such formulae for tree-level and planar one loop gluon amplitudes.