Positive Geometries, Corolla Polynomial and Gauge Theory Amplitudes

2024/05/17 by Laddha, Alok, Suthar, Amit · 2 citations
#FOS: Physical sciences #High Energy Physics - Theory (hep-th)

paper · doi:10.48550/arxiv.2405.10601

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 ϕ³ 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 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 \textrmTrϕ³ amplitude discovered in [3,4] and give an explicit construction of such formulae for tree-level and planar one loop gluon amplitudes.

Cited by

• Gauge Theory Amplitudes from Cubic Scalar Feynman diagrams
• BCFW like recursion for Deformed Associahedron

Related

• The corolla polynomial: a graph polynomial on half-edges
• Properties of the corolla polynomial of a 3-regular graph
• Towards Positive Geometries of Massive Scalar field theories
• The Positive orthogonal Grassmannian and loop amplitudes of ABJM
• Loops of Loops Expansion in the Amplituhedron