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. ABHY associahedron of dimension k determines a unique meromorphic k-form in the kinematic space of Mandelstam invariants which is k + 3 point tree-level amplitude in bi-adjoint ϕ3 theory. As ABHY further proved in [1], while the color-ordered amplitude is a form in the kinematic space, the (double) color-dressed amplitudes are scalars obtained by the so called color-form duality. Color-form duality can be implemented by contracting a d-log form with a multi-vector field (MVF) whose coefficients are fixed by the color factors of the theory. We extend the latter result to identify tree-level S-matrix of Yang Mills theory with a scalar obtained by contracting the canonical form of ABHY associahedron with a MVF in the kinematic space. Components of this MVF are also determined by the combinatorial structures that underlie associahedron and Corolla polynomial which 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. The map from latter integral to the former was generated by the Corolla polynomial. 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 recently. 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 Tr(ϕ3) amplitude discovered in [3, 4] and give an explicit construction of such formulae for tree-level and planar one loop gluon amplitudes.
