Towards positive geometry of multi scalar field amplitudes. Accordiohedron and effective field theory

Abstract

Abstract The geometric structure of S-matrix encapsulated by the “Amplituhedron program” has begun to reveal itself even in non-supersymmetric quantum field theories. Starting with the seminal work of Arkani-Hamed, Bai, He and Yan [1] it is now understood that for a wide class of scalar quantum field theories, tree-level amplitudes are canonical forms associated to polytopes known as accordiohedra. Similarly the higher loop scalar integrands are canonical forms associated to so called type-D cluster polytopes for cubic interactions or recently discovered class of polytopes termed pseudo-accordiohedron for higher order scalar interactions.In this paper, we continue to probe the universality of these structures for a wider class of scalar quantum field theories. More in detail, we discover new realisations of the associahedron in planar kinematic space whose canonical forms generate (colour-ordered) tree-level S matrix of external massless particles with n − 4 massless poles and one massive pole at m2. The resulting amplitudes are associated to λ1$$ {\phi}_1^3 $$ ϕ 1 3 + λ2$$ {\phi}_1^2 $$ ϕ 1 2 ϕ2 potential where ϕ1 and ϕ2 are massless and massive scalar fields with bi-adjoint colour indices respectively. We also show how in the “decoupling limit” (where m → ∞, λ2 → ∞ such that g :$$ \frac{\uplambda_2}{m} $$ λ 2 m = finite) these associahedra project onto a specific class of accordiohedron which are known to be positive geometries of amplitudes generated by λ$$ {\phi}_1^3 $$ ϕ 1 3 + g$$ {\phi}_1^4 $$ ϕ 1 4 .