Towards Positive Geometry of Multi Scalar Field Amplitudes : Accordiohedron and Effective Field Theory

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 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 $m^2$. The resulting amplitudes are associated to ${\lambda }_1{\varphi }_1^3+{\lambda }_2{\varphi }_1^2{\varphi }_2$ potential where ${\varphi }_1$ and ${\varphi }_2$ are massless and massive scalar fields with bi-adjoint colour indices respectively. We also show how in the "decoupling limit" (where $m\to \infty ,{\lambda }_2\to \infty$ such that $g:=\frac{{\lambda }_2}{m}=\text{finite}$) these associahedra project onto a specific class of accordiohedron which are known to be positive geometries of amplitudes generated by $\lambda {\varphi }_1^3+g{\varphi }_1^4$.

43 pages, 11 Figures

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Cited by in corpus (1)

• BCFW like recursion for Deformed Associahedron