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

TL;DR

This paper explores the geometric structure of scalar quantum field theory amplitudes, revealing new associahedron realizations that incorporate massive particles and connecting them to known positive geometries in the decoupling limit.

Contribution

It introduces novel associahedron realizations in kinematic space for scalar theories with mixed mass and interaction terms, extending the geometric understanding of amplitudes.

Findings

New associahedron realizations generate amplitudes with one massive and multiple massless poles.

Decoupling limit projects these associahedra onto known accordiohedra related to pure cubic and quartic interactions.

The geometric structures unify various scalar field theories within the positive geometry framework.

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 $\lambda_{1}\, \phi_{1}^{3}\, +\, \lambda_{2}\, \phi_{1}^{2}\phi_{2}$ potential where $\phi_{1}$ and $\phi_{2}$ are massless and massive scalar fields with bi-adjoint colour indices respectively. We also show how in the "decoupling limit" (where $m \rightarrow \infty, \lambda_{2} \rightarrow \infty$ such that $g := \frac{\lambda_{2}}{m} = \textrm{finite}$) these associahedra project onto a specific class of accordiohedron which are known to be positive geometries of amplitudes generated by $\lambda \phi_{1}^{3} + g \phi_{1}^{4}$.