BCFW like recursion for Deformed Associahedron

TL;DR

This paper extends BCFW-like recursion relations to deformed associahedra and D-type cluster polytopes, enabling recursive computation of scattering amplitudes in certain cubic scalar theories.

Contribution

It introduces a generalized recursion framework for deformed positive geometries, including associahedra and D-type cluster polytopes, applicable to tree and one-loop amplitudes.

Findings

Recursion relations are adapted for deformed associahedra.

The formalism captures one-loop amplitudes in cubic theories.

Projective triangulation corresponds to recursion terms.

Abstract

In this paper, we explore the applicability of the BCFW-like recursion relations \cite{He:2018svj,Yang:2019esm} to a wider class of positive geometries. Previously it was found in \cite{Jagadale:2022rbl}, the tree level scattering amplitude of a theory with more than one type of scalar particles interacting via cubic couplings of different strength can be captured by a deformed realization of the ABHY-associahedorn in the kinematic space. In the literature, we explore the adaptation of the recursion relations for the case of deformed associahedron. The formalism is further generalized to the deformed realization of the D-type cluster polytopes which captures the one-loop amplitudes in this class of cubic theories. These recursion terms correspond to projective triangulation of the associahedron (or D-type cluster polytopes). Towards the end, we briefly mention the idea of recovering EFT amplitudes from the cubic theory in terms of recursion relations.