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BCFW like recursion for Deformed Associahedron

Sujoy Mahato, Sourav Roychowdhury

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.

BCFW like recursion for Deformed Associahedron

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.

Paper Structure

This paper contains 20 sections, 97 equations, 11 figures.

Table of Contents

  1. Introduction and summary
  2. Review of the associahedron as the amplituhedron of general scalar theory
  3. Deformed realization of the associahedron
  4. Interaction couplings and deformation parameters
  5. A specific case of recursion
  6. Example I : Four-point amplitude in massless theory
  7. Example II : Six point deformed amplitude in massless theory
  8. Example III: Loop in deformed case
  9. Interpretation of the recursion relation
  10. Geometric Picture : Projective Triangulations
  11. Four point amplitude in massless theory
  12. Six point massless amplitude
  13. Three point loop amplitude
  14. EFT amplitudes
  15. Conclusions and outlook
  16. ...and 5 more sections

Figures (11)

  • Figure 1: The projectection of $1d$ associahedron on s-axis via recursion
  • Figure 2: Six point deformed associahedron and projection of $X_{36}$ facet onto $X_{13}X_{14}$ edge
  • Figure 3: Prism formed by projecting $X_{36}$ facet onto $X_{13}X_{14}$ edge
  • Figure 4: Geometric realisation of $D_3$ polytope and projection of $X_{2\bar{3}}$ facet onto $Y_1Y_2$ edge
  • Figure 5: Prism formed by projecting $X_{2\bar{3}}$ facet onto $Y_{1}Y_{2}$ edge
  • ...and 6 more figures