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 $\lambda {\varphi }_1^3+g{\varphi }_1^4$.

arXiv comment43 pages, 11 Figures

References in corpus (22)

• Scattering Equations and Matrices: From Einstein To Yang-Mills, DBI and NLSM
• On Tree Amplitudes in Gauge Theory and Gravity
• Scattering Forms and the Positive Geometry of Kinematics, Color and the Worldsheet
• Positive Geometries and Canonical Forms
• Combinatorics and Topology of Kawai-Lewellen-Tye Relations
• BCFW Recursion Relation with Nonzero Boundary Contribution
• Scattering Equations and Feynman Diagrams
• Stokes Polytopes : The positive geometry for ${\varphi }^4$ interactions
• Accordiohedra as positive geometries for generic scalar field theories
• On Positive Geometries of Quartic Interactions II : Stokes polytopes, Lower Forms on Associahedra and Worldsheet Forms
• The positive geometry for ${\varphi }^p$ interactions
• Geometric realizations of the accordion complex of a dissection
• Amplituhedra, and Beyond
• Associahedra for finite type cluster algebras and minimal relations between $\mathbf{g}$-vectors
• Stringy canonical forms and binary geometries from associahedra, cyclohedra and generalized permutohedra
• Weights and recursion relations for ${\varphi }^p$ tree amplitudes from the positive geometry
• String-Like Dual Models for Scalar Theories
• On Positive Geometry and Scattering Forms for Matter Particles
• Open associahedra and scattering forms
• Constraining the weights of Stokes Polytopes using BCFW recursions for Phi^4
• An empirical $G$-Wishart prior for sparse high-dimensional Gaussian graphical models
• CHY Theory for Several Fields

Cited by in corpus (1)

• BCFW like recursion for Deformed Associahedron