Hyperbolic Geometry and Amplituhedra in 1+2 dimensions

Abstract

Recently, the existence of an Amplituhedron for tree level amplitudes in the bi-adjoint scalar field theory has been proved by Arkhani-Hamed et al. We argue that hyperbolic geometry constitutes a natural framework to address the study of positive geometries in moduli spaces of Riemann surfaces, and thus to try to extend this achievement beyond tree level. In this paper we begin an exploration of these ideas starting from the simplest example of hyperbolic geometry, the hyperbolic plane. The hyperboloid model naturally guides us to re-discover the moduli space Associahedron, and a new version of its kinematical avatar. As a by-product we obtain a solution to the scattering equations which can be interpreted as a special case of the two well known solutions in terms of spinor-helicity formalism. The construction is done in $1+2$ dimensions and this makes harder to understand how to extract the amplitude from the dlog of the space time Associahedron. Nevertheless, we continue the investigation accommodating a loop momentum in the picture. By doing this we are led to another polytope called Halohedron, which was already known to mathematicians. We argue that the Halohedron fulfils many criteria that make it plausible to be understood as a 1-loop Amplituhedron for the cubic theory. Furthermore, the hyperboloid model again allows to understand that a kinematical version of the Halohedron exists and is related to the one living in moduli space by a simple generalisation of the tree level map.

arXiv comment28 pages, 15 figures

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• Loop Integrands for Scattering Amplitudes from the Riemann Sphere
• Positive Geometries and Canonical Forms
• Scattering Amplitudes from Intersection Theory
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• CHY Loop Integrands from Holomorphic Forms
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Cited by in corpus (13)

• Stokes Polytopes : The positive geometry for ${\varphi }^4$ interactions
• 1-loop Amplitudes from the Halohedron
• Integrands of loop amplitudes within loop-tree duality
• Positive geometry, local triangulations, and the dual of the Amplituhedron
• The positive geometry for ${\varphi }^p$ interactions
• The SAGEX Review on Scattering Amplitudes, Chapter 7: Positive Geometry of Scattering Amplitudes
• Positive Geometries for all Scalar Theories from Twisted Intersection Theory
• Sign Flip Triangulations of the Amplituhedron
• Amplituhedra, and Beyond
• On Positive Geometry and Scattering Forms for Matter Particles
• Emergent unitarity from the amplituhedron
• Triangular tessellations of one-loop scattering amplitudes in ${\varphi }^3$ theory
• Superstring amplitudes meet surfaceology