Abstract
In [1], two of the present authors along with P. Raman attempted to extend the Amplituhedron program for scalar field theories [2] to quartic scalar interactions. In this paper we develop various aspects of this proposal. Using recent seminal results in Representation theory [3, 4], we show that projectivity of scattering forms and existence of kinematic space associahedron completely capture planar amplitudes of quartic interaction. We generalise the results of [1] and show that for any n-particle amplitude, the positive geometry associated to the projective scattering form is a convex realisation of Stokes polytope which can be naturally embedded inside one of the ABHY associahedra defined in [2, 5]. For a special class of Stokes polytopes with hyper-cubic topology, we show that they have a canonical convex realisation in kinematic space as boundaries of kinematic space associahedra.We then use these kinematic space geometric constructions to write world-sheet forms for \U0001d7194 theory which are forms of lower rank on the CHY moduli space. We argue that just as in the case of bi-adjoint \U0001d7193 scalar amplitudes, scattering equations can be used as diffeomorphisms between certain frac{n-4}{2} forms on the world-sheet and frac{n-4}{2} forms on ABHY associahedron that generate quartic amplitudes.
Highlights
In [1], two of the present authors along with P
Using recent seminal results in Representation theory [3, 4], we show that projectivity of scattering forms and existence of kinematic space associahedron completely capture planar amplitudes of quartic interaction
For a special class of Stokes polytopes with hyper-cubic topology, we show that they have a canonical convex realisation in kinematic space as boundaries of kinematic space associahedra
Summary
[2] brought the Amplituhedron program and the world of polytopes in contact with scattering amplitudes for non-supersymmetric Quantum field theories. These results fit neatly into the basic paradigm of CHY formula for scattering amplitudes where the underlying moduli space (moduli space of punctured Riemann sphere) is universal but the integrand (which is an n − 3 form for an n particle scattering) depends on the theory This appears to be in contrast with the emerging picture in kinematic space where for every scalar interaction the corresponding polytope is a unique member of the “accordiohedron family” [18] and the form that corresponds to the amplitude is the unique canonical form associated to the accordiohedron. We attempt to re-write the pushforward maps for the lower forms as integral formulae for certain sub-manifolds in the moduli space We show that this is true for forms which are labelled by quadrangulations that consist of parallel diagonals of an n-gon and the corresponding sub-manifold in the moduli space is precisely the image of kinematic space Stokes polytope under CHY scattering equations.
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