Abstract
Building on the seminal work of Arkani-Hamed, He, Salvatori and Thomas (AHST) [1] we explore the positive geometry encoding one loop scattering amplitude for quartic scalar interactions. We define a new class of combinatorial polytopes that we call pseudo-accordiohedra whose poset structures are associated to singularities of the one loop integrand associated to scalar quartic interactions. Pseudo-accordiohedra parametrize a family of projective forms on the abstract kinematic space defined by AHST and restriction of these forms to the type-D associahedra can be associated to one-loop integrands for quartic interactions. The restriction (of the projective form) can also be thought of as a canonical top form on certain geometric realisations of pseudo-accordiohedra. Our work explores a large class of geometric realisations of the type-D associahedra which include all the AHST realisations. These realisations are based on the pseudo-triangulation model for type-D cluster algebras discovered by Ceballos and Pilaud [2].
Highlights
Work on exploring the relationship between 1-loop φ3 integrand and positive geometry, we refer the reader to [10].) The catalyst for this development was the fact that associahedra are combinatorial models for type-A cluster algebras
Building on the seminal work of Arkani-Hamed, He, Salvatori and Thomas (AHST) [1] we explore the positive geometry encoding one loop scattering amplitude for quartic scalar interactions
We define a new class of combinatorial polytopes that we call pseudo-accordiohedra whose poset structures are associated to singularities of the one loop integrand associated to scalar quartic interactions
Summary
We briefly review the essential ideas from [3, 6, 7, 13, 14]. We only review the results which are central to our analysis in this paper. It was shown that given any triangulation T of an n-gon, the following system of linear constraints generate a class of realisations which include ABHY realisations. The choice of T 0 is arbitrary and the form Ωn is independent of such a reference.4 It was shown in [3] that this form projects to the canonical top form on the (any) realisation of the associahedron and this canonical form is precisely the planar amplitude in massless φ3 theory. Cpq , p=i q=j dXij + dXk = dXi + dXkj. On the other hand if the reference triangulation T does not have any diagonal with one end point in Vk,j and other end point in V ,i, i−1 j−1.
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