Hyperbolic geometry and amplituhedra in 1+2 dimensions

We recently proposed the Halohedron to be the 1-loop Amplituhedron for planar φ 3 theory. Here we prove this claim by showing how it is possible to extract the integrand for the partial amplitude m 1 n (1,. .. , n|1,. .. , n) from the canonical form of an Halohedron which lives in an abstract space. This space is just a step away from ordinary kinematical space at 1-loop, because it is composed by abstract variables associated to propagators of 1-loop Feynman diagrams. Such variables, however, are unbound from momentum conservation relations that would give problems such as double poles. As an application of our construction, we exploit a well known recursion formula for the canonical form of a polytope in order to produce an expression for the 1-loop integrand which would not be evident starting from Feynman diagrams.

Section: Introductionmentioning

confidence: 99%

1-loop amplitudes from the Halohedron

Salvatori

2019

“…We shall call them [1], [2],...,[k], where k = p 2 . The primitives are all shown in the above figure (6). We shall now show that these are the only primitives.…”

mentioning

confidence: 74%

“…• We also require the accordiohedron AC (P ) p,n to have dimension n, which is the number of propagators. 6 We shall first introduce some notations and variables that we shall use to describe the kinematic space.…”

Section: Positive Geometry For φ P Interactionsmentioning

confidence: 99%

“…We could try to recursively construct these graphs from the n = 1 graphs. So without loss of generality we consider only Feynman graphs in which two of the vertices lie in a line as shown in the figure (6). The 3rd vertex can then be made to lie on the central line connecting the first two vertices and it can be either above or below this line.…”

Section: Primitives and Weights For N = 2 Casementioning

confidence: 99%

See 1 more Smart Citation

The positive geometry for 𝜙p interactions

Raman¹

2019

Starting with the seminal work of Arkani-Hamed et al [1], in [2], the "Amplituhedron program" was extended to analyzing (planar) amplitudes in massless φ 4 theory. In this paper we show that the program can be further extended to include φ p (p > 4) interactions. We show that tree-level planar amplitudes in these theories can be obtained from geometry of polytopes called accordiohedron which naturally sits inside kinematic space. As in the case of quartic interactions the accordiohedron of a given dimension is not unique, and we show that a weighted sum of residues of the canonical form on these polytopes can be used to compute scattering amplitudes. We finally provide a prescription to compute the weights and demonstrate how it works in various examples.1

“…Another natural question is to see how our construction is related to other polytopes in the literature, such as generalized permutohedra [23] (beyond the Cayley cases) and cluster associahedra [25]. Very recently, there have been explorations of mathematical structures related to scattering forms, worldsheet forms and the geometries (see for example [26][27][28][29][30][31][32]). It would be interesting to see how some of them fit in our picture as well.…”

Section: Discussionmentioning

confidence: 99%

Scattering forms, worldsheet forms and amplitudes from subspaces

Yan

Zhang

et al. 2018

We present a general construction of two types of differential forms, based on any (n−3)-dimensional subspace in the kinematic space of n massless particles. The first type is the so-called projective, scattering forms in kinematic space, while the second is defined in the moduli space of n-punctured Riemann spheres which we call worldsheet forms. We show that the pushforward of worldsheet forms, by summing over solutions of scattering equations, gives the corresponding scattering forms, which generalizes the results of [1]. The pullback of scattering forms to subspaces can have natural interpretations as amplitudes in terms of Bern-Carrasco-Johansson double-copy construction or Cachazo-He-Yuan formula. As an application of our formalism, we construct in this way a large class of d log scattering forms and worldsheet forms, which are in one-to-one correspondence with non-planar MHV leading singularities in N = 4 super-Yang-Mills. For every leading singularity function, we present a new determinant formula in moduli space, as well as a (combinatoric) polytope and associated scattering form in kinematic space. These include the so-called Cayley cases, where in each case the scattering form is the canonical forms of a convex polytope in the subspace, and scattering equations admit elegant rewritings as a map from the moduli space to the subspace.