Abstract
The relationship between certain geometric objects called polytopes and scattering amplitudes has revealed deep structures in QFTs. It has been developed in great depth at the tree- and loop-level amplitudes in mathcal{N} = 4 SYM theory and has been extended to the scalar ϕ3 and ϕ4 theories at tree-level. In this paper, we use the generalized BCFW recursion relations for massless planar ϕ4 theory to constrain the weights of a class of geometric objects called Stokes polytopes, which manifest in the geometric formulation of ϕ4 amplitudes. We see that the weights of the Stokes polytopes are intricately tied to the boundary terms in ϕ4 theories. We compute the weights of N = 1, 2, and 3 dimensional Stokes polytopes corresponding to six-, eight- and ten-point amplitudes respectively. We generalize our results to higher-point amplitudes and show that the generalized BCFW recursions uniquely fix the weights for an n-point amplitude.
Highlights
Relation between so-called planar scattering forms on kinematic space and a polytope known as Associahedron
We use the generalized BCFW recursion relations for massless planar φ4 theory to constrain the weights of a class of geometric objects called Stokes polytopes, which manifest in the geometric formulation of φ4 amplitudes
It was shown that the planar φ4 amplitudes can be obtained from the geometry of an object known as Stokes polytopes
Summary
We give an overview of the key results of [3], where the relationship between planar Feynman graphs in φ4 theory and positive geometries was established. We focus on the construction of planar massless φ4 amplitudes at tree-level by summing over the Stokes polytopes. N = (2I + 2), there are FI number of Stokes polytopes whose weighted sum gives the full amplitude. The weighted sum of these functions over all Stokes polytopes gives the full planar scattering amplitude. All the other quadrangulations can be obtained by cyclic permutations of the quadrangulations of this subset What this implies is that, for any given n, once the rational canonical functions for a given set of primitives {Q1, . There is a unique choice of αQ’s such that the amplitude determined by the weighted sum over all primitives Mn = Mn, where Mn is the tree-level planar φ4 amplitude. Where I and J corresponds to all the allowed splitting as in figure 1
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