Constraining the weights of Stokes Polytopes using BCFW recursions for Phi^4

TL;DR

This paper uses generalized BCFW recursion relations to determine the weights of Stokes polytopes in planar $4$ theory, revealing their connection to boundary terms and providing a recursive method for higher-point amplitudes.

Contribution

It introduces a recursive approach to constrain Stokes polytope weights in $4$ theory, extending geometric amplitude formulations beyond previously studied theories.

Findings

Weights of Stokes polytopes are fixed by boundary terms.

Explicit weights computed for 6, 8, and 10-point amplitudes.

Recursive method generalizes to higher-point amplitudes.

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 $\mathcal{N}=4~\text{SYM}$ theory and has been extended to the scalar $\phi^3$ and $\phi^4$ theories at tree-level. In this paper, we use the generalized BCFW recursion relations for massless planar $\phi^4$ theory to constrain the weights of a class of geometric objects called Stokes polytopes, which manifest in the geometric formulation of $\phi^4$ amplitudes. We see that the weights of the Stokes polytopes are intricately tied to the boundary terms in $\phi^4$ 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.