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

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.