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.
Constraining the weights of Stokes Polytopes using BCFW recursions for Phi^4 · Around
H. Elvang and Y.-t. Huang, Scattering Amplitudes in Gauge Theory and Gravity. Cambridge: Cambridge University Press. (2015), d o i : 10.1017 / C B O 9781107706620 doi:10.1017/CBO9781107706620