Constraining the weights of Stokes polytopes using BCFW recursions for ϕ 4

Abstract

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

Keywords

• Scattering Amplitudes
• Differential and Algebraic Geometry