Connecting scalar amplitudes using the positive tropical Grassmannian

Abstract

The biadjoint scalar partial amplitude, \( {m}_n\left(\mathbbm{I},\mathbbm{I}\right) \), can be expressed as a single integral over the positive tropical Grassmannian thus producing a Global Schwinger Parameterization. The first result in this work is an extension to all partial amplitudes m_n(α, β) using a limiting procedure on kinematic invariants that produces indicator functions in the integrand. The same limiting procedure leads to an integral representation of ϕ⁴ amplitudes where indicator functions turn into Dirac delta functions. Their support decomposes into C_n/2−1 regions, with C_q the q^th-Catalan number. The contribution from each region is identified with a m_n/2+1(α, \( \mathbbm{I} \)) amplitude. We provide a combinatorial description of the regions in terms of non-crossing chord diagrams and propose a general formula for ϕ⁴ amplitudes using the Lagrange inversion construction. We start the exploration of ϕ^p theories, finding that their regions are encoded in non-crossing (p – 2)-chord diagrams. The structure of the expansion of ϕ^p amplitudes in terms of ϕ³ amplitudes is the same as that of Green functions in terms of connected Green functions in the planar limit of Φ^p−1 matrix models. We also discuss possible connections to recent constructions based on Stokes polytopes and accordiohedra.