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 mn(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 Cn/2−1 regions, with Cq the qth-Catalan number. The contribution from each region is identified with a mn/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 ϕp theories, finding that their regions are encoded in non-crossing (p – 2)-chord diagrams. The structure of the expansion of ϕp 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 Φp−1 matrix models. We also discuss possible connections to recent constructions based on Stokes polytopes and accordiohedra.
