Connecting Scalar Amplitudes using The Positive Tropical Grassmannian

The biadjoint scalar partial amplitude, $m_n(\mathbb{I},\mathbb{I})$, 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 $m_n(\alpha,\beta)$ using a limiting procedure on kinematic invariants that produces indicator functions in the integrand. The same limiting procedure leads to an integral representation of $\phi^4$ amplitudes where indicator functions turn into Dirac delta functions. Their support decomposes into $\textrm{C}_{n/2-1}$ regions, with $\textrm{C}_q$ the $q^{\rm th}$-Catalan number. The contribution from each region is identified with a $m_{n/2+1}(\alpha,\mathbb{I})$ amplitude. We provide a combinatorial description of the regions in terms of non-crossing chord diagrams and propose a general formula for $\phi^4$ amplitudes using the Lagrange inversion construction. We start the exploration of $\phi^p$ theories, finding that their regions are encoded in non-crossing $(p-2)$-chord diagrams. The structure of the expansion of $\phi^p$ amplitudes in terms of $\phi^3$ amplitudes is the same as that of Green functions in terms of connected Green functions in the planar limit of $\Phi^{p-1}$ matrix models. We also discuss possible connections to recent constructions based on Stokes polytopes and accordiohedra.