Connecting scalar amplitudes using the positive tropical Grassmannian

A bstract The biadjoint scalar partial amplitude, $$ {m}_n\left(\mathbbm{I},\mathbbm{I}\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>m</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mfenced> <mml:mi>I</mml:mi> <mml:mi>I</mml:mi> </mml:mfenced> </mml:math> , 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 ϕ 4 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 , $$ \mathbbm{I} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>I</mml:mi> </mml:math> ) amplitude. We provide a combinatorial description of the regions in terms of non-crossing chord diagrams and propose a general formula for ϕ 4 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 ϕ 3 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.