Tropical Amplitudes For Colored Lagrangians

TL;DR

This paper extends a tropical geometric framework for scattering amplitudes from $\

Contribution

It introduces tropical numerator functions and combinatorial methods to compute amplitudes for general Lagrangians with complex interactions.

Findings

Tropical numerators can compute amplitudes for Lagrangians with infinitely many interactions.

New combinatorial descriptions of diagrams for arbitrary valence interactions.

Methods for formulating amplitudes using tropical geometry and Wick contraction.

Abstract

Recently a new formulation for scattering amplitudes in Tr($\Phi^3$) theory has been given based on simple combinatorial ideas in the space of kinematic data. This allows all-loop integrated amplitudes to be expressed as ''curve integrals'' defined using tropical building blocks - the ''headlight functions''. This paper shows how the formulation extends to the amplitudes of more general Lagrangians. We will present a number of different ways of introducing tropical ''numerator functions'' that allow us to describe general Lagrangian interactions. The simplest family of these ''tropical numerators'' computes the amplitudes of interesting Lagrangians with infinitely many interactions. We also describe methods for tropically formulating the amplitudes for general Lagrangians. One uses a variant of ''Wick contraction'' to glue together numerator factors for general interaction vertices. Another uses a natural characterization of polygons on surfaces to give a novel combinatorial description of all possible diagrams associated with arbitrary valence interactions.