Circles and Triangles, the NLSM and Tr($\Phi^3$)

A surprising connection has recently been made between the amplitudes for Tr($\Phi^3$) theory and the non-linear sigma model (NLSM). A simple shift of kinematic variables naturally suggested by the associahedron/stringy representation of Tr$(\Phi^3$) theory yields pion amplitudes at all loops. In this note we provide an elementary motivation and proof for this link going in the opposite direction, starting from the non-linear sigma model and discovering its formulation as a sum over triangulations of surfaces with simple numerator factors. This uses an ancient connection between "circles" and "triangles", interpreting the equation $y = \sqrt{1 - x^2}$ both as parametrizing points on a circle as well as generating the number of triangulations of polygons. A further simplification of the numerator factors exposes them as arising from the kinematically shifted Tr($\Phi^3$) theory, and gives rise to novel tropical representations of NLSM amplitudes. The connection to Tr$(\Phi^3)$ theory defines a natural notion of "surface-soft limit" intrinsic to curves on surfaces. Remarkably, with this definition, the soft limit of pion amplitudes vanishes directly at the level of the integrand, via obvious pairwise cancellations. We also give simple, explicit expressions for the multi-soft factors for tree and loop-level integrands in the limit as any number of pions are taken "surface-soft".