Scattering Equations and Feynman Diagrams

TL;DR

The paper shows that individual Feynman diagrams in scalar theories can be systematically mapped to CHY integrands using polygon (and triangle) decompositions, first for $\varphi^3$ theory and then generalized to $\varphi^p$ with mixed vertices. It proves a direct correspondence via a polygon-graph construction and the Baadsgaard integration rules, and explains how two-cycle CHY integrands yield a unique diagrammatic decomposition while connecting to string theory in the $\alpha'\to 0$ limit. It also contrasts the polygon-based CHY construction with the Pfaffian CHY formulation for $\varphi^4$, illustrating how different yet equivalent representations can organize the same amplitudes. The work lays groundwork for extending CHY representations to more general theories and loop-level amplitudes, with potential links to BCJ relations and gauge-invariance considerations in Yang-Mills theory.

Abstract

We show a direct matching between individual Feynman diagrams and integration measures in the scattering equation formalism of Cachazo, He and Yuan. The connection is most easily explained in terms of triangular graphs associated with planar Feynman diagrams in $φ^3$-theory. We also discuss the generalization to general scalar field theories with $φ^p$ interactions, corresponding to polygonal graphs involving vertices of order $p$. Finally, we describe how the same graph-theoretic language can be used to provide the precise link between individual Feynman diagrams and string theory integrands.