All Loop Scattering As A Counting Problem

TL;DR

The paper develops a curve-integral formalism for all-loop amplitudes in a colored scalar theory, replacing the conventional diagrammatic sum with a combinatorial curve-counting problem on fatgraphs. Central to the approach are the g-vector Feynman fan, headlight functions, and Mirzakhani kernels, which mod out the mapping class group to yield finite curve-space integrals governed by surface-like Symanzik polynomials. The authors establish two key miracles: a universal curve-based description that factorizes large-n amplitudes into tree plus low-point loop data, and a local transfer-matrix method that computes headlight functions efficiently from mountainscapes, enabling all-order constructions and recursion. They demonstrate explicit all-loop formulations and compute concrete amplitudes from tree level up to genus-one 2-loop, while outlining recursive strategies and extensions toward gravity, strings, and emergent spacetime, signaling a new combinatorial foundation for quantum field theory.

Abstract

This is the first in a series of papers presenting a new understanding of scattering amplitudes based on fundamentally combinatorial ideas in the kinematic space of the scattering data. We study the simplest theory of colored scalar particles with cubic interactions, at all loop orders and to all orders in the topological 't Hooft expansion. We find a novel formula for loop-integrated amplitudes, with no trace of the conventional sum over Feynman diagrams, but instead determined by a beautifully simple counting problem attached to any order of the topological expansion. These results represent a significant step forward in the decade-long quest to formulate the fundamental physics of the real world in a radically new language, where the rules of spacetime and quantum mechanics, as reflected in the principles of locality and unitarity, are seen to emerge from deeper mathematical structures.

Figures (34)

• Figure 1: Fat graphs at tree-level, 1-loop single trace, 1-loop double trace, and 2-loop single trace, respectively.
• Figure 2: A tree fat graph with momenta assigned to all edges.
• Figure 3: Every curve, $C$, drawn on the fat graph inherits a momentum, $P_C^\mu$, from the momenta assigned to the fat graph itself.
• Figure 4: Describing a curve on a fatgraph (left) using a mountainscape diagram (right).
• Figure 5: The five (non boundary) curves that we can draw on a 5-point tree fatgraph.