All Loop Scattering For All Multiplicity

TL;DR

The paper advances a curve integral framework for scattering amplitudes in colored trφ^3 theory, revealing a striking decoupling of dependence on particle multiplicity n and loop order L. By focusing on L-loop tadpole-like subgraphs and leveraging a telescopic headlight structure, it derives all-n formulas at tree level and extends them to higher loops, including non-planar cases, up to two loops. A central feature is Tree-Loop factorization, which reduces complex all-n amplitudes to data computable from tadpole graphs, via Mirzakhani kernels and a finite set of curves; this yields explicit, polynomially growing, all-multiplicity expressions for planar and non-planar amplitudes. The work lays out a rigorous, scalable program for computing multi-loop, multi-trace amplitudes in this theory and outlines promising directions for large-n, large-L behavior and renormalization within a tropical/Teichmüller geometry framework.

Abstract

This is part of a series of papers describing the new curve integral formalism for scattering amplitudes of the colored scalar tr$φ^3$ theory. We show that the curve integral manifests a very surprising fact about these amplitudes: the dependence on the number of particles, $n$, and the loop order, $L$, is effectively decoupled. We derive the curve integrals at tree-level for all $n$. We then show that, for higher loop-order, it suffices to study the curve integrals for $L$-loop tadpole-like amplitudes, which have just one particle per color trace-factor. By combining these tadpole-like formulas with the the tree-level result, we find formulas for the all $n$ amplitudes at $L$ loops. We illustrate this result by giving explicit curve integrals for all the amplitudes in the theory, including the non-planar amplitudes, through to two loops, for all $n$.

Figures (22)

• Figure 1: The $n$-point tree-level comb graph.
• Figure 2: An $n$-point 1-loop planar graph.
• Figure 3: An incomplete curve can be extended in one of two ways: to the left, or to the right.
• Figure 4: A Tree-Loop fatgraph for the planar 1-loop amplitudes.
• Figure 5: The Planar One-loop tadpole (left), and its two curves: $S$ and $P$.