All Loop Scattering For All Multiplicity

TL;DR

This paper introduces a new curve integral formalism for scattering amplitudes in colored scalar $ ext{tr}\u03b6^3$ theory, revealing a decoupling of particle number and loop order, and provides explicit formulas for all $n$-particle amplitudes up to two loops.

Contribution

It develops a universal curve integral approach for all multiplicities and loop orders, simplifying the computation of scattering amplitudes in the theory.

Findings

Decoupling of $n$ and $L$ in amplitude dependence.

Explicit formulas for all $n$-particle amplitudes at up to two loops.

Non-planar amplitudes are included in the formalism.

Abstract

This is part of a series of papers describing the new curve integral formalism for scattering amplitudes of the colored scalar tr$\phi^3$ 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 $n$.