Behavior of Cross Sections for Large Numbers of Particles

TL;DR

This paper investigates the behavior of scattering cross sections involving large numbers of particles, showing that non-perturbative effects prevent factorial growth and challenging previous perturbative predictions in high-energy physics.

Contribution

It introduces a systematic approximation scheme for large N particle scattering in $b4 b4$ theory and demonstrates that cross sections do not grow factorially as perturbation theory suggests.

Findings

Scattering amplitudes grow more slowly than perturbative predictions.

Partial and total cross sections do not exhibit factorial growth.

No systematic large N approximation exists for $2 o N$ processes.

Abstract

It has been suggested that scattering cross sections at very high energies for producing large numbers of Higgs particles may exhibit factorial growth, and that curing this growth might be relevant to other questions in the Standard Model. We point out, first, that the question is inherently non-perturbative; low orders in the formal perturbative expansion do not give a good approximation to the scattering amplitude for sufficiently large N for any fixed, small value of the coupling. Focusing on $\lambda \phi^{4}$ theory, we argue that there may be a systematic approximation scheme for processes where N particles near threshold scatter to produce N particles, and discuss the leading contributions to the scattering amplitude and cross sections in this limit. Scattering amplitudes do not grow as rapidly as in perturbation theory. Additionally, partial and total cross sections do not show factorial growth. In the case of cross sections for $2 \to N$ particles, there is no systematic large N approximation available. That said, we provide evidence that non-perturbatively, there is no factorial growth in partial or total cross sections.