Nonequilibrium dynamics in the O(N) model to next-to-next-to-leading order in the 1/N expansion

TL;DR

This paper develops advanced equations for the nonequilibrium dynamics of the O(N) model at next-to-next-to-leading order in the 1/N expansion, and numerically investigates their behavior in quantum mechanics to improve understanding of convergence beyond mean-field approximations.

Contribution

It derives the real and causal evolution equations for the O(N) model at NNLO in the 2PI-1/N expansion, enabling systematic convergence studies.

Findings

Numerical solutions in 0+1 dimensions show improved accuracy over previous approximations.

Comparison with exact solutions demonstrates the effectiveness of the NNLO approach.

The equations highlight the complexity introduced by internal vertices in higher-order expansions.

Abstract

Nonequilibrium dynamics in quantum field theory has been studied extensively using truncations of the 2PI effective action. Both 1/N and loop expansions beyond leading order show remarkable improvement when compared to mean-field approximations. However, in truncations used so far, only the leading-order parts of the self energy responsible for memory loss, damping and equilibration are included, which makes it difficult to discuss convergence systematically. For that reason we derive the real and causal evolution equations for an O(N) model to next-to-next-to-leading order in the 2PI-1/N expansion. Due to the appearance of internal vertices the resulting equations appear intractable for a full-fledged 3+1 dimensional field theory. Instead, we solve the closely related three-loop approximation in the auxiliary-field formalism numerically in 0+1 dimensions (quantum mechanics) and compare to previous approximations and the exact numerical solution of the Schroedinger equation.