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arXiv:0809.3390 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Effective convergence of the two-particle irreducible 1/N expansion for nonequilibrium quantum fields

Gert Aarts A, Nathan S. Laurie, Anders Tranberg B

Published Dec 29, 2008• 17 citations
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Abstract

The $1/N$ expansion of the two-particle irreducible effective action offers a powerful approach to study quantum field dynamics far from equilibrium. We investigate the effective convergence of the $1/N$ expansion in the $O(N)$ model by comparing results obtained numerically in $1+1$ dimensions at leading, next-to-leading and next-to-next-to-leading order in $1/N$ as well as in the weak coupling limit. A comparison in classical statistical field theory, where exact numerical results are available, is made as well. We focus on early-time dynamics and quasiparticle properties far from equilibrium and observe rapid effective convergence already for moderate values of $1/N$ or the coupling.

DOI
10.1103/physrevd.78.125028

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We study the nonequilibrium dynamics of the O(N)\mathrm{O}(N)O(N) model in classical and quantum field theory in 1+1 dimensions, for N>1.N>1.N>1. We compare numerical results obtained using the Hartree approximation and two next to leading order approximations: the bare vertex approximation and the two-particle irreducible 1/N1/N1/N expansion. The latter approximations differ through terms of order g2,{g}^{2},g2, where g is the scaled coupling constant, $g=\ensuremath{\lambda}/N.$ In this paper we investigate the statement regarding the convergence with respect to g. The differences between these two approximation schemes are computed and found to be already small for relatively small values of N, when $\ensuremath{\lambda}$ is fixed.
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I review the use of the 2PI effective action in nonequilibrium quantum field theory. The approach enables one to find approximation schemes which circumvent long-standing problems of non-thermal or secular (unbounded) late-time evolutions encountered in standard loop or 1/N expansions of the 1PI effective action. It is shown that late-time thermalization can be described from a numerical solution of the three-loop 2PI effective action for a scalar ϕ4ϕ^4ϕ4--theory in 1+1 dimensions (with Jurgen Cox, hep-ph/0006160). Quantitative results far from equilibrium beyond the weak coupling expansion can be obtained from the 1/N expansion of the 2PI effective action at next-to-leading order (NLO), calculated for a scalar O(N) symmetric quantum field theory (hep-ph/0105311). Extending recent calculations in classical field theory by Aarts et al. (hep-ph/0007357) and by Blagoev et al. (hep-ph/0106195) to N>1N>1N>1 we show that the NLO approximation converges to exact (MC) results already for moderate values of NNN (with Gert Aarts, hep-ph/0107129). I comment on characteristic time scales in scalar quantum field theory and the applicability of classical field theory for sufficiently high initial occupation numbers.
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