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arXiv:hep-th/0604156 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

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

Gert Aarts A, Anders Tranberg B

Published Jul 6, 2006• 29 citations
arXivHTMLPDF

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. Because of 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 Schr\"odinger equation.

DOI
10.1103/physrevd.74.025004

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Nonequilibrium dynamics in quantum field theory has been studied extensively using truncations of the 2PI effective action. Both 1/N1/N1/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)O(N)O(N) model to next-to-next-to-leading order in the 2PI-1/N1/N1/N expansion. Because of the appearance of internal vertices the resulting equations appear intractable for a full-fledged 3+13+13+1 dimensional field theory. Instead, we solve the closely related three-loop approximation in the auxiliary-field formalism numerically in 0+10+10+1 dimensions (quantum mechanics) and compare to previous approximations and the exact numerical solution of the Schr"odinger equation.
We derive the nonequilibrium real-time evolution of an O(N)O(N)O(N)-invariant scalar quantum field theory in the presence of a nonvanishing expectation value of the quantum field. Using a systematic 1/N1/N1/N expansion of the 2PI effective action to next-to-leading order, we obtain nonperturbative evolution equations which include scattering and memory effects. The equivalence of the direct method, which requires the resummation of an infinite number of skeleton diagrams, with the auxiliary-field formalism, which involves only one diagram at next-to-leading order, is shown.
We calculate the far-from-equilibrium dynamics and thermalization both for the quantum and the classical O(N)–model. The early and late-time behavior can be described from the 2PI–loop expansion for weak couplings or the nonperturbative 2PI–1/N expansion of the effective action beyond leading order. A comparison with exact simulations in 1+1 dimensions in the classical limit shows that the 2PI–1/N expansion at next-to-leading order gives quantitatively precise results already for moderate values of N. We derive a criterion for the validity of the classical approximation and verify it by comparing far-fromequilibrium quantum and classical dynamics. At late times one observes the expected deviations due to the difference between classical and quantum thermal equilibrium. In recent years we have witnessed an enormous increase of interest in the dynamics of quantum fields out of equilibrium. Strong motivation comes from a wide range of applications including current and upcoming relativistic heavy-ion collision experiments, phase transitions in the early universe or the dynamics of Bose-Einstein condensation. Directly simulating quantum fields in real time, such as solving the Schrödinger equation for the wave functional is prohibitively difficult and one has to find suitable approximations. The
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.
The 1/N1/N1/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/N1/N1/N expansion in the O(N)O(N)O(N) model by comparing results obtained numerically in 1+11+11+1 dimensions at leading, next-to-leading and next-to-next-to-leading order in 1/N1/N1/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/N1/N1/N or the coupling.
We consider the nonequilibrium evolution of an O(N)-symmetric scalar quantum field theory using a systematic two-particle irreducible 1/N-expansion to next-to-leading order, which includes scattering and memory effects. The corresponding ``full Kadanoff-Baym equations'' are solved numerically without further approximations. This allows one to obtain a controlled nonperturbative description of far-from-equilibrium dynamics and the late-time approach to quantum thermal equilibrium. Employing in addition a first-order gradient expansion for the Wigner transformed correlators we derive kinetic equations. In contrast to standard descriptions based on loop expansions, our equations remain valid for nonperturbatively large fluctuations. As an application, we discuss the fluctuation dominated regime following parametric resonance in quantum field theory.
Transport or kinetic equations are often derived assuming a quasiparticle (on-shell) representation of the spectral function. We investigate this assumption using a three-loop approximation of the 2PI effective action in real time, without a gradient expansion or on-shell approximation. For a scalar field in 1+11+11+1 dimensions the nonlinear evolution, including the integration over memory kernels, can be solved numerically. We find that a spectral function approximately described by a nonzero width emerges dynamically. During the nonequilibrium time evolution the Wigner transformed spectral function is slowly varying, even in the presence of strong qualitative changes in the effective particle distribution. These results may be used to make further analytical progress toward a quantum Boltzmann equation including off-shell effects and a nonzero width.