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arXiv:hep-ph/0305220 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Quantum dynamics of Φ4 field theory beyond leading order in 1+1 dimensions

Jürgen Baacke, Andreas Heinen

Published Dec 23, 2003• 11 citations
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Abstract

We consider the out-of-equilibrium evolution of a classical condensate field $\ensuremath{\varphi}=〈\ensuremath{\Phi}〉$ and its quantum fluctuations for a ${\ensuremath{\Phi}}^{4}$ model in $1+1$ dimensions with a double well potential. We use the two-particle point-irreducible (2PPI) formalism in the two-loop approximation. We compare our results to those obtained in the Hartree approximation, in the bare vertex approximation, and in the two-particle irreducible next-to-leading order large-$N$ approach, with thermal initial conditions. In the 2PPI scheme, we find that the system tends to the symmetric configuration at late times, as expected in the absence of spontaneous symmetry breaking.

DOI
10.1103/physrevd.68.127702

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We consider the out-of-equilibrium evolution of a classical condensate field \ensuremathφ=〈\ensuremathΦ〉\ensuremath{\varphi}=〈\ensuremath{\Phi}〉\ensuremathφ=〈\ensuremathΦ〉 and its quantum fluctuations for a \ensuremathΦ4{\ensuremath{\Phi}}^{4}\ensuremathΦ4 model in 1+11+11+1 dimensions with a double well potential. We use the two-particle point-irreducible (2PPI) formalism in the two-loop approximation. We compare our results to those obtained in the Hartree approximation, in the bare vertex approximation, and in the two-particle irreducible next-to-leading order large-NNN approach, with thermal initial conditions. In the 2PPI scheme, we find that the system tends to the symmetric configuration at late times, as expected in the absence of spontaneous symmetry breaking.
We consider the out-of-equilibrium evolution of a classical condensate field and its quantum fluctuations for a ${\ensuremath{\Phi}}^{4}$ model in 1+11+11+1 dimensions with a symmetric and a double well potential. We use the two-particle point-irreducible (2PPI) formalism and go beyond the Hartree approximation by including the sunset term. In addition to the mean field $\ensuremath{\varphi}(t)=〈\ensuremath{\Phi}〉$ the 2PPI formalism uses as a variational parameter a time dependent mass M2(t){\mathcal{M}}^{2}(t)M2(t) which contains all local insertions into the Green's function. We compare our results to those obtained in the Hartree approximation. In the symmetric ${\ensuremath{\Phi}}^{4}$ theory we observe that the mean field shows a stronger dissipation than the one found in the Hartree approximation. The dissipation is roughly exponential in an intermediate time region. In the theory with spontaneous symmetry breaking, i.e., with a double well potential, the field amplitude tends to zero, i.e., to the symmetric configuration. This is expected on general grounds: in (1+1)(1+1)(1+1)-dimensional quantum field theory there is no spontaneous symmetry breaking for T>0,T>0,T>0, and so there should be none at finite energy density (microcanonical ensemble), either. Within the time range of our simulations the momentum spectra do not thermalize and display parametric resonance bands.
We perform a detailed numerical investigation of the dynamics of a single component ``explicitly broken symmetry'' $\ensuremath{\lambda}{\ensuremath{\varphi}}^{4}$ field theory in 1+1 dimensions, using a Schwinger-Dyson equation truncation scheme based on ignoring vertex corrections. In an earlier paper, we called this the bare vertex approximation (BVA). We assume here that the initial state is described by a Gaussian density matrix peaked around some nonzero value of $〈\ensuremath{\varphi}(0)〉,$ and characterized by a single particle Bose-Einstein distribution function at a given temperature. We compute the evolution of the system using three different approximations: the Hartree approximation, the BVA, and a related two-particle irreducible (2PI) 1/N1/N1/N expansion, as a function of coupling strength and initial temperature. In the Hartree approximation, the static phase diagram shows that there is a first order phase transition for this system. As we change the initial starting temperature of the system, we find that the BVA relaxes to a new final temperature and exhibits behavior consistent with a second order phase transition. We find that the average fields equilibrate for arbitrary initial conditions in the BVA, unlike the behavior exhibited by the Hartree approximation, and we illustrate how $〈\ensuremath{\varphi}(t)〉$ and $〈\ensuremath{\chi}(t)〉$ depend on the initial temperature and on the coupling constant. The Fourier transform of the two-point functions at late times can be fitted by a Bose-Einstein distribution function whose temperature is independent of momentum. We interpret this as evidence for thermalization.
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.
This work is devoted to the study of relaxation--dissipation processes in systems described by Quantum Field Theory. In the first part, I focus on the phi^4 scalar quantum field theory in finite volume in the large N limit. I find that the zero-mode quantum fluctuations cannot grow macroscopically large starting from microscopic initial conditions, while the long-wavelength fluctuations do scale with the linear size of the system, signalling dynamical infrared properties quite different from the equilibrium ones. I also introduce a non-gaussian Hartree-Fock approximation (tdHF) and I derive the mean-field coupled time-dependent Schroedinger equations for the modes of the scalar field. The dynamical evolution of a uniform condensate is studied in a further controlled gaussian approximation, for N=1. Results similar to the large N case show an internal inconsistency of this approximation. I also study the dynamics of the system in infinite volume in the broken symmetry phase. The data show that the fixed points of the evolution cover at most the classically metastable part of the static effective potential. I finally study the dynamical evolution for the O(N+1) nonlinear sigma model in 1+1 dimensions in the large N limit. I first of all verify that the large coupling limit of the O(N+1) phi^4 model, which renders the model non linear, commutes with the large N limit, so that the O(infty) nonlinear sigma model is uniquely defined. I then study the evolution of several observables, like the spectrum of produced particles, finding no evidence for parametric resonance. I conclude with some remarks on the ``state of art'' in gauge theories and some comments about the open issues in the subject.
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.
The quantum time evolution of ${\ensuremath{\varphi}}^{4}$-field theory for a spatially homogeneous system in 2+1 space-time dimensions is investigated numerically for out-of-equilibrium initial conditions on the basis of the Kadanoff-Baym equations including the tadpole and sunset self-energies. Whereas the tadpole self-energy yields a dynamical mass, the sunset self-energy is responsible for dissipation and an equilibration of the system. In particular we address the dynamics of the spectral (``off-shell'') distributions of the excited quantum modes and the different phases in the approach to equilibrium described by Kubo-Martin-Schwinger relations for thermal equilibrium states. The investigation explicitly demonstrates that the only translation invariant solutions representing the stationary fixed points of the coupled equation of motions are those of full thermal equilibrium. They agree with those extracted from the time integration of the Kadanoff-Baym equations for $\stackrel{\ensuremath{\rightarrow}}{t}\ensuremath{\infty}.$ Furthermore, a detailed comparison of the full quantum dynamics to more approximate and simple schemes such as that of a standard kinetic (on-shell) Boltzmann equation is performed. Our analysis shows that the consistent inclusion of the dynamical spectral function has a significant impact on relaxation phenomena. The different time scales that are involved in the dynamical quantum evolution towards a complete thermalized state are discussed in detail. We find that far off-shell $1\ensuremath{\leftrightarrow}3$ processes are responsible for chemical equilibration, which is missed in the Boltzmann limit. Finally, we briefly address the case of (bare) massless fields. For sufficiently large couplings $\ensuremath{\lambda}$ we observe the onset of Bose condensation, where our scheme within symmetric ${\ensuremath{\varphi}}^{4}$ theory breaks down.