Nonequilibrium evolution in scalarO(N)models with spontaneous symmetry breaking

Abstract

We consider the out-of-equilibrium evolution of a classical condensate field and its quantum fluctuations for a scalar O(N) model with spontaneously broken symmetry. In contrast to previous studies we do not consider the large N limit, but the case of finite N , including N = 1, i.e., plain λφ 4 theory. The instabilities encountered in the one-loop approximation are prevented, as in the large-N limit, by back reaction of the fluctuations on themselves, or, equivalently, by including a resummation of bubble diagrams. For this resummation and its renormalization we use formulations developed recently based on the effective action formalism of Cornwall, Jackiw and Tomboulis. The formulation of renormalized equations for finite N derived here represents a useful tool for simulations with realistic models. Here we concentrate on the phase structure of such models. We observe the transition between the spontaneously broken and the symmetric phase at low and high energy densities, respectively. This shows that the typical structures expected in thermal equilibrium are encountered in nonequilibrium dynamics even at early times, i.e., before an efficient rescattering can lead to thermalization.

Renormalisability of the 2PI-Hartree approximation of multicomponent scalar models in the broken symmetry phase

“…At tree (classical) level this background results in different multiplet structures in the two sectors, but solving (81) iteratively one can show that the emerging exact mass spectra will have eventually the same structure in both sectors due to the coupling realised by the tensor H. To be specific, when v 3 = 0, v 8 = 0 one has 3 degenerate doublets in the "planes" [1,2], [4,5] and [6,7] and one coupled set with unequal eigenvalues in the [3,8] "plane". These "planes" correspond to the pairs of fields…”

“…Making use of the gap equation (8) for the sigma field in the equation of state (10), one immediately sees that for the consistent renormalisation of the two equations one has to require…”

“…In quantum field theory it corresponds to the momentum independent two-loop truncation of the two-particle irreducible effective action. It is used extensively both in equilibrium [2,3,4] and out-of-equilibrium [5,6,7,8] non-perturbative investigations of phase transition phenomena. Its non-perturbative renormalisability was demonstrated as particular case of the general proof of renormalisability of the physical quantities computed in various 2PI approximations [9,1,10].…”

Renormalized and renormalization-group invariant Hartree-Fock approximation

“…These approximations have been extensively studied and their features are well known [11][12][13][14][15][16]: they do provide a backreaction term on the evolution of quantum fluctuations that stabilize dynamics at late time but, on the other hand, they fail to properly describe an important aspect of late-time dynamics such as thermalization. More elaborate approaches going beyond mean field have been put forward by considering the 2-particle-irreducible (2PI) or the two-point-particle-irreducible effective action [17,18] at two (or more) loops or at next-to-leading order in 1=N expansion [19][20][21][22], yielding indeed approximate numerical thermalization at strong coupling.…”

Goldstone excitations from spinodal instability

“…It was found that the squared mass of the "pion" modes starts to oscillate around the asymptotic zero value rather early, but the oscillations are damped only for asymptotically large times as t −1 . In a recent paper [8] renormalised non-equilibrium gap equations were solved for the effective mass-squares of the longitudinal and transversal modes propagating on a time dependent background. These masses were in earlier papers [9,10] with a self-consistent parametrisation of the corresponding propagators.…”