PHYSICAL REVIEW D, VOLUME 63, 045021 Nonequilibrium dynamics in quantum field theory at high density: The ‘‘tsunami’’ F. J. Cao and H. J. de Vega LPTHE, Universite´ Pierre et Marie Curie (Paris VI) et Denis Diderot (Paris VII), Tour 16, 1er. e´tage, 4, Place Jussieu, 75252 Paris, Cedex 05, France ~Received 29 November 1999; published 29 January 2001! The dynamics of a dense relativistic quantum fluid out of thermodynamic equilibrium is studied in the framework of theF4 scalar field theory in the largeN limit. The time evolution of a particle distribution in momentum space~the tsunami! is computed. The effective mass felt by the particles in such a high density medium equals the tree level mass plus the expectation value of the squared field. The case of negative tree level squared mass is particularly interesting. In such a case dynamical symmetry restoration as well as dynamical symmetry breaking can happen. Furthermore, the symmetry may stay broken with a vanishing asymptotic squared mass showing the presence of out of equilibrium Goldstone bosons. We study these phenomena and identify the set of initial conditions that lead to each case. We compute the equation of state which turns out to depend on the initial state. Although the system does not thermalize, the equation of state for asymptotically broken symmetry is of radiation type. We compute the correlation functions at equal times. The two point correlator for late times is the sum of different terms. One stems from the initial particle distribution. Another term accounts for the out of equilibrium Goldstone bosons created by spinodal unstabili- ties when the symmetry is asymptotically broken. Both terms are of the order of the inverse of the coupling for distances where causal signals can connect the two points. The contribution of the out of equilibrium Gold- stones exhibits scaling behavior in a generalized sense. DOI: 10.1103/PhysRevD.63.045021 PACS number~s!: 11.15.Pg, 11.30.Qc on e er o a ic sta ris th e iv de k t te ab n d gh c e - eld er - tive time ast li- is a- pa- al ical nc- - m ly, om- of be- nt i- n- I. INTRODUCTION Physical phenomena in high energy density situati cannot be treated with the usual perturbation methods. S consistent nonperturbative methods are necessary in ord describe the out-of-equilibrium dynamics of relaxation quantum fields in such situations. The largeN limit is a par- ticularly powerful tool for scalar models. The need for self-consistent method stems from the fact that the part propagation in such situations depends on the detailed of the system. Relevant physical situations at high energy densities a in the hadron~quark-gluon! plasma as in ultrarelativistic heavy ion collisions, in the interior of dense stars, and in early universe@1,2#. We consider a dense relativistic quantum gas out of th modynamic equilibrium. We investigate the nonperturbat dense regime in which the energy density is of the or mR 4/l wherel is the scalar self-coupling andmR the physical mass in vacuum. In these conditions, even for very wea coupled theories non-linear effects are important and mus treated nonperturbatively. We thus use the largeN limit that provides a nonperturbative scheme which respects the in nal as well as the space-time symmetries, is renormaliz and permits explicit calculations. More precisely, we co sider theO(N) vector model with quartic self-interaction an the scalar field in the vector representation ofO(N). We study the physics of the in-medium effects at hi energy density. In such out of equilibrium regime the effe tive mass felt by the particles is time-dependent and differ from the tree level massmR 2. In the large-N approximation, it takes the form M2~ t !5mR 21 l 2N ^FW 2&~ t !, ~1.1! 0556-2821/2001/63~4!/045021~19!/$15.00 63 0450 s lf- to f le te e e r- e r ly be r- le - - nt whereFW (x) stands for aN-component scalar field with self couplingl. We see that the quantum fluctuations of the scalar fi directly contribute to the effective mass. We shall consid for simplicity translationally invariant situations. That is, in variant under spatial translations. Therefore, the effec mass and all one-point expectation values depend on but not on space coordinates. Obviously, these are not realistic descriptions for the v and complex problems in ultrarelativistic heavy-ion col sion. However, the physics of such a simplified problem rich enough to get understanding of more involved situ tions. We work in the small coupling regimel!1. The reason being that the different dynamical time scales are well se rated in thel!1 regime. For larger couplings the dynamic time scales become of the same order and different phys phenomena get mixed up. We choose as initial conditions a Gaussian wave fu tional and^FW &(0)5^FẆ &(0)50. The calculation of correla- tion functions is done in the general case^FW &(0)Þ0. As initial distribution of particles we will choose a ‘‘tsu nami’’ @3,4#. That is, a distribution of particles in momentu space that we choose for simplicity spherical. Typical such distribution is a shell peaked in a wave numberk0 with a large density of particles;mR 3/l. We have seen that these conditions do not determine c pletely the initial state, and that the remaining degrees freedom can be interpreted as the quantum coherence tween differentk-modes. We choose two highly cohere initial states that we will call case I and case II. As we see in Eq.~1.1! the in-medium effects give a pos tive contribution to the effective mass. In general, this co ©2001 The American Physical Society21-1 e. hi th e y u- ol nd ac o- si th - e an r es ou x ta nd a or ua t o ss s- m- a- o es te e r s f an r. - F. J. CAO AND H. J. de VEGA PHYSICAL REVIEW D63 045021 tribution to the effective mass initially oscillates with tim These oscillations can produce parametric resonance w is shut off here after a few oscillations by the damping of oscillations due to decoherence. This resonant effect is n ligible for small l. ~Parametric unstabilities are shut off b the back reaction for no particle initial conditions@9–11#!. We present analytic solutions for initial particle distrib tions narrow in momentum space. Such self-consistent s tions express in close form in terms of elliptic functions, a reproduce the numerical solutions with very good accur till the damping of the oscillations becomes important~due to decoherence phenomena!. The narrower is the initial dis- tribution in momentum, the longer in time this effective s lution holds. We investigate in the present paper the case wheremR 2 .0 as well as the case wheremR 2,0. The more interesting case corresponds tomR 2,0. We have in such case spontaneously broken symmetry for low den and low energy states. For large initial energy density, positive definite term (l/2)^FW 2&(t) may overcome the nega tive squared massmR 2,0 in Eq.~1.1! and the symmetry may be restored:M2(t).0 ~see Table II!. This happens att50 for the case we call II provided th energy densityE satisfies E. umRu2 lR k0 2. For late times the symmetry is restored in both cases, I II, provided E. umRu4 lR 12 umRu2 lR k0 2. Moreover, we find in case II an interval of energies whe the symmetry is initially unbroken@M2(0).0# and where it is broken for asymptotic times: umRu2 lR k0 2,E, umRu4 lR 12 umRu2 lR k0 2. That is, we have a dynamical symmetry breaking for th situations~see Table II!. We compute the asymptotic equation of state for this of equilibrium system. That is, we derive for late times e plicit formulas for the pressure as a function of the to energy. The equation of state we obtain explicitly depe on the initial state,l andmR ~see Table III!. Notice that even for asymptotic times the system does not thermalize. In p ticular, the distribution functions reach nonthermal limits f t→`. When the symmetry is asymptotically broken, the eq tion of state takes the radiation form P5 E 3 in spite of the fact that the system is out of equilibrium Moreover, the Goldstone theorem is valid here in this ou 04502 ch e g- u- y ty e d e e t - l s r- - . f equilibrium situation. Namely, the effective squared ma M2(t) asymptotically vanishes when the symmetry is a ymptotically broken. The equation of state for asymptotically unbroken sy metry@Eqs.~7.7!# has the cold matter and the radiation equ tion of state as limiting cases. We explicitly compute the tw point correlation functions. For late times, the two point correlator at equal tim C(uxW u,t) expresses as a sum of two or three terms: C~ uxW u,t !5Corigin~ uxW u!1Cp~ uxW u,t !1Cs~ uxW u,t !. ~1.2! There is the time-independent pieceCorigin(uxW u) concentrated around the origin. The pulse term,Cp(uxW u,t), is due to the particles in the initial distribution that effectively propaga as free particles with massM` , Cp~ uxW u,t !5 1 luxW u F~ uxW u22vgt2c!, wherevg is the group velocity@vg,1 for unbroken symme- try and vg51 for dynamically broken symmetry wher M`50#. F(u) is nonzero only aroundu50. The last term in Eq.~1.2! is only present for dynamically broken symmetry, Cs~ uxW u,t !5 K luxW u QS uxW u 2t D , where K is a constant. The functionQ(u) is of the order O~1! only for 0,u,1 due to causality. When the orde parameter̂ FW &(t) is identically zero we haveQ(u)5u(1 2u). When the order parameter is nonzero,Q(u) oscillates with u. At a given timet, the number of oscillations ofQ(u) in the interval 0,u,1 equals the number of oscillation performed by the order parameter^FW &(t) from time t50 till time t. That is, scaling exists for̂FW &(0)Þ0 in a generalized sense since the functionQ(u) changes each timêFW &(t) performs an oscillation. This is due to the appearance o extra length scale, the initial value of the order paramete II. THE MODEL We consider theO(N)-invariant scalar field model with quartic self-interaction@3–5# with the scalar field in the vec tor representation ofO(N). The action and Lagrangian density are given by S5E d4x L, L5 1 2 @]mFW ~x!#22 1 2 m2FW 22 l 8N ~FW 2!2. ~2.1! The canonical momentum conjugate toFW (x) is PW ~x!5FẆ ~x!, ~2.2! 1-2 sla o to yp s e- ers a- e in the NONEQUILIBRIUM DYNAMICS IN QUANTUM FIELD . . . PHYSICAL REVIEW D 63 045021 and the Hamiltonian is given by H5E d3xH 1 2 PW 2~x!1 1 2 @¹FW ~x!#2 1 1 2 m2FW 21 l 8N ~FW 2!2J . ~2.3! In the present case we will restrict ourselves to a tran tionally invariant situation, i.e., eigenstates of the total m mentum operator. In this case the order parameter^FW (xW ,t)& is independent of the spatial coordinatesxW and only depends on time. The Heisenberg equations of motion for the field opera take the form F]21m21 l 2N FW 2~x!GFW ~x!50. ~2.4! The couplingl is chosen to remain fixed in the largeN limit. It is convenient to write the field in the Schro¨dinger picture as FW ~x!5„s~x!,hW ~x!…5„ANf~ t !1x~x!,hW ~x!…, ~2.5! with ^hW (xW ,t)&50 where hW represents theN21 ‘‘pions,’’ andf(t)5^s(x)&. Thus,^x(x)&50. A. The wave functional „Schrödinger picture… We shall consider Gaussian wave functionals of the t @3,5–7,9# C@FW ~ .,t !#5N 1/2~ t !PkW expF2 AkW~ t ! 2 hW kW•hW 2kWG , ~2.6! where hW kW~ t !5E d3xhW ~xW ,t !eikW•xW. ~2.7! @Hence, we can assumeA2kW(t)5AkW(t) without loss of gen- erality.# As shown below, such Gaussian wave functionals are lutions of the Schro¨dinger equation in theN5` limit. The Hamiltonian ~2.3! in the largeN limit is essentially a har- monic oscillator Hamiltonian with self-consistent, tim dependent frequencies: H~ t !5NVhcl~ t !2 l 8N S ( kW ^hW kW•hW 2kW& D 2 1( kW F2 1 2 d2 dhW kW•dhW 2kW 1 1 2 v kW 2 ~ t !hW kW•hW 2kWG , wherehcl(t) stands for the classical Hamiltonian 04502 - - r e o- hcl~ t !5 1 2 F ḟ2~ t !1m2f2~ t !1 l 4 f4~ t !G , v kW 2 ~ t !5k21m21 l 2 Ff2~ t !1 1 N ^hW 2~x!&G . ~2.8! The functional Schro¨dinger equation is then given by (\ 51) i ]C ]t 5HC. ~2.9! More explicitly, i ]C ]t 5H NVhcl~ t !2 l 8N S ( kW ^hW kW•hW 2kW& D 2 1( kW F2 1 2 d2 dhW kW•dhW 2kW 1 1 2 v kW 2 ~ t !hW kW•hW 2kWG J C@FW #, ~2.10! which then leads to a set of differential equations forAkW(t). The evolution equations forAkW(t) andN(t) are obtained by taking the functional derivatives and comparing pow of hkW on both sides. We obtain the following evolution equ tions: iȦkW~ t !5A kW 2 ~ t !2v kW 2 ~ t !, ~2.11! N~ t !5N~0!expH 2 i E 0 t dt8F2NVhcl~ t8! 2 l 4N S ( kW ^hW kW•hW 2kW&~ t8!D 2 1N( kW AkW~ t8!G J ~2.12! with AkW(t)5ARkW(t)1 iAIkW(t). The time dependence of the normalization factorN(t) is completely determined by that of theAkW(t) as a consequenc of unitary time evolution. Notice that the first two terms the right-hand side~RHS! of Eq. ~2.10! were missing in Ref. @3# as well as the corresponding first two terms inside phase in Eq.~2.12!. Using the expression for the wave functional~2.6! we have ^hW kW•hW 2kW&5 **PqWDhW qWe 2ARqW ~ t !hW qW•hW 2qW~hW kW•hW 2kW ! **PqWDhW qWe 2ARqW ~ t !hW qW•hW 2qW 5 N 2ARkW~ t ! , ~2.13! and ^hW kW•hW kW8&5^hW kW•hW 2kW&dkW ,2kW8 , ~2.14! 1-3 ol- o - th y ld re la : he nta- me n F. J. CAO AND H. J. de VEGA PHYSICAL REVIEW D63 045021 leading to the self-consistency condition in the infinite v ume limit, ^hW 2~x!& N 5 1 N E d3k ~2p!3 ^hW kW~ t !•hW 2kW~ t !&5E d3k ~2p!3 1 2ARkW~ t ! . ~2.15! The expectation value of the Heisenberg equation of m tion ~2.4! yields the equation of motion for the order param eterf(t) @3#: f̈~ t !1m2f~ t !1 l 2 Ff2~ t !1 ^hW 2~x!& N Gf~ t !50, ~2.16! where we used that̂hW &50 and^hW 2hW &50. Equations~2.11!–~2.12! together with Eq.~2.16! define the time evolution of this quantum state in the infiniteN limit. B. The field modes„Heisenberg picture… The Ricatti equation~2.11! can be linearized by writing AkW(t) in terms of the functionswkW(t) as AkW~ t !52 i ẇ kW * ~ t ! w kW * ~ t ! , ~2.17! leading to the mode equations@3# ẅ kW * 1v kW 2 ~ t !w kW * 50. ~2.18! The relation~2.17! defines the mode functionswkW up to an arbitrary multiplicative constant that we choose such that Wronskian takes the value WkW[wkWẇkW * 2ẇkWwkW * 52i . ~2.19! The functionswkW have a very simple interpretation: the obey the Heisenberg equations of motion for the pion fie obtained from the Hamiltonian~2.3!. Therefore, we can write the Heisenberg field operators as hW ~xW ,t !5E d3kW &~2p!3 @aW kWwkW~ t !eikW•xW1aW kW † w kW * ~ t !e2 ikW•xW#, ~2.20! whereaW kW ,aW kW † are the time independent annihilation and c ation operators with the usual canonical commutation re tions. Thus, thewkW(t) are the mode functions of the field. From Eq.~2.17! we obtain the following useful relations ARkW~ t !5 1 uwkW~ t !u2 , AIkW~ t !52 1 2 d dt lnuwkW~ t !u2. ~2.21! Then, using these relations and Eq.~2.15! we express the fluctuation in terms of the mode functions: 04502 - e s - - ^h2&~ t ![ ^hW 2~x!& N 5 1 2 E d3k ~2p!3 uwkW~ t !u2. ~2.22! C. Definition of the particle number Since in a time dependent situation the definition of t particle number is not unique, we choose todefinethe par- ticle number with respect to the eigenstates of the insta neous Hamiltonian~2.8! at theinitial time, i.e., n̂kW5 1 NvkW F2 1 2 d2 dhW kW•dhW 2kW 1 1 2 v kW 2 hW kW•hW 2kWG2 1 2 . ~2.23! Here,vkW is the frequency~2.8! evaluated att50. The expectation value of the number operator in the ti evolved state is then nkW~ t !5^Cun̂kWuC& 5 @ARkW~ t !2vkW# 21A IkW 2 ~ t ! 4vkWARkW~ t ! 5 D kW 2 ~ t !1d kW 2 ~ t ! 4@11DkW~ t !# , ~2.24! whereDkW(t) anddkW(t) are defined through the relations ARkW~ t !5vkW@11DkW~ t !#, AIkW~ t !5vkWdkW~ t !. ~2.25! In terms of the mode functionswkW(t) and ẇkW(t), the expec- tation value of the number operator is given by nkW~ t !5 1 4vkW @ uẇkW~ t !u21v kW 2uwkW~ t !u2#2 1 2 . ~2.26! For initially broken symmetry the frequencies in Eq.~2.23! are modified for lowk according to Eq.~3.17!. D. Initial conditions We will take initial conditions for the field expectatio value of the form f~0!5f0 , ḟ~0!50, ~2.27! since we can makeḟ(0) to vanish by a shift in time. The initial quantum state defined by Eq.~2.6! is deter- mined givingARkW(0) andAIkW(0) for all kW or equivalently, using the relations~2.25!, giving DkW[DkW(0) and dkW [dkW(0): ARkW~0!5vkW@11DkW#, AIkW~0!5vkWdkW . ~2.28! We can invert Eq.~2.24! expressingDkW(t) in terms ofdkW(t) and the particle numbernkW(t), DkW~ t !52FnkW~ t !6An kW 2 ~ t !1nkW~ t !2 d kW 2 ~ t ! 4 G . ~2.29! ~This formula was derived in Ref.@3# for the case where dkW(t)[0.) 1-4 a th - nd al - fie e s er he lu n l an i- fs. the ed. tion fini- itial NONEQUILIBRIUM DYNAMICS IN QUANTUM FIELD . . . PHYSICAL REVIEW D 63 045021 The 6 sign in Eq.~2.29! is not the sign ofmR 2. The 6 possibility is due to the fact that the values ofnkWÞ0 anddkW do not fix completely the initial state for the modes. We sh call case I to the upper sign in Eq.~2.29! and case II to the lower sign. Thus, we can express the initial conditions in terms of initial distribution of particlesnkW(0), thephase of the wave functionaldkW and the choice of sign in Eq.~2.29!. For sim- plicity, we will consider spherically symmetric particle dis tributions and with Gaussian form, nkW~0!5 N̂0 I expF2 ~k2k0!2 ŝ2 G , ~2.30! whereN̂0 is the total number of particles per unit volume a I is a normalization factor. We shall always considerN̂0 @m3. As we shall see below~Sec. IV A! it will be conve- nient to considerN̂0;m3/l. That is, our initial distribution of particles is a spheric shell in momentum space peaked around a momentumk0 . However, as seen in Ref.@3# and below features of the dy namics contain essential ingredients even for this simpli problem. @Nonspherically symmetrical distributions can b also treated in this scheme. Just the number of equation doubled or tripled and the momentum integrals in Eq.~2.22! become double or triple integrals to be computed num cally at each time step.# As the initial conditions are spherically symmetric and t evolution equations are invariant under rotations the so tions will be spherically symmetric. So, the dependence okW is only through the modulusk. The initial conditions on the mode functions follow from the relation~2.17! and the initial conditions~2.28! on Ak(t) plus the Wronskian constraint~2.19!. Thus, the initial condi- tions on the mode functions are wk~0!5 1 AVk , ẇk~0!52 iVk1vkdk ADk , ~2.31! with Vk defined by Vk[ARk~0!5vk@11Dk#. ~2.32! We consider initial conditions with dk~0!50. ~2.33! The effect of takingdkÞ0 is discussed below. We then see from Eq.~2.29! that for modes with large occupation number (nk@1), case I corresponds to initia conditions with uwk(0)u!1 and uẇk(0)u@1; while case II corresponds touwk(0)u@1 anduẇk(0)u!1. See for more de- tails Secs. IV A and IV B. That is, the physical meaning of the6 signs in Eq.~2.29! for nk@1 anddk50 is as follows. The upper sign~1! cor- responds to an initialk-mode with small initial amplitude and large initial velocity and vice versa for the lower~2! sign. If we neglect the interaction during early times we c write 04502 ll e d is i- - wk~ t !5 1 AVk cos~vkt !2 i AVk vk sin~vkt !, which describes an ellipse in the complexw plane~see Fig. 1!. The ratio of the half-axis of the ellipse goes as vk Vk 5 1 11Dk . This ratio is!1 ~@1! in case I~II ! when nk@1 as follows from Eq. ~2.29! and as can be seen in Fig. 1. Taking dkÞ0 keeps the elliptic trajectory with the add tional effect of rotating its axis clockwise for increasingdk . For an initial state with no particles (Dk50, dk50), the initial conditions become: wk(0)51/Avk and ẇk(0) 52 iAvk. These were the initial conditions used in Re @9–11#. III. EVOLUTION EQUATIONS The evolution equations for the expectation value of field f(t) and for the mode functionswk(t), Eq. ~2.16! and Eq. ~2.18!, respectively, are: d2f~ t ! dt2 1H mB 21 l 2 @f2~ t !1^h2&B~ t !#J f~ t !50, ~3.1! d2wk~ t ! dt2 1H k21mB 21 l 2 @f2~ t !1^h2&B~ t !#J wk~ t !50, ~3.2! with the self-consistent condition ^h2&B~ t !5 1 2 E d3k ~2p!3 uwk~ t !u2. ~3.3! The above evolution equations must be renormaliz This is achieved by demanding that the equations of mo be finite. The divergent pieces are absorbed into a rede tion of the mass and coupling constant: FIG. 1. The mode functions in the complexw plane for short times and initial conditionsnkW;1/g@1, dkW50. Case I is on the left and case II on the right. The dot and the arrow represents the in values ofwk and its derivative, respectively. 1-5 re e - th ec m th s as r r- ons les a r n le gle re F. J. CAO AND H. J. de VEGA PHYSICAL REVIEW D63 045021 Md 2~ t ![mB 21 l 2 @^h2&B~ t !1f2~ t !# 5mR 21 lR 2 @^h2&R~ t !1f2~ t !#. ~3.4! A detailed derivation of the renormalization prescriptions quires a WKB analysis of the mode functionswk(t) that reveals their ultraviolet properties. Such an analysis has b performed in Refs.@3,5,9#. In summary, quadratic and loga rithmic divergences are absorbed in the mass term while coupling constant absorbs a logarithmically divergent pi @5,8#. The renormalized quantum fluctuations take the for ^h2&R~ t !5 1 2 E d3k ~2p!3 F uwk~ t !u22 1 k 1 u~k2k! 2k3 Md 2~ t !G ~3.5! with k an arbitrary renormalization scale. We define now dimensionless quantities choosing physical mass in vacuumumRu as unit of mass: q[ k umRu ; t[umRut; z2~t![ lRf2~ t ! 2umRu2 ; wq~t![AumRuwk~ t !; g[ lR 8p2 ; vq~t![ vk~ t ! umRu ; Vq[ Vk umRu ; gS~t![ lR 2umRu2 ^h2&R~ t !. ~3.6! In terms of these dimensionless quantities the equation motion ~3.1!,~3.2! become F d2 dt2 611z2~t!1gS~t!Gz~t!50, ~3.7! F d2 dt2 1q2611z2~t!1gS~t!Gwq~t!50; ~3.8! gS~t!5gE 0 ` q2 dqH uwq~t!u22 1 q 1 u~q21! 2q3 Md 2~t! umRu2 J , ~3.9! where we have chosen the renormalization scalek5umRu for simplicity and the sign6 corresponds here to the sign ofmR 2. In Eqs.~3.7!,~3.8!, M2~t![ Md 2~t! umRu2 5a1z2~t!1gS~t! ~3.10! plays the role of a time dependent effective squared m a[sgn(mR 2)561. Depending on whetherM2(0) is positive or negative the symmetry will be initially unbroken or broken. In Ref. @3# a different definition for the mass unit and fo S~t! were used. The advantage to useumRu here stems from 04502 - en e e e of s. the fact that it is independent of the initial number of pa ticles. The present choice yields much simpler expressi for the final results. The initial conditions become in dimensionless variab for M2(0).0: z~0!5z0 , ż~0!50, ~3.11! wq~0!5 1 AVq , ẇq~0!52 iVq1vqdq AVq , ~3.12! with Vq[vq@11Dq#, vq5Aq21M2~0!, Dq52Fnq~0!6Anq 2~0!1nq~0!2 dq 2 4 G . ~3.13! From Eqs.~2.33! and ~2.30! we have dq50, nq~0!5 N0 I expF2 ~q2q0!2 s2 G , ~3.14! with N05N̂0 /umRu3 ands5ŝ/umRu. Therefore, gS~t!5gE 0 ` q2 dqH uwq~t!u22 1 q 1 u~q21! 2q3 M2~t!J . ~3.15! To perform the numerical evolution, we have introduced momentum cutoffL. The quantum fluctuations become fo finite cutoff, gS~t!5 1 12 g 2 logL H E 0 L q2 dquwq~t!u22 g 2 L2 1 g 2 logL@a1z2~t!#J 1OS g L2D . ~3.16! This is a positive quantity@up to O(g)#. a5sgn(mR 2)561. In the case where the symmetry is initially broke (M2(0),0) the only change in Eqs.~3.11!–~3.16! is that the initial frequenciesvq are modified for lowq as follows @10#: vq5HAq21uM2~0!u for q2,11uM2~0!u, Aq21M2~0! for q2.11uM2~0!u. ~3.17! IV. EARLY TIME DYNAMICS We study now the time evolution of a narrow partic distributionnq(0) peaked atq5q0 . We show below that we can approximate the dynamics for early times using a sin modeweff(t). We solve the time evolution ofweff(t) in close form in terms of elliptic functions. Moreover, we compa 1-6 s h so n I e ion r ge en e o- s NONEQUILIBRIUM DYNAMICS IN QUANTUM FIELD . . . PHYSICAL REVIEW D 63 045021 these analytic results with the full numerical solution of Eq ~3.7!–~3.9!. For mR 2,0, we only consider initial particle peaks wit q0 2.2. Thus, they are well outside possible spinodally re nant bands. For simplicity, we considerz050 in this section and in Secs. V, VI, and VII. A. Case I In this case, we haveDq52@nq(0)1Anq 2(0)1nq(0)#. We see from Eq.~3.13! that the modes withnq(0)!1 have Vq.vq . Therefore,wq(0) and ẇq(0) are of orderg0 @see Eq. ~3.12!#. Thus, their contribution togS(t) will be of orderg for early times. On the other hand, we have for modes withnq(0) 5O(1/g) @asg!1 this impliesnq(0)@1# Dq.4nq~0!@1, Vq.4vqnq~0!@1. ~4.1! Therefore, Eqs.~3.12! imply that these modes haveuwq(0)u !1. Thus, gS I~0!5O~g!, ~4.2! M2~0!5a1O~g!5611O~g!, ~4.3! where a5sgn(mR 2)561. For these modes withnq(0) 5O(1/g), uẇq(0)u@1. Therefore, these modes will the grow and their contribution will dominategS(t) for early times. We always consider thatnq(0)5O(1/g) for some inter- val in q. Hence, its contribution togS(t) will be of order one and will have an important effect on the dynamics. such conditions, the total number of particlesN0 per unit volume is also of the orderO(1/g). For the initial conditions considered, Eq.~3.14!, these dominant modes are in a small interval centered atq5q0 , and they are in phase at least for short times. More precis the q-modes will stay in phase for (q2q0)t!2p. The modes which are in phase contribute coherently togS(t) ~each one with a contribution proportional to its occupat number!. Hence, for smallt a good approximation is to conside that all the particles are in a single mode withq5q0 . This approximation will apply as long as the modes with lar occupation number stay in phase. In these conditions Eq.~3.16! yields gS~t!5gq0 2uweff~t!u2Dq1O~g!1O~gs!, ~4.4! whereDq's @see Eq.~3.14!#. Thus d2weff dt2 1vq0 2 weff~t!1gDqq0 2uweff~t!u2weff~t!50, ~4.5! with vq05Aq0 21a @recall a5sgn(mR 2)561 and we choose q0 2.2#. We also have 04502 . - n ly, N054pq0 2Dqneff ⇒ Veff5 N0vq0 pq0 2Dq . ~4.6! Solving the nonlinear differential equation~4.5! with the ini- tial conditions~3.12! and ~4.6!, we obtainweff(t). We find, from Eq. ~4.4!, gS I~t!5~q0 21a!S 11 1 122k2D 3F 1 12k2sn2SA q0 21a 122k2 t,kD 21G , ~4.7! with sn(z,k) the Jacobi sine function and k5A1 2 S 12 1 11 gS I max q0 21a D . ~4.8! The function ~4.7! is non-negative and oscillates betwe zero and gS I max5~q0 21a!FA11 2gN0 p~q0 21a!3/221G , ~4.9! with period T52A122k2 q0 21a K~k!, ~4.10! whereK(k) stands for the complete elliptic integral of th first kind. @Notice that Eq.~4.8! implies 122k2.0#. The elliptic solution~4.7! correctly predicts the amplitude of the first oscillation and the oscillation period~if the initial distribution of particles is not too wide aroundq5q0). See Fig. 2. The amplitude of the numerical solution is well repr duced by the elliptic solution~4.7! till damping becomes FIG. 2. Case I.mR 2.0. Unbroken symmetry;gN05250, q0 55, z050, andg51027. Comparison between numerical solution and the early time approximation~4.7!. 1-7 b on - e th i e th e im tri n t e th o c m d he ith e a F. J. CAO AND H. J. de VEGA PHYSICAL REVIEW D63 045021 important. The oscillation period keeps well reproduced Eq. ~4.7! for longer times than the amplitude.~See Fig. 2.! We see from the numerical solution of the exact equati ~3.7!–~3.9! that gS(t) exhibit significantly damped oscilla tions after a few periods. The integral overq for gS(t) @Eq. ~3.15!# gets damped with time due to the loss of coherence between the differ q modes in the distribution peak. We can apply here adiabatic approximation~see Appendix A!. In the late time limit, gS(t) therefore tends to the valuegS I max/2. In addi- tion, the narrower is the peak, the slower the oscillations gS(t) are damped. See Fig. 2. The time scale where the numerical and the early tim solution deviate in amplitude, essentially depends on width s of the initial distribution. The smaller iss, the latest the early time solution~4.7! holds. Notice that Eq.~4.7! givesgS(t) to zero order ins, whereas the damping of th oscillations is given by higher orders ins. In addition, we have seen from the numerical resolution that the early t evolution depends ong only throughgN0 as predicted by Eqs.~4.8!,~4.9!. When there is particle production through parame resonance, particles in the initial peak distribution are an hilated ~in order to conserve the total energy!. This reduces the contribution of the initial peak togS(t) whereas the oscillations due to the created particles give a contribution gS(t). These oscillations are due to the coherence betw the created particles at differentq. Such changes in the distribution of particles influence asymptotic value ofgS(t) for late times. Contrary to the vacuum initial conditions, parametric resonance shuts here by the damping of the oscillations and not due to ba reaction as in Ref.@9#. Therefore, for smallg the influence of parametric resonances is highly suppressed. When para ric resonance is appreciable, it makes the dynamics to pend ong and not only through the combinationgN0 . In case I, asM2(0)5a5sgn(mR 2)561. The symmetry is initially spontaneously broken or not depending on whet mR 2 ~the squared tree level mass! is negative or positive. B. Case II In this case,Dq52@nq(0)2Anq 2(0)1nq(0)#. As in case I, gS(t) is dominated for short times by the modes w large occupation numbers which are in phase at smallt due to the initial conditions~3.14!. Thus, we can do the sam approximation as in case I, considering that all particles in a single mode withq5q0 . We have for the modes withq>q0 , nq(0)5O(1/g)@1, 04502 y s nt e n s e e c i- o en e ff k- et- e- r re Dq.211 1 4nq~0! , Vq. vq 4nq~0! !1. ~4.11! Thus, fort50, gS II ~0!5 gN0 pvq0 1O~g!1O~gs!, ~4.12! M2~0!5a1gS II ~0!, ~4.13! wherea5sgn(mR 2)561. Equation ~4.12! is a third degree equation ingS II (0) sincevq05Aq0 21a1gS II (0). Theexplicit solution is given in Appendix D. We find, in the limiting cases, gS II ~0!gN0@~q0 2 = 1a!3/2S gN0 p D 2/3 2 1 3 ~q0 21a!1OS q0 21a gN0 D 2/3 , gS II ~0!gN0!~q0 2 = 1a!3/2 gN0 pAq0 21a 1OS gN0 q0 21a D 2 . ~4.14! For early times Eq.~3.16! becomes gS~t!5gq0 2Dquweff~t!u21O~g!1O~gs!. ~4.15! Therefore, d2weff dt2 1~q0 21a!weff~t!1gq0 21Dquweff~t!u2weff~t!50. ~4.16! Using the initial conditions given by Eq.~3.12! and deff(0) 50 we get from Eq.~4.11!, Veff5 pq0 2Dqvq0 N0 , ~4.17! where we have used thatneff5N0/4pq0 2Dq. Thus, the solution of Eq.~4.16! with the specified initial conditions can be written as gS II ~t!5gS I~t1T/2!5~q0 21a!S 11 1 122k2D F 1 12k2sn2SA q0 21a 122k2~t1T/2!,kD 21G ~4.18! 1-8 d - n nd on n f o - vior ass. and g ge he lue for As ou- e of to la- ith r of ce im NONEQUILIBRIUM DYNAMICS IN QUANTUM FIELD . . . PHYSICAL REVIEW D 63 045021 wheresn(z,k) is the Jacobi sine function,T is the real period T52A122k2 q0 21a K~k!, K(k) stands for the complete elliptic integral of the first kin and k5A1 2 S 12 1 11 gS II max q0 21a D . ~4.19! The expression forgS II max is gS II max5gS II ~0!5 gN0 pAq0 21a1gS II ~0! . ~4.20! Notice that the relationgS II (t)5gS I(t1T/2) is true for given values ofq0 , a, andk. Here gS(t) oscillates between zero andgS II max, while the effective squared massM2(t) oscillates between its ini tial valueM2(0)5611gS II max and its tree level value61 at the minima ofgS(t). We see that this approximatio gives us correctly the amplitude of the first oscillation a the oscillation period; but not the damping of the oscillati that is due to the dephasing of the initial particle distributio The broader is the initial particle distribution, the more e fective the dephasing works and the faster the damping curs. As in case I, the smaller iss ~width of the initial particle distribution!, the later the early time solution~4.18! holds ~see Figs. 2 and 3!. Recall that the damping of the oscilla tions is given by higher orders ins while Eq. ~4.18! gives gS(t) to zero order ins. These higher orders ins will also break the relation:gS II (t)5gS I(t1T/2); that we have found at zeroth order. FIG. 3. Case II.mR 2,0. gS(t) as a function oft. Dynamically broken symmetry;gN054.478, q051.3083,s50.05233, andz0 50. Comparison between numerical solutions and the early t approximation~4.18!. For late timesgS(t) tends to 1, thusM` 2 50 ~see Fig. 6!. 04502 . - c- In case II, M2(0)5a1gS II max. @Recall a5sgn(mR 2) 561.# As gS II max>0, for mR 2.0 the symmetry is initially unbroken. Instead, formR 2,0, the symmetry is initially un- broken forgS II max.1. ForgS II max,1 the symmetry is ini- tially spontaneously broken. V. INTERMEDIATE AND LATE TIME DYNAMICS We discuss here the intermediate and late time beha of the quantum fluctuations and the effective squared m We mean by late time, times later than the spinodal time than the damping time. We considerz050. We observe from the numerical results the followin common features in the late time behavior for a wide ran of initial conditions. The asymptotic constant values of t magnitudes~mass, pressure, number of particles! depend on g, for small g, only through the combinationgN0 ~except when parametric resonance is important!. Energy is con- served to one part in 107 confirming the precision of our numerical calculations. We have also seen that the mass tends to its limiting va oscillating with an amplitude that decays at least as;1/t. A similar asymptotic decay has been found in this model zero particle initial conditions andz(0)Þ0 @10#. In Sec. VII we derive the asymptotic equation of state. A. mR 2Ì0 „aÄ¿1… We have for cases I and II that M2~t!511gS~t!, ~5.1! althoughgS(t) has a different expression in each case. gS(t)>0 the symmetry is unbroken for allt. The adiabatic approximation holds~see Appendix A! since parametric resonance is negligible in the weak c pling regime considered here. Hence, the asymptotic valu gS(t) is gSmax/2 and the asymptotic squared mass goes ~see Fig. 2! M` 2 511 gSmax 2 .0. ~5.2! This result is in good agreement with the numerical calcu tions ~see Table I!. We find that the mass tends to this value oscillating w an amplitude that decays at least as;1/t. The initial peak of particles becomes lower and wider, and the total numbe particles slightly decreases compared to its initial value. B. mR 2Ë0„aÄÀ1… We have for both case I and case II that M2~t!5211gS~t!. ~5.3! Recall that in case I the symmetry is initially broken sin S I(0)50. In case II,S II (0)5S II max and the symmetry is initially broken ~unbroken! for gS II max,1 (gSII max.1). We have two different asymptotic regimes~see Table II!. e 1-9 n ou al e te . an s l re he s. par- so- ance late ime F. J. CAO AND H. J. de VEGA PHYSICAL REVIEW D63 045021 ~a! gSmax/2.1. Asymptotically unbroken symmetry. I this regime the results are similar to those in the previ subsection V A, with M` 2 5211 gSmax 2 .0. ~5.4! This is indeed the value we obtain numerically~see Fig. 4!. We find that for smallg there is no appreciable spinod resonance here. This is so becauseM2(t) oscillates around the positive valueM` 2 , although for some intervals of tim M2(t),0. In this regime the symmetry is restored for la times due to the presence of a high density of particles particular, we have already noticed that for case I,M2(0) 521. This does not change appreciably the dynamics the symmetry gets restored providedgSmax/2.1. ~b! gSmax/2,1. Asymptotically broken symmetry. In thi regime for intermediate times~times earlier than the spinoda time ts) M2(t) oscillates around the negative value, 2m25211 gSmax 2 ,0, ~5.5! giving rise to spinodal resonances.~See Figs. 3, 5, and 6.! The dynamics turns to be similar as for a constant squa mass2m2. That is, the spinodally resonant band is in t FIG. 4. Case II.mR 2,0. M2(t) as a function oft. The symme- try is unbroken.gN0567.96, q057.071, s50.4243, andz050. Thus, gS II max53.000 @see Eq.~4.20!# and M` 2 5211gS II max/2 50.5000@See Eq.~5.4!#. TABLE I. Initial and late effective mass and symmetry formR 2 .0. mR 2.0 Case I Case II M2(0) 11 111gS II max Initial symmetry unbroken unbroken M` 2 111 gSI max 2 111 gSII max 2 Late time symmetry unbroken unbroken 04502 s In d d q-interval from q50 to q5m and the spinodal timets is the same as for a constant squared mass of2m2 ~see Appendix C!. We can further distinguish~1! whengSmax,1, M2(t) is always negative for timest,ts , and ~2! when gSmax .1, M2(t) can be temporarily positive due to oscillation This happens in case I whereM2(0)521. In case II,M2(0)5211gSmax.0, but after a time of order T/2, M2(t),0, and M2(t) continues to oscillate around the negative value given by Eq.~5.5!. Thus, in case II we have dynamical symmetry breaking~see Table II!. At time ts the spinodal resonance has created enough ticles @of the orderO(1/g)# to give an important contribution to gS(t). This finally makesgS(t) oscillate around 1. Thus,M2(t) oscillates around zero and the spinodal re nance stops. The particles created by the spinodal reson are coherent. This gives new oscillations toM2(t). These oscillations get damped and the squared mass goes for times to its asymptotic value, FIG. 5. Case II.mR 2,0. gS(t) as a function oft. Dynamically broken symmetry;gN055.101, q051.5336, s50.2191, andz0 50. Comparison between numerical solutions and the early t approximation ~4.18!. For late times~not shown in the figure! gS(t) tends to 1. FIG. 6. Case II.mR 2,0. M2(t) as a function oft. The symme- try is dynamically broken.gN054.478,q051.3083,s50.05233, andz050. ~The same initial conditions andg as in Fig. 3.! 1-10 NONEQUILIBRIUM DYNAMICS IN QUANTUM FIELD . . . PHYSICAL REVIEW D 63 045021 TABLE II. Initial and late effective mass and symmetry formR 2,0. mR 2,0 Case I Case II M2(0) 21 211gS II max Initial symmetry broken if gS II max,1 broken if gS II max.1 unbroken if gS I max 2 ,1 if gS I max 2 .1 if gS II max 2 ,1 if gS II max 2 .1 M` 2 0 211 g(I max 2 0 211 g( II max 2 Late time symmetry broken unbroken broken unbroken Change of symmetry no change dynamical symmetry restoration no change dynamical symmetry breaking no change th t e d n rg si ta. s es he M` 2 50. ~5.6! The vanishing of the effective mass is accompanied by presence of Goldstone bosons out of equilibrium as in@3,9#. Particles created by spinodal resonances remain in q-interval from 0 tom. This creation of particles depletes th initial peak of particles keeping the total energy conserve For t,ts the dynamic depends ong, for small g, only through the combinationgN0 . However,ts depends explic- itly on g. VI. ENERGY After renormalization, taking dimensionless variables a introducing a momentum cut-off, the renormalized ene can be written as@9# Eren5 2umRu4 lR e, ~6.1! e5 1 2 ż2~t!1 a 2 z2~t!1 1 4 z4~t!1 1 4 12a 2 1 g 2 E0 L q2 dq@ uẇq~t!u21vq 2~t!uwq~t!u2# 2 1 4 @gS~t!#21O~g!, ~6.2! wherea5sgn(mR 2). One can easily check that the energy is conserved u the renormalized equations of motion~3.7!–~3.9!. Using the initial conditions~3.11!, ~3.12!, ~3.14!, andz0 50 we obtain for the energy att50 e5 1 4 12a 2 1 g 2 E0 L q2dqS vq1 vq 2 Vq D 2 1 4 @gS~0!#21O~g!. ~6.3! 04502 e he . d y ng This equation gives the energy in terms of the initial da e21/4(12a)/2 is always positive because we considerz0 50 andż050. A. Case I Let us consider an initial narrow distribution of particle which is peaked atq5q0 . Thus, we can use Eqs.~4.1! and ~6.3! and consider all particles in a single mode withq5q0 as in previous sections.~Recall that in this casevq0 2 5q0 2 1a.) The energy is then given by e5 gN0 2p ~q0 21a!1 1 4 12a 2 1O~g!1O~gs!. ~6.4! B. Case II Under the same approximations@now using Eq.~4.11!#, the energy is given by e5~q0 21a! gS II ~0! 2 1FgS II ~0! 2 G2 1 1 4 12a 2 1O~g!1O~gs!, ~6.5! wherevq0 2 5q0 21a1gS(0) andgS II (0) is given by the re- lation ~4.12!, i.e., gS II (0)5gN0 /pAq0 21a1gS II (0). C. gSmax in terms of the energy The energy andgSmax have different expressions in cas I and II @see Eq.~4.9! vs Eq.~4.20!#. However, it is important to notice thatgSmax has the same expression in terms of t energy both for cases I and II, gSmax5A~q0 21a!214S e2 1 4 12a 2 D2~q0 21a!. ~6.6! We have verified that Eqs.~6.4!–~6.6! are valid numerically for initial particle distributions with widths,1. 1-11 s iti ng e er d im at xi- en- II. en - e bro- ect the f in e I se- e F. J. CAO AND H. J. de VEGA PHYSICAL REVIEW D63 045021 VII. EQUATION OF STATE We derive here the equation of state~i.e., the pressure a a function of the energy! for asymptotic times. As shown below, the asymptotic equation of state depends on the in state. We considerz050 anddq50. Notice thatz050 and ż050 impliesz(t)50 for all t. A. Sum rule A further expression for the energy follows by evaluati the right-hand side of Eq.~6.1! in thet→` limit. Using Eqs. ~B10! and ~B13! yields e5 1 4 12a 2 1E 0 L q4 dq Mq 2~`!1M` 2 gS` 2 1 4 ~gS`!21O~g!. ~7.1! @The cosine terms in Eqs.~B10! and ~B13! for late time are fastly oscillant inq and thus they do not contribute to th q-integral in the infinite time limit.# Equation ~7.1! allows us to express the integral ov Mq 2(`) in terms of known quantities: the initial data an M` 2 , E 0 L q4 dq Mq 2~`!5e2 1 4 12a 2 2M` 2 gS` 1 1 4 ~gS`!21O~g!. ~7.2! This sum rule holds forz050 anddq50. B. Pressure and equation of state The renormalized pressure can be written as Pren~t!5 2umRu4 lR p~t!, ~7.3! p~t!52e1 ż21gE 0 L q2 dqF uẇq~t!u21 q2 3 uwq~t!u2G 1O~g!. ~7.4! We analogously evaluate the pressure in the infinite t limit with the result p`52e1 4 3 E0 L q4 dq Mq 2~`!1M` 2 gS` . ~7.5! Using now the sum rule~7.2! andM` 2 5a1gS` yields p`5 1 3 e2 a 3 gS`2 1 3 12a 2 . ~7.6! @Recalla5sgn(mR 2).# 04502 al e We consider narrow distributions of particles centered q5q0 . To obtain the equation of state, we make the appro mation of considering all particles in a single mode withq 5q0 , and we express the pressure as a function of the ergy. The equation of state is the same for case I and case We have to distinguish between unbroken and brok symmetry fort5`. ~1! M` 2 .0 In this regimegS`5gSmax/2 with gSmax given by Eq. ~6.6!. Thus, the equation of state~7.6! becomes p`5 1 3 e2 a 6 FA~q0 21a!214S e2 1 4 12a 2 D2~q0 21a!G 2 1 3 12a 2 . ~7.7! This equation givesp` as a function ofe for an initial momentum distribution centered atq0 . Therefore, the equa tion of state explicitly depends on the initial conditions. Let us consider some limiting cases. In the limit, e21/4 (12a)/2!(q0 21a)2, we have the equation of state p`5 1 3 S e2 1 4 12a 2 D S 12 a q0 21a D 2 1 4 12a 2 . ~7.8! For a511 (mR 2.0) this equation reduces to p`5 e 3 S 12 1 q0 211D , ~7.9! and interpolates between a cold matter~for q0!1), and a radiation~for q0@1) equation of state. In the opposite limite21/4 (12a)/2@(q0 21a)2.1 we have a radiation type equation of statep`51/3e. ~2! M` 2 50 This can only happen ifa521 (mR 2,0). M` 2 521 1gS`50 implies gS`51 and we have a radiation typ equation of state, p`5 e 3 . ~7.10! The pressure is continuous at the boundary between ken and unbroken symmetry, but its derivative with resp to the energy have a discontinuity 1/3 1/(q0 211). @See Eq. ~7.10! vs Eq.~7.7!.# We see that the equation of state is determined by energy, the momentumq0 around which the initial peak o particles is centered and the sign of the physical mass vacuummR 2. But it does not depend whether we are in cas or II. ~See Table III.! VIII. THE INFINITE TIME LIMIT The spectral distribution does not take here the Bo Einstein form in the infinite time limit. This was already th 1-12 NONEQUILIBRIUM DYNAMICS IN QUANTUM FIELD . . . PHYSICAL REVIEW D 63 045021 TABLE III. Asymptotic equation of state in the different situations. M` 2 50 M` 2 .0 mR 2,0 mr 2.0 Equation of state p`5 e 3 p`5 e 3 1 q0 221 6 FA11 4S e2 1 4D ~q0 221!221G2 1 3 p`5 e 3 1 q0 211 6 FA11 4e ~q0 211!221G is on b en n u on he tio el es he at ly - a s ex - we i- se d f 7. case in Refs.@3,9–11# in the infiniteN limit. It is not obvious that subdominant 1/N corrections can render the spectral d tribution of the Bose-Einstein type since the distributi functions in theN5` limit differ from Bose-Einstein distri- butions on an amount of order one. On the other hand, the quantum modes do interact tween themselves for all finite times in the high energy d sity situations considered here and in Refs.@3#, @9–11#. We are in a different situation than for quantum modes arou the vacuum where the modes are free in theN5` limit. That is, in the vacuum case equilibration cannot be achieved less 1/N contributions are included. One may think that the lack of asymptotic thermalizati is linked to the translationally invariant character of t states considered since an infinite number of conserva laws holds in such situation@12#. Then, the equilibration time scale can be infinity in theN5` limit. That is, it can grow with N, and being infinity forN5`, it cannot be re- covered by 1/N corrections. However, classical calculations in a two-fields scalar fi theory averaging over space-dependent plane-wave-type lutions yielded also nonthermal distributions for late tim @13#. These distributions ressemble qualitatively to t asymptotic distributions in the largeN limit @9–11#. In the other hand, quantum calculations with inhomogeneous st for fermions in 111 dimensions yielded approximate Fermi-Dirac distributions@15#. However, as noticed in a dif ferent context, Boltzmann-like distributions do not necess ily imply thermal equilibrium@16#. Last but not least, a model may reach for late time stationary situation which is nonthermal but turbulent as plicitly shown in Ref.@14#. The issue of late time thermali zation or nonthermalization is clearly very important and hope to deal with these deep problems in the future. IX. CORRELATION FUNCTIONS AND BOSE CONDENSATE The equal time correlation function is given for an arb trary timet by ^ha~rW,t!hb~0W ,t!&2^ha~rW,t!&^hb~0W ,t!& 5da,bC~rW,t! 5da,bE d3q 2~2p!3 uwq~t!u2eiqW •rW. ~9.1! 04502 - e- - d n- n d so- es r- a - The initial conditions are specified in Eqs.~3.11!–~3.14!; and as they are rotationally invariant, C~r ,t!5 1 4pr E0 ` q dquwq~t!u2 sin~qr !. ~9.2! In this section we considerz050 andz0Þ0 with z0!1. A. Early time We have the same results forz050 and forz0!1. 1. Case I At t50 the correlation function is of order one, becau uwq(0)u2&1. @uwq(0)u2;1 for the nonoccupied modes an uwq(0)u2!1 for the highly occupied modes, see Eq.~4.1!.# For the highly occupied modes we haveuẇq(0)u25O(1/g). Thus, for early timest5O(1) these modes will have uwq(t)u25O(1/g). This makes the correlation function be o order 1/g near the origin for early times, as we see in Fig. 2. Case II In this case att50, the modes haveuwq(0)u25O(1/g). This makes the correlation function be of order 1/g near the origin for t50, as we see in Figs. 10 and 12. FIG. 7. Case I.mR 2.0. Unbroken symmetry.grC(r ,t52) for g51027 and initial conditions:z050, gN05250, q054.0, ands 50.3. 1-13 d w uls ta m . n, ri- so de- ere s re the oef- of ibu- e nc- ty . y ti- so- F. J. CAO AND H. J. de VEGA PHYSICAL REVIEW D63 045021 B. Late time The late time behavior of the correlation function depen whether the symmetry is dynamically broken or not. Ho ever, its behavior is the same for both cases, I and II. We have to distinguish the two regimes. 1.M` 2 Ì0 In this regime we have the same results forz050 and for z0!1. We observe that at intermediate times a spherical p develops. This spherical pulse propagates with a cons radial speed given by the radial group velocity forq5q0 and an amplitude that decreases as 1/r . The radial width of the pulse,L, remains approximately constant~see Figs. 8 and 9!. The group velocity is asymptotically given by vg5 dvq` dq U q5q0 5 d dq Aq21M` 2U q5q0 5 q0 Aq0 21M` 2 . ~9.3! We recall that hereM` 2 5a1gSmax/2.1 @see Eqs.~5.2! and ~5.4!#. @a5sgn(mR 2)561.# FIG. 8. Case I.mR 2.0. Unbroken symmetry.grC(r ,t510) with the sameg and initial conditions as in Fig. 7. FIG. 9. Case I.mR 2.0. Unbroken symmetry.grC(r ,t550) with the sameg and initial conditions as in Fig. 7. 04502 s - e nt Asymptotically, the correlation function becomes the su of two terms C~r ,t!5Corigin~r !1Cp~r ,t!, ~9.4! whereCorigin(r ) is the correlation function near the origin This term is asymptotically time-independent. The pulse contribution to the correlation functio Cp(r ,t), has approximately the asymptotic form Cp~r ,t!5 1 gr P~r 22vgt2c!. ~9.5! Here c is a constant of order one, andP(u) is of the order O(1) only for 2L/2,u,L/2 whereL is the width of the pulse~i.e., the pulse is localized aroundr .2vgt1c). The pulse term is due to the particles in the initial dist bution that effectively propagate as free particles. This is since the effective mass in the mode equations~3.2! becomes asymptotically constant and hence the modes effectively couple from each other. The pulse term is absent when th are no particles in the initial state@11#. In summary, the correlator is of orderO(1/g)@1 for 2vgt1c2L/2,r ,2vgt1c1L/2 whereas causality make it to fall to O(1) values forr .2vgt1c1L/2. 2.M` 2 Ä0 In this regime the symmetry is broken and there we spinodal resonances for earlier times. Since the effective mass vanishes asymptotically mode withq50 behaves as w0~t!5t→`L1Kt. ~9.6! The Wronskian guarantees that neither of the complex c ficients L,K can vanish@11#. This linear growth with time can be interpreted as an out of equilibrium novel form Bose Einstein condensation. We analyze below the contr tion of this condensateCs(r ,t) to the correlation function. We have studied bothz050 andz0Þ0 ~with z0!1). ~a! z050. For late time (t@ts), the particles created du to the spinodal resonance contribute to the correlation fu tion with a termCs(r ,t) of order 1/g whenr is in the inter- val ~0,2t!. This term decays as 1/r . There is in addition a pulse term,Cp(r ,t), in the cor- relator that moves away from the origin with unit veloci @remember Eq.~9.3! and thatM` 2 50#. See Figs. 10 and 11 Thus, the correlation function is asymptotically given b C~r ,t!5Corigin~r !1Cs~r ,t!1Cp~r ,t!. ~9.7! Here,Corigin(r ), the correlation near the origin is asympto cally time-independent. The contribution of the pulse,Cp(r ,t), has the form Cp~r ,t!5 1 gr P~r 22t2c!. ~9.8! The contribution from the particles created by spinodal re nance,Cs(r ,t), has the form 1-14 les . n- d of tted ally on ight tion n the e ld- ss- tion - . . . NONEQUILIBRIUM DYNAMICS IN QUANTUM FIELD . . . PHYSICAL REVIEW D 63 045021 Cs~r ,t!5 K gr QS r 2t D , ~9.9! whereK is a constant andQ(u)5u(12u). Introducing the dynamical correlation lengthj(t);2t and defining the variableu5r /j(t), Cs(r ,t) can be written in the form Cs~r ,t!5 K guj~t! u~12u!. ~9.10! Using the customary notation for the scaling regime Cs~r ,t!5 1 @j~t!#2~12z! I ~u!, ~9.11! with the anomalous dynamical exponentz51/2 ~the naive scaling length dimension of the field is 1!. In this case the scaling function is given by FIG. 10. Case II. mR 2,0. Dynamically broken symmetry grC(r ,t50) for g51027 and initial conditions: z050, gN0 54.478, q051.3083, and s50.07850. For r .5, we have grC(r ,t50).0. FIG. 11. Case II. mR 2,0. Dynamically broken symmetry grC(r ,t) for t57.850 andt526.17 with the sameg and initial conditions as in Fig. 10. We see that for late timesgrCs(r ,t) .Ku(12r /2t) @see Eq.~9.9!#. 04502 I ~u!5 K gu u~12u!. ~9.12! An analogous spinodal termCs(r ,t) in the correlator have been obtained for initially broken symmetry and no partic in the initial state in@11#. The spinodal termCs(r ,t) can be interpreted as follows For timest later thants there is a zero momentum conde sate formed by Goldstone bosons travelling at the spee light and back-to-back. That is, massless particles emi from the points ~0,t! and (r ,t) form propagating fronts which at timet are at a distancet2ts from the origin and from r, respectively. These space-time points are caus connected for 2(t2ts)>r . Otherwise,Cs(r ,t) is not of or- der 1/g but of order one. An alternative interpretation of the causality step functi goes as follows. Signals are emitted at the speed of l from all points in the condensate. We have causal connec between the points 0 andr once signals starting from a give point arrive to both points. The earlier this happens is for signals emitted from the half-away point atr /2. These sig- nals need a timet5r /2 to reach both points. Hence, th correlator is of the order 1/g for 2t.r . Moreover, the analytical derivation of Eq.~9.12! from the low-q behavior of the mode functions given in Ref.@11# also applies here. Notice that the scaling contribution of the Go stone bosons to the correlator is different from a free ma less propagator. We have here a 1/r falloff whereas a free massless scalar field has a 1/r 2 falloff. ~b! z0Þ0 ~with z0!1). For late time (t@ts), the par- ticles created by spinodal resonance give a contribu Cs(r ,t) of order 1/g to the correlation function in the inter val 0,r ,2t. See Figs. 12–14. The correlation function is asymptotically given by C~r ,t!5Corigin~r !1Cs~r ,t!1Cp~r ,t!, ~9.13! where the time-independent pieceCorigin(r ) is the correlation near the origin.Cp(r ,t) is the contribution of the pulse, FIG. 12. Case II. mR 2,0. Dynamically broken symmetry grC(r ,t50) for g51027 and initial conditions: z050.2617 31025, gN054.478,q051.3083, ands50.2617. Forr .1.5, we havegrC(r ,t50).0. 1-15 by de ion he te of ll- rk a he um- e- el se ns ant. of ing in try gh tum re icle ic y a e of on of tion ld- for ze old- ese nt the is n- .C. io . F. J. CAO AND H. J. de VEGA PHYSICAL REVIEW D63 045021 Cp~r ,t!5 1 gr P~r 22t2c!, ~9.14! andCs(r ,t) is the contribution from the particles created spinodal resonance, Cs~r ,t!5 1 @j~t!#2~12z! I r~u!, ~9.15! with j(t);2t the correlation length,u5r /j(t), and z 51/2. I t(u) is no longer given by Eq.~9.12! when z0Þ0. Qt(u)5uIt(u) now oscillates withu. At a given timet, the number of oscillations ofQt(u) in the interval 0,u,1 equals the number of oscillations performed by the or parameterz~t! from timet50 till time t. That is, the scaling exists for z0Þ0 in a generalized sense since the funct FIG. 13. Case II. mR 2,0. Dynamically broken symmetry grC(r ,t) for t5104.66 andt5261.7 with the sameg and initial conditions as in Fig. 12. For timest such thatz(t).0 we see that grCs(r ,t).KQt(r /2t) @see Eq.~9.15!#. FIG. 14. Case II.mR 2,0. Dynamically broken symmetry.z(t) with the sameg and initial conditions as in Fig. 12. Fort P@100,262#, z(t).0. 04502 r Qt(u) changes each timez~t! performs an oscillation. This is due to the appearance of the extra length scalez0 . See Figs. 12–14. As for unbroken symmetry, the pulse term is due to t particles in the initial distribution that effectively propaga as free particles. Both forz050 andz0Þ0 causality makes the correlator of orderO(1) for r .2t1c. For r ,2t1c the correlations are of orderO(1/g)@1. X. SUMMARY AND OVERVIEW We have studied the time evolution of strongly out equilibrium states in QFT of the tsunami type. That is, she type distributions of particles in momentum space. We wo in the (FW 2)2 theory in the largeN limit. We thoroughly explore the dynamics for all times as function of the initial state and of the tree level mass in t vacuum. We concentrate on the dependence on initial n ber of particles for a thin shell momentum distribution. B sides solving numerically the time evolution we found nov analytic solutions for early times in the thin shell limit. The solutions express in closed form in terms of elliptic functio and they are valid till decoherence effects become import We show in this way the presence of a whole variety possible evolution patterns including symmetry break and/or unbreaking or restoring of symmetry as depicted Tables I and II. The driving mechanism in the symme changes during time evolution is particle production throu spinodal unstabilities and/or decoherence of the quan modes. The rich manifold of physical evolutions uncovered he provide novel and nonperturbative mechanisms for part production and relaxation that are out of thermodynam equilibrium and cannot be described in the early stages b hydrodynamic evolution. These new mechanisms could b potential interest in the physics of ultrarelativistic heavy i collisions. Especially interesting in this context is the emergence generalized scaling behaviors in the late times correla functions as a landmark for the abundant production of Go stone bosons out of equilibrium~due to the breaking of the symmetry and the vanishing of the effective mass asymptotic times!. These new scaling behaviors generali the standard scaling found in previous works@11# and may constitute physical signals for the presence of massless G stone bosons produced through spinodal unstabilities in th out of equilibrium situations. Although our investigations with translationally invaria states clearly show a rich variety of physical phenomena, necessity of more realistic states to describe collisions fairly obvious. We defer for future research the very inte sive numerical work needed to deal with such problems. ACKNOWLEDGMENTS We thank D. Boyanovsky for useful discussions. F.J thanks the Ministerio de Educacio´n y Cultura ~Spain! for financial support through the program Becas de Formac´n de Profesorado Universitario en el Extranjero. 1-16 e . d u e o q rs on- as rs es q. fol- NONEQUILIBRIUM DYNAMICS IN QUANTUM FIELD . . . PHYSICAL REVIEW D 63 045021 APPENDIX A: ADIABATIC APPROXIMATION FOR THE MODES In this Appendix we use the adiabatic form of the mod to evaluate the quantum fluctuationsS~t!. The modeswq(t) can be represented as@7# wq~t!5 1 APq~t! @aqe2 i *0 tdx Pq~x!1bqei *0 tdx Pq~x!#, ~A1! whereaq andbq are constants andPq(t) depends on time Inserting Eq.~A1! into Eq.~3.8! yields the following nonlin- ear differential equation forPq(t): P̈q~t! 2Pq~t! 2 3 4 S Ṗq~t! Pq~t! D 2 1Pq 2~t!5q21M2~t!. ~A2! The initial conditions~3.12! combined with the Wronskian conservation yields uaqu22ubqu251. ~A3! As long asPq(t) is a real function, we have for the square modulus uwq~t!u25 1 Pq~t! F uaqu21ubqu212uaqbqu 3cosS 2E 0 t dxPq~x!1aqD G , ~A4! whereaq is a time independent phase. Whenq belongs to a parametric resonant bandPq(t) gets an imaginary part. We consider here the case where s unstabilities have a negligible effect. We are interested in the modes with large amplitud uaqu@1,ubqu@1. That is, those in an interval of widths around the peak momentumq0 . For such modes, thanks t Eq. ~A3! we can approximateuaqu.ubqu. Inserting Eq.~A4! into the integral~3.15! for S~t!, and approximating the slowly varying factor 1/Pq(t) by its av- erage on a period~that we will denote 1/P̂q) yields S~t!.2E q2 dq P̂q uaqu2F11cosS 2E 0 t dxPq~x!1aqD G . ~A5! It follows from here the bounds 0