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February1,2008 20:16 WSPC/TrimSize: 9inx6inforProceedings sewm2002 3 0 0 NONEQUILIBRIUM DYNAMICS OF THE O(N) LINEAR 2 SIGMA MODEL IN THE HARTREE APPROXIMATION n a J 7 S.MICHALSKI 1 Institut fu¨r Physik Theoretische Physik III 1 Otto-Hahn-Str. 4 v 4 D–44221 Dortmund, Germany 3 E-mail: [email protected] 1 1 Weinvestigatetheout-of-equilibriumevolutionofaclassicalbackgroundfieldand 0 its quantum fluctuations in the scalar O(N) model with spontaneous symmetry 3 breaking 1. We consider the 2-loop 2PI effective action in the Hartree approx- 0 imation, i.e. including bubble resummation but without non-local contributions / h tothe Dyson-Schwinger equation. Weconcentrate onthe(nonequilibrium)phase p structureofthemodelandobserveafirst-ordertransitionbetweenaspontaneously - broken and a symmetric phase at low and high energy densities, respectively. So p typical structures expected inthermal equilibriumareencountered innonequilib- e riumdynamicsevenatearlytimesbeforethermalization. h : v i X 1. The model r 1.1. Applications a Scalar models have a wide range of applications in quantum field theory. Normally they are parts of more complex models like e.g. the Standard Model or Grand Unified Theories but they often serve as toy models for a simplifieddescriptionofcomplexphenomenasuchasinflationarycosmology or meson interactions in relativistic heavy ion collisions. 1.2. Nonequilibrium 2PI effective potential We consider the O(N) model with spontaneous symmetry breaking whose classical action is 1 λ 2 [Φ~]= d4x L[Φ~]= d4x ∂ Φ~ ∂µΦ~ Φ~2 v2 . (1) µ S Z Z (cid:26)2 · − 4 (cid:16) − (cid:17) (cid:27) Following Refs. 1,2 we can compute the 2PI effective action 3 in the Hartree approximation. Furthermore, we diagonalize the Green function 1 February1,2008 20:16 WSPC/TrimSize: 9inx6inforProceedings sewm2002 2 by an O(N)-symmetric ansatz and by restricting the background field to one direction. Since the Greenfunction is local, it canbe describedby two (time-dependent) mass parameters 2 . For nonequilibrium purposes it Mσ,π can be factorized into mode functions G (x,t ;x′,t )= d3k f (k,t )f¯ (k,t ) eik·(x−x′) , (2) σ,π > < Z (2π)32ωσ,π σ,π > σ,π < where ω = k2+ 2 (0). One constructs an expression for the total σ,π Mσ,π q (conserved) energy density of the system in the Hartree approximation 1 1 λ v2 = φ˙2+ 2φ2 φ4 2 +(N 1) 2 E 2 2Mσ − 2 − 2(N +2)(cid:20)Mσ − Mπ(cid:21) 1 (N +1) 4 +3(N 1) 4 2(N 1) 2 2 (3) −8λ(N +2)(cid:20) Mσ − Mπ− − Mσ Mπ +2Nλ2v4 + σ(t)+(N 1) π(t) , (cid:21) Efl − Efl where the fluctuation energy densities σ,π are the nonequilibrium analogs Efl of the one-loop “log det” terms expressed by mode functions f(k,t) ~ d3k Efl∗(t)= 2 Z (2π)32ω∗ (cid:20)|f˙∗(k,t)|2+(k2+M2∗) |f∗(k,t)|2(cid:21), ∗=σ,π .(4) 1.3. Equations of motion The equations of motion follow from the conservation of the energy (3). The backgroundfield obeys φ¨+ 2(t) 2λ φ2(t) φ(t)=0 , (5) Mσ − (cid:2) (cid:3) the mass parameters are solutions of the gap equations 2 = λ 3φ2 v2+3~ +(N 1)~ (6) Mσ (cid:16) 0− Fσ − Fπ(cid:17) 2 = λ φ2 v2+~ +(N +1)~ , (7) Mπ (cid:16) 0− Fσ Fπ(cid:17) where ∗ is the fluctuation integral F d3k ∗(t)= f∗(k,t)2 with =σ,π F Z (2π)32ω∗| | ∗ which equals the usual tadpole integral at t = 0 (cf. section 1.4). The equation for the mode functions is f¨∗(k,t)+ k2+ 2∗(t) f∗(k,t)=0 . (8) (cid:20) M (cid:21) February1,2008 20:16 WSPC/TrimSize: 9inx6inforProceedings sewm2002 3 Thefactthatthemodeequationiscoupledtothegapequations(6)and(7) hasanimportantinfluenceonthedynamics. Whenatime-dependentmass square 2(t) acquires a negative value, eq. (8) will imply an exponential M growth of the modes which reacts back on the mass squares via the gap equations by drivingthem back to positive values. Inthe one-loopapprox- imation (with no gap equations) the system shows unphysical behavior 5 because the modes never stop growing exponentially. 1.4. Initial conditions At the beginning of the nonequilibrium evolution we fix the classicalback- ground field to a certain value φ(0)=φ . The mode functions are those of 0 free fields: f (k,0)=1, f˙(k,0)= iω , and the mass parameters and i i i σ − M are solutions of the gap equations (6) and (7) at t=0. π M 2. Results and discussion 2.1. Time evolutions HerewewillpresenttheresultsforN =4,λ=1andfortherenormalization scale set equal to the tree level sigma mass µ2 = 2λv2. We have only consideredinitialvaluesφ >v forthebackgroundfieldbecauseforsmaller 0 values the initial value of the parameter (0) is imaginary. This means π M that the region v <φ<v can only be explored dynamically. We display − time evolutions of the background field for two different initial conditions in Fig. 1 φ(t) φ(t) 1.2 1 1 0 0.8 −1 t t 0.6 0 50 100 0 50 100 150 200 Figure1. Timeevolutionsofthebackgroundfieldforφ0=1.3v,φ0=1.6v. Amplitudes areinunitsofv andtimeismeasuredinunitsof(√λv)−1. It can be seen that there are two phases depending on the value of February1,2008 20:16 WSPC/TrimSize: 9inx6inforProceedings sewm2002 4 φ : a symmetric phase when the field φ(t) is oscillating about zero, and 0 a phase of broken symmetry when φ(t) is oscillating about a finite value. The“critical”valueofφ seemstobeclosetotheclassicallyexpectedvalue 0 √2λv where the total energy is equal to the height of the barrier. 2.2. Phase structure InordertoanalyzethephasestructureofthemodelintheHartreeapprox- imation we define φ∞ as a time averaged amplitude at late times, i.e. the valueaboutwhichthefieldoscillates. φ∞ playstheroleofanorderparam- eter whose dependence on the total energy of the system is investigated. The total energy of the system is given here by the initial value φ which 0 is analogous to the temperature in equilibrium field theory. φ∞ 1 0,5 φ0 0 1 1,2 1,4 1,6 1,8 Figure2. Thelate-timeamplitudeasfunctionoftheinitialamplitudeforN =4. Fig. 2 clearly shows a discontinuous jump of φ∞ at φ0 √2v — a ≃ typical sign of a first order phase transition as found in equilibrium (see e.g. Refs. 4,2). 2.3. A dynamical effective potential This nonequilibrium system shows a phase structure which is comparable toasysteminthermalequilibrium,soitwouldbeniceiftherewasanother correspondence. We define a dynamical, i.e. time-dependent, potential which can be compared to the finite temperature potential in equilibrium 1 V (t)= φ˙2(t) . (9) pot E − 2 This potential can only be measured within the oscillation range of the background field φ(t). For two different initial conditions it is shown in February1,2008 20:16 WSPC/TrimSize: 9inx6inforProceedings sewm2002 5 Fig. 3. In the broken symmetry phase the minimum at φ = v moves 8 −0.8 Vpot7 Vpot 6 −0.85 5 4 −0.9 3 2 −0.95 1 0 φ −1 φ −1 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 Figure 3. Evolution of the potential energy (9) for φ0 = 2v (unbroken phase) and φ0=1.2v (brokenphase). to smaller values but eventually remains different from zero, so the system settlesatafiniteexpectationvalue. Inthesymmetricphasethetwominima entirelydisappearafter afewoscillationsandanew (symmetric)minimum at φ=0 appears. 3. Conclusions and summary The analysis of the nonequilibrium dynamics of the O(N) model in the Hartree approximation allowed us to study new features of the system which are not accessible in the one-loop or infinite component (N ) → ∞ approximations. Though thermalization is only expected at approximations beyond the Hartree level, i.e., when including nonlocal corrections, thenonequilibriumsystematlatetimesshowsstrikingsimilaritiestoasys- tem in thermal equilibrium. One can define an order parameter which is dependentonthetotalenergyofthesystem,givenbytheinitialconditions. Analyzing the dependence of the order parameter on the initial conditions one finds a first order phase transition. References 1. J. Baacke and S. Michalski, Phys. Rev. D 65, 065019 (2002) [arXiv:hep-ph/0109137]. 2. Y. Nemoto, K. Naito and M. Oka, Eur. Phys. J. A 9, 245 (2000) [arXiv:hep-ph/9911431]. 3. J.M.Cornwall, R.JackiwandE.Tomboulis,Phys.Rev.D 10,2428(1974). 4. H. Verschelde and J. de Pessemier, Eur. Phys. J. C 22, 771 (2002) [arXiv:hep-th/0009241]. 5. J. Baacke, K. Heitmann and C. Pätzold, Phys. Rev. D 55, 2320 (1997) [arXiv:hep-th/9608006].