1e Temperature

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1 Equilibration Processes in Sphaleron Transitions D.Yu.Grigoriev Institute for Nuclear Research of Russian Academy of Sciences, 7 Moscow, Russia Abstract Sphaleron transitions in the course of thermalization of highly nonequilibrium initial eld congurations are numerically investigated in (+)-dimensional Abelian two-higgs model. It is shown that the transition rate is sensitive to deviations from thermal equilibrium. The process of thermalization can be very slow and is getting even more slower for congurations with low total energies. In recent years, an equilibrium rate of thermal sphaleron transitions has been thoroughly studied in wide temperature range both analytically and numerically [-7]. However, many modern scenarios for baryogenesis are based on rather complicated dynamical phenomena (e.g. propagating domain walls) that by its nature are strongly non-equilibrium. Better understanding of when we really can use equilibrium approaches is also required in the study of parametric resonance and its applications to cosmology, where at the end of resonance at least some of the elds involved are left in highly excited states [8-]. Here we numerically investigate the process of thermalization for two nonequilibrium initial congurations. While the energy of the rst conguration is localised in low-momentum modes, the second one represents relatively high-momentum state (in respect to the spatial scale of sphaleron). It is shown that the thermalization process can tae a considerable amount of time this amount strongly depends on the total energy of the system, or, in other words, the equilibrium temperature nally reached. The sphaleron

2 transition rate is very sensitive to deviations of the system from equilibrium, although the rate approaches its equilibrium value generally on the same time scale as the eld spectra tae their normal thermal shape. More detailed quantitative study of the transition rate variations and its relation to evolution of the spectra will be discussed in a separate paper []. The simulations presented below were carried out in the same (+)-dimensional two-higgs Abelian model used in ref. []: L = ; F +(D ) y (D )+(D ) y (D ) ; V ( ) V ( ) = ( y ; v ) + ( y ; v ) + ( y + y ; v ; v ) + 5 (Re( y ) ; v v cos ) + 6 (Im( y ) ; v v sin ) where = = =, 5 = 6 =:5, v = v =, =. In fact, the second Higgs eld has no serious eect on equilibration dynamics however, it's presence is useful for ltering out the thermal noise in the process of counting the sphaleron transitions, because real transitions correspond to simultaneous changes in winding numbers of both scalar elds coupled with the gauge eld A. The rst conguration is studied on the lattice with N = 5 sites and the size of L = 5, while for the second conguration the size was reduced to L =. The temperature dependence of equilibrium transition rate ; for both L =5andL =ispresented on Figs. and. Note that direct sphaleron counting technique used throughout this study prevents us from getting reliable data for high-t region where the dependence ;(T ) can no longer be approximated by simple Boltzmann exponent. Initially almost all energy of the rst conguration is localised at the Higgs eld sin(x=l) (one half-wave). The number of sphaleron transitions accumulated in the course of real-time evolution is presented on Fig. as a function of time t. The transition to equilibrium regime (solid line on Fig. ) occurs at t about :::. The evolution of spectrum of scalar elds is presented on Figs. -7.

3 . exp(-6.68/t+.6).. Rate/volume e-5 e-6 e-7 e Temperature Figure : Sphaleron transition rate ; as a function of dimensionless temperature T for lattice volumes L = 5 and L = 5 (combined). Solid line is the best exponential approximation to data. The rst run presented below on Figs. -7 has the (conserving) energy E = 79 which corresponds to T = :75, ; = :7 ;7 and expected time between transitions =(L ;) ; = 8. In the second conguration the scalar eld is close to sin(5 x=l), while the characteristic spatial momentum of the sphaleron for actual parameters of Higgs potential (v = v etc.) can be estimated as sph v p :5 =L. Because both the spatial volume of this conguration and its energy (controlled by the initial amplitude of ) are lower than in the rst run (Figs. -7), the equilibrium transition rate decreases correspondingly, and the time interval between two subsequent transitions is expected to be of order 8. These purely thermal transitions are too rare to be observed using available computing resources, so the major goal of the second example is to show that a slight (6%) decrease in energy (i.e. temperature) may greatly increase the thermalization time. Fig. 8 presents the dynamics of sphaleron transitions for

4 . L= fit: exp(-6.67/t+.) L=5/5 fit: exp(-6.68/t+.6). Rate/volume. e-5 e-6 e Temperature Figure : ;(T ) for L =. The exponential t (solid line) is compared to that one from Fig.. The second run (Figs. 8-8) has E = 7 corresponding to T =:5 and = 8. the second conguration it's important to note that because the equilibrium transition rate is virtually negligible (less than one transition per the whole plot of Fig. 8), all the transitions present on Fig. 8 are of non-equilibrium nature. The evolution of eld spectra is presented on Figs. 9-8 note that their equilibration is also considerably slower than that one on Figs. -7. Finally, it's interesting to notice how dierently the sphaleron transitions occur on early stages of evolution in both congurations (Figs. 9 and ): for short-wave initial conguration, the transitions are suppressed at the beginning of evolution. A probable reason is slow propagation of energy to sphaleron-related spectral modes in this case. The author is indebted to J.Baace, L.McLerran, V.Rubaov, A.Smilga, M.Shaposhniov and M.Tsypin for stimulating questions and discussions. This study was supported by RBRF grant a.

5 N_transitions 6 8 e+6.e+6 Time Figure : Sphaleron transition dynamics for the rst run (see also Fig. 9). Solid line corresponds to time interval between transitions =. 5

6 5.5 phi_() phi_() phi_() phi_() Figure : Energy spectrum of scalar elds and at time t = (initial spectrum not shown). Mode number N (axis x) is related to physical momentum as = N. Note that because L of periodicity inn the right-hand part of the plot corresponds to lowmomentum modes with negative. Figure 5: Energy spectrum at t = 8.5 phi_() phi_().5 phi_() phi_() Figure 6: Energy spectrum at t = Figure 7: Energy spectrum at t = 5 6

7 7 6 5 N_transitions Time Figure 8: Sphaleron transition dynamics for the second run (see also Fig. ). Estimated thermal rate corresponds to average time between transitions 8. 5 phi_() phi_() 8 phi_() phi_() Figure 9: Energy spectrum + E(;) of scalar elds and at t = (initial state). Mode number N (axis x) is related to physical momentum as = L N Figure : Energy spectrum at t = 7

8 5 phi_() phi_() 8 phi_() phi_() Figure : Energy spectrum at t = 7 Figure : Energy spectrum at t = 6 phi_() phi_() 5.5 phi_() phi_() Figure : Energy spectrum at t = Figure : Energy spectrum at t = phi_() phi_().5 phi_() phi_() Figure 5: Energy spectrum at t = Figure 6: Energy spectrum at t = 8

9 phi_() phi_().8.6 phi_() phi_() Figure 7: Energy spectrum at t = 7 Figure 8: Energy spectrum at t = 56 9

10 5 5 5 N_transitions Time 6 Figure 9: Initial part of plot Fig.. N_transitions Time Figure : Initial part of plot Fig. 8.

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