Strong and Electroweak Matter 2002

PREFACE.

LOCAL ORGANIZERS AND ADVISORY BOARD.

PARTICIPANTS.

CONTENTS.

We propose a method to study lattice QCD at non-vanishing temperature (T) and chemical potential (µ). We use n_f=2+1 dynamical staggered quarks with semi-realistic masses on L_t=4 lattices. The critical endpoint (E) of QCD on the Re(µ)-T plane is located. We calculate the pressure (p), the energy density (∊) and the baryon density (n_B) of QCD at non-vanishing T and µ.

The talk summarizes results of lattice investigations of QCD at finite temperature. The topics discussed include the flavor dependence of the phase transition, the critical temperature and the equation-of-state as well as hadronic correlation functions.

When thermal QCD crosses the critical temperature from below, its pressure density rises drastically, consistent with the picture of deconfinement and the release of partons as light degrees of freedom. On the other hand, the concept of partons is a perturbative one, whereas interactions with the infrared modes in the plasma always introduce non-perturbative contributions. Here I show how partonic correlators can be defined in a gauge invariant and non-perturbative manner which applies to all energy scales. In particular, I compute the magnetic mass for hot SU(2) gauge theory and find m_A = 0.36(2)g²T, whose inverse for large T is the largest correlation length in the system.

At nonzero temperature, an SU(N) gauge theory without dynamical quarks has a deconfining phase transition. Besides the usual, charge-one Polyakov loop, loops with Z(N) charge greater than one can affect non-universal features of the transition. This is investigated for two and three colors. I also introduce operators for “quarkless baryons”, which are neutral under Z(N), and characterize how baryon anti-baryon pairs deconfine.

We consider color-superconducting quark matter where quarks of the same flavor form Cooper pairs of spin one. We discuss the value of the color-superconducting gap parameter at zero temperature, the transition temperature to the normal conducting state, and whether such color superconductors exhibit an electromagnetic Meissner effect.

I provide new arguments supporting the validity of the t’Hooft anomaly conditions at non zero quark chemical potential. These constraints strengthen the quark-hadron continuity scenario. Finally I review the 2SC effective Lagrangian for color superconductivity.

Lattice simulations of QCD at non-zero baryon density suffer from a severe complex action problem. When QCD is reduced to a Potts model for static quarks, this problem can be solved completely using a meron-cluster algorithm. In models of static baryons on which chiral symmetry is nonlinearly realized, the complex action problem does not even arise both for two and for three flavors. Numerical simulations of such models are expected to yield qualitative (and perhaps even semi-quantitative) information about the QCD phase diagram.

The major goal of the RHIC experimental program at Brookhaven National Laboratory is to make and study the Quark Gluon Plasma. Another new form of matter, the Color Glass Condensate may be formed in these collisions. The recent results from RHIC are reviewed in this context.

We review recent calculations of the probability that a hard parton radiates an additional energy fraction ΔE due to scattering in spatially extended matter, and we discuss its applications for the suppression of leading hadrons in heavy ion collisions at collider energies.

Various aspects of transport coefficients in quantum field theory are reviewed. We describe recent progress in the calculation of transport coefficients in hot gauge theories using Kubo formulas, paying attention to the fulfillment of Ward identities. We comment on why the color conductivity in hot QCD is much simpler to compute than the electrical conductivity. The nonperturbative extraction of transport coefficients from lattice QCD calculations is briefly discussed.

We review recent developments for the description of far-from-equilibrium dynamics of quantum fields and subsequent thermalization.

Hot non-abelian gauge fields with typical momenta of order g²T, the so called magnetic screening scale, are strongly coupled. In the electroweak theory they determine the rate for anomalous baryon number violation in the very early universe. We discuss an effective theory which describes their dynamics at leading order in the gauge coupling g.

The dimensionally reduced action is believed to provide for a theoretically consistent and numerically precise effective description of the thermodynamics of the quark-gluon plasma, once the temperature is above a few hundred MeV. Although dramatically simpler than the original QCD it is, however, still a strongly interacting, confining theory. In this talk I speculate on whether there could exist a further simplified recipe within that theory, for physically relevant temperatures, which would already lead to a phenomenologically satisfactory description of the free energy and various correlation lengths of hot QCD, but with only a minimal amount of numerical non-perturbative input needed.

I review what is required to compute transport coefficients in ultra-relativistic, weakly coupled gauge theories, at leading order in g, using kinetic theory. Then I discuss how the calculation would look in alternative approaches: the 2PI method, and direct diagrammatic analysis. I argue that the 2PI method may be a good way to derive the kinetic theory, but is not very useful directly (in a gauge theory). The diagrammatic approach is almost hopeless.

We review the proposal to resum the physics of hard thermal loops in the thermodynamics of the quark-gluon plasma through nonperturbative expressions for entropy and density obtained from a Φ-derivable two-loop approximation. A comparison with the recently solved large-N_f limit of hot QCD is performed, and some updates, in particular on quark number susceptibilities, are made.

Quasiparticle dynamics in relativistic plasmas associated with hot, weakly-coupled gauge theories (such as QCD at asymptotically high temperature T) can be described by an effective kinetic theory, valid on sufficiently large time and distance scales. This effective kinetic theory may be used to evaluate observables which are dominantly sensitive to the dynamics of typical ultrarelativistic excitations, to leading order in the running coupling g(T) and all orders in 1/log g(T)⁻¹. Suitable observables include transport coefficients (viscosities and diffusion constants) and energy loss rates. This summary sketches the form of the effective theory and outlines its domain of applicability.

After recalling the fundamental importance of understanding the high-energy limit of QCD, I will use the first part of this talk to discuss a number of different theoretical approaches to this complicated problem. In the second part of the talk, I will explain the relevance of new and precise HERA data to the above theoretical question, emphasizing in particular diffractive deep-inelastic scattering (DIS) at small x.

I review the description of the hadron wavefunction at small x as a Colour Glass Condensate. In this context, I discuss the phenomenon of gluon saturation and some of its remarkable consequences: a new “geometric scaling” for F₂, which has been recently identified at HERA, and the unitarization of the hadronic cross-sections at high energy. I show that by combining saturation and confinement one obtains cross-sections which saturate the Froissart bound.

After a short review of baryogenesis mechanisms, we focus on the charge transport mechanism at the electroweak scale, effective at strong electroweak phase transitions. Starting from the one-loop Schwinger-Dyson equations for fermions coupled to bosons, we present a derivation of the relevant kinetic equations in the on-shell and gradient approximations, relevant for the thick wall baryogenesis regime. We then discuss the CP-violating source from the semiclassical force in the flow term, and compare it with the source arising in the collision term of the kinetic equation. Finally, we summarize the results concerning the chargino mediated baryogenesis in the Minimal Supersymmetric Standard Model.

After a short review of inlationary preheating, we discuss the development of equilibrium in the frameworks of massless λΦ⁴ model. It is shown that the process is characterised by the appearance of Kolmogorov spectra and the evolution towards thermal equilibrium follows self-similar dynamics. Simplified kinetic theory gives values for all characteristic exponents which are close to what is observed in lattice simulations. This allows estimation of the resulting reheating temperature.

In this talk, I present a recent calculation of one-loop vacuum polarization in a de-Sitter inflationary background. This provides possibly the first example of an analytical result from a calculation by hand of radiative corrections in an out-of-equilibrium situation. The model considered is massless, minimally coupled scalar QED. Gauge invariance remains manifest, but as a result of the photon coupling to the scalar, the conformal invariance of electromagnetism is broken. An effective photon field equation is obtained which, to leading order in the number of inflationary e-folds, is consistent with the existence of a dynamically generated photon mass.

This work has been done in collaboration with Tomislav Prokopec at Heidelberg University and Richard Woodard at the University of Florida.

We discuss cosmological implications of the time variation of fundamental mass scales. A particular model of crossover quintessence is compatible with all present cosmological observations. This model can also reconcile the reported time variation of the fine structure constant from quasar absorption lines with the bounds from archeo-nuclear physics and tests of the equivalence principle.

We apply a complex chiral random matrix model as an effective model to QCD with a small chemical potential at zero temperature. In our model the correlation functions of complex eigenvalues can be determined analytically in two different limits, at weak and strong non-Hermiticity. We compare them to the distribution of the smallest Dirac operator eigenvalues from quenched QCD lattice data for small values of the chemical potential, appropriately rescaled with the volume. This confirms the existence of two different scaling regimes from lattice data.

A discussion of the overlap problem of reweighting approaches to evaluating critical phenomenon in fermionic systems is motivated by highlighting the divergence of the joint probability density function of a general ratio. By identifying the bounds for which this integral can be expressed in closed form, we establish criteria for accurately mapping the joint ratio distribution of two disjoint ensembles through interpolation. The approach is applied to QCD with four staggered flavours to evaluate the critical line in the β − µ plane.

We comment on the reweighting method in the chemical potential (µ_q) direction. We study the fluctuation of the reweighting factor during Monte-Carlo steps. We find that it is the absolute value of the reweighting factor that mainly contributes to the shift of the phase transition line (β_c) by the presence of µ_q. The phase fluctuation is a cause of the sign problem, but the effect on β_c seems to be small. We also discuss β_c for Iso-vector chemical potential and β_c determined from simulations with imaginary chemical potential.

We investigate the singlet, triplet and colour average heavy quark free energies in SU(2) pure gauge theory at various temperatures T. We focus on the long distance behaviour of the free energies, studying in particular the temperature dependence of the string tension and the screening masses. The results are qualitatively similar to the SU(3) scenario, except near the critical temperature T_c of the deconfining transition. Finally we test a recently proposed method to renormalize the Polyakov loop.

We review our results for the QCD phase diagram at baryonic chemical potential µ_B ≤ πT. Our simulations are performed with an imaginary chemical potential µ_I for which the fermion determinant is positive. For 2 flavors of staggered quarks, we map out the phase diagram and identify the pseudo-critical temperature T_c(µ_I). For µ_I/T ≤ π/3, this is an analytic function, whose Taylor expansion is found to converge rapidly, with truncation errors far smaller than statistical ones. The truncated series may then be continued to real µ, yielding the corresponding phase diagram for . This approach provides control over systematics and avoids reweighting. We outline our strategy to find the (2+1)-flavor critical point.

Spatial ’t Hooft loops of strength k measure the qualitative change in the behaviour of electric colour flux in confined and deconfined phase of SU(N) gauge theory. They show an area law in the deconfined phase, known analytically to two loop order with a “k-scaling” law k(N − k). In this paper we compute the O(g³) correction to the tension. It is due to neutral gluon fields that get their mass through interaction with the wall. The simple k-scaling is lost in cubic order. The generic problem of non-convexity shows up in this order. The result for large N is explicitely given.

We present results from a lattice Monte Carlo study of the Nambu – Jona-Lasinio model in 3+1 dimensions with a baryon chemical potential µ ≠ 0. As µ is increased there is a transition from a chirally-broken phase to relativistic quark matter, in which baryon number symmetry appears spontaneously broken by a diquark condensate at the Fermi surface, implying a superfluid ground state. Finite volume corrections to this relativistic BCS scenario, however, are anomalously large.

In this talk, I try to show that the sign problem of dense QCD is due to modes whose frequency is higher than the chemical potential. An effective theory of quasi-quarks near the Fermi surface has a positive measure in the leading order. The higher-order corrections make the measure complex, but they are suppressed as long as the chemical potential is sufficiently larger than Λ_QCD. As a consequence of the positivity of the effective theory, we can show that the global vector symmetries except the U(1) baryon number are unbroken at asymptotic density.

We discuss simple characteristic properties of two macroscopic quantum liquids which can exist in the context of QCD matter at nonzero baryon density, and develop spin-one condensates: (1) Relativistic fermionic quantum fluids in the deconfined 3-color QCD. (2) Nonrelativistic bosonic quantum fluids with Bose-Einstein condensate of colorless spin-1 baryons in the confined 2-color QCD.

We present results on the renormalized quark-anti-quark free energy in SU(3) gauge theory at finite temperatures. We discuss results for the singlet, octet and colour averaged free energies and comment on thermal relations which allow to extract separately the potential energy and entropy from the free energy.

The transition from the hadronic phase to the phase of color-superconductivity at large densities is addressed by an effective theory which incorporates the Yang-Mills dynamics in addition to the di-quark degree of freedom. A toy version of this theory is studied by lattice simulations. A first order phase transition separates the regime of broken color-electric flux tubes from the color superconducting phase. My findings suggest that the quark and gluon liberation occurs at the same critical chemical potential.

We present our recent results for the ρ and σ mesons considered as resonances in pion-pion scattering in a thermal bath. We use chiral perturbation theory to order p⁴ for the low energy behaviour, then extend the analysis via the unitarization method of the Inverse Amplitude into the resonance region. The width of the rho broadens about twice the amount required by phase space considerations alone, its mass staying practically constant up to temperatures of order 150 MeV. The sigma meson behaves in accordance to chiral symmetry restoration expectations.

We discuss the principles underlying higher spin glueball calculations on the lattice. We measure glueball angular wave functions, decompose them in Fourier modes and extrapolate the Fourier coefficients to the continuum. This provides a reliable labelling of the continuum states. We present preliminary results on the glueball spectrum of the SU(2) gauge theory in 2+1 dimensions for spins ranging from 0 to 6 inclusive.

The computational cost of numerical simulations of QCD with light dynamical Wilson-quarks is estimated by determining the autocorrelation of various quantities. In test runs the expected qualitative behaviour of the pion mass and coupling at small quark masses is observed.

Semiclassical background fields in SU(2) lattice gauge theory at temperatures below but close to the deconfinement transition are carefully (re)investigated. By the help of the ‘cooling’ method and analyzing the localization behaviour of the real modes of the Dirac-Wilson fermion operator we find evidence for (anti)selfdual Kraan-van Baal caloron solutions typically occuring with non-trivial holonomy.

Using quasi-particle models, lattice data can be mapped to finite chemical potential. By comparing a simple and an HTL quasi-particle model, we derive the general trend that a full inclusion of the plasmon effect will give.

We compute derivatives of thermodynamic quantities with respect to µ, at µ = 0 for 2 and 3 flavors of degenerate quark masses. This allows us to estimate the phase transition line in the T, µ plane and quantify the influence of a non vanishing chemical potential on the equation of state by computing lines of constant energy, pressure and density. Moreover we evaluate the order of the QCD phase transition by measuring the Binder Cumulant of the chiral condensate. This gives access to the chiral critical point on the phase transition line.

We numerically investigate the three-dimensional O(6) model on 12³ to 120³ lattices. From Binder’s cumulant at vanishing magnetic field we obtain the critical coupling J_c = 1.42865(5) and verify this value with the χ²-method. The universal value of Binder’s cumulant at this point is g_r(J_c) = −1.94456(10). At the critical coupling we find the critical exponents ν = 0.818(5), β = 0.425(2) and γ = 1.604(6) from a finite size scaling analysis. We also determine the finite-size-scaling function on the critical line and the equation of state. Our O(6)-result for the equation of state is compared to the Ising, O(2) and O(4) results.

We examine the chiral phase transition of strong interaction within a linear σ-model with two light quark flavors. Going beyond previous mean field analyses of the phase diagram, we derive an analytic renormalization group flow equation for the thermodynamic potential including mesonic fluctuations.

Quark-antiquark 4-point correlators in the mesonic channels at T > 0 are studied on the lattice to obtain insight into the hadronic and plasma physics of QCD.

We discuss modifications of hadron properties in the heat bath extracted from Euclidian correlation functions and spectral functions reconstructed with the Maximum Entropy Method. To investigate the cut-off dependencies we perform simulation on various lattice sizes and present the result of an analytic calculation of the mesonic spectral functions in the infinite temperature limit.

We discuss a non-perturbative renormalization of n-point Polyakov loop correlation functions by explicitly introducing a renormalization constant for the Polyakov loop operator on a lattice deduced from the short distance properties of 2-pint correlators. We calculate this constant for the SU(3) gauge theory.

I discuss how the charged hadron multiplicities recently measured at RHIC energies are described in terms of gluon degrees of freedom, starting from the initial conditions of central heavy ion collisions as given by the saturation scenario, and followed by “bottom-up” equilibration before final hadronization.

We study the collective quark excitations in an extremely anisotropic system of two interpenetrating streams of the quark-gluon plasma. In contrast to the gluon modes, all quark ones appear to be stable in such a system. Even more, the quark modes in the two-stream system are very similar to those in the isotropic plasma.

The effect of thermally excited electron-positron pairs on the bulk properties of the color-flavor locked quark phase inside compact stars is studied.

I discuss the effects and quantities that influence the decoupling of particles from the fireball. The crucial role is played by the scattering rate. I show the results for the scattering rate at SPS and RHIC and discuss their implications.

Quantum gauge and (fermionic) matter fields, with microscopic dynamics described by Relativistic Quantum Field Theory (RQFT), are studied in out-of-equilibrium situations. Possible initial non-equilibrium states are represented by suitable initial density operators ρ_Q,in. Suitable functional integrals for ρ_Q,in, which generalize those in Equilibrium Thermal Field Theory, are considered. The subsequent time evolution of the system is formulated in some model.

The conventional weak-coupling expansion for thermodynamic quantities in hot field theories shows poor convergence unless the coupling constant is tiny. I discuss screened perturbation theory (SPT) which is a way of reorganizing the perturbative expansion for scalar theories and hard-thermal-loop perturbation theory (HTLPT), which is its generalization to gauge theories. I present results for the pressure to three loops in SPT and to two loops in HTLPT. We compare the latter with three-and four-dimensional lattice simulations of pure-glue QCD.

We discuss the method of Φ-derivable approximations in gauge theories. There, two complications arise, namely the violation of Bose symmetry in correlation functions and the gauge dependence. For the latter we argue that the error introduced by the gauge dependent terms is controlled, therefore not invalidating the method.

We study the onset of Goldstone phenomenon in a hybrid inflation scenario. The physically motivated range of parameters is analyzed in order to meet the cosmological constraints. Classical equations of motion are solved and the evolution through the spontaneous symmetry breaking is followed. We emphasize the role of topological defects that partially maintain the disordered phase well after the waterfall. We study the emergence of the Goldstone excitations and their role in the onset of the radiation dominated universe.

A real-time path integral for ultrasoft QCD is formulated. It exhibits a Feynman’s influence functional. The statistical properties of the theory and the gauge symmetry are explicit. The correspondence is established with the alternative version, where a noise term enters a transport equation.

Because of IR (pinch) singularities a resummation is necessary in non-equilibrium field theories, that can be performed by using Kadanoff–Baym equations. Taking Landau prescription correctly into account, Kadanoff–Baym equations reduce to Boltzmann equations only in a restricted kinematical range; in other cases a new equation (the former constraint equation) has to be considered. In relaxation time approximation this new equation results in the shifting and smearing of multiparticle thresholds.

Transport theory is an efficient approach to derive an effective theory for the soft modes of QCD at high temperature. It is known that the leading order operators of this theory can be obtained from (semi-classical) kinetic equations of quasiparticles carrying classical or quantum color charges. Higher order operators can also be obtained. Discrepancy between these quasiparticle models starts for dimension 4 operators, which converge in the limit of high dimensional color representations. These quasiparticle models are reviewed and compared.

We investigate the out-of-equilibrium evolution of a classical background field and its quantum fluctuations in the scalar O(N) model with spontaneous symmetry breaking ¹. We consider the 2-loop 2PI effective action in the Hartree approximation, i.e. including bubble resummation but without non-local contributions to the Dyson-Schwinger equation. We concentrate on the (nonequilibrium) phase structure of the model and observe a first-order transition between a spontaneously broken and a symmetric phase at low and high energy densities, respectively. So typical structures expected in thermal equilibrium are encountered in nonequilibrium dynamics even at early times before thermalization.

The problem of maintaining gauge invariance in the 2PI formulation of QED is discussed. A modified form of the the 2PI effective action is suggested in which Ward identities for external (background field) and internal (quantum field) gauge transformations are both satisfied, but in different manners. The resdidual gauge-fixing dependence in this modified 2PI formulation vanishes in a certain low momentum limit, which allows it to be used reliably for calculating quantities such as transport coeffcients and soft field relaxation in hot gauge theories.

We show that CP-violation can lead to an asymmetric diffusion of the Chern-Simons number in thermal equilibrium. This asymmetry leads to a linearly growing expectation value of the third power of the Chern-Simons number. In the long-time limit all expectation values of powers of Chern-Simon numbers are determined by their appropriate disconnected parts.

We study the spectral function of the sigma meson at finite temperature in O(4) linear sigma model adopting the resummation technique called optimized perturbation theory to 2-loop. As a first step of this study, we calculate the “sunset diagram” for a scalar field theory at finite temperature, which is one of two-loop self-energy diagrams, by reducing it to an expression that involves one-loop self-energy. The result can be easily evaluated numerically. We also study the structure of its imaginary part. As an application, we evaluate the effect of the thermal sunset diagram on the spectral function for the sigma meson to 1-loop.

We consider a scalar theory at finite temperature in the 2PI resummation scheme, including ϕ³ and ϕ⁴ interactions. Already at the one loop level in this scheme, we have to deal with a non local approximation. We carry out the renormalization and obtain finite equations for the propagator. Within this model we can explore the effect of non local contributions to the self-energy in the evaluation of thermodynamic quantities.

We discuss the critical properties of the three-dimensional NJL model at nonzero temperature. We show that the Z₂-symmetric model undergoes a second order phase transition with 2d Ising exponents and its critical region is suppressed by a factor . We also provide numerical evidence that the U(1)-symmetric model undergoes a BKT transition in accordance with the dimensional reduction scenario.

The enhancement of the scalar-isoscalar spectral function near the two-pion threshold is studied in the framework of an effective linear σ model, using a large N approximation in the number of the Goldstone bosons. The effect is rather insensitive to the detailed T = 0 characteristics of the σ pole, it is accounted by a pole moving with increasing T along the real axis of the second Riemann sheet towards the threshold location from below.

If SU(N) gauge fields live in a world with a circular extra dimension, coupling there only to adjointly charged matter, the system possesses a global Z(N) symmetry. If the radius is small enough such that dimensional reduction takes place, this symmetry is spontaneously broken. It turns out that its fate at high temperatures is not easily decided with straightforward perturbation theory. Utilising non-perturbative lattice simulations, we demonstrate here that the symmetry does get restored at a certain temperature T_c, both for a 3+1 and a 4+1 dimensional world (the latter with a finite cutoff). To avoid a cosmological domain wall problem, such models would thus be allowed only if the reheating temperature after inflation is below T_c.

QCD (SU(3)) in 2+1 space - time dimensions near the light cone becomes a critical theory in the limit of x → 0 with a diverging correlation length where the exponent λ₂ = 2.52 is obtained from the center group Z(3) of SU(3). We derive geometrical scaling for the structure function in deep inelastic scattering at low x from the correlation length ξ(x) of Wilson lines near the light cone.

The contribution of QCD-instantons to the phenomenon of saturation in deep-inelastic scattering (DIS) at small Bjorken-x is investigated.

We study the evolution of a stochastic helical magnetic field generated in the early Universe after the electroweak phase transition, using standard magnetohydrodynamics (MHD). We find how the coherence length ξ, magnetic energy E_M and magnetic helicity H evolve with time. We show that the self-similarity of the magnetic power spectrum alone implies that ξ ~ t^1/2. This in turn implies that magnetic helicity decays as H ~ t^−2s, and that the magnetic energy decays as E_M ~ t^−0.5−2s, where s inversely proportional to the magnetic Reynolds number Re_M. These laws improve on several previous estimates.

In this talk we review the status of possible models for electroweak baryogenesis. We make special emphasis on the analysis of the available parameter space for the Minimal Supersymmetric Standard Model. We also comment on NMSSM, R_p violating and modified cosmological models.

In this talk, I discuss the formation of magnetic monopoles in a phase transition from the confining SU(2) phase to the Coulomb phase in a hot Georgi-Glashow model. I argue that monopoles are formed from long-wavelength thermal fluctuations, which freeze out after the phase transition.

A measurement of the big bang relic neutrinos would open a new window to the early universe. We review various possibilities to detect this cosmic neutrino background and substantiate the assertion that – apart from the rather indirect evidence to be gained from cosmology and large-scale structure formation – the annihilation of ultrahigh energy cosmic neutrinos with relic anti-neutrinos (or vice versa) on the Z-resonance is a unique process having sensititivy to the relic neutrinos, if a sufficient flux at exists. The associated absorption dips in the ultrahigh energy cosmic neutrino spectrum may be searched for at forthcoming neutrino and air shower detectors. The associated protons and photons may have been seen already in form of the cosmic ray events above the Greisen-Zatsepin-Kuzmin cutoff.

Results are presented of a numerical study of the distribution of W-bosons generated in a tachyonic electroweak pre-heating transition.

We consider the creation of non-zero Chern-Simons number in a model of the early Universe, where the Higgs field experiences a fast quench at the end of inflation. We perform numerical lattice simulations in the Abelian Higgs model in 1+1 dimensions and in the SU(2)-Higgs model in 3+1 dimensions with an added effective CP-violating term. We also comment on the appropriate choice of vacuum initial conditions for classical simulations.