The bubble wall velocity is essential for the phase transition dynamics in the early universe and its cosmological implications, such as the energy budget of phase transition gravitational wave and electroweak baryogenesis. One key factor to determine the wall velocity is the collision term that quantifies the interactions between the massive particles in the plasma and the bubble wall. We improve the calculations of the collision term beyond the leading-log approximation, and further obtain more precise bubble wall velocity for a representative effective model.
We present an entirely analytic, leading log order determination of the friction an electroweak bubble wall feels during a first order electroweak phase transition. The friction is dominated by W bosons, and gives a wall velocity parametrically ~ alpha_w, and numerically small, ~ .01 -- 0.1 depending on the Higgs mass.
The electroweak phase transition, if proceeding through nucleation and growth of bubbles, should generate large scale turbulent flow, which in turn generates magnetic turbulence and hence magnetic fields on the scale of turbulent flow. We discuss the seeding of this turbulent field by the motion of the dipole charge layers in the phase transition bubble walls, and estimate the strength of the produced fields.
Assuming that the electroweak and QCD phase transitions are first order, upon supercooling, bubbles of the new phase appear. These bubbles grow to macroscopic sizes compared to the natural scales associated with the Compton wavelengths of particle excitations. They propagate by burning the old phase into the new phase at the surface of the bubble. We study the hydrodynamic stability of the burning and find that for the velocities of interest for cosmology in the electroweak phase transition, the shape of the bubble wall is stable under hydrodynamic perturbations. Bubbles formed in the cosmolog…
We discuss aspects of poor infrared behaviour of the perturbation expansion for the effective potential of the Higgs mode near the electroweak phase transition, and enlarge on the discovery that higher order effects weaken the transition. In addition, we outline our recent attempts at understanding the dynamics involved in the propagation of bubbles formed in the first order transition.
The electroweak baryogenesis depends on the profile of the CP-violating bubble wall created at the first order phase transition. We attempt to determine it by solving the coupled equations of motion for the moduli and phases of the two Higgs doublets at the transition temperature. A variety of CP-violating bubble walls are classified by boundary conditions. We point out that a sufficiently small explicit CP violation gives nonperturbative effects to yield the baryon asymmetry of the universe.
We investigate the motion of fermions in the presence of an electro\-weak phase transition bubble wall. We derive and solve the Dirac equation for such fer\-mions, and compute the transmission and reflection coefficients for fermions traveling from the symmetric to the asymmetric phases separated by the domain wall.
We compute the net electric current generated during a first order electroweak phase transition when fermions transit from the false to the true vacuum. This current is generated by the CP-violating fermion interaction with the Higgs during the phase transition and is quantified in terms of a CP-violating phase in the bubble wall separating the symmetric from the symmetry-broken phases. We comment on the seed magnetic field that this current is able to generate and it is possible implications for magnetogenesis in the early universe during the electroweak phase transition.
We compute the wall velocity in the MSSM. We therefore generalize the SM equations of motion for bubble walls moving through a hot plasma at the electroweak phase transition and calculate the friction terms which describe the viscosity of the plasma. We give the general expressions and apply them to a simple model where stops, tops and W bosons contribute to the friction. In a wide range of parameters including those which fulfil the requirements of baryogenesis we find a wall velocity of order v = 0.05-0.1 much below the SM value.
We consider the role played by subcritical bubbles during the electroweak phase transition, estimate their average size, amplitude and formation rate taking into account the crucial role played by thermalization. We also study the influence of subcritical bubbles on the formation of critical bubbles in the thin wall regime and show that, contrary to some recent claims, subcritical bubbles do not affect the nucleation of critical bubbles in an appreciable way. From this fact we conclude that the electroweak baryogenesis scenarios associated with a first order electroweak phase transition still …
We analyze Higgs condensate bubble expansion during a first-order electroweak phase transition in the early Universe. The interaction of particles with the bubble wall can be accompanied by the emission of multiple soft gauge bosons. When computed at fixed order in perturbation theory, this process exhibits large logarithmic enhancements which must be resummed to all orders when the wall velocity is large. We perform this resummation both analytically and numerically at leading logarithmic accuracy. The numerical simulation is achieved by means of a particle shower in the broken phase of the e…
We consider non-perturbative solutions of the Weinberg-Salam model at finite temperature. We employ an effective temperature-dependent potential yielding a first order phase transition. In the region of the phase transition, there exist two kinds of static, spherically symmetric solutions: sphalerons and bubbles. We analyze these solutions as functions of temperature. We consider the most general spherically symmetric fluctuations about the two solutions and construct the discrete modes in the region of the phase transition. Sphalerons and bubbles both possess a single unstable mode. We presen…
We calculate the velocity and thickness of a bubble wall at the electroweak phase transition in the Minimal Standard Model. We model the wall with semiclassical equations of motion and show that friction arises from the deviation of massive particle populations from thermal equilibrium. We treat these with Boltzmann equations in a fluid approximation in the background of the wall. Our analysis improves on the previous work by using the two loop effective potential, accounting for particle transport, and determining the wall thickness dynamically. We find that the wall is significantly thicker …
Baryogenesis at the electroweak scale depends on divers but identifiable details of bubble wall dynamics and the particle physics. We show that inclusion of the dynamics of relative phase in two-Higgs-doublet model (2HDM) enhances the adiabatic order of the mechanism proposed by McLerran-Shaposhnikov-Turok-Voloshin where the scalar-scalar-vector triangle diagram with top quark in the loop gives rise to a significant contribution to the effective chemical potential biasing the Chern-Simons number. We also show that in 2HDM with less stringent constraints on Yukawa couplings than those imposed b…
The first order phase transition proceeds via nucleation and growth of true vacuum bubbles. When charged particles collide with the bubble they could radiate electromagnetic wave. We show that, due to an energy loss of the particles by the radiation, the damping pressure acting on the bubble wall depends on the velocity of the wall even in a thermal equilibrium state.
In any scenario of the electroweak baryogenesis, the profile of the CP violating bubble wall, created at the first-order phase transition, plays an essential role. We attempt to determine it by solving the equations of motion for the scalars in the two-Higgs-doublet model at the transition temperature. According to the parameters in the potential, we found three solutions. Two of them smoothly connect the CP-violating broken phase and the symmetric phase, while the other connects CP-conserving vacua but violates CP in the intermediate region. We also estimate the chiral charge flux, which will…
By treating CP-violating interaction as a perturbative term, we solve the Dirac equation in the background of electroweak bubble wall (the distorted wave Born approximation). We obtain the transmission and reflection coefficients for a chiral fermion incident from the symmetric-phase region and for the one from the broken-phase region respectively. There hold the respective sets of unitarity relations and also reciprocity relations among them. These relations enable us to rigorously derive quantum-number flux through the bubble wall, which is the first order quantity of the CP violation. The f…
A general prescription to solve the Dirac equation in the presence of CP-violating electroweak bubble wall is presented. The profile of the bubble wall is not specified except that the wall height is $m_0$ and zero deep in the broken- and the symmetric-phase regions, respectively, where $m_0$ is a fermion mass given by the Higgs-vacuum-expectation value and the Yukawa coupling. The CP-violating effects are evaluated by regarding CP-violating part of the bubble wall as a perturbation to CP-conserving solutions. The basic quantity, $R_{R\rightarrow L}-\bar R_{R\rightarrow L}$, which would contri…
If cosmic strings are present at the electroweak phase transition, they can act as seeds on which bubbles of true vacuum nucleate. We explore the nature of such a phase transition, in particular the wall velocity and thickness of the bubbles. {}From the viewpoint of electroweak baryogenesis, adiabatic conditions exist in the expanding bubble walls, and such models of baryogenesis can be successfully applied. In the present mechanism, the nature of the electroweak phase transition is insensitive to the other details of the model, thus reducing the uncertainties in the estimate of net baryon asy…
We present results for the bubble wall velocity and bubble wall thickness during a cosmological first-order phase transition in a condensed form. Our results are for minimal extensions of the Standard Model but in principle are applicable to a much broader class of settings. Our first assumption about the model is that only the electroweak Higgs is obtaining a vacuum expectation value during the phase transition. The second is that most of the friction is produced by electroweak gauge bosons and top quarks. Under these assumptions the bubble wall velocity and thickness can be deduced as a func…
We study the subcritical bubble formation near the phase space domain wall. We take into account that the phase of the scalar field can vary using complex U(1) symmetric field and a phenomenological potential with cubic term responsible to symmetry breaking. We show that the presence of the domain wall induces subcritical bubbles so that their formation rate near the wall is considerably larger than far of it. The allowed deviations of the phases of new bubbles are so large that they prevent the system from induced nucleation.
We study the CP-violating bubble wall by solving the coupled equations of motion derived from the two-Higgs-doublet model for the moduli $ρ_i$ and phases $θ_ i$ of the two scalars $(i=1,2)$ at the transition temperature. We find a solution that the CP-violating phase $θ\equivθ_1-θ_2$ connecting the CP-conserving vacua strongly violates CP in the intermediate region while the moduli largely deviate from the kink shape there. We estimate the chiral charge flux through the wall surface, which may be efficient enough to produce the baryon asymmetry by the sphaleron process.
Bubbles in a first-order electroweak phase transition are nucleated with radii $R_0$ and expand with velocity $v$. If $v$ is subsonic, a bubble becomes unstable to non-spherical perturbations when its radius is roughly $10^4\, R_0$. These perturbations accelerate the transition, and the effective velocity of bubble growth rapidly becomes supersonic. The transition should subsequently proceed spherically via detonation. If for some reason the onset of detonation is postponed, the surface area of the bubbles may be enhanced by $10^5$. We discuss consequences for electroweak baryogenesis.
Using the liquid instanton model continued into Minkowski space and modified for finite temperature, the energy momentum tensor and the surface tension for buble walls during the QCD phase transition are derived. The resulting surface tension is in agreement with lattice calculations. Application to bubble collisions is discussed.
We analyse the dynamics of the relativistic bubble expansion during the first order phase transition focusing on the ultra relativistic velocities $γ\gg 1$. We show that fields much heavier than the scale of the phase transition can significantly contribute to the friction and modify the motion of the bubble wall leading to interesting phenomenological consequences. NLO effects on the friction due to the soft vector field emission are reviewed as well.
The limiting bubble wall velocity during a first-order electroweak phase transition is of interest in scenarios for electroweak baryogenesis. Khlebnikov has recently proposed an interesting method for computing this velocity based on the fluctuation-dissipation theorem. I demonstrate that at one-loop order this method is identical to simple, earlier techniques for computing the wall velocity based on computing the friction from particles reflecting off or transmitting through the wall in the ideal gas limit.
Proceeding from WKB quantization conditions, we derive a semiclassical expression for the current of fermions scattered off a propagating bubble wall in the presence of longitudinal gauge field. It agrees with the expression used by Nasser and Turok in semiclassical analysis of instability of electroweak bubble walls with respect to longitudinal $Z$ condensation. We discuss the resulting dispersion relation for longitudinal $Z$ field and show that light species are important for the analysis of stability, because of their large contribution to plasma frequency.
We present a general numerical approach to solve the equations for bubble wall profiles in models with more than one scalar field and CP violating phases. We discuss the algorithm as well as several problems associated with it and show some profiles for demonstration found with our method.
In extensions of the Standard Model with extra scalars, the electroweak phase transition can be very strong, and the bubble walls can be highly relativistic. We revisit our previous argument that electroweak bubble walls can "run away," that is, achieve extreme ultrarelativistic velocities $γ\sim 10^{14}$. We show that, when particles cross the bubble wall, they can emit transition radiation. Wall-frame soft processes, though suppressed by a power of the coupling $α$, have a significance enhanced by the $γ$-factor of the wall, limiting wall velocities to $γ\sim 1/α$. Though the bubble walls ca…
Standard electroweak baryogenesis in the context of a first order phase transition is effective in generating the baryon asymmetry of the universe if the broken phase bubbles expand at subsonic speed, so that CP asymmetric currents can diffuse in front of the wall. Here we present a new mechanism for electroweak baryogenesis which operates for supersonic bubble walls. It relies on the formation of small bubbles of the symmetric phase behind the bubble wall, in the broken phase, due to the heating of the plasma as the wall passes by. We apply the mechanism to a model in which the Higgs field is…
We study the dynamics of electroweak bubbles in the scenario of a strong first order inverse electroweak phase transition at the TeV scale involving the global structure of the nonlinear sigma field in the littlest Higgs model with T parity. Employing the one-loop order finite temperature effective potential, we find that the pressure in the symmetric phase i.e., inside the bubble is always greater than that in the asymmetric phase i.e., outside the bubble, so that the bubbles are expanding. By calculating the fluid velocities in the two phases we arrive at the condition of a supersonic deflag…
We discuss the critical bubbles of the electroweak phase transition using an effective high-temperature 3-dimensional action for the Higgs field $φ$. The separate integration of gauge and Goldstone boson degrees of freedom is conveniently described in the 't Hooft-Feynman covariant background gauge. The effective dimensionless gauge coupling $g_3(T)^2$ in the broken phase is well behaved throughout the phase transition. However, the behavior of the one-loop $Z(φ)$ factors of the Higgs and gauge kinetic terms signalizes the breakdown of the derivative expansion and of the perturbative expansion…
One mechanism for generating a baryon asymmetry at the electroweak phase transition involves propagation of particle asymmetries generated by reflection from the bubble walls into the unbroken phase. Hitherto attention has focussed on top quarks because of their large mass and thus effective scattering from bubble walls. In this paper we point out that leptons may be more efficient mediators of this type of electroweak baryogenesis, particularly for thicker bubble walls favored by perturbative calculations. We carry out an analytic calculation of each stage of the mechanism and conclude it pro…
By treating CP-violating interaction as a perturbative term, we solved in a previous paper the Dirac equation in the background of electroweak bubble wall, and obtained the transmission and reflection coefficients for a chiral fermion incident from the symmetric-phase region. We give the transmission and reflection coefficients under the other boundary condition, that is, the case of the fermion incident from the broken-phase region. There hold the respective sets of unitarity relations and also reciprocity relations among them. These relations enable us to obtain a simple form of quantum-numb…
First order phase transitions in the early universe naturally lead to the production of a stochastic background of gravitational waves and to the generation of a matter-antimatter asymmetry. The dynamics of the phase transition is affected by the density perturbations in the hot plasma. We address this topic by providing, for the first time, a full numerical solution to the linearized Boltzmann equation for the top quark species coupled to the Higgs field during a first order phase transition at the electroweak scale. Differently from the traditional approaches, our results do not depend on an…
Electroweak Baryogenesis from Collapsing Domain Walls
University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；Argonne National Laboratory（阿贡国家实验室）；University of Oklahoma（俄克拉荷马大学）；The Hong Kong University of Science and Technology（香港科技大学）
In models of thick wall electroweak baryogenesis a common assumption is that the plasma interacting with the expanding Higgs bubble wall during the electroweak phase transition is in kinetic equilibrium (or close to it). We point out that, in addition to the requirement of low wall velocity, kinetic equilibrium requires that the change in the momentum of the particles due to the force exerted by the wall should be much less than that due to scattering as the plasma passes through the wall. We investigate whether this condition is satisfied for charginos and neutralinos participating in thick w…
We discuss the generation of the baryon asymmetry by a strong first order electroweak phase transition in the early universe, particularly in the context of the MSSM. This requires a thorough numerical treatment of the bubble wall profile in the case of two Higgs fields. CP violating complex particle masses varying with the Higgs field in the wall are essential. Since in the MSSM there is no indication of spontaneous CP violation around the critical temperature (contrary to the NMSSM) we have to rely on standard explicit CP violation. Using the WKB approximation for particles in the plasma we …
We revisit the computation of bubble wall friction during a cosmological first-order phase transition, using an extended fluid Ansatz to solve the linearized Boltzmann equation. A singularity is found in the fluctuations of background species as the wall approaches the speed of sound. Using hydrodynamics, we argue that a discontinuity across the speed of sound is expected on general grounds, which manifests itself as the singularity in the solution of the linearized system. We discuss this result in comparison with alternative approaches proposed recently, which find a regular behaviour of the…
The dynamical growth rate of bubbles nucleating in relativistic plasma in thermal first-order phase transitions is analyzed. The framework is a hydrodynamical model which consists of relativistic fluid and an order parameter field. The results of analytical approximations and numerical simulations coincide well.
We solve the equations of motion for a CP violating phase between the two Higgs doublets at the bubble wall of the MSSM electroweak phase transition. Contrary to earlier suggestions, we do not find indications of spontaneous ``transitional'' CP violation in the MSSM. On the other hand, in case there is explicit CP violation in the stop and chargino/neutralino sectors, the relative phase between the Higgses does become space dependent, but only mildly even in the maximal case. We also demonstrate that spontaneous CP violation within the bubble wall could occur, e.g., if the Higgs sector of the …
The electroweak phase transition and the sphaleron decoupling condition in the MSSM are revisited taking the latest experimental data into account. The light Higgs boson scenario and the ordinary decoupling limit which are classified by the relative size between the CP-odd Higgs boson mass and Z boson mass are considered within the context of electroweak baryogenesis. We investigate v/T at not only the critical temperature at which the effective potential has two degenerate minima but also the nucleation temperature of the critical bubbles, where v is a vacuum expectation value of the Higgs bo…
We investigate the gravitational behavior of spherical domain walls (bubbles) arising during the phase transitions in the early Universe. In the thin-wall approximation we show the existence of the new solution of Einstein equations with negative gravitational mass of bubbles and the reversed direction of time flow on the shell. This walls exhibit gravitational repulsion just as the planar walls are assumed to do. The equilibrium radius and critical mass of such objects are found for realistic models.
In extensions of the Standard Model with SU(2) singlet scalar fields, there can be regions of parameter space for which the electroweak phase transition is first order already at the mean-field level of analysis. We show that in this case the phase interface (bubble wall) can become ultra-relativistic, with the relativistic gamma factor gamma = (1-v_{wall}^2)^{-1/2} growing linearly with the wall's propagation distance. We provide a simple criterion for determining whether the bubble wall "runs away" in this way or if gamma approaches a terminal value.
We investigate the role of large amplitude sub-critical thermal fluctuations in the dynamics of first order phase transitions. In particular, we obtain a kinetic equation for the number density of sub-critical fluctuations of the broken-symmetric phase within the symmetric phase, modeled as spherical bubbles, and solve it analytically for temperatures above the critical temperature. We study the approach to equilibrium and obtain the equilibrium distribution of sub-critical bubbles of the unstable phase by examining three possible mechanisms responsible for their removal; their shrinking, thei…
We present solutions to the equations of motion for bubble wall profiles in the minimal and a non minimal supersymmetric extension of the Standard model. We discuss the method of the numerical approach and present results for the two models (MSSM and NMSSM). Calculations for the NMSSM were done in collaboration with S.J. Huber and M.G. Schmidt.
We compute a net electric current during a first order EWPT arising from the asymmetric propagation of fermion chiral modes due to a CP-violating interaction with the Higgs. The interaction is quantified in terms of a CP-violating phase in the bubble wall that separate both false and true vacuum phases. We comment on the possibility of this current to generate a seed magnetic field and its implications for primordial magne- togenesis in the early Universe.
Extensions of the Standard Model are being considered as viable settings for a first-order electroweak phase transition which satisfy Sakharov's three conditions for the generation of the baryon asymmetry of the Universe. These extensions provide a sufficiently strong phase transition and remove the main obstacles which appear in the context of the Standard Model: A far-too-high lower bound on the Higgs mass, immediate wipeout of the newly-created baryon asymmetry, and insufficient CP violation. We describe the Universe hydrodynamically as a fluid coupled to the Higgs field via a phenomenologi…
We perform large-scale real-time simulations of a bubble wall sweeping through an out-of-equilibrium plasma. The scenario we have in mind is the electroweak phase transition, which may be first order in extensions of the Standard Model, and produce such bubbles. The process may be responsible for baryogenesis and can generate a background of primordial cosmological gravitational waves. We study thermodynamic features of the plasma near the advancing wall, the generation of Chern-Simons number/Higgs winding number and consider the potential for CP-violation at the wall generating a baryon asymm…
We consider the question of whether cosmic ray catalysed false vacuum decay can be phenomenologically more important than spontaneous decay via quantum tunnelling. We extend the zero bubble wall width Landau-WKB analysis of catalysed false vacuum decay to include the leading order effects of finite wall width and derive an expression for the thin-wall bubble action. Using this we calculate the exponential suppression factor for the catalysed decay rate at the critical bubble energy, corresponding to the largest probability of catalysed decay. We show that, in general, cosmic ray catalysed deca…
By treating CP-violating interaction of the electroweak bubble wall as a perturbative term, chiral charge flux through the bubble wall is estimated. It is found that the absolute value of the flux $F_Q$ has a sharp peak at $m_0 \sim a \sim T$ with $F_Q/(u T^3) \sim 10^{-3}\, (Q_L-Q_R)\,Δθ$. Here $m_0$ is the fermion mass, $1/a$ is the wall thickness, $T$ is the temperature at which the bubbles are growing, $u$ is the wall velocity, $Q_{L(R)}$ is the chiral charge of the relevant left(right)-handed fermion and $Δθ$ is the measure of CP violation.
It has been recently shown that meta-stable domain walls exist in high-density QCD ($μ\neq 0$) as well as in QCD with large number of colors ($N_c\to\infty$), with the lifetime being exponentially long in both cases. Such metastable domain walls may exist in our world as well, especially in hot hadronic matter with temperature close to critical. In this paper we discuss what happens if a bubble made of such wall is created in heavy ion collisions, in the mixed phase between QGP and hadronic matter. We show it will further be expanded to larger volume $\sim 20 fm^3$ by the pion pressure, before…
Fluctuation-Induced Friction in Bubble-Wall Dynamics of Cosmological First-Order Phase Transitions
Hangzhou Institute for Advanced Study, University of Chinese Academy of Sciences（杭州高等研究院，中国科学院大学）；School of Physical Sciences, University of Chinese Academy of Sciences（中国科学院大学物理科学学院）；Institute of Theoretical Physics, Chinese Academy of Sciences（中国科学院理论物理研究所）
I discuss the applicability of classical techniques to the study of the dynamics of infrared, bosonic fields at the electroweak phase transition. I present the lattice as a natural means of cutting off hard, nonclassical modes, and discuss the problem of integrating out the excluded modes perturbatively. I then apply the classical lattice technique to study the dynamics of the phase interface (bubble wall) and the generation of Chern-Simons number in each phase and on the bubble wall.
