We study the dynamics of confinement-deconfinement (C-D) phase transition in the context of relativistic heavy-ion collisions within the framework of effective models for the Polyakov loop order parameter. We study the formation of $Z(3)$ walls and associated strings in the initial transition from the confining (hadronic) phase to the deconfining (QGP) phase via the so called Kibble mechanism. Essential physics of the Kibble mechanism is contained in a sort of domain structure arising after any phase transition which represents random variation of the order parameter at distances beyond the typical correlation length. We implement this domain structure by using the Polyakov loop effective model with a first order phase transition and confine ourselves with temperature/time ranges so that the first order C-D transition proceeds via bubble nucleation, leading to a well defined domain structure. The formation of $Z(3)$ walls and associated strings results from the coalescence of QGP bubbles expanding in the confining background. We investigate the evolution of the $Z(3)$ wall and string network. We also calculate the energy density fluctuations associated with $Z(3)$ wall network and strings which decay away after the temperature drops below the quark-hadron transition temperature during the expansion of QGP. We discuss evolution of these quantities with changing temperature via Bjorken's hydrodynamical model and discuss possible experimental signatures resulting from the presence of $Z(3)$ wall network and associate strings.
Starting from the QCD Lagrangian and the surface tension of QCD bubbles we derive the critical size of bubbles, the nucleation probability and the nucleation site separation distance. We find the separation of sites is more than an order of magnitude larger than previous QCD model estimates, which could lead to observable effects.
Localised heterogeneities have been recently discovered to act as bubble-nucleation sites in nonlinear field theories. Vacuum decay seeded by black holes is one of the most remarkable applications. This article proposes a simple and exactly solvable $ϕ^4$ model exhibiting bubble nucleation around localised heterogeneities. Bubbles with a rich dynamical behaviour are observed depending on the topological properties of the heterogeneity. The linear stability analysis of soliton-bubbles predicts the formation of oscillating bubbles and the insertion of new bubbles inside an expanding precursor bu…
The standard vacuum bounce formalism suffers from inconsistencies when applied to thermal bubble nucleation, for which ad hoc workarounds are commonly adopted. Identifying the length scales on which nucleation takes place, we demonstrate how the construction of an effective description for these scales naturally resolves the problems of the standard vacuum bounce formalism. Further, by utilising high-temperature dimensional reduction, we make a connection to classical nucleation theory. This offers a clear physical picture of thermal bubble nucleation, as well as a computational framework whic…
On the basis of Borel resummation, we propose a systematical improvement of bounce calculus of quantum bubble nucleation rate. We study a metastable super-renormalizable field theory, $D$ dimensional O(N) symmetric $ϕ^4$ model ($D<4$) with an attractive interaction. The validity of our proposal is tested in D=1 (quantum mechanics) by using the perturbation series of ground state energy to high orders. We also present a result in D=2, based on an explicit calculation of vacuum bubble diagrams to five loop orders.
We study the probability for nucleation of quark matter droplets in the dense cold cores of old neutron stars induced by the presence of a self-annihilating dark matter component, $χ$. Using a parameterized form of the equation of state for hadronic and quark phases of ordinary matter, we explore the thermodynamic conditions under which droplet formation is facilitated by the energy injection from $χ$ self-annihilations. We obtain the droplet nucleation time as a function of the dark matter candidate mass, $m_χ$. We discuss further observational consequences.
We calculate the bubble nucleation rate in a fourth-order gravity theory whose action contains an $R^2$ term, under the thin-wall approximation, by two different methods. First the bounce solution is found for the Euclidean version of the original action. Next, we use a conformal transformation to transform the theory into general relativity with an additional scalar field and then proceed to find the bounce solution. The same results are obtained from both calculations.
I calculate the first corrections to the dynamical pre-exponential factor of the bubble nucleation rate for a relativistic first-order phase transition in an expanding cosmological background by estimating the effects of the Hubble expansion rate on the critical bubbles of Langer's statistical theory of metastability. I also comment on possible applications and problems that arise when one considers the field theoretical extensions of these results (the Coleman-De Luccia and Hawking-Moss instantons and decay rates).
The effect of subcritical hadron bubbles on a first-order quark-hadron phase transition is studied. These subcritical hadron bubbles are created due to thermal fluctuations, and can introduce a finite amount of phase mixing (quark phase mixed with hadron phase) even at and above the critical temperature. For reasonable choices of surface tension and correlation length, as obtained from the lattice QCD calculations, we show that the amount of phase mixing at the critical temperature remains below the percolation threshold. Thus, as the system cools below the critical temperature, the transition…
In field theories where a metastable false vacuum state arises as a result of radiative corrections, the calculation of the rate of false vacuum decay by bubble nucleation depends on the effective potential and the other functions that appear in the derivative expansion of the effective action. Beginning with the Nielsen identity, we derive a series of identities that govern the gauge dependence of these functions. Using these, we show, to leading nontrivial order, that even though these functions are individually gauge-dependent, one obtains a gauge-independent result for the bubble nucleatio…
We present an evaluation of the 1-loop prefactor in the lifetime of a metastable state which decays at finite temperature by bubble nucleation. Such a state is considered in one-component phi^4 model in three space dimensions. The calculation serves as a prototype application of a fast numerical method for evaluating the functional determinants that appear in semiclassical approximations.
We present a study of the role of fermions in the decay of metastable states of a scalar field via bubble nucleation. We analyze both one and three-dimensional systems by using a gradient expansion for the calculation of the fermionic determinant. The results of the one-dimensional case are compared to the exact results of previous work.
We estimate bubble-nucleation rates for cosmological phase transitions. We concentrate on the evaluation of the pre-exponential factor, for which we give approximate analytical expressions. Our approach relies on the use of a real coarse-grained potential. We show how the coarse-graining scale can be determined in the studies of high-temperature phase transitions. We discuss the metastability bound on the Higgs-boson mass and the electroweak phase transition. We find that the saddle-point approximation is reliable in the first case and breaks down in the second case.
Two equations for the calculation of the critical energy required for homogeneous nucleation in a superheated liquid, and the related critical radius of the nucleated vapour bubble, are obtained, the former by the direct application of the first law of thermodynamics, the latter by considering that the bubble formation implies the overcoming of a barrier of the free enthalpy potential. Comparisons with the currently used relationships demonstrate that the sensitivity of the bubble chambers employed in dark matter searches can be sometimes notably overestimated.
We summarize recent work on the consistent calculation of bubble-nucleation rates. Our approach is based on the notion of a real coarse-grained potential. The bubble-nucleation rate is calculated through an expansion around the semiclassical saddle point associated with tunnelling. We resolve outstanding problems related to the convexity of the potential, the double-counting of the effect of fluctuations and the inherent ultraviolet divergences. We determine the region of validity of the expansion around the saddle point. We find that this expansion fails near the spinodal line, and for weak o…
A gauge-invariant framework for computing bubble nucleation rates at finite temperature in the presence of radiative barriers was presented and advocated for model-building and phenomenological studies in an accompanying article arXiv:2112.05472. Here, we detail this computation using the Abelian Higgs Model as an illustrative example. Subsequently, we recast this approach in the dimensionally-reduced high-temperature effective field theory for nucleation. This allows for including several higher order thermal resummations and furthermore delineate clearly the approach's limits of validity. Th…
A formalism is developed to calculate the bubble nucleation rate in theories where the symmetry breaking is by radiative corrections. Although to leading and next-to-leading order the result can be expressed in terms of the effective action of the theory, this is not true of the higher order corrections. In particular, the imaginary part of the effective potential does not explicitly enter the calculation and the problems of interpretation which it would entail are avoided. Based on a talk given at the NATO Advanced Research Workshop Electroweak Physics and the Early Universe, Sintra, Portugal…
We study induced nucleation by considering the accumulation rate of shrinking subcritical bubbles. We derive the probability for a collection of subcritical bubbles to form a critical bubble, and argue that this mechanism could well play a role in electroweak phase transitions if the Higgs is heavy.
SISSA, International School for Advanced Studies（国际高等研究院）；INFN Sezione di Trieste（意大利国家核物理研究所的里雅斯特分部）；IFPU, Institute for Fundamental Physics of the Universe（宇宙基础物理研究所）
At first order phase transitions the transition proceeds through droplet nucleation and growth. We discuss a lattice method for calculating the droplet nucleation rate, including the complete dynamical factors. The method is especially suitable for very strongly suppressed droplet nucleation, which is often the case in physically interesting transitions. We apply the method to the 3-dimensional cubic anisotropy model in a parameter range where the model has a radiatively induced strong first order phase transition, and compare the results with analytical approaches.
We develop a method to calculate the prefactor in the expression for the bubble nucleation rate. A fermion with Yukawa coupling is considered where a step potential can be used as a good approximation in the thin wall limit. Corrections due to thicker walls are considered by perturbing about the thin wall case. We derive the thermal one loop effective action, calculating it numerically, and find that the prefactor in the nucleation rate can both suppress and enhance, for a given temperature, when the usual renormalisation conditions are applied to the effective potential.
The thermal nucleation of quark matter bubbles inside neutron stars is examined for various temperatures which the star may realistically encounter during its lifetime. It is found that for a bag constant less than a critical value, a very large part of the star will be converted into the quark phase within a fraction of a second. Depending on the equation of state for neutron star matter and strange quark matter, all or some of the outer parts of the star may subsequently be converted by a slower burning or a detonation.
Modern particle physics and cosmology support the idea that a background of invisible material pervades the whole universe, and identify in the cosmic vacuum the ultimate source of matter-energy, both seen and unseen. Within the framework of the theory of fundamental relativistic membranes, we suggest a self-consistent, vacuum energy-driven mechanism for dark matter creation through gauge symmetry rearrangement.
One possible way in which phase transitions in the early universe may have ocurred is via nucleation of bubbles of the new phase (true vacuum) in the old phase (false vacuum). The technique most widely used to compute the probability of bubble nucleation is based on instanton methods in the context of the semiclassical approximation. At zero temperature in (3+1) dimensions the nucleation rate is dominated by the O(4) symmetric instanton, a sphere of radius $R$, while at temperatures $T >> R^{-1}$, the decay is dominated by a ``cylindrical'' (static) instanton wtih O(3) invariance. There has be…
The fluctuation determinant which determines the preexponential factor of the transition rate for minimal bubbles is computed for the electroweak theory with $\sin Θ_W = 0$. As the basic action we use the three-dimensional high-temperature action including, besides temperature dependent masses, the $T Φ^3$ one-loop contribution which makes the phase transition first order. The results show that this contribution (which has then to be subtracted from the exact result) gives the dominant contribution to the one-loop effective action. The remaining correction is of the order of, but in general la…
We investigate the scenario of homogeneous nucleation for a first order quark-hadron phase transition in a rapidly expanding background of quark gluon plasma. Using an improved preexponential factor for homogeneous nucleation rate, we solve a set of coupled equations to study the hadronization and the hydrodynamical evolution of the matter. It is found that significant supercooling is possible before hadronization begins. This study also suggests that spinodal decomposition competes with nucleation and may provide an alternative mechanism for phase conversion particularly if the transition is …
We determine the fermionic corrections to the nucleation rate of bubbles at the electroweak phase transition. The fermion determinant is evaluated exactly and by using the gradient expansion. The gradient expansion is found to be a reliable approximation and is used to extrapolate to the large values of $ν_n = (2n+1)πT$ needed in the Matsubara sum. The contribution to effective action is found to be negative and to be given, essentially, by the gradient terms, the finite part of the wave function renormalization. Only the top quark contribution is evaluated, it is of the same order as the Higg…
We include the electromagnetic currents from fermion degrees of freedom in the equations of motion for electroweak MSSM with a right-handed Stop that we have recently investigated. It is found that near the surface of the bubble walls there are important effects on the electromagnetic fields produced during bubble nucleation.
Impact of Bubble Nucleation History and Friction on Primordial Black Hole Formation
Regional Centre for Accelerator based Particle Physics, Harish-Chandra Research Institute（哈里斯-钱德拉研究所加速器粒子物理区域中心）；Homi Bhabha National Institute（霍米·巴巴国立研究所）
Classical-statistical lattice simulations provide a useful approximation to out-of-equilibrium quantum field theory, but only for systems exhibiting large occupation numbers, and only for phenomena that are not intrinsically quantum mechanical in nature. In certain special circumstances, it can be appropriate to initialize such real-time simulations with quantum-like zero-point fluctuations. We will revisit these points, and investigate reports that quantum bubble nucleation rates in 1+1 dimensions can be computed through the classical evolution of such a quantum-like initial condition. We fin…
We derive the equations of motion for electroweak MSSM with a right-handed Stop, from which we derive the equations for the electromagnetic field that arises from bubble nucleation and collisions during the first order electroweak phase transition that can occur in this MSSM. Introducing an isospin ansatz we derive e.o.m. for the electrically charged W fields uncoupled from all other fields. These serve as the current for the Maxwell-like e.o.m. for the em field. The resulting electromagnetic field arising during EWPT bubble nucleation is found. This electromagnetic structure, along with that …
We present numerical results on bubble profiles, nucleation rates and time evolution for a weakly first-order quark-hadron phase transition in different expansion scenarios. We confirm the standard picture of a cosmological first-order phase transition, in which the phase transition is entirely dominated by nucleation. We also show that, even for expansion rates much lower than those expected in heavy-ion collisions nucleation is very unlikely, indicating that the main phase conversion mechanism is spinodal decomposition.
We study gravitational-wave production from bubble collisions in a cosmic first-order phase transition, focusing on the possibility of model separation by the bubble nucleation rate dependence of the resulting gravitational-wave spectrum. By using the method of relating the spectrum with the two-point correlator of the energy-momentum tensor $\left< T(x)T(y) \right>$, we first write down analytic expressions for the spectrum with a Gaussian correction to the commonly used nucleation rate, $Γ\propto e^{βt}\rightarrow e^{βt-γ^2t^2}$, under the thin-wall and envelope approximations. Then we quant…
We present a consistent calculation of bubble-nucleation rates in theories of two scalar fields. Our approach is based on the notion of a coarse-grained free energy that incorporates the effects of fluctuations with momenta above a given scale k. We establish the reliability of the method for a variety of two-scalar models and confirm the conclusions of previous studies in one-field theories: Langer's theory of homogeneous nucleation is applicable as long as the expansion around the semiclassical saddle point associated with tunnelling is convergent. This expansion breaks down when the exponen…
This paper extends classical results by Langer and Kramers and combines them with modern methods from high-temperature field theory. Assuming Langevin dynamics, the end-product is an all-orders description of bubble-nucleation at high temperatures. Specifically, it's shown that equilibrium and non-equilibrium effects factorize to all orders, and that the nucleation rate splits into a statistical and a dynamical prefactor. The derivation clarifies, and incorporates, higher-order corrections from zero-modes. The rate is also shown to be real to all orders in perturbation theory. The methods are …
We investigate the dynamics of a strong first-order quark-hadron transition driven by cubic interaction via homogeneous bubble nucleation in the Friedberg-Lee model. The one-loop effective thermodynamics potential of the model and the critical bubble profiles have been calculated at different temperatures and chemical potentials. By taking the temperature and the chemical potential as the variables, the evolutions of the surface tension, the typical radius of the critical bubble and the shift in the coarse-grained free energy in the presence of a nucleation bubble are obtained and the limit on…
We study the quark deconfinement phase transition in hot $β$-stable hadronic matter. Assuming a first order phase transition, we calculate the enthalpy per baryon of the hadron-quark phase transition. We calculate and compare the nucleation rate and the nucleation time due to thermal and quantum nucleation mechanisms. We compute the crossover temperature above which thermal nucleation dominates the finite temperature quantum nucleation mechanism. We next discuss the consequences for the physics of proto-neutron stars. We introduce the concept of limiting conversion temperature and critical mas…
In this paper I discuss how to consistently incorporate higher-order corrections to the bubble-nucleation rate at finite temperature. Doing so I examine the merits of different approaches, with the goal of reducing uncertainties for gravitational-wave calculations. To be specific, the region of applicability and accuracy of the derivative expansion is discussed. The derivative expansion is then compared to a numerical implementation of the Gelfand-Yaglom theorem. Both methods are applied to popular first-order phase transition models, like a loop-induced barrier and a SM-EFT tree-level barrier…
We discuss finite-size effects on homogeneous nucleation in first-order phase transitions. We study their implications for cosmological phase transitions and to the hadronization of a quark-gluon plasma generated in high-energy heavy ion collisions. Very general arguments allow us to show that the finite size of the early universe has virtually no relevance in the process of nucleation and in the growth of cosmological bubbles during the primordial quark-hadron and the electroweak phase transitions. In the case of high-energy heavy ion collisions, finite-size effects play an important role in …
We present a lattice method to compute bubble nucleation rates at radiatively induced first order phase transitions, in high temperature, weakly coupled field theories, nonperturbatively. A generalization of Langer's approach, it makes no recourse to saddle point expansions and includes completely the dynamical prefactor. We test the technique by applying it to the electroweak phase transition in the minimal standard model, at an unphysically small Higgs mass which gives a reasonably strong phase transition (lambda/g^2 =0.036, which corresponds to m(Higgs)/m(W) = 0.54 at tree level but does no…
We have recently suggested that tiny black holes can act as nucleation seeds for the decay of the metastable Higgs vacuum. Previous results applied only to the nucleation of thin-wall bubbles, and covered a very small region of parameter space. This paper considers bubbles of arbitrary profile and reaches the same conclusion: black holes seed rapid vacuum decay. Seeded and unseeded nucleation rates are compared, and the gravitational back reaction of the bubbles is taken into account. The evolution of the bubble interior is described for the unseeded nucleation. Results are presented for the r…
We compute the probability distribution of the invariant separation between nucleation centers of colliding true vacuum bubbles arising from the decay of a false de Sitter space vacuum. We find that even in the limit of a very small nucleation rate per unit Hubble volume the production of widely separated bubble pairs is suppressed. This distribution is of particular relevance for the recently proposed ``colliding bubble braneworld'' scenario, in which the value of Omega_k (the contribution of negative spatial curvature to the cosmological density parameter) is determined by the invariant sepa…
We calculate gravitational wave power spectra from first order early Universe phase transitions using the Sound Shell Model. The model predicts that the power spectrum depends on the mean bubble separation, the phase transition strength, the phase boundary speed, with the overall frequency scale set by the nucleation temperature. There is also a dependence on the time evolution of the bubble nucleation rate. The gravitational wave peak power and frequency are in good agreement with published numerical simulations, where bubbles are nucleated simultaneously. Agreement is particularly good for d…
False vacuum decay in quantum mechanical first order phase transitions is a phenomenon with wide implications in cosmology, and presents interesting theoretical challenges. In the standard approach, it is assumed that false vacuum decay proceeds through the formation of bubbles that nucleate at random positions in spacetime and subsequently expand. In this paper we investigate the presence of correlations between bubble nucleation sites using a recently proposed semi-classical stochastic description of vacuum decay. This procedure samples vacuum fluctuations, which are then evolved using class…
We present a consistent picture of tunnelling in field theory. Our results apply both to high-temperature field theories in four dimensions and to zero-temperature three-dimensional ones. Our approach is based on the notion of a coarse-grained potential U_k that incorporates the effect of fluctuations with characteristic momenta above a given scale k. U_k is non-convex and becomes equal to the convex effective potential for k --> 0. We demonstrate that a consistent calculation of the nucleation rate must be performed at non-zero values of k, larger than the typical scale of the saddle-point co…
