Quark matter nucleation in hot hadronic matter

We study the quark deconfinement phase transition in hot $\beta$-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 mass $M_{cr}$ for proto-hadronic stars, and we show that proto-hadronic stars with a mass $M<M_{cr}$ could survive the early stages of their evolution without decaying to a quark star.


Introduction
In the last few years there has been a growing interest in the study of the nucleation process of quark matter (QM) in the core of massive neutron stars. In particular, it has been shown [1][2][3][4][5][6][7][8] that above a threshold value of the central pressure a pure hadronic compact star (HS) is metastable to the decay (conversion) to a quark star (QS) (i.e., to a hybrid neutron star or to a strange star [9,10], depending on the details of the equation of state (EOS) for quark matter used to model the phase transition [11][12][13][14]). This stellar conversion process liberates a huge amount of energy (a few 10 53 erg) [15] and it could be the energy source of some of the long Gamma Ray Bursts (GRBs).
The research reported in Refs. [1][2][3][4][5][6][7][8] has focused on the quark deconfinement phase transition in cold (T = 0) and neutrino-free neutron stars. In this case the formation of the first drop of QM could take place solely via a quantum nucleation process.
A neutron star at birth (proto-neutron star) is very hot (T = 10-30 MeV) with neutrinos being still trapped in the stellar interior [16,17]. Subsequent neutrino diffusion causes deleptonization and heats the stellar matter to an approximately uniform entropy * Corresponding author. Depending on the stellar composition, during this stage neutrino escape can lead the more "massive" stellar configurations to the formation of a black hole [17,18]. However, if the mass of the star is sufficiently small, the star will remain stable and it will cool to temperatures well below 1 MeV within a cooling time t cool ∼ a few 10 2 s, as the neutrinos continue to carry energy away from the stellar material [16,17]. Thus in a proto-neutron star, the quark deconfinement phase transition will be likely triggered by a thermal nucleation process. In fact, for sufficiently high temperatures, thermal nucleation is a much more efficient process with respect to the quantum nucleation mechanism. Some of the earlier studies on quark matter nucleation (see e.g. [19][20][21][22][23]) have already dealt with thermal nucleation in hot and dense hadronic matter. In these studies, it was found that the prompt formation of a critical size drop of quark matter via thermal activation is possible above a temperature of about 2-3 MeV. As a consequence, it was inferred that pure hadronic stars are converted to quark stars within the first seconds after their birth. However, these works [19][20][21][22] reported an estimate of the thermal nucleation based on "typical" values for the thermodynamic properties characterizing the central part of neutron stars.
Our main objective in this and next related works, is to establish if a newborn hadronic star (proto-hadronic star) could survive the early stages of its evolution without "decaying" to a quark star. In the present Letter, we calculate the thermal nucleation rate of quark matter in hot (T = 0) and neutrino-free hadronic matter using a finite temperature EOS for hadronic and quark matter. In addition, we calculate the quantum nucleation rate at finite temperature, and compare the thermal and quantum nucleation time at different temperatures and pressures characterizing the central conditions of metastable proto-hadronic compact stars. We compute the crossover temperature above which thermal nucleation dominates above the finite temperature quantum nucleation mechanism. Finally we briefly discuss some consequences for the physics of proto-neutron stars.

Phase equilibrium
For a first-order phase transition 1 the conditions for phase equilibrium are given by the Gibbs' phase rule are the Gibbs' energies per baryon (average chemical potentials) for the hadron and quark phase, respectively, ε H (ε Q ), P H ( P Q ), s H (s Q ) and n H (n Q ) denote respectively the total (i.e., including leptonic contributions) energy density, total pressure, total entropy density, and baryon number density for the hadron (quark) phase. Above the "transition point" ( P 0 ) the hadronic phase is metastable, and the stable quark phase will appear as a results of a nucleation process. Small localized fluctuations in the state variables of the metastable hadronic phase will give rise to virtual drops of the stable quark phase. These fluctuation are characterized by a time scale ν −1 0 ∼ 10 −23 s. This time scale is set by the strong interactions (which are responsible for the deconfinement phase transition), and it is many orders of magnitude shorter than the typical time scale for the weak interactions. Therefore quark flavor must be conserved during the deconfinement transition. We will refer to this form of deconfined matter, in which the flavor content is equal to that of the β-stable hadronic system at the same pressure and temperature, as the Q * -phase. Soon afterward a critical size drop of quark matter is formed, the weak interactions will have enough time to act, changing the quark flavor fraction of the deconfined droplet to lower its energy, and a droplet of β-stable quark matter is formed (hereafter the Q -phase).
This first seed of quark matter will trigger the conversion [15,24,25] of the pure hadronic star to a hybrid star or to a strange star. Thus, pure hadronic stars with values of the central pressure larger than P 0 are metastable to the decay (conversion) to hybrid stars or to strange stars [1][2][3][4][5]. The mean lifetime of the metastable stellar configuration is related to the time needed to nucleate the first drop of quark matter in the stellar center and depends dramatically on the value of the stellar central pressure [1][2][3][4][5].

Quantum and thermal nucleation rates
The main effect of finite temperature on the quantum nucleation mechanism of quark matter is to modify the energy barrier 1 We assume the quark deconfinement phase transition to be of the first order.
This assumption is common in most of the studies of quark deconfinement in compact stars.
separating the quark phase from the metastable hadronic phase. This energy barrier, which represents the difference in the free energy of the system with and without a Q * -matter droplet, can be written as where R is the radius of the droplet (supposed to be spherical), and σ is the surface tension for the surface separating the hadron from the Q * -phase. The energy barrier has a maximum at the . Notice that we have neglected the term associated with the curvature energy, and also the terms connected with the electrostatic energy, since they are known to introduce small corrections [2,29]. The value of the surface tension σ for the interface separating the quark and hadron phase is poorly known, and typically values used in the literature range within 10-50 MeV fm −2 [26,28,29]. We assume σ to be temperature independent and we take σ = 30 MeV fm −2 .
The quantum nucleation time τ q can be straightforwardly evaluated within a semi-classical approach [27][28][29]. First one computes, in the WKB approximation, the ground state energy E 0 and the oscillation frequency ν 0 of the drop in the potential well U (R, T ). Then, the probability of tunneling is given by where A(E) is the action under the potential barrier, which in a relativistic framework reads being R ± the classical turning points and m(R) the droplet effective mass. The quantum nucleation time is then equal to with N c ∼ 10 48 being the number of nucleation centers expected in the innermost part (r R nuc ∼ 100 m) of the hadronic star, where the pressure and temperature can be considered constant and equal to their central values.
The thermal nucleation rate can be written [32] as where κ is the so-called dynamical prefactor, which is related to the growth rate of the drop radius R near the critical radius (R c ), Ω 0 is the so-called statistical prefactor, which measures the phasespace volume of the saddle-point region around R c , and U (R c , T ) is the activation energy, i.e., the change in the free energy of the system required to activate the formation of a critical size droplet. The Langer theory [30][31][32][33] of homogeneous nucleation has been extended in Refs. [34,35] to the case of first order phase transitions occurring in relativistic systems, as in the case of the quark deconfinement transition. The statistical prefactor, can be written [34] as where ξ Q is the quark correlation length, which gives a measure of the thickness of the interface layer between the two phases (the droplet "surface thickness"). In the present calculation we take ξ Q = 0.7 fm according to the estimate given in Refs. [22,34].
For the dynamical prefactor we have used a general expression which has been derived by Venugopalan and Vischer [35] (see also Refs. [34,36] where w = w Q * − w H is the difference between the enthalpy density of the two phases, λ the thermal conductivity, η and ζ are the shear and bulk viscosities respectively of hadronic matter. Notice that the nucleation prefactor used in the present work differs significantly from the one used in previous works [19][20][21] where, based on dimensional grounds, the prefactor was taken to be equal to T 4 . There are not many calculations of the transport properties of dense hadronic matter. With a few exceptions (see e.g. [37,38]), most of them are relative to nuclear or pure neutron matter [39][40][41][42][43]. These quantities have been calculated by Danielewicz [40] in the case of nuclear matter. According to the results of Ref. [40], the dominant contribution to the prefactor κ comes from the shear viscosity η. Therefore, we take λ and ζ equal to zero, and we use for the shear viscosity the following relation [40]:

Equation of state
Over the last decade, it has been realized that strong interacting matter at high density and low temperature may possess a large assortment of phases. Different possible patterns for color superconductivity have been conjectured (see e.g. [44,45] and references therein quoted). Very recently, a new phase of QCD, named quarkyonic phase, has been predicted [46,47]. This hypothetical matter phase is characterized by chiral symmetry and confinement [46][47][48].
In the present work, we have adopted a more traditional view, assuming a single first order phase transition between the confined (hadronic) and deconfined phase of dense matter, and we have used rather common models for describing them. For the hadronic phase we have used models which are based on a relativistic Lagrangian of hadrons interacting via the exchange of σ , ρ and ω mesons. We have used one of the parameters sets given in Ref. [49]: hereafter we refer to this model as the GM1 equation of state. For the quark phase we have adopted a phenomenological EOS [11] which is based on the MIT bag model for hadrons. In this work, we have used the following set of parameters: m u = m d = 0, m s = 150 MeV for the masses of the up, down and strange quark, respectively, B = 85 MeV/fm 3 for the bag constant, and α s = 0 for the QCD structure constant. The two models for the EOS have been generalized to the case of finite temperature.

Results and discussion
In Fig. 1 we plot the Gibbs' energies per baryon for the hadronphase and the Q * -phase at different temperatures, T = 0, 10, 20, 3 MeV; lines with the larger slope refer to the hadron-phase. As we see, the transition pressure P 0 (indicated by a full dot) decreases when the hadronic matter temperature is increased. The  phase equilibrium curve P 0 (T ) for the hadron-quark phase transition (within the present schematic model for the EOS) is shown in Fig. 2. As it is well known, for a first-order phase transition the derivative dP 0 /dT is related to the specific (i.e., per baryon) latent heat Q of the phase transition by the Clapeyron-Clausius equation (13) whereW H (W Q * ) andS H (S Q * ) denote respectively the enthalpy per baryon and entropy per baryon for the hadron (quark) phase. The specific latent heat Q and the phase numbers densities n H and n Q * at phase equilibrium are reported in Table 1. As expected for a first order phase transition one has a discontinuity jump in the phase number densities: in our particular case n Q * (T , P 0 ) > n H (T , P 0 ). This result, together with the positive value of Q (i.e., the deconfinement phase transition absorbs heat) tell us (see Eq. (12)), that the transition temperature decreases with pressure (as in the melting of ice).
In Fig. 3, we represent the energy barrier for a virtual drop of the Q * -phase as a function of the droplet radius and for different temperatures at a fixed pressure P = 57 MeV/fm 3 . As expected, from the results plotted in Fig. 1, the energy barrier U (R, T ) and Table 1 The specific latent heat Q and the phase numbers densities n H and n Q * at phase equilibrium.   the droplet critical radius R c decrease as the matter temperature is increased. This effect favors the Q * -phase formation, and in particular increases (decreases) the quantum nucleation rate (nucleation time τ q ) with respect to the corresponding quantities calculated at T = 0.
In Fig. 4 we plot the quantum and thermal nucleation times of the Q * -phase in β-stable hadronic matter as a function of temperature and at a fixed pressure P = 57 MeV/fm 3 . As expected, we find a crossover temperature T co above which thermal nucle-  will nucleate a Q * -matter drop during the early stages of their evolution, and will finally evolve to cold and deleptonized quark stars, or will collapse to black holes. The lines labeled T S represent the stellar matter temperature as a function of pressure at fixed entropies per baryonS/k B = 1 (dashed line) and 2 (solid line).
ation is dominant with respect to the quantum nucleation mechanism. For the case reported in Fig. 4, we have T co = 7.05 MeV and the corresponding nucleation time is log 10 (τ /s) = 54.4. The crossover temperature, for different values of the pressure of βstable hadronic matter, is reported in Table 2 (second column) together with the nucleation time calculated at T = T co (third column).
Having in mind the physical conditions in the interior of a proto-hadronic star [16,17] (see Section 1 of the present Letter), to establish if this star will survive the early stages of its evolution without "decaying" to a quark star, one has to compare the quark matter nucleation time τ = min(τ q , τ th ) with the cooling time t cool ∼ a few 10 2 s. If τ t cool then quark matter nucleation will not likely occur in the newly formed star, and this star will evolve to a cold deleptonized configuration. We thus introduce the concept of limiting conversion temperature Θ for the proto-hadronic star and define it as the value of the stellar central temperature T c for which the Q * -matter nucleation time is equal to 10 3 s.
The limiting conversion temperature Θ will clearly depend on the value of the stellar central pressure (and thus on the value of the stellar mass).
The limiting conversion temperature Θ is plotted in Fig. 5 as a function of the stellar central pressure. A proto-hadronic star with a central temperature T c > Θ will likely nucleate a Q * -matter drop during the early stages of its evolution, and will finally evolve to a cold and deleptonized quark star, or will collapse to a black hole (depending on the particular model adopted for the matter EOS).
For an isoentropic stellar core [16,17], the central temperature of the proto-hadronic star is given, for the present EOS model, by the lines labeled by T S in Fig. 5, relative to the caseS = 1k B (dashed curve) andS = 2k B (continuous curve). The intersection point ( P S , Θ S ) between the two curves Θ(P ) and T S (P ) thus gives the central pressure and temperature of the configuration that we denote as the critical mass configuration of the protohadronic stellar sequence. The value of the gravitational critical mass M cr = M(P S , Θ S ) and baryonic critical mass M B,cr are reported in Table 3, for three different choices of the entropy per baryon,S/k B = 0 (corresponding to a cold hadronic star), 2 1 and 2.
In the same table, we also report the value of the gravitational mass M of the cold hadronic star with baryonic mass equal to M B,cr . This configuration is stable (τ = ∞) with respect to Q *matter nucleation in the caseS/k B = 2, and it is essentially stable (having a nucleation time enormously larger than the age of the universe) in the caseS/k B = 1. Note that these numbers are model dependent and, therefore, one must take them just as indicative values. A careful analysis is beyond the scope of the present work and it will be addressed in a future work.
In summary, in this work we have studied the quark deconfinement phase transition in hot β-stable hadronic matter, and we have explored some of its consequences for the physics of neutron stars at birth. Our main finding is that proto-hadronic stars with a mass lower that the critical value M cr could survive the early stages of their evolution without decaying to a quark star. However, the prompt formation of a critical size drop of quark matter could take place when M > M cr . These proto-hadronic stars evolve to cold and deleptonized quark stars, or collapse to a black holes. Finally, if quark matter nucleation occurs during post-bounce stage of core-collapse supernova, then the quark deconfinement phase transition could trigger a delayed supernova explosion characterized by a peculiar neutrino signal [50].