Critical spectrum of fluctuations for deconfinement at proto-neutron star cores
We study the deconfinement of hadronic matter into quark matter in a protoneutron star focusing on the effects of the finite size on the formation of just-deconfined color superconducting quark droplets embedded in the hadronic environment. The hadronic phase is modeled by the non-linear Walecka model at finite temperature including the baryon octet and neutrino trapping. For quark matter we use an Nambu-Jona-Lasinio model including color superconductivity. The finite size effects on the just deconfined droplets are considered in the frame of the multiple reflection expansion. In addition, we consider that just deconfined quark matter is transitorily out of equilibrium respect to weak interaction, and we impose color neutrality and flavor conservation during the transition. We calculate self-consistently the surface tension and curvature energy density of the quark hadron inter-phase and find that it is larger than the values typically assumed in the literature. The transition density is calculated for drops of different sizes, and at different temperatures and neutrino trapping conditions. Then, we show that energy-density fluctuations are much more relevant for deconfinement than temperature and neutrino density fluctuations. We calculate the critical size spectrum of energy-density fluctuations that allows deconfinement as well as the nucleation rate of each critical bubble. We find that drops with any radii smaller than 800 fm can be formed at a huge rate when matter achieves the bulk transition limit of 56 times the nuclear saturation density.
pacs:12.39.Fe, 25.75.Nq, 26.60.Kp
The cores of massive neutron stars have densities that may favor the nucleation of small droplets of quark matter, that under appropriate conditions may grow converting a large part of the star into quark matter (for recent work see (1); (2); (3); (4) and references therein). Since the the equation of state above the nuclear saturation density is still uncertain the problem is usually analyzed within a two-phase description in which a hadronic model valid around the nuclear saturation density is extrapolated to larger densities and a quark model that is expected to be valid only for asymptotically large densities is extrapolated downwards. Within this kind of analysis some work has been performed recently in order to determine the effect of different hadronic and quark equations of state, as well as the effect of temperature, neutrino trapping and color superconductivity (1).
An important characteristic of the deconfinement transition in neutron stars, is that quark and lepton flavors must be conserved during the process leaving just deconfined quark matter that is transitorily out of equilibrium with respect to weak interactions (see (1) and references therein). When color superconductivity is included in the analysis together with flavor conservation, it is found that the most likely configuration of the just deconfined phase is two-flavor color superconductor (2SC) provided the pairing gap is large enough (5).
Since our previous studies were made in bulk, we shall now include finite size effects in the description of the just deconfined drops. To this end we employ the Nambu-Jona-Lasinio (NJL) model for quark matter and include finite size effects within the multiple reflection expansion (MRE) framework (see (6); (7); (8) and references therein). Through this analysis we can determine the density of hadronic matter at which deconfinement is possible for different radii of the just formed quark drops using typical conditions expected in the interior of protoneutron stars (PNSs), i.e. temperatures in the range of MeV and chemical potentials of the trapped neutrino gas up to MeV. Notice that to form a finite size drop it is needed some over-density with respect to the bulk transition density in order to compensate the surface and curvature energy cost. Since this energy cost depends on the drop radius , so does the necessary over-density necessary to nucleate it. Thus, we can derive a critical fluctuation spectrum ( versus ) delimiting which over-densities can deconfine and grow unlimitedly and which ones will shrink back to hadronic matter.
The paper is organized as follows. In Section II we briefly describe main features of the hadronic model. In Sect. III we present the quark model we use taking into account finite size effects and derive self-consistently the surface tension and the curvature energy from the MRE thermodynamic potential. In Sect IV we the find the density at which color superconducting droplets of different radii can be nucleated in protoneutron star matter. In Section V we find the critical spectrum of density fluctuations as well as the nucleation rate of the critical size droplets. In Sect. V we present our conclusions.
Ii The hadronic phase
where the indices , and refer to baryons, mesons and leptons respectively. For the baryons we have
with extending over the nucleons , and the following hyperons , , , , , and . The contribution of the mesons , and is given by
where the coupling constants are
with , and equal to for the nucleons and 0.9 for hyperons. Electrons and neutrinos are included as a free Fermi gas in chemical equilibrium with all other particles.
There are five constants in the model that are determined by the properties of nuclear matter, three that determine the nucleon couplings to the scalar, vector and vector-isovector mesons , , , and two that determine the scalar self interactions and . It is assumed that all hyperons in the octet have the same coupling than the . These couplings are expressed as a ratio to the nucleon couplings mentioned above, that we thus simply denote , and .
In the present work we use the following values for the constants: , , , and . This model has been labelled as GM4 in previous work (1). The equation of state is rather stiff and gives a maximum mass of 2 M, that seems to be more adequate in light of the recently determined mass of the pulsar PSR J1614-2230 with (13). This is the largest mass reported ever for a pulsar with a high precision. For more details on the explicit form of the equation of state the reader is referred to Ref. (1).
Iii The quark matter phase
The just deconfined quark matter phase is implemented by using the NJL effective model with the inclusion quark-quark interactions, which are the responsible for color superconductivity.
The corresponding Lagrangian is given by
where is the current mass matrix in flavor space. In the present paper we work in the isospin symmetric limit . Moreover, and with are the Gell-Mann matrices corresponding to the flavor and color groups respectively, and . Finally, the charge conjugate spinors are defined as follows: and , where is the Dirac conjugate spinor and .
In order to determine the relevant thermodynamic quantities we have to obtain the grand canonical thermodynamic potential at finite temperature and chemical potentials . Here, and denotes flavor and color indices respectively. In the following we present the thermodynamic potential for the bulk system and then, in Sect. III B, we discuss the effects of finite size in the effective potential, for the spherical droplets.
iii.1 Quark matter in bulk
Starting from Eq. (5), we perform the usual bosonization of the theory. We introduce the scalar and pseudoscalar meson fields and respectively, together with the bosonic diquark field . In what follows we will work within the mean field approximation (MFA), in which these bosonic fields are expanded around their vacuum expectation values and the corresponding fluctuations neglected. For the meson fields this implies and . Concerning the diquark mean field, we will assume that in the density region of interest only the 2SC phase might be relevant. Thus, we adopt the ansatz , . Integrating out the quark fields and working in the framework of the Matsubara and Nambu-Gorkov formalism we obtain the following MFA quark thermodynamic potential per unit volume (further calculation details can be found in Refs. (14); (15); (16); (17))
where is the cut-off of the model and is defined by
Here, and , where . Note that in the isospin limit we are working and, thus, .
The total thermodynamic potential of the quark matter phase (super-index Q) is obtained by adding to the contribution of the leptons and a vacuum constant. Namely,
where and are the thermodynamic potentials of the electrons and neutrinos respectively. For them we use the expression corresponding to a free gas of ultra-relativistic fermions
where stands for the pressure, , and the degeneracy factor is for electrons and for neutrinos. Notice that in Eq. (9) we have subtracted the constant in order to have a vanishing pressure at vanishing temperature and chemical potentials. More details are shown below in Sec. III.3.
iii.2 Finite size effects: inclusion of MRE
In the present work we consider the formation of finite size droplets of quark matter. The effect of finite size is included in the thermodynamic potential adopting the formalism of multiple reflection expansion (MRE; see Refs. (6); (7); (8) and references therein).
In the MRE framework, the modified density of states of a finite spherical droplet is given by
are the surface and curvature contributions to the new density of states respectively. For we employ the Madsen ansatz (6) because its functional form for any finite quark mass has not yet been derived in the MRE frame.
As shown in (8), the density of states of MRE for massive quarks is reduced compared with the bulk one, and for a range of small momentum becomes negative. The way of excluding this non physical negative density of states is to introduce an infrared cutoff in momentum space (see (8) for details). Thus, in our quark model, which includes the MRE formalism for finite spherical droplets of color superconducting quark matter, we have to perform the following replacement
In order to obtain the value of , we have to solve the equation with respect to the momentum and take the larger root as the IR cut-off. Note that depends on the quark mass and the radius of the droplets and consequently the has the same dependence (more details are given below in Sec. III.3).
Therefore, the full thermodynamic potential reads:
Multiplying on both sides of the last equation by the volume of the quark matter drop and rearranging terms we arrive to the following form for
where the pressure is given by
the surface tension is
and the curvature energy density is
Here we are considering a spherical drop, i.e. the area is and the curvature is .
From the grand thermodynamic potential we can readily obtain the number density of quarks of each flavor and color , the number density of electrons , and the number density of electron neutrinos . The corresponding number densities of each flavor, , and of each color, , in the quark phase are given by and respectively. The baryon number density reads . Finally, the Gibbs free energy per baryon is
For the NJL model, the values of the quark masses and the coupling constant can be obtained from the meson properties in the vacuum. Here we use a set of parameters taken from (18), but without the ’t Hooft flavor mixing interaction. The procedure, obtained from (19), is to keep and fixed and then tune the remaining parameters and in order to reproduce MeV and MeV at zero temperature and density. An estimate of can be obtained from Fierz transformation of the one-gluon-exchange interactions in which case one gets 0.75. The resulting parameter sets are given in Table 1. For this set of parameters we get MeV/fm and MeV/fm (for set 1 and 2 respectively).
As we previously mentioned, the value of is the largest root when solving with respect to , depending on and R. For a given mass, we find numerically that can be fitted as follows
with in fm and in MeV. The coefficients and are found to be and for MeV; 227.72 and for MeV; 228.60 and for MeV.
iii.4 Flavor conservation, color neutrality and other conditions
In order to derive a quark matter EOS from the above formulae it is necessary to impose a suitable number of conditions on the variables and . Three of these conditions are consequences from the fact that the thermodynamically consistent solutions correspond to the stationary points of with respect to , , and . Thus, we have
To obtain the remaining conditions one must specify the physical situation in which one is interested in. As in previous works (6); (20); (21); (1), we are dealing here with just deconfined quark matter that is temporarily out of chemical equilibrium under weak interactions. The appropriate condition in this case is flavor conservation between hadronic and deconfined quark matter. This can be written as
being and the abundances of each particle in the hadron and quark phase respectively. It means that the just deconfined quark phase must have the same “flavor” composition than the -stable hadronic phase from which it has been originated. Notice that, since the hadronic phase is assumed to be electrically neutral, flavor conservation ensures automatically the charge neutrality of the just deconfined quark phase. The conditions given in Eq. (23) can be combined to obtain
where is the particle number density of the -species in the quark phase. The quantities , and are functions of the pressure and temperature, and they characterize the composition of the hadronic phase. These expressions are valid for any hadronic EOS. For hadronic matter containing , , , , , , and , we have
Additionally, the deconfined phase must be locally colorless; thus it must be composed by an equal number of red, green and blue quarks
Also, , , , and pairing will happen provided that is nonzero, leading to
In order to have all Fermi levels at the same value, we consider (5)
Finally, including the conditions Eqs.(22) we have 13 equations involving the 14 unknowns (, , , , and ). For given value of one of the chemical potentials (e.g. ), the set of equations can be solved once the values of the parameters , , , the temperature and the radius of the drop are given. Instead of , we can provide a value of the Gibbs free energy per baryon and solve simultaneously Eqs. (24)-(33) together with Eqs. (22) in order to obtain , , , , and .
Iv Deconfinement of color superconducting droplets in proto-neutron star matter
The total thermodynamic potential of a quark matter drop immersed in an homogeneous hadronic environment is , where the indexes and refer to the hadronic and the quark phase respectively. For the hadronic phase we have and for the quark phase we have .
Since is constant, we have:
Thus, for a spherical droplet the condition for mechanical equilibrium reads (c.f. (6)):
Notice that in the bulk limit we find the standard condition .
Additionally, we assume thermal and chemical equilibrium, i.e. the Gibbs free energy per baryon are the same for both hadronic matter and quark matter at a given common temperature. Thus, we have
If we fix the radius of the deconfined drop for a given temperature and neutrino chemical potential of the trapped neutrinos in the hadronic phase , there is an unique hadronic pressure at which the equilibrium conditions are fulfilled. Notice that, differently than in the bulk case studied in (1), and are not equal because of the appearance of a surface and a curvature term in the condition for mechanical equilibrium. Also, the mass-energy density and at the equilibrium point are different in general. Similarly, while the abundance of neutrinos is the same in both the hadronic and just deconfined quark phases, the chemical potentials and are different.
In the present work we want to describe thermodynamic conditions analogous to those encountered in protoneutron stars. From numerical simulations of the first tens of seconds of evolution (23), we know that the protoneutron star cools from MeV to temperatures below 2-4 MeV in about one minute. In that period, the chemical potential of the trapped neutrinos decreases from MeV to essentially zero. Then, taking into account typical protoneutron star conditions, we have solved Eqs. (36) and (37) together with the conditions showed in Sections II and III, for temperatures in the range MeV, in the range MeV and for different radii of the color superconducting droplets. The results are displayed in Figs. 1 and 2 for the parameterizations of the equations of state given in previous sections.
In Fig. 1 we show the mass-energy density of hadronic matter (in units of the nuclear saturation density g cm) above which it is energetically favorable the conversion of a portion of hadronic matter into an identical deconfined drop (i.e. with the same , , , and radius ). For the calculations we considered the following possibilities: set 1 of the NJL model in the left panels and set 2 in the right panels, temperatures in the range 0-60 MeV, and neutrino chemical potential MeV. Within each plot we consider different values of ; from right to left the curves correspond to . Notice that, in the six panels of Fig. 1, the curves for the bulk case () are almost coincident with the curves for fm. The main observed effect in Fig. 1 is that the transition density increases considerably for small radii. This is a consequence of the significant increase of the surface tension and the curvature energy for small (see Fig. 2).
In Fig. 2 we display the surface tension and the curvature energy as a function of temperature for different radii of the droplet and with the set of that arises from the equilibrium condition discussed before. Notice that and do not depend explicitly on the radius of the drop (see Eqs. (18) and (19)), but for a given , and there is a unique set of satisfying the equilibrium conditions, i.e. and depend on through . From Fig. 2 we see that while both and are roughly independent of the temperature, there is a larger dependence with the radius leading to the behavior of the transition density explained in the caption of Fig. 1. Notice also that the here-found values of are larger that those found within the MIT Bag model which are typically 30-60 MeV fm for drops larger than few fm (24).
V Fluctuations and deconfinement
According to the theory of homogeneous nucleation, the free energy involved in the formation of a spherical quark bubble of radius is given by (25)
where is the pressure difference between internal and external parts of the bubble. For given , and , the extremal points (maximum or minimum) of are obtained from , which leads to Eq. (36). Thus, the critical radii are given by:
For both solutions are complex and is a monotonically decreasing function of . This means that any small fluctuation of one phase into the other will gain energy by expanding and a rapid phase transition is likely to occur. For , has a local minimum at and a local maximum at . For we have again that any small fluctuation is energetically favored. For bubbles with radii larger than gain energy by growing unlimitedly, while those below the critical size gain energy by shrinking to zero (if ) or to (if ). In this case, the standard assumption in the theory of bubble nucleation in first order phase transitions is that bubbles form with a critical radius (25).
The approach we adopted in the previous section is closely related to what we explained in the above paragraph. Instead of finding the critical radius for arbitrary values of , and , we fixed and found the corresponding , and that satisfy the conditions presented in Secs. II, III and IV. Since Eq. (36) is satisfied by construction, the radius is precisely the critical radius introduced in Eq. (39), given that we choose the solution that verifies .
In the light of the previous discussion we may give another interpretation to the results presented in Fig. 1. Let us consider, for simplicity, a uniform hadronic system with trapped neutrinos having a chemical potential and a constant mass-energy density infinitesimally to the right of the curve with in the plane corresponding to the same . Since this density corresponds to the bulk transition we shall refer to it as . For such a density, it is favorable to convert the system into quark matter in bulk, but this is not possible in practice due to the surface and curvature energy cost.
However, fluctuations in the independent thermodynamic variables of the hadronic fluid may drive some portion of it to a state described by . Nevertheless, notice that the curves of Fig. 1 are quite vertical, i.e. the transition is not very sensitive to changes in . The same holds for variations in . Thus, we shall consider only energy-density fluctuations with radius that drive some part of the hadron fluid to a density to the right of a curve with a given . As explained in the caption of Fig. 1, quark drops with radii larger than are energetically favored. Thus, if the fluctuation has a size larger than it will be energetically favorable for it to grow indefinitely. In order to quantify this, we calculate the difference between and the hadronic density of the point that allows nucleation with radius or larger. In such a way we can construct a critical spectrum as a function of for different values of and as seen in Fig. 3. Fluctuations of a given over-density must have a size larger than the critical value given in Fig. 3 in order to grow. Equivalently, fluctuations of a given size must have an over-density larger than the critical one for that size.
We can calculate the formation rate of critical bubbles through
where in our case is the work required to form a quark bubble with the critical radius from hadronic matter at the bulk transition point
Instead of , different prefactors are used for in other works (see e.g. (2)). However, this fact does not affect significantly the results because is largely dominated by the exponent in Eq. (40); i.e. we always have with the second term much larger than the first.
The results are given in Fig. 4 and show that critical bubbles with fm are strongly disfavored while those with fm have a huge rate. In practical situations, i.e. at neutron star cores, this means that if fluctuations lead hadronic matter to the bulk transition point, quark drops with fm will nucleate instantaneously.
Vi Summary and Conclusions
In the present paper we have studied the deconfinement of quark matter in protoneutron stars employing a two-phase description of the first order phase transition. For the hadronic phase we used the non-linear Walecka model (Sect. II) which includes the whole baryon octet, electrons and trapped electron neutrinos in equilibrium under weak interactions. For the just deconfined quark matter we used a NJL model including color superconducting quark-quark interactions. In this phase, finite size effects are included via the multiple reflection expansion framework (Sect. III). We considered that color superconducting quark droplets are formed in mechanical and thermal equilibrium with the hadronic environment when the Gibbs free energy of both phases are equal. Also, since deconfinement is a strong interaction process, we consider that the just formed quark phase has the same flavor composition than the hadronic -stable phase, and consequently it is out of chemical equilibrium under weak interactions. The further -equilibration of such quark phase is not addressed within the present paper.
Through this model we determined the density of hadronic matter at which deconfinement is possible for different radii of the just formed quark drops, as well as for different values of the temperature and the chemical potential of the trapped neutrinos (see Fig. 1). We have used typical conditions expected in the interior of protoneutron stars, i.e. temperatures in the range of MeV and chemical potentials of the trapped neutrino gas up to MeV. As expected, we find that the transition density increases significantly when the radius of the droplet decreases. This is a consequence of the increase of the surface tension and the curvature energy for small radii. Additionally, we found that and at the deconfinement point are almost independent of the temperature and the neutrino chemical potential (see Fig. 2).
When the bulk transition density is reached in the core of a neutron star, it is energetically favored to convert a macroscopically large portion of hadronic matter into quark matter. But in practice, it is needed some over-density with respect to the bulk transition density in order to compensate the surface and curvature energy cost of a finite drop. Since this energy cost depends on the drop radius, so does the necessary over-density and over-pressure necessary to nucleate it. Thus, we can derive a critical fluctuation spectrum versus delimiting which fluctuations are able to grow unlimitedly and which will shrink (see Fig. 3). Typically, fluctuations of above the bulk point are needed for the nucleation of drops with fm. However, the nucleation rates vary over several orders of magnitude. Our results show that drops with fm have a huge nucleation rate while those with fm are strongly suppressed (see Fig. 4).
We have also shown that fluctuations in the temperature and in the chemical potential of trapped neutrinos are not very important for deconfinement. This is in contrast with previous results found within the frame of the MIT Bag model (e.g. in (26) it is argued that nucleation is suppressed at MeV and in (20) it is found that neutrino trapping precludes deconfinement). Instead, fluctuations in the energy density are the more efficient way to trigger the transition.
Notice that the nucleation rate and the typical radii of deconfined drops are also very different from the values found within the MIT Bag model (see e.g. (2) and references therein). The drops studied in (2) have typically radii less than 10 fm and a long nucleation time, in contrast ours may have much larger radii and nucleate almost instantaneously. This is due to the use of different equations of state for the quark phase as well as for the different treatments of the surface and curvature terms. While in (2) the surface tension is assumed to be constant ( = 30 MeV fm), in our work and are calculated self consistently within the MRE formalism resulting non-constant values around 140 MeV fm and 110 MeV fm respectively. Since our surface tension is larger, larger critical drops are obtained. The typical is also larger and results in huge nucleation rates. In the context of protoneutron stars the main conclusion is that if the bulk transition point is attained near the star centre, quark matter drops with fm will nucleate instantaneously. Since the bulk transition density is , this should happen for stars with masses larger than .
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