Quark matter nucleation with a microscopic hadronic equation of state
Abstract
The nucleation process of quark matter in cold () stellar matter is investigated using the microscopic Brueckner–Hartree–Fock approach to describe the hadronic phase, and the MIT bag model, the Nambu–Jona–Lasinio, and the Chromo Dielectric models to describe the deconfined phase of quark matter. The consequences of the nucleation process for neutron star physics are outlined. Hyperonic stars are metastable only for some of the quark matter equations of state considered. The effect of an hyperonic three body force on the metastability of compact stars is estimated, and it is shown that, except for the Nambu–Jona–Lasinio model and the MIT bag model with a large bag pressure, the other models predict the formation of hybrid stars with a maximum mass not larger than .
PACS number(s): 97.60.s, 97.60.Jd, 26.60.Dd, 26.60.Kp
I Introduction
Quark matter (QM) nucleation in neutron stars has been studied by many authors both at zero b0 (); b1 (); b2 (); b3 (); b4 (); b5 (); b6 (); b7 (); b8 () and finite temperature h1 (); h2 (); h3 (); h4 (); me (); me1 (), due to its potential connection with explosive astrophysical events such as supernovae and gamma ray bursts. In all these works, the hadronic phase was described using phenomenological models, such as e.g., the wellknown relativistic mean field (RMF) model based on effective Lagrangian densities where the baryonbaryon interaction is described in terms of meson exchanges serot86 (). Among the different RMF models, one of the most popular parametrizations is the one of Glendenning and Moszkowski gm91 () of the nonlinear Walecka model which have been widely used to study the effect of the hadronic equation of state (EoS) on the QM nucleation process. In particular, the effect of different hyperon couplings on the critical mass b0 (); b1 () for pure hadronic stars (HSs, i.e. neutron stars in which no fraction of QM is present) and the stellar conversion energy grb () was studied in Ref. b6 (). It was found that increasing the value of the hyperon coupling constants, increases the stellar metastability threshold mass and the value of the critical mass, thus making the formation of quark stars (QSs, i.e. hybrid stars or strange stars depending on the details of the EoS for quark matter used to model the phase transition) less likely. In that work, the hadronic phase was also described using the quarkmesoncoupling model qmc (), concluding, in that case, that the formation of a quark star was only possible with using a small value of the bag pressure. In all these works the MIT bag model mit () was used to describe the quark matter phase. In a recent work b9 (), two models that contain explicitly the chiral symmetry were applied to describe the quark phase, namely the Nambu–JonaLasinio (NJL) model nambu () (see also bba2 (); njl1 ()) and the Chromo Dielectric model (CDM) cdm (); cdm1 (). It was shown there that it is very difficult to populate the quark star branch with the NJL model and, therefore, all compact stars would be pure hadronic stars in that case. On the contrary, with the CDM, both hadronic and quark star configurations can be formed.
In the present work we study the nucleation of quark matter using an hadronic EoS based on microscopic calculations. In particular, we employ two hadronic EoS based on microscopic Brueckner–Hartree–Fock (BHF) calculations of hypernuclear matter. The first one (hereafter called Model ) is the recent parametrization provided by Schulze and Rijken hans () which uses the Argonne V18 nucleonnucleon (NN) Ar18 () supplemented by the microscopic threebody force (TBF) of Ref. hans_NNN () between nucleons (NNN), and the recent Nijmegen extended softcore ESC08b hyperonnucleon (YN) potentials nij08 (). The second one (hereafter called Model ) is based on our recent work of Ref. isaac2011 () where we used the Argonne V18 NN force and the Nijmegen softcore NSC89 YN one ny () in a microscopic BHF calculation of hyperonic matter supplemented with additional simple phenomenological densitydependent contact terms, that mimic the effect of NNN, NNY and NYY TBFs, to establish numerical lower and upper limits to the effect of hyperonic TBF on the maximum mass of neutron stars. To describe the quark phase, in the present work, we use the three different models already mentioned, the MIT bag model mit (), the NJL model nambu () and the CDM model cdm ().
The paper is organized in the following way. In sections II and III, we briefly review the BHF approach and the main features of quark matter nucleation in hadronic stars, respectively. Our results are presented in Sec. IV. Finally, a summary and the main conclusions of this work are given in Sec. V.
Ii The BHF approach
The BHF approach is the lowest order of the Brueckner–Bethe–Goldston (BBG) manybody theory bbg (). In this theory, the ground state energy of nuclear matter is evaluated in terms of the socalled holeline expansion, where the perturbative diagrams are grouped according to the number of independent holelines. The expansion is derived by means of the inmedium twobody scattering matrix. The matrix, which takes into account the effect of the Pauli principle on the scattered particles and the inmedium potential felt by each nucleon, has a regular behavior even for shortrange repulsions, and it describes the effective interaction between two nucleons in the presence of a surrounding medium. In the BHF approach, the energy is given by the sum of only twoholeline diagrams including the effect of twobody correlations through the matrix. It has been shown by Song et al. song98 () that the contribution to the energy from threeholeline diagrams (which account for the effect of threebody correlations) is minimized when the socalled continuous prescription jeneuke76 () is adopted for the inmedium potential, which is a strong indication of the convergence of the holeline expansion. The BHF approach has been extended to hyperonic matter by several authors hans (); isaac2011 (); bhf (). The interested reader is referred to these works for the specific details of the BHF calculation of hyperonic matter, and to Ref. bbg () for an extensive review of the BBG manybody theory.
Iii Quark matter nucleation in hadronic stars
The conditions of phase equilibrium, in the case of a firstorder phase transition footnote (), are given by the Gibbs’ phase rule, which in the case of cold () matter can be written as:
(1) 
where
(2) 
are the Gibbs energies per baryon for the hadron (H) and quark (Q) phases, respectively, and the quantities , , and denote respectively the total (i.e., including leptonic contributions) energy density, total pressure, and baryon number density of the two phases. Above the transition pressure the hadronic phase is metastable, and the stable quark phase will appear as a result of a nucleation process. Quantum fluctuations will form virtual drops of quark matter. The characteristic oscillation time of these drops, in the potential energy barrier separating the metastable hadronic phase and quark phase, is set by strong interactions, which are responsible of the deconfinement transition, thus . This time is many orders of magnitude smaller than the weak interaction characteristic time ( s), consequently quark flavor must be conserved forming a virtual drop of quark matter. We call phase this deconfined quark matter, in which the flavor content is equal to that of the stable hadronic phase at the same pressure and temperature. Soon after 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 stable quark matter is formed. (hereafter the Qphase).
This first seed of quark matter will trigger the conversion oli87 (); hbp91 (); grb () of the pure hadronic star to a quark star. Thus, pure hadronic stars with values of the central pressure higher than and corresponding masses , are metastable to the decay (conversion) to quark stars b0 (); b1 (); b2 (); b3 (); b4 (); b5 (); b6 (); b7 (). 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 it depends dramatically on the value of the stellar central pressure.
As in Refs. b0 (); b1 (); b2 (), we define as the critical mass of the hadronic star sequence, the value of the stellar gravitational mass for which the nucleation time of a matter droplet is equal to one year: . Pure hadronic stars with are thus very unlikely to be observed. plays the role of an effective maximum mass b2 () for the hadronic branch of compact stars.
In a cold and neutrinofree hadronic star the formation of the first drop of quark matter could take place solely via a quantum nucleation process. The basic quantity needed to calculate the nucleation time is the energy barrier separating the Q*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 iida98 (); b2 ()
(3) 
where is the radius of the droplet (supposed to be spherical), and is the surface tension for the surface separating the hadron from the Q*phase. The energy barrier has a maximum at the critical radius .
The quantum nucleation time can be straightforwardly evaluated within a semiclassical approach iida98 (); b1 (); b2 (); b3 () and it can be expressed as
(4) 
where is the probability of tunneling the energy barrier in its ground state, is the oscillation frequency of a virtual drop of the Q*phase in the potential well, and is the number of nucleation centers expected in the innermost part ( m) of the hadronic star, where the pressure and temperature (in finite T case) can be considered constant and equal to their central values.
Iv Results and discussion
We will now discuss the results obtained with the two microscospic hadronic EoS considered and, in particular, we will comment whether the possible discussed scenarios are compatible with the recent measurement Demorest10 () of the mass of the pulsar PSR J16142230 with a mass M = (1.97 0.04) .
In Fig. 1 we plot the Gibbs energy per baryon as a function of pressure using the microscopic approach of Ref. hans () (Model ) for the hadronic phase and one of the following models for the phase: MIT bag (top panels), CDM (left bottom panel) and NJL (right bottom panel) model. It is interesting to note that the formation of the phase is possible only in the case of the MIT bag model EoS with a low value of the bag constant ( MeV fm). In all the other cases considered in Fig. 1, the curve for Gibbs energy per baryon for the phase never crosses the one for the hadronic phase, consequently, the hadronic phase will always remain stable with respect to the formation of phase droplets. For these three QM models, this result implies that the pure hadronic stars (hyperonic stars) described by Model are stable up to their maximum mass configuration .
By numerical integration of the Tolman–Oppenheirmer–Volkov equations shapiro83 (), we have calculated the structural properties for pure hadronic and quark star sequences. The main results, in the case of Model EoS for the hadronic phase, are summarized in Table 1, where we report the maximum gravitational mass (third column), the gravitational threshold mass for metastable configurations (fourth column), and the gravitational (baryonic) critical mass () (fifth (sixth) column) for the pure hadronic star sequence. (seventh column) is the gravitational mass of the hybrid star formed by the stellar conversion process of the HS with and assuming baryon number conservation in the process grb () (i.e. assuming ). Finally is the total energy liberated in the stellar conversion. It is interesting to note that hybrid star configurations can be obtained with the MIT bag model with MeV fm, and with the CDM. In the latter case, however, the transitory nonstable phase is not energetically achievable (see left bottom panel Fig. 1). Thus in this case the hadronic star sequence is stable up to the maximum mass configuration (thus we have no entries in Tab. 1 for the quantities , , , and ). In the case of the MIT bag model with MeV fm the matter nucleation is possible and one has . The conversion of this star will produce an hybrid star with . If this object is a member of a binary stellar system, eventual accretion of matter from the companion will allow it to reach a maximum mass of 1.544 . In this case the pulsar PSR J16142230 will neither be an hyperonic star nor an hybrid star.
We have also artificialy turned off the hyperonic degrees of freedom in Model 1 EoS, and considered pure nucleonic stars. In this case we have , and a critical mass for all the quark matter EoS considered in Tab. 1. In all cases, however, the critical mass configuration will collapse to a black hole (BH entry in seventh column in Tab. 1) and thus the hybrid star sequence can not be populated. In this case the pulsar PSR J16142230 would be an hadronic star containing only nucleons and leptons.
nucleons+hyperons  MIT ()  1.37  1.227  1.272  1.397  1.233  71.26  1.544 


CDM  1.37            1.591 
only nucleons 
MIT ()  2.27  2.193  2.226  2.677  BH    1.544 
MIT ()  2.27  2.242  2.254  2.720  BH    1.471  
NJL  2.27  2.229  2.246  2.708  BH    1.879  
CDM  2.27  2.242  2.255  2.722  BH    1.592 
MIT  1.48  1.245  1.368  1.526  1.335  60.32  1.574  
MIT  1.38  1.210  1.230  1.356  1.203  47.62  1.574  
MIT  1.60  1.293  1.504  1.688  1.456  85.57  1.574  
CDM  1.60  1.397  1.472  1.648  1.440  56.96  1.624 
We next discuss the results obtained with the hadronic EoS based on our recent work of Ref. isaac2011 () (Model ) where, as we said, a microscopic BruecknerHartreeFock approach of hyperonic matter based on the Argonne V18 NN and the NSC89 NY forces is supplemented with additional simple phenomenological densitydependent contact terms that mimic the effect of nucleonic and hyperonic threebody forces. In particular, we consider three different parametrizations of this model corresponding to different values of the incompressibility coefficient, , of symmetric nuclear matter at saturation, and the parameter , which characterizes the strengh of the hyperonic threebody forces: ( MeV and ), ( MeV and ) and ( MeV and ). The interested reader is referred to isaac2011 (), and particularly to Tables 1 and 2 of this reference, for details. The results for this model are shown in Table 2 and Fig. 2. As in the previous case, we summarize in Table 2 the main stellar properties obtained with this model in combination with the MIT bag and the CDM. Note that in this case with the NJL model no transition occurs for any of the three parametrizations and . In Fig. 2 we plot the Gibbs energy per baryon as a function of pressure using: the MIT bag model for the phase and an hadronic parametrization with the incompressibility MeV () and MeV () with in both cases. Results for the parametrization , and the CDM and NJL models are not shown for conciseness. It is interesting to note that the parameter , does not influence much the mass and radius of the hybrid star maximum mass configurations.
Note also (see Table 2) that, similarly to Model , for the parametrizations and , the formation of the phase is possible only in the case of the MIT bag model EoS with a low value of the bag constant ( MeV fm). Nevertheless, for the parametrization , the stable hybrid sequence may be populated from the stellar conversion of the critical mass hadronic star if the quark phase is described either with the MIT bag model (with MeV/fm) or with the CDM model. As already said, no transition is found for the NJL model. We also note that in the case of the parametrization plus the MIT bag model with MeV/fm the phase nucleation time at the center of the maximum mass () hadronic star is much larger than the age of the Universe, and thus, it is extremely unlikely to populate the hybrid star branch in this case. In the most favorable scenario the possible largest star mass would be 1.624 (for CDM). This mass could occur if after, the conversion, the star accretes mass from an eventual companion star if the object is in a binary system.
V Summary and Conclusions
Using the microscopic Brueckner–Hartree–Fock approach to describe the EoS of dense hadronic matter we have studied the possibility of occurrence of a deconfinement phase transition into quark matter in neutron star cores. Quark matter has been described with three different models, namely, the MIT bag, the CDM and the NJL models. We have concluded that hyperonic hadronic stars will not suffer a deconfinement phase transition except if the quark EOS is obtained using the MIT bag model with a value of the bag pressure MeV/fm. In this case, however, it is not possible to get a star with a mass above 1.54 . On the other hand, we have found that if the hadronic matter has no hyperons then deconfinement will occur only in very massive stars, with and the stars will decay into a blackhole. Within this microscopic approach to the hadronic phase the pulsar PSR J16142230 would be an hadronic star containing only nucleons and leptons.
We have also studied the possible effect of a hyperonic TBF using the model proposed in isaac2011 (). It was shown that for the hardest EOS with MeV and an hybrid star could be formed. Only NJL and the MIT with MeV/fm did not predict a metastable star in this case. Within this scenario a maximum mass of 1.624 was predicted, very far from the mass of the PSR J16142230. However, it was also shown that it was not so much the hyperonic TBF strengh but more the incompressibility of the nucleonic part of the EoS that defines the possible deconfinement transition.
Acknowledgments
This work has been partially supported by the initiative QREN financed by the UE/FEDER throught the Programme COMPETE under the projects PTDC/FIS/113292/2009 and CERN/FP/116366/2010, the grant SFRH/BD/62353/2009, and by COMPSTAR, an ESF Research Networking Programme.
References
 (1) Z. Berezhiani, I. Bombaci, A. Drago, F. Frontera, A. Lavagno, Nucl. Phys. B  Proceedings Supplements 113, 269 (2002).
 (2) Z. Berezhiani, I. Bombaci, A. Drago, F. Frontera, and A. Lavagno, Astrophys. J. 586, 1250 (2003).
 (3) I. Bombaci, I. Parenti, and I. Vidaña, Astrophys. J. 614, 314 (2004).
 (4) A. Drago, A. Lavagno, and G. Pagliara, Phys. Rev. D 69, 057505 (2004).
 (5) G. Lugones and I. Bombaci, Phys. Rev. D 72, 065021 (2005).
 (6) I. Bombaci, G. Lugones, and I. Vidaña, Astron. and Astrophys. 462, 1017 (2007).
 (7) I. Bombaci, P.K. Panda, C. Providência, and I. Vidaña, Phys. Rev. D 77, 083002 (2008).
 (8) C. Bambi and A. Drago, Astropart. Phys. 29, 223 (2008).
 (9) A. Drago, G. Pagliara, and J. SchaffnerBielich, J. Phys. G 35, 014052 (2008).
 (10) D. Logoteta, I. Bombaci, C. Providência, and Isaac Vidaña, Phys. Rev. D 85, 023003 (2012).
 (11) J. E. Horvath, O. G. Benvenuto, and H. Vucetich, Phys. Rev. D 45, 3865 (1992).
 (12) J. E. Horvath, Phys. Rev. D 49, 5590 (1994).
 (13) M. L. Olesen and J. Madsen, Phys. Rev. D 49, 2698 (1994).
 (14) H. Heiselberg, in Strangeness and Quark Matter, Ed. G. Vassiliadis, World Scientific, 338 (1995); arXiv:hepph/9501374.
 (15) I. Bombaci, D. Logoteta, P.K. Panda, C. Providência, and I. Vidaña, Phys. Lett. B 680, 448 (2009).
 (16) I. Bombaci, D. Logoteta, C. Providência, and I. Vidaña, Astron. and Astrophys. 528, A71 (2011).
 (17) B. D. Serot, and J. D. Walecka, Adv. Nucl. Phys. 16, (1986); Int. J. Mod. Phys. E 6, 515 (1997).
 (18) N. K. Glendenning and S. A. Moszkowski, Phys. Rev. Lett. 67, 2414 (1991).
 (19) I. Bombaci and B. Datta, Astrophys. J. 530, L69 (2000).
 (20) P. A. M. Guichon, Phys. Lett. B 200, 235 (1988); K. Saito and A. W. Thomas, Phys. Lett. Bb 327, 9 (1994); P. A. M, Guichon, K. Saito, E. Rodionov, and A. W. Thomas, Nucl. Phys. A 601, 349 (1996); K. Saito, K. Tsushima, and A. W. Thomas, Nucl. Phys. A 609, 339 (1996); P. K. Panda, A. Mishra, J. M. Eisenberg, and W. Greiner, Phys. Rev. C 56, 3134 (1997).
 (21) E. Farhi and R. L. Jaffe, Phys. Rev. D 30, 2379 (1984).
 (22) Y. Nambu and G. JonaLasinio, Phys Rev. 122, 345 (1961).
 (23) M. Buballa, Phys. Rep. 407, 205 (2005).
 (24) D.P. Menezes and C. Providência, Phys. Rev. C 68, 035804 (2003); M. Baldo, M. Buballa, G. F. Burgio, F. Neumann, M. Oertel, and H.J. Schulze Phys. Lett. B 562, 153 (2003).
 (25) H. J. Pirner, G. Chanfray, and O. Nachtmann, Phys. Lett. B 147, 249 (1984).
 (26) J. A. Mcgovern, M. C. Birse, and D. Spanos, J. Phys. G 16, 1561 (1990); W. Broniowski, M. Cibej, and M. Kutschera and M. Rosina, Phys Rev. D 41, 285 (1990); S. K. Gosh and S. C. Pathak, J. Phys. G 18, 755 (1992); T. Neuber, M. Fiolhais, K. Goeke, and J. N. Urbano, Nucl.Phys. A 560, 909 (1993).
 (27) H.J. Schulze and T. Rijken, Phys. Rev. C 84, 035801 (2011).
 (28) R. B. Wiringa, V. G. J. Stoks, and R. Schiavilla, Phys. Rev. C, 51, 38 (1995).
 (29) Z. H. Li, U. Lombardo, H.J. Schulze, and W. Zuo, Phys. Rev. C 77, 034316 (2008).
 (30) T. Rijken, M. Nagels, and Y. Yamamoto, Nucl. Phys. A 835, 160 (2010); Y. Yamamoto, E. Hiyama, and T. Rijken, ibid. 835, 350 (2010); T. Rijken, M. Nagels, and Y. Yamamoto, Prog. Theor. Phys. Suppl. 185, 14 (2010); Y. Yamamoto, T. Motoba, and T. Rijken, ibid. 185, 72 (2010).
 (31) I. Vidaña, D. Logoteta, C. Providência, A. Polls, and I. Bombaci, EPL 94, 11002 (2011).
 (32) P. M. M. Maessen, Th. A. Rijken, and J. J. de Swart, Phys. Rev. C 40, 2226 (1989).
 (33) B. D. Day, Rev. Mod. Phys. 39, 719 (1967); M. Baldo, in Nuclear Methods and the Nuclear Equation of State (World Scientific, Singapore, 1999); M. Baldo, and G. F. Burgio, Reports on Progress in Physics, Volume 75, Issue 2, pp. 026301 (2012).
 (34) H. Q. Song, M. Baldo, G. Giansiracusa, and U. Lombardo, Phys. Rev. Lett. 81, 1584 (1998); Phys. Lett. B 411, 237 (1999).
 (35) J. P. Jeneuke, A. Lejeune, and C. Mahaux, Phys. Rep. 25, 83 (1976).
 (36) H.J. Schulze, A. Lejeune, J. Cugnon, M. Baldo, and U. Lombardo, Phys. Lett. B 355, 21 (1995); H.J. Schulze, M. Baldo, U. Lombardo, J. Cugnon, and A. Lejeune, Phys. Rev. C 57, 704 (1998); M. Baldo, G. F. Burgio, and H.J. Schulze, Phys. Rev. C 58, 3688 (1998); I. Vidaña, A. Polls, A. Ramos, and V. G. J. Stoks, Phys. Rev. C 61, 025802 (2000); M. Baldo, G. F. Burgio, and H.J. Schulze, Phys. Rev. C 61, 055801 (2000); I. Vidaña, A. Polls, A. Ramos, L. Engvik, and M. HjorthJensen, Phys. Rev. C, 62, 035801 (2000) 035801; H.J. Schulze, A. Polls, A. Ramos, and I. Vidaña, Phys. Rev. C, 73 (2006) 058801.
 (37) The deconfinement transition in the high density region relevant for neutron stars is assumed to be of first order.
 (38) A. V. Olinto, Phys. Lett. B 192, 71 (1987).
 (39) H. Heiselberg, G. Baym, and C. J. Pethick, Nucl. Phys. B (Proc. Suppl.) 24, 144 (1991).
 (40) K. Iida and K. Sato, Phys. Rev. C 58, 2538 (1998).
 (41) P. B. Demorest, T. Pennucci, S. M. Ransom, H. S. E. Roberts, and J. W. T. Hessel, Nature 467, 1081 (2010).
 (42) S. L. Shapiro and S. A. Teukolsky, Black Holes, White Dwarfs and Neutron Stars (Wiley, New York, 1983); N. K. Glendenning, Compact Stars: Nuclear Physics, Particle Physics and General Relativity, 2nd ed. (Springer, Berlin, 2000); P. Haensel, A. Y. Potekhin, and D. G. Yakovlev, Neutron Stars 1: Equation of State and Structure (Astrophysics and Space Science Library, Springer, New York, 2007).