1 On the importance of nucleation for the formation of quark cores inside compact stars Bruno Werneck Mintz* Eduardo Souza Fraga Universidade Federal do Rio de Janeiro

2 Introduction: deconfined matter in compact stars The density in the interior of a solar-mass proto-neutron star may reach 2-5 n 0 or even more [1]. Such high densities can be enough to favor a deconfined phase instead of a hadronic one in the core of the star. In this case, a hybrid star is formed. (F. Weber, 2000)

3 Given a certain stellar mass, the presence of a deconfined core can, e.g., lead to a different radius in comparison with a purely hadronic star, making it even more compact. Hybrid stars can be powerful particle accelerators due to their intense magnetic fields, as well as natural “laboratories” to study the dense phases of QCD. The formation of a deconfined core through a first-order phase transition is highly energetic and may be one mechanism for the generation of gamma ray bursts. Pure quark stars (“strange stars”) are also possible in principle, but we will not consider them here.

4 Tolman-Oppenheimer-Volkoff equations The Einstein equations can be exactly solved inside a static and spherically symmetric matter distribution. The resulting equations for the matter, energy density and pressure distributions are the TOV equations [1]: A third equation (the Equation of state) is needed!

5 Nuclear matter equation of state We assume that deep inside the star nuclear matter may be described by the EoS given in [2] (parameter set TM1). The corresponding EoS considers all the hyperons in the octet, but no interactions between them.

6 Deconfined matter equation of state The coupling of the strong interactions is a monotonically decreasing function of the energy. In 1st order in pert. theory: Due to asymptotic freedom, strongly interacting matter becomes deconfined from hadrons at a certain energy scale (temperature or chemical potential). (E.S.F., 2003)

7 The environment found deep inside neutron stars is very dense (  ~ 400 MeV) and relatively cold (in comparison with the QCD energy scale -- T < MeV) [3]. Color superconducting (deconfined) phases are expected under these conditions. We consider the CFL phase with (T = 0) [4] where  is the chemical potential, m s is the strange quark mass (u and d quarks are massless), B eff is an effective bag constant,  is the superconducting gap and c=0.3 is a correction to the number of effective degrees of freedom due to strong interactions (at 2 loop level) [5].

8 First order phase transition Given the nuclear and the deconfined EoS, at a certain density inside the star the highest pressure phase should be favored over the other. The discontinuity in p’(  ) indicates a 1st order transition. For  close to  c, the mechanism of phase conversion is bubble nucleation, but for larger values of chemical potential it should be spinodal decomposition [6].

9 Nucleation rate The probability of nucleation of a bubble per unit volume per unit time can be overestimated by [7] with where  F is the free energy shift when a critical bubble is created,  is the surface tension and  p is the pressure difference between both phases.

10 Nucleation rates for dense neutron stars For the chosen hadronic EoS (with fixed parameters), we (over)estimate the nucleation rate for reasonable values of the parameters of the CFL EoS. For example, for c=0.2, m s =200MeV,  =120MeV, B eff =(180MeV) 4 and T=50MeV: Our next steps will be to check if nucleation is also suppressed for other possible EoS and to analyze the possible role of spinodal decomposition in the deconfinement transition in hybrid stars [8]. (extremely suppressed!)

11 Conclusion These estimates show that although the environment found inside neutron stars is very dense and quite hot for a stellar object, its energy density seems not to be enough to trigger an efficient phase conversion to a deconfined phase through bubble nucleation. The authors thank Juergen Schaffner-Bielich and Giuseppe Pagliara for valuable discussions. This work was partially supported by CAPES, CNPq, FAPERJ and FUJB/UFRJ. Acknowledgments

12 References [1] N. K. Glendenning, Compact Stars: nuclear physics, particle physics and general relativity. 2nd ed. (Springer, 2000). [2] Y. Sugahara and H. Toki, Nucl. Phys. A 579, 557 (1994). [3] J. A. Pons et al., Astrophys. J. 513, 780 (1999). [4] M. Alford et al, Astrophys. J. 629, 969 (2005). [5] E. S. Fraga, R. D. Pisarski and J. Schaffner-Bielich, Phys. Rev. D 63, (2001). [6] J. D. Gunton, M. San Miguel and P. S. Sahni, in Phase Transitions and Critical Phenomena, edited by C. Domb and J. L. Lebowitz (Academic Press, London, 1983), vol. 8. [7] L. P. Csernai and J. I. Kapusta, Phys. Rev. D 46, 1379 (1992). [8] E. S. Fraga and B. W. Mintz. Work in progress.