Download On the importance of nucleation for the formation of quark cores

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
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Introduction: deconfined matter in
compact stars
• The density in the interior of a solar-mass proto-neutron
star may reach 2-5 n0 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)
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• 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.
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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!
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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.
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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)
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• The environment found deep inside neutron stars is very dense
(m ~ 400 MeV) and relatively cold (in comparison with the
QCD energy scale -- T < 20-50 MeV) [3].
• Color superconducting (deconfined) phases are expected under
these conditions. We consider the CFL phase with (T = 0) [4]
where m is the chemical potential, ms is the strange quark mass
(u and d quarks are massless), Beff is an effective bag constant,
D 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].
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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’(m)
indicates a 1st order transition.
• For m close to mc, the mechanism of phase conversion is bubble
nucleation, but for larger values of chemical potential it should
be spinodal decomposition [6].
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Nucleation rate
• The probability of nucleation of a bubble per unit volume
per unit time can be overestimated by [7]
with
where DF is the free energy shift when a critical bubble is
created, s is the surface tension and Dp is the pressure
difference between both phases.
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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, ms=200MeV,
D=120MeV, Beff=(180MeV)4 and T=50MeV:
(extremely suppressed!)
• 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].
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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.
Acknowledgments
The authors thank Juergen Schaffner-Bielich and Giuseppe Pagliara for
valuable discussions. This work was partially supported by CAPES,
CNPq, FAPERJ and FUJB/UFRJ.
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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, 121702 (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.
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