Quark-hadron phase transition

H. Reeves, in F. Sanchez, M. Collados and R. Rebolo (eds.), Observational and Physical Cosmology, Cambridge University Press 1990, p. 73, includes a description of the physics of the quark-hadron phase transition. 

The analysis of Ref. [42] as well as the NJL-type model investigation of Ref. [43] are based on a comparison of homogeneous phases. The neutrality conditions can, however, also be fulfilled giving up the requirement of separately neutral phases and to consider mixed phases in chemical equilibrium which are only neutral in total. This procedure has been pushed forward by Glendenning in the context of the quark-hadron phase transition in neutron stars where a similar problem related to electrical neutrality occurs [44], For the case of electrically and color neutral quark matter the phase boundaries are... 

The influence of the appearance of such exotic states like quarks in stellar matter is topic of the study of quasi-stationary simulations of the evolution of isolated compact stars [15, 12, 7, 23] and accreting systems, where one companion is a superdense compact object [9,27], In this work we investigate the observability of the hadron-quark deconfinement phase transition in the dynamical evolution of a neutron star merger. 

Burgio, G. F., Baldo, M., Schulze, H.-J., Sahu, P. K. (2002). The hadron-quark phase transition in dense matter and neutron stars. Phys.Rev., C66 025802-025815. 

However, a more realistic model for the phase transition between baryonic and quark phase inside the star is the Glendenning construction [16], which determines the range of baryon density where both phases coexist. The essential point of this procedure is that both the hadron and the quark phase are allowed to be separately charged, still preserving the total charge neutrality. This implies that neutron star matter can be treated as a two-component system, and therefore can be parametrized by two chemical potentials like electron and baryon chemical potentials [if. and iin. The pressure is the same in the two phases to ensure mechanical stability, while the chemical potentials of the different species are related to each other satisfying chemical and beta stability. The Gibbs condition for mechanical and chemical equilibrium at zero temperature between both phases reads... 

Concerning the quark matter EOS, we found that a density dependent bag parameter B p) is necessary in order to be compatible with the CERN-SPS findings on the phase transition from hadronic to quark matter. Joining the corresponding EOS with the baryonic one, maximum masses of about 1.6 M are reached, in line with other recent calculations of neutron star properties employing various phenomenological RMF nuclear EOS together with either effective mass bag model [39] or Nambu-Jona-Lasinio model [40] EOS for quark matter. 

The value of the maximum mass of neutron stars obtained according to our analysis appears rather robust with respect to the uncertainties of the nuclear and the quark matter EOS. Therefore, the experimental observation of a very heavy (M > 1.6M ) neutron star, as claimed recently by some groups [41] (M ss 2.2 M ), if confirmed, would suggest that either serious problems are present for the current theoretical modelling of the high-density phase of nuclear matter, or that the assumptions about the phase transition between hadron and quark phase are substantially wrong. In both cases, one can expect a well defined hint on the high density nuclear matter EOS. 

From the above estimations we conclude that is it at least a good approximation to consider only homogeneous phases to describe the quark matter phase. In Fig. 4 we display the pressure as a function of fi for neutral homogeneous quark matter phases. We see that at small // the 2SC phase (dashed line) is favored whereas at large // we find a CFL phase (solid line). Normal quark matter (dotted line) turns out to be never favored. This will be our input for the description of the quark matter phase. Of course, in order to construct a compact star, we also have to take into account the possibility of a hadronic component in the equation of state (EOS). To this end, we take a given hadronic EOS and construct a phase transition to quark matter from the requirement of maximal pressure. This is shown in the left panel of Fig. 5 for an example hadronic EOS [53], At the transition point to the quark-matter phase we directly enter the CFL phase and normal or 2SC quark matter is completely irrelevant in this... 

It is interesting to notice that the three pressure surfaces in Figure 7 form a characteristic swallowtail structure. As one could see, the appearance of this structure is directly related to the fact that the phase transition between color superconducting and normal quark matter, which is driven by changing parameter //,. is of first order. In fact, one should expect the appearance of a similar swallowtail structure also in a self-consistent description of the hadron-quark phase transition. Such a description, however, is not available yet. 

Thus in this approximation the magnetic field 7 enters from the hadronic phase into the CEL quark phase in the form of quark magnetic vortices. The transition zone is of the order Xq + Xp which entails that quantity D is small, of the order (Xp + Xq) /a, so that the condition D = 0 is well fulfilled. 

At nonzero temperatures the mass gap decreases as a function of the chemical potential already in the phase with broken chiral symmetry. Hence the model here gives unphysical low-density excitations of quasi-free quarks. A systematic improvement of this situation should be obtained by including the phase transition construction to hadronic matter. However, in the present work we circumvent the confinement problem by considering the quark matter phase only for densities above the nuclear saturation density no, i.e. ub > 0.5 no. 

We have investigated the influence of diquark condensation on the thermodynamics of quark matter under the conditions of /5-equilibrium and charge neutrality relevant for the discussion of compact stars. The EoS has been derived for a nonlocal chiral quark model in the mean field approximation, and the influence of different form-factors of the nonlocal, separable interaction (Gaussian, Lorentzian, NJL) has been studied. The model parameters are chosen such that the same set of hadronic vacuum observable is described. We have shown that the critical temperatures and chemical potentials for the onset of the chiral and the superconducting phase transition are the lower the smoother the momentum dependence of the interaction form-factor is. 

According to Quantum Chromodynamics (QCD) a phase transition from hadronic matter to a deconfined quark phase should occur at a density of a few times nuclear matter saturation density. Consequently, the core of the more massive neutron stars is one of the best candidates in the Universe where such deconfined phase of quark matter (QM) could be found. Since /3-stable hadronic matter posses two conserved charges (i.e., electric charge and baryon... 

In bulk matter the quark-hadron mixed phase begins at the static transition point defined according to the Gibbs criterion for phase equilibrium... 

Figure 1. Chemical potentials of the three phases of matter (H, Q, and Q ), as defined by Eq. (2) as a function of the total pressure (left panel) and energy density of the H- and Q-phase as a function of the baryon number density (right panel). The hadronic phase is described with the GM3 model whereas for the Q and Q phases is employed the MIT-like bag model with ms = 150 MeV, B = 152.45 MeV/fm3 and as = 0. The vertical lines arrows on the right panel indicate the beginning and the end of the mixed hadron-quark phase defined according to the Gibbs criterion for phase equilibrium. On the left panel P0 denotes the static transition point.

A quark-hadron mixed phase between 5 x 1014g/cm3 and 1015g/cm3 3.5po where both quark and hadrons are present. In this phase transition region, the EoS substantially softens with T 1 — 1.5. 

Had this process proceeded unchecked it would have by-passed several important stages in cosmic evolution between t = 10 and Is, such as baryogenesis, electroweak symmetry breaking, combination of free quarks to form hadrons and interconversion between protons and neutrons. Not to interrupt this orderly evolution it would therefore be useful to have the phase transition postponed for a while. Many phase transitions are indeed known to be delayed by the phenomenon of supercooling. Why not this one ... 

Figure 5. Left Pressure as a function of /lib for a hadronic EOS [53] (dash-dotted line) and homogeneous quark matter in the normal phase (dotted line), the 2SC phase (dashed line) and the CFL phase (solid line). Right Corresponding energy densities as a function of pressure. The closed (open) circles connected by thin dashed lines indicate the discontinuities at the transition points from hadronic to CFL (normal) quark matter.

In our scenario, we consider a purely hadronic star whose central pressure is increasing due to spin-down or due to mass accretion, e.g., from the material left by the supernova explosion (fallback disc), from a companion star or from the interstellar medium. As the central pressure exceeds the threshold value Pq at static transition point, a virtual drop of quark matter in the Q -phase can be formed in the central region of the star. As soon as a real drop of Q -matter is formed, it will grow very rapidly and the original Hadronic Star will be converted to and Hybrid Star or to a Strange Star, depending on the detail of... 

To summarize, in the present scenario pure hadronic stars having a central pressure larger than the static transition pressure for the formation of the Q -phase are metastable to the decay (conversion) to a more compact stellar configuration in which deconfined quark matter is present (i. e., HyS or SS). These metastable HS have a mean-life time which is related to the nucleation time to form the first critical-size drop of deconfined matter in their interior (the actual mean-life time of the HS will depend on the mass accretion or on the spin-down rate which modifies the nucleation time via an explicit time dependence of the stellar central pressure). We define as critical mass Mcr of the metastable HS, the value of the gravitational mass for which the nucleation time is equal to one year Mcr = Miis t = lyr). Pure hadronic stars with Mh > Mcr are very unlikely to be observed. Mcr plays the role of an effective maximum mass for the hadronic branch of compact stars. While the Oppenheimer-Volkov maximum mass Mhs,max (Oppenheimer Volkov 1939) is determined by the overall stiffness of the EOS for hadronic matter, the value of Mcr will depend in addition on the bulk properties of the EOS for quark matter and on the properties at the interface between the confined and deconfined phases of matter (e.g., the surface tension a). 

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