The hadronization phase

The characteristic line emanating from Tc is naturally related to the critical line Tc(fi) enclosing the hadronic phase. The comparison, in Figure 4, of our result for the curvature of the critical line at g - 0, which can be calculated in lattice QCD [13], is a nontrivial and successful test of the extension of the quasiparticle approach to g > 0. 

Figure 6. Mass-radius relation of different compact star configurations. The left panels correspond to calculations with parameter set RKH for the quark matter phase and the right panels to parameter set HK, respectively. From the upper panel downwards the hadronic phase is described by a BHF calculation without hyperons [55], a relativistic mean field calculation [57] and a chiral SU(3) model [53].

In the hadronic phase due to the entrainment effect around each neutron vortex appears a cluster of proton vortices which generates mean magnetic field B = 4 1014G. The magnetic flux of such a cluster is... 

To conclude this section we mention that the electromagnetic potential in the hadronic phase of a neutron star can be found from the solution (51) by replacing the penetration depth for quark matter q with that for hadronic matter Xp. 

We can find the magnetic field in the hadronic matter phase from the solution (51) by taking into account that proton vortices in this phase generate a homogeneous mean magnetic field with amplitude B and direction parallel to the axis of rotation of the star [22], For the components of the magnetic field Bp in the hadronic phase (for a [Pg.273]

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. 

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.

Figure 2. Nucleation time as a function of the maximum gravitational mass of the hadronic star. Solid lines correspond to a value of a = 30 MeV/fm2 whereas dashed ones are for a = 10 MeV/fm2. The nucleation time corresponding to one year is shown by the dotted horizontal line. The different values of the bag constant (in units of MeV/fm3) are plotted next to each curve. The hadronic phase is described with the GM1 model.

Table 1. The critical mass and energy released in the conversion process of an HS into a QS for several values of the Bag constant and the surface tension. Column labeled MQs,max denotes the maximum gravitational mass of the final QS sequence. The value of the critical gravitational mass of the initial HS is reported on column labeled Mcr whereas those of the mass of the final QS and the energy released in the stellar conversion process are shown on columns labeled Mfi and Econv respectively. BH denotes those cases in which the baryonic mass of the critical mass configuration is larger than the maximum baryonic mass of the QS sequence (M r > MQS>max). In these cases the stellar conversion process leads to the formation of a black hole. Units of B and a are MeV/fm3 and MeV/fm2 respectively. All masses are given in solar mass units and the energy released is given in units of 10B1 erg. The hadronic phase is described with the GM1 model, ms and as are always taken equal to 150 MeV and 0 respectively. The GM1 model predicts a maximum mass for the pure HS of 1.807 M .

Mhs of the HS corresponding to the given value of the central pressure, as implied by the solution of the Tolmann-Oppeneimer-Volkov equations for the pure Hadronic Star sequences. The results of our calculations are reported in Fig. 2 which is relative to the GM1 EOS for the hadronic phase. Each curve refers to a different value of the bag constant and the surface tension. 

In Fig. 3, we show the MR curve for pure HS within the GM1 model for the EOS of the hadronic phase, and that for hybrid stars or strange stars for different values of the bag constant B. The configuration marked with an asterisk on the hadronic MR curves represents the hadronic star for which the central pressure is equal to Pq. The full circle on the hadronic star sequence represents the critical mass configuration, in the case a = 30 MeV/fm2. The full... 

Figure Jt. The maximum mass MQs,max for the quark star configurations (HS or SS), the critical mass Mcr and the mass Mfi of the stable QS to which it evolves are plotted as a function of the bag constant B. The vertical doted fines labeled B1 — BIV mark the boundary of different ranges of the bag constant which give a different astrophysical output for our scenario, as discussed in the text. The dashed horizontal line gives the value of the maximum mass for the pure hadronic star sequence. All the results are relative to the GM3 model for the EOS for the hadronic phase, the surface tension a is taken equal to 30 MeV/fm2.

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. 

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. 

In the original MIT bag model the bag constant B 55 MeV fm-3 is used, while values B 210 MeV fm-3 are estimated from lattice calculations [34], In this sense B can be considered as a free parameter. We found, however, that a bag model involving a constant (density independent) bag parameter B, combined with our BHF hadronic EOS, will not yield the required phase transition in symmetric matter at pr 6po 1/fm3 [28]. This can only be accomplished by introducing a density dependence of the bag parameter. (The dependence on asymmetry is neglected at the current level of investigation). In practice we use a Gaussian parameterization,... 

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... 

From the intersection of the two surfaces representing the hadron and the quark phase one can calculate the equilibrium chemical potentials of the mixed phase,... 

The EOS resulting from this procedure is shown in Fig. 10(b), where the pure hadron, mixed, and pure quark matter portions are indicated. The mixed phase begins actually at a quite low density around po- Clearly the outcome of the mixed phase construction might be substantially changed, if surface and Coulomb energies were taken into account [36], For the time being these are, however, unknown and have been neglected. 

Abstract We investigate the phase structure of color superconducting quark matter at intermediate densities for two- and three flavor systems. We thereby focus our attention on the influence of charge neutrality conditions as well as /3-equilibrium on the different phases. These constraints are relevant in the context of quark matter at the interior of compact stars. We analyze the implications of color superconductivity on compact star configurations using different hadronic and quark equations of state. 

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... 

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... 

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.

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. 

In our case, nearly equal volume fractions of the two quark phases are likely to form alternating layers (slabs) of matter. The energy cost per unit volume to produce such layers scales as a2/3(r 2SC — niN )2/3 where a is the surface tension [25], Therefore, the quark mixed phase is a favorable phase of matter only if the surface tension is not too large. Our simple estimates show that max < 20 MeV/fm2. However, even for slightly larger values, 20 < a < 50 MeV/fm2, the mixed phase is still possible, but its first appearance would occur at larger densities, 3po < Pn < 5po. The value of the maximum surface tension obtained here is comparable to the estimate in the case of the hadronic-CFL mixed phase obtained in Ref. [26], The thickness of the layers scales as a1 /3(r/i2 SY -) — niN ) 2/3 [25], and its typical value is of order 10 fm in the quark mixed phase. This is similar to the estimates in various hadron-quark and hadron-hadron mixed phases [25, 26], While the actual value of the surface tension in quark matter is not known, in this study we assume that it is... 

As can be seen from obtained solutions, the magnetic field in both quark and hadronic phases depends on r only very close to the phase boundary at r = a. So we conclude that in the main part of the volume of the quark and hadron phases the magnetic field is constant and directed parallel to the rotation axis of the star, see the solutions (67) - (70). In this approximation the condition... 

Besides the crust and the hadron shell, the hybrid star contains also a quark core. Both the nucleon shell and the quark core can be in superconducting phases, in dependence on the value of the temperature. Fluctuations affect transport coefficients, specific heat, emissivity, masses of low-lying excitations and respectively electromagnetic properties of the star, like electroconductivity and magnetic field structure, e.g., renormalizing critical values of the magnetic field (/ ,, Hc, Hc2). 

Contribution of pairing fluctuations to the specific heat in the hadron shell is minor for the case of the neutron pairing due to a small value of Tc < IMeV compared to the value of the neutron chemical potential f//, > 50 MeV). Therefore in the neutron channel fluctuations of the gap are relevant only in a very narrow vicinity of the critical point. However this effect might be not so small for protons, for which the chemical potential is of the order of several MeV, whereas the gap is of the order of one MeV. Therefore it seems that fluctuations may smear the phase transition in a rather broad vicinity of the critical point of the proton superconductivity. 

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. 

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