Scalar-isovector δ-meson mean-field and mixed phase structure in compact stars

The deconfinement phase transition from hadronic matter to quark matter in the interior of compact stars is investigated. The hadronic phase is described in the framework of relativistic mean-field (RMF) theory, when also the scalar-isovector δ-meson effective field is taken into account. The MIT bag model for describing a quark phase is used. The changes of the parameters of phase transition caused by the presence of δ-meson field are investigated. Finally, alterations in the integral and structure parameters of hybrid stars due to deconfinement phase transitions are discussed.


INTRODUCTION
Study of the structure characteristics and composition of the matter constituents at extremely high density region is of great interest in both nuclear and neutron star physics.The RMF theory [1] has been effectively applied to describe the structure of finite nuclei, the features of heavy-ion collisions, and the equation of state (EOS) of nuclear matter.Inclusion of the scalar-isovector  meson in this scheme and investigation of its influence on low density asymmetric nuclear matter was realized in [2,3].At sufficiently high density, different exotic degrees of freedom, such as pion and kaon condensates, also deconfined quarks, may appear in the strongly interacting matter.The modern concept of hadron-quark phase transition is based on the feature of that transition, that is the presence of two conserved quantities in this transition: baryon number and electric charge [4].It is known that, depending on the value of surface tension, s  , the phase transition of nuclear matter into quark matter can occur in two scenarios [5,6]: ordinary first order phase transition with a density jump (Maxwell construction), or formation of a mixed hadron-quark matter with a continuous variation of pressure and density (Glendenning construction) [4].Uncertainty of the surface tension values does not allow to determine the phase transition scenario, taking place in realty.In our recent paper [7] in the assumption that the transition to quark matter is a usual first-order phase transition, described by Maxwell construction, we have shown that the presence of the  -meson field leads to the decrease of transition pressure 0 P , of baryon number densities In this article we investigate the hadron-quark phase transition of neutron star matter, when the transition proceeds through a mixed phase.Influence of  -meson field on such phase transition characteristics and of compact star structure is discussed.
Five other constants, , b and c , then can be numerically determined:  phase transition parameters and, thus, alter the structural characteristics of neutron stars.The results of our analysis show that the scalar -isovector  -meson field inclusion leads to the increase of the EOS stiffness of nuclear matter due to the splitting of proton and neutron effective masses, and also due to the increase of asymmetry energy (for details see Ref. [9]).

Quark Matter
To describe the quark phase an improved version of the MIT bag model is used, in which the interactions between d u， and s quarks inside the bag are taken into account in the one-gluon exchange approximation [10].We choose for the strong interaction constant.

Deconfinement Phase Transition Parameters
There


. The mechanical and chemical equilibrium conditions (Gibbs conditions) for mixed phase are The volume fraction of quark phase is / ( ) where QM V and NM V are volumes occupied by quark matter and nucleonic matter, respectively.We applied the global electrical neutrality condition for mixed quark-nucleonic matter, according to Glendenning [4,8], (1 ) ( , ) ( ) 0.
The baryon number density in the mixed phase is determined as and the energy density is In case of 0 The system of Eqs.3, 5, 6 and 7 makes it possible to determine EOS of mixed phase between these critical states.
Note, that in the case of an ordinary first-order phase transition both nuclear and quark matter are assumed to be separately electrically neutral, and at some pressure 0 P , corresponding to the coexistence of the two phases, their baryon chemical potentials are equal, i.e., Such phase transition scenario is known as phase transition with constant pressure (Maxwell construction).Table 1 represents the parameter sets of the mixed phase both with and without  -meson field.It is shown that the presence of  -field alters threshold characteristics of the mixed phase.The lower threshold parameters, , ,  are slowly decreased.
In Figure 3 we plot the species number densities as a function of baryon density n for Glendenning construc- tion.Quarks appear at the critical density 0.077 fm .The hadronic matter completely disappears at 1.083 fm , where the pure quark phase occurs.
Using the obtained EOS of nuclear matter, we have integrated the Tolman-Oppenheimer Volkoff equations and obtained the mass M and the radius R of compact stars for the different values of central pressure c P .Figure 4 illustrate the ( ) M R dependence of neutron stars.We can see, that the behavior of mass-radius dependence significantly differs for the two types of phase transitions.

CONCLUSIONS
In this paper we have studied the deconfinement phase transition of neutron star matter, when the nuclear matter is described in the RMF theory with  -meson effective field.We show that the inclusion of scalar -isovector  -meson field terms leads to the stiff nuclear matter

In Figure 1 Figure 1 .
Figure 1.Energy per baryon b E as a function of the baryon number density n and the asymmetry parameter  in case of a  -equilibrium charged npe -plasma.

Figure 2 .
Figure 2. Re-denoted meson mean-fields as a function of the baryon number density n in case of a β-equilibrium chargeneutral npe -plasma with and without  -field.
are two independent conserved charges in hadron quark phase transition: baryonic charge and electric charge.The constituents chemical potentials of the npe -plasma in  -equilibrium are expressed through two potentials, b


, corresponding to the lower threshold of a mixed phase, are determined by solving Eq.3 and Eq.5.This allows to find the lower boundary parameters N P , N  and N n .Similarly, we calculate the upper boundary values of mixed phase parameters, Q P , Q

Figure 4
shows that for 60  B MeV/fm 3 there are unstable regions, where / 0 c dM dP  between two stable branches of compact stars, corresponding to configurations with and without quark matter.In this case, there is a nonzero minimum value of the quark phase core radius.Accretion of matter on a critical neutron star configuration will then result in a catastrophic rearrangement of the star, forming a star with a quark matter core.The range of mass values for stars, containing the mixed phase, is [0.085 ; 1.853 ] M M   .In case of Maxwellian type phase transition, the analogous range is [0

Figure 3 .
Figure 3. Constituents number density vs. baryon number density in case of Glendenning construction.Vertical dotted lines represent the mixed phase boundaries.

Figure 4 .
Figure 4.The mass-radius relation of neutron star with different deconfinement phase transition scenarios.Open circles and squares denote the critical configurations for Glendenning and Maxwellian type transitions, respectively.Solid circles and squares denote hybrid stars with minimal and maximal masses, respectively.EOS.In a nucleonic star both the gravitational mass and corresponding radius of the maximum mass stable configuration increases with the inclusion of the  -field.The presence of scalar -isovector  -meson field alters the threshold characteristics of the mixed phase.The lower threshold parameters, , , N N N n P  are increased, while the upper thresholds, , , Q Q Q n P 

Table 1 .
The mixed phase parameters with and without meson field.