^{1,2}

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^{3}

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Corresponding author.

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We investigate the equation of state (EoS) and the effect of the hadron-quark phase transition of strong interaction matter in compact stars. The hadron matter is described with the relativistic mean field theory, and the quark matter is described with the Dyson-Schwinger equation approach of QCD. The complete EoS of the hybrid star matter is constructed with not only the Gibbs construction but also the 3-window interpolation. The mass-radius relation of hybrid stars is also investigated. We find that, although the EoS of both the hadron matter with hyperon and

It has been well known that, when one has an equation of state (EoS) for the dense matter in a star, one can calculate the mass-radius relation of the star by solving the Tolman-Oppenheimer-Volkov (TOV) equation, and compare the result with astronomical observations. Compact stars are then regarded as wonderful laboratories in the Universe, which have the extreme condition impossible to reach on Earth, to test the theories for not only cold-dense matter, but also thermal-dense matter

The EoS has been known as essential to astronomical research (for reviews, see, e.g., Refs.

One way to solve the hyperon puzzle is the modification of the interactions at high density. There have been phenomenological models for the matter consisting of nucleons and hyperons and corresponding leptons (hereafter we refer to the star composed of such a kind of matter simply as a neutron star) which predict neutron stars of mass exceeding

However, the possible appearance of quark matter should also be taken into consideration, and is believed to be a straightforward way to solve the hyperon puzzle (see, e.g., the discussion in Ref.

A simple but widely implemented model for the quark matter is the MIT bag model (see, e.g., Refs.

It has been known that the Dyson-Schwinger equations (DSEs) (see, e.g., Ref.

For the EoS of the matter in hybrid stars, one should take into account the transition from hadron phase to quark phase. An ideal theory for describing the hadron-quark phase transition is that all the properties are depicted with a unified Lagrangian for the system, but such a theory has definitely not yet been established. At present, one has to describe the quark phase and the hadron phase with separate approaches, and derive the complete EoS by construction.

One of the commonly used methods of the construction is the Gibbs construction

However, apart from its success, the Gibbs construction has its limitations. For example, near the hadron-quark phase transition density, the distance between quarks in one hadron and that in different hadrons is of the same order; the hadrons should then not be regarded as point particles. Though the hadron models are accurate for the matter near the saturation density due to the calibrations coming from plenty of experiments, they are unreliable and different from each other greatly in the higher density region, the phase transition regions (see, e.g., Refs.

To fix these problems, the 3-window construction model

In this paper, we investigate the EoS of compact star matter in a quite large baryon density region and the mass-radius relation of compact stars to solve the hyperon puzzle (as well as the “

This paper is organized as follows. After this introduction, we describe briefly the models for the hadron matter and the quark matter in Secs.

In order to calculate the mass-radius relation of the star composed of mainly strong interaction matter, we need to calculate the EoS of the matter (hadron matter, quark matter, or their mixture). For the hadron matter in which the quark degrees of freedom do not appear, we adopt the relativistic mean field theory.

The RMF theory

The Lagrangian of the TW-99 model for the hadron matter including hyperons and

In this work, we consider not only the baryon octet

The Lagrangian for the baryon octet reads

The

In some other literatures, the self-interaction of the

Parameters of the mesons and their couplings (taken from Ref.

For hyperons, we represent them with the relation between the hyperon coupling and the nucleon coupling as

The

The field equations can be derived by differentiating the Lagrangian. Under RMF approximation, the system is assumed to be in the static, uniform ground state. The partial derivatives of the mesons all vanish; only the 0-component of the vector meson and the third component of the isovector meson survive and can be replaced with the corresponding expectation values. The field equations of the mesons are then

The equation of motion (EoM) of the baryon is

Under the EoM of Eq.

One can then get the baryon (number) density,

The expression of the density of leptons is the same as those for baryons, except that the effective mass and the effective chemical potential should be replaced with the corresponding mass and chemical potential of the leptons,

The matter in the star composed of hadrons should be in

Then, combining Eqs.

The EoS of hadron matter can be calculated from the energy-momentum tensor,

The contribution of the leptons to the energy density can be written in the similar form as baryons with a spin degeneracy parameter

As for the pressure of the system, we can determine that with the general formula,

To describe the properties of the matter in quark phase, we adopt the DSE approach of QCD

The starting point of the DSE approach is the gap equation,

At finite chemical potential, the quark propagator can be decomposed according to the Lorentz structure as

At zero chemical potential, a commonly used ansatz for the dressed gluon propagator and the dressed quark-gluon interaction vertex is

For the interaction part, we adopt the Gaussian-type effective interaction (see, e.g., Refs.

In case of finite chemical potential, an exponential dependence of the

Moreover, Ref.

With the above equations, we can get the quark propagator, and derive the EoS of the quark matter in the same way as taken in Refs.

The number density of quarks as a function of its chemical potential is

The pressure of each flavor of quark at zero temperature can be obtained by integrating the number density,

The total pressure of the quark matter is the sum of the pressure of each flavor of quark,

Theoretically, the starting point of the integral

The quark matter in a compact star should also be in

Therefore, we can calculate the properties of the quark matter with a given baryon chemical potential (baryon density).

After having the EoSs of both the hadron matter and the quark matter, we derive the complete EoS of the hybrid star matter by construction.

A widely used construction is the Gibbs construction

In the mixed region, the pressure of the two phases is the same. And though the two phases may not be charge neutral separately, there still exists a global electric charge neutral constraint. If we define the quark fraction

Combining Eqs.

The calculated relation between the pressure and the baryon chemical potential of different phases is shown in Fig.

Calculated result of the relation between the pressure and the baryon chemical potential of the matter with different particle components. The solid, dashed curve corresponds to that of the hadron matter without and with hyperons, respectively. The star curve denotes that of the hadron matter with both hyperons and

In the scheme of the 3-window interpolation construction, as the baryon density increases, the compact star matter goes through three regions. At low density, the matter is in hadron phase composed of hadrons which are approximated as point particles. At high density, quarks, the components of hadrons, are no longer confined, so that the properties of the matter are governed by the quark degrees of freedom. In the middle density, there should be a transition from hadron matter to quark matter, where hadrons percolate, and the boundary of any hadron gradually disappears.

The EoSs of both the hadron and the quark phases are based on models. The hadron model results are accurate near the saturation density, but differ greatly in the high density region, while the quark models are appropriate in the extremely high density region and lose accuracy at low density. In the transition region, neither the hadron model nor the quark model represent the nature individually. Therefore, an interpolation between the quark and the hadron phases should be taken.

Here we adopt the

Note that the interpolating function in Eq.

Variation behaviors of the interpolation functions

The pressure of the transition region can be determined with the thermodynamic relation,

The calculated results of the relation between the pressure and the energy density (the EoS in convention) of the pure hadron matter and that of the pure quark matter are shown in Fig.

Calculated EoS of the hadron matter with different particle compositions and the pure quark matter via different parameters in the interaction kernel. The solid curve and dashed curve correspond to that of the hadron phase without and with hyperons, respectively. The star curve denotes that of the hadron phase with both hyperons and

The calculated results of the EoS of the hybrid star matter under the Gibbs construction with quark phase fixed via different parameters in the DSE approach of QCD are shown in Fig.

Calculated EoS of the hybrid star matter under the Gibbs construction and that of the pure hadron matter without including hyperons and

Recalling the scheme of the Gibbs construction, one can know that the point at which the curve is not smooth corresponds to the appearance of quark matter (the lower unsmooth point) and the disappearance of hadron matter (the upper unsmooth point), where the quark fraction

From Eqs.

Calculated

It can be easily seen from Fig.

The calculated results of the EoS of the hybrid star matter under the 3-window interpolation construction with parameters

Calculated EoS of the hybrid star matter under the 3-window interpolation construction with parameters

The same as Fig.

Comparing with the results in Gibbs construction shown in Fig.

A more significant difference between the results under the 3-window interpolation and the Gibbs construction is the stiffness of the EoS. Even though the EoS of the quark phase is generally softer than that of the hadron phase, the EoS of the hybrid star matter under the 3-window interpolation construction can be stiffer than the hadron EoS in the transition region, especially for the hadron matter with the inclusion of hyperons and

The calculated results of the relation between the pressure and the baryon chemical potential (

Calculated

Recalling the scheme of the 3-window interpolation [Eqs.

Calculated

The same as Fig.

To show the characteristics of the EoS of the hybrid star matter more explicitly, we display the calculated results of the relation between the pressure and the baryon density (

Calculated relation between the pressure and the baryon number density (

Figures

Calculated

The same as Fig.

The mass-radius relation of compact stars can be calculated by solving the TOV equation,

The obtained mass-radius relation for the pure hadron star and pure quark star is shown in Fig.

Calculated mass-radius relation of a compact star with pure hadron matter or pure quark matter. The

The obtained mass-radius relation of the pure nucleon star whose composing matter does not include either hyperons or

Calculated mass-radius relation of a neutron star without the inclusion of hyperons (solid line) and those of hybrid stars with the EoS of the composing matter being fixed using the Gibbs construction (lines with symbols). The Nq

Figure

The obtained results of the mass-radius relation of the hybrid star with the EoS of the matter being constructed by the 3-window interpolation with parameters

Calculated mass-radius relation of a neutron star without the inclusion of hyperons (solid line) and those of hybrid stars with the EoS of the composing matter being constructed using the 3-window interpolation with parameters

The same as Fig.

One can see from the figures that, just like those in the case of the Gibbs construction, the maximum mass and the corresponding radius of the hybrid star decrease with the increasing of the parameter

As for the results in the case with hyperons and

Comparing Figs.

To summarize the main results of the properties of the stars and for the convenience of further discussion, we list our obtained maximum mass, the corresponding radius, and center density of the hybrid star whose EoS of the composing matter is fixed with different construction schemes in Table

Calculated results of the maximum mass, the corresponding radius and center density of the pure hadron star (NS) and pure quark star (QS), and those of the hybrid stars (HS) whose EoS of the composing (hybrid) matter is determined with different construction schemes and different parameters for the quark sector.

One can notice easily from Table

At first we show the calculated results of the particle fraction as a function of baryon density for the hadron matter in Figs.

Calculated result of the particle fraction of the hadron matter including nucleons and hyperons.

Calculated result of the particle fraction of the hadron matter including nucleons, hyperons, and

One can see further from Figs.

The calculated result of the baryon density dependence of the particle fraction of the hybrid matter under the Gibbs construction and the variation behavior of the particle fraction of the matter in terms of the distance from the center of the maximum mass hybrid star are illustrated in Figs.

Calculated result of the baryon density dependence of the particle fraction in the hybrid star matter under the Gibbs construction, where the hadron matter does not include either hyperons or

Calculated variation behavior of the particle fraction of the matter in terms of the distance from the center of the maximal mass hybrid star under the Gibbs construction. The other items are the same as Fig.

We have also calculated the baryon density dependence of the particle fraction of the hybrid matter under the 3-window interpolation construction, as well as the variation behavior of the particle fraction of the matter in terms of the distance from the center of the maximum mass hybrid star. The obtained result of the baryon density dependence of the particle fraction in the hybrid matter whose hadron sector consists of only nucleons is shown in Fig.

Calculated result of the baryon density dependence of the particle fraction in the hybrid star matter under the 3-window interpolation construction, where the hadron matter does not include either the hyperons or the

The same as Fig.

The same as Fig.

Calculated variation behavior of the particle fraction of the matter in terms of the distance from the center of the maximum mass hybrid star under the 3-window interpolation construction. The hadron phase in the hybrid matter does not include hyperons and

The same as Fig.

The same as Fig.

From Figs.

It is also remarkable that as the compact star with maximum mass exceeding

We have investigated the mass-radius relation of hybrid stars with both the Gibbs construction and the 3-window interpolation construction for the EoS of the composing matters in this paper. For that of the hadron phase we adopt the result of the relativistic mean field theory, and for that of the quark phase we take the result via the Dyson-Schwinger equation approach of QCD.

Our calculation manifests that the Gibbs construction results in a rather soft EoS for the hybrid star matter, while for the 3-window interpolation, the EoS of the hybrid star matter can be stiffer than those in both the hadron phase and the quark phase separately. Therefore, for a hybrid star whose hadron matter sector includes hyperons and

It indicates that taking the hadron-quark phase transition into account with the 3-window interpolation scheme to construct the EoS of the hybrid star matter can solve the hyperon puzzle and the

For the radius of the compact stars, our result of the hybrid star seems larger than some of the estimates from observational data at first glance. For example, the radius of the neutron star with canonical mass,

Analyzing the detail of our calculation and the obtained results, we confirm that the maximum mass of the neutron star is determined mainly by the stiffness of the EoS at density above

The work was supported by the National Natural Science Foundation of China under Contracts No. 11435001, No. 11305144, and No. 11475149, and the National Key Basic Research Program of China under Contracts No. G2013CB834400 and No. 2015CB856900.