#### 2.1. Hadronic Matter

In the most primitive conception, the matter in the core of a neutron star is constituted from neutrons. At a slightly more accurate representation, the cores consist of neutrons and protons whose electric charge is balanced by leptons (

$\lambda =\{{e}^{-},{\mu}^{-}\}$). Other particles, like hyperons (

$B=\{n,p,\mathrm{\Lambda},\mathrm{\Sigma},\mathrm{\Xi}\}$) and the Δ resonance, may be present if the Fermi energies of these particles become large enough so that the existing baryon populations can be rearranged and a lower energy state be reached. To model this hadronic phase, we make use of the relativistic mean-field (RMF) approximation, in which the interactions between baryons are described by the exchange of scalar (

σ), vector (

ω), and isovector (

ρ) mesons [

26]. The standard mean-field Lagrangian is given by [

4,

5,

27,

28,

29,

30].

with the meson-baryon vortices given by

The

σ and

ω mesons in Equation (

1) are responsible for nuclear binding while the

ρ meson is required to obtain the correct value for the empirical symmetry energy. The cubic and quartic

σ terms in Equation (

1) are necessary, at the relativistic mean-field level, to obtain the empirical incompressibility of nuclear matter [

28,

29]. The field tensors

${\omega}_{\mu \nu}$ and

${\mathit{\rho}}_{\mu \nu}$ are defined as

${\omega}_{\mu \nu}={\partial}_{\mu}{\omega}_{\nu}-{\partial}_{\nu}{\omega}_{\mu}$ and

${\mathit{\rho}}_{\mu \nu}={\partial}_{\mu}{\mathit{\rho}}_{\nu}-{\partial}_{\nu}{\mathit{\rho}}_{\mu}$.

The parameters (i.e., coupling constants) of the theory must reproduce the bulk properties of infinite nuclear matter at saturation density,

${\rho}_{0}=0.16\phantom{\rule{3.33333pt}{0ex}}{\mathrm{fm}}^{-3}$. These are the binding energy

$E/N$, effective nucleon mass

${m}_{N}^{*}/{m}_{N}$, nuclear incompressibility

K, and the symmetry energy

${a}_{\mathrm{sy}}$ and its density derivative

L. Of the six, the values of

K,

${a}_{\mathrm{sy}}$, and

L carry some uncertainty. The

K value is believed to lie in the range between about 220 and 260 MeV [

31], or between 250 and 315 MeV, as recently suggested in Reference [

32]. The values for

${a}_{\mathrm{sy}}$ and

L are in the ranges of 29 to 35 MeV and 43 to 70 MeV, respectively [

33,

34,

35]. We have chosen two parameter sets, denoted GM1 and DD2, which approximately cover the uncertainties in the nuclear matter properties just above. They are listed in

Table 1.

The coupling constants associated with GM1 are

${g}_{\sigma B}=9.572$,

${g}_{\omega B}=10.618$,

${g}_{\rho B}=8.198$,

${b}_{\sigma}=0.002936$, and

${c}_{\sigma}=-0.00107$ [

36].

The field equations for the baryon fields

${\psi}_{B}$, which follow from Equation (

1), are given by [

4,

5,

8,

27].

The meson fields in Equation (

3) are solutions of the following field equations [

4,

5,

8,

27].

In the standard RMF limit, the meson fields of Equations (

4)–(

6) simplify to [

4,

5,

8,

27].

where

$\overline{\sigma}$,

$\overline{\omega}$, and

${\overline{\rho}}_{03}$ denote the mean-field limits of

σ,

ω, and

$\overline{\rho}$, respectively, and the effective baryon masses are given by

${m}_{B}^{*}\left(\overline{\sigma}\right)={m}_{B}-{g}_{\sigma B}\overline{\sigma}$.

${J}_{B}$ and

${I}_{3B}$ denote the spin respectively isospin of a baryon of type

B, and

${\rho}_{B}$ stands for the number density of baryon

B.

In addition to the standard RMF theory discussed just above, we also consider a lagrangian where the meson-baryon vortices

${\mathrm{\Gamma}}_{MB}$ (where

$M=\sigma ,\omega ,\rho $) of Equation (

2) are no longer constant but rather depend on density [

38,

39,

40]. In that case the values of the vortices are derived from relativistic Dirac-Brueckner calculations of nuclear matter, which use one-boson-exchange interactions as an input. A characteristic feature of the density-dependent theory is the occurrence of rearrangement terms in the expression for the baryon chemical potential, which leads to a more complex condition for chemical equilibrium compared to the standard RMF approximation [

40]. The parameter set of the density-dependent (DD) treatment adopted in this paper is denoted DD2 (

${G}_{V}=0$), where a vanishing vector coupling constant

${G}_{V}=0$ among quarks has been chosen [

37]. For DD2, the coupling constants at saturation density are

${g}_{\sigma N}=10.687$,

${g}_{\omega N}=13.342$, and

${g}_{\rho N}=3.627$ [

37], which lead to the saturation properties of infinite nuclear matter shown in

Table 1.

The equation of state of the standard mean-field treatment (for the DD formalism, see References [

39,

40,

41,

42]) is obtained by solving Equations (

7)–(

9) together with the charge conservation conditions (baryonic, electric) given by [

4,

5,

27].

where

${\rho}_{b}$ is the total baryonic density and

${q}_{B}$ and

${q}_{\lambda}$ are the electric charges of baryons and leptons, respectively. Particles in the hadronic phase are subject to the chemical equilibrium condition

where

${\mu}_{B}$ is the chemical potential and

${b}_{B}$ is the baryon number of baryon

B.

${\mu}_{n}$ and

${\mu}_{e}$ denote the linearly independent chemical potentials of neutrons and electrons, respectively, which reflect baryon number and electric charge conservation on neutron star matter. New baryon or lepton states are populated when the right side of Equation (

11) is greater than the particle’s chemical potential. The baryonic and leptonic number densities (

${\rho}_{B}$ and

${\rho}_{\lambda}$) are both given by

${\rho}_{i}=(2{J}_{i}+1){k}_{i}^{3}/6{\pi}^{2}$. The unknowns of the theory are the meson mean-fields (

σ,

ω,

ρ), and the neutron and electron fermi momenta (

${k}_{n}$,

${k}_{e}$). Finally, the energy density and pressure of the hadronic phase are given by [

4,

5,

27].

#### 2.2. Deconfined Quark Phase

A popular model widely used to describe deconfined 3-flavor (up, down, strange) quark matter is the Nambu–Jona-Lasinio model [

43,

44,

45,

46,

47]. Here we use a non-local extension of this model (n3NJL) [

48,

49], whose effective action is given by

where

$\psi \equiv {(u,d,s)}^{T}$,

$\widehat{m}=\mathrm{diag}({m}_{u},{m}_{d},{m}_{s})$ is the current quark mass matrix,

${\lambda}_{a}$ (

$a=1,...,8$) denote the Gell-Mann matrices–generators of SU(3), and

. The currents

${j}_{a}^{S,P}\left(x\right)$ and

${j}_{V}^{\mu}\left(x\right)$ are given by

where

$\tilde{g}\left(z\right)$ is a form factor responsible for the non-local character of the interaction. Finally, the constants

${T}_{abc}$ in the t’Hooft term accounting for flavor-mixing are defined by

The current quark mass

$\overline{m}$ of up and down quarks and the coupling constants

${G}_{S}$ and

H in Equation (

14), have been fitted to the pion decay constant,

${f}_{\pi}$, and meson masses

${m}_{\pi}$,

${m}_{\eta}$, and

${m}_{{\eta}^{\prime}}$, as described in [

50,

51]. The result of this fit is

$\overline{m}=6.2$ MeV,

$\mathsf{\Lambda}=706.0$ MeV,

${G}_{S}{\mathsf{\Lambda}}^{2}=15.04$,

$H{\mathsf{\Lambda}}^{5}=-337.71$. The strange quark current mass is treated as a free parameter and was set to

${m}_{s}=140.7$ MeV. The strength of the vector interaction

${G}_{V}$ is expressed in terms of the strong coupling constant

${G}_{S}$. To account for the uncertainty in the theoretical predictions for the ratio

${G}_{V}/{G}_{S}$, we treat the vector coupling constant as a free parameter [

52,

53,

54], which varies from 0 to

$0.09\phantom{\rule{0.166667em}{0ex}}{G}_{S}$.

For the mean-field approximation, the thermodynamic potential following from

${S}_{E}$ of Equation (

14) is given by

where

${\overline{\sigma}}_{f}$,

${\overline{\omega}}_{f}$, and

${\overline{S}}_{f}$ are the quark scalar, vector, and auxiliary mean fields, respectively. Moreover,

${E}_{f}$ is given by

${E}_{f}=\sqrt{{\mathit{p}}^{2}+{m}_{f}^{2}}$ and we have defined

${\omega}_{f}^{2}={({p}_{0}+i{\mu}_{f})}^{2}+{\mathit{p}}^{2}$. The momentum dependent quark masses are given by

${M}_{f}\left({\omega}_{f}^{2}\right)={m}_{f}+{\overline{\sigma}}_{f}g\left({\omega}_{f}^{2}\right)$. The quantities

$g\left({\omega}_{f}^{2}\right)=\mathrm{exp}(-{\omega}_{f}^{2}/{\mathsf{\Lambda}}^{2})$ are the Gaussian form factors, responsible for the nonlocal nature of the interaction among quarks [

49]. The auxiliary mean fields are given by

The vector interactions taken into account in the treatment shift the quark chemical potentials as ${\widehat{\mu}}_{f}={\mu}_{f}-g\left({w}_{f}^{2}\right){\overline{\omega}}_{f}$ and ${\widehat{\omega}}_{f}^{2}={({p}_{0}+i{\widehat{\mu}}_{f})}^{2}+{p}^{2}$. The scalar and vector mean fields are obtained by minimizing the grand thermodynamic potential, $\partial \mathsf{\Omega}/\partial {\overline{\sigma}}_{f}=0$ and $\partial \mathsf{\Omega}/\partial {\overline{\omega}}_{f}=0$. Finally, the quark number densities are obtained from ${\rho}_{f}=\partial \mathsf{\Omega}/\partial {\mu}_{f}$.

To determine the equation of state one must solve a nonlinear system of equations for the fields

${\overline{\sigma}}_{f}$ and

${\overline{\omega}}_{f}$, and the neutron and electron chemical potentials

${\mu}_{n}$ and

${\mu}_{e}$. This system of equations consists of the mean field equations

with cyclic permutations over the quark flavors,

${\overline{\omega}}_{f}-2{G}_{V}\partial \mathsf{\Omega}/\partial {\overline{\omega}}_{f}=0$, and the baryonic and electric charge conservation equations

${\sum}_{f=u,d,s}\phantom{\rule{0.277778em}{0ex}}{\rho}_{f}-3{\rho}_{b}=0$ and

${\sum}_{f=u,d,s}\phantom{\rule{0.277778em}{0ex}}{\rho}_{f}\phantom{\rule{0.166667em}{0ex}}{q}_{f}+{\sum}_{\lambda ={e}^{-},{\mu}^{-}}\phantom{\rule{0.277778em}{0ex}}{\rho}_{\lambda}\phantom{\rule{0.166667em}{0ex}}{q}_{\lambda}=0$, respectively. Finally, the pressure

${p}_{Q}$ and energy density

${\u03f5}_{Q}$ are given by

and

where

${\mathsf{\Omega}}_{0}$ was chosen by the condition that

${P}_{Q}$ vanishes at

$T=\mu =0$ [

48,

49].

#### 2.3. Quark-Hadron Mixed Phase

To model the mixed phase region of quarks and hadrons in neutron stars, we use the Gibbs condition for phase equilibrium between hadronic (

H) and quark (

Q) matter,

where

${P}_{H}$ and

${P}_{Q}$ denote the pressures of hadronic matter and quark matter, respectively [

55,

56]. The quantity

$\left\{\varphi \right\}$ in Equation (

22) stands collectively for the field variables (

$\overline{\sigma}$,

$\overline{\omega}$,

$\overline{\rho}$) and Fermi momenta (

${k}_{B}$,

${k}_{\lambda}$) that characterize a solution to the equations of confined hadronic matter (

Section 2). We use the symbol

$\chi \equiv {V}_{Q}/V$ to denote the volume proportion of quark matter,

${V}_{Q}$, in the unknown volume

V. By definition,

χ varies between 0 and 1, depending on how much confined hadronic matter has been converted to quark matter. Equation (

22) is to be supplemented with the conditions of global baryon charge conservation and global electric charge conservation. The global conservation of baryon charge is expressed as [

55,

56].

where

${\rho}_{Q}$ and

${\rho}_{H}$ denote the baryon number densities of the quark phase and hadronic phase, respectively. The global neutrality of electric charge is given by [

55,

56].

with

${q}_{Q}$ and

${q}_{H}$ denoting the electric charge densities of the quark phase and hadronic phase, respectively. We have chosen global rather than local electric charge neutrality. Local NJL studies carried out for local electric charge neutrality have been reported recently in References [

57,

58,

59,

60].

In

Figure 2, we show the GM1 and DD2 equations of state (EoS) used in this work to study the quark-hadron composition of rotating neutron stars. The solid dots mark the beginning and the end of the quark-hadron mixed phases for these equations of state. Since the Gibbs condition is used to model the quark-hadron phase transition, pressure varies monotonically with the proportion of the phases in equilibrium. This would not be the case if the Maxwell construction were used to model the phase equilibrium between quarks and hadrons, in which case the pressure throughout the mixed phase is constant. For that reason, the mixed phase would strictly be excluded from neutron stars, but small cores made entirely of quark matter may still be possible in neutron stars close the maximum-mass model (see, for instance, Refrences [

61,

62], and references therein).

A Maxwell-like phase transition is generally supported by larger surface tensions,

σ, of quark matter (see, however, Reference [

63]). Values of

$\sigma \sim 30\phantom{\rule{0.166667em}{0ex}}{\mathrm{MeV}/\mathrm{fm}}^{2}$ have recently been suggested in the literature [

64,

65,

66,

67,

68], but its actual value is an open issue.

The GM1 and DD2 equations of state are compared in

Figure 2 with models for the equation of state that have recently been suggested in the literature. ’HLPS’ and ’Neutron matter’ show the constraints on the equation of state established by Hebeler, Lattimer, Pethick, and Schwenk [

33,

69]. The curves labeled ’EoS I’, ’EoS II’, and ’EoS III’ show the compact star matter equations of state determined by Kurkela et al. [

70], which are based on an interpolation between the regimes of low-energy chiral effective theory and high-density perturbative QCD. One sees that the GM1 and DD2 models are well within these limits. The only difference concerns the behavior of the equation of state at sub-nuclear densities (labeled “Neutron matter” in

Figure 2), where our models provide slightly more pressure. This, however, does in no way impact the results for the quark-hadron compositions shown in

Section 4, because of the large masses of these stars.

Figure 3 shows the gravitational mass versus central neutron stars density (left panel) and gravitational mass versus radius of non-rotating neutron stars for the equations of state discussed in this section. The maximum masses of these neutron stars are between 1.9 and

$2.1\phantom{\rule{0.166667em}{0ex}}{M}_{\odot}$. Stellar rotation, which will be discussed in the next section, increases the masses of these stars to values that are between 2.2 and

$2.4\phantom{\rule{0.166667em}{0ex}}{M}_{\odot}$, while their central densities drops by around 20%.