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τ$_{b}$ is given by 

where p$_{Fτ}$$_{b}$ is their Fermi momentum, and U$_{τ}$$_{b}$ and m$_{τ}$$_{b}$ are, respectively, the single-particle potential and mass of the baryon. For leptons ( l = µ , e ) their chemical potential is given by µ$_{l}$ = ( m 2 l + p 2 $_{Fl}$) 1 / 2 with p$_{Fl}$ = (3 π $^{2}$ρ$_{l}$ ) 1 / 3 being their Fermi momentum. The relative abundances of various hadrons and leptons for a given total baryon density are then obtained by solving above equations. 

In terms of the densities of various particles, the total energy density of the hadron phase can be written as 

where V$_{H}$ and V$_{L}$ are the contributions from baryons and leptons, respectively. The former can be written as 

where V$_{HP}$ = (1 / 2) ∑ b,b ′ V$_{bb}$ ′ is the potential energy den- sity of baryons with V$_{bb}$ ′ calculated from Eq. (2), and V$_{HK}$ and V$_{HM}$ are, respectively, the kinetic energy and mass contributions given by 

The contribution V$_{L}$ from leptons is calculated by treat- ing them as a free Fermi gas, i.e., 

The pressure of the hadron phase is obtained from the thermodynamical relation 

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where b and l run over all species of baryons and lep- tons, respectively. We note that the thermodynamical consistency condition 

is satisfied. 

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As the nuclear matter density increases, such as in the core of neutron stars, not only hyperons appear but also a quark matter could exist [15]. To take into consideration possible transition between the hadronic matter and the quark matter, we follow many previous studies to use in the present study the MIT bag model [16, 29] to describe the cold quark matter that might exist in the dense core of neutron stars. 

For the quark phase consisting of quarks and leptons, the baryon number conservation and charge neutrality conditions are given by expressions similar to Eqs. (9) and (10) with i denoting now quarks and leptons. For the β -equilibrium condition in the quark phase, it is given by µ$_{i}$ = µ Q b b$_{i}$ − µ Q c q$_{i}$ with µ Q b and µ Q c being the baryon and charge chemical potentials of the quark phase, re- spectively. 

The total energy density and pressure of the quark phase can be calculated from 

where V$_{Q}$ and P$_{Q}$ are the energy density and pressure of quarks, which can be calculated from the MIT bag model as described in the following, and V$_{L}$ and P$_{L}$ are the energy density and pressure of leptons given by same expressions as those in the hadron phase. 

At zero temperature, the density ρ$_{f}$ , chemical potential µ$_{f}$ , and energy density V$_{f}$ of quarks of flavor f = u, d, s in the quark matter are given, respectively, by 

where p$_{Ff}$ is the Fermi momentum of quarks of flavor f . For the quark masses, they are taken to be m$_{u}$ = m$_{d}$ = 0 and m$_{s}$ = 150 MeV. In the bag model, the energy density is modified by a bag constant B , resulting in an energy density given by 

This leads to the following pressure for the quark matter: 

where ρ is the total baryon density of the quark phase 

## B. The quark phase 

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