# Quark-quark interaction and quark matter in neutron stars

Y. Yamamoto1<EMAIL_ADDRESS>N. Yasutake2 Th.A. Rijken31 1RIKEN Nishina
Center, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan
2Department of Physics, Chiba Institute of Technology, 2-1-1 Shibazono
Narashino, Chiba 275-0023, Japan
3IMAPP, Radboud University, Nijmegen, The Netherlands

###### Abstract

Hyperon ($Y$) mixing in neutron-star matter brings about a remarkable
softening of the equation of state (EoS) and the maximum mass is reduced to a
value far less than $2M_{\odot}$. One idea to avoid this “hyperon puzzle in
neutron stars” is to assume that the many-body repulsions work universally for
every kind of baryons. The other is to take into account the quark
deconfinement phase transitions from a hadronic EoS to a sufficiently stiff
quark-matter EoS. In the present approach, both effects are handled in a
common framework. As well as the hadronic matter, the quark matter with the
two-body quark-quark interactions are treated within the Brueckner-Bethe-
Goldstone theory beyond the mean field frameworks, where interaction
parameters are based on the terrestrial data. The derived mass-radius
relations of neutron stars show that maximum masses reach over $2M_{\odot}$
even in the cases of including hadron-quark phase transitions, being
consistent with the recent observations for maximum masses and radii of
neutron stars by the NICER measurements and the other multimessenger data.

###### pacs:

21.30.Cb, 21.30.Fe, 21.65.+f, 21.80.+a, 12.39.Jh, 25.75.Nq, 26.60.+c

##  I Introduction

In studies of neutron stars, the fundamental role is played by the equation of
state (EoS) for dense nuclear matter. The observed masses of neutron stars
J1614$-$2230 Demorest10 , J0348+0432 Antoniadis13  and J0740+6620
Cromartie2020  are given as $(1.97\pm 0.04)M_{\odot}$, $(2.01\pm
0.04)M_{\odot}$ and $(2.17^{+0.11}_{-0.10})M_{\odot}$, respectively, being
important conditions for the stiffness of the EoS of neutron-star matter. In
non-relativistic approaches, the stiff EoS giving the maximum mass of
$2M_{\odot}$ can be derived from the existence of strongly repulsive effects
such as three-nucleon repulsions in the high-density region APR98 .

The hyperon ($Y$) mixing in neutron-star matter brings about a remarkable
softening of the EoS and a maximum mass is reduced to a value far less than
$2M_{\odot}$. The mechanism of EoS softening is understood as follows: With
increasing of baryon density toward centers of neutron stars, chemical
potentials of neutrons become high so that neutrons at Fermi surfaces are
changed to hyperons ($Y$) via strangeness non-conserving weak interactions
overcoming rest masses of hyperons. Then, it should be noted that naively such
a mechanism of EoS softening works also for mixing of any exotic particles
such as quarks into neutron matter.

One of the ideas to avoid this “hyperon puzzle in neutron stars” is to assume
that the many-body repulsions work universally for every kind of baryons NYT .
In Refs.YFYR14 YFYR16 YTTFYR17 , the multi-pomeron exchange potential (MPP)
was introduced as a model of universal repulsions among three and four baryons
on the basis of the extended soft core (ESC) baryon-baryon interaction model
developed by two of the authors (T.R. and Y.Y.) and M.M. Nagels ESC16 .

Another solution for the hyperon puzzle has been suggested by taking into
account quark deconfinement phase transitions from a hadronic-matter EoS
(H-EoS) to a sufficiently stiff quark-matter EoS (Q-EoS) in the neutron-star
interiors, namely by studying hybrid stars having quark matter in their cores
Schaffner99  Baldo2006  Lastowiecki2012  Shahrbaf1  Shahrbaf2  Maslov19  Xia19
Kojo2015  Baym2018  Otto2020 . It is well known that repulsive effects in
quark phases are needed to result in massive neutron stars of $2M_{\odot}$. In
the Nambu-Jona-Lasinio (NJL) model, for instance, repulsions to stiffen EoSs
are given by vector interactions Kunihiro  , strengths of which are treated as
phenomenological parameters to stiffen the EoSs. Note that NJL models,
including extended ones, are mainly based on mean-field approximations, in
which two-body quark-quark interactions are not used explicitly. In spite of
many works for hadron-quark phase transitions in neutron-star matter, there is
not yet a unified theory of both the hadronic and quark phases.

In this work, our approach to hadron-quark phase transitions is different from
the usual methods in which the deconfined quark phases are treated in mean-
field approximations. We handle here the quark matter with the two-body quark-
quark ($QQ$) potentials derived as follows: The meson-exchange quark-quark
potentials are derived from the ESC baryon-baryon ($BB$) potentials in the
framework of the constituent quark model (CQM). The quark-quark-meson ($QQM$)
vertices are defined such that, upon folding with the Gaussian ground-state
baryonic quark wave functions, the $BB$ potentials are reproduced QQint . In
this process the $QQM$ couplings are related to the $BBM$ couplings, and the
extra interactions at the quark level necessary to achieve this connection are
completely determined. (Like in the ESC16 $BB$-potentials, relativistic
effects are included in the $QQ$-potentials via the small components of the
Dirac-spinors and a $1/M_{Q}$-expansion.) The quark-quark instanton-exchange
potential is derived from tuning the baryon masses ($N,\Lambda,\Sigma,\Xi$),
and $\Delta_{33}$ in the CQM. Here, also the one-gluon-exchange (OGE) and the
confining potential are included.

With use of these $QQ$ potentials together with the ESC $BB$-potentials,
baryonic matter and quark matter are treated in the common framework of the
Brueckner-Bethe-Goldstone (BBG) theory, where the transitions between them are
described in a reasonable way. It should be emphasized here that our $QQ$
potentials are determined on the basis of the terrestrial data and do not
include parameters only for the purpose of stiffening the quark-matter EoS.

Recently, the radius measurement has been performed for the most massive
neutron star PSR J0740+6620: The two analyses have been done independently for
the X-ray data taken by the Neutron Star Interior Composition Explorer (NICER)
and the X-ray Multi-Mirror (XMM-Newton) observatory. The radius and mass are
$12.39^{+1.30}_{-0.98}$ km and $2.072^{+0.067}_{-0.066}$ M⊙ Riley2021  or
$13.7^{+2.6}_{-1.5}$ km ($68\%$) and $2.08\pm 0.07$ M⊙ Miller2021 . The radius
of a typical 1.4M⊙ neutron star $R_{1.4M_{\odot}}$ has been estimated by
combining the NICER measurements and the other multimessenger data Raaij2021
Peter2021 . In Ref.Raaij2021 , the two values of
$R_{1.4M_{\odot}}=12.33^{+0.76}_{-0.81}$ km and
$R_{1.4M_{\odot}}=12.18^{+0.56}_{-0.79}$ km are obtained for the two different
high-density EoSs of a piecewise-polytrophic (PP) model and a model based on
the speed of sound, respectively. In Ref.Peter2021 , the estimated value is
$R_{1.4M_{\odot}}=11.94^{+0.76}_{-0.87}$ km at $90\%$ confidence. These values
of radii are rather similar to each other. On the other hand, when the
implication of PREX-II for the neutron skin thickness of heavy nuclei are
taken into account on the neutron-star EoS, they obtain 13.33 km
$<R_{1.4M_{\odot}}<14.26$ km Brendan2021 . Our obtained EoSs in this work are
investigated in the light of these new data.

This paper is organized as follows: In Sect.II, the hadronic-matter EoS
(H-EoS) is recapitulated on the basis of our previous works. In Sect.III, on
the basis of realistic $QQ$ interaction models, the BBG theory is applied to
quark matter: In III-A, the G-matrix framework is outlined for quark matter.
In III-B, our $QQ$ potentials are explained, which are composed of the
extended meson-exchange potential, the multi-pomeron potential, the instanton
potential and the one-gluon exchange potential. In III-C, the $QQ$ G-matrix
interactions in coordinate space are parameterized as density-dependent
interactions. In Sect.IV, there are obtained the quark-matter EoSs (Q-EoS) and
$MR$ diagrams of hybrid stars: In IV-A, Q-EoSs are derived. In IV-B, hadron-
quark phase transitions in hybrid stars are investigated on the basis of the
obtained EoSs. In IV-C, the $MR$ relations of hybrid stars are obtained by
solving the TOV equation. The conclusion of this paper is given in Sect.V.

##  II Hadronic-matter EoS

Here, the hadronic matter is defined exactly as $\beta$-stable baryonic matter
including leptons. On the basis of the BBG theory, the hadronic-matter EoS
(H-EoS) is derived with use of the ESC baryon-baryon interaction model YFYR14
YFYR16 YTTFYR17 . Then, the EoS is stiff enough to assure the neutron-star
masses of $2M_{\odot}$, if the strong three-nucleon repulsion is taken into
account. However, the hyperon ($Y$) mixing results in remarkable softening of
the EoS canceling this repulsive effect. In order to avoid this “hyperon
puzzle”, it is assumed that the repulsions work universally for $Y\\!N\\!N$,
$Y\\!Y\\!N$ $Y\\!Y\\!Y$ as well as for $N\\!N\\!N$. In YFYR14 YFYR16 YTTFYR17
, such universal repulsions are modeled as the multi-pomeron exchange
potential (MPP). In Ref.YTTFYR17  they proposed three versions of MPP: MPa,
MPa+, MPb. MPa and MPa+ (MPb) include the three- and four-body (only three-
body) MPPs, where mixings of $\Lambda$ and $\Sigma^{-}$ hyperons are taken
into account. The three-body part of MPa (MPa+) is less repulsive than (equal
to) that of MPb, and the four-body parts of MPa and MPa+ are equal to each
other. The EoSs for MPa and MPa+ are stiffer than that for MPb, because of
which radii of neutron stars obtained from the formers are larger than those
from the latter. Our ESC $BB$ interactions including MPb, MPa and MPa+ are
named as H1, H2, H3, for simplicity. In addition, we introduce two versions H0
and H1’ for comparative studies: H0 is the nucleon-nucleon part of H1, being
used in nuclear-matter EoSs with no hyperons. H1’ is the $BB$ interaction H1
in which MPP works only among nucleons. In the case of H1’, the remarkable
softening of the EoS is brought about by hyperon mixing.

As shown later, neutron-star radii $R$ for masses lower than about
$1.5M_{\odot}$ are determined by H-EoSs even in our $MR$ diagrams including
hadron-quark transitions. For the H-EoSs derived from the above $BB$
interactions, the obtained values of radii at $1.4M_{\odot}$
($R_{1.4M_{\odot}}$) are 12.4 km (H1), 13.3 km (H2) and 13.6 km (H3).

##  III Quark-Quark interaction and quark matter

###  III.1 G-matrix framework

The BBG theory is adopted for studies of quark matter on the basis of two-body
$QQ$ potentials given in Ref.QQint . Here, correlations induced by $QQ$
potentials are renormalized into coordinate-space G-matrix interactions, being
considered as effective $QQ$ interactions to derive the Q-EoS. In this stage
to construct G-matrix interactions in quark matter, color quantum numbers are
not taken into account.

We start from the G-matrix equation for the quark pair $f_{1}f_{2}$ in quark
matter, where $f_{1}$ and $f_{2}$ denote flavor quantum numbers ($u,d,s$):

$\displaystyle
G_{cc_{0}}=v_{cc_{0}}+\sum_{c^{\prime}}v_{cc^{\prime}}{Q_{y^{\prime}}\over\omega-\epsilon_{f^{\prime}_{1}}-\epsilon_{f^{\prime}_{2}}}G_{c^{\prime}c_{0}}$
(1)

where $c$ denotes a relative state $(y,T,L,S,J)$ with $y=f_{1}f_{2}$, $S$ and
$T$ being spin and isospin quantum numbers, respectively. Orbital and total
angular momenta are denoted by $L$ and $J$, respectively, with ${\bf J}={\bf
L}+{\bf S}$: A two-quark state is specified by ${}^{2S+1}L_{J}$. In Eq. (1),
$\omega$ gives the starting energy in the starting channel $c_{0}$. The Pauli
operator $Q_{y}$ acts on intermediate quark states with $y=f_{1}f_{2}$. We
adopt for simplicity the gap choice for the intermediate states in the
G-matrix equation, meaning that an intermediate energy $\epsilon_{f}$ is
replaced by a kinetic-energy operator. The G-matrix equation (1) is
represented in the coordinate space, whose solutions give rise to G-matrix
elements.

The quark single particle (s.p.) energy $\epsilon_{f}$ in quark matter is
given by

$\displaystyle\epsilon_{f}(k_{f})={\hbar^{2}k_{f}^{2}\over
2m_{f}}+U_{f}(k_{f})$ (2)

where $k_{f}$ is a $f$-quark momentum ($f=u,d,s$). The potential energy
$U_{f}$ is obtained self-consistently in terms of the G-matrix as

$\displaystyle U_{f}(k_{f})=\sum_{|{\bf k}_{f^{\prime}}|}\langle{\bf
k}_{f}{\bf k}_{f^{\prime}}\mid
G_{ff^{\prime}}(\omega=\epsilon_{f}(k_{f})+\epsilon_{f^{\prime}}(k_{f^{\prime}}))\mid{\bf
k}_{f}{\bf k}_{f^{\prime}}\rangle$ (3)

where $(TLSJ)$ quantum numbers are implicit. Then, the potential energy per
particle $\langle U\rangle=\sum_{f}\langle U_{f}\rangle$ is obtained by
averaging $U_{f}(k_{f})$ over $f$:

$\displaystyle\langle U\rangle=\frac{3}{2}\
\sum_{f}\omega_{f}\int_{0}^{k_{F}^{f}}\frac{d^{3}k_{f}}{(2\pi)^{3}}\
U_{f}(k_{f})$ (4)

where $\omega_{f}=\rho_{f}/(\sum_{f^{\prime}}\rho_{f^{\prime}})$ with a
$f$-quark density $\rho_{f}$. Making a partial wave reduction of Eq. (3) with
explicit use of $TLSJ$ quantum numbers, $U_{f}(k_{f})$ is represented as a sum
of $U^{TLSJ}_{f}(k_{f})$ obtained from G-matrix elements
$G_{ff^{\prime}}^{TLSJ}$.

###  III.2 Quark-Quark interactions

Our $QQ$ interaction is given by

$\displaystyle V_{QQ}$ $\displaystyle=$ $\displaystyle
V_{EME}+V_{INS}+V_{OGE}+V_{MPP}$ (5)

where $V_{EME}$, $V_{INS}$, $V_{OGE}$ and $V_{MPP}$ are the extended meson-
exchange potential, the instanton exchange potential, the one-gluon exchange
potential and the multi-pomeron potential, respectively QQint . The included
parameters in our $QQ$ potential are chosen so as to be consistent with
physical observables as much as possible. The contributions of the confining
potential ($V_{conf}$) to $V_{QQ}$ are minor in quark matter, being omitted in
this work.

The $V_{EME}$ $QQ$ potential is derived from the ESC16 $BB$ potential ESC16
so that the $QQM$ couplings are related to the $BBM$ couplings through folding
procedures with Gaussian baryonic quark wave functions. Then, the $V_{EME}$
$QQ$ potential is basically of the same functional expression as the ESC16
$BB$ potential. The explicit expressions for $V_{EME}$ $QQ$ potentials are
given in Ref.QQint , In the ESC modeling, the strongly repulsive components in
$BB$ potentials are described mainly by vector-meson and pomeron exchanges
between baryons. It should be noted that this feature persists in the
$V_{EME}$ $QQ$ potential, which includes the strongly repulsive components
originated from vector-meson and pomeron exchanges between quarks.

Multi-pomeron exchanges are expected to work not only among baryons but also
among quarks, in which the baryon mass $M_{B}$ is replaced by the quark mass
$M_{Q}=M_{B}/3$ and the pomeron-baryon-baryon coupling constant $g_{PBB}$ is
replaced by the pomeron-quark-quark coupling constant $g_{PQQ}$. In this work,
the $QQ$ multi-pomeron potential $V_{MPP}$ is derived from the version MPa for
the MPP among baryons.

The included parameters included in $V_{INS}$ and $V_{OGE}$ are chosen so as
to reproduce basic features of baryon mass spectra. The form of the one-gluon
exchange potential is given as

$V_{OGE}(r)=\frac{1}{4}\,({\bf\lambda^{C}_{1}\cdot\lambda^{C}_{2}})\,\alpha_{S}\,V_{vector}(m_{G};r)$
(6)

where $\lambda^{C}_{a}$, $a=1,,,8$ are the Gell-Mann matrices in color SU(3)
space and $V_{vector}(m_{G};r)$ is the vector-type one boson exchange
potential. Its explicit form is given by Eq.(E9a) in Ref.QQint . The strength
of $V_{OGE}$ is determined by the quark-gluon coupling constant $\alpha_{S}$,
being fixed as $\alpha_{S}=0.25$ in this work. The gluon mass $m_{G}$ is taken
as 420 MeV Hut95 . In quark matter,
$({\bf\lambda^{C}_{1}\cdot\lambda^{C}_{2}})$= $-8/3,+4/3,+4/3,-8/3$ in states
of $(S,T)$=$(0,0),(0,1),(1,0),(1,1)$, respectively.

The instanton potential $V_{INS}$ is based on the SU(3) generalization of the
’t Hooft interaction for (u,d,s) quarks. In the configuration space, with the
addition of the Gaussian cut-off $\exp(-k^{2}/\Lambda_{I}^{2})$, the local
instanton potential is given as QQint

$\displaystyle
V_{INS}(r)=-(4/3-\mbox{\boldmath$\lambda^{F}_{1}\cdot\lambda^{F}_{2}$})\,G_{I}\,\left(\frac{\Lambda_{I}}{2\sqrt{\pi}}\right)^{3}$
$\displaystyle\times\left[1+\frac{\Lambda_{I}^{2}}{2m_{Q}^{2}}\left(3-\frac{1}{2}\Lambda_{I}^{2}r^{2}\right)\left(1-\frac{1}{3}\mbox{\boldmath$\sigma_{1}\cdot\sigma_{2}$}\right)\right]$
$\displaystyle\times\exp\left(-\frac{1}{4}\Lambda_{I}^{2}r^{2}\right)$ (7)

where $\lambda^{F}_{a}$, $a=1,,,8$ are the Gell-Mann matrices in flavor SU(3)
space and $m_{Q}$ is the quark mass. In two-quark states, $\lambda$F operators
are reduced to $\tau$ operators of isospin. The strength of $V_{INS}$ is
determined by coupling constant $G_{I}$ and cut-off mass $\Lambda_{I}$. They
are taken as $G_{I}=2.5$ GeV-2 and $\Lambda_{I}=0.55$ GeV, being estimated
from the $\pi-\rho$ mass splitting.

Figure 1: Averaged single particle potentials $\langle U\rangle$ in quark
matter as a function of the baryon number density
$\rho_{B}=\frac{1}{3}\rho_{Q}$ in the case of $\rho_{u}=\rho_{d}=\rho_{s}$.
The solid, short-dashed, long-dashed and dot-dashed curves are the
contributions to $\langle U\rangle$ from $V_{EME}$, $V_{MPP}$, $V_{OGE}$ and
$V_{INS}$, respectively. The bold-solid curve is obtained by
$V_{EME}+V_{MPP}+V_{INS}+V_{OGE}$.  Figure 2: Averaged single particle
potentials $\langle U\rangle$ in quark matter as a function of the baryon
number density $\rho_{B}=\frac{1}{3}\rho_{Q}$ in the case of
$\rho_{u}=\rho_{d}=\rho_{s}$. The solid curve is obtained from $V_{EME}$. The
even- and odd-state contributions $\langle U_{even}\rangle$ and $\langle
U_{odd}\rangle$ are given by the dashed and short-dashed curves. The dotted
curves are the corresponding values of $\langle U_{even}\rangle$ and $\langle
U_{odd}\rangle$ in neutron matter.

In order to demonstrate the features of our $QQ$ interaction, we show the
averaged s.p. potentials $\langle U\rangle$ given by Eq. (4) as a function of
the baryon number density $\rho_{B}=\frac{1}{3}\rho_{Q}$ in the case of
$\rho_{u}=\rho_{d}=\rho_{s}$. In Fig. 1, the solid, short-dashed, long-dashed
and dot-dashed curves are the contributions to $\langle U\rangle$ from
$V_{EME}$, $V_{MPP}$, $V_{OGE}$ and $V_{INS}$, respectively. The bold-solid
curve is obtained by the sum of $V_{EME}+V_{MPP}+V_{INS}+V_{OGE}$. The
strongly-repulsive nature of $\langle U(\rho_{B})\rangle$ is the key point in
this work, which leads to the quark-matter EoS stiff enough to reproduce
neutron-star masses over $2M_{\odot}$. In the figure, the repulsive
contribution of $V_{EME}$ is found to be essential the repulsive nature of
$\langle U(\rho_{B})\rangle$, where the repulsive contributions of $V_{OGE}$
and $V_{MPP}$ are considerably canceled by the attractive contribution of
$V_{INS}$. It is worthwhile to say that the repulsive components in $V_{EME}$
are from the vector-meson and pomeron exchanges. This feature persists from
the ESC $BB$ interaction model.

In neutron matter, $\langle U\rangle$ includes repulsive contributions from
the multi-pomeron potential MPP, being quite large in high density regions.
The strengths of three- and four-body parts of MPP are proportional to
$(g_{PBB})^{3}$ and $(g_{PBB})^{4}$, respectively, $g_{PBB}$ being the
pomeron-baryon-baryon coupling constant. In $QQ$ potentials, $g_{PBB}$ is
replaced by the pomeron-quark-quark coupling constant $g_{PQQ}$. Because of
the relation $g_{PQQ}=\frac{1}{3}g_{PBB}$, the strengths of three- and four-
body parts of MPP among quarks are far smaller than those among baryons.
Therefore, MPPs among quarks are not so remarkable in comparison with those
among baryons.

The even- and odd-state contributions to $\langle U\rangle$ are denoted as
$\langle U_{even}\rangle$ and $\langle U_{odd}\rangle$, respectively, In Fig.
2, the solid curve shows $\langle U(\rho_{B})\rangle$ obtained from $V_{EME}$,
and $\langle U_{even}(\rho_{B})\rangle$ and $\langle U_{odd}(\rho_{B})\rangle$
are given by the dashed and short-dashed curves. The dotted curves are the
even- and odd-state contributions of averaged neutron potentials $\langle
U_{even}\rangle$ and $\langle U_{odd}\rangle$ in neutron matter. The
remarkable feature of $\langle U_{even}\rangle$ and $\langle U_{odd}\rangle$
given by $V_{EME}$ is that they are attractive and repulsive, respectively.
This feature of $\langle U_{even}\rangle$ and $\langle U_{odd}\rangle$ is
similar to the corresponding one in neutron matter qualitatively.

When our $QQ$ potentials are used in quark-matter calculations, it is
reasonable to assume the constituent quark masses originated from the chiral
symmetry breaking as the QCD non-perturbative effect. Then, it is probable
that the constituent quark masses in quark matter become smaller than those in
vacuum and move to current masses in the high-density limit. At the mean-field
(MF) level usually, the density-dependent quark masses in matter have been
derived from the MF-Lagrangian such as that of the NJL model. In the present
approach, we introduce phenomenologically the density-dependent quark mass

$\displaystyle
M_{Q}^{*}(\rho_{Q})=M_{0}/[1+\exp\\{\gamma(\rho_{Q}-\rho_{c}\\}]+m_{0}+C$ (8)

with $C=M_{0}-M_{0}/[1+\exp(-\gamma\rho_{c})]$ assuring
$M_{Q}^{*}(0)=M_{0}+m_{0}$, where $\rho_{Q}$ is number density of quark
matter, and $M_{0}$ and $m_{0}$ are taken as 300 (360) MeV and 5 (140) MeV for
$u$ and $d$ ($s$) quarks. Then, we have $M_{Q}^{*}(0)=$ 305 (500) MeV for $u$
and $d$ ($s$) quarks. The adjustable parameters $\rho_{c}$ and $\gamma$ are
used to control mainly the onset densities of quark phases into hadronic
phases.

Furthermore, because the quark mass reduction has to bring about an increase
of the vacuum energy $B$, we assume simply

$\displaystyle B(\rho_{Q})=M_{Q}^{*}(0)-M_{Q}^{*}(\rho_{Q})\ .$ (9)

It is well known that there are three schemes for the density-dependent quark
mass Blaschke20 : (i) a constant quark mass, (ii) a linear density dependence
(Brown-Rho scaling Brown ), (iii) a density-dependence within a higher-order
NJL model Kashiwa  Benic . Eq. (8) includes these schemes, representing (i)
for $\gamma=0$, (ii) for small values of $\gamma$ and (iii) for large values
of $\gamma$. The parameter $\rho_{c}$ is chosen as $6\rho_{0}$ by referring to
forms of (iii) derived from the higher-order NJL models.

We define the following five sets with different values of $\gamma$ of $QQ$
interactions for deriving Q-EoSs.

Q0 : $V_{EME}$ with $\gamma$=1.2

Q1 (Q1e) : $V_{EME}+V_{INS}+V_{OGE}$ with $\gamma$=1.0 ($\gamma$=2.6)

Q2 (Q2e) : $V_{EME}+V_{MPP}+V_{INS}+V_{OGE}$ with $\gamma$=1.6 ($\gamma$=2.2)

In the cases of Q0, Q1 and Q2, the values of $\gamma$ are chosen so that the
critical chemical potentials and densities for phase transitions are as small
as possible. In the cases of Q1e and Q2e, they are chosen so that critical
densities are near crossing points of hadronic and quark energy densities.

In Fig. 3 the quark mass $M_{Q}^{*}(\rho_{Q})$ ($Q=u,d$) as a function of the
baryon number density $\rho_{B}=\rho_{Q}/3$ is plotted in the cases of (a) Q1,
(b) Q1e, (c) Q2 and (d) Q2e. The density-dependent quark masses in these cases
(especially Q1e and Q2e) are found to be rather close to (ii) with the Brown-
Rho scaling.

Figure 3: Quark mass as a function of the baryon number density $\rho_{B}$ for
the (a) Q1, (b) Q1e, (c) Q2 and (d) Q2e models. As a reference, also the
Brown-Rho scaling is shown by the dashed line.

It is quite important to use the density-dependent quark masses together with
our $QQ$ potentials. When constant quark masses are used, hadron-quark
transitions derived from our $QQ$ potentials occur in density regions over
$5\rho_{0}$ in hadronic matter giving $2M_{\odot}$ masses. In such a case,
quark phases have no effect on masses and radii of neutron stars, even if they
exist in inner cores.

For instance, when the baryon-baryon interaction H2 is used together with
quark-quark interaction Q2, the combined set is denoted as H2+Q2. Hereafter,
combinations of $BB$ and $QQ$ interactions are expressed like this.

###  III.3 Effective Quark-Quark interactions

For applications to quark-matter calculations, we construct density-dependent
effective local interactions ${\cal G}_{QQ}(\rho_{Q};r)$ simulating G-matrices
in coordinate space, where $\rho_{Q}$ is number density of quark matter. We
use here the method given in Ref.Yama10 .

The effective interactions are written as ${\cal G}_{QQ}={\cal G}_{EME}+{\cal
G}_{MPP}+{\cal G}_{INS}+{\cal G}_{OGE}$ approximately corresponding to
$V_{QQ}=V_{EME}+V_{MPP}+V_{INS}+V_{OGE}$. Though they can be obtained for each
$(ff^{\prime},T,L,S,J)$ state, for simplicity, the dependence on $L$ is
approximated by that on parity $P$ and the dependence on $J$ is averaged:
Quantum numbers $TLSJ$ are reduced to $TSP$. The respective interactions are
represented in two- or one-range Gaussian forms, and coefficients are adjusted
so that s.p. potentials $U^{TSP}_{f}$ obtained from ${\cal
G}_{ff^{\prime}}^{TSP}$ simulate the original G-matrix results. It is far
easier to derive quark-matter EoSs with use of these density-dependent
interactions ${\cal G}_{QQ}$ than derivations by G-matrix calculations with
$V_{QQ}$.

The density-dependent effective interactions ${\cal G}_{EME}$ and ${\cal
G}_{OGE}$ derived from $V_{EME}$ and $V_{OGE}$. respectively, are
parameterized in a two-range Gaussian form as

$\displaystyle{\cal
G}_{EME,OGE}(\rho,r)=(a\rho^{\alpha}+b\rho^{\beta})\cdot\exp(-(r/0.8)^{2})$
$\displaystyle\hskip 56.9055pt+c\cdot\exp(-(r/1.6)^{2}).$ (10)

The parameter set $(a,\alpha,b,\beta,c)$ in Eq.(III.3) is given for each
$(y,T,S,P)$ state with $y=qq,qs,ss$ ($q=u,d$). In Tables 1 and 2, the values
of parameters are tabulated for ${\cal G}_{EME}$ and ${\cal G}_{OGE}$,
respectively.

${\cal G}_{INS}$ derived from $V_{INS}$ is parameterized in an one-range
Gaussian form as

$\displaystyle{\cal
G}_{INS}(\rho,r)=(a\rho^{\alpha}+b\rho^{\beta})\cdot\exp(-(r/0.6)^{2})\ .$
(11)

The parameter set $(a,\alpha,b,\beta,c)$ in Eq.(11) is given for each
$(y,T,S,P)$ state with $y=qq,qs$ ($q=u,d$). The values of them are given in
Table 3.

${\cal G}_{MPP}$ derived from $V_{MPP}$ is parameterized in an one-range
Gaussian form as

$\displaystyle{\cal G}_{MPP}(\rho,r)=(a+b\rho^{\beta})\cdot\exp(-(r/1.3)^{2})$
(12)

being independent of $(y,T,S)$ and given only for $P$. The values of
parameters $(a,b,\beta)$ are given in Table 4.

Table 1: ${\cal G}_{EME}(\rho,r)=(a\rho^{\alpha}+b\rho^{\beta})\cdot\exp(-(r/0.8)^{2})\\\ \qquad\qquad+c\cdot\exp(-(r/1.6)^{2})$. $y=qq,qs,ss$ ($q=u,d$).  $y$ |  $T$ $S$ $P$ | $a$ | $\alpha$ | $b$ | $\beta$ | $c$
---|---|---|---|---|---|---
$qq$ | 1 0 + |  $-$3.520 | $-1$ |  $-$17.94 | 0 |  $-$0.9978
| 0 1 + |  $-$2.871 | $-1$ |  $-$30.59 | 0 |  $-$0.8389
| 0 0 $-$ | 43.34 | $-1$ | 192.8 | 0 | 3.896
| 1 1 $-$ | 6.621 | $-1$ | 102.5 | 0 | 1.595
$qs$ | 1/2 0 $+$ |  $-$0.5716 | $-1$ |  $-$28.27 | 0 |  $-$0.4530
| 1/2 1 $+$ |  $-$0.6959 | $-1$ |  $-$24.58 | 0 |  $-$0.1993
| 1/2 0 $-$ |  $-$1.597 | $-1$ | 149.0 | 0 | 1.568
| 1/2 1 $-$ | 1.183 | $-1$ | 75.98 | 0 | 1.217
$ss$ | 0 0 $+$ |  $-$2.755 | $-1$ |  $-$26.37 | 0 |  $-$0.1212
| 0 1 $-$ |  $-$1.651 | $-1$ | 51.06 | 0 | 0.3558
Table 2: ${\cal G}_{OGE}(\rho,r)=(a\rho^{\alpha}+b\rho^{\beta})\cdot\exp(-(r/0.8)^{2})\\\ \qquad\qquad+c\cdot\exp(-(r/1.6)^{2})$. $y=qq,qs,ss$ ($q=u,d$).  $y$ |  $T$ $S$ $P$ | $a$ | $\alpha$ | $b$ | $\beta$ | $c$
---|---|---|---|---|---|---
$qq$ | 1 0 + | 8.565 | 1 | 3.892 | 0.3742 | 0.5185
| 0 1 + | 7.543 | 1 | 1.977 | 0.6431 | 1.142
| 0 0 $-$ | -0.8959 | 1 | 9.982 | 0.2741 | 0.5027
| 1 1 $-$ | 8.094 | 1 | 11.64 | 0.2881 | 1.147
$qs$ | 1/2 0 $+$ |  $-$4.733 | 1 |  $-$1.359 | 0.4161 |  $-$0.2593
| 1/2 1 $+$ |  $-$3.658 | 1 |  $-$0.7316 | 0.6347 |  $-$0.5709
| 1/2 0 $-$ |  $-$1.282 | 1 |  $-$3.141 | 0.3675 |  $-$0.2514
| 1/2 1 $-$ |  $-$5.645 | 1 |  $-$3.831 | 0.3727 |  $-$0.5736
$ss$ | 0 0 $+$ | 10.48 | 1 | 1.478 | 0.5930 | 0.5185
| 0 1 $-$ | 11.23 | 1 | 7.743 | 0.3250 | 1.147
Table 3: ${\cal G}_{INS}(\rho,r)=\
(a\rho^{\alpha}+b\rho^{\beta})\cdot\exp(-(r/0.6)^{2})$.
$y=qq,qs$ ($q=u,d$). $y$ |  $T$ $S$ $P$ | $a$ | $\alpha$ | $b$ | $\beta$
---|---|---|---|---|---
$qq$ | 0 1 $+$ | 0.2132 | $-1$ |  $-$130.0 | 0
| 0 0 $-$ | 124.8 | 0 |  $-$38.82 | 0.4227
$qs$ | 1/2 0 $+$ |  $-$1.504 | $-1$ |  $-$61.35 | 0.0705
| 1/2 1 $+$ |  $-$0.1638 | $-1$ |  $-$63.47 | 0
| 1/2 0 $-$ | 59.03 | 0 |  $-$17.06 | 0.4966
| 1/2 1 $-$ |  $-$36.95 | 0 |  $-$5.749 | 0.3192
Table 4: ${\cal G}_{MPP}(\rho,r)=(a+b\rho^{\beta})\cdot\exp(-(r/1.3)^{2})$.  $P$ | $a$ | $b$ | $\beta$
---|---|---|---
$+$ | 0.3597 | 1.600 | 1.490
$-$ | 0.4338 | 2.618 | 1.384

##  IV EoS and $MR$ diagram of hybrid star

###  IV.1 Derivation of quark-matter EoS

Let us derive the EoS of quark matter composed of quarks with flavor
$f=u,d,s$. In this derivation, we use the density-dependent $QQ$ interactions
Eq.(III.3), Eq.(11), Eq.(12) based on the non-relativistic formalism.
Relativistic expressions are used only for kinetic energies.

A single $f$ quark potential in quark matter composed of $f^{\prime}$ quarks
is given by

$\displaystyle U_{f}(k)$ $\displaystyle=$
$\displaystyle\sum_{f^{\prime}}U_{f}^{(f^{\prime})}(k)=\sum_{f^{\prime}}\sum_{k^{\prime}<k_{F}^{f^{\prime}}}\langle
kk^{\prime}|{\cal G}_{ff^{\prime},ff^{\prime}}|kk^{\prime}\rangle$

with $f,f^{\prime}=u,d,s$, where spin and isospin quantum numbers are
implicit. The quark energy density is given by

$\displaystyle\varepsilon_{f}$ $\displaystyle=$ $\displaystyle
2N_{c}\sum_{f}\int_{0}^{k_{F}^{f}}\frac{d^{3}k}{(2\pi)^{3}}\left\\{\sqrt{\hbar^{2}k^{2}+M_{f}^{2}}+\frac{1}{2}U_{f}(k)\right\\}$
(14) $\displaystyle+B(\rho_{Q})$

where $N_{c}=3$ is the number of quark colors. The quark number density is
given as $\rho_{Q}=\sum_{f}\rho_{f}$ with
$\rho_{f}=N_{c}\frac{(k_{F}^{f})^{3}}{3\pi^{2}}$. The chemical potential
$\mu_{f}$ and pressure $P_{Q}$ are expressed as

$\displaystyle\mu_{f}=\frac{\partial\varepsilon_{Q}}{\partial\rho_{f}}\ ,$
(15) $\displaystyle
P_{Q}=\rho_{Q}^{2}\frac{\partial(\varepsilon_{Q}/\rho_{Q})}{\partial\rho_{Q}}=\sum_{f}\mu_{f}\rho_{f}-\varepsilon_{Q}\
.$ (16)

Here, we consider the EoS of $\beta$-stable quark matter composed of $u$, $d$,
$s$, $e^{-}$. The equilibrium conditions are summarized as follows:

(1) chemical equilibrium conditions,

$\displaystyle\mu_{d}=\mu_{s}=\mu_{u}+\mu_{e}$ (17)

(2) charge neutrality,

$\displaystyle 0=\frac{1}{3}(2\rho_{u}-\rho_{d}-\rho_{s})-\rho_{e}$ (18)

(3) baryon number conservation,

$\displaystyle\rho_{B}=\frac{1}{3}(\rho_{u}+\rho_{d}+\rho_{s})=\frac{1}{3}\rho_{Q}$
(19)

In the parabolic approximation, the following relation can be derived:

$\displaystyle\mu_{e}=\mu_{d}-\mu_{u}=4\beta E_{sym}$ (20)

where $x=\rho_{u}/(\rho_{u}+\rho_{d})$ and $\beta=1-2x$. $E_{sym}$ is the
symmetric energy of $ud$ part.

When the chemical potentials (15) are substituted into (17), the chemical
equilibrium conditions are represented as equations for densities $\rho_{u}$,
$\rho_{d}$, $\rho_{s}$ and $\rho_{e}$. Then, equations (17) $-$ (19) are
solved iteratively, and densities and chemical potentials in equilibrium are
obtained. Finally, energy densities (14) and pressures (16) can be calculated.

An example of solution is demonstrated in Fig. 4: The number fractions of
quarks and electrons in $\beta$-stable quark matter are plotted as a function
of the baryon density $\rho_{B}$ in the case of using Q2, where solid (dashed)
curves are for $u$, $d$ and $s$ quarks (electrons). In the figue, the electron
fractions are not visible below the $s$-quark onset. The reason of such small
values are because the symmetry energies $E_{sym}$ in Eq.(20) are not so large
in the case of our $QQ$ interactions.

Figure 4: The number fractions of quarks and electrons in $\beta$-stable quark
matter as a function of the baryon density $\rho_{B}$ in the case of using Q2.
The fractions of $u$, $d$ and $s$ quarks are given by solid curves, and that
of electrons $e^{-}$ are by dashed curve.

###  IV.2 Phase transition from hadronic matter to quark matter

The EoSs are shown in Fig. 5, where pressures of quark matter are given as a
function of the energy density $\epsilon$ and compared to those of hadronic
matter. Steeper slopes of curves correspond to stiffer EoSs: The Q-EoSs are
stiffer than the H-EoSs, and the EoSs for (b) Q1 and (c) Q2 are stiffer than
that for (a) Q0 owing to the repulsive contributions of $V_{OGE}$ and
$V_{MPP}$. As shown later, these features are clearly reflected in the $MR$
curves of hybrid stars.

Figure 5: Pressures of hadronic matter and quark matter as a function of the
energy density $\epsilon$. Short-dashed, long-dashed and dot-dashed curves are
for hadronic matter for H1, H2 and H3, respectively. Solid curves are for
quark matter obtained from (a) Q0, (b) Q1 and (c) Q2.

In order to construct the hybrid EoS including a transition from hadronic
phase and quark phase, we use the replacement interpolation method RIM
Shahrbaf2 , being a simple modification of the Maxwell and the Glendenning
(Gibbs) constructions Glendenning . In our actual calculations, we follow the
interpolation formula given in Ref.Shahrbaf2 . Then, interpolated regions can
be considered as mixed phases. Both of H-EoSs and Q-EoSs are assumed to
fulfill separately the charge-neutrality and $\beta$-equilibrium conditions.
The EoSs of hadronic and quark phases and that of mixed phase are described
with the relations between pressures and chemical potentials $P_{H}(\mu)$,
$P_{Q}(\mu)$ and $P_{M}(\mu)$, respectively. The critical chemical potential
$\mu_{c}$ for the transition from the hadronic phase to the quark phase is
obtained from the Maxwell condition

$\displaystyle P_{Q}(\mu_{c})=P_{H}(\mu_{c})=P_{c}\ .$ (21)

The pressure of the mixed phase is represented by a polynomial ansatz

$\displaystyle
P_{M}(\mu)=\sum^{N}_{q=1}\alpha_{q}(\mu-\mu_{c})^{q}+P_{c}+\Delta P$ (22)

where the pressure shift $\Delta P$ at $\mu_{c}$ is treated as a free
parameter. The pressure of the mixed phase at $\mu_{c}$ is determined by
$P_{M}(\mu_{c})=P_{c}+\Delta P=(1+\Delta_{P})P_{c}$ with $\Delta_{P}=\Delta
P/P_{c}$. Then, the matching chemical potential $\mu_{H}$ ($\mu_{Q}$) of
$P_{M}(\mu)$ to $P_{H}(\mu)$ ($P_{Q}(\mu)$) can be obtained from the
continuity condition. The corresponding matching densities $\rho_{H}$ and
$\rho_{Q}$ are obtained with use of $\rho(\mu)=dP(\mu)/d\mu$. The finite
values of $\Delta_{P}=0.05-0.07$ corresponds to the Glendenning construction
Shahrbaf2 . We choose $\Delta_{P}=0.07$ in this work.

Figure 6: Pressures as a function of the chemical potential $\mu_{B}$. Short-
dashed, long-dashed and dot-dashed curves are pressures of hadronic matter for
H1, H2 and H3, respectively. In the left panel, solid curves are pressures of
quark matter obtained from (a) Q0, (b) Q1 and (c) Q2. In the right panel, they
are obtained by using constant quark masses without density dependences Eq.
(8) and vacuum energies Eq. (9).

In Fig. 6, pressures are drawn as a function of the chemical potential
$\mu_{B}$, where short-dashed, long-dashed and dot-dashed curves are pressures
of hadronic matter for H1, H2 and H3, respectively. In the left panel, solid
curves are pressures of quark matter obtained from (a) Q0, (b) Q1 and (c) Q2.
The crossing of the hadronic and the quark-matter curves is considered to be a
condition for phase transition to occur. The values of $P$ at crossing points
give the critical pressures $P_{c}$ for phase transitions. The hadronic and
quark-matter curves are connected smoothly by Eq. (22). Then, the effective-
mass parameter $\gamma$ in Eq. (8) is adjusted so that cross points appear at
similar values of $\mu_{B}\sim 1200$ MeV. In the right panel, on the
otherhand, solid curves are obtained from Q0, Q1 and Q2 by using constant
quark masses $M_{Q}^{*}(\rho_{Q}=0)$ without density dependences Eq. (8) and
vacuum energies Eq. (9). It is found that there is no crossing point in this
region of $\mu_{B}$. Thus, the density-dependent quark mass plays a decisive
role in the occurrence of phase transition.

Figure 7: Pressures as a function of the chemical potential $\mu_{B}$ in the
transition region. The short-dashed curve is obtained by the H-EoS for H2 and
the solid curve is by the Q-EoS for Q2. The bold-solid curve is the
interpolated one.

In Fig. 7, pressures are given as a function of the chemical potential
$\mu_{B}$ in the transition region. The short-dashed curve is obtained by the
H-EoS for H2 and the solid curve is by the Q-EoS for Q2. The bold-solid curve
is obtained by the interpolation method.

Table 5: Pressures $P_{c}$ at critical chemical potentials $\mu_{c}$ in phase transitions from the hadronic phases for H1, H2 and H3 to the quark-matter phases for Q0, Q1 and Q2. Values of $\mu_{H}$ ($\mu_{Q}$) are chemical potentials at matching points between mixed phases and hadron (quark) phases.  | $P_{c}$ | $\mu_{c}$ | $\mu_{H}$ | $\mu_{Q}$
---|---|---|---|---
| MeV/fm3 | MeV | MeV | MeV
H1+Q0 | 125.7 | 1241 | 1186 | 1373
H1+Q1 | 92.55 | 1183 | 1141 | 1277
H1+Q2 | 110.4 | 1215 | 1095 | 1368
H2+Q0 | 139.6 | 1254 | 1199 | 1386
H2+Q1 | 102.3 | 1193 | 1149 | 1282
H2+Q2 | 138.2 | 1252 | 1141 | 1415
H2+Q1e | 132.9 | 1189 | 1243 | 1446
H2+Q2e | 209.6 | 1360 | 1261 | 1600
H3+Q0 | 136.0 | 1251 | 1198 | 1382
H3+Q1 | 101.2 | 1191 | 1148 | 1279
H3+Q2 | 131.6 | 1243 | 1142 | 1412
Table 6:  Critical densities (fm-3) of phase transitions: $\rho_{H}$ and $\rho_{Q}$ are densities at matching points in phase transitions from the hadronic phases for H1, H2 and H3 to the quark-matter phases for Q0, Q1, Q2, Q1e and Q2e. $\rho^{c}_{H}$ and $\rho^{c}_{Q}$ are critical densities for the Maxwell construction defined by $P_{H}(\rho^{c}_{H})=P_{Q}(\rho^{c}_{Q})=P_{c}$. Values of $\rho_{E}$ are densities at crossing points of energy densities $\epsilon_{H}(\rho)$ and $\epsilon_{Q}(\rho)$. There is no crossing point in the case of H1+Q2.  | $\rho_{H}$ | $\rho_{Q}$ | $\rho^{c}_{H}$ | $\rho^{c}_{Q}$ | $\rho_{E}$
---|---|---|---|---|---
H1+Q0 | 0.566 | 0.904 | 0.661 | 0.673 | 0.784
H1+Q1 | 0.490 | 0.703 | 0.574 | 0.544 | 0.918
H1+Q2 | 0.407 | 0.721 | 0.623 | 0.584 | —
H2+Q0 | 0.521 | 0.930 | 0.664 | 0.694 | 0.702
H2+Q1 | 0.446 | 0.712 | 0.573 | 0.561 | 0.753
H2+Q2 | 0.433 | 0.776 | 0.661 | 0.620 | 0.716
H2+Q1e | 0.506 | 1.02 | 0.650 | 0.707 | 0.643
H2+Q2e | 0.608 | 0.987 | 0.790 | 0.722 | 0.695
H3+Q0 | 0.482 | 0.922 | 0.616 | 0.689 | 0.659
H3+Q1 | 0.416 | 0.706 | 0.568 | 0.559 | 0.692
H3+Q2 | 0.407 | 0.772 | 0.608 | 0.612 | 0.660

Our phase transition is specified by the pressures $P_{c}$ at critical
chemical potentials $\mu_{c}$ and boundary values of chemical potentials and
densities for mixed phases. They are shown in the cases of phase transitions
from the H-EoS for H1, H2 and H3 to the Q-EoSs for Q0, Q1, Q2, Q1e and Q2e. In
Table 5, the chemical potentials at matching points are given by values of
$\mu_{H}$ and $\mu_{Q}$. In Table 6, $\rho_{H}$ and $\rho_{Q}$ are densities
at matching points in phase transitions, and $\rho^{c}_{H}$ and $\rho^{c}_{Q}$
are critical densities defined by the
$P_{H}(\rho^{c}_{H})=P_{Q}(\rho^{c}_{Q})=P_{c}$ in the case of $\Delta_{P}=0$.
It is reasonable that that the values of $\rho^{c}_{H}$ and $\rho^{c}_{Q}$ are
between the values of $\rho_{H}$ and $\rho_{Q}$. The Maxwell construction is
conditioned by $\rho^{c}_{H}<\rho^{c}_{Q}$. As found in Table 6, however, the
values of $\rho^{c}_{H}$ are larger than those of $\rho^{c}_{Q}$ in some
cases, meaning that first-order phase transitions do not appear in
$\Delta_{P}=0$ limits.

The values of $\rho_{E}$ are densities at crossing points of energy densities
$\epsilon_{H}(\rho)$ and $\epsilon_{Q}(\rho)$. In the case of H2+Q1e (H2+Q2e),
the value of $\rho_{E}$ is between (smaller than) $\rho^{c}_{H}$ and
$\rho^{c}_{Q}$.

In Fig. 8, pressures are given as a function of energy density $\epsilon$ in
the transition region. The short-dashed curve is obtained by the H-EoS for H2
and the solid curve is by the Q-EoS for Q2. The bold-solid curve is pressure
in the interpolated region.

Figure 8: Pressures as a function of as a function of the energy density
$\epsilon$ in the region of phase transitions. The short-dashed curves are
obtained by the H-EoS for H2 and the solid curve is by the Q-EoS for Q2. The
bold-solid curve is the interpolated ones in the mixed phase.

It is worthwhile to point out that our hybrid-EoSs are consistent with the
picture of hadron-quark continuity Kojo2015 Baym2018 . In these references,
the interpolated pressures are given in the density region of
$2<\rho_{B}/\rho_{0}<(4-7)$, where quark degrees of freedom gradually emerge.
Correspondingly, our mixed phases are given in the region of
$(2.4-3.3)<\rho_{B}/\rho_{0}<(4.1-6.0)$, as found in Table 6.

Figure 9: Pressure $P$ as a function of density $\rho_{B}$ in the case of
H2+Q1e, where the dashed (short-dashed) curves are for hadronic (quark)
matter. The horizontal solid line shows the range of the hadron-quark mixed
phases in case of $\Delta_{P}=0$. The dot-dashed curve is in the case of
$\Delta_{P}=0.07$.

Let us demonstrate that the Maxwell construction appears in the the
$\Delta_{P}=0$ limit. In Fig. 9, we show pressure $P$ as a function of density
$\rho_{B}$ in the case of using H2+Q1e, where the dashed (short-dashed) curves
are for the hadronic (quark) matter. The horizontal solid lines show the range
of the hadron-quark mixed phase. It is well known that the Maxwell
construction is specified by the horizontal line in the $P-\rho$ diagram,
where the density values at ends of horizontal and vertical lines are given by
$\rho^{c}_{H}$ and $\rho^{c}_{Q}$. The difference of the curve for
$\Delta_{P}=0$ from the dot-dashed curve for $\Delta_{P}=0.07$ is found to be
small, the appearance of which is seen in the corresponding $MR$ curves as
shown later. Not only in the case of He+Q1e, there appear the similar curves
specifying the Maxwell construction in the cases of
$\rho^{c}_{H}<\rho^{c}_{Q}$ in Table 6.

Our hybrid-star EoS is composed of H-EoS and Q-EoS, being combined by the
interpolation formula including the parameter $\Delta_{P}$. The $MR$ relations
of hybrid stars can be obtained by solving the Tolmann-Oppenheimer-Volkoff
(TOV) equation, where our hybrid EoSs are connected smoothly to the crust EoS
Baym1 Baym2  in the low-density side.

Figure 10: Hybrid-star masses as a function of radius $R$, where the short-
dashed curves are obtained by the H-EoS for H2. In the left panel, the solid
and dashed curves are obtained by H2+Q1e in cases of $\Delta_{P}=0.07$ and
$\Delta_{P}=0$, respectively. In the right panel, the upper (lower) solid
curves are obtained by H2+Q2 (H2+Q1), and the upper (lower) dashed curves are
by H2+Q2e (H2+Q1e).

In Fig. 10, hybrid-star masses are shown as a function of radius $R$ in the
cases of using Q1e or Q2e (Q1 or Q2), where the short-dashed curves are
obtained by the H-EoS for H2.

In the left panel, the solid and dashed curves are obtained by H2+Q1e in cases
of $\Delta_{P}=0.07$ and $\Delta_{P}=0$ (Maxwell construction), respectively.
The slight reduction of the latter compared to the former is due to the
difference between the $\Delta_{P}=0.07$ and $\Delta_{P}=0$ curves in in
Fig.9.

In the right panel, the upper (lower) solid curves are obtained by H2+Q2
(H2+Q1) in the case of $\Delta_{P}=0.07$, and the upper (lower) dashed curves
are by H2+Q2e (H2+Q1e). The maximum masses are $2.15M_{\odot}$
($2.25M_{\odot}$) for H2+Q2e (H2+Q2), and $2.07M_{\odot}$ ($2.16M_{\odot}$)
for H2+Q1e (H2+Q1). The quark-phase onset values of the central baryon
densities are 0.54 fm-3 (0.48 fm-3) in the case of H2+Q1e (H2+Q1), and 0.65
fm-3 (0.46 fm-3) in the case of H2+Q2e (H2+Q2). Thus, Q2e (Q1e) leads to the
larger onset density and the smaller maximum mass than Q2 (Q1): It is a
general trend that maximum masses become smaller as onset densities of quark
phases become larger.

In our approach, there is no clear criteria to decide which of Q2 (Q1) and Q2e
(Q1e) is more appropriate. In the following section, we use Q1 and Q2 because
they seem to be more suitable than Q1e and Q2e in the light of the recent
observations for the maximum masses.

###  IV.3 $MR$ diagrams of hybrid stars

Figure 11: Hybrid-star masses as a function of radius $R$ (left panel) and
central density $\rho_{Bc}$ (right panel). The solid curves are obtained by
the Q-EoSs for (a) Q0, (b) Q1 and (c) Q2. The short-dashed curves are by the
H-EoS for H2. The deviations from the latter to the formers are by phase
transitions from hadronic-matter to quark-matter. The rectangle indicates the
region of mass $2.072^{+0.067}_{-0.066}$M⊙ and radius $12.39^{+1.30}_{-0.98}$
km Riley2021  for PSR J0740+6620. Figure 12: Hybrid-star masses as a function
of radius $R$ (left panel) and central density $\rho_{Bc}$ (right panel),
where the Q-EoS for Q2 is used. Short-dashed, long-dashed and dot-dashed
curves are obtained with H-EoSs for (a) H1, (b) H2 and (c) H3, respectively.
Solid curves show the deviations by transitions from hadronic-matter to quark-
matter phases. In the left panel, the rectangle indicates the region of mass
$2.072^{+0.067}_{-0.066}$M⊙ and radius $12.39^{+1.30}_{-0.98}$ km Riley2021 .
Dotted and solid line segments indicate
$R_{1.4M_{\odot}}=12.33^{+0.76}_{-0.81}$ km (PP model) and
$R_{1.4M_{\odot}}=12.18^{+0.56}_{-0.79}$ km (CS model) Raaij2021 , and dashed
and dot-dashed ones do $R_{1.4M_{\odot}}=11.94^{+0.76}_{-0.87}$ km Peter2021
and $R_{1.4M_{\odot}}=13.80\pm 0.47$ km Brendan2021 , respectively.

In Fig.11, hybrid-star masses are given as a function of radius $R$ (left
panel) and central baryon density $\rho_{Bc}$ (right panel). The curves
obtained by the Q-EoSs for (a) Q0, (b) Q1 and (c) Q2 are given by solid
curves, and those by the H-EoS for H2 are given by the short-dashed curves.
The former curves are connected from the latter curves by the hadron-quark
phase transitions. The maximum masses for (b) Q1 and (c) Q2 are over
$2M_{\odot}$ substantially. It is noted that the Q-EoS for (a) Q0 derived from
$V_{EME}$ is still stiff enough to reach $2M_{\odot}$ without help of the
repulsive contributions of $V_{OGE}$ and $V_{MPP}$. This repulsive components
in $V_{EME}$ come from vector-meson and pomeron exchanges between quarks. The
rectangle in the left panel indicates the region of mass
$2.072^{+0.067}_{-0.066}$M⊙ and radius $12.39^{+1.30}_{-0.98}$ km Riley2021
for the most massive neutron star PSR J0740+6620. The $MR$ curves for Q1 and
Q2 are found to pass through this rectangle.

In the left panel of Fig.12, hybrid-star masses are drawn as a function of
radius $R$, where the Q-EoSs for Q2 and H-EoSs for (a) H1, (b) H2 and (c) H3
are used. Short-dashed, long-dashed and dot-dashed curves are obtained with
H-EoSs for H1, H2 and H3, respectively. Solid curves show deviations by
transitions from hadronic-matter to quark-matter phases. The maximum masses in
the figures are as follows: In the cases of H-EoSs, they are $1.82M_{\odot}$
(H1), $1.94M_{\odot}$ (H2) and $2.07M_{\odot}$ (H3). In the cases of including
hadron-quark transitions, they are $2.25M_{\odot}$ (H1+Q2), $2.25M_{\odot}$
(H2+Q2) and $2.28M_{\odot}$ (H3+Q2). The maximum masses are noted to be
determined by the Q-EoSs, being larger than those given by the H-EoSs. The
rectangle in the left panel is the same as that in Fig.11, indicating the
mass-radius region obtained from the observation Riley2021 . The $MR$ curves
for the Q-EoSs pass through the rectangle, though those for the H-EoSs (dashed
curves) are below this rectangle.

The radii $R$ at $1.4M_{\odot}$ ($R_{1.4M_{\odot}}$) are given as follows: The
values of $R_{1.4M_{\odot}}$ are 12.5 km (H1+Q2), 13.3 km (H2+Q2) and 13.6 km
(H3+Q2), being obtained in the cases of including hadron-quark transitions.
The similar values are obtained by using H-EoSs only, which means that the
values of $R_{1.4M_{\odot}}$ are determined by H-EoSs. In the figure, dotted
and solid line segments indicate $R_{1.4M_{\odot}}=12.33^{+0.76}_{-0.81}$ km
(PP model) and $R_{1.4M_{\odot}}=12.18^{+0.56}_{-0.79}$ km (CS model)
Raaij2021 , and dashed and dot-dashed ones do
$R_{1.4M_{\odot}}=11.94^{+0.76}_{-0.87}$ km Peter2021  and
$R_{1.4M_{\odot}}=13.80\pm 0.47$ km Brendan2021 , respectively. The former
three line segments (dotted, solid and dashed lines) are similar with each
other, and the $MR$ curve for H1 intersects them. On the other hand, the $MR$
curves for H2 and H3 intersect the dot-dashed line, but do not intersect the
other three lines. In the present stage of the observations for radii of
neutron stars, it is difficult to determine which one of H1, H2 and H3 lead to
the most reasonable EoS.

In the right panel of Fig.12, hybrid-star masses are drawn as a function of
central baryon density $\rho_{Bc}$, where the Q-EoS for Q2 is used. Short-
dashed, long-dashed and dot-dashed curves are obtained with H-EoSs for H1, H2
and H3, respectively. Solid curves show the deviations by transitions from
hadronic-matter to quark-matter phases. In the cases of including hadron-quark
transitions, the onset values of $\rho_{Bc}$ for quark phases are 0.40 fm-3
(H1+Q2), 0.46 fm-3 (H2+Q2) and 0.43 fm-3 (H3+Q2).

It is useful to compare our results for $MR$ diagrams with those in
Ref.Shahrbaf2 , because we employ the method in this reference for the hadron-
quark phase transitions. Though their quark-matter EoS is based on the
nonlocal Nambu-Jona-Lasinio (nlNJL) model differently from ours, it is found
that the quark-phase regions of the $MR$ curves in Fig.4 of Shahrbaf2  are
similar to ours qualitatively. Especially, maximum masses of $2M_{\odot}$ are
reproduced well, namely the Q-EoSs are stiff similarly in both cases of ours
and Shahrbaf2 . However, the hadronic-matter regions are different from each
other, since softer H-EoSs are used in Shahrbaf2  than ours.

As stated before, the hyperon mixing results in remarkable softening of the
EoS. In order to avoid this “hyperon puzzle”, the universal repulsions modeled
as MPP are included in our derivations of our H-EoSs. Here, let us try to use
the H-EoS for H1’ in which the MPP repulsions work only among nucleons. The
$BB$ interaction used in Shahrbaf2  is of this type. In Fig.13, hybrid-star
masses are given as a function of radius $R$ (left panel) and of central
density $\rho_{Bc}$ (right panel). The top dashed curve (a) is obtained from
the H-EoS for H0 without hyperons. The middle dashed curve (b) is from the
H-EoS for H1. The bottom dashed curve (c) is from the H-EoS for H1’ including
hyperons, in which the MPP repulsions work only among nucleons. The solid
curves show the deviations by transitions from hadronic phase to quark-matter
phase for Q1. It should be noted that the large difference from the top dashed
curve to the bottom dashed curve demonstrates the softening of the EoS by
hyperon mixing. Then, the solid curves show the deviations by transitions from
hadronic phases to quark-matter phases for Q1. The lowering of the maximum
mass by the EoS softening turns out to be recovered by the transition to the
quark-matter phase given by the stiff EoS. It is interested that the curves
for H0+Q1 are similar to those for H0. The basic feature of the $MR$ curve for
H1’+Q1 is similar to those of the curves in Shahrbaf2 .

Figure 13: hybrid-star masses are given as a function of radius $R$ (left
panel), and as a function of central density $\rho_{Bc}$ (right panel). The
top dashed curve (a) is obtained from the H-EoS for H0 without hyperons. The
middle dashed curve (b) is from the H-EoS for H1. The bottom dashed curve (c)
is from the H-EoS for H1’ including hyperons, in which the MPP repulsions work
only among nucleons. The solid curves show the deviations by transitions from
hadronic phase to quark-matter phase for Q1.

Our $MR$ diagrams of hybrid stars are derived from H-EoSs for $BB$
interactions (H1, H2, H3) and Q-EoSs for $QQ$ interactions (Q0, Q1, Q2): There
are nine combinations of H-EoSs and Q-EoSs, among which some combinations are
used in the above results. In Table 7, features of the obtained $MR$ diagrams
in all combinations of (H1, H2, H3) and (Q0, Q1, Q2) are demonstrated by
showing the calculated values of maximum masses $M_{max}$ and radii
$R_{M_{max}}$, and radii at $1.4M_{\odot}$ ($R_{1.4M_{\odot}}$). For
comparison, those for H1’ and H1’+Q1 are added. Here, the important features
are as follows: (1) In all cases, the Q-EoSs combined with the H-EoSs are
stiff enough to reproduce maximum masses over $2M_{\odot}$. (2) The values of
$R_{1.4M_{\odot}}$ are specified by the H-EoSs.

Table 7: Maximum masses $M_{max}$ and radii $R_{M_{max}}$, radii at $1.4M_{\odot}$ $R_{1.4M_{\odot}}$, dimensionless tidal deformability at $1.4M_{\odot}$ $\Lambda_{1.4M_{\odot}}$ | $M_{max}/M_{\odot}$ | $R_{M_{max}}$ | $R_{1.4M_{\odot}}$ | $\Lambda_{1.4M_{\odot}}$
---|---|---|---|---
|  | (km) | (km) |
H1 | 1.82 | 10.4 | 12.4 | 422
H1+Q0 | 1.99 | 10.0 | 12.4 | 422
H1+Q1 | 2.14 | 10.3 | 12.4 | 422
H1+Q2 | 2.25 | 10.7 | 12.5 | 422
H1’ | 1.52 | 10.4 | 12.1 | 334
H1’+Q1 | 2.10 | 10.0 | 12.2 | 337
H2 | 1.94 | 10.3 | 13.3 | 671
H2+Q0 | 2.01 | 10.4 | 13.3 | 671
H2+Q1 | 2.16 | 10.6 | 13.3 | 671
H2+Q2 | 2.25 | 10.9 | 13.3 | 671
H3 | 2.07 | 10.7 | 13.6 | 771
H3+Q0 | 2.04 | 10.7 | 13.6 | 771
H3+Q1 | 2.18 | 10.8 | 13.6 | 771
H3+Q2 | 2.28 | 11.2 | 13.6 | 771

Another constraint for the EoS is given by the tidal deformability, being the
induced quadruple polarizability. The dimensionless tidal deformability
$\Lambda$ is defined as $\Lambda=(2/3)k_{2}(c^{2}R/GM)^{5}$ Abbott2017 , where
$c$ is the speed of light, $R$ and $M$ are radius and mass of a neutron star
and $G$ is the gravitational constant. $k_{2}$ is the tidal Love number
describing the response of each star to the external disturbance. The binary
neutron star merger GW170817 give the upper limit on the tidal deformability
of a neutron star with mass $1.4M_{\odot}$: $\Lambda_{1.4M_{\odot}}\leq 800$
Piekarewitz2019 . In Table 7 are given the calculated values of
$\Lambda_{1.4M_{\odot}}$ for our EoSs, where all values are less than the
upper limit of 800. It should be noted that the values of
$\Lambda_{1.4M_{\odot}}$ are determined by the H-EoSs, even if the Q-EoSs are
combined with them.

##  V Conclusion

The EoSs and $MR$ diagrams of hybrid stars are obtained on the basis of our
$QQ$ interaction model composed of the extended meson exchange potential
($V_{EME}$), the multi-pomeron exchange potential ($V_{MPP}$), the instanton
exchange potential ($V_{INS}$) and the one gluon exchange potential
($V_{OGE}$), whose strengths are determined on the basis of terrestrial data
with no adhoc parameter to stiffen EoSs. The repulsive nature of our $QQ$
interaction in high density region are basically given by $V_{EME}$ including
strongly repulsive components owing to vector-meson and pomeron exchanges.
Additional repulsions (attractions) are given by $V_{MPP}$ and $V_{OGE}$
($V_{INS}$). The resultant repulsions included in our $QQ$ interaction are so
strong that the quark-matter EoSs become stiff enough to give maximum masses
of hybrid stars over $2M_{\odot}$.

Hadronic-matter EoSs (H-EoS) and quark-matter EoSs (Q-EoS) are derived in the
same framework based on the BBG theory. In quark matter, density-dependent
quark masses are introduced phenomenologically, playing a decisive role in the
occurrence of phase transition. Parameters of density dependences are taken so
that hadron-quark phase transitions are able to occur at reasonable density
region owing to the reduction of the quark masses and chemical potentials in
quark matter. Our resulting density dependence of effective quark mass is
similar to the Brown-Rho scaling.

Our H-EoSs are still not stiff enough to give maximum masses of neutron stars
over $2M_{\odot}$ due to the softening by hyperon mixing, although the
stiffness is recovered substantially by universal many-body repulsions. In the
case of using our $QQ$ interaction model, the Q-EoSs are stiffer than the
H-EoSs and $MR$ curves of hybrid stars shift above those of stars with
hadronic phases only. The maximum masses of the formers including quark phases
become larger than those of the latters, and $MR$ curves are characterized by
Q-EoSs in the mass region higher than about $1.5M_{\odot}$. Our Q-EoSs for Q1
and Q2 are stiff enough to give a maximum mass over $2M_{\odot}$. The derived
mass and radius are consistent with the recent measurement for the most
massive neutron star PSR J0740+6620, obtained by the combining analysis for
the NICER and the other multimessenger data.

In our approach, star radii $R_{1.4M_{\odot}}$ given by hadronic-matter EoSs
do not changed by the hadron-quark phase transitions, namely they are
determined by H-EoSs regardless of Q-EoSs. There are three estimates of
$R_{1.4M_{\odot}}$ based on the NICER measurements and the other
multimessenger data. Two of them give $R_{1.4M_{\odot}}=11.1-13.1$ km, and the
other $R_{1.4M_{\odot}}=13.1-14.4$ km. Our H-EoS for H1 (H2 or H3) is
consistent with the former (latter).

Our H-EoSs and Q-EoSs lead to $MR$ diagrams of hybrid stars consistent with
the recent observations for masses and radii.

## Acknowledgments

The authors would like to thank D. Blaschke for valuable comments and fruitful
discussions. This work was supported by JSPS KAKENHI (No.20K03951 and
No.20H04742).

## References

  * (1) P.B. Demorest, T. Pennucci, S.M. Ransom, M.S.E. Roberts, and J.W. Hessels, Nature (London) 467, 1081 (2010).
  * (2) J. Antoniadis et al., Science 340, 6131 (2013).
  * (3) H.T. Cromartie et al., Nat. Astron. 4, 72 (2020).
  * (4) A. Akmal, V.R. Pandharipande, and D.G. Ravenhall, Phys. Rev. C58, 1804 (1998).
  * (5) S. Nishizaki, Y. Yamamoto, and T. Takatsuka, Prog. Theor. Phys. 105, 607 (2001); 108, 703 (2002).
  * (6) Y. Yamamoto, T. Furumoto, N. Yasutake and Th.A. Rijken, Phys. Rev. C90, 045805 (2014).
  * (7) Y. Yamamoto, T. Furumoto, N. Yasutake and Th.A. Rijken, Eur. Phys. J. A (2016) 52: 19.
  * (8) Y. Yamamoto, H. Togashi, T. Tamagawa, T. Furumoto, N. Yasutake, and Th. A. Rijken, Phy. Rev. C96, 065804 (2017).
  * (9) M. M. Nagels, Th. A. Rijken, and Y. Yamamoto, Phys. Rev. C99, 044002 (2019); C99, 044003 (2019).
  * (10) K. Schertler, S. Leupold, and J. Schaffner-Bielich, Phys.Rev. C60, 025801 (1999).
  * (11) M.Baldo, G.F. Burgio, and H-J. Schulze, Superdense QCD Matter and Compact stars (NATO Science Series 2: Mathematics, Physics and Chemistry vol.197) (2006) ed D.Blaschke and D. Sedrakian.
  * (12) R. Lastowiecki, D. Blaschke, H. Grigorian, and S. Typel, Acta Phys. Pol. B5, 535 (2012).
  * (13) M. Shahrbaf, D. Blaschke, A.G. Grunfeld, and H.R. Moshfegh, Phys. Rev. C101, 025807 (2020).
  * (14) M. Shahrbaf, D. Blaschke, and S. Khanmohamadi, J. Phys. G: Nucl. Part. Phys. 47, 115201 (2020).
  * (15) K. Maslov, N. Yasutake, D. Blaschke, A. Ayriyan, H. Grigorian, T. Maruyama, T. Tatsumi, and D. N. Voskresensky, Phys. Rev. C100, 025802 (2019).
  * (16) Cheng-Jun Xia, Toshiki Maruyama, Nobutoshi Yasutake, and Toshitaka Tatsumi, Phys. Rev. D99, 103017 (2019).
  * (17) T, Kojo, P.D. Powell, Y. Song, and G. Baym, Phys. Rev. D91, 045003 (2015).
  * (18) G. Baym, T. Hatsuda, T. Kojo, T.D. Powell, Y. Song, and T. Takatsuka, Rept. Prog. Phys. bf 81 056902 (2018),
  * (19) K. Otto, M. Oertel, and B-J Schaefer, Phys. Rev. D101, 103021 (2020); arXiv:1910.11929 (2020)
  * (20) T. Kunihiro, Phys. Letters B271, 395 (1991).
  * (21) Th.A. Rijken and Y. Yamamoto, “Extended-soft-core Quark-Quark Model Constituent Quark Meson-exchange Interactions”,
http://nn-online.org/eprints/pdf/20.06.pdf

  * (22) T.E. Riley, et al.. Astrophys. J. Lett. 918 L27 (2021), arXiv:2105.06980 [astro-ph.HE]
  * (23) M.C. Miller, et al.. Astrophys. J. Lett. 918 L28 (2021), arXiv:2105.06979 [astro-ph.HE]
  * (24) G. Raaijmakers, et al.. “Constraints on the dence matter equation of state and neutron star properties from NICER’s mass-radius estimate of PSR J0740+6620 and multimessenger observations”, (2021), arXiv:2105.06981 [astro-ph.HE]
  * (25) T. Peter, et al.. “Nuclear-Physics Multi-Messenger Astrophysics Constraints on the Neutron-star Equation of State: Adding NICER’s PSR J0740+6620 Measurement”, (2021); arXiv:2105.08688 [astro-ph.HE]
  * (26) Brendan T. Reed, F. Fattoyev, C.J. Horowitz, and J. Piekarewicz, Phys. Rev. Lett. 126, 172503 (2021); arXiv:2101.03193
  * (27) M. Hutter, “Interaction in QCD, Theory and Application of the Instanton Liquid Model”, Ph.D.-thesis, University of Munich, 1995.
  * (28) D. Blaschke, H. Grigorian, and G. Röpke, Particles 2020, 3, 477; doi:10.3390/particles3020033
  * (29) G.E. Brown and M. Rho, Phys. Rev. Lett. 66, 2720 (1991).
  * (30) K. Kashiwa, H. Kouno, T. Sakaguchi, M. Matsuzaki, and M. Yahiro, Phys. Lett. B 647, 446 (2007).
  * (31) S. Benic, Eur. Phys. J. A 50, 111 (2014).
  * (32) Y.Yamamoto, T.Motoba, Th.A. Rijken, Prog. Theor. Phys. Suppl. No.185, 72 (2010).
  * (33) V. Abgaryan, D. Alvarez-Cstillo, A. Ayriyan, D. Blaschke and H. Grigorian, Universe 4, no, 9, 94 (2018).
  * (34) N.K. Glendenning, Phys. Rev. D46, 1274 (1992).
  * (35) G. Baym, A. Bethe, and C. Pethick, Nucl. Phys. A175, 225 (1971).
  * (36) G. Baym, C.J. Pethick, and P. Sutherland, Astrophys. J.170, 299 (1971).
  * (37) B.P. Abbott et al. (LIGO Scientific and Virgo Collaborations), Phys. Rev. Lett. 119, 161101 (2017).
  * (38) J. Piekarewitz and F.J. Fattoyev, Phys. Rev. C99, 045802 (2019).