# Neutron star properties with careful parameterization in the (axial)vector
meson extended linear sigma model

Péter Kovács<EMAIL_ADDRESS>Institute for Particle and Nuclear
Physics, Wigner Research Centre for Physics, 1121 Budapest, Hungary  Institute
of Physics, Eötvös University, 1117 Budapest, Hungary  János Takátsy
Institute for Particle and Nuclear Physics, Wigner Research Centre for
Physics, 1121 Budapest, Hungary  Institute of Physics, Eötvös University, 1117
Budapest, Hungary  Jürgen Schaffner-Bielich  Institut für Theoretische Physik,
Goethe Universität Frankfurt, D-60438 Frankfurt, Germany  György Wolf
Institute for Particle and Nuclear Physics, Wigner Research Centre for
Physics, 1121 Budapest, Hungary

###### Abstract

The existence of quark matter inside the cores of heavy neutron stars is a
possibility which can be probed with modern astrophysical observations. We use
an (axial)vector meson extended quark-meson model to describe quark matter in
the core of neutron stars. We discover that an additional parameter constraint
is necessary in the quark model to ensure chiral restoration at high
densities. By investigating hybrid star sequences with various parameter sets
we show that low sigma meson masses are needed to fulfill the upper radius
constraints, and that the maximum mass of stable hybrid stars is only slightly
dependent on the parameters of the crossover-type phase transition. Using this
observation and results from recent astrophysical measurements a constraint of
$2.6<g_{V}<4.3$ is set for the constituent quark – vector meson coupling. The
effect of a nonzero bag constant is also investigated and we observe that its
effect is small for values adopted in previous works.

##  I Introduction

The theory of strong interaction, Quantum Chromodynamics (QCD) is notoriously
difficult to tackle, especially at high but not very high densities or
temperatures (around the chiral phase boundary). Although lattice Monte-Carlo
calculations in the last decades achieved immense progress by solving QCD at
low densities and revealing the nature of strongly interacting matter Aoki
_et al._ (2006a); Borsanyi  _et al._ (2014), the Sign Problem hinders their
application at high densities de Forcrand (2009). On the other hand,
perturbative methods only become reliable at very high energies, not relevant
for most physical scenarios involving dense nuclear matter Kurkela  _et al._
(2010). From the experimental side ALICE at CERN Grelli (2018), and PHENIX and
STAR at RHIC Sakaguchi (2019); Tlusty (2018) also managed to explore QCD at
low density and high temperature. Up to this day experimental data at high
densities is scarce and have rather bad statistics Tlusty (2018); Larsen
(2018), however, multiple experimental facilities that are under construction
are designed to explore this region with higher precision in the near future
Ablyazimov  _et al._ (2017); Kekelidze (2017).

One possibility to explore this area is the usage of effective theories, which
can be applied at finite density and provide important insight about certain
aspects of strongly interacting matter. In our approach the underlying
principle of constructing such a model is to require that the Lagrangian –
involving composite particles (mesons and constituent quarks) instead of the
fundamental quarks and gluons – has the same global symmetries as QCD itself.
One group of these models consists of chiral effective field theories, which
are designed to describe chiral restoration at high temperatures and
densities, and are expected to be reliable in the vicinity of chiral phase
transition.

Recent studies, based on astrophysical measurements in the last few years,
argue that deconfined quark matter might also exist inside the core of neutron
stars (NSs), and in dense stellar remnants (see e.g. Annala  _et al._ (2020);
Blaschke  _et al._ (2020a)). The emergence of increasingly robust predictions
were made possible by major advances in the observation of NSs in the previous
decade, which have already put multiple constraints on the equation of state
(EoS) of dense strongly interacting matter. These constraints stem from a
variety of astrophysical observations, ranging from the discovery of NSs with
masses of $2$ $M_{\odot}$ Demorest  _et al._ (2010); Antoniadis  _et al._
(2013); Cromartie  _et al._ (2019), through gravitational-wave measurements of
the inspirals of NS–NS systems Abbott  _et al._ (2018, 2020), to the X-ray
pulse-profile measurements of pulsars with NICER Riley  _et al._ (2019);
Miller  _et al._ (2019); Riley  _et al._ (2021); Miller  _et al._ (2021)
together with qualitative improvements in X-ray radius measurements (see e.g.
Ozel  _et al._ (2016)).

The description of quark matter in connection with neutron stars has been
investigated using differing chiral approaches. The chiral mean field model is
based on a Yukawa-type scalar and vector meson exchange for nucleons and
quarks obeying chiral symmetry Dexheimer and Schramm (2010); Negreiros  _et
al._ (2010); Dexheimer  _et al._ (2013, 2019); Roark and Dexheimer (2018);
Dexheimer  _et al._ (2021). The chiral quark-meson model uses the linear sigma
model and has been extended by implementing vector meson exchange and a vacuum
term Zacchi  _et al._ (2015, 2017). Both additions to the linear sigma model
turned out to be of having a significant impact on the properties of compact
star configurations. The quark-meson model has been also investigated for
compact stars by going beyond mean-field using renormalization for the quark
part Zacchi and Schaffner-Bielich (2019) and by adopting the functional
renormalization group method Otto  _et al._ (2020a, b). Confronting recent
astrophysical data on neutron stars with chiral models for quark matter has
been also performed within a nonlocal Nambu-Jona-Lasinio model Alvarez-
Castillo  _et al._ (2019); Shahrbaf  _et al._ (2020); Blaschke  _et al._
(2020b) and a unified quark-meson-nucleon model Marczenko  _et al._ (2020).

In this paper we use an (axial)vector meson extended linear sigma model with
constituent quarks (or quark-meson model) at zero temperature and finite quark
(or baryon) chemical potential to describe the properties of hybrid stars. The
advantage of this model – altogether with the parameterization procedure and
the approximations that were used – is that it reproduces the meson spectrum
(and also various decay widths) quite well at $T=\mu_{q}=0$ Parganlija  _et
al._ (2013) and moreover its finite temperature version also agrees well with
various lattice results Kovács  _et al._ (2016). Since we think that
parameterization plays a crucial role – i.e. depending on the starting
position of the parameter space the system shows very different behavior at
finite $T$ and/or $\mu_{q}$ – in the description of properties at finite
densities, we investigate the consequences of the asymptotic behavior of the
system of equations on the parameterization. It turns out that the system does
not behave as expected with every parameterization.

Using two different hadronic models at low densities we construct hybrid star
EoS’s, which fulfill all the current expectations coming from astrophysical
measurements, providing some of the parameters, like the $g_{V}$ vector
coupling, are set properly.

The paper is organized as follows. Section II is devoted to the introduction
of the model, set up of the $\beta$-equilibrium and charge neutrality
conditions, calculation of the pressure and field equations, and the
parameterization procedure. In Section III the hadronic EoS’s and
interpolation methods (between the hadronic and the quark EoS’s) are
demonstrated together with a brief summary on compact star observables.
Section IV contains our results, where the EoS’s, the $M-R$ curves, the tidal
deformabilities ($\Lambda$) and their dependence on various parameters are
analyzed. Finally, we summarize the implications of our work in Section V.
Some additional details can be found in Appendices A-B.

##  II The vector meson extended linear sigma model

The Lagrangian of the model is a version of the three flavored (axial)vector
meson extended linear sigma model introduced in Parganlija  _et al._ (2013),
where it was thoroughly investigated at zero temperature. A slightly modified
version of that model was used for finite temperature investigations in Kovács
_et al._ (2016). Here we use the latter with an additional (axial)vector
Yukawa type term. Consequently, the total Lagrangian of the model reads as

$\displaystyle\mathcal{L}$
$\displaystyle=\mathop{\mathrm{Tr}}[(D_{\mu}M)^{\dagger}(D_{\mu}M)]-m_{0}^{2}\mathop{\mathrm{Tr}}(M^{\dagger}M)-\lambda_{1}[\mathop{\mathrm{Tr}}(M^{\dagger}M)]^{2}-\lambda_{2}\mathop{\mathrm{Tr}}(M^{\dagger}M)^{2}+c_{1}(\det
M+\det M^{\dagger})+\mathop{\mathrm{Tr}}[H(M+M^{\dagger})]{}$
$\displaystyle-\frac{1}{4}\mathop{\mathrm{Tr}}(L_{\mu\nu}^{2}+R_{\mu\nu}^{2})+\mathop{\mathrm{Tr}}\left[\left(\frac{m_{1}^{2}}{2}+\Delta\right)(L_{\mu}^{2}+R_{\mu}^{2})\right]+i\frac{g_{2}}{2}(\mathop{\mathrm{Tr}}\\{L_{\mu\nu}[L^{\mu},L^{\nu}]\\}+\mathop{\mathrm{Tr}}\\{R_{\mu\nu}[R^{\mu},R^{\nu}]\\}){}$
$\displaystyle+\frac{h_{1}}{2}\mathop{\mathrm{Tr}}(M^{\dagger}M)\mathop{\mathrm{Tr}}(L_{\mu}^{2}+R_{\mu}^{2})+h_{2}\mathop{\mathrm{Tr}}(|L_{\mu}M|^{2}+|MR_{\mu}|^{2})+2h_{3}\mathop{\mathrm{Tr}}(L_{\mu}MR^{\mu}M^{\dagger})+\bar{\Psi}\left[i\gamma_{\mu}D^{\mu}-\mathcal{M}\right]\Psi{}$
$\displaystyle-
g_{V}\bar{\Psi}\left[\gamma_{\mu}V^{\mu}+\frac{g_{A}}{g_{V}}\gamma^{5}\gamma_{\mu}A^{\mu}\right]\Psi\,,$
(1)

where, as it is described in detail in Kovács  _et al._ (2016),
$M=M_{S}+M_{PS}$, $L^{\mu}=V^{\mu}+A^{\mu}$, $R^{\mu}=V^{\mu}-A^{\mu}$ and
$M_{S},M_{PS},V^{\mu},A^{\mu}$ stand for the scalar, the pseudoscalar, the
vector and the axial vector nonets, respectively, while
$\Psi=(q_{u},q_{d},q_{s})^{T}$ for the constituent quark fields. Some comments
are in order: there are two new unknown parameters, the $g_{V}$ vector
coupling, which has a direct impact on the value of the maximal mass on the
$M-R$ curve of compact stars. Although the $g_{A}$ axial coupling (or the
$g_{A}/g_{V}$ ratio) is also unknown it will not appear in any of the
following expressions, thus its value is irrelevant for the current
investigation. In Kovács  _et al._ (2016) we also introduced Polyakov-loop
variables, which vanish at zero temperature, thus they will not be present in
the equations relevant for compact stars, however, since in the
parameterization the pseudocritical temperature is used, it still affects the
parameter set.

As a standard procedure for theories with spontaneous symmetry breaking (SSB),
nonzero vacuum expectation values (vev) are assumed for the non-strange and
strange isoscalar fields $\sigma_{N}$ and $\sigma_{S}$ and for the temporal
component of the three vector fields: the charge neutral $\rho_{0}^{\mu}$,
$\omega^{\mu}$, and $\Phi^{\mu}$. These expectation values are denoted as
follows:

$\displaystyle\langle\sigma_{N}\rangle$
$\displaystyle\equiv\phi_{N},\quad\langle\sigma_{S}\rangle\equiv\phi_{S},{}$
$\displaystyle\langle\rho_{0}^{0}\rangle$ $\displaystyle\equiv
v_{\rho},\quad\langle\omega^{0}\rangle\equiv
v_{\omega},\quad\langle\Phi^{0}\rangle\equiv v_{\Phi}.$ (2)

It should be noted that we neglect here the small effect of isospin breaking,
which would require the introduction of a nonzero expectation value for the
scalar $a_{0}^{0}$ field.

Hereafter the fields are shifted with their nonzero expectation values, which
subsequently results in the tree-level expressions for the meson and
constituent quark masses and the tree-level decay widths. Moreover, the
nonzero vector vev’s shift the $\mu_{u}$, $\mu_{d}$, and $\mu_{s}$ quark
chemical potentials of the constituent quark fields subsequently leading to
the following effective quark chemical potentials for the different flavors,

$\displaystyle\tilde{\mu}_{u}$
$\displaystyle=\mu_{u}-\frac{1}{2}g_{V}(v_{\omega}+v_{\rho}),{}$
$\displaystyle\tilde{\mu}_{d}$
$\displaystyle=\mu_{d}-\frac{1}{2}g_{V}(v_{\omega}-v_{\rho}),$ (3)
$\displaystyle\tilde{\mu}_{s}$
$\displaystyle=\mu_{s}-\frac{1}{\sqrt{2}}g_{V}v_{\Phi}.{}$

These shifts stem from the vector Yukawa term, that is the last term of the
Lagrangian (Eq. (1)).

###  II.1 $\beta$-equilibrium and charge neutrality

We add a free electron gas to our system with some $\mu_{e}$ electron chemical
potential and assume $\beta$-equilibrium, that is

$\mu_{d}=\mu_{s}=\mu_{u}+\mu_{e}{}$

after neutrinos have left the system. Thus using

$\mu_{q}\equiv\frac{1}{3}\mu_{B}=\frac{1}{3}(\mu_{u}+\mu_{d}+\mu_{s})$ (4)

for the quark chemical potential, the chemical potentials for the different
flavors are given by

$\displaystyle\mu_{u}$ $\displaystyle=\mu_{q}-\frac{2}{3}\mu_{e},{}$
$\displaystyle\mu_{d}$ $\displaystyle=\mu_{q}+\frac{1}{3}\mu_{e},$ (5)
$\displaystyle\mu_{s}$ $\displaystyle=\mu_{q}+\frac{1}{3}\mu_{e}.{}$

Charge neutrality is also applied, which can be written as

$\frac{2}{3}n_{u}-\frac{1}{3}n_{d}-\frac{1}{3}n_{s}-n_{e}=0,$ (6)

where $n_{u/d/s}$ and $n_{e}$ are the number densities for the $u,d,s$ quarks
and the electron, respectively. They can be calculated as

$n_{f}=\frac{\partial p}{\partial\mu_{f}},\,f\in(u,d,s),\quad
n_{e}=\frac{\partial p}{\partial\mu_{e}},$ (7)

where $p$ is the pressure.

Finally, the effective quark chemical potentials for the different flavors can
be written as,

$\displaystyle\tilde{\mu}_{u}$
$\displaystyle=\mu_{q}-\frac{2}{3}\mu_{e}-\frac{1}{2}g_{V}(v_{\omega}+v_{\rho}),{}$
$\displaystyle\tilde{\mu}_{d}$
$\displaystyle=\mu_{q}+\frac{1}{3}\mu_{e}-\frac{1}{2}g_{V}(v_{\omega}-v_{\rho}),$
(8) $\displaystyle\tilde{\mu}_{s}$
$\displaystyle=\mu_{q}+\frac{1}{3}\mu_{e}-\frac{1}{\sqrt{2}}g_{V}v_{\Phi}.{}$

It should be noted here that while $\mu_{u/d/s}$ play the role of the usual
chemical potentials, in the calculation of the grand potential (or the
pressure) the $\tilde{\mu}_{u/d/s}$ effective chemical potentials appear.

###  II.2 Pressure and the field equations

The pressure is given by

$p(\mu_{f},\mu_{e})=\Omega_{0}-\Omega(T=0,\mu_{f},\mu_{e}),$ (9)

where $\Omega$ is the grand potential. The grand potential is calculated in a
hybrid approximation used in Kovács  _et al._ (2016) at zero temperature with
additional vector condensates introduced above. In this hybrid approach we
consider only quark fluctuations, while all the mesons are at tree-level. The
grand potential consist of the following terms,

$\Omega_{\textrm{tot}}=U_{\textrm{meson}}(\phi_{N},\phi_{S},v_{\rho},v_{\omega},v_{\Phi})+\Omega_{\bar{q}q}^{\textrm{vac}}+\Omega_{\bar{q}q}^{\textrm{mat}}(\mu_{f})+\Omega_{\textrm{el}},$
(10)

where $U_{\textrm{meson}}$ stands for the tree - level meson potential,
$\Omega_{\bar{q}q}^{\textrm{vac}}$ and $\Omega_{\bar{q}q}^{\textrm{mat}}$ for
the vacuum and matter part of the one - loop constituent quark contributions,
while $\Omega_{\textrm{el}}$ for the electron contribution. Its explicit form
can be found in Eq. (49) of Appendix A. The field equations (FE) are the
stationary points of the grand potential, i.e.

$\frac{\partial\Omega_{\textrm{tot}}}{\partial\phi_{N}}=\frac{\partial\Omega_{\textrm{tot}}}{\partial\phi_{S}}=\frac{\partial\Omega_{\textrm{tot}}}{\partial
v_{\omega}}=\frac{\partial\Omega_{\textrm{tot}}}{\partial
v_{\rho}}=\frac{\partial\Omega_{\textrm{tot}}}{\partial v_{\Phi}}=0.$ (11)

These are five coupled equations for the $\mu_{q}$ (or $\mu_{B}$) dependence
of the scalar and vector condensates. There is another unknown, the $\mu_{e}$
electron chemical potential, which is determined through the charge neutrality
condition, which is also coupled to the preceding five equations. The explicit
form of the system of six equations that needs to be solved reads as

$\displaystyle\frac{\partial\Omega_{\textrm{tot}}}{\partial\phi_{N}}$
$\displaystyle=m_{0}^{2}\phi_{N}+\lambda_{1}\left(\phi_{N}^{2}+\phi_{S}^{2}\right)\phi_{N}+\frac{1}{2}\lambda_{2}\phi_{N}^{3}-\frac{c_{1}}{\sqrt{2}}\phi_{N}\phi_{S}-h_{N}-\frac{1}{2}\left(h_{1}+h_{2}+h_{3}\right)\phi_{N}\left(v_{\omega}^{2}+v_{\rho}^{2}\right)-\frac{1}{2}h_{1}\phi_{N}v_{\Phi}^{2}{}$
$\displaystyle-$
$\displaystyle\frac{3g_{F}}{8\pi^{2}}m_{u}^{3}\left[1+4\log\frac{m_{u}}{M_{0}}\right]+\frac{3g_{F}}{4\pi^{2}}\sum_{f\in(u,d)}m_{f}^{3}\left[\gamma_{f}\sqrt{\gamma_{f}^{2}-1}-\log\left(\gamma_{f}+\sqrt{\gamma_{f}^{2}-1}\right)\right]=0$
(12) $\displaystyle\frac{\partial\Omega_{\textrm{tot}}}{\partial\phi_{S}}$
$\displaystyle=m_{0}^{2}\phi_{S}+\lambda_{1}\left(\phi_{N}^{2}+\phi_{S}^{2}\right)\phi_{S}+\lambda_{2}\phi_{S}^{3}-\frac{c_{1}}{2\sqrt{2}}\phi_{N}^{2}-h_{S}-\frac{1}{2}h_{1}\phi_{S}\left(v_{\omega}^{2}+v_{\rho}^{2}\right)-\frac{1}{2}\left(h_{1}+2h_{2}+2h_{3}\right)\phi_{S}v_{\Phi}^{2}{}$
$\displaystyle-$
$\displaystyle\frac{3g_{F}}{8\sqrt{2}\pi^{2}}m_{s}^{3}\left[1+4\log\frac{m_{u}}{M_{0}}\right]+\frac{3g_{F}}{2\sqrt{2}\pi^{2}}m_{s}^{3}\left[\gamma_{s}\sqrt{\gamma_{s}^{2}-1}-\log\left(\gamma_{s}+\sqrt{\gamma_{s}^{2}-1}\right)\right]=0$
(13) $\displaystyle\frac{\partial\Omega_{\textrm{tot}}}{\partial v_{\omega}}$
$\displaystyle=-m_{\rho}^{2}v_{\omega}+\frac{m_{u}^{3}}{2\pi^{2}}g_{V}\sum_{f\in(u,d)}\operatorname{sgn}(\tilde{\mu}_{f})\left(\gamma_{f}^{2}-1\right)^{\frac{3}{2}}=0$
(14) $\displaystyle\frac{\partial\Omega_{\textrm{tot}}}{\partial v_{\rho}}$
$\displaystyle=-m_{\rho}^{2}v_{\rho}+\frac{m_{u}^{3}}{2\pi^{2}}g_{V}\left[\operatorname{sgn}(\tilde{\mu}_{u})\left(\gamma_{u}^{2}-1\right)^{\frac{3}{2}}-\operatorname{sgn}(\tilde{\mu}_{d})\left(\gamma_{d}^{2}-1\right)^{\frac{3}{2}}\right]=0$
(15) $\displaystyle\frac{\partial\Omega_{\textrm{tot}}}{\partial v_{\Phi}}$
$\displaystyle=-m_{\Phi}^{2}v_{\Phi}+\frac{m_{s}^{3}}{\sqrt{2}\pi^{2}}g_{V}\operatorname{sgn}(\tilde{\mu}_{s})\left(\gamma_{s}^{2}-1\right)^{\frac{3}{2}}=0$
(16) $\displaystyle 2\operatorname{sgn}$
$\displaystyle(\tilde{\mu}_{u})\left(\tilde{\mu}_{u}^{2}-m_{u}^{2}\right)^{\frac{3}{2}}-\operatorname{sgn}(\tilde{\mu}_{d})\left(\tilde{\mu}_{d}^{2}-m_{d}^{2}\right)^{\frac{3}{2}}-\operatorname{sgn}(\tilde{\mu}_{s})\left(\tilde{\mu}_{s}^{2}-m_{s}^{2}\right)^{\frac{3}{2}}-\operatorname{sgn}(\mu_{e})\left(\mu_{e}^{2}-m_{e}^{2}\right)^{\frac{3}{2}}=0,$
(17)

where $\gamma_{f}$ is defined in Eq. (52), while the $m_{\rho}=m_{\omega}$,
and $m_{\Phi}$ masses are given in Eqs. (50), (51). It should be noted that
each term that contains $\sqrt{\gamma_{f}^{2}-1}$ or
$\sqrt{\tilde{\mu}_{f}^{2}-m_{f}^{2}}$ is only present if
$\tilde{\mu}_{f}>m_{f}$.

###  II.3 Model parameters and asymptotic behavior of the equations

In order to solve our system of equations (Eqs. (12)-(17)) at finite
$\mu_{q}$, the parameters of the model should be determined first. The 15
unknown parameters are $m_{0}$, $\lambda_{1}$, $\lambda_{2}$, $c_{1}$,
$m_{1}$, $g_{1}$, $g_{2}$, $h_{1}$, $h_{2}$, $h_{3}$, $\delta_{S}$,
$\phi_{N}$, $\phi_{S}$, $g_{F}$, and $g_{V}$. As described in detail in
Section IV. of Kovács  _et al._ (2016) we calculate meson masses and decay
widths at $\mu_{q}=T=0$, moreover $T_{c}$ at $\mu_{q}=0$ and compare them to
their experimental value – taken from the PDG Zyla  _et al._ (2020), while for
$T_{c}$ the value $150$ MeV was used, which was taken from the lattice Aoki
_et al._ (2006b); Borsanyi  _et al._ (2010) – through a $\chi^{2}$ fit. Since
in the current approximation we use tree - level (axial)vector masses the
$g_{V}$ vector coupling does not appear in any of the expressions, thus it
remains a free parameter. It should be noted, however, that if we include
fermionic fluctuations in the curvature masses of the (axial)vectors, $g_{V}$
will also be fixed through the $\chi^{2}$ minimization Kovács  _et al._
(2021).

Figure 1: The dependence of the $\phi_{N/S}$ scalar condensates on the quark
chemical potential with (solid lines) and without (dashed lines) the
asymptotic condition of Eq. (30). The parameter sets for these results are
shown in Table 1, and in Table IV. of Kovács  _et al._ (2016), respectively,
and $g_{V}=5$ in both cases. This shows that a naive parameterization of the
model results in an increase of the scalar condensates at higher densities and
therefore the recurrence of chiral symmetry breaking.

During the investigation of different parameter sets we noticed some unusual
behaviour of the $\phi_{N/S}$ scalar condensates as a function of $\mu_{q}$,
namely for large $\mu_{q}$ values $\phi_{N/S}$ started to increase
substantially, as shown in Fig. 1. This behavior would mean the recurrence of
chiral symmetry breaking for large $\mu_{q}$, which seems unphysical. To avoid
this unwanted behavior we investigated the asymptotic solution for large
$\mu_{q}$ and demanded the disappearance of the $\phi_{N/S}$ condensates at
these asymptotic values. Thus, more explicitly it is assumed that for large
$\mu_{q}$

$\displaystyle\phi_{N}\sim\mu_{q}^{-\alpha},\;\phi_{S}\sim\mu_{q}^{-\beta},{}$
$\displaystyle
v_{\omega}\sim\mu_{q}^{k},\;v_{\rho}\sim\mu_{q}^{l},\;v_{\Phi}\sim\mu_{q}^{n},{}$
$\displaystyle\textrm{with}\;\alpha,\beta,k,l,n\geq 0.$ (18)

It is also assumed that $\tilde{\mu}_{u/d/s}>0$. With these assumptions in
leading order

$\displaystyle\operatorname{sgn}(\tilde{\mu}_{f})m_{f}^{3}(\gamma_{u}^{2}-1)^{\frac{3}{2}}$
$\displaystyle\approx\tilde{\mu}_{f}^{3},$ $\displaystyle
m_{\omega/\rho/\Phi}^{2}$ $\displaystyle\approx m_{1}^{2}.$

Consequently, Eqs. (14)-(17) will have the forms – neglecting the electron’s
mass as well,

$\displaystyle m_{1}^{2}v_{\omega}$
$\displaystyle\approx\frac{g_{V}}{2\pi^{2}}\left(\tilde{\mu}_{u}^{3}+\tilde{\mu}_{d}^{3}\right),$
(19) $\displaystyle m_{1}^{2}v_{\rho}$
$\displaystyle\approx\frac{g_{V}}{2\pi^{2}}\left(\tilde{\mu}_{u}^{3}-\tilde{\mu}_{d}^{3}\right),$
(20) $\displaystyle m_{1}^{2}v_{\Phi}$
$\displaystyle\approx\frac{g_{V}}{\sqrt{2}\pi^{2}}\tilde{\mu}_{s}^{3},$ (21)
$\displaystyle\mu_{e}^{3}$ $\displaystyle\approx
2\tilde{\mu}_{u}^{3}-\tilde{\mu}_{d}^{3}-\tilde{\mu}_{s}^{3}.$ (22)

Rearranging Eq. (21) and substituting Eqs. (8) for $\tilde{\mu}_{s}$ it can be
written that

$\frac{g_{V}}{\sqrt{2}\pi^{2}}\left(\mu_{q}+\frac{1}{3}\mu_{e}-\frac{1}{\sqrt{2}}g_{V}v_{\Phi}\right)^{3}-m_{1}^{2}v_{\Phi}\approx
0,$ (23)

where the first part of the left hand side is a third order polynomial in
$v_{\Phi}$ that scales with $\sim\mu_{q}^{3n}$, thus the second term
$m_{1}^{2}v_{\Phi}$ which scales only with $\sim\mu_{q}^{n}$ can be neglected
compared to that since $n\geq 0$. Consequently, the expression in the
parentheses – which is simply $\tilde{\mu}_{s}$ – is zero in leading order
($\tilde{\mu}_{s}\approx 0$). Similar arguments lead to
$\tilde{\mu}_{u}\approx 0,\;\tilde{\mu}_{d}\approx 0$. Accordingly, from Eq.
(22) we conclude that $\mu_{e}\approx 0$. Thus the effective quark chemical
potentials for the flavors in leading order are zero,

$\tilde{\mu}_{u}\approx 0,\;\tilde{\mu}_{d}\approx 0,\;\tilde{\mu}_{s}\approx
0\;\mu_{e}\approx 0.$ (24)

Using Eq. (8) in Eqs. (24) and rearranging for the vector condensates results
in

$\displaystyle v_{\omega}$ $\displaystyle\approx\frac{2}{g_{V}}\mu_{q}$ (25)
$\displaystyle v_{\rho}$ $\displaystyle\approx 0$ (26) $\displaystyle
v_{\Phi}$ $\displaystyle\approx\frac{\sqrt{2}}{g_{V}}\mu_{q}.$ (27)

Using these approximations in Eq. (12) gives

$-h_{N}-\frac{1}{2}(h_{1}+h_{2}+h_{3})\phi_{N}\frac{4}{g_{V}^{2}}\mu_{q}^{2}-\frac{1}{2}h_{1}\phi_{N}\frac{2}{g_{V}^{2}}\mu_{q}^{2}\approx
0.$ (28)

Expressing $\phi_{N}$ finally results in

$\phi_{N}=-\frac{h_{N}g_{V}^{2}}{3h_{1}+2h_{2}+2h_{3}}\mu_{q}^{-2},$ (29)

Since $\phi_{N}>0$, and $h_{N}>0$, this implies the following condition:

$3h_{1}+2h_{2}+2h_{3}<0.$ (30)

Investigating the asymptotic behavior of Eq. (13) for $\phi_{S}$ leads to the
same condition. Consequently, the asymptotic exponents of the condensates are:

$\alpha=\beta=2,\quad k=n=1,\quad l=0.$ (31)

Note: we also checked this numerically with different parameterizations and
found that the condition in (30) is valid.

It is worth to note that to reach that conclusion we assumed in Eq. (18) that
$k,l$ and $n$ exponents are all non-negative. Let us now assume for instance
that contrary to that $n<0$. In this case in $\tilde{\mu}_{s}$ (Eq. (8)) the
$v_{\Phi}$ term could be neglected compared to $\mu_{q}$ and we would end up
with
$m_{1}^{2}v_{\Phi}\approx\frac{g_{V}}{\sqrt{2}\pi^{2}}(\mu_{q}+1/3\mu_{e})^{3}$,
where the left hand side tends to zero, while the right hand side tends to
infinity. Similarly, the assumptions $k<0$ or $l<0$ would also result in
contradictions.

We should add that although with this condition the scalar condensates tend to
zero at asymptotically large densities, they can still show an increase at
intermediately large densities where the asymptotic behaviour does not yet
apply. However, this only happens at large densities, which are not present
inside neutron stars, and where our model would already lose its predictive
power.

Returning to the determination of the parameters, we use the same procedure as
described in Kovács  _et al._ (2016) with the additional condition of Eq.
(30). The resulting parameter set is given in Table 1.

Table 1: Parameter values for $m_{\sigma}=290$ MeV  Parameter | Value | Parameter | Value
---|---|---|---
$\phi_{N}$ [GeV] | $0.1290$ | $g_{1}$ | $5.3296$
$\phi_{S}$ [GeV] | $0.1406$ | $g_{2}$ | $-1.0579$
$m_{0}^{2}$ [GeV2] | $-1.2370{\scriptscriptstyle E\negthinspace-2}$ | $h_{1}$ | $5.8467$
$m_{1}^{2}$ [GeV2] | $0.5600$ | $h_{2}$ | $-12.3456$
$\lambda_{1}$ | $-1.0096$ | $h_{3}$ | $3.5755$
$\lambda_{2}$ | $25.7328$ | $g_{F}$ | $4.9571$
$c_{1}$ [GeV] | $1.4700$ |  $M_{0}$ [GeV] | $0.3935$
$\delta_{S}$ [GeV2] | $0.2305$ |  |

In this case the $\sigma$ (or $f_{0}$) mass is quite low, namely
$m_{\sigma}=290$ MeV. We may also study the dependence of our results on the
$\sigma$ mass, since our fit prefers a rather small value for that, and
experimentally it is not very well defined either. Indeed it is a very broad
resonance: $m_{f_{0}(500)}=400\,\textrm{to}\,800$ MeV,
$\Gamma_{f_{0}(500)}=100\,\textrm{to}\,800$ MeV Zyla  _et al._ (2020). For any
chosen $\sigma$ mass we have to find a parameter set, which reproduces that
value. To achieve this we have increased the contribution of the $\sigma$ mass
to the $\chi^{2}$ by a factor of 1000. This way, by minimizing the $\chi^{2}$,
the obtained fit reproduces the prescribed $\sigma$ mass with less than
$0.2\%$ error. The resulting parameter sets can be found in Appendix B.

##  III Compact stars

###  III.1 Hadronic equation of state

To be able to construct the sequence of stable NSs we need a reliable equation
of state covering many orders of magnitudes in density from subsaturation
densities up to about $n\approx 5-6\,n_{0}$, with $n_{0}$ being the nuclear
saturation density. Below saturation nuclear methods such as hadron resonance
gas models Huovinen and Petreczky (2010); Bazavov  _et al._ (2012) and chiral
effective field theories Lynn  _et al._ (2016); Tews  _et al._ (2018) offer a
robust way for describing nuclear matter. The uncertainties of chiral
effective field theories mainly stem from the truncation of the nuclear
Hamiltonian within the expansion, as well as the regulariztaion scheme and
scale, resulting in the margins of error for the pressure in state-of-the-art
models increasing rapidly above $n\approx 2\,n_{0}$ Tews  _et al._ (2019).
Looking at the other side of the density spectrum, at extremely high densities
we expect QCD to become asymptotically free, thus enabling the use of
perturbative QCD (pQCD) calculations Mogliacci  _et al._ (2013). However,
these methods are only reliable at densities much higher than the ones
expected to be present in the center of the most massive NSs. Nevertheless,
the EoS of quark matter models should converge to those obtained from pQCD
calculations at asymptotically high densities.

In the intermediate-density region therefore no fully reliable theory exists,
and indeed there is a large selection of nuclear theories using various
approaches that range from variational methods to relativistic mean field
(RMF) theories , which results in a high variation in the calculated nuclear
EoS’s. These approaches are all based on extrapolations from experimentally
measurable nuclear properties at saturation. The most important of these
quantities are the binding energy per nucleon ($E_{0}\approx-16.3$ MeV),
compressibility ($K_{0}=240\pm 20$ MeV), symmetry energy ($S_{0}=31.6\pm 2.7$
MeV) and the slope of symmetry energy ($L=58.9\pm 16$ MeV) Shlomo  _et al._
(2006); Piekarewicz (2010); Chen  _et al._ (2010); Li  _et al._ (2019).

We use two RMF models, the Steiner–Fischer–Hempel (SFHo) model Steiner  _et
al._ (2013) and the density-dependent RMF model of Typel et al. (DD2) Typel
_et al._ (2010); Hempel and Schaffner-Bielich (2010). They are both consistent
with the aforementioned nuclear constraints, the only major difference being
the different values for the slope of the symmetry energy $L$. This results in
the DD2 EoS being stiffer than the SFHo EoS. Some basic properties of the two
models are included in Table 2. We assume NSs have hadronic crusts described
by the EoS’s of Baym et al. Baym  _et al._ (1971) and Negele & Vautherin
Negele and Vautherin (1973).

Table 2: Nuclear properties of symmetric nuclear matter described by the SFHo and DD2 RMF models as well as some properties of neutron stars described by these models.  Property |  SFHo |  DD2
---|---|---
Saturation density, $n_{0}$ [fm-3] |  0.16 |  0.15
Binding energy per baryon, $E_{0}$ [MeV] |  -16.17 |  -16.02
Compressibility, $K_{0}$ [MeV] |  245.2 |  242.7
Symmetry energy, $S_{0}$ [MeV] |  31.2 |  32.73
Slope of symmetry energy, $L$ [MeV] |  45.7 |  57.94
Maximum mass neutron star [$M_{\odot}$] |  2.06 |  2.42
Radius of $M=1.4\,M_{\odot}$ neutron star [km] |  11.97 |  13.26

###  III.2 Hadron–quark phase transition

Since the hadronic and quark phases are described by qualitatively different
models, we need to match them ’by hand’ and find an appropriate interpolation
method to fix the EoS at intermediate densities. One method uses a simple
Maxwell construction, assuming that the hadronic and quark models describe
strongly interacting matter correctly below/above the phase transition point,
where $p_{H}(\mu_{B})=p_{Q}(\mu_{B})$. This construction results in a first-
order phase transition and is limited to cases where the hadronic EoS is
stiffer than the quark EoS.

Instead of the Maxwell construction here we use two methods that interpolate
between the hadronic and quark EoS’s on a finite density range, and hence
result in crossover phase transitions. The idea of a smooth interpolation can
be supported by the argument that both models lose their validity in the
intermediate density region, and therefore a strict extrapolation of the two
EoS’s is generally not justified.

One of these methods interpolates between the pressures, $p(\mu_{B})$, on a
finite range of chemical potential (see e.g. Abgaryan  _et al._ , 2018; Baym
_et al._ , 2019). Here the hadronic EoS is restricted to the domain below
$\mu_{BL}$, and the quark EoS to the domain above $\mu_{BU}$. These chemical
potentials correspond to baryon number densities $n_{L}$ and $n_{U}$,
respectively. In the intermediate region a reasonable choice for the
interpolating function is a polynomial that smoothly connects the two parts:

$p(\mu_{B})=\sum_{m=0}^{N}C_{m}\mu_{B}^{m}\>,\quad\mu_{BL}<\mu_{B}<\mu_{BU},$
(32)

where $C_{m}$ are coefficients that we may fix by matching the pressure and
its derivatives at the boundary points. We use a fifth-order polynomial, and
match the pressure, the number density, and the sound speed at both boundary
points (this is equivalent to matching the pressure together with its first
and second derivatives).

The energy density interpolation method, introduced in Masuda  _et al._
(2013), applies a smooth interpolation between the $\varepsilon(n_{B})$
curves:

$\varepsilon(n_{B})=\varepsilon_{H}(n_{B})f_{-}(n_{B})+\varepsilon_{Q}(n_{B})f_{+}(n_{B}),$
(33)

where $f_{\pm}$ are hyperbolic tangent interpolating functions:

$f_{\pm}(n_{B})=\frac{1}{2}\left(1\pm\tanh\left(\frac{n_{B}-\bar{n}_{B}}{\Gamma}\right)\right),$
(34)

with $\bar{n}_{B}$ and $\Gamma$ parameterizing the center and width of the
phase transition. The pressure can then be calculated from the thermodynamic
relation $p=n_{B}^{2}\,\partial(\varepsilon/n_{B})/\partial n_{B}$. This
induces the following expression:

$p(n_{B})=p_{H}(n_{B})f_{-}(n_{B})+p_{Q}(n_{B})f_{+}(\rho_{B})+\Delta p,$ (35)

with

$\Delta p=n_{B}(p_{H}(n_{B})-p_{Q}(n_{B}))g(n_{B}),$ (36)
$g(n_{B})=\frac{1}{2\Gamma}\cosh^{-2}\left(\frac{n_{B}-\bar{n}_{B}}{\Gamma}\right).$
(37)

Both methods enable us to set the onset and length of the phase transition,
which grants us additional degrees of freedom compared to a Maxwell
construction. On the other hand, the applicability of these methods is also
constrained to a limited range of parameters, since unphysical EoS’s may also
appear, where the sound speed exceeds unity or the energy density decreases
with increasing chemical potential.

Throughout our investigation we will use the SFHo EoS together with the energy
density interpolation as our standard choice to construct hybrid stars,
although we will investigate the effect of using different hadronic EoS’s and
concatenation methods in Section IV.3.

###  III.3 Compact star observables

NS masses and radii can be obtained from general relativistic calculations.
The line element for a spherically symmetric configuration can be expressed
the following way:

$\mathrm{d}s^{2}=e^{\nu(r)}\mathrm{d}t^{2}-e^{\lambda(r)}\mathrm{d}r^{2}-r^{2}(\mathrm{d}\vartheta^{2}+\sin^{2}\vartheta\,\mathrm{d}\varphi^{2}).$
(38)

Assuming that the matter inside NSs can be considered an approximately
spherically symmetric perfect fluid with zero temperature, and introducing the
variable $m(r)$ as

$e^{\lambda(r)}=\left[1-\frac{2m(r)}{r}\right]^{-1},$ (39)

we can obtain the Tolman–Oppenheimer–Volkoff (TOV) equations Tolman (1939);
Oppenheimer and Volkoff (1939):

$\displaystyle\frac{\mathrm{d}m(r)}{\mathrm{d}r}$ $\displaystyle=4\pi
r^{2}\varepsilon(r),$ (40) $\displaystyle\frac{\mathrm{d}p(r)}{\mathrm{d}r}$
$\displaystyle=-[\varepsilon(r)+p(r)]\dfrac{m(r)+4\pi
r^{3}p(r)}{r^{2}-2m(r)r},$ (41)

where $p(r)$ is the pressure related to the energy density $\varepsilon(r)$ by
the EoS. Generally these equations are integrated numerically, and the
boundary conditions $\varepsilon(r=0)=\varepsilon_{c}$, $p(R)=0$ and $m(R)=M$
determine the total mass ($M$) and radius ($R$) of the NS for a certain
central energy density $\varepsilon_{c}$. Varying this energy density creates
a sequence of NSs, and thus we obtain the $M-R$ relation for the specific EoS.
Ideally one would measure the masses and radii of NSs and hence gradually
constrain the $M-R$ curve and the nuclear EoS. Unfortunately NS radii are
extremely difficult to measure precisely and up until now the most accurate
measurements managed to achieve an accuracy of $\sim 10\%$ meaning $\sim
1-1.5$ km. The masses of NSs in binary systems, however, can be measured with
remarkable precision, and in fact the most robust constraints originate from
the measurements of the most massive NSs. The 2 $M_{\odot}$ constraint gives a
powerful lower limit on the stiffness of the EoS.

Gravitational waves (GW) provide an independent way to observe NSs through
their inspiral and merger with another compact object. In the final stages of
the inspiral NSs are distorted through tidal interactions and this shifts the
phase of the emitted GW signal. One of the measurable parameters of NS–NS
mergers is the dimensionless quadrupole tidal deformability parameter
$\Lambda=\lambda/M^{5}$, with the $\lambda$ tidal deformability being related
to the $l=2$ tidal Love number:

$k_{2}=\frac{3}{2}\lambda R^{-5}.$ (42)

Allowing small perturbations on the spherical metric, one can show that
$k_{2}$ can be expressed the following way Hinderer (2008); Damour and Nagar
(2009):

$\displaystyle k_{2}$
$\displaystyle=\frac{8}{5}(1-2\beta)^{2}\beta^{5}[2\beta(y_{R}-1)-y_{R}+2]$
$\displaystyle\times\\{2\beta[4(y_{R}+1)\beta^{4}+(6y_{R}-4)\beta^{3}+(26-22y_{R})\beta^{2}$
$\displaystyle+3(5y_{R}-8)\beta-3y_{R}+6]+3(1-2\beta)^{2}$
$\displaystyle\times[2\beta(y_{R}-1)-y_{R}+2]\ln\left(1-2\beta\right)\\}^{-1},$
(43)

where $\beta=M/R$ is the compactness parameter of the NS and
$y_{R}=y(R)=[rH^{\prime}(r)/H(r)]_{r=R}$ with $H(r)$ being a function related
to the quadrupole metric perturbation. $y_{R}$ can be obtained by solving the
following first-order differential equation:

$\displaystyle ry^{\prime}(r)$ $\displaystyle+y(r)^{2}+r^{2}Q(r)$
$\displaystyle+y(r)e^{\lambda(r)}\left[1+4\pi
r^{2}(p(r)-\varepsilon(r))\right]=0,$ (44)

where

$\displaystyle Q(r)=4\pi
e^{\lambda(r)}\left(5\varepsilon(r)+9p(r)+\frac{\varepsilon(r)+p(r)}{c_{s}^{2}(r)}\right)$
$\displaystyle-6\frac{e^{\lambda(r)}}{r^{2}}-(\nu^{\prime}(r))^{2}.$ (45)

Here $c_{s}^{2}=\mathrm{d}p/\mathrm{d}\varepsilon$ is the sound speed squared,
while $\nu^{\prime}(r)$ is given by

$\nu^{\prime}(r)=\dfrac{2[m(r)+4\pi r^{3}p(r)]}{r^{2}-2m(r)r}.$ (46)

In case there is a discontinuity in the EoS (e.g. due to a first-order phase
transition), owing to a $\Delta\varepsilon$ jump in the energy density at
constant pressure, the term in Eq. (45) containing $1/c_{s}^{2}$ will diverge.
One then needs to add an extra term when solving for the $y$ variable at the
$r_{d}$ location of the discontinuity Postnikov  _et al._ (2010); Takátsy and
Kovács (2020):

$y(r_{d}^{+})-y(r_{d}^{-})=-\frac{4\pi
r_{d}^{3}\Delta\varepsilon}{m(r_{d})+4\pi r_{d}^{3}p(r_{d})}.$ (47)

The analysis of GW170817 inferred a value of $\Lambda<800$ for $1.4$
$M_{\odot}$ NSs in the low-spin limit Abbott  _et al._ (2017). A thorough
investigation of this constraint performed by Annala et al. using a generic
family of EoS’s found an upper radius limit of $13.6$ km for $1.4$ $M_{\odot}$
NSs Annala  _et al._ (2018). A lower radius constraint was inferred by
Bauswein et al. from the absence of prompt collapse during this event Bauswein
_et al._ (2017), while an upper mass limit of $2.16^{+0.17}_{-0.15}$
$M_{\odot}$ was proposed by Rezzolla et al. using a quasi-universal relation
between the maximum mass of static and uniformly rotating NSs Rezzolla  _et
al._ (2018).

##  IV Results

Solving simultaneously the system of six equations (Eqs. (12)-(17)) for some
parameter set we get the $\mu_{q}$ quark chemical potential dependence of all
the condensates. In Fig. 2 the $\phi_{N/S}$ condensates are shown as a
function of $\mu_{q}$ for different values of the $g_{V}$ vector coupling. For
lower $g_{V}$ values an unstable part is present, causing a first order phase
transition, which disappears for larger vector couplings.

Figure 2: The chemical potential dependence of the $\phi_{N}$ (blue) and
$\phi_{S}$ (green) scalar condensates. The parameter set with $m_{\sigma}=290$
MeV was used for each curve. Increasingly bright tones correspond to
increasing vector couplings of $0$ (solid), $2$ (dashed), $4$ (dashed-dotted)
and $6$ (dotted).

If the phase transition is of first order at $T=0$ as a function of
$\mu_{B}\;(=3\mu_{q})$, then there is a critical end point (CEP) somewhere on
the chiral phase boundary on the $T-\mu_{B}$ plane. Since, as it is known and
will also be seen here, nonzero vector coupling is needed in order to fulfill
the two solar mass criteria for the $M(R)$ curves, it seems that if
$g_{V}\gtrsim 3.1$ (for $m_{\sigma}=290$ MeV) in this framework the existence
of a CEP is unlikely. This is based on the observation that if in a linear
sigma model the phase transition is not of first order as a function of
$\mu_{B}$ at $T=0$, then there is no CEP on the $T-\mu_{B}$ plane (see e.g.
(Kovacs and Szep, 2007))

###  IV.1 Equation of state with different vector couplings and compact star
properties

Along the solution (of Eqs. (12)-(17)) the pressure $p$ (Eq. (9)) and its
derivatives can be calculated, from which one can construct the equation of
state (EoS). The EoS, which is the pressure as the function of the energy
density $\varepsilon$, can be seen in Fig. 3 for the SFHo hadronic model, for
the pure quark and for the hybrid stars for two different values of $g_{V}$.

Figure 3: Top: the EoS ($p$ as a function of $\varepsilon$) of the SFHo model
(yellow solid line), as well as for quark (solid) and hybrid (dashed) stars
using the eLSM with the parameter set corresponding to $m_{\sigma}=290$ MeV
and vector couplings $g_{V}=0$ (green) and $g_{V}=5$ (blue). The inset
contains the same curves for the $n_{B}(\mu_{B})$ dependence, while the
circles correspond to the central conditions inside the maximum mass neutron
stars. For the hybrid EoS’s the energy density interpolation was used with
$\bar{n}_{B}=3.5n_{0}$ and $\Gamma=1.5n_{0}$. Bottom: the speed of sound
squared for the same EoS’s.

For the case $g_{V}=0$ the hybrid EoS smoothly connects the hadronic and quark
phases. However, for the case $g_{V}=5$, even though the quark EoS is softer
than the hadronic one, an intermediate region appears where the hybrid EoS
becomes stiffer than both the quark and the hadronic ones. This results in an
increase of the maximum compact star mass (see Fig. 4). We note that this
behaviour is not the consequence of the specific choice for the concatenation
method, since it appears for the $p(\mu)$ interpolation as well (see the
comparison of the concatenation methods in IV.3). The same results were
reported already in Masuda  _et al._ (2013), while other studies investigating
a hadron–quark continuity also found similar results Kojo (2021).

As described in Sec. III.3 the EoS’s are needed to calculate the $M-R$ curves
and $\Lambda$ tidal deformability parameters for a compact star. In Fig. 4 the
$M-R$ curves can be seen for the EoS’s of the SFHo model, of the pure quark
model and of hybrid stars with various $g_{V}$ vector couplings.

Figure 4: $M-R$ relations for the SFHo model (yellow) and for different quark
(blue) and hybrid (green) EoS’s. Increasingly bright tones correspond to
increasing vector couplings of $0$ (solid), $2$ (dashed), $4$ (dashed-dotted)
and $6$ (dotted). The different shaded areas correspond to the lower and upper
radius constraints of Bauswein et al. Bauswein  _et al._ (2017) and Annala et
al. Annala  _et al._ (2018), respectively, different credibility limits of the
upper mass constraint of Rezzolla et al. Rezzolla  _et al._ (2018), and the
mass of PSR J0348+0432 Antoniadis  _et al._ (2013). The yellow shaded area
corresponds to the region excluded by causality.

We see that while the low-density (low-mass) behaviour of the relations for
hybrid stars is determined by the hadronic EoS, the maximum mass region is
characterized by the quark EoS. For $g_{V}=0$, where the hybrid EoS smoothly
interpolates between the two phases (see Fig. 3), the quark and hybrid models
describe maximum mass compact stars with approximately the same masses. For
larger vector couplings, on the other hand, the maximum mass is greatly
increased due to the intermediate stiffening of the hybrid EoS. The radii of
quark stars with larger vector couplings are also greatly increased owing to
the absence of the first order phase transition and an incorrect low-density
behaviour – due to the lack of proper degrees of freedom, i.e. the baryons, at
low densities.

In Fig. 5 the tidal deformability parameter versus the compact star mass can
be seen for the same EoS’s as in the case of Fig. 4.

Figure 5: The same curves as in Fig. 4, but for the tidal deformabilities
$\Lambda$ as function of the compact star masses $M$. The red bar corresponds
to the region excluded by the constraint $\Lambda(1.4M_{\odot})<800$ deduced
from the measurement of GW170817 Abbott  _et al._ (2017). Figure 6: The
maximum mass of stable neutron star sequences as a function of the vector
coupling $g_{V}$, for different sigma masses. The parameters of the
concatenation are $\bar{n}=3.5n_{0}$ and $\Gamma=1.5n_{0}$. For higher sigma
masses this results in EoS’s that violate either thermodynamic stability or
causality.

Note that the excluded region in the figure corresponds to the upper radius
constraint of Annala et al. Annala  _et al._ (2018) on the $M-R$ plot, and it
excludes the same models. Currently the measurements of the tidal
deformabilities can not provide stricter constraint on the EoS’s than mass and
radius measurements, however, both are expected to be improved with upcoming
measurements. From the two figures we can conclude that considering hybrid
stars the vector coupling should be in the following range to meet all the
requirements,

$2.6<g_{V}<4.3.$ (48)

This range is valid in case of $m_{\sigma}=290$ MeV and can be directly seen
in Fig. 6, where the maximum mass of stable neutron stars is shown as a
function of $g_{V}$ for different $m_{\sigma}$. The lower value of the range
is given by the intersection point of the curve with bottom of the green band
(titled PSR J0348+0432), while the upper one is from the intersection point of
the curve with the top of the light rose-colored band (titled Rezzola et al.)

###  IV.2 Dependence on the sigma meson mass

Beside $g_{V}$ there is another very important parameter, the mass of the
$f_{0}$ or $\sigma$ meson, that substantially changes the behavior of the
solution to the field equations and consequently the behavior of the EoS
itself. Its very important role comes from the fact that the $\sigma_{N}$ non-
strange and the $\sigma_{S}$ strange scalar fields acquire nonzero condensates
– which are the $\phi_{N}$ and $\phi_{S}$ – in the meson sector. However, it
is worth noting that other condensates, like pion or kaon condensates, are
also considered in the literature, see e.g. Herpay and Kovacs (2008); Thorsson
_et al._ (1994), but that is out of the scope of the current investigation. In
Fig. 7 one can see the $\phi_{N/S}$ condensates as a function of the $\mu_{q}$
quark chemical potential for different values of $m_{\sigma}$.

Figure 7: The chemical potential dependence of the $\phi_{N}$ (blue solid) and
$\phi_{S}$ (green dotted) scalar condensates. Increasingly bright tones
correspond to parameter sets with increasing sigma meson masses with $g_{V}=3$
for each curve. For larger $m_{\sigma}$ masses the first order phase
transition (indicated by the slight back-bending of the $\phi(\mu_{q})$
curves) disappears, and for $m_{\sigma}=700$ MeV the vacuum expectation value
of $\phi_{N}$ becomes larger than the value of $\phi_{S}$.

As it can be seen, the phase transition is first order only for very low
values of $m_{\sigma}$ , which is indicated by the slight back-bending of the
$m_{\sigma}=290$ Mev curve for $\phi_{N/S}$. If $m_{\sigma}\gtrsim 300$ MeV
the transition becomes crossover and the pseudocritical chemical potential –
defined by the inflection point of the $\phi_{N}(\mu_{q})$ curve – shifts
toward larger values.

The effect of changing $m_{\sigma}$ for three different $g_{V}$ values can be
seen in Fig. 8.

Figure 8: $M-R$ relations of hybrid stars for different vector couplings and
sigma meson masses, using the SFHo hadronic EoS and the energy density
interpolation with $\bar{n}_{B}=3.5n_{0}$ and $\Gamma=2n_{0}$. Increasingly
bright tones correspond to the five parameter sets with increasing
$m_{\sigma}$ (the same as in Fig. 7), while the different types of lines
correspond to different vector couplings of $1$ (solid), $3$ (dashed) and $5$
(dashed-dotted). The curves for $m_{\sigma}=700$ MeV with $g_{V}=1$ and $3$
are omitted, since the hybrid EoS’s produced with these parameters are not
stable.

It can be observed that generally the larger $m_{\sigma}$ and $g_{V}$ are, the
larger the compact star masses and radii are. However, the value of
$m_{\sigma}$ moderately modifies the slope of the $M-R$ curve in the mid-mass
region. For a given $g_{V}$ value the change in the value of the maximum mass
is about $15-17\%$ for the total range of the $\sigma$ mass studied here. On
the other hand, if for a fixed $m_{\sigma}$ we change $g_{V}$ from 1 to 5 we
get an approximately $40\%$ change in the maximum mass.

###  IV.3 Dependence on hadronic EoS and concatenation method and the role of
the bag constant

We also investigated the effects of changing the hadronic EoS and the method
of concatenation. In Fig. 9 the $M-R$ curves are shown for different
$\bar{n}_{B}$ and $\Gamma$ values with the SFHo and DD2 hadronic models.

Figure 9: $M-R$ relations of hybrid stars with the SFHo (green) and the DD2
(blue) hadronic EoS’s and the energy density interpolation method with
different values for $\bar{n}_{B}$ (solid) and for $\Gamma$ (dashed). Brighter
tones correspond to larger values in both cases. $g_{V}=3$ for each curve. For
the DD2 case only the $M-R$ curves of stable EoS’s are included.

Even though the radii of hybrid stars are dependent on the choice of the phase
transition parameters and the hadronic EoS, the maximum mass allowed by the
different models is encompassed within a small range.

As it is discussed in Sec. III.2 two different kinds of interpolation were
used in our investigations, one that interpolates between the energy densities
as a function of the baryon density and another one that uses the pressure as
a function of the baryochemical potential. The comparison of the two
approaches can be seen in Fig. 10 for given values of $g_{V}$ and
$m_{\sigma}$.

Figure 10: Top: hybrid EoS’s produced by the two concatenation methods using
the SFHo model (yellow solid line) and the eLSM with the parameter set
corresponding to $m_{\sigma}=290$ MeV and $g_{V}=3$ (green solid line). The
black crosses correspond to $n_{L}\equiv n_{B,H}(\mu_{BL})$ and $n_{U}\equiv
n_{B,Q}(\mu_{BU})$, while the circles correspond to the central conditions
inside the maximum mass neutron stars. The parameters of the two types of
interpolation methods were chosen so that both arrive to the hadronic and
quark EoS’s at approximately the same densities. Bottom: the speed of sound
squared for the same EoS’s.

The $\varepsilon(n)$ (dashed) and the $p(\mu)$ (dotted) interpolations both
show similar features with stiffenings in the intermediate-density region,
although this starts at lower densities for the $p(\mu)$ case. Even if the two
kind of interpolation methods use very different functions – a polynomial and
a tangent hyperbolic – we see similar behavior in the intermediate region. It
is also worth noting that, even though the EoS’s look similar, more pronounced
difference in the speed of sound are apparent for the two different
interpolation methods.

Figure 11: $M-R$ relations of hybrid stars for different concatenations as
well as for the SFHo (yellow) and DD2 (blue) hadronic EoS’s. The parameters
for the $\varepsilon(n)$ (dashed) and $p(\mu)$ (dotted) concatenations are the
same as in Fig. 10.

In Fig. 11 the effect of the different interpolation methods are shown for the
two types of hadronic EoS’s, the SFHo and the DD2. In both cases there is a
slight change in the values of the maximal mass that happens oppositely for
the two hadronic EoS’s. In the middle mass range there is also a slight change
in the radii, which act in the same way for the two hadronic curves by
increasing the radius, which shows a smaller effect for the case of DD2 . With
the current observations neither of the scenarios depicted here can be
excluded.

Finally, we have also investigated the role of the bag constant $B$ in the
current framework. We have taken different values for $B^{1/4}$ from $0$ to
$110$ MeV (similarly to Zacchi and Schaffner-Bielich (2019)). The bag constant
represents a vacuum contribution and it is simply an additional constant for
the pressure and the energy density.

Figure 12: $M-R$ relations in case of a nonzero bag constant for the
$\varepsilon(n)$ (top) and the $p(\mu)$ (bottom) interpolation methods. In the
$p(\mu)$ case the interpolation limits were chosen so that larger values for
$B$ could be accommodated as well without producing unstable EoS’s.

Many previous studies have also investigated the effect of changing the bag
constant only using a Maxwell construction for modeling the phase transition
(e.g. Zacchi and Schaffner-Bielich (2019); Cierniak and Blaschke (2020)). Some
of these studies found that the M–R curves cross each other in the vicinity of
a single point, the so-called ”special point” Yudin  _et al._ (2014); Cierniak
and Blaschke (2020, 2021). In our case, due to the differing nature of our
phase transition construction, we do not expect such a special point to
appear.

In Fig. 12 the effect of a nonzero $B$ term can be seen for the two kind of
interpolation methods. On the upper figure the $M-R$ relations for the
$\varepsilon(n)$ concatenation are shown for two $g_{V}$ and four $B^{1/4}$
values. As $B^{1/4}$ increases, the low mass neutron stars will develop small
radii – as if they were pure quark stars – due to the incorrect low-density
behaviour. This behavior can not be observed in case of the $p(\mu_{B})$
concatenation in the lower figure. Beside this change the effect of the bag
constant is not so dramatic in our case.

##  V Conclusions

We investigated hybrid star properties using the concept of hadron-quark
crossover, in which we took a hadronic EoS together with a quark one and
connected them with some smooth interpolation method in an intermediate region
where both models are inaccurate. For the hadronic EoS we have used two
different relativistic mean field models, the SFHo and the DD2, while for the
quark part an (axial)vector meson extended linear sigma model was used with
additional constituent quarks. The latter model reproduces the meson spectrum
in vacuum well and also agrees with various lattice results at finite
temperature and zero density.

We argued that the changes in the values of the parameters of the Lagrangian
can have a significant effect on the properties of the EoS and consequently on
the properties of hybrid stars themselves. For this very reason, we
investigated the asymptotic behavior of our system of equations as a function
of the $\mu_{q}$ quark chemical potential and found a condition among a set of
parameters of our Lagrangian that should be fulfilled in order to acquire
vanishing chiral condensates for very large $\mu_{q}$, as it is expected
physically.

The interpolation method was also altered and its effect on the $M-R$ curves
was shown to be moderate in the mid-mass range, while the maximum hybrid star
mass remained approximately constant. Moreover the parameters of the
interpolation – like position and width –, the $g_{V}$ vector coupling and the
$m_{\sigma}$ sigma meson mass were also changed in some range and their effect
were analyzed in detail. We found that for a given value of the sigma meson
mass there is a relatively small acceptable range in $g_{V}$, imposing
constraints from astrophysical observations. The consequences of the
constraints on the tidal deformabilities for different EoS’s and $g_{V}$
vector couplings was also discussed. Finally we found in connection with the
bag constant that its introduction does not affect the $M-R$ curves
significantly if its value is not too high for a given $g_{V}$ value.

In conclusion, all the current astrophysical constraints from observation are
compatible with the investigated phenomenological model if the relevant
parameters – like the vector coupling or the sigma meson masses – are within a
certain range and the parameters also satisfy a condition that comes from the
investigation of the asymptotic behavior of the field equations. Turning the
argument around, one sees that data from neutron stars considerably constrain
the parameters of the chiral quark-meson model for bulk quark matter.

It is worth to note that the $g_{V}$ vector coupling can also be determined
from the parameterization procedure if one uses one-loop order curvature
masses for the vector and axial vector meson masses Kovács  _et al._ (2021).
In Kovács  _et al._ (2021) its value was found to be around $5$ for a sigma
mass around $300$ MeV, which is a little higher than the upper bound of the
acceptable range found here. To resolve this tension further investigation is
needed.

###### Acknowledgements.

J. T. and P. K. acknowledge support by the National Research, Development and
Innovation (NRDI) fund of Hungary, financed under the FK_19 funding scheme,
Project No. FK 131982. J. T., P. K. and Gy. W. acknowledge the support of the
NRDI fund K138277. P. K. also acknowledges support by the János Bolyai
Research Scholarship of the Hungarian Academy of Sciences. J. T. was supported
by the ÚNKP-21-3 New National Excellence Program of the Ministry for
Innovation and Technology from the source of NRDI fund. J. S.-B. acknowledges
support by the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation) – project number 315477589 – TRR 211.

##  Appendix A Grand potential in the hybrid approximation

The grand potential in the current (hybrid) approximation reads

$\displaystyle\Omega_{\textrm{tot}}$
$\displaystyle=\frac{1}{2}m_{0}^{2}\left(\phi_{N}^{2}+\phi_{S}^{2}\right)+\frac{\lambda_{1}}{4}\left(\phi_{N}^{2}+\phi_{S}^{2}\right)^{2}+\frac{\lambda_{2}}{8}\left(\phi_{N}^{4}+2\phi_{S}^{4}\right)-\frac{c_{1}}{2\sqrt{2}}\phi_{N}^{2}\phi_{S}-h_{N}\phi_{N}-h_{S}\phi_{S}{}$
$\displaystyle-\frac{1}{2}m_{\rho}^{2}\left(v_{\omega}^{2}+v_{\rho}^{2}\right)-\frac{1}{2}m_{\Phi}^{2}v_{\Phi}^{2}-\frac{3}{8\pi^{2}}\left[m_{u}^{4}\log\left(\frac{m_{u}}{M_{0}}\right)+m_{d}^{4}\log\left(\frac{m_{d}}{M_{0}}\right)+m_{s}^{4}\log\left(\frac{m_{s}}{M_{0}}\right)\right]{}$
$\displaystyle-\frac{3}{8\pi^{2}}\sum_{f\in(u,d,s)}m_{f}^{4}\left[\log\left(\gamma_{f}+\sqrt{\gamma_{f}^{2}-1}\right)+\frac{1}{3}\gamma_{f}\sqrt{\gamma_{f}^{2}-1}\left(2\gamma_{f}^{2}-5\right)\right]{}$
$\displaystyle-\frac{m_{e}^{4}}{8\pi^{2}}\left[\log\left(\gamma_{e}+\sqrt{\gamma_{e}^{2}-1}\right)+\frac{1}{3}\gamma_{e}\sqrt{\gamma_{e}^{2}-1}\left(2\gamma_{e}^{2}-5\right)\right]$
(49)

where the vector masses are given by

$\displaystyle m_{\rho}^{2}$
$\displaystyle=m_{\omega}^{2}=m_{1}^{2}+\frac{1}{2}h_{1}\left(\phi_{N}^{2}+\phi_{S}^{2}\right)+\frac{1}{2}\left(h_{2}+h_{3}\right)\phi_{N}^{2},$
(50) $\displaystyle m_{\Phi}^{2}$
$\displaystyle=m_{1}^{2}+\frac{1}{2}h_{1}\left(\phi_{N}^{2}+\phi_{S}^{2}\right)+\left(h_{2}+h_{3}\right)\phi_{S}^{2}+2\delta_{S},$
(51)

and

$\gamma_{f}=\frac{|\tilde{\mu}_{f}|}{m_{f}},\,f\in(u,d,s),\quad\gamma_{e}=\frac{|\mu_{e}|}{m_{e}}$
(52)

have also been introduced.

##  Appendix B Parameter sets

In Table 3 we present all the parameter sets that were used for the different
sigma masses.

Table 3: Parameter values for different sigma meson masses Parameter |  $m_{\sigma}=400$ MeV |  $m_{\sigma}=500$ MeV |  $m_{\sigma}=600$ MeV |  $m_{\sigma}=700$ MeV
---|---|---|---|---
$\phi_{N}$ [GeV] | 0.1210 | 0.1218 | 0.1231 | 0.1212
$\phi_{S}$ [GeV] | 0.1367 | 0.1315 | 0.1255 | 0.1185
$m_{0}^{2}$ [GeV2] | -3.5902${\scriptscriptstyle E\negthinspace-2}$ | -8.6251${\scriptscriptstyle E\negthinspace-2}$ | -0.1780 | -0.2777
$m_{1}^{2}$ [GeV2] | 0.5600 | 0.5600 | 0.5600 | 0.5600
$\lambda_{1}$ | 0.1757 | 0.9634 | 1.3876 | 2.3572
$\lambda_{2}$ | 23.0159 | 25.9249 | 29.7608 | 34.9895
$c_{1}$ [GeV] | 1.5884 | 1.6273 | 1.6407 | 1.7110
$\delta_{S}$ [GeV2] | 0.2371 | 0.2323 | 0.2305 | 0.2351
$g_{1}$ | 5.3250 | 5.4668 | 5.6116 | 5.8237
$g_{2}$ | -2.5367 | -2.1499 | -1.6939 | -1.4069
$h_{1}$ | 5.3887 | 3.9904 | 0.6246 | -1.1022
$h_{2}$ | -10.3257 | -7.4554 | -1.9978 | 1.4249
$h_{3}$ | 1.8646 | 1.4698 | 1.0551 | 0.2283
$g_{F}$ | 5.0050 | 5.0396 | 5.3835 | 5.7157
$M_{0}$ [GeV] | 0.4522 | 0.4522 | 0.5579 | 0.6357

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2.0pt{\rm d}^{3}q}{(2\pi)^{9}}\frac{B_{\rm eq}({\vec{p}},{\vec{q}}\hskip
1.0pt)B_{\rm eq}(-{\vec{p}},-{\vec{q}}\hskip
1.0pt)}{P(k)P(q)P(p)}\,(2\pi)^{3}\delta_{\rm
D}({\vec{p}}+{\vec{q}}-{\vec{k}})\,,$ (A.28)

which agrees precisely with the bispectrum result (A.26).

A similar comparison applies for $b_{2}$, where the associated bispectrum is
given by (4.2). Substituting this into (A.26), the bispectrum Fisher matrix
becomes

$\displaystyle F^{\rm(B)}_{2,2}$ $\displaystyle=\frac{V}{6}\int\frac{{\rm
d}^{3}k_{1}{\rm d}^{3}k_{2}{\rm
d}^{3}k_{3}}{(2\pi)^{9}}\frac{4\left(P(k_{1})P(k_{2})+P(k_{1})P(k_{3})+P(k_{2})P(k_{3})\right)^{2}}{P(k_{1})P(k_{2})P(k_{3})}(2\pi)^{3}\delta_{\rm
D}({\vec{k}}_{1}+{\vec{k}}_{2}+{\vec{k}}_{3})$ $\displaystyle=V\int\frac{{\rm
d}^{3}k_{1}{\rm
d}^{3}k_{2}}{(2\pi)^{9}}\left(\frac{2P(k_{1})P(k_{2})}{P(k_{3})}+4P(k_{1})\right).$
(A.29)

For comparison, evaluating the correlators in the map-level Fisher matrix
(A.22), we get

$\displaystyle F_{2,2}^{\rm(M)}$ $\displaystyle=V\int\frac{{\rm
d}^{3}k}{(2\pi)^{3}}\frac{[({\delta^{\rm
obs}_{g}})^{2}]({\vec{k}})[({\delta^{\rm
obs}_{g}})^{2}](-{\vec{k}})+4\sigma^{2}(2[({\delta^{\rm
obs}_{g}})]({\vec{k}})[({\delta^{\rm obs}_{g}})](-{\vec{k}})-P(k))}{P(k)}$
$\displaystyle=V\int\frac{{\rm d}^{3}k\hskip 2.0pt{\rm
d}^{3}p}{(2\pi)^{6}}\left(\frac{2P(p)P(|{\vec{k}}-{\vec{p}}\hskip
1.0pt|)}{P(k)}+4P(p)\right),$ (A.30)

where we used $\sigma^{2}=\int{\rm d}^{3}p\,P(p)/(2\pi)^{3}$. After relabeling
the integration variables, it is easy to see that the map-level and bispectrum
Fisher matrix elements are indeed the same.

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