# Constraints for the X17 boson from compacts objects observations

A. Kanakis-Pegios<EMAIL_ADDRESS>V. Petousis<EMAIL_ADDRESS>M.
Veselský<EMAIL_ADDRESS>Jozef Leja<EMAIL_ADDRESS>Ch. C.
Moustakidis<EMAIL_ADDRESS>
###### Abstract

We investigate the hypothetical X17 boson on neutron stars and Quark Stars
(QSs) using various hadronic Equation of States (EoSs) with phenomenological
or microscopic origin. Our aim is to set realistic constraints on its coupling
constant and the mass scaling, with respect to causality and various possible
upper mass limits and the dimensionless tidal deformability $\Lambda_{1.4}$.
In particular, we pay special attention on two main phenomenological
parameters of the X17, the one is related to the coupling constant
$\mathrm{g}$ that it has with hadrons or quarks and the other with the in-
medium effects through the regulator $\mathrm{C}$. Both are very crucial
concerning the contribution on the total energy density and pressure. In the
case of considering the X17 as a carrier of nuclear force in Relativistic Mean
Field (RMF) theory, an admixture into vector boson segment was constrained by
20% and 30%. In our investigation, we came to the general conclusion that the
effect of the hypothetical X17 both on neutron and QSs constrained mainly by
the causality limit, which is a specific property of each EoS. Moreover, it
depends on the interplay between the main two parameters that is the
interaction coupling $\mathrm{g}$ and the in-medium effects regulator
$\mathrm{C}$. These effects are more pronounced in the case of QSs concerning
all the bulk properties.

###### keywords:

X17 Boson; Neutron star; Quark Stars; Equation of State

††journal: Physics Letters B

[lebel1]organization=Department of Theoretical Physics, Aristotle University
of Thessaloniki,city=Thessaloniki, postcode=54124, country=Greece
[lebel2]organization=Institute of Experimental and Applied Physics, Czech
Technical University,city=Prague, postcode=110 00, country=Czechia

[lebel3]organization=Faculty of Mechanical Engineering - Slovak University of
Technology in Bratislava,city=Bratislava, postcode=812 31, country=Slovakia

##  1 Introduction

In 2016 an article of Krasznahorkay et al. appeared [1], where an anomaly in
angular correlation of $e^{-}e^{+}$ decay of the 1+ excited level of 8Be
nucleus at ${18.15~{}}$MeV reported, and specifically observed enhancement at
folding angles. Since first report till today, Krasznahorkay and his group,
reported in addition the same anomaly - observed in the angular correlation of
the $e^{-}e^{+}$ emission - in the excited states of 4He and 12C [2, 3, 4].
The reported anomalies at a folding angles was interpreted as a signature of a
new neutral boson with a mass of about ${m_{X}=17~{}}$MeV.

These reported observations placed the hypothetical X17 boson as a dark matter
candidate and in that spirit since then, several theoretical works pursued
this claim [5, 6]. However, an explanation relating this particle to the QCD
vacuum was also proposed [7], in the conjecture that the 17 MeV particle could
mediate the nucleon-nucleon interactions at large distances in an unbound
cluster configuration. Since the assumption that the 17 MeV boson is the only
carrier of nuclear interactions is somewhat extreme, we investigated the
possible influence of the hypothetical 17 MeV boson on nuclear matter and its
influence on the structure of the compact astrophysical objects like neutron
stars [8].

A further investigation [9], explores the hypothetical 17 MeV boson in the
frame of the Relativistic Mean Field (RMF) theory, constructing a universal
Equation of State (EoS) that satisfy all of the well known experimental
constraints, from finite nuclei, heavy ion collisions all the way to the
neutron stars, allowing a reproduction in masses from ${\cong
1.4\;\mathrm{M_{\odot}}}$ up to ${\cong 2.5\;\mathrm{M_{\odot}}}$. The values
of the radius show an agreement with the recent measurement by NICER [10, 11].
Also the value of the maximum mass is in a good agreement with the recently
reported mass of pulsar $2.35\;\mathrm{M_{\odot}}$ [12] and potentially also
with the mass of the secondary component of the gravitational wave event
GW190814 [13]. A previous investigation [14], following an alternative
direction, tried to set the ratio ${g{{}^{2}}}$/${\mu{{}^{2}}}$ for a Weakly
Interactive Light Boson (WILB) inside the neutron star. In this investigation,
the range of the ratio ${g{{}^{2}}}$/${\mu{{}^{2}}}$ was estimated to be less
than 2 GeV-2. A value that looks to be suitable for the neutron star
environment talking in account also the experimental constraints for
consistency of symmetry and binding energy. Also using the fact that the
presence of WILB does not effect the crust properties of neutron star matter
they obtaining a quite restricted constraint of the WILB characteristic scale.
They also show that the in-medium modification effect indeed cannot be
neglected and for their investigation they assumed that the WILB mass follows
the same scaling as the one of Brown–Rho[15].

Another similar research [16], provides a ratio ${g{{}^{2}}}$/${\mu{{}^{2}}}$
of a WILB to be less than 50 GeV-2 and above than 25 GeV-2. They also
investigated an upper value of ${g{{}^{2}}}$/${\mu{{}^{2}}}$ = 100 GeV-2 which
deviates a lot from the must fulfilled restrictions concerning the binding
energy and the symmetry energy, even though reproducing an acceptable mass
according to the GW190814 observation [13] close to ${\cong
2.5\;\mathrm{M_{\odot}}}$, which is comparable with the astrophysical
observation. In the case of Quark Star (QS) a research [17], considering the
non-Newtonian gravity effects, for current quark masses of $m_{u}$ = 2.16 MeV,
$m_{d}$ = 4.67 MeV, and $m_{s}$ = 93 MeV, they estimate a range of
${g{{}^{2}}}$/${\mu{{}^{2}}}$ between 4.58 GeV-2 and 9.32 GeV-2 reaching a
maximum mass ${\cong 2.4\;\mathrm{M_{\odot}}}$.

In all the aforementioned investigations, the ratio
${g{{}^{2}}}$/${\mu{{}^{2}}}$ for a WILB and only the ratio, was thoroughly
investigated. No attempt was made to estimate separately the coupling
$\mathrm{g}$ from the boson mass $\mu$ for the WILB, and that was reasonable,
because the mass of a possible candidate boson was not known or guessed at
that time.

In general, concerning the WILB case scenario inside a neutron star and it’s
connection with the hypothetical U boson, which in our case could be also
represented by the hypothetical X17 boson, has been investigated also in the
past [18, 19, 20, 21, 22, 23, 24, 25].

In our present work, having as a candidate a 17 MeV boson (hereafter X17), we
investigating the effects of the non-Newtonian gravity together with a series
of different models - hadronic EoSs with phenomenological and realistic or
microscopic origin, from the RMF to the Momentum Dependent Interaction (MDI)
model and QSs. Our main motivation is to set realistic constraints on the X17
coupling constant $\mathrm{g}$ and the scaling of its mass, which is affected
by the changes in the baryon density well above its saturation value inside
the neutron star and QS. All of our research is compared with astrophysical
observations, reported from LIGO-VIRGO and pulsar observations.

The paper is organized as follows: in Section 2, we briefly describe the non-
Newtonian gravity model while in Section 3, we present the nuclear models in
the context of which we are investigating our constraints for the X17 boson.
In Section 4, we present the basic formalism of the tidal deformability while
in Section 5, we display and discuss the results of the present study. In
Section 6, we finalize with our concluding remarks.

##  2 The non-Newtonian gravity model

The deviation of the Newton’s gravitational potential, known as non-Newtonian
gravity [26], usually parameterized in the form:

$V(r)=-\frac{Gm_{1}m_{2}}{r}\left(1+\alpha_{\tiny{G}}e^{-r/\lambda}\right)=V_{N}(r)+V_{Y}(r)$
(1)

where $V_{N}(r)$ is the Newtonian potential, $V_{Y}(r)$ is the Yukawa
correction, $G=6.67\times 10^{-11}$ N m2/kg2, is the universal gravitational
constant, $\alpha_{\rm G}$ is the dimensionless coupling constant of the
Yukawa force and $\lambda$ represents the range of the Yukawa force mediated
by the exchange of a boson with mass $\mu$. The above quantities related
according to the following relations:

$\alpha_{\rm G}=\pm\frac{{\rm g}^{2}\hbar c}{4\pi
Gm_{b}^{2}},\qquad\lambda=\frac{\hbar}{\mu c}$ (2)

where the $\pm$ sign refers to scalar(+) and vector(-) boson, $\mathrm{g}$ is
the boson-baryon coupling constant and $m_{b}$ is the baryon mass.

##  3 The nuclear models

###  3.1 The Relativistic Mean Field (RMF) model

In the case of the RMF theory using the extended Dirac-Hartree approximation,
the energy density and pressure of neutron matter is given by the following
expressions [27]:

$\displaystyle{\cal E}$ $\displaystyle=$ $\displaystyle\frac{(\hbar
c)^{3}g_{v}^{2}}{2(m_{v}c^{2})^{2}}n_{b}^{2}+\frac{(\hbar
c)^{3}(\frac{g_{\rho}}{2}){{}^{2}}}{2(m_{\rho}c^{2})^{2}}\rho_{I}^{2}+\frac{(m_{s}c^{2})^{2}}{2g_{s}^{2}(\hbar
c)^{3}}(m_{b}c^{2}-m_{b}^{*}c^{2})^{2}$ (3) $\displaystyle+$
$\displaystyle\frac{\kappa}{6g_{s}^{3}}(m_{b}c^{2}-m_{b}^{*}c^{2})^{3}+\frac{\lambda}{24g_{s}^{4}}(m_{b}c^{2}-m_{b}^{*}c^{2})^{4}$
$\displaystyle+$
$\displaystyle\sum_{i=n,p}\frac{\gamma}{(2\pi)^{3}}\int_{0}^{k_{F}i}4\pi
k^{2}\sqrt{(\hbar ck)^{2}+(m_{i}^{*}c^{2})^{2}}dk$ $\displaystyle P$
$\displaystyle=$ $\displaystyle\frac{(\hbar
c)^{3}g_{v}^{2}}{2(m_{v}c^{2})^{2}}n_{b}^{2}+\frac{(\hbar
c)^{3}(\frac{g_{\rho}}{2}){{}^{2}}}{2(m_{\rho}c^{2})^{2}}\rho_{I}^{2}-\frac{(m_{s}c^{2})^{2}}{2g_{s}^{2}(\hbar
c)^{3}}(m_{b}c^{2}-m_{b}^{*}c^{2})^{2}$ (4) $\displaystyle+$
$\displaystyle\frac{\kappa}{6g_{s}^{3}}(m_{b}c^{2}-m_{b}^{*}c^{2})^{3}+\frac{\lambda}{24g_{s}^{4}}(m_{b}c^{2}-m_{b}^{*}c^{2})^{4}$
$\displaystyle+$
$\displaystyle\sum_{i=n,p}\frac{1}{3}\frac{\gamma}{(2\pi)^{3}}\int_{0}^{k_{F}i}\frac{4\pi
k^{2}}{\sqrt{(\hbar ck)^{2}+(m_{i}^{*}c^{2})^{2}}}dk$

where ${\cal E}$ is the energy density, $P$ is the pressure, $g_{s}$, $g_{v}$
and $g_{\rho}$ are the couplings of the scalar boson, vector boson, and iso-
vector $\rho$-meson respectively, $m_{s}$, $m_{v}$ and $m_{\rho}$ are the rest
masses of scalar and vector bosons and $\rho$-meson respectively, the term
$\rho_{I}$ involves the difference between the proton and neutron densities
(important for finite nuclei), also $\kappa$ and $\lambda$ are the couplings
of the cubic and quartic self-interaction of the scalar boson, $m_{b}$ and
$m_{b}^{*}$ are the rest mass and the effective mass of the nucleon, $n_{b}$
is the nucleonic density, $k_{F}$ is the Fermi momentum of nucleons at zero
temperature and $\gamma$ is the degeneracy, with value $\gamma=4$ for
symmetric nuclear matter and $\gamma=2$ for neutron matter (used in this
investigation).

Considering the possibility that the hypothetical 17 MeV boson [1], could
contribute as a second vector boson, we can write an effective mass term in
the following form [9]:

$m_{v}^{*2}=a_{X}^{2}m_{X}^{2}+(1-a_{X})^{2}m_{\omega}^{2}$ (5)

where $a_{X}$ is the admixture coefficient of the $m_{X}=17$ MeV boson to the
total vector potential. Depending on the value of $a_{X}$, the effective mass
can range from $m_{\omega}=782.5$ MeV to 17 MeV.

Using two different values for the admixture coefficient ($a_{X}$): 0.2(20%,
$m^{*}_{v}$ = 626 MeV) and 0.3(30%, $m^{*}_{v}$ = 547.8 MeV) and increasing
the value on the standard $\rho$-coupling by 5% and 10% (effective
$\rho$-coupling: g${}^{*}_{\rho}$), respectively, we constructed a set of 13
EoSs, fulfilling experimental constraints in analogy to [9], where properties
of nuclear matter and finite nuclei were considered.

All combinations and parameters that were used are shown in Table.1 and
Table.2. The corresponding Mass-Radius diagrams are depicted in Fig.1.

Table 1: Constrained parameter sets for eight EoSs with incompressibilities K0=245-260 MeV, $a_{X}$ = 0.2 (20%, $m^{*}_{v}$ = 626 MeV), standard $\rho$-coupling = 4.47 [28] and effective $\rho$-coupling g${}^{*}_{\rho}$ increased (+) by 5% and 10% compared to its standard value. EoS |  g${}^{*}_{\rho}$ |  gv |  gs |  m${}_{s}[MeV]$ |  $\kappa$[MeV] | $\lambda$
---|---|---|---|---|---|---
E1 | +5% | 7.61 | 6.78 | 406.6 | 19.0 | -60.0
E2 | +5% | 8.00 | 6.76 | 391.4 | 17.0 | -63.3
E3 | +5% | 8.00 | 7.03 | 405.6 | 19.5 | -80.0
E4 | +10% | 7.23 | 7.27 | 451.9 | 25.0 | -33.3
E5 | +10% | 7.23 | 7.27 | 451.9 | 25.5 | -46.7
E6 | +10% | 7.23 | 7.51 | 467.0 | 28.5 | -56.7
E7 | +10% | 7.61 | 7.03 | 421.7 | 21.0 | -60.0
E8 | +10% | 7.61 | 7.03 | 421.7 | 21.5 | -73.3
Table 2: Constrained parameter sets for five EoSs with incompressibilities K0=245-260 MeV, $a_{X}$ = 0.3 (30%, $m^{*}_{v}$ = 547.8 MeV), standard $\rho$-coupling = 4.47 [28] and effective $\rho$-coupling g${}^{*}_{\rho}$ increased (+) by 5% and 10% compared to its standard value. EoS |  g${}^{*}_{\rho}$ |  gv |  gs |  ms[MeV] |  $\kappa$[MeV] | $\lambda$
---|---|---|---|---|---|---
E9 | +5% | 7.61 | 8.08 | 451.9 | 19.0 | -103.3
E10 | +5% | 8.00 | 8.35 | 451.9 | 18.5 | -123.3
E11 | +5% | 7.23 | 8.33 | 482.2 | 26.0 | -150.0
E12 | +10% | 6.47 | 5.77 | 346.1 | 14.5 | -33.3
E13 | +10% | 7.23 | 8.07 | 467.0 | 23.0 | -123.3

###  3.2 The Momentum Dependent Interaction (MDI) model

The Momentum Dependent Interaction (MDI) model used here, was already
presented and analyzed in a previous paper [29, 30]. The MDI is designed to
reproduce the results of the microscopic calculations of both nuclear and
neutron rich matter at zero temperature and can be extended to finite
temperature. The energy density of the baryonic matter, is given by:

$\displaystyle{\cal E}(u,I)$ $\displaystyle=$
$\displaystyle\frac{3}{10}E_{F}^{0}n_{0}\left[(1+I)^{5/3}+(1-I)^{5/3}\right]u^{5/3}$
(6) $\displaystyle+$ $\displaystyle\frac{1}{3}{\cal
A}n_{0}\left[\frac{3}{2}-\left(\frac{1}{2}+x_{0}\right)I^{2}\right]u^{2}$
$\displaystyle+$ $\displaystyle\frac{\frac{2}{3}{\cal
B}n_{0}\left[\frac{3}{2}-\left(\frac{1}{2}+x_{3}\right)I^{2}\right]u^{\sigma+1}}{1+\frac{2}{3}{\cal
B}^{\prime}n_{0}\left[\frac{3}{2}-\left(\frac{1}{2}+x_{3}\right)I^{2}\right]u^{\sigma-1}}$
$\displaystyle+$ $\displaystyle u\sum_{i=1,2}\left[\frac{}{}{\cal C}_{i}({\cal
J}_{n}^{i}+{\cal J}_{p}^{i})+\frac{({\cal C}_{i}-8Z_{i})}{5}I({\cal
J}_{n}^{i}-{\cal J}_{p}^{i})\right]$

In Eq. (6), $E_{F}^{0}$ is the Fermi energy of symmetric nuclear matter at the
equilibrium density $n_{0}=0.16$ fm-3, $I=(n_{n}-n_{p})/n$ and $u=n/n_{0}$.
The parameters ${\cal A}$, ${\cal B}$, $\sigma$, ${\cal C}_{1}$, ${\cal
C}_{2}$, and ${\cal B}^{\prime}$ which appear in the description of symmetric
nuclear matter and the additional parameters $x_{0}$, $x_{3}$, $Z_{1}$, and
$Z_{2}$ used to determine the properties of asymmetric nuclear matter, are
treated as parameters constrained by empirical knowledge [29]. By suitable
choice of the above parameters we can regulate the stiffness of the
corresponding EoS. This stiffness is well reflected by the values of the slope
parameter $L$ which defined as:

$L=3n_{0}\left(\frac{\partial E_{sym}(n)}{\partial n}\right)_{n=n_{0}}$ (7)

where the symmetry energy, in general is defined as,

$E_{sym}(n)=\frac{1}{2!}\left(\frac{\partial^{2}E(n,I)}{\partial
I^{2}}\right)_{I=0}$ (8)

and $E(n,I)={\cal E}(u,I)/n$ is the energy per baryon. Moreover, the quantity
${\cal J}_{\tau}^{i}(n,I,T)$ is defined as:

${\cal J}_{\tau}^{i}(n,I,T)=2\int\frac{d^{3}k}{(2\pi)^{3}}{\rm
g}(n,\Lambda_{i})f_{\tau}$ (9)

where $f_{\tau}$, (for $\tau$ = n, p) is the Fermi-Dirac distribution
function. The function ${\rm{\cal G}}(k,\Lambda_{i})$ ) suitably chosen to
simulate finite range effects is of the following form:

${\rm{\cal
G}}(k,\Lambda_{i})=\left[1+\left(\frac{k}{\Lambda_{i}}\right)^{2}\right]^{-1}$
(10)

where the finite-range parameters are $\Lambda_{1}=1.5k_{F}^{0}$ and
$\Lambda_{2}=3k_{F}^{0}$ and $k_{F}^{0}$ is the Fermi momentum at the
saturation point $n_{0}$.

###  3.3 Contribution on energy and pressure of the X17 boson

The energy density of the WILB boson in neutron star matter is given by [16,
31]:

${\cal E}_{\rm B}=\pm\frac{(\hbar c)^{3}}{2}\left(\frac{{\rm
g}}{m_{B}c^{2}}\right)^{2}n_{b}^{2}$ (11)

Where ${\rm g}$ is the coupling constant of the interaction and $m_{B}c^{2}$
the mass of the boson. The sign (+) corresponds to a vector boson (repulsive
interaction) and (-) to a scalar boson (attractive interaction). The energy
per baryon then is given by $E_{bar}={\cal E}_{\rm B}/n_{b}$.

The corresponding contribution on the pressure is defined as:

$\displaystyle P_{B}$ $\displaystyle=$ $\displaystyle
n^{2}\frac{\partial({\cal E_{B}}/n)}{dn}$ (12) $\displaystyle=$
$\displaystyle\frac{(\hbar c)^{3}}{2}\left(\frac{{\rm
g}}{m_{B}c^{2}}\right)^{2}n_{b}^{2}\left(1-\frac{2n_{b}}{m_{B}c^{2}}\frac{\partial(m_{B}c^{2})}{\partial
n_{b}}\right)$

In the specific case where the the mass does not depends on the density the
pressure contribution is identical to that of the energy density. Now, the
total equation of state is just the sum of the baryons and bosons that is:

${\cal E}={\cal E}_{\rm bar}\pm{\cal E}_{\rm B},\quad P=P_{\rm bar}\pm P_{\rm
B}$ (13)

In previous work [14, 16, 31] the range of the ratio $\left({\rm
g}/{m_{B}c^{2}}\right)^{2}$ was varying between $[0-200]\;{\rm GeV}^{-2}$.
However, in the present study we consider that the coupling varies in the
interval $[0.01-0.022]$ which corresponds (for $m_{B}c^{2}=17$ MeV) the
interval for $(g/m_{B}c^{2})^{2}$ $[0.346,1.675]$ GeV-2.

According to Brown and Rho [15], the in-medium modification of the mesons
follows the linear scaling:

$m_{\rm B}^{*}\equiv m_{\rm B}\left(1-C\frac{n_{b}}{n_{0}}\right)\ ({\rm
MeV})$ (14)

We consider, following the suggestion in Ref.[14] that, at least at low
densities, in-medium modification of X17 mass takes a similar form to Eq.14.
The parameter $\mathrm{C}$ is fixed in order the predicted EoS to be
compatible with the bulk properties in symmetric nuclear matter ($E_{\rm
bind}=-16$ MeV and $n_{0}=0.16$ fm-3).

A suitable parametrization is also the following:

$m_{\rm B}^{*}\equiv\frac{17}{1-C}\left(1-C\frac{n_{b}}{n_{0}}\right)\ ({\rm
MeV})$ (15)

which ensures that at the saturation density $n_{0}$, $m_{\rm B}^{*}=17$ MeV.
To good approximation the contribution of the WILB begins after the crust-core
interface.

In previous studies, it was found that in the case of vector boson, the total
equation of state becomes very stiff leading to a very large values of the
maximum mass and corresponding larges radius. On the other hand in the case of
scalar boson the EoS is so soft leading to a very small values (out of the
observations) of the maximum mass. In any case the majority of the studies
focuses only on the maximum mass and not on the radius and the tidal
deformability. The aforementioned quantity directly related with information
obtained from the detection of gravitational waves.

###  3.4 Quark Stars (QSs)

In the present work we use the Color-Flavor Locked (CFL) model for the Quark
Stars (QSs), following the previous work of Lugones and Horvath [32]. In
particular, we use the lowest order approximation, in which the EoSs are given
in a simple and analytical form and in particular the contribution of the
pairing correlations and the mass of the strange quark are provided
explicitly. The pressure, energy density and the baryon density are given by
the expressions [32] (see also [33]):

$P_{Q}=\frac{3\mu^{4}}{4\pi^{2}(\hbar
c)^{3}}-\frac{3(m_{s}c^{2})^{2}\mu^{2}}{4\pi^{2}(\hbar
c)^{3}}+\frac{3\Delta^{2}\mu^{2}}{\pi^{2}(\hbar c)^{3}}-B$ (16) ${\cal
E}_{Q}=\frac{9\mu^{4}}{4\pi^{2}(\hbar
c)^{3}}-\frac{3(m_{s}c^{2})^{2}\mu^{2}}{4\pi^{2}(\hbar
c)^{3}}+\frac{3\Delta^{2}\mu^{2}}{\pi^{2}(\hbar c)^{3}}+B$ (17)

and

$n_{b}=\frac{\mu^{3}}{\pi^{2}(\hbar
c)^{3}}-\frac{(m_{s}c^{2})^{2}\mu}{2\pi^{2}(\hbar
c)^{3}}+\frac{2\Delta^{2}\mu}{\pi^{2}(\hbar
c)^{3}}=\frac{\mu^{3}}{\pi^{2}(\hbar c)^{3}}+\frac{3\mu\alpha}{\pi^{2}(\hbar
c)^{3}}$ (18)

where

$\mu^{2}=-3\alpha+\sqrt{9\alpha^{2}+\frac{4}{3}\pi^{2}(P_{Q}+B)(\hbar c)^{3}}$
(19)

and

$\alpha=-\frac{(m_{s}c^{2})^{2}}{6}+\frac{2\Delta^{2}}{3}$ (20)

In this study we use the parametrization of the EoS according to the paper
[34]. In particular each equation of state is denoted as
CFLX[$B$,$\Delta$,$m_{s}$], where $X$, is the numbering of the specific EoS,
$B$ (in MeV fm-3), $\Delta$ (in MeV ), $m_{s}$ (in MeV) are the bag constant,
the pairing and the mass of the strange quark respectively. It is worthwhile
to notice here that, in general, the lower the value of $B$ and $m_{s}$ the
stiffer the EoS as well as the lower values of $\Delta$ the softer the EoS. In
the present study we use the sets CFL2[60,50,150] (intermediate stiffness),
CFL10[80,150,150] (stiff) and CFL13[100,100,150] (soft) in order to cover a
large region of stiffness.

The energy density of the bosons in the case of quark stars is given by [17,
24]:

${\cal E}_{\rm B}=\pm\frac{9(\hbar c)^{3}}{2}\left(\frac{{\rm
g}}{m_{B}c^{2}}\right)^{2}n_{b}^{2}$ (21)

where the contribution of the pressure is given by an expression similar to
(12). The total equation of state is just the sum of the quarks and bosons
that is:

${\cal E}={\cal E}_{\rm Q}\pm{\cal E}_{\rm B},\quad P=P_{\rm Q}\pm P_{\rm B}$
(22)

It is worth clarifying here that the main difference with the case of neutron
stars is the extra factor of 9 in terms of the contribution to pressure and
energy density. This additional contribution plays, as we shall see, an
important role in the effect of the boson on the basic properties of compact
objects.

###  3.5 Speed of sound

The speed of sound is a very crucial quantity since is directly related with
the stiffness of the EoS. In particular, the adiabatic speed of sound is
defined as:

$v_{s}/c=\sqrt{\left(\frac{dP}{d{\cal E}}\right)_{S}}$ (23)

where $S$ is the entropy per baryon. More importantly, the speed of sound
introduce an upper limit on stiffness of the EoS according to which the speed
of sound cannot exceed the speed of light. Bedaque and Steiner [35] have
provided simple arguments that support as an upper limit the value
$c/\sqrt{3}$ in nonrelativistic and/or weakly coupled theories. These authors
pointed out that the existence of neutron stars with masses about two solar
masses combined with the knowledge of the EoS of hadronic matter at low
densities is not consistent with this bound. In any case, in studies related
with the predictions of bulk properties of compact object special attention
must be given on the density dependence of the speed of sound and the upper
limits must be carefully taken into account.

##  4 Tidal Deformability

Very important sources of gravitational waves are those produced during the
phase of the inspiral on a binary system of neutron stars before they finally
merge [36, 37, 38]. These kind of sources leads to the measurement of various
properties of neutron stars. During the inspiral phase the tidal effects can
be detected [37].

The $k_{2}$ parameter, also known as tidal Love number, depends on the EoS and
describes the response of a neutron star to the tidal field $E_{ij}$ with
respect to the induced quadrupole field $Q_{ij}$ [37, 38]. Their exact
relation is given below:

$Q_{ij}=-\frac{2}{3}k_{2}\frac{R^{5}}{G}E_{ij}\equiv-\lambda E_{ij},$ (24)

where $R$ is the neutron star radius and $\lambda=2R^{5}k_{2}/3G$ is the tidal
deformability. Also, another quantity that is well measured is the effective
tidal deformability $\tilde{\Lambda}$ which is given by [39]

$\tilde{\Lambda}=\frac{16}{13}\frac{(12q+1)\Lambda_{1}+(12+q)q^{4}\Lambda_{2}}{(1+q)^{5}},$
(25)

where the mass ratio $q=m_{2}/m_{1}$ lies within the range $0\leq q\leq 1$ and
$\Lambda_{i}$ is the dimensionless deformability [39]

$\Lambda_{i}=\frac{2}{3}k_{2}\left(\frac{R_{i}c^{2}}{M_{i}G}\right)^{5}\equiv\frac{2}{3}k_{2}\beta_{i}^{-5},\quad
i=1,2.$ (26)

The effective tidal deformability $\tilde{\Lambda}$ plays important role on
the neutron star merger process and is one of the main quantities that can be
inferred by the detection of the corresponding gravitation waves.

##  5 Results and Discussion

Starting with the RMF nuclear model of Pure Neutron Matter (PNM), we see that
its parameterization makes the EoSs to present two different behaviors. The
general tendency is, as we can see in the Fig.1, that the increase of the
admixture fraction coefficient $a_{X}$ leads to stiffer EoS and on the other
hand the increase of the $\rho$-coupling leads to a softer EoS. The
combination between these two parameters can be optimized, resulting a
reasonable EoS which fits inside the accepted limits with respect the
observational data.

Figure 1: The M-R diagram corresponds to the RMF model of PNM for various EoSs
(for more details see the text). The shaded regions from bottom to top
represent the HESS J1731-347 remnant [40], the GW170817 event [39], PSR
J1614-2230 [41], PSR J0348+0432 [42], PSR J0740+6620 [43], and PSR J0952-0607
[44] pulsar observations for the possible maximum mass.

Performing this optimization we can deduce that an admixture of $30\%$ in the
$a_{X}$ coefficient and using a $\rho$-coupling of +$5\%$ results a stiff EoS
with maximum mass of $\mathrm{2.33\;M_{\odot}}$. Furthermore an admixture of
$20\%$ in $a_{X}$ in combination with the $\rho$-coupling varying between
+$5\%$ and +$10\%$ results EoSs with maximum masses from
$\mathrm{2.1\;M_{\odot}}$ to $\mathrm{2.27\;M_{\odot}}$ and radius between
12.5 Km to 13 Km respectively. We notice that none of the total EoSs of the
RMF model that we used in our work can describe the HESS J1731-347 [40].

Figure 2: The $\tilde{\Lambda}-q$ dependence corresponds to the RMF model of
PNM. The shaded region shows the excluded values derived from the GW170817
event [39].

In addition as we can see from Fig.2, in the ${\tilde{\Lambda}}-q$ diagram
concerning the effective tidal deformability with respect to the binary mass
ratio $q$, the aforementioned optimization can exist well below the specified
upper limit of the first gravitational wave GW170817 observation given by LIGO
[39]. In our study we adopted the estimations of GW170817 detection concerning
the component masses and the chirp mass of the system [39]. As one can observe
from Fig.2, the observational upper limit on $\tilde{\Lambda}$ favors in
general the $20\%$ admixture, contrary to the $30\%$ admixture, as the first
one provides softer EoSs. Across the same amount of admixture, the EoSs with
higher $\rho$-coupling provide lower values of $\tilde{\Lambda}$. Though, this
is not an all-out decisive behavior for the exclusion of the $30\%$ admixture
since there are cases that fulfill the observational upper limit on
$\tilde{\Lambda}$.

When considering $\beta$-equilibrated (npe) matter instead of PNM in the RMF
model and the admixture into vector boson segment constrained by 20% and 30%
we see not much of a difference of neutron stars radius. A appreciable
reduction appears in the maximum mass as shown in Fig.3. Looking to the
relative $\tilde{\Lambda}$ we see a good agreement with the constraints from
GW170817 observational data as depicted in Fig.4.

Figure 3: The M-R diagram corresponds to the RMF model of $\beta$-equilibrated
(npe) matter for various EoSs (for more details see the text). The shaded
regions from bottom to top represent the HESS J1731-347 remnant [40], the
GW170817 event [39], PSR J1614-2230 [41], PSR J0348+0432 [42], PSR J0740+6620
[43], and PSR J0952-0607 [44] pulsar observations for the possible maximum
mass.

Furthermore, we studied the effect of the X17 boson using the MDI model. To be
more specific, we used as a base three main MDI EoSs with different slope
parameter ${\mathrm{L}}$: $\mathrm{L=65\;MeV}$, $\mathrm{L=72.5\;MeV}$, and
$\mathrm{L=80\;MeV}$, with blue, green, and red color in the corresponding
figures respectively. The coupling constant $\mathrm{g}$ is allowed to take
values up to $\mathrm{g\simeq 0.022}$ to fulfill the properties of symmetric
nuclear matter at $\mathrm{n_{0}}$, while the parameter $\mathrm{C}$ ranges in
the interval $\mathrm{C\in[0,C_{max}]}$, where $\mathrm{C_{max}}$ is the
higher allowed value, unique for each EoS, derived from the non-violation of
causality. To be more specific, the value of $\mathrm{g_{max}\simeq 0.022}$
arises from the deviation of the binding energy of the symmetric nuclear
matter at the saturation density, i.e. $\mathrm{-17\;Mev\leq
E_{bind}(n_{0})\leq-15\;MeV}$.

Figure 4: The $\tilde{\Lambda}-q$ dependence corresponds to the RMF model of
$\beta$-equilibrated (npe) matter. The shaded region shows the excluded values
derived from the GW170817 event [39].

In Fig, 5 the mass-radius dependence for all the cases of the MDI model is
displayed. The solid curves correspond to the three initial EoSs without the
contribution of the X17 boson, while the dashed and dash-dotted curves
correspond to the EoSs within the presence of the X17 boson for
$\mathrm{g=0.011}$, and $\mathrm{g=0.022}$ respectively. Among the same set of
EoS and $\mathrm{g}$, the darker colored curve corresponds to $\mathrm{C=0}$,
while the lighter one to $\mathrm{C=C_{max}}$. At first sight, the higher
value of $\mathrm{L}$ provides stiffer EoS, nevertheless the amount of
contribution of X17. The EoSs with $\mathrm{g=0.011}$ (dashed curves) lie
closely to their corresponding initial EoS (solid curves) leading to a slight
increment of $\mathrm{M_{max}}$ and radius $\mathrm{R}$, while the bigger
differentiation from the initial EoSs is for $\mathrm{g=0.022}$. Also, the
effect from the contribution of the X17 is stronger on the coupling constant
$\mathrm{g}$ than the parameter $\mathrm{C}$. We notice that all the EoSs lie
outside the estimated region for the HESS J1731-347 remnant [40].

Figure 5: The M-R diagram corresponds to the MDI models. The shaded regions
from bottom to top represent the HESS J1731-347 remnant [40], the GW170817
event [39], PSR J1614-2230 [41], PSR J0348+0432 [42], PSR J0740+6620 [43], and
PSR J0952-0607 [44] pulsar observations for the possible maximum mass. Figure
6: The $\tilde{\Lambda}-q$ dependence corresponds to the MDI models. The
shaded region shows the excluded values derived from the GW170817 event [39].

Figure 7: (a) The maximum mass, and (b) the tidal deformability
$\Lambda_{1.4}$ of a $\mathrm{1.4\;M_{\odot}}$ neutron star related to the
parameter C for the three MDI EoSs. The horizontal shaded areas on the left
panel correspond to those of Figure 1. The green shaded area on the right
panel indicates the constraints from GW170817 [45], while the colored diagonal
lines show the excluded regions from the violation of causality for each EoS.

The expansion of the aforementioned EoSs by their application to the
observational data of the GW170817 event, allowed us to study directly their
behavior through the $\tilde{\Lambda}-q$ dependence, as one can observe in
Fig. 6. In this figure, the first two parametrizations for $\mathrm{L}$ (blue
and green curves) are in a first instance in good agreement with the
observational upper limit on $\tilde{\Lambda}$. The third and more stiff
branch of EoSs (red curves) also lies inside the acceptance area (imposed by
the upper limit of $\tilde{\Lambda}$), but it us much closer to the limit,
with the more stiff EoS of the total MDI EoSs to be almost exact on it.
Additionally, as we increase the slope parameter $\mathrm{L}$, the EoSs spread
out more, e.g. for $\mathrm{L=65\;MeV}$ the effect of the X17 is smaller
compared to the other EoSs in total. At a second level, across EoSs with the
same $\mathrm{L}$, this effect depends more on the coupling constant
$\mathrm{g}$, as we already observed from Fig. 5. Therefore, a more detailed
examination of the dependence of the EoSs on the two main parameters
$\mathrm{g}$, and $\mathrm{C}$ is needed.

In Figure 7, we examined the dependence of the EoSs from the $\mathrm{C}$
parameter using observational constraints on the $\mathrm{M_{max}}$ and
$\mathrm{\Lambda_{1.4}}$. By focusing on the left panel of Figure 7, we notice
that at the most high contribution of the X17, the maximum mass increased
$\mathrm{\Delta M_{max}\simeq 0.066\;M_{\odot}}$ for $\mathrm{L=65\;MeV}$,
$\mathrm{\Delta M_{max}\simeq 0.068\;M_{\odot}}$ for $\mathrm{L=72.5\;MeV}$,
and $\mathrm{\Delta M_{max}\simeq 0.069\;M_{\odot}}$ for $\mathrm{L=80\;MeV}$.
However, for the same value of $(\mathrm{g,C})$ bewteen the three different
set of MDI EoSs, the $\mathrm{\Delta M}$ is higher for the set with the lower
$\mathrm{L}$; e.g. for $\mathrm{g=0.022}$ and $\mathrm{C=0.06}$ the
corresponding values are $\mathrm{\Delta M_{max}^{L=65}\simeq
0.06\;M_{\odot}}$, $\mathrm{\Delta M_{max}^{L=72.5}\simeq 0.045\;M_{\odot}}$,
and $\mathrm{\Delta M_{max}^{L=80}\simeq 0.043\;M_{\odot}}$. Hence, the effect
of the X17 under this perspective is as a relative amount higher in the softer
EoSs. Moreover, the EoS with $\mathrm{L=65\;MeV}$ fails to provide a neutron
star with $\mathrm{M_{max}\geq 2.0\;M_{\odot}}$ and requires high value of the
coupling constant $\mathrm{g}$ in order to provide a neutron star with
$\mathrm{M_{max}\simeq 1.9\;M_{\odot}}$, such as the PSR J1614-2230 [41]. In
the other two cases of $\mathrm{L}$, the presence of the X17 could lead them
to predict a $\mathrm{\geq 2.0\;M_{\odot}}$ neutron star (for a variety of
combinations of $\mathrm{g}$ and $\mathrm{C}$) but not higher than the
$\mathrm{\sim 2.05\;M_{\odot}}$ value. The ”shark-fin” shaded region arises
from the constraints that the non-violation of the causality implies on the
$\mathrm{C_{max}}$, with the peak corresponding to the
$\mathrm{(g=0.022,C=C_{max})}$ pair of values for each one of the three MDI
set of EoSs. In addition the shaded regions for $\mathrm{L=72.5\;MeV}$ and
$\mathrm{L=80\;MeV}$ are hardly be distinguished.

By examining further the way in which the contribution dependence of the X17
and the corresponding EoSs on the $\mathrm{C}$ parameter evolves, we
constructed the Fig. 7(b), in which the behavior of the dimensionless tidal
deformability $\Lambda_{1.4}$ of a $\mathrm{1.4\;M_{\odot}}$ neutron star
related to the $\mathrm{C}$ for all the MDI EoSs is shown. The shape of the
shaded regions is similar to that of the left panel of Fig. 7 and each peak
corresponds to the $\mathrm{(g=0.022,C=C_{max})}$ combination, unique for each
set of EoSs. In both panels, as we move to higher values of $\mathrm{L}$ the
peak, meaning the higher contribution of X17 for each EoS, is shifted to
higher values of $\mathrm{C}$. The diagonal colored lines indicate the area of
values of $\mathrm{g}$ and $\mathrm{C}$ that violates the causality.

Contrary to the behavior that we observed in the $\mathrm{M_{max}-C}$ diagram,
the $\mathrm{L=72.5\;MeV}$ and $\mathrm{L=80\;MeV}$ families of EoSs are
clearly distinguished. Especially, the $\mathrm{L=80\;MeV}$ set of EoSs lies
almost inside the observational data from LIGO (green shaded area) with a
small peak violating the observational upper limit of $\Lambda_{1.4}$. The
other two families of MDI EoSs lie inside the observational data for all the
combinations of $\mathrm{g}$ and $\mathrm{C}$ that we used. Therefore, even if
the $\mathrm{L=72.5\;MeV}$ and $\mathrm{L=80\;MeV}$ set of EoSs provided
almost identical maximum masses as we showed in Fig. 7(a), their dependence on
$\Lambda_{1.4}$ (i.e. the radius $\mathrm{R_{1.4}}$) highlights their
differences, and especially the sensitivity of the radius to the X17
contribution. Another issue that arises is the tension between the favor of
softer EoS from the tidal deformability perspective, and the requirement for a
stiffer EoS in order to provide a sufficient high maximum neutron star mass.
This tension, through the corresponding observations, we used as a tool in our
study to impose constraints on the contribution of X17.

As we mentioned before, the study of the X17 related to radius
$\mathrm{R_{1.4}}$ is of interest. In Fig. 8 we show the behavior of
$\mathrm{R_{1.4}}$ with respect to the coupling constant $\mathrm{g}$ for all
the three sets of MDI EoSs. The horizontal shaded regions indicate various
estimations of $\mathrm{R_{1.4}}$ [46, 47, 48]. The $\mathrm{L=72.5\;MeV}$ and
$\mathrm{L=80\;MeV}$ EoSs lie inside only on the estimated area of Ref. [48].
On the other hand, the EoSs with $\mathrm{L=65\;MeV}$ lie inside the values
predicted by Refs. [46, 47], and above the coupling constant $\mathrm{g\simeq
0.011}$ the corresponding EoSs insert the region of Ref. [48], but violate the
upper limit of Ref. [47]. As a general remark, as we move to higher values of
$\mathrm{g}$ the effect of the X17 boson becomes bigger, leading to bigger
radii. Also, for higher values of $\mathrm{g}$ the dependence from the
$\mathrm{C}$ parameter is stronger (thickening of the curved shaded region),
contrary to the lower values of $\mathrm{g}$ where the impact of different
$\mathrm{C}$ on the radius $\mathrm{R_{1.4}}$ vanishes.

Figure 8: The radius $\mathrm{R_{1.4}}$ of a $\mathrm{1.4\;M_{\odot}}$ neutron
star related to the coupling constant $\mathrm{g}$ for all the MDI EoSs that
we used in our study. The horizontal shaded regions correspond to different
constraints on $\mathrm{R_{1.4}}$ [46, 47, 48].

So far we studied the two main parameters, $\mathrm{g}$ and $\mathrm{C}$,
through their dependence on the macroscopic properties of neutron stars,
derived from the EoS, and by exploiting the available observations. In
general, only for higher coupling constant $\mathrm{g}$ ($\mathrm{g\gtrapprox
0.009}$) the EoS becomes more sensitive to the $\mathrm{C}$. In order to
examine further how $\mathrm{g}$ and $\mathrm{C}$ affects the EoSs, we
introduced a $\mathrm{g-C}$ parameter space, displayed in Fig. 9. This figure
is suitable for an overall view on these parameters and the way that the
constraints affect them. The green shaded region shows the allowed parameter
space with respect to causality, for all three set of MDI EoSs. The green
arrows show the direction of the accepted region. The green solid curve
indicates the limit for the $\mathrm{L=80\;MeV}$ EoS, the dashed one indicates
the limit for the $\mathrm{L=72.5\;MeV}$ EoS, while the dash-dotted curve
indicates the limit for the $\mathrm{L=65\;MeV}$ EoS.

As one can observe, the constraint imposed from the non-violation of
causality, cuts off the very high values of $\mathrm{g}$ and $\mathrm{C}$
(grey shaded area). Additionally, as we move to EoSs with lower $\mathrm{L}$,
this bound is shifted to even lower values of the parameter space, minimizing
further the ”window”. The red solid curve shows the maximum mass limit of
$\mathrm{M_{max}=2\;M_{\odot}}$ for the $\mathrm{L=80\;MeV}$ EoS; for the
parameter space on the left of this curve, the combination of $(\mathrm{g,C})$
leads to neutron stars with $\mathrm{M_{max}<2\;M_{\odot}}$. The light orange
curve, shows the observational upper limit of $\mathrm{\Lambda_{1.4}}$ for the
$\mathrm{L=80\;MeV}$ EoS; only the combination of parameters on the left side
of this curve can be accepted, without violating the observational value.
Therefore, for the $\mathrm{L=80\;MeV}$ EoS, if we looking for a neutron star
with at least $\mathrm{M_{max}\simeq 2\;M_{\odot}}$ with respect to
$\mathrm{\Lambda_{1.4}}$ and causality, then we must search inside the curved
triangular parameter space that the three curves form. We notice that this
space is unique for each set of EoS, and the corresponding $\mathrm{M_{max}}$
requirement. Summarizing, strong constraints are introduced both by
observational data and causality leading to a significant limitation of the
allowed range of parameters; a behavior that is present correspondingly in the
case of QSs, as we will demonstrate below.

Figure 9: Constraints for $\mathrm{g}$ and $\mathrm{C}$ for the three MDI EoSs
with respect to causality, the possible upper mass limit
$\mathrm{M_{max}=2\;M_{\odot}}$ and the dimensionless tidal
$\Lambda_{1.4}=580$ for the MDI (L=80 MeV) EoS.

Beyond the hadronic EoSs, we expanded our study of the contribution of the X17
boson to QSs. The effect of this contribution is depicted on the mass-radius
dependence in Fig. 10. In particular, we used three different -concerning the
parametrization- sets of EoSs running from soft (CFL13 EoS) and medium (CFL2
EoS) to stiff (CFL10 EoS) behavior, covering the most possible cases. We
notice that in the case of QSs we chose the same upper value for $\mathrm{g}$
so that the comparison with the hadronic EoSs of the MDI model to be more
clear (the quark EoSs does not have to respect the properties of symmetric
nuclear matter at the saturation density). The dashed curves correspond to
coupling constant $\mathrm{g}=0.011$ while the dash-dotted ones correspond to
$\mathrm{g}=0.022$. For the same value of $\mathrm{g}$ the lighter colored
curves correspond to higher $\mathrm{C}$. As we underlined in the hadronic
EoSs, the contribution of X17 in general depends stronger on $\mathrm{g}$ than
on $\mathrm{C}$. It is worth noticing that in the case of a pure QS the
effects of X17 are more pronounced compared to the neutron star case, beucase
of the additional contribution of factor of 9 on energy density and pressure
(see Eq. 21). Especially, the possible existence of X17 affects dramatically
the provided maximum mass in each set of EoSs. The effect on the radius of a
$\mathrm{1.4\;M_{\odot}}$ star is moderate but enough to affect the tidal
deformability (see Fig. 12). Also, it is interesting and noteworthy that the
CFL10 (stiff) and CFL13 (medium) EoSs satisfy simultaneously both the
observational data and the corresponding constraints.

Figure 10: The M-R diagram of the three CFL quark EoSs. The shaded regions
from bottom to top represent the HESS J1731-347 remnant [40], the GW170817
event [39], PSR J1614-2230 [41], PSR J0348+0432 [42], PSR J0740+6620 [43], and
PSR J0952-0607 [44] pulsar observations. Figure 11: The speed of sound as a
function of the pressure for the three quark EoSs and various parametrization
for the X17 boson. The square points indicate the pair of values which provide
the $\mathrm{M_{max}}$ for each CFL EoS.

An important microscopic quantity to study the properties of the EoS is the
speed of sound (see Eq. 23). In the context of this study we used so far the
upper bound on the speed of sound, derived from the causality; the speed of
sound should never exceed the speed of light $\mathrm{(v_{s}/c)^{2}\leq 1}$.
In Fig. 11 we demonstrate the relation between $\mathrm{(v_{s}/c)^{2}}$ and
the pressure $\mathrm{P}$ for all the CFL EoSs. The square points indicate the
pair of values which provide the $\mathrm{M_{max}}$ for each CFL EoS. All EoSs
respect the upper bound of causality. We notice the high dependence of the
speed of sound on the value of coupling constant $\mathrm{g}$. Additionally,
the dependence on $\mathrm{C}$ can be observed; this dependence becomes
stronger for higher values of $\mathrm{g}$.

By applying the study of quark EoSs to the observational estimations of the
properties of the GW170817 event [39], we examined further the behavior of X17
in the CFL EoSs, as shown in Fig. 12, through the $\mathrm{\tilde{\Lambda}-q}$
dependence. As one can observe, the effect of the X17 depends on the
parametrization of both the coupling constant $\mathrm{g}$ and the in-medium
effects described by the parameter $\mathrm{C}$, leading to an appreciably
increase of $\tilde{\Lambda}$. As in the case of hadronic EoSs, the dependence
on $\mathrm{g}$ is stronger than on $\mathrm{C}$. In general, the stiffness on
the EoS magnifies gradually this deviation on $\tilde{\Lambda}$ from the
initial (without X17) EoS.

Figure 12: The dimensionless tidal defomability $\tilde{\Lambda}$ for the
three quark EoSs and various parametrization for the X17 boson. The shaded
region shows the acceptance values derived from the GW170817 event [39].

Figure 13: (a) The maximum mass, and (b) the tidal deformability
$\Lambda_{1.4}$ corresponding to $\mathrm{1.4\;M_{\odot}}$ related to the
parameter C for the three CFL EoSs. The horizontal shaded areas on the left
panel correspond to those of Fig. 10. The green shaded area on the right panel
indicates the constraints from GW170817 [45], while the colored diagonal lines
show the excluded regions from the violation of causality for each EoS.

Figure 14: Constraints for $\mathrm{g}$ and $\mathrm{C}$ for (a) the CFL13
quark EoS (soft), the possible upper mass limits $\mathrm{1.8\;M_{\odot}}$ and
$\mathrm{2.0\;M_{\odot}}$ and the dimensionless tidal $\Lambda_{1.4}=580$, (b)
the CFL2 quark EoS (intermediate), the possible upper mass limits
$\mathrm{2.2\;M_{\odot}}$ and $\mathrm{2.4\;M_{\odot}}$ and the dimensionless
tidal $\Lambda_{1.4}=580$, and (c) the CFL10 quark EoS (stiffer), the possible
upper mass limit $\mathrm{2.4\;M_{\odot}}$ and the dimensionless tidal
$\Lambda_{1.4}=580$ [45]. In all panels, the green shaded region indicates the
allowed parameter space derived by the non-violation of causality, while the
yellow ones indicate the regions where we not include in our study.

In order to take a deeper look on the behavior of the quark EoSs which include
the X17 boson, we studied the dependence of their properties related directly
to the parameter $\mathrm{C}$, by imposing also available constraints from
observations. In Fig. 13 we display the maximum mass $\mathrm{M_{max}}$ (left
panel) and the tidal deformability $\Lambda_{1.4}$ (right panel) related to
$\mathrm{C}$ respectively. The blue color corresponds to the CFL13 set of EoSs
(soft), the green one to the CFL2 (intermediate), and the purple color
corresponds to the CFL10 (stiff). In both panels the colored regions of the
EoSs are defined by the non-violation of the causality. In Fig. 13(b) the
diagonal colored lines indicate the violation of causality. In both
$\mathrm{M_{max}-C}$ and $\mathrm{\Lambda_{1.4}-C}$ diagrams, the
corresponding peak of each set of EoSs occurs when $\mathrm{g=0.022}$ and
$\mathrm{C=C_{max}}$, where $\mathrm{C_{max}}=0.0456$ for CFL10,
$\mathrm{C_{max}}=0.0595$ for CFL2, and $\mathrm{C_{max}}=0.0653$ for CFL10.
From softer to stiffer EoS this peak shifts to higher values of $\mathrm{C}$,
with a corresponding grow on $\mathrm{M_{max}}$ and $\mathrm{\Lambda_{1.4}}$.
Across the same value of $\mathrm{g}$, i.e. $\mathrm{g=const.}$ as the
$\mathrm{C}$ increases, the effect of the X17 becomes higher; we clarify that
the upper limit of each shaded region on the left side of each peak
corresponds to $\mathrm{g=0.022}$. The imprint of the high contribution of the
X17 to the quark EoSs can be observed clearly from the deviation of the
$\mathrm{M_{max}}$ and the $\mathrm{\Lambda_{1.4}}$ from the corresponding
value of the initial EoS; e.g. for the CFL2 EoS, $\mathrm{g=0.022}$, and
$\mathrm{C=C_{max}}$ it is $\mathrm{\Delta M_{max}\simeq 0.417\;M_{\odot}}$.
Regarding the $\mathrm{\Lambda_{1.4}}$, the corresponding deviation for the
CFL13 EoS is $\mathrm{\Delta\Lambda_{1.4}\simeq 85}$, for the CFL2 EoS it is
$\mathrm{\Delta\Lambda_{1.4}\simeq 99}$, while for the CFL10 EoS it is
$\mathrm{\Delta\Lambda_{1.4}\simeq 167}$. Hence, there is an agreement with
the gradual change of stiffness. From the Fig. 13(b) we notice that all EoSs
lie inside the GW170817 accepted region, except some cases with high
$\mathrm{g}$ and $\mathrm{C}$ for the CFL10 set of EoSs. So far, the available
observational data offered us useful constraints on the two parameters
$\mathrm{g}$ and $\mathrm{C}$, but a more detailed view is needed.

For the purpose of the aforementioned examination, we constructed the
$\mathrm{g-C}$ parameter space for each set of quark EoS, by applying various
upper limits on $\mathrm{M_{max}}$, $\mathrm{\Lambda_{1.4}}$ and including the
non-violation of causality. In Fig. 14 the three parameter spaces are
demonstrated; the change on the stiffness (soft to stiff) of the EoS
corresponds to the direction from the left to the right panel. This kind of
diagrams clarify specifically the role of the two parameters. As we move from
softer to stiffer EoS, the range of pair values $\mathrm{(g,C)}$ that satisfy
the mentioned constraints increases. Therefore, since the range of values for
the coupling constant $\mathrm{g}$ is common for all cases, the stiffer the
EoS the higher the value of $\mathrm{C_{max}}$, meaning higher contribution of
X17. We underline at this point that across the same set of EoSs the
contribution of X17 is characterized firstly by the value of $\mathrm{g}$. The
displayed points correspond to the crossed colored curves’ values with the
causality limit. The curves of $\mathrm{M_{max}(C)}$ and
$\mathrm{\Lambda_{1.4}(C)}$ are unique for each set of CFL EoSs. The
$\mathrm{M_{max}(C)}$ curves define that the $\mathrm{(g,C)}$ pair of values
on the left of each one of these curves do not provided the desired
$\mathrm{M_{max}}$. These points are a) $\mathrm{g=0.0105,\;C_{max}=0.05325}$
with $\mathrm{M_{max}=1.8\;M_{\odot}}$ and
$\mathrm{g=0.012,\;C_{max}=0.04581}$ with $\mathrm{M_{max}=2.0\;M_{\odot}}$
for the CFL13 EoS (soft case), b) $\mathrm{g=0.0189,\;C_{max}=0.0608}$ with
$\mathrm{M_{max}=2.2\;M_{\odot}}$ for the CFL2 EoS (intermediate case), and c)
$\mathrm{g=0.0102,\;C_{max}=0.079}$ with $\mathrm{M_{max}=2.4\;M_{\odot}}$ and
$\mathrm{g=0.017,\;C_{max}=0.06948}$ with $\mathrm{\Lambda_{1.4}=580}$ for the
CFL10 EoS (stiff case). In the latter case, the observational limit on
$\mathrm{\Lambda_{1.4}}$ excludes all the $\mathrm{(g,C)}$ values on the right
of its curve. We observe that as we move to stiffer EoS only higher values on
$\mathrm{M_{max}}$ can provide further constraints. Concerning the constraints
imposed by the $\mathrm{\Lambda_{1.4}}$, these can be useful in stiffer EoSs.

Concluding, the more soft the equation of state the more limited is the range
of parameters, for both hadronic and quark case. In this context, additional
observational data concerning the maximum mass as well as more strict upper
and even lower limits on $\Lambda_{1.4}$ may lead to much stringent
constraints regarding the coupling constant $\mathrm{g}$ and the in-medium
effect regulator $\mathrm{C}$.

##  6 Concluding Remarks

We studied the effect of the hypothetical X17 boson on the EoS of neutron star
matter as well as on QS and the corresponding bulk properties including mass,
radius and tidal deformability. In particular, we payed attention on two main
phenomenological parameters of the X17: a) the coupling constant $\mathrm{g}$
of its interaction with hadrons or quarks, and b) the in-medium effects
through a regulator $\mathrm{C}$. Both are very crucial concerning the
contribution on the total energy density and pressure. We suggest that it is
possible to provide constraints on these parameters with respect to causality
and various possible upper mass limits and the dimensionless tidal
deformability $\Lambda_{1.4}$. Moreover, we found that the more stiff is the
EoS (hadronic or quark), the more indiscernible are the effects on the
properties of compact objects. In particular, we found that the effect of the
existence of the hypothetical X17 boson, are more pronounced, in the case of
QSs, concerning all the bulk properties. This is due mainly on the extra
factor 9 both on the energy and pressure contribution to the total ones. It
must be emphasized that in the present study, special attention was paid to
maintain the non violation of causality on the EoSs, while systematically
taken into account the in-medium effects on the mass of the hypothetical boson
(which usually had been omitted in similar works so far). In addition, an
attempt was made to find possible constraints on the hypothetical X17 boson
with the help of observational data and mainly those derived from the
detection of gravitational waves.

It would be also of great interest to perform similar calculations in the
context of modified gravity theories in order to fully demonstrate the
implications of the X17 boson to the properties of various compact objects
[49, 50, 51]. Finally, it is worth to notice that the present study can form
the framework for similar studies concerning the possible existence of bosons
in nuclear matter and their consequences on the structure and basic properties
of compact objects. Likely, in this case it will be possible from both
terrestrial and astrophysical observations to make the best possible estimate
of the properties of these particles, which will concern both individual
properties and those related to the interaction with the environment in which
they are found.

## Acknowledgments

This work is supported by the Czech Science Foundation (GACR Contract No.
21-24281S) and by the Hellenic Foundation for Research and Innovation (HFRI)
under the 3rd Call for HFRI PhD Fellowships (Fellowship Number: 5657). One of
the authors (Ch.C.M) would like to thank Prof. M.I. Krivoruchenko and Prof. F.
Simkovic for useful discussion and correspond.

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