# Dense matter equation of state of a massive neutron star with anti-kaon
condensation

Vivek Baruah Thapa<EMAIL_ADDRESS>Monika Sinha<EMAIL_ADDRESS>Indian
Institute of Technology Jodhpur, Jodhpur 342037, India

###### Abstract

Recent measurements of neutron star mass from several candidates (PSR
J$1614-2230$, PSR J$0348+0432$, MSP J$0740+6620$) set the lower bound on the
maximum possible mass for this class of compact objects $\sim 2$ M⊙. Existence
of stars with high mass brings the possibility of existence of exotic matter
(hyperons, meson condensates) at the core region of the objects. In this work,
we investigate the (anti)kaon ($K^{-},\bar{K}^{0}$) condensation in
$\beta-$equilibrated nuclear matter within the framework of covariant density
functional theory. The functionals in the kaonic sector are constrained by the
experimental studies on $K^{-}$ atomic, kaon-nucleon scattering data fits. We
find that the equation of state softens with the inclusion of (anti)kaon
condensates, which lowers the maximum mass of neutron star. In one of the
density-independent coupling cases, the $K^{-}$ condensation is through a
first-order phase transition type, which produces a $2$ M⊙ neutron star. The
first-order phase transition results in mixed phase region in the inner core
of the stars. While $\bar{K}^{0}$ condensation appears via second-order phase
transition for all the models we consider here.

neutron stars; equation of state; anti-kaon condensates; mixed phase

##  I Introduction

Being born as a consequence of supernova explosion, the neutron stars provide
an idiosyncratic environment to study the exotic phases of matter at high
densities which may range from a few to several times nuclear saturation
density Glendenning (1996); Weber  _et al._ (2007). Such high density ranges
could never be achieved in any terrestrial laboratories now or in the near
future at low temperature as present inside these compact objects.
Consequently till now the true nature of matter above nuclear saturation
density is a matter of speculation. The neutron star being very compact is
subject to tremendous inward gravitational pull which is supported by the
degeneracy pressure mainly of neutrons with small admixture of other baryon
and lepton species. The exact nature of inter-particle interaction at such
high densities is not known precisely, though many authors have provided
various phenomenological models based on density functional theories Vautherin
and Brink (1972); Shen  _et al._ (1998); Douchin and Haensel (2001); Sun  _et
al._ (2008); Bao and Shen (2014) and realistic nuclear potentials Akmal  _et
al._ (1998); Li  _et al._ (2006); Carlson  _et al._ (2015); Hu  _et al._
(2017); Logoteta (2019) to explain and understand the nature of highly dense
matter and internal structure of these compact stars. Relativistic-Mean Field
(RMF) theory, also known as Walecka model is one of the foremost models
Walecka (1974). This model produces the saturation properties of nuclear
matter and finite nuclei to a good extent. However, in order to explain the
saturation properties viz. compression modulus and effective mass, non-linear
self-interaction terms for the scalar meson fields has been incorporated in
this model Walecka (1986); Schaffner and Mishustin (1996). In addition, other
self-interacting vector meson fields are also brought into consideration to
explain the interaction in high density regimes Bodmer (1991); Gmuca (1992);
Furnstahl  _et al._ (1996). But at those regimes due to the dependence on
higher orders of fields, instabilities may arise. The density-dependence of
the meson-baryon couplings (DDRH model) is another possible approach to reduce
the instabilities as in the later case the higher orders field terms are not
brought into consideration Typel (2005); Typel  _et al._ (2010). For
thermodynamic consistency, a re-arrangement term is considered which
contributes explicitly to the matter pressure consequently influencing the
equation of state at higher densities.

Nuclear matter is composed of mainly neutrons with small admixture of protons
and electrons in $\beta$-equilibrium condition and fraction of protons are
electrons are equal to keep the condition of charge neutrality. With the
increase of neutron density the electron density and hence the Fermi momentum
increases to keep the matter in $\beta$-equilibrium. When the Fermi momentum
of electron reaches the mass of muons, muons appear. With further increase of
density when the electron Fermi energy reaches the vacuum mass of mesons (pion
or kaon), condensate of negatively charged mesons starts to appear which in
turn help to maintain the charge neutrality. However, $s$-wave $\pi N$
scattering potential being repulsive the effective ground state mass of $\pi$
meson increases (Glendenning, 1985) opposing the possibility of $\pi$-meson
appearing. But, effective ground state mass of $K$-meson decreases due to its
attractive interaction with nucleons opening the possibility of $K$-meson
appearing. Kaplan and Nelson Kaplan and Nelson (1988); Nelson and Kaplan
(1987) for the very first time demonstrated that antikaon $K^{-}$ may undergo
Bose-Einstein condensation in dense matter formed in heavy-ion collisions.
Furthermore, other evidences such as the $K^{-}$ atomic data, kaon-nucleon
scattering data Lee  _et al._ (1995); Brown  _et al._ (1994); Lee  _et al._
(1994) studied by several authors in chiral perturbation theory also
encouraged the concept of $K^{-}$ condensed phase presence in the interior of
neutron star. The in-medium energy of (anti)kaon $K^{-}$ mesons decreases in
the dense matter due to the lowering of effective mass. Finally, the onset of
$s$-wave $K^{-}$ condensation occurs when the chemical potential of $K^{-}$
($\omega_{K^{-}}$) equates the electron chemical potential ($\mu_{e}$). The
$s$-wave $\bar{K}^{0}$ appears when its chemical potential
($\omega_{\bar{K}^{0}}$) equates to zero. The threshold density of (anti)kaon
appearance is very sensitive to the optical potential in nuclear symmetric
matter. Studies Brown  _et al._ (1994); Lee  _et al._ (1994); Knorren  _et
al._ (1995); Schaffner and Mishustin (1996) reveal that $K^{+}$ mesons develop
a repulsive optical potential nature in the nuclear matter. Thus, it may be
concluded that kaon $K^{+}$ condensation is not favored inside the neutron
star. Phase transitions from hadronic to kaonic phases in dense matter may be
either a first (mixed phase) or second order type depending on the (anti)kaon
optical potential depths. Various observational features of neutron star
evolution such as the spin down rates, cooling, glitches may be affected by
the alterations of weak interaction rates, transport properties of matter
interior to neutron star due to phase transitions Heiselberg and Hjorth-Jensen
(2000); Kubis and Kutschera (2003). The first order phase transition cannot be
explained by merely the Maxwell’s construction accounting for only one charge
conservation because neutron stars have two conserved charges viz. baryon
number conservation and global charge neutrality. The Gibbs conditions are
employed to adequately explain the mixed phase regime of the neutron star
interior Glendenning (1996).

The composition and equation of state (EOS) of matter inside the compact stars
are constrained by recent observations of compact objects in wide range of
electromagnetic spectra and in gravitational wave. Most fundamental property
of compact objects which constrain the EOS is the observed mass of neutron
stars (NSs) which sets the limit of maximum possible mass of NS family. One of
the most accurately measured pulsar masses was done by Hulse and Taylor of
binary pulsar PSR $1913+16$ with mass $1.4408\pm 0.0003$ M⊙ Hulse and Taylor
(1975). In recent years, several new neutron star mass measurements $>2$ M⊙
were accomplished viz. the millisecond pulsar (MSP) PSR J$1614-2230$ of mass
$1.97\pm 0.04$ M⊙ Demorest  _et al._ (2010); Özel  _et al._ (2010), PSR
J$0348+0432$ ($2.01\pm 0.04$ M⊙) Antoniadis  _et al._ (2013) and MSP
J$0740+6620$ ($2.14^{+0.20}_{-0.18}$ M⊙ with 95$\%$ credibility) Cromartie
_et al._ (2020). Gravitational wave observations have been also successful in
providing bounds on the compact object mass. The recent ‘GW190814’ event
observed by the LIGO-Virgo Collaboration (LVC) from a coalescence of a black
hole and a lighter companion sets the mass of the former to be
$23.2^{+1.1}_{-1.0}$ M⊙ and that of the latter to be $2.59^{+0.08}_{-0.09}$ M⊙
Abbott  _et al._ (2020). The nature of the lighter companion to be a heavy
neutron star or a light black hole is still not clear. Recently, the space
mission NICER (Neutron star Interior Composition ExploreR) also provided mass-
radius measurements of PSR J$0030+0451$ to be $1.44^{+0.15}_{-0.14}$ M⊙,
$13.02^{+1.24}_{-1.06}$ km Miller  _et al._ (2019) and $1.34^{+0.15}_{-0.16}$
M⊙, $12.71^{+1.14}_{-1.19}$ km Riley  _et al._ (2019) respectively.

The existence of massive NSs open the possibility of exotic matter appearance
in the inner core of the star. The addition of exotic degrees of freedom
enhances the softening of equation of states (EOSs). But with the observation
of neutron stars possessing mass greater than $2$ M⊙, the softer EOSs may be
mostly discarded as they lead to stars with lower maximum mass Glendenning
(1996). Various non-linear density-independent parametrizations viz. GM1, GM2,
GM3 Glendenning and Moszkowski (1991); TM1 Sugahara and Toki (1994) evaluates
relatively softer EOSs which cannot develop $2$ M⊙ neutron stars with
(anti)kaon condensation Banik and Bandyopadhyay (2001a); Pal  _et al._ (2000a)
undergoing $1^{\text{st}}-$order phase transition. GMT parametrization Pal
_et al._ (2000a) generates a relatively stiffer EOS thus producing a mixed
phase region in the neutron star interior. The GMT set is generated by
reproducing the saturation properties of TM1 parametrization with the
exclusion of non-linear self-interaction $\omega$-meson (vector) term
exclusion. Density-dependent model parametrizations such as DD2 are able to
produce stiff EOSs with baryon octet and (anti)kaon condensates as composition
leading to neutron stars possessing mass $\geq$ $2$ M⊙ Char and Banik (2014).

In this work, we explore the possibility of (anti)kaon condensation in
$\beta$-equilibrated nuclear matter in the inner core within the non-linear
and density-dependent covariant density functional (CDF) model which shows
consistent results with recent astrophysical observations.

The paper is organized in the following manner. In Sec. II we briefly discuss
the non-linear CDF and density-dependent relativistic hadron (DDRH) field
theory formalism incorporated in this work. Our results are shown in Sec. III
and our conclusions are summarized in Sec. IV.

##  II Formalism

###  II.1 Model

In this section, we introduce the non-linear and density dependent CDF model
to study the phase transition from hadronic to anti-kaon condensed matter
which could be either of the first-order or second-order form. Throughout the
work, for the hadronic matter we have considered nucleons ($N\equiv n,p$)
alongside electrons and muons. The strong interactions between the baryons as
well as the anti-kaons are mediated by the scalar $\sigma$, isoscalar-vector
$\omega^{\mu}$ and isovector-vector $\rho^{\mu\nu}$ meson fields. We have
considered the mean field model with non-linear (NL) scalar meson self
interaction as well as density-dependent coupling constants. Throughout the
model, the implementation of natural units is incorporated ($\hbar=c=1$). In
general, the total Lagrangian density of the matter is given by Glendenning
and Schaffner-Bielich (1999); Pal  _et al._ (2000a); Banik and Bandyopadhyay
(2001b, a); Pal  _et al._ (2000a); Glendenning and Schaffner-Bielich (1999)

$\displaystyle\mathcal{L}$
$\displaystyle=\sum_{N}\bar{\psi}_{N}(i\gamma_{\mu}D^{\mu}-m^{*}_{N})\psi_{N}+\sum_{l}\bar{\psi}_{l}(i\gamma_{\mu}\partial^{\mu}-m_{l})\psi_{l}$
(1)
$\displaystyle+D^{*}_{\mu}\bar{K}D^{\mu}K-m^{*^{2}}_{K}\bar{K}K+\frac{1}{2}(\partial_{\mu}\sigma\partial^{\mu}\sigma-
m_{\sigma}^{2}\sigma^{2})$
$\displaystyle-\frac{1}{4}\omega_{\mu\nu}\omega^{\mu\nu}+\frac{1}{2}m_{\omega}^{2}\omega_{\mu}\omega^{\mu}-\frac{1}{4}\boldsymbol{\rho}_{\mu\nu}\cdot\boldsymbol{\rho}^{\mu\nu}+\frac{1}{2}m_{\rho}^{2}\boldsymbol{\rho}_{\mu}\cdot\boldsymbol{\rho}^{\mu}$
$\displaystyle-\text{U}(\sigma),\quad\quad\text{[Only for NL model}].$

Here, U$(\sigma)$ stands for the self interactions of scalar mesons which is
present in case of NL model, but not considered in case of density-dependent
couplings, the fields $\psi_{N}$, $\psi_{l}$ correspond to the baryon and
lepton fields with their bare masses, $m_{N}$ and $m_{l}$ respectively. The
covariant derivative is given by

$D_{\mu}=\partial_{\mu}+ig_{\omega j}\omega_{\mu}+ig_{\rho
j}\boldsymbol{\tau}_{j}\cdot\boldsymbol{\rho}_{\mu}$ (2)

with ‘$j$’ denoting the nucleons and (anti)kaons. The isospin doublets for
kaons are denoted by $K\equiv(K^{+},K^{0})$ and that for anti-kaons by,
$\bar{K}\equiv(K^{-},\bar{K}^{0})$. The effective nucleon (Dirac) and anti-
kaon masses in the mean-field approximation are given by

$\displaystyle m_{N}^{*}=m_{N}-g_{\sigma N}\sigma,\quad
m_{K}^{*}=m_{K}-g_{\sigma K}\sigma$ (3)

where $m_{N}$, $m_{K}$ are the bare nucleon and kaon masses respectively. The
field strength tensors for the vector fields in eq.(1) are given by

$\displaystyle\omega_{\mu\nu}$
$\displaystyle=\partial_{\mu}\omega_{\nu}-\partial_{\mu}\omega_{\nu},$ (4)
$\displaystyle\boldsymbol{\rho}_{\mu\nu}$
$\displaystyle=\partial_{\nu}\boldsymbol{\rho}_{\mu}-\partial_{\mu}\boldsymbol{\rho}_{\nu}$

The scalar self-interaction terms Boguta and Bodmer (1977) required in NL
model are given by

$\text{U}(\sigma)=\frac{1}{3}g_{2}\sigma^{3}+\frac{1}{4}g_{3}\sigma^{4}$ (5)

where, $g_{2}=bm_{N}g^{3}_{\sigma N}$ and $g_{3}=cg^{4}_{\sigma N}$. In the
same approximation, the meson fields acquire the ground state expectation
values as

$\displaystyle\sigma$
$\displaystyle=-\frac{1}{m_{\sigma}^{2}}\frac{\partial\text{U}}{\partial\sigma}+\sum_{b}\frac{1}{m_{\sigma}^{2}}g_{\sigma
b}n_{b}^{s}+\sum_{\bar{K}}\frac{1}{m_{\sigma}^{2}}g_{\sigma
K}n_{\bar{K}}^{s},$ (6) $\displaystyle\omega_{0}$
$\displaystyle=\sum_{b}\frac{1}{m_{\omega}^{2}}g_{\omega
b}n_{b}-\sum_{\bar{K}}\frac{1}{m_{\omega}^{2}}g_{\omega K}n_{\bar{K}},$
$\displaystyle\rho_{03}$ $\displaystyle=\sum_{b}\frac{1}{m_{\rho}^{2}}g_{\rho
b}\boldsymbol{\tau}_{b3}n_{b}+\sum_{\bar{K}}\frac{1}{m_{\rho}^{2}}g_{\rho
K}\boldsymbol{\tau}_{\bar{K}3}n_{\bar{K}}$

The first term in right hand side of eq.(6) ($\sigma$-meson field) is required
only in the case of NL model. In case of DDRH model, the scalar self-
interaction terms are absent Hofmann  _et al._ (2001a, b). The meson-baryons
(anti-kaon) couplings are denoted by $g_{ij}$, where $i$ goes over the mesons
and $j$ over the baryons and (anti)kaons. $\tau_{j}$ represents the iso-spin
operator. The scalar and baryon(vector) number densities are defined for the
baryons as $n_{N}^{s}=\langle\bar{\psi}_{N}\psi_{N}\rangle$,
$n_{N}=\langle\bar{\psi}_{N}\gamma^{0}\psi_{N}\rangle$ respectively. In case
of the $s$-wave (anti)kaons condensates, the number density is given by
Glendenning and Schaffner-Bielich (1999)

$\displaystyle n_{K^{-},\bar{K}^{0}}$
$\displaystyle=2\left(\omega_{\bar{K}}+g_{\omega
K}\omega_{0}\pm\frac{1}{2}g_{\rho K}\rho_{03}\right)$ (7)
$\displaystyle=2m^{*}_{K}\bar{K}K$

The scalar density and vector number density of the baryon-$N$ at zero
temperature are given by

$\displaystyle n^{s}_{N}$
$\displaystyle=\frac{m^{*}_{N}}{2\pi^{2}}\left[p_{{F}_{N}}E_{F_{N}}-m_{N}^{*^{2}}\ln\left(\frac{p_{{F}_{N}}+E_{F_{N}}}{m_{N}^{*}}\right)\right],$
(8) $\displaystyle n_{N}$ $\displaystyle=\frac{p_{{F}_{N}}^{3}}{3\pi^{2}},$

respectively, where $p_{{F}_{N}}$ and $E_{F_{N}}$ are the Fermi momentum and
fermi energy of the $N$-th nucleon respectively. The in-medium energies of
$\bar{K}\equiv(K^{-},\bar{K}^{0})$ for s-wave condensation are provided by

$\omega_{K^{-},\bar{K}^{0}}=m^{*}_{K}-g_{\omega
K}\omega_{0}\mp\frac{1}{2}g_{\rho K}\rho_{03}$ (9)

with the isospin projections for $K^{-},\bar{K}^{0}$ being $-1/2,+1/2$
respectively.

The chemical potential of the $N$-th nucleon is

$\displaystyle\mu_{N}=\sqrt{p_{F_{N}}^{2}+m_{N}^{*2}}+g_{\omega
N}\omega_{0}+g_{\rho N}\boldsymbol{\tau}_{N3}\rho_{03}+\Sigma^{r},$ (10)

where, the rearrangement term $\Sigma^{r}$ is introduced to maintain the
thermodynamic consistency in case of DDRH model Hofmann  _et al._ (2001b)
which is given by

$\displaystyle\Sigma^{r}$ $\displaystyle=\sum_{N}\left[\frac{\partial
g_{\omega N}}{\partial n}\omega_{0}n_{N}-\frac{\partial g_{\sigma N}}{\partial
n}\sigma n_{N}^{s}+\frac{\partial g_{\rho N}}{\partial
n}\rho_{03}\boldsymbol{\tau}_{N3}n_{N}\right],$ (11)

where $n=\sum_{N}n_{N}$ is the total baryon number density. This re-
arrangement term contributes explicitly only to the matter pressure. In case
of NL CDF model, this term is not required.

The total energy density from the baryonic and leptonic matter is given by

$\displaystyle\begin{aligned}
\varepsilon_{f}&=\frac{1}{2}m_{\sigma}^{2}\sigma^{2}+\frac{1}{2}m_{\omega}^{2}\omega_{0}^{2}+\frac{1}{2}m_{\rho}^{2}\rho_{03}^{2}+\sum_{N}\frac{1}{\pi^{2}}\left[p_{{F}_{N}}E^{3}_{F_{N}}-\frac{m_{N}^{*2}}{8}\left(p_{{F}_{N}}E_{F_{N}}+m_{N}^{*2}\ln\left(\frac{p_{{F}_{N}}+E_{F_{N}}}{m_{N}^{*}}\right)\right)\right]\\\
&+\frac{1}{\pi^{2}}\sum_{l}\left[p_{{F}_{l}}E^{3}_{F_{l}}-\frac{m_{l}^{2}}{8}\left(p_{{F}_{l}}E_{F_{l}}+m_{l}^{2}\ln\left(\frac{p_{{F}_{l}}+E_{F_{l}}}{m_{l}}\right)\right)\right]\end{aligned}$
(12)

For the anti-kaons condensates, the energy density contribution to the total
one is given by

$\varepsilon_{\bar{K}}=m^{*}_{K}(n_{K^{-}}+n_{\bar{K}^{0}})$ (13)

Anti-kaons being Bose condensates, there is no direct contribution to the
total matter pressure from their ends. The matter pressure is related to the
energy density via the thermodynamic relation (Gibbs-Duhem) as

$p_{m}=\sum_{N}\mu_{N}n_{N}+\sum_{l}\mu_{l}n_{l}-\varepsilon_{f},$ (14)

as there will be no contribution from the anti-kaons. The total energy density
is given as $\varepsilon=\varepsilon_{f}+\varepsilon_{\bar{K}}$.

Initially at the lower density, matter is composed of neutrons ($n$), protons
($p$) and electrons ($e$) in beta-equilibrium and charge neutrality condition.
Beta-equilibrium at this stage is satisfied by the chemical potential balance,

$\mu_{n}=\mu_{p}+\mu_{e}$ (15)

With increasing density, when the chemical potential of electrons becomes
equal to the rest mass of muons, the muons appear. Hence the threshold
equilibrium condition for the onset of muons is, $\mu_{e}=m_{\mu}$.

Studies Prakash  _et al._ (1997); Glendenning (1996) shows that strangeness
changing processes such as, $N\rightleftharpoons N+\bar{K}$ and
$e^{-}\rightleftharpoons K^{-}$ may come into picture inside the neutron star
core. Hence kaons may appear by these reactions, when the threshold conditions
are satisfied as

$\displaystyle\mu_{n}-\mu_{p}=\omega_{K^{-}}=\mu_{e},\quad\omega_{\bar{K}^{0}}=0$
(16)

The charge neutrality conditions in the hadronic and kaon condensed phases are
given by

$\displaystyle Q^{h}$ $\displaystyle=\sum_{N}q_{N}n^{h}_{N}-n_{e}-n_{\mu}=0,$
(17) $\displaystyle Q^{\bar{K}}$
$\displaystyle=\sum_{N}q_{N}n^{\bar{K}}_{N}-n_{K^{-}}-n_{e}-n_{\mu}=0$

respectively, where $n^{h}_{N}$ and $n^{\bar{K}}_{N}$ represents the number
densities in hadronic and kaon phases respectively, both having the same form.

The appearance of (anti)kaon condensate may occur either through first order
or through second order phase transition from the hadronic to kaon phase
depending on the optical potential depths of $K^{-}$ at nuclear saturation
density. In case the transition sets through first order form, the mixed phase
comes into picture: the two phases of pure hadronic matter without condensate
and with condensate co-exist. Then for this case, the Gibbs conditions
alongside global baryon number conservation and charge neutrality can be
enforced to determine the mixed phase state Glendenning and Schaffner-Bielich
(1999, 1998); Glendenning (1992). The Gibbs conditions for this state are,

$\displaystyle p^{h}_{m}$ $\displaystyle=p^{\bar{K}}_{m},$ (18)
$\displaystyle\mu^{h}_{N}$ $\displaystyle=\mu^{\bar{K}}_{N}$ (19)

where $h$ and $\bar{K}$ superscripts represent the respective quantities in
hadronic and anti-kaon condensed phase respectively. In addition, two
additional global constraints (viz. global baryon number conservation and
charge neutrality) are enforced via the relations,

$\displaystyle n_{N}$ $\displaystyle=(1-\chi)n^{h}_{N}+\chi n^{\bar{K}}_{N},$
(20) $\displaystyle(1$ $\displaystyle-\chi)Q^{h}+\chi Q^{\bar{K}}=0$ (21)

respectively. Here, $\chi$ is the fraction of anti-kaon ($K^{-}$) condensed
phase in the mixed phase regime. $\chi\sim 0,1$ represents the initiation and
termination of mixed phase region respectively. Region with $\chi<0$ is the
pure hadronic phase and $\chi>1$ is the pure (anti)kaon condensed phase. In
the mixed phase, the total energy density also changes the form to,

$\varepsilon=(1-\chi)\varepsilon^{h}+\chi\varepsilon^{\bar{K}}$ (22)

where, $\varepsilon^{h}$, $\varepsilon^{\bar{K}}$ denotes the total energy
density in hadronic and kaon phases respectively.

###  II.2 Coupling parameters

In the NL CDF model, we adopt the GMT Pal  _et al._ (2000a) and NL3 Lalazissis
_et al._ (1997) parametrizations for meson-nucleon couplings. For the DDRH
model, DD-ME2 Lalazissis  _et al._ (2005), DD2 Typel  _et al._ (2010) and PKDD
Long  _et al._ (2004) parametrizations are considered for meson-nucleon
couplings. Table-1 provides the NL coupling parameters implemented in this
work.

Table 1: The coupling constants for the CDF models GMT, NL3, at nuclear saturation density $n_{0}$. Model | $g_{\sigma N}$ | $g_{\omega N}$ | $g_{\rho N}$ |  $g_{2}$ (fm-1) | $g_{3}$
---|---|---|---|---|---
GMT | 9.9400 | 12.2981 | 9.2756 | 10.5745 | $-24.1907$
NL3 | 10.2170 | 12.8680 | 8.9480 | 10.4310 | $-28.8850$

In case of the density-dependent model, the meson-nucleon coupling constants
are functions of density as

$g_{iN}(n)=g_{iN}(n_{0})f_{i}(x)\quad\quad\text{for }i=\sigma,\omega$ (23)

where, $x=n/n_{0}$, $n_{0}$ is the nuclear saturation density and

$f_{i}(x)=a_{i}\frac{1+b_{i}(x+d_{i})^{2}}{1+c_{i}(x+d_{i})^{2}}$ (24)

For the $\rho$-meson, the density-dependent coupling constant is given by

$g_{\rho N}(n)=g_{\rho N}(n_{0})e^{-a_{\rho}(x-1)}$ (25)

The density-dependent hadronic model parameters implemented in this work are
listed in table-2.

Table 2: The coupling constants for the DDRH models DD-ME2, DD2, PKDD, at $n_{0}$. Model | $g_{\sigma N}$ | $g_{\omega N}$ | $g_{\rho N}$
---|---|---|---
DD-ME2 | 10.5396 | 13.0189 | 7.3672
DD2 | 10.686681 | 13.342362 | 7.25388
PKDD | 10.7385 | 13.1476 | 8.5996
Table 3: The coefficient values for the density-dependent coupling parametrizations, at $n_{0}$. Model | $a_{\sigma}$ | $a_{\omega}$ | $a_{\rho}$ | $b_{\sigma}$ | $b_{\omega}$ | $c_{\sigma}$ | $c_{\omega}$ | $d_{\sigma}$ | $d_{\omega}$
---|---|---|---|---|---|---|---|---|---
DD-ME2 | 1.3881 | 1.3892 | 0.5647 | 1.0943 | 0.9240 | 1.7057 | 1.4620 | 0.4421 | 0.4775
DD2 | 1.35763 | 1.369718 | 0.518903 | 0.634442 | 0.496475 | 1.005358 | 0.817753 | 0.57581 | 0.638452
PKDD | 1.327423 | 1.342170 | 0.183305 | 0.435126 | 0.371167 | 0.691666 | 0.611397 | 0.694210 | 0.738376

The bare nucleon masses are considered to be identical for both the NL mean
field model cases as $m_{N}=939$ MeV. In the DDRH models we consider
$m_{N}=938.9,~{}938.565,~{}939.573$ MeV for DD-ME2, DD2, PKDD parametrizations
respectively. The bare mass of the (anti)kaons in this calculation is
considered to be 493.69 MeV.

The coefficients in eqs.(24)-(25) are fixed by several procedures. For details
we refer the readers to the references.-Typel  _et al._ (2010); Lalazissis
_et al._ (2005); Long  _et al._ (2004). The parameter values employed in the
DDRH model to get the density dependence is given in table-3. The meson-masses
for the GMT, NL3, DD-ME2, DD2, PKDD models of the coupling constants
incorporated in the calculations are listed in table-4.

Table 4: Mass values (in units of MeV) of the $\sigma$, $\omega$, $\rho$-mesons in GMT, NL3, DD-ME2, DD2, PKDD parametrizations. Model | $m_{\sigma}$ | $m_{\omega}$ | $m_{\rho}$
---|---|---|---
GMT | 511.198 | 783.000 | 770.000
NL3 | 508.194 | 782.501 | 763.000
DD-ME2 | 550.124 | 783.000 | 763.000
DD2 | 546.212459 | 783.000 | 763.000
PKDD | 555.5112 | 783.000 | 763.000

The respective parametrizations satisfy the values of the quantities at
$n_{0}$ as shown in the table-5. In the table, $E/A$, $K_{0}$, $a_{sym}$
denote the binding energy per nucleon, compression modulus, symmetry energy
coefficient.

Table 5: The nuclear properties of the CDF models at respective nuclear saturation densities. Model | $n_{0}$ | $E/A$ | $K_{0}$ | $a_{sym}$ | $m^{*}_{N}/m_{N}$
---|---|---|---|---|---
| (fm-3) | (MeV) | (MeV) | (MeV) |
GMT | 0.145 | $-16.3$ | 281.00 | 36.90 | $0.634$
NL3 | 0.148 | $-16.299$ | 271.76 | 37.40 | $0.600$
DD-ME2 | 0.152 | $-16.14$ | 250.89 | 32.30 | 0.572
DD2 | 0.149065 | $-16.02$ | 242.70 | 32.73 | 0.5625
PKDD | 0.149552 | $-16.267$ | 262.181 | 36.79 | 0.5712

The meson-(anti)kaon couplings are not considered to be density-dependent Char
and Banik (2014) in DDRH model. The vector coupling parameters in the kaon
sector are evaluated from the iso-spin counting rule and quark model Banik and
Bandyopadhyay (2001b) as

$g_{\omega K}=\frac{1}{3}g_{\omega N},\quad g_{\rho K}=g_{\rho N}$ (26)

and for the scalar coupling parameters, they are calculated at nuclear
saturation density from the real part of $K^{-}$ optical potential depth as

$U_{\bar{K}}(n_{0})=-g_{\sigma K}\sigma(n_{0})-g_{\omega
K}\omega_{0}(n_{0})+\Sigma^{r}_{N}(n_{0})$ (27)

where, $\Sigma^{r}_{N}(n_{0})$ is the contribution from the nucleons alone and
is not considered in case of NL CDF model.

Experimental studies Li  _et al._ (1997); Pal  _et al._ (2000b) show that the
kaons experience a repulsive interaction in nuclear matter whereas antikaons
experience an attractive potential. Several model calculations Koch (1994);
Waas and Weise (1997); Lutz (1998); Ramos and Oset (2000); Schaffner-Bielich
_et al._ (2000) provide a very broad range of optical potential values as
$-120\leq U_{\bar{K}}\leq-40$ MeV. Another calculation from hybrid model
Friedman  _et al._ (1999) suggests the value of $K^{-}$ optical potential to
be in the range $180\pm 20$ MeV at nuclear saturation density. In this work,
we have considered a $K^{-}$ potential range of $-160\leq U_{\bar{K}}\leq-120$
MeV and the scalar meson-(anti)kaon couplings evaluated for the above
potential depth range are listed in table-6.

Table 6: Parameter values of the scalar $\sigma$ meson-(anti)kaon couplings in GMT, NL3, DD-ME2 and DD2 parametrizations at $n_{0}$. Model |  $U_{\bar{K}}$ (MeV)
---|---
| $-120$ | $-140$ | $-160$
GMT | 0.8217 | 1.4006 | 1.9796
NL3 | 0.4707 | 1.0088 | 1.5469
DD-ME2 | 0.4311 | 0.9553 | 1.4796
DD2 | 0.3155 | 0.8359 | 1.3562
PKDD | 0.4309 | 0.9639 | 1.4970

##  III Results

We model the matter to be composed of nucleons with the possibility of
(anti)kaon condensate appearance with increasing density. We compare the
matter properties and the resultant star structure with different approach in
calculating the EOS of the matter within relativistic mean field theory
considering different (anti)kaon optical potentials
$U_{\bar{K}}=-120,-140,-160$ MeV.

###  III.1 Non-Linear CDF model

Figure 1: The variation of matter pressure with energy density. Left panel:
GMT parameterization, right panel: NL3 parameterization cases with several
(anti)kaon potential depths ($U_{\bar{K}}$). The solid lines represent the
composition of only nucleons ($n,p$) and leptons ($e^{-},\mu^{-}$). The other
curves represent the composition with (anti)kaons in addition to nucleons and
leptons. The dashed curve depicts $U_{\bar{K}}=-120$ MeV, dash-dotted curve
with $U_{\bar{K}}=-140$ MeV and dotted curve with $U_{\bar{K}}=-160$ MeV.

In the case of GMT parametrization, the transition from hadronic to kaonic
phase is observed to be of second order for $U_{\bar{K}}=$ $-120,-140$ MeV but
of first order for $U_{\bar{K}}=-160$ MeV. While in case of NL3
parameterization, the phase transition is of second order for the whole range
of $K^{-}$ potential adopted here.

Fig.1 shows the matter pressure as a function of energy density for both the
GMT and NL3 models. The appearance of (anti)kaons to a great extent softens
the EOS. The two kinks in the EOSs marks the onset of $K^{-}$ and
$\bar{K}^{0}$ respectively. The two kinks are observed to be in higher
densities for the NL3 model in comparison to the GMT case referring the delay
of (anti)kaons into the matter for the former case. The appearance of
(anti)kaon condensation is through first order transition only for $K^{-}$
with $U_{\bar{K}}=-160$ MeV in GMT parametrization. In other cases it is
second order phase transition.

Fig.2 shows the results of the mass-radius (M-R) relationship for static
spherical stars from solution of the Tolman-Oppenheimer-Volkoff (TOV)
equations Glendenning (1996) corresponding to the EOSs discussed here and
shown in fig.1. For the crust, we have considered the EOS of Baym, Pethick and
Sutherland Baym  _et al._ (1971). Table-7 provides the set of maximum mass
values, corresponding radius and central density for the nucleons and
(anti)kaon EOSs with various values of $U_{\bar{K}}$. Inclusion of (anti)kaons
leads to reduction of maximum mass of the compact stars.

Table 7: Parameter values of the maximum mass stars. Here, maximum mass, Mmax (in units of M⊙), radius (in km), corresponding central density $n_{c}$ (in units of $n_{0}$) of nucleon compact stars for different values of $K^{-}$ optical potential depths $U_{\bar{K}}$ (in units of MeV) at $n_{0}$ in GMT and NL3 parametrization models. | GMT | NL3
---|---|---
$U_{\bar{K}}$ | Mmax | R | $n_{c}$ | Mmax | R | $n_{c}$
(MeV) | (M⊙) | (km) | ($n_{0}$) | (M⊙) | (km) | ($n_{0}$)
$0$ | 2.66 | 12.80 | 4.91 | 2.77 | 13.14 | 4.52
$-120$ | 2.44 | 13.31 | 4.76 | 2.59 | 13.56 | 4.37
$-140$ | 2.27 | 13.36 | 4.75 | 2.47 | 13.59 | 4.47
$-160$ | 2.00 | 13.22 | 4.78 | 2.28 | 13.48 | 4.53
Figure 2: The M-R relations corresponding to the EOSs shown in fig.1. Left
panel: GMT, right panel: NL3 parameterization. The solid curve corresponds to
the pure nucleonic star, while the dashed, dash-dotted, dotted curves
correspond to stars with (anti)kaon condensation and optical potentials,
$U_{\bar{K}}=-120,-140,-160$ MeV respectively. The mass constraints from the
various astrophysical observations are represented by the region enclosed by
dotted corresponding to the GW190814 observation Abbott  _et al._ (2020),
dashed-lines to the MSP J0740+6620 Cromartie  _et al._ (2020). The mass-radius
limits obtained for PSR J0030+0451 from the NICER experiment Miller  _et al._
(2019); Riley  _et al._ (2019) are represented by the squared dashed regions.

The variation of particle fractions in the matter with total baryonic number
density is shown in the fig.3 for $U_{\bar{K}}(n_{0})=-160$ MeV for both the
parametrization sets. For GMT parametrization the mixed phase initiates with
the onset of $K^{-}$ at $\sim 2.22~{}n_{0}$. In addition to the global
conservation rule of baryon number (eq.(20)) and charge neutrality (eq.(21)),
pressure and chemical potential equilibrium conditions between two phases
(eqs.(18),(19)) determines the mixed phase region. Due to higher rest mass of
(anti)kaons compared to the lepton species and being bosons, the former
condense in the lowest energy state and so are preferred to maintain the
global charge neutrality condition. This results in decrease in electron and
muon populations as can be clearly visualized in fig.3. The mixed phase
terminates at $\sim 2.90~{}n_{0}$. Further, with the appearance of
$\bar{K}^{0}$ at $\sim 3.49~{}n_{0}$ and ceasing of electron population around
$4-4.5~{}n_{0}$ the proton and $K^{-}$ populations becomes equal following the
charge neutrality condition. $\bar{K}^{0}$ condensates through the second
order phase transition. However, for NL3 parametrization the phase transition
occurs via second order for both the (anti)kaons ($K^{-},\bar{K}^{0}$)
implying the absence of mixed phase regime. Even though we are fixing the
$U_{\bar{K}}$ identical to GMT model cases, the (anti)kaons appear at a
slightly higher densities compared to the former case.

The threshold densities for the onset of $K^{-},\bar{K}^{0}$ in the dense
nuclear matter for different $K^{-}$ potentials are provided in Table-8. It
can be observed that the threshold densities shifts towards lower densities
with the increase in strength of $U_{\bar{K}}$ at $n_{0}$.

Table 8: Threshold densities, $n_{cr}$ (in units of $n_{0}$) for antikaon condensation in dense nuclear matter for different values of $K^{-}$ optical potential depths $U_{\bar{K}}$ (in units of MeV) at $n_{0}$. | GMT | NL3
---|---|---
$U_{\bar{K}}$ |  $n_{cr}$($K^{-}$) |  $n_{cr}$($\bar{K}^{0}$) |  $n_{cr}$($K^{-}$) |  $n_{cr}$($\bar{K}^{0}$)
(MeV) | ($n_{0}$) | ($n_{0}$) | ($n_{0}$) | ($n_{0}$)
$-120$ | 2.87 | 4.45 | 2.77 | 4.35
$-140$ | 2.56 | 3.96 | 2.49 | 3.94
$-160$ | 2.22 | 3.49 | 2.24 | 3.53
Figure 3: Population densities $n_{i}$ (in units of $n_{0}$) of various
species as a function of baryon number density. Left panel: GMT, right panel:
NL3 model and $K^{-}$ potential depth of $-160$ MeV. Solid lines denote
neutron ($n$), long-dashed lines proton ($p$), dash-dotted curves electron
($e^{-}$), dotted lines muon ($\mu^{-}$), short-dashed lines $K^{-}$ and dash-
double dotted lines denote $\bar{K}^{0}$ population.

Fig.4 shows the extent of mixed phase region inside the neutron star modelled
with GMT parametrization. It is evident that the mixed phase regime starts
($\chi\sim 0$) from around matter density of $\sim 2.22~{}n_{0}$ which
corresponds to star radius $7.34$ km, and terminates ($\chi\sim 1$) at around
matter density of $2.9~{}n_{0}$ or corresponding stellar radius of, $\sim
6.13$ km.

Figure 4: Baryon number density ($n$), radial distance ($r$) inside the
compact star as a function of volume fraction of the (anti)kaon condensate
($\chi$) for the GMT model case with $U_{\bar{K}}=-160$ MeV. The solid curve
denotes $\chi(r)$, while dashed curve represents $\chi(n)$.

The interactions between (anti)kaons and nucleons alters the nucleon effective
mass in the mixed phase regime where both the hadronic and kaonic phase co-
exist. This effect in shown in fig.5. With a difference of $\sim 100$ MeV, the
effective nucleon masses are observed to increase in kaonic phase while it
decreases in the pure hadronic phase as we move interior towards the pure
kaonic phase regime.

Figure 5: Effective nucleon mass, $m^{*}_{N}$ as a function of baryon number
density for the GMT model case with $U_{\bar{K}}=-160$ MeV. The solid curve
denotes $m^{*}_{N}(n_{B})$ in hadronic phase, while dash-dotted curve
represents $m^{*}_{N}(n_{B})$ in kaonic phase. Region with $n<2.2~{}n_{0}$ is
the pure hadronic phase, between $2.2\leq n\leq 2.9~{}n_{0}$ is the mixed
phase and $n>2.9~{}n_{0}$ is the pure (anti)kaonic phase.

The charge densities of each normal and kaon condensed phase in the mixed
phase region for the GMT model with $U_{\bar{K}}=-160$ MeV as a function of
the kaon volume fraction is shown in fig.6. The central solid black line
represents the global charge neutrality condition as provided in eq.(21).

Figure 6: Charge densities in the pure hadronic phase and (anti)kaon condensed
phase as a function of volume fraction of the latter for the GMT model case
with $U_{\bar{K}}=-160$ MeV. $Q^{h},Q^{\bar{K}}$ denote the charge in hadronic
and kaonic phases respectively.

The (anti)kaon energies as a function of baryon number density with
$U_{\bar{K}}=-120$, $-140$, $-160$ MeV for the two parameter sets (GMT, NL3)
are shown in fig.7. The in-medium energies for both the (anti)kaons decrease
with density. The dashed curve representing $\mu_{e}$ crossing over the
$\omega_{\bar{K}}$ curves marks the end of pure hadronic phase and initiation
of pure kaonic phase. $K^{-}$ condensations initiates once the value of
$\omega_{K^{-}}$ reaches that of the electron chemical potential and
$\bar{K}^{0}$ condensation starts when the value of $\omega_{\bar{K}^{0}}$
equates to zero. From fig.7, it is observed that the threshold condition,
$\omega_{K^{-}}=\mu_{e}$ is achieved way early than the
$\omega_{\bar{K}^{0}}=0$ one, leading to earlier appearance of $K^{-}$.

Figure 7: The effective energy of (anti)kaons as a function of baryon number
density, left panels: GMT, right panels: NL3 parametrization. Upper panels:
in-medium energies of $\bar{K}^{0}$, lower panels: in-medium energies of
$K^{-}$ with several $U_{\bar{K}}$. The dashed lines represent the respective
electron chemical potentials for each model case. The solid curve exhibits the
$U_{\bar{K}}$ strengths of $-120$ MeV, dash-dotted lines exhibits $-140$ MeV
case and dotted lines exhibits the $-160$ MeV case.

The difference in the energy densities of the hadronic and kaonic phases for
GMT parameterization with $U_{\bar{K}}=-160$ MeV is shown in fig.8. The total
energy density in the mixed phase region is evaluated from eq.(22) and grows
monotonically with density.

Figure 8: Energy density as a function of baryon number density for the GMT model with $U_{\bar{K}}=-160$ MeV for the $3$ phases. The solid curve denotes the total energy density variation, while dash-dotted curves represent $\varepsilon^{h}$ and dash-double dotted curve represent $\varepsilon^{\bar{K}}$. The three regions demarcation are similar as fig.5. Figure 9: The equation of states, Left panel: DD-ME2, center panel: DD2, right panel: PKDD parametrization case with several (anti)kaon potential depths ($U_{\bar{K}}$). The solid lines represent the composition of only nucleons ($n,p$) and leptons ($e^{-},\mu^{-}$). The dashed curves depict $U_{\bar{K}}=-120$ MeV, dash-dotted curves with $U_{\bar{K}}=-140$ MeV and dotted curves with $U_{\bar{K}}=-160$ MeV represent the composition with (anti)kaons. Table 9: Threshold densities, $n_{cr}$ (in units of $n_{0}$) for antikaon condensation in dense nuclear matter for different values of $K^{-}$ optical potential depths $U_{\bar{K}}$ (in units of MeV) at $n_{0}$ with density-dependent DD-ME2, DD2, PKDD parametrizations. | DD-ME2 | DD2 | PKDD
---|---|---|---
$U_{\bar{K}}$ (MeV) |  $n^{K^{-}}_{cr}$($n_{0}$) |  $n^{\bar{K}^{0}}_{cr}$($n_{0}$) |  $n^{K^{-}}_{cr}$($n_{0}$) |  $n^{\bar{K}^{0}}_{cr}$($n_{0}$) |  $n^{K^{-}}_{cr}$($n_{0}$) |  $n^{\bar{K}^{0}}_{cr}$($n_{0}$)
$-120$ | 3.00 | 4.89 | 3.08 | 5.11 | 3.16 | 4.75
$-140$ | 2.72 | 4.42 | 2.79 | 4.62 | 2.82 | 4.29
$-160$ | 2.47 | 3.96 | 2.53 | 4.14 | 2.53 | 3.84

###  III.2 Density-Dependent CDF model

For DDRH model, the phase transition from hadronic to kaonic phase for all
parametrizations and all (anti)kaon optical potential depths considered in the
present work is through the second-order phase transition.

The matter pressure as a function of energy density (EOS) for DDRH model is
shown in fig.9. It is observed that the stiffest EOS results in for the DD-ME2
parametrization. Incorporation of $\bar{K}$ softens the EOS and more
pronounced effects are seen with deeper $K^{-}$ optical potential depths. The
two kinks in the EOS marks the onsets of $K^{-}$ and $\bar{K}^{0}$
respectively. While the first kink appears to be $\sim 400$ MeV/fm3 for all
the three parametrizations, the onset of $\bar{K}^{0}$ denoted by the second
kink is delayed the most in DD2 parametrization case.

Table-9 provides the threshold densities of (anti)kaon condensation in
different potential depths of (anti)kaons. It can be observed from table-9
that the appearance of $K^{-}$-mesons is foremost in case of DD-ME2 model
irrespective of the optical potential. On the other hand, the appearance of
$\bar{K}^{0}$ is the earliest in PKDD model among others. With the increasing
potential depth of (anti)kaons in symmetric nuclear matter, the onset of
(anti)kaons happens earlier.

Fig.10 presents the results of mass-radius (M-R) relationship for static
spherically symmetric stars by solving the TOV equations corresponding to the
EOSs in fig.9. The crust EOS is the same as considered in the NL CDF model.
Table-10 provides the set of maximum mass, and corresponding radius and
central density with different values of $U_{\bar{K}}$. For the stars with
only nucleons and leptons, the maximum masses are $2.48$, $2.42$, $2.33$ M⊙
with DD-ME2, DD2 and PKDD parametrization respectively. The maximum mass of
the compact stars decreases substantially with the inclusion of (anti)kaons.
It is observed that configurations with DD-ME2 satisfy the observed maximum
mas neutron star constraint of $\sim 2$ M⊙ uptil $U_{\bar{K}}=-160$ MeV. For
the other two model cases, DD2 satisfy the constraint uptil $U_{\bar{K}}=-140$
MeV, while PKDD providing a further softer EOS satisfy the constraint only
upto $U_{\bar{K}}=-120$ MeV.

Fig.11 represents the (anti)kaon energies as a function of baryon number
density with $U_{\bar{K}}=-120$ MeV for the density-dependent coupling models.
The onset of $K^{-}$ and $\bar{K}^{0}$ occurs with $\omega_{K^{-}}$ crossing
over $\mu_{e^{-}}$ and $\omega_{\bar{K}^{0}}$ being equal to zero
respectively. Similar behavior is observed in case of DD-ME2 and DD2
parametrization models, while for PKDD model, the $K^{-}$ in-medium energy is
higher compared to the other two cases and $\bar{K}^{0}$ in-medium energy is
observed to be lower.

Figure 10: The M-R relations corresponding to the EOSs in fig.9. Left panel: DD-ME2, center panel: DD2, right panel: PKDD parametrization. The mass-radius constraints are similar as in fig.2. The solid curve denotes the case with only nucleons and leptons, while the dashed curves correspond to $U_{\bar{K}}=-120$ MeV, dash-dotted curves represents $U_{\bar{K}}=-140$ MeV and dotted curves denote $U_{\bar{K}}=-160$ MeV case. Table 10: Maximum mass, Mmax (in units of M⊙), radius (in km), corresponding central density (in units of $n_{0}$) of nucleon compact stars for different values of $K^{-}$ optical potential depths $U_{\bar{K}}$ (in units of MeV) at $n_{0}$ in DD-ME2, DD2, PKDD parametrization models. |  | DD-ME2 | DD2 | PKDD
---|---|---|---|---
Configuration |  $U_{\bar{K}}$ (MeV) | Mmax(M⊙) | R (km) |  $n_{c}$ ($n_{0}$) | Mmax(M⊙) | R (km) |  $n_{c}$ ($n_{0}$) | Mmax(M⊙) | R (km) |  $n_{c}$ ($n_{0}$)
$npe\mu$ | 0 | 2.48 | 11.96 | 5.36 | 2.42 | 11.77 | 5.69 | 2.33 | 11.63 | 5.94
| $-120$ | 2.29 | 12.28 | 5.37 | 2.21 | 12.14 | 5.68 | 2.10 | 12.31 | 5.54
$npe\mu\bar{K}$ | $-140$ | 2.18 | 12.37 | 5.33 | 2.09 | 12.23 | 5.62 | 1.95 | 12.44 | 5.36
| $-160$ | 2.01 | 12.43 | 5.22 | 1.92 | 12.29 | 5.47 | 1.73 | 12.56 | 5.05

The population densities of different species, $n_{i}$ (in units of $n_{0}$)
in the neutron star interior as a function of baryon number density for the
density-dependent models are provided in fig.12. (Anti)kaons, being bosons are
not constrained by Pauli blocking, resulting in lepton fraction suppression at
high density regions. The population behavior in two cases of DD-ME2 and DD2
are observed to be similar with slight difference in the threshold density of
$\bar{K}$-meson appearance. The proton population (subsequently electron and
muon) is higher at lower densities for PKDD model case compared to DD-ME2, DD2
models due to higher symmetry energy. For PKDD model, in addition to the early
onset of $\bar{K}^{0}$ particles, the eradication of lepton species is quite
delayed compared to the former parametrizations. This results in further
softening of EOS.

Figure 11: The effective energy of (anti)kaons as a function of baryon number
density with $U_{\bar{K}}=-120$ MeV. Upper panel: for $\bar{K}^{0}$ and lower
panel: for $K^{-}$ with solid lines denoting DD-ME2, dash-dotted curves
representing DD2 and dotted lines representing PKDD parametrization. Figure
12: Same as fig.3. Upper panel: DD-ME2, center panel: DD2, lower panel: PKDD
parametrization model and $K^{-}$ potential depth of $-120$ MeV. Solid curves
denote neutron ($n$), long-dashed curves proton ($p$), dash-dotted lines
electron ($e^{-}$), dotted lines muon ($\mu^{-}$), short-dashed curves
$K^{-}$, dash-double dotted lines denote $\bar{K}^{0}$ population.

The inclusion of hyperons as well as $\Delta$-resonances alongwith
(anti)-kaons in neutron star matter in case of DD-ME2 coupling parametrization
model is shown in fig.13. In this case, the meson-baryon as well as the
meson-(anti-kaon) interactions are considered to be mediated by
$\sigma,\omega,\rho,\sigma^{*},\phi$-mesons. The coupling parameters for the
meson-hyperon and meson-$\Delta$ baryon interactions are adapted from
reference-Thapa  _et al._ (2020). The EOS with matter composition as nucleons,
hyperons, $\Delta$-resonances, (anti)-kaons (NY$\Delta$K) is observed to be
stiffer than the one without $\Delta$-baryons (NYK). This happens because of
the early appearance of $\Delta$-resonance particles which pushes the advent
of hyperons to higher densities. The phase transition of (anti)-kaons is
observed to be second-order form in this matter composition indicating the
absence of mixed phase. Detailed discussions on this matter composition will
be provided in future works.

##  IV Conclusions and Outlook

We investigate the appearance of the (anti)kaon condensation in
$\beta$-equilibrated charge neutral nucleonic matter within the framework of
mean field theory with density-independent (NL CDF model) as well as dependent
(DDRH model) coupling constants. Observations constrain the maximum mass of
neutron star family to be above $2$ M⊙. Stars with mass near and above $2$ M⊙
must contain central density above $4~{}n_{0}$. At such high density the phase
transition to boson condensate within the nuclear matter is highly probable.
However, the appearance of (anti)kaon condensation softens the equation of
state lowering the maximum mass of the neutron star family. We discuss here
certain parametrizations of EOS within RMF model which are stiff enough to
provide maximum mass above $2$ M⊙ with appearance of (anti)kaon condensation
at the inner core of the star. We obtain the result that with NL model, GMT
and NL3 parametrizations provide the required stiff EOS to have maximum mass
$2$ M⊙ with all the kaon optical potential depth $U_{\bar{K}}=-120,-140,-160$
MeV. The observational constraint from GW190814 is uncertain as the star has
not been identified as NS unambiguously Abbott  _et al._ (2020); Sedrakian
_et al._ (2020); Jie Li  _et al._ (2020). The (anti)kaon condensed star with
$U_{\bar{K}}=-120$ MeV and NL3 coupling model persuades this gravitational
wave observation mass constraint. For DDRH model, the DD-ME2 parametrization
gives the observationally possible M-R relation for all the considered values
of $U_{\bar{K}}$.

Figure 13: Left panel: Equation of states, right panel: M-R relations of the
neutron star matter with composition as, Nucleons, Hyperons, (anti)-kaons
(NYK) and Nucleons, Hyperons, $\Delta$-resonances, (anti)-kaons (NY$\Delta$K)
represented by solid and dashed lines respectively within DD-ME2 model. The
mass-radius constraints are similar as in fig.2. The (anti)-kaon optical
potential is considered to be $-140$ MeV.

In most of the cases, the transition to condensed phase is through second
order phase transition. First order phase transition occur only with
$U_{K}=-160$ MeV for NL model with GMT parametrization. In this case the star
possesses a mixed phase region containing both hadronic and condense phase
simultaneously. To explain the mixed phase regime, Gibbs rules alongside
global baryon number conservation and charge neutrality are exploited. The
mixed phase region ranges for a length of $\sim 1.21$ km from stellar radius
of 6.13 to 7.34 km. The outer core region upto 7.34 km is the pure hadronic
phase while the region in the inner core from 6.13 km to center of the star is
the pure (anti)kaon condensed phase. Moreover, the $\bar{K}^{0}$ condensation
is a second-order phase transition. The compact star with the mixed phase
regime satisfies the bounds set on mass-radius by the various recent
astrophysical observations. The EOS (mixed phase) evaluated with GMT model
incorporating the $\bar{K}$ condensation which satisfies $2$ M⊙ criteria can
be employed to study the glitch phenomena in pulsars.

In case of density-dependent parametrization models (DD-ME2, DD2, PKDD), the
(anti)kaon condensation is through second-order phase transition. Among these
parametrizations, DD-ME2 produces the stiffest equation of state for both the
cases with only nucleons as well as nucleons and (anti)kaons. All parameter
sets explain the $2$ M⊙ neutron star without the inclusion of (anti)kaons.
Higher optical potential leads to early appearance of $K^{-}$ in the star
interior. In case of DD2 model, the configuration with $U_{\bar{K}}=-160$ MeV
doesn’t satisfy the $\sim 2$ M⊙ maximum mass star. For the PKDD model
producing the softest EOS among the considered coupling models, the
astrophysical maximum mass constraint is not satisfied with
$U_{\bar{K}}\geq-140$ MeV. Within the framework model considered in this work,
the upper limit for $U_{\bar{K}}$ in case of DD-ME2 model is $-160$ MeV. The
likelihood of hyperon and $\Delta$-baryons in the neutron star matter
alongside (anti)-kaons are also studied with DD-ME2 model and
$U_{\bar{K}}=-140$ MeV. The (anti)-kaons tend to appear at the higher density
for the matter with $\Delta$-baryons in comparison to the one without
$\Delta$-resonances in the neutron star matter. The mass-radius constraints
are observed to be satisfied by this matter composition as well. Further
analysis on the effects due to the presence of hyperons as well as
$\Delta$-baryons with the appearance of (anti)-kaonic condensation in neutron
star matter is beyond the scope of this work.

###### Acknowledgements.

The authors thank the anonymous referee for the constructive comments which
have contributed to ameliorate the quality of the manuscript significantly.
The authors acknowledge the funding support from Science and Engineering
Research Board, Department of Science and Technology, Government of India
through Project No. EMR/2016/006577 and Ministry of Education, Government of
India. The authors are also thankful to Sarmistha Banik and Debades
Bandyopadhyay for fruitful discussions.

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