# Hybrid Strangeon Stars

Chen Zhang<EMAIL_ADDRESS>The HKUST Jockey Club Institute for Advanced
Study, The Hong Kong University of Science and Technology, Hong Kong S.A.R.,
P.R.China  Yong Gao<EMAIL_ADDRESS>Max-Planck-Institut für
Gravitationsphysik (Albert-Einstein-Institut), Am Mühlenberg 1, D-14476
Potsdam-Golm, Germany  Department of Astronomy, School of Physics, Peking
University, Beijing 100871, China  Kavli Institute for Astronomy and
Astrophysics, Peking University, Beijing 100871, China  Cheng-Jun Xia
<EMAIL_ADDRESS>Center for Gravitation and Cosmology, College of Physical
Science and Technology, Yangzhou University, Yangzhou 225009, China  Renxin Xu
<EMAIL_ADDRESS>Department of Astronomy, School of Physics, Peking
University, Beijing 100871, China  Kavli Institute for Astronomy and
Astrophysics, Peking University, Beijing 100871, China

###### Abstract

It was conjectured that the basic units of the ground state of bulk strong
matter may be strange-clusters called strangeons, and they can form self-bound
strangeon stars that are highly compact. Strangeon stars can develop a strange
quark matter (SQM) core at high densities, particularly in the color-flavor-
locking phase, yielding a branch of hybrid strangeon stars. We explore the
stellar structure and astrophysical implications of hybrid strangeon stars. We
find that hybrid strangeon stars can meet various astrophysical constraints on
pulsar masses, radii, and tidal deformabilities. Finally, we show that the
strangeon-SQM mixed phase is not preferred if the charge-neutrality condition
is imposed at the strangeon-SQM transition region.

##  I Introduction

The detection of gravitational waves (GWs) from the coalescence of compact
binaries by LIGO/Virgo collaborations LIGOScientific:2016aoc ;
LIGOScientific:2017bnn ; LIGOScientific:2018mvr ; TheLIGOScientific:2017qsa ;
Abbott:2018wiz ; Abbott:2020uma ; Abbott:2020khf  has greatly improved our
knowledge of black holes and compact stars. They offer unique opportunities to
probe unconventional QCD matter phases, such as quark matter and strangeon
matter.

Quark matter (QM), a state comprised of deconfined free-flowing quarks, can
possibly exist inside the neutron star core (i.e. conventional hybrid stars
Alford:2004pf ; Alford:2013aca ). If they are stable at zero pressure, either
in form of strange quark matter (SQM) Bodmer:1971we ; Witten ; Terazawa:1979hq
; Farhi:1984qu  or up-down quark matter ($ud$QM) Holdom:2017gdc , they can
constitute entire quark star Haensel:1986qb ; Alcock:1986hz ; Xu:1999bw ;
Yang:2023haz ; Zhang:2019mqb ; Zhao:2019xqy ; Ren:2020tll ; Cao:2020zxi ;
Wang:2021byk ; Yuan:2022dxb ; Restrepo:2022wqn ; Xia:2020byy ; Xia:2022tvx  or
the crust (i.e. inverted hybrid stars Ren:2022loa ), both with potentially
distinct astrophysical implications Xu:1998ur ; Xu:2000sr ; Ren-Xin:2002cvr ;
Wu:2006de ; Kankiewicz:2016dha ; Kuerban:2019prr ; Usov:1997ff ;
Glendenning:1994zb ; Geng:2015uja ; Zou:2022voz ; Geng:2021apl ; Wang:2021jyo
; Zhang:2018zdn ; Huang:2017ocl ; Zhang:2023zth . Effects from QCD
interactions such as color-superconductivity and perturbative QCD (pQCD)
corrections can help quark stars meet various astrophysical constraints
Weissenborn:2011qu ; Zhou:2017pha ; Zhang:2020jmb . Generally, it is expected
that at very high densities quark matter should be in the color-flavor-locking
phase (CFL), where $u,d,s$ quarks form cooper pairs antisymmetrically in
color-flavor space with equal fractions by the attractive one-gluon exchange
channel, providing a lowered energy state.

Strangeon matter (SM) is similar to strange quark matter where both are
composed of a nearly equal number of $u,d,s$ quarks Xu:2003xe ; Lai:2009cn ;
Lai:2017ney ; Miao:2020cqj . However, strangeon matter has quarks localized as
clusters in a globally solid state due to the large masses of and the strong
coupling between strangeons. Strangeon stars Xu:2003xe ; Zhou:2004ue ;
Zhou:2014tba ; Lai:2009cn ; Lai:2017ney ; Miao:2020cqj ; Lai:2017mjv ;
Lai:2018ugk ; Lai:2020mlu ; Gao:2021uus  composed of strangeon matter have
intrinsic stiff equation of state (EOS) and large compactness, and they had
already been proposed to support massive pulsars ($\gtrsim 2M_{\odot}$
Lai:2009cn ) before the announcement of the first massive pulsar PSR
J1614-2230 Demorest:2010bx . Recently, we have shown that all strangeon stars
are compact enough to feature a photonsphere that is essential to the
generation of GW echoes Zhang:2023mzb .

The transition from strangeon matter to strange quark matter is likely to
occur, considering such “deconfinement” originates from a shrinking of
strangeon lattice spacing as density or pressure increases so that the lattice
constant becomes smaller than the radius of individual quark bags, as
described by the linked bag model in Ref. Miao:2020cqj . This gives rise to a
new type of stellar objects, the _Hybrid Strangeon Stars_ , consisting of a
strangeon crust and a strange quark matter core. Pure strangeon stars can form
from neutron stars absorbing strangeon nuggets, or quantum nucleation in the
interior. If SQM is more stable than SM at some density, then the same process
can take place and form hybrid strangeon stars directly or through the SQM
quantum nucleation inside strangeon stars. Such first-order phase transition
needs the center pressure to be larger than some critical value at the
corresponding central chemical potential. Such lift of center pressure beyond
critical point can happen from spin-down, accretion or merger of strangeon
stars.

As for the organization of this paper, we first introduce the EOSs of SM and
SQM, and constrain the EOS parameters from the stability considerations. Then,
with Maxwell constructions where a sharp interface is assumed, we solve the
hybrid stellar structures and study their compatibility with astrophysical
constraints. Finally, we explore the possibility of mixed phase (Gibbs
construction) where the transition region is with mixed SM and SQM rather than
a sharp interface.

##  II Equations of States

For the quark matter sector, we adopt the unified treatment of interacting
quark matter that recently developed in Zhang:2020jmb  and later applied in
several studies Zhang:2021fla ; Zhang:2021iah ; Blaschke:2021poc ;
Blaschke:2022egm ; Oikonomou:2023otn ; Gammon:2023uss .

Referring to Zhang:2020jmb , we first rewrite the thermodynamic potential
$\Omega$ of the superconducting quark matter Alford:1998mk ; Rajagopal:2000ff
; Lugones:2002va ; Alford:2002kj ; Alford:2004pf  in a general form with the
pQCD correction Fraga:2001id  included:

$\displaystyle\Omega=$
$\displaystyle-\frac{\xi_{4}}{4\pi^{2}}\mu^{4}+\frac{\xi_{4}(1-a_{4})}{4\pi^{2}}\mu^{4}-\frac{\xi_{2a}\Delta^{2}-\xi_{2b}m_{s}^{2}}{\pi^{2}}\mu^{2}$
(1) $\displaystyle-\frac{\mu_{e}^{4}}{12\pi^{2}}+B,$

where $\mu$ and $\mu_{e}$ are the respective average quark and electron
chemical potentials. The first term represents the unpaired free quark gas
contribution. The second term with $(1-a_{4})$ represents the pQCD
contribution from one-gluon exchange for gluon interaction to
$O(\alpha_{s}^{2})$ order. To phenomenologically account for higher-order
contributions, we can vary $a_{4}$ from $a_{4}=1$, corresponding to a
vanishing pQCD correction, to very small values where these corrections become
large Fraga:2001id ; Alford:2004pf ; Weissenborn:2011qu . The term with
$m_{s}$ accounts for the correction from the finite strange quark mass if
applicable, where $m_{s}=95\pm 5\rm\,MeV$ ParticleDataGroup:2014cgo , and we
choose $m_{s}=95\rm\,MeV$ as its benchmark value. The term with the gap
parameter $\Delta$ represents the contribution from color superconductivity.
$(\xi_{4},\xi_{2a},\xi_{2b})$ represents different state of color-
superconducting phases. $B$ is the effective bag constant that accounts for
the non-perturbative contribution from the QCD vacuum.

The corresponding equation of state was derived in Ref. Zhang:2020jmb :

$P=\frac{1}{3}(\rho-4B)+\frac{4\lambda^{2}}{9\pi^{2}}\left(-1+\text{sgn}(\lambda)\sqrt{1+3\pi^{2}\frac{(\rho-B)}{\lambda^{2}}}\right),$
(2)

where

$\lambda=\frac{\xi_{2a}\Delta^{2}-\xi_{2b}m_{s}^{2}}{\sqrt{\xi_{4}a_{4}}}.$
(3)

Note that $\rm sgn(\lambda)$ represents the sign of $\lambda$. The chemical
potential (per baryon number) has the following form:

$\mu_{\rm
QM}=\frac{3\sqrt{2}}{(a_{4}\xi_{4})^{1/4}}\sqrt{[(P+B)\pi^{2}+\lambda^{2}]^{1/2}-\lambda}\,.$
(4)

Taking the zero pressure limit of $\mu_{\rm QM}$, we obtain the energy per
baryon number, which can be converted into the following form:

$\left(\frac{E}{A}\right)_{\rm
QM}=\frac{3\sqrt{2}\pi}{(\xi_{4}a_{4})^{1/4}}\frac{{B}^{1/4}}{\sqrt{(\lambda^{2}/B+\pi^{2})^{1/2}+\lambda/\sqrt{B}}},$
(5)

where we see a larger $\lambda$ lowers the energy as expected.

We have examined that hybrid strangeon star with a core of unpaired strange
quark matter ($\Delta=0$) can not support $2M_{\odot}$ while retaining radial
stability that requires $\partial M/\partial P_{c}>0$. This is not a surprise
considering strangeon EOS is much stiffer than that of unpaired SQM, and a
transition to a much softer EOS is likely to induce radial instabilities due
to insufficient degenerate pressure to resist the gravitational pulling. We
can thus stabilize the hybrid strangeon stars by introducing color-
superconductivity effects to stiffen the SQM EOS 111Such instabilities can
also be cured by considering the scenario of slow SM-SQM conversions (with
respect to radial-oscillation timescale) Pereira:2017rmp ; Lugones:2021zsg .
Considering both SM and CFL have the three-flavor symmetry, we expect that the
surface tension of SM-SQM is not large, thus a fast conversion is preferred
and correspondingly the stability criteria should retain to be $\partial
M/\partial P_{c}>0$, i.e. the star mass increases with center pressure..
Therefore, in the following discussions, we specify the SQM phase to be CFL
$(\xi_{4}=3,\xi_{\rm 2a}=3,\xi_{\rm 2b}=3/4)$, considering the shared flavor
composition and the fact that color superconductivity stiffens the EOSs.
Besides, we set $a_{4}=1$ (no extra QCD corrections) for simplicity.

Following previous studies Xu:2003xe ; Zhou:2004ue ; Zhou:2014tba ; Lai:2009cn
; Lai:2017ney ; Miao:2020cqj ; Lai:2017mjv ; Lai:2018ugk ; Lai:2020mlu ;
Gao:2021uus , we assume the interaction potential between two strangeons is
described by the Lennard-Jones potential Jones(1924) :

$U(r)=4\epsilon\left[\left(\frac{\sigma}{r}\right)^{12}-\left(\frac{\sigma}{r}\right)^{6}\right],$
(6)

where $r$ is the distance between two strangeons, and $\sigma$ is the distance
when $U(r)=0$. The parameter $\epsilon$ describes the depth of the interaction
potential between strangeons. A larger $\epsilon$ will then indicate a larger
repulsive force at short range and thus maps to a stiffer EOS.

The mass density $\rho$ and pressure density $p$ of zero-temperature dense
matter composed of strangeons derived from Lennard-Jones potential Lai:2009cn
reads

$\displaystyle\rho$ $\displaystyle=$ $\displaystyle
2\epsilon\left(A_{12}\sigma^{12}n^{5}-A_{6}\sigma^{6}n^{3}\right)+nN_{\rm
q}m_{\rm q}\,,$ (7) $\displaystyle P$ $\displaystyle=$ $\displaystyle
n^{2}\frac{\mathrm{d}(\rho/n)}{\mathrm{d}n}=4\epsilon\left(2A_{12}\sigma^{12}n^{5}-A_{6}\sigma^{6}n^{3}\right)\,,$
(8)

where $A_{12}=6.2$, $A_{6}=8.4$, and $n$ is the number density of strangeons.
$N_{\rm q}m_{\rm q}$ is the mass of a strangeon with $N_{\rm q}$ being the
number of quarks in a strangeon and $m_{q}$ being the average constituent
quark mass. The contributions from degenerate electrons and vibrations of the
lattice are neglected due to their expected smallness.

At the surface of strangeon stars, the pressure becomes zero, and we obtain
the surface number density of strangeons as
$\big{[}A_{6}/(2A_{12}\sigma^{6})\big{]}^{1/2}$. For convenience, it is
transformed into baryon number density, i.e.,

$n_{\rm s}=\left(\frac{A_{6}}{2A_{12}}\right)^{1/2}\frac{N_{\rm
q}}{3\sigma^{3}}\,,$ (9)

so that the EOS can be rewritten into the following simpler form

$\displaystyle\frac{\rho}{n_{s}}$
$\displaystyle=\frac{a}{9}\tilde{\epsilon}\left(\frac{1}{18}\bar{n}^{5}-\bar{n}^{3}\right)+m_{q}\bar{n},$
(10) $\displaystyle\frac{P}{n_{s}}$
$\displaystyle=\frac{2\,a}{9}\tilde{\epsilon}\left(\frac{1}{9}\bar{n}^{5}-\bar{n}^{3}\right),$

where $a=A_{6}^{2}/A_{12}=8.4^{2}/6.2\approx 11.38$,
$\tilde{\epsilon}=\epsilon/N_{q}$ and $\bar{n}=N_{q}\,n/n_{s}$. Note that
$\bar{n}=3$ at star surface where $P=0$.

The chemical potential of strangeon matter can be derived via the
thermodynamic relation $\mu=(\rho+P)/n$. Note that to study its crossings with
$\mu_{\rm QM}$, one needs to further convert it to the chemical potential per
baryon number

$\mu_{\rm
strangeon}=\frac{3\mu}{N_{q}}=3\frac{\rho/n_{s}+P/n_{s}}{\bar{n}}=3m_{q}+{a}\tilde{\epsilon}(\frac{5}{54}\bar{n}^{4}-\bar{n}^{2}).$
(11)

Referring to Eq. (10), we see that both the EOS $P(\rho)$ and $\mu_{B}(P)$ for
strangeons only depends on parameters $n_{s}$ and $\tilde{\epsilon}$ with the
dependence on $N_{q}$ absorbed. Taking the zero pressure limit of $\mu_{\rm
strangeon}$, we obtain the energy per baryon number:

$\left(\frac{E}{A}\right)_{\rm
strangeon}=3m_{q}-\frac{3a}{2}\tilde{\epsilon},$ (12)

where we see that $E/A$ has no dependence on $n_{s}$, decreases as
$\tilde{\epsilon}$ increases, and is always smaller than that of normal
nucleons. Requiring a positive $E/A$ (or a non-negative $\rho$ at zero
pressure) sets a theoretical bound: $\epsilon/N_{q}\leq 2m_{q}/a\approx
54.5\rm\,MeV.$

The transition pressure or density can be determined by the crossings of their
chemical potentials. A necessary condition for such chemical potential
crossing is that the zero-pressure chemical potential (i.e., energy per baryon
number $E/A$) of strangeon (Eq. (12)) is smaller than that of CFL (Eq. (5)).
We show the related parameter space as the blue-shaded bands of Fig. 1. We see
that overall, the existence of such a hybrid configuration prefers a
relatively stiff strangeon EOS (large $\tilde{\epsilon}$) but a relatively
soft CFL phase (large $B$ or small $\Delta$).

On the other hand, the hybrid configuration would become radially unstable
($\partial M/\partial P_{\rm c}<0$) if the transition pressure $P_{\rm trans}$
is too large Alford:2013aca , as we have also examined explicitly. Referring
to Eq. (4) and Eq. (11), we see that the strangeon matter to SQM transition is
more likely to occur at smaller $P_{\rm trans}$ in the case of a smaller $B$,
a larger $\Delta$ (stiffer SQM EOS), a smaller $\tilde{\epsilon}$ (softer SM
EOS) or smaller $n_{s}$. These conditions compete with those from the
stability condition mentioned in the previous paragraph, constraining the
allowed parameter space.

Figure 1: Allowed parameter space (blue-shaded) for the existence of hybrid
strangeon stars from stability consideration $(E/A)_{\rm Strangeon}<(E/A)_{\rm
CFL}$. (top) CFL bag constant $B$ and (bottom) CFL superconductivity gap
$\Delta$ versus parameter $\epsilon/N_{q}$ of strangeon matter. For the bottom
sub-figure, the shaded region with lighter-colored contour lines represents
larger bag constant, sampling $B=60,80,100,120\,\rm MeV/fm^{3}$ (bottom to
top).

Figure 2: $M$-$R$ (left) and $\Lambda$-$M$ (right) of hybrid strangeon stars
(solid lines) with $\epsilon/N_{q}=80/9\approx 8.9$ (black), $120/9\approx
13.3$ (blue) MeV, $n_{s}=0.22$ (thin), $0.30$ (thick) for the strangeon
composition, and $B=60$ (top), $80$ (bottom) $\rm\,MeV/fm^{3}$ for the CFL
composition. Lines with darker colors denote larger $\Delta$, sampling 60, 80
MeV for top panels and 60, 100, 120 MeV for bottom panels, respectively. (no
large-$\Delta$ lines in top panels due to stability constraints referring to
Fig. 1.) Dashed lines are pure strangeon star configurations. Shaded regions
are constraints with $90\%$ credibility from the NICER mission PSR J0030+0451
(green colored) NICER1 ; NICER2 , PSR J0740+6620 (cyan colored)NICER3 ; NICER4
. The cyan-dotted vertical line in the right panels denotes the GW170817’s
$\Lambda(1.4M_{\odot})\leq 800$ constraint TheLIGOScientific:2017qsa .

##  III astrophysical implications

The stellar structure can be solved via the Tolman-Oppenheimer-Volkoff (TOV)
equation Tolman:1939jz ; Oppenheimer:1939ne ,

$\displaystyle\begin{aligned} \frac{dm}{dr}&=4\pi\rho r^{2}\,,\\\
\frac{dP}{dr}&=(\rho+P)\frac{m+4\pi Pr^{3}}{2mr-r^{2}},\,\\\ \end{aligned}$
(13)

where the profiles $P(r)$ and $m(r)$ are solved as functions of the center
pressure $P_{\rm c}$. The radius $R$ and physical mass $M$ of the compact
stars are determined by $P(R)=0$ and $M=m(R)$, respectively. One then obtains
the mass-radius relation $M(R)$ of hybrid strangeon stars by solving the TOV
equations together with the EOSs of the two matter phases, where the
transition point is determined by the crossing of their chemical potentials,
as introduced in the last section.

To compare with gravitational wave observations, we can further compute the
dimensionless tidal deformability $\Lambda=2k_{2}/(3C^{5})$, where $C=M/R$ is
the compactness and $k_{2}$ is the Love number that characterizes the stars’
response to external disturbances AELove ; Hinderer:2007mb ; Hinderer:2009ca ;
Postnikov:2010yn . The Love number $k_{2}$ can be determined by solving a
function $y(r)$ from a specific differential equation Postnikov:2010yn  and
the TOV equation Eq. (13), with the boundary condition $y(0)=2$. For hybrid
configurations, the matching condition Damour:2009vw ; Takatsy:2020bnx
$y(r_{d}^{+})-y(r_{d}^{-})=-4\pi r_{d}^{3}\Delta\rho_{d}/(m(r_{d})+4\pi
r_{d}^{3}\,P(r_{d}))$ should be imposed at $r_{d}$ (i.e., the core radius and
the star radius), where an energy density jump $\Delta\rho_{d}$ occurs.

For illustration, we show various benchmark TOV solutions and corresponding
tidal deformabilities in Fig. 2 for $B=60,80$ $\rm MeV/fm^{3}$ with
$\tilde{\epsilon}$ and $\Delta$ choices satisfying $(E/A)_{\rm
Strangeon}<(E/A)_{\rm CFL}$ (shaded bands in Fig. 1).

We see that all the benchmark examples shown in Fig. 2 satisfy NICER
constraints 222Note that we show the NICER X-ray constraints in the graph but
neglect them in the table considering the X-ray analyses of hybrid strangeon
stars may be different from those of neutron stars., while the GW170817
constraints ($\Lambda_{1.4M_{\odot}}\leq 800$) can be met by hybrid strangeon
stars with $(\tilde{\epsilon},n_{s}\rm/fm^{-3},\Delta/MeV)=(80/9,0.22,80)$ for
$B=60\rm\,MeV/fm^{3}$ (upper panels), and
$(\tilde{\epsilon},n_{s}\rm/fm^{-3},\Delta/MeV)=(80/9,0.22,120)$,
$(80/9,0.3,120)$ for $B=80\rm\,MeV/fm^{3}$ (lower panels).

The general features of correlations between constraints and parameters are
summarized in Table 1. For example, as the second row of Table 1 summarizes,
hybrid strangeon stars with small $\tilde{\epsilon}$ (black lines) and $n_{s}$
(thin lines), or large $\Delta$ (darker colored lines) and small B (such as
upper panels) tend to be radially unstable ($\partial M/\partial P_{\rm
c}<0$), which means radial stabilities require CFL to be not too soft compared
to the stiffness of strangeon EOS, considering $\Delta$ and $\epsilon/N_{q}$
signal the stiffness of each of the two matter phases. However, we also see
that the $M_{\rm TOV}\gtrsim 2M_{\odot}$ constraint Demorest:2010bx  prefers
overall stiff EOSs for both two matter phases (a large $\tilde{\epsilon}$ or
$\Delta$), while GW170817 tidal deformability constraint
($\Lambda_{1.4M_{\odot}}\leq 800$ TheLIGOScientific:2017qsa ) prefers the
opposite at low center densities. These together set bounds on the allowed
parameter space.

| $\tilde{\epsilon}$ | $n_{s}$ | $\Delta$ | $B$ |
---|---|---|---|---|---
$(E/A)_{\rm Strangeon}<(E/A)_{\rm CFL}$ |  $+$ |  $\char 92\relax$ |  $-$ |  $+$ |
$\partial M/\partial P_{\rm c}>0$ |  $-$ |  $-$ |  $+$ |  $-$ |
$M_{\rm TOV}\gtrsim 2M_{\odot}$ |  $+$ |  $-$ |  $+$ |  $-$ |
$\Lambda_{1.4M_{\odot}}\leq 800$ |  $-$ |  $+$ |  $-$ |  $+$ |

Table 1: Correlations of constraints and the EOS parameters. Plus(minus) sign
means positive (negative) correlation, while slash sign means no correlation.

To elaborate on the explicit layer structure, we dissect hybrid strangeon
stars by showing the masses and radii of their CFL cores as functions of the
centre pressure in Fig. 3. Here, we choose two benchmark examples of bag
constant $B=60,80\,\rm MeV/fm^{3}$ with different CFL gaps and a fixed
strangeon phase ($n_{s}=0.22\rm/fm^{3},\tilde{\epsilon}=80/9\,MeV$). We find
that, as a general feature, the compact stars are pure strangeon stars at low
$P_{c}$, and then develop a CFL core as $P_{c}$ increases. At the maximum mass
points, the strangeon crusts have widths of $1\sim 5$ km and masses of
$0.1\sim 1\,M_{\odot}$, where a smaller bag constant or a smaller $\Delta$
maps to a thicker crust. At the $M=2\,M_{\odot}$ point, all cases map to
hybrid strangeon stars, with a core of mass $1.13\,(1.86)\,M_{\odot}$ for
$\Delta=100\,(120)$ MeV case when $B=80$ $\rm MeV/fm^{3}$, and a core of mass
$0.27\,(1.50)\,M_{\odot}$ for $\Delta=60\,(80)$ MeV case when $B=60\,\rm
MeV/fm^{3}$. At the $M=1.4\,M_{\odot}$ point, for $B=60\,(80)\,\rm
MeV/fm^{3}$, the $\Delta=60\,(100)$ MeV case is a pure strangeon star, while
the $\Delta=80\,(120)$ MeV case is a hybrid strangeon star with a core of mass
$0.64\,(1.16)M_{\odot}$.

Figure 3: Mass (left) and radius (right) versus center pressure, $P_{\rm c}$,
for hybrid strangeon stars with strangeon crusts of
{$n_{s}=0.22\rm/fm^{3},\tilde{\epsilon}=80/9\,MeV$}, and CFL cores of
{$B=60\,\rm MeV/fm^{3}$, $\Delta=60,80$ MeV} (top) and {$B=80\,\rm
MeV/fm^{3}$, $\Delta=100,120$ MeV} (bottom). Darker color denotes larger
$\Delta$ values. Dashed lines denote pure strangeon stars. Solid lines denote
hybrid strangeon stars. The dot-dashed lines denote the CFL cores. The right
ends of the solid and dot-dashed lines are truncated at the corresponding
maximum mass points.

##  IV Mixed Phase

A strangeon-quark mixed phase is possible around the intersurface of the
strangeon crust and quark matter core, in analogy to the hadron-quark mixed
phase in the conventional hybrid neutron stars.

To construct mixed phase of hybrid strangeon stars, we need $\mu_{e}\neq 0$,
thus the strange quark matter sector should be in either the normal unpaired
phase ($\Delta=0$)333 As aforementioned for the sharp transition, normal
strange quark matter core with a strangeon crust are not likely to be
radially-stable. We expect the situation may be alleviated in the mixed phase
scenario. or the charged CFL phase Alford:2004pf , where $s$ quarks no longer
have an equal fraction as $u,d$ quarks. We keep the flavor-symmetry in the
strangeon sector considering its solid state with charge-neutrality being
enforced, since the Compton wavelength of dilute electrons is much larger the
scale of a strangeon.

We adopt here the Gibbs construct for the mixed phase as outlined in Ref.
Schertler:1999 . In this case, one may achieve the charge neutrality where the
pressures of both strangeon and quark matter are functions of the baryon and
electron chemical potentials $\mu_{B}$ and $\mu_{e}$. The Gibbs condition for
the equilibrium between the two phases (at zero temperature) is

$P_{\rm{SnP}}(\mu_{B},\mu_{e})=P_{\rm{QkP}}(\mu_{B},\mu_{e})=P_{\rm{MxP}}(\mu_{B},\mu_{e}),$
(14)

where the pressure function for strangeon phase $P_{\rm{SnP}}(\mu_{B})$ can be
inferred from Eq. (10) and Eq. (11) with addition of background electrons
$P_{\rm{SnP}}(\mu_{B},\mu_{e})=P_{\rm{SnP}}(\mu_{B})+\mu_{e}^{4}/(12\pi^{2})$.
Besides, for quark matter phase $P_{\rm{QkP}}$ can be inferred from Eq. (1)
with the identities $p=-\Omega$, $\mu=\mu_{B}/3$. Their intersection yields
the mixed phase $P_{\rm{MxP}}(\mu_{B},\mu_{e})$, as shown in Fig. 4.

Figure 4: Pressure is plotted as a function of $\mu_{B}$ and $\mu_{e}$ for
strangeon phase (green) and strange quark matter (red) of normal unpaired (top
panel) and charged CFL phase (bottom panel) of $\Delta=100$ MeV. The mixed
phase sits in the intersection of the two surfaces. For illustration, here
$B=80\rm\,MeV/fm^{3}$, $m_{s}=95\rm\,MeV$ for the SQM phase and
$\epsilon/N_{q}=80/9,n_{s}=0.3\rm\,/fm^{3}$ for strangeon phase.

The global charge neutrality condition reads

$\displaystyle(1-\chi)\leavevmode\nobreak\ n_{c}^{\rm
SnP}+\chi\leavevmode\nobreak\ n_{c}^{\rm QkP}=0,$ (15)

where, $n_{c}^{\rm SnP}$ and $n_{c}^{\rm QkP}$ denote the total charge
densities in strangeon phase and quark matter phase (either unpaired SQM or
CFL) respectively, with

$\displaystyle n_{c}^{\rm SnP}$ $\displaystyle=$ $\displaystyle-n_{c}^{\rm
e},$ (16) $\displaystyle n_{c}^{\rm QkP}$ $\displaystyle=$
$\displaystyle\frac{2}{3}n_{u}-\frac{1}{3}n_{d}-\frac{1}{3}n_{s}-n_{c}^{\rm
e},$ (17)

where $n_{c}^{\rm e}=\mu_{e}^{3}/(3\pi^{2})$, $n_{u,d}=\mu_{u,d}^{3}/\pi^{2}$,
$n_{s}=(\mu_{s}^{2}-m_{s}^{2})^{3}/\pi^{2}$ with $\mu_{i}$ the quark chemical
potential of flavor $i$. $\chi$ defines the volume fraction of the quark
matter in mixed phase defined as $\chi=V_{\rm QkP}/(V_{\rm QkP}+V_{\rm SnP}).$

We have examined various combinations of parameter sets for SQM in either the
normal unpaired phase or CFL phase, finding that the mixed phase
$P_{\rm{MxP}}(\mu_{B},\mu_{e})$ that satisfies the global charge neutrality
condition only resides in a very tiny segment of the intersection line, with
variations of $\mu_{B}$ smaller than $1$ MeV range near the zero $\mu_{e}$
point, where $\mu_{e}$ lift to $6\sim 8$ MeV. Thus the mixed-phase region for
hybrid strangeon stars is negligible, and all results should approximately be
the same as those obtained from the Maxwell construction studied in the last
section, i.e., the system is effectively reduced into one conserved charge due
to the negligible contribution of electrons. As we have examined, introducing
QCD corrections ($a_{4}<1$) will lift the intersecting $\mu_{B}$ but does not
help enlarge the charge-neutral region of the mixed phase. This matches the
expectation that the flavor-symmetry breaking effects are small in both
strangeon and SQM sectors, resulting in a very small $\mu_{e}$ and its limited
variation range when considering charge neutrality.

##  V Summary and discussion

We have explicitly shown the new possibility of the hybrid configuration of
strangeon stars with a strange quark matter core and a thick strangeon crust.
We also demonstrated their compatibility with astrophysical constraints with
selected benchmark examples. It is shown that mixed phase is not preferred for
hybrid strangeon stars with a CFL core.

Hybrid strangeon stars can naturally accommodate the pulsar glitch phenomena
as a result of the star quakes in the thick strangeon crust Lai:2017xys ;
Wang:2020xsm ; Lai:2023axr ; Lu:2023dwi . Besides, the large shear modulus
change and density continuities at the crust-core interface are likely to
result in large and distinct crust-core interfacial modes that can be probed
by gravitational-wave observations Zhu:2022pja . We leave these for future
studies.

###### Acknowledgements.

Acknowledgments.  C. Zhang is supported by the Institute for Advanced Study at
The Hong Kong University of Science and Technology. C.J Xia is supported by
the National Natural Science Foundation of China (Grant No. 12275234 and No.
12342027) and the National SKA Program of China (No. 2020SKA0120300). R.-X Xu
is supported by the National SKA Program of China (2020SKA0120100).

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