1 1 0 Academic Editor: Firstname Lastname 25 March 2023 9 June 2023 11 July
2023 https://doi.org/ Impact of Multiple Phase Transitions in Dense QCD on
Compact Stars Impact of Multiple Phase Transitions in Dense QCD on Compact
Stars Armen Sedrakian 1,2 Armen Sedrakian Sedrakian, A.

###### Abstract

This review covers several recent developments in the physics of dense QCD
with an emphasis on the impact of multiple phase transitions on astrophysical
manifestations of compact stars. To motivate the multi-phase modeling of dense
QCD and delineate the perspectives, we start with a discussion of the
structure of its phase diagram and the arrangement of possible color-
superconducting and other phases. It is conjectured that pair-correlated quark
matter in $\beta$-equilibrium is within the same universality class as spin-
imbalanced cold atoms and the isospin asymmetrical nucleonic matter. This then
implies the emergence of phases with broken space symmetries and tri-critical
(Lifshitz) points. The beyond-mean-field structure of the quark propagator and
its non-trivial implications are discussed in the cases of two- and three-
flavor quark matter within the Eliashberg theory, which takes into account the
frequency dependence (retardation) of the gap function. We then construct an
equation of state (EoS) that extends the two-phase EoS of dense quark matter
within the constant speed of sound parameterization by adding a conformal
fluid with a speed of sound $c_{\rm conf.}=1/\sqrt{3}$ at densities $\geq
10~{}n_{\rm sat}$, where $n_{\rm sat}$ is the saturation density. With this
input, we construct static, spherically symmetrical compact hybrid stars in
the mass–radius diagram, recover such features as the twins and triplets, and
show that the transition to conformal fluid leads to the spiraling-in of the
tracks in this diagram. Stars on the spirals are classically unstable with
respect to the radial oscillations but can be stabilized if the conversion
timescale between quark and nucleonic phases at their interface is larger than
the oscillation period. Finally, we review the impact of a transition from
high-temperature gapped to low-temperature gapless two-flavor phase on the
thermal evolution of hybrid stars.

###### keywords:

QCD matter; phase diagram; compact stars

##  1 Introduction

The astrophysics of compact stars entered the era of multimessenger astronomy
in 2017 with the discovery of the neutron star binary merger event GW170817
Abbott et al. (2017). Combined with radio observations of massive pulsars in
binaries with white dwarfs Cromartie et al. (2020) and X-ray observations of
nearby solitary neutron stars Riley et al. (2021); Miller et al. (2021),
compact star astrophysics nowadays offers important insights into their global
properties and potentially into the phase structure of dense matter Weber
(2005); Alford et al. (2008); Fukushima and Hatsuda (2011); Anglani et al.
(2014); Blaschke and Chamel (2018); Baym et al. (2018); Burgio et al. (2021);
Lattimer (2023); Sedrakian et al. (2023). Studies of matter at high densities
are fundamental to our understanding of the strong force of the Standard Model
and underlying concepts such as confinement, spontaneous chiral symmetry
breaking, and dynamical mass generation Ding et al. (2023).

This work studies the impact of multiple phase transitions in dense QCD matter
on the physics of compact stars. We partially review the relevant physics but
also provide a novel discussion of the mass–radius diagram of compact stars in
the case where a conformal fluid is added to the two-phase, constant speed of
sound parametrization of the EoS of quark matter Alford and Sedrakian (2017).
This is motivated by the recent work that showed that even though the high-
central-density stars are typically unstable toward radial oscillation (see
Ref. Bardeen et al. (1977), hereafter BTM), i.e., when $dM/d\rho_{c}<0$, where
$M$ is the star’s gravitational mass and $\rho_{c}$ is the central density,
they can be stabilized if the conversion between nucleonic and quark phases is
slow compared to the characteristic period of radial oscillations Pereira et
al. (2018); Curin et al. (2021); Gonçalves and Lazzari (2022); Rau and
Sedrakian (2023).

To motivate the modeling, Section 2 provides a brief overview of the phase
diagram of dense QCD matter as we understand it from the studies of the
thermodynamics of various high-density phases, such as the color-
superconducting phases Alford et al. (2008); Fukushima and Hatsuda (2011);
Anglani et al. (2014); Blaschke and Chamel (2018); Baym et al. (2018) or
quarkionic phases McLerran and Pisarski (2007); Fukushima and Kojo (2016);
Pisarski et al. (2019); McLerran and Reddy (2019); Margueron et al. (2021).
Utilizing the knowledge gained from the studies of imbalanced cold atoms
Strinati et al. (2018) and isospin asymmetrical nuclear matter Stein et al.
(2014); Sedrakian and Clark (2019), the possible phase structure of pair-
correlated quark matter is conjectured based on the universal features of
imbalanced pair-correlated fermionic systems. Section 3 discusses the
computations of Green’s functions in the two- and three-flavor phases
Sedrakian et al. (2018); Alford et al. (2018) and potential new effects
arising beyond the adiabatic (frequency-independent) approximation of the gap.
In Section 4, the constant speed-of-sound parameterization of the EoS of quark
matter phases Zdunik and Haensel (2013); Alford et al. (2013) is used to
explore the mass–radius ($M$-$R$) diagram of compact stars with deconfinement
and multiple phase transitions. For two-phase transitions, one from nucleonic
to two-flavor quark matter and another from two-flavor to three-flavor phase
of quark matter, we recover the fourth family of compact stars,  which is
separated from the third family by the instability region Alford and Sedrakian
(2017). Here we show that a high-density phase of conformal fluid at densities
$\geq 10~{}n_{\rm sat}$, where $n_{\rm sat}=0.16$ fm-3 is the saturation
density, modifies the classically unstable tracks in the $M$-$R$ diagram
compared to the case when such transition does not occur Li et al. (2023).
Such modification is phenomenologically important because of the possible
stabilization mechanism of radial oscillation modes of hybrid stars Pereira et
al. (2018); Curin et al. (2021); Gonçalves and Lazzari (2022); Rau and
Sedrakian (2023), which is discussed in Section 6. In Section 5, we discuss
the cooling of compact stars with quark cores. We then simulate the thermal
evolution of these stars on a time scale on the order of million years with a
focus on the impact of the phase transition from the gapped to the gapless
phase of 2SC matter in the core of the star. Our conclusions are given in
Section 7.

##  2 A Brief Review of the Phase Diagram of Dense QCD

Matter in compact stars covers the large number density ($n\geq n_{\rm sat}$),
large isospin, and relatively low-temperature ($0\leq T\leq 100$ MeV) portion
of the phase diagram of strongly interacting matter. The extremely low
temperature ($T\leq 0.1$) The MeV regime is relevant for mature compact stars,
whereas the higher temperature domain is relevant for supernovas and binary
neutron star mergers. The complexity of the phase diagram arises due to the
multiple order parameters describing (interrelated) phenomena, which include
deconfinement phase transition (with the Polyakov loop as the order parameter
of the center symmetry), chiral phase transition (and its condensate as the
order parameter), the color-superconducting phases (with the anomalous
correlator as the order parameter). Depending on the non-zero value of one or
several order parameters, distinct phases may arise: an extensively studied
case is color-superconducting phases with various pairing patterns Alford et
al. (2008); Fukushima and Hatsuda (2011); Anglani et al. (2014); Blaschke and
Chamel (2018); Baym et al. (2018). A more recent suggestion is a confined but
chirally symmetric quarkyonic phase at compact star densities McLerran and
Pisarski (2007); Pisarski et al. (2019); McLerran and Reddy (2019). A crude
sketch of the phase diagram of strongly interacting matter is shown Figure 1,
along with the regions that are covered by current and future facilities
(RHIC, NICA, and FAIR). The low-density and low-temperature region of the
phase diagram contains nuclei embedded into a sea of charge-neutralizing
electrons and neutrons at higher densities. As the density increases, a first-
order phase transition to bulk nuclear matter occurs at around $0.5~{}n_{\rm
sat}$. A further increase in density can lead to the deconfinement of nucleons
to form quark matter for $n\geq(2-3)\times n_{\rm sat}$.

The transition from nuclear matter to deconfined quark matter could be of the
first or second order, or a crossover Alford et al. (2008); Fukushima and
Hatsuda (2011); Anglani et al. (2014); Blaschke and Chamel (2018); Baym et al.
(2018). The first-order phase transition leaves a marked imprint on the
macroscopic properties of compact stars because the EoS contains a density
jump, which may give rise to new stable branches of compact stars (i.e., their
third family) separated from nucleonic stars by a region of instability
Gerlach (1968); Glendenning and Kettner (2000); Schertler et al. (2000);
Alvarez-Castillo et al. (2019). Smooth crossover without changes in the values
of the order parameter or the wave function of the three-quark states would be
a less dramatic change in the slope of the EoS, best visualized in terms of
the speed of sound Baym et al. (2018); Fukushima and Kojo (2016); Kojo (2019).
As mentioned above, two sequential first-order phase transitions can lead to
the appearance of a new branch of compact stars—fourth family—separated from
the third family by an instability region Alford and Sedrakian (2017); Li et
al. (2020, 2021, 2023), assuming the classical stability criterion
$dM/d\rho_{c}>0$ is valid. In the case of slow phase transition between the
nuclear and quark matter phases, the two families are not separated by an
instability region; i.e., they form a continuous branch where the regions with
$dM/d\rho_{c}<0$ are stabilized Rau and Sedrakian (2023) (see Section 6).

Figure 1: Sketch of the phase diagram of strongly interacting matter in the
temperature and baryonic density plain. Compact stars cover the low-
temperature and high-density regimes of this phase diagram. The parameter
ranges covered by the FAIR, NICA, and RHIC facilities are also indicated.

Deconfined quark matter at low temperature and high density is expected to
have characteristic features of degenerate Fermi systems, which are familiar
from condensed matter physics. Therefore, emergent phenomena such as (color)
superconductivity are expected in channels where gluon exchange is attractive
Alford et al. (2008); Fukushima and Hatsuda (2011); Anglani et al. (2014).
Various color superconducting phases may arise, depending on the number of
flavors and colors involved in the pairing, the ratio of $\Delta/\delta\mu$,
where $\Delta$ is the gap in the quasiparticle spectrum,
$\delta\mu=(\mu_{d}-\mu_{d})/2$ is the difference in the chemical potentials
of down ($d$) and up ($u$) quarks, and the mass of strange quark $m_{s}$ in
the three-flavor quark matter. The two-flavor candidate phases classified
according to these parameters are as follows:

  * (a)

The “2SC” phase (where the abbreviation refers to two-superconducting-colors)
Alford et al. (2008)

$\displaystyle\Delta_{2SC}\propto\langle\psi^{T}(x)C\gamma_{5}\tau_{2}\lambda_{2}\psi(x)\rangle\neq
0,\quad 0\leq\delta\mu<\Delta/\sqrt{2},$ (1)

where $C=i\gamma^{2}\gamma^{0}$ is the charge conjugation operator, $\tau_{2}$
is the second component of the Pauli matrix acting in the SU$(2)_{f}$ flavor
space, and $\lambda_{A}$ is the antisymmetric Gell–Mann matrix acting in the
SU$(3)_{c}$ color space. The properties of the 2SC phase resemble that of the
ordinary BCS theory, including vanishing resistivity and vanishing heat
capacity because the quarks near the Fermi surface remain gapped.

  * (b)

Phases with broken space symmetries, which are associated with a finite
momentum of the condensate Alford et al. (2008); Anglani et al. (2014)
(hereafter FF phase) or deformation of the Fermi surface Müther and Sedrakian
(2003) (hereafter DFS phase):

$\displaystyle\Delta_{2SC}\neq
0,\quad\delta\mu>\Delta/\sqrt{2},\quad\vec{P}\neq 0\quad\textrm{(FF)}$ (2)
$\displaystyle\Delta_{2SC}\neq
0,\quad\delta\mu>\Delta/\sqrt{2},\quad\delta\epsilon\neq 0\quad\textrm{(DFS)}$
(3)

where $\vec{P}$ is the center of mass momentum of a Cooper pair and
$\delta\epsilon$ quantifies the quadrupole deformation of the Fermi surfaces
of $u$ and $d$ quarks.

  * (c)

Mixed phase(s) Bedaque et al. (2003)

$\displaystyle\Delta_{2SC}\propto\langle\psi^{T}(x)C\gamma_{5}\tau_{2}\lambda_{2}\psi(x)\rangle\neq
0,\quad\delta\mu=0,\quad 0\leq x_{s}\leq 1,$ (4)

which corresponds to a mixture between a perfectly symmetrical “2SC”
superconductor and a normal system accommodating the excess number of $d$
quarks. Here, $x_{s}$ is the filling factor defined as the ratio of the
superconducting and total volumes. (We assume that there is an excess of $d$
over $u$ quarks, as is expected in quark matter in compact stars under
$\beta$-equilibrium.)

The color-flavor-locked (CFL) phase Alford et al. (1999) is expected to be the
ground state of three-flavor quark matter at asymptotically large densities
where the strange quark is massless, Fermi surfaces of quarks coincide, and,
therefore, the pairing among quarks occurs in a particularly symmetrical
manner. At densities relevant for neutron stars, the perfect CFL phase is
unlikely to be realized; rather, some of its variants have $m_{s}\neq 0$
and/or $\delta\mu\neq 0$—chemical potential shifts between various flavors of
quarks Alford et al. (2005). Therefore, the phases listed above can be
replicated with an allowance of additional non-zero $us$ and $ds$ pairings

$\displaystyle\Delta_{ud}\neq 0,\quad\Delta_{sd}\neq 0,\quad\Delta_{su}\neq
0,\quad(m_{s}\neq 0;\,\delta\mu\neq 0).$ (5)

A complete phase diagram of quark matter that includes most, if not all, of
the phases mentioned above, is not available to date. However, various
imbalanced superfluids, such as cold atoms, isospin asymmetrical nuclear
matter, and flavor-imbalanced quark matter show a high degree of universality.
Thus, possible structures of the phase diagram of quark matter can be
conjectured by extrapolating from the detailed studies of the phase diagrams
of cold atomic gases Strinati et al. (2018) and isospin asymmetrical nuclear
matter Stein et al. (2014). These are, clearly, speculative and need to be
confirmed using explicit computations of relevant quark phases.

Figure 2 shows two schematic phase diagrams of color-superconducting matter in
the density–temperature plane. For sufficiently large temperatures, the
unpaired normal phase is the preferred state of matter, ignoring any other
correlation beyond the pairing. The phases with broken symmetries, the FF and
the DFS phases, are preferable in temperature–density strips at low
temperatures and high densities. At lower temperatures, the PS phase is the
preferred one. At higher temperatures, the spatially symmetric 2SC phase
dominates. It is seen that the phase diagram contains two tri-critical points,
i.e., the points where three different phases coexist. The critical point,
which has the FF state at the intersection, is a Lifshitz point as, per
construction, it is a meeting point of the modulated (FF), ordered (PS/2SC),
and disordered (unpaired) states. Of course, this is the case if the
transition temperature to the CFL phase is below the tri-critical temperature;
otherwise, the unpaired state should be replaced by a variant of the CFL
phase. Note that depending on the parameters of the model, two or one
tricritical points may be located on the unpairing line or the line of
transition to the CFL phase, as illustrated in Figure 2, left and right
panels, respectively. The model can be tuned to produce a four-critical point
if both points coincide. We also note that the low-density limit corresponds
to the strong coupling regime where the pairs are tightly bound, whereas the
high-density limit corresponds to the weak-coupling regime. Therefore, one can
anticipate signatures of BCS–BEC crossover. These can be seen by examining
several characteristic quantities, for example, the ratio of the coherence
length to the interparticle distance $\xi/d$, where $\xi/d\gg 1$ corresponds
to the BCS and $\xi/d\ll 1$ corresponds to the BEC limit, or the ratio of the
gap to the (average) chemical potential $\Delta/\mu$, where $\Delta/\mu\ll 1$
corresponds to the BCS and $\Delta/\mu\gg 1$ corresponds to the BEC limit. For
discussions of BCS=-BEC crossover in dense quark matter, see Refs. Sun et al.
(2007); Sedrakian and Rischke (2009); Ferrer et al. (2015); Duarte et al.
(2017). This phenomenon shows a high degree of universality as well; see for
example, the studies of nuclear matter Lombardo et al. (2001); Sedrakian and
Clark (2006); Jin et al. (2010) and cold atoms Giorgini et al. (2008);
Strinati et al. (2018).

Figure 2: Sketch of the phase diagram of strongly interacting matter in the
temperature and baryonic density plain, including (collectively indicated)
modulated FF-phase and deformed Fermi surface DFS phase. The tri-critical
points are shown with dots; the Lifshitz point is adjacent to the FF,
unpaired/CFL phases, and homogenous (PS) phases. In the left panel, it is
located on the unpairing or CFL-transition (solid line). The dashed lines
correspond to the phase-separation lines among various phases. Signatures of
BCS–BEC crossover/transition may emerge when moving from high to low
densities.

##  3 Structure of Green Functions in Two-Flavor Quark Matter

Order-by-order computations of the magnitude of the gaps in the
superconducting phases can be carried out in the weak-coupling (extreme high-
density) regime, where the one-gluon exchange is the dominant interaction.
Approximate Eliashberg-type equations for the flavor-symmetric 2SC phase were
solved within one-gluon exchange approximation in Refs. Son (1999); Schäfer
and Wilczek (1999), showing that the pairing gap scales with the coupling $g$
as $\Delta\sim\mu g^{-5}\exp(-1/g)$. Such a scaling also applies to the high-
density CFL phase, where the perturbative approach is more reliable than at
densities relevant to the 2SC phase. More recently, Eliashberg-type equations
were solved for two-flavor Sedrakian et al. (2018) and three-flavor
superconductors Alford et al. (2018). The first study used the quark–meson
coupling model, keeping only the frequency dependence of the gap, whereas the
second study kept frequency and momentum dependences but ignored the imaginary
part of the pairing gap. These theories not only improve the description of
quark matter but also lead to phenomenologically important implications, such
as the presence of electrons in the CFL phase Alford et al. (2018), which are
not allowed when the gap is constant Rajagopal and Wilczek (2001).

We now briefly outline these approaches following Ref. Sedrakian et al.
(2018). The inverse Nambu–Gorkov quark propagator is given by

$S^{-1}(q)=\left(\begin{array}[]{cc}{\not{q}}+{\mu}\gamma_{0}-m&\bar{\Delta}\\\
\Delta&({\not{q}}-{\mu}\gamma_{0}+m)^{T}\end{array}\right),$ (6)

where $q$ is the four-momentum and $\Delta$ is the gap with
$\bar{\Delta}\equiv\gamma_{0}\Delta^{{\dagger}}\gamma_{0}$. Equation (6) is
written for the case of equal number densities of up and down quarks with a
common chemical potential $\mu$ and mass $m$. The bare quark–meson vertices
$\Gamma_{\pi}^{i}(q)$ and $\Gamma_{\sigma}(q)$ are given by

$\displaystyle\Gamma_{\pi}^{i}(q)=\left(\begin{array}[]{cc}\frac{\tau^{i}}{2}\gamma_{5}&0\\\
0&-(\frac{\tau^{i}}{2}\gamma_{5})^{T}\end{array}\right),\quad\Gamma_{\sigma}(q)=\left(\begin{array}[]{cc}\mathbb{I}&0\\\
0&-\mathbb{I}\end{array}\right),$ (11)

where pions couple to quarks using a pseudo-scalar coupling, whereas $\sigma$s
couple via a scalar coupling, with $\mathbb{I}$ being a unit matrix in the
Dirac and isospin spaces. Their propagators are given by

$\displaystyle D_{\pi}(q)=\frac{1}{q_{0}^{2}-\bm{q}^{2}-m_{\pi}^{2}},\qquad
D_{\sigma}(q)=\frac{1}{q_{0}^{2}-\bm{q}^{2}-m_{\sigma}^{2}},$ (12)

where $m_{\pi/\sigma}$ values are the meson masses. The equation for the gap
in the Fock approximation is given via

$\displaystyle\Delta(k)$ $\displaystyle=$ $\displaystyle
ig_{\pi}^{2}\int\frac{d^{4}q}{(2\pi)^{4}}\left(-\frac{\tau^{i}}{2}\gamma_{5}\right)^{T}S_{21}(q)\frac{\tau^{j}}{2}\gamma_{5}\delta_{ij}D_{\pi}(q-k)$
(13) $\displaystyle+$ $\displaystyle
ig_{\sigma}^{2}\int\frac{d^{4}q}{(2\pi)^{4}}(-\mathbb{I})^{T}S_{21}(q)\mathbb{I}D_{\sigma}(q-k),$

where $g_{\pi}$ and $g_{\sigma}$ are the coupling constants. Adopting the
color–flavor structure of the gap function corresponding to a 2SC
superconductor, one then finds

$\displaystyle\Delta^{ab}_{ij}(k)=(\lambda_{2})^{ab}(\tau_{2})_{ij}C\gamma_{5}[\Delta_{+}(k)\Lambda^{+}(k)+\Delta_{-}(k)\Lambda^{-}(k)],\quad$
(14)

where $a,b\dots$ refer to the color space, $i,j,\dots$ refer to the flavor
space, and the projectors onto the positive and negative states are defined in
the standard fashion as
$\Lambda^{\pm}(k)=(E_{k}^{\pm}+{\bm{\alpha}}\cdot{\bm{k}}+m\gamma_{0})/2E_{k}^{\pm},$
where $E_{k}^{\pm}=\pm\sqrt{{\bm{k}}^{2}+m^{2}}$ and
${\bm{\alpha}}=\gamma_{0}{\bm{\gamma}}$. The coupled Equations (6)–(14) must
be solved for the gap function, which is a function of three-momentum and the
frequency. In the low-temperature limit, the relevant momenta are close to the
Fermi momentum and the dependence on the magnitude of the three-momentum can
be eliminated by fixing it at the Fermi momentum. The gap Equation (13) then
depends only on the energy, which reflects the fact that the pairing
interaction is not instantaneous—a common feature of the Fock self-energies in
ordinary many-body perturbation theory. The solutions for the positive energy
projection of the gap function are shown in Figure 3 as a function of
frequency. The structure of the real and imaginary components of the gap
function shows a maximum around frequencies at which the meson spectral
functions are peaked. Thus, it is important to include the retardation effect
when the color superconductor is probed at such frequencies. In the low-
frequency limit, it is sufficient to use the BCS approximation where the
interaction is instantaneous so that the imaginary part vanishes
Im$\,\Delta(\omega)=0$ and the real part is a constant
Re$\,\Delta=\Delta(\omega=0)$.

Figure 3: Dependence of the real (solid) and imaginary (dashed) components of
the positive energy projection of the gap function on frequency for two
different values of the coupling shown by blue and red lines Sedrakian et al.
(2018). The BCS theory predicts a constant on-shell value Re$\,\Delta=$ Const.
and a vanishing Im$\,\Delta(\omega)$.

Ref. Alford et al. (2018) considered the full momentum and energy dependence
of the gap in the Fock approximation within the Yukawa model but neglected the
imaginary part of the anomalous self-energy. Their work shows that the
retardation implies a CFL phase that is not a perfect insulator, as charge
neutrality requires some electrons to be present in matter. This is not the
case in the treatment based on the BCS model Rajagopal and Wilczek (2001).
Thus, the phenomenology of the CFL phase is modified: its specific heat,
thermal conductivity, and magnetic response will change due to the
contribution of electrons. This example, in which the simple BCS ansatz for
the gap is replaced by a more complete gap function, demonstrates some
unexpected features of color superconductors, which may be important for their
transport and dynamic response to various probes.

##  4 Equation of State and Mass–Radius diagram

We have seen in the previous section that the phase diagram of quark matter
may have a complicated structure. At minimum, there are two robust phases of
color superconducting matter: the low-density two-flavor 2SC phase and the
high-density three-flavor CFL phase. See, for example, Ref. Bonanno and
Sedrakian (2012) for a Nambu–Jona-Lasinio study and compact stars with two
phase transitions in this model. However, additional phases are very likely
because it is energetically favorable to break the rotational and
translational symmetries due to the stress on the paired state induced by the
finite mass of the strange quark and $\beta$-equilibrium, which induces
disparity in the chemical potentials of $u$ and $d$ quarks. In addition, or
alternatively, quarkyonic phases may interfere.

For the specific computation below, we adopt a covariant density functional
EoS of nuclear matter in the nucleonic phase Li and Sedrakian (2019); Li et
al. (2020, 2021). This EoS, in the absence of the phase transition to quark
matter, produces nucleonic compact stars with a maximum $m\equiv
M/M_{\odot}\simeq 2.6$, where $M$ is the gravitational mass of the star and
$M_{\odot}$ is the solar mass. Allowing for phase transition to quark matter,
we consider a straightforward extension of the constant speed of sound EoS of
Ref. Alford and Sedrakian (2017) that allows for a conformal phase of quark
matter at high densities with a constant speed of sound, i.e.,

$\displaystyle
p({\epsilon})=\left\\{\begin{array}[]{ll}p_{1},&{\epsilon}_{1}<{\epsilon}<{\epsilon}_{1}\\!+\\!\Delta{\epsilon}_{1}\\\\[2.15277pt]
p_{1}+s_{1}\bigl{[}{\epsilon}-({\epsilon}_{1}\\!+\\!\Delta{\epsilon}_{1})\bigr{]},&{\epsilon}_{1}\\!+\\!\Delta{\epsilon}_{1}<{\epsilon}<{\epsilon}_{2}\\\\[2.15277pt]
p_{2},&{\epsilon}_{2}<{\epsilon}<{\epsilon}_{2}\\!+\\!\Delta{\epsilon}_{2}\\\\[2.15277pt]
p_{2}+s_{2}\bigl{[}{\epsilon}-({\epsilon}_{2}\\!+\\!\Delta{\epsilon}_{2})\bigr{]},&{\epsilon}_{2}\\!+\\!\Delta{\epsilon}_{2}<{\epsilon}<{\epsilon}_{3}\\\
p_{3},&{\epsilon}_{3}<{\epsilon}<{\epsilon}_{3}\\!+\\!\Delta{\epsilon}_{3}\\\\[2.15277pt]
p_{3}+s_{3}\bigl{[}{\epsilon}-({\epsilon}_{3}\\!+\\!\Delta{\epsilon}_{3})\bigr{]},&{\epsilon}>{\epsilon}_{3}\end{array}\right.$
(21)

where the three pairs of the pressure and energy density $p_{1,2,3}$ and
${\epsilon}_{1,2,3}$ correspond, respectively, to the transition from hadronic
to quark matter, from a low-density (2SC) quark phase to a high-density (CFL)
quark phase, and from the high-density quark phase to the conformal fluid. The
squared sound speeds in the quark phases are denoted by $s_{1}$, $s_{2}$, and
$s_{3}=c^{2}_{\rm conf.}$. Note that we assume that the 2SC and CFL quark
phases are separated by a jump at the phase boundary, as it follows from the
study of Ref. Bonanno and Sedrakian (2012). At high densities, the CFL phase
reaches the “conformal limit” where the interactions are dominated by the
underlying conformal symmetry of QCD. In this limit, the speed of sound
squared is $s_{3}=1/3$ (in units of speed of light), whereas the effects of
the pairing gap of the CFL phase can be neglected in a first approximation.
Note that we allow for a small jump between proper CFL and conformal zero-gap
fluid, but its effect on the observables is marginal, i.e., a smooth
interpolation would not change the results.

According to Equation (21), the modeling of the EoS of quark phases involves
the following parameters:

  * •

The three (energy) densities at which the sequential transitions between the
nucleonic phase, 2SC phase, CFL phase, and conformal fluid take place.

  * •

The magnitudes of the jumps in the energy density at the points of the
transition from nuclear to the 2SC phase, $\Delta\varepsilon_{1}$, from the
2SC to the CFL phase, $\Delta\varepsilon_{2}$, and from the CFL to the
conformal fluid phase $\Delta\varepsilon_{3}$.

  * •

The speeds of sound in the 2SC and CFL phases $s_{1}$ and $s_{2}$. The speed
of sound of the conformal fluid is held fixed at $s_{3}=1/3$. Note that for
any phase, $s\leq 1$ by causality.

Our model EoS is constructed using the following parameters. The transition
pressure and energy density from nuclear and quark matter are $p_{1}=1.7\times
10^{35}$ dyn/cm2 and ${\epsilon}_{1}=8.4\times 10^{14}$ g cm-3, respectively.
The magnitude of the first jump $\Delta{\epsilon}_{1}=0.6~{}{\epsilon}_{1}$.
The upper range of the energy density of the 2SC phase is determined as
${\epsilon}_{1}^{\rm max}=\delta_{2SC}({\epsilon}_{1}+\Delta{\epsilon}_{1})$,
where $\delta_{2SC}$ is a dimensionless parameter measuring the width of the
2SC phase. The magnitude of the second jump is parametrized in terms of the
ratio parameter $r=\Delta{\epsilon}_{2}/\Delta{\epsilon}_{1}$. The extent of
the CFL phase is determined by limiting its energy density range to
${\epsilon}_{2}^{\rm max}=\delta_{CFL}({\epsilon}_{2}+\Delta{\epsilon}_{2})$.
The transition to the conformal fluid is assumed to be of the first order with
a small (compared to other scales) energy-density jump equal to $0.1r$. The
transition to the conformal fluid phase occurs at densities
${\epsilon}_{3}=2.25$–$2.57\times 10^{15}$ g cm-3, i.e., by about a factor of
10 larger than the saturation density. The speeds of sound squared are fixed
as

$\displaystyle s_{1}=0.7,\qquad s_{2}=1,\qquad s_{3}=\frac{1}{3}.$ (22)

The values of $s_{1}$ and $s_{2}$ are chosen to obtain triplet configurations
with large enough masses of hybrid stars. The magnitudes of jumps between the
nuclear, 2SC, and CFL phases were chosen suitably to produce twin and triplet
configuations Alford and Sedrakian (2017).

Figure 4 shows a collection of EoS constructed based on Equation (21), which
shares the same low-density nuclear EoS. In this collection, we vary the
parameter $r$ (as indicated in the plot) for fixed values
$\delta_{2SC}=\delta_{CFL}=0.27$. The corresponding $M$–$R$ relations for
static, spherically symmetrical stars obtained from solutions to
Tolman–Oppenheimer–Volkoff equations are shown in Figure 5.

Figure 4: The pressure vs. energy density (EoS) for nucleonic matter (long-
dash-dotted curve) and a series of EoSs that contain two sequential phase
transitions via Maxwell construction manifest in the jumps of the energy
density. The models differ by the magnitude of the second jump measured in
terms of the ratio $r=\Delta\epsilon_{2}/\Delta\epsilon_{1}$.

For the chosen magnitude of the first jump $\Delta{\epsilon}_{1}$, the $M$–$R$
curves show the phenomenon of twins—two stars of the same mass but different
radii. The radii of twins differ by about 1 km. The more compact configuration
is a hybrid star, i.e., a star with a quark core and nuclear envelope, whereas
the less compact counterpart is a purely nucleonic star. The second phase
transition may or may not result in a classically stable sequence depending on
the value of the parameter $r$ parameterizing the magnitude of the second
jump. For small jumps $r=0.1$ and 0.23, new stable branches arise, which are
continuously connecting to the stable 2SC branch ($r=0.1$) or are separated by
a region where the stars are unstable ($r=0.23$). It is seen that, in this
case, triplets of stars with different radii but the same masses appear. The
densest stars contain, in addition to the 2SC phase, a layer of the CFL phase,
whereby the central density on the stable branch can exceed the onset density
of the conformal fluid. This implies that the densest member of a triplet will
contain in its center conformal fluid with $c_{\rm conf.}=1/\sqrt{3}$. For
each $M$-$R$ curve in Figure 5, the star with a central density at which the
conformal fluid first appears is shown by a dot (this density is fixed at
$10~{}n_{\rm sat}$). The stable branch of conformal fluid containing stars is
followed by a classically unusable branch with $dM/d\rho_{c}<0$. For
asymptotically large central densities, the masses and radii increase again.
The family of the EoSs that differ only in the value of the parameter $r$
cross at a “special point”. This type of crossing has been observed for twin
star configurations with a variation in a particular parameter of the EoS
Cierniak and Blaschke (2020); however, the EoS excluded two sequential phase
transitions. The behavior of $M$–$R$ curves at very high central densities
differs from the ones that were found in Ref. Li et al. (2023), where a branch
of ultracompact twin stars with masses of the order of $1~{}M_{\odot}$ and
radii in the range of $6$–$7$ km were found for a single phase transition from
the nuclear matter to the quark phase. Thus, we conclude that the high-density
asymptotics of the EoS modifies the behavior of the $M$–$R$ curves if the
conformal limit is achieved at densities of the order of $10~{}n_{\rm sat}$.

Figure 5: The $M$–$R$ relations corresponding to the EoS shown in Figure 4 for
several ratios of the second jump. The right panel enhances the high-mass
range to demonstrate the emergence of the triplets and the fourth family of
compact stars. Note that the different MR curves cross each other at the
special point located in the low-mass and low-radius region, in analogy to the
single-phase transition case; see Ref. Cierniak and Blaschke (2020). The blue
circles indicate the stars in which the central density corresponds to
$10~{}n_{\rm sat}$ at which the conformal fluid sets in.

The observation above may have phenomenological implications for the following
reason. The stability of stellar configurations is commonly determined by the
requirement that the star’s mass must increase with increasing central density
(or central pressure), i.e., $\partial M/\partial\rho_{c}>0$. An alternative
and physically more transparent method is to compute the radial modes of
oscillation of a star and determine the stable configurations from the
requirement that their frequencies are real. Ref. Pereira et al. (2018) showed
that the classical stability conditions fail if the conversion rate is slow,
i.e., if its characteristic timescale is longer than the period of
oscillations. In that case, the fundamental modes are stable even when
$\partial M/\partial\rho_{c}<0$; i.e., stars with central densities larger
than the one corresponding to the maximum-mass star (which lie to the left
from the maximum on the $M$–$R$ diagram in Figure 5) will be stable. This
observation also applies to configurations with two-phase transitions, as
shown in Refs. Gonçalves and Lazzari (2022); Rau and Sedrakian (2023).
Furthermore, Ref. Rau and Sedrakian (2023) shows that, in this case, the
classically unstable stars contribute to the count of same-mass stars, which
leads to the appearance of higher-order multiplets such as quadruplets,
quintuplets, and sextuplets. We will return to the stability of hybrid stars
in Section 6.

##  5 Cooling of Compact Stars with Quark Matter Cores

The cooling of compact stars may provide indirect information about quark
phases in hybrid stars. The properties of phases of dense quark matter affect
both neutrino emission and the specific heat content that determine the
cooling rate of a compact object in general; see Refs. Lyra et al. (2023);
Zapata et al. (2022); Sedrakian (2017, 2016); de Carvalho et al. (2015);
Grigorian et al. (2015); Noda et al. (2013); Negreiros et al. (2012); Hess and
Sedrakian (2011); Sedrakian (2013); Stejner et al. (2009); Blaschke and
Grigorian (2007); Anglani et al. (2006); Popov et al. (2006); Alford et al.
(2005); Shovkovy and Ellis (2003); Blaschke et al. (2001).

Non-superconducting relativistic quark matter cools predominantly via the
direct Urca processes involving $d$, $u$, and $s$ quarks Duncan et al. (1984)

$\displaystyle\begin{aligned} &d\rightarrow u+e+\bar{\nu}_{e},\\\
&u+e\rightarrow d+\nu_{e},\\\ &s\rightarrow u+e+\bar{\nu}_{e},\\\
&u+e\rightarrow s+\nu_{e},\end{aligned}$ (23)

where $\nu_{e}$ and $\bar{\nu}_{e}$ are the electon neutrino and
antineutrions. The neutrino emissivity through the direct Urca process for
non-strange quarks is given via Iwamoto (1980)

$\displaystyle\epsilon_{\beta}=8.8\times
10^{26}\alpha_{s}\left(\frac{n}{n_{\text{sat
}}}\right)Y_{e}^{1/3}T_{9}^{6}\quad\mathrm{ergs}~{}\mathrm{cm}^{-3}~{}\mathrm{~{}s}^{-1},$
(24)

where $n$ is the baryon density, $Y_{e}$ is the electron fraction, $T_{9}$ is
the temperature in units of $10^{9}\mathrm{~{}K}$, and $\alpha_{s}$ is the
running strong coupling constant. The emissivity given by Equation (24)
implies that the stars containing unpaired quark matter would cool quickly via
this direct Urca process. The cooling would be slower if the quark spectrum
contains a gap. In the case of the phenomenologically relevant 2SC phase, two
alternatives are possible, depending on whether the Fermi surfaces of quarks
are a full gap or they contain zero-gap segments (nodes). The latter feature
arises in the case of pairing between fermions on different Fermi surfaces, as
discussed in Section 2.

Ref. Jaikumar et al. (2006) studied a generic case where the quark spectrum is
gapped if the parameter $\zeta=\Delta_{0}/\delta\mu$ associated with the new
scale $\delta\mu=\left(\mu_{d}-\mu_{u}\right)/2$, where $\mu_{u,d}$ are the
chemical potentials of light quarks and $\Delta_{0}$ is the gap for
$\delta\mu=0$. The suppression of emissivity by pairing is qualitatively
different in the cases in which $\zeta>1$ and $\zeta<1$. The novelty arises in
the second case, where Fermi surfaces have nodes and particles can be excited
around these nodes without any energy cost (which is not the case for gapped
Fermi surfaces). Note that in the case of the FF phase, the shift in the
chemical potential is replaced by a more general function—the anti-symmetric
in the flavor part of the single particle spectrum of up and down quarks. This
new physics can be captured by adopting a generic parameterization of the
suppression factor of the quark Urca process with pairing suggested in Ref.
Jaikumar et al. (2006). The neutrino emissivity of the 2SC phase
$\epsilon_{2SC}^{rg}$ can be related to the Urca rate in the normal phase (24)
as

$\displaystyle\epsilon_{2SC}^{rg}\left(\zeta;T\leq
T_{c}\right)=2f(\zeta)\epsilon_{\beta},\quad
f(\zeta)=\frac{1}{\exp\left[(\zeta-1)\frac{\delta\mu}{T}-1\right]+1},$ (25)

where the parameters $\zeta$ and $\delta\mu$ were introduced above, $T$ is the
temperature, and $T_{c}$ is the critical temperature of the phase transition
from normal to the 2SC phase. Furthermore, the parameter $\zeta(T)$ is
temperature-dependent and we adopt the parametrization

$\displaystyle\zeta(T)=\zeta_{i}-\Delta\zeta g(T),$ (26)

where $\zeta_{i}$ is the initial value, $\Delta\zeta$ is the constant change
in this function, and the function $g(T)$ describes the transition from the
initial value $\zeta_{i}$ to the asymptotic final value
$\zeta_{f}=\zeta_{i}-\Delta\zeta$. The transition is conveniently modeled by
the following function

$\displaystyle g(T)=\frac{1}{\exp\left(\frac{T-T^{*}}{w}\right)+1},$ (27)

which allows one to control the temperature of transition by adjusting the
parameter $T^{*}$ and the smoothness of the transition via the width parameter
$w$. An additional issue to address is the role of the blue quarks that do not
participate in the 2SC pairing. Blue quarks may pair among themselves due to
the attractive component of the strong force as in the ordinary BCS case (as
both members of the Cooper pair are on the same Fermi surface). Then, the
emissivity of blue quarks in the superfluid state is given by

$\displaystyle\epsilon_{BCS}^{b}\left(T\leq
T_{cb}\right)\simeq\epsilon_{\beta}^{b}\left(T>T_{cb}\right)\exp\left(-\frac{\Delta_{b}}{T}\right),$
(28)

where $\Delta_{b}$ is the gap in the blue quark spectrum, $T_{cb}$ is the
corresponding critical temperature, and $\epsilon_{\beta}^{b}$ is the neutrino
emissivity of blue quarks in the normal state. As discussed in Section 4, the
densest members of the triplets contain cores of CFL matter that is fully
gapped. In this case, the excitations are the Goldstone modes of the CFL
phase. Their emissivity, as well as the specific heat, is rather small
compared to other phases due to their very small number density Jaikumar et
al. (2002). In the following discussion, we will ignore the role of the CFL
phase in the cooling of hybrid stars. In the conformal fluid phase, we expect
three-flavor pairing gap $\Delta\sim\mu g^{-5}\exp(-1/g)$,
$g=\sqrt{4\pi\alpha_{s}}$, with a spin–flavor structure of the CFL phase.

Let us turn to the cooling simulations of hybrid stars with a gapless 2SC
superconductor. The cooling tracks are shown in Figure 6, and the input
physics beyond the emissivities is discussed elsewhere Hess and Sedrakian
(2011); Sedrakian (2013, 2016, 2017). The key parameter regulating the
behavior of the cooling curves in Figure 6 is the temperature $T^{*}$, which
controls the transition from the gapped to ungapped 2SC phase. Similar results
were obtained in the context of rapid cooling of the compact star in
Cassiopeia (Cas) A remnant in Ref. Sedrakian (2013, 2016, 2017). The model has
a second parameter, the gap for blue-colored quarks $\Delta_{b}$, which
prohibits rapid cooling via the Urca process involving only blue quarks. The
third parameter $w$ in Equation (27) accounts for the finite time scale of the
phase transition—see Refs. Sedrakian (2013, 2016)—but it is important only for
the fine-tuning of the cooling curves close to the age of the Cas A. The
various cooling tracks shown in Figure 6 correspond to various values of
$T^{*}$ for fixed values of $w$ and $\Delta_{b}$ and stellar configuration of
mass $1.93~{}M_{\odot}$. It is seen that if $T^{*}$ is small, then the quark
core does not influence the cooling, because during the entire evolution
$T>T^{*}$; therefore the neutrino emission is suppressed by the fully gapped
Fermi surfaces of red-green quarks. For large $T^{*}$, early transition to the
gapless phase occurs, and the star cools fast via the direct Urca process.
Note that the value of $T^{*}$ can be fine-tuned to reproduce not only the
current temperature of Cas A but also the fast decline claimed to be observed
during the last decade or so; see Ref. Shternin et al. (2023) and references
therein. From the brief discussion above, one may conclude that the phase
transitions within the cold QCD phase diagram may induce interesting and
phenomenologically relevant changes in the cooling behavior of compact stars.
Although we will not discuss in any depth the dependence of cooling tracks on
the stellar mass, it should be pointed out that the onset of new phases in the
interiors of compact stars, for example, hyperonization, meson condensation,
and phase transition to quark matter, lead to mass hierarchy in the cooling
curves Raduta et al. (2018, 2019); Anzuini et al. (2022); Tsuruta et al.
(2023). Typically, one finds that heavier stars that have central densities
beyond the threshold for the onset of the new phase cool faster than the light
stars containing only nucleonic degrees of freedom. This is also the case for
models of stars studied here. For example, stars with masses $M\sim 1.1--
1.6\,M_{\odot}$ remain warm over longer time scales and are thus hotter than
their heavy analogs, which develop large quark cores.

Figure 6: Cooling tracks of compact stars with quark cores in the surface-
temperature–age diagram. The masses of the stars are the same,
$M=1.93~{}M_{\odot}$, and the different curves correspond to different values
of the parameter $T^{*}$ in units of keV, except the dotted line, which
corresponds to 1 $M_{\odot}$ mass nucleonic compact star without a quark core.
The observational points with error bars are shown by green circles; the
arrows show the upper limits on surface temperatures of known objects.

##  6 Stability Criteria for Hybrid Stars

The oscillation modes of a compact star are important probes of their internal
structure, as has been shown in the case of $g$ modes, which are sensitive to
the size of the density jumps at a first-order phase transition between
hadronic and quark matter Orsaria et al. (2019); Wei et al. (2020); Jaikumar
et al. (2021). They are expected to leave an imprint on the emitted
gravitational wave signal during the binary inspiral of a neutron star, as
well as in the post-merger phase Bauswein et al. (2019); Weih et al. (2020);
Liebling et al. (2021); Prakash et al. (2021).

As discussed briefly in Section 4, high-central density stars on the
descending branch of $M$–$R$ diagram can have phenomenological implications if
they are stabilized by some mechanism, which we discuss in this section. The
main mode of instability for non-rotating, spherically symmetrical fluid stars
in general relativity is the instability against the radial $f$-mode of
oscillations Chandrasekhar (1964). If the $f$-mode frequency
$\omega_{f}^{2}>0$, the stellar configuration is dynamically stable, and it is
unstable if $\omega_{f}^{2}<0$. The location of this instability point on the
$M$-$R$ diagram agrees well with the turning point of the mass–central-density
($M-\rho_{c}$) curve. The stars on the ascending branch are stable, whereas
those on the descending branch are unusable. The maximum mass is the point of
marginal stability. Numerical simulations found some violations of this
criterion Gourgoulhon et al. (1995); Takami et al. (2011), but quantitative
deviations are insignificant. However, recent work found that the agreement
between these criteria is strongly violated for stars with first-order phase
transition, as we review below.

Early work on stellar oscillations with phase transitions inside the star was
carried out in the Newtonian theory assuming uniform phases Bisnovatyi-Kogan
and Seidov (1984); Haensel et al. (1989). Two possibilities arise depending on
the interplay between the scales in the problem: (a) when the conversion rate
from one phase to another is fast, the interface between phases oscillates as
a whole when perturbed; (b) if, however, conversion is slow, then the
interface is fixed over the period of characteristic oscillations. The second
case is interesting because, as shown in Ref. Pereira et al. (2018), the sign
of $\omega_{f}^{2}$ does not change at the maximum mass $M_{\rm max}$ but
stays positive over a segment where $\partial M/\partial\rho_{c}<0$. This
implies that the classically unstable branch becomes stable against $f$-mode
oscillations. Several subsequent studies confirmed this feature in the case of
single- Curin et al. (2021) and two-phase transitions Gonçalves and Lazzari
(2022); Rau and Sedrakian (2023). The case of two-phase transition was
extended in several directions in Ref. Rau and Sedrakian (2023) by focusing on
EoS, which supported classical twin and triplet star configurations, as
discussed in Section 2. It was shown that in the case of slow conversion,
higher-order multiplet stars arise, since now the stars on the $\partial
M/\partial\rho_{c}<0$ segments of the mass–central-density curve are located
on the stable branch. Also, the properties of the reaction mode of a compact
star Haensel et al. (1989), which arises in case (a) with one or more rapid
phase transitions, were studied.

The fundamental modes of hybrid stars are obtained from the set of equations
Chanmugam (1977); Gondek et al. (1997)

$\displaystyle\frac{\textrm{d}\xi}{\textrm{d}r}$ $\displaystyle=$
$\displaystyle\left(\frac{\textrm{d}\nu}{\textrm{d}r}-\frac{3}{r}\right)\xi-\frac{\Delta
P}{r\Gamma P},$ (29) $\displaystyle\frac{\textrm{d}\Delta P}{\textrm{d}r}$
$\displaystyle=$
$\displaystyle\left[\textrm{e}^{2\lambda}\left(\omega^{2}\textrm{e}^{-2\nu}-8\pi
P\right)+\frac{\textrm{d}\nu}{\textrm{d}r}\left(\frac{4}{r}+\frac{\textrm{d}\nu}{\textrm{d}r}\right)\right]\left(\rho+P\right)r\xi$
(30) $\displaystyle-$
$\displaystyle\left[\frac{\textrm{d}\nu}{\textrm{d}r}+4\pi(\rho+P)r\textrm{e}^{2\lambda}\right]\Delta
P,$

where $\xi=\xi_{\rm dim}/r$, with $\xi_{\rm dim}$ being the Lagrangian
displacement, $r$ the radial coordinate, $\Delta P$ the Lagrangian
perturbation of pressure, $\rho$ the mass–energy-density, $\omega$ the angular
frequency, $\Gamma$ the adiabatic index, and $\text{e}^{2\nu}$ and
$\text{e}^{2\lambda}$ the metric coefficients entering the
Tomann–Oppenheimer–Volkoff equations. In a first approximation, the adiabatic
index for a chemically equilibrated relativistic fluid can be taken as that of
the matter in $\beta$-equilibrium $\Gamma=[(\rho+P)/P]\left(dP/d\rho\right).$
The set of Equations (29) and (30) can be solved provided the boundary
conditions are known. These are specified by assuming that the displacement
field is divergence-free at the center and that the Lagrangian variation of
the pressure vanishes at the surface of the star:

$\Delta P(r=0)=-3\Gamma P\xi(r=0),\qquad\Delta P(r=R)=0.$ (31)

The $\omega^{2}$ values obtained in this manner are usually labeled according
to the number of radial nodes in $\xi$ and the $f$ mode corresponds to the
nodeless mode.

In the case of multiple phase transitions in the QCD phase diagram, one needs
junction conditions that relate the values of Lagrangian perturbations on both
sides of the interface between phases. Such junction conditions already appear
in the work of Ref. Haensel et al. (1989) in the Newtonian cases, whereas the
the general relativistic case is treated in Ref. Karlovini et al. (2004). For
the slow conversion rate  one has the junction condition

$\left[\Delta P\right]^{+}_{-}=0,\qquad\left[\xi\right]^{+}_{-}=0;$ (32)

for rapid conversion rate, one has

$\left[\Delta P\right]^{+}_{-}=0,\qquad\left[\xi-\frac{\Delta
P}{r}\left(\frac{\textrm{d}P}{\textrm{d}r}\right)^{-1}\right]^{+}_{-}=0,$ (33)

where $+/-$ refer to the high- and low-density sides of the transition,
respectively. At present, it is not possible to state with confidence which
limit is realized in quark matter, as the conversion rate varies significantly
over the parameter space; see Ref. Bombaci et al. (2016) for a discussion and
earlier references. Ref. Rau and Sedrakian (2023) considered modified junction
conditions that smoothly interpolate between the two limiting cases.

Phenomenologically, the most interesting implication of the modified stability
criteria is the existence of new stable configurations beyond those that are
classically stable. In particular, in the case where twins and triplets exist
according to classical criteria of stability, additional configurations will
arise when conversion between phases at the interface is slow. These can form
quadruplets (the maximum number in the case of twins) and quintuplets and
sextuplets in the case of triplets. A particular case that allows for
classical triplet stars is illustrated in Figure 7, adapted from Ref. Rau and
Sedrakian (2023). The fundamental mode frequency $\omega_{f}$ is shown as a
function of the central pressure of the stars in two cases when both
interfaces (i.e., nucleonic to 2SC and 2SC-CFL) feature rapid or slow
conversion. (The case of rapid–slow and slow–rapid conversions are
intermediate cases, and we omitted them.) To recover the classical case, one
needs to assume that the conversion at each interface is rapid: in this case,
the instability region is characterized by the vanishing of the real part of
$\omega_{f}$, as seen in Figure 7. In the case of slow conversions at both
interfaces, one finds a continuous positive solution across the values of
central densities of the stellar sequences, thus indicating that the stars are
always stable, even on the descending branch of the mass–central-pressure
curve.

Figure 7: The fundamental mode of triplet stars as a function of the central
pressure of the configuration in the cases when both nucleonic-2SC and 2SC-CFL
interfaces feature slow (solid line) or rapid conversion (dotted line). In the
case of rapid conversion, the classical stability criteria apply; i.e., there
are no real $f_{0}$ solutions in the region where stars are unstable. The
corresponding curves are not shown. For more details, see Ref. Rau and
Sedrakian (2023).

To summarize the recent findings regarding the stability of the hybrid stars,
we have seen that their stability against the fundamental oscillation modes
strongly depends on the junction conditions at the interfaces between the
phases. These are determined by the rate of conversion between phases at the
phase boundary. In the case of slow phase transitions (i.e., when the
conversion time scale is larger than the characteristic period of the
oscillations), the usual stability criteria are modified and new stable
segments appear that were previously unstable. Alternative variants of
junction conditions that are intermediate between slow and rapid conversion
were also considered, but the resulting radial modes do not differ
significantly from the slow conversion case, with corresponding implications
for the stability of the stars Rau and Sedrakian (2023).

##  7 Conclusions

The investigation of dense QCD through the astrophysics of compact stars is an
actively pursued subject. This is due to the substantial observational
progress, which includes measurements of the masses and radii of pulsars and
gravitational wave signals from mergers of two neutron stars and neutron-
star–black-hole binaries. A more thorough comprehension of the thermodynamics
of dense QCD, weak interactions, and the dynamics of phase transitions would
greatly enhance our ability to model astrophysical phenomena relevant to
current observational programs.

This work gave an overview of the phase diagram of cold and dense QCD
appropriate for compact stars. We stressed that the universality of the phase
diagram of imbalanced fermionic superfluids, such as cold atomic gases and
nuclear matter, provides a valuable guide to the possible arrangement of the
color-superconducting phases in neutron stars, the presence of tri-critical
points, and BCS–BEC crossovers. The universality allows one to conjecture the
possible structures of the phase diagram in the density–temperature plane
including such phases, such as the Fulde–Ferrel phase, deformed Fermi surface
phase, and the phase separation.

As a novel contribution, the previously proposed parametrization of the EoS of
dense quark matter with sequential phase transitions was extended to include a
conformal fluid at large densities ($n\geq 10~{}n_{\rm sat}$) with the speed
of sound $c_{\rm conf.}=1/\sqrt{3}$. The part of the $M$-$R$ diagram that
contains twins and triplets remains intact because the transition to conformal
fluid occurs at larger central densities than those achieved in these objects.
Nevertheless, for large central densities, we find behavior that is
qualitatively different from earlier studies of this regime: the $M$–$R$
curves spiral in; i.e., after reaching a minimum, they turn to the right
(larger radius region), thus avoiding the region of ultra-compact stars.
Therefore, if the conformal limit is reached for densities much larger than
those considered here, the ultracompact region with radii 6–7 km can be
populated Li et al. (2023). In the opposite case of the early onset of the
conformal limit (as discussed in Section 4), the radii will remain large, but
small-mass regions can be populated if the stability criteria are modified by
the slow conversion at the interface(s) between the phases. Another
interesting new observation is that the change in the magnitude of the jump
from 2SC to the CFL phase induces a special point on the $M$–$R$ diagram at
which all the curves meet in analogy to the case of single-phase transition;
see Ref. Cierniak and Blaschke (2020). The importance of studying this
asymptotically large central density regime is phenomenologically relevant if
the conversion between various quark and nuclear phases is slow compared to
the characteristic timescale of oscillations, as discussed in Section 6. In
this case, the stars on the descending branch of mass–central-density (and its
counterpart on the $M$-$R$ diagram) may be stable Pereira et al. (2018); Curin
et al. (2021); Gonçalves and Lazzari (2022); Rau and Sedrakian (2023),
contrary to the classical requirement $dM/d\rho_{c}>0$ for the branch to be
stable, which in turn leads to higher multipole (beyond triplets) stars on the
$M$–$R$ diagram.

This research was funded by Deutsche Forschungsgemeinschaft Grant No. SE
1836/5-2 and the Polish NCN Grant No. 2020/37/B/ST9/01937.

The data presented in this study are available on request from the author.

###### Acknowledgements.

The author is grateful to M. Alford, J.-J. Li, and P. B. Rau for collaboration
on modeling compact stars with quark cores and the referees for helpful
comments. The author declares no conflict of interest. References

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