# Topology and emergent symmetries in dense compact star matter

Yong-Liang Ma<EMAIL_ADDRESS>School of Fundamental Physics and Mathematical
Sciences, Hangzhou Institute for Advanced Study, UCAS, Hangzhou, 310024, China
Wen-Cong Yang  School of Fundamental Physics and Mathematical Sciences,
Hangzhou Institute for Advanced Study, UCAS, Hangzhou, 310024, China  College
of Physics, Jilin University, Changchun, 130012, China

###### Abstract

It has been found that the topology effect and the possible emergent scale and
hidden local flavor symmetries at high density reveal a novel structure of the
compact star matter. The $N_{f}\geq 2$ baryons can be described by the
skyrmion in the large $N_{c}$ limit and there is a robust topology change in
the skyrmion matter approach to dense nuclear matter. The hidden scale and
local flavor symmetries which are sources introducing the lightest scalar
meson—dilaton—and lowest lying vector mesons into to nonlinear chiral
effective theory are seen to play important roles in understanding the nuclear
force. We review in this paper the generalized nuclear effective theory
(G$n$EFT), which applicable to nuclear matter from low density to the compact
star density, constructed with the robust conclusion from the topology
approach to dense matter and emergent scale and hidden local flavor
symmetries. The topology change at density larger than two times saturation
density $n_{0}$ encoded in the parameters of the effective field theory is
interpreted as the hadron-quark continuity in the sense of Cheshire Cat
Principle. A novel feature predicted in this theory that has not been found
before is the precocious appearance of the conformal sound velocity in the
cores of massive stars, although the trace of the energy-momentum tensor of
the system is not zero. That is, in contrast to the usual picture, the cores
of massive stars are composed of quasiparticles of fractional baryon charges,
neither baryons nor deconfined quarks. Hidden scale and local flavor
symmetries emerge and give rise a resolution of the longstanding $g_{A}$
quench problem in nuclei transition. To illustrate the rationality of the
GnEFT, we finally confront the generalized effective field theory to the
global properties of neutron star and the data from gravitational wave
detections.

##  I Introduction

Although it has been investigated for several decades, there is no consensus
on how to describe the equation of state (EoS) of dense nuclear matter
relevant to compact stars Brown and Rho (2002); Holt  _et al._ (2016); Drews
and Weise (2017); Baym  _et al._ (2018); Ma and Rho (2020a); Li  _et al._
(2019); Ma and Rho (2021a); Lovato  _et al._ (2022). We do not certainly know
what are the constituents involved and how do the symmetries of quantum
chromodynamics (QCD) evolve in medium. The resolution of these questions has
strong impacts on the most fundamental issues of particle and nuclear physics
that have defied theorists, for example, the mechanism of chiral symmetry
breaking and emergence of nucleon mass.

In the past decade, we studied the dense nuclear matter using a generalized
nuclear effective field theory (G$n$EFT) including the lowest lying vector
mesons $\rho$ and $\omega$ and the lightest iso-scalar scalar meson $\sigma$,
in addition to the nucleon and pion considered in the standard chiral effect
field theory (S$\chi$EFT) Paeng  _et al._ (2016, 2017); Ma  _et al._ (2019a);
Ma and Rho (2019a). Implemented with the robust conclusions of the medium
modified hadron properties obtained from the topology structure of QCD at
large $N_{c}$ limit, we found that at density relevant to the cores of massive
stars, the sound velocity saturates the approximate conformal limit although
the scale symmetry is not restored and the system is still in the confined
phase Ma and Rho (2019b, 2020b). That is, there could be a pseudoconformal
structure in the cores of massive stars (see Refs. Ma and Rho (2020a, 2021a);
Rho and Ma (2021); Ma and Rho (2021b); Lee  _et al._ (2022); Rho (2022a) for
reviews). Although the existing of the conformal sound velocity in massive
neutron stars is in stark contrast to the previous beliefs Bedaque and Steiner
(2015); Tews  _et al._ (2018); Moustakidis  _et al._ (2017); Alsing  _et al._
(2018), it is now observed in more and more models McLerran and Reddy (2019);
Jeong  _et al._ (2020); Kapusta and Welle (2021); Zhao and Lattimer (2020);
Margueron  _et al._ (2021) and allowed by the constraints from the neutron
star data Fujimoto  _et al._ (2022); Marczenko  _et al._ (2022a).

To obtain the pseudoconformal structure of dense nuclear matter, two
significant ingredients are considered in the G$n$EFT, the hidden symmetries
which are invisible in the matter-free space—hidden scale symmetry and hidden
local flavor symmetry—and topology of QCD at large $N_{c}$.

The local flavor symmetry—hidden local symmetry (HLS) Bando  _et al._ (1985,
1988); Harada and Yamawaki (2003)—provides an effective field theory (EFT)
approach of vector mesons $\rho$ and $\omega$ in the framework of the
nonlinear realization of chiral symmetry. And, the breaking scale symmetry
offers a source for introducing the scalar meson—regarded as dilaton—á la
Crewther and Tunstall Crewther and Tunstall (2015); Catà  _et al._ (2019);
Crewther (2020). In the compact star matter, due to the strong correlations
among its constituents, these hidden symmetries may emerge and these emergent
symmetries affect the compact star properties in either direct or indirect
way.

When considered in the large $N_{c}$ limit, baryon can be regarded as the
topology soliton—skyrmion—in a nonlinear field theory Skyrme (1961). This is
an alternative approach to nuclear physics other than the widely used models
including nucleon as an explicit fermionic field nowdays. Using the skyrmion
approach, people found that when the nuclear matter is squeezed to a certain
high density, the constituent of matter is changed from the baryon number-1
object to the baryon number-1/2 object, i.e., there is a topology change
Kugler and Shtrikman (1988, 1989). The existing of this topology change is
robust although the density where it happens—denoted as $n_{1/2}$—is model
dependent Lee  _et al._ (2003); Park  _et al._ (2004, 2008); Ma  _et al._
(2013, 2014a); Ma and Rho (2017). An interesting conclusion which has not been
observed in other approaches is that, after $n_{1/2}$, some hadron properties
such as the pion decay constant in medium $f_{\pi}^{\ast}$ and effective
nucleon mass $m_{N}^{\ast}$ become density independent Ma  _et al._ (2013,
2014a). Moreover, it is found that at $n\mathrel{\hbox to0.0pt{\lower
3.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$>$}}n_{1/2}$, the sound
velocity saturates the conformal limit $v_{s}^{2}/c^{2}\simeq 1/3$ Shao and Ma
(2022) although $f_{\pi}^{\ast}$ and $m_{N}^{\ast}$ are not zero.

Since the skyrmion approach to nuclear matter takes tremendous numerical
simulations and includes obscure mathematics, the approach to nuclear matter
using the EFTs including baryon fields as explicit degrees of freedom are
widely used now. We implement the model independent observations from the
skyrmion approach and the effects of the emergent symmetries to the medium
modified parameters in G$n$EFT through the (extended) Brown-Rho scaling Brown
and Rho (1991). By using the $V_{lowk}$ RG approach implementing the strategy
of Wilsonian renormalization group flow Bogner  _et al._ (2003), with respect
to the constraints from the nuclear matter around saturation density
$n_{0}\approx 0.16~{}$fm-3, we construct the equation of state (EoS) of
compact star matter which has pseudoconformal symmetry, i.e., the sound
velocity at the density relevant to the compact stars satisfies the conformal
limit $v_{s}^{2}/c^{2}\simeq 1/3$ but the conformal symmetry is not restored.
The predictions of such constructed pseudoconformal model (PCM) satisfy all
the constraints from terrestrial experiments and astrophysical observations.

In this contribution, complementary to Refs. Ma and Rho (2020a, 2021a); Rho
and Ma (2021); Ma and Rho (2021b); Lee  _et al._ (2022); Rho (2022a), we will
review the key points of the pseudoconformal structure of dense nuclear matter
with special interests on the emergent scale and local symmetries and topology
constituents of the compact star matter.

##  II Hidden symmetries and hadron resonances

In the S$\chi$EFT of nuclear physics, the hadron degrees of freedoms are
nucleon and pions. However, it has been recognized long time ago that the
hadron resonances are crucial for reproducing the empirical date of nuclear
matter around saturation density, such as the iso-scalar scalar meson $\sigma$
and vector mesons $\rho$ and $\omega$ in the Walecka model Serot and Walecka
(1986).

Another reason to include the hadron resonances in G$n$EFT is that although
the finite nuclei as well as infinite nuclear matter can be fairly accurately
accessed by nuclear EFTs—“pionless or pionful”—anchored on relevant symmetries
they are expected to break down at some high density relevant to, say, the
interior of massive stars. For example, when applying the S$\chi$EFT to
nuclear matter where the power counting in density is $O(k_{F}^{q})$ people
found that even for the normal nuclear matter, the expansion requires going to
$\sim q=5$ Holt  _et al._ (2016) therefore the more loops should be considered
and more parameters are involved. This makes the calculation involved and
ambiguities hard to control.

With the above considerations, we include the iso-scalar scalar meson $\sigma$
and vector mesons $\rho$ and $\omega$ in G$n$EFT. Note that, it is recently
found that the iso-vector scalar mesons $\delta$ (denoted as $a_{0}(980)$ in
particle physics) also affects the EoS through the symmetry energy in a
sizable way Li  _et al._ (2022); Miyatsu  _et al._ (2022). However, since we
do not have any idea on how to construct an EFT for them and their structures
are still under debate, we will not consider them in the present work.

In the literature, the hidden symmetries which are not visible in the matter-
free space provide sources for including hadron resonances in the effective
theories. Explicitly, the observed hadron resonances list in particle data
group booklet Workman (2022) indicates that the approximate chiral symmetry in
QCD breaks to the vector channel and pions can be regarded as the Nambu-
Goldstone bosons of the broken axial symmetry. The local flavor
symmetry—hidden local symmetry (HLS) Bando  _et al._ (1985, 1988); Harada and
Yamawaki (2003)—suggests an EFT approach of vector mesons in the framework of
the nonlinear realization of chiral symmetry. And, the hidden scale symmetry
offers a source for introducing the iso-scalar scalar meson to chiral EFTs á
la Crewther and Tunstall Crewther and Tunstall (2015); Catà  _et al._ (2019).
In the compact star matter, due to the strong correlations among its
constituents, these hidden symmetries may emerge and these emergent symmetries
affect the compact star properties in either a direct or an indirect way.

###  II.1 Hidden scale symmetry

The QCD Lagrangian in chiral limit is invariant under scale transformation.
Therefore, the trace of the momentum-energy tensor $\theta_{\mu}^{\mu}$
vanishes at classical level, i.e, $\langle\theta_{\mu}^{\mu}\rangle=0$. This
invariance is broken by trace anomaly at quantum level

$\displaystyle\theta_{\mu}^{\mu}$ $\displaystyle=$
$\displaystyle\frac{\beta(\alpha_{s})}{4\alpha_{s}}G_{\mu\nu}^{a}G^{a\mu\nu}+(1+\gamma_{5})\sum_{q=u,d,s}m_{q}\bar{q}{q},$
(1)

where $m_{q}$ is the quark mass. Since the trace anomaly has the quantum
number of vacuum, it has long been regarded as the source of the iso-scalar
scalar meson in effective models Isham  _et al._ (1970); Ellis (1970);
Schechter (1980); Crewther and Tunstall (2015); Catà  _et al._ (2019).

In the construction of the effective model of the iso-scalar scalar meson
using the trace anomaly as its source, the only constraint on the effective
Lagrangian comes from the anomaly match. However, to build an EFT of scalar
meson á la Weinberg, one should set up the power counting mechanism. This was
finalized by the pioneer work of Crewther and Tunstall Crewther and Tunstall
(2015); Catà  _et al._ (2019).

An alternative to the CT scheme is the framework proposed by Golterman and
Shamir (GS) in the large $N_{c}$ and large $N_{f}$ Veneziano limit Golterman
and Shamir (2016). Although the IR structure is presumably different in both
approaches, the GS scheme and CT scheme are found of the same form to NLO once
$\beta^{\prime}$ in CT and $\Delta n_{f}=|n_{f}^{c}-n_{f}|$ in GS are related.
In this work, we follow the CT scheme.

The basic idea of Crewther and Tunstall (CT for short) is that, the iso-scalar
scalar meson can be regarded as the pseudo-Nambu-Goldstone boson of the
spontaneous breaking of the scale symmetry driven by the explicit scale
symmetry breaking. Provided that there is a nonperturbative infrared fixed
point (IRFP) $\alpha_{IR}$ in QCD, the mass of the scalar meson, here dilaton,
is generated by the explicit scale symmetry breaking which is encoded in the
departure from the IRFP and the current quark mass. The magnitude of the mass
is proportional to the deviation from the IRFP
$(\Delta\alpha=\alpha_{IR}-\alpha_{s})$ and the current quark mass therefore
the situation is very similar to the chiral perturbation theory where the
masses of the Nambu-Goldstone bosons are proportional to the current quark
masses which measure the magnitude of the explicit chiral symmetry breaking.
Note that whether the nonperturbative IRFP which the CT approach is anchored
on exists in QCD is not yet confirmed. Among a variety of approaches, we
simply refer to the positive arguments given in Refs. Brodsky  _et al._
(2010); Horsley  _et al._ (2014); Yu  _et al._ (2022). Moreover, the lattice
QCD indicates that in the thermal system, the scale symmetry may exists in IR
region and therefore leads to the exists of massless glueballs Alexandru and
Horváth (2019). It seems not strange to expect such a scale invariance may
emerge in medium.

Following the procedure of CT Crewther and Tunstall (2015); Catà  _et al._
(2019), introducing the conformal compensator field $\chi$ which has the scale
dimension 1 and
$\langle\theta_{\mu}^{\mu}\rangle=\langle\chi^{4}\rangle=f_{\chi}^{4}$, one
can write the chiral-scale effective Lagrangian in terms of the scale
dimensions in the chiral-scale power counting in momentum (derivative), quark
mass and $\Delta\alpha=\alpha_{IR}-\alpha_{s}$

$\displaystyle\mathcal{O}(p^{2})\sim\mathcal{O}(m_{q})\sim\mathcal{O}(\Delta\alpha).$
(2)

In terms of the pseudoscalar pions $U(x)=e^{2i\pi(x)/f_{\pi}}$ and
$\chi=e^{\sigma(x)/f_{\chi}}$ with $\sigma$ being the dilaton field which will
be identified with the lightest scalar meson $f_{0}(500)$, one can write the
effective Lagrangian at leading chiral-scale order as

$\displaystyle{\cal L}_{\rm LO}$ $\displaystyle=$ $\displaystyle{\cal L}_{\rm
inv}^{d=4}+{\cal L}_{\rm anom}^{d>4}+{\cal L}_{\rm mass}^{d<4}$ (3)

where $d$ stands for the scale dimension and

$\displaystyle{\cal L}_{\rm inv}^{d=4}$ $\displaystyle=$ $\displaystyle
c_{1}\frac{f_{\pi}^{2}}{4}\left(\frac{\chi}{f_{\chi}}\right)^{2}{\rm
Tr}\left(\partial_{\mu}U\partial^{\mu}U^{\dagger}\right)+\frac{1}{2}c_{2}\partial_{\mu}\chi\partial^{\mu}\chi$
(4a) $\displaystyle{}+c_{3}\left(\frac{\chi}{f_{\chi}}\right)^{4},$
$\displaystyle{\cal L}_{\rm anom}^{d>4}$ $\displaystyle=$
$\displaystyle(1-c_{1})\frac{f_{\pi}^{2}}{4}\left(\frac{\chi}{f_{\chi}}\right)^{2+\beta^{\prime}}{\rm
Tr}\left(\partial_{\mu}U\partial^{\mu}U^{\dagger}\right)$ (4b)
$\displaystyle{}+\frac{1}{2}(1-c_{2})\left(\frac{\chi}{f_{\chi}}\right)^{\beta^{\prime}}\partial_{\mu}\chi\partial^{\mu}\chi$
$\displaystyle{}+c_{4}\left(\frac{\chi}{f_{\chi}}\right)^{4+\beta^{\prime}},$
$\displaystyle{\cal L}_{\rm mass}^{d<4}$ $\displaystyle=$
$\displaystyle{}\frac{f_{\pi}^{2}}{4}\left(\frac{\chi}{f_{\chi}}\right)^{3-\gamma_{m}}{\rm
Tr}\left(\mathcal{M}^{\dagger}U+U^{\dagger}\mathcal{M}\right),$ (4c)

where $\mathcal{M}$ stands for the current quark matrix with $\mathcal{M}={\rm
diag}(m_{\pi}^{2},m_{\pi}^{2},2m_{K}^{2}-m_{\pi}^{2})$, $\gamma_{m}$ is the
anomalous dimension of the quark mass operator $\bar{q}q$, $c_{i}$’s are
unknown constants. It should be noted that, different from the chiral
perturbation theory, $c_{3}$ and $c_{4}$ have scale-chiral order $O(p^{2})$
since, as will be seen latter, they are proportional to the dilaton mass
square, similarly to $\mathcal{M}$ for the pseudo-scalar Nambu-Goldstone
mesons.

We next consider the dilaton potential $V(\chi)$

$\displaystyle V(\chi)$ $\displaystyle=$
$\displaystyle{}-c_{3}\left(\frac{\chi}{f_{\chi}}\right)^{4}-c_{4}\left(\frac{\chi}{f_{\chi}}\right)^{4+\beta^{\prime}}.$
(5)

The saddle point equation in the matter-free space yields

$\displaystyle V(\chi)$ $\displaystyle=$
$\displaystyle{}-(4+\beta^{\prime})c\left(\frac{\chi}{f_{\chi}}\right)^{4}+4c\left(\frac{\chi}{f_{\chi}}\right)^{4+\beta^{\prime}},$
(6)

where

$\displaystyle c={}-\frac{1}{4}c_{4}=\frac{1}{4+\beta^{\prime}}c_{3}>0.$ (7)

We see that, with $\beta^{\prime}\neq 0$, that is, the dilaton potential
breaks scale symmery, the dilaton potential is in the Nambu-Goldstone (NG)
mode, i.e., the minima of the potential appears at
$\langle\chi\rangle=f_{\chi}$. However, if $\beta^{\prime}=0$, $V(\chi)=0$ and
dilaton potential is scale invariant, the scale symmetry cannot spontaneous
break. This simple observation illustrates that the spontaneous breaking and
explicit breaking of scale symmetry are correlated and the spontaneous
breaking is triggered by explicit breaking, which agrees with that unlike
chiral symmetry, spontaneous breaking of scale symmetry cannot take place in
the absence of explicit symmetry breaking Freund and Nambu (1968). We refer to
this as “Freund-Nambu theorem.”

Using the definition of the dilaton mass $m_{\sigma}$, from the dilaton
potential one has

$\displaystyle c$ $\displaystyle=$
$\displaystyle\frac{m_{\sigma}^{2}f_{\chi}^{2}}{4\beta^{\prime}(4+\beta^{\prime})}.$
(8)

So that the constant $c$ and therefore $c_{3}$ and $c_{4}$ through relation
(7), has chiral-scale dimension $O(p^{2})$. We finally obtained the dilaton
potential as

$\displaystyle V(\chi)$ $\displaystyle=$
$\displaystyle{}\frac{m_{\sigma}^{2}f_{\chi}^{2}}{4\beta^{\prime}(4+\beta^{\prime})}\left(\frac{\chi}{f_{\chi}}\right)^{4}\left[-(4+\beta^{\prime})+4\left(\frac{\chi}{f_{\chi}}\right)^{\beta^{\prime}}\right].$
(9)

When $\beta^{\prime}\ll 1$, the dilaton potential is approximated to
Goldberger  _et al._ (2008)

$V(\chi)=\frac{m_{\sigma}^{2}f_{\chi}^{2}}{4}\left(\frac{\chi}{f_{\chi}}\right)^{4}\left(\ln\frac{\chi}{f_{\chi}}-\frac{1}{4}\right).$
(10)

This yields the scale Ward-Takahashi identity

$\displaystyle\langle\theta_{\mu}^{\mu}\rangle=\langle\partial_{\mu}D^{\mu}\rangle={}-\frac{m_{\chi}^{2}}{4f_{\chi}^{2}}\left\langle\chi^{4}\right\rangle$
(11)

which is the partially conserved dilatonic current (PCDC) relation, the
counterpart to the PCAC for the pion.

Along the reasoning of CT, one can set up a systematic higher-order expansion
and write down the higher order terms Li  _et al._ (2017); Catà and Müller
(2020). In the general Lagrangian, there are so many unknown parameters that
it is difficult to give any prediction in practice, even at the leading order.
However, one can make a substantial progress and arrive at a manageable form
by taking the so called “leading-order scale symmetry (LOSS)” approximation
that corresponds to

$\displaystyle c_{1}\approx c_{2}\approx 1.$ (12)

That is, in LOSS, the scale symmetry breaking—in the chiral limit—is lodged
entirely in the dilaton potential $V(\chi)$. The resulting Lagrangian is

$\displaystyle{\cal L}_{{\rm LO}}^{\chi{\rm
limit}}=\frac{f_{\pi}^{2}}{4}\left(\frac{\chi}{f_{\chi}}\right)^{2}{\rm
Tr}\left(\partial_{\mu}U\partial^{\mu}U^{\dagger}\right)+\frac{1}{2}\partial_{\mu}\chi\partial^{\mu}\chi-V(\chi).$
(13)

Whether the LOSS approximation is valid cannot be justified from the first
principle. The numerical analysis shows that it works for light nuclei Li  _et
al._ (2018a, b) and compact star matter Ma and Rho (2020a) but it violates
around saturation density Ma and Rho (2020c). In this contribution, without
specification, we work with LOSS.

###  II.2 Hidden local flavor symmetry

To include the vector mesons into the chiral effective theory, among a variety
of approaches, we use the hidden local symmetry (HLS) Bando  _et al._ (1985,
1988); Harada and Yamawaki (2003). Explicitly, considering the chiral symmetry
$G_{\rm global}=[SU(2)_{L}\times SU(2)_{R}]_{\rm global}$, following the
convention of Ref. Harada and Yamawaki (2003), we decompose the field $U(x)$
as

$\displaystyle U(x)$ $\displaystyle=$
$\displaystyle\xi_{L}^{\dagger}\xi_{R}(x).$ (14)

Therefore, one can sandwich a local unitary transformation $H_{\rm
local}=[U(2)_{V}]_{\rm local}$ between this decomposition. Under
transformation $G_{\rm global}\times H_{\rm local}$, $\xi_{L,R}$ transform as

$\displaystyle\xi_{L,R}(x)\mapsto\xi_{L,R}^{\prime}(x)=h(x)\xi_{L,R}(x)g_{L,R}^{\dagger},$
(15)

where $h(x)\in H_{\rm local}$ and $g_{L,R}\in SU(2)_{L,R}$. The variables
$\xi_{L,R}$ can be parameterized as

$\displaystyle\xi_{L,R}(x)=e^{i\sigma(x)/(2f_{\sigma})}e^{\pm
i\pi(x)/(2f_{\pi})},$ (16)

where $\pi(x)=\pi^{a}X^{a}$ and $\sigma(x)=\sigma^{\alpha}S^{\alpha}$ with
$X^{a}$ being the generators of the broken chiral symmetry and $S^{a}$ as the
generators of the unbroken subgroup $H$. Note that here $\sigma$ is the Nambu-
Goldstone boson which becomes the longitudinal part of gauge boson $V_{\mu}$
of symmetry $H_{\rm local}$ with $f_{\sigma}$ being its decay constant.

With quantities $\xi_{L,R}$ one can define the following two $1$-forms:

$\displaystyle\hat{\alpha}_{\parallel\mu}$ $\displaystyle=$
$\displaystyle\frac{1}{2i}(D_{\mu}\xi_{R}\cdot\xi_{R}^{\dagger}+D_{\mu}\xi_{L}\cdot\xi_{L}^{\dagger}),$
$\displaystyle\hat{\alpha}_{\perp\mu}$ $\displaystyle=$
$\displaystyle\frac{1}{2i}(D_{\mu}\xi_{R}\cdot\xi_{R}^{\dagger}-D_{\mu}\xi_{L}\cdot\xi_{L}^{\dagger})\
,$ (17)

where the covariant derivative is defined as
$D_{\mu}\xi_{R,L}=(\partial_{\mu}-iV_{\mu})\xi_{R,L}$, and both of these
quantities transform as $\hat{\alpha}_{\parallel,\perp}^{\mu}\rightarrow
h(x)\hat{\alpha}_{\parallel,\perp}^{\mu}h(x)^{\dagger}$. For the gauge field
$V_{\mu}$ we have the field strength tensor

$\displaystyle
V_{\mu\nu}(x)=\partial_{\mu}V_{\nu}(x)-\partial_{\nu}V_{\mu}(x)-i[V_{\mu}(x),V_{\nu}(x)],$
(18)

with the transformation $V_{\mu\nu}(x)\rightarrow
h(x)V_{\mu\nu}(x)h(x)^{\dagger}$.

In terms of the two 1-forms defined by Eq. (17) and field strength tensor
(18), one can construct a Lorentz invariant Lagrangian with the minimal number
of derivatives as

$\displaystyle{\cal L}_{\rm HLS}$ $\displaystyle=$ $\displaystyle
f_{\pi}^{2}{\rm
Tr}[\hat{a}_{\perp\mu}\hat{a}_{\perp}^{\mu}]+f_{\sigma}^{2}{\rm
Tr}[\hat{a}_{\parallel\mu}\hat{a}_{\parallel}^{\mu}]-\frac{1}{2g^{2}}{\rm
Tr}[V_{\mu\nu}V^{\mu\nu}].$

To generate the masses of the gauge bosons therefore to identify them as the
physical vector mesons $\rho$ and $\omega$, we use the Higgs mechanism and
take the unitary gauge

$\displaystyle\xi_{L}^{\dagger}$ $\displaystyle=$
$\displaystyle\xi_{R}\equiv\xi=e^{i\pi/(2f_{\pi})},\;\;\;\;U(x)=\xi^{2}(x).$
(20)

Then, the gauge bosons $V_{\mu}$ acquires mass

$\displaystyle m_{V}^{2}=m_{\rho}^{2}=m_{\omega}^{2}=g^{2}f_{\sigma}^{2},$
(21)

which has the standard form of the gauge boson mass from the Higgs mechanism.

So far, we use $H_{\rm local}=[U(2)_{V}]_{\rm local}$ therefore the rho meson
and omega meson have the same mass (21). This approximation works well in the
matter-free space. However, in medium, it is found that this approximation
breaks Paeng  _et al._ (2016, 2012) and it is reasonable to take the HLS
$H_{\rm local}=[SU(2)_{V}\times U(1)_{V}]_{\rm local}$.

In the HLS, considering that the masses of $\rho$ mesons are smaller than the
chiral symmetry breaking scale $\Lambda_{\chi}$, one can make a systematic
expansion including vector meson loops due to the gauge invariance Georgi
(1989, 1990) and, set up a self-consistent power counting mechanism, the
essential character of effective theory Harada and Yamawaki (2003). Since in
the nuclear matter, due to the strong correlation among hadrons, the effective
mass of $\rho$ meson is reduced, the convergence of the expansion is enhanced.

Based on the Wilsonian renormalization group (RG) approach, it is found that
at high energy scale $f_{\pi}\to 0$ and $m_{\rho}\to m_{\pi}\to 0$, i.e.,
there is a vector manifestation (VM) fixed point in the hidden local symmetry
(HLS) Harada and Yamawaki (2003, 2001). It is not strange to expect that the
VM appears at (super-)high density. We will see latter that this VM, although
happens at the supper high density beyond the cores of massive stars, it
affects the equation of state of neutron star matter in an indirect way.

In addition, in the approach to the baryonic matter using the dilaton
compensated chiral effective theory, people found that there is a dilaton
limit fixed point (DLFP) which states that the medium modified decay constant
$f_{\chi}^{\ast}\to 0$ in theory at high density Beane and van Kolck (1994);
Paeng  _et al._ (2012). When the DLFP is approached, the vector meson $\rho$
becomes massless and the HLS is emerged Suzuki (2017). Although the DLFP is
saturated at the density beyond the core of massive neutron stars $\sim
10n_{0}$, it affects the properties of the equation of state of neutron star,
for example the sound velocity Paeng  _et al._ (2016, 2017); Yang and Ma .

##  III Topology change and hadron-quark continuity

It is recognized long time ago that in the large $N_{c}$ limit, baryon
properties share the same $N_{c}$ scaling as the soliton properties in the
nonlinear mesonic theories Witten (1979). This gives an alternative approach
to nuclear physics by using the topology properties of QCD at large $N_{c}$
limit, that is, regarding the baryon as a skyrmion in the Skyrme model Skyrme
(1961), other than the standard EFT approach including the baryon fields as
explicit degrees of freedom.

Using the skyrmion approach and regarding baryon as skyrmion, one can study
the single baryon, multibaryon and nuclear matter in a unified way Ma and Rho
(2017). Since the skyrmion approach is only based on the topology structure of
QCD, some qualitative conclusions obtained in the approach, such as the
existence of the topology change and density dependence of some parameters in
nuclear matter that will be illustrated later should be model independent.

###  III.1 Baryons as topology objects and topology change

In the nonlinear realization of chiral symmetry, the pion figures as the
Nambu-Goldstone boson of the spontaneous breaking of chiral symmetry and it is
expressed in the polar parameterization through $U(x)$.

Since the unitary field $U(x)$ satisfies
$U(x)U(x)^{\dagger}=U(x)^{\dagger}U(x)=1$, for any fixed time, say, $t_{0}$,
the matrix $U(\mathbf{x},t_{0})$ defines a map from the manifold $R^{3}$ to
the manifold $S^{3}$ in isospin space, that is

$\displaystyle U(\mathbf{x},t_{0}):R^{3}\to S^{3},$ (22)

for the static configuration $U(\mathbf{x},t_{0})$. At low energy limit, QCD
goes to the vacuum, i.e.,

$\displaystyle U(|\mathbf{x}|\to\infty,t_{0})$ $\displaystyle=$
$\displaystyle\mathbf{1},$ (23)

therefore, all the points at $|\mathbf{x}|\to\infty$ are mapped onto the north
pole of $S^{3}$ and energy of the system is finite.

In the language of topology, maps (22) constitute the third homotopy group
$\pi_{3}(S^{\,3})\sim Z$ with $Z$ being the additive group of integers which
accounts for the times that $S^{3}$ is covered by the mapping
$U(\mathbf{x},t_{0})$, i.e., winding numbers. Because a change of the time
coordinate can be regarded as a homotopy transformation which cannot transit
between the field configurations in homotopically distinct classes, the
winding number is a conserved quantity in the homotopy transformation by the
unitary condition of the field $U(x)$ and condition (23). In skyrmion models,
the conserved winding number represent the conserved baryon number in QCD. The
baryon arises as a topological soliton with the topology lodged in the chiral
field $U(x)$. Therefore, in the construction of the skyrmion-type model, only
the unitary condition of the field $U(x)$ and the condition (23) are essential
characteristics that should be taken into account Ma and Rho (2017).

In the skyrmion approach, one can simulate the nuclear matter by putting
skyrmions onto the crystal lattice, first put forward by Klebanov Klebanov
(1985), and regarding the skyrmion matter as baryonic matter. The density
effect enters when the crystal size is changed. This approach suggests a
method to study the nuclear matter at densities higher than the dilute density
using the topology of QCD. In practice, we do not know which crystalline the
nature favors. So far, the face-centered-cubic (FCC) crystal is the known
configuration which yields the lowest energy Kugler and Shtrikman (1988,
1989).

Among a variety of properties revealed in the crystal approach to dense
matter, the most important one is the existence of half-skyrmion—a winding
number-1/2 object—configurations at some higher density. Being topological,
its presence is a robust prediction Goldhaber and Manton (1987). Its does not
depend on what degrees of freedom other than the pions are involved. What is
significant is that it involves a topology change from skyrmions to half-
skyrmions, which is responsible for a dramatic change in the properties of the
dense matter at a density $n_{1/2}\mathrel{\hbox to0.0pt{\lower
3.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$>$}}2n_{0}$, a feature
which has not been observed in other approaches in the literature. We will see
that it plays significant roles in describing the equation of state for
compressed baryonic matter relevant for massive compact stars.

Fig. 1 shows how the skyrmion FCC crystal configuration transforms to the
half-skyrmion configuration in terms of the distribution of baryon number
density. In the left panel, one can easily see that besides the corners and
the center of the square where the skyrmions are originally put, the baryon
number density emerges at the middles of the lines connecting the corners.
That is, in the half-skyrmion phase, the vertices where the baryon number
accumulates forms the CC crystal. After integration, each blue area has
winding number-1/2 (for a detailed explanation, see, e.g., Ref. Ma and Harada
(2016)).

Figure 1: The distribution of the baryon number density in the skyrmion (left
panel) matter and half-skyrmion matter (right panel). The crystal size $2L$ is
denoted by the blue quare.

Due to this topology change, a variety of novel phenomena which have not been
observed in the standard nucleon EFT approach emerges. Although the locations
of the density $n_{1/2}$ where these phenomena start depend on model, their
existences are model independent. Some of them which closely relate the
present review are summarized as the following:

  * •

Quark condensate: In skyrmion matter, the space-average of the normalized
quark condensate $\phi_{0}=\frac{1}{2}{\rm Tr}(U)$ is

$\displaystyle\langle\phi_{0}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{V}\int^{2L}_{0}d^{3}x\,\frac{1}{2}{\rm Tr}(U),$ (24)

with $V$ being the volume of the crystal cell. It is found that in the
skyrmion matter $\langle\phi_{0}\rangle\neq 0$ but $\langle\phi_{0}\rangle\to
0$ in the half-skyrmion state. This implies that the quark condensate
$\langle\bar{q}q\rangle^{\ast}\to 0$ in the half-skyrmion matter when space-
averaged.

  * •

Pion decay constant: In the skyrmion crystal approach, it is found that the
medium modified pion decay constant $f_{\pi}^{\ast}$ first decreases with
density until $n_{1/2}$ but after $n_{1/2}$ $f_{\pi}^{\ast}$ stays as a
constant. We plot $f_{\pi}^{\ast}$ as a function of crystal size in Fig. 2.
This means the in the half-skyrmion matter, although the space averaged quark
condensate vanishes, the chiral symmetry is not restored and it is still in
the Nambu-Goldstone mode. Actually, in the half-skyrmion matter, the
inhomogeneous quark condensate persists Harada  _et al._ (2015).

  * •

Nucleon mass: By using the medium modified pion decay constant
$f_{\pi}^{\ast}$, one can calculate the density dependence of nucleon mass
$m_{N}^{\ast}$ and obtain the scaling relation

$\displaystyle\frac{m_{N}^{\ast}}{m_{N}}$ $\displaystyle\approx$
$\displaystyle\frac{f_{\pi}^{\ast}}{f_{\pi}},$ (25)

which, as discussed later, is consistent with the Brown-Rho scaling from the
LOSS Brown and Rho (1991). It is found that, as shown in Fig. 2, similar to
$f_{\pi}^{\ast}$, $m_{N}^{\ast}$ first decreases with density until $n_{1/2}$
after which it keeps as a constant. This is predominantly, if not entirely,
due to the space-averaged quark condensate going to zero at $n_{1/2}$. Since
in the half-skyrmion matter, $\langle\bar{q}q\rangle^{\ast}\to 0$, this
observation indicates that the nucleon mass as decomposition

$\displaystyle m_{N}=m_{0}+\Delta(\bar{q}q),$ (26)

that is, there is chiral invariant part in the nucleon mass and the parity
doubling of the nucleons may emerge in dense nuclear matter Detar and Kunihiro
(1989); Motohiro  _et al._ (2015).

Figure 2: The medium modified $f_{\pi}^{\ast}$ and $m_{N}^{\ast}$ vs. lattice
size $L$ in HLS Ma  _et al._ (2014b).

We want to emphasize that the tendencies discussed above are robust but the
location of $n_{1/2}$ is highly model dependent. So far we cannot pin down the
value of $n_{1/2}$. The combination of the terrestrial experiments and
astrophysical obserevations leads to the constraint, as see later,
$2.0n_{0}\mathrel{\hbox to0.0pt{\lower 3.0pt\hbox{\hskip
1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$<$}}n_{1/2}\mathrel{\hbox to0.0pt{\lower
3.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$<$}}4.0n_{0}$.

###  III.2 Topological baryon for $N_{f}=1$

As stated above that the skyrmion approach is anchored on the map (22). How to
or if it is possible to study baryon such as $\Delta$ resonance in one-flavor
QCD using the topology approach is a problem since $\pi_{3}(U(1))=0$. In one-
flavor QCD case, the chiral effective theory is dominated by the axial $U(1)$
anomaly for the $\eta^{\prime}$ meson, and the soliton construction no longer
applies since, for instance, the standard topological charge cannot be
identified.

In 2018, Komargodski Komargodski (2018) noted that the effective theory has a
conserved topological current
$J_{\alpha\beta\gamma}=\epsilon_{\alpha\beta\gamma\lambda}\partial^{\lambda}\eta^{\prime}/2\pi$
which are carried by $(1+2)$-dimensional charged sheets with the
$\eta^{\prime}$ field undergoing a $2\pi$ jump across the sheet. When these
sheets are finite dimensional with a boundary, they can carry massless edge
excitations with baryon quantum numbers therefore can be identified with fast
spinning baryons. These sheets are described by a topological field theory
through a level-rank duality argument Hsin and Seiberg (2016); Gaiotto  _et
al._ (2018); Benini (2018), much like in the fractional quantum Hall (FQH)
effect Tong (2016). The baryons are analogous to the gapless edge excitations
in quantum Hall (QH) droplets.

By using the HLS approach, Karasik Karasik (2020) pointed that the vector
mesons play the role of the Chern-Simons vector fields living on the QHD that
forms the $N_{f}=1$ baryon. This proposal gives a unified picture for the two
types of baryons and allows them to continuously transform one to the other.
Recently, Bigazzi et al proposed a string theory description of the QH sheet
using the Witten-Sakai-Sugimoto model Bigazzi  _et al._ (2022).

###  III.3 Cheshire Cat Principle and Quark-Hadron Continuity

Based on what we discussed above and will develop below, it is found that the
topology change is significant for developing the pseudo-conformal model (PCM)
of dense nuclear matter, especially for the existence of the conformal sound
velocity in compact star matter. Because of this topology change, there is a
cusp structure in the symmetry energy $E_{sym}$ Ma  _et al._ (2019a); Liu  _et
al._ (2019) which provides a simple mechanism for the putative soft-to-hard
change in the EoS for compact stars at $n\sim 2n_{0}$ needed to account for
the observed massive $\sim 2M_{\odot}$. In the models that resort to hadron-
quark continuity in terms of specific quark degrees of freedom that are
strongly coupled, the hardening of the EoS at $n\mathrel{\hbox to0.0pt{\lower
3.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$>$}}2n_{0}$ is
associated with “deconfinement” of quarks Baym  _et al._ (2018); McLerran and
Reddy (2019). The question is whether or how the topology change represents
the “quark deconfinement” process. Here, we give a conjecture on this issue.

We first consider one-flavor QCD in which case, as we discussed above, the
topological baryon an be interpreted as the fractional Quantum Hall (FQH)
droplet Komargodski (2018). In this case, the connection between the topology
change and the quark deconfinement can be made by using the Cheshire Cat
mechanism Ma  _et al._ (2019b).

Explicitly, considering a $(1+2)$-dimensional chiral bag surrounding a QH
droplet as shown in Fig. 3. The bag is an annulus of width $2R$ clouded by an
$\eta^{\prime}$ with a monodromy of $2\pi$. The bag is filled in by $N_{c}$
quarks. In the limit of the zero bag radius, the chiral bag reduces to a
vortex string with unit baryon number—the simile is left.

Figure 3: $(1+2)$-dimensional chiral bag surrounding a QH droplet. Ma  _et
al._ (2019b).

It is shown that a current transverse to the smile embodying the Callan-Harvey
anomaly outflow Callan and Harvey (1985) appears. This transverse current is
analogous to the Hall current of the QH effect through the emergence of an
effective $U(1)$ gauge field. This $U(1)$ gauge field lives in the disk
enclosed by the Cheshire Cat smile and is described by a purely topological
field theory in $1+2$ dimensions and the emergent action of the emergent
$U(1)$ gauge field is of FQH droplet. The quantum numbers of this baryon as a
QH droplet follow readily from the chiral bag construction. This argument can
be extended to the case where $N_{f}=2,3$ that we are concerned with in this
review.

Now suppose the $\eta^{\prime}$ becomes light as is expected at high density.
Then the FQH pancakes could become relevant as density increases and figure in
dense matter in a form of a stack of FQH pancakes. Interactions must then
induce the $N_{c}$ quarks with the fractional (1/$N_{c}$) baryon charge living
on the boundary of the pancakes could tunnel between the pancakes. This could
lead to sheets of fractional baryon-charged topological objects in (3+1)
dimensions. In fact in recent analysis of dense matter using skyrmion crystal
approach, one finds certain configurations unstable at low density but
stabilized at high density of sheets with half-baryon charged objects called
“lasagnes” Park  _et al._ (2019) and also with $1/q$-charged baryons in tube
configurations with baryons living on the surface of the tube Canfora (2018).
In addition, it is recently found that this QHD sheet exists in the string
theory description of single-flavor QCD Bigazzi  _et al._ (2022). Anyway, it
seems not impossible that the layers of FQH droplets in (3+1) dimensions give
rise to deconfined quasiparticles dual to quarks of fractional charges, e.g,
half-skyrmions and this FQH droplet may be explored in superdense compact star
matter Rho (2022b). Such deconfinement can take place in the presence of
domain walls as in some condensed matter systems Sulejmanpasic  _et al._
(2017) and the half-skyrmions probed in the density regime $n>n_{1/2}$ could
be deconfined as in the Néel-VBS deconfined quantum critical transition
Senthil  _et al._ (2004); Sulejmanpasic  _et al._ (2017).

##  IV Generalized nuclear effective field theory

Equipped by the discussion of the hidden symmetries, following the procedure
discussed above, one can write down the generalized nuclear effective field
theory (G$n$EFT). Here, for simplicity, we only consider the leading order
scale symmetry (LOSS). In this limit, the effective Lagrangian is expressed as

$\displaystyle{\cal L}_{GnEFT}$ $\displaystyle=$ $\displaystyle{\cal
L}_{GnEFT}^{M}+{\cal L}_{GnEFT}^{B}-V(\chi)$ (27)

where

$\displaystyle{\cal L}_{GnEFT}^{M}$ $\displaystyle=$ $\displaystyle
f_{\pi}^{2}\left(\frac{\chi}{f_{\sigma}}\right)^{2}{\rm
Tr}\left[\hat{\alpha}_{\perp\mu}\hat{\alpha}_{\perp}^{\mu}\right]$
$\displaystyle{}+af_{\pi}^{2}\left(\frac{\chi}{f_{\sigma}}\right)^{2}{\rm
Tr}\left[\hat{\alpha}_{\parallel\mu}\hat{\alpha}_{\parallel}^{\mu}\right]+\frac{1}{2}\partial_{\mu}\chi\partial^{\mu}\chi+\cdots,$
$\displaystyle{\cal L}_{GnEFT}^{B}$ $\displaystyle=$
$\displaystyle\bar{N}i\gamma_{\mu}D^{\mu}N-\frac{\chi}{f_{\sigma}}m_{0}\bar{N}N$
$\displaystyle{}+g_{A}\bar{N}\gamma_{\mu}\gamma_{5}\hat{\alpha}_{\perp}^{\mu}N+g_{V}\bar{N}\gamma_{\mu}\hat{\alpha}_{\parallel}^{\mu}N,$
$\displaystyle V(\chi)$ $\displaystyle=$ $\displaystyle
h_{5}\left(\frac{\chi}{f_{\sigma}}\right)^{4}+h_{6}\left(\frac{\chi}{f_{\sigma}}\right)^{4+\beta^{\prime}},$
(28)

with $N$ being the iso-doublet of the baryon fields. Using the saddle-point
equations and in terms of the dilaton mass $m_{\sigma}$, the dilaton potential
is reexpressed as

$\displaystyle V(\chi)$ $\displaystyle=$
$\displaystyle{}\frac{m_{\sigma}^{2}f_{\chi}^{2}}{\beta^{\prime}(4+\beta^{\prime})}\left(\frac{\chi}{f_{\chi}}\right)^{4}\left[\left(\frac{\chi}{f_{\chi}}\right)^{\beta^{\prime}}-\left(1+\frac{\beta^{\prime}}{4}\right)\right].$
(29)

Lagrangian (27) is the starting point of the PCM of the compact star matter.

###  IV.1 Generalized Brown-Rho Scaling

From G$n$EFT, one can derive the generalized BR scaling Brown and Rho (1991)
which mimics the medium modified hadron properties at LOSS. The most general
scaling including the corrects to LOSS can be found in Ref. Li and Ma (2017).
At LOSS, one finds

$\displaystyle\frac{f_{\pi}^{\ast}}{f_{\pi}}$ $\displaystyle=$
$\displaystyle\frac{m_{V}^{\ast}}{m_{V}}=\frac{m_{N}^{\ast}}{m_{N}}=\Phi(n),\quad\frac{m_{\sigma}^{\ast}}{m_{\sigma}}=\Phi(n)^{1+\beta^{\prime}},$
(30)

where $\Phi(n)=\langle\chi\rangle^{\ast}/\langle\chi\rangle$.

Note that, different from the BR scaling originally proposed in Ref. Brown and
Rho (1991), here, the density scaling of the sigma mass depends on
$\beta^{\prime}$. When $\beta^{\prime}\ll 1$, the dilaton potential reduces to
the logarithm form Li  _et al._ (2017) and scaling of sigma mass becomes
$m_{\sigma}^{\ast}/m_{\sigma}\to\Phi(n)$, the form suggested in Ref. Brown and
Rho (1991).

So far, we do not have any priority to fix $\beta^{\prime}$ without ambiguity.
What we learned is that when using chiral-scale EFT to dense skyrmion matter
$1\lesssim|\beta^{\prime}|\lesssim 3.5$ Ma and Rho (2018); Shao and Ma (2022).
This magnitude is consistent with the phenomenological Lagrangian approach to
nuclear matter where the six-point interaction of the sigma meson—roughly
$\beta^{\prime}=2$ in the present framework—is found significant Motohiro  _et
al._ (2015).

It should be noted that the scaling relation (30) is obtained from the LOSS.
In the following explicit calculation, to fit the nuclear matter properties
around saturation density $n_{0}$, we should fine tuning the scaling
parameters which is attributed to the corrections to LOSS.

###  IV.2 Quenching of $g_{A}$ in nuclei transition

Let us put the discussion of compact star matter aside for a moment. We show
how the scale symmetry manifests in nuclei by looking at the $g_{A}$ quench
problem in the nuclear Gamow-Teller transitions, that is, the axial coupling
constant $g_{A}^{\rm free}=1.276$ in the vacuum should be taken as $g_{A}^{\rm
eff}\to 1$ in the nuclear Gamow-Teller transitions Wilkinson (1973); Suhonen
(2017); Engel and Menéndez (2017). Some results are reviewed in Ref. Rho
(2022c) recently.

In G$n$EFT, the axial current relevant to the nuclear Gamow-Teller transitions
is expressed as

$\displaystyle q_{\rm SSB}g_{A}\bar{\psi}\tau^{+}\gamma_{\mu}\gamma_{5}\psi$
(31)

where

$\displaystyle q_{\rm SSB}$ $\displaystyle=$ $\displaystyle
c_{A}+(1-c_{A})\Phi^{\beta^{\prime}}.$ (32)

In the LOSS, $c_{A}=1$. So that, the value $c_{A}<1$ accounts for the breaking
of LOSS and reflects the patterns of the scale symmetry in nuclei system.
Using (31), the quench factor is finally expressed as Ma and Rho (2020c)

$\displaystyle q_{\rm GnEFT}^{\rm ESPM}$ $\displaystyle=$ $\displaystyle
q_{\rm SSB}\times q_{\rm SNC},$ (33)

where $q_{\rm SNC}$ accounts for strong nuclear many-body correlations. By
using the Fermi-liquid fixed point theory Friman and Rho (1996), one can work
out $q_{\rm SNC}$ Li  _et al._ (2018a). With the value $\Phi(n_{0})\simeq 0.8$
Kienle and Yamazaki (2004), one obtains Ma and Rho (2020c)

$\displaystyle q_{\rm SNC}\simeq 0.79$ (34)

In nuclei upto $A\sim 60$, $g_{A}^{\rm eff}$ in shell model comes out to be
Suhonen (2017); Engel and Menéndez (2017)

$\displaystyle g_{A}^{\rm eff}=q_{\rm light}g_{A}^{\rm free}=0.98-1.18$ (35)

with $g_{A}^{\rm free}=1.276$. In the range $q_{\rm light}=0.76-0.93$ implied
by this equation, let us pick what gives $g_{A}^{\rm eff}\simeq 1$

$\displaystyle q_{\rm light}\simeq 0.78.$ (36)

With respect to (34), one concludes that $q_{\rm SSB}\simeq 1$ in Eq. (34).
This indicates that $c_{A}=1$, LOSS work well for the light nuclei system.

However, as the mass number of nuclei goes up above $A\sim 60$, the scenario
is different. A more stringent recent experiment from RIKEN on the
superallowed GT decay of the doubly magic nucleus 100Sn Lubos  _et al._ (2019)
yields Ma and Rho (2020c)

$\displaystyle q_{\rm RIKEN}^{\rm ESPN}=0.46-0.55.$ (37)

This means that in the heavy nuclei system, only (34) is not enough to account
for the quench factor. This discrepancy can be interpreted by assuming
$c_{A}\simeq 0.15$ and $\beta^{\prime}\simeq 2.5$—the same values that resolve
the HWZ problem in Refs. Ma and Rho (2018); Shao and Ma (2022). This choice
gives

$\displaystyle q_{\rm SSB}=0.64$ (38)

which leads to

$\displaystyle q_{GnEFT}^{\rm ESPM}=q_{\rm SSB}\times q_{\rm SNC}=0.64\times
0.79\simeq 0.5$ (39)

that consists with the RIKEN data well.

What we learned from this analysis can be summarized as follows: At very low
energy and density, at the unitarity limit (in the framework of pionless EFT),
conformal symmetry emerges in light nuclei and in the EOS of baryonic matter
van Kolck (2019); Tews  _et al._ (2017). At the normal nuclear matter density,
on the contrary, such symmetry is evidently absent, but at high density
approaching the dilaton-limit fixed point (DLFP), as we will see later, the
symmetry reappears. This tells us how the scale symmetry manifests in nuclear
system.

##  V Equation of state of nuclear matter

Let us come back to the nuclear matter properties by using the G$n$EFT with
generalized BR scaling. Since the G$n$EFT includes, in addition to the Nambu-
Goldstone bosons pions, the effects from the hadron resonances $\sigma,\rho$
and $\omega$, the obtained equation of state is expected to applicable to the
core of massive stars, i.e., $\sim 10n_{0}$. To take the meson fluctuation
effects into account, we use the $V_{lowk}$ approach Bogner  _et al._ (2003).
Therefore, the density effect come from both the intrinsic density dependence
inherits from the BR scaling and hadron correlations.

Considering that the density dependence of the medium modified hadron
properties are categorized into two regions due to the topology change
delimited by density $n_{1/2}$, we denote the region $n<n_{1/2}$ as R-I and
region $n>n_{1/2}$ as R-II. The density scaling of the medium modified hadron
properties are summarized as follows:

  * •

R-I: In this region, the scaling function $\Phi$ in the master formulism (30)
decreases with density. Without first principle information on the explicit
form of $\Phi$, we parameterize it as

$\displaystyle\Phi_{I}=\frac{1}{1+c_{I}\frac{n}{n_{0}}}$ (40)

with $c_{I}$ being a constant. With respect to the nuclear matter properties
Ma and Rho (2021b) and the measured pion decay constant Kienle and Yamazaki
(2004), the range of $c_{I}$ is found to be

$\displaystyle c_{I}\approx 0.13-0.20.$ (41)

In practice, to reproduce the nuclear matter properties around saturation
density, it is easy to imagine that there should be fine-tuning within the
range (41).

  * •

R-II: Due to the topology change at $n_{1/2}\mathrel{\hbox to0.0pt{\lower
3.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$>$}}2n_{0}$, the scaling
behaviours of some parameters in R-II are drastically different from that in
R-I. The existence of such a topology change is one of the most robust inputs
from skyrmion matter. The scaling behaviours of the parameters are quite
involved.

    * –

$g_{\rho}$ and $\rho$ mass: The hidden local gauge coupling $g_{\rho}$ related
to the $\rho$ mass through the KSRF relation. Combined with the vector
manifestation(VM) fixed-point structure of HLS leads to that for $n>n_{1/2}$
the coupling $g_{\rho}$ should drop to zero toward the putative VM fixed point
$n_{VM}$. We take the simple form Yang and Ma

$\displaystyle\frac{g^{*}_{\rho NN}}{{g_{\rho
NN}}}=\begin{cases}1-0.1\frac{n}{n_{0}},&for\quad n\in(n_{1/2},3.5n_{0})\\\
0.65-0.04\frac{(n-3.5n_{0})}{n_{0}},&for\quad
n\in(3.5n_{0},n_{VM})\end{cases}$

which gives $m_{\rho}^{\ast}/m_{\rho}=g^{*}_{\rho NN}/g_{\rho NN}\to 0$ at
$n_{\rm VM}\approx 20n_{0}$. Where $n_{\rm VM}$ is located is not known in
QCD. In compact stars, whether it is $\sim 6n_{0}$ or $\mathrel{\hbox
to0.0pt{\lower 3.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise
1.0pt\hbox{$>$}}20n_{0}$ does not make noticeable differences with one
possible exception, namely, the star sound velocity as we will see below.

    * –

Nucleon mass: As we learn from the 1/2-skyrmion phase that the parity doubling
emerges giving rise to the chiral-invariant mass $m_{0}$ and the pion decay
constant $f_{\pi}^{\ast}$ becomes density invariant. In the chiral-scale
effective theory, they both locked to the dilaton condensate
$f_{\chi}^{\ast}$. Therefore we have

$\displaystyle\frac{m_{N}^{\ast}}{m_{N}}\approx\frac{f_{\chi}^{\ast}}{f_{\chi}}\approx\frac{f_{\pi}^{\ast}}{f_{\pi}}\equiv\kappa\sim(0.6-0.9).$
(43)

    * –

Dilaton mass: The dilaton mass is also proportional to the dilaton condensate
which follows from the partially conserved dilatation current (PCDC) Crewther
and Tunstall (2015), we then have

$\displaystyle\frac{m_{\sigma}^{\ast}}{m_{\sigma}}\approx\kappa.$ (44)

    * –

$\omega$ meson: The nuclear matter density dependences of the $\omega$ meson
properties are subtle. Using the HLS, the $\omega$ mass is locked to the
hidden gauge coupling constant. Since the $U(2)$ HLS which works well in R-I
breaks in R-II Paeng  _et al._ (2012, 2016), some sort of fine-tuning is
needed in the density-scaling of $\omega$ mass and hidden gauge coupling
constant. We take it as

$\displaystyle\frac{m_{\omega}^{\ast}}{m_{\omega}}\approx\kappa\frac{g_{\omega}^{\ast}}{g_{\omega}}=\kappa\Phi_{\omega}(n).$
(45)

In the numerical calculation, we take

$\displaystyle\Phi_{\omega}\equiv\frac{g_{\omega}^{\ast}}{g_{\omega}}\approx
1-d\frac{n-n_{1/2}}{n_{0}}$ (46)

with $d\approx 0.05$.

After the above discussions, one can make a numerical calculation of the
equation of state of the nuclear matter once the vacuum values of the
parameters are fixed. It is found that, with the only parameter $c_{I}$, all
the nuclear matter properties $\leq 2n_{0}\lesssim n_{1/2}$ can be well
reproduced Ma and Rho (2021b).

The density $n_{1/2}$ where the topology change happens and how the R-I and
R-II are delineated changes density dependence of the hadron properties
drastically and therefore impact the EoS in a qualitative way. However, as we
discussed above, the location of the topology change is model dependent so
that we cannot pin down its value theoretically. With respect to constraints
from the various astrophysical observations so far available, the maximum mass
of neutron star and the gravity-wave data, we constrain $n_{1/2}$ as
$2.0n_{0}<n_{1/2}<4.0n_{0}$ Ma and Rho (2019a).

###  V.1 Vector manifestation

Where the vector manifestation fixed point $n_{\rm vm}$ is located is known
neither theoretically nor empirically. While most of the global properties of
compact stars do not seem to depend much on where $n_{\rm vm}$ lies since its
value is above the possible central density of massive compact stars, it seems
that it affects the sound velocity of compact star matter in an indirect way.

Here, to show the effect of $n_{\rm vm}$ on the sound velocity, we fix the
typical value $n_{1/2}=2.5n_{0}$. We choose $n_{\rm VM}=6.75n_{0}$ and
$20n_{0}$. The lower value of the density is about the central density of
massive stars, and the upper value represents an “asymptotic density” where
perturbative QCD is expected to be applicable. The $n_{\rm VM}$ dependences of
the sound velocity are plotted in Fig. 4.

Figure 4: Sound velocity for $n_{\rm vm}=6.75n_{0}$ (upper panel) and
$20n_{0}$ (lower panel), both computed in $V_{lowk}$ RG with
$n_{1/2}=2.5n_{0}$ Yang and Ma .

From Fig. 4 one can easily see that the location of $n_{\rm vm}$ drastically
affects the behaviour of the sound velocity. When $n_{\rm vm}$ is big, the
sound velocity converges to the “conformal velocity” $v_{s}^{2}\approx 1/3$
after $n_{1/2}$. However, for a smaller $n_{\rm vm}$, e.g., $n_{\rm vm}\approx
7n_{0}$, it increases steadily after $n_{1/2}$ and overshoots conformal
velocity. We will see later that the conformal sound velocity is locked to the
(pseudo-)conformality of the matter.

###  V.2 Pseudoconformal structure

Now, let us understand what is the implication of the conformal velocity in
nuclear matter.

In the matter system, the sound velocity is defined by

$\displaystyle v_{s}^{2}=\frac{\partial P(n)}{\partial
n}/\frac{\partial\epsilon}{\partial n},$ (47)

where $\epsilon$ and $P$ are, respectively, the energy density and the
pressure density. We then have

$\displaystyle\int\frac{\partial P(n)}{\partial n}dn=\int
v_{s}^{2}\frac{\partial\epsilon}{\partial n}dn-\frac{1}{3}C_{0}$ (48)

with $C_{0}$ being a constant independent of density. For a constant sound
velocity, one obtains

$\displaystyle P(n)$ $\displaystyle=$ $\displaystyle
v_{s}^{2}\epsilon(n)-\frac{1}{3}C_{0}.$ (49)

And for $v_{s}^{2}/c^{2}=1/3$, we obtain

$\displaystyle\epsilon(n)-3P(n)$ $\displaystyle=$ $\displaystyle C_{0}.$ (50)

For an ideal liquid system where the nuclear matter is assumed to work, the
trace of the energy-momentum tensor (TEMT) is expressed as

$\displaystyle\theta_{\mu}^{\mu}=\epsilon-3P.$ (51)

Therefore, when $v_{s}^{2}/c^{2}=1/3$, one has

$\displaystyle\langle\theta_{\mu}^{\mu}\rangle=C_{0},$ (52)

which is a density independent quantity. When $C_{0}=0$, the TEMT vanishes so
that the scale symmetry in dense matter is restored. The system with this
property can be regarded as that where both the scale symmetry and chiral
symmetry are restored, quarks are deconfined and perturbative QCD applies.
Since this scenario happens at the density much beyond that in the core of
massive stars, we are not interested in it.

Let us focus on the scenario $C_{0}\neq 0$, that is, the TEMT is a density
independent quantity. This scenario does happen in the chiral-scale EFT
approach to nuclear matter. In the mean field approach it is shown that, going
toward the DLFP, the TEMT $\langle\theta_{\mu}^{\mu}\rangle$ is a function of
only the dilaton condensate Paeng  _et al._ (2012). Now if the condensate goes
to a constant $\sim m_{0}$ due to the emergence of parity-doubling which is
found in the skyrmion crystal approach to nuclear matter Ma  _et al._ (2013),
then the $\langle\theta_{\mu}^{\mu}\rangle$ will be independent of density.
This chain of reasoning is confirmed in the full $V_{lowk}$ RG formalism
specifically for the case of $n_{1/2}=2n_{0}$. In Fig. 5 is shown the TEMT
(left panel) that gives the conformal velocity for $n\mathrel{\hbox
to0.0pt{\lower 3.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise
1.0pt\hbox{$>$}}3n_{0}$ (right panel).

Figure 5: $\langle\theta_{\mu}^{\mu}\rangle$ (upper panel) and $v_{s}$ (lower
panel) vs. density for $\alpha=0$ (nuclear matter) and $\alpha=1$ (neutron
matter) in $V_{lowk}$ RG for $n_{1/2}=2n_{0}$ and $v_{\rm vm}=25n_{0}$.

One can easily see that, the scenario
$\langle\theta_{\mu}^{\mu}\rangle=C_{0}\neq 0$ means that, the sound velocity
saturated the conformal limit but the conformal symmetry is not restored. We
call this matter as pseudoconformal matter in which the nucleon has an
effective constant mass and pion decay constant is not zero.

A recent analysis combining astrophysical observations and model independent
theoretical ab initio calculations Annala  _et al._ (2020) shows that in the
core of massive stars the sound velocity approaches the comformal limit
$v_{s}^{2}/c^{2}\to 1/3$ and the polytropic index takes the value
$\gamma<1.75$ — the value close to the minimal one obtained in hadronic
models. Therefore the core of the massive stars is populated by “deconfined”
quarks. An explicit calculation shows that the polytropic index $\gamma<1.75$
in the PCM (see Fig. 6) but we are still in the confined phased. Therefore the
smallness of the polytropic index and conformal velocity cannot be regarded as
sufficient criterions for the appearance of the deconfined quark.

Finally we compare in Fig. 7 our prediction for $P/\epsilon$ with the
conformality band obtained by the sound velocity interpolation method Annala
_et al._ (2020). We see that our prediction is close to, and parallel with,
the conformality band, but most significantly, it lies above this band. The
parallelism and location of our prediction come from the fact that in PCM the
trace of the energy-momentum tensor is a position constant. The predicted
results of GnEFT as a whole resemble the “deconfined” quark structure of
Annala  _et al._ (2020). There are, however, basic differences between the
two. First of all, in our theory, conformality is broken, though perhaps only
slightly at high density, in the system. Most important of all, the confined
half-skyrmion fermion in the half-skyrmion phase is not deconfined. It is a
quasiparticle of fractional baryon charge, neither purely baryonic nor purely
quarkonic. In fact it can be anyonic lying on a (2+1) dimensional sheet Ma and
Rho (2021a).

Figure 6: Density dependence of the polytropic index $\gamma={d\ln
P}/{d\ln\epsilon}$ in neutron matter from the pseudo-conformal model. Figure
7: Comparison of $(P/\epsilon)$ between the PCM velocity and the band
generated with the sound velocity interpolation method used in Annala  _et
al._ (2020). The gray band is from the causality and the green band from the
conformality. The red line is the PCM prediction. The dash-dotted line
indicates the location of the topology change.

###  V.3 Equation of state

We next compute the equation of state of the pseudoconformal dense nuclear
matter and compare it to the constraint from the astrophysical observation and
gravitational wave detection. We also vary the last parameter in the model
$n_{1/2}$ to see its effect on the EoS.

We should say that, whatever the topology change density
$n_{1/2}\mathrel{\hbox to0.0pt{\lower 3.0pt\hbox{\hskip
1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$>$}}2n_{0}$ is, the properties of ordinary
nuclear matter are fixed as stated already. In addition, we assume and
actually numerically checked that for $n_{1/2}\geq 2n_{0}$, slightly above
that transition density, the sound velocity must be $v_{s}^{2}/c^{2}\approx
1/3$.

It turns out that the feature of the EoS at $n>n_{1/2}$ can be captured by a
simple two-parameter formula for the energy per particle

$\displaystyle E/A=-m_{N}+X^{\alpha}x^{1/3}+Y^{\alpha}x^{-1}$ (53)

where $X$ and $Y$ are parameters to be fixed and $\alpha=(N-Z)/(N+Z)$ and
$x\equiv n/n_{0}$. From (53), one concludes that the sound velocity satisies

$\displaystyle\frac{v_{s}^{2}}{c^{2}}=\frac{1}{3},$ (54)

independently of $X^{\alpha}$ and $Y^{\alpha}$.

What we refer to as the pseudo-conformal model for the EoS is then $E/A$ given
by the union of that given by $V_{lowk}$ in R-I ($n<n_{1/2}$) and that given
by Eq. (53) in R-II ( $n\geq n_{1/2}$) with the parameters $X^{\alpha}$ and
$Y^{\alpha}$ fixed by the continuity at $n=n_{1/2}$ of the chemical potential
and pressure

$\displaystyle\mu_{I}=\mu_{II},\ P_{I}=P_{II}\ \ {\rm at}\ \ n=n_{1/2}.$ (55)

This formulation is found to work very well for both $\alpha=0$ and $1$ in the
entire range of densities appropriate for massive compact stars, say up to
$n\sim(6-7)n_{0}$, for the case $n_{1/2}\mathrel{\hbox to0.0pt{\lower
3.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$>$}}2n_{0}$ where the
full $V_{lowk}$RG calculation is available Paeng  _et al._ (2017).

We plot the sound velocity in Fig. 8 by varying $n_{1/2}$. From this figure
one can easily see that after $n_{1/2}$, the PC sound velocity
$v_{s}^{2}/c^{2}=1/3$ emerges which indicates the emergence of pseudoconformal
symmetry. It is clear from Fig. 8 that the sound velocity for the case of
$n_{1/2}=4n_{0}$ violates the causality bound $v_{s}^{2}/c^{2}<1$. The spike
structure could very well be an artifact of the sharp connection made at the
boundary. What is however physical is the rapid increase of the sound speed at
the transition point signaling the changeover of the degrees of freedom
Hippert  _et al._ (2021) and the derivative contribution from the trace
anomaly Fujimoto  _et al._ (2022); Marczenko  _et al._ (2022a). Significantly,
this allows us to set the constraint for $n_{1/2}$

$\displaystyle 2n_{0}\mathrel{\hbox to0.0pt{\lower 3.0pt\hbox{\hskip
1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$<$}}n_{1/2}\mathrel{\hbox to0.0pt{\lower
3.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$<$}}4n_{0}.$ (56)

What is important of this constraint is that the emergence of the conformal
sound velocity is an order of magnitude lower than the asymptotic density
$\mathrel{\hbox to0.0pt{\lower 3.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise
1.0pt\hbox{$>$}}50n_{0}$ perturbative QCD predicts. This is signals the
precocious emergence of pseudo-conformality in compact stars. A recent
detailed analysis of currently available data in the quarkyonic madel does
confirm the onset density of $v_{c}^{2}\approx 1/3$ at $\sim 4n_{0}$ Zhao and
Lattimer (2020); Kapusta and Welle (2021); Margueron  _et al._ (2021)

Figure 8: Sound velocity as a function of density in neutron matter with
different $n_{1/2}$.  Figure 9: Predicted pressure for neutron matter
($\alpha=1$) vs density compared with the available experimental bound
(shaded) given by Ref. Tsang  _et al._ (2018) and the bound at $6n_{0}$ given
by the blue band.

Plotted below in Fig. 9 is the predicted pressure $P$ for $n_{1/2}/n_{0}=3,4$
compared with the presently available heavy-ion data Tsang  _et al._ (2018).
The case of $n_{1/2}=4n_{0}$, while consistent with the bound at $n\sim
6n_{0}$, goes outside of the presently available experimental bound at $n\sim
4n_{0}$. This may again be an artifact of the sharp matching, but that it
violates the causality bound seems to put it in tension with Nature.
Nonetheless it may be too hasty to rule out the threshold density
$n_{1/2}=4n_{0}$.

##  VI Star properties and gravitational waves

The final topic we want to discuss is to confront the PCM with the star
properties and the gravitational wave detections for the purpose to show the
rationality of the PCM.

The solution of the TOV equation with the pressures of leptons in beta
equilibrium duly taken into account as in Ref. Paeng  _et al._ (2017) yields
the results for the star mass $M$ vs. the radius $R$. It is found that the
maximum mass comes out to be roughly $2.04M_{\odot}\sim 2.23M_{\odot}$ for
$2.0\leq n_{1/2}/n_{0}\leq 4.0$ which is consistent with the present
astrophysical observation Demorest  _et al._ (2010); Antoniadis  _et al._
(2013); Cromartie  _et al._ (2019). We plot in Fig. 10 for the M-R relation
with typical value $n_{1/2}=2.5n_{0}$. One can easily see that the present
calculation is consistent with the observations.

Figure 10: M-R relation from PCM with observed mass of pulsar
J0348+0432Antoniadis  _et al._ (2013) and radius constraintsRaaijmakers  _et
al._ (2019)Miller  _et al._ (2019) from NICER.

Next, let us turn to how our theory fares with what came out of the LIGO/Virgo
gravitational observations. The quantities that we will consider are the
dimensionless tidal deformability $\Lambda_{i}$ for the star $M_{i}$ and
$\tilde{\Lambda}$ defined by

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

for $M_{1}$ and $M_{2}$ constrained to the well-measured “chirp mass”

$\displaystyle{\cal M}$ $\displaystyle=$
$\displaystyle\frac{(M_{1}M_{2})^{3/5}}{(M_{1}+M_{2})^{1/5}}=1.188M_{\odot}.$
(58)

To confront the LIGO/Virgo data, we plot our prediction for $\Lambda_{1}$ vs.
$\Lambda_{2}$ in Fig. 11. As it stands, our prediction is compatible with the
LIGO/Virgo constraint for $n_{1/2}\mathrel{\hbox to0.0pt{\lower
3.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$>$}}2n_{0}$. Although
there seems to be some tension with the pressure, the result for
$n_{1/2}=4n_{0}$ is of quality comparable to that of $n_{1/2}=3n_{0}$.

Figure 11: Tidal deformabiliitiess $\Lambda_{1}$ and $\Lambda_{2}$ associated
with the high-mass $M_{1}$ and low mass $M_{2}$ components of the binary
neutron star system GW170817 with chirp mass $1.188M_{\odot}$. The constraint
from GW170817 at the 90% probability contour is also indicated. We quote
“FSUGarnet (0.16)” Fattoyev  _et al._ (2018) as a presently available “state-
of-art” theoretical prediction.

##  VII Summary and perspective

We reviewed in this work the possible emergent symmetries and topology change
in dense compact star matter. The information of the medium modified hadron
properties obtained from the skyrmion crystal approach, in addition to the
presumed emergent scale and flavor symmetry, inspired the construction of the
pseudoconformal model of dense nuclear matter relevant to compact stars.

In the pseudoconformal model, a peculiar feature that has not been observed by
any other models previously is that in compact star matter the trace of the
energy-momentum tensor is a nonzero density independent quantity and
therefore, induce the precious appearance of the conformal limit
$v_{s}^{2}/c^{2}=1/3$, in stark contrast to what was widely accepted in the
society Tews  _et al._ (2018). That is, there is a pseudoconformal symmetry in
the compact star matter. The predictions of the pseudoconformal model are
consistent with all the constraints from all the terrestrial experimental and
astrophysical observations.

We finally devote ourselves to the possible extensions and revisions of the
model.

The idea of the chiral-scale EFT á la Crewther and Tunstall which G$n$EFT is
based on is anchored on three-flavor QCD. Therefore $f_{0}(500)$ can be taken
as the same footing as the pseudoscalar mesons pion and kaon. However, in the
present approach, we only included the up and down quarks and ignored the
strange degrees of freedom for simplicity. So that it is interesting to extend
the present framework to see the effect of strangeness on compact star matter
Weise (2019). In addition, it is also interesting to extend the present
approach to include the $\Delta$ baryon Takeda  _et al._ (2018); Marczenko
_et al._ (2022b).

Another issue should be addressed in the future is to include the corrections
to the LOSS applied so far such that, in addition to the mass parameters, the
coupling constants also have IDD. This procedure may change the stiffness of
the EOS and the tidal the deformability of the compact star. Meanwhile, the
sound velocity after the topology change may deviate from the conformal limit
because of the explicit breaking of the conformal limit. It should be noted
that since the correction from the explicit breaking of the conformal limit is
taken as a perturbative one, the global picture of the compact star discussed
is intact.

Finally, it is interesting to pin down the density at which the hidden scale
and local flavor symmetry emerge. This is encoded in the IDDs of the hadron
parameters such as pion decay constant, dilaton decay constant, rho-N-N
coupling and rho meson mass. By checking the effect of the location of the
emergent symmetries on the star properties as well ths waveforms of the
gravitational waves, one can also extract the information of the emergent
symmetries and so phase structure of QCD at low temperature Yang  _et al._
(2021).

###### Acknowledgements.

The work of Y. L. M. was supported in part by National Science Foundation of
China (NSFC) under Grant No. 11875147 and No. 12147103 and National Key R&D
Program of China (2021YFC2202900).

## References

  * Brown and Rho (2002) G. E. Brown and M. Rho, Phys. Rept. 363, 85 (2002), arXiv:hep-ph/0103102 .
  * Holt  _et al._ (2016) J. W. Holt, M. Rho, and W. Weise, Phys. Rept. 621, 2 (2016), arXiv:1411.6681 [nucl-th] .
  * Drews and Weise (2017) M. Drews and W. Weise, Prog. Part. Nucl. Phys. 93, 69 (2017), arXiv:1610.07568 [nucl-th] .
  * Baym  _et al._ (2018) G. Baym, T. Hatsuda, T. Kojo, P. D. Powell, Y. Song, and T. Takatsuka, Rept. Prog. Phys. 81, 056902 (2018), arXiv:1707.04966 [astro-ph.HE] .
  * Ma and Rho (2020a) Y.-L. Ma and M. Rho, Prog. Part. Nucl. Phys. 113, 103791 (2020a), arXiv:1909.05889 [nucl-th] .
  * Li  _et al._ (2019) B.-A. Li, P. G. Krastev, D.-H. Wen, and N.-B. Zhang, Eur. Phys. J. A 55, 117 (2019), arXiv:1905.13175 [nucl-th] .
  * Ma and Rho (2021a) Y.-L. Ma and M. Rho, Symmetry 13, 1888 (2021a), arXiv:2009.09219 [nucl-th] .
  * Lovato  _et al._ (2022) A. Lovato _et al._ , (2022), arXiv:2211.02224 [nucl-th] .
  * Paeng  _et al._ (2016) W.-G. Paeng, T. T. S. Kuo, H. K. Lee, and M. Rho, Phys. Rev. C 93, 055203 (2016), arXiv:1508.05210 [hep-ph] .
  * Paeng  _et al._ (2017) W.-G. Paeng, T. T. S. Kuo, H. K. Lee, Y.-L. Ma, and M. Rho, Phys. Rev. D 96, 014031 (2017), arXiv:1704.02775 [nucl-th] .
  * Ma  _et al._ (2019a) Y.-L. Ma, H. K. Lee, W.-G. Paeng, and M. Rho, Sci. China Phys. Mech. Astron. 62, 112011 (2019a), arXiv:1804.00305 [nucl-th] .
  * Ma and Rho (2019a) Y.-L. Ma and M. Rho, Phys. Rev. D 99, 014034 (2019a), arXiv:1810.06062 [nucl-th] .
  * Ma and Rho (2019b) Y.-L. Ma and M. Rho, Phys. Rev. D 100, 114003 (2019b), arXiv:1811.07071 [nucl-th] .
  * Ma and Rho (2020b) Y.-L. Ma and M. Rho, (2020b), arXiv:2006.14173 [nucl-th] .
  * Rho and Ma (2021) M. Rho and Y.-L. Ma, Mod. Phys. Lett. A 36, 2130012 (2021), arXiv:2101.07121 [nucl-th] .
  * Ma and Rho (2021b) Y.-L. Ma and M. Rho, AAPPS Bull. 31, 16 (2021b), arXiv:2103.00744 [nucl-th] .
  * Lee  _et al._ (2022) H. K. Lee, Y.-L. Ma, W.-G. Paeng, and M. Rho, Mod. Phys. Lett. A 37, 2230003 (2022), arXiv:2107.01879 [nucl-th] .
  * Rho (2022a) M. Rho, Symmetry 14, 994 (2022a), arXiv:2203.06998 [nucl-th] .
  * Bedaque and Steiner (2015) P. Bedaque and A. W. Steiner, Phys. Rev. Lett. 114, 031103 (2015), arXiv:1408.5116 [nucl-th] .
  * Tews  _et al._ (2018) I. Tews, J. Carlson, S. Gandolfi, and S. Reddy, Astrophys. J. 860, 149 (2018), arXiv:1801.01923 [nucl-th] .
  * Moustakidis  _et al._ (2017) C. C. Moustakidis, T. Gaitanos, C. Margaritis, and G. A. Lalazissis, Phys. Rev. C 95, 045801 (2017), [Erratum: Phys.Rev.C 95, 059904 (2017)], arXiv:1608.00344 [nucl-th] .
  * Alsing  _et al._ (2018) J. Alsing, H. O. Silva, and E. Berti, Mon. Not. Roy. Astron. Soc. 478, 1377 (2018), arXiv:1709.07889 [astro-ph.HE] .
  * McLerran and Reddy (2019) L. McLerran and S. Reddy, Phys. Rev. Lett. 122, 122701 (2019), arXiv:1811.12503 [nucl-th] .
  * Jeong  _et al._ (2020) K. S. Jeong, L. McLerran, and S. Sen, Phys. Rev. C 101, 035201 (2020), arXiv:1908.04799 [nucl-th] .
  * Kapusta and Welle (2021) J. I. Kapusta and T. Welle, Phys. Rev. C 104, L012801 (2021), arXiv:2103.16633 [nucl-th] .
  * Zhao and Lattimer (2020) T. Zhao and J. M. Lattimer, Phys. Rev. D 102, 023021 (2020), arXiv:2004.08293 [astro-ph.HE] .
  * Margueron  _et al._ (2021) J. Margueron, H. Hansen, P. Proust, and G. Chanfray, Phys. Rev. C 104, 055803 (2021), arXiv:2103.10209 [nucl-th] .
  * Fujimoto  _et al._ (2022) Y. Fujimoto, K. Fukushima, L. D. McLerran, and M. Praszalowicz, (2022), arXiv:2207.06753 [nucl-th] .
  * Marczenko  _et al._ (2022a) M. Marczenko, L. McLerran, K. Redlich, and C. Sasaki, (2022a), arXiv:2207.13059 [nucl-th] .
  * Bando  _et al._ (1985) M. Bando, T. Kugo, S. Uehara, K. Yamawaki, and T. Yanagida, Phys. Rev. Lett. 54, 1215 (1985).
  * Bando  _et al._ (1988) M. Bando, T. Kugo, and K. Yamawaki, Phys. Rept. 164, 217 (1988).
  * Harada and Yamawaki (2003) M. Harada and K. Yamawaki, Phys. Rept. 381, 1 (2003), arXiv:hep-ph/0302103 .
  * Crewther and Tunstall (2015) R. J. Crewther and L. C. Tunstall, Phys. Rev. D 91, 034016 (2015), arXiv:1312.3319 [hep-ph] .
  * Catà  _et al._ (2019) O. Catà, R. J. Crewther, and L. C. Tunstall, Phys. Rev. D 100, 095007 (2019), arXiv:1803.08513 [hep-ph] .
  * Crewther (2020) R. J. Crewther, Universe 6, 96 (2020), arXiv:2003.11259 [hep-ph] .
  * Skyrme (1961) T. H. R. Skyrme, Proc. Roy. Soc. Lond. A 260, 127 (1961).
  * Kugler and Shtrikman (1988) M. Kugler and S. Shtrikman, Phys. Lett. B 208, 491 (1988).
  * Kugler and Shtrikman (1989) M. Kugler and S. Shtrikman, Phys. Rev. D 40, 3421 (1989).
  * Lee  _et al._ (2003) H.-J. Lee, B.-Y. Park, M. Rho, and V. Vento, Nucl. Phys. A 726, 69 (2003), arXiv:hep-ph/0304066 .
  * Park  _et al._ (2004) B.-Y. Park, M. Rho, and V. Vento, Nucl. Phys. A 736, 129 (2004), arXiv:hep-ph/0310087 .
  * Park  _et al._ (2008) B.-Y. Park, M. Rho, and V. Vento, Nucl. Phys. A 807, 28 (2008), arXiv:0801.1374 [hep-ph] .
  * Ma  _et al._ (2013) Y.-L. Ma, M. Harada, H. K. Lee, Y. Oh, B.-Y. Park, and M. Rho, Phys. Rev. D 88, 014016 (2013), [Erratum: Phys.Rev.D 88, 079904 (2013)], arXiv:1304.5638 [hep-ph] .
  * Ma  _et al._ (2014a) Y.-L. Ma, M. Harada, H. K. Lee, Y. Oh, B.-Y. Park, and M. Rho, Phys. Rev. D 90, 034015 (2014a), arXiv:1308.6476 [hep-ph] .
  * Ma and Rho (2017) Y.-L. Ma and M. Rho, Sci. China Phys. Mech. Astron. 60, 032001 (2017), arXiv:1612.06600 [nucl-th] .
  * Shao and Ma (2022) L.-Q. Shao and Y.-L. Ma, Phys. Rev. D 106, 014014 (2022), arXiv:2202.09957 [nucl-th] .
  * Brown and Rho (1991) G. E. Brown and M. Rho, Phys. Rev. Lett. 66, 2720 (1991).
  * Bogner  _et al._ (2003) S. K. Bogner, T. T. S. Kuo, and A. Schwenk, Phys. Rept. 386, 1 (2003), arXiv:nucl-th/0305035 .
  * Serot and Walecka (1986) B. D. Serot and J. D. Walecka, Adv. Nucl. Phys. 16, 1 (1986).
  * Li  _et al._ (2022) F. Li, B.-J. Cai, Y. Zhou, W.-Z. Jiang, and L.-W. Chen, Astrophys. J. 929, 183 (2022), arXiv:2202.08705 [nucl-th] .
  * Miyatsu  _et al._ (2022) T. Miyatsu, M.-K. Cheoun, and K. Saito, Astrophys. J. 929, 82 (2022), arXiv:2202.06468 [nucl-th] .
  * Workman (2022) R. L. Workman (Particle Data Group), PTEP 2022, 083C01 (2022).
  * Isham  _et al._ (1970) C. J. Isham, A. Salam, and J. A. Strathdee, Phys. Rev. D 2, 685 (1970).
  * Ellis (1970) J. R. Ellis, Nucl. Phys. B 22, 478 (1970), [Erratum: Nucl.Phys.B 25, 639–639 (1971)].
  * Schechter (1980) J. Schechter, Phys. Rev. D 21, 3393 (1980).
  * Golterman and Shamir (2016) M. Golterman and Y. Shamir, Phys. Rev. D 94, 054502 (2016), arXiv:1603.04575 [hep-ph] .
  * Brodsky  _et al._ (2010) S. J. Brodsky, G. F. de Teramond, and A. Deur, Phys. Rev. D 81, 096010 (2010), arXiv:1002.3948 [hep-ph] .
  * Horsley  _et al._ (2014) R. Horsley, H. Perlt, P. E. L. Rakow, G. Schierholz, and A. Schiller, Phys. Lett. B 728, 1 (2014), arXiv:1309.4311 [hep-lat] .
  * Yu  _et al._ (2022) Q. Yu, H. Zhou, X.-D. Huang, J.-M. Shen, and X.-G. Wu, Chin. Phys. Lett. 39, 071201 (2022), arXiv:2112.01200 [hep-ph] .
  * Alexandru and Horváth (2019) A. Alexandru and I. Horváth, Phys. Rev. D 100, 094507 (2019), arXiv:1906.08047 [hep-lat] .
  * Freund and Nambu (1968) P. G. O. Freund and Y. Nambu, Phys. Rev. 174, 1741 (1968).
  * Goldberger  _et al._ (2008) W. D. Goldberger, B. Grinstein, and W. Skiba, Phys. Rev. Lett. 100, 111802 (2008), arXiv:0708.1463 [hep-ph] .
  * Li  _et al._ (2017) Y.-L. Li, Y.-L. Ma, and M. Rho, Phys. Rev. D 95, 114011 (2017), arXiv:1609.07014 [hep-ph] .
  * Catà and Müller (2020) O. Catà and C. Müller, Nucl. Phys. B 952, 114938 (2020), arXiv:1906.01879 [hep-ph] .
  * Li  _et al._ (2018a) Y.-L. Li, Y.-L. Ma, and M. Rho, Chin. Phys. C 42, 094102 (2018a), arXiv:1710.02840 [nucl-th] .
  * Li  _et al._ (2018b) Y.-L. Li, Y.-L. Ma, and M. Rho, Phys. Rev. C 98, 044318 (2018b), arXiv:1804.00310 [nucl-th] .
  * Ma and Rho (2020c) Y.-L. Ma and M. Rho, Phys. Rev. Lett. 125, 142501 (2020c), arXiv:2002.03310 [nucl-th] .
  * Paeng  _et al._ (2012) W.-G. Paeng, H. K. Lee, M. Rho, and C. Sasaki, Phys. Rev. D 85, 054022 (2012), arXiv:1109.5431 [hep-ph] .
  * Georgi (1989) H. Georgi, Phys. Rev. Lett. 63, 1917 (1989).
  * Georgi (1990) H. Georgi, Nucl. Phys. B 331, 311 (1990).
  * Harada and Yamawaki (2001) M. Harada and K. Yamawaki, Phys. Rev. Lett. 86, 757 (2001), arXiv:hep-ph/0010207 .
  * Beane and van Kolck (1994) S. R. Beane and U. van Kolck, Phys. Lett. B 328, 137 (1994), arXiv:hep-ph/9401218 .
  * Suzuki (2017) M. Suzuki, Phys. Rev. D 96, 065010 (2017), arXiv:1707.01589 [hep-ph] .
  * (73) W.-C. Yang and Y.-L. Ma, In preparation .
  * Witten (1979) E. Witten, Nucl. Phys. B 160, 57 (1979).
  * Klebanov (1985) I. R. Klebanov, Nucl. Phys. B 262, 133 (1985).
  * Goldhaber and Manton (1987) A. S. Goldhaber and N. S. Manton, Phys. Lett. B 198, 231 (1987).
  * Ma and Harada (2016) Y.-L. Ma and M. Harada, (2016), arXiv:1604.04850 [hep-ph] .
  * Harada  _et al._ (2015) M. Harada, H. K. Lee, Y.-L. Ma, and M. Rho, Phys. Rev. D 91, 096011 (2015), arXiv:1502.02508 [hep-ph] .
  * Detar and Kunihiro (1989) C. E. Detar and T. Kunihiro, Phys. Rev. D 39, 2805 (1989).
  * Motohiro  _et al._ (2015) Y. Motohiro, Y. Kim, and M. Harada, Phys. Rev. C 92, 025201 (2015), [Erratum: Phys.Rev.C 95, 059903 (2017)], arXiv:1505.00988 [nucl-th] .
  * Ma  _et al._ (2014b) Y.-L. Ma, M. Harada, H. K. Lee, Y. Oh, and M. Rho, Int. J. Mod. Phys. Conf. Ser. 29, 1460238 (2014b), arXiv:1312.2290 [hep-ph] .
  * Komargodski (2018) Z. Komargodski, (2018), arXiv:1812.09253 [hep-th] .
  * Hsin and Seiberg (2016) P.-S. Hsin and N. Seiberg, JHEP 09, 095 (2016), arXiv:1607.07457 [hep-th] .
  * Gaiotto  _et al._ (2018) D. Gaiotto, Z. Komargodski, and N. Seiberg, JHEP 01, 110 (2018), arXiv:1708.06806 [hep-th] .
  * Benini (2018) F. Benini, JHEP 02, 068 (2018), arXiv:1712.00020 [hep-th] .
  * Tong (2016) D. Tong (2016) arXiv:1606.06687 [hep-th] .
  * Karasik (2020) A. Karasik, SciPost Phys. 9, 008 (2020), arXiv:2003.07893 [hep-th] .
  * Bigazzi  _et al._ (2022) F. Bigazzi, A. L. Cotrone, and A. Olzi, (2022), arXiv:2211.05147 [hep-th] .
  * Liu  _et al._ (2019) X.-H. Liu, Y.-L. Ma, and M. Rho, Phys. Rev. C 99, 055808 (2019), arXiv:1811.10012 [nucl-th] .
  * Ma  _et al._ (2019b) Y.-L. Ma, M. A. Nowak, M. Rho, and I. Zahed, Phys. Rev. Lett. 123, 172301 (2019b), arXiv:1907.00958 [hep-th] .
  * Callan and Harvey (1985) C. G. Callan, Jr. and J. A. Harvey, Nucl. Phys. B 250, 427 (1985).
  * Park  _et al._ (2019) B.-Y. Park, W.-G. Paeng, and V. Vento, Nucl. Phys. A 989, 231 (2019), arXiv:1904.04483 [hep-ph] .
  * Canfora (2018) F. Canfora, Eur. Phys. J. C 78, 929 (2018), arXiv:1807.02090 [hep-th] .
  * Rho (2022b) M. Rho, (2022b), arXiv:2211.14890 [nucl-th] .
  * Sulejmanpasic  _et al._ (2017) T. Sulejmanpasic, H. Shao, A. Sandvik, and M. Unsal, Phys. Rev. Lett. 119, 091601 (2017), arXiv:1608.09011 [hep-th] .
  * Senthil  _et al._ (2004) T. Senthil, A. Vishwanath, L. Balents, S. Sachdev, and M. P. A. Fisher, Science 303, 1490 (2004), arXiv:cond-mat/0311326 .
  * Li and Ma (2017) Y.-L. Li and Y.-L. Ma, (2017), arXiv:1710.02839 [hep-ph] .
  * Ma and Rho (2018) Y.-L. Ma and M. Rho, Phys. Rev. D 97, 094017 (2018), arXiv:1612.04079 [nucl-th] .
  * Wilkinson (1973) D. H. Wilkinson, Phys. Rev. C 7, 930 (1973).
  * Suhonen (2017) J. T. Suhonen, Front. in Phys. 5, 55 (2017), arXiv:1712.01565 [nucl-th] .
  * Engel and Menéndez (2017) J. Engel and J. Menéndez, Rept. Prog. Phys. 80, 046301 (2017), arXiv:1610.06548 [nucl-th] .
  * Rho (2022c) M. Rho, (2022c), arXiv:2212.05558 [nucl-th] .
  * Friman and Rho (1996) B. Friman and M. Rho, Nucl. Phys. A 606, 303 (1996), arXiv:nucl-th/9602025 .
  * Kienle and Yamazaki (2004) P. Kienle and T. Yamazaki, Prog. Part. Nucl. Phys. 52, 85 (2004).
  * Lubos  _et al._ (2019) D. Lubos _et al._ , Phys. Rev. Lett. 122, 222502 (2019).
  * van Kolck (2019) U. van Kolck, Nuovo Cim. C 42, 52 (2019).
  * Tews  _et al._ (2017) I. Tews, J. M. Lattimer, A. Ohnishi, and E. E. Kolomeitsev, Astrophys. J. 848, 105 (2017), arXiv:1611.07133 [nucl-th] .
  * Annala  _et al._ (2020) E. Annala, T. Gorda, A. Kurkela, J. Nättilä, and A. Vuorinen, Nature Phys. 16, 907 (2020), arXiv:1903.09121 [astro-ph.HE] .
  * Hippert  _et al._ (2021) M. Hippert, E. S. Fraga, and J. Noronha, Phys. Rev. D 104, 034011 (2021), arXiv:2105.04535 [nucl-th] .
  * Tsang  _et al._ (2018) C. Y. Tsang, M. B. Tsang, P. Danielewicz, W. G. Lynch, and F. J. Fattoyev, (2018), arXiv:1807.06571 [nucl-ex] .
  * Demorest  _et al._ (2010) P. Demorest, T. Pennucci, S. Ransom, M. Roberts, and J. Hessels, Nature 467, 1081 (2010), arXiv:1010.5788 [astro-ph.HE] .
  * Antoniadis  _et al._ (2013) J. Antoniadis _et al._ , Science 340, 6131 (2013), arXiv:1304.6875 [astro-ph.HE] .
  * Cromartie  _et al._ (2019) H. T. Cromartie _et al._ (NANOGrav), Nature Astron. 4, 72 (2019), arXiv:1904.06759 [astro-ph.HE] .
  * Raaijmakers  _et al._ (2019) G. Raaijmakers _et al._ , Astrophys. J. Lett. 887, L22 (2019), arXiv:1912.05703 [astro-ph.HE] .
  * Miller  _et al._ (2019) M. C. Miller _et al._ , Astrophys. J. Lett. 887, L24 (2019), arXiv:1912.05705 [astro-ph.HE] .
  * Fattoyev  _et al._ (2018) F. J. Fattoyev, J. Piekarewicz, and C. J. Horowitz, Phys. Rev. Lett. 120, 172702 (2018), arXiv:1711.06615 [nucl-th] .
  * Weise (2019) W. Weise, JPS Conf. Proc. 26, 011002 (2019), arXiv:1905.03955 [nucl-th] .
  * Takeda  _et al._ (2018) Y. Takeda, Y. Kim, and M. Harada, Phys. Rev. C 97, 065202 (2018), arXiv:1704.04357 [nucl-th] .
  * Marczenko  _et al._ (2022b) M. Marczenko, K. Redlich, and C. Sasaki, Astrophys. J. Lett. 925, L23 (2022b), arXiv:2110.11056 [nucl-th] .
  * Yang  _et al._ (2021) W.-C. Yang, Y.-L. Ma, and Y.-L. Wu, Sci. China Phys. Mech. Astron. 64, 252011 (2021), arXiv:2011.03665 [nucl-th] .