\section{The pseudo-conformal model}
\sect{\bf The pseudo-conformal model}
We now rephrase the two basic observations $\bf A$ and $\bf B$  listed above in a form that encapsulates them in a simple EoS. We do this by replacing what is found in  the $V_{lowk}$RG in the density regime $n\mathrel{\rlap{\lower3pt\hbox{\hskip1pt$\sim$} n_{1/2}$  in a manifestly pseudo-conformal form. This approach can be considered as belonging to the class of what is referred to as ``Energy Density Functional (EDF)" popular in nuclear theory community, in particular in the form of relativistic mean field theory. What makes our approach however distinctly different from the usual EDF approaches found in the literature is the intrinsic density dependence  inherited from QCD and the impact of the topology change.

The arguments we develop below rely on the ``master formula" for
 the behavior of  hadron masses in nuclear medium established in \cite{PKLMR}, $\frac{m^\ast_N}{m_N} \approx \frac{m^\ast_\sigma}{m_\sigma} \approx \frac{g_V}{g_V^\ast} \frac{m^\ast_V}{m_V} \approx \frac{f^\ast_\pi}{f_\pi} \approx \frac{\langle \chi \rangle^\ast}{\langle \chi \rangle}\equiv \Phi$ (where $V=(\rho,\omega)$),
 that we assume holds up to the central density of massive stars, i.e., $n_{\rm cen}\approx (5-7)n_0$.
The asterisk stands for the intrinsic density dependence inherited from QCD at the matching scale and the quantities without asterisk are free-space quantities. We denote the two regions divided at $n_{1/2}$ as R(egion)-I for $n< n_{1/2}$ and R(egion{-II for $n\geq n_{1/2}$.

The master scaling relation given above dictates the density dependence of the bare parameters of the effective Lagrangian $bs$HLS, with which the $V_{lowk}$RG is to be performed.

First the region R-I:

Up to $n_0$, there is only one parameter $\Phi$. This is because at low density  $g_V^\ast/g_V=1$. This follows from that the vector meson mass accurately satisfies the KSRF low-energy theorem $m_V^\ast=2 f_\pi^\ast g_V^\ast$ and in the $bs$HLS Lagrangian the vector meson mass scales with $\langle}\def\ra{\rangle\chi\ra^\ast$.  In \cite{PKLMR}, it was shown that the $V_{lowk}$RG with the density scaling of $f_\pi^\ast$ or equivalently $\Phi$  inferred from deeply bound pionic systems describes very well the bulk properties of normal nuclear matter at $n_0$.

We assume that this one parameter EoS can be extrapolated via the density dependence of $\Phi$  up to the  higher topology change density $n_{1/2} > n_0$.



We now turn to the region R-II.

First we would like to capture the observations made in the $V_{lowk}$ in R-II in an EoS that is manifestly pseudo-conformal, that is, possessing a non-vanishing trace of energy-momentum tensor which is independent of density, and exactly reproduces the $V_{lowk}$RG result. In \cite{PKLMR}, such an EoS has been found in the extremely simple form of the  energy per nucleon (minus the vacuum rest mass of the matter written) of the forrm
\begin{equation}
\frac{E}{A} = X^\alpha \left( \frac{n}{n_0} \right)^{1/3} + Y^\alpha \left( \frac{n}{n_0} \right)^{-1} -939\, {\rm MeV} \label{RII}
\end{equation}
with the  coefficients $X^\alpha$ and $Y^\alpha$ for $\alpha=(n_n-n_p)/(n_n+n_p)=(0,1)$ (where $n_{n,p}$ are the neutron and proton densities).  It is easy to verify that for any values of $X$ and $Y$,
\begin{eqnarray}}\def\ee{\end{eqnarray}
\frac{\partial}{\partial n} \langle}\def\ra{\rangle\theta_\mu^\mu\ra=\frac{\partial \epsilon (n)}{\partial n} (1-3{v_s^2})=0\label{derivTEMT}
\ee
with $v_s^2=\frac{\partial P(n)}{\partial n}/\frac{\partial \epsilon(n)}{\partial n}$.  Since $\frac{\partial \epsilon (n)}{\partial n} \neq 0$ in the range of densities involved, we immediately obtain the sound velocity $v_s^2 = 1/3$.

Now we compute the EoS in R-II using (\ref{RII}) with the coefficients determined by the continuity at $n=n_{1/2}$ between R-I given by \cite{PKLMR} and R-II, (\ref{RII}),  for the chemical potential $\mu$ and the pressure $P$
\begin{eqnarray}}\def\ee{\end{eqnarray}
\mu_I=\mu_{II},\ \ P_I=P_{II} \ \ {\rm at} \ \ n=n_{1/2}
\ee
where the subscripts $I$ and $II$ represent the regions R-I and R-II respectively. The union of the EoSs of R-I and R-II so computed  --  that we call ``pseudo-conformal model (PCM)"  -- reproduces  {\it precisely} the results of the $V_{lowk}$RG for massive stars.  This verifies that the EoS from (\ref{RII}) exactly captures the physics of the $V_{lowk}$RG in R-II for $n_{1/2}=2n_0$. We interpret this as saying that the pseudo-conformality given in the extremely simple EoS  Eq. (\ref{RII}) embodies  in R-II the physics of the full $V_{lowk}$RG that results from an intricate interplay of the observables {\bf A} and {\bf B}..




Now we would like to look at the EoS for a system that is built with the topology change at a density higher than $2n_0$, i.e., $n_{1/2} > 2n_0$.  This has to do with the information on the tidal deformability inferred from the recent observation of gravitational waves coming from coalescing neutron stars~\cite{Ligo/Virgo}. It is found that the dimensionless tidal deformability $\Lambda$ -- as defined by the GW170817 ~\cite{Ligo/Virgo} -- calculated in the  $V_{lowk}$RG~\cite{PKLMR} with $n_{1/2}\simeq 2.0n_0$ is found to be $\sim 790$, close to the reported upper bound 800 for $\sim 1.4M_\odot$~\cite{Ligo/Virgo}. Furthermore the more recent analysis of the GW170817 data indicates that the bound  could be tightened to a lower value, with the presently quoted range being between 720 and 170~\cite{abbottetal}. This seems to require that  since $\Lambda$ is sensitive to the symmetry energy in R-I and with the cusp structure entailing the lowering of the symmetry energy going toward $n_{1/2}$ from below,  the topology change density  $n_{1/2}$ be increased to above  $2.0n_0$ so as to bring $\Lambda$ to a lower value.  This seems feasible  since the global properties of  massive stars given within the model are verified to differ very little between $1.5 n_0$ and $2.0n_0$~\cite{dongetal,PKLMR}. Thus we expect  one should be able  to bring $n_{1/2}$ higher than $2.0n_0$,  keeping  unaffected the good overall star properties obtained for $n_{1/2}=2n_0$ in \cite{PKLMR}.



To do this, we assume that the PCM as defined above can be applied to a system with $n_{1/2} > 2n_0$ without affecting the standard star properties seriously. As a first trial, we pick $n_{1/2}=2.6n_0$ and explore what transpires with this value.




\sect{\bf Massive star properties}
First we look at the standard star properties given by the PCM with $n_{1/2}=2.6n_0$. By construction, the sound velocity converges to $v_s^2=1/3$ at $n_{1/2}\mathrel{\rlap{\lower3pt\hbox{\hskip1pt$\sim$} 3n_0$ as in the case of $n_{1/2}=2.0n_0$.


The crucial question is: Does this PCM EoS for $n_{1/2}=2.6n_0$ give  correct global star properties as it did for $n_{1/2}=2.0n_0$ reproducing exactly the $V_{lowk}$RG?

The EoSs for both $\alpha=1$ and $0$ are found to come out quite close to the $V_{lowk}$RG results for $n_{1/2}=2n_0$. They are all consistent with currently available experimental bounds.
The particularly important quantity for our discussion is the symmetry energy $E_{sym}$,  which is plotted in Fig.~\ref{EA}.  It comes out consistent with the available experimental bounds, which are at present too broad to be stringent. There is a constraint based on combined data from neutron stars and gravitational waves given in \cite{bal}, $E_{sym} (2n_0)= 46.9\pm 10.1$ MeV, with which the predicted value $E_{sym}^{pred} (2n_0)\simeq 49$ MeV is consistent. However the density probed there is in R-I, so it is not directly relevant to the pseudo-conformal structure.  What  is  remarkable is that the cross-over from soft to hard in $E_{sym}$ -- which mimics  the cusp in the skyrmion description-- resembles closely that of the {\it full} $V_{lowk}$ RG result~\cite{PKLMR}.
\begin{figure}[h]
\begin{center}
\includegraphics[width=5.5cm]{Esym4.pdf}
\caption{$E_{sym}$ (vs density $n$) predicted by the PCM (solid line), which is consistent with the (heavy-ion) experimental bounds quoted from B.A. Li and L.W. Chen, Phy. Rev. C{\bf 72}, 064611 (2005) and M.B. Tsang {\it et al}, Int. J. Mod. Phys. E {\bf 19}, 1631 (2010). }\label{EA}
\end{center}
\end{figure}


In Fig.~\ref{MvsR} are given M vs. R and the central density of the compact star predicted by the PCM with $n_{1/2}=2.6$. They are shown to indicate that the overall structure of the star is insensitive to the density $n_{1/2}$. The maximum mass comes out at  $M_{\rm max}= 2.02\, M_\odot$ with radius at $R = 11.86$ km. The central density is found to be  $n_{\rm cent} = 5.6\, n_0$. They are to be compared with the $V_{lowk}$ RG results of \cite{PKLMR} (for $n_{1/2}=2n_0$):  $M_{\rm max}=2.05 M_\odot$, $R = 12.19$ km, and $n_{\rm cent} = 5.1\, n_0$. There is very little difference -- and perhaps {\it no difference} within the approximations involved --  from the case of $n_{1/2}=2.0 n_0$ (and also that of $n_{1/2}=1.5 n_0$~\cite{dongetal}).
\begin{figure}[h]
\begin{center}
\includegraphics[width=4.3cm]{MvsR.pdf}\includegraphics[width=4.2cm]{nc.pdf}
\caption{ The mass vs. radius of the neutron star (left panel) in beta equilibrium and the mass vs. the central density of the neutron star (right panel).}\label{MvsR}
\end{center}
\end{figure}

These results are convincingly verifying that the pseudo-conformal EoS correctly captures the full $V_{lowk}$RG physics. They also
reinforce relative insensitivity of the standard star properties to the location of topology change $n_{1/2}$.

\sect{\bf Tidal deformability}
We now turn to the case of the tidal deformability.
 In a stark contrast,  the location of the topology change density  is found to strongly influence the tidal deformability in  the PCM.\footnote{This contrasts with other energy density functional models in which $\Lambda$ could be varied arbitrarily by fine-tuning parameters without affecting the basic structure of EoS.}

\begin{table}[h]
\begin{center}
\caption{Dimensional and dimensionless tidal  deformability:  $\lambda$ and $\Lambda$ }
\label{tidaldef}
\begin{center}
\begin{tabular}{c c c c c}
\hline
\hline
$M/M_{\odot}$ & $n_{cent}/n_0$   &  $\lambda/(10^{36}{\rm gcm^2s^2})$   & $\Lambda/100$  &   $ R/{\rm km}$ \\
\hline
1.12 & 2.0 & 4.16 & 22.5 & 12.64 \\
1.22 & 2.1 & 4.02 & 14.4 & 12.69 \\
1.31 & 2.2 & 3.86 & 9.45 & 12.76 \\
1.40 & 2.3 & 3.67 & 6.44 & 12.79 \\
1.49 & 2.4 & 3.46 & 4.52 & 12.83 \\
1.57 & 2.5 & 3.26 & 3.26 & 12.85 \\
\hline
\hline
\end{tabular}
\end{center}
\end{center}
\end{table}



 The results for the $\Lambda$ given by the PCM for $n_{1/2}=2.6 n_0$ are summarized  in Table.~\ref{tidaldef}  for the range of star masses involved.

Based on the effective mean-field cusp structure as predicted,  we expected $\Lambda$ to decrease,   when $n_{1/2}$ is increased to $2.6n_0$, from $\sim 800$ that was  found for $n_{1/2}=2.0 n_0$.  This expectation is confirmed: One indeed obtains a significant drop in $\Lambda$.  For $1.4\, M_\odot$, we find $\Lambda\sim 640$  with $n_{\rm cent} = 2.3\, n_0$. The radius for the $1.4 M_\odot$ star comes out to be $\sim 12.8$ km.


 It is noteworthy for assessing the principal mechanism in question  that the radius for the relevant star mass in the PCM is  insensitive to the location of $n_{1/2}$, hence to the cusp structure of the symmetry energy.\footnote{The radius is likely to depend on other details of the  star structure -- such as the crust etc. --  than the high density properties focused on in this paper. While the tidal deformability for the $1.4 M_\odot$ star is changed by roughly  25\% in going from $n_{1/2}=2n_0$ to $n_{1/2}=2.6n_0$, the radius is found to change by less than 1.5\%.}

 %

\sect{\bf Remarks}
 It is intriguing that the simple $E/A$, Eq.~(\ref{RII}), efficiently captures the complex structure of $\bf A$ \& $\bf B$ of $V_{lowk}$ RG in R-II. It seems remarkable also that  the appearance of the pseudo-conformal sound velocity $v_s^2\simeq 1/3$ at $n\mathrel{\rlap{\lower3pt\hbox{\hskip1pt$\sim$} n_{1/2}$ as found in the $V_{lowk}$ RG calculation and confirmed by the pseudo-conformal model is  fully compatible with the observed properties of massive compact stars.   Given that converging to and staying at $v_s^2=1/3$ is impossible at non-asymptotic density unless there is a change of degrees of freedom~\cite{tews}, we are lead to that  the half-skyrmions  {\it must} play the role of non-hadronic degrees of freedom in the range of density in which strongly-coupled quarks, such as quarkyonic phase, are considered to figure~\cite{baymetal}.  The half-skyrmions in $n> n_{1/2}$ are confined by the magnetic monopole, an intrinsic property in QCD, hence a potential link to QCD degrees of freedom~\cite{cho}. This invites us to conjecture that the pseudo-conformal sound velocity is a signal in dense matter both for the emergence of scale symmetry and local flavor symmetry hidden in QCD~\cite{CCT,yamawaki2018} {\it and} for the manifestation of a Cheshire Cat phenomenon. This adds one more  to the growing evidence of Cheshire Cat phenomenon in dense hadronic matter, along with the superqualitons~\cite{qualiton},  vortices~\cite{alfordetal} etc. in the color-flavor-locked phase.

When the value of $\Lambda$ is measured  with a precision, it will provide  theoretically crucial information on the topology change density $n_{1/2}$, a.k.a, the hadron-quark crossover density.  At present there are no known clues, experimental or theoretical, as to how to locate it precisely. Should  the bound be tightened to a lower value than $\sim 600$ in the future, the PCM will provide a simple means to probe the range of $n_{1/2}$ compatible with the bound.  This will be the first impact of neutron-star physics on nuclear and particle physics in a quantity that cannot be accessed by reliable theories or experiments.




We are very grateful to Tom Kuo for suggesting the expression (\ref{RII}) and many useful  discussions. Y.L. Ma is supported in part by National Science Foundation of China (NSFC) under grant No. 1147571, 1147308 and Seeds Funding of Jilin University. Two of the authors, YLM and MR, would like to acknowledge the hospitality of APCTP, Korea where this paper was finalized.





