Dichotomy of Baryons as Quantum Hall Droplets and Skyrmions:
Topological Structure of Dense Matter

To address the dichotomy problem (DP) in highly compressed baryonic matter, we incorporate the $\eta^{\prime}$ field, $\eta^{\prime}\in U_{A}(1)$, in addition to the pseudo-scalar NG bosons $\pi\in SU(2)$ and the vectors $\rho_{\mu}\in SU(2)$ and $\omega\in U(1)$. The reason for this, as we will argue, is that although $\eta^{\prime}$ is massive compared with the mesons we will take into account, it goes massless in the limit that $N_{c}\to\infty$ and plays a crucial role for bringing in Chern-Simons topological field theory structure in (possibly) dense baryonic systems. In the modern development, it is being suggested that the role of $\eta^{\prime}$ in the guise of fractional quantum Hall “pancake” for the flavor singlet baryon $B^{(0)}$ plays an indispensable role for QCD phase structure at high density. We will comment on this matter later although it is not well understood at present.

The Crewther’s GD approach to scale symmetry crewther adopted in this paper MR-review – which will turn out to play a crucial role in our theory – necessitates the kaons on par with the dilaton. For our purpose, however, we can ignore the strange quark – given the presence of the $\eta^{\prime}$ meson – and focus on the two light flavors. For the reason that will become clear later, unless otherwise specified, the $\rho$ and $\omega$ fields will be treated in $U(2)$-nonsymmetric way.³³3In the Seiberg-dual approach to HLS, the $\omega$ meson is not pure $U(1)$ of $U(N_{f})$ that contains $\rho$ but a mixture of $U(1)$s. This feature will appear later in Sect. V where baryonic matter with $\eta^{\prime}$ present is treated.

We write the chiral field $U$ as ⁴⁴4In this paper, we use the convention of HY:PR

$\displaystyle U=\xi^{2}=e^{i{\eta^{\prime}}/{f_{\eta}}}e^{i\tau_{a}\pi_{a}/f}$

(1)

and the HLS fields as

$\displaystyle V_{\mu}^{\rho}$

$\displaystyle=$

$\displaystyle{}\frac{1}{2}g_{\rho}\rho_{\mu}^{a}\tau^{a},\quad V_{\mu}^{\omega}={}\frac{1}{2}g_{\omega}\omega_{\mu}.$

(2)

Expressed terms of the Maurer-Cartan 1-forms

$\displaystyle\hat{\alpha}_{\parallel,\perp}^{\mu}$

$\displaystyle=$

$\displaystyle\frac{1}{2i}\left(D^{\mu}\xi\cdot\xi^{\dagger}\pm D^{\mu}\xi^{\dagger}\cdot\xi\right)$

(3)

where $D_{\mu}\xi=(\partial_{\mu}-iV_{\mu}^{\rho}-iV_{\mu}^{\omega})\xi$, the HLS Lagrangian we are concerned with is of the same form as the HLS Lagrangian for 3 flavors HY:PR with the parity-anomalous homogeneous Wess-Zumino (hWZ) Lagrangian composed of 3 terms (in the absence of external fields). For the $SU(2)\times U(1)$ case we are dealing with, there is no five-dimensional Wess-Zumino (WZ) term.

To implement scale symmetry à la GD crewther , we are to use as explained below the conformal compensator field $\chi=f_{\chi}e^{\chi/f_{\chi}}$ which has both mass dimension and scale dimension 1. The structure of the Lagrangian that we use in this discussion was written down before in our reviews MR-review and we shall write it down here again with $\eta^{\prime}$ incorporated. In oder to justify its extension in the density domain where FQH droplets could play a role, an additional ingredient to what figures in MR-review is needed. It will be given below.

The scale-symmetrized Lagrangian that we denote as sHLS is of the form

	$\displaystyle{{\cal L}}_{sHLS}$	$\displaystyle=$	$\displaystyle f^{2}\Phi^{2}\text{Tr}\left(\hat{\alpha}^{\mu}_{\perp}\hat{\alpha}_{\perp\mu}\right)+f_{\sigma\rho}^{2}\Phi^{2}\text{Tr}\left(\hat{\alpha}^{\mu}_{\parallel}\hat{\alpha}_{\parallel\mu}\right)$		(4)
			$\displaystyle{}+f_{0}^{2}\Phi^{2}\text{Tr}\left(\hat{\alpha}^{\mu}_{\parallel}\right)\text{Tr}\left(\hat{\alpha}_{\parallel\mu}\right)\ +{{\cal L}}_{hWZ}$
			$\displaystyle{}-\frac{1}{2g_{\rho}^{2}}\text{Tr}(V_{\mu\nu}V^{\mu\nu})-\frac{1}{2g_{0}^{2}}\text{Tr}(V_{\mu\nu})\text{Tr}(V^{\mu\nu})$
			$\displaystyle{}+\frac{1}{2}\partial_{\mu}\chi\partial^{\mu}\chi+V(\chi)$

where ${{\cal L}}_{hWZ}$ is the hWZ term that conserves parity and charge conjugation but violates intrinsic parity, $\Phi$ is defined as

$\displaystyle\Phi=\chi/f_{\chi}$

(5)

and $V_{\chi}$ is the dilaton potential, the explicit form of which is not needed for our purpose. $V^{\mu\nu}$ is the usual field tensor

$\displaystyle V^{\mu\nu}$

$\displaystyle=$

$\displaystyle\partial^{\mu}V^{\nu}-\partial^{\nu}V^{\mu}-i[V^{\mu},V^{\nu}]$

(6)

with $V^{\mu}=V_{\rho}^{\mu}+V_{\omega}^{\mu}$. In (4)

$\displaystyle f_{0}^{2}$

$\displaystyle=$

$\displaystyle\frac{f_{\sigma\omega}^{2}-f_{\sigma\rho}^{2}}{2},\qquad\frac{1}{g_{0}^{2}}=\frac{1}{2}\left(\frac{1}{g_{\omega}^{2}}-\frac{1}{g_{\rho}^{2}}\right),$

(7)

where $f_{\sigma V}$ figures in the mass formula $m_{V}^{2}=f^{2}_{\sigma V}g_{V}^{2}$⁵⁵5The familiar example is for $V_{\mu}=\frac{1}{2}g_{\rho}\rho_{\mu}^{a}\tau^{a}$, $f^{2}_{\sigma\rho}=2f_{\pi}^{2}$, giving the KSRF relation, $m_{\rho}^{2}=2f_{\pi}^{2}g_{\rho}^{2}$.. The $U(2)$ symmetry is recovered at the classical level by setting $g_{\omega}=g_{\rho}$ and $f_{0}=1/g_{0}=0$ in (4). The hWZ Lagrangian that will be found to unify the $N_{f}=1$ and $N_{f}\geq 2$ baryons consists of three terms (ignoring the external fields not needed in our case)

$\displaystyle{{\cal L}}_{hWZ}=\frac{N_{c}}{16\pi^{2}}\sum_{i=1}^{3}c_{i}{{\cal L}}_{i}$

(8)

with

	$\displaystyle{{\cal L}}_{1}$	$\displaystyle=$	$\displaystyle i{\rm Tr}[\hat{\alpha}_{L}^{3}\hat{\alpha}_{R}-\hat{\alpha}_{R}^{3}\hat{\alpha}_{L}],$
	$\displaystyle{{\cal L}}_{2}$	$\displaystyle=$	$\displaystyle i{\rm Tr}[\hat{\alpha}_{L}\hat{\alpha}_{R}\hat{\alpha}_{L}\hat{\alpha}_{R}],$
	$\displaystyle{{\cal L}}_{3}$	$\displaystyle=$	$\displaystyle i{\rm Tr}V[\hat{\alpha}_{L}\hat{\alpha}_{R}-\hat{\alpha}_{R}\hat{\alpha}_{L}].$		(9)

We should make two remarks on the Lagrangian (4) that need to be kept in mind in what follows. First it is $O(p^{2})$ in power counting HY:PR except for the hWZ term which while quartic is indispensable for unifying FQH droplets and skyrmions karasik . Second it is made scale-invariant by the conformal compensator except for the dilaton potential $V(\chi)$ which could contain scale-symmetry explicit breaking, e.g., quark mass terms. The rationale for this strategy for scale symmetry is explained below.

In accordance with the GD scheme crewther with the IR fixed point specified above, even if explicit symmetry breaking is ignored, scale symmetry can be spontaneously broken by dilaton condensate generating masses to the hadrons. The scheme follows roughly the line of ideas based on “hidden quantum scale invariance” QSI . The underlying reason is that sHLS that we are exploiting is connected with strong-weak dualities à la Seiberg, typically associated with supersymmetric gauge theories Y ; Komargodski ; abel-bernard ; kitanoetal . This connection allows us to exploit the possible duality of HLS to the gluons that will figure in the problem. In addition, the applicability of duality in non-supersymmetric case as ours is made feasible if scale symmetry is broken by the dilaton in terms of the conformal compensator abel-bernard .

From the point of view of our bottom-up approach, it is important to note that the HLS we are dealing with is dynamically generated HLS84 . This means that the coefficients $c_{i}$s in the hWZ term that play a key role in the unification of solitonic description of both $N_{f}=1$ and $N_{f}\geq 2$ baryons are constants that cannot be fixed by the theory. For low density, therefore, they are to be determined by experiments⁶⁶6They could be fixed by holographic QCD, but there are no known holographic QCD models that possess possible “orange” karasik – not to mention ultraviolet – completion that would allow approaching the density.. However as we will see later, in approaching QH droplet baryons bottom-up in density, the coefficients will be “quantized” by topology karasik ; karasik2 . This takes place because there can be a phase transition from a Higgs mode to a topological phase in which the HLS fields are supposed to be (Seiberg-)dual to the gluons of QCD Y .

To exploit the mapping of topological inputs of the sHLS Lagrangian into a mean-field approximation with G$n$EFT, we add the nucleon coupling to the sHLS fields implementing both HLS and scale symmetry as

	$\displaystyle{\mathcal{L}}_{N}$	$\displaystyle=$	$\displaystyle\bar{N}(i\ooalign{\hfil/\hfil\crcr$D$}-\Phi m_{N})N{}+g_{A}\bar{N}\ooalign{\hfil/\hfil\crcr$\hat{\alpha}$}_{\perp}\gamma_{5}N$		(19)
			$\displaystyle{}+\bar{N}\left(g_{V\rho}\ooalign{\hfil/\hfil\crcr$\hat{\alpha}$}_{\parallel}+g_{V0}{\rm Tr}[\ooalign{\hfil/\hfil\crcr$\hat{\alpha}$}_{\parallel}]\right)N+\cdots\,,$

with the covariant derivative $D_{\mu}=\partial_{\mu}-iV^{\rho}_{\mu}-iV^{\omega}_{\mu}$ and dimensionless parameters $g_{A}$, $g_{V\rho}$ and $g_{V0}\equiv\frac{1}{2}(g_{V\omega}-g_{V\rho})$. The ellipsis stands for higher derivative terms that will not be taken into account in what follows. The Lagrangian concerned that we shall refer to as $bs$HLS is

$\displaystyle{\mathcal{L}}_{bsHLS}={{\cal L}}_{sHLS}+{\mathcal{L}}_{N}.$

(20)

With the explicit presence of the baryon field, the role of the hWZ terms is relegated to the baryon-field coupling to the vector and scalar fields that takes over the $\omega$ repulsion in dense baryonic matter.

We consider what happens when the density goes up and approaches the DLFP first considered by Beane and van Kolck bira-dlfp and apply it to the model we are considering in WGP-interplay ; PKLMR . To do this we assume that approaching the DLFP at high density amounts to going toward the IR fixed point à la CT crewther ; CT described above where both chiral symmetry and scale symmetry are realized in the NG mode with the dilaton mass and pion mass going to zero in the chiral limit.

Starting from the vacuum where chiral symmetry is realized nonlinearly, as density increases, one would like to arrive, at some point near $n_{0}$, at the linear realization of chiral symmetry, say, in the form of the Gell-Mann-Lévy (linear) sigma model gell-mann which qualitatively captures nuclear dynamics as the Walecka mean-field model does. This means transforming the nonlinear structure of sHLS that is the habitat of the skyrmion structure to a form more adapted to dense matter, namely the half-skyrmion structure developed for the EoS of massive compact stars in PKLMR ; MR-review . This feature of transformation is encoded in the hidden scale symmetry as pointed out by Yamawaki yamawaki . This point will be further elaborated on in Section VII.

To see how the $bs$HLS Lagrangian behaves as density is increased, we follow Beane and van Kolck bira-dlfp and transform $bs$HLS to a linear basis, $\Sigma=\frac{f_{\pi}}{f_{\chi}}U\chi\propto\sigma^{\prime}+i\vec{\tau}\cdot{\vec{\pi}}^{\prime}$. We interpret taking the limit ${\mathcal{S}}\equiv{\rm Tr}(\Sigma^{\dagger}\Sigma)\to 0$ as approaching the DLFP. Now how to take the dilaton limit requires a special interpretation. In mapping the key information of topological structure of baryonic matter to G$n$EFT as explained in MR-review , it is essential to interpret the limiting ${\mathcal{S}}\to 0$ in the same sense as in going from the skyrmion phase to the half-skyrmion phase at a density above that of normal nuclear matter. In going from skyrmions to half-skyrmions in skyrmion crystal simulation, the quark condensate $\langle\bar{q}q\rangle$ is found to globally go to zero at a density denoted $n_{1/2}>n_{0}$.⁸⁸8To give an idea, $n_{1/2}$ in massive compact stars comes in MR-review at $\sim 3n_{0}$. The condensate however is non-zero locally, thereby supporting a chiral density wave in skyrmion crystal CDW . This seems to be the case in general as observed in various models buballa . As a consequence, the pion decay constant is non-zero, hence the state is in the NG mode. The same is true for the dilaton condensate with inhomogeneity in consistency with the GDS. This feature resembles the “pseudogap” structure in condensed matter physics. As there the issue is subtle and highly controversial (see pseudogap for a comprehensive discussion on this matter). In what follows we interpret the limiting ${\mathcal{S}}\to 0$ in this sense. The order parameters for the symmetries involved in medium up to the possible IR fixed point of crewther are more complicated involving higher dimensional field operators Harada-Sasaki . Approaching the DLFP, the quantities involved will be denoted by asterisk $\ast$ as $\langle\chi\rangle^{\ast}\propto f_{\pi}^{\ast}$ and $\langle\chi\rangle^{\ast}\propto f_{\chi}^{\ast}$ with $f_{\pi}^{\ast}\sim f_{\chi}^{\ast}\neq 0$. The matter in the half-skyrmion phase going toward the DLFP then has a resemblance to the pseudo-gap phase with fractional skyrmions present in SYK superconductivity gorskyetal .

One finds that in the limit ${\mathcal{S}}\to 0$ there develop singularities in the thermodynamic potential. Imposing that there be no singular terms in that limit gives what we identify as DLFP “constraints” PKLMR

$\displaystyle f_{\pi}\to f_{\chi}\neq 0,\ g_{A}\to g_{V\rho}\to 1.$

(21)

Furthermore since the $\rho$-meson coupling to the nucleon is given by

$\displaystyle g_{\rho NN}=g_{\rho}(g_{V\rho}-1),$

(22)

one sees that the $\rho$ meson decouples – independently of the “vector manifestation (VM)” with $g_{\rho}\to 0$ HY:PR – from the nucleons as the DLFP is approached. On the other hand, the $\omega$-NN coupling $g_{\omega NN}=g_{\omega}(g_{V\omega}-1)$ remains nonzero for $g_{\omega}\neq 0$ because $g_{V\omega}-1\not\to 0$ in the DLFP limit. This has been verified at one-loop order in the RGE WGP-interplay . In fact the EoS at high density relevant to massive compact stars requires this for the stability of the matter. It is well-known that there is a delicate balance between the dilaton condensate which enters in the dilaton mass $m_{\chi}^{\ast}$ and the $\omega$-NN coupling $g_{\omega NN}$. In fact this balance is the well-known story of the roles of the scalar attraction and vector repulsion in nuclear physics at normal nuclear matter density. It becomes more acute at higher densities.

The broad phase structure involved is depicted in Fig. 1. Apart from the nuclear matter equilibrium density $n_{0}$ and the topology change density $n_{1/2}$, other densities are not precisely pinned down. What’s given in the review MR-review does not represent precise values, hence Fig. 1 should be taken at best highly schematic.

The nucleon in-medium mass is connected to the $\omega$-nucleon coupling by the equations of motion for $\chi$ and $\omega$ and the in-medium property of the $\chi$ condensate, $\langle\chi\rangle^{\ast}$, or more appropriately the in-medium dilaton decay constant $f_{\chi}^{\ast}$ which controls the in-medium mass of the dilaton, hence the nucleon mass, at high density. This means that up to the DLFP, the effective nucleon mass will remain constant proportional to the dilaton condensate $\langle\chi\rangle^{\ast}$. This is seen in Fig. 2. This $\langle\chi\rangle^{\ast}$ comes out to be equal to the scale-chiral invariant mass of the nucleon $m_{0}$ that figures in the parity-doubling model for the nucleon sasakietal . This then suggests that $m_{0}$ can show up signaling the presence at a higher density of the DLFP through strong nuclear correlations even if it is not explicit in the QCD Lagrangian.

We claim that this is in accord with the GDS (“genuine dilaton” scenario) with the nucleon mass remaining massive together with the non-zero dilaton decay constant.

The remarkable interplay between the dilaton condensate $\langle\chi\rangle^{\ast}$ and the $\omega$-NN coupling has an important impact on the EoS for density $n\mathrel{\hbox to0.0pt{\lower 3.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$>$}}n_{1/2}$ at which $\langle\chi\rangle^{\ast}$ flattens in density⁹⁹9The flattening to a density-independent constant of $\langle\chi\rangle^{\ast}/f_{\chi}$ at $n_{1/2}$ arising from an intricate interplay between $g_{\omega NN}$ and $\langle\chi\rangle^{\ast}$ in Fig. 2 is related to that of $\langle\bar{q}q\rangle^{\ast}/f_{\pi}$ in a skyrmion-crystal simulation of HLS ohetal . It is not obvious how to correctly implement the dilaton field in the crystal simulation, so the relation between the dilaton and quark condensates does not seem to come out correctly on skyrmion crystals. But in the genuine dilaton scenario incorporated in G$n$EFT, we believe they should be tightly related – as we argued – as density approaches the IR fixed point density.. As noted above, the induced density dependence for the $\rho$-NN coupling $\propto(g_{V\rho}-1)$ drops rapidly such that the $\rho$ decouples from nucleons at the DLFP whereas ($g_{V\omega}-1$) remains non-zero. How this impacts on nuclear tensor forces and consequently on the symmetry energy $E_{sym}$ deserves to be investigated in nuclear structure. Furthermore the vector manifestation leads to the gauge coupling $g_{\rho}\to 0$ HY:PR whereas that for $\omega$ coupling $g_{\omega}$ drops only slightly. The delicate balance between the attraction due to the scalar (dilaton) exchange and the repulsion due to the $\omega$ exchange plays a crucial role for the EoS for $n\mathrel{\hbox to0.0pt{\lower 3.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$>$}}n_{1/2}$ of massive neutron stars PKLMR .

A striking consequence of the interplay between the $g_{\omega NN}$ coupling and the condensate $\langle\chi\rangle^{\ast}$ at $n\mathrel{\hbox to0.0pt{\lower 3.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$>$}}n_{1/2}$ in the G$n$EFT framework, not shared by other models in the literature, is the precocious emergence of hidden scale symmetry in nuclear interactions. The details are involved but the phenomenon can be clearly seen in the mean-field approximation with the $bs$HLS Lagrangian (20).

The vacuum expectation value of the trace of the energy-momentum tensor $\theta_{\mu}^{\mu}$ is given by

$\displaystyle\langle\theta_{\mu}^{\mu}\rangle=4V(\langle\chi\rangle)-\langle\chi\rangle\left.\left(\frac{\partial V(\chi)}{\partial\chi}\right)\right|_{\chi=\langle\chi\rangle^{\ast}}+\cdots$

(23)

where the ellipsis stands for chiral symmetry breaking (quark mass) terms. Now if one ignores the quark mass terms, then given that the $\langle\chi\rangle^{\ast}$ which should be identified with the dilaton decay constant is independent of density MR-review , we have

$\displaystyle\frac{\partial\langle\theta_{\mu}^{\mu}(n)\rangle}{\partial n}=\frac{\partial\epsilon(n)}{\partial n}\big{(}1-3v_{s}^{2}/c^{2}\big{)}=0.$

(24)

One expects that $\frac{\partial\epsilon(n)}{\partial n}\neq 0$ and hence, within the range of density where (24) holds, say, $\sim(3-7)n_{0}$, we arrive at what is commonly associated with the “conformal sound speed”

$\displaystyle v_{s}^{2}/c^{2}=1/3.$

(25)

Since however the trace of the energy-momentum tensor is not zero at the density involved which is far from asymptotic, it should be more appropriately called “pseudo-conformal” velocity.

This prediction made in the mean field for a neutron star of mass $M\simeq 2M_{\odot}$ has been confirmed – modulo of course quark-mass terms — in going beyond the mean-field approximation using the $V_{lowk}$RG approach PKLMR ; MR-PCM . Needless to say the quark mass terms could affect this result bringing in possible deviation from (25), but it seems reasonable to assume that the corrections cannot be significant.

In following Karasik’s arguments karasik ; karasik2 , we take the sHLS Lagrangian (4) and focus on the terms that involve the $\eta^{\prime}$ field in that Lagrangian. In doing this manipulation, the role of the conformal compensator present to provide (Seiberg-)duality plays the crucial role in contrast to what’s done in karasik ; karasik2 where the the role of dilaton effects is doing double trace which is ignored. With the baryons generated as solitons in sHLS, the parameters of the Lagrangian contain the “intrinsic density dependence” (IDD) inherited from QCD at the matching between EFT and QCD. First we ask what one should “dial” in the parameters of G$n$EFT – in the spirit of the strategy used – to access compact stars so that the system approaches the $\eta^{\prime}$ sheet. Next we ask whether high baryon density supplied by gravity makes the $\eta^{\prime}$ ring “visible.”

Suppose we increase density beyond $n_{1/2}$. Recalling from what we have learned in the mean-field result with (20) (with the nucleon explicitly included), it seems reasonable to assume the $\rho$ decouples first at some density above $n_{1/2}$ before reaching the DLFP. Since the gauge coupling $g_{\rho}$ goes to zero in approaching the vector manifestation fixed point $n_{VM}$ (say, $n\mathrel{\hbox to0.0pt{\lower 3.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$>$}}25n_{0}$) PKLMR ), the mass $m_{\rho}\sim f_{\pi}g_{\rho}$ goes to zero independently of whether or not $f_{\pi}$ goes to zero and the $\rho$ decouples from the pions. The Lagrangian ${{\cal L}}_{sHLS}$ (4) will then reduce to what was written down by Karasik karasik ; karasik2

	$\displaystyle{{\cal L}}_{sHLS}$	$\displaystyle=$	$\displaystyle\frac{1}{2}(\partial_{\mu}\chi)^{2}+V(\chi)+\frac{1}{2}{\Phi}^{2}(\partial_{\mu}\eta^{\prime})^{2}$		(26)
			$\displaystyle{}-\frac{1}{4}(\omega_{\mu\nu})^{2}+\frac{1}{2}m_{\omega}^{2}\Phi^{2}\omega_{\mu}\omega^{\mu}$
			$\displaystyle{}-\kappa\frac{N_{c}}{8\pi^{2}}\epsilon^{\mu\nu\alpha\beta}\omega_{\mu}\partial_{\nu}\omega_{\alpha}\partial_{\beta}\eta^{\prime}+\cdots,$

where we have written $f=f_{\eta}$ and $f_{\sigma\rho}^{2}=af_{\pi}^{2}$. The coefficient $\kappa$ is related to the coefficient $c_{3}$ in the hWZ term (8) for the HLS dynamically generated, appropriate for low density. We assume that it will become quantized as in karasik when the $\eta^{\prime}$ ring is “probed” at some high density as explained below.¹⁰¹⁰10In karasik2 , it has been argued that the well-known vector dominance in the presence of electroweak fields leads to the same constraint. In this operation, we assumed that the limit ${\mathcal{S}}\to 0$ is equivalent to having, as in the crystal simulation, both the dilaton and chiral condensates – space averaged globally – go to zero while they locally support density waves with their decay constants remaining nonzero. As density increases further beyond the DLFP, the condensate will vanish locally and the kinetic energy term of $\eta^{\prime}$ field gets killed and the $\omega$ mass $m_{\omega}\propto\langle\chi\rangle$ goes to zero. Indeed, it is explicitly shown in CrystalHLS in the crystal approach that at a density $n\gg n_{1/2}$ the phase becomes homogeneous – without density waves – so that $f_{\pi}\propto f_{\chi}\propto\langle\chi\rangle=0$. We interpreted this phase as deconfined since both chiral and scale symmetries are restored. We will be left with the massless $\chi$ and $\omega$ fields and the $\omega\eta^{\prime}$ coupling coming from ${{\cal L}}_{3}$ in the hWZ term, (8).

It is here that our approach to scale symmetry à la GD crewther ; QSI brings the role of the sliding dilaton condensate at high density into contact via the Seiberg-type duality with the $\eta^{\prime}$ ring structure.

The last term of (26) can be written as

$\displaystyle{{\cal L}}_{CS\eta^{\prime}}=-\kappa\frac{N_{c}}{4\pi}J_{\mu\nu\alpha}\omega^{\mu}\partial^{\nu}\omega^{\alpha}.$

(27)

with the topological $U(1)$ 2-form symmetry current

$\displaystyle J_{\mu\nu\alpha}=\frac{1}{2\pi}\epsilon_{\mu\nu\alpha\beta}\partial^{\beta}\eta^{\prime}.$

(28)

Now a highly pertinent observation is that the charged objects under these symmetries get metamorphosed to infinitely extended sheets that interpolate from $\eta^{\prime}=0$ on one side to $\eta^{\prime}=2\pi$ on the other zohar ; karasik , involving a sheet $\eta^{\prime}=\pi$. The current is conserved because $\eta^{\prime}$ in the space of $\eta^{\prime}$ configuration is a circle and $\pi_{1}(S^{1})={\cal Z}$. The Lagrangian (27) corresponds to the CS field identified with the $\omega$ field coupling to the baryon charge. The CS field is a topologically non-trivial gauge field and hence gauge invariance requires that the $\kappa$ be quantized $\kappa=1$.¹¹¹¹11 This argument holds if we assume that the matter is in the topological phase where HLS is Seiberg-dual to QCD. And also with the vector dominance karasik2 . Karasik identifies the hWZ term with constrained coefficients as “hidden WZ” term in contrast to the “homogenous WZ term” HY:PR relevant at lower density. Note that going from “homogeneous hWZ” to “hidden hWZ” by density would require the unhiding of hidden scale invariance crewther ; QSI . This follows from the presumed duality between the gluon fields in QCD and the HLS fields in EFT Y . Now going beyond the DLFP as the system is brought toward the putative density at which $\langle\chi\rangle\to 0$ and $\langle\bar{q}q\rangle\to 0$, $f_{\pi}\sim f_{\chi}\to 0$ and $m_{\chi}\to 0$, $m_{\omega}\to 0$ etc, then one winds up with a quantum Hall baryon with $B=1$ and $J=N_{c}/2$ zohar . This essentially rephrases Karasik’s argument in terms of the G$n$EFT to arrive at the $N_{f}=1$ baryon from the $N_{f}=2$ baryons.

In what’s described above, we have assumed that the $\rho$ field decouples first before reaching the DLFP as indicated in Section IV.1. This is what seems to take place in compact-star matter studied in PKLMR ; MR-review . Instead of $U(1)$ CS field theory, however, one can generalize the discussion to nonabelian CS field theory from the sHLS Lagrangian (4). The Lagrangian (27) is modified to Y

$\displaystyle{{\cal L}}^{\prime}_{CS\eta^{\prime}}=-\kappa\frac{N_{c}}{4\pi}J_{\mu\nu\alpha}{\rm Tr}\left(V^{\mu}\partial^{\nu}V^{\alpha}+\frac{2}{3}V^{\mu}V^{\nu}V^{\alpha}\right)$

(29)

where $V_{\mu}=\frac{1}{2}(\tau\cdot\rho_{\mu}+\omega_{\mu})$, assuming $U(2)$ symmetry, is restored at the high density concerned. We now identify the source of the baryon number as $B=(N_{c}/N_{f}){\rm Tr}V$ and differentiating the action $\int{{\cal L}}^{\prime}_{CS\eta^{\prime}}$ with respect to $B$, we get, with $\kappa=1$, the baryon density

$\displaystyle\rho_{B}=\frac{1}{4\pi^{2}}\epsilon^{ijk}{\rm Tr}(\partial_{i}V_{j})\partial_{k}\eta^{\prime}+\cdots.$

(30)

With the configuration $\eta^{\prime}=0$ at $x_{3}=-\infty$ and $\eta^{\prime}=2\pi$ at $x_{3}=\infty$, the baryon number is gotten $B=1$ by integrating over $x_{3}$. In Y , an interpretation of this phenomenon is made in terms of an anyon excitation of the quantum Hall droplet with baryon number $B=1/N_{c}$ leading to a quark described as a soliton made of hadrons, what one might interpret as a novel manifestation of hadron-quark duality.

This exposes the $\eta^{\prime}$ ring in the $N_{f}=2$ setting. This observation is relevant to the possible decay of the $\eta^{\prime}$ ring to a pionic sheet described in the next subsection.

Above we have seen that in some density regime, one arrives at a CS theory coupled to a baryon-charge one object that could be identified with the $\eta^{\prime}$ ring. This is done, we suggest, by what amounts to going beyond the DLFP in the G$n$EFT Lagrangian¹²¹²12For doing this more realistically, it may be necessary to include higher-lying vectors and scalars as in holographic models Y . This is beyond our scheme so we won’t go into the matter further.. What’s interesting is to view the process in terms of the $N_{f}=2$ skyrmion given by the sHLS Lagrangian, namely, how the hedgehog ansatz in the background of the $\eta^{\prime}$ field is “deformed” as density goes up. It seems plausible, as suggested in karasik , that high density first impacts on the EoS such that

$\displaystyle\pi_{1}=\pi_{2}=0,\ V_{\mu}^{1}=V_{\mu}^{2}=0,V_{\mu}^{3}=\omega_{\mu}$

(31)

and then distorts the hedgehog configuration to

$\displaystyle(\sigma+i\pi_{3})/\sqrt{\sigma^{2}+\pi_{3}^{2}}=e^{i\eta^{\prime}/2f_{\eta}}.$

(32)

This suggests that while at low density $\eta^{\prime}$ in $U=e^{i\eta^{\prime}/f_{\eta}}e^{i\tau\cdot\pi/f}$ present in the $\eta^{\prime}$ ring plays no significant role, except perhaps, giving an $O(N_{c}^{0})$ correction to the $\Delta-N$ mass difference which could not be significant, the $\eta^{\prime}$ ring becomes important as density increases.

We consider the density regime where the $\rho$ mesons are decoupled from the nucleons and the $\eta^{\prime}$ ring is unstable, so decays to skyrmions. Noting that the $\eta^{\prime}$ ring, i.e., ${{\cal L}}_{CS\eta^{\prime}}$, is embedded in the full hWZ term, we should look at the hWZ term (8). Following karasik , we write (in the unitary gauge $\xi_{R}=\xi_{L}^{\dagger}=\xi$)

	$\displaystyle{{\cal L}}_{hWZ}$	$\displaystyle=$	$\displaystyle\frac{N_{c}}{24\pi^{2}}\epsilon^{\mu\nu\rho\sigma}g_{\omega}\omega_{\mu}$		(33)
			$\displaystyle\times{\rm Tr}\Bigg{[}\left(\frac{3}{8}\kappa_{1}\right)2\partial_{\nu}\xi\xi^{\dagger}\partial_{\rho}\xi\xi^{\dagger}\partial_{\sigma}\xi\xi^{\dagger}$
			$\displaystyle\qquad\quad\;+\left(\frac{1}{2}\kappa_{2}\right)3iV_{\nu}(\partial_{\rho}\xi\partial_{\sigma}\xi^{\dagger}-\partial_{\rho}\xi^{\dagger}\partial_{\sigma}\xi)$
			$\displaystyle\qquad\quad\;+\left(\frac{1}{2}\kappa_{3}\right)3i\partial_{\nu}V_{\rho}(\partial_{\sigma}\xi\xi^{\dagger}-\partial_{\sigma}\xi^{\dagger}\xi)\Bigg{]}$

where only the terms contributing to the $N_{f}=2$ completion of the topological term ${\cal L}_{CS\eta^{\prime}}$ are retained. The coefficients $\kappa_{i}$s can be identified with $c_{i}$s of the hWZ term (8)¹³¹³13The coefficients $c_{i}$ have been fixed in karasik2 by imposing vector dominance (for which the $c_{4}$ term corresponding to the photon field is included). As will be remarked in the last Section, VD does not work well in nuclear physics unless the infinite tower of vector mesons of holographic QCD is included but it would be interesting to study their impact on dense matter.

$\displaystyle\kappa_{1}=c_{1}-c_{2},\ \kappa_{2}=c_{1}+c_{2},\kappa_{3}=\ c_{3}.$

(34)

Under gauge transformation $\omega_{\mu}\to\omega_{\mu}-\frac{1}{g_{\omega}}\partial_{\mu}\lambda$, one has

	$\displaystyle\delta S$	$\displaystyle=$	$\displaystyle\frac{N_{c}}{12\pi^{2}}\epsilon^{\mu\nu\rho\sigma}\partial_{\mu}\lambda{\rm Tr}\Bigg{[}\left(\frac{3}{8}\kappa_{1}\right)\partial_{\nu}\xi\xi^{\dagger}\partial_{\rho}\xi\xi^{\dagger}\partial_{\sigma}\xi\xi^{\dagger}\Bigg{]}$
			$\displaystyle+\frac{iN_{c}}{8\pi^{2}}\epsilon^{\mu\nu\rho\sigma}\partial_{\mu}\lambda\partial_{\nu}{\rm Tr}\Bigg{[}\left(\frac{1}{2}\kappa_{2}\right)V_{\rho}(\partial_{\sigma}\xi\xi^{\dagger}-\partial_{\sigma}\xi^{\dagger}\xi)\Bigg{]}.$

Then the gauge invariance yields the constraints

	$\displaystyle\int d\phi\left[\left(\frac{1}{2}\kappa_{2}\right)\left(\omega_{\phi}+V_{\phi}^{3}\right)+\left(\frac{3}{8}\kappa_{1}\right)4\pi_{1}\partial_{\phi}\pi_{2}\right]$
	$\displaystyle=const.$		(36)

So that, on the world-sheet for the $N_{f}=1$ baryon, one has karasik

$\displaystyle\frac{1}{2}\kappa_{2}\int d\phi(\omega_{\phi}+V_{\phi}^{3})=2\pi,\ \pi_{1,2}=0.$

(37)

The $\eta^{\prime}$ ring is thereby “seen.”¹⁴¹⁴14We note that the mixing of two $U(1)$s $(\omega_{\phi}+V_{\phi})$ in this formula (and also the equality $V_{3}=\omega$ in (31)) follows from the assumed duality of HLS kitanoetal mentioned in footnote 2.

Now suppose the $\eta^{\prime}$ sheet structure, a background buried in the system of $N_{f}=2$ skyrmions, is unstable and could subsequently decay into skyrmions in a different sheet structure containing the isovector degrees of freedom

$\displaystyle\omega_{\phi}+V_{\phi}^{3}=0,\ \ \frac{3}{8}\kappa_{1}\int d\phi\pi_{1}\partial_{\phi}\pi_{2}=\frac{\pi}{2}.$

(38)

The question is: What is the structure of the matter encoded in the condition $\frac{3}{8}\kappa_{1}\int d\phi\pi_{1}\partial_{\phi}\pi_{2}=\pi/2$ to which the $\eta^{\prime}$ decays? Could this be a sort of droplets that can be described in a topological field theory, involving isovector degrees of freedom, e.g., the $\pi^{\pm}$, the $\rho$ vectors etc. as in the form of a nonabelian CS Lagrangian that seems to arise in the Cheshire Cat for $N_{f}=2$ baryons? We have no answer to this question. Clearly isovector mesons must figure. This has to do with the quantization of other coefficients than the one giving the $\eta^{\prime}$ ring.

We return to the skyrmion crystal simulation on which the G$n$EFT for massive compact stars is anchored PKLMR ; MR-review . As detailed there, the topological structure of the skyrmions simulated on crystal is translated into the parameters of the G$n$EFT Lagrangian, which is then treated in an RG-approach to many-nucleon interactions. The key role played in this procedure is the topological feature encoded in the skyrmion structure of hidden scale symmetry and local symmetry of sHLS. Notable there are the cusp in the symmetry energy of dense matter due to the “heavy” degrees of freedom¹⁵¹⁵15An intriguing observation is that the “cusp” in the symmetry energy (e.g., Fig.4 in MR-review ) arises at the next-to-the leading order in $1/N_{c}$ in the rotational quantization of the skyrmion energy in the Skyrme model (with pions only) as found first in Esym-cusp . This observation was discussed in some detail in PREX . The cusp in the $\eta^{\prime}$ potential in the Veneziano-Witten Lagrangian for $\eta^{\prime}$ is also higher order in $1/N_{c}$. Now when the HLS vectors are “integrated in” as done in karasik , the cusp is no longer present. Similarly when the HLS vectors are present in $Gn$EFT, the cusp also disappears as seen Fig. 7 in MR-review . The symmetry energy involves the nuclear tensor forces and the cusp may be indicative of possible singularity in the one-pion exchange tensor force when given in terms of higher pion derivative terms in $Sn$EFT bira-tensor . The singularity could very well be eliminated by the $\rho$ meson as in $Gn$EFT. It would be interesting if this observation had something to do in $N_{f}\geq 2$ baryons similarly to what the HLS fields do to the $\eta^{\prime}$ singularity for the $N_{f}=1$ pancake baryon. This could lead to a possible resolution of the dichotomy problem., the parity doubling in the baryon spectra, and a “pseudo-gap” structure of the half-skyrmion phase. These properties encapsulated in the RG-approach with G$n$EFT led to the prediction of possible precocious emergence of scale symmetry in massive-star matter with the pseudo-conformal sound velocity of star $v_{pcs}^{2}/c^{2}\simeq 1/3$ at a density $n\mathrel{\hbox to0.0pt{\lower 3.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$>$}}3n_{0}$.

Let us explore what this skyrmion crystal structure suggests for a possible sheet structure of dense matter.

It is observed in molecular dynamics simulation of nuclear matter expected in neutron-star crust and core-collapse supernova at a density a packing fraction of $\sim 5/16$ of nuclear saturation density $n_{0}\sim 2\times 10^{14}$g/cm³ that a system of “sheets” of lasagne, among a variety of complex shapes of so-called “nuclear pasta,” could be formed and play a significant role in the EoS in low-density regime of compact star matter pasta . Involved here are standard nuclear interactions between neutrons and protons in addition to electromagnetic interactions.

At higher densities, say, at densities $\sim(2-7)n_{0}$, it is seen in skyrmion crystal simulations that a stack of lasagne sheets PPV or of tubes or spaghettis  canfora is energetically favored over the homogenous structure. Involved here are fractionalized skyrmions, 1/2-baryon-charged for the former and $1/q$-charged for the latter with $q$ odd integer. Those fractionalized skyrmions can be considered “dual” to (constituent) quarks in the sense of baryon-quark duality in QCD. There is an indication that the sheet structure of stack of lasagnes could give a consistent density profiles of even finite nuclei PPV . In fact there seems little doubt that an inhomogeneity is favored in dense matter at non-asymptotic densities CDW ; buballa . Thus it could be considered robust.

The two phenomena at low and high densities involve basically different aspects of strong interactions, but there is a tantalizing hint that something universal is in action in both cases. We are tempted to consider that topology is involved there. This is particularly plausible at high density given that the “pasta” structure, be that lasagne or tubes (or spaghetti), is found to be strikingly robust. Up to date, the analysis has been made with an ansatz for the pion field, i.e., the Atiyah-Manton ansatz for the lasagne sheet and a special ansatz allowing analytical treatment for the tubes. The robustness must have to do with the fact that what is crucially involved is the topology and it is the pion field that carries the topology. What is striking is the resulting structure does not seem to depend on the presence of other massive degrees of freedom such as the vector mesons or scalar canfora2 ; CrystalHLS . There, adding infinite number of higher derivative terms to the Skyrme Lagrangian is found not to modify the ansatz for the tubes. It is therefore highly likely that the same structure would arise from the presence of the hidden scale-local degrees of freedom manifested in different forms of sHLS.

Our sHLS Lagrangian could contain the unified descriptions of both $N_{f}=1$ droplet – $\eta^{\prime}$ ring – and $N_{f}=2$ skyrmions in an EoS, but we have not been able to capture both in a unified way. That is, how are the infinite-hotel and the FQH structures combined in the EoS and whether and how does the latter structure figure in compact-star physics?

A significant recent development, relevant conceptually to this matter, is the work treating the fractional quantum Hall phenomenon in the Kohn’s functional density approach à la Kohn-Sham jain . The key ingredient in this approach is the weakly interacting composite fermions (CF) formed as bound states of electrons and (even number of) quantum vortices. Treated in Kohn-Sham density functional theory one arrives at the FQH state that captures certain strongly-correlated electron interactions. The merit of this approach is that it maps the Kohn-Sham density functional, a microscopic description, to the CS Lagrangian, a coarse-grained macroscopic description, for the fractional quantum Hall effect.

Now the possible relevance of this development to our problem is as follows. First, Kohn-Sham theory more or less underlies practically all nuclear EFTs employed with success in nuclear physics, as for instance, energy density functional approaches to nuclear structure. Second, our G$n$EFT approach belongs to this class of density-functional theories in the strong-interaction regime. Third, the successful working of the G$n$EFT model backed by robust topology and implemented with intrinsic density dependence inherited from QCD could very well be attributed to the power of the (Kohn-Sham-type) density functional in baryonic matter at high density $n\mathrel{\hbox to0.0pt{\lower 3.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$>$}}3n_{0}$. These three observations combined suggest to approach the dichotomy problem in a way related to what was done for FQHE.

The first indication that G$n$EFT anchored on the topology change could be capturing the weak CF structure of jain in FQHE is seen in the nearly non-interacting quasiparticle behavior in the chiral field configuration $U$ in the half-skyrmion phase (see Fig. 8 in MR-review ). This feature may be understood as follows. Due to hidden $U(1)$ gauge symmetry in the hedgehog configuration, the half-skyrmion carries a magnetic monopole associated with the dual $\omega$ cho ; nitta . The energy of the “bare” monopole in the half-skyrmion diverges when separated, but the divergence is tamed by interactions in the skyrmion as a bound state of two half-skyrmions where the divergence is absent. In a way analogous to what happens in the Kohn-Sham theory of FQHE jain , there could intervene the gauge interactions between the skyrmions pierced by a pair of monopoles in sHLS– as composite fermions – possibly induced by the Berry phases due to the magnetic vortices. Thus it is possible that the topological structure of the FQH is buried in the bound half-skyrmion structure at high density. A possible avenue to the problem is to formulate the EoS in terms of a stack of ordered coupled sheets of CS droplets. We hope to work this out in the future work.

In the dilaton limit where the constraints (21) set in, there are NG excitations and the nucleon mass is $O(m_{0})$ with $m_{0}\simeq ym_{N}$, $y\sim(0.6-0.9)$. In compact stars treated in PKLMR , the core of the massive stars with $M\sim 2M_{\odot}$ has density $\sim(6-7)n_{0}$. A natural question one raises is whether the core of the star contains “deconfined” quarks either co-existing with or without baryons. In the framework of MR-review , the constituents of the core are fractional-baryon-charged quasiparticles. They are neither baryons nor quarks. The fractional-charged phase arises without order-parameter change and hence considered evolving continuously from baryonic phase with a certain topology change. In certain models having domain walls, those fractional-charged objects can be “deconfined” on the domain wall domainwall . If the sheets in the skyrmion matter discussed above are domain walls, then it is possible that the fractional-charged objects are “deconfined” on the sheets in the sense discussed in domainwall .

There are two significant issues raised here.

One is the possible observation of an evidence for “quarks” in the core of massive neutron stars evidence . Very recently, combining the astrophysical observations and theoretical ab initio calculations, Annal et al. concluded that inside the maximally massive stars there could very well be a quark core consisting of “deconfined quarks” evidence . Their analysis is based on the observation that in the core of the massive stars, the sound velocity approaches the conformal limit $v_{s}/c\to 1/\sqrt{3}$ and the polytropic index takes the value $\gamma<1.75$, 1.75 being the value close to the minimal one obtained in the hadronic model. It has been found vs-core that in the pseudo-conformal structure of our G$n$EFT, the sound velocity becomes nearly conformal $v_{pcs}/c\sim 1/\sqrt{3}$ and the $\gamma$ goes near 1 at $n\mathrel{\hbox to0.0pt{\lower 3.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$>$}}3n_{0}$. Thus at least at the maximum density relevant for $\sim 2M_{\odot}$ stars, what could be interpreted as “deconfined quarks” can be more appropriately fractionally charged quasipartlcles. Are these “deconfined” objects on domain walls as in domainwall or confined two half-skyrmions as mentioned in Section VI.1?

The other is what is referred to as “baryon-quark continuity” in MR-review in the domain of density relevant to compact-star phenomenology. This is not in the domain of density relevant to the color-flavor locking which is to take place at asymptotic density wilczek . It seems more appropriate to say that the gauge degrees of freedom we are dealing with should be considered as “dual” to the gluons in QCD Y .

It has recently been argued that the hadron-quark continuity in the sense of wilczek is ruled out on the basis of the existence of a nonlocal order parameter involving a (color-)vortex holonomy CFLruledout . But such a “theorem,” perhaps holding at asymptotic density, could very well be irrelevant even at the maximum possible density observable in nature, whatever the maximum mass of the star stable against gravitational collapse might be. The argument of CFLruledout cannot rule out the baryon-quark duality argument given in MR-review and in this paper which is far below asymptotic density. The presence of the scale-chiral-invariant nucleon mass $m_{0}$ testifies for this assertion.

The hadron-quark duality for which the hidden scale symmetry – together with the HLS – figures crucially in our discussion of resolving the dichotomy, we argue, leaves a trail of other observables where its effect has impacts on. One prominent case is the long-standing mystery described above of the “quenched” axial-vector coupling constant $g_{A}$ in nuclear medium reviewed in the note that accompanies this note multifarious-gA .

We recall as the density approaches the DLFP density, the constraints (21) require that the effective $g_{A}\to 1$. Surprisingly, as has been recently shown gA , the effective $g_{A}^{\rm ss}$ in Gamow-Teller transitions (most accurately measurable in doubly-magic nuclei) – which is denoted as $g_{A}^{\rm Landau}$ in gA – is predicted to be “quenched” in the presence of emerging scale invariance from $g_{A}=1.27$ in free space to $g_{A}^{\rm ss}\approx 1$ at a much lower density, say, in finite nuclei.

Now here is an intriguing possibility. While the precocious onset of the conformal sound speed $v_{s}^{2}/c^{2}=1/3$ in massive stars at density $n\mathrel{\hbox to0.0pt{\lower 3.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$>$}}3n_{0}$ is a signal for an emergence of an albeit approximate scale symmetry, there has been up to date no evident indication of the pseudo-conformal structure at low density $n\mathrel{\hbox to0.0pt{\lower 3.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$<$}}3n_{0}$. As suggested in WS-MPL , it is an appealing possibility that $g_{A}^{\rm eff}\approx 1$ in nuclei (modulo possible, as yet unknown, $\beta^{\prime}$ corrections) and $g_{A}=1$ at the DLFP is the continuity of the “emergent” scale symmetry reflecting hidden quantum scale invariance between low and high densities.

Regardless of whether the hidden nature of scale symmetry is appropriate for the “genuine dilaton” of Crewther crewther or quantum scale invariance QSI or the dilaton in the conformal window appelquist , scale symmetry is intrinsically hidden. This point has been clearly illustrated in Yamawaki’s argument yamawaki . Yamawaki starts with the $SU(2)_{L}\times SU(2)_{R}$ linear sigma model with two parameters which corresponds to the Standard Model Higgs Lagrangian, makes a series of field re-parameterizations and writes the SM Higgs model in two terms, a scale invariant term and a potential term which breaks scale symmetry which depends on one dimensionless parameter $\lambda$. By dialing $\lambda\to\infty$, he gets the nonlinear sigma model with the scale symmetry breaking shoved into the NG boson field kinetic energy term, and by dialing $\lambda\to 0$ he gets scale-invariant theory going toward the conformal window. This suggests that one can think of what’s happening in baryonic matter as dialing the parameter $\lambda$ in terms of nuclear dynamics. For compact-star physics, it’s the density that does the dialing.

The work of YLM was supported in part by the National Science Foundation of China (NSFC) under Grant No. 11875147 and 11475071.