Pseudo-Conformal Sound Speed in the Core of Compact Stars

Abstract

By implementing the putative “hadron-quark continuity” conjectured in QCD in terms of skyrmion-half-skyrmion topological change in an effective field theory for dense matter, we argue that (quasi-)baryons could “masquerade” deconfined quarks in the interior of compact stars. We interpret this phenomenon as a consequence of possible interplay between hidden scale symmetry and hidden local symmetry at high density. A surprising spin-off of the emerging symmetry that we call “pseudo-conformality” is that the long-standing puzzle of the quenched $g_A\approx 1$ in nuclei can be given a simple resolution by the way the hidden symmetries impact nuclear dynamics at low density.

1. The Approach: G$n$EFT

It was found in [1] that when two hidden symmetries presumed to be encoded in QCD, scale symmetry and flavor local symmetry, were suitably incorporated into chiral symmetric effective field theory to capture low-energy nuclear dynamics and extended to a chiral-scale symmetric EFT—coined as GnEFT in [2]—to address the equation of state of compact-star matter, the sound speed (SS) of massive stars came out precociously close to the conformal speed $v_s^2/c^2=1/3$ in the interior of the stars at a density of $n\sim (3-6)n_0$. The trace of the energy momentum tensor (TEMT) was found to be not zero (even in the chiral limit) at that density, so the speed cannot be truly conformal. It was therefore referred to—in the absence of a better terminology—as “pseudo-conformal.” In this article, we explain how this surprising prediction could account for the possible presence of “deconfined quarks” in the core of compact stars updating recent developments confronting microscopic approaches anchored on QCD proper being discussed in the current literature [3,4,5,6]. It turns out that the notion of pseudo-conformality can figure equally importantly in nuclear physics at low density. In Appendix A, we show how the long-standing mystery of the quenched $g_A\simeq 1$ in Gamow–Teller transitions in nuclei can be resolved in GnEFT.

Given that the nuclear dynamics involved at non-asymptotic densities is intrinsically nonperturbative from the QCD point of view, the approach is inevitably anchored in effective field theory.

The hidden symmetries, suitably incorporated, involve “heavy degrees of freedom (HdFs)” that encompass the density range from the regime of normal nuclear matter density $n_0\approx 0.16$ fm${}^{-3}$, where standard chiral EFT applies, to $\sim 7n_0$, where it is most likely broken down. In the framework of GnEFT that we have formulated, the hidden local symmetry is represented by the light-quark vector mesons $\mathcal{V}=(\rho ,\omega )$ “conjectured” to be (Seiberg-)dual to the gluons [7]. It turns out to be significant that this local symmetry is “hidden” in the sense defined in [8,9]: they are to emerge from pion clouds as composite gauge bosons. For this phenomenon to take place, it is indispensable that there be the “vector manifestation (VM)” [10] at some high density with the vector mesons becoming massless. Where the VM density $n_{\mathrm{VM}}$ is located cannot be calculated in the effective theory. It turns out, however, that the sound velocity behaves qualitatively differently depending on whether $n_{\mathrm{VM}}$ is near or asymptotically higher than the density of the star.

Our approach relies closely on that the scale symmetry enters via the “genuine dilaton (GD)” formulated by Crewther and Tunstall [11,12,13]. The GD is associated with an infrared (IR) fixed point at which the QCD $\beta$ function vanishes, i.e., $\beta \left({\alpha }_{\mathrm{IR}}\right)=0$ (where ${\alpha }_s$ is the strong gauge coupling), realized in the Nambu–Goldstone mode with massless pions $\pi$ and dilaton (that we denote as ${\sigma }_d$) with nonzero pion decay constant $f_{\pi }$ and dilaton decay constant, $f_{{\sigma }_d}$, $f_{\pi }=f_{{\sigma }_d}\ne 0$. Most significantly for us, it supports massive matter fields at the IR fixed point. A somewhat similar idea was put forward in [14]. The difference, if any, between these two ideas, we believe, would not affect our arguments made for nuclear matter at non-asymptotic density.

It should be stressed that this GD IR structure appears to be drastically different [13,14] from the IR structure of the dilatonic Higgs models, which purport to go beyond the standard model. This, we believe, makes a basic difference that leads us to the pseudo-conformality, not the conformality [4]—which in our picture should set at a much higher density than in compact stars—that prevails in nuclear dynamics in a density from $\sim n_0$ to the densities relevant to compact stars and possibly beyond before the dilaton limit, defined below, sets in [15]. It should be remarked, however, that the “conformal dilaton phase” of [14] which borders between the conformal window and QCD could support the conformality discussed in [4]. How the QCD phase moves into the conformal phase, if present, is not clear.

The scale symmetry is incorporated into the HLS Lagrangian ${\mathcal{L}}_{\mathrm{HLS}}$ [8] with the “conformal compensator (CC)” field $\chi$ [16]

$$\begin{array}{c}\hfill \chi =f_{\chi }{e}^{{\sigma }_d/f_{\chi }}\end{array}$$

(1)

which linearly transforms both in mass and length scales. (Note that ${\sigma }_d$, like the pion in chiral symmetry transforms nonlinearly under scaling.) One can formally construct a scale-invariant Lagrangian from ${\mathcal{L}}_{\mathrm{HLS}}$ by multiplying a suitable power of the CC field $\chi$ such that the action is scale-invariant. The resulting scale-chiral Lagrangian can be written as

$$\begin{array}{c}\hfill {\mathcal{L}}_{\chi \mathrm{HLS}}={\mathcal{L}}_{\mathrm{inv}}+V\end{array}$$

(2)

where all scale-symmetry-breaking terms, including quark mass terms, are included in the “dilaton potential” V. One can rewrite this in terms of the power counting in both chiral and scale symmetries with the trace anomaly taken into account [11,17] by implementing the standard power-counting in chiral symmetry [18] with the departure from the IR fixed point taken as

$$\begin{array}{c}\hfill \Delta {\alpha }_s\sim O\left(p^2\right)\sim O\left({\partial }^2\right).\end{array}$$

(3)

The resulting scale-chiral expansion [17,19,20]—that we call $\chi$PT${}_{{\sigma }_d}$ expansion—is a lot more complicated than the standard chiral perturbation theory (SchiPT), but to the leading order (LO) in the scale-chiral expansion to which we will restrict, it turns out to be relatively simple.

To access baryonic matter, the baryon fields ${\psi }^{†}=\left(pn\right)$ are coupled into ${\mathcal{L}}_{\chi \mathrm{HLS}}$. We will be dealing with $N_f=2$ although the GD approach of [11] is for $N_f=3$. While skipping details that can be found in the reviews [15], we briefly mention here that the HdFs brought in by the hidden symmetries are to play the role in GnEFT the putative “hadron-quark continuity” that plays the crucial role in going, in what we consider to be hadronic variables, from nuclear matter at low density to compact-star matter at high densities. We call the resulting Lagrangian ${\mathcal{L}}_{\psi \chi \mathrm{HLS}}$ with which GnEFT is formulated, as we will subsequently explain. In the scale-chiral power counting involving the vector mesons [10] and the scalar $\chi$ [17], this Lagrangian will then be of $O\left({\partial }^n\right)\sim O\left(p^n\right)$ with $n=1,2,...$. We will be mainly working with the leading order (LO) $n\le 2$.

Now, instead of doing the systematic power expansion, highly successful in SchiPT, to high-order in $\chi$PT${}_{{\sigma }_d}$ expansion, which is in principle doable but cumbersome at best with uncontrollable paramers, we will instead develop an EFT that maps the resulting LO Lagrangian denoted ${\mathcal{L}}_{\psi \chi \mathrm{HLS}}^{\mathrm{LO}}$ to a density functional theory (DFT) built on the renormalization-group (RG) treatment of baryons on the Fermi sea along the line developed in condensed matter physics [21]. We apply the approach of how to go from a chiral Lagrangian (in LO with suitable BR-scaling taken into account) to the “Fermi-liquid fixed point (FLFP)” theory, first formulated in in [22], to the Lagrangian ${\mathcal{L}}_{\psi \chi \mathrm{HLS}}^{\mathrm{LO}}$. This is the principal tool in GnEFT that we will employ. Going beyond the FLFP can also be and will be done in “ring-diagram approximations” in $V_{lowk}$ RG, as described in [1]. What is needed to access the compact-star densities in this GnEFT approach, not worked out in [22], is how the HdFs effectuate the putative hadron-quark continuity (HQC) at $n\gtrsim 2n_0$ in the Landau (fixed-point) parameters extracted from the Lagrangian.

In this approach, there is only one unique—not a hybrid—Lagrangian, an effective Landau(–Migdal) Lagrangian, that is to be valid over the whole range of involved densities with the fixed-point quasi-particle (QP) mass or Landau mass $m_L$, two-body QP–QP interaction parameters F, G, etc., built from ${\mathcal{L}}_{\psi \chi \mathrm{HLS}}^{\mathrm{LO}}$. It should be reminded to the readers that it is Migdal who reformulated Landau theory for nuclear processes [23]. Therefore from here on, by “Landau,” we will mean “Landau–Migdal.” We should point out that the Fermi-liquid structure described here focuses on bulk properties of matter, in particular, the equation of state (EoS), whereas in condensed matter physics, what is currently of high interest in the EFT of Fermi liquid is gapless Fermi-surface fluctuations, potentially leading to non-Fermi-liquid states [24].

The key points developed in [1] (refined in [15] with errors corrected) are as follows:

• Up to the density of $n_0$, the parameters of the (Landau–Migdal) Lagrangian are controlled by the known properties of the normal nuclear matter. Treated at the mean-field level, which corresponds to what is known as the Landau Fermi-liquid fixed point (FLFP) approximation (with $\overline{N}=k_F/({\Lambda }_{\mathrm{FS}}-k_F)\to \infty$) [21], it reproduces more or less all global properties of nuclear matter that are reliably described, to N${}^3$LO, by SchiEFT. How the FLFP approximation fares in nature is aptly illustrated by the anomalous proton orbital gyromagnetic ratio $\delta g_l^p$, the quenched $g_A$ in nuclei, and enhanced axial-charge transitions in heavy nuclei [22].
• The effect of the putative hadron-quark (HQ) continuity is brought into the Lagrangian by what is given by the topology change at $n\gtrsim 2n_0$ from skyrmions to half-skyrmions when the topological baryons of the Lagrangian ${\mathcal{L}}_{\chi \mathrm{HLS}}$ are put on the crystal lattice [1]. Putting skyrmions on the crystal lattice to describe baryon matter is evidently a very poor procedure at low densities (e.g., at $\sim n_0$, which is in Fermi liquid), however, it is justified at high density and in the large $N_c$ limit which underlies the Landau Fermi-liquid effective field theory. The precise densities involved cannot be pinned down in theory. However, the transition involves robust features that topology brings in. The most crucial feature for us is that at the HQ changeover, the condensate of the bilinear quark fields $\left(q^{†}q\right)$, while non-zero locally and supporting chiral waves, goes to zero when space-averaged at a density of $n=n_{1/2}\gtrsim 2n_0$. However, the pion condensate $f_{\pi }$ remains nonzero, thus the quark condensate is not an order parameter of the chiral phase transition. It somewhat resembles the “pseudo-gap” phenomenon in condensed matter physics [25].
• There arise various, highly remarkable, consequences when this topology change is incorporated into the Lagrangian ${\mathcal{L}}_{\psi \chi \mathrm{HLS}}$ and treated at the mean field (FLFP) approximation.
  First: There is a cusp in the nuclear symmetry energy $E_{\mathrm{sym}}$ wrapped by the $\rho$-meson cloud at the skyrmion-to-half-skyrmion density $n_{1/2}\gtrsim 2n_0$. This brings in the soft-to-hard changeover in the EoS at that cross-over density [26] that naturally accounts for $K_0$, $E_{\mathrm{sym}}\left(n_0\right)$ and $L=3n\frac{d{E}_{\mathrm{sym}}}{dn}{|}_{n=n_0}$ to be consistent with nature (e.g., [27], $K_0\approx 240$ MeV, $E_{\mathrm{sym}}\left(n_0\right)\approx 31.7$ MeV, $L\left(n_0\right)=57.7\pm 19$ MeV.)
  Second: The coupling between the dilaton $\chi$ and $\omega$, both playing a crucial role in the in-medium nucleon mass, leads, in the chiral limit, to the extremely simple result for the TEMT for $n\gtrsim n_{1/2}$ [1]

  $$\begin{array}{c}\hfill \langle {\theta }_{\mu }^{\mu }\rangle =ϵ-3P=4V_d\left(\langle \chi \rangle \right)-\langle \chi \rangle \frac{\partial V_d\left(\chi \right)}{\partial \chi }{|}_{\chi =\langle \chi \rangle }.\end{array}$$

  (4)
  In this formula, the dilaton potential $V_d$ contains also the baryon fields. In doing this calculation, it is imperative that the density dependence of the parameters inherited from matching with QCD (via relevant current correlators and the VEV with the dilaton field) be treated in accordance with the thermodynamic consistency [28]. Unless this is done correctly, one fails to arrive at (4) [1].
  Third: $\langle \chi \rangle$ becomes density-independent for $n\gtrsim n_{1/2}$ only for high VM density, $n_{\mathrm{VM}}\gtrsim 25n_0$, and $\langle \chi \rangle \to c{m}_0$ (with c a constant), where $m_0$ is the chiral invariant nucleon mass. This result, due to a close interplay—with the $\rho$ decoupled from the nucleons—between the dilaton $\chi$ and the $\omega$, signals the emergence of parity doubling in the baryon spectrum [29,30]. In our approach, the parity doubling is not put in $\stackrel{⌢}{\mathrm{ab}}$ initio as is done in the literature. It emerges from the interactions. It was verified to remain valid when $\mathcal{O}(1/\overline{N})$ corrections to the FLFP approximation are made in the $V_{lowk}$ formalism. That the TEMT (4) becomes density-independent for $n\gtrsim n_{1/2}$ is the key ingredient of the pseudo-conformality.

2. Predictions on Compact Stars

We now turn to the predictions on compact stars given in the theory GnEFT.

First, we look at the density dependence of the TEMT. One obtains a qualitatively correct answer by looking at the mean field (a.k.a., FLFP) approximation result (4). Taking the derivative with respect to density, we have

$$\begin{array}{c}\hfill \frac{\partial }{\partial n}\langle {\theta }_{\mu }^{\mu }\rangle =\frac{\partial ϵ\left(n\right)}{\partial n}\left(1-3\frac{v_s^2\left(n\right)}{c^2}\right)\end{array}$$

(5)

where the sound speed $v_s$ is inserted using $v_s^2/c^2=\frac{\partial P\left(n\right)}{\partial n}/\frac{\partial ϵ\left(n\right)}{\partial n}$. Since the condensate $\langle \chi \rangle$ changes as density changes for the density regime $n\lesssim n_{1/2}$, with $n_{1/2}$ being the topology change density, the TEMT changes with density. However, as noted, the dilaton condensate tends towards a density-independent constant $\propto m_0$, as n goes above $n_{1/2}$, the right-hand side of (5) will go to zero

$$\begin{array}{c}\hfill \frac{\partial ϵ\left(n\right)}{\partial n}\left(1-3\frac{v_s^2\left(n\right)}{c^2}\right)\to 0\mathrm{for}n\to n_{1/2}.\end{array}$$

(6)

There is no reason to expect $\frac{\partial ϵ\left(n\right)}{\partial n}\to 0$ (e.g., no Lee–Wick state), hence, we arrive at what we call pseudo-conformal speed

$$\begin{array}{c}\hfill v_{\mathrm{pcs}}^2/c^2\to 1/3.\end{array}$$

(7)

One arrives at the same result in the $V_{lowk}$ calculation, going beyond the FLFP approximation [1]. What was found was that with the HQ changeover from below to above $n_{1/2}$, strong nuclear correlations intervene in the way that the TEMT varies over the changeover density, with a “huge” bump produced with the speed going towards the causality limit $v_s/c=1$ and rapidly converging at an increasing density close to, though not necessarily on top of, $v_s^2/c^2=1/3$ [1]. This becomes notably more prominent if the changeover density is $n\gtrsim 4n_0$—in fact, at this $n_{1/2}$, even the causality is violated [15]. The mechanism for producing such a big bump may be indicative of how the changeover from hadronic degrees of freedom in EFT to quark–gluon degrees of freedom in nonperturbative QCD may be signaling the complexity involved in that region. (This resembles, curiously, how the HLS degrees of freedom (via the homogeneous or hidden Wess–Zumion term) wraps the cusp structure for the ${\eta }^{\prime }$ potential associated with the $U_A\left(1\right)$ anomaly [31]. It would be interesting to see whether there is any connection between the two phenomena.)

The detailed structure depends on the location of the changeover density. It turns out, however, to be qualitatively the same in the range of phenomenology places, say, at $2\lesssim n_{1/2}/n_0<4$ [15]. The typical structure is illustrated in Figure 1 for $n_{1/2}=2n_0$.

It is important to note that, in the range of densities involved in compact stars

$$\begin{array}{c}\hfill \langle {\theta }_{\mu }^{\mu }\rangle >0\end{array}$$

(8)

which gives

$$\begin{array}{c}\hfill \Delta =\frac{1}{3}-\frac{P}{ϵ}>0\end{array}$$

(9)

going independently of density for $n\gtrsim n_{1/2}$. It can be seen in Figure 2 that it is parallel and very close, with small deviation, to the band generated with the “sound velocity interpolation method” used in [3].

It should of course be stressed that the approach to the pseudo-conformal speed $v_{\mathrm{pcs}}^2/c^2\approx 1/3$ is not conformal $v_s^2/c^2=1/3$. As noted below, at a higher density, approaching what is called a “dilaton-limit fixed point (DLFP)” [33], where $\langle \chi \rangle$ goes to zero, the sound speed should approach the true conformal speed. We believe that the DLFP density should be close to the VM fixed point $n_{\mathrm{VM}}\gtrsim 25n_0$, way outside of the range of densities involved in the stars. It is difficult to directly relate what is described in this GnEFT with the analysis made in [4]. It is, however, tantalizing that the sound speed given in Figure 2 in [4] resembles the pseudo-conformal sound speed (Figure 1). It indicates that we are dealing here with strongly coupled (pseudo-)conformal matter.

There are further indications that the pseudo-conformal matter resembles “deconfined quark” matter. Let us look at the polytropic index defined by

$$\begin{array}{c}\hfill \gamma =dlnP/dlnϵ.\end{array}$$

(10)

Plotted in Figure 3 is the prediction for $\gamma$. It shows a large $\gamma \sim 3$ below the $n_{1/2}$ expected in nuclear matter, and drops below 1.75 at the topology change and then goes to near 1 at the core density ∼$6n_0$ of the star. This reproduces what was identified as a signal for “deconfined quark matter" in [3]. However, there are basic differences between our system and what is described in [3]. In our theory, conformality is broken, though perhaps only slightly at a high density, in the system. There can also be fluctuations around $v_{pcs}^2/c^2=1/3$ coming from the effects by the anomalous dimension ${\beta }^{\prime }$. Thus, the $v_{pcs}^2/c^2$ must fluctuate around 1/3, not on top of it.

3. Consistency with pQCD Top–Down?

The degrees of freedom that enter above the hadron–quark continuity simulated by the topology change with what appear to be fractionally charged baryonic quasiparticles are not so outlandish if one imagines strong nuclear correlations in action as in the way electrons behave in strongly correlated condensed matter systems. In fact, some albeit speculative but novel ideas along that line were recently discussed in [34,35].

Now, let us see how these – somewhat unorthodox – ideas can mesh with what is predicted by QCD.

There have been tour-de-force efforts to arrive at the EoS relevant to the core of compact stars in perturbative QCD coming as top–down. Some of the results obtained thus far appear to be quite relevant to what we obtained in GnEFT. Among them are the results of the pQCD calculation given in [6] that render feasible even a semi-quantitative comparison with the GnEFT prediction.

In Figure 4 (left panel) is shown the GnEFT prediction for $n_{1/2}\approx 3n_0$, corresponding to the hadron–quark crossover density. (Shown also is for $n_{1/2}=4n_0$ which was ruled out in our scheme because the sound speed exceeds the causality limit.) The prediction for $n_{1/2}=2n_0$ is slightly different from that of $n_{1/2}=3n_0$ but they both are essentially indistinguishable with the uncertainty involved. What is characteristic of this prediction is that the cusp in the symmetry energy $E_{\mathrm{sym}}$, “wrapped” by $\rho$ meson cloud with dropping mass, is first hardened by approaching $n_{1/2}$. This makes the symmetry energy at $\sim 2n_0$ bigger than what is given by standard chiral perturbation theory, then softened before reaching the density of the star core at $\sim 6n_0$; this feature is reflected in Figure 4 with $E_{\mathrm{sym}}$ bending after $n_{1/2}$, making the EoS softer toward the central density of the star. It is easy to see how this softening behavior sets in in the Fermi-liquid fixed point approximation in GnEFT. As noted, as one approaches the dilaton limit fixed point, the $\rho$ meson decouples from the baryonic matter (before the VM fixed point) and the remaining (heavy meson) degrees of freedom, the scalar (dilaton ${\sigma }_d$), and the isoscalar vector meson ($\omega$) interplay to keep the nucleon mass stay at $\sim m_0$. The quasiparticles involved are more or less free of interactions, hence preserving scale invariance (as also seen in a dense half-skyrmion simulation, Figure (11) (right panel) in [1]).

In Figure 4 (right panel) is plotted the pQCD result of [6]. Here, one sees the way the QCD asymptotically high-density calculations are propagated down to lower densities subjected to “thermodynamic consistency, stability and causality.” The calculation provides rather stringent constraints, as indicated in the figure, to the EoS at densities relevant to the core of the star. The result clearly shows that pQCD softens the EoS of the most massive neutron stars going toward the central density of ∼$6n_0$. Note that the stiffening and then softening in the EoS take place roughly at the same densities as in GnEFT. Precise matching would not make much sense given the approximations involved in both pQCD and GnEFT, but there is a clear qualitative consistency. It would be interesting to understand the stiffening–softening of the EoS in pQCD as in GnEFT, where the interplay of scale symmetry and hidden local symmetry is found to play a crucial role in the emerging parity-doubling symmetry.

4. Conclusions with a Bit of Speculation

In this brief review is recounted what has taken place, initiated in 2007, in the five-year “World-Class University Program” at Hanyang University in Seoul supported by the Korean Government, then continued at IBS (Korea) and at Jilin University in Changchun (China), with the objective of understanding ultradense matter stabie against gravitational collapse. This issue has currently become a hot topic in physics with the advent of gravitational waves. The resulting approach of GnEFT with only hadronic degrees of freedom, implemented with hidden symmetries of QCD with the parameters of the Lagrangian endowed with the vacuum sliding by density, leads to the results are globally consistent with the available data.

What is striking of this approach is that although no explicit QCD degrees of freedom are involved, the property of the core of massive stars predicted in this approach is uncannily similar to that of “deconfined quarks” with one apparent difference in the nature of the sound speed.

How can this be?

We have no convincing argument but one possible answer is this.

In the skyrmion–half-skyrmion crystal simulations of dense matter, one can imagine how the half-skyrmions can turn into fractionally charged baryons. Two half-skyrmions are confined by a pair of monopoles, so they are not separable. However, surprisingly, the bound two-1/2-skyrmions are found to propagate scale-invariantly, as observed in [1]. Suppose that it is feasible to liberate the two half-skyrmions by suppressing the monopoles. Then, it may be feasible to transform two half-skyrmions into three 1/3-charged objects [36]. One can then think of fractionally charged baryons populating the dense matter inside the core. Indeed, in condensed matter physics, with domain walls, there can be stacks of sheets containing deconfined fractionally charged objects behaving like “deconfined quarks” coming from the bulk in which the objects are confined [37]. In [38], highly speculative ideas along this line are entertained.

In conclusion, we propose that what is seen in the core of compact stars could be fractionalized quasiparticles masquerading “deconfined quarks.” As a test, we offer the “duck test” [39]: If it looks like a duck, swims like a duck, and quacks like a duck, then it probably is a duck.