# Dense Baryonic Matter Predicted in “Pseudo-Conformal Model”

## Abstract

**:**

## 1. Introduction

## 2. G$\mathit{n}$EFT

## 3. Predictions

#### 3.1. Density Regime $n\lesssim {n}_{0}$

#### 3.2. Density Regime $n>{n}_{1/2}$

**I**)–(

**IV**) incorporated into GnEFT for the PCM is that the trace of the energy-momentum-tensor ${\theta}_{\mu}^{\mu}$ for $n\gtrsim {n}_{1/2}$ goes as

#### 3.3. Predictions vs. Observables

**Smoothed cusp of ${E}_{sym}\left(n\right)$ at $n\gtrsim {n}_{1/2}$:**The bending-over of ${E}_{sym}$ influences the slope L and induces the “soft-to-stiff” changeover. It also plays a crucial role in giving rise to the pseudo-conformal sound velocity (to be addressed below). Although as stressed the detailed structure and magnitude cannot be precisely pinned down, its simplicity with intricate topology change in the jungle of theories (as depicted in [9]) is a distinctive prediction of the PCM. It is at odds with the PREX-II/Jefferson data, which give generally stiff EoS, although there are some caveats [18,19]. To date, there are no trustworthy experimental data to quantitatively compare with.**Maximum mass star: ${M}^{\mathrm{max}}$**:$$\begin{array}{ccc}\hfill \mathrm{PCM}\phantom{\rule{4pt}{0ex}}\mathrm{prediiction}:& & {M}^{max}\approx 2.05{M}_{\odot},\hfill \\ & & \phantom{\rule{4pt}{0ex}}{R}_{2.0}\approx 12.8\phantom{\rule{4pt}{0ex}}\mathrm{km},\hfill \\ & & ({n}_{central}\approx 5.1{n}_{0}),\hfill \end{array}$$$$\begin{array}{ccc}\hfill \mathrm{PSR}\mathrm{J}0740+6620:& & {M}^{max}=2.08\pm 0.07{M}_{\odot},\hfill \\ & & {R}_{2.0}=12.35\pm 0.75\phantom{\rule{4pt}{0ex}}\mathrm{km},\hfill \\ & & ({n}_{central}=??),\hfill \end{array}$$No empirical data are known to be available at present for the central density ${n}_{central}$. The only information on this quantity inferred—not extracted—from PSR J0740 + 6620 is violently at odds with the PCM prediction. I will address this issue below.**1.44 ${M}_{\odot}$ star**:$$\begin{array}{ccc}& \mathrm{PCM}\phantom{\rule{4pt}{0ex}}\mathrm{prediction}& :{R}_{1.44}\approx 12.8\phantom{\rule{4pt}{0ex}}\mathrm{km}\hfill \\ & \mathrm{PSR}\mathrm{J}0030+0451& :{R}_{1.44}=12.45\pm 0.65\phantom{\rule{4pt}{0ex}}km.\hfill \end{array}$$The stunning agreements between the PCM predictions and the NICER and XMM-Newton measurements—with the exception of the sound velocity to be addressed below—could not be accidental. Not only does the maximum star mass come out the same, but also the radii agree. Furthermore, the difference $\Delta R={R}_{2.0}-{R}_{1.4}\approx 0$, in agreement with the data. We will note later that this support of the PCM by the NICER/XMM-Newton has an even more surprising implication on scale-chiral symmetry in nuclear medium so far unsuspected.

**Tidal deformability ${\mathsf{\Lambda}}_{1.4}$**The ${\mathsf{\Lambda}}_{1.4}$ predicted in the PCM comes out to be ∼550, to be compared with ${\mathsf{\Lambda}}_{1.4}={190}_{-120}^{+390}$ (GW1700817). This may seem to signal a tension. However, there is a basic difficulty in theoretically pinning down ${\mathsf{\Lambda}}_{1.4}$. In the PCM, the density at which ${\mathsf{\Lambda}}_{1.44}$ is measured is ∼$2.4{n}_{0}$. This density sits very close to where the topology change takes place. It is here the SchiEFT is most likely to start breaking down as the cusp in ${E}_{sym}$ indicates and the pQCD cannot access. This is an “uncharted wilderness” for theory. As can be seen in [4], a small increase in the central density, say, from $2.3{n}_{0}$ to $2.5{n}_{0}$ (or increase in corresponding star mass), makes $\mathsf{\Lambda}$ drop to 420 while involving no change at all in radius. This means that the location of the HQC will strongly influence the $\mathsf{\Lambda}$. One can associate this behavior with the increase in attraction in going from ${n}_{0}$ toward ${n}_{1/2}$ in the cusp structure as one can see in the schematic plot Figure 1. This clearly suggests that it would be extremely difficult to theoretically pin down $\mathsf{\Lambda}$ in the vicinity of the crossover regime.As noted below, the sound velocity has a complex “bump” structure in the vicinity of the topology change density. This is due to the interplay, encoding the putative HQC, between the hadronic degrees of freedom and the “dual quark-gluon” degrees of freedom. This would complicate significantly the linking of ${\mathsf{\Lambda}}_{1.4}$ to the structure of the sound velocity below or near ${n}_{1/2}$. To give an example, let me quote [23] where the bump structure—“the slope, the hill, the drop, the swoosh, etc.”—associated with the possible phase structure of QCD is proposed to pin down ${\mathsf{\Lambda}}_{1.4}$ by up-coming measurements. The hope here is to determine the possible phase transition near the HQC density. Given the theoretical wilderness inevitably involved, this seems a far-fetched endeavor.In short, contrary to what is claimed by some workers in the field, ruling out an EoS based on the precise value of ${\mathsf{\Lambda}}_{1.4}$ would be premature.**Sound speed ${v}_{s}$**The most striking prediction of the PCM, so far not shared by other models, is the sound speed for $n\gtrsim {n}_{1/2}$. It predicts the pseudo-conformal sound speed$$\begin{array}{c}\hfill {v}_{s}^{pcss}/{c}^{2}\approx 1/3\phantom{\rule{4pt}{0ex}}\mathrm{for}\phantom{\rule{4pt}{0ex}}n\gtrsim {n}_{1/2}.\end{array}$$It is not to be identified with the conformal sound speed ${v}_{s}^{conform}/{c}^{2}=1/3$ because the energy-momentum tensor is not traceless, i.e., scale symmetry is spontaneously broken.This prediction can be understood as follows.As noted above, the quasiparticle mass ${m}_{Q}^{\ast}$ goes $\propto {\langle \chi \rangle}^{\ast}$ as the density goes above ${n}_{1/2}$ and the dilaton condensate becomes independent of density, reaching ${m}_{0}$. This has to do with a delicate interplay between the attraction associated with the dilaton exchange and the $\omega $ repulsion, which leads to the parity doubling. Where this interplay starts taking place cannot be pinned down precisely but it must be in the density regime where the symmetry energy is involved, going from ${n}_{1/2}$ to the core of massive stars, say, $\gtrsim 6{n}_{0}$. In this density regime, the Landau fixed-point approximation with ${\overline{N}}^{-1}=({\mathsf{\Lambda}}_{F}-{k}_{f})/{k}_{F}\sim 1/{k}_{F}\to 0$ can be taken to be reliable. One can then calculate the trace of the energy-momentum tensor in the mean-field approximation of GnEFT, i.e., LFL fixed-point approximation, which will become density-independent as given by (11). In this density range we will have$$\begin{array}{c}\hfill \frac{\partial}{\partial n}\langle {\theta}_{\mu}^{\mu}\rangle =\frac{\partial \u03f5\left(n\right)}{\partial n}\left(1-3\frac{{v}_{s}^{2}}{{c}^{2}}\right)\approx 0\end{array}$$$$\begin{array}{c}\hfill \left(1-3\frac{{v}_{s}^{2}}{{c}^{2}}\right)\approx 0\end{array}$$$$\begin{array}{c}\hfill {({v}_{s}^{pcs}/c)}^{2}\approx 1/3.\end{array}$$The “approximate zero” here stands for the fact that it is pseudo-conformal with scale symmetry broken both explicitly and spontaneously, the dilaton mass and the $\omega $ mass balancing so as to lead to parity doubling in the dense system. The true conformal velocity, within the model, should be reached only at a density much higher than that of the core density of the massive stars. Where precisely the conformality sets in is not relevant to the compact star physics.In Figure 3 is shown the sound speed ${v}_{s}/c$ for $\alpha =0$ (nuclear matter) and 1 (neutron matter) calculated in ${V}_{lowK}$ RG for ${n}_{1/2}$. They are of the same form for $2<{n}_{1/2}/{n}_{0}<4$ except for the slight shift in the density and the height of ${v}_{s}$. This result serves as an illustration of the arguments to follow.What is noticeable is the large bump in ${v}_{s}$ in the vicinity of ${n}_{1/2}$ and the rapid convergence to the speed 1/3. The approximation involved on top of the pseudo-conformality would of course give fluctuations on top of ${v}_{pcs}^{2}/{c}^{2}\approx 1/3$ but the point here is it is the pseudo-conformality that “controls” the general structure. The large bump signals a complex interplay between hadronic and non-hadronic degrees of freedom manifested through the pseudo-gap structure of the chiral condensates. I will discuss below how the degrees of freedom in the core of the massive stars could masquerade as “deconfined quarks”.**Quenched ${g}_{A}$ in nuclei**Though it is not directly connected with the star properties, a relevant and intriguing observation is what I would call “quasibaryon” ${g}_{A}$ in nuclear matter. It follows from the possible existence of the IR fixed point associated with the “genuine dilaton (GD)." The effective ${g}_{A}$ in the Gamow–Teller transitions in nuclei, ${g}_{A}^{eff}$, is observed to be ${g}_{A}^{eff}\approx 1$ from light nuclei to heavy nuclei and even to the dilaton-limit fixed point at $n\gtrsim 25{n}_{0}$. It has been argued that an approximate scale invariance “emerges” in nuclear interactions [24], in a way most likely related to the way ${({v}_{s}^{pcs}/c)}^{2}\approx 1/3$ sets in precociously.Returning to ${v}_{s}$, is there any indication in recent astrophysical observations for such a precocious onset of the pseudo-conformal sound velocity?To date, there is no known “smoking-gun” signal for the sound velocity from observations. In the literature, however, there are a gigantic number of articles on the structure of sound velocity deduced from the gravity wave data as well as theoretically. Some argue for phase transitions or continuous ones or simply no crossovers, etc. Some extreme cases are discussed in [9]. I will no go into this wilderness here. Let me just describe one case which illustrates most transparently what can very well be involved.Let us take the case of NICER and XMM-Newton observables (NXN for short) discussed, namely (13) and (14). This case brings out how puzzling the problem can be.In [25], the properties of high-density matter were inferred in most detailed analyses of the NXN data. Ruling out essentially all other scenarios, with or without phase changes, the authors arrive at the sound velocity (“H-bump”) plotted in Figure 4.The central density and the maximum sound velocity inferred were$$\begin{array}{ccc}\hfill {n}_{\mathrm{cent}}/{n}_{0}& =& 3.{0}_{-1.6}^{+1.6}\phantom{\rule{4pt}{0ex}},\hfill \\ \hfill {v}_{s}^{2}/{c}^{2}& =& 0.{79}_{-0.20}^{+021}\phantom{\rule{4pt}{0ex}}.\hfill \end{array}$$While the star properties they took into account are exactly those reproduced by the PCM, i.e., (13) and (14), the central density and the sound velocity are totally different from the PCM predictions. One can understand the low central density accounting for the sound speed overshooting the conformal bound, characteristic of strongly interacting hadronic phase. In fact there are in the literature numerous scenarios anchored on a variety of density-functional approaches giving rise to the wilderness of one form or other in the sound velocity—including bumps similar to the H-bump—but I am not aware of any that can survive the battery of bona fide constraints coming from the current observations both in theory and experiment as claimed by [25].A puzzle immediately raised is this: How can the PCM with an emergent (pseudo-)conformality and the strong H-bump with no hint of conformal symmetry give the almost identical global star properties (13) and (14)? The only statement one can make at this point is (A) either the sound velocity and the global star properties are totally unrelated or (B) there is something wrong either in the strong H-bump scenario or in the simple PCM structure. Option (A) is hard to accept, so perhaps option (B) is a plausible possibility. My bet is option (B) and the H-bump scenario are at odds with nature.**Conformailty**

## 4. Conclusion: The Duck Story

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Schematic illustration of the symmetry energy ${E}_{sym}\left(n\right)$ by the skyrmion crystal (red dashed line)) and by nucleon correlations dominated by the nuclear tensor forces (solid line).

**Figure 2.**${E}_{sym}$ (solid circle) obtained in the full ${V}_{lowk}$ RG approach for ${n}_{1/2}=2{n}_{0}$. It is reproduced exactly by the pseudo-conformal model (solid line). Idem for ${n}_{1/2}\sim (2\u20134){n}_{0}.$.

**Figure 3.**${v}_{s}$ vs. density for $\alpha =0$ (nuclear matter) and $\alpha =1$ (neutron matter) in ${V}_{lowk}$ RG for ${n}_{1/2}=2{n}_{0}$ and ${v}_{\mathrm{vn}}=25{n}_{0}$.

**Figure 4.**$\rho $ in unit of g/cm${}^{3}$ (the “H-bump” scenario) taken from [25]. The red contour stands for 50% and 90% inferred sound speed and central density.

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**MDPI and ACS Style**

Rho, M.
Dense Baryonic Matter Predicted in “Pseudo-Conformal Model”. *Symmetry* **2023**, *15*, 1271.
https://doi.org/10.3390/sym15061271

**AMA Style**

Rho M.
Dense Baryonic Matter Predicted in “Pseudo-Conformal Model”. *Symmetry*. 2023; 15(6):1271.
https://doi.org/10.3390/sym15061271

**Chicago/Turabian Style**

Rho, Mannque.
2023. "Dense Baryonic Matter Predicted in “Pseudo-Conformal Model”" *Symmetry* 15, no. 6: 1271.
https://doi.org/10.3390/sym15061271