Scale-invariant hidden local symmetry, topology change, and dense baryonic matter

When scale symmetry is implemented into hidden local symmetry in low-energy strong interactions to arrive at a scale-invariant hidden local symmetric (HLS) theory, the scalar $f_0\left(500\right)$ may be interpreted as pseudo-Nambu-Goldstone (pNG) boson, i.e., dilaton, of spontaneously broken scale invariance, joining the pseudoscalar pNG bosons $\pi$ and the matter fields $V=(\rho ,\omega )$ as relevant degrees of freedom. Implementing the skyrmion–half-skyrmion transition predicted at large $N_c$ in QCD at a density roughly twice the nuclear matter density found in the crystal simulation of dense skyrmion matter, we determine the intrinsically density-dependent “bare parameters” of the scale-invariant HLS Lagrangian matched to QCD at a matching scale ${\mathrm{\Lambda }}_M$. The resulting effective Lagrangian, with the parameters scaling with the density of the system, is applied to nuclear matter and dense baryonic matter relevant to massive compact stars by means of the double-decimation renormalization-group $V_{\mathrm{low}k}$ formalism. We satisfactorily postdict the properties of normal nuclear matter and more significantly predict the equation of state of dense compact-star matter that quantitatively accounts for the presently available data coming from both the terrestrial and space laboratories. We interpret the resulting structure of compact-star matter as revealing how the combination of hidden-scale symmetry and hidden local symmetry manifests itself in compressed baryonic matter.

Figure 1

${\tilde{V}}_T\left(r\right)\equiv V_T\left(r\right){\left({\tau }_1{\tau }_2{S}_{12}\right)}^{-1}$. For illustration, we take $n_{1/2}=2n_0,{\mathrm{\Phi }}_I={\mathrm{\Phi }}_{II}^{\rho }=1-0.15\frac{n}{n_0}$ and $\kappa =1$.

Figure 2

Diagrams included in the $pphh$ ring-diagram summation for the ground-state energy of nuclear matter. Included are self-energy insertions on the single-particle propagator as indicated by (a), and $pphh$ ring diagrams by (b)–(d).

Figure 3

Ground-state energy $E_0$ per nucleon of symmetric nuclear matter (left panel) and neutron matter (right panel).

Figure 4

Comparison of our calculated symmetry energies $E_{\mathrm{sym}}\left(n\right)$ with the empirical ones of Li et al. [43] and Tsang et al. [44]. It is worth noting that the predicted symmetry energy manifests a shift from soft to hard at $n_{1/2}$ reflecting the classical cusp structure in the skyrmion description of the topology change.

Figure 5

Calculated pressure in symmetric nuclear matter (left panel) and the same in neutron matter (right panel) compared with the Danielewicz constraints [42].

Figure 6

Mass-radius relation of pure neutron stars calculated from the EoS of Fig. 5 (right panel). Our EoS does not contain the part of EoS of low densities appropriate for the surface region and hence cannot account for the low-mass stars.

Figure 7

Central densities of pure neutron stars of Fig. 6.

Figure 8

The sound velocity obtained from a fitting formula that reproduces the EoS of the neutron matter in Fig. 3. Near the crossover density $n_{1/2}=2n_0$, there is some discontinuity—most likely artificial—in velocity caused by the sharp demarcation of the parameter scaling, which turns out to be sensitive to the way the fitting is done.

Figure 9

Moment of inertia of neutron stars of Fig. 6.

Figure 10

Tadpole diagram for self-energies for the nucleon $N$ and the hyperon $\mathrm{\Lambda }$ in medium. The loop corresponds to the nucleon scalar density $n_s=\langle \overline{N}N\rangle$ for coupling to $\sigma$ and the nucleon number density $n=\langle N^{†}N\rangle$ for coupling to $\omega$.

Figure 11

${\mu }_{\mathrm{\Lambda }}-m_{\mathrm{\Lambda }}$ vs $n/n_0$ calculated with the scaling parameters determined in the theory and summarized in Table 4. The demarcation density was chosen for $n_{1/2}=(1.5,2.0)n_0$. The density at which the mass shift crosses zero corresponds to $\mathrm{\Delta }$ of [49].

Figure 12

${\tilde{V}}_T^{\pi }\left(r\right)\equiv V_T^{\pi }\left(r\right){\left({\tau }_1{\tau }_2{S}_{12}\right)}^{-1}$ with $n_{1/2}=2n_0$.

Figure 13

Ground-state energy $E_0(n,0)$ of symmetric nuclear matter with one parameter $c_I$ in the range $0.13–0.2$.

Figure 14

Ground-state energy $E_0(n,0)$ of symmetric nuclear matter (left panel) and neutron matter (right panel) with $\text{U}\left(2\right)$ symmetry for $(\rho ,\omega )$.