Hagedorn bag-like model with a crossover transition meets lattice QCD

Thermodynamic functions, the (higher-order) fluctuations and correlations of conserved charges at ${\mu }_B=0$, and the Fourier coefficients of net-baryon density at imaginary ${\mu }_B$, are considered in the framework of a Hagedorn bag-like model with a crossover transition. The qualitative behavior of these observables is found to be compatible with lattice QCD results. Fair quantitative description of the lattice data is obtained when quasiparticle-type quarks and gluons with nonzero masses are introduced into the bag spectrum. The equation of state of the model exhibits a smooth and wide crossover transition.

Figure 1

Temperature dependence of (a) scaled pressure $p/T^4$ and (b) scaled energy density $\varepsilon /T^4$ calculated in the Hagedorn model with quark-gluon bags filled with massless quarks and gluons. Lattice QCD data of the Wuppertal-Budapest collaboration [45] are shown by the blue bands. The short horizontal lines depict the Stefan-Boltzmann limiting values.

Figure 2

The temperature dependence of (a) the filling fraction (f.f.), (b) the average particle mass $\langle m\rangle$, (c) the average particle volume $\langle v\rangle$, and (d) the particle number density $n$, calculated in the Hagedorn model with quark-gluon bags filled with massless quarks and gluons.

Figure 3

The temperature dependencies of the second order diagonal susceptibilities, ${\chi }_2^B, {\chi }_2^Q$, and $-{\chi }_2^S$, calculated in the Hagedorn model with quark-gluon bags filled with massless quarks and gluons at ${\mu }_B=0$, and compared with the lattice QCD data from Refs. [47, 48].

Figure 4

The temperature dependence of the scaled pressure $p/T^4$, calculated within the bag model equation of state [Eq. (34)] for massive quarks [Eqs. (53) and (54)] with $B^{1/4}=200\mathrm{MeV}$ (solid line), and for massless quarks [Eq. (14)] with $B^{1/4}=250\mathrm{MeV}$ (dashed line). The lattice QCD data of the Wuppertal-Budapest collaboration [45] are shown by the blue band.

Figure 5

The temperature dependence of (a) the scaled pressure $p/T^4$, (b) the scaled energy density $\varepsilon /T^4$, (c) the trace anomaly $(\varepsilon -3p)/T^4$, and (d) the speed of sound squared $c_s^2$, computed in the Hagedorn model with quark-gluon bags filled with massive quarks and gluons. The dashed and dash-dotted lines in (a) depict, respectively, the first and second terms of Eq. (38) for the total pressure whereas the green line depicts the pressure from the bag model equation of state. The dotted curves in (c) and (d) depict calculation results where the eigenvolumes of the PDG hadrons were taken to be four times smaller, i.e., ${\varepsilon }_0=16B$. The lattice QCD data of the Wuppertal-Budapest collaboration [45] are shown by the blue bands.

Figure 6

The temperature dependence of the second order conserved charge susceptibilities: (a) ${\chi }_2^B$, (b) ${\chi }_2^Q$, (c) ${\chi }_2^S$, (d) ${\chi }_{11}^{BQ}$, (e) ${\chi }_{11}^{QS}$, (f) $-{\chi }_{11}^{BS}$, computed in the Hagedorn model with quark-gluon bags filled with massive quarks and gluons. Lattice QCD data of the Wuppertal-Budapest [47] and HotQCD collaborations [48] are shown by the blue and green bands, respectively.

Figure 7

The temperature dependence of the baryon-strangeness correlator ratio $C_{BS}=-3\frac{{\chi }_{11}^{BS}}{{\chi }_2^S}$, computed in the Hagedorn bag-like model with quark-gluon bags filled with massive quarks and gluons. Lattice QCD data of the Wuppertal-Budapest [47] and HotQCD collaborations [48] are shown by the blue and green bands, respectively.

Figure 8

The temperature dependence of the conserved charges susceptibilities: (a) ${\chi }_4^B/{\chi }_2^B$, (b) ${\chi }_4^S/{\chi }_2^S$, (c) ${\chi }_6^B/{\chi }_2^B$, and (d) ${\chi }_8^B$, computed in the Hagedorn model with quark-gluon bags filled with massive quarks and gluons, and compared to the lattice QCD data of the Wuppertal-Budapest [62, 65] and HotQCD [63, 64] collaborations.

Figure 9

Temperature dependence of the leading four Fourier coefficients of the net baryon density at imaginary ${\mu }_B$ calculated within the Hagedorn model with quark-gluon bags filled with massive quarks and gluons (solid lines) compared with the lattice QCD data [70] (dots). The arrows depict the Stefan-Boltzmann limiting values [70].

Figure 10

Temperature dependence of the available volume fraction, $1-\text{f.f.}$, calculated within the Hagedorn model with quark-gluon bags filled with massive quarks and gluons (black solid line). The blue symbols depict the lattice QCD data of the Wuppertal-Budapest collaboration [7] for the subtracted chiral condensate ${\mathrm{\Delta }}_{l,s}$.

Figure 11

The ${\mu }_B$ dependence of the Hagedorn temperature, $T_H$ (55), computed for the massive quarks and gluons (53) and the bag constant $B^{1/4}=200\mathrm{MeV}$.

Figure 12

The mass spectrum of the PDG hadrons (blue line) and of the QGP bags, using the Hagedorn mass spectrum form [Eq. (15)] (orange line), or evaluated numerically through Eq. (16) for three different values of $M_0$ (in $\mathrm{GeV}/c^2$): 0.1 (red), 0.5 (green), and 1 (brown). Evaluations are done for the “massive quarks” parameter set [Eq. (54)] at the temperature $T=T_H\simeq 167.7\mathrm{MeV}$.

Figure 13

The temperature dependence of the ratio $p\left(\mathrm{Approx}\right)/\mathrm{p}\left(\mathrm{Full}\right)$, evaluated for the massive quarks parameter set for three different values of $m_0$ (in $\mathrm{GeV}/c^2$): 0.1 (red), 0.5 (green), and 1 (brown).