Hadron resonance gas equation of state from lattice QCD

The Monte Carlo results in lattice QCD for the pressure and energy density at small temperature $T<155$ MeV and zero baryonic chemical potential are analyzed within the hadron resonance gas model. Two extensions of the ideal hadron resonance gas are considered: the excluded volume model, which describes a repulsion of hadrons at short distances, and the Hagedorn model with an exponential mass spectrum. Considering both of these models we do not find conclusive evidence in favor of either of them. The controversial results appear because of rather different sensitivities of the pressure and energy density to both excluded volume and Hagedorn mass spectrum effects. On the other hand, we have found clear evidence for a simultaneous presence of both of them. They lead to rather essential contributions: suppression effects for thermodynamical functions of the hadron resonance gas due to the excluded volume effects and enhancement due to the Hagedorn mass spectrum.

Figure 1

The lattice results from Ref. [3] for $3p/T^4$ (circles) and $\varepsilon /T^4$ (squares) at zero baryonic chemical potential.

Figure 2

(a) The Id-HRG pressure, $p^{\mathrm{id}}\left(T\right)/T^4$, as a function of temperature at $\mu =0$ (dotted line) and the Boltzmann approximation ${\eta }_i=0$ (solid line). (b) The Id-HRG pressure and EV-HRG pressure functions for several different values of hard-core radius $r$. The Stefan-Boltzmann limit for the deconfined quark-gluon phase, $p_{\mathrm{SB}}/T^4={\sigma }_{\mathrm{SB}}/3\cong 5.2$, is indicated by the horizontal dotted line.

Figure 3

The results of the EV-HRG model for different values of $r$ compared to the lattice data for $p/T^4$ (a) and $\varepsilon /T^4$ (c). The values of ${\chi }_p^2/N_{\mathrm{df}}$ (b) and ${\chi }_{\varepsilon }^2/N_{\mathrm{df}}$ (d) shown as functions of $r$; the shaded gray area corresponds to $r\le 0.13$ fm.

Figure 4

The results of the EV-HRG-H model for different values of $r$ compared to the lattice data for $p/T^4$ and $\varepsilon /T^4$ in (a) and (b), respectively. The value of $C$ is fixed as $C=0.05{\text{GeV}}^{3/2}$.

Figure 5

(a) Parameter $C$ which minimizes ${\chi }^2/N_{\mathrm{df}}$ at each value of $r$ shown as a function of $r$. (b) The quantity ${\chi }^2/N_{\mathrm{df}}$ as a function of $r$. For each value of $r$, parameter $C$ is fitted in order to minimize ${\chi }^2/N_{\mathrm{df}}$. The shaded gray area corresponds to $r\le 0.13$.