Matching excluded-volume hadron-resonance gas models and perturbative QCD to lattice calculations

We match three hadronic equations of state at low energy densities to a perturbatively computed equation of state of quarks and gluons at high energy densities. One of them includes all known hadrons treated as point particles, which approximates attractive interactions among hadrons. The other two include, in addition, repulsive interactions in the form of excluded volumes occupied by the hadrons. A switching function is employed to make the crossover transition from one phase to another without introducing a thermodynamic phase transition. A ${\chi }^2$ fit to accurate lattice calculations with temperature $100<T<1000$ MeV determines the parameters. These parameters quantify the behavior of the QCD running gauge coupling and the hard core radius of protons and neutrons, which turns out to be $0.62\pm 0.04$ fm. The most physically reasonable models include the excluded-volume effect. Not only do they include the effects of attractive and repulsive interactions among hadrons, but they also achieve better agreement with lattice QCD calculations of the equation of state. The equations of state constructed in this paper do not result in a phase transition, at least not for the temperatures and baryon chemical potentials investigated. It remains to be seen how well these equations of state will represent experimental data on high-energy heavy-ion collisions when implemented in hydrodynamic simulations.

Figure 1

Pressure normalized by $T^4$. The dotted curve represents the parameter-free, point-particle hadron-resonance gas. The solid curve represents perturbative QCD with two parameters adjusted to fit the lattice result taken from Ref. [3].

Figure 2

Trace anomaly normalized by $T^4$. The dotted curve represents the parameter-free, point-particle hadron-resonance gas. The solid curve represents perturbative QCD with two parameters adjusted to fit the lattice result taken from Ref. [3].

Figure 3

Pressure of crossover models using those parameters from Table 1 that minimize ${\chi }^2$ per degree of freedom. Lattice data is from Ref. [3].

Figure 4

Trace anomaly of crossover models using those parameters from Table 1 that minimize ${\chi }^2$ per degree of freedom. Lattice data is from Ref. [3].

Figure 5

Switching function.

Figure 6

Pressure of crossover models. Lattice data is from Ref. [15].

Figure 7

Trace anomaly of crossover models. Lattice data is from Ref. [15].

Figure 8

Pressure of crossover models.

Figure 9

Trace anomaly of crossover models.