Examination of the sensitivity of the thermal fits to heavy-ion hadron yield data to the modeling of the eigenvolume interactions

The hadron-resonance gas (HRG) model with the mass-proportional eigenvolume (EV) corrections is employed to fit the hadron yield data of the NA49 Collaboration for central Pb+Pb collisions at $\sqrt{s_{NN}}=6.3$, 7.6, 8.8, 12.3, and 17.3 GeV, the hadron midrapidity yield data of the STAR Collaboration for Au+Au collisions at $\sqrt{s_{NN}}=200$ GeV, and the hadron midrapidity yield data of the ALICE Collaboration for Pb+Pb collisions at $\sqrt{s_{NN}}=2760$ GeV. At given bombarding energy, for a given set of radii, the EV HRG model fits do not just yield a single $T-{\mu }_B$ pair, but a whole range of $T-{\mu }_B$ pairs, each with similarly good fit quality. These pairs form a valley in the $T-{\mu }_B$ plane along a line of nearly constant entropy per baryon, $S/A$, which increases nearly linearly with bombarding energy $E_{\mathrm{lab}}$. The entropy per baryon values extracted from the data at the different energies are a robust observable: it is almost independent of the details of the modeling of the eigenvolume interactions and of the specific $T-{\mu }_B$ values obtained. These results show that the extraction of the chemical freeze-out temperature and chemical potential is extremely sensitive to the modeling of the short-range repulsion between the hadrons. This implies that the ideal point-particle HRG values are not unique. The wide range of the extracted $T$ and ${\mu }_B$ values suggested by the eigenvolume HRG fits, as well as the approximately constant $S/A$ at freeze-out, are consistent with a nonequilibrium scenario of continuous freeze-out, where hadrons can be chemically frozen out throughout the extended space-time regions during the evolution of the system. Even when the EV HRG fits are restricted to modest temperatures suggested by lattice QCD, the strong systematic effects of EV interactions are observed.

Figure 1

The temperature dependence of ${\chi }^2/N_{\mathrm{dof}}$ of fits to data of the NA49, STAR, and ALICE collaborations on hadron yields in central Pb+Pb and Au+Au collisions within the point-particle HRG model (solid black curves), the diagonal eigenvolume model (thin dotted blue lines), and the crossterms eigenvolume model (thick dotted blue lines). The constant ${\varepsilon }_0$ is fixed in order to reproduce the effective hard-core proton radius of 0.5 fm.

Figure 2

The temperature dependence of ${\chi }^2/N_{\mathrm{dof}}$ of fits to data of the NA49 Collaboration on hadron yields in central Pb+Pb collisions at $\sqrt{s_{NN}}=8.8$ GeV within the point-particle HRG model (solid black curve), the diagonal eigenvolume model (differently styled thin lines), and the crossterms eigenvolume model (differently styled thick lines). The constant ${\varepsilon }_0$ is fixed in order to reproduce the effective hard-core proton radii of 0.4, 0.5, and 0.6 fm.

Figure 3

Regions in the $T-{\mu }_B$ plane where the crossterms eigenvolume HRG model with bag-like constant ${\varepsilon }_0$ fixed to reproduce the hard-core radius of $r_p=0.5$ fm yields a better fit to the NA49 and STAR data as compared to the point-particle HRG model. The locations of the local ${\chi }^2$ minima are depicted by the diamonds for the point-particle HRG case, and by the dots for the eigenvolume HRG. For RHIC energy the light shaded area additionally depicts the wide $T-{\mu }_B$ region where ${\chi }^2$ of the fit to the STAR data within the eigenvolume model has a broad minimum. The solid lines show the isentropic curves for the eigenvolume model, which go through the global ${\chi }^2$ minima.

Figure 4

The collision energy dependence of the entropy per baryon ratio $S/A$ calculated from the HRG model at the global minimum of the ${\chi }^2$ fit to the SPS hadron yield data. Calculations are done within the point-particle HRG and within the eigenvolume HRG with mass-proportional eigenvolumes. The parameter ${\varepsilon }_0$ in Eq. (10) is fixed to reproduce the effective hard-core proton radii of 0.4, 0.5, and 0.6 fm.

Figure 5

Temperature dependence of the scaled pressure at zero baryon chemical potential calculated within the crossover equation of state (solid line), and the crossterms eigenvolume HRG with $v_i=m_i/{\varepsilon }_0$ and ${\varepsilon }_0=704\mathrm{MeV}/{\mathrm{fm}}^3$ (dashed line). The lattice QCD data of the Wuppertal-Budapest Collaboration [51] is shown by the symbols with error bars.

Figure 6

The extracted freeze-out parameters within the point-particle HRG, and the case when eigenvolumes of hadrons are proportional to their mass with ${\varepsilon }_0=704\mathrm{MeV}/{\mathrm{fm}}^3$, modeled by the crossterms eigenvolume HRG. The parameterized freeze-out curves from Refs. [11, 30], obtained within the point-particle-like (including constant eigenvolume for all hadrons) HRG models, are depicted by lines.