Chemical equilibrium study in nucleus-nucleus collisions at relativistic energies

We present a detailed study of chemical freeze-out in nucleus-nucleus collisions at beam energies of $11.6A$, $30A$, $40A$, $80A$, and $158A\text{GeV}$. By analyzing hadronic multiplicities within the statistical hadronization approach, we have studied the strangeness production as a function of center-of-mass energy and of the parameters of the source. We have tested and compared different versions of the statistical model, with special emphasis on possible explanations of the observed strangeness hadronic phase space undersaturation. We show that, in this energy range, the use of hadron yields at midrapidity instead of in full phase space artificially enhances strangeness production and could lead to incorrect conclusions as far as the occurrence of full chemical equilibrium is concerned. In addition to the basic model with an extra strange quark nonequilibrium parameter, we have tested three more schemes: a two-component model superimposing hadrons coming out of single nucleon-nucleon interactions to those emerging from large fireballs at equilibrium, a model with local strangeness neutrality and a model with strange and light quark nonequilibrium parameters. The behavior of the source parameters as a function of colliding system and collision energy is studied. The description of strangeness production entails a nonmonotonic energy dependence of strangeness saturation parameter ${\gamma }_S$ with a maximum around $30A\text{GeV}$. We also present predictions of the production rates of still unmeasured hadrons including the newly discovered ${\Theta }^{+}\left(1540\right)$ pentaquark baryon.

Figure 1

Above: measured vs fitted multiplicities in the statistical model supplemented with ${\gamma }_S$ parameter [SHM$\left({\gamma }_S\right)$] in Au-Au collisions at a beam energy of $11.6A\text{GeV}$ in the fit A; also quoted are the best-fit parameters. Below: residual distribution.

Figure 2

Above: measured vs fitted multiplicities in the statistical model supplemented with ${\gamma }_S$ parameter in Pb-Pb collisions at a beam energy of $40A\text{GeV}$ in the fit A; also quoted are the best-fit parameters. Below: residual distribution.

Figure 3

Above: measured vs fitted multiplicities in the statistical model supplemented with ${\gamma }_S$ parameter in Pb-Pb collisions at a beam energy of $80A\text{GeV}$ in the fit A; also quoted are the best-fit parameters. Below: residual distribution.

Figure 4

Above: measured vs fitted multiplicities in the statistical model supplemented with ${\gamma }_S$ parameter in Pb-Pb collisions at a beam energy of $158A\text{GeV}$ in the fit A; also quoted are the best-fit parameters. Below: residual distribution. Note that $\Lambda \left(1520\right)$ was not used in the fit (see text).

Figure 5

Comparison between measured and calculated (in fits A and B) $⟨K^{+}⟩∕⟨{\pi }^{+}⟩$ and $⟨{\pi }^{-}⟩∕⟨{\mathrm{N}}_p⟩$ ratios as a function of the center-of-mass energy in the examined collisions. For the SPS energy points the statistical errors are indicated with solid lines, while the contribution of the common systematic error is shown as a dotted line. The lines connect the fitted values.

Figure 6

Spectrum of known light-flavored hadronic species up to a mass of $1.8\text{GeV}$.

Figure 7

Left: primary yields of various particles as a function of the cutoff on the hadronic mass spectrum. Right: fitted ${\gamma }_S$, baryon-chemical potential, and temperature as a function of the cutoff on the hadronic mass spectrum.

Figure 8

Rapidity distributions of ${\pi }^{\pm }$, $K^{\pm }$, $\varphi$, and $\Omega$ emitted from a single fireball at rest at a kinetic freeze-out temperature of $T=125\text{MeV}$.

Figure 9

Above: measured vs fitted multiplicities in the statistical model supplemented with strangeness suppression in $pp$ collisions at a beam energy of $158\text{GeV}$ corresponding to $\sqrt{s}=17.2\text{GeV}$; also quoted the best-fit parameters. Below: residual distribution.

Figure 10

Minimum ${\chi }^2$ of multiplicity fits in Pb-Pb collisions at a beam energy of $158A\text{GeV}$ as a function of a fixed light quark nonequilibrium parameter ${\gamma }_q$. The round dots are the ${\chi }^2$’s obtained with the main data sample, while triangular dots are those obtained with pion multiplicities lowered by one standard deviation and all others unchanged. The vertical dashed line indicates the condensation point.

Figure 11

Chemical freeze-out points in the ${\mu }_B\text{−}T$ plane in various heavy ion collisions. The full round dots refer to Au-Au at $11.6A$ and Pb-Pb collisions at $40A$, $80A$, $158A\text{GeV}$ obtained in the analysis A, while the hollow square dot has been obtained in Ref. 11 by using particle ratios measured at midrapidity in Au-Au collisions at ${\sqrt{s}}_{NN}=130\text{GeV}$. The hollow round dot without error bars refers to Pb-Pb collisions at $30A\text{GeV}$ and has been obtained by forcing $T$ and ${\mu }_B$ to lie on the parabola fitted to the full round dots.

Figure 12

Strangeness nonequilibrium parameter ${\gamma }_S$ as a function of the nucleon-nucleon center-of-mass energy. Full dots refer to fit A, hollow dots to fit B.

Figure 13

Measured $⟨K^{+}⟩∕⟨{\pi }^{+}⟩$ ratio as a function of the fitted baryon-chemical potential. The full square dot is a preliminary full phase space measurement in Au-Au collisions at ${\sqrt{s}}_{NN}=200\text{GeV}$ [37] and the error is only statistical; the arrow on the left signifies that its associated baryon chemical potential is lower than that estimated at ${\sqrt{s}}_{NN}=130\text{GeV}$ [11] used here. For the SPS energy points the statistical errors are indicated with solid lines, while the contribution of the common systematic error is shown as a dotted line. Also shown are the theoretical values for a hadron gas along the fitted chemical freeze-out curve shown in Fig. 11, for different values of ${\gamma }_S$.

Figure 14

${\lambda }_S$ estimated from the fits A (full dots) and B (hollow dots) as a function of the fitted baryon-chemical potential. Also shown are the theoretical values for a hadron gas along the fitted chemical freeze-out curve shown in Fig. 11, for different values of ${\gamma }_S$.