Baryon number fluctuations from a crossover equation of state compared to heavy-ion collision measurements in the beam energy range $\sqrt{s_{\mathit{NN}}}=7.7$ to 200 GeV

Fluctuations of the proton number distribution in central Au-Au collisions have been measured by the STAR Collaboration in a beam energy scan at the Relativistic Heavy Ion Collider (RHIC). The motivation is a search for evidence of a critical point in the equation of state. It was found that the skewness and kurtosis display an interesting energy dependence. We compare these measurements to an equation of state which smoothly interpolates between an excluded volume hadron resonance gas at low energy density to a perturbative plasma of quarks and gluons at high energy density. This crossover equation of state agrees very well with the lattice QCD equation of state. The crossover equation of state can reproduce the data if the fluctuations are frozen at a temperature significantly lower than the average chemical freeze-out.

Figure 1

The square of the sound speed as a function of temperature for zero baryon chemical potential. The points are from lattice QCD [11]. The dashed line is for a noninteracting massive pion gas. The top panel shows the sound speed for hadronic resonance gases, including those for point hadrons and for the two excluded volume models. The bottom panel shows the full crossover equations of state.

Figure 2

The square of the sound speed as a function of temperature for a baryon chemical potential of 400 MeV. The points are from lattice QCD [12]. The dashed line is for a noninteracting massive pion gas. The top panel shows the sound speed for hadronic resonance gases, including those for point hadrons and for the two excluded volume models. The bottom panel shows the full crossover equations of state.

Figure 3

The kurtosis as a function of temperature for zero baryon chemical potential. The points are from lattice QCD [13]. The top panel shows the kurtosis for hadronic resonance gases, including those for point hadrons and for the two excluded volume models. It also shows the purely perturbative QCD result for quarks and gluons. The bottom panel shows the full crossover equations of state.

Figure 4

The top panel shows the kurtosis as a function of temperature for zero baryon chemical potential. The points are from lattice QCD [13]. The curve is the same as the crossover from the previous figure. The shaded region shows the uncertainty when fitting the crossover equation of state parameters to lattice QCD at zero chemical potential. The bottom panel shows the susceptibility as a function of temperature for zero baryon chemical potential. The dotted curve is the parametrization from the lattice calculations in Ref. [13] while the points are from the HotQCD Collaboration [14].

Figure 5

The skewness as a function of collision energy per nucleon pair. The points are from STAR measurements. The top panel shows the skewness for excluded volume hadronic resonance gases. It also shows the purely perturbative QCD result for quarks and gluons. The bottom panel shows the full crossover equations of state. The energy dependence of the temperature and chemical potential are determined by the conditions at average chemical freeze-out as in Eq. (14).

Figure 6

The kurtosis as a function of collision energy per nucleon pair. The points are from STAR measurements. The top panel shows the kurtosis for excluded volume hadronic resonance gases. It also shows the purely perturbative QCD result for quarks and gluons. The bottom panel shows the full crossover equations of state. The energy dependence of the temperature and chemical potential are determined by the conditions at average chemical freeze-out as in Eq. (14).

Figure 7

The skewness (top panel) and kurtosis (bottom panel) for the crossover model exI. The shaded region shows the uncertainty when fitting the crossover equation of state parameters to lattice QCD at zero chemical potential. The energy dependence of the temperature and chemical potential are determined by the conditions at average chemical freeze-out as in Eq. (14).

Figure 8

Ratio of variance to mean for the crossover equation of state compared to the measurements by the STAR Collaboration. The energy dependence of the temperature and chemical potential are determined as in Eq. (14) but with a temperature, which is 26 MeV lower than in Figs. 5 to 7 ($a=140$ MeV).

Figure 9

The skewness (top panel) and kurtosis (bottom panel) for the crossover model exI. The shaded region shows the uncertainty when fitting the crossover equation of state parameters to lattice QCD at zero chemical potential. The energy dependence of the temperature and chemical potential are determined as in Eq. (14) but with a temperature which is 26 MeV lower than in Figs. 5 to 7 ($a=140$ MeV).

Figure 10

Ratio of kurtosis to the square of skewness for the two crossover equations of state compared to the measurements by the STAR Collaboration.

Figure 11

A fit to the STAR measurements of the temperature and chemical potential at each beam energy using the crossover equation of state exI.