Hadron resonance gas and mean-field nuclear matter for baryon number fluctuations

I give an estimate for the skewness and the kurtosis of the baryon number distribution in two representative models; i.e., models of a hadron resonance gas and relativistic mean-field nuclear matter. I emphasize formal similarity between these two descriptions. The hadron resonance gas leads to a deviation from the Skellam distribution if quantum statistical correlation is taken into account at high baryon density, but this effect is not strong enough to explain fluctuation data seen in the beam-energy scan at RHIC/STAR. In the calculation of mean-field nuclear matter, the density correlation with the vector $\omega$ field rather than the effective mass with the scalar $\sigma$ field renders the kurtosis suppressed at higher baryon density so as to account for the experimentally observed behavior of the kurtosis. Finally, I discuss the difference between the baryon number and the proton number fluctuations from correlation effects in isospin space. The numerical results suggest that such effects are only minor even in the case of complete randomization of isospin.

Figure 1

Skewness of the baryon number estimated in the HRG (thermus2.3) is shown by the (red) fine mesh. The (blue) sparse mesh represents the Skellam expectation, $tanh({\mu }_{\text{B}}/T)$.

Figure 2

Kurtosis of the baryon number estimated in the HRG (thermus2.3) is shown by the (red) fine mesh. The (blue) sparse mesh represents the Skellam expectation, which is unity.

Figure 3

HRG-estimated baryon density (including not only nucleons but also all baryonic resonances of the particle data contained in the thermus2.3 package) as a function of $T$ and ${\mu }_{\text{B}}$. The nucleon contribution is nearly a half of the results shown. The vertical lines represent the collision energy with spacing of $1\text{GeV}$. The extremal point corresponds to $\sqrt{s_{{}_{NN}}}\simeq 8\text{GeV}$. The chemical freeze-out line is drawn according to Eqs. (3) and (4).

Figure 4

Skewness of the baryon number distribution. The red dot, the green dotted line, and the blue dashed line represent the results from the BES-STAR, the HRG, and the RMF, respectively. The bands represent uncertainty from the freeze-out ${\mu }_{\text{B}}$ by $\pm 10\%$.

Figure 5

Kurtosis of the baryon number distribution. The legend convention is the same as in Fig. 4. The bands represent uncertainty from the freeze-out ${\mu }_{\text{B}}$ by $\pm 10\%$.

Figure 6

Kurtosis calculated in the RMF for various vector couplings.

Figure 7

Ratio of ${\chi }_p^{\left(4\right)}$ to ${\chi }_B^{\left(4\right)}$ which is close to $1/2$ and the deviation from $1/2$ is less than 10% at small $\sqrt{s_{{}_{NN}}}$.