Cumulants of the baryon number from central Au+Au collision at $E_{\mathrm{lab}}=1.23$ GeV/nucleon reveal the nuclear mean-field potentials

Fluctuations of the baryon number in relativistic heavy-ion collisions are a promising observable to explore the structure of the QCD phase diagram. The cumulant ratios in heavy-ion collisions at intermediate energies ($\sqrt{s_{\mathit{NN}}}<5$ GeV) have not been studied to date. We investigate the effects of mean-field potential and clustering on the cumulant ratios of baryon and proton number distributions in Au+Au collisions at beam energy of 1.23 GeV/nucleon as measured by the HADES Collaboration at GSI. Ultrarelativistic quantum molecular dynamics (UrQMD) and the JAM model are used to calculate the cumulants with different mean-field potentials. It is found that the cumulant ratios are strongly time dependent. At the early stage, the effects of the potentials on the fluctuations of the particle multiplicity in momentum space are relatively weak. The mean fields enhance the fluctuations during the expansion stage, especially for small rapidity acceptance windows. The enhancement of cumulant ratios for free protons is strongly suppressed as compared to that for all baryons. The mean-field potentials and the clustering play an important role for the measured cumulant ratios at intermediate energy.

Figure 1

Yield distributions of all baryons as functions of the reduced longitudinal ($y_z$/$y_b$, filled squares) and transverse ($y_x$/$y_b$, open circles) rapidities from central Au+Au collisions at beam energy of 1.23 GeV/nucleon. Calculations with the soft momentum-dependent (SM), the hard momentum-dependent (HM), and the hard without momentum-dependent (H) mean-field potentials are compared to calculation without mean-field potential (cascade mode). The corresponding values of $varxz$ are also shown. The error bars are smaller than the symbols size.

Figure 2

Rapidity dependence for the cumulants ($C_1$–$C_3$) of all baryons produced in central Au+Au collisions at beam energy of 1.23 GeV/nucleon. Results have been calculated with SM, HM, and H potentials as well as in the cascade mode (solid symbols). The results are compared to a baseline from the binomial distribution (open symbols). Error bars for all data shown, are smaller than the symbol sizes.

Figure 3

Rapidity dependence for the cumulant ratios ($C_2$/$C_1,C_3$/$C_1,S\sigma$) of all baryons produced in central Au+Au collisions at beam energy of 1.23 GeV/nucleon. Results have been calculated with SM, HM, and H potentials as well as in the cascade mode (solid symbols). The results are compared to a baseline from the binomial distribution (open symbols). Error bars for all data shown, are smaller than the symbol sizes.

Figure 4

Time evolution for the cumulants ($C_1,C_2$, and $C_3$) of all baryons at midrapidity produced from central Au+Au collisions at beam energy of 1.23 GeV/nucleon. Results have been calculated with HM and SM potentials as well as in the cascade mode (solid symbols). Error bars are smaller than symbols size.

Figure 5

Time evolution for the cumulants ratios ($C_2$/$C_1,C_3$/$C_1$, and $S\sigma$) of all baryons at midrapidity produced from central Au+Au collisions at beam energy of 1.23 GeV/nucleon. Results have been calculated with HM and SM potentials as well as in the cascade mode (solid symbols). Error bars are smaller than symbols size. The results are compared to the baseline of the binomial distribution (open symbols).

Figure 6

Time evolution for $C_2/C_1$ of all baryons from JAM. Calculations with momentum-dependent hard (MH) and soft (MS) potentials are compared to the result obtained with cascade mode.

Figure 7

Rapidity dependence for the cumulants ($C_1$–$C_3$) of free baryons, i.e. excluding all baryons that are in a cluster, produced in central Au+Au collisions at beam energy of 1.23 GeV/nucleon. Results have been calculated with SM, HM and H potentials as well as in the cascade mode (solid symbols). Error bars for all data shown, are smaller than the symbol sizes.

Figure 8

Rapidity dependence for the cumulant ratios ($C_2$/$C_1,C_3$/$C_1,S\sigma$) of free baryons, i.e., excluding all baryons that are in a cluster, produced in central Au+Au collisions at beam energy of 1.23 GeV/nucleon. Results have been calculated with SM, HM, and H potentials as well as in the cascade mode (solid symbols). Error bars for all data shown, are smaller than the symbol sizes.

Figure 9

The cumulant ratios ($C_2/C_1,C_3/C_1,S\sigma$) of free baryons (solid symbols) and all baryons (open symbols) as a function of $C_1$. When plotted as function of $C_1$, the cumulant ratios for HM and SM parametrizations essentially agree with one another.

Figure 10

Comparison of the cumulant ratios for all baryons vs free baryons. The diagonals are drawn to guide the eye. Mostly, an enhancement of the cumulant ratio with respect to all baryons is observed.

Figure 11

Rapidity dependence for the cumulant ratios of free protons (protons that are not bound in clusters) produced in central Au+Au collisions at beam energy of 1.23 GeV/nucleon. Results have been calculated with SM, HM, and H potentials as well as in the cascade mode. Again, the differences between the cumulants for the different model setups is strongly decreased due to the almost random isospin distribution.