Importance of separated efficiencies between positively and negatively charged particles for cumulant calculations

We show the importance of separated efficiency corrections between positively and negatively charged particles for cumulant calculations by toy models and analytical calculations. Our results indicate that $S\sigma$ in published net-proton results from the STAR experiment will be suppressed about 10% in central collisions and 20% in peripheral collisions at the beam energy of $\sqrt{s_{NN}}=200\mathrm{GeV}$ if the separated efficiencies are used to perform the efficiency correction, while no sizable efffect can be seen for $\kappa {\sigma }^2$.

Figure 1

(a) Correlation between positively and negatively charged particles. (b) Net-charge distribution calculated from (a). (c) Correlation between positively and negatively charged particles, which are randomly sampled from (a) according to binomial efficiency. (d) Net-charge distribution calculated from (c).

Figure 2

Deviation of cumulants from input value assuming (a) $\sqrt{s_{NN}}=200\mathrm{GeV}$ and (b) $\sqrt{s_{NN}}=7.7\mathrm{GeV}$, where ${\mu }_{\pm }$ and ${\varepsilon }_{\pm }$ were taken from [4, 10]. $K_n$ represents the $n\mathrm{th}$-order corrected cumulant and $B_n$ represents the Skellam baseline of the $n\mathrm{th}$-order cumulant.

Figure 3

Deviation of (a) $K_1$, $K_2$, and $K_3$; (b) $S\sigma$ and $\kappa {\sigma }^2$; (c) $S\sigma /\mathrm{Skellam}$ from input values as a function of beam energy. They are defined in Eqs. (31, 32, 33, 34). Closed symbols represent the result assuming 0–5% central collisions, and open symbols represent the result assuming 70–80% peripheral collisions. Dashed lines represent the baseline for the efficiency correction ($=0$).

Figure 4

Skellam baselines of (a) odd- and even-order cumulants and (b) $S\sigma$, $\kappa {\sigma }^2$, and $S\sigma /\mathrm{Skellam}$ as a function of beam energy. Closed symbols represent the results assuming 0–5% central collisions, and open symbols represent the results assuming 70–80% peripheral collisions. Baselines of $\kappa {\sigma }^2$ and $S\sigma /\mathrm{Skellam}$ are represented as a solid blue line since they are unity by definition.

Figure 5

Relative deviation of (a) $K_1$, $K_2$, and $K_3$; (b) $S\sigma$ and $\kappa {\sigma }^2$; (c) $S\sigma /\mathrm{Skellam}$ from input values as a function of beam energy, which corresponds to the ratio of Fig. 3 to Fig. 4. Closed symbols represent the results assuming 0–5% central collisions, and open symbols represent the result assuming 70–80% peripheral collisions. Dashed lines represent the baseline for efficiency correction ($=0$).

Figure 6

Relative deviation of results corrected by using averaged efficiency and weighted averaged efficiency as a function of beam energy. The panels show (a) $K_1$, $K_2$, and $K_3$ and (b) $S\sigma$ and $\kappa {\sigma }^2$. For $S\sigma$ and $\kappa {\sigma }^2$ the results of weighted averaged efficiency are overlapped with the results of averaged efficiency. Dashed lines represent the baseline for efficiency correction ($=0$).