Event-by-Event Efficiency Fluctuations and Efficiency Correction for Cumulants of Superposed Multiplicity Distributions in Relativistic Heavy-ion Collision Experiments.

We performed systematic studies for the effects of event-by-event efficiency fluctuations on efficiency correction for cumulant analysis in relativistic heavy-ion collision experiments. Experimentally, particle efficiencies of events measured with different experimental conditions should be different. For fluctuation measurements, the final event-by-event multiplicity distributions should be the superposed distributions of various type of events measured under different conditions.We demonstrate the efficiency fluctuation effects in a numerical simulation, in which we construct an event ensemble consists of events with two different efficiencies. By using the mean particle efficiencies, we find the efficiency corrected cumulants show large deviations from the original inputs when the discrepancy of the two efficiencies are large. We further studied the effects of efficiency fluctuations for the cumulants of net-proton distributions by implementing the UrQMD events of Au+Au collisions at $\sqrt{s_{NN}}=7.7$ GeV in a realistic STAR detector acceptance. We consider the unequal efficiency in two sides of the Time Projection Chamber (TPC), multiplicity dependent efficiency and the event-by-event variations of the collision vertex position along the longitudinal direction ($V_\mathrm{z}$). When the efficiencies fluctuate dramatically within the studied event sample, the effects of efficiency fluctuations have significant impacts on the efficiency corrections of cumulants with the mean efficiencies. We find that this effect can be effectively suppressed by binning the entire event ensemble into various sub-event samples, in which the efficiency variations are relative small. The final efficiency corrected cumulants can be calculated from the weighted average of the corrected factorial moments of the sub-event samples with the mean efficiencies.