More efficient formulas for efficiency correction of cumulants and effect of using averaged efficiency

We derive formulas for the efficiency correction of cumulants with many efficiency bins. The derivation of the formulas is simpler than the previously suggested method, but the numerical cost is drastically reduced from the naïve method. From analytical and numerical analyses in simple toy models, we show that use of the averaged efficiency in the efficiency correction might cause large deviations in some cases and should not be used, especially for high order cumulants. These analyses show the importance of carrying out the efficiency correction without taking the average.

Figure 1

Number of terms (factorial moments or cumulants) to be calculated in the conventional method based on the factorial moments [14] and the new method in this paper, Eqs. (74) and (75), respectively, for the efficiency correction of the fourth and sixth order cumulants as a function of efficiency bins.

Figure 2

Deviation of the efficiency corrected values of cumulants using averaged efficiency $\mathrm{\Delta }{K}_m$ assuming (a) Gauss distribution and (b) distribution that has 5% smaller cumulants than Poisson distribution.

Figure 3

Deviation of the efficiency corrected $m\mathrm{th}$ order cumulant from input value $\mathrm{\Delta }{K}_m$ up to fourth order. Blue circles represent the results from separated efficiencies, and red squares are results with the averaged efficiency. Parameters for the distribution are $C_1=20$ and $C_2=25$. Efficiencies are ${\varepsilon }_A=0.3$ and ${\varepsilon }_B=0.7$. The result can be directly compared with analytical calculation in Sec. 4.

Figure 4

Relative deviation of efficiency corrected $m\mathrm{th}$ order cumulant from input value $\left(\mathrm{\Delta }{K}_m/K_m\right)$. Blue circles and red squares represent the results from separated and averaged efficiencies, respectively.

Figure 5

Relative deviation of efficiency corrected $m\mathrm{th}$ order cumulant from input value $\mathrm{\Delta }{K}_m/K_m$.