Calculating particle correlators with the account of detector efficiency

The formulae for m-order correlators K_m of a given particle observable (e.g., energy, transverse momentum or a conserved discrete quantum number) accounting for the track reconstruction efficiencies in a real detector are presented. The calculation of second- to fourth-order correlators is considered in some detail. Similar to the case of an ideal detector, the correlators can be expressed through the event-by-event fluctuation measures of the observable single event mean, the pseudocorrelators (determined by the pseudo-central moments of the observable distribution) and their cross terms. It allows one to avoid the combinatorics and essentially reduce the computer time when calculating the higher-order correlators in high multiplicity events. Compared with the case of ideal detector, this reduction is somewhat smaller due to the increased number of pseudocorrelators and additional calculations of the moments of the distribution of the track weights. For a constant track reconstruction efficiency, the correlator formulae reduce to those for an ideal detector. However, in real experiments the efficiencies are usually essentially dependent on particle momenta and may lead to substantial corrections of momentum correlators on the level of tens of percent.