Efficiency corrections for factorial moments and cumulants of overlapping sets of particles

Abstract

In this note we discuss subtleties associated with the efficiency corrections for measurements of off-diagonal cumulants and factorial moments for a situation when one deals with overlapping sets of particles, such as correlations between numbers of protons and positively charged particles. We present the efficiency correction formulas for the case when the detection efficiencies follow a binomial distribution. I. INTRODUCTIONFluctuations of conserved charges as a probe of the phase structure of strongly interacting matter have recently received considerable interest theoretically as well as experimentally (for a recent review, see [1]). For example, higher order cumulants of the net baryon density are sensitive to the existence of a critical point [2], and may also provide insights about the chiral criticality governing the cross-over transition at vanishing baryochemical potential [3]. The so-called off-diagonal cumulants, i.e. correlations between two different conserved charges, such as baryon number and strangeness, on the other hand, provide insight about the effective degrees of freedom in the medium [4,5].Experimentally, these cumulants are measured by analyzing event-by-event distributions of particles produced in heavy-ion collisions. For these measurements to reveal the true fluctuations of the system created in these collisions, one needs to take into account and remove fluctuations induced by the detector measurement process itself. These detector induced fluctuations, often referred to as efficiency fluctuations [6-8], arise from the finite detection probability W D (n, N ) of an actual detector, where W D (n, N ) is the probability to observe n particles given N ≥ n particles in an event. The probability distribution of observed particles, p(n), is related to the distribution of true particles, P (N ), by(1)Consequently, the cumulants of the observed distribution p(n) differ from those of the true distribution, P (N ). Therefore, an unfolding procedure is needed, which mathematically corresponds to finding the inverse of W D (n, N ). This is not an easy task in general (see discussion e.g. in [9]). However, if W D (n, N ) can be approximated by a binomial distribution -a reasonably good approximation in a number of cases (see [10,11]) -the relevant formulas for the efficiency corrections of cumulants can and have been derived [6,7,12,13]. However, certain subtleties arise when efficiency corrections are performed for off-diagonal cumulants, such as the correlation of net-proton or net-kaon number with the net-charge number. These subtleties have not yet been addressed in the literature. It is the purpose of this note to discuss and provide the necessary efficiency correction formulas. These may be useful for the ongoing and future heavy-ion experiments, in particular as an effort to measure such off-diagonal cumulants is underway (see e.g. [14]).

Overview of intermittency analysis in heavy-ion collisions

“…Here W (n|N ) is the probability of experimentally measured n particles, given N originally produced particles in the event. In general, W (n|N ) can be approximated by a binomial distribution as: [72][73][74]…”