Properties and uses of factorial cumulants in relativistic heavy-ion collisions

We discuss properties and applications of factorial cumulants of various particle numbers and for their mixed channels measured by event-by-event analysis in relativistic heavy-ion collisions. After introducing the factorial cumulants for systems with multiparticle species, their properties are described. The uses of factorial cumulants in the study of critical fluctuations are discussed. We point out that factorial cumulants play useful roles in understanding fluctuation observables when they have underlying physics approximately described by the binomial distribution. As examples, we suggest novel utilization methods of the factorial cumulants in the study of the momentum cut and rapidity window dependences of fluctuation observables.

Figure 1

Relation between moments, cumulants, factorial moments, and factorial cumulants. The relations between moments and factorial moments are equivalent to those between cumulants and factorial cumulants [arrows (a)]. Also, the relations between moments and cumulants are equivalent to those between factorial moments and factorial cumulants [arrows (b)].

Figure 2

Function $F_m\left(X\right)$ defined in Eq. (121), expressing the dependence of all factorial cumulants on the experimental accept-ance $\mathrm{\Delta }y$.

Figure 3

The upper panel shows $F_m\left(X\right)$ divided by $\mathrm{\Delta }{y}^{m-1}$. In the lower panel the same function is shown in the normalization $[F_m\left(X\right)/\mathrm{\Delta }{y}^{m-1}]/F_m\left(0\right)$. Dependences of all factorial cumulants on $\mathrm{\Delta }y$ are proportional to $F_m\left(X\right)$. The expansions of $F_m\left(X\right)$ up to $X^2$, Eq. (122), are also shown by dashed lines.