Factorial moments may be used to search for intermittency in events [Bia86]. The whole field was much studied in the late eighties and early nineties, and a host of different measures have been proposed. We only implement one of the original prescriptions.
To calculate the factorial moments, the full rapidity (or pseudorapidity) and azimuthal ranges are subdivided into bins of successively smaller size, and the multiplicity distributions in bins is studied. The program calculates pseudorapidity with respect to the axis; if desired, one could first find an event axis, e.g. the sphericity or thrust axis, and subsequently rotate the event to align this axis with the direction.
The full rapidity range
(or pseudorapidity range
are subdivided into and
equally large bins. In fact, the whole analysis is
performed thrice: once with and the (or )
divided into 1, 2, 4, 8, 16, 32, 64, 128, 256 and 512 bins, once
with and the range subdivided as above, and
finally once with
according to the same binary
sequence. Given the multiplicity in bin , the :th
factorial moment is defined by
The as given here are defined for the individual event, and have to be averaged over many events to give a reasonably smooth behaviour. If particle production is uniform and uncorrelated according to Poisson statistics, one expects for all moments and all bin sizes. If, on the other hand, particles are locally clustered, factorial moments should increase when bins are made smaller, down to the characteristic dimensions of the clustering.