Ghost Track Probability

If one is tracking to the surface of a detector and one assigns the closest hit to the track and there are random extra pileup hits then there is a chance that an extraneous hit will be assigned to the track rather than the true hit. The following is a calculation of the probability that this happens, producing a ghost track. This is for the case that the extra hits are randomly spread uniformly over the surface and that the pointing uncertainty is a Gaussian distribution.

First the bottom line, the probability of false association:

is the projection resolution

is the random hit density on the surface, or

The derivation:

We consider a projected point on a detector surface where the tracker thinks the true hit is located and use this as the origin for the distribution of true hits. In this approach consider that for every true hit there is a probability that there was one or more background hits closer to the projection point (the origin) and sum over all true hits times this probability. This can be done by expressing the true hit distribution as a probability per area and integrating this times the probability for a background hit over the full search area.

For a radial symmetric Gaussian distribution we can reduce the integral to one dimension where the probability of finding the true hit per unit radius as a function of r is:

eq. 1

This distribution is normalized to yield 1 when integrated from 0 to

For a more complicated true hit distribution one would retain the integral over two dimensions.

We now need a probability for finding one or more background hits closer to the origin than the true hit.

For a uniform random distribution of points on a plane the following gives the probability for finding one or more points in an area A, where r is the point density.

( A bit convoluted derivation of this expression can be found at:

http://www-rnc.lbl.gov/~wieman/PileupProbabilityDerivation.htm )

The above can be expressed as the probability of finding one or more points in a circle centered on the origin

eq. 2

Combining eq. 1, the probability density per r for finding the true hit, times eq.2, the probability that inside that true hit one will find one or more background hits, and integrating gives the probability that a background hit gets assigned to the track rather than the correct hit:

eq 3

solving the integral

re-expressing

Let

Also check note:

http://www-rnc.lbl.gov/~wieman/GhostTracksMCsearchRlimited.htm

This gives the formula for a finite search cone plus a Monte Carlo check of the result

An example using the highlighted formula above:

The ideal pointing precision from the SVT to the MVD outer barrel, 1.2 GeV/c p

The pileup hit density on the outer MVD barrel for 4 ms integration with luminosity,

So, although the probability of one or more pileup hits in a window large enough to encompass the true hit with 98% certainty is 10%, the misidentification is only 2.5%