The most interesting of the vertex finding methods tested here is based on correlations between the pattern of hits on the inner and outer detectors. This method uses a single row of chips on the inner barrel and the corresponding coverage on the outer barrel, or 1/6th of the total circumference.
Figure 11: Correlation function for 7 adjacent chips in the vicinity of the vertex. Realistic case --- multiple strips turned "on" by slanting tracks, multiple scattering on, R2/R1 = 1.5. The peak now occupies more than one channel, but it is still easily found.
When an interaction occurs, tracks project outward from the vertex, producing a pattern of hit strips on the inner and outer barrel. To first order, the pattern on the outer barrel is equal to that on the inner barrel except that all distances between hit strips are increased by a factor of (R2/R1). If we take the pattern on the outer barrel, and shrink it by a factor of R1/R2, we would be able to take this new pattern, and slide it along the inner barrel until there is a perfect match between the hit patterns on the two barrels. We search for the match by forming the correlation function between the patterns as a function of relative position z. For any z we multiply ("and" in hardware) the value of the strip on the inner barrel (1=hit, 0=no hit) by the value of the overlying strip of the outer barrel, and sum these values for all strips. For a randomly chosen relative offset this sum will be small (to first order equal to the multiplicity in the segment of the outer barrel tested times the occupancy of the inner barrel), but when the patterns match, the sum will be equal to the multiplicity in the ideal case. The value of z in this case translates directly into the event vertex position with a resolution equal to one strip width.
A hardware implementation to deliver this vertex position would execute the calculation described above on each pair of inner/outer chips in parallel. Thus one of the 20 pairs finds the vertex, all others turn up null answers. This algorithm works for central Au+Au events, but for the lower multiplicities of p+p and p+Au collisions, there are not enough hits per chip to reliably find the vertex. However, offline the algorithm could be extended to use all hits in all chips, and the method would also work for p+Au and p+p collisions. In this limit the algorithm's efficiency would be similar to that of pseudo-tracking.
A series of tests were done for R1 = 6cm and R2 = 12cm. Using central Au+Au events for the ideal case where each track turns on only one strip and ignoring multiple scattering, the algorithm finds the correct vertex in 20/20 events with a (peak)/(average background) ratio of about 3/1. Allowing each track to turn on multiple strips due to its angle of incidence increases the apparent occupancy far from the vertex, and the algorithm never finds the vertex in this case. However, when clusters of contiguous hits were replaced by a single hit, the algorithm finds the vertex in 20/20 cases again, still assuming no multiple scattering. The peak/background ratio remained around 3/1. Including multiple scattering spreads hits across neighboring strips, and reduces the signal without changing the background. In this case the algorithm found the correct vertex 17/20 times with a typical (peak)/(average background) ratio of about 2/1. The last step was to reduce the radius of the outer barrel to R2= 1.5R1, which matches the current detector design. This reduces the effect of multiple scattering, and the correct vertex is found in 19/20 cases with a typical (peak)/(average background) ratio slightly larger than 2. An example of the resulting correlation function is shown in fig. 11, for 7 chips centered on the chip over the vertex. The channel corresponding to the vertex appears as the maximum value. Increasing the number of adjacent channels used to calculate the correlation function would improve the peak to background ratio.
The correlation method is much faster than the pseudo-tracking method. However, because it requires the patterns of hits to line up exactly in the two barrels, it is more sensitive to multiple scattering than the pseudo-tracking method. The on-line version of the correlation method requires at least a few tracks going into the chip over the vertex, which does not generally happen for p+p and p+Au events. The offline version of the correlation method, using all strips, found the correct vertex in 18/20 events for p+Au, which is similar to the efficiency of the pseudo-tracking method.