Vertex from Pseudo-tracking

Another method used to find the vertex is based on "pseudo-tracking." This method treats all pairs of hits in the inner and outer barrels as potential tracks and calculates a vertex position from them. The real vertex appears as the most probable value of the vertex position.

Figure 6: Probability vs. the difference in the parallel strip in the outer barrel which was hit and the strip that would have been hit with no multiple scattering. All charged particles which hit both barrels are included.

There are two stages in the pseudo-tracking vertex search. First, only the parallel strips are used, obviating the need to test all pairs of hits as each parallel strip covers a small (azimuthal angle). Particles from the central axis of the vertex detector have the same at each barrel, except for small variations due to multiple scattering. Fig. 6 shows a distribution of the change in the strip number hit in the outer barrel due to multiple scattering in the inner barrel; most particles hit the outer barrel within 2 strips of the expected position. For each hit in the outer barrel, all hits in the inner barrel which are within 2 strips are used to calculate a possible vertex position. The top part of fig. 7 shows the resulting distribution of vertices for a sample Au+Au event. An estimate of the event vertex appears as a peak. The peak height gives the number of particles hitting both barrels of this half of the detector. The background comes from random pairs of hits. Limiting the hits tested to 2 strips reduces the number of pairs that must be tested by a factor of = . This reduces the background in the histograms shown in the upper half of fig. 7 by while only reducing the counts in the peak by about 12% (see fig. 6).

Figure 7: Au+Au - pseudo-tracking example. Top shows the distribution of vertices using parallel strips only. Bottom shows the distribution of vertices using the perpendicular strips. Arrows mark the true vertex and the vertex found in each stage.

Because the peak found using pseudo-tracking with parallel strips is broad, the vertex position is estimated by taking the center of gravity of three bins around the maximum. Because the strips are long (5cm) in the z direction, this method can not give very good vertex resolution. Fig. 8 shows the vertex resolution for p+p, p+Au, and Au+Au using pseudo-tracking with parallel strips only. For p+p and p+Au, this method gives better resolution than the center of (CG) and is much less noise sensitive. However, pseudo-tracking with parallel strips alone still does not give a vertex resolution for p+p which is significantly better than the variation in the vertex position itself⁵.

Figure 8: The vertex resolution using pseudo-tracking with parallel strips only for p+p, p+Au minimum bias, and Au+Au central. The horizontal axis is the difference the true vertex and the vertex found.

This first stage using parallel strips is used to estimate a vertex position. The second stage of the vertex search uses perpendicular strips which are short in the z direction (100), and determine the vertex much more accurately. Beginning with an approximate vertex reduces the range of vertex positions to be searched and increases the speed of the algorithm. The second stage of the vertex search is similar to the first, but as each perpendicular strip occupies a large , the number of pairs of hits that must be tested is about N₁N₂/3, where 3 is the number of different azimuthal segments with perpendicular strips and N₁and N₂ are the number of hits on the perpendicular strips in the inner and outer barrels, respectively. For central Au+Au events, this number is large, so the algorithm is slow. The vertex from the first stage of pseudo-tracking is used to restrict the pairs of strips which are tested; for Au+Au, only those pairs of strips which point to a vertex position within 5cm of the vertex found in the first stage of pseudo-tracking are tested. For p+p and p+Au, this range is expanded to 10cm. A histogram of vertex positions is calculated from the pairs of hits. An example of one of these histograms is shown on the bottom part of fig. 7. The vertex position appears as the peak in this distribution.

Figure 9: Notice the difference in scale from fig. 8. The vertex resolution using pseudo-tracking with both parallel and perpendicular strips for p+p, p+Au minimum bias, and Au+Au central. ^*Due to the bin size used, the RMS deviations are upper limits.

Fig. 9 shows the vertex resolution using both stages of pseudo-tracking. The correct vertex is found in all events tested for central Au+Au collisions. For p+p and Au+Au the correct vertex is usually found. Table 3 summarizes the efficiency of pseudo-tracking vertex search for different assumed levels of noise for the three systems. "Total events" and "triggers" are the total number of Monte Carlo events and the number of those that satisfied the "trigger" condition - at least two charged particles hitting both cylinders of the vertex detector. The column labeled "% of triggers" gives the efficiency of the vertex search algorithm - the fraction of the events for which the vertex found was within 5mm of the true vertex. The last column gives the resolution of the vertex finding algorithm based on the widths of the peaks in fig. 9. These widths are upper limits due to the size of the bins used in the pseudo-tracking algorithm. Especially for p+p collisions, noise has a significant effect on the vertex finding efficiency.

system	Pnoise	Total events	Triggers	% of triggers	(mm)
p+p	0.0001	2000	1699	91%	0.4
p+p	0.0003	2000	1699	86%	0.4
p+p	0.0010	2000	1699	71%	0.4
p+Au	0.0001	2000	1921	97%	0.3
p+Au	0.0003	2000	1921	94%	0.3
p+Au	0.0010	2000	1921	90%	0.3
Au+Au	0.001	150	150	100%	0.2

Table 3: Efficiency of pseudo-tracking vertex search vs. assumed level of noise for p+p, p+Au, Au+Au. Interaction diamond assumes = 5.7, 16, 20cm for p+p, p+Au, Au+Au, respectively. See text for explanation of columns.

Figure 10: p+Au minimum-bias events from Fritiof. The probability that the correct vertex will be found vs. the number of charged particles that hit both layers of the vertex detector for three assumptions about the noise level. Notice the differences in efficiency for the lowest multiplicity, where the effects of noise are most important

Fig. 10 demonstrates one source of problems with the pseudo-tracking vertex search --- when the multiplicity is very low, the probability of finding the vertex is also low and this probability gets smaller for higher noise levels. Higher noise levels have a significant effect on the vertex finding efficiency at low multiplicity (below 10), but the efficiency reaches 100% in each case for sufficiently high multiplicity. In an ideal case, with no noise, 100% efficiency, and no multiple scattering, the algorithm would work for even one charged particle hitting both layers of the detector.

Table 4: Efficiency of pseudo-tracking vertex search for p+p, p+Au, Au+Au with only 2 azimuthal segments of perpendicular strips used and a length of 64cm. Interaction diamond assumes = 5.7, 16, 20cm for p+p, p+Au, Au+Au, respectively. Compare this to table 4, using the full detector.

A vertex detector like the one described in the Tales/Sparhc letter of intent⁷ which had only 2 azimuthal segments of strips perpendicular to the beam, instead of 3, and was 64 cm long, instead of 100cm, could still find the vertex, but with reduced efficiency. Table 4 shows the expected vertex finding efficiency for this detector configuration. The efficiencies are smaller (compare to table 3), especially for p+p, but if the cost savings are large enough, the efficiency loss may be acceptable. Some efficiency is lost when the vertex is outside the shorter detector. However, tests with the full detector configuration show that the pseudo-tracking algorithm can find the vertex in central Au+Au collisions in 94 out of 100 events even when it is 50cm outside of the detector (100cm from the center of the detector), although the resolution falls to 2mm.