<html><head><title></title><style type="text/css"> p { margin: 0 0 0 0; font-family: Arial, sans-serif; font-size:10pt; font-weight: normal; font-style: normal; vertical-align: baseline; color: black; text-decoration: none; } div { border-width:0px; border-style: solid; border-color: red; } body { background-color: #fff; } p img { vertical-align: middle; } p b, li b { font-weight : bold; } p i, li i { font-style : italic; } p u, li u { text-decoration : underline; } p so, li so { text-decoration : line-through; } p sub, p sub { font-size : 70%; vertical-align: sub; } p sup, p sup { font-size : 70%; vertical-align: super; } ul { padding-left : 2em; /* for mozilla list marker box */ margin-left : 0; /* for IE list marker box */ } ol { padding-left : 2em; /* for mozilla list marker box */ margin-right : -0.5em; margin-left : 0; /* for IE list marker box */ } p.Style1, li.Style1 /*Normal*/ { margin-left: 0pt; margin-right: 0pt; text-align: left; font-family: 'Arial', sans-serif;; font-size: 10pt;font-weight: normal; font-style: normal; vertical-align: baseline; } p.Style2, li.Style2 /*Heading 1*/ { margin-left: 0pt; margin-right: 0pt; text-align: left; font-family: 'Arial', sans-serif;; font-size: 13.26315789473684pt;font-weight: bold; font-style: normal; vertical-align: baseline; } p.Style3, li.Style3 /*Heading 2*/ { margin-left: 0pt; margin-right: 0pt; text-align: left; font-family: 'Arial', sans-serif;; font-size: 12pt;font-weight: bold; font-style: italic; vertical-align: baseline; } p.Style4, li.Style4 /*Heading 3*/ { margin-left: 0pt; margin-right: 0pt; text-align: left; font-family: 'Arial', sans-serif;; font-size: 12pt;font-weight: normal; font-style: normal; vertical-align: baseline; } p.Style5, li.Style5 /*Paragraph*/ { margin-left: 0pt; margin-right: 0pt; text-indent: 21pt; text-align: left; font-family: 'Arial', sans-serif;; font-size: 10pt;font-weight: normal; font-style: normal; vertical-align: baseline; } p.Style6, li.Style6 /*List*/ { margin-left: 14.25pt; margin-right: 0pt; text-indent: -14.25pt; text-align: left; font-family: 'Arial', sans-serif;; font-size: 10pt;font-weight: normal; font-style: normal; vertical-align: baseline; } p.Style7, li.Style7 /*Indent*/ { margin-left: 108pt; margin-right: 0pt; text-align: left; font-family: 'Arial', sans-serif;; font-size: 10pt;font-weight: normal; font-style: normal; vertical-align: baseline; } p.Style8, li.Style8 /*Title*/ { margin-left: 0pt; margin-right: 0pt; text-align: center; font-family: 'Times New Roman', sans-serif;; font-size: 24pt;font-weight: bold; font-style: normal; vertical-align: baseline; } p.Style9, li.Style9 /*Subtitle*/ { margin-left: 0pt; margin-right: 0pt; text-align: center; font-family: 'Times New Roman', sans-serif;; font-size: 18pt;font-weight: normal; font-style: normal; vertical-align: baseline; } </style></head><body><div id="mc-region-773" style="position: absolute; top: 11.25pt; left: 18pt; width: 416pt; "><a name="" /><p class="Style1"><span style="font-size: 13.26315789473684pt; "><b>Probability for correct hit association using minimum weighted chi-square, a Monte Carlo check of the formula, and application to the inner IST layer</b></span></p></div><div id="mc-region-775" style="position: absolute; top: 74.25pt; left: 18pt; width: 57.5pt; "><a name="" /><p class="Style1">H. Wieman</p><p class="Style1">18-Jan-2007</p></div><div id="mc-region-779" style="position: absolute; top: 104.25pt; left: 6pt; width: 48.5pt; "><a name="" /><p class="Style1"><span style="font-size: 10.266666666666667pt; "><b>Abstract</b></span></p></div><div id="mc-region-778" style="position: absolute; top: 122.25pt; left: 6pt; width: 432.5pt; "><a name="" /><p class="Style1">We compare two different Monte Carlo assumptions with a previously derived formula for the probability of correct hit association using the minimum weighted chi-square method of hit finding used in STAR tracking. The comparison is done using parameters which match a possible configuration of the inner IST layer. </p></div><div id="mc-region-781" style="position: absolute; top: 186.75pt; left: 12pt; width: 74.75pt; "><a name="" /><p class="Style1"><span style="font-size: 12pt; "><b>Introduction</b></span></p></div><div id="mc-region-786" style="position: absolute; top: 212.25pt; left: 12pt; width: 424.25pt; "><a name="" /><p class="Style1">This addresses the situation where one has track projections onto a detector surface which has multiple hits. One associates the track with the hits by finding the minimum weighted chi-square when the uncertainly differs in the two directions on the detector plane. This can be done two different ways. </p><p class="Style1"></p><p class="Style1">1. One can simply associate each track with the hit that gives the minimum chi-square distance. This gives the following expression which was <b><u><span style="color: #00f; text-decoration: underline ;"><a href="http://www-rnc.lbl.gov/~wieman/HitFinding2DXsq.htm">derived earlier</a></span></u></b> </p></div><div id="mc-region-821" style="position: absolute; top: 338.25pt; left: 30pt; width: 103.5pt; "><a name="" /><img border="0" src=".\HitFinding2DXsqMC_images/IMG0002_25494203.PNG" id="generatedImage821" path=".\HitFinding2DXsqMC_images/IMG0002_25494203.PNG" style="width: 103.5pt;height: 32.25pt;"></img></div><div id="mc-region-820" style="position: absolute; top: 344.25pt; left: 270pt; width: 161.75pt; "><a name="" /><p class="Style1">probability for correct hit association</p></div><div id="mc-region-826" style="position: absolute; top: 350.25pt; left: 180pt; width: 26.75pt; "><a name="" /><p class="Style1">eq. 1</p></div><div id="mc-region-790" style="position: absolute; top: 393pt; left: 36pt; width: 24.75pt; "><a name="" /><img border="0" src=".\HitFinding2DXsqMC_images/IMG0003_25494203.PNG" id="generatedImage790" path=".\HitFinding2DXsqMC_images/IMG0003_25494203.PNG" style="width: 24.75pt;height: 12.75pt;"></img></div><div id="mc-region-791" style="position: absolute; top: 422.25pt; left: 42pt; width: 12.75pt; "><a name="" /><img border="0" src=".\HitFinding2DXsqMC_images/IMG0004_25494203.PNG" id="generatedImage791" path=".\HitFinding2DXsqMC_images/IMG0004_25494203.PNG" style="width: 12.75pt;height: 13.5pt;"></img></div><div id="mc-region-792" style="position: absolute; top: 422.25pt; left: 84pt; width: 233.75pt; "><a name="" /><p class="Style1">is the density of hits per area on the detector surface</p></div><div id="mc-region-793" style="position: absolute; top: 452.25pt; left: 42pt; width: 10.5pt; "><a name="" /><img border="0" src=".\HitFinding2DXsqMC_images/IMG0005_25494203.PNG" id="generatedImage793" path=".\HitFinding2DXsqMC_images/IMG0005_25494203.PNG" style="width: 10.5pt;height: 17.25pt;"></img></div><div id="mc-region-794" style="position: absolute; top: 452.25pt; left: 84pt; width: 313.25pt; "><a name="" /><p class="Style1">is the Gaussian distribution parameter in the x direction ( x uncertainty)</p></div><div id="mc-region-795" style="position: absolute; top: 482.25pt; left: 42pt; width: 12pt; "><a name="" /><img border="0" src=".\HitFinding2DXsqMC_images/IMG0006_25494203.PNG" id="generatedImage795" path=".\HitFinding2DXsqMC_images/IMG0006_25494203.PNG" style="width: 12pt;height: 17.25pt;"></img></div><div id="mc-region-796" style="position: absolute; top: 482.25pt; left: 84pt; width: 313.25pt; "><a name="" /><p class="Style1">is the Gaussian distribution parameter in the y direction ( y uncertainty)</p></div><div id="mc-region-800" style="position: absolute; top: 506.25pt; left: 84pt; width: 128.75pt; "><a name="" /><p class="Style1">The search window is infinite</p></div><div id="mc-region-799" style="position: absolute; top: 536.25pt; left: 12pt; width: 427.25pt; "><a name="" /><p class="Style1" style="text-indent: 0pt; list-style-type: none;">Note, in this case multiple track projections can share the same hit. When using this method the probability for correct hit association depends only on the hit density, not the track density. This is the preferred method for association since it gives the highest correct probability.</p></div><div id="mc-region-802" style="position: absolute; top: 584.25pt; left: 12pt; width: 428.75pt; "><a name="" /><p class="Style1">2. A second slightly different method, pairs only one hit with one track projection. Namely if a track projection - hit pair gives a minimum chi-square then that pair absorbs the hit and it does not get used in another pair. The expression for this case is:</p></div><div id="mc-region-824" style="position: absolute; top: 656.25pt; left: 30pt; width: 132pt; "><a name="" /><img border="0" src=".\HitFinding2DXsqMC_images/IMG0007_25494218.PNG" id="generatedImage824" path=".\HitFinding2DXsqMC_images/IMG0007_25494218.PNG" style="width: 132pt;height: 32.25pt;"></img></div><div id="mc-region-828" style="position: absolute; top: 662.25pt; left: 198pt; width: 26.75pt; "><a name="" /><p class="Style1">eq. 2</p></div><div id="mc-region-823" style="position: absolute; top: 662.25pt; left: 276pt; width: 161.75pt; "><a name="" /><p class="Style1">probability for correct hit association</p></div><div id="mc-region-804" style="position: absolute; top: 711pt; left: 24pt; width: 24.75pt; "><a name="" /><img border="0" src=".\HitFinding2DXsqMC_images/IMG0008_25494218.PNG" id="generatedImage804" path=".\HitFinding2DXsqMC_images/IMG0008_25494218.PNG" style="width: 24.75pt;height: 12.75pt;"></img></div><div id="mc-region-805" style="position: absolute; top: 740.25pt; left: 30pt; width: 12.75pt; "><a name="" /><img border="0" src=".\HitFinding2DXsqMC_images/IMG0009_25494218.PNG" id="generatedImage805" path=".\HitFinding2DXsqMC_images/IMG0009_25494218.PNG" style="width: 12.75pt;height: 13.5pt;"></img></div><div id="mc-region-806" style="position: absolute; top: 740.25pt; left: 72pt; width: 233.75pt; "><a name="" /><p class="Style1">is the density of hits per area on the detector surface</p></div><div id="mc-region-807" style="position: absolute; top: 776.25pt; left: 30pt; width: 9.75pt; "><a name="" /><img border="0" src=".\HitFinding2DXsqMC_images/IMG0010_25494218.PNG" id="generatedImage807" path=".\HitFinding2DXsqMC_images/IMG0010_25494218.PNG" style="width: 9.75pt;height: 13.5pt;"></img></div><div id="mc-region-808" style="position: absolute; top: 776.25pt; left: 72pt; width: 249.5pt; "><a name="" /><p class="Style1">is the density of projected tracks on the detector surface</p></div><div id="mc-region-809" style="position: absolute; top: 812.25pt; left: 30pt; width: 10.5pt; "><a name="" /><img border="0" src=".\HitFinding2DXsqMC_images/IMG0011_25494218.PNG" id="generatedImage809" path=".\HitFinding2DXsqMC_images/IMG0011_25494218.PNG" style="width: 10.5pt;height: 17.25pt;"></img></div><div id="mc-region-810" style="position: absolute; top: 812.25pt; left: 72pt; width: 313.25pt; "><a name="" /><p class="Style1">is the Gaussian distribution parameter in the x direction ( x uncertainty)</p></div><div id="mc-region-811" style="position: absolute; top: 842.25pt; left: 30pt; width: 12pt; "><a name="" /><img border="0" src=".\HitFinding2DXsqMC_images/IMG0012_25494218.PNG" id="generatedImage811" path=".\HitFinding2DXsqMC_images/IMG0012_25494218.PNG" style="width: 12pt;height: 17.25pt;"></img></div><div id="mc-region-812" style="position: absolute; top: 842.25pt; left: 72pt; width: 313.25pt; "><a name="" /><p class="Style1">is the Gaussian distribution parameter in the y direction ( y uncertainty)</p></div><div id="mc-region-816" style="position: absolute; top: 878.25pt; left: 72pt; width: 99.5pt; "><a name="" /><p class="Style1">search window infinite</p></div><div id="mc-region-818" style="position: absolute; top: 908.25pt; left: 18pt; width: 426.5pt; "><a name="" /><p class="Style1">Since this method (see <b><u><span style="color: #00f; text-decoration: underline ;"><a href="http://www-rnc.lbl.gov/~wieman/HitFinding2DXsqTrackDenTemp.htm">derivation</a></span></u></b>) depends on the sum of both hit and track density it always gives a lower value for the correct association. For this reason we do not consider this further and work now only with the first method.</p></div><div id="mc-region-830" style="position: absolute; top: 966.75pt; left: 12pt; width: 365pt; "><a name="" /><p class="Style1"><span style="font-size: 12pt; "><b>Parameters for Monte Carlo check of eq. 1, using IST inner layer example</b></span></p></div><div id="mc-region-832" style="position: absolute; top: 1004.25pt; left: 12pt; width: 432.5pt; "><a name="" /><p class="Style1">In checking the formula, eq. 1, we use the inner IST layer as an example, namely a detector plane composed of 60 <span style="font-size: 10pt; font-family: 'Symbol', sans-serif;"><span style="font-weight: normal; font-style: normal;">m</span></span><span style="font-size: 10pt; font-family: 'Arial', sans-serif;"><span style="font-weight: normal; font-style: normal;">m wide strips that are 3.8 cm long. There are no pads. The analysis determines the probability of correctly associating hits in this detector plane with projected tracks. This is done as a function of hit density on the detector plane.</span></span> </p></div><div id="mc-region-833" style="position: absolute; top: 1077pt; left: 18pt; width: 29.25pt; "><a name="" /><img border="0" src=".\HitFinding2DXsqMC_images/IMG0013_25494218.PNG" id="generatedImage833" path=".\HitFinding2DXsqMC_images/IMG0013_25494218.PNG" style="width: 29.25pt;height: 12.75pt;"></img></div><div id="mc-region-834" style="position: absolute; top: 1077pt; left: 84pt; width: 12pt; "><a name="" /><img border="0" src=".\HitFinding2DXsqMC_images/IMG0014_25494218.PNG" id="generatedImage834" path=".\HitFinding2DXsqMC_images/IMG0014_25494218.PNG" style="width: 12pt;height: 12.75pt;"></img></div><div id="mc-region-974" style="position: absolute; top: 1106.25pt; left: 6pt; width: 433.25pt; "><a name="" /><p class="Style1">The Monte Carlo hits are distributed on a detector plane that is square with edge length L.</p></div><div id="mc-region-975" style="position: absolute; top: 1154.25pt; left: 6pt; width: 128.75pt; "><a name="" /><p class="Style1">The hit density is varied from</p></div><div id="mc-region-976" style="position: absolute; top: 1155pt; left: 150pt; width: 12pt; "><a name="" /><img border="0" src=".\HitFinding2DXsqMC_images/IMG0015_25494234.PNG" id="generatedImage976" path=".\HitFinding2DXsqMC_images/IMG0015_25494234.PNG" style="width: 12pt;height: 12.75pt;"></img></div><div id="mc-region-977" style="position: absolute; top: 1155pt; left: 186pt; width: 9pt; "><a name="" /><img border="0" src=".\HitFinding2DXsqMC_images/IMG0016_25494234.PNG" id="generatedImage977" path=".\HitFinding2DXsqMC_images/IMG0016_25494234.PNG" style="width: 9pt;height: 12.75pt;"></img></div><div id="mc-region-978" style="position: absolute; top: 1155pt; left: 228pt; width: 5.25pt; "><a name="" /><img border="0" src=".\HitFinding2DXsqMC_images/IMG0017_25494234.PNG" id="generatedImage978" path=".\HitFinding2DXsqMC_images/IMG0017_25494234.PNG" style="width: 5.25pt;height: 12.75pt;"></img></div><div id="mc-region-979" style="position: absolute; top: 1148.25pt; left: 276pt; width: 19.5pt; "><a name="" /><img border="0" src=".\HitFinding2DXsqMC_images/IMG0018_25494234.PNG" id="generatedImage979" path=".\HitFinding2DXsqMC_images/IMG0018_25494234.PNG" style="width: 19.5pt;height: 33.75pt;"></img></div><div id="mc-region-967" style="position: absolute; top: 1226.25pt; left: 30pt; width: 43.5pt; "><a name="" /><img border="0" src=".\HitFinding2DXsqMC_images/IMG0020_25494250.PNG" id="generatedImage967" path=".\HitFinding2DXsqMC_images/IMG0020_25494250.PNG" style="width: 43.5pt;height: 29.25pt;"></img></div><div id="mc-region-970" style="position: absolute; top: 1233pt; left: 78pt; width: 12pt; "><a name="" /><img border="0" src=".\HitFinding2DXsqMC_images/IMG0021_25494250.PNG" id="generatedImage970" path=".\HitFinding2DXsqMC_images/IMG0021_25494250.PNG" style="width: 12pt;height: 12.75pt;"></img></div><div id="mc-region-852" style="position: absolute; top: 1232.25pt; left: 132pt; width: 312.5pt; "><a name="" /><p class="Style1">The resolution along the strip. The strip length sets the resolution since it is much larger than the track projected resolution in this dimension.</p></div><div id="mc-region-859" style="position: absolute; top: 1286.25pt; left: 132pt; width: 312.5pt; "><a name="" /><p class="Style1">The resolution in the transverse direction is dominated by track projection errors which are larger than the width of the IST detector strips. This track projection error of 300 <span style="font-size: 10pt; font-family: 'Symbol', sans-serif;"><span style="font-weight: normal; font-style: normal;">m</span></span><span style="font-size: 10pt; font-family: 'Arial', sans-serif;"><span style="font-weight: normal; font-style: normal;">m on IST inner comes from Jim Thomas. It is for 750 MeV/c Kaons using a detector geometry of TPC, SSD, IST outer at r = 17cm and IST inner at r = 12 cm </span></span></p></div><div id="mc-region-966" style="position: absolute; top: 1292.25pt; left: 30pt; width: 37.5pt; "><a name="" /><img border="0" src=".\HitFinding2DXsqMC_images/IMG0022_25494250.PNG" id="generatedImage966" path=".\HitFinding2DXsqMC_images/IMG0022_25494250.PNG" style="width: 37.5pt;height: 17.25pt;"></img></div><div id="mc-region-971" style="position: absolute; top: 1293pt; left: 78pt; width: 12pt; "><a name="" /><img border="0" src=".\HitFinding2DXsqMC_images/IMG0023_25494250.PNG" id="generatedImage971" path=".\HitFinding2DXsqMC_images/IMG0023_25494250.PNG" style="width: 12pt;height: 12.75pt;"></img></div><div id="mc-region-862" style="position: absolute; top: 1376.25pt; left: 18pt; width: 410pt; "><a name="" /><p class="Style1">The Monte Carlo is thrown in two different ways. In one case the hits are distributed uniformly along the length of the 3.8 cm strips (flat). In the other case a normal distribution (normal) is used with </p></div><div id="mc-region-864" style="position: absolute; top: 1406.25pt; left: 84pt; width: 52.5pt; "><a name="" /><img border="0" src=".\HitFinding2DXsqMC_images/IMG0024_25494250.PNG" id="generatedImage864" path=".\HitFinding2DXsqMC_images/IMG0024_25494250.PNG" style="width: 52.5pt;height: 29.25pt;"></img></div><div id="mc-region-961" style="position: absolute; top: 1462.5pt; left: 12pt; width: 188.25pt; "><a name="" /><img border="0" src=".\HitFinding2DXsqMC_images/IMG0025_25494250.PNG" id="generatedImage961" path=".\HitFinding2DXsqMC_images/IMG0025_25494250.PNG" style="width: 188.25pt;height: 48pt;"></img></div><div id="mc-region-964" style="position: absolute; top: 1478.25pt; left: 246pt; width: 179pt; "><a name="" /><p class="Style1">weighted chi-square function used for hit association</p></div><div id="mc-region-764" style="position: absolute; top: 1561.5pt; left: 18pt; width: 370.5pt; "><a name="" /><img border="0" src=".\HitFinding2DXsqMC_images/IMG0028_25494265.PNG" id="generatedImage764" path=".\HitFinding2DXsqMC_images/IMG0028_25494265.PNG" style="width: 370.5pt;height: 396pt;"></img></div><div id="mc-region-972" style="position: absolute; top: 1976.25pt; left: 12pt; width: 432.5pt; "><a name="" /><p class="Style1">Fig. 1 A comparison of estimates for probability of correct hit to track association as a function of hit density. This is for a structure composed of 3.8 cm long strips where the track projection limits the resolution in one direction and the strip length determines the resolution in the other direction. Note, the two Monte Carlo checks agree with the simple formula of eq. 1. The formula is derived assuming a normal distribution for the hits, but this does not seem to matter in this application. The Monte Carlos, one with a flat distribution and one with a normal distribution, gives the same result in agreement with the formula. This may be because the low resolution direction has very little effect and differences are hidden. It will be interesting to do the test with a less asymmetrical case.</p><p class="Style1"></p><p class="Style1">The vertical lines mark two different estimates (see below) for the hit density at r = 12 cm for central U+U at 200 GeV. In one case the probability for correct hit association is 88% and in the other the association drops to 82%. This in any case implies that simple strips for IST inner is a viable solution for tracking.</p></div><div id="mc-region-973" style="position: absolute; top: 2168.25pt; left: 6pt; width: 203.75pt; "><a name="" /><p class="Style1">The estimates of hit densities for central U+U:</p></div><div id="mc-region-910" style="position: absolute; top: 2210.25pt; left: 6pt; width: 47.75pt; "><a name="" /><p class="Style1">The result</p></div><div id="mc-region-911" style="position: absolute; top: 2204.25pt; left: 96pt; width: 85.5pt; "><a name="" /><img border="0" src=".\HitFinding2DXsqMC_images/IMG0029_25494265.PNG" id="generatedImage911" path=".\HitFinding2DXsqMC_images/IMG0029_25494265.PNG" style="width: 85.5pt;height: 33.75pt;"></img></div><div id="mc-region-908" style="position: absolute; top: 2258.25pt; left: 6pt; width: 388.25pt; "><a name="" /><p class="Style1">is based on <b><u><span style="color: #00f; text-decoration: underline ;"><a href="http://www.star.bnl.gov/~fisyak/star/Alignment/Pass49/TpcSsd/RevisedOccupancy.html">Yuri Fisyak's work</a></span></u></b> with the SVT in Cu+Cu measurements.</p></div><div id="mc-region-923" style="position: absolute; top: 2300.25pt; left: 6pt; width: 72.5pt; "><a name="" /><p class="Style1">The other result</p></div><div id="mc-region-922" style="position: absolute; top: 2300.25pt; left: 120pt; width: 43.5pt; "><a name="" /><img border="0" src=".\HitFinding2DXsqMC_images/IMG0030_25494265.PNG" id="generatedImage922" path=".\HitFinding2DXsqMC_images/IMG0030_25494265.PNG" style="width: 43.5pt;height: 17.25pt;"></img></div><div id="mc-region-921" style="position: absolute; top: 2294.25pt; left: 180pt; width: 19.5pt; "><a name="" /><img border="0" src=".\HitFinding2DXsqMC_images/IMG0031_25494265.PNG" id="generatedImage921" path=".\HitFinding2DXsqMC_images/IMG0031_25494265.PNG" style="width: 19.5pt;height: 33.75pt;"></img></div><div id="mc-region-920" style="position: absolute; top: 2300.25pt; left: 222pt; width: 56pt; "><a name="" /><p class="Style1">comes from</p></div><div id="mc-region-924" style="position: absolute; top: 2336.25pt; left: 12pt; width: 41.25pt; "><a name="" /><img border="0" src=".\HitFinding2DXsqMC_images/IMG0032_25494265.PNG" id="generatedImage924" path=".\HitFinding2DXsqMC_images/IMG0032_25494265.PNG" style="width: 41.25pt;height: 28.5pt;"></img></div><div id="mc-region-926" style="position: absolute; top: 2342.25pt; left: 96pt; width: 314.75pt; "><a name="" /><p class="Style1">for 200 GeV Au+Au and a <b><u><span style="color: #00f; text-decoration: underline ;"><a href="http://arxiv.org/pdf/nucl-th/0411054">1.5 multiplier</a></span></u></b> for U+U and a radius of 12 cm</p></div><div id="mc-region-931" style="position: absolute; top: 2383.5pt; left: 12pt; width: 234pt; "><a name="" /><img border="0" src=".\HitFinding2DXsqMC_images/IMG0033_25494296.PNG" id="generatedImage931" path=".\HitFinding2DXsqMC_images/IMG0033_25494296.PNG" style="width: 234pt;height: 36pt;"></img></div><div id="mc-region-930" style="position: absolute; top: 2390.25pt; left: 288pt; width: 44pt; "><a name="" /><p class="Style1">for z = 0</p></div><div id="mc-region-731" style="position: absolute; top: 2459.25pt; left: 24pt; width: 361.5pt; "><a name="" /><img border="0" src=".\HitFinding2DXsqMC_images/IMG0036_25494296.PNG" id="generatedImage731" path=".\HitFinding2DXsqMC_images/IMG0036_25494296.PNG" style="width: 361.5pt;height: 339pt;"></img></div><div id="mc-region-937" style="position: absolute; top: 2804.25pt; left: 12pt; width: 432.5pt; "><a name="" /><p class="Style1">Fig. 2 Hit and track positions as they appear in the Monte Carlo with a flat distribution in the z direction. This piece of the plane is magnified in the r phi direction, the direction in which the resolution is good. This is the direction that is instrumental in making the hit - track projection association. The hit density is 0.5 hits/cm<sup>2</sup> </p></div><div id="mc-region-939" style="position: absolute; top: 2874.75pt; left: 12pt; width: 77.75pt; "><a name="" /><p class="Style1"><span style="font-size: 12pt; "><b>Conclusions</b></span></p></div><div id="mc-region-941" style="position: absolute; top: 2900.25pt; left: 12pt; width: 412.25pt; "><a name="" /><p class="Style1">The formula works well for determining probability of correct hit association using the minimum weighted chi-square method.</p></div><div id="mc-region-944" style="position: absolute; top: 2942.25pt; left: 12pt; width: 428pt; "><a name="" /><p class="Style1">Correct hit association is improved by not limiting hits to a single pair, i.e. by using the method for which formula, eq. 1, applies.</p></div><div id="mc-region-948" style="position: absolute; top: 2984.25pt; left: 12pt; width: 432.5pt; "><a name="" /><p class="Style1">The formula for association probability works for cases where the hit distribution is uniform over a long dimension. </p></div><div id="mc-region-952" style="position: absolute; top: 3020.25pt; left: 12pt; width: 407.75pt; "><a name="" /><p class="Style1">Hit to track association can be expected to work reasonably well at the inner IST layer for the highest expected hit densities.</p></div><div id="mc-region-954" style="position: absolute; top: 3060.75pt; left: 12pt; width: 74pt; "><a name="" /><p class="Style1"><span style="font-size: 12pt; "><b>Future work</b></span></p></div><div id="mc-region-956" style="position: absolute; top: 3080.25pt; left: 12pt; width: 422.75pt; "><a name="" /><p class="Style1">Check the probability formula for cases with uniform distributions where the dimensions are more uniform, that is for squares instead of long strips.</p></div><div id="mc-region-958" style="position: absolute; top: 3122.25pt; left: 12pt; width: 428pt; "><a name="" /><p class="Style1">Investigate a pad strip configuration for the IST where ghosting increases the density of hits, but the pads significantly improve the resolution in the z direction. This may require a calculation with hit densities that are different in the two directions.</p></div><div></div></body></html>