Документ взят из кэша поисковой машины. Адрес оригинального документа : http://zebu.uoregon.edu/~uochep/talks/talks06/vlcws-sinev.pdf Дата изменения: Tue Aug 1 23:43:08 2006 Дата индексирования: Tue Oct 2 11:41:45 2012 Кодировка: Поисковые слова: equinox

LCWS06, Vancouver, BC, July 19-23, 2006

Tracking resolution,fitting and detector optimization
Nikolai Sinev, University of Oregon


Topics to be discussed
Limits on resolution for track parameters Different track fitting methods Discussion of results of track fitting with SLD-like fitter (weight matrix based) Comparison with simple circle fitter Effects of detector design Conclusions
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Limits on resolution
We may have perfect spatial resolution of tracking sensors, however still have limited resolution of the tracking system because of multiple scattering. According to Keisuke Fujii ,
/=(C/LB)10/7(X/X0) Here =333.56 cm·T·Gev-1, C=0.0141 GeV, (X/X0) total amount of material expressed in rad. len. L ­ lever arm (cm) B ­ magnetic field (T).
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SiD detector parameters
On the right amount of material and lever arm is shown for our SiD detector (sid00) as function of Tangent Lambda

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Limits on curvature resolution, applying Keisuke Fujii formula

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Numerical values of hit displacement due to MS (mm)
Total track momentum Barrel middle Barrel outer End cap End cap mid.out mid.in End cap out out End cap out in

1 GeV 5 GeV 20 GeV

2.8 0.44 0.1

6.7 0.91 0.21 0.04

4.0 0.78 0.19

3.5 0.68 0.16

7.5 1.47 0.36 0.072

6.4 1.25 0.6 0.12

100 GeV 0.02
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0.038 0.032

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Different track fitting methods
Simple circle fitting with weights defined by MS + detector resolution Simple circle fitting with equal weights for all layers Weight matrix based fitting (SLD fitter)
With MS+detector resolution weights With equal weights include / exclude correlations btw layers Include dE/dx energy loss. IP constrained Best
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Effect of IP constraining

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Best Fit Procedure
From the previous slide we can derive procedure for best estimation of track parameters at point of origin for tracks originated at IP:
1.Use unconstrained fit to estimate track parameters outside beam pipe, and use this parameters to calculate full momentum of the track. 2.Use constrained fit for best estimate of track's direction at IP. Combine full momentum from first fit and direction from second to get best estimation of track at IP
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Comparison of different fitting methods.
For long time I was puzzled by the fact, that weight matrix fitter does not improve curvature measurement compare to circle drawn through 3 points. I suspected bug in the program. So I compared it's results with results of non-iterative circle fitter algorithm, developed by V.Karimaki in 1991, and encoded into JAS by Norman Graf.
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Comparison of different fitting methods - continue
It appeared, that both methods give absolutely the same results if we disable energy loss corrections in WM based fitter (as it is not available in simple circle fitter) and remove correlation terms in weight matrix (again as it is not included in simple circle fitter). And if measurement errors are MS dominated, such fitting gives worse estimation of the curvature, than just circle drawn through 3 points (no fitting case).
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Comparison of different fitting methods - continue
The weight matrix based fitter gives better curvature estimation if we include energy loss correction (that is true only for very low momentum tracks), and correlation between measurements in different layers because of MS. However, at best this leads to the same accuracy in curvature, as in no-fitting case. The best accuracy in the curvature both fitters can achieve if we set equal weights for all layers. In that case curvature estimation is better than no-fit case (though not much, by 10-20 % only). Other track parameters (like impact parameter and directions) have much worse errors in that case, however.
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Comparison of different fitting methods - continue
Fitting with equal weights gives best results only when errors are dominated by MS. As soon as detector resolution became comparable with MS errors, equal weights fitting looses it's advantage. (Practically at 50GeV total momentum) .

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Best fit momentum accuracy

Comparison of fitted Pt/Pt with expected from K.Fujii formula

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Discussion of momentum resolution
As seen from previous slide, our real resolution appears better than expected. There may be a couple of reason for this:
1.K.Fujii assumed scattering media filling detector volume uniformly. Results may be different for material concentrated in few dense shells. 2.As we used method based on track measurement outside beam pipe, we should exclude beam pipe material. It is not done in these calculations, all materials in tracking volume are included.
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Effect of sensors resolution
Next few slides will present residuals in different track parameters for different track momentums and deep angles. As can be seen from hits displacement due to MS calculations (see slide 4), even at 100 GeV we still have MS as a major contributor to momentum measurements inaccuracy. Vertex detector resolution starts affecting impact parameters residuals at as low track momentum as ~3 Gev. Because of lack of time, I will present results mostly for "ideal" detector with perfect sensors resolution, and only for 50 and 100 GeV give comparison with real SiD detector parameters.

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Pt and full momentum residuals for 1 GeV tracks
Tangent lambda / unfitted / fitted P/P unfitted P/P fitted

<1.2

1.2-2.

2.-3.

3.-5.

0.0028 0.0069 0.01

0.014

0.0022 0.0031 0.0043 0.0074 0.0027 0.004 0.0054 0.0077

0.0022 0.0031 0.0043 0.0071
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Pt and full momentum residuals for 5 GeV tracks
Tangent lambda / unfitted / fitted P/P unfitted P/P fitted

<1.2

1.2-2.

2.-3.

3.-5.

0.0021 0.0033 0.0046 0.0072 0.0020 0.0031 0.004 0.0066

0.0021 0.0031 0.0042 0.0064 0.0021 0.0031 0.0041 0.0066
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Pt and full momentum residuals for 20 GeV tracks
Ideal sensors Tangent lambda / unfitted / fitted P/P unfitted P/P fitted SiD real sensors

1.3-2

2.-3.

1.3-2.

2.-3.

0.003

0.0039 0.003

0.004

0.0029 0.0038 0.0029 0.004 0.0029 0.0039 0.0029 0.0039 0.0029 0.0038 0.0029 0.0038
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Pt and full momentum residuals for 100 GeV tracks
Ideal sensors Tangent lambda / unfitted / fitted P/P unfitted P/P fitted SiD real sensors

0.75

0.75

0.0019 0.0022 0.0019 0.0022
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0.0042 0.0032 0.0042 0.0032
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Impact parameters and direction residuals for 1 GeV tracks
Tangent lambda d0 unfitted d0 fitted 0 unfitted 0 fitted Tan() unfitted Tan() fitted

<1.2

1.2-2.

2.-3.

3.-5.

0.028 0.09 0.01 0.021 0.0025 0.006 0.0023 0.01 0.001

0.12 0.037 0.01 0.019

0.13 0.073 0.02 0.05

0.0008 0.0017 0.0027 0.0053 0.0031 0.0073 0.017
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Impact parameters and direction residuals for 5 GeV tracks
Tangent lambda d0 unfitted d0 fitted 0 unfitted 0 fitted Tan() unfitted Tan() fitted

<1.2

1.2-2.

2.-3.

3.-5.

0.007 0.018 0.024 0.036 0.002 0.004 0.007 0.017 0.0005 0.0013 0.0017 0.0024 0.0001 0.0003 0.0005 0.0012 0.0005 0.0025 0.0047 0.0092 0.0002 0.0006 0.0014 0.005
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Impact parameters and direction residuals for 20 GeV tracks
Ideal sensors Tangent lambda d0 unfitted d0 fitted 0 unfitted 0 fitted Tan() unfitted Tan() fitted SiD real sensors

1.2-2.

2.-3.

1.2-2.

2.-3.

0.0045 0.0011 0.0003 0.00007 0.0006 0.0001

0.0061 0.0019 0.0004 0.00013 0.0011 0.0003

0.0056 0.0043 0.0003 0.0001 0.0006 0.0002

0.0071 0.0058 0.0004 0.0002 0.0011 0.0004
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Impact parameters and direction residuals for 100 GeV tracks
Ideal sensors Tangent lambda d0 unfitted d0 fitted 0 unfitted 0 fitted Tan() unfitted Tan() fitted SiD real sensors

0.75

0.75

0.00045 0.0003 0.000018 0.000009 0.000025 0.00001
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0.0035 0.0019 0.000036 0.000025 0.0036 0.00009
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Conclusions
SiD detector tracking performance is not limited by spatial resolution of silicon tracker sensors for up to 100 GeV tracks. From the point of view of momentum resolution there is no benefit in extra layers. Rather smaller number of layers would be beneficial as it reduces amount of material. Of course, pattern recognition and reconstruction of lower momentum tracks may benefit from larger number of layers.In any case all efforts should be made to reduce amount of material inside tracking volume. Track fitter has little effect on the tracking resolution in case of multiple scatter dominated errors.There is no need to invest heavily in better fitter algorithm.

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