A track fitting method for multiple scattering

1
A track fitting method for multiple scattering

• Peter Kvasnicka,

5th SILC meeting, Prague 2007
2
Introduction

• This talk is about a track fitting method that
  explicitly takes into account multiple
  scattering.
• IT IS NOT NEW It was invented long time ago and,
  apparently re-invented several times.
• Helmut Eichinger Manfred Regler, 1981
• Gerhard Lutz 1989
• Volker Blobel 2006
• A.F.Zarniecki, 2007 (EUDET report)
• as I learned after I re-invented it myself.

3
Motivation is obvious

• Multiple scattering is a notorious complication
  and is particularly serious
• For low-energy particles
• For very precise detectors
• For systems of many detectors

4
Motivation (continued)

• Remedies
• Thinner detectors
• Higher energies
• Extrapolation of fits to infinite energies
• Better methods of fitting (Kalman filter)
• The Kalman filter is the best known method able
  to account for multiple scattering, yet it is
  used relatively little
• High cost / benefit ratio
• The Kalman filter requires just the parameters
  that one would want to compute (detector
  resolutions), and does not offer an affordable
  way of computing them.
• I will introduce a different method, which is
  simpler and behaves similarly to the Kalman
  filter in many repsects.

5
Outline

• Fitting tracks with lines
• Probabilistic model of a particle track
• Scattering algebra and statistics
• Some results
• Conclusions

6
Fitting tracks with lines
To keep things simple, I only consider a 2D
situation
7
Fitting tracks with lines (contd)
8
Fitting tracks with lines (contd)

• Information on detector resolutions can be used
  to weight the fit
• Multiple scattering violates the assumption of
  independence of regression residuals the
  covariance matrix is no longer diagonal.
  Therefore, direct calculation of detector
  resolutions is not possible.

9
Behind the lines

• As a rule, information on multiple scattering is
  at hand
• Formulas describing the distribution of
  scattering angles are well-known
• Simulations (GEANT) are commonplace in particle
  experiments
• Last but not least, we can try to intelligently
  use the data to estimate scattering distributions

10
Outline

• Fitting tracks by line
• Probabilistic model of a particle track
• Scattering algebra and statistics
• Some results
• Conclusions

11
Probabilistic model of a particle track
z0
z1
z2
z3
z4
f3
f0
f1
f2
We use paraxial approximation tg f f

and the distribution of fs is Moliere, i.e.,
approximately Gaussian
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Probabilistic model of a particle track

• To put this to some use, we set up the task as
  follows

We have a system of n scatterers and N particle
tracks. k of the scatterers are in fact
detectors that provide us with information about
the impact point xi of a particle, alas with
errors di. The task is to estimate impact points
on the scatterers (some, or all).
13
Outline

• Fitting tracks by line
• Probabilistic model of a particle track
• Scattering algebra and statistics
• Some results
• Conclusions

14
Scattering algebra and statistics

• We have to reconstruct impact points on
  scatterers, given by

• Our observables are

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Scattering algebra and statistics

• In matrix form, this can be written as

Both the hidden parameters and observables are
expressed as products of some matrices with the
(approximately jointly Gaussian) vector of random
variables. Note that A? are selected rows from
AX, with 1s added for measurement errors. We
want to estimate X based on ?.
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Scattering algebra and statistics

• We estimate rows of A? as the best linear
  combinations of rows of AX, that is, we seek a
  matrix T such that

The solution is
The solution is
Covariance of weights
17
Outline

• Fitting tracks by line
• Probabilistic model of a particle track
• Scattering algebra and statistics
• Some results
• Conclusions

18
Comparison with line fit DEPFET simulation
Zbynek will say more on simulations in his talk

• Data GEANT4 simulations of DEPFET detectors
  (Zbynek Drasal)
• 5 identical detectors with identical distances of
  36, 45, or 120 mm, the middle DEPFET is the DUT
• Detector resolution simulated by Gaussian
  randomization of impact points (sigma 0.5, 1.2 a
  3 micron)
• Particles 80, 140, and 250 GeV pions
• Fit without the use of DUT data
• RMS residuals plotted against
• scattering parameter (RMS scattering
  angle)(jtypical distance between detectors) /
  detector resolution

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Comparison - DEPFET (contd)
Line fit Kinked fit
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Comparison SILC TB simulation

• Same type of simulation by Zbynek, two
  geometries
• The one we use in October (3 telescopes 32 and 8
  cm apart, DUT (CMS) 1 meter behind
• The one planned for the June testbeam, with DUT
  in between the more distant telescopes.
• Beam energies 1, 2, 6 GeV
• Resolutions 1.5 um (tels) and 9 um (DUT)
• Scattering parameter defined from the point of
  view of the DUT

21
Comparison SILC TB simulation
Line fit Kinked fit
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Outline

• Fitting tracks by line
• Probabilistic model of a particle track
• Scattering algebra and statistics
• Some results
• Conclusions

23
Conclusions

• The method switches between linear regression and
  interpolation between points similarly to the
  Kalman filter
• The method is useful in the moderate scattering
  regime at low scattering, line fits give
  basically the same results, and at high
  scattering, there is little help. So use line
  fits where applicable.
• We can plug in experimental uncertainties of
  impact point measurements (e.g., from
  eta-correction)
• We can also estimate other combinations of
  parameters, for example, the scattering angles or
  detector resolutions themselves. Nice, but so far
  not too useful
• Alignment Work in progress