1. A track fitting method for multiple scattering Peter Kvasnička, 5th SILC meeting, Prague 2007

2. Peter Kvasnicka, 5 th SILC meeting, Prague 2007 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. Peter Kvasnicka, 5 th SILC meeting, Prague 2007 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. Peter Kvasnicka, 5 th SILC meeting, Prague 2007 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. Peter Kvasnicka, 5 th SILC meeting, Prague 2007 Outline Fitting tracks with lines Probabilistic model of a particle track Scattering algebra and statistics Some results Conclusions

6. Peter Kvasnicka, 5 th SILC meeting, Prague 2007 Fitting tracks with lines To keep things simple, I only consider a 2D situation

7. Peter Kvasnicka, 5 th SILC meeting, Prague 2007 Fitting tracks with lines (cont’d)

8. Peter Kvasnicka, 5 th SILC meeting, Prague 2007 Fitting tracks with lines (cont’d) 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. Peter Kvasnicka, 5 th SILC meeting, Prague 2007 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. Peter Kvasnicka, 5 th SILC meeting, Prague 2007 Outline Fitting tracks by line Probabilistic model of a particle track Scattering algebra and statistics Some results Conclusions

11. Peter Kvasnicka, 5 th SILC meeting, Prague 2007 Probabilistic model of a particle track φ0φ0 φ1φ1 φ2φ2 φ3φ3 … z0z0 z1z1 z2z2 z3z3 z4z4 and the distribution of φ’s is Moliere, i.e., approximately Gaussian We use paraxial approximation tg φ ~ φ

12. Peter Kvasnicka, 5 th SILC meeting, Prague 2007 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. Peter Kvasnicka, 5 th SILC meeting, Prague 2007 Outline Fitting tracks by line Probabilistic model of a particle track Scattering algebra and statistics Some results Conclusions

14. Peter Kvasnicka, 5 th SILC meeting, Prague 2007 Scattering algebra and statistics We have to reconstruct impact points on scatterers, given by Our observables are

15. Peter Kvasnicka, 5 th SILC meeting, Prague 2007 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 ξ are selected rows from A X, with 1’s added for measurement errors. We want to estimate X based on ξ.

16. Peter Kvasnicka, 5 th SILC meeting, Prague 2007 Scattering algebra and statistics We estimate rows of A ξ as the best linear combinations of rows of A X, that is, we seek a matrix T such that The solution is Covariance of weights:

17. Peter Kvasnicka, 5 th SILC meeting, Prague 2007 Outline Fitting tracks by line Probabilistic model of a particle track Scattering algebra and statistics Some results Conclusions

18. Peter Kvasnicka, 5 th SILC meeting, Prague 2007 Comparison with line fit – DEPFET simulation 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 Zbynek will say more on simulations in his talk

19. Peter Kvasnicka, 5 th SILC meeting, Prague 2007 Comparison - DEPFET (cont’d) * Line fit * Kinked fit

20. Peter Kvasnicka, 5 th SILC meeting, Prague 2007 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. Peter Kvasnicka, 5 th SILC meeting, Prague 2007 Comparison – SILC TB simulation * Line fit * Kinked fit

22. Peter Kvasnicka, 5 th SILC meeting, Prague 2007 Outline Fitting tracks by line Probabilistic model of a particle track Scattering algebra and statistics Some results Conclusions

23. Peter Kvasnicka, 5 th SILC meeting, Prague 2007 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