1. LCWS 06 Bangalore, India, March 9-13 29961 Track fitting using weight matrix Nick Sinev, University of Oregon

2. LCWS 06 Bangalore, India, March 9-13 20062 Topics: History Method description What is in the package Additional facilities WMTrackPropagator performance Fitter performance

3. LCWS 06 Bangalore, India, March 9-13 29963 History Algorithm described here was originally developed by Mours Benoit for SLD. In 1999 I have ported this algorithm for use in JAS environment for ILC simulations in the framework of hep.lcd After new JAS package for ILC simulations was introduced, need to port all hep.lcd facilities into org.lcsim package appeared. At Snowmass meeting I was assigned a task to port fitter.

4. LCWS 06 Bangalore, India, March 9-13 29964 Method description (following Mours Benoit) dχ 2 /dP m = 0 (m = 1,2…5) (1) χ2= Σ ij [R i *M ij *R j ] (R is residual for i,j = 1… nmeas) M ij is weight matrix element) (2) R i = R i0 +Σ m [dR i /dP m *(P m -P m0 )] (3) From (1) and (2) Σ ij [dR i /dP m *M ij *R j ] = 0 using (3) Σ ij [dR i /dP m *M ij *(R j0 +Σ n [dR j /dP n (P n -P n0 )])] Defining WP mn =Σ ij [dR i /dP m *M ij *dR j /dP n ] And Y m = - Σ ij [dR i /dP m *M ij *R j0 ] we get system of eq. Σ n [WP mn *(P n -P n0 ) = Y m

5. LCWS 06 Bangalore, India, March 9-13 29965 Difference with Kalman Fitter Kalman fitter always fit 2 objects: fitted track from previous points and 1 next layer measurement. So, it is fast, but, it does not take into account correlations between measurements. Weight matrix method fits many measurements simultaneously. Weight matrix takes into account correlations between measurements in different layers. But it is slow, especially if number of layers is high (inverting of large rank matrix takes time).

6. LCWS 06 Bangalore, India, March 9-13 29966 How weight matrix is calculated Weight matrix is the inverse of error matrix. Errors composing error matrix are calculated as combination of hit displacement resulting from multiple scattering and from detector resolution error. Multiple scattering in each layer of material causes correlated displacement in all subsequent layers. So, elements of the error matrix m,n are calculated as sum of products of displacements from preceding layers for layers m and n. Displacements due to detectors resolution are added only to diagonal elements, as they are independent.

7. LCWS 06 Bangalore, India, March 9-13 29967 What is in the package Package org.lcsim.recon.tracking.wmfitter consists of 2 classes: WeightMatrix SLDTrackFitter In addition, few classes of more common interest are created in org.lcsim.recon.tracking.util: CMTransform to transform Covariance Matrix from 3D carthesian into 2D on cylinder or perpendicular to Z plane, 1 coordinate in Rphi direction, another Z or R, WMTrackPropagator – propagate track through all layers of tracking system, ThreePointCircle and ThreePointTrkPar used to built track passing through 3 points ReconstractedTrack – object implementing Track interface and including access function to set track parameters.

8. LCWS 06 Bangalore, India, March 9-13 29968 Additional facilities To be able to test fitter, I needed to be able to generate smeared hits. To do this, 2 classes were added to org.lcsim.event.util package: SmearedTrackerHit extends tracker hit, and has access function to smear coordinates. And it also inherits from SimTrackerHit getMCParticle() function. SmearTrackerHits is the driver to generate SmearedTrackerHits from the list of SimTrackerHit. And has settable smear parameters for different detectors. As TrackFinder is still not ported into org.lcsim, I have used TrackChiter, provided by M. Ronan. However, I needed to modify it to use SmearedTrackerHit in place of SimTrackerHit.

9. LCWS 06 Bangalore, India, March 9-13 29969 WMTrackPropagator features At the first event program reads geometry and material information from detector description and makes list of all objects in the tracking volume. For each track all crossing points with objects inside tracking volume are calculated, amount of material passed by track is calculated and deviations from initial track position for each sensitive layer, along with correlations of such deviations in the form of error matrix is generated. The control options include possibility to take into account air inside detector and energy loss of the track in the process of passing through detector

10. LCWS 06 Bangalore, India, March 9-13 299610 WMTrackPropagator performance WMTrackPropagator traces track through all materials inside tracking region. Here is the plot of total amount of material passed by 5 GeV muon before it exits tracking detector

11. LCWS 06 Bangalore, India, March 9-13 299611 WMTrackPropogator performance - continued To have correct track positions at large radius correction for energy loss in Tracker material need to be made. WMTrackPropogator has the option to ignore Energy loss (residual for positive and negative tracks for such case are shown at Left) or include it (residuals shown at right)

12. LCWS 06 Bangalore, India, March 9-13 299612 Fitter features Fitter works with both barrel and endcap detectors Option to do IP constrained fit is included Fitter fits track parameters at closest to IP point of the track. However the momentum of fitted track at any point along track can be extracted. Any subset of the layers can be excluded from the fit (may be useful if there is a kink on the track, so excluding outer most layer may give better fit).

13. LCWS 06 Bangalore, India, March 9-13 299613 Fitter performance Distribution of  2 for unfitted (just connecting 3 hits) tracks and fitted tracks for all tracks in the event. This distribution varies a little for different Pt regions

14. LCWS 06 Bangalore, India, March 9-13 299614 Fitter performance - continue Impact parameter (microns) and Pt residual/Pt 2 vs. Pt for unfitted and fitted tracks.

15. LCWS 06 Bangalore, India, March 9-13 299615 Fitter problems While first impression was that fitter worked reasonably well, some questions appeared. For example, if I ignore correlations in the measurements, I would expect that fitter performance will became worse. However, it was opposite. So, I can’t say that this software is ready for release right now. I need to sort out all unexpected behavior. And this is the reason why I do not show here more plots – for example, for endcap region, effect of IP constrain and so on.

16. LCWS 06 Bangalore, India, March 9-13 299616 Conclusions Weight matrix based fitter was ported from lcd.hep into org.lcsim environment In process new features have been added and ability to fit tracks in endcap disks is now included However, not all aspects of fitter performance are understood. Some more debugging is needed.