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

LCWS 06 Bangalore, India, March 9-13 2996 1

Track fitting using weight matrix

Nick Sinev, University of Oregon


Topics:

History Method description What is in the package Additional facilities WMTrackPropagator performance Fitter performance


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.


Method description (following Mours Benoit)

dχ2/dPm = 0 (m = 1,2…5) (1) χ2= Σij[Ri*Mij*Rj] (R is residual for i,j = 1… nmeas)

Mij is weight matrix element) (2) Ri = Ri0+Σm[dRi/dPm*(Pm-Pm0)] (3) From (1) and (2) Σij[dRi/dPm*Mij*Rj] = 0 using (3) Σij[dRi/dPm*Mij*(Rj0+Σn[dRj/dPn(Pn-Pn0)])] Defining WPmn=Σij[dRi/dPm*Mij*dRj/dPn] And Ym = - Σij[dRi/dPm*Mij*Rj0] we get system of eq. Σn[WPmn*(Pn-Pn0) = Ym


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).


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.


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.


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.


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


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 itexits tracking detector


WMTrackPropogator performance - continued

To have correct track positions at large radius correction for energy loss inTracker material need to be made. WMTrackPropogator has the option to ignore Energy loss (residual for positive and negative tracks for such case are shown atLeft) or include it (residuals shown at right)


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).


Fitter performance

Distribution of 2 for unfitted (just connecting 3 hits) tracks and fittedtracks for all tracks in the event. This distribution varies a little for differentPt regions


Fitter performance - continue

Impact parameter (microns) and Pt residual/Pt2 vs. Pt for unfitted and fittedtracks.


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.


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.

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

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