Track fitting study

S. Gupta, W. daSilva, T. Mibe

14

th

muon g-2/EDM Collaboration meeting

June 9, 2017

S. Gupta, W. daSilva, T. Mibe Track fitting study June 9, 2017 1 / 21

Overview

1 Track fitting

2 Kalman filter

3 Track fitting results

4 Validation of results

5 Conclusion

S. Gupta, W. daSilva, T. Mibe Track fitting study June 9, 2017 2 / 21

Status of track reconstruction studies

As already mentioned by colleagues, a ROOT and GEANT basedsimulation framework and Hough transform based track finding toolhas been developed.We discuss the work done in developing a track fitting framework

S. Gupta, W. daSilva, T. Mibe Track fitting study June 9, 2017 3 / 21

Track fitting

Elements of Track fitting

The basic ”ingredients” required for track fitting are-

The track model- ”How does the particle move in the given fields?”.Eg- Helix track in constant B field

Material e↵ects and Energy loss-”How does particle loose its energyin the detector/medium?”. Eg- Bremsstrahlung, Multiple scattering,Ionization loss etc.

Track fitting method - ”A choice of an appropriate algorithm”. Eg-Kalman Filter

Error propagation- The error in measurement of trajectory propertiesis propagated to get the final error estimate. We use a Covariancematrix Ck and propagate it linearly through every detector layer.

Track parametrization- An appropriate choice of coordinates

S. Gupta, W. daSilva, T. Mibe Track fitting study June 9, 2017 4 / 21

Track fitting

Non-Gaussian distribution

The track fitting algorithms assume a Gaussian distribution of errors.However, the PDF of multiple scattering and Bremsstrahlung have anon-Gaussian component with long tail.

1 Multiple scattering- Distribution of scattering angle ✓ has a Gaussiancore with a non-Gaussian tail as shown in Figure a).

2 Bremsstrahlung- It follows Bethe-heitler distribution which is clearlynon-Gaussian. Figure b) shows the PDF of z = Efinal

Einitialfor di↵erent

materia thickness t.

(a) PDF of scattering angle (b) Bremsstrahlung PDFS. Gupta, W. daSilva, T. Mibe Track fitting study June 9, 2017 5 / 21

Track fitting

Problem statement

We intend to develop a GENFIT based framework to perform track fittingin the positron tracker of the g-2 experiment-

Use GENFIT package for track fitting and study its performance fortracking low momentum positrons, for which it has not been tested.

Choose an appropriate track fitting algorithm.

Study the track fitting e�ciency and momentum resolution fordi↵erent momentum range of positrons

S. Gupta, W. daSilva, T. Mibe Track fitting study June 9, 2017 6 / 21

Track fitting

GENFIT

GENFIT o↵ers many track fitting algorithm. Ideal choice for smallercollaborations which lack the manpower to develop dedicated track fittingpackage.We input the silicon tracker geometry and a simulated positron event inGENFIT for track fitting.

(a) GENFIT GUI (b) Typical fitted positron event

S. Gupta, W. daSilva, T. Mibe Track fitting study June 9, 2017 7 / 21

Kalman filter

Kalman Track filter

The measurements m are related to the true track parameters xt throughthe measurement model H

m = Hxt + ✏ (1)

Where V = Cov(✏), is the Covariance matrix of the measurement errors.Kalman filter works as follows-

Seed state- Choose a starting state of the track.Filtering

1 Prediction- Predict the state and error matrix in k th detector layerusing the state in k � 1th layer.

xk|k�1

= fk|k�1

(xk�1|k�1

) (2)

Ck|k�1

= FkCk�1|k�1

FTk + PkQkP

Tk (3)

Here Pk and Qk take into account the e↵ect of material between k th &k � 1th layer on the uncertainty in state vector.

S. Gupta, W. daSilva, T. Mibe Track fitting study June 9, 2017 8 / 21

Kalman filter

Kalman Track filter

2 Update- Account for the measurement made in k th layer and assignthe state in k th layer as the weighted mean of the two

Ck|k = (C�1

k|k�1

+ HTk V�1

k Hk)�1 (4)

xk|k = Ck|k(C�1

k|k�1

xk|k�1

+ HTk V�1

k mk) (5)

Smoothing- The filter is run backwards from nth to 1st measurementand state xk|k is set as a mean of forward and backward filter.

xsm,k = C�1

f ,k|k�1

xf ,k|k�1

+ C�1

b,k|kxb,k|k (6)

S. Gupta, W. daSilva, T. Mibe Track fitting study June 9, 2017 9 / 21

Kalman filter

Modified Kalman filter

Despite its advantages, Kalman filter has following shortcomings-1 E↵ect of bias due to initial seed hit x

0|02 Presence of outliers

These issues can be mitigated by the following ”tweaks”

1 Assign weight to each hit- Deterministic annealing filter (DAF)2 Using a reference track- Kalman filter with a Reference track

DAF with reference Kalman filter can be the ideal choice of algorithm

S. Gupta, W. daSilva, T. Mibe Track fitting study June 9, 2017 10 / 21

Kalman filter

The choice of fitting algorithm

Fitting was carried out in GENFIT on a sample of 250 events.

The number of positrons track fitted by each fitting algorithm werecounted.

Clearly, DAF with ref. Kalman is the best choice, owing to its abilityto reject outliers as well as bad initial hits.

Further, we optimized the parameters of DAF ref. Kalman filter toensure convergence with minimal processing time.

Filter Simp. Kalman Ref. Kalman DAF with Ref. Kalman

% Track fitted 83.93 87.95 99.59

Table: Comparison of the performance of di↵erent track fitting algorithm

S. Gupta, W. daSilva, T. Mibe Track fitting study June 9, 2017 11 / 21

Track fitting results

Track fitting e�ciency

The fraction of events for which the fitting was successful and convergedwas calculated for track fitting e�ciency.The preliminary plot shows that the track fitting e�ciency improves withthe positron momentum & is ⇡ 95 % in the desired region.

Figure: Track fitting e↵ vs momentum

S. Gupta, W. daSilva, T. Mibe Track fitting study June 9, 2017 12 / 21

Track fitting results

Why doesn’t the fit converge?

Although the fitting e�ciency is high in the desired momentum range,there are 2 primary reasons why the fitting fails for certain events-

1 Multiple hits on the same vane-for low momentum e+

2 Kinks in positron trajectory- for all momentum e+

(a) 2 hits in the same vane (b) Kink in e+ trajectory

S. Gupta, W. daSilva, T. Mibe Track fitting study June 9, 2017 13 / 21

Track fitting results

Cause of fit failure

As shown in the histogram, low momentum regime is dominated bymultiple hits on the same vane

Low momentum e+ are tangent to the vane, leading to multiple hits.High momentum e+ are nearly perpendicular to the vane, thereforewe don’t observe any multiple hit.Kinks in e+ trajectory can be observed at the entire momentumrange.

(a) Low momentum events (b) High momentum eventsS. Gupta, W. daSilva, T. Mibe Track fitting study June 9, 2017 14 / 21

Track fitting results

Momentum resolution

Fitted track is extrapolated to the first hit and estimated momentum iscompared with the true momentumMomentum resolution is ⇡ 1� 1.5 % and shows improvement withpositron momentum.

(a) Momentum resolution histogram (b) Resolution vs momentum

S. Gupta, W. daSilva, T. Mibe Track fitting study June 9, 2017 15 / 21

Track fitting results

Improvement of momentum resolution

Can we explain the improvement in momentum resolution theoretically ?The uncertainty in momentum resolution due to multiple scattering on canbe approximated as

�p

p=

0.053

BLsin(✓)

pt (7)

RMS pt (⇡ 100 MeV/c) >RMS pz (⇡ 15 MeV/c)Therefore, at high momenta ✓ ⇡ ⇡

2

.

Figure: At high momentum ✓ ⇡ ⇡2

S. Gupta, W. daSilva, T. Mibe Track fitting study June 9, 2017 16 / 21

Validation of results

Goodness of fit

GENFIT has been used in Belle II which studies e+, e� collision at 10.3GeV. However, there has not been any specific study of positron in thesame kinetic region as g-2Goodness of fit must be checked to ensure GENFIT performs well in theMeV/c momentum range.We use p-value distribution of the �2/NDF for the purpose

S. Gupta, W. daSilva, T. Mibe Track fitting study June 9, 2017 17 / 21

Validation of results

P value distribution

Ideal p-value distribution must be flat with a peak around 0 formed due tooutliersWe can see a large number of outliers in the plots which could beattributed to the long tail in multiple scattering & Bremsstrahlungdistribution.

(a) p-value plot for [125,150] MeV/c (b) p-value plot for [250,275] MeV/c

S. Gupta, W. daSilva, T. Mibe Track fitting study June 9, 2017 18 / 21

Validation of results

Checking GENFIT performance

E↵orts have been started to test GENFIT performance for low-energye/e+ tracking.The plot (credits ZEFIZEF Mohamed) shows the relative error as afunction of pixel size for a 105 MeV electron.Similar study will be be carried out for positrons.

S. Gupta, W. daSilva, T. Mibe Track fitting study June 9, 2017 19 / 21

Conclusion

Summary & Outlook

DAF with ref. Kalman was chosen as the best choice owing to itscapability to cope with bias initial hits and outliers.

The preliminary results suggest that the momentum resolution andtrack fitting e�ciency is good in the detector’s designed range ofpositron momentum.

The presence of large number of outliers in the p-value plot impliesthe need to test GENFIT models as well as cross sections in the lowmomentum regime for positrons.

GENFIT’s linear tracking mechanism needs to be changed to fit thetrack for the events with 2 hits on the same vane.

S. Gupta, W. daSilva, T. Mibe Track fitting study June 9, 2017 20 / 21

Conclusion

Summary & Outlook

We now have a framework to perform track fitting and analyze the results

A track fitting code and analysis codehas been prepared and committed toGitlab (https://gitlab.in2p3.fr/shobhit/Genfitg-2).

Presently the studies have beenperformed on the events simulatedusing ICEDUST (modified COMETframework)

With minor modification in the readingroutines, the framework could beadapted to read the Kyushu simulationfile and geometry as well as interfacedwith the track finding code.

S. Gupta, W. daSilva, T. Mibe Track fitting study June 9, 2017 21 / 21