August, 7 ILC 1

Kudzanayi Munetsi-Mugomba

supervised by

Dr. Caroline Milstene

Software Infrastructure for the Charged Particles Reconstruction

at the International Linear Collider

August, 7 ILC 2

OUTLINE

The HELIX- The dependence of the trajectory of the charged particle on the Magnetic Field.

The KALMAN Filter (Part 1-Transport Matrix) The dependence of the trajectory on both the

Magnetic Field and the Material the particle goes through.

The comparison of the Transport Matrix to the Helix at different Energies -RESULTS

The KALMAN Filter (Part 2-Update using the Data)

The Update of the Particle Trajectory-RESULTS Discussion Conclusion

August, 7 ILC 3

Assuming a vacuum (no interaction)

Particle only changes direction, therefore velocity remains constantWhich is a circle.

Pz ≠0, - Circle Helix

HelixThe Motion of a Particle in a Magnetic Field

Py Px

Pz Bz

PT

P

Px

Pz

Py

Bz

Bz, is parallel to Pz but acts on the PyPx -plane

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KALMAN FILTER- Part 1

XYZPxPyPz

Transport Matrix Φ,R

Covk-1 Covk

Bz dE/dx

PHASE SPACE POINT

Transport Model

The Kalman Filter, contains a theoretical part( transport matrix), and is dependent upon Bz (magnetic field), and the dEdx (loss of energy due to the ionization of the medium by the particle)

Theoretical part

Since the detector contains material a more elaborate algorithm is needed to representthe motion of the particle in the detector.

Xk-1 Xk

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2-122

max222

cm ; ]2

)(2ln

2

1[

12

2 gMeVI

cem

A

ZKz

dx

dE T

• The mean rate of energy-loss by ionization of relativistic charged particles(protons, alpha particles, atomic ions, but not electrons) in material theyTraverse.• Charged particles moving through matter interact with the electrons ofatoms in the material, Tmax is the maximum kinetic energy imparted to a freeElectron in a single collision• The interaction excites or ionizes the atoms. This leads to an energy loss of the traveling particle which is function of β. Particles of the sameVelocity have similar rates of energy loss.K=4π me

re2 c2 , where me and re are the mass and radius of the electron, c the light velocity

Z,A are the atomic number and the atomic mass of the absorber, me is the electron mass, c the speed of light, I the mean excitation energy(eV)δ(βγ) is the density effect correction

Bethe-Block Equation

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0.5 GeV Muon - Comparison of the Particle Motion to the Helix, dE/dx(Fe) versus dE/dx~0

0.5 GeV dE/dx(Fe)

0.5 GeVdE/dx~0

Helix accounts only on the effect of the magnetic field on a charged particle.We want to find/ measure the trajectory of a particle in the Detector hence; we account for the energy loss in the material

Velocity changesDirection changes

Velocity unchangedDirection changes

We realize that the Transport Model is better than the Helix, but in the material there are random. processes. which still are not accounted for, e.g. Bremstrahlung, Multiple Scattering, decays …

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5 GeV Muon Motion Comparison With the Helix

For dE/dx=Fe and dE/dx~0

5 GeV dE/dx(Fe)

5 GeVdE/dx=0

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0.5 GeV Muon Comparison for 2 different Materials

dE/dx(Al) versus dE/dx(Fe)

0.5 GeV dE/dx(Al)

0.5 GeV dE/dx=Fe

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Uncertainityin xyz Meas

KALMAN FILTER- Part 2

A

Transport Matrix

Uncertainitycorrection =

B

Bz dE/dx

position and momentum of a particle

Transport ModelUpdate

Contains a theoretical part( transport matrix), and the actual data that we combine in the updateTo improve the Model by correcting the trajectory obtained using our measurement points.

Theoretical part

+

XyzMeas

Uncertainitycorrection

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Explaining the Transport of the Precision Information

Update

Transport

AbsorberTungsten (W)

Active materialSilicon (Si)

Absorber Tungsten (W)

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Part 2- Description

In order to be able to test the software for various detector setups while insuring the stability of the software, I developed a set of tests for the Kalman Filter.

To make it simple and without any external input, from now on, I will be using the Helix as data on which to correct the particle trajectory at update.

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Following Results

The Results are given for SLIDE 13 for the Hadron

Calorimeter HCal setup. - assuming that the absorber thickness of iron(Fe)=2cm - the number of steps in the absorber =4 - number of tracks =1The Results in all the other SLIDES for the Muon

Detector, MuDetsetup; - the absorber is Iron(Fe) – 10 cm thick - the number of steps = 10 - the number of tracks = 1

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0.5 GeV Muon Kalman Filter (Material -Fe) using the HELIX asDATAHCal absorber 2cm- Fe

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0.5 GeVIn Fe

1 GeVIn Fe

5 GeVIn Fe

10GeVIn Fe

Muon-x versus y: Kalman Stepper updated on the Helix as Data with dE/dx(Fe). At 0.5 GeV the track stopped. Using 10cm thick Fe absorber as in MuDet

Kalman Filter (Material -Fe) Dependance on Energy

using the HELIX asDATA

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5 GeV Muon- Kalman Filter with Helix( Data)

Includes Field and dE/dx(Fe)5GeV Muon- Kalman Stepper updated on the Helix as Data

In both planes: y vs x (mm) Z vs x (mm)

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Kalman Filter Another Application

The nonlinear problem of tracking and correcting the trajectory of a satellite over a time is difficult with the recognition of modeling errors and ground site radar tracking errors.

an accurate modeling program with the fidelity to correct for any errors in orbital motion and predict the most accurate position at some future time is required.

Here the correction of the trajectory of a satellite during its motion in space done using the Kalman Filter

Multi-functional Transport Satellite

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RESULTS which FOLLOW

The Following Results the Kalman Filter on 20 tracks at 2 different energies;

The 20 tracks were obtained by smearing the hits of the Helix in a gaussian.

- the trajectories of the particles are starting from an interaction point smeared in a gaussian

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Kalman Filter for 20 Muon Tracks- at 2 Energies -

Data(Helix)x vs. y

0.5 GeV 5 GeV0.5 GeV

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Kalman Filter at 5 GeV for 20 Muon Tracks- Data(Helix) dE/dx(Fe) in both planes y vs x (mm)z vs x (mm)

5 GeV 5 GeV

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Conclusion The helix was a good approximation for high

momenta particles or low dE/dx materials. Applying the transport matrix alone, did take care

of both the magnetic field effects and material effects through dE/dx and is a better approximation than the Helix.

However other random effects are not accounted for, e.g. multiple-scattering, Bremsstrahlung…

But we have shown that after the inclusion of the update one is able to correct the trajectory of the reconstructed track to fit the actual data and account for all the random processes.

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A SPECIAL THANKS TO!

Caroline Milstene Marcel Demarteau A special thanks to Fermilab for granting me this

opportunity to conduct research in the dynamic world of High Energy Physics.

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QUESTIONS

That’s right! NO!

www.fnal.gov www.kudzie.com

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Bonus Material

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5 GeV Muon Helix and Equations of MotionsdE/dx(Si) versus dE/dx=0

5 GeV dE/dx(Si)

5 GeV dE/dx=0

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Comparison of Kalman Filter High versus Low Energies

Muon-x versus y: Kalman Stepper updated on the Helix as Data with

dE/dx(Fe). At 0.5 GeV the track stopped.

0.5 GeV 10 GeV

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Xk-1 Φ,R Xk

Covk-1 Covk

Transport

Zk

mCovUpdate

Kalman- Filter Principle

μx

X