August, 7ILC1 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 constant Which is a circle. Pz ≠0, - Circle Helix Helix The Motion of a Particle in a Magnetic Field Py Px Pz Bz PTPT P Px Pz Py Bz Bz, is parallel to Pz but acts on the PyPx -plane

August, 7 ILC 4 KALMAN FILTER- Part 1 X Y Z Px Py Pz Transport Matrix Φ,R Covk-1 Covk BzdE/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 represent the motion of the particle in the detector. Xk-1Xk

August, 7 ILC 5 The mean rate of energy-loss by ionization of relativistic charged particles (protons, alpha particles, atomic ions, but not electrons) in material they Traverse. Charged particles moving through matter interact with the electrons of atoms in the material, T max is the maximum kinetic energy imparted to a free Electron 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 same Velocity have similar rates of energy loss. K=4π m e r e 2 c 2, where m e and r e 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, m e 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

August, 7 ILC 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 GeV dE/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 changes Direction changes Velocity unchanged Direction 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 …

August, 7 ILC 7 5 GeV Muon Motion Comparison With the Helix For dE/dx=Fe and dE/dx~0 5 GeV dE/dx(Fe) 5 GeV dE/dx=0

August, 7 ILC 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

August, 7 ILC 9 Uncertainity in xyz Meas KALMAN FILTER- Part 2 A Transport Matrix Uncertainity correction = B BzdE/dx position and momentum of a particle Transport Model Update Contains a theoretical part( transport matrix), and the actual data that we combine in the update To improve the Model by correcting the trajectory obtained using our measurement points. Theoretical part + Xyz Meas Uncertainity correction

August, 7 ILC 10 Explaining the Transport of the Precision Information Update Transport Absorber Tungsten (W) Active material Silicon (Si) Absorber Tungsten (W)

August, 7 ILC 11 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.

August, 7 ILC 12 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 =1 The Results in all the other SLIDES for the Muon Detector, MuDet setup; - the absorber is Iron(Fe) – 10 cm thick - the number of steps = 10 - the number of tracks = 1

August, 7 ILC GeV Muon Kalman Filter (Material -Fe) using the HELIX asDATA HCal absorber 2cm- Fe

August, 7 ILC GeV In Fe 1 GeV In Fe 5 GeV In Fe 10GeV In 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

August, 7 ILC 15 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)

August, 7 ILC 16 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

August, 7 ILC 17 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

August, 7 ILC 18 Kalman Filter for 20 Muon Tracks- at 2 Energies - Data(Helix) x vs. y 0.5 GeV 5 GeV 0.5 GeV

August, 7 ILC 19 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

August, 7 ILC 20 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.

August, 7 ILC 21 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.

August, 7 ILC 22 QUESTIONS That’s right! NO!

August, 7 ILC 23 Bonus Material

August, 7 ILC 24 5 GeV Muon Helix and Equations of Motions dE/dx(Si) versus dE/dx=0 5 GeV dE/dx(Si) 5 GeV dE/dx=0

August, 7 ILC 25 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

August, 7 ILC 26 X k-1 Φ,R X k Cov k-1 Cov k Transport Z k mCov Update Kalman- Filter Principle μ x X