In the Muon package under study, the tracks are extrapolated using an
algorithm which accounts for the magnetic field and the ionization (dE/dx). We
improved the calculation of the field dependent term to increase the muon
detection efficiency at lower momenta using a Runge-Kutta method. The muon
identification and hadron separation in b-bbar jets is reported with the
improved software. In the same framework, the utilization of the Kalman filter
is introduced. The principle of the Kalman filter is described in some detail
with the propagation matrix, with the Runge-Kutta term included, and the effect
on low momenta single muons particles is described.Comment: PDF,5pages,2 Figures,1 Table,Presented at the 2005 International
  Linear Collider Physics and Detectors Workshop,Snowmass,Colorado,14-27 Aug.
  2005, PSN1011 in the proceedin

/Fermilab

Fisk, G.

Milstene, C.

Para, A.

English

arXiv

In the Muon package under study, the tracks are extrapolated using an algorithm which accounts for the magnetic field and the ionization (dE/dx). We improved the calculation of the field dependent term to increase the muon detection efficiency at lower momenta using a Runge-Kutta method. The muon identification and hadron separation in b-bbar jets is reported with the improved software. In the same framework, the utilization of the Kalman filter is introduced. The principle of the Kalman filter is described in some detail with the propagation matrix, with the Runge-Kutta term included, and the effect on low momenta for low momenta single muons particles is described

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                                                                                                       ALCPG1006 12005 ALCPG & ILC Workshops  Snowmass, U.S.A.  Muon ID- Taking Care of Lower Momenta Muons  C. Milsténe, G. Fisk, A. Para  FNAL, Batavia, IL 60510, USA In the Muon package under study, the tracks are extrapolated using an algorithm which accounts for the magnetic field and the ionization (dE/dx). We improved the calculation of the field dependent term to increase the muon detection efficiency at lower momenta using a Runge-Kutta method. The muon identification and hadron separation in b-bbar jets is reported with the improved software. In the same framework, the utilization of the Kalman filter is introduced. The principle of the Kalman filter is described in some detail with the propagation matrix, with the Runge-Kutta term included, and the effect on low momenta for low momenta single muons particles is described  1. INTRODUCTION Muons in the 3-5 GeV range are a substantial part of the  produced at 500 GeV center of mass energy, for example they constitute ~14% of the muons produced in bbar-b events.  This note describes improvements in the  identificationat the lower momentum limit. The improvements are made by applying two techniques: first, second order corrections to the propagation algorithm are introduced using the Runge-Kutta method, second, using a Kalman filter, multiple scattering and random processes are added to the propagation.  Multiple scattering is particularly important for low momenta tracks, hence of particular interest in the region explored,   The note describes the improvements achieved by these methods 2. THE RUNGE-KUTTA CORRECTION The Muon package has been described in detail Ref.[1].  Also, the Muon ID algorithm has been applied to 10000 b-bbar Events [2].  This note focuses mostly on the simulated muons of lower momenta which still reach the muon detector (3GeV p <5GeV) and for which the approximation dtdptp TT /~/  , is insufficient.  Here pT (GeV/c) is the variation in the transverse momentum of a particle going through a magnetic field B (Tesla) for a time t(s) and   lightqcE.withBBqdtdp )/1(303.0/____, pv     E is the particle energy in GeV, q its charge in electron units and clight (m/s).  In a 5 Tesla magnetic field, for low momenta, one has to calculate the integral in order to obtain the finite difference equation of motion:  .0_zzTpdtBp_Tp and the magnetic field dependent change in momentum is now given by the equations /5.0;25.01))/(2222_tBtBpBppzzTzTT (_ This introduces correction terms in the magnetic field dependent momentum change. FERMILAB-CONF-05-556-E   ALCPG1006 2The term - · pi ; (i=x ,y)  is a friction term. The energy loss due to ionization in matter (for step n of the trajectory) is given by x,y,zidsnpnppnEdxndEp iMatteri ,... ))(/)(()/)(()/)((    Here ds is the distance traversed by the particle. 10000 bbar-b jets events were analyzed with the improved software.  3. MUON-IDENTIFICATION WITH THE IMPROVED METHOD 3.1. The Algorithm   The analysis starts with reconstructed and well fitted tracks from the tracker with energy deposit in the Electro-Magnetic Calorimeter, EMCal, the Hadron Calorimeter, HDCal and MUDet within a ( , ) angular width around their path.  The angular width is optimized for each sub-detector [1],[2]. The Muon Detector trajectories are selected if #hits 12 in 12 or more layers within the ( , ) width.  For tracks below 10 GeV a (1/p) dependent bin has been chosen. Also, tracks which do not leave a signal in the last 5 layers of HDCal are rejected. This last cut takes advantage of the fact that most of the hadrons interact well before the end of HDCal and that the hadron showers are short whereas the muons have a penetrating power. Therefore the muons, reach the end of HDCal and leave hits in the 5 last layers.  An additional cut requires the layers with the minimum number of hits to have 1 or 2 hits, another characteristic of minimum ionizing particles(MIP). This scans the track for regions of low activity in which MIP pattern can be checked. 3.2. Performance of Muon ID . The effect of the cuts is shown in Table I. The detection Efficiency is actually better than reported because some of the undetected tracks share hits between barrel and end-cap and will be detected in a 4 detector. Also, the normalization is to the total number of reconstructed-final-fitted tracks, whereas part of these do not reach 12 layers (see above) and should be subtracted.                     Table I: Muon Identification in 10000 b-bbar with at least 12 layers/12 Hits in the Muon Barrel  Detector Muons Pions Kaons Protons Tracker Recons. Final 739 18024 4303  1712 Tracker Good Fit  715 17120 4072 1579 HDCal     1 min Hits  2                  5 Last Layers > 0 Hit   700  700 588 357 247 204 26 15 MUDet,   12 hits  12 layers 671 77 50 5 MUDet -  Min Hits  2 ; Max Hits  7 670 59 39 5 Efficiencies of Detection And  Hadrons Rejection Det. 94% Rej. 1/305 Rej. 1/110 Rej. 1/342 The overall muon detection efficiency obtained without discounting for the effects described above is ~94%. This covers the entire momentum range. It is ~100% at, and above, 10 GeV. For muon momentum at or above 3GeV the    ALCPG1006 3detection efficiency increased with the Runge-Kutta correction from ~30% to 66%.  A Kalman filter is used next in order to account for multiple scattering and other random processes and get a better separation of muons from hadrons.  3.3. The Principle The Kalman Filter is a method used to construct processes for which the information at each step can be fully derived from the information at the preceding step and where a covariant matrix of error is available at each step. The Kalman Filter is composed of two components.  The propagation component, using the propagation matrix, propagates the  information and the error information at each step.  The filtering component combines the information received at the end of the step to the information from data measurements.         Figure 1: Stepping in the Passive Material (1, 2, 3,4, ) and recording in the Active Material step 5  The Kalman Filter operates by using a State Vector that describes the processes history. For the present case, a phase space point is used as the State Vector. The State Vector change includes the effects of dE/dx and of the magnetic field in a step by step procedure.  It is calculated using the transport matrix (steps 1, 2, 3, 4 shown in Figure 1) in the passive material. The same analytical form is used in the stepper, but is now translated into a transport matrix, with the Runge-Kutta term included. The state vector at location k-1 is propagated using the propagation matrix, together with the covariant matrix, represented by the cone, which provides the information on random processes,. Following the propagation step, the State Vector is updated using Kalman filtering. The filtering combines the information derived from the propagation with the measurement made at the new point to produce an optimal state vector. It takes place when the hit is recorded in the active material (step 5) as shown in figure 1. The procedure defines, for each radius, a ( , ) dynamic path in which the hits left by the particle in the active material of the sub-detectors are the data collected.    3.4. The Equations  The stages of the Kalman Filter are described by the equations below.     ALCPG1006 4materiel passive in nPropagatio    )1()()()()(111111kTkkkkkkkQPPxxrecorded is signal the where                             or,scintillat in K.Filter,       )2()(]1[)()]([)()(kkkkkkkkkkPHKPxHzKxx   IMatrix (3)K.Gain   011||])([)(pQRHPHHPKkkTkkkTkkkQk is the noise from Multiple scattering-  a [6x6] matrix, Rk  is the measurement error,  dx, dy, dz  -  a [3x3] matrix Hk is the measurement matrix -  a [3x6] matrix, k  = propagation matrix, applied in passive material- [6x6] The matrix, Xk(-) is the extrapolated vector state (x, y, z, px, py, pz)  -[6x1] , Zk is the measured quantities ( , ,r) translated into (x,y,z)-[3x1] matrix, Xk(+) is the state vector after applying the Kalman filter-[6x1] Matrix,Kk is the Kalman Gain matrix - a [6x3] Matrix The transport matrix is given below in some detail.  The matrix represent the propagation equations given in [1] and [2], dT is the time taken by step n., I is a [6x6] matrix Unity. The term -  · pi ; i=x ,y  is a friction term is in the diagonal in the Transport Matrix   )           dz000dy000dxRc/Pabs100*dE/dxcdedx ;          bb ; bbbaabaa   )/(pz)/(py)/(px)/(z)/( y)/(xbbbaabaa pzpypxz (-)yx01-k1-k1k-1k-1-k1-kkkkk kkeslan GeV, B(T, energy ice in (cm)disBclightnEnpqBzfEclightdxyzctgrzryrxcXxlanXxcPMeVdxyzdxyzdxyzcdedxcdedxBzfBzfcdedxabIdTIdTzxn tan**)()(**3.0)(;/*100;cos;sin1);0/(038.01(0/)/6.13(;000000)(000)()(0)()(*;**)()()()()(0The state vector at location k-1, is propagated using the propagation matrix, to location k. The choice of the state vector as the phase space point allows the use the stepper Algorithm translated into the propagation matrix. In the notation adopted the (-) indicates the extrapolation and the (+) the Kalman weighting.     3.5. The Results The results for single muons at 4GeV before and after the application of the Kalman Filter are shown in Figure 2.      ALCPG1006 5The figure below shows  (track-hit)  between the extrapolated track and the hit at different depths (different layer) in the Muon Detector. In the LHS of the figure the width of the distribution is driven by the resolution at 4 GeV and the hit collection path covers ~ 3bins in . The RHS of the figure shows that one can restrict the hit collection to one bin in . The Filter helps to master random effects such as multiple-scattering when particles lose sufficient amount of momentum and enters the domain where these effects are dominant. As a result, the Filter allows a better separation between muons and neighboring hadrons in jets.                                            Figure 2: (track-hit)  at 4 GeV Before and After applying Kalman Filter 4. CONCLUSIONS The second order corrections and the Kalman filter were implemented in the Muon code to improve the efficiency in the low momentum range. The improved analysis includes multiple scattering, the magnetic field effect integrated using the Runge-Kutta method and the loss of energy through ionization, dE/dx.  The error at starting point has been chosen to be the angle bin size in the calorimeter ECAL. Prior to the entry into ECAL the paths are constructed by the stepper. The Kalman filter reconstructs realistic propagation at each step and allows hits to be collected in a narrower kinematic band. As a result of applying the Kalman Filter, the cuts around the track path may be restricted to a minimum even for low momenta. This was previously possible only at higher momenta. This new capability is particularly important to allow for an effective separation of particles in jets.                               References [1]C. Milsténe, G. Fisk, A. Para, Test of the Charged Particle Stepper With Muons , presented at the American Linear Collider Physics Group Workshop- at Victoria-BC-Canada- Fermilab,28Aug.04-Tm-2274-E -C04-07-28. [2] C. Milsténe, G. Fisk, A. Para,   Improvements to the Muon Tracking and Identification Software at the Linear Collider , Fermilab-Conf-C05-314-E,presented at the Linear Collider Workshop at Stanford-USA-19March05, Under PSN1011 of the Workshop Proceeding.      This document was created with Win2PDF available at http://www.daneprairie.com.The unregistered version of Win2PDF is for evaluation or non-commercial use only.

Muon ID- Taking Care of Lower Momenta Muons

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