Material Effects and Error Propagation in STEP

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Material Effectsand Error Propagation in STEP

• Esben Lund, University of Oslo

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Track Parameters

• To reconstruct tracks we need to agree on a
  common set of track parameters.
• Since our track measurements are always done in
  some known part of the detector it is useful to
  recycle this information.
• Tracks are defined by two local positions on a
  plane or a line, corresponding to some active
  part of the detector.

Full set of track parameters

• In addition, tracks have two globally defined
  angles, the azimuthal angle f, and the polar
  angle q. f is the projection angle into the x-y
  plane, and q is the angle between the track and
  z-axis (beam).
• Finally, tracks have momentum and charge, q/p.

x1 x2 j q q/p
s1 c12 cl3 cl4 cl5
c21 s2 c23 c24 c25
c31 c32 s3 c34 c35
c41 c42 c43 s4 c45
c51 c52 c53 c54 s5
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Track Fitting with a Kalman Filter

• This method is the basis for track fitting in
  much of the ATLAS tracking.
• Track fitting produces a number, the chi-square,
  indicating the quality of the track.
• The Kalman filter starts with a track state on a
  measurement surface A.
• It then predicts the intersection with the next
  measurement surface B along the track.
• The measurement on surface B is used to update
  the predicted state.
• This method does not involve big matrix
  inversions, and material effects are easily
  included.
• Measurements close to the predictions lowers the
  chi-square of the fit, indicating a well
  understood track.

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Material Effects

• In a 7000-ton detector like ATLAS, material
  effects must be included in the propagation,
  especially for the muons going through the whole
  detector.
• The detector material and distribution is defined
  in the detector geometry.
• At ATLAS energies the main energy loss comes from
  ionization of the material and the radiative
  effects of bremsstrahlung, direct ee- pair
  production and photonuclear interactions.
• Only ionization and bremsstrahlung are treated in
  STEP.
• In addition to the energy loss, multiple
  scattering is included into the error
  propagation.

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Ionization Energy Loss, Bethe-Bloch

• The mean energy loss from ionization is given by
  the Bethe-Bloch equation
• K is a constant, z is the charge of the incident
  particle, Z is the atomic number and A is the
  atomic mass of the material.
• I is the mean excitation energy, r is the density
  of the material, Tmax is the maximum kinetic
  energy which can be imparted to a free electron
  in a single collision and d is the density effect.

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Bremsstrahlung Energy Loss,Bethe-Heitler

• The mean energy loss to relativistic particles
  from bremsstrahlung is given by the Bethe-Heitler
  equation
• X0 is the radiation length of the material
  traversed, M is the rest mass and E is the energy
  of the incident particle.

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Energy Loss Calculated by STEP
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Efficiencies WhenIncluding Energy Loss
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Multiple Scattering

• Since STEP is a track estimator it always uses
  the mean values of energy loss and multiple
  scattering.
• The mean deflection of the track by multiple
  scattering is zero, but new correlations are
  introduced to the covariance matrix. These
  correlations are of the same order of magnitude
  as those coming from the error propagation
  itself.
• The multiple scattering is calculated separately
  and added to the covariance matrix after the
  parameter and error propagation is done
• J is the jacobian transporting the covariance
  matrix (described later).

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Multiple Scattering Layers

• Multiple scattering is calculated in ten layers,
  each given by this matrix
• L is the thickness of the layer, D is the
  remaining distance to the target surface.

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Testing the Multiple Scattering

• Multiple scattering is a stochastic process which
  can be tested by using a Monte Carlo simulation.
• This simulation is done by slicing the volume
  into 100 layers normal to the track. After going
  through each layer the track is deflected by a
  random polar angle, qms, taken from a Gaussian
  distribution with a standard deviation
• L is the thickness of the layer.

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Statistical Tools, Pulls and the c2

• The differences between the final track
  parameters of the undisturbed and the scattered
  tracks are called residuals. These can be
  compared to the multiple scattering covariance
  calculated by STEP.
• For this purpose we use the normalized residuals,
  or pull values
• and the chi-square
• i indicating the simulated tracks, j the track
  parameters and m the mean values.

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Evaluating Pulls and Chi-squares

• The beauty of the pulls and chi-square is that no
  prior knowledge of the test is needed to evaluate
  them.
• Since the pulls are offset by the average value
  m, and normalized by the square root of the final
  variance, their peaks should be at zero and their
  standard deviation equal to one.
• When integrating the standard chi-square
  distribution from the test chi-square value to
  infinity we get a probability value for the
  chi-square. Doing this many times we will get a
  flat distribution of p-values if our test
  chi-squares are distributed according to the
  standard chi-square.
• In short, the pulls should be Gaussians centered
  around zero with a standard deviation of one, and
  the p-values (calculated from the chi-square)
  should be flat from zero to one.

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Scattering Pulls and Chi-square
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Error Propagation

• The covariance matrix, indicated by the ellipses,
  is transported along the track from the initial
  surface to the target surface.

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The Covariance Matrix

• The covariance matrix is a symmetric 5x5 matrix
  containing the measurement uncertainties and
  correlations introduced by the limited resolution
  of the detector and the multiple scattering
• x are the local track parameters and lt...gt are
  the expectation values.
• The propagated covariance is given by a
  similarity transformation
• To do the error propagation we need the
  derivatives of the final track parameters with
  respect to the initial parameters, the so-called
  Jacobian, J.

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Finding the Jacobian Using the Bugge-Myrheim
Method

• The Jacobian is a 5x5 matrix
• We use the Bugge-Myrheim method for finding the
  Jacobian. The idea is to simply differentiate the
  Runge-Kutta-Nystrøm recursion formulaes with
  respect to the initial track parameters, and use
  these differentiated recursion formulaes for
  transporting the Jacobian in parallel with the
  track parameters.

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The RKN Recursion Formulaes

• The Runge-Kutta-Nystrøm recursion formulaes are
  given by
• h is the step length, k is the equation of
  motion, u and u are the global track parameters

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Differentiating the Recursion Formulaes

• Differentiating the recursion formulaes with
  respect to the initial local track paramers, xi,
  we get
• Dk is an 8x8 matrix

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Differentiating the Recursion Formulaes

• The derivatives of the Dk matrix contain the
  equation of motion, k, differentiated with
  respect to the global track parameters, u and u

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Derivatives with Respect to u

• Differentiating the equation of motion, k, with
  respect to the global track parameters u, we get

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Derivatives with Respect to u

• Differentiating the equation of motion, k, with
  respect to the global track parameters u, we
  get

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Testing the Error Propagation

• To test the analytical error propagation we use
  the pull and p-values introduced in the multiple
  scattering.
• This time the residuals are generated by varying
  the initial track parameters according to the
  initial covariance matrix.

• These residuals are compared to the covariance
  matrix propagated by STEP in the pulls and
  chi-square.

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Error Propagation Pullsand Chi-square
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Combined Testing of Error Propagation and
Material Effects

• The covariance matrix is propagated with energy
  loss and multiple scattering.
• Residuals are found by varying the initial
  parameters according to the initial covariance
  matrix and simulating the multiple scattering as
  shown earlier.