High accuracy momentum reconstruction with orthogonal polynomial sets E.P. Akishina, V.V. Ivanov and E.I. Litvinenko

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High accuracy momentum reconstruction with
orthogonal polynomial setsE.P. Akishina, V.V.
Ivanov and E.I. Litvinenko
Laboratory of Information Technologies,Joint
Institute for Nuclear Research, Russia
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An algorithm for a high accuracy momentum
reconstruction applying the points of a charged
particle trajectory through an inhomogeneous
field of the CBM dipole magnet is presented. A
set of representative trajectories is computed
and used to construct a multidimensional function
which gives the momentum in terms of observable
quantities.
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Uniform magnetic field approximation
Reaction AuAu at 25 AGeV
The momentum p is calculated by a formula
- constant
- deflection angle
Root Mean Square
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Method of accurate momentum restoration
In inhomogeneous magnetic field the deflection
angle f is a function
- the point in the first STS detector
- tangents of trajectory in this point
We have to construct the inverse function
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Method of accurate momentum restoration

• The procedure includes two steps
• A representative set of all relevant trajectories
  through the magnet is computed, and the
  corresponding set of angles is formed.
• This set is used to construct the explicit
  function, which gives the momentum in terms of
  observables.
• Each trajectory is defined by five variables
• - the coordinates
  of a point in the first STS
• - the tangents of
  a particle trajectory in the point
• - is the inverse of particle
  momentum.

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Method of accurate momentum restoration
Let be a range of i-th
variable.
We normalize i-th variable to the range -1,1
We choose the discrete points due to the
Tchebysheff distribution
The set of determines
the collection of fixed trajectories, which are
traced through the magnetic field and the set of
corresponding deflections
is calculated.
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Method of accurate momentum restoration
Let the range of f() be f() is
normalized to -1,1
discrete number of is chosen
Applying the inverse interpolation, we calculate
corresponding values of
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Method of accurate momentum restoration
Let be in the form
where T(x) is the orthogonal Tchebysheff
polynomial
The coefficients are calculated using
the formula
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Accuracy of momentum restoration
We choose
- number of points for which the trajectories
are calculated.
We choose
- the momentum values in the range 1-10 GeV/c
Total number of 5 x 5 x 5 x 5 x 7 4
375
Total number of coefficients can be decreased
without accuracy loss. The significant
coefficients are selected from the whole set
using the Fisher test 89 significant
coefficients. Here we consider only positively
charged particles.
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Accuracy of momentum restoration
Top plots (all 4375 coefficients)
show distribution of (left, in
MeV/c) distribution of
(right)
Bottom plots (89 coefficients) shows distribution
of (left, in MeV/c) distribution
of (right)
RMS 0.147
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Accuracy of momentum restoration
In order to estimate the accuracy of the method
on data close to real data, we used the GEANT
data (multiple scattering is included) s 0.263
Distribution of for GEANT data
(for positively charged particles).
Distribution of (in MeV/c) for GEANT
data (for positively charged particles).
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Accuracy of momentum restoration
The accuracy of the momentum restoration for all
(positive and negative) tracks s 0.342
Distribution of for GEANT
data (for all tracks).
Distribution of (in MeV/c) for GEANT
data (for all tracks).
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Accuracy of momentum restoration
For a small part of tracks (approximately 10),
the parameters of which were out of the ranges of
variables , we used
the approximation of the uniform magnetic field.
Figure shows the distribution of
(in MeV/c) for such events.
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Accuracy of momentum restoration
The accuracy of the method with using a Kalman
filter (both a multiple scattering and coordinate
resolution of the STS detectors 20 µm were taken
into account)
Distribution of (for negatively
charged particles)
Distribution of (for positively charged
particles)
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Conclusion

• Main results of this work can be summurized as
  follows
• The possibility to reconstruct charged particle
  momenta registered by coordinate STS-detectors
  with a high accuracy is demonstrated (on the
  basis of the GEANT data which include multiple
  scattering)
• The estimation of the momentum reconstruction
  accuracy applying the procedure of particle
  trajectory fitting on the basis of Kalman filter
  
• The algorithm for momenta restoration is
  elaborated. With the help of the later the
  estimation was realized both the accuracy of the
  mathematical method (which includes inaccuracies
  of the magnetic field) and the accuracy of the
  the momentum reconstruction on the basis of the
  GEANT data and data prepared by Kalman filter

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Our next steps are as follows

• To increase the accuracy of the algorithm by
  separate restoration of momenta for paricles with
  different charges.
• To increase the accuracy of the algorithm by
  dividing the whole interval of momenta on a few
  subintervals.