1. Sergey Gorbunov Ivan Kisel DESY, Zeuthen KIP, Uni-Heidelberg
An analytic formula for track extrapolation in an inhomogeneous magnetic field
Sergey Gorbunov Ivan Kisel
DESY, Zeuthen KIP, Uni-Heidelberg
KIP
ACAT 2005, DESY, Zeuthen
May 22-27, 2005

2. Extrapolation in Magnetic Field
Differential equation of charged particle motion in a magnetic field
Along Z direction :
ACAT05 DESY, Zeuthen
S. Gorbunov, I. Kisel

3. Analytic Expression Coefficients: Equation of motion:
Extrapolation error:
ACAT05 DESY, Zeuthen
S. Gorbunov, I. Kisel

4. Analytic Expression Features
B(z)
B( (z0+z)/2 )
Real track (theoretically)
Track guess (practically)
B(z0) z0
(z0+z)/2 z
Features :
Field is taken as a function of z coordinate.
ACAT05 DESY, Zeuthen
S. Gorbunov, I. Kisel

5. Analytic Expression Features
The precision of extrapolation does not depend on a shape of the magnetic field.
One can cut off the higher-order terms in the series.
ACAT05 DESY, Zeuthen
S. Gorbunov, I. Kisel

6. Example: Third order extrapolation in a homogeneous By field
Analytic 3
Example: Third order extrapolation in a homogeneous By field
Small parameter:
ACAT05 DESY, Zeuthen
S. Gorbunov, I. Kisel

7. Coefficients in the Analytic Formula (CBM)
Analytic Light
Analytic 1 cx
= 1.13e-04 cy
= 5.64e-04 cz
= 3.09e-04
cxx
= 3.72e-07
cxy
= 2.57e-06
cxz
= 1.25e-06
cyx
= 3.95e-06
cyy
= 2.39e-04
cyz
= 3.04e-05
czx
= 5.21e-07
czy
= 4.64e-06
czz
= 3.74e-06
cxxx
= 5.15e-09
cxxy
= 2.47e-08
cxxz
= 8.73e-09
cxyx
= 3.36e-08
cxyy
= 3.83e-07
cxyz
= 6.57e-08
cxzx
= 5.28e-09
cxzy
= 3.20e-08
cxzz
= 1.78e-08
cyxx
= 4.81e-08
cyxy
= 5.46e-07
Cyxz
= 9.34e-08
cyyx
= 7.67e-07
cyyy
= 5.85e-05
Cyyz
= 3.75e-06
cyzx
= 1.30e-07
cyzy
= 1.35e-06
Cyzz
= 4.03e-07
czxx
= 8.33e-09
czxy
= 3.35e-08
Czxz
= 1.47e-08
czyx
= 4.76e-08
czyy
= 8.63e-07
Czyz
= 1.33e-07
czzx
= 1.59e-08
czzy
= 9.91e-08
Czzz
= 5.49e-08
Analytic 2
Analytic 3
ACAT05 DESY, Zeuthen
S. Gorbunov, I. Kisel

8. Performance of Extrapolation (CBM)
Method
Residuals
Pulls
p/p x y tx ty
q/p
Runge-Kutta 4
0.64 27 24
1.5
1.17
1.05
1.01
1.02
1.00
Analytic 3
1.18
Analytic 2
0.68
1.30
1.08
1.03
Analytic 1
0.94 30 25
1.90
1.37
1.10
Analytic Light
1.19
Analytic Central
2.49 38
1.7
3.77
2.23
1.33
ACAT05 DESY, Zeuthen
S. Gorbunov, I. Kisel

9. Implementation in CBM CA Track Finder
ALL MC TRACKS
RECONSTRUCTABLE TRACKS
Number of hits >= 4
REFERENCE TRACKS
Momentum > 1 GeV
Track category
Efficiency, %
Reference set
99.45
All set
96.98
Extra set
89.46
Clone
0.01
Ghost
0.61
ACAT05 DESY, Zeuthen
S. Gorbunov, I. Kisel

10. Conclusion
An analytic formula of extrapolation in magnetic field has been derived
Precision of extrapolation does not depend on a shape of the field.
The implementation in the CBM track fitting procedure has the same accuracy as the fourth-order Runge-Kutta method.
ACAT05 DESY, Zeuthen
S. Gorbunov, I. Kisel