1. Sept. 2003 G. Zech, Iterative alignment, PHYSTAT 2003, SLAC. 1 A simple iterative alignment method using gradient descending minimum search G. Zech and T. Zeuner What is the problem? In modern HEP experiments some 10^3 ionizing particles / event are generated which are measured in position sensitive detectors with high precision. The posoition of the detectors of size meters has to be inferred from track measurements to a precision of order micrometer. Typical numbers ( CMS experiment): Silicon tracker: 10 000 000 channels 16 000 modules  96 000 parameters + vertex detector + muon detector + magnet parameters

2. Sept. 2003 G. Zech, Iterative alignment, PHYSTAT 2003, SLAC. 2 General Remarks Some scaling constants cannot be inferred from single tracks or straight tracks from a common vertex (squeezing, rotations, sharing) Magnet bending has to be disentangled from misalignment Tracks relate mostly projective moduls and not lateraly positioned modules. Residual distributions may look perfect (swamped by tracks passing through well aligned sequential modules). Badly aligned detectors produce artificial residuals centered at zero. Consequences use preferentially tracks through overlap region use kinematically constrained multitrack events

3. Sept. 2003 G. Zech, Iterative alignment, PHYSTAT 2003, SLAC. 3 Technical Solutions - Collect residuals of many tracks, infer parameters from overall least square fit. (V. Blobel et al. Durham 2002) successfully applied to H1, HERA-B vertex detector, - Iterative, using single events and gradient descending minimum search. (G.Z. +T.Z. applied to Hera-B Inner Tracker. use kinematically constrained multitrack events like V 0 or tracks from a common vertex.

4. Sept. 2003 G. Zech, Iterative alignment, PHYSTAT 2003, SLAC. 4 Properties of the iterative method Applicable to full events containing several tracks  connects lateral detectors. Updating, can be used online. Simple, robust and fast, no large matrices involved.

5. Sept. 2003 G. Zech, Iterative alignment, PHYSTAT 2003, SLAC. 5 Outline of the method Fit track or event (V 0 )    C ompute dependence of   on position parameters k of the detectors. Modify k in proportion to the gradient of   u i : measured coordinate u i f : fitted coordinate  k : learning constant (correction is small compared to deviation)  k : change applied to parameter

6. Sept. 2003 G. Zech, Iterative alignment, PHYSTAT 2003, SLAC. 6 Qualitatively: Move all detectors in such a way that the deviations are reduced

7. Sept. 2003 G. Zech, Iterative alignment, PHYSTAT 2003, SLAC. 7 How to get the correlation coefficient a)Computed by the reconstruction program: - modify k by small amout  k, - let program compute change  (u i -u i f ), b) Compute analytically c) Approximate by leading terms (Gradient descending minimum search is insensitive to approximations, you just have to move downhill)

8. Sept. 2003 G. Zech, Iterative alignment, PHYSTAT 2003, SLAC. 8 Leading terms for planar chambers: C u shift along measured coordinate C z shift perpendicular to plane C  azimutal rotation For example shift in z direction:

9. Sept. 2003 G. Zech, Iterative alignment, PHYSTAT 2003, SLAC. 9 Remarks To start with, a rough alignment is required to fit the tracks. The learning constants can be reduced according to improvement of the alignment. Updating the alignment constants after each event may not be practical. Averaging over a bunch of events is more efficient. Sometimes it is useful to adjust only part of the parameters or to check the influence of one of them. This is easily steered by setting learning constants equal to zero. The same events can be used repeatedly because a single gradient descending step does not exhaust the information. Drift time information should be excluded as long as the adjustment is not good enough to resolve the ambiguities. Once drift time is included short times should be eliminated.

10. Sept. 2003 G. Zech, Iterative alignment, PHYSTAT 2003, SLAC. 10 Gradient descending minimum search is depending on the metric. Changing the metric is equivalent to changing the learning constants. Convergence is slow if the gradient is low. Fortunately, the gradients in our case increse with distance from the minimum. There are no secondary minima. Efficiency is increased when large deviations are excluded. (truncated mean)

11. Sept. 2003 G. Zech, Iterative alignment, PHYSTAT 2003, SLAC. 11 Simple toy example (about 500 tracks, 100 iterations) 8 sequential planar detectors 4 vertical strips, x measurement 2 rotated by +5 0, 2 rotated by –5 0

12. Sept. 2003 G. Zech, Iterative alignment, PHYSTAT 2003, SLAC. 12 Change of position parameters with iteration

13. Sept. 2003 G. Zech, Iterative alignment, PHYSTAT 2003, SLAC. 13 Hera-B, one position parameter

14. Sept. 2003 G. Zech, Iterative alignment, PHYSTAT 2003, SLAC. 14 Residuals before and after alignment

15. Sept. 2003 G. Zech, Iterative alignment, PHYSTAT 2003, SLAC. 15 Residual distribution of one MSGC chamber after alignment

16. Sept. 2003 G. Zech, Iterative alignment, PHYSTAT 2003, SLAC. 16 Summary Gradient descending optimization of detector alignment is simple, easy to program and very general, applicable to an arbitrary number of detector parameters, it can be used online, it has successfully been applied to the Inner Tracker of Hera-B.