Global Chi2 alignment OF TGC chambers

1. YairMahalalel, Feb. 17th MMX Global Chi2 alignment OF TGC chambers

2. Complementary to optical/mechanical alignment. Advantages – 6D alignment Generated from experimental data Disadvantages – Module alignment is only relative to other modules (only option for most modules) Requires available tracks, ideally a working collider Track based alignment

3. Relies on /Tracking/TrkAlignment , a generic reimplementation of the inner detector toolset which has been developed for many years Provides all the necessary services, database interfaces etc. out of the box Uses ATHENA reconstruction and tracking for state of the art handling of data Current implementation

4. Minimization of Where V is the covariance matrix and are the residuals Requires calculating either analytically or numerically. Theoretical summary

5. Is implicit in the analytical derivative calculation. Incorrect when large non-uniformity exists along a chamber, e.g. in the MDTs. Can neither be used when aligning compound modules (e.g. wheels). In these cases the derivatives are calculated numerically by shifting and rotating the module and refitting the track. Adds computation complexity and non-trivial parameter dependence. Assumption of constant magnetic field

6. Currently a software design flaw prevents chamber shifting because TGC hits are combined to form a CompetingRIOsOnTrack measurement, generating a new surface which is the average of individual hit surfaces. The new surface is not connected to the module so the shifting the module doesn’t move the hit. A new approach is currently being tested. Numerical derivatives and TGC

7. Implementation complete – Level 3 (single chamber) align modules Using analytical derivatives Geometry transformations Debug ntuples Alignment DB I/O TGC alignment implementation

8. AMDB geometry description difference between TGC and MDTs Various enhancements to generic tools to support 2D chambers and second coordinate measurements Off by 2 chamber phi indexing Bug fixes

9. Found in AnalyticalDerivCalcTool by CSC aligners Triggers when a single chamber has one 1D measurement (wires) and one 2D (strips) Might explain strange behavior when running 6D alignment Patch received a few days ago. Will be validated by us. Known bug– derivative calculation

10. GeoModel support for aligning individual endcap TGC modules doesn’t exist yet – no code to read chamber eta index. A fix should be comitted in a couple of weeks by Stefania Spagnolo. Known problem – endcap support

11. Station description In ASZT file Typical A-line in ASZT file – W Stat jff jzz job Translations Rotations A T1E 1 2 0 1.234 1.234 1.234 0.00123 0.00123 0.00123

12. Convergence slow and erratic, and to the wrong point. Results not satisfactory

13. Implementation complete but validation is proving more confusing and slow than expected Currently known problems are in external services (but are being fixed) Hopefully once we finish validating these fixes we’ll start seeing more reasonable results Then – more serious validation of our code using MC and real data summary

14. Backup slides

15. Corrections to nominal chamber locations are collected from various sources – Optical sensors Resistive sensors Track based alignment The corrections specific for every run are applied by GeoModel at ATHENA initialization. Huge problem – thousands of modules to align How alignment works

16. Many approaches – Robust method – fits distributions to tracks Local fit – needs many iterations Global fit – few iterations, but potentially huge memory requirement. Algorithms also vary by their ability to handle magnetic fields (curved tracks), multiple Coulomb scattering, etc. Track based alignment

17. Main methodology – minimization of global error Where V is the covariance matrix and the Are the measurement residuals, defined as the distance between measurements and intersection of track with the sensor plane. Least squares linear expansion

18. The intersection of the extrapolated track with the sensor plane depends on two parameter sets – The track parameters The align parameters Least squares linear expansion

19. Should be diagonal, or at least block diagonal within the module but rarely is because of – Multiple Coulomb scattering Additional constraints (event main vertex) The covariance matrix V

20. The equation which needs to be solved is In order to solve it we need to calculate the full derivative Under the linearity assumption that for small enough changes. Solution condition

21. The linear expansion around the original yields the equation With the solution Solution for the track parameters

22. We can look at this expression for as function of and use its derivative to rewrite the full derivative as Where Solution for the track parameters

23. Similarly to our track parameter treatment, we can write the equation for the align parameter derivative – Where Solution of align parameters

24. We define the unit vectors crossing the strips, along the strips and perpendicular to the sensor plane. The two residuals are then And their derivatives with respect to a general parameter are Analytical residual derivatives

25. The intersection of the path length with the detector plane, , is given by the equation Which is solved iteratively. Can now be written as . Since is an implicit function of , The path length

26. The residual derivatives can now be written as where Or using local track direction of , General Residual derivatives