1. Detector Alignment with Tracks Wouter Hulsbergen (Nikhef, BFYS)

2. 2 Detector alignment LHC silicon detectors provide <20 micron coordinate resolution resolution of a point inside a silicon wafer challenge: maintain such resolution in structures that contain many wafers requires accurate positioning of detector-elements, both active (when installing) and passive (in software) alignment is the procedure that calibrates detector positions input to the alignment survey: a measuring rod, RASNIK,... tracks: reconstructed trajectories from charged particles

3. 3 Alignment parameters alignment parameters: rotations and translations of solid objects in space alignment performed at different levels of granularity, with different constraints from assembly or survey smallest granularity: silicon wafer, driftchamber module etc other DOFs (e.g. deformations of wafers) usually considered later number of parameters considered in alignment of tracking detectors see e.g. proceedings of 1 st LHC alignment workshop

4. 4 Tracks and residuals * * * * * * x z detector plane hit: strip/wire/pad with fixed coordinate x track model, e.g. x(z) = a 0 + a 1 z track fitting and alignment is all about hit 'residuals' track parameters

5. 5 Track fitting * * * * * * the track fit is a 'least-squares-estimator' → minimizes track 'chi-square' x z minimization performed with (semi-) analytic method 'minimize' means Newton-Raphson method for finding the 'zero' of a non-linear function it is not so different from what's happening inside MINUIT

6. 6 Perfect detector: residuals are unbiased * * * * * * x z x_hit - x_track residual RMS ~ detector position resolution ⊕ multiple scattering etc

7. 7 Misaligned detector: biased residuals x z note: one layer was misaligned … but next layer has biased residuals as well typical problem in detector alignment: residuals from track fit are correlated * * * * * * x_hit - x_track

8. 8 Alignment using residuals simple alignment method: extract misalignments from residual histograms not easy to extend to detector displacements other than measurement direction no straightforward method to deal with correlations, especially in 'segmented' detectors tracks constrain correlated movements alignment becomes a bookkeeping problem: 'residuals in module A1' with respect to tracks in B1, C2, D3' etc most popular solution: 'minimum chisquare method for alignment' consider the chi-square of a sample of tracks minimize this chi-square simultaneously with respect to alignment parameters and track parameters

9. 9 Minimum chi-square method for alignment the solution to this minimum chi-square problem can again be written as 'the big matrix' 'the big vector' average residuals correlations between elements change in alignment parameters eliminating the track-parameters from this problem is actually not totally trivial need to exploit that different tracks only correlated via alignment parameters best known implementation of this idea: MILLIPEDE by Viktor Blobel

10. 10 Weak modes: poorly constrained common movements special complication in alignment with tracks: some (linear) combinations of alignment parameters are unconstrained global translation z-scale shearing more dangerous than unconstrained modes are so-called 'weak modes' 'statistically-underconstrained' common movements in a track samples with finite size extremely sensitive to mistakes ('outliers') in track reconstruction can lead to poor convergence of alignment procedure weak modes are the major concern in detectors that require alignment of many elements LHCb inner tracker: silicon tracker with O(500) ladders LHCb outer tracker: drift chamber with 216 modules CMS/Atlas inner detectors

11. 11 Example: weak modes in central Si tracker (from C. Escobar, Vertex 2008) weak modes affect physics.... but almost everything that affects track, vertex or momentum resolution can be extracted from data

12. 12 Constraining 'weak modes' design: overlap, redundancy overlaps constrain radial expansion and clamshell effects different data sets: cosmics, beam-halo, magnet-off off-axis events constrain twist and eleptical distortions survey constrains scale, like z-scale of LHCb VELO survey measurements multi-track constraints: vertex, invariant mass, beam kinematics mass constraints fix curvature bias vertex constraints fix clamshell before after

13. 13 Example of unconstrained mode in LHCb spectrometer LHCb spectrometer measures 'kink' of particle around magnet axis: kink ~ Q/p xz shearing of the tracking stations leads to bias in the kink → momentum bias magnet forward tracker velo-TT Q/p bias → bias in mass as function of asymmetry of decay can be used to extract shearing with <100 micron precision from J/ ψ→μ + μ ­ decays shearing → Q/p bias J/psi mass peak versus p( μ + ) - p( μ ­ ) 10 m

14. 14

15. 15