1. This document is a working document and a base for discussions. Everybody who thinks to have better ideas is warmly invited to change it ! Muon alignment strategy straight tracks optical alignment high Pt track alignment (small/large, EC/barrel, BIS8, BEE) MS/ID alignment C. Amelung, F.Bauer, P.F. Giraud, C.Guyot Review on the different alignment steps and fitting procedures

2. There are also B-lines (MDT deformations), which will not be considered here (reference frame independent). Definition of A-lines A-line: difference of chamber stations position & orientation with respect to nominal. These differences are given within the SZT frame (local sector frame) defined in AMDB. Potentially 5 different type of A-lines (or intermediate geometries) have been identified. Each of this A-line type contains the information of the previous A-line. A 0 : Initial geometry using straight tracks A 1 : after 1 st optical alignment A 2 : after optical+ high Pt track fit (small-large, BIS8, BEE) A 3 : after Barrel-EndCap alignment using high Pt tracks A 4 : after MS-ID fit Different approaches are thinkable: - Start with A 1 (instead of A 0 ) during an absolute optical alignment - combine A 2, A 3, A 4 in one single fit For each of the A-lines an error Matrix M should be provided.

3. A 0 : Alignment with straight tracks Goal: provide a reference geometry G 0 (  sag <40  m), when the magnet is off. If the optical absolute mode is good enough (see EndCap), straight tracks are not necessary to get G 0. Otherwise it might be interesting to mix the knowledge coming from the straight tracks with the knowledge coming from the absolute calibration of the optic Data type: - Data set are (1) cosmics, (2) B=0T tracks coming from the IP and (3) halo beam. - Required statistics for (2): ~20.000 tracks (p T >10 GeV) per tower - 400 towers => 8 10 6 tracks - Expected rate at L=10 31 : 20Hz => 4.10 5 s = 100 hours = ~ 4 days Data taking: - In parallel the optical sensor response should be recorded: a) in order to determine a sensor response R 0 for the data-taking period b) and to follow time variation in the geometry Output: - A 0 -lines stored in a database - Error matrix M 0 which should be combined to all further relative mode error matrices

4. Data treatment: Use best known geometry as input (survey, absolute optical alignment?). 2 methods thinkable: a) i) make an absolute optical alignment and determine a 1 st geometry (A abs -lines) ii) use this 1 st geometry as external loose constrains in the the straight track fit Production of A o -lines b) make a combined fit using straight tracks and absolute optical alignment and put degrees of freedom in such a way that missing or weakly determined optical calibration can be determined. Further use of A 0 -lines optional, the initial geometry is in the alignment constants. 2 ways to perform the fits in the (a) and (b) method: 1) sector by sector fit (i) perform sub-fits by sector (production of A sector -lines and error matrix M sector ) (ii) select overlap tracks between sectors (small/large and EC/Barrel) and refit all A lines for these overlap tracks using constraints coming from the A sector lines and the error matrix M sector 2) global fit: solving a huge linear system A 0 : Alignment with straight tracks (2)

5. The optical system can work in two modes: –Absolute: provide A-lines in the Atlas coordinate frame A abs with an error matrix M abs (after  2 rescaling) –Relative: Starting with a reference geometry G 0 (corresponding to A-lines A 0 with error matrix M 0 and to sensor response R 0 ) The difference  A=A-A 0 is fitted using the difference in sensor response R-R 0. The new A-lines are A 1 = A 0 +  A with a error matrix M rel (M rel2 = M 02 +M (  A)2 ). In both fit, the nominal position of the MDT stations is entered via constraints with a rather large error (e.g. 20mm/20 mrad). Thus the coordinate frame (i.e. the Atlas reference frame) is defined with respect to the average nominal position of all MDT stations. The Muon spectrometer can be connected to the (LHC or ATLAS reference frame) by using the survey targets in the fit (either as sensors or as additional constraints) The alignment fit is processed after each readout loop over the sensors (~every 30’). The constants A abs and A rel are stored in Oracle and COOL for this IoV. The error matrices are big (e.g. ~4000x4000 for the barrel or the EC). They should be calculated and stored only when the situation dramatically changes (i.e. breakdown of sensor) A 1 : 1 st optical alignment

6. A 2 : small-large High Pt alignment 3 methods: 1 st method: Pseudo track sensor + optic fit Pseudo track sensor: difference of the momentum reconstructed in the small versus momentum reconstructed in the large sector (called 3+3 sensor). Determination of pseudo track sensors done at Tier2 in Munich and then stored on condition database at Cern. In order to limit the number of Pseudo Track Sensors it might be necessary to combine PTS passing through the same  -  bin. 2 nd method: track fit using optic as constraint Set constraints using A 1 -lines and error matrix M 1 coming from optical fit. Make a fit using these constraints and the tracks in the overlap region. 3 rd method: track + optic fit Fit using optic and tracks using Milepede like method. 2 ways to do these fits: a) sector by sector (keep large sector parameter fixed and leave free parameters for small sector chambers) b) global fit (might be the simplest way..)

7. A 2 : BEE and BIS8 High Pt alignment No combination with optic necessary. The fits are done using tracks only.

8. A 34 : EndCap-Barrel-ID fit We say that the internal deformation of the 3 sub-detectors is best determined by their own internal alignment. => Fit with 3*6=18 parameter => 12 free parameter. More sophisticated fits have one potential danger, here an extremistic version: f.e. if ID deduce by error that the ID is round while reality is eggshape, we could deduce that the Barrel is eggshape, while reality is round. Otherwise we have to proceed in 2 steps: distinct production of A 3 and of A 4

9. 2 Procedures similar to the previous step: –Determine pseudo track sensors for tracks crossing BIS7/BML6/BOL6 and EIL4 (diff between segment in EIL4 and segment extrapolated from the barrel track). –Refit all A-lines from the EC (A A +A C : ~3500 variables) + 16 sets of 6 variables for each global barrel sector positions (  r i,   ), using the pseudo tracks sensors + the error matrices from the previous step (optical+small/large). The barrel error matrix applies to the transformed A-lines T(  r i,   )A i –The new A-lines are the fitted EC ones and the transformed barrel A-lines. A new global (~7000x7000) error matrix has to be provided –A full fit of all A-lines a la Millepede can also be considered. The error matrix from the previous step would be used as a constraint The calculation of the final error matrix comes automatically from the fit. This fit would be done at the calibration centre. A 3 : EndCap-Barrel High Pt alignment

10. Results of the MS/ID alignment: –48 (or 64 if we distinguish A and C sides for the barrel) sets of 6 parameters (  r i,   ) with error matrices M i describing the positions of the barrel and EC sectors w.r.t. the ID considered as perfectly aligned Combination with A-lines from the previous step A 3 with error matrix M 3 –The A-lines of a sector from the MS/ID fit are given in the ID coordinate frame by T(  r i,   ) A oi –The ID position in the Atlas coordinate frame is described by the unknown set (  r ID,  ID ) which has to be fitted –Do a full fit of all A-lines in the Atlas coordinate frame and of the displacements parameters of the ID and of all MS sectors with the following terms in the  2 :  2 = t M o (  i (T -1 (  r i,   )T(  r ID,  ID ) A i - A oi )) 2 M o +  i t M i (  r i,   ) 2 M i –The final A-lines in the ID coordinate frame are given by T(  r ID,  ID ) A A 4 : MS-ID alignment

11. Appendix: error matrix for a transformed set of A-lines Q: Given A-lines A with error matrix M, what is the error matrix M T for the transform set T(  r,  )A with a error matrix E on (  r,  )? A i = (  s i,  i ) for station i located at L i w.r.t. the centre of rotation of transformation T TA i = (  s i +  r+L i x ,  i +  ) = M ij +  ij. xxxx

12. A-lines quality and control plot A good way to see the quality of the alignment are: contour plot in z-  showing the expected momentum resolution. This plots should be saved in form of table in the database, in order to give weights for the momentum measurement of different muon tracks.