The strategy for track based alignment of the CMS tracking and muon detectors is presented. After an overview over the used data samples, the general alignment strategy is presented, with a focus of the procedures envisaged at the start of data taking in 2008. The three currently used alignment algorithms are discussed and first results on their application to the CMS tracker are presented, as well as studies on the alignment of the Muon detector with tracks.The strategy for track based alignment of the CMS tracking and muon detectors is presented. After an overview over the used data samples, the general alignment strategy is presented, with a focus of the procedures envisaged at the start of data taking in 2008. The three currently used alignment algorithms are discussed and first results on their application to the CMS tracker are presented, as well as studies on the alignment of the Muon detector with tracks

Schilling, Frank-Peter

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Track-based alignment in the CMS detectorF.-P. Schillingafor the CMS Collaborationa CERN, Geneva, SwitzerlandAbstractThe strategy for track-based alignment of the CMS tracking and muon detectors ispresented. After an overview of the data samples used, the general alignment strategyis presented, with a focus on the procedures envisaged at the start of data-taking in2007/8. The three currently used alignment algorithms are discussed and the firstresults of their application on the CMS tracker are presented, as well as studies on thealignment of the Muon detector with tracks.20.1 IntroductionThe alignment of the CMS tracking and muon detectorsrepresents a demanding challenge, because the numberof alignment parameters to be determined with high ac-curacy is very large. In particular, aligning the∼ 15 000silicon modules of the pixel and strip trackers with aprecision which is comparable to or better than their in-trinsic resolution of 10–50 µm, requires solving a prob-lem with O(100 000) unknowns. In addition to surveymeasurements at construction time and optical (laser)alignment [1, 2], track-based alignment will be a neces-sary ingredient to approach this huge challenge (see alsoRef. [3]).At the time of this workshop, a lot of activitiesrelated to track-based alignment were already ongoingin CMS, and are summarized in Ref. [4]. Three dif-ferent alignment algorithms have been implemented inthe CMS software within a common framework. Align-ment studies applying these algorithms to various MonteCarlo data sets have been performed. The advantage ofhaving several algorithms at your disposal is that the re-sults obtained can be cross-checked using different al-gorithms with different systematics (see Section 20.2 onthe data sets considered for alignment). Other activitiesrelated to track-based alignment are the developmentof a dedicated alignment stream produced during theprompt reconstruction at the Tier-0, which uses a specialreduced data format (‘AlCaReco’). It will enable us torun track-based alignment with a short turn-around timeafter the data have been taken. Moreover, a frameworkis being developed in order to combine the results oftrack-based and laser alignment with survey data. Thebenefits of using mass and vertex constraints, as well asof overlapping tracker modules are also studied. Furtherwork is ongoing in order to establish observables sensi-tive to misalignment other than χ2, in order to fix globaltransformations which leave the χ2 unchanged.20.2 Data samplesThe following data samples are currently considered foralignment:– High-pt muons from Z,WThese constitute the primary source of high-quality tracks for alignment, because of their hightransverse momentum and small amount of mul-tiple scattering in the tracker material. A compi-lation of the expected event rates after the HLT isshown in Table 20.1.– Cosmic muonsCosmics are a valuable source of tracks since theyare available before the first beams are in theLHC machine. They are particularly useful forthe alignment of the barrel tracker and muon de-tectors. Estimates show that the rate for cosmicmuons accepted by the L1 trigger and stand-alonemuon reconstruction is around ∼ 400 Hz [5].– Beam halo muonsAs soon as the LHC is commissioned with sin-gle beams, near-horizontal beam halo muons con-stitute a valuable source of tracks for the align-ment of the tracker and muon end-caps. A rate of∼ 5 kHz is expected per side accepted by the L1muon trigger and stand-alone muon reconstruc-tion [5]. A problem arises for the tracker sincethe muon end-cap trigger has no acceptance forradii covered by the tracker end-cap. In order totrigger beam halo muons within the tracker accep-tance, dedicated scintillators will be installed.181– Muons from J/Ψ and b hadron decaysEven though having a comparatively small trans-verse momentum spectrum, these will be veryuseful, in particular at the start-up of the LHC,when luminosities will be modest and muonsfrom Z,W decays not yet available. In addition,for muons from J/Ψ→ µ+µ− an invariant massconstraint can be used.– Isolated tracks from QCD eventsAt low luminosities this will be the only source ofcollision tracks which might be useful for align-ment. Multiple scattering in the tracker material isclearly an issue, but studies show that these eventscould be beneficial at least for the alignment of thepixel detector.20.3 Alignment strategyThe earliest information on the tracker and muon align-ment will come from survey measurements carried outduring construction, as well as from the laser align-ment systems. The alignment can then be improvedupon with cosmic and beam halo muons. The currentLHC start-up schedule foresees a 2–3 week calibrationrun at√s = 900 GeV and luminosities not exceeding1029 cm−2 s−1 at the end of 2007. Since no high-ptmuons from W,Z decays will be available under theseconditions, minimum-bias QCD events will constitutethe only source of tracks for alignment. In addition, thepixel detector will not yet be installed. If possible, a firsttrack-based alignment of composite strip tracker struc-tures (layers, etc.) would be performed.During the winter shutdown 2007/8 the pixel de-tector will be installed and the subsequent pilot physicsrun will strive for luminosities up to 1032 cm−2 s−1.This will allow large statistics of high-quality muontracks from W,Z decays to be accumulated in a shorttime (Table 20.1). A two-step procedure is currentlyforeseen to align the tracker: first, a stand-alone align-ment of the pixel detector will be carried out. Second,the strip tracker will be aligned using the pixel detectoras a reference system.20.4 Alignment algorithmsTrack-based alignment has proven to be the optimalmethod for the alignment of large tracking detectors inprevious experiments. However, it represents a majorchallenge at the CMS because the number of degrees offreedom involved is very large: considering 3+3 transla-tional and rotational degrees of freedom for each of the∼ 15 000 modules leads to O (100 000) alignment pa-rameters, which have to be determined with a precisionof ∼ 10 µm. Moreover, the full covariance matrix is ofsize O (100 000 × 100 000).In CMS, three different track-based alignmentalgorithms are implemented in the reconstruction soft-ware, using a common software framework. Some ofthem have been successfully used at other experiments,others are newly developed. In the following sections,the main features and initial results of using these algo-rithms in CMS are summarized.20.4.1 General software frameworkWithin the CMS software it is not necessary to apply(mis)alignment corrections to the geometry at the simu-lation step. Instead, (mis)alignment can be applied ‘onthe fly’ during reconstruction. Dedicated software toolshave been implemented to move and rotate parts of thetracking or muon detectors in a hierarchical way [6].In addition, a so-called alignment position error can beadded to the intrinsic uncertainty of reconstructed hitsin order to take into account the effects of misalignmentin the track reconstruction. Two dedicated misalign-ment scenarios [7] have been implemented which em-ulate the expected misalignment for different phases ofdata taking: the First Data Taking scenario and the LongTerm scenario (for details see Ref. [8]). A fast track refithas been implemented in the reconstruction software, insuch a way that redoing the full pattern recognition isavoided 1.Alignment studies are performed using the re-duced AlCaReco format, in which only the tracks usedfor alignment are kept in the event (e.g., the two muontracks in the case of Z0 → µ+µ− events). This signif-icantly improves both the disk space needed as well asthe alignment algorithm performance.The alignment algorithms have been imple-mented in the standard CMS reconstruction software us-ing a common layer of software, which provides all fea-tures that are common to all algorithms, for instance themanagement of alignment parameters and covariancematrices, the calculation of derivatives with respect totrack or alignment parameters, input/output, and an in-terface to the CMS offline conditions database.20.4.2 HIP algorithmAn iterative alignment algorithm using the Hits and Im-pact Points (HIP) method was developed [9]. It is able todetermine the alignment of individual sensors by mini-mizing a local χ2 function depending on the alignmentparameters, constructed from the track-hit residuals onthe sensor. Correlations between different sensors arenot explicitly included, but taken care of implicitly byiterating the method, which involves consecutive cyclesof calculating the alignment parameters and refitting thetracks. The algorithm is computationally light becauseno inversion of large matrices is involved. An alterna-1The assumption that misalignment does not change the assignment of hits to tracks was verified for the case of not too large misalignments.182tive implementation of the algorithm is designed to aligncomposite detector structures for a common translationand rotation [10], for example pixel ladders or layers.The composite alignment involves only a small numberof parameters, and therefore a rather small number oftracks is sufficient to carry out alignment already in thebeginning at data-taking.The HIP algorithm has been used [10] for thealignment of the pixel barrel modules using the FirstData Taking misalignment scenario. The pixel end-capsand the strip tracker are not misaligned. The proce-dure has been iterated 10 times using 200 000 simulatedZ0 → µ+µ− events. Figure 20.1 (left) shows the differ-ences between the true and estimated alignment param-eters. The convergence is good, with r.m.s. values of7(23) µm for the x, y(z) coordinates, respectively. Thealgorithm was also applied to a test beam set-up [11].20.4.3 Kalman filter algorithmA method for global alignment using charged tracks canbe derived from the Kalman filter. The method is itera-tive, so the alignment parameters are updated after eachtrack. It can be formulated in such a way that no largematrices have to be inverted [12]. In order to achieve aglobal alignment the update is not restricted to the de-tector elements that are crossed by the track, but can beextended to those elements that have significant corre-lations with the ones in the current track. This requiressome bookkeeping, but keeps the computational load toan acceptable level. It is possible to use prior informa-tion about the alignment obtained from mechanical sur-vey measurements as well as from laser alignment. Thealgorithm can also be extended to deal with kinemati-cally constrained track pairs (originating from particledecays).The algorithm has been implemented in the CMSsoftware and studied in two small subsets of the silicontracker: a telescope-like section of the inner and outerbarrel, and a wheel-like subset of the inner barrel, con-sisting of 156 modules in 4 layers. The tracks used weresimulated single muons with pt = 100 GeV. Randommisalignment with a standard deviation of σ = 100 µmwas applied to the local x and y positions of the mod-ules. Results from the alignment of the wheel-like set-up are shown in Figure 20.1 (right). It shows the evo-lution of the differences between true and estimated x-shifts for layers 1 and 2. A total of 100 000 tracks wereprocessed. As can be seen, the speed of convergencedepends on the layer. For more details, see Ref. [13].20.4.4 Millepede-II algorithmMillepede [14] is a well established and robust programpackage for alignment which has been used successfullyat other experiments, for example at H1, CDF, LHCb.Being a non-iterative method, it has been shown that itcan improve the alignment precision considerably withrespect to other algorithms.Millepede is a linear least-squares algorithmwhich is fast, accurate, and can take into account corre-lations among parameters. In the least-squares fit, localtrack parameters and global alignment parameters arefitted simultaneously. The solution for the alignmentparameters is obtained from a matrix equation for theglobal parameters only. For N alignment parametersthis requires the inversion of a N ×N matrix. However,this method can only be used up to N ∼ 10 000 be-cause of CPU and memory constraints. The alignmentof the CMS tracker exceeds this limit by one order ofmagnitude. Therefore, a new version, Millepede-II [15]was developed, which offers different solution methodsand is applicable for N much larger than 10 000. InMillepede-II, in addition to the matrix inversion and adiagonalization method, a new method for the solutionof very large matrix equations is implemented. Thisminimum residual method applicable for sparse matri-ces determines a good solution by iteration in acceptabletime even for large N . For more details, see Ref. [16].Millepede-II has been interfaced to the CMSsoftware and the alignment of parts of the CMS trackerhas been carried out using different scenarios [15]. Asan example, Fig. 20.2 (left) shows hit residuals in rφ forthe new iterative method. Each individual sensor of thetracker was misaligned. The alignment procedure wascarried out in the barrel region (|η| < 0.9) of the striptracker using 1.8 million Z0 → µ+µ− events. The pixellayers and the outermost barrel layer were kept fixed,resulting in ∼ 8400 alignment parameters. The conver-gence is very good, and the results obtained are identicalto those using the matrix inversion method, but the newmethod is faster by about three orders of magnitude.Figure 20.2 (right) shows the required CPU timeas a function of the number of alignment parametersfor the diagonalization and matrix inversion methods, aswell as for the new method used in Millepede-II. It canbe seen that Millepede-II is expected to be capable ofsolving the full CMS tracker alignment problem withinreasonable CPU time.Millepede-II has also been used [17] to investi-gate the global correlations between alignment param-eters for the CMS tracker. It turns out that in certaincases these correlations can be very high (above 99%).Studies show that it is very important to combine sam-ples of tracks with different topologies, such as collisiontracks and cosmics, in order to reduce these global cor-relations.20.5 Muon alignmentThe CMS Muon system consists of 790 individualchambers with an intrinsic resolution in the range 75–100 µm. Excellent alignment of the muon system is par-183ticularly important to ensure efficient muon triggeringand good track momentum resolution at large momenta,where the resolution is dominated by the muon detector.For optimal performance of the muon spectrom-eter over the entire momentum range up to 1 TeV, thedifferent muon chambers must be aligned with respectto each other and to the central tracking system to within100–500 µm. To control misalignment during commis-sioning and to monitor further displacements during op-eration, which can be of the order of 1 mm, the CMSwill combine measurements from an optical-mechanicalsystem with the results of track-based alignment [18].Two approaches are pursued: alignment using trackswhich are extrapolated from the tracker, and a stand-alone muon alignment (Fig. 20.3).20.6 ConclusionsThe alignment of the CMS tracker and muon detectorsconstitutes a significant challenge because of the largenumber of parameters (∼ 100 000 in the tracker), as wellas the high intrinsic resolution of the detectors.Even though LHC operation is still more thanone year away, a lot of activities related to track-basedalignment in the CMS are already ongoing. The ini-tial results obtained with the three alignment algo-rithms considered are very promising, although a realis-tic alignment of the full tracker at the sensor level is yetto be demonstrated. In addition, real data from trackertest set-ups and from the Magnet Test and Cosmic Chal-lenge are being studied for alignment. Work is ongoingon improving further the alignment software and strat-egy, in order to be well prepared once the first collisionsare delivered by the LHC.References[1] I. Vila, The CMS hardware alignment system,these proceedings.[2] P. Arce, COCOA: CMS Object-oriented Code forOptical Alignment, these proceedings.[3] M. Weber, The CMS alignment challenge, theseproceedings.[4] CMS Collaboration, CERN/LHCC 2006-001(2006).[5] E. Barberis et al., CMS Note 2006/012 (2006).[6] I. Belotelov et al., CMS Note 2006/008 (2006).[7] P. Vanlaer et al., CMS Note 2006/029 (2006).[8] G. Steinbrueck, Impact of misalignment onphysics, these proceedings.[9] V. Karimaki et al., CMS CR 2003/022(2003), presented at CHEP 2003, La Jolla[arXiv:physics/0306034].[10] V. Karimaki et al., CMS Note 2006/018 (2006).[11] T. Lampen et al., CMS Note 2006/006 (2006).[12] R. Fruehwirth et al., CMS Note 2006/022 (2006).[13] R. Fruehwirth, Track-based alignment using aKalman filter technique, these proceedings.[14] V. Blobel, Millepede program descriptionand code, http://www.desy.de/~blobel/wwwmille.html.[15] P. Schleper et al., CMS Note 2006/011 (2006).[16] V. Blobel, Alignment algorithms, these proceed-ings.[17] M. Stoye, private communication.[18] A. Calderon et al., CMS Note 2006/016 (2006).Table 20.1: Anticipated rates of W± → µ±ν and Z0 → µ+µ− events after HLT in 2008Luminosity 1032 cm−2 s−1 2× 1033 cm−2 s−1Time Few weeks 6 months 1 day Few weeks One yearIntegrated luminosity 100 pb−1 1 fb−1 1 fb−1 10 fb−1W± → µ±ν 700 k 7 M 100 k 7 M 70 MZ0 → µ+µ− 100 k 1 M 20 k 1 M 10 M184 Iteration 0 2 4 6 8 10m]µx [∆ -400-300-200-1000100200300400 Entries  720Mean    1.964RMS     7.159m] µx [∆ -400 -300 -200 -100 0 100 200 300 400 N( Al i gn ab l es )0100200300400500 After I Iterations:I=0I=1I=5I=10 Iteration 0 2 4 6 8 10m]µy  [∆-400-300-200-1000100200300400 Entries  720Mean    -2.55RMS     6.843m] µy [∆ -400 -300 -200 -100 0 100 200 300 400 N( Al i gn ab l es )0100200300400500 Iteration 0 2 4 6 8 10m]µz  [∆-400-300-200-1000100200300400 Entries  720Mean    9.151RMS     23.13m] µz [∆ -400 -300 -200 -100 0 100 200 300 400 N( Al i gn ab l es )020406080100120140160180number of updates5000 10000 15000 20000 25000 30000m]µ [xδ-200-150-100-50050100150200number of updates5000 10000 15000 20000 25000 30000m]µ [xδ-200-150-100-50050100150200Fig. 20.1: Left: Alignment of the Pixel barrel modules with the HIP algorithm. The residuals in global coordinates are shownas a function of iteration, and projected for 0, 1, 5 and 10 iterations. Right: Kalman filter alignment. Residuals in local x forTIB layers 1 (top) and 2 (bottom) as a function of the number of processed tracks.m]µtrue - estimated position [-100 -80 -60 -40 -20 0 20 40 60 80 100# of events020406080100120140160180200220φ r∆ Entries  3661Mean   -0.1166RMS     4.597Fig. 20.2: Millepede II: Left: Residuals in rφ in the strip tracker barrel before (red) and after (black) alignment using Mille-pede II. Right: CPU time as a function of alignment parameters for matrix inversion (blue) and Millepede II.Mean  0.005919RMS    0.08903 (cm)1δ-0.6 -0.4 -0.2 0 0.2 0.4 0.6020004000600080001000012000 Mean   0.07451RMS   0.08907AlignedMisalignedEstimator for MB1 Mean  -0.003876RMS    0.06399 (cm)2δ-0.6 -0.4 -0.2 0 0.2 0.4 0.602000400060008000100001200014000Mean  -0.08079RMS   0.06887AlignedMisalignedEstimator for MB2Fig. 20.3: Muon alignment: rφ estimator for muon chambers in different wheels for aligned (solid line) and ±1 mm displacedsamples of W±185

Track-based alignment in the CMS detector

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