Alignment of the CMS tracker with LHC and cosmic ray data

24 July 2021

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Alignment of the CMS tracker with LHC and cosmic ray data / Chatrchyan S; Khachatryan V; Sirunyan AM; Tumasyan A; Adam W; Bergauer T; Dragicevic M; Ero J; Fabjan C; Friedl M; Fruhwirth R; Ghete VM; Hartl C; Hormann N; Hrubec J; Jeitler M; Kiesenhofer W; Knunz V; Krammer M; Kratschmer I; Liko D; Mikulec I; Rabady D; Rahbaran B; Rohringer H; Schofbeck R; Strauss J; Taurok A; Treberer-Treberspurg W; Waltenberger W; Wulz CE; Mossolov V; Shumeiko N; Gonzalez JS; Alderweireldt S; Bansal M; Bansal S; Beaumont W; Cornelis T; De Wolf EA; Janssen X; Knutsson A; Luyckx S; Mucibello L; Ochesanu S; Roland B; Rougny R; Van Haevermaet H; Van Mechelen P; Van Remortel N; Van Spilbeeck A; Blekman F; Blyweert S; D'Hondt J; Devroede O; Heracleous N; Kalogeropoulos A; Keaveney J; Kim TJ; Lowette S; Maes M; Olbrechts A; Python Q; Strom D; Tavernier S; Van Doninck W; Van Lancker L; Van Mulders P; Van Onsem GP; Villella I; Caillol C; Clerbaux B; De Lentdecker G; Favart L; Gay APR; Leonard A; Marage PE; Mohammadi A; Pernie L; Reis T; Seva T; Thomas L; Vander Velde C; Vanlaer P; Wang J; Adler V; Beernaert K; Benucci L; Cimmino A; Costantini S; Dildick S; Garcia G; Klein B; Lellouch J; Mccartin J; Rios AAO; Ryckbosch D; Diblen SS; Sigamani M; Strobbe N; Thyssen F; Tytgat M; Walsh S; Yazgan E; Zaganidis N; Basegmez S; Beluffi C; Bruno G; Castello R; Caudron A; Ceard L; Da Silveira GG; De Callatay B; Delaere C; Du Pree T; Favart D; Forthomme L; Giammanco A; Hollar J; Jez P; Komm M; Lemaitre V; Liao J; Michotte D; Militaru O; Nuttens C; Pagano D; Pin A; Piotrzkowski K; Popov A; Quertenmont L; Selvaggi M; Marono MV; Garcia JMV; Beliy N; Caebergs T; Daubie E; Hammad GH; Alves GA; Martins MC; Martins T; Pol ME; Souza MHG; Alda WL; Carvalho W; Chinellato J; Custodio A; Da Costa EM; Damiao DD; Martins CD; De Souza SF; Malbouisson H; Malek M; Figueiredo DM; Mundim L; Nogima H; Da Silva WLP; Santaolalla J; Santoro A; Sznajder A; Manganote EJT; Pereira AV; Bernardes CA; Dias FA; Tomei TRFP; Gregores EM; Mercadante PG; Novaes SF; Padula SS; Genchev V; Iaydjiev P; Marinov A; Piperov S; Rodozov M; Sultanov G; Vutova M; Dimitrov A; Glushkov I; Hadjiiska R; Kozhuharov V; Litov L; Pavlov B; Petkov P; Bian JG; Chen GM; Chen HS; Chen M; Du R; Jiang CH; Liang D; Liang S; Meng X; Plestina R; Tao J; Wang X; Wang Z; Asawatangtrakuldee C; Ban Y; Guo Y; Li Q; Li W; Liu S; Mao Y; Qian SJ; Wang D; Zhang L; Zou W; Avila C; Montoya CAC; Sierra LFC; Florez C; Gomez JP; Moreno BG; Sanabria JC; Godinovic N; Lelas D; Polic D; Puljak I; Antunovic Z; Kovac M; Brigljevic V; Kadija K; Luetic J; Mekterovic D; Morovic S; Tikvica L; Attikis A; Mavromanolakis G; Mousa J; Nicolaou C; Ptochos F; Razis PA; Finger M; Finger M; Abdelalim AA; Assran Y; Elgammal S; Kamel AE; Mahmoud MA; Radi A; Kadastik M; Muntel M; Murumaa M; Raidal M; Rebane L; Tiko A; Eerola P; Fedi G; Voutilainen M; Harkonen J; Karimaki V; Kinnunen R; Kortelainen MJ; Lampen T; Lassila-Perini K; Lehti S; Linden T; Luukka P; Maenpaa T; Peltola T; Tuominen E; Tuominiemi J; Tuovinen E; Wendland L; Tuuva T; Besancon M; Couderc F; Dejardin M; Denegri D; Fabbro B; Faure JL; Ferri F; Ganjour S;

Givernaud A; Gras P; de Monchenault GH; Jarry P; Locci E; Malcles J; Nayak A; Rander J; Rosowsky A; Titov M; Baffioni S; Beaudette F; Busson P; Charlot C; Daci N; Dahms T; Dalchenko M; Dobrzynski L; Florent A; de Cassagnac RG; Mine P; Mironov C; Naranjo IN; Nguyen M; Ochando C; Paganini P; Sabes D; Salerno R; Sauvan JB; Sirois Y; Veelken C; Yilmaz Y; Zabi A; Agram JL; Andrea J; Bloch D; Bonnin C; Brom JM; Chabert EC; Charles L; Collard C; Conte E; Drouhin F; Fontaine JC; Gele D; Goerlach U; Goetzmann C; Gross L; Juillot P; Le Bihan AC; Van Hove P; Gadrat S; Baulieu G; Beauceron S; Beaupere N; Boudoul G; Brochet S; Chasserat J; Chierici R; Contardo D; Depasse P; El Mamouni H; Fan J; Fay J; Gascon S; Gouzevitch M; Ille B; Kurca T; Lethuillier M; Lumb N; Mathez H; Mirabito L; Perries S; Alvarez JDR; Sgandurra L; Sordini V; Donckt MV; Verdier P; Viret S; Xiao H; Zoccarato Y; Tsamalaidze Z; Autermann C; Beranek S; Bontenackels M; Calpas B; Edelhoff M; Esser H; Feld L; Hindrichs O; Karpinski W; Klein K; Kukulies C; Lipinski M; Ostapchuk A; Perieanu A; Pierschel G; Preuten M; Raupach F; Sammet J; Schael S; Schulte JF; Schwering G; Sprenger D; Verlage T; Weber H; Wittmer B; Wlochal M; Zhukov V; Ata M; Caudron J; Dietz-Laursonn E; Duchardt D; Erdmann M; Fischer R; Guth A; Hebbeker T; Heidemann C; Hoepfner K; Klingebiel D; Knutzen S; Kreuzer P; Merschmeyer M; Meyer A; Olschewski M; Padeken K;

Original Citation:

Alignment of the CMS tracker with LHC and cosmic ray data

Published version:

DOI:10.1088/1748-0221/9/06/P06009 Terms of use:

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Yang ZC; York A; Bouhali O; Eusebi R; Flanagan W; Gilmore J; Kamon T; Khotilovich V; Krutelyov V; Montalvo R; Osipenkov I; Pakhotin Y; Perloff A; Roe J; Safonov A; Sakuma T; Suarez I; Tatarinov A; Toback D; Akchurin N; Cowden C; Damgov J; Dragoiu C; Dudero PR; Faulkner J; Kovitanggoon K; Kunori S; Lee SW; Libeiro T; Volobouev I; Appelt E; Delannoy AG; Greene S; Gurrola A; Johns W; Maguire C; Mao Y; Melo A; Sharma M; Sheldon P; Snook B; Tuo S; Velkovska J; Arenton MW; Boutle S; Cox B; Francis B; Goodell J; Hirosky R; Ledovskoy A; Lin C; Neu C; Wood J; Gollapinni S; Harr R; Karchin PE; Don CKK; Lamichhane P; Belknap DA; Borrello L; Carlsmith D; Cepeda M; Dasu S; Duric S; Friis E; Grothe M; Hall-Wilton R; Herndon M; Herve A; Klabbers P; Klukas J; Lanaro A; Levine A; Loveless R; Mohapatra A; Ojalvo I; Palmonari F; Perry T; Pierro GA; Polese G; Ross I; Sakharov A; Sarangi T; Savin A; Smith WH. -In: JOURNAL OF INSTRUMENTATION. - ISSN 1748-0221. - 9:6(2014), pp. P06009-P06009.

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EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

CERN-PH-EP/2013-037 2014/08/22

CMS-TRK-11-002

Alignment of the CMS tracker with LHC and cosmic ray

data

The CMS Collaboration

Abstract

The central component of the CMS detector is the largest silicon tracker ever built. The precise alignment of this complex device is a formidable challenge, and only achievable with a significant extension of the technologies routinely used for track-ing detectors in the past. This article describes the full-scale alignment procedure as it is used during LHC operations. Among the specific features of the method are the simultaneous determination of up to 200 000 alignment parameters with tracks, the measurement of individual sensor curvature parameters, the control of system-atic misalignment effects, and the implementation of the whole procedure in a multi-processor environment for high execution speed. Overall, the achieved statistical ac-curacy on the module alignment is found to be significantly better than 10 µm.

Published in the Journal of Instrumentation as doi:10.1088/1748-0221/9/06/P06009.

c 2014 CERN for the benefit of the CMS Collaboration. CC-BY-3.0 license ⇤_{See Appendix A for the list of collaboration members}

Contents 1

Contents

1 Introduction . . . 1

2 Tracker layout and coordinate system . . . 3

3 Global position and orientation of the tracker . . . 5

4 Methodology of track based internal alignment . . . 7

4.1 Track parameterisation . . . 8

4.2 Alignment parameterisation . . . 9

4.3 Hierarchical and differential alignment by using equality constraints . . 11

4.4 Weak modes . . . 12

4.5 Computing optimisation . . . 13

5 Strategy of the internal alignment of the CMS tracker . . . 14

6 Monitoring of the large structures . . . 17

6.1 Monitoring of the strip tracker geometry . . . 17

6.2 Monitoring of the pixel detector geometry with tracks . . . 18

7 Statistical accuracy of the alignment . . . 19

8 Sensor and module shape parameters . . . 23

9 Control of systematic misalignment . . . 26

9.1 Monitoring of the tracker geometry with Z! µµevents . . . 28

9.2 Monitoring of the tracker geometry with the CMS calorimeter . . . 28

9.3 Sensitivity to systematic misalignment . . . 31

10 Summary . . . 32

A The CMS Collaboration . . . 39

1 Introduction

The scientific programme of the Compact Muon Solenoid (CMS) experiment [1] at the Large Hadron Collider (LHC) [2] covers a very broad spectrum of physics and focuses on the search for new phenomena in the TeV range. Excellent tracking performance is crucial for reaching these goals, which requires high precision of the calibration and alignment of the tracking de-vices. The task of the CMS tracker [3, 4] is to measure the trajectories of charged particles (tracks) with very high momentum, angle, and position resolutions, in combination with high reconstruction efficiency [5]. According to design specifications, the tracking should reach a resolution on the transverse momentum, pT, of 1.5% (10%) for 100 GeV/c (1000 GeV/c) momen-tum muons [5]. This is made possible by the precise single-hit resolution of the silicon detectors of the tracker and the high-intensity magnetic field provided by the CMS solenoid.

The complete set of parameters describing the geometrical properties of the modules compos-ing the tracker is called the tracker geometry and is one of the most important inputs used in the reconstruction of tracks. Misalignment of the tracker geometry is a potentially limiting factor for its performance. The large number of individual tracker modules and their arrangement over a large volume with some sensors as far as_⇡6 m apart takes the alignment challenge to a new stage compared to earlier experiments. Because of the limited accessibility of the tracker inside CMS and the high level of precision required, the alignment technique is based on the tracks reconstructed by the tracker in situ. The statistical accuracy of the alignment should remain significantly below the typical intrinsic silicon hit resolution of between 10 and 30 µm [6, 7]. Another important aspect is the efficient control of systematic biases in the alignment of

the tracking modules, which might degrade the physics performance of the experiment. Sys-tematic distortions of the tracker geometry could potentially be introduced by biases in the hit and track reconstruction, by inaccurate treatment of material effects or imprecise estimation of the magnetic field, or by the lack of sensitivity of the alignment procedure itself to such de-grees of freedom. Large samples of events with different track topologies are needed to identify and suppress such distortions, representing a particularly challenging aspect of the alignment effort.

The control of both the statistical and systematic uncertainties on the tracker geometry is cru-cial for reaching the design physics performance of CMS. As an example, the b-tagging perfor-mance can significantly deteriorate in the presence of large misalignments [5, 8]. Electroweak precision measurements are also sensitive to systematic misalignments. Likewise, the imper-fect knowledge of the tracker geometry is one of the dominant sources of systematic uncer-tainty in the measurement of the weak mixing angle, sin2(qeff)[9].

The methodology of the tracker alignment in CMS builds on past experience, which was in-strumental for the fast start-up of the tracking at the beginning of LHC operations. Following simulation studies [10], the alignment at the tracker integration facility [11] demonstrated the readiness of the alignment framework prior to the installation of the tracker in CMS by align-ing a setup with approximately 15% of the silicon modules by usalign-ing cosmic ray tracks. Before the first proton-proton collisions at the LHC, cosmic ray muons were recorded by CMS in a dedicated run known as “cosmic run at four Tesla” (CRAFT) [12] with the magnetic field at the nominal value. The CRAFT data were used to align and calibrate the various subdetectors. The complete alignment of the tracker with the CRAFT data involved 3.2 million cosmic ray tracks passing stringent quality requirements, as well as optical survey measurements carried out be-fore the final installation of the tracker [13]. The achieved statistical accuracy on the position of the modules with respect to the cosmic ray tracks was 3–4 µm and 3–14 µm in the central and forward regions, respectively. The performance of the tracking at CMS has been studied during the first period of proton-proton collisions at the LHC and proven to be very good already at the start of the operations [14–16].

While the alignment obtained from CRAFT was essential for the early physics programme of CMS, its quality was still statistically limited by the available number of cosmic ray tracks, mainly in the forward region of the pixel detector, and systematically limited by the lack of kinematic diversity in the track sample. In order to achieve the ultimate accuracy, the large track sample accumulated in the proton-proton physics run must be included. This article de-scribes the full alignment procedure for the modules of the CMS tracker as applied during the commissioning campaign of 2011, and its validation. The procedure uses tracks from cosmic ray muons and proton-proton collisions recorded in 2011. The final positions and shapes of the tracker modules have been used for the reprocessing of the 2011 data and the start of the 2012 data taking. A similar procedure has later been applied to the 2012 data.

The structure of the paper is the following: in section 2 the tracker layout and coordinate system are introduced. Section 3 describes the alignment of the tracker with respect to the magnetic field of CMS. In section 4 the algorithm used for the internal alignment is detailed with em-phasis on the topic of systematic distortions of the tracker geometry and the possible ways for controlling them. The strategy pursued for the alignment with tracks together with the datasets and selections used are presented in section 5. Section 6 discusses the stability of the tracker geometry as a function of time. Section 7 presents the evaluation of the performance of the aligned tracker in terms of statistical accuracy. The measurement of the parameters describing the non-planarity of the sensors is presented in section 8. Section 9 describes the techniques

3

adopted for controlling the presence of systematic distortions of the tracker geometry and the sensitivity of the alignment strategy to systematic effects.

2 Tracker layout and coordinate system

The CMS experiment uses a right-handed coordinate system, with the origin at the nominal collision point, the x-axis pointing to the centre of the LHC ring, the y-axis pointing up (per-pendicular to the LHC plane), and the z-axis along the anticlockwise beam direction. The polar angle (q) is measured from the positive z-axis and the azimuthal angle (j) is measured from the positive x-axis in the x-y plane. The radius (r) denotes the distance from the z-axis and the pseudorapidity (h) is defined as h= ln[tan(q/2)].

A detailed description of the CMS detector can be found in [1]. The central feature of the CMS apparatus is a 3.8 T superconducting solenoid of 6 m internal diameter.

Starting from the smallest radius, the silicon tracker, the crystal electromagnetic calorimeter (ECAL), and the brass/scintillator hadron calorimeter (HCAL) are located within the inner field volume. The muon system is installed outside the solenoid and is embedded in the steel return yoke of the magnet.

Figure 1: Schematic view of one quarter of the silicon tracker in the r-z plane. The positions of the pixel modules are indicated within the hatched area. At larger radii within the lightly shaded areas, solid rectangles represent single strip modules, while hollow rectangles indicate pairs of strip modules mounted back-to-back with a relative stereo angle. The figure also illus-trates the paths of the laser rays (R), the alignment tubes (A) and the beam splitters (B) of the laser alignment system.

The CMS tracker is composed of 1440 silicon pixel and 15 148 silicon microstrip modules organ-ised in six sub-assemblies, as shown in figure 1. Pixel modules of the CMS tracker are grouped into the barrel pixel (BPIX) and the forward pixel (FPIX) in the endcap regions. Strip mod-ules in the central pseudorapidity region are divided into the tracker inner barrel (TIB) and the tracker outer barrel (TOB) at smaller and larger radii respectively. Similarly, strip modules in the endcap regions are arranged in the tracker inner disks (TID) and tracker endcaps (TEC) at smaller and larger values of z-coordinate, respectively.

The BPIX system is divided into two semi-cylindrical half-shells along the y-z plane. The TIB and TOB are both divided into two half-barrels at positive and negative z values, respectively. The pixel modules composing the BPIX half-shells are mechanically assembled in three con-centric layers, with radial positions at 4 cm, 7 cm, and 11 cm in the design layout. Similarly,

four and six layers of microstrip modules compose the TIB and TOB half-barrels, respectively. The FPIX, TID, and TEC are all divided into two symmetrical parts in the forward (z>0) and backward (z < 0) regions. Each of these halves is composed of a series of disks arranged at different z, with the FPIX, TID, and TEC having two, three, and nine such disks, respectively. Each FPIX disk is subdivided into two mechanically independent half-disks. The modules on the TID and TEC disks are further arranged in concentric rings numbered from 1 (innermost) to 3 (outermost) in TID and from 1 up to 7 in TEC.

Pixel modules provide a two-dimensional measurement of the hit position. The pixels com-posing the modules in BPIX have a rectangular shape, with the narrower side in the direction of the “global” rj and the larger one along the ”global” z-coordinate, where “global” refers to the overall CMS coordinate system introduced at the beginning of this section. Pixel modules in the FPIX have a finer segmentation along global r and a coarser one roughly along global rj, but they are tilted by 20 around r. Strip modules positioned at a given r in the barrel generally measure the global rj coordinate of the hit. Similarly, strip modules in the endcaps measure the global j coordinate of the hit. The two layers of the TIB and TOB at smaller radii, rings 1 and 2 in the TID, and rings 1, 2, and 5 in the TEC are instrumented with pairs of microstrip mod-ules mounted back-to-back, referred to as “rj” and “stereo” modmod-ules, respectively. The strip direction of the stereo modules is tilted by 100 mrad relative to that of the rj modules, which provides a measurement component in the z-direction in the barrel and in the r-direction in the endcaps. The modules in the TOB and in rings 5–7 of the TEC contain pairs of sensors with strips connected in series.

The strip modules have the possibility to take data in two different configurations, called “peak” and “deconvolution” modes [17, 18]. The peak mode uses directly the signals from the analogue pipeline, which stores the amplified and shaped signals every 25 ns. In the de-convolution mode, a weighted sum of three consecutive samples is formed, which effectively reduces the rise time and contains the whole signal in 25 ns. Peak mode is characterised by a better signal-over-noise ratio and a longer integration time, ideal for collecting cosmic ray tracks that appear at random times, but not suitable for the high bunch-crossing frequency of the LHC. Therefore, the strip tracker is operated in deconvolution mode when recording data during the LHC operation. The calibration and time synchronization of the strip modules are optimized for the two operation modes separately.

w u v γ _α β u w track ψ utrk _v w track ζ vtrk

Figure 2: Sketch of a silicon strip module showing the axes of its local coordinate system, u, v, and w, and the respective local rotations a, b, g (left), together with illustrations of the local track angles y and z (right).

A local right-handed coordinate system is defined for each module with the origin at the ge-ometric centre of the active area of the module. As illustrated in the left panel of figure 2, the u-axis is defined along the more precisely measured coordinate of the module (typically along the azimuthal direction in the global system), the v-axis orthogonal to the u-axis and in the module plane, pointing away from the readout electronics, and the w-axis normal to the

5

module plane. The origin of the w-axis is in the middle of the module thickness. For the pixel system, u is chosen orthogonal to the magnetic field, i.e. in the global rj direction in the BPIX and in the radial direction in the FPIX. The v-coordinate is perpendicular to u in the sensor plane, i.e. along global z in the BPIX and at a small angle to the global rj direction in the FPIX. The angles a, b, and g indicate right-handed rotations about the u-, v-, and w-axes, respectively. As illustrated in the right panel of figure 2, the local track angle y (z) with respect to the module normal is defined in the u-w (v-w) plane.

3 Global position and orientation of the tracker

While the track based internal alignment (see section 4) mainly adjusts the positions and angles of the tracker modules relative to each other, it cannot ascertain the absolute position and ori-entation of the tracker. Survey measurements of the TOB, as the largest single sub-component, are thus used to determine its shift and rotation around the beam axis relative to the design values. The other sub-components are then aligned relative to the TOB by means of the track based internal alignment procedure. The magnetic field of the CMS solenoid is to good ap-proximation parallel to the z-axis. The orientation of the tracker relative to the magnetic field is of special importance, since the correct parameterisation of the trajectory in the reconstruction depends on it. This global orientation is described by the angles qxand qy, which correspond to rotations of the whole tracker around the x- and y-axes defined in the previous section. Un-corrected overall tilts of the tracker relative to the magnetic field could result in biases of the reconstructed parameters of the tracks and the measured masses of resonances inferred from their charged daughter particles. Such biases would be hard to disentangle from the other sys-tematic effects addressed in section 4.4. It is therefore essential to determine the global tracker tilt angles prior to the overall alignment correction, because the latter might be affected by a wrong assumption on the magnetic field orientation. It is not expected that tilt angles will change significantly with time, hence one measurement should be sufficient for many years of operation. The tilt angles have been determined with the 2010 CMS data, and they have been used as the input for the internal alignment detailed in subsequent sections of this article. A repetition of the procedure with 2011 data led to compatible results.

The measurement of the tilt angles is based on the study of overall track quality as a function of the qx and qy angles. Any non-optimal setting of the tilt angles will result, for example, in incorrect assumptions on the transverse field components relative to the tracker axis. This may degrade the observed track quality. The tilt angles qx and qyare scanned in appropriate intervals centred around zero. For each set of values, the standard CMS track fit is applied to the whole set of tracks, and an overall track quality estimator is determined. The track quality is estimated by the total c2_{of all the fitted tracks, Â c}2_{. As cross-checks, two other track quality} estimators are also studied: the mean normalised track c2 _{per degree of freedom,hc}2_/N

dofi, and the mean p value, hP(c2, Ndof)i, which is the probability of the c2 to exceed the value obtained in the track fit, assuming a c2 _{distribution with the appropriate number of degrees} of freedom. All methods yield similar results; remaining small differences are attributed to the different relative weight of tracks with varying number of hits and the effect of any remaining outliers.

Events are considered if they have exactly one primary vertex reconstructed by using at least four tracks, and a reconstructed position along the beam line within ±24 cm of the nominal centre of the CMS detector. Tracks are required to have at least ten reconstructed hits and a pseudorapidity of|h| < 2.5. The track impact parameter with respect to the primary vertex must be less than 0.15 cm (2 cm) in the transverse (longitudinal) direction. For the baseline

analysis that provides the central values, the transverse momentum threshold is set to 1 GeV/c; alternative values of 0.5 and 2 GeV/c are used to check the stability of the results. Only tracks with c2_/N

dof < 4 are selected in order to reject those with wrongly associated hits. For each setting of the tilt angles, each track is refitted by using a full 3D magnetic field model [19, 20] that also takes tangential field components into account. This field model is based on measure-ments obtained during a dedicated mapping operation with Hall and NMR probes [21]. Each tilt angle is scanned at eleven settings in the range±2 mrad. The angle of correct align-ment is derived as the point of maximum track quality, corresponding to minimum total c2_, determined by a least squares fit with a second-order polynomial function. The dependence of the total c2_{divided by the number of tracks on the tilt angles q}

xand qyis shown in figure 3.

[mrad] x θ -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 track / N track 2 χ Σ 20.54 20.56 20.58 CMS 2010 [mrad] y θ -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 track / N track 2 χ Σ 20.54 20.56 20.58 CMS 2010

Figure 3: Dependence of the total c2_{of the track fits, divided by the number of tracks, on the} assumed qx(left) and qy(right) tilt angles for|h| < 2.5 and pT > 1 GeV/c. The error bars are purely statistical and correlated point-to-point because the same tracks are used for each point. In each plot, only one angle is varied, while the other remains fixed at 0. The second-order polynomial fit describes the functional dependence very well. There is no result for the scan point at qy = 2 mrad, because this setting is outside the range allowed by the track recon-struction programme. While the qydependence is symmetric with a maximum near qy⇡0, the qx dependence is shifted towards positive values, indicating a noticeable vertical downward tilt of the tracker around the x-axis with respect to the magnetic field. On an absolute scale, the tilt is small, but nevertheless visible within the resolution of beam line tilt measurements [15]. Figure 4 shows the resulting values of qxand qyfor five intervals of track pseudorapidity, for tracks with pT > 1 GeV/c for the total c2estimator. The statistical uncertainties are taken as the distance corresponding to a one unit increase of the total c2_{; they are at most of the order} of the symbol size and thus hardly visible. The outer error bars show the root mean square (RMS) of the changes observed in the tilt angle estimates when changing the track quality estimators and the pT threshold. The right plot shows the results of the method applied to simulated events without any tracker misalignment. They are consistent with zero tilt within the systematic uncertainty. The variations are smaller in the central region within|h| < 1.5, and they are well contained within a margin of±0.1 mrad, which is used as a rough estimate of the systematic uncertainty of this method. The left plot in figure 4 shows the result obtained from data; the qxvalues are systematically shifted by⇡0.3 mrad, while the qyvalues are close to zero. The nominal tilt angle values used as alignment constants are extracted from the central

h region of figure 4 (left) as qx = (0.3±0.1)mrad and qy = (0±0.1)mrad, thus eliminating an important potential source of systematic alignment uncertainty. These results represent an important complementary step to the internal alignment procedure described in the following

7 η -2 -1 0 1 2 [mrad] opt x,y θ -0.2 0 0.2 0.4 0.6 0.8 η -2 -1 0 1 2 [mrad] opt x,y θ -0.2 0 0.2 0.4 0.6 0.8 x θ y θ CMS 2010 Data η -2 -1 0 1 2 [mrad] opt x,y θ -0.2 0 0.2 0.4 0.6 0.8 η -2 -1 0 1 2 [mrad] opt x,y θ -0.2 0 0.2 0.4 0.6 0.8 x θ y θ CMS Simulation MC (no misalignment)

Figure 4: Tracker tilt angles qx (filled circles) and qy (hollow triangles) as a function of track pseudorapidity. The left plot shows the values measured with the data collected in 2010; the right plot has been obtained from simulated events without tracker misalignment. The statis-tical uncertainty is typically smaller than the symbol size and mostly invisible. The outer error bars indicate the RMS of the variations which are observed when varying several parameters of the tilt angle determination. The shaded bands indicate the margins of±0.1 discussed in the text.

sections.

4 Methodology of track based internal alignment

Track-hit residual distributions are generally broadened if the assumed positions of the sili-con modules used in track resili-construction differ from the true positions. Therefore standard alignment algorithms follow the least squares approach and minimise the sum of squares of normalised residuals from many tracks. Assuming the measurements mij, i.e. usually the re-constructed hit positions on the modules, with uncertainties sijare independent, the minimised objective function is c2(p, q) = tracks

Â

Â

where fij is the trajectory prediction of the track model at the position of the measurement, depending on the geometry (p) and track (qj) parameters. An initial geometry description p0is usually available from design drawings, survey measurements, or previous alignment results. This can be used to determine approximate track parameters qj0. Since alignment corrections can be assumed to be small, fij can be linearised around these initial values. Minimising c2 after the linearisation leads to the normal equations of least squares. These can be expressed as a linear equation system Ca = b with aT _{= (Dp, Dq), i.e. the alignment parameters Dp} and corrections to all parameters of all n used tracks DqT _{= (Dq}

1, . . . , Dqn). If the alignment corrections are not small, the linear approximation is of limited precision and the procedure has to be iterated.

For the alignment of the CMS tracker a global-fit approach [22] is applied, by using the MILLE -PEDE II program [23]. It makes use of the special structure of C that facilitates, by means of

block matrix algebra, the reduction of the large system of equations Ca = b to a smaller one

for the alignment parameters only,

Here C0 _{and b}0 _{sum contributions from all tracks. To derive b}0_{, the solutions Dq}

jof the track fit equations CjDqj =bjare needed. For C0, also the covariance matrices Cj 1have to be calcu-lated. The reduction of the matrix size from C to C0_{is dramatic. For 10}7_{tracks with on average} 20 parameters and 105 _{alignment parameters, the number of matrix elements is reduced by} a factor larger than 4⇥106_{. Nevertheless, no information is lost for the determination of the} alignment parameters Dp.

The following subsections explain the track and alignment parameters Dq and Dp that are used for the CMS tracker alignment. Then the concept of a hierarchical and differential alignment by using equality constraints is introduced, followed by a discussion of “weak modes” and how they can be avoided. The section closes with the computing optimisations needed to make MILLEPEDEII a fast tool with modest computer memory needs, even for the alignment of the

CMS tracker with its unprecedented complexity.

4.1 Track parameterisation

In the absence of material effects, five parameters are needed to describe the trajectory of a charged particle in a magnetic field. Traversing material, the particle experiences multiple scat-tering, mainly due to Coulomb interactions with atomic nuclei. These effects are significant in the CMS tracker, i.e. the particle trajectory cannot be well described without taking them into account in the track model. This is achieved in a rigorous and efficient way as explained below in this section, representing an improvement compared to previous MILLEPEDEII

align-ment procedures [10, 11, 13] for the CMS silicon tracker, which ignored correlations induced by multiple scattering.

A rigorous treatment of multiple scattering can be achieved by increasing the number of track parameters to npar =5+2nscat, e.g. by adding two deflection angles for each of the nscat thin scatterers traversed by the particle. For thin scatterers, the trajectory offsets induced by multi-ple scattering can be ignored. If a scatterer is thick, it can be approximately treated as two thin scatterers. The distributions of these deflection angles have mean values of zero. Their stan-dard deviations can be estimated by using preliminary knowledge of the particle momentum and of the amount of material crossed. This theoretical knowledge is used to extend the list of measurements, originally containing all the track hits, by “virtual measurements”. Each scat-tering angle is virtually measured to be zero with an uncertainty corresponding to the respec-tive standard deviation. These virtual measurements compensate for the degrees of freedom introduced in the track model by increasing its number of parameters. For cosmic ray tracks this complete parameterisation often leads to npar > 50. Since in the general case the effort to calculate C 1

j is proportional to n3par, a significant amount of computing time would be spent to calculate C 1

j and thus C0and b0. The progressive Kalman filter fit, as used in the CMS track reconstruction, avoids the n3

parscaling by a sequential fit procedure, determining five track pa-rameters at each measurement. However, the Kalman filter does not provide the full (singular) covariance matrix C 1

j of these parameters as needed in a global-fit alignment approach. As shown in [24], the Kalman filter fit procedure can be extended to provide this covariance ma-trix, but since MILLEPEDE II is designed for a simultaneous fit of all measurements, another

approach is followed here.

The general broken lines (GBL) track refit [25, 26] that generalises the algorithm described in Ref. [27], avoids the n3

par scaling for calculating Cj 1 by defining a custom track parameteri-sation. The parameters are qj = (Dqp, u1, . . . , u(nscat+2)), where D

q

p is the change of the inverse momentum multiplied by the particle charge and uiare the two-dimensional offsets to an initial

4.2 Alignment parameterisation 9

reference trajectory in local systems at each scatterer and at the first and last measurement. All parameters except Dq_pinfluence only a small part of the track trajectory. This locality of all track parameters (but one) results in Cjbeing a bordered band matrix with band width m 5 and border size b=1, i.e. the matrix elements ckl_j are non-zero only for k_b, l _b or_|k l_{| }m.

By means of root-free Cholesky decomposition (Cband

j = LDLT) of the band part Cbandj into a diagonal matrix D and a unit left triangular band matrix L, the effort to calculate C_j 1and qjis reduced to µ n2

par· (m+b)and µ npar· (m+b)2, respectively. This approach saves a factor of 6.5 in CPU time for track refitting in MILLEPEDEII for isolated muons (see section 5) and of 8.4

for cosmic ray tracks in comparison with an (equivalent) linear equation system with a dense matrix solved by inversion.

The implementation of the GBL refit used for the MILLEPEDE II alignment of the CMS tracker

is based on a seed trajectory derived from the position and direction of the track at its first hit as resulting from the standard Kalman filter track fit. From the first hit, the trajectory is prop-agated taking into account magnetic-field inhomogeneities, by using the Runge–Kutta tech-nique, and the average energy loss in the material as for muons. As in the CMS Kalman filter track fit, all traversed material is assumed to coincide with the silicon measurement planes that are treated as thin scatterers. The curvilinear frame defined in [28] is chosen for the local coor-dinate systems at these scatterers. Parameter propagation along the trajectory needed to link the local systems uses Jacobians assuming a locally constant magnetic field between them [28]. To further reduce the computing time, two approximations are used in the standard process-ing: material assigned to stereo and rj modules that are mounted together is treated as a single thin scatterer, and the Jacobians are calculated assuming the magnetic field~_{B to be parallel to}

the z-axis in the limit of weak deflection, |~B|

p ! 0. This leads to a band width of m= 4 in the matrix Cj.

4.2 Alignment parameterisation

To first approximation, the CMS silicon modules are flat planes. Previous alignment approaches in CMS ignored possible deviations from this approximation and determined only corrections to the initial module positions, i.e. up to three shifts (u, v, w) and three rotations (a, b, g). However, tracks with large angles of incidence relative to the silicon module normal are highly sensitive to the exact positions of the modules along their w directions and therefore also to local w variations if the modules are not flat. These local variations can arise from possible cur-vatures of silicon sensors and, for strip modules with two sensors in a chain, from their relative misalignment. In fact, sensor curvatures can be expected because of tensions after mounting or because of single-sided silicon processing as for the strip sensors. The specifications for the con-struction of the sensors required the deviation from perfect planarity to be less than 100 µm [1]. To take into account such deviations, the vector of alignment parameters Dp is extended to up to nine degrees of freedom per sensor instead of six per module. The sensor shape is parame-terised as a sum of products of modified (orthogonal) Legendre polynomials up to the second order where the constant and linear terms are equivalent to the rigid body parameters w, a and

b:

w(ur, vr) = w

+w10·ur +w01·vr

+w20· (u2r 1/3) +w11· (ur·vr) +w02· (v2r 1/3).

(3) Here ur 2 [ 1, 1] (vr 2 [ 1, 1]) is the position on the sensor in the u- (v-) direction, nor-malised to its width lu (length lv). The coefficients w20, w11 and w02 quantify the sagittae of the sensor curvature as illustrated in figure 5. The CMS track reconstruction algorithm treats

r u -1 -0.5 0 0.5 1 r v -1 -0.5 0 0.5 1 w -0.2 0 0.2 0.4 0.6 - 1/3 2 r : u 20 w r u -1 -0.5 0 0.5 1 r v -1 -0.5 0 0.5 1 w -1 -0.5 0 0.5 1 r v ⋅ r : u 11 w r u -1 -0.5 0 0.5 1 r v -1 -0.5 0 0.5 1 w -0.2 0 0.2 0.4 0.6 - 1/3 2 r : v 02 w

Figure 5: The three two-dimensional second-order polynomials to describe sensor deviations from the flat plane, illustrated for sagittae w20=w11 =w02 =1.

the hits under the assumption of a flat module surface. To take into account the determined sensor shapes, the reconstructed hit positions in u and (for pixel modules) v are corrected by

( w(ur, vr)·tan y)and( w(ur, vr)·tan z), respectively. Here the track angle from the sensor normal y (z) is defined in the u-w (v-w) plane (figure 2), and the track predictions are used for urand vr.

To linearise the track-model prediction fij, derivatives with respect to the alignment parameters have to be calculated. If fijis in the local u (v) direction, denoted as fu( fv) in the following, the derivatives are 0 B B B B B B B B B B B B B B B B B B B @ ∂fu ∂u ∂∂fuv ∂fu ∂v ∂fv ∂v ∂fu ∂w ∂fv ∂w ∂fu ∂w10 ∂fv ∂w10 ∂fu ∂w01 ∂fv ∂w01 ∂fu ∂g0 ∂_∂gfv0 ∂fu ∂w20 ∂fv ∂w20 ∂fu ∂w11 ∂fv ∂w11 ∂fu ∂w02 ∂fv ∂w02 1 C C C C C C C C C C C C C C C C C C C A = 0 B B B B B B B B B B B B B B B B B B B @ 1 0 0 1 tan y tan z ur·tan y ur·tan z vr·tan y vr·tan z vrlv/(2 s) urlu/(2 s) (u2_r 1/3)·tan y (u2_r 1/3)·tan z ur·vr·tan y ur·vr·tan z (v2_r 1/3)·tan y (v2_r 1/3)·tan z 1 C C C C C C C C C C C C C C C C C C C A . (4)

Unlike the parameterisation used in previous CMS alignment procedures [29], the coefficients of the first order polynomials w01 = l₂v ·tan a and w10 = ₂lu ·tan b are used as alignment parameters instead of the angles. This ensures the orthogonality of the sensor surface param-eterisation. The in-plane rotation g is replaced by g0 _{= s·g with s =} lu+lv

2 . This has the advantage that all parameters have a length scale and their derivatives have similar numerical size.

The pixel modules provide uncorrelated measurements in both u and v directions. The strips of the modules in the TIB and TOB are parallel along v, so the modules provide measure-ments only in the u direction. For TID and TEC modules, where the strips are not parallel, the hit reconstruction provides highly correlated two-dimensional measurements in u and v. Their covariance matrix is diagonalised and the corresponding transformation applied to the derivatives and residuals as well. The measurement in the less precise direction, after the diag-onalisation, is not used for the alignment.

4.3 Hierarchical and differential alignment by using equality constraints 11

4.3 Hierarchical and differential alignment by using equality constraints

The CMS tracker is built in a hierarchical way from mechanical substructures, e.g. three BPIX half-layers form each of the two BPIX half-shells. To treat translations and rotations of these substructures as a whole, six alignment parameters Dp_lfor each of the considered substructures can be introduced. The derivatives of the track prediction with respect to these parameters, d fu/v/dDpl, are obtained from the six translational and rotational parameters of the hit sensor

Dp_sby coordinate transformation with the chain rule d fu/v dDp_l = dDps dDp_l · d fu/v dDp_s, (5) where dDps

dDp_l is the 6⇥6H Jacobian matrix expressing the effect of translations and rotations of the large structure on the position of the sensor.

These large-substructure parameters are useful in two different cases. If the track sample is too small for the determination of the large number of alignment parameters at module level, the alignment can be restricted to the much smaller set of parameters of these substructures. In addition they can be used in a hierarchical alignment approach, simultaneously with the alignment parameters of the sensors. This has the advantage that coherent misplacements of large structures in directions of the non-sensitive coordinate v of strip sensors can be taken into account.

This hierarchical approach introduces redundant degrees of freedom since movements of the large structures can be expressed either by their alignment parameters or by combinations of the parameters of their components. These degrees of freedom are eliminated by using linear equality constraints. In general, such constraints can be formulated as

Â

Dp_i=s, (6)

where the index i runs over all the alignment parameters. In MILLEPEDE II these constraints

are implemented by extending the matrix equation (2) by means of Lagrangian multipliers. In the hierarchical approach, for each parameter Dpl of the larger structure one constraint with s=0 has to be applied and then all constraints for one large structure form a matrix equation,

components

Â

where Dps,i are the shift and rotation parameters of component i of the large substructure. Similarly, the technique of equality constraints is used to fix the six undefined overall shifts and rotations of the complete tracker.

The concept of “differential alignment” means that in one alignment step some parameters are treated as time-dependent while the majority of the parameters stays time-independent. Time dependence is achieved by replacing an alignment parameter Dpt in the linearised form of equation (1) by several parameters, each to be used for one period of time only. This method allows the use of the full statistical power of the whole dataset for the determination of pa-rameters that are stable with time, without neglecting the time dependence of others. This is especially useful in conjunction with a hierarchical alignment: the parameters of larger struc-tures can vary with time, but the sensors therein are kept stable relative to their large structure.

4.4 Weak modes

A major difficulty of track based alignment arises if the matrix C0 _{in equation (2) is} ill-condi-tioned, i.e. singular or numerically close to singular. This can result from linear combinations of the alignment parameters that do not (or only slightly) change the track-hit residuals and thus the overall c2_{(Dp, Dq) in equation (1), after linearisation of the track model f}

ij. These linear combinations are called “weak modes” since the amplitudes of their contributions to the solution are either not determinable or only barely so.

Weak modes can emerge if certain coherent changes of alignment parameters Dp can be com-pensated by changes of the track parameters Dq. The simplest example is an overall shift of the tracker that would be compensated by changes of the impact parameters of the tracks. For that reason the overall shift has to be fixed by using constraints as mentioned above. Other weak modes discussed below influence especially the transverse momenta of the tracks. A specific problem is that even very small biases in the track model fijcan lead to a significant distortion of the tracker if a linear combination of the alignment parameters is not well determined by the data used in equation (1). As a result, weak modes contribute significantly to the systematic uncertainty of kinematic properties determined from the track fit.

The range of possible weak modes depends largely on the geometry and segmentation of the detector, the topology of the tracks used for alignment, and on the alignment and track param-eters. The CMS tracker has a highly segmented detector geometry with a cylindrical layout within a solenoidal magnetic field. If aligned only with tracks passing through the beam line, the characteristic weak modes can be classified in cylindrical coordinates, i.e. by module dis-placements Dr, Dz, and Dj as functions of r, z, and j [30]. To control these weak modes it is crucial to include additional information in equation (1), e.g. by combining track sets of differ-ent topological variety and differdiffer-ent physics constraints by means of

• cosmic ray tracks that break the cylindrical symmetry,

• straight tracks without curvature, recorded when the magnetic field is off,

• knowledge about the production vertex of tracks,

• knowledge about the invariant mass of a resonance whose decay products are ob-served as tracks.

Earlier alignment studies [13] have shown that the usage of cosmic ray tracks is quite effective in controlling several classes of weak modes. However, for some types of coherent deforma-tions of the tracker the sensitivity of an alignment based on cosmic ray tracks is limited. A prominent example biasing the track curvature k µ _pq_T (with q being the track charge) is a twist deformation of the tracker, in which the modules are moved coherently in j by an amount directly proportional to their longitudinal position (Dj= t·z). This has been studied exten-sively in [31]. Other potential weak modes are the off-centring of the barrel layers and endcap rings (sagitta), described by(Dx, Dy) = s·r· (sin js, cos js), and a skew, parameterised as

Dz = w·sin(j+jw). Here s and w denote the amplitudes of the sagitta and skew weak

modes, whereas jsand jware their azimuthal phases.

As a measure against weak modes that influence the track momenta, such as a twist deforma-tion, information on the mass of a resonance decaying into two charged particles is included in the alignment fit with the following method. A common parameterisation for the two tra-jectories of the particles produced in the decay is defined as in [32]. Instead of 2_·5 parameters (plus those accounting for multiple scattering), the nine common parameters are the position of the decay vertex, the momentum of the resonance candidate, two angles defining the