Track Reconstruction in the CMS Tracker - DESY

Track Reconstruction in theCMS Tracker

T. SpeerUniversity of Zurich

X International Workshop on Advanced Computing and Analysis Techniques in Physics Research

DESY Zeuthen25th May 2005

ACAT05 -25th May 2005 - p. 2

Tracking at LHC

� proton-proton collisions at �s = 14 TeV� bunch spacing of 25 ns� Luminosity:

� low-luminosity: 2·1033 cm-2 s-1 (10 fb-1 per year - for the first 3 years)� high-luminosity: 1034 cm-2 s-1 (100 fb-1 per year)

~ 20 minimum bias events per bunch crossing~ 5000 charged tracks per event

Radius: 2cm 10cm 25cm 60cmN

Tracks/(cm2*25ns) 10.0 1.0 0.10 0.01

� Fast response time to resolve bunch crossing� Fine granularity to resolve nearby tracks

� Reconstruction of narrow heavy objects: ~ 1-2 % pT resolution at ~100 GeV/c

4 T field - ~1.1m Tracker radius: 1.80 mm sagitta for 100 GeV/c pT track!

� Ability to tag b/� jets through secondary vertices: good impact parameter resolution

ACAT05 -25th May 2005 - p. 3

The CMS Experiment

ECAL & HCAL inside solenoid

13x6 m Solenoid: 4 Tesla Field

� Tracking up to � � 2.5Muon system in return yoke

First muon chamber just after solenoid� extend lever arm for p

T

measurement

� 22m Long, 15m Diameter, 14'000 Ton Detector

ACAT05 -25th May 2005 - p. 4

The CMS Tracker

� CMS has chosen an all-silicon configuration

Rely on �few� measurement layers, each able to provide robust (clean) and precise coordinate determination:

� Pixel detector: 2 - 3 points� Silicon Strip Tracker: 10 - 14 points

ACAT05 -25th May 2005 - p. 5

The Pixel system

Pixel-size: 100 �m x 150 �mHit-resolution:

� r-� : � � 10 �m (Lorentz angle 23° in 4 T field)� r-z : � � � �m

Active area ~ 1m2 - 66·106 pixelsGeometry:� 3 Barrel layers

r = 4.4 cm, 7.3 cm, 10.2 cm� 2 Pairs of Forward/Backward Disks

r = 6 cm-15 cm ; z = 34.5cm, 46.5cm� 3 high resolution measurement points for |�| < 2.2

ACAT05 -25th May 2005 - p. 6

The CMS Silicon Strip Tracker

blue: double-sided detectorsred: single-sided detectors

Endcap (TEC): 9 disks pairs - r<60cm: Thin sensors - r>60cm: Thick sensors

Inner Disks (TID): 3 disks pairs- Thin sensors

Outer Barrel (TOB): 6 layers - Thick (500 μm) sensors - Long Strips

Inner Barrel (TIB): 4 layers - Thin (320 μm)sensors - Short Strips

Material budget of the tracker

ACAT05 -25th May 2005 - p. 7

The CMS Silicon Strip Tracker

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� 14 different sensor geometries

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Number of SST hits by tracks:

ACAT05 -25th May 2005 - p. 8

Track reconstruction

� The Combinatorial Kalman Filter is the main algorithm used to reconstruct charged tracks

� Local method: one track reconstructed at a time, starting from an initial trajectory.

� Recursive procedure: track parameters estimated from a set of reconstructed hits

� Takes into account the energy loss and multiple-scattering between layers

� Integrates pattern recognition and track fitting

� The Kalman Filter is mathematically equivalent to a global least square minimization (LSM), optimal when

� model is linear

� random noise Gaussian

� For non-linear models or non-Gaussian noise, it is still the optimal linear estimator

ACAT05 -25th May 2005 - p. 9

The Combinatorial Kalman Filter

� Inside-out tracking: start in the first Pixel layers, grow tracks layer by layer to the outer layer of the SST.

� Initial trajectories (seeds): pixel hit pairs (every combination of 2 pixel layers, compatible with the beam spot and a minimum p

T cut)

Cross section of a quadrant of the barrel of the tracker

ACAT05 -25th May 2005 - p. 10


� Inside-out tracking: start in the first Pixel layers, grow tracks layer by layer to the outer layer of the SST.

� Initial trajectories (seeds): pixel hit pairs (every combination of 2 pixel layers, compatible with the beam spot and a minimum p

T cut)

� Reasonable number of seeds, with good quality

� Fine granularity: low occupancy, high purity

� High precision

� Hadrons: high interaction probability in the tracker (~20% of 1 GeV pions do not reach the outer layer)

� Favour tracks with pixel hits: precision on track parameter at the vertex (needed for vertex reconstruction, b-tagging, etc...)

� Outside-in tracking:

� Muon reconstruction: seeds in the outer layers based on muon-chamber seeds

� Electrons from conversions: seeds in the outer layers based on ECAL clusters

ACAT05 -25th May 2005 - p. 11


Track reconstruction is decomposed in 4 modular, independent, components:� Generation of seeds� Trajectory Building: construction of trajectories for a given seed

� Trajectories are extrapolated from layer to next layer, accounting for multiple scattering and energy loss

� On the new layer, new trajectories are constructed, with updated parameters (and errors) for each compatible hit in the layer.

� All trajectories are grown to the next layer in parallel to avoid bias.� The number of trajectories to grow is limited according to their �2 and the

number of missing hits.� Trajectory Cleaning: hit assignment ambiguity resolution� Trajectory Smoothing: final fit of trajectories

� Obtain optimal estimates at every measurement point along the track. � In addition to providing tracks accurate at both ends this procedure provides

more accurate rejection of outliers

ACAT05 -25th May 2005 - p. 12

The Combinatorial Kalman FilterTrack reconstruction efficiency for single tracks:

muons, pT = 1, 10, 100 GeV/c pions, p

T = 10 GeV/c

For pions: lower efficiency due to nuclear interactions in the tracker

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ACAT05 -25th May 2005 - p. 13

The Combinatorial Kalman FilterTrack reconstruction efficiency for b-jets (E

T=200 GeV), tracks with p

T > 0.9 GeV/c,

min. 8 hits:

Loss dominated by � interactionEfficient and robust pattern recognition:

� Low contamination from spurious hits, even with PU� Reconstruction ambiguities are solved after the first few layers.

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ACAT05 -25th May 2005 - p. 14


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σ(p

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pT resolution

Dominated by the lever-armTransverse impact parameter resolutionDominated by the resolution of the hits inthe pixel detector.

Track parameter resolutions (muons, pT = 1, 10, 100 GeV/c)

ACAT05 -25th May 2005 - p. 15

Partial reconstructionCombinatorial KF also suitable for usage in the High-Level Trigger (HLT):� Track parameter resolutions reach an asymptotic value after using only first 5/6 hits

Resolutions as a function of the number of hits used: (b-jets, 2.5<pT<5, |�|<0.9)

(a)

Reconstructed Hits

σ(p

trec -

p tsim

)[G

eV/c]

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(a)

Reconstructed Hitsσ

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pT resolution Transverse impact parameter resolution

(“0 hits” indicates full track reconstruction!)

ACAT05 -25th May 2005 - p. 16

Partial reconstruction

No. of hits

Effi

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No. of hits

Fak

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ate

bb − jets ET=200 GeV L=2x1033 cm-2s-1

Efficiency 0.<|η|<1.4Efficiency 1.4<|η|<2.4Fake Rate 0.<|η|<1.4Fake Rate 1.4<|η|<2.4

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Partial reconstruction: stop track reconstruction once enough information is available to answer a specific question

Same components, algorithms used.Precision sufficient for most HLT applications: vertex reconstruction, b-tagging

ACAT05 -25th May 2005 - p. 17

Adaptive filters

Several adaptive algorithms have been implemented:� LSM optimal when

� Model is linear� Random noise Gaussian (measurement errors, process noise)

� Pdf involved are usually non-Gaussian:� Measurement errors have Gaussian core, with tails� Energy loss and multiple scattering (tails)� Gaussian-sum Filter

� Large background noise (electronic noise, low pT tracks, electrons, etc.)

� Hit degradation� Hit assignment errors� Deterministic Annealing Filter & Multi-Track Fit

ACAT05 -25th May 2005 - p. 18

The Gaussian-sum Filter

� Pdf involved are usually non-Gaussian:

� Measurement errors have Gaussian core, with tails

� Energy loss and multiple scattering (tails)

� Gaussian-sum Filter (GSF): instead of single Gaussian, model the pdfs involved by mixture of Gaussians:

� Main component of the mixture would describe the core of the distribution

� Tails would be described by one or several additional Gaussians.

� For electrons, above ~100MeV/c, energy loss dominated by bremsstrahlung

� Bethe and Heitler energy loss model is highly non-Gaussian

� In the standard KF, distribution approximated by single Gaussian

� Model the Bethe-Heitler distribution by a mixture of Gaussians

ACAT05 -25th May 2005 - p. 19


� All involved distributions are Gaussian mixtures

� State vector is also distributed according to a mixture of Gaussians

� GSF: Non-linear generalization of the Kalman Filter

� Weighted sum of several Kalman Filters

� GSF is implemented as a number of Kalman filters run in parallel

� The weights of the components are calculated separately

� Exponential growth: combinatorial combination of the state vector components with energy-loss components

� Number of components have to be limited to a predefined number at each step

� Cluster (collapse) components with the smallest 'distance' (Distance measurements: Kullback-Leibler Distance or Mahalanobis Distance)

� Output is full Gaussian mixture of state vector

� Can be used in subsequent application (GSF vertex fit already implemented)

ACAT05 -25th May 2005 - p. 20

p / pΔ-0.6 -0.4 -0.2 0 0.2 0.4 0.6

Tra

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KF

GSF

Residuals

Full simulation = 10 GeV/ctp

mixture6CDF12 components

Q(68%) = 0.092Q(95%) = 0.632

Q(68%) = 0.119Q(95%) = 0.522


(q/p)σ(q/p) / Δ-6 -4 -2 0 2 4 6

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GSF

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Pull quantities

Full simulation = 10 GeV/ctp

mixture6CDF12 components

Mean: 0.12RMS: 1.32

Mean: 0.06RMS: 1.25

Probability transform for q/p0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

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GSF

� Improvement of the core of the residual distribution� Little reduction of the tails:

� Radiation in the first layer can not be detected � can be compensated by vertex constraint

� Non-Gaussian measurement errors in the Pixel detectors� Incorporate Gaussian mixtures of measurement errors (also for non-electron fits!)� Most efficient for low energy electrons (a few tens of GeV), little gain at 100 GeV� Pattern recognition needs to be evaluated

ACAT05 -25th May 2005 - p. 21

Adaptive filters: the DAF and the MTF

� In very dense environments (e.g. high ET b-jets, � jets), degradation due to

large background noise

� High track density: hit degradation due to contamination of nearby tracks

� High hit density: wrong hit assignment

� Kalman Filter: hard hit assignment

� Soft hit assignment may be more suitable

� Global approach of hit assignation, using full track information

� Part of the hit assignment done in the final track fit

� Expect better hit assignment

ACAT05 -25th May 2005 - p. 22


� Deterministic Annealing Filter (DAF): single track fit

� Competition between hits: on a same surface, several hits may compete for a track

� hit weights (assignment probability) based on hit-track distance (residual) and competing measurements

� Multi-Track Fit (MTF): concurrent multi-track fit on collection of hits

� Competition between tracks and hits

� Each hit on a layer can belong to each of several tracks

Iterative Kalman Filter with annealing

Both need initial hit collection and track seed(s): basic pattern recognition and track parameters from KF

� DAF: initial hit collection around a KF track.

� MTF: collection of tracks from KF (or even DAF), close in momentum space, hits collected around these tracks

With this seeding, track finding efficiencies can not be improved w.r.t. KF

ACAT05 -25th May 2005 - p. 23

The Deterministic Annealing Filter

� For “isolated tracks”, even at high luminosity, the DAF does not provide a measurable improvement in track quality

� “dense environment”: b-jet with ET=200 GeV,

Tracks with pT

> 15 GeV/c, min. 8 hits:� Better track quality (�2)

DAF

χ2 probability

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�2 probability - |�|<0.7

ACAT05 -25th May 2005 - p. 24

The Deterministic Annealing Filter

DAF

Δx

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RMS 0.2479E-02 151.9 / 83

P1 -0.1110E-03 0.2429E-04P2 308.8 7.327P3 0.1219E-02 0.2615E-04P4 15.59 1.570P5 0.5345E-02 0.2260E-03

cm

σ = 25μm

KF

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RMS 0.2997E-02 91.09 / 86

P1 -0.1166E-03 0.2688E-04P2 264.6 7.213P3 0.1156E-02 0.2909E-04P4 27.08 1.876P5 0.5165E-02 0.1550E-03

cm

σ = 30μm

DAF

Δx/σ(x)

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RMS 1.745 112.0 / 84

P1 -0.8696E-01 0.1791E-01P2 279.2 6.543P3 0.9091 0.1863E-01P4 14.14 1.495P5 3.818 0.1758

σ = 1.8

KF

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RMS 2.272 105.9 / 92

P1 -0.9311E-01 0.2046E-01P2 234.2 6.178P3 0.9064 0.2184E-01P4 20.06 1.546P5 4.291 0.1682

σ = 2.4

Transverse IP resolution - |�|<0.7

Transverse IP pull - |�|<0.7

2022242628303234363840

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ACAT05 -25th May 2005 - p. 25


Reconstruction of � tracks from the decay of high-p

T �:

H0��, m(H0) = 500 GeV/c2

� KF: Kalman Filter alone� DAF: DAF with seed from KF� KF+MTF: MTF tracks, seeded with KF

tracks� DAF+MTF: MTF tracks, seeded with

DAF tracks KF

χ2 probability

0200400600800

1000120014001600

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0.2371 0.3150

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0255075

100125150175200

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MeanRMS

0.4413 0.2915

KF+MTF

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MeanRMS

0.4492 0.3126

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020406080

100120140160

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MeanRMS

0.4806 0.2906

�2 probability

ACAT05 -25th May 2005 - p. 26


KF

Δx/σ(x)

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RMS 2.988 83.91 / 92

P1 0.7313E-01 0.2627E-01P2 172.5 6.183P3 0.8812 0.2859E-01P4 20.08 1.439P5 4.602 0.1731

σ = 2.9

DAF

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RMS 2.200 77.95 / 86

P1 0.3948E-01 0.1974E-01P2 252.3 7.247P3 0.8280 0.1991E-01P4 14.19 1.599P5 4.041 0.2429

σ = 2

KF+MTF

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RMS 2.056 66.79 / 81

P1 0.2044E-01 0.2082E-01P2 232.3 6.722P3 0.8839 0.2151E-01P4 9.770 1.467P5 4.283 0.3685

σ = 1.9

DAF+MTF

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RMS 1.966 58.31 / 80

P1 0.2682E-01 0.1944E-01P2 262.1 7.233P3 0.8578 0.2009E-01P4 12.61 1.713P5 3.836 0.2704

σ = 1.8

KF

Δx

050

100150200250300350400

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RMS 0.3567E-02 109.3 / 92

P1 0.8368E-04 0.3491E-04P2 162.1 6.150P3 0.1066E-02 0.3940E-04P4 26.48 1.671P5 0.5313E-02 0.1550E-03

cm

σ = 36μm

DAF

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RMS 0.2914E-02 154.0 / 89

P1 0.4792E-04 0.2709E-04P2 232.4 6.946P3 0.1069E-02 0.2925E-04P4 16.10 1.345P5 0.5550E-02 0.2155E-03

cm

σ = 30μm

KF+MTF

050

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RMS 0.2941E-02 109.1 / 88

P1 0.1243E-04 0.2967E-04P2 204.3 6.385P3 0.1118E-02 0.3250E-04P4 15.22 1.351P5 0.5529E-02 0.2241E-03

cm

σ = 30μm

DAF+MTF

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RMS 0.2839E-02 99.43 / 86

P1 0.3102E-04 0.2854E-04P2 224.2 6.532P3 0.1162E-02 0.3186E-04P4 14.64 1.382P5 0.5605E-02 0.2389E-03

cm

σ = 29μm

Transverse IP resolution Transverse IP pull

Little improvement with the MTF over the DAF

ACAT05 -25th May 2005 - p. 27


� For “isolated tracks”, even at high luminosity, the DAF and/or the MTF do not provide a measurable improvement in track quality

� DAF: “dense environment”, e.g. b-jet with ET=200 GeV , � jets:

� Better track parameter resolutions and error estimates

� Better track quality (�2)

� MTF: little improvement over DAF at the expected track densities!

� little improvement on track parameter resolution

� slightly better error estimate

� slightly better overall track quality

� Better hit assignment (slightly lower fake rate)

� Seeding delicate (esp. MTF)

� Better seeding methods would be needed

� Slower then standalone KF, use where appropriate

ACAT05 -25th May 2005 - p. 28

Conclusion

� CMS has a very robust and versatile tracker and track reconstruction algorithms, with sufficient redundancy to operate in a very challenging environment

� Very capable tracker

� Low occupancy� Reconstructed hits have a high purity

� Combinatorial Kalman filter shown to give very good results even in difficult environments:

� good performance, suitable for high luminosity or heavy ion collisions� high efficiency, low fake rate

� Efficient and robust pattern recognition

� Low contamination from spurious hits, even with PU� Reconstruction ambiguities are solved after the first few layers

� Fast enough for to be used in the HLT

� Good track parameter resolutions after using only the first five to six hits

� More sophisticated methods available for specific applications (GSF, DAF, MTF): adaptive algorithms show improvements w.r.t. LSM in difficult situations

Documents

Track Reconstruction in the CMS Tracker - DESY