track reconstruction in the cms tracker - desy
TRANSCRIPT
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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barrel layersforward rings
� 14 different sensor geometries
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s Black: Total number of hits Green: double-sided hits Red: double-sided hits in thin detectors Blue: double sided hits in thick detectors.
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
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)
� 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
The Combinatorial Kalman Filter
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.
η
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bb jets Et=200 GeV
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ACAT05 -25th May 2005 - p. 14
The Combinatorial Kalman Filter
10-2
10-1
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η
σ(p
T)/
p T
μ, pT = 100 GeV/cμ, pT = 10 GeV/cμ, pT = 1 GeV/c
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0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5
η
σ(d
0) (μ
m)
μ, pT = 100 GeV/cμ, pT = 10 GeV/cμ, pT = 1 GeV/c
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
(d0
rec -
d0
sim
)[μ
m]
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0 2 4 6 8 10 12
pT resolution Transverse impact parameter resolution
(“0 hits” indicates full track reconstruction!)
ACAT05 -25th May 2005 - p. 16
Partial reconstruction
No. of hits
Effi
cien
cy
No. of hits
Fak
e R
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
(rr)
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No. of hitsE
ffici
ency
No. of hits
Fak
e R
ate
bb − jets ET=200 GeV L=1034 cm-2s-1
Efficiency 0.<|η|<1.4Efficiency 1.4<|η|<2.4Fake Rate 0.<|η|<1.4Fake Rate 1.4<|η|<2.4
(rr)
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b-jets, ET = 200GeV, low luminosity b-jets, E
T = 200GeV, high luminosity
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
The Gaussian-sum Filter
� 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
cks
/ bin
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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
The Gaussian-sum Filter
(q/p)σ(q/p) / Δ-6 -4 -2 0 2 4 6
Tra
cks
/ bin
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GSF
KF
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
Fra
ctio
n o
f tr
acks
/ b
in
0
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0.015
0.02
0.025
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0.035
0.04
0.045KF
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
Adaptive filters: the DAF and the MTF
� 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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250
0 0.25 0.5 0.75 1
MeanRMS
0.4303 0.2847
KF
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MeanRMS
0.3349 0.3183
�2 probability - |�|<0.7
ACAT05 -25th May 2005 - p. 24
The Deterministic Annealing Filter
DAF
Δx
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700
-0.01 0 0.01
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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600
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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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600
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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
0 0.5 1 1.5 2 2.5η
Δx μ
m1
1.251.5
1.752
2.252.5
2.753
0 0.5 1 1.5 2 2.5η
Δx/σ
(x)
KFDAF
ACAT05 -25th May 2005 - p. 25
Adaptive filters: the DAF and the MTF
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
0 0.25 0.5 0.75 1
MeanRMS
0.2371 0.3150
DAF
0255075
100125150175200
0 0.25 0.5 0.75 1
MeanRMS
0.4413 0.2915
KF+MTF
050
100150200250300350400
0 0.25 0.5 0.75 1
MeanRMS
0.4492 0.3126
DAF+MTF
020406080
100120140160
0 0.25 0.5 0.75 1
MeanRMS
0.4806 0.2906
�2 probability
ACAT05 -25th May 2005 - p. 26
Adaptive filters: the DAF and the MTF
KF
Δx/σ(x)
050
100150200250300350400
-10 -5 0 5 10
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
0
100
200
300
400
500
-10 -5 0 5 10
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
0
100
200
300
400
500
-10 -5 0 5 10
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
0
100
200
300
400
500
-10 -5 0 5 10
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
-0.01 0 0.01
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
0
100
200
300
400
500
-0.01 0 0.01
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
100150200250300350400450
-0.01 0 0.01
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
0
100
200
300
400
500
-0.01 0 0.01
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
Adaptive filters: the DAF and the MTF
� 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