The document discusses analyzing the underlying event in proton-proton collisions using tracks reconstructed in the ATLAS inner detector. It describes selecting tracks based on requirements on hits, transverse momentum, distance of closest approach to the beam spot or primary vertex, and fit quality. Tracks are categorized as primary vertex tracks, used to reconstruct vertices, and beam-spot tracks, used to study trigger and vertex reconstruction efficiencies independently of vertexing. The analysis aims to isolate the low-energy QCD contribution to events using measurements in the transverse region.

1. Measuring a Known Unknown of
QCD
The Underlying Event
in Proton-Proton Collisions
at 900 GeV & 7 TeV
Gabriel Hare
University of California, Santa Cruz
30th June 2011
1

2. Outline
• Introduction to the Underlying Event
- The LHC, Parton Distributions, Color Rules, Event Topology
• Analysis Summary:
- ATLAS Inner Detector
- Track Selection & Weights
- Event Selection & Weights
• Analysis Details
• Underlying Event Measurements:
- Particle Number
- Transverse Momentum Density
- Mean Particle Transverse Momentum
★ Conclusions
• Analysis Minutiae
➡ References
2

3. Introduction to the
Underlying Event
3

4. The Large Hadron Collider
• Protons are produced by ionizing hydrogen.
• Accelerated sequentially in LinAc2 (50 MeV) Booster (1.4 GeV) PS (26 GeV) SPS (450 GeV) LHC (7 TeV)
• Protons are grouped in “bunches” in beams circulating in both directions that intersect at the center of ATLAS.
• “Events” are bunch crossings in which there is at least one collision of protons.
• “Pile-up” describes the situation in which there is more than one proton collision in an event.
Map overlay from : http://upload.wikimedia.org/wikipedia/commons/0/06/Location_Large_Hadron_Collider.PNG
Accelerator layout from : http://public.web.cern.ch/public/en/research/AccelComplex-en.html
4

5. Proton Contents up
charm
anti-charm
up
down
down
up
proton up
• “Parton” = any particle found in a
proton. gluon
• Mostly “Quarks” & “Gluons”. proton
• Quarks radiate gluons.
• Gluons split into a pairs of gluons
or a quark & anti-quark.
• At high energies a proton is gluon
described by a “Parton
Distribution”
proton
5

6. Color Charge Mnemonic
• The “strong force” (which
communicates an SU(3)
orientation) is quantized as
gluons.
quark quark • The charge carried by quarks
that interacts with the strong
force can be in one of 3 states of
quark charges referred to (by
analogy) as “Colors”.
• Charges are represented by
displacements in a plane.
• “Confinement” of the strong
force requires that only color-
neutral particles can break free
from protons.
• Baryons are color neutral
combinations of 3 quarks,
quark such as a proton.
6

7. Color Charge Mnemonic
• Anti-quarks carry negative
anti-quark charges, described as the same
color in the opposite direction.
• “Confinement” of the strong
force requires that only color-
neutral particles can break free
from protons.
• Mesons are color neutral
combinations of a quark and
an anti-quark. Pions are the
most common instance.
• Hadrons are either color-
neutral combinations of
three quarks or three anti-
quarks.
anti-quark anti-quark
7

8. Color Charge Mnemonic
• Colors are conserved!
radiated
• Gluons bind quarks & anti-
quarks together by exchanging
green + anti-red units color.
gluon
• There are 8 charge
combinations for gluons:
• 6 gluon charges describe
displacements between
quark or anti-quark color
states.
• 2 gluon non-colored states
change the relative wave-
front phase of quarks of
different colors.
radiated • The wave-front phase of quarks
determines whether a pair of
green + anti-red
quarks can combine to a color
gluon
neutral hadron or one of the 2
non-colored gluon states.
8

9. Underlying Event
Outgoing Transverse Toward
Parton
ISR
Hard Scatter
Incoming Incoming
Parton Parton Incoming Incoming
Proton Proton
Outgoing
FSR Parton
Beam Remnants Beam Remnants
Away Transverse
In the context of event simulation the “Underlying
Incoming Incoming Event” refers to everything that does not originate
Proton Proton from the Hard Scatter outgoing partons.
Model dependent contributions include:
pp Collision
Outgoing
FSR
• Multiple Parton Interactions (MPI):
Parton
- Associated with higher multiplicity events.
- Angular distribution that is independent of the Hard Scatter.
MPI Incomming
Parton • Initial State Radiation (ISR):
Incomming
Parton - Angular distribution that is nearly independent of the Hard
Scatter.
Outgoing
• Final State Radiation (FSR):
ISR Parton - Yields jets of particles in the Toward and Away regions.
9

10. Particle Jets
1. Radiated particles also radiate (or split) so
FSR results in a “shower” of quarks and
gluons.
• Particles produced in a shower are
generally close together in angle.
2. Quarks and gluons are joined into color-
neutral “strings”.
• Energy is distributed along string.
3. Strings “fragment” into pieces with the
masses of hadrons.
• Pions are the most frequently produced
hadron.
4. The string fragment hadrons form clusters
of higher transverse momentum particles
that are described as “jets”.
• The are many possible definitions of jets...
• This analysis avoids the problem of
choosing a jet definition.
Event Display from : http://www.atlas.ch/photos/atlas_photos/selected-photos/events/Atlantis-dijet-highpt-159224_3533152.png
10

11. Underlying Event
• Goal: Isolate the low-energy QCD contribution Toward
T1
to events (in a Minimum Bias sample) that is
independent of the Hard Scatter energy.
• Assume a Di-Jet structure for events. +ϕ
- The ϕ intervals that are nearly transverse to the Di-
Jets is assumed to be principally filled by the Transverse Transverse
Underlying Event.
- The energy of each of the jets is correlated to the
hard scatter energy.
Away
➡ At low energies it is sufficient to use the highest pT
(leading) track T1, rather than the highest ET T2
(leading) jet.
• Define ϕ with respect to the leading track.
➡ π/3 < |ϕ| < 2π/3 defines the “Transverse” region.
• In the context of measurements, the content Toward
of the Transverse region of events will be
identified as the “Underlying Event”. Transverse Transverse
- Correspondence between the measured Away
Underlying Event and MPI or ISR is determined
when generators are tuned.
11

12. Underlying Event:
Analysis Summary
12

13. Tracks & Particles
• Transverse Momentum: Jet 1
Transverse
-
Momentum
Partons with equal and opposite momentum Jet 1
generally yield jets with equal and opposite Parton
momentum. Center of Mass
- Generally parton momenta are not balanced Rest Frame
resulting in a “boosted” collision. Incoming Incoming
Parton Parton
‣ Jet momenta along the incoming parton axis
are not equal. Jet 2
Jet 2 Transverse
‣ Jet “Transverse Momentum” (pT) with respect to Momentum
the incoming parton axis remains equal.
• Particles:
Jet 1
• Ionize detector material yielding currents in a Transverse
Momentum
cluster of responsive detector elements which are
individually recorded as “hits”. Jet 1
Detector
• Tracks: Rest Frame
• Estimated trajectories of particles “reconstructed” Incoming
Parton
Incoming
Parton
from hits.
➡ Some particles might not be successfully
Jet 2
Jet 2
reconstructed. Transverse
Momentum
13

14. The ATLAS Inner Detector
MBTS
MBTS
TRT
• Tracker: |η| < 2.5 iRad • Trigger: 2.1 < |η| < 3.8
- Pixel Detectors: 3 barrel cylinders, 3 - Minimum Bias Trigger Scintillator (MBTS):
disks in each end-cap. 16 cell disks in each end-cap.
- Inner-most pixel layer is “B-Layer”. - Event Trigger: Hit in any cell of the MBTS.
- Stereo-Strip Tracker (SCT): 4 barrel
cylinders, 9 disks in each end-cap.
• Reconstruction:
- Transition Radiation Tracker (TRT): Axial
- Space-points are defined by Pixel hits, and
by hits on crossing strips in the SCT.
straws in the barrel, radial straws in the end-
caps. (Coverage for |η| < 2.1 only) - Tracks are seeded using space-points from
Pixel and SCT, and are extrapolated to include hits
- 2 Tesla Solenoid encloses the inner from Pixel, SCT, and TRT.
detector. Central charged particles require ~500
MeV pT to pass through entire Tracker.
14

15. Number of Pixel hits per track
4.6
Track Selection
MC ND s=7 TeV
4.4
Data s=7 TeV
4.2
100 < p < 500 MeV
T
4
ATLAS Preliminary
3.8
3.6
• Primary Vertex Tracks: 3.4
3.2
➡ Used to fill profiles. 3
Average Number of Pixel Hits
-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
ATLAS
- “Inside-Out” or “Low-pT” reconstruction methods. 4.5 Preliminary
s = 7 TeV
track
- pT ≥ 100 MeV, |η| < 2.5 iRad, 4 500 MeV ≤ pT
Data 2010
- |d0Vtx| < 1.5 mm, |z0Vtx · Sin(θ)| < 1.5 mm. 3.5 Minimum Bias MC
- The track is not required to have been used when 3
constructing the primary vertex.
-
2.5
B-Layer hit if expected. -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
- ≥ 1 Pixel Hit, including B-Layer.
Number of SCT hits per track
10
MC ND s=7 TeV
-
9.5
SCT hit requirement depends on pT Data s=7 TeV
9
100 < p < 500 MeV
‣
T
pT ≥ 100 MeV : ≥ 2 SCT Hits 8.5 ATLAS Preliminary
‣ pT ≥ 200 MeV : ≥ 4 SCT Hits 8
7.5
‣ pT ≥ 300 MeV : ≥ 6 SCT Hits 7
- Fit requirement to suppress high pT fakes: 10.5
Average Number of SCT Hits
-2.5 -2 -1.5 -1 -0.5 0 ATLAS
0.5 1 1.5 2 2.5
10 Preliminary
s = 7 TeV track
‣ pT ≥ 10 GeV : Prob(χ2, NDofF) ≥ 0.01 9.5
9
500 MeV ≤ pT
Data 2010
Minimum Bias MC
8.5
8
Plots compare average hit counts in 7.5
Measured & Simulated events. 7
6.5
Orange plots from: https://atlas.web.cern.ch/Atlas/GROUPS/PHYSICS/CONFNOTES/ATLAS-CONF-2010-024/ -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
Yellow plots from : https://atlas.web.cern.ch/Atlas/GROUPS/PHYSICS/CONFNOTES/ATLAS-CONF-2010-046/
15

16. Tracks/0.2 mm
nch 2, | | < 2.5, 100 < p < 150 MeV
T
Track Selection
s = 900 GeV Data
MC ND:
106 ATLAS Preliminary all
primaries
non-electrons
electrons
105
• Preliminary Tracks: 104
➡ Used to reconstruct primary vertices & identify pile-up
vertices. -10 -8 -6 -4 -2 0 2 4 6 8 10
Vertex d0 [mm]
- All reconstruction methods, (+ “Outside-In”, + “Very-Low-pT”)
Tracks/0.2 mm
nch 2, | | < 2.5, 200 < p < 250 MeV
-
T
≥ 1 Pixel Hit, ≥ 4 SCT Hits, ≥ 6 Pixel+SCT Hits, 107 s = 7 TeV Data
MC ND:
-
ATLAS Preliminary all
pT > 100 MeV, |η| < 2.5 iRad, 106
primaries
non-electrons
electrons
- |d0BS| < 4 mm, |σd0BS| < 0.9 mm, |σz0BS| < 10 mm.
105
• Beam-Spot Tracks:
➡ Used to characterize the trigger and vertex reconstruction
104
efficiencies.
-10 -8 -6 -4 -2 0 2 4 6 8 10
- Intended to be similar to Preliminary Tracks. Vertex d0 [mm]
-
Tracks/0.2 mm
Dependency on the vertex reconstruction is avoided by nch 2, | | < 2.5, 400 < p < 450 MeV
T
107
selecting with respect to the beam spot perigee. s = 7 TeV
Data
ATLAS Preliminary MC ND:
- “Inside-Out”, “Low-pT” reconstruction methods. 106 all
primaries
-
secondaries
pT > 100 MeV, |η| < 2.5 iRad, 105
- |d0BS| < 1.8 mm, 104
- Same hit & fit requirements as Primary Vertex Tracks
103
-15 -10 -5 0 5 10 15
Plots from : https://atlas.web.cern.ch/Atlas/GROUPS/PHYSICS/CONFNOTES/ATLAS-CONF-2010-046/ Vertex z0 [mm]
16

17. 1
< Track fit prob. >
generated particle p [GeV]
ATLAS Preliminary
Track pT Migration
140
Simulation (non-diffractive)
T
120
-1
June 14, 2010 – 13 : 45 100 DRAFT 31 10
80
600 60
R [mm]
10-2
• Track reconstruction concludes with a χ2 fit to
500
40
20
all hits associated with a track. 400 0 10-3
20 40 60 80 100 120 140 160 180 200
• Top: there is a clear difference in the mean fit
300
reconstructed track p [GeV]
T
generated particle p [GeV]
probabilities Prob(χ2, NDofF) for correct &
Nsel
ATLAS Preliminary 107
140
Simulation (non-diffractive) 106
incorrect pT.
200
T
120
105
•
100 100
Middle: pT migration from low pT tracks yields 80
104
103
the majority of the tracks above 40 -3000
GeV. -2000
0
-1000 60 0 1000 2000 3000
z [mm] 102
• Bottom: When a particle scatters Issue: Mis-measured high-pT tracks (II)
off of 10 40
20
detector material (cryostat) the of badly measured tracks. The black boxes indicate the end vertex position of the matched generated par- 10
fit can yield a
Figure 31: MC distribution of badly-reconstructed tracks in the detector zR-plane. See text for definition 1
Hadronic interaction can fake high40 T tracks 100 120 140 160 180 200
p 60 80 0 -1
very high track pT.
20
ticles, and the red (blue) boxes show the position of the SCT (Pixel) hits associated to the reconstructed
tracks. - topology observed in MC±2.35, and η = ±2.55 . reconstructed track p [GeV]
The gray dashed lines highlight η = / data T
detector material in MC: O(1%) are decays in flight
charged particle
Alternative Requirements: reconstructed track
• Require d0Vtx < 0.2 - preferred at high |#|
interaction with material
✓ Used for systematics
June 14, 2010 – 13 : 45 DRAFT 31
Figure 32: Illustration of a low momentum charged particle (blue line) that is reconstructed with high
Simulation Simulation
•
momentum (red line). The black filled dots with the vertical lines represent the silicon measurements.
Require TRT hits
600
R [mm]
Entries
90
-
500 80
1
TRT covers |η| < 2.1 only
< Track fit prob. >
mc-truth-track p [GeV]
Entries
103 70
140
400
0.9
•
60
0.8
T
Wald-Wolfowitz (check for
120
300 50
0.7
100 102 40
SCT Hits
0.6
residual runs)
200 30
80 0.5
True End 20
-
0.4
Pixel Hits
100
60 10
Correlated to fit probability
10
40
0.3
0 0
-3000 -2000 -1000 0 1000 2000
0.2 3000 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
z [mm] !
20
0.1 1
- current track reconstruction setup seems0.2 0.3to0.4 0.5 much discriminative
Migration plots from : https://atlas.web.cern.ch/Atlas/GROUPS/PHYSICS/CONFNOTES/ATLAS-CONF-2010-046/ 0 not have 0.6 0.7 0.8 0.9 1
Figure 31: MC distribution60 badly-reconstructed tracks in the detector 200
0
20 40 of 80 100 120 140 160 180 zR-plane. See text for definition 0.1 0
power in this region (long extrapolation distances between constraining hits)
2
prob(Track-fit ! )
Explanation from : http://indico.cern.ch/getFile.py/access?contribId=4&resId=0&materialId=slides&confId=102382
of badly measured tracks. The black boxes indicate the end vertex position of the matched generated par-
reco-track p [GeV]
T 5
ticles, and the red (blue) boxes show the position of the SCT (Pixel) hits associated to the reconstructed
tracks. The gray dashed lines highlight η = ±2.35, and η = ±2.55 .
Wednesday, August 4, 2010 17
detector material 2

18. Arbitrary units
False Tracks
106 pT ≤ 500 MeV
all
105 with common hit
1. Fake tracks are defined to be tracks that cannot 104
be matched to some true charged particle
according March 15, 2010 – 11 : 25
to the following matching criteria. DRAFT 103 2
- Cone matched if pT > 500 MeV & ΔR < 0.05 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Minimal ∆R(track,particle)
- Cone matched if pT ≤ 500 MeV & ΔR < 0.15 & one
Fraction of Tracks
Tracks
common hit in the pixel detector.
10-1 10 -1
Primaries
Figure 20: Minimal ∆R between a truth particle
ATLAS Preliminary
Strange decays
and a reconstructed track. In red, Interactions
Had. one common
2. Secondary tracks resulting from decays (or
10 -2
10-2
hit in the pixel detector is required.
material interactions) that are identified instead as
10 -3
10-3
primary tracks. (Secondary fraction ~ 0.02.)
10-4
3. Out of Kinematic Range tracks whose matched
-5
10-4
10
true particles are outside of the kinematic range,383 closest match is kept. The cone based method w
10-5
because either their pT is too low 0or 2 4 η is 8too384
-10 -8 -6 -4 -2 their 6 10 is nearby but did not generate the track. In thes
-10 -8 -6 -4 -2 0 2 4 6 8 10
high. (OKR fraction ~0.2, but only at pT & η
d0 [mm]
385 any hit in common and will have veryd0[mm] different
Nevts/bin
edges.) 386 matching in the following. InGenerated 2nd pile-up vertex
Pile-up
the previous analy
4
387 of 10 sources of used. The in the Monte matc
Figure 1: Shape (left) and distribution (right) for the different < 0.05 wassecondariesObserved 2nd vertex Carlo
∆R effect of fake
- A charged primary stable particlegreen ηtrue > shaded) long lived particles and in bluelike for minimum bias e
In black the primaries, in with (light 2.5 can be multiplicity environment particles from hadronic
388
103
ATLAS Preliminary
reconstructed if the are shown.
interactions vertex is displaced towards -z, and track direction resolution dramatically degrades a
389
s= 7 TeV
will pass the selection criteria if ηrec < 2.5. 2
10
4. Pile-up 48
yieldsfor the primaries (0 < barcode < 200000) and 390 secondaries Cone Plus 200000 and barcode = 01 )
additional vertices that can merge the 4.2 The (barcode > Hit Based Matching
10
with the49primary vertex are used reconstruction. in the data leaving the normalisation B for the secondaries
These templates in the to fit the distribution
391 If a large cone is used, the fraction of fake match
50 free
Matching plot (top) from : ATL-COM-PHYS-2010-682
1
Secondaries plot (middle) from : ATL-COM-INDET-2010-011
392 fake matching by 15 20 25 common hit40
5 10
requiring a 30 35 betwee
Pile-Up plot (bottom) from : ATL-CONF-2010-046 393 required to be in the pixel detector. After requirin
# tracks @ vertex
f (d0 ) = A × ( f p (d0 ) + B/A × fs (d0 )) 18
394 shown in Fig. 20.

19. Track
Systematic Uncertainty Size Region
Track Selection ±1% flat in pT and
Material ±2 15% decreases with pT , increases with | |
Resolution ±5% 100 < pT < 150 MeV only, flat in
⇥2 prob. cut 10% flat, only for pT > 10 GeV
Weighting
10-100% Only for pT > 10 GeV,
Alignment and other high pT
strong dependence, larger for the negative end-cap
Table 1: The systematic uncertainties on the tracking e ciency. Stable uncertainties are quoted relative to
7 TeV : Efficiency of Charged Primary
All Particle Reconstruction
the track reconstruction e ciency. 2.5 1
η (iRad)
2
1
w trk (p T ,η ) = * (1- fFake ) * (1- fSec )2010(1-NDOKR ) ATLAS Preliminary
s = 7 TeV s = 7 TeV
1.003 1.003
KS fitted mass ratio
KS fitted mass ratio
Data * / MC f (nominal)
1.5 ATLAS Preliminary
ε trk
Data 2010 / MC ND (nominal)
1.002
MC ND (+5%) / MC ND (nominal) 1.002 MC ND (+5%) / MC ND (nominal)
1
MC ND (+10%) / MC ND (nominal) MC ND (+10%) / MC ND (nominal)
• The aggregation weight wtrk includes 1.001 0.5 1.001
corrections for the reconstruction Efficiency, 1 0 1 10-1
0
0
and for the fractions of Fakes, Secondaries, 0.999
-0.5 0.999
and tracks from Outside the Kinematic Range. 0.998
-1
0.998
-1.5
• Charged Primary Stable (CPS) Particles:
-2.5 -2 -1.5 -1 -0.5 0
0.997
0.5 1 1.5
-2 2 + 2.5
η (π )
0.997
-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
-
η (π )
-2.5 -1 10-2
‣ Stable Particles: τ > 3*10-2 ns (a) 10 1 (b) 10
pT (GeV)
‣ Primary Particles: no Stable 8: Fitted K 0 mass ratios as a function of
Figure
predecessor 7 TeV : Track Reconstruction Efficiency : Absolute Systematic Uncertainty
for data and various MC simulated material descriptions
s
2.5 1
over to the nominal MC sample. The values are obtained from the positive (a) and negative (b) track.
η (iRad)
‣ pT > 100 MeV & |η| < 2.5 iRad 0
The Ks candidates considered for these plots are required to have a reconstructed decay radius smaller
2
• than 25 mm, an 1.5 0
The reconstruction efficiency hasi.e. before the beam pipe. Furthermore, the two pion tracks of all Ks candidates are required
to have at least four silicon hits. The vertical error bars show the statistical uncertainty only (data and
uncertainty σtrk due principally to the horizontal and bands indicate the uncertainty due to the magnetic field strength. 10-1
MC), while material orange 1
0.5
interaction or decay uncertainties.
0
• This uncertainty’s effect is estimated in figure 8. From this-0.5in terms of radiation length and interaction length. The mass
Detector has been increased by 10%, both
versus is shown
by study, one can see that the material description in the nominal
10 -2
making three versions of MC sample models the observed masses in the barrel (| | . 1.3) well; one can conclude that in the region
the corrected -1
profiles, one corrected using εby , and two is a good estimate for the possible amount of extra material present in the
probed trk this study, 10% -1.5
detector relative to the MC. -2
others using εtrk±σtrk. The track length method is also similar to that used in [2]; tracks are reconstructed using the Pixel
-2.5 10 -3
-1
10 1 10
detector only and are matched to
our good tracks that have the full track selection cuts (GeV)
Table from: https://atlas.web.cern.ch/Atlas/GROUPS/PHYSICS/CONFNOTES/ATLAS-CONF-2010-046/ pT applied. The
fraction of Pixel only tracks with a successful match to a full track defines the SCT extension rate.19

20. Event Selection & Weighting
• Selected Events require:
MBTS_1 Trigger Efficiency
1
pT > 100 MeV, | | < 2.5, nBS
-
2
Data Quality: Stable colliding bunches, solenoid ON, and 0.99
sel
nominal inner detector performance.
0.98 ATLAS Preliminary
- Trigger: At least one hit in the MBTS. → εtrig(nBS)
0.97
s = 7 TeV
-
Data 2010
Single Primary Vertex. → εvert(nBS, pTMin, Δz)
0.96
- The Primary Vertex is identified as the candidate with the
0.95
highest Σ(pT2) of its preliminary tracks. 2 4 6 8 10 12 14 16 18 20
n BS
-
sel
No “Pile-Up”: At most one Primary Vertex Candidate with 4 1.02
Vertex reconstruction efficiency
or more associated preliminary tracks. 1
- When there are only 2 beam-spot tracks εvert depends on 0.98 ATLAS Preliminary
the lowest track pTMin and the Δz distance between the Data 2010
0.96
tracks.
s = 7 TeV
-
0.94 BS
At least two selected tracks. → εevent p > 100 MeV, | | < 2.5, n
T sel
2
-
0.92
Two selected tracks guarantees two beam-spot tracks.
0.9
-
2 3 4 5 6 7 8 9
Variation of the track reconstruction efficiency by σtrk also nBS
sel
varies εevent.
Candidate Events: Selected Events/
1 1 1 ∫L
wev = * * Candidate Events
ε trig (nBS ) ε vert (nBS ,p TMin ,Δz ) ε lead
s= 900 GeV 9.12 μb-1 357511/449666
Luminosity at 900 GeV from Liquid Argon Forward Calorimeter: ATL-COM-LUM-2010-002
Luminosity at 7 TeV from LUCID: ATLAS-CONF-2010-046 s= 7 TeV 190 μb-1 10033043/12805094
Plots from: https://atlas.web.cern.ch/Atlas/GROUPS/PHYSICS/CONFNOTES/ATLAS-CONF-2010-046/
20

21. Migration Effects
• There are two migration effects, both due to the possibility that the
reconstruction will fail to identify the true highest pT Primary Stable
Charged (PSC) particle.
1. If the highest pT PSC particle is incorrectly reconstructed. Or, the
highest pT PSC particle is missed, the second highest pT PSC particle
may be identified as the leading track instead.
- The effect is a reduction in the pT scale that characterizes the event.
- In the rise preceding the plateau the migration yields densities that are too high.
2. If the highest pT PSC particle is missed, the orientation of the
reconstructed event will not be consistent with the orientation of the
true event.
- The effect in this case is that the Transverse region may receive contributions
from the Toward & Away regions where there is jet-like activity.
- This is most significant in the plateau region of the profile, and yields an
increase in the track number & summed pT densities.
• These effects are corrected by a final bin-by bin unfolding.
- This unfolding assumes that migration in data and simulated events is similar.
- An associated systematic uncertainty is estimated by comparing MC09 Pythia
and PhoJet unfolding factors.
21

22. Underlying Event:
Analysis Details
22

23. Track Correction
• Define P(T,pT) to be the distribution for track momentum pT, and track
number T, with P(T) the pT distribution normalized to T.
• A sample for the n event drawn from this distribution yields T [n]
tracks, where the t track has momentum p T [n,t ] .
• Suppose that we are interested in the total pT of tracks y in (a region
of) an event. Using the distribution P(T,pT) this is simply:
N T
M1 ( y ) =
Meas ∫ p T * P(T,p T ) ≈ ∑ ∑ p T [n,t ]
T,pT n t
• Suppose that there is pT dependent track finding efficiency E p T and( )
a T dependent vertex finding efficiency V T . ()
• A sample E[ t ] or V [n] drawn from an efficiency is ∈{0,1}.
• If no corrections are applied the measured distribution converges
to E(T)*V(pT)*P(T,pT).
( ) ()
-1 -1
 
• Applying the weight E p T to each track, and V T
to each event,
the measured distribution converges to a function of the measured
track number T and the measured momentum p T normalized to the

corrected number of tracks: P T,p T ( )
23

24. Track Correction
• Making a measurement of y, which is the total track pT corrected for
the efficiency, is simply a matter of including the correction weights.
( ) ( )
N T
M1 ( y ) = ∑ V T [n] ∑ E p T [n,t ] * p T [n,t ]
-1 -1
  
n t
• The event-to-event variation of M1(y) is used in the definition of the
statistical error of a measurement of M1(y). In this case, simply square
the result of the weighted sum over t, and weight by V  T -1.
2
()
 2 ( y ) ≈ V T [n] ⎛ E p [n,t ]
( ) ( ) ⎞
N T
∑ ∑ T * p T [n,t ]⎟
-1 -1
M ⎜
n ⎝ t ⎠
• In the case of the event-mean track pT, the weighted sum of pT is
divided by the weighted track count.
• CONCLUSION: Weighting by 1/ε is correct!
• In the case of the mean track pT versus track number, the track
number migration is not corrected, so the corrected mean track pT
refers to the (non-integer) average of the weighted track count, but the
x axis bins will still refer to the integer count of measured tracks which
receives contributions from higher true track number events.
24

25. Stat. Errors for Std. Dev.
• In general we are working with 2-dimensional distributions S(x,y)
defined by a counting a events from a finite sample.
- x : the event scale. (e.g. lead track pT)
- y : a region characterization. (e.g. scalar-summed track pT)
• Define the additive 1-dimensional moment curves [MN(x)](y) by filling
each bin weighted by zN.
∀
⎡MN ( x ) ⎤ ( y ) = ∫ S ( x,y ) * yN
⎣ ⎦
y
• S is the distribution of sampled events, including track and event
correction weights (wev ≥ 1) and sample weights (ωev ≤ 1).
• We need M1, M2, M3, M4.
• For the statistical errors we also need X0 : a count of events without
correction weights, and X1 a count of events with correction weights.
- Simulated events can have X0 sample weights ωev < 1 when a
region of phase space is over-produced.
- The sample weight ensures a proportionate estimated uncertainty
despite having a large sample.
25

26. Stat. Errors for Std. Dev.
• After combining all of the weighted samples, the normalized moments
can be defined (combining histogram bins if desired).
mN ( y ) = MN ( y ) / X1 ( y )
• To begin with, we are interested in the mean v1 value of a probability
distribution P(y), and it’s standard deviation v2 with respect to event-
to-event variations.
v1 ( y ) = c1 ( y ) = m1 ( y )
v 2 ( y ) = c 2 ( y ) = m1 ( y ) - m2 ( y )
• The statistical uncertainty for a measure of v1 is:
( ) ( )
U v1 ( y ) = U m1 ( y ) = c 2 ( y ) X 0 - 1 ( )
• In order to define a statistical for v2 a new sampled value y2 can be
defined whose mean value is c2:
(
y2 ( x ) = y - m ( x ) )
1 2
• A sample of y2 is reduced by one, since m was used in the definition.
( ) ( )
c 2 ( y2 ) = m ( y ) - 4 * m ( y ) * m ( y ) - m ( y ) + 8 * m ( y ) m ( y ) - 4 * m ( y ) ( )
4 3 1 2 2 2 1 2 1 4
( )
U v 2 ( y ) = c 2 ( y2 ) X0 - 2 ( ) ( 2 * v ( y ))
2
26

27. Stat. Errors for Std. Dev.
• All of the profiles considered here can be considered to be derived by
a finite sample from a 2 dimensional probability density P(x,y).
• In the absence of migration with respect to X-axis bins, and in the
absence of event selection bias, individual track weights wtrk are
sufficient to correct only the mean values of the Y-axis distributions.
- For the mean transverse pT density the relevant distribution has
the sum of the track pT (y1) in the transverse region as the Y-axis.
- For the standard deviation the relevant distribution has the square
of the sum of the track pT less the mean squared (y2) as the Y-axis.
• In the entire event the individual CPS particle pT probability, and the
number and pT densities as functions of eta are entirely corrected by
track and event weights.
• The CPS particle number probability must be corrected for migration.
• The mean individual CPS particle pT as a function of the CPS particle
number also must be corrected for migration, and in this case there is
a correlation with the mean pT that must also be accounted for.
27

28. Migration Correction
• All corrections are derived from a sample of events generated using
the ATLAS MC09 tune of Pythia 6.4 and simulated in GEANT 4.
• Similar detector conditions (disabled modules) to those of the runs
during which the data would be collected.
• A comparable misalignment is included in the simulation.
• However, the simulated events have a wider distribution of the
primary vertex z position so it is necessary to assign a sample
weight ωev(z0Vtx) to the simulated events.
• These simulated events were used to derive the reconstruction
efficiency and false track fractions.
• These events are also used to derive the final correction factors,
expressed as bin multipliers, to account for migration effects.
• The bin multiplier is simply defined to be the ratio of the values in the
true profiles over the reconstructed & corrected values.
v true ( x )
mmult ( x ) = corr
v reco ( x )
• An alternative set of correction factors derived using PhoJet was
found to yield a difference of at most 2%.
28

29. Migration Uncertainty
• If there are insufficient statistics the correction factor mmult ( x ) will have
a significant associated uncertainty.
• Assuming that the bin-by-bin unfolding only corrects migration, the
extent of the migration is:
v migr ( x ) = v true ( x ) − v reco ( x )
corr
• Assuming that v true ( x ) and v reco ( x ) are uncorrelated the uncertainty on
corr
the value of v migr ( x ) is:
( )
U v migr ( x ) = U( v true ( x )) +U v
2
( corr
reco ( x ))
2
• The sample yielding v reco ( x ) is principally a subset of v true ( x )
corr
determined by the events lost due to inefficiencies.
• Thus, there is actually a correlation between v true ( x ) and v reco ( x ) so
corr
the uncertainty is overestimated for v migr ( x ).
• The uncertainty for the bin multiplier mmult ( x ) expressed in terms of the
statistical uncertainties for v true ( x ) and v migr ( x ) is:
( )
v true ( x ) * U v migr ( x ) +U( v true ( x )) * v migr ( x )
2 2 2 2
U(mmult ( x )) =
(v ( x ) - v migr ( x ))
2
true
29