Top Quark Mass and Width using the Template Method

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Top Quark Mass and Widthusing the Template Method
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Why study tops?

• So far, the top quark is the heaviest
  fundamental (point) particle observed in nature
• Decays before hadronizing
• Helps pin down the mass of the Higgs boson
• Can constrain new physics
• Because we can

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The Tevatron
CDF
36 bunches of 1010 anti-protons
Bunches cross 2.5 million times per second
The Tevatron (center of mass energy 1.96 TeV)
DZero
36 bunches of 1011 protons
Berkeley
(2000 miles)
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CDF

• SVX Silicon important for good tracking,
  necessary for b-tagging
• Leptons Best-measured objects in events
• COT drift chamber w/ coverage hlt1, s(PT)/PT
  0.15PT
• 1.4 Tesla superconducting solenoid outside
  tracking system
• EM cal sE / E 14 /ÖE
• Hadronic calorimeter crucial for difficult jet
  measurements
• Had cal sE / E 80 /ÖE
• Muons scintillator chamber, coverage out to
  h1.5
• Need entire detector to get a good measurement
  of momentum imbalance

SVX
EM cal
Muon
Had cal
COT
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Tevatron performance at CDF
Top Mass analysis 1.7 fb-1 of data
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Top quark phenomenology

• Tops always decay via t-gtWb
• Event topology then depends on W decays
• Hadronic (quarks)
• Leptonic (electron or muon neutrino)
• This analysis uses the LeptonJets channel
• One W decays to hadrons, the other to leptons
• Signature 4 quarks, 1 charged lepton
  undetected neutrino
• OK, so just take the invariant mass and youre
  done. Right?
• Not so simple

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Why Mtop is difficult

• With 4 (and only 4 jets!), there are 12
  different ways of assigning jets to partons at
  hard scattering
• Neutrino from W decay
• Non-negligible backgrounds
• Jets are difficult

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Jet Energy Scale
?c unit of combined nominal CDF JES calibration
uncertainty
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Top Quark Mass Some handles

• Instead of taking the invariant mass of the
  system, we will have to make a measurement by
  comparing data to Monte Carlo simulation
• Find the parent top mass distribution most
  consistent with our data
• We want to measure a variable thats correlated
  to the top mass
• System is over-constrained (helps choose from 12
  possible jet-parton assignments)

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Event Selection

• Use b-tagging in SVX to reduce combinatorics and
  increase SB
• Divide events into 2 exclusive subsamples with
  different SB and different reconstructed mass
  shapes

Top Event Tag Efficiency 60 False Tag Rate
(per jet) 0.5
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Reconstructed mass templates
Reconstructed mass is correlated to (but not the
same thing as) the true top quark mass
2-tag templates more sharply peaked than 1-tag
templates
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In-situ calibration using dijet mass templates

• Kinematic fitter works well, but distributions
  are highly correlated not only to top quark mass,
  but also to calibration of jets in the detector
• Introduce the dijet mass from the hadronically
  decaying W
• Use as in situ calibration of jet energy scale
  (JES)
• We use the dijet mass closest to the well-known
  W mass from any pair of untagged jets among
  leading 4 jets (studied other choices, this was
  best)

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How do we use the templates?

• How do we get the probability to observe an event
  with mtreco and mjj?
• Previously, assume the two observables are
  uncorrelated, and parameterize as a function of
  mtop and shifts in JES
• Near-impossible to account for correlation
  between observables
• Parameterizations difficult, mathematically bad

New Use a Kernel Density Estimate-based approach
to form PDFs that are two-dimensional in
observables
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A 1d KDE pictorial tutorial
Probability
Mtreco
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Adaptive KDE

• Want smoothing to depend on statistics within
  sample
• First pass fixed kernel width
• Second pass Varying width kernels
• Narrower kernel in high-stats region ? sharper
  peak
• Wider kernel in low-stats region ? smoother tail

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2d KDE
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Backgrounds

• Must deal with non-negligible backgrounds
  arising mostly from Wjets (real heavy flavor and
  mistags)
• Estimate Wjets, QCD normalizations from data
• Wbb/Wcc/Wc fractions from MC
• MC estimates for single-top and dibosons

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Backgrounds (1d)
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Backgrounds (2d)
More correlation between observables for
background events
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Likelihood Fit

• Maximize likelihood for expected number of 1-tag
  and 2-tag signal and background events in a grid
  of over 2000 Mtop-JES points
• Fit a 2d parabola (including correlation
  cross-term) to the minimized negative
  log(likelihood) values

Example pseudoexperiment
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Tests of machinery
Machinery works and shows no significant bias
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Systematics
Even with in situ calibration, JES systematic
dominates
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Measurement!
Mtop 171.6 ? 2.0 GeV/c2
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Differential pulls
Scale all uncertainties by 4.7
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Cross-checks
All errors uncorrected
would have 3 GeV JES systematic
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1d projections
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Plans for top mass measurement

• Add more data (2 fb-1 measurement) for
  publication
• Our group also has a template-based measurement
  in the dilepton channel using KDE
• Will be extended to two observables and 2d KDE
  for better statistical power
• We plan on combining the measurements in the
  same likelihood
• Will move away from fitting a 2d quadratic and
  instead smooth out the likelihood using more
  non-parametric statistical tricks (local
  polynomial smoothing)

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Switching gears
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Top quark decays

• In SM, tops decay with a lifetime of 4x10-25
  seconds
• Vtb 1, and Mtop gtgt MWMb
• Total width 1.5 GeV ? (Mtop)3 and calculated to
  1
• Only measurement so far is unpublished CDF
  result c? lt 52.5 ?m at 95 CL
• Lower limit on top quark width, ?tgt 0.002 eV

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The idea

• Use the kinematic fitter with templates that
  vary not as a function of top quark mass, but as
  a function of top quark width
• Use a likelihood fit gives the measured top
  quark width
• Use the likelihood output to set a 95 CL on the
  top quark width

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In practice

• Selection is the same EXCEPT
• No ?2 cut (was found to reduce sensitvity)
• Allow extra jets in 1-tag events
• Use only 1 fb-1 of data

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Trusting the Monte Carlo?
?top 50 GeV
?top 1.5 GeV
?top 30 GeV
MwMb
General consensus from theorists Can trust MC to
30 GeV
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More on the Monte Carlo
Parton level mean, before event selection
Parton level RMS, before event selection
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Reconstructed mass templates
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Likelihood fit output for ?t
How might we use these to set limits?
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How to set upper limits

• Lets say we want to set a 95 upper limit on
  the top quark width. We have likelihood output
  from MC samples with varying ?t 1.5, 5.0, 10.0,
  . (GeV)

95 of curve
Prob
?true
Data
?L-fit
?L-fit
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How to set lower limits

• Lets say we want to set a 95 lower limit on
  the top quark width. We have likelihood output
  from MC samples with varying ?t 1.5, 5.0, 10.0,
  . (GeV)

95 of curve
Prob
Data
?true
?L-fit
?L-fit
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Does setting limits always work?

• Sometimes with this technique, you dont set a
  limit!
• And what about two-sided limits?

Data
?true
?L-fit
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Feldman-Cousins machinery

• Frequentist approach to setting limits that
  guarantees that we can always make a measurement
• Tells us how to choose our confidence bands (aka
  an ordering principle)
• Choose bands based on likelihood ratios. For
  every MC point, define a likelihood ratio

x output of likelihood fit
?i true width being examined
Width with max prob at x
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Use of likelihood ratio functions

• Use the likelihood ratio to select the 95
  confidence region for a particular true width
• Order (select) by the most likelihood ratio

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Likelihood ratio functions

• We parameterize the likelihood output so that we
  have a likelihood ratio and confidence band for
  arbitrary ?t

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Does it work?
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Systematics

• Move to Bayesian approach, as is typical
• Easy to incorporate into Feldman-Cousins
• Systematics change our confidence bands
• Unfortunately, systematics in limit-setting
  procedures are a bit more tricky than in simple
  measurements (such as the top quark mass)
• Systematics study of the unknown

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Reiterating Systematics

• Typically, we have a PDF of the systematic
  parameter (such as scale of jets, background
  fraction, etc.), which we assume is Gaussian
• Typically, we assume that linear shifts in the
  systematic parameter cause linear shifts in the
  parameter we are measuring

Jet Energy Scale
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But

• What if linear shifts in JES do not cause linear
  shifts in the top quark width out of the
  likelihood?

This is the function we use to smear out the
likelihood output (its Gaussian for normal
systematics)
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Systematics summary
Jet resolution smear jets to worsen resolution
by extra 5 Systematics studied at different top
widths when possible, systematic taken to be
conservative non-Gaussian systematics
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Smeared likelihood output
Systematics included Original PDF
Make new parameterizations, new L ratios all over
again after convoluting with systematics
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Likelihood fit!
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Likelihood fit!
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Confidence bands with data
?top lt 12.7 GeV at 95 CL
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What about assuming Mtop 175?

• All our MC was generated with Mtop 175 GeV/c2,
  but world average is Mtop 170.9 ? 1.8 GeV/c2
• (How) does this affect our measurement?

Mass 171
Mass 168
Mass 175
Prob
mtreco
To accommodate a different mass, likelihood fit
prefers a larger width - we conservatively do NOT
take a systematic
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Plans for top width measurement

• Worlds first direct limit on the top quark
  width
• Set an upper limit with 95 confidence only 1
  order of magnitude from the SM prediction
• Still statistics limited
• Result to be published - PRL going through
  internal review process

?top lt 12.7 GeV at 95 CL
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Conclusion
Mtop (GeV/c2) 171.6 ? 2.1 (statJES) ? 1.1
(syst)
?top lt 12.7 GeV at 95 CL
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Backup
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The top quark

• The top quark
• Discovered only 13 years ago at the Tevatron
• Weak isospin partner of the bottom quark
• Charge 2/3 (-2/3 for anti-top)

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Boundary cuts

• KDE doesnt know about hard/soft cutoffs in the
  observables
• Probability leaks into unpopulated regions
• Easiest fix is to explicitly set boundaries and
  force kernels to stay inside via renormalization
  inside the boundary
• Amounts to extra selection cut on mtreco, mjj
• Efficiency high for signal events passing ?2
  cut, somewhat worse for background events

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Fit for JES (just a cross-check)
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Did we get lucky?
p-value 23.4
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Likelihood fit output mean vs ?t
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On adaptive density estimates

• The adaptive density estimates have smoothing
  that varies from MC point to MC point
• Tails have larger smoothing, core of
  distribution has less smoothing
• Intuitively makes some sense, but can lead to
  trouble - if function changes faster than
  sqrt(x), can have bizarre nonlocal behavior

Evaluating estimate for f(x) MC point at x1
farther away than MC point at x0 Point at x1
contributes weight but point at x0 does not!
Counterintuitive
x
x0
x1
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Clipped adaptive density estimates

• Try to remove some of the non-locality of the
  adaptive estimates by not allowing the smoothing
  to get too large
• Previously, h could be larger than the entire
  width of the template!
• Clip the pilot density estimates fpilot(xi) -gt
  max0.1fpilot(x0), fpilot(xi)
• fpilot(x0) is pilot density estimate with
  maximum value
• Equivalent to setting hmax sqrt(10)hmin
• Dont let h get too large!
• Recommended in one of the very early adaptive
  density estimate papers

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Other weird systematics
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Are we statistics or systematics limited?
Limited by statistics
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An outline of the steps ahead

• Start out with signal and background Monte Carlo

• Run MC through kinematic fitter to get an
  estimator of top quark mass
• Same reconstructed top quark mass as before

• Parameterize discrete Monte Carlo to form
  density functions of reconstructed mass at any
  arbitrary top quark width

• Run Monte Carlo through an extended maximum
  likelihood fit just as before in mass analysis
• Measured top quark width one number per event

• Parameterize likelihood output to get expected
  distribution of measured top quark width for at
  any arbitrary top quark width
• Defines the limit-setting machinery

• Run data through kinematic fitter and likelihood
  fit
• Set limit!

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ISR/FSR

• In Run I, switch ISR on/off
• using PYTHIA, ?Mtop 1.3GeV
• In Run II systematic approach
• ISR/FSR effects are governed
• by DGALP evolution eq.
• ltPtgt of the DY(ll) as a function of Q2

m
m-
qq -gt tt vs mm-
Pt(tt) at generator
(2Mt)2
log(M2)
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Ensuring continuity

• Sometimes (due to fluctuations in tails), can
  have discontinuities in confidence bands. Be
  conservative and ensure that things stay
  continuous