From LEP to LHC
1. Physics of the Z boson (I)2. Physics of the Z boson (II)3. Physics of the W boson4. Physics of the top quark5. Tests of the Standard Model6. Search for the Higgs boson (I)7. Search for the Higgs boson (II)
Salvatore MeleCERN
From LEP to LHC
Salvatore MeleCERN
1. Physics of the Z boson (I)• LEP (Statistics interlude)
2. Physics of the Z boson (II)• LEP (Statistics interlude)
3. Physics of the W boson• LEP, Tevatron, LHC (Statistics interlude)
4. Physics of the top quark• Tevatron, LHC
5. Tests of the Standard Model• LEP, Tevatron
6. Search for the Higgs boson (I)• LEP (Statistics interlude)
7. Search for the Higgs boson (II)• Tevatron, LHC
From LEP to LHC
Salvatore MeleCERN
1. Physics of the Z boson (I)• LEP (Statistics interlude)
2. Physics of the Z boson (II)• LEP (Statistics interlude)
3. Physics of the W boson• LEP, Tevatron, LHC (Statistics interlude)
4. Physics of the top quark• Tevatron, LHC
5. Tests of the Standard Model• LEP, Tevatron
6. Search for the Higgs boson (I)• LEP (Statistics interlude)
7. Search for the Higgs boson (II)• Tevatron, LHC
Search for the Higgs boson at LEP
Introductory remarksSearches at LEP IStatistical interlude: setting limitsSearches at LEP 2: event selectionSeaches at LEP2: statistical methodSearches at LEP2: resultsMuch ado about nothing?What if...?
Salvatore Mele | From LEP to LHC | Troisième cycle 5
Higgs at LEP2: comprehensive bibliography
First resultsALEPH Phys. Lett. B495 (2000) 1 arXiv:hep-ex/0011045DELPHI Phys. Lett. B499 (2001) 23 arXiv:hep-ex/0102036L3 Phys. Lett. B495 (2000) 18 arXiv:hep-ex/0011043OPAL Phys. Lett. B499 (2001) 38 arXiv:hep-ex/0101014
Final resultsALEPH Phys. Lett. B526 (2002) 191 arXiv:hep-ex/0201014DELPHI Eur. Phys. J. C32 (2004) 145 arXiv:hep-ex/0303013L3 Phys. Lett. B517 (2001) 319 arXiv:hep-ex/0107054OPAL Eur. Phys. J. C26 (2003) 479 arXiv:hep-ex/0209078
Final Combined resultsLEP Phys. Lett. B565 (2003) 61 arXiv:hep-ex/0306033
Salvatore Mele | From LEP to LHC | Troisième cycle 6
The missing link…•The Standard Model does not explain why particleshave a mass (nor predicts the values)
•Invoke the Higgs mechanism to make particlesmassive, through their interaction with a field whichpermeates the whole Universe(“Was it aether, Mr. Michelson? Mr. Morley?”)
•Elegant solution… which implies we should observe“the ripples in this field” in the form of anadditional particle, the Higgs boson.
•A decade of indirect hints and direct searches…
Salvatore Mele | From LEP to LHC | Troisième cycle 7
Radiative corrections
Electroweak processes “feel” bosons and fermions
Salvatore Mele | From LEP to LHC | Troisième cycle 8
The Standard Model black-box
ααS
mZmtmH
GF
StandardModel
mZΓZσhad
RlAFBALRmWmt
χ2 fit
mZ ΓZ σhad Rl AFB ALR
mW mt
mH
Use input parameters...to predict values...
to compare to measurements...
...to predict mH
Salvatore Mele | From LEP to LHC | Troisième cycle 9
Measured vs. predicted: mW and mt
Repeat the fit leaving mW and mt as a free parameters
Salvatore Mele | From LEP to LHC | Troisième cycle 10
*THE* plot
mH =84+34-26 GeV
mH < 154 GeV @ 95% C.L.
Salvatore Mele | From LEP to LHC | Troisième cycle 11
Searches at LEP2
Salvatore Mele | From LEP to LHC | Troisième cycle 12
Direct Higgs-boson searches at the Z pole
Salvatore Mele | From LEP to LHC | Troisième cycle 13
Higgs-boson decays
Decay dominated by b-quark pairs
Salvatore Mele | From LEP to LHC | Troisième cycle 14
Searching for the Higgs at the Z pole
Salvatore Mele | From LEP to LHC | Troisième cycle 15
Upper limits and confidence intervals
• Confidence intervals• Basic methods:
likelihood integrationand confidence belts
• Limits: dealing withsmall samples withoutand with background
Salvatore Mele | From LEP to LHC | Troisième cycle 16
Confidence IntervalsGiven some measurements, apply a prescription to define twonumbers ta and tb such that
P(ta≤t≤tb) = 100γ%
Frequentistic: in n experiments γn would give a best estimate Tfor t in [ta, tb]
Bayesian: our posterior belief that t∈[ta, tb] is 100γ%
Another formulation: there is a probability 100γ% that [ta, tb]“covers” t.
Prescriptions: “integrating the likelihood” and “confidencebelts”
Salvatore Mele | From LEP to LHC | Troisième cycle 17
Integrating the likelihood
Tta tb
γ/2γ/2
Leaves the opend doubt of where to startintegrating from, several intervals are
possible but they all satisfy the definition
Salvatore Mele | From LEP to LHC | Troisième cycle 18
The confidence belt
γx1(µ)
x2(µ)
µ0
x2(µ0)x1(µ0) x
µ2(x)
µ1(x)•Measure x to infer µ
•∀µ x can be predictedthrough a known p.d.f.
•∀µ an integration of thelikelihood gives a 100γ%range for x, delimited bythe curves x1(µ) and x2(µ)
•Chosen a µ0 it holds:x1(µ0)<x<x2(µ0) andµ2(x)<µ0<µ1(x)
•For the “true” value ofµ it holds by constructionP(µ2≤µ≤µ1) = 100γ%
Salvatore Mele | From LEP to LHC | Troisième cycle 19
Upper and lower limits• We may want to express the result of an experiment
as a limit– We measured a quantity, and quantify the “maximum”
deviation from it still allowed by the data– We searched for something that was not found (the
limitino!)• Which is the probability that it is not there?• Which fraction of identical experiments will not find it either?
• A particular case of the confidence interval. Insteadof t∈[ta, tb], look for tlim: t∈[tlim,∞ ) or tlim: t∈(-∞ ,tlim]
• Basic example, measure x to infer µ. The maximumdeviation from data still allowed is 1/2 of theGaussian probability for these values
Salvatore Mele | From LEP to LHC | Troisième cycle 20
A particular case
µlim
pdf(n|µ) •Observe few or noevents n when lookingfor a signal
•Which is the largestallowed value of µcompatible with nobserved events?
•Build the pdf(n|µ) for the µ expected signal events forn observed signal events and integrate it to therequested confidence level
•Maybe the most controversial statistical problem ofinference from physical observations
Salvatore Mele | From LEP to LHC | Troisième cycle 21
Poisson process without background
•Case in which n=2 events are observed•Read off the axes the 95% CL at 6.30•Extension to the non-trivial case of non-integer µ: 1-γ = ∫∞µlim pdf(n|µ)dµ
pdf(n=2|µ)
cumulativeΣk=0
n pdf(k|µ)
9.2710.515
7.899.154
6.687.753
5.326.302
3.894.741
2.303.000
µlim (90%)µlim (95%)n
•If n is observed, which is the maximum value of µ•Straightforward application of the p.d.f. integration case•Poisson process: pdf(n|µ) = e-µ µn/n!•Integrate the Poissonian pdf for a fixed value of n and extract µ
How far can one go quoting a limit and
not a measurement?
Salvatore Mele | From LEP to LHC | Troisième cycle 22
Poisson processes with background•If n events are observed, and µb are expected from backgroundwhich is the maximum value of the signal µs•Naive approach
• Poisson process: pdf(n|µ) = e-µ µn/n!• µ = µs+ µb• Apply the previous method for µ and extract µs = µ - µb
4.893.682.320.89-0.703
5.894.683.321.890.302
6.895.684.322.891.301
7.496.184.823.391.800.5
7.896.685.323.892.300
4321n = 0µs
Unhealthy...
23
Poisson processes with background•If n events are observed, and µb are expected from backgroundwhich is the maximum value of the signal µs we can imagine toobserve ns events due to signal and nb events due to background
!
P(n | µs+ µ
b) = P(n
s| µ
s)P(n
b| µ
b)
n s = 0
n"nb
#n b = 0
n
#
•This is a pdf which is properly normalized. If n events areobserved and nb≤n the normalization is lost.•Recover it through the substitution
!
P(nb
| µb)"
P(nb
| µb)
P(nb
| µb)
n b = 0
n
#
•Find confidence levels integrating a normalised pdf to find theµs which considers the probability of finding n or less events
!
1" # = 1-
P(ni| µ
s+ µ
b)
n i = 0
n
$
P(nb
| µb)
n b = 0
n
$
Salvatore Mele | From LEP to LHC | Troisième cycle 24
Poisson processes with background (cont.)
6.635.494.383.302.303
7.055.864.663.462.302
7.496.244.963.652.301
7.756.465.133.762.300.5
7.896.685.323.892.300
4321n = 0µs
Expecting 3 and seeing 0, is this still normal?
Salvatore Mele | From LEP to LHC | Troisième cycle 25
Unphysical regions•Bayes theorem: probability for the “real value” α given theobservation A:
!
f (" | A) = f (A |") f (")
f (A |") f (")d"#
•An application of the Bayes theorem is the determination oflimits in presence of unphysical regions: imagine measuring aparameter which is defined positive (|Vub|, Δm2, ...) what to doif the pdf allows negative values?•Introduce a prior for α>0 and exclude α<0... but which prior?
Posterior probability for α
Prior probability for α Likelihood for A given α
!
f (" | A)
!
" !
f (A |")
!
A!
f (")
!
"
Posterior probability for α Likelihood for A given α Prior probability for α
Salvatore Mele | From LEP to LHC | Troisième cycle 26
Poisson processes with background
5.354.343.522.832.303
6.084.923.873.002.302
6.995.714.443.262.301
7.496.174.833.502.300.5
7.896.685.323.892.300
4321n = 0µs
The definition of CL asprobability other experimentsgive same results is notanymore true by construction
µlim
pdf(n|µ)
γ 100%(1-γ) 100%
Cutting the unphysical region
Salvatore Mele | From LEP to LHC | Troisième cycle 27
How to set the limits?• Open question between different approaches
(dependence on prior?)• No approach has exact coverage everywhere nor
allows limits to be combined as they are• Always need access to likelihood for a proper
combination• LEP 2 Higgs search was the first groundbreaking
example of analyses conceived with combination(helas of limits) in mind.
Salvatore Mele | From LEP to LHC | Troisième cycle 28
Direct search for the Higgs boson
Higgs boson produced inassociation with a Z boson
Accessible up tomH ~ √ s-mZ ~(206-91)GeV =115 GeV
Decays in the heaviestavailable fermions
e
e
q !
q !
Z
"#
b
b
H
Salvatore Mele | From LEP to LHC | Troisième cycle 29
Extremely rare process
Salvatore Mele | From LEP to LHC | Troisième cycle 30
Tagging through the b lifetime
• b-hadrons have a lifetime τ~1.5ps• ...and a boost in Z->bb γ~6
• Decay length <L>=<γβ>cτ~2.7mm• Impact parameter D= γβcτ sinϕ
– <sinϕ>=1/βγ– <D>=cτ=450 µm
• Effects ~10 times the resolutions!
Salvatore Mele | From LEP to LHC | Troisième cycle 31
Impact parameter and secondary vertices significance
Salvatore Mele | From LEP to LHC | Troisième cycle 32
Powerful b-tag variables including all information
Salvatore Mele | From LEP to LHC | Troisième cycle 33
Can we observe such a rare process?Try to detect a similar (and rare) process
Salvatore Mele | From LEP to LHC | Troisième cycle 34
Typical selection variables, 4jet channel
Salvatore Mele | From LEP to LHC | Troisième cycle 35
Missing energy channel
36
Leptonic channel
Salvatore Mele | From LEP to LHC | Troisième cycle 37
Mass spectraSignal “pops up” whiletightening the cuts
Background mostly from ZZ final states
The mass spectra are notall the story
Combine the informationwith the b-tag probability
Salvatore Mele | From LEP to LHC | Troisième cycle 38
Building a discriminant
Var1
Var1
Var2
Var2
Prob
abili
ty
Prob
abili
ty
P(b)2
P(s)2P(s)1
P(b)1
P(s)=P(s)1xP(s)2 P(b)=P(b)1xP(b)2
Calculate for each event the probability of beingsignal or background
Salvatore Mele | From LEP to LHC | Troisième cycle 39
The likelihood-ratio techniqueCombine 4 channels x 4 experiments x 15 energies
= 240 analyses
Start from the distribution of the S/B of each data,signal and background Monte Carlo event of eachanalysis
Salvatore Mele | From LEP to LHC | Troisième cycle 40
Plotting and understanding the likelihood ratio
Salvatore Mele | From LEP to LHC | Troisième cycle 41
LEP Combined Results
There is some dip around 116GeV......where the sensitivity is quite small.
Salvatore Mele | From LEP to LHC | Troisième cycle 42
Results per experiment
Salvatore Mele | From LEP to LHC | Troisième cycle 43
Results per channel
Salvatore Mele | From LEP to LHC | Troisième cycle 44
Talking percents
For each mH use -2lnQto extract the values of:CLs+b Probability of data observation in the s+b hypothesis (signal overfluctuation)
1-CLbProbability of data observation in the b-only hypothesis (background underfluctuation)
Salvatore Mele | From LEP to LHC | Troisième cycle 45
Assess the probability of a background under fluctuation
Salvatore Mele | From LEP to LHC | Troisième cycle 46
Lower mass limit on mH
The ratio CLs=CLs+b/CLb as a function of mH defines a lower mass limit on mH
Salvatore Mele | From LEP to LHC | Troisième cycle 47
“Combining the four experiments, a slight excess is present inthe region above 115 GeV. This deviation, although of lowsignificance, is compatible with the expectation for a Higgs
boson in this mass range, while being also in agreement withthe background hypothesis. Indeed, for a Higgs boson of such a
high mass only a faint signal could be expected from the amountof data collected”
ALEPH: excess compatible with mH= 115 GeVL3 and OPAL: consistent with background but slightly favourpresence of signalDELPHI: deficit with respect to background expectations
Was there a Higgs boson in LEP data?
No definite conclusion could be drawn:if it was there it was too heavy to give a strong signal.
mH > 114.4 GeV (@ 95% C.L.)
Time will tell us if the observed effect was real.
Phys. Lett. B 565 (2003) 61
Salvatore Mele | From LEP to LHC | Troisième cycle 48
What was all this fuss about?
Salvatore Mele | From LEP to LHC | Troisième cycle 49
What was all this fuss about?
Salvatore Mele | From LEP to LHC | Troisième cycle 50
Report by theLEP experimentsfew hours after
the end ofdata taking
Nov. 3, 2000
Salvatore Mele | From LEP to LHC | Troisième cycle 51
What if...?• LEP did not find evidence for a Higgs boson... or
rather for the reactionBut... what if...
1. ...the ZH cross section is less than that we expectand there were far less Higgs boson produced?
2. ...the Higgs boson decays into fermions other thanb quarks?
3. ..the Higgs boson is fermiophobic and decays intobosons rather than fermions?
4. ...we do not assume anything about the Higgsboson decays?
Salvatore Mele | From LEP to LHC | Troisième cycle 52
Anomalous Higgs coupling ξ1. What if the ZH cross section is less than that we
expect and there were far less Higgs boson produced?
No excess for a cross section up to 20 times smaller
Salvatore Mele | From LEP to LHC | Troisième cycle 53
Flavour-independent searches2. What if the Higgs boson decays into fermions other
than b quarks?
Replicate analyses without the b-tag information
Salvatore Mele | From LEP to LHC | Troisième cycle 54
Fermiophobic Higgs bosons3. What if the Higgs boson is fermiophobic and decays
into bosons rather than fermions?
Photons would be the only available channels at LEP
Salvatore Mele | From LEP to LHC | Troisième cycle 55
Just look for e+e- -> ZH4. What if we do not assume anything about the Higgs
boson decays?
Select events with a Z boson decaying into leptons andstudy the recoil-mass spectrum
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