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Experience from Experience from Searches Searches
at the at the TevatronTevatron
Harrison B. Prosper
Florida State University
18 January, 2011
PHYSTAT 2011
CERN
OutlineOutline
h Introduction
h Case Studies
h Search for a rare decay (D0)
h Search for single top (D0)
h Search for Bs0 oscillations (CDF)
h Search for the Higgs (CDF/D0)
h Conclusions
PHYSTAT 2011 Harrison B. Prosper 2
IntroductionIntroduction
PHYSTAT 2011 Harrison B. Prosper 3
The Tevatron (1991 – 2011)The Tevatron (1991 – 2011)
Goals:
1. To test the Standard Model (SM)
2. To find hints of new physics
A few key SM predictions:
h jet spectra ✓h existence of top quark ✓h creation of top quarks singly ✓h creation of di-bosons (WW/ZZ/WZ/Wγ/Zγ) ✓h properties of B mesons ✓h existence of Higgs
PHYSTAT 2011 Harrison B. Prosper 4
PHYSTAT 2011 Harrison B. Prosper 5
“There are known knowns… There are known unknowns… But there are also unknown unknowns.”
Donald Rumsfeld
The Standard Model in ActionThe Standard Model in Action
The observed transverse
momentum spectrum
of jets agrees with
SM predictions over
10 orders of magnitude
This illustrates why we
take our null hypothesis,
the Standard Model,
seriously.
PHYSTAT 2011 Harrison B. Prosper 6
CDF & D0CDF & D0
PHYSTAT 2011 Harrison B. Prosper 7
Particle Physics DataParticle Physics Data
Each collision event yields ~ 1MB of data. However, these data are compressed by a factor of ~103 – 104 during
event reconstruction:
PHYSTAT 2011 Harrison B. Prosper 8
CourtesyCDF
Particle Physics DataParticle Physics Data
PHYSTAT 2011 Harrison B. Prosper 9
CDF (24 September 1992)proton + anti-proton
3 positron (e+)2 neutrino (ν)3 Jet13 Jet23 Jet33 Jet4
A total of 17 measurements,after event reconstruction
Case StudiesCase Studies
PHYSTAT 2011 Harrison B. Prosper 10
Search for a Rare Decay (D0)Search for a Rare Decay (D0)
PHYSTAT 2011 Harrison B. Prosper 11
Search for a Rare Decay (D0)Search for a Rare Decay (D0)
PHYSTAT 2011 Harrison B. Prosper 12
The goal: test the Standard Model prediction
BF =Bs0 → μ+μ−
Bs0 → everything
=(3.6 ±0.3)×10−9
Bs0
Search for a Rare Decay (D0)Search for a Rare Decay (D0)
PHYSTAT 2011 Harrison B. Prosper 13
Compress data to the unitinterval using a Bayesian neural network
β = BNN(Data)
Cuts1. β > 0.92. 5.0 ≤ mμμ ≤ 5.8 GeV
define the signal region
Phys.Lett. B693 (2010) 539-544 e-Print: arXiv:1006.3469 [hep-ex]
Search for a Rare Decay (D0)Search for a Rare Decay (D0)
PHYSTAT 2011 Harrison B. Prosper 14
D0 results (6.1 fb-1) observed backgroundRunIIa na = 256 Ba = 264 ± 13 eventRunIIb nb = 823 Bb = 827 ± 23 events
The likelihood for these data is the 2-count model
p(n|s, μ) = Poisson(na|sa+ μa) Poisson(nb|sb + μb)
where the s and μ are the expected signal and background counts, respectively. The evidence-based prior for the backgrounds istaken to be the product of two normal distributions.
A Search for a Rare DecayA Search for a Rare Decay
PHYSTAT 2011 Harrison B. Prosper 15
For D0, the branching fraction (BF) is related to be the signals as follows
BF = (4.90 ± 1.00) × 10-9 × sa (RunIIa)
BF = (1.84 ± 0.36) × 10-9 × sb (RunIIb)
The limit BF < 5.1 x 10-8 @ 95% C.L. is derived using CLs, based on the statistic x = log[p(n|BF) / p(n|0)], where p(n|BF) is the likelihood marginalized over all nuisance parameters.
[Recap CLs (Luc’s talk): define p1(BF) = P[x < x0| H1(BF)], reject all BF for which p1(BF) < γ p1(0), and define a (1 – γ) C.L. upper limit as the smallest rejected value of BF.]
Search for Single Top (D0)Search for Single Top (D0)
PHYSTAT 2011 Harrison B. Prosper 16
Search for Single TopSearch for Single Top
The goal: test the Standard Model prediction that the process
exists and has a total cross section of 3.46 ± 0.18 pb (assuming a top quark mass of mtop=170 GeV).
This corresponds to a production rate of ~ 1 in 1010 collisions.
PHYSTAT 2011 Harrison B. Prosper 17
p + p→ t+ X
Search for Single TopSearch for Single Top
PHYSTAT 2011 Harrison B. Prosper 18
S/B ~ 1/260
Search for Single TopSearch for Single Top
PHYSTAT 2011 Harrison B. Prosper 19
The data are reduced to
M counts described by the
likelihood
where σ (the cross section)
is the parameter of interest
and the εi and μi are nuisance
parameters.
p(n |σ,ε,μ)
= Poisson(ni |ε iσ + μi )i=1
M
∏
Search for Single TopSearch for Single Top
D0 (and CDF) compute the posterior p(σ | n) assuming:
1. a flat prior for π(σ)
2. an evidence-based prior for π(ε, μ)
PHYSTAT 2011 Harrison B. Prosper 20
Search for Single TopSearch for Single Top
Estimate of “signal significance” using a p-value:
p0 = P[t > t0| H0]
The statistic t is the mode of the the posterior density.
PHYSTAT 2011 Harrison B. Prosper 21
Search for BSearch for Bss00 Oscillations (CDF) Oscillations (CDF)
PHYSTAT 2011 Harrison B. Prosper 22
Search for BSearch for Bss00 Oscillations Oscillations
PHYSTAT 2011 Harrison B. Prosper 23
The goal: test the Standard Model prediction that the oscillation process
exists and is governed by the time-dependent probabilities
with A = 1
Bs0 ↔ Bs
0
Nino T. Leonardo (PhD Dissertation, MIT, 2006)
pBs
0 → Bs0 (t |A,Δm) =
12τ
e−t/τ [1−Acos(Δmt)]
pBs0→ Bs
0 (t |A,Δm) =12τ
e−t/τ [1+ Acos(Δmt)]
There are (at least) two complications:
1. the time of decay t of a B particle is measured with some uncertainty
2. there is background
The probability model is therefore a convolution of a signal plus
background mixture and a resolution function.
The latter is modeled as
a normal with a
variance σ2 that
depends on t.
Search for BSearch for Bss00 Oscillations Oscillations
PHYSTAT 2011 Harrison B. Prosper 24
Nino T. Leonardo (PhD Dissertation, MIT, 2006)
The likelihood is a product of these functions, one for each measured decay time:
Finding the amplitude A.
For a given oscillation frequency, Δm, a maximum likelihood fit is performed for the amplitude.
It is found that at Δm = 17.8/ps, A = 1.21 ± 0.20, which is consistent with A = 1 and inconsistent with A = 0.
Search for BSearch for Bss00 Oscillations Oscillations
PHYSTAT 2011 Harrison B. Prosper 25
p(t | A,Δm) ~ N(ti | ′t ,σ i2 )⊗[αP( ′t ) + (1−α)B( ′t )]
i=1
M
∏
Estimating the “signal significance”.
This is done using the likelihood ratio test statistic
Λ = log[p(t | A=0) / p(t | A=1, Δm)],
The significance is
defined to be the
p-value:
p0 = P[Λ < Λ0| H0]
= 8 x 10-8
Search for BSearch for Bss00 Oscillations Oscillations
PHYSTAT 2011 Harrison B. Prosper 26
CDF, PRL 97, 242003 (2006)
Search for the HiggsSearch for the Higgs
PHYSTAT 2011 Harrison B. Prosper 27
Search for the HiggsSearch for the Higgs
PHYSTAT 2011 Harrison B. Prosper 28
Here is all available evidence about the Higgs:
π(s) ≡π(s|H1) = p(s|m,H1)p(m|
100
200
∫ H1)dm p(s | m, H
1) p(m | H
1)
Given the evidence-based prior, π(s), that encodes what we know about the Higgs from the Tevatron and LEP, we could test the Higgs hypothesis with current LHC data by computing a Bayes factor (see Jim Berger’s talk):
or by computing the expected loss (d(N)) (see José Bernardo’s talk)
…just a thought!
Higgs @ CERNHiggs @ CERN
PHYSTAT 2011 Harrison B. Prosper 29
B01=
Poisson(N |b)π(b)db0
∞
∫Poisson(N |s+b)π(b)π(s)dbds
0
∞
∫0
∞
∫
ConclusionsConclusions
PHYSTAT 2011 Harrison B. Prosper 30
h Discoveries can be had, in spite of our eclectic, and sometimes muddled, approach to statistics.
h “We” remain ferociously fond of exact frequentist coverage.
h p-values remain king! But Bayes is tolerated.
h CLs still lives…alas!
h Physicists can be taught!