Transcript what Higgs xs

Why do Wouter (and
ATLAS) put asymmetric
errors on data points ?
What is involved in the CLs
exclusion method and what
do the colours/lines mean ?
ATLAS J/Ψ peak (muons)
Excluding SM
Higgs masses
LEP exclusion
Tevatron
exclusion
Why do you put an error on a data-point anyway ?
ATLAS J/Ψ peak (muons)
Estimate of underlying truth (model value)

Poisson distribution
P(n | ) 
n e  
n!
Probability to observe n events
when λ are expected
Poisson distribution
P(0 | 4.9)  0.00745
P(2 | 4.9)  0.08940
P(3 | 4.9)  0.14601
P(4 | 4.9)  0.17887
#observed
varying
Lambda
hypothesis
fixed
Number of observed events
λ=4.90

Poisson distribution: properties
P(n | ) 
n e  
n!
Poisson distribution
http://www.nikhef.nl/~ivov/Statistics/Poisson.pdf
properties
(1) Mean:
(2) Variance:
x  
(x  x)2   
(3) Most
likely value:
first integer ≤ λ
 the famous √N
Lambda known  expected # events
λ=0.00
λ=4.90
λ=1.00
λ=5.00
Large number of events
λ=40.0
Unfortunately this is not what you wanted to know …
What you have:
P(Nobs | )
What you want:
P( | Nobs)
From data to theory
P( | Nobs)  P(Nobs | )P()
Likelihood: Poisson distribution
“what can I say about the measurement (Number
of observed events) given an expectation from an
underlying theory ?”
This is what you want to know:
“what can I say about the underlying theory given my observation
of a given number of events ?”
Nobs known (4)  information on lambda
“Given a number of observed events (4):
 what is the most likely / average / mean underlying true vanue of λ ?”
P(4 | 0)  0.00000
P(4 | 2)  0.09022
P(4 | 4)  0.19537
P(4 | 6)  0.13385
P(Nobs=4|λ)
Likelihood:
λ (hypothesis)
#observed
fixed
Lambda
hypothesis
varying
Normally you plot -2log(Likelihood)

Properties of P(λ|N) for flat P(λ)
P( | Nobs)  P(Nobs | )P()
http://www.nikhef.nl/~ivov/Statistics/Poisson.pdf
Assuming P(λ) is flat
properties
  x 1
(1) Mean:
(2) Variance:
(  )2   x 1
(3) Most likely
 value:

λmost likely = x
P(Nobs=4|λ)
This is normally presented as likelihood curve
Pdf for λ
68.4%
-2Log(P(Nobs=4|λ))
-2Log(Prob)
λ (hypothesis)
-1.68
Likelihood
+2.35
ΔL=+1
sigma: ΔL=+1
2.32
4.00
6.35
4

2.35
1.68
So, if you have observed 4 events
your best estimate for λ is … :
ATLAS J/Ψ peak (muons)
CLS method
http://www.nikhef.nl/~ivov/Statistics/thesis_I_v_Vulpen.pdf
Chapter 7.4
Your Higgs analysis
Scaled to correct cross-sections and 100 pb-1
SM+Higgs
Higgs
Higgs
SM
SM
Discriminant variable
Discriminant variable
Hebben we nou de Higgs gezien of niet ?
Can also be an invariant mass plot
Approach 1: counting
Experiment 1
Experiment 2
tellen
tellen
Discriminant variable
Discriminant variable
Origin
# events
Origin
# events
SM
12.2
SM
12.2
Higgs
5.1
Higgs
5.1
MC total
17.3
MC total
17.3
Data
11
Data
17
Expectations
If the Higgs is there:
On average 17.2 events
If the Higgs is NOT there:
On average 12.2 events
SM
Experiment 1:
11 events observed
SM + Higgs
Experiment 2:
17 events observed
Discovery

P
(N | N SM )dN  5.7 10
7
poisson
N obs

- Only look at what you expect from Standard Model background
- Given the SM expectation: if probability to observe as many
events you have observed (or more) is smaller than 5.7 10-7
 SM hypothesis is very unlikely  reject SM  discovery !
Test hypotheses: rules for discovery
Integrate this plot
SM
SM + Higgs
In the hypothesis that there is NO
Higgs (SM hypothesis):
What is the probability to observe
as many events as I have
observed …OR EVEN MORE
If P < 5.7 10-7  reject SM
P(N≥33|12.2) = 6.35 10-7
P(N≥34|12.2) = 2.24 10-7
Question 1: did you make a discovery ?

P
See previous slide:
7
(11
(or17)
|12.2)dN

5.7
10
poisson
11 (or 17)


P
7
(N
|
N
)dN

5.7
10
poisson
SM
N obs

Yes
Discovery
No
No discovery
Question 2: did you expect to make a discovery:
If the Higgs is there:
On average 17.2 events
If the Higgs is NOT there:
On average 12.2 events

P
(N |12.2)dN  0.07
poisson
17

SM
SM + Higgs
If you observe exactly the
number of events you
expect (assuming the
Higgs is there), it is not
unlikely enough to be
explained by the SM
 NO discovery expected
Question 3: At what luminosity do you expect to
make a discovery ?
Lumi x 1
SM
SM + Higgs
NSM = 12.2
NHiggs = 5.1
no

P
(N |12.2)dN  0.07
poisson
17
Lumi x 10

NSM = 122.0
NHiggs = 51.0

SM
SM + Higgs
no
P
6
(N
|122)dN

5.5
10
poisson
173

Lumi x 12.5
NSM = 152.5
NHiggs = 63.75

P
(N |152.5)dN  5.2 10 7
poisson
216
yes
Discovery or not
It is not likely you get exactly the number of events you
expect.
 You can be lucky … or unlucky.
From simple counting to the
real thing in 3 steps
1) Introduce X (Likelihood ratio) test statistic
2) From simple counting to weighted counting
(a real analysis)
3) Toy Monte-Carlo (fake experiments)
From simple counting to the
real thing in 3 steps
1) Introduce X (Likelihood ratio) test statistic
2) From simple counting to weighted counting
(a real analysis)
3) Toy Monte-Carlo (fake experiments)
Hypothesis testing: likelihood ratio
Hypothesis 1: the Standard Model without the Higgs boson
Hypothesis 2: the Standard Model with the Higgs boson
Definieer een statistic (= variabele) die onderscheid maakt
tussen de 2 hypotheses.
Note: kan vanalles zijn: # events of Neural net output.
Ls b
Q
Lb
Likelihood ratio

 frequently used: X=-2ln(Q)
Ex: counting experiment
Ppoisson(n | sb )
Q
Ppoisson(n | b )
Likelihood ratio: counting
14 events observed
Counting experiment
N events left after some a
selection of cut on discriminant
P(N | s b)
Q
P(N | b)


e (sb)(s b) n /n!

e bb n /n!
s (s b)
e
e bb n
n
Q
Variabele transformatie

More SM+Higgs like
More SM like
Used in plots:

X  2ln( Q)
Note: X = 0 means hypoteses
equally likely
100.000 SM
experiments
100.000 SM + Higgs
experiments
Likelihood ratio: counting
Counting experiment
N events left after some a
selection of cut on discriminant
P(N | s b)
Q
P(N | b)


e (sb)(s b) n /n!


e bb n /n! 
n
(s
b)
 e s b n
e b
15 events observed
14 events observed
P(15 |12.2)  0.076
P(15 |17.3)  0.087
P(14 |12.2)  0.093
X  0.278

X  0.420

More SM+Higgs like

P(14 |17.3)  0.076
More SM like

Used in plots:

X  2ln( Q)
Note: X = 0 means hypoteses
equally likely
100.000 SM
experiments
100.000 SM + Higgs
experiments
From simple counting to the
real thing in 3 steps
1) Introduce X (Likelihood ratio) test statistic
2) From simple counting to weighted counting
(a real analysis)
3) Toy Monte-Carlo (fake experiments)
Likelihood ratio
Counting experiment
N events left after some a
selection of cut on discriminant
Weighted counting experiment
Eveny event has a weight according to a
NN output or discriminant called pi :
Signal: S(pi) and Background B(pi)
tellen
B(pi)
S(pi)+B(pi)
Q
e
(sb)
(s b) /n!
e bb n /n!
n
Q
e
(sb)
(s b) /n!

e bb n /n!
n

sS( pi )  bB( pi )
i1
sb
n
 B( pi )
n
i1
From simple counting to the
real thing in 3 steps
1) Introduce X (Likelihood ratio) test statistic
2) From simple counting to weighted counting
(a real analysis)
3) Toy Monte-Carlo (fake experiments)
Many possible experiments
Experiment 1
Experiment 2
tellen
Discriminant variable
tellen
Discriminant variable
1) Experiment condensed in 1 variable
Note: Each experiment (read ATLAS) yields only ONE value of Q
see 2 slides ago for counting example
2) Do Toy-MC experiments to study distribution of Q
Note: Two distributions: for SM and SM+Higgs hypothesis
Toy Monte Carlo experiment
λSM(i)+ λSM+Higgs(i)
λSM(i)
SM
toy experiment: Draw for each bin i a random number from
Poisson with μ= λSM (i)
SM+Higgs toy experiment: Draw for each bin i a random number from
Poisson with μ= λSM(i)+ λSM+Higgs(i)
The Higgs does not exist: 100,000 toy-experiments (SM)
The Higgs exists:
100,000 toy-experiments (SM+Higgs)
With 1 and 2 sigma bands for SM hypothesis
Note (again): each experiment will produce 1 (one) number in this plot
Different masses … different cross-sections
Small Higgs cross-section
Large Higgs cross-section
Two hypotheses are more apart if:
1) cross-section of Higgs is larger
2) Higgs is more different from SM
LEP plots
dummy
Cross-section
drops as
function of mass
LEP paper Fig 1
dummy
dummy
Expectation for Q or -2ln (Q): toy experiments
CL b = Pb (X  X obs) =


X obs
Pb (X)dX
Clb = confidence level in the background
Probability that background results
in the numer observed or less
1- CL b = Pb (X  X obs) =


X obs
SM
SM+Higgs
Pb (X)dX
Probability that background results
in the numer observed or (even) more
If 1-CLb < 5.7 10-7 we can say we
reject the SM hypothesis  discovery !
The famous 5 sigma
Discovery

7
P
(N
|
N
)dN

5.7
10
 poisson
SM
N obs

1  CL b  5.7 10
7
Do you expect to discover Higgs with at this mass ?
Average SM+Higgs experiment: 1-CLb = 2 10^-7
So yes, you expect to make a discovery IF 10xSM
The one 2-sigma is not the other 2-sigma
2.X sigma discrepancy at mh ~ 97 GeV
Far away form what you expect from Higgs
1.X sigma away at mh = 114 GeV
Exactly what you expect from Higgs
No 5 sigma discovery  what Higgs hypotheses can we reject
No discovery
No 5 sigma deviation found … what now ?
Trying to say something on the hypothesis
that the Higgs exists  exclusion
Exclusion
- Look at what you expect from Standard Model +Higgs
- Given the SM + Higgs expectation:
if probability to observe as many events you have observed
(or less) is smaller than 5%
 SM+Higgs hypothesis is not very likely  reject SM+Higgs
CL s+b
CL s 
 0.05
CL b
Expectation for Q or -2ln (Q): toy experiments
CL b = Pb (X  X obs) =

CL s+b = Ps+b (X  X obs) =


X obs

X obs
Pb (X)dX
Ps+b (X)dX
SM
Probability that signal hypothesis
results in the numer observed or less
SM+Higgs
Extra Normalisation:
CL s+b
CL s 
CL b
This is why it is called
modified frequentist
Cls = confidence level in the signal
If CLs < 0.05 we are allowed to reject
the SM+Higgs at 95% confidence level
The famous 95% confidence level
Question 2: did you expect to be able to exclude ?
CLs mean SM-only expeciment is 0.13  > 0.05 so NO !
Question 3: At what luminosity do you expect to
make a discovery ?
Lumi = 1x normal lumi
CLs = 0.13  no exclusion for
average SM-only experiment
#SM = 100 #H = 10
Lumi = 2x normal lumi
CLs = 0.034  exclusion for
average SM-only experiment
#SM = 200 #H = 20
A scan:
2 sigma up
CLs = 0.66
CLs = 0.13
CLs = 0.046
CLs
1 sigma down
Si: If you would have a 1 sigma
downward fluctuation, i.e. you see less
events than you expect there is less
room for a SM+Higgs hypothesis. In
this case you would have been able to
exclude it.
CLs = 0.05
Luminosity / nominal luminosity
You expect to be able to exclude at Lumi / Lumi nominal = 1.70
Question 4: At what Higgs xs do you expect to
make a discovery ?
Higgs XS = 1x normal Higgs XS
CLs = 0.13  no exclusion for
average SM-only experiment
#SM = 100 #H = 10
Higgs XS = 2x normal Higgs XS
CLs = 0.006  exclusion for
average SM-only experiment
#SM = 100 #H = 20
A scan:
2 sigma up
CLs = 0.66
CLs = 0.13
CLs = 0.046
CLs
1 sigma down
CLs = 0.05
Higgs XS / nominal Higgs XS
You expect to be able to exclude at Higgs XS / Higgs XS nominal = 1.40
A projection along the CLs = 0.05 line
Higgs XS / nominal Higgs XS
At what Higgs XS scale factordo you expect to be able to exclude the Higgs
hypothesis ?
SM only (2 sigma up)
SM only (1 sigma up)
1.4
SM only (mean)
SM only (1 sigma down)
SM only (2 sigma down)
Nominal luminosity
Higgs XS / nominal Higgs XS
You can now scan over Higgs masses
1.4
The important thing is of course what you actually measured
Finito!