1. Status of the Higgs to tau tau
Carlos Solans
26th March 2010

2. Status of the Higgs to tau tau - 26th March 2010
Introduction
The aim of this analysis is to set the exclusion limit and expected significance of a signal for the neutral MSSM Higgs in the ditau decay channel (and finally to lepton hadron) using a bayesian technique
We will use visible mass instead of invariant mass to be able to analyze first data
Similar analysis have been already presented in ATLAS for the CSC notes and in the Statistics forum
We will innovate by incorporating template morphing
Similar tools are available: RooStats, BAT
We will use our own minimization package based on the very well known Minuit implementation in ROOT.
Carlos Solans
Status of the Higgs to tau tau - 26th March 2010

3. Status of the Higgs to tau tau - 26th March 2010
Bayesian analysis
Starting from Bayes’ theorem for conditional probability, where the whole information for the event is contained in the posterior…
The posterior probability for a set of parameters given the data is proportional to the likelihood of those parameters to the data times the prior for the parameters.
If we assume that the prior for the parameters is uncorrelated, we can split it in a product of priors for each parameter
We will incorporate systematic uncertainties via nuisance parameters (as priors).
Another possibility will be to incorporate our knowledge about the constraints on the nuisance parameters into the likelihood, leaving the priors as non-informative.
Anyways is accepted by the ATLAS Statistics forum
Parameter of interest
Nuisance parameters
Data
Carlos Solans
Status of the Higgs to tau tau - 26th March 2010

4. Status of the Higgs to tau tau - 26th March 2010
Profiling
In order to eliminate the dependence of the posterior on the nuisance parameters, we maximize the likelihood times the prior with respect to the nuisance parameters.
The alternative is to marginalize the prior
We obtain the maximum likelihood estimator of the nuisance parameters through and iterative procedure (maximization) called profiling
In fact what we do is maximize the likelihood times the prior for the nuisance parameters.
Where we consider the prior for the parameter of interest to be non-informative (flat) in the range
As we will see, there is no need to include the prior for the data in the maximization. In both cases considered here it won’t affect the result.
Maximum likelihood estimator x
f(x)
Carlos Solans
Status of the Higgs to tau tau - 26th March 2010

5. Status of the Higgs to tau tau - 26th March 2010
Setting limits
Given the posterior probability for the parameters given the data normalized to a given value, we can do a scan in the parameter of interest (cross-section) and compute the cross-section value at which 95% of the probability lies.
We use the bayesian credibility limit (p-value) definition to define the upper exclusion limit to the Higgs cross section (θL) with p=0.05
For each cross section value we compute the posterior independent of the nuisance parameters which is proportional to value of the likelihood times the prior.
Note the proportionality constant is the same for all values of the cross section.
We obtain the upper limit cross section by numerically integrating the posterior normalized to zero cross-section.
Avoid computing the proportionality constant
We repeat this operation for many pseudo-experiments under the null signal hypothesis and compute the upper exclusion limit as the median of the distribution.
θL for this pseudo-experiment
Carlos Solans
Status of the Higgs to tau tau - 26th March 2010

6. Setting limits summary
For each pseudo-experiment we compute the value of the cross-section under which 95% of the integral lies
The upper exclusion limit at 95% CL will be the median of the distribution for many pseudo-experiments
Carlos Solans
Status of the Higgs to tau tau - 26th March 2010

7. Estimating significance
Given the definition of the Bayes factor as the ratio of posterior to prior odds of full signal strength hypothesis (θ1) to null signal hypothesis (θ0).
Which is independent of the prior for the parameter of interest (and the prior for the data).
This test statistic is commonly used in bayesian analysis in ATLAS
To estimate the significance of a discovery, we generate many pseudo-experimental data under the background only (B) assumption and the signal plus background (S+B) assumption.
We compute the p-value of the S+B data being a fluctuation of B data
And convert the p-value into number of sigmas of a double sided gaussian distribution by means of the inverse normal cumulative distribution (Φ-1)
Carlos Solans
Status of the Higgs to tau tau - 26th March 2010

8. Estimating significance summary
For a given cross section, we compute the B10 statistic for many pseudo-experimental data with signal and without signal.
We compute the probability of S+B data being a fluctuation of B as the percentage of B10B that is over the median of the B10S+B distribution.
Carlos Solans
Status of the Higgs to tau tau - 26th March 2010

9. Status of the Higgs to tau tau - 26th March 2010
PHYSICS
Carlos Solans
Status of the Higgs to tau tau - 26th March 2010

10. Status of the Higgs to tau tau - 26th March 2010
Signal production
Direct production
Associated production
Carlos Solans
Status of the Higgs to tau tau - 26th March 2010

11. Status of the Higgs to tau tau - 26th March 2010
Background processes
Z  tau tau
t tbar
QCD
Carlos Solans
Status of the Higgs to tau tau - 26th March 2010

12. Status of the Higgs to tau tau - 26th March 2010
MSSM Higgs
Carlos Solans
Status of the Higgs to tau tau - 26th March 2010

13. Status of the Higgs to tau tau - 26th March 2010
MSSM Higgs
Carlos Solans
Status of the Higgs to tau tau - 26th March 2010

14. Status of the Higgs to tau tau - 26th March 2010
Hands on profiling
To start working on our analysis we need a set of efficiency distributions (templates) for our observable (Mvis).
It is commonly accepted that the likelihood function is a poisson distribution where the mean value is the expected number of counts.
In our case, the likelihood function is a binned likelihood function where the expected number of counts in each bin is a function of the parameter of interest and the nuisance parameters
Our parameter of interest is the signal cross section: σA
Nuisance parameters can be:
Those that modify the number of counts in each bin: L
Those that modify the shape of the template (In our case by taking into account two other templates): fj
In both cases the prior for the nuisance parameters is a Gaussian p.d.f.
Carlos Solans
Status of the Higgs to tau tau - 26th March 2010

15. Common nuisance parameters
The expected number of counts in each bin that will be used in the likelihood core is given by (without morphing):
A pseudo-experiment is a Poisson fluctuation of contributing templates
We fluctuate each bin of each template using a poisson PDF
So that the observed number of counts (xi) can be built the same way as μi
The un-normalized posterior takes into account systematic uncertainties on the nuisance parameters
Carlos Solans
Status of the Higgs to tau tau - 26th March 2010

16. Energy scale uncertainties
Systematic uncertainties in the energy scale can be taken into account by the usage of “template morphing”.
Search for Higgs bosons predicted in two-Higgs-doublet models via decays to tau lepton pairs in 1.96 TeV proton-antiproton collisions. e-Print: arXiv: [hep-ex]
Apart from the nominal template, two new templates are computed that take into account the positive and negative energy scale shifts.
i.e. To compute the jet energy shift, all jet energies have to be shifted
Carlos Solans
Status of the Higgs to tau tau - 26th March 2010

17. Status of the Higgs to tau tau - 26th March 2010
Template morphing
The efficiency distributions are now a combination of nominal plus shifted templates much alike the first term of a Taylor’s expansion series
In the expected number of events in each bin, we replace the efficiency distributions by the efficiency distributions with morphing (jet and electron energy scales)
The morphing parameter controls the strength of the energy scale uncertainty.
The prior for the morphing parameter is a Gaussian PDF with mean zero and standard deviation one.
Therefore the posterior probability shall take into account these systematic uncertainties
Carlos Solans
Status of the Higgs to tau tau - 26th March 2010

18. Status of the Higgs to tau tau - 26th March 2010
DATASETS
Carlos Solans
Status of the Higgs to tau tau - 26th March 2010

19. Status of the Higgs to tau tau - 26th March 2010
Datasets
For the current analysis we use MC datasets produced for Athena 12 in the MSSM mH-max scenario
Signal datasets
Four different values of mA: ( 150, 300, 450, 600 ) GeV
With different values of tanβ
Smearing of the templates for different values of tanβ should be considered. We only play with normalization here.
Background datasets
Z to tau tau
ttbar
QCD dijets (J1-J5) used for fake rate computation
Carlos Solans
Status of the Higgs to tau tau - 26th March 2010

20. Identification efficiency
Carlos Solans
Status of the Higgs to tau tau - 26th March 2010

21. Identification efficiency
Carlos Solans
Status of the Higgs to tau tau - 26th March 2010

22. Identification efficiency
Carlos Solans
Status of the Higgs to tau tau - 26th March 2010

23. Status of the Higgs to tau tau - 26th March 2010
Fake rates
With the current event selection, a real Jet that is reconstructed as a tau and passes the tau selection will be removed from the Jet container by the overlap removal.
This effect “is” very small when tight tau cuts are applied
We define the fake rate of taus as
# tau candidates after the identification cuts
# jet without overlap removal
We can define the fake rate of taus for loose or tight tau cuts
Carlos Solans
Status of the Higgs to tau tau - 26th March 2010

24. Status of the Higgs to tau tau - 26th March 2010
Fake rates
Carlos Solans
Status of the Higgs to tau tau - 26th March 2010

25. Status of the Higgs to tau tau - 26th March 2010
MC event
Carlos Solans
Status of the Higgs to tau tau - 26th March 2010

26. Status of the Higgs to tau tau - 26th March 2010
RESULTS
Carlos Solans
Status of the Higgs to tau tau - 26th March 2010

27. Status of the Higgs to tau tau - 26th March 2010
Exclusion limit
Setting exclusion limits for two background model for all masses
Uncertainty in jet and electron energy scale shifts
ΔE(electron)=1%, ΔE(jet)=5%,
Carlos Solans
Status of the Higgs to tau tau - 26th March 2010

28. Upper exclusion limit at 95% CL
150 GeV
300 GeV
450 GeV
600 GeV
No morphing
Morphing
Carlos Solans
Status of the Higgs to tau tau - 26th March 2010

29. Upper exclusion limit at 95% CL
Carlos Solans
Status of the Higgs to tau tau - 26th March 2010

30. Status of the Higgs to tau tau - 26th March 2010
Coverage
We repeat the example but this time adding signal to each pseudo-experiment
100% of the predicted cross-section by MSSM for the given parameters (mA=150 GeV, tanβ=45, mhmax scenario, σA=24500 fb)
The method to set exclusion limits is slightly conservative for the MSSM model presented
For conservative limit we expect < 5% of S+B pseudo-experiments to lie below the true value of S
In our case, ~3.4% (morphing) and ~3.8% (no morphing) of the pseudo-experiments lie under the signal cross-section.
~3.4%
~3.8%
Carlos Solans
Status of the Higgs to tau tau - 26th March 2010

31. Status of the Higgs to tau tau - 26th March 2010
Signal significance
Signal significance for two background model for all masses and different values of the cross-section
Uncertainty in jet and electron energy scale shifts
ΔE(electron)=1%, ΔE(jet)=5%,
Carlos Solans
Status of the Higgs to tau tau - 26th March 2010

32. Status of the Higgs to tau tau - 26th March 2010
Signal significance
Carlos Solans
Status of the Higgs to tau tau - 26th March 2010

33. Status of the Higgs to tau tau - 26th March 2010
Signal significance
Carlos Solans
Status of the Higgs to tau tau - 26th March 2010