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Kyle Cranmer (NYU) OKC Prospects, Stockholm, Sept. 16, 2010

Cosmology and Particle Physics

Kyle Cranmer,

New York University

The Theory/Experiment Interface:

Publishing the Likelihood Function with the RooFit/RooStats Workspace

1

(2)

Kyle Cranmer (NYU)

Center for Cosmology and Particle Physics

OKC Prospects, Stockholm, Sept. 16, 2010

Some Personal History

2

(3)

Kyle Cranmer (NYU)

Cosmology and Particle Physics

OKC Prospects, Stockholm, Sept. 16, 2010

Some Personal History

2

Archbishop of Canterbury Thomas Cranmer (born: 1489, executed:

1556) author of the “Book of

Common Prayer”

(4)

Kyle Cranmer (NYU)

Center for Cosmology and Particle Physics

OKC Prospects, Stockholm, Sept. 16, 2010

Some Personal History

2

Archbishop of Canterbury Thomas Cranmer (born: 1489, executed:

1556) author of the “Book of Common Prayer”

Two centuries later (when this Book had become an official prayer book of the Church of England) Thomas Bayes was a non-conformist minister

(Presbyterian) who refused to use

Cranmerʼs book

(5)

Kyle Cranmer (NYU)

Particle Physics

OKC Prospects, Stockholm, Sept. 16, 2010 3

“Bayesians address the question everyone is interested in, by using assumptions no-one believes”

“Frequentists use impeccable logic to deal with an issue of no interest to anyone”

-L. Lyons

(6)

Kyle Cranmer (NYU)

Center for Cosmology and Particle Physics

OKC Prospects, Stockholm, Sept. 16, 2010

A few points

Objective part of Bayesian inference is encoded in Likelihood

‣ improvements in Likelihood is not Bayesian vs. Frequentist

Prior may be based on data

‣ but it also depends on the initial prior

In the same way that the “Bayesian calculus” allows for

propagation of belief, the measurements can be combined with the likelihood function

4

P (theory |data) = L(data |theory)π(theory) P (data)

π(theory) ∝ L (data’ |theory)η(theory) η(theory)

L tot (data’ |theory) = L(data’|theory)L (data’ |theory)

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Kyle Cranmer (NYU)

Cosmology and Particle Physics

OKC Prospects, Stockholm, Sept. 16, 2010

LHC Data Likelihood

Functions

Interpretation

Fundamental Lagrangian

Data Modeling Data Modeling

CMB

Dark Matter Searche

Data Modeling

Ideal scenario

The ideal scenario for the interface between the data and the inference to the fundamental lagrangian parameters is through a likelihood

function that accurately incorporates all the experimental systematics and retains as much power in the data as possible

Is this feasible?

‣ It is the basic model on which Zfitter, GFitter, SFitter, Fittino, MasterCode, Kismet, SuperBayes, etc. are based

‣ unfortunately, likelihood functions are usually simplistic and based on a few 1-d measurements

5

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Kyle Cranmer (NYU)

Center for Cosmology and Particle Physics

OKC Prospects, Stockholm, Sept. 16, 2010

Current scenario

Taken from the GFitter paper

6

Eur. Phys. J. C (2009) 60: 543–583 557

information available for 10 discrete data points in the mass range 155 ≤ M H ≤ 200 GeV based on prelimi- nary searches using data samples of up to 3 fb −1 inte- grated luminosity [73]. For the mass range 110 ≤ M H ≤ 200 GeV, Tevatron results based on 2.4 fb −1 are provided for −2 ln Q [ 72], however not for the corresponding con- fidence levels.

To include the direct Higgs searches in the complete SM fit we interpret the −2 ln Q results for a given Higgs mass hypothesis 23 as measurements and derive a log- likelihood estimator quantifying the deviation of the data from the corresponding SM Higgs expectation. For this purpose we transform the one-sided CL s +b into two- sided confidence levels 24 using CL 2 - sided

s +b = 2CL s +b for CL s +b ≤ 0.5 and CL 2 - sided

s +b = 2(1 − CL s +b ) for CL s +b >

0.5. The contribution to the χ 2 estimator of the fit is then obtained via δχ 2 = 2·[Erf −1 (1 −CL 2 - sided

s +b ) ] 2 , where Erf −1 is the inverse error function, 25 and where the under- lying probability density function has been assumed to be symmetric (cf. footnote 21 on p. 556).

For the complete mass range available for the LEP searches (M H ≤ 120 GeV), and for the high-mass region of the Tevatron searches (155 ≤ M H ≤ 200 GeV), we em- ploy the CL s +b values determined by the experiments. For the low-mass Tevatron results (110 ≤ M H ≤ 150 GeV), where the CL s +b values are not provided, they are esti- mated from the measured −2 ln Q values that are com-

23 This procedure only uses the M H value under consideration, where Higgs-mass hypothesis and measurement are compared. It thus ne- glects that in the SM a given signal hypothesis entails background hy- potheses for all M H values other than the one considered. An analysis accounting for this should provide a statistical comparison of a given hypothesis with all available measurements. This however would re- quire to know the correlations among all the measurement points (or better: the full experimental likelihood as a function of the Higgs-mass hypothesis), which are not provided by the experiments to date. The difference to the hypothesis-only test employed here is expected to be small at present, but may become important once an experimental Higgs signal appears, which however has insufficient significance yet to claim a discovery (which would allow one to discard all other Higgs- mass hypotheses). We thank Bill Murray (RAL) for bringing this point to our attention.

24 The experiments integrate only the tail towards larger −2 ln Q values of the probability density function to compute CL s +b (corresponding to a counting experiment with to too few observed events with respect to the s + b hypothesis), which is later used to derive CL s in the modified frequentest approach. They thus quantify Higgs-like (not necessarily SM Higgs) enhancements in the data. In the global SM fit, however, one is interested in the compatibility between the SM hypothesis and the experimental data as a whole, and must hence account for any devi- ation, including the tail towards smaller −2 ln Q values (corresponding to a counting experiment with too many Higgs candidates with respect to the s + b hypothesis where, s labels the SM Higgs signal).

25 The use of Erf −1 provides a consistent error interpretation when (re)translating the χ 2 estimator into a confidence level via CL = 1 − Prob(χ 2 , 1) = Erf( !

χ 2 /2).

pared with those expected for the s +b hypothesis, and us- ing the errors derived by the experiments for the b hypoth- esis. We have tested this approximation in the high-mass region, where the experimental values of CL s +b from the Tevatron are provided, and found a systematic overesti- mation of the contribution to our χ 2 test statistics of about 30%, with small dependence on the Higgs mass. We thus rescale the test statistics in the mass region where the CL s +b approximation is used (i.e. 110 ≤ M H ≤ 150 GeV) by the correction factor 0.77. 26 Once made available by the TEVNPH Working Group, this approximation will be replaced by the published CL s +b values.

Our method follows the spirit of a global SM fit and takes advantage from downward fluctuations of the back- ground in the sensitive region to obtain a more restrictive limit on the SM Higgs production as is obtained with the modified frequentest approach. The resulting χ 2 curves versus M H are shown in Fig. 4.1. The low-mass exclu- sion is dominated by the LEP searches, while the infor- mation above 120 GeV is contributed by the Tevatron experiments. Following the original figure, the Tevatron measurements have been interpolated by straight lines for the purpose of presentation and in the fit which deals with continuous M H values.

Constraints on the weak mixing angle can also be derived from atomic parity violation measurements in caesium, thal- lium, lead and bismuth. For heavy atoms one determines the weak charge, Q W ≈ Z(1 − 4 sin 2 θ W ) − N. Because the present experimental accuracy of 0.6% (3.2%) for Q W from Cs [77, 78] (Tl [79, 80]) is still an order of magnitude away from a competitive constraint on sin 2 θ W , we do not include it into the fit. (Including it would reduce the error on the fitted Higgs mass by 0.2 GeV.) Due to the same reason we do not include the parity violation left-right asymmetry measurement using fixed target polarised Møller scattering at low Q 2 = 0.026 GeV 2 [81]. 27

The NuTeV Collaboration measured ratios of neutral and charged current cross sections in neutrino–nucleon scatter- ing at an average Q 2 $ 20 GeV 2 using both muon neutrino and muon anti-neutrino beams [82]. The results derived for the effective weak couplings are not included in this analy- sis because of unclear theoretical uncertainties from QCD effects such as next-to-leading order corrections and nu- clear effects of the bound nucleon parton distribution func- tions [83] (for reviews see, e.g., Refs. [84, 85]).

Although a large number of precision results for α S at various scales are available, including recent 3NLO deter- minations at the τ -mass scale [17, 18, 86, 87], we do not

26 The correction factor reduces the value of the χ 2 test statistics. As described in footnote 32, its application has little impact on the fit re- sults.

27 The main success of this measurement is to have established the run-

ning of the weak coupling strength at the 6.4σ level.

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Kyle Cranmer (NYU)

Cosmology and Particle Physics

OKC Prospects, Stockholm, Sept. 16, 2010

The situation 10 years ago...

7

Origins I: The First “Statistics in HEP” conference

But a practical problem remained: How to communicate multi-D likelihood?

!"#$%&'()*%+',)-.+'./'0%')01%'./'2)1231).%'2/4%$)5%6'7/-'8$//*',)-.+'./')01%'./'2)1231).%'5//&-%++'/9'9:.;!

http://indico.cern.ch/conferenceDisplay.py?confId=100458

(10)

Kyle Cranmer (NYU)

Center for Cosmology and Particle Physics

OKC Prospects, Stockholm, Sept. 16, 2010

Outline

Information:

‣ What is the RooStats Project?

‣ What the workspace can do for SUSY/BSM Fits

‣ Real-life examples from the LHC

Example Use cases

‣ A critical look at the weak points in our current chain

Moving forward:

‣ Hard problems that can be solved with planning

‣ Making a clear request to the experiments (discussion)

‣ Preparing toy benchmark examples (discussion)

8

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Kyle Cranmer (NYU)

Cosmology and Particle Physics

OKC Prospects, Stockholm, Sept. 16, 2010

RooStats: Project info

Started in 2005, when René Brun asked me to help organize statistical tools in ROOT

‣ Main goals are to provide a common

framework for various statistical techniques (Frequentist, Bayesian, Likelihood based,...)

We want tools to work with probability models of arbitrary complexity (which implies interfaces, etc.)

‣ Decided to base tools on RooFit’s data modeling language and core interfaces

Initially an ATLAS/CMS project, but other experiments are interested (LHCb, Fermi, ...)

‣ core developers

● K. Cranmer (ATLAS), Lorenzo Moneta (ROOT), Gregory Schott (CMS), Wouter Verkerke (RooFit)

‣ open project, you are welcome to contribute

● ~10 others contributing now, growing fast

Included since ROOT v5.22 (we are now on 5.27)

9 https://twiki.cern.ch/twiki/bin/view/RooStats/WebHome

RooStats has been a topic of

conversation in every combined ATLAS/

CMS statistics forum meeting

‣ In July, we showed the first toy

ATLAS/CMS Higgs combination using the tools of RooFit/RooStats.

‣ see agenda:

http://indico.cern.ch/conferenceDisplay.py?confId=100458

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Kyle Cranmer (NYU)

Center for Cosmology and Particle Physics

OKC Prospects, Stockholm, Sept. 16, 2010

Major Goals and Status

Goal: Standardize interface for major statistical procedures so that they can work on an arbitrary RooFit model & dataset and handle many parameters of interest and nuisance parameters (systematics).

Status: Done

ConfIntervalCalculator & HypoTestCalculator interface for tools

they return ConfidenceInterval and HypoTestResult

Goal: Implement most accepted techniques from Frequentist, Bayesian, and Likelihood-based approaches

Status: Done / Ongoing

ProfileLikelihoodCalculator: (Likelihood) the method of MINUIT/MINOS

FeldmanCousins: (Frequentist) a generalization of F-C that can incorporate systematics

MCMCCalculator: (Bayesian) uses Metropolis-Hastings algorithm (native or BAT)

HybridCalculator: (Bayesian/Frequentist Hybrid) used at LEP and Tevatron

Goal: Provide utilities to perform combined measurements

Status: Partially done / Ongoing

RooWorkspace allows one to save arbitrary RooFit model (even with custom code) into a .root file. PDFs and DataSets have been extended to facilitate combinations.

● Same technology can aid in digital publishing

10

today’s focus

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Kyle Cranmer (NYU)

Cosmology and Particle Physics

OKC Prospects, Stockholm, Sept. 16, 2010

What goes in a Workspace

The workspace stores the full probability model and any data necessary to evaluate the likelihood function

‣ it is the code necessary to evaluate the likelihood function at an arbitrary point in the parameter space. It is not a big table of likelihood values!

‣ we are using the same ROOT technology that the LHC experiments are using to save their data

● well supported, and supports “schema evolution” / backwards compatibility

‣ the probability model also allows you to generate toy data for any given parameter point

● necessary for frequentist methods, goodness of fit, coverage

‣ PDFs and functions can be extended by the user (source stored in workspace) I will show some visualization of real-life LHC probability models. Let’s start with a simple example:

11

G(x |µ, σ)

RooRealVar : µ

RooRealVar : x RooRealVar : σ

RooGaussian : G

Figure 2: test

2 Parameter Estimation

3 Test Statistics and Sampling Distributions

3.1 TestStatistic interface and implementations

We added a new interface class called TestStatistic. It defines the method Evaluate(data, parameterPoint), which returns a double. This class can be used in conjunction with the ToyMCSampler class to generate sampling distributions for a user-defined test statistic.

The following concrete implementations of the TestStatistic interface are currently available ProfileLikelihoodTestStatReturns the log of profile likelihood ratio. Generally a powerful test statistic. NumEventsTestStatReturns the number of events in the dataset. Useful for number counting experiments. DebuggingTestStat Simply returns a uniform random number between 0,1. Useful for debugging. SamplingDistribution and the TestStatSampler interface and implementations

We introduced a “result” or data model class called SamplingDistribution, which holds the sampling distribution of an arbitrary real valued test statistic. The class also can return the inverse of the cumulative distribution function (with or without interpolation).

We introduced an interface for any tool that can produce a SamplingDistribution, called TestStatSampler. The interface is essentially GetSamplingDistribution(parameterPoint) which returns a SamplingDistribution based on a given probability density function. We foresee a few versions of this tool based on toy Monte Carlo, importance sampling, Fourier transforms, etc.

The following concrete implementation of the TestStatSampler interface are currently available ToyMCSamplerUses a Toy Monte Carlo approach to build the sampling distribution. The pdf’s generate method to generate is used to generate toy data, and then the test statistic is evaluated at the requested parameter point. DebuggingSampler Simply returns a uniform distribution between 0,1. Useful for debugging. NeymanConstruction and FeldmanCousins

A flexible framework for the Neyman Construction was added in this release. The Ney- manConstruction is a concrete implementation of the IntervalCalculator interface, but it needs several additional components to be specified before use. The design factorizes the choice

8

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Kyle Cranmer (NYU)

Center for Cosmology and Particle Physics

OKC Prospects, Stockholm, Sept. 16, 2010

RooFit: A data modeling toolkit

12

Wouter Verkerke, UCSB

Building realistic models

– Composition (‘plug & play’)

– Convolution

g(x;m,s) m(y;a 0 ,a 1 )

=

! =

g(x,y;a0,a1,s)

Possible in any PDF

No explicit support in PDF code needed

Wouter Verkerke, UCSB

Building realistic models

• Complex PDFs be can be trivially composed using operator classes – Addition

– Multiplication

+ =

* =

Wouter Verkerke, UCSB

Parameters of composite PDF objects

RooAddPdf

sum

RooGaussian

gauss1

RooGaussian

gauss2

RooArgusBG

argus

RooRealVar

g1frac

RooRealVar

g2frac

RooRealVar

x

RooRealVar

sigma

RooRealVar

mean1

RooRealVar

mean2

RooRealVar

argpar

RooRealVar

cutoff

RooArgSet *paramList = sum.getParameters(data) ; paramList->Print("v") ;

RooArgSet::parameters:

1) RooRealVar::argpar : -1.00000 C 2) RooRealVar::cutoff : 9.0000 C 3) RooRealVar::g1frac : 0.50000 C 4) RooRealVar::g2frac : 0.10000 C 5) RooRealVar::mean1 : 2.0000 C 6) RooRealVar::mean2 : 3.0000 C 7) RooRealVar::sigma : 1.0000 C

The parameters of sum are the combined parameters

of its components

A major tool at BaBar. Fit complicated models with >100 parameters!

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Kyle Cranmer (NYU)

Cosmology and Particle Physics

OKC Prospects, Stockholm, Sept. 16, 2010 13

The RooFit/RooStats workspace

RooStat’s Workspace can save in a file the full likelihood model and the minimal data necessary to reproduce likelihood function.

The technology is generic, we decide how to parametrize the model.

Being used by ATLAS/CMS for very complicated models

Need this for combinations, exciting potential

for publishing results.

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Kyle Cranmer (NYU)

Center for Cosmology and Particle Physics

OKC Prospects, Stockholm, Sept. 16, 2010

Extracting Contours from these results

• One can plot 2-d contours, 1-d likelihood functions.

• One can evaluate likelihood in N-d and use to evaluate a theoretical model

• If the model has nuisance parameters for systematics, they will be included!

• Easy to combine multiple measurements

14

• The workspace can represent arbitrary models with many parameters of interest and many nuisance parameters

Taken from Wouter Verkerke, NIKHEF

This contour is NOT an ellipse!

(17)

Kyle Cranmer (NYU)

Particle Physics

OKC Prospects, Stockholm, Sept. 16, 2010

Examples of Real-Life LHC Models

15

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Kyle Cranmer (NYU)

Center for Cosmology and Particle Physics

OKC Prospects, Stockholm, Sept. 16, 2010

ATLAS H->γγ

16

RooSimultaneoussimPdf_reparametrized_hgg RooSuperCategoryfitCatRooCategoryetaCat

RooCategoryconvCat

RooCategoryjetCat

RooAddPdfsumPdf_{etaGood;noConv;noJets}_reparametrized_hgg RooProdPdfBackground_{etaGood;noConv;noJets} Hfitter::MggBkgPdfBackground_mgg_{etaGood;noConv;noJets}

RooRealVaroffset

RooRealVarxi_noJets

RooRealVarmgg RooAddPdfBackground_cosThStar_{etaGood;noConv;noJets}

RooGaussianBackground_bump_cosThStar_2RooRealVarbkg_csth02

RooRealVarbkg_csthSigma2 RooRealVarcosThStar

RooRealVarbkg_csthRelNorm2 RooGaussianBackground_bump_cosThStar_1_{etaGood;noConv;noJets}

RooRealVarbkg_csth01_{etaGood;noJets}

RooRealVarbkg_csthSigma1_{etaGood;noJets} Hfitter::HftPeggedPolyBackground_poly_cosThStar_{etaGood;noConv;noJets}

RooRealVarbkg_csthCoef3

RooRealVarbkg_csthCoef5 RooRealVarbkg_csthPower_{etaGood;noJets}

RooRealVarbkg_csthCoef1_{etaGood;noJets}

RooRealVarbkg_csthCoef2_{etaGood;noJets}

RooRealVarbkg_csthCoef4_{etaGood;noJets}

RooRealVarbkg_csthCoef6_{etaGood;noJets}

RooRealVarbkg_csthRelNorm1_{etaGood;noJets} RooAddPdfBackground_pT_{etaGood;noConv;noJets}

Hfitter::HftPolyExpBackground_pT_3

RooRealVarbkg_ptM0

RooRealVarbkg_ptPow3

RooRealVarbkg_ptExp3 RooRealVarpT RooRealVarbkg_ptTail2Norm

RooGaussianBackground_pT_TC

RooRealVarbkg_ptTCS RooRealVarbkg_ptTailTCNorm Hfitter::HftPolyExpBackground_pT_2_{etaGood;noConv;noJets}

RooRealVarbkg_ptPow2_noJets

RooRealVarbkg_ptExp2_noJets Hfitter::HftPolyExpBackground_pT_1_{etaGood;noConv;noJets}

RooRealVarbkg_ptPow1_noJets

RooRealVarbkg_ptExp1_noJets RooRealVarbkg_ptTail1Norm_noJets RooProdPdfSignal_{etaGood;noConv;noJets} RooAddPdfSignal_mgg_{etaGood;noConv;noJets}

RooGaussianSignal_mgg_T

ail

RooRealVarmTail

RooRealVarsigTail RooCBShapeSignal_mgg_Peak_{etaGood;noConv;noJets}

RooFormulaVarmHiggsFormula_{etaGood;noConv;noJets}

RooRealVarmHiggs RooRealVardmHiggs_{etaGood;noConv} RooRealVarmRes_{etaGood;noConv}

RooRealVartailAlpha_{etaGood;noConv}

RooRealVartailN_{etaGood;noConv}

RooRealVarmggRelNorm_{etaGood;noConv} RooAddPdfSignal_cosThStar_{etaGood;noConv;noJets} RooGaussianSignalcsthstr_1_{etaGood;noConv;noJets}

RooRealVarsig_csthstrSigma_1RooRealVarsig_csthstr0_1_{etaGood;noJets} RooGaussianSignalcsthstr_2_{etaGood;noConv;noJets}

RooRealVarsig_csthstrSigma_2RooRealVarsig_csthstr0_2_{etaGood;noJets} RooRealVarsig_csthstrRelNorm_1_{etaGood;noJets}

RooAddPdfSignal_pT_{etaGood;noConv;noJets}

RooBifurGaussSignalpT_4

RooRealVarsig_pt0_4

RooRealVarsig_ptSigmaL_4

RooRealVarsig_ptSigmaR_4 RooRealVarsig_ptRelNorm_4 RooBifurGaussSignalpT_1_{etaGood;noConv;noJets}

RooRealVarsig_pt0_1_noJets

RooRealVarsig_ptSigmaL_1_noJets

RooRealVarsig_ptSigmaR_1_noJets RooGaussianSignalpT_2_{etaGood;noConv;noJets}

RooRealVarsig_pt0_2_noJets

RooRealVarsig_ptSigma_2_noJets RooGaussianSignalpT_3_{etaGood;noConv;noJets}

RooRealVarsig_pt0_3_noJets

RooRealVarsig_ptSigma_3_noJets RooRealVarsig_ptRelNorm_1_noJets

RooRealVarsig_ptRelNorm_2_noJets RooRealVarn_Background_{etaGood;noConv;noJets}

RooFormulaVarnCat_Signal_{etaGood;noConv;noJets}_reparametrized_hggRooRealVarf_Signal_{etaGood;noConv;noJets}

RooProductnew_n_Signal

RooRealVarmu

RooRealVarn_Signal

RooAddPdfsumPdf_{etaMed;noConv;noJets}_reparametrized_hgg

RooProdPdfBackground_{etaMed;noConv;noJets} Hfitter::MggBkgPdfBackground_mgg_{etaMed;noConv;noJets}

RooAddPdfBackground_cosThStar_{etaMed;noConv;noJets} RooGaussianBackground_bump_cosThStar_1_{etaMed;noConv;noJets}

RooRealVarbkg_csth01_{etaMed;noJets}

RooRealVarbkg_csthSigma1_{etaMed;noJets} Hfitter::HftPeggedPolyBackground_poly_cosThStar_{etaMed;noConv;noJets}

RooRealVarbkg_csthPower_{etaMed;noJets}

RooRealVarbkg_csthCoef1_{etaMed;noJets}

RooRealVarbkg_csthCoef2_{etaMed;noJets}

RooRealVarbkg_csthCoef4_{etaMed;noJets}

RooRealVarbkg_csthCoef6_{etaMed;noJets}

RooRealVarbkg_csthRelNorm1_{etaMed;noJets} RooAddPdfBackground_pT_{etaMed;noConv;noJets}

Hfitter::HftPolyExpBackground_pT_2_{etaMed;noConv;noJets}

Hfitter::HftPolyExpBackground_pT_1_{etaMed;noConv;noJets} RooProdPdfSignal_{etaMed;noConv;noJets} RooAddPdfSignal_mgg_{etaMed;noConv;noJets}

RooCBShapeSignal_mgg_Peak_{etaMed;noConv;noJets}

RooFormulaVarmHiggsFormula_{etaMed;noConv;noJets}

RooRealVardmHiggs_{etaMed;noConv} RooRealVarmRes_{etaMed;noConv}

RooRealVartailAlpha_{etaMed;noConv}

RooRealVartailN_{etaMed;noConv}

RooRealVarmggRelNorm_{etaMed;noConv} RooAddPdfSignal_cosThStar_{etaMed;noConv;noJets}RooGaussianSignalcsthstr_1_{etaMed;noConv;noJets}

RooRealVarsig_csthstr0_1_{etaMed;noJets} RooGaussianSignalcsthstr_2_{etaMed;noConv;noJets}

RooRealVarsig_csthstr0_2_{etaMed;noJets} RooRealVarsig_csthstrRelNorm_1_{etaMed;noJets}

RooAddPdfSignal_pT_{etaMed;noConv;noJets}

RooBifurGaussSignalpT_1_{etaMed;noConv;noJets}

RooGaussianSignalpT_2_{etaMed;noConv;noJets}

RooGaussianSignalpT_3_{etaMed;noConv;noJets} RooRealVarn_Background_{etaMed;noConv;noJets} RooFormulaVarnCat_Signal_{etaMed;noConv;noJets}_reparametrized_hgg

RooRealVarf_Signal_{etaMed;noConv;noJets}

RooAddPdfsumPdf_{etaBad;noConv;noJets}_reparametrized_hgg RooProdPdfBackground_{etaBad;noConv;noJets} Hfitter::MggBkgPdfBackground_mgg_{etaBad;noConv;noJets}

RooAddPdfBackground_cosThStar_{etaBad;noConv;noJets} RooGaussianBackground_bump_cosThStar_1_{etaBad;noConv;noJets}

RooRealVarbkg_csth01_{etaBad;noJets}

RooRealVarbkg_csthSigma1_{etaBad;noJets} Hfitter::HftPeggedPolyBackground_poly_cosThStar_{etaBad;noConv;noJets}

RooRealVarbkg_csthPower_{etaBad;noJets}

RooRealVarbkg_csthCoef1_{etaBad;noJets}

RooRealVarbkg_csthCoef2_{etaBad;noJets}

RooRealVarbkg_csthCoef4_{etaBad;noJets}

RooRealVarbkg_csthCoef6_{etaBad;noJets} RooRealVarbkg_csthRelNorm1_{etaBad;noJets}

RooAddPdfBackground_pT_{etaBad;noConv;noJets}

Hfitter::HftPolyExpBackground_pT_2_{etaBad;noConv;noJets}

Hfitter::HftPolyExpBackground_pT_1_{etaBad;noConv;noJets} RooProdPdfSignal_{etaBad;noConv;noJets}

RooAddPdfSignal_mgg_{etaBad;noConv;noJets} RooCBShapeSignal_mgg_Peak_{etaBad;noConv;noJets}

RooFormulaVarmHiggsFormula_{etaBad;noConv;noJets}

RooRealVardmHiggs_{etaBad;noConv} RooRealVarmRes_{etaBad;noConv}

RooRealVartailAlpha_{etaBad;noConv}

RooRealVartailN_{etaBad;noConv}

RooRealVarmggRelNorm_{etaBad;noConv} RooAddPdfSignal_cosThStar_{etaBad;noConv;noJets}RooGaussianSignalcsthstr_1_{etaBad;noConv;noJets}

RooRealVarsig_csthstr0_1_{etaBad;noJets} RooGaussianSignalcsthstr_2_{etaBad;noConv;noJets}

RooRealVarsig_csthstr0_2_{etaBad;noJets} RooRealVarsig_csthstrRelNorm_1_{etaBad;noJets}

RooAddPdfSignal_pT_{etaBad;noConv;noJets}

RooBifurGaussSignalpT_1_{etaBad;noConv;noJets}

RooGaussianSignalpT_2_{etaBad;noConv;noJets}

RooGaussianSignalpT_3_{etaBad;noConv;noJets} RooRealVarn_Background_{etaBad;noConv;noJets}

RooFormulaVarnCat_Signal_{etaBad;noConv;noJets}_reparametrized_hgg

RooRealVarf_Signal_{etaBad;noConv;noJets}

RooAddPdfsumPdf_{etaGood;Conv;noJets}_reparametrized_hgg RooProdPdfBackground_{etaGood;Conv;noJets} Hfitter::MggBkgPdfBackground_mgg_{etaGood;Conv;noJets}

RooAddPdfBackground_cosThStar_{etaGood;Conv;noJets}

RooGaussianBackground_bump_cosThStar_1_{etaGood;Conv;noJets}

Hfitter::HftPeggedPolyBackground_poly_cosThStar_{etaGood;Conv;noJets}

RooAddPdfBackground_pT_{etaGood;Conv;noJets}

Hfitter::HftPolyExpBackground_pT_2_{etaGood;Conv;noJets}

Hfitter::HftPolyExpBackground_pT_1_{etaGood;Conv;noJets} RooProdPdfSignal_{etaGood;Conv;noJets}

RooAddPdfSignal_mgg_{etaGood;Conv;noJets} RooCBShapeSignal_mgg_Peak_{etaGood;Conv;noJets}

RooFormulaVarmHiggsFormula_{etaGood;Conv;noJets}

RooRealVardmHiggs_{etaGood;Conv} RooRealVarmRes_{etaGood;Conv}

RooRealVartailAlpha_{etaGood;Conv}

RooRealVartailN_{etaGood;Conv} RooRealVarmggRelNorm_{etaGood;Conv}

RooAddPdfSignal_cosThStar_{etaGood;Conv;noJets}RooGaussianSignalcsthstr_1_{etaGood;Conv;noJets}

RooGaussianSignalcsthstr_2_{etaGood;Conv;noJets}

RooAddPdfSignal_pT_{etaGood;Conv;noJets}

RooBifurGaussSignalpT_1_{etaGood;Conv;noJets}

RooGaussianSignalpT_2_{etaGood;Conv;noJets}

RooGaussianSignalpT_3_{etaGood;Conv;noJets} RooRealVarn_Background_{etaGood;Conv;noJets}RooFormulaVarnCat_Signal_{etaGood;Conv;noJets}_reparametrized_hgg

RooRealVarf_Signal_{etaGood;Conv;noJets}

RooAddPdfsumPdf_{etaMed;Conv;noJets}_reparametrized_hgg RooProdPdfBackground_{etaMed;Conv;noJets} Hfitter::MggBkgPdfBackground_mgg_{etaMed;Conv;noJets}

RooAddPdfBackground_cosThStar_{etaMed;Conv;noJets}

RooGaussianBackground_bump_cosThStar_1_{etaMed;Conv;noJets}

Hfitter::HftPeggedPolyBackground_poly_cosThStar_{etaMed;Conv;noJets}

RooAddPdfBackground_pT_{etaMed;Conv;noJets}

Hfitter::HftPolyExpBackground_pT_2_{etaMed;Conv;noJets}

Hfitter::HftPolyExpBackground_pT_1_{etaMed;Conv;noJets} RooProdPdfSignal_{etaMed;Conv;noJets}

RooAddPdfSignal_mgg_{etaMed;Conv;noJets}RooCBShapeSignal_mgg_Peak_{etaMed;Conv;noJets}

RooFormulaVarmHiggsFormula_{etaMed;Conv;noJets}

RooRealVardmHiggs_{etaMed;Conv} RooRealVarmRes_{etaMed;Conv}

RooRealVartailAlpha_{etaMed;Conv}

RooRealVartailN_{etaMed;Conv} RooRealVarmggRelNorm_{etaMed;Conv}RooAddPdfSignal_cosThStar_{etaMed;Conv;noJets}

RooGaussianSignalcsthstr_1_{etaMed;Conv;noJets}

RooGaussianSignalcsthstr_2_{etaMed;Conv;noJets}

RooAddPdfSignal_pT_{etaMed;Conv;noJets}

RooBifurGaussSignalpT_1_{etaMed;Conv;noJets}

RooGaussianSignalpT_2_{etaMed;Conv;noJets}

RooGaussianSignalpT_3_{etaMed;Conv;noJets} RooRealVarn_Background_{etaMed;Conv;noJets}RooFormulaVarnCat_Signal_{etaMed;Conv;noJets}_reparametrized_hgg

RooRealVarf_Signal_{etaMed;Conv;noJets}

RooAddPdfsumPdf_{etaBad;Conv;noJets}_reparametrized_hgg RooProdPdfBackground_{etaBad;Conv;noJets} Hfitter::MggBkgPdfBackground_mgg_{etaBad;Conv;noJets}

RooAddPdfBackground_cosThStar_{etaBad;Conv;noJets} RooGaussianBackground_bump_cosThStar_1_{etaBad;Conv;noJets}

Hfitter::HftPeggedPolyBackground_poly_cosThStar_{etaBad;Conv;noJets}

RooAddPdfBackground_pT_{etaBad;Conv;noJets}

Hfitter::HftPolyExpBackground_pT_2_{etaBad;Conv;noJets}

Hfitter::HftPolyExpBackground_pT_1_{etaBad;Conv;noJets} RooProdPdfSignal_{etaBad;Conv;noJets} RooAddPdfSignal_mgg_{etaBad;Conv;noJets}

RooCBShapeSignal_mgg_Peak_{etaBad;Conv;noJets}

RooFormulaVarmHiggsFormula_{etaBad;Conv;noJets}

RooRealVardmHiggs_{etaBad;Conv} RooRealVarmRes_{etaBad;Conv}

RooRealVartailAlpha_{etaBad;Conv}

RooRealVartailN_{etaBad;Conv} RooRealVarmggRelNorm_{etaBad;Conv}

RooAddPdfSignal_cosThStar_{etaBad;Conv;noJets}RooGaussianSignalcsthstr_1_{etaBad;Conv;noJets}

RooGaussianSignalcsthstr_2_{etaBad;Conv;noJets}

RooAddPdfSignal_pT_{etaBad;Conv;noJets}

RooBifurGaussSignalpT_1_{etaBad;Conv;noJets}

RooGaussianSignalpT_2_{etaBad;Conv;noJets}

RooGaussianSignalpT_3_{etaBad;Conv;noJets} RooRealVarn_Background_{etaBad;Conv;noJets}

RooFormulaVarnCat_Signal_{etaBad;Conv;noJets}_reparametrized_hgg RooRealVarf_Signal_{etaBad;Conv;noJets}

RooAddPdfsumPdf_{etaGood;noConv;jet}_reparametrized_hgg RooProdPdfBackground_{etaGood;noConv;jet} Hfitter::MggBkgPdfBackground_mgg_{etaGood;noConv;jet}

RooRealVarxi_jet RooAddPdfBackground_cosThStar_{etaGood;noConv;jet} RooGaussianBackground_bump_cosThStar_1_{etaGood;noConv;jet}

RooRealVarbkg_csth01_{etaGood;jet}

RooRealVarbkg_csthSigma1_{etaGood;jet} Hfitter::HftPeggedPolyBackground_poly_cosThStar_{etaGood;noConv;jet}

RooRealVarbkg_csthRelNorm1_{etaGood;jet}

RooAddPdfBackground_pT_{etaGood;noConv;jet}

Hfitter::HftPolyExpBackground_pT_2_{etaGood;noConv;jet}

Hfitter::HftPolyExpBackground_pT_1_{etaGood;noConv;jet}

RooRealVarbkg_ptPow1_jet

RooRealVarbkg_ptExp1_jet RooProdPdfSignal_{etaGood;noConv;jet}RooAddPdfSignal_mgg_{etaGood;noConv;jet}

RooCBShapeSignal_mgg_Peak_{etaGood;noConv;jet}

RooFormulaVarmHiggsFormula_{etaGood;noConv;jet} RooAddPdfSignal_cosThStar_{etaGood;noConv;jet}RooGaussianSignalcsthstr_1_{etaGood;noConv;jet}

RooGaussianSignalcsthstr_2_{etaGood;noConv;jet}

RooAddPdfSignal_pT_{etaGood;noConv;jet}

RooBifurGaussSignalpT_1_{etaGood;noConv;jet}

RooRealVarsig_pt0_1_jet

RooRealVarsig_ptSigmaL_1_jet

RooRealVarsig_ptSigmaR_1_jet RooGaussianSignalpT_2_{etaGood;noConv;jet}

RooRealVarsig_pt0_2_jet

RooRealVarsig_ptSigma_2_jet RooGaussianSignalpT_3_{etaGood;noConv;jet}

RooRealVarsig_pt0_3_jet

RooRealVarsig_ptSigma_3_jet RooRealVarsig_ptRelNorm_1_jet

RooRealVarsig_ptRelNorm_2_jet RooRealVarn_Background_{etaGood;noConv;jet}

RooFormulaVarnCat_Signal_{etaGood;noConv;jet}_reparametrized_hgg

RooRealVarf_Signal_{etaGood;noConv;jet}

RooAddPdfsumPdf_{etaMed;noConv;jet}_reparametrized_hgg RooProdPdfBackground_{etaMed;noConv;jet} Hfitter::MggBkgPdfBackground_mgg_{etaMed;noConv;jet}

RooAddPdfBackground_cosThStar_{etaMed;noConv;jet} RooGaussianBackground_bump_cosThStar_1_{etaMed;noConv;jet}

RooRealVarbkg_csth01_{etaMed;jet}

RooRealVarbkg_csthSigma1_{etaMed;jet} Hfitter::HftPeggedPolyBackground_poly_cosThStar_{etaMed;noConv;jet}

RooRealVarbkg_csthPower_{etaMed;jet}

RooRealVarbkg_csthCoef1_{etaMed;jet}

RooRealVarbkg_csthCoef2_{etaMed;jet}

RooRealVarbkg_csthCoef4_{etaMed;jet}

RooRealVarbkg_csthCoef6_{etaMed;jet} RooRealVarbkg_csthRelNorm1_{etaMed;jet}

RooAddPdfBackground_pT_{etaMed;noConv;jet}

Hfitter::HftPolyExpBackground_pT_2_{etaMed;noConv;jet}

Hfitter::HftPolyExpBackground_pT_1_{etaMed;noConv;jet} RooProdPdfSignal_{etaMed;noConv;jet} RooAddPdfSignal_mgg_{etaMed;noConv;jet}

RooCBShapeSignal_mgg_Peak_{etaMed;noConv;jet}

RooFormulaVarmHiggsFormula_{etaMed;noConv;jet} RooAddPdfSignal_cosThStar_{etaMed;noConv;jet} RooGaussianSignalcsthstr_1_{etaMed;noConv;jet}

RooRealVarsig_csthstr0_1_{etaMed;jet} RooGaussianSignalcsthstr_2_{etaMed;noConv;jet}

RooRealVarsig_csthstr0_2_{etaMed;jet} RooRealVarsig_csthstrRelNorm_1_{etaMed;jet}

RooAddPdfSignal_pT_{etaMed;noConv;jet}

RooBifurGaussSignalpT_1_{etaMed;noConv;jet}

RooGaussianSignalpT_2_{etaMed;noConv;jet}

RooGaussianSignalpT_3_{etaMed;noConv;jet} RooRealVarn_Background_{etaMed;noConv;jet}

RooFormulaVarnCat_Signal_{etaMed;noConv;jet}_reparametrized_hggRooRealVarf_Signal_{etaMed;noConv;jet}

RooAddPdfsumPdf_{etaBad;noConv;jet}_reparametrized_hgg

RooProdPdfBackground_{etaBad;noConv;jet} Hfitter::MggBkgPdfBackground_mgg_{etaBad;noConv;jet}

RooAddPdfBackground_cosThStar_{etaBad;noConv;jet} RooGaussianBackground_bump_cosThStar_1_{etaBad;noConv;jet}

RooRealVarbkg_csth01_{etaBad;jet}

RooRealVarbkg_csthSigma1_{etaBad;jet} Hfitter::HftPeggedPolyBackground_poly_cosThStar_{etaBad;noConv;jet}

RooRealVarbkg_csthPower_{etaBad;jet}

RooRealVarbkg_csthCoef1_{etaBad;jet}

RooRealVarbkg_csthCoef2_{etaBad;jet}

RooRealVarbkg_csthCoef4_{etaBad;jet}

RooRealVarbkg_csthCoef6_{etaBad;jet} RooRealVarbkg_csthRelNorm1_{etaBad;jet}

RooAddPdfBackground_pT_{etaBad;noConv;jet}

Hfitter::HftPolyExpBackground_pT_2_{etaBad;noConv;jet}

Hfitter::HftPolyExpBackground_pT_1_{etaBad;noConv;jet} RooProdPdfSignal_{etaBad;noConv;jet}RooAddPdfSignal_mgg_{etaBad;noConv;jet}

RooCBShapeSignal_mgg_Peak_{etaBad;noConv;jet}

RooFormulaVarmHiggsFormula_{etaBad;noConv;jet} RooAddPdfSignal_cosThStar_{etaBad;noConv;jet}RooGaussianSignalcsthstr_1_{etaBad;noConv;jet}

RooRealVarsig_csthstr0_1_{etaBad;jet} RooGaussianSignalcsthstr_2_{etaBad;noConv;jet}

RooRealVarsig_csthstr0_2_{etaBad;jet} RooRealVarsig_csthstrRelNorm_1_{etaBad;jet}

RooAddPdfSignal_pT_{etaBad;noConv;jet}

RooBifurGaussSignalpT_1_{etaBad;noConv;jet}

RooGaussianSignalpT_2_{etaBad;noConv;jet}

RooGaussianSignalpT_3_{etaBad;noConv;jet} RooRealVarn_Background_{etaBad;noConv;jet} RooFormulaVarnCat_Signal_{etaBad;noConv;jet}_reparametrized_hgg

RooRealVarf_Signal_{etaBad;noConv;jet}

RooAddPdfsumPdf_{etaMed;Conv;jet}_reparametrized_hgg RooProdPdfBackground_{etaMed;Conv;jet} Hfitter::MggBkgPdfBackground_mgg_{etaMed;Conv;jet}

RooAddPdfBackground_cosThStar_{etaMed;Conv;jet} RooGaussianBackground_bump_cosThStar_1_{etaMed;Conv;jet}

Hfitter::HftPeggedPolyBackground_poly_cosThStar_{etaMed;Conv;jet}

RooAddPdfBackground_pT_{etaMed;Conv;jet}

Hfitter::HftPolyExpBackground_pT_2_{etaMed;Conv;jet}

Hfitter::HftPolyExpBackground_pT_1_{etaMed;Conv;jet} RooProdPdfSignal_{etaMed;Conv;jet}

RooAddPdfSignal_mgg_{etaMed;Conv;jet}RooCBShapeSignal_mgg_Peak_{etaMed;Conv;jet}

RooFormulaVarmHiggsFormula_{etaMed;Conv;jet} RooAddPdfSignal_cosThStar_{etaMed;Conv;jet}RooGaussianSignalcsthstr_1_{etaMed;Conv;jet}

RooGaussianSignalcsthstr_2_{etaMed;Conv;jet}

RooAddPdfSignal_pT_{etaMed;Conv;jet}

RooBifurGaussSignalpT_1_{etaMed;Conv;jet}

RooGaussianSignalpT_2_{etaMed;Conv;jet}

RooGaussianSignalpT_3_{etaMed;Conv;jet} RooRealVarn_Background_{etaMed;Conv;jet}

RooFormulaVarnCat_Signal_{etaMed;Conv;jet}_reparametrized_hgg

RooRealVarf_Signal_{etaMed;Conv;jet}

RooAddPdfsumPdf_{etaBad;Conv;jet}_reparametrized_hgg RooProdPdfBackground_{etaBad;Conv;jet} Hfitter::MggBkgPdfBackground_mgg_{etaBad;Conv;jet}

RooAddPdfBackground_cosThStar_{etaBad;Conv;jet} RooGaussianBackground_bump_cosThStar_1_{etaBad;Conv;jet}

Hfitter::HftPeggedPolyBackground_poly_cosThStar_{etaBad;Conv;jet}

RooAddPdfBackground_pT_{etaBad;Conv;jet}

Hfitter::HftPolyExpBackground_pT_2_{etaBad;Conv;jet}

Hfitter::HftPolyExpBackground_pT_1_{etaBad;Conv;jet} RooProdPdfSignal_{etaBad;Conv;jet}

RooAddPdfSignal_mgg_{etaBad;Conv;jet}RooCBShapeSignal_mgg_Peak_{etaBad;Conv;jet}

RooFormulaVarmHiggsFormula_{etaBad;Conv;jet} RooAddPdfSignal_cosThStar_{etaBad;Conv;jet}RooGaussianSignalcsthstr_1_{etaBad;Conv;jet}

RooGaussianSignalcsthstr_2_{etaBad;Conv;jet}

RooAddPdfSignal_pT_{etaBad;Conv;jet}

RooBifurGaussSignalpT_1_{etaBad;Conv;jet}

RooGaussianSignalpT_2_{etaBad;Conv;jet}

RooGaussianSignalpT_3_{etaBad;Conv;jet} RooRealVarn_Background_{etaBad;Conv;jet}

RooFormulaVarnCat_Signal_{etaBad;Conv;jet}_reparametrized_hgg

RooRealVarf_Signal_{etaBad;Conv;jet}

RooAddPdfsumPdf_{etaGood;noConv;vbf}_reparametrized_hgg RooProdPdfBackground_{etaGood;noConv;vbf} Hfitter::MggBkgPdfBackground_mgg_{etaGood;noConv;vbf}

RooRealVarxi_vbf RooAddPdfBackground_cosThStar_{etaGood;noConv;vbf}

RooGaussianBackground_bump_cosThStar_1_{etaGood;noConv;vbf}RooRealVarbkg_csth01_{etaGood;vbf}

RooRealVarbkg_csthSigma1_{etaGood;vbf} Hfitter::HftPeggedPolyBackground_poly_cosThStar_{etaGood;noConv;vbf}

RooRealVarbkg_csthPower_{etaGood;vbf}

RooRealVarbkg_csthCoef1_{etaGood;vbf}

RooRealVarbkg_csthCoef2_{etaGood;vbf}

RooRealVarbkg_csthCoef4_{etaGood;vbf}

RooRealVarbkg_csthCoef6_{etaGood;vbf}

RooRealVarbkg_csthRelNorm1_{etaGood;vbf} RooAddPdfBackground_pT_{etaGood;noConv;vbf}

Hfitter::HftPolyExpBackground_pT_2_{etaGood;noConv;vbf}

RooRealVarbkg_ptPow2_vbf

RooRealVarbkg_ptExp2_vbf Hfitter::HftPolyExpBackground_pT_1_{etaGood;noConv;vbf}

RooRealVarbkg_ptPow1_vbf

RooRealVarbkg_ptExp1_vbf RooRealVarbkg_ptTail1Norm_vbf RooProdPdfSignal_{etaGood;noConv;vbf}RooAddPdfSignal_mgg_{etaGood;noConv;vbf}

RooCBShapeSignal_mgg_Peak_{etaGood;noConv;vbf}

RooFormulaVarmHiggsFormula_{etaGood;noConv;vbf} RooAddPdfSignal_cosThStar_{etaGood;noConv;vbf}RooGaussianSignalcsthstr_1_{etaGood;noConv;vbf}

RooRealVarsig_csthstr0_1_{etaGood;vbf} RooGaussianSignalcsthstr_2_{etaGood;noConv;vbf}

RooRealVarsig_csthstr0_2_{etaGood;vbf} RooRealVarsig_csthstrRelNorm_1_{etaGood;vbf}

RooAddPdfSignal_pT_{etaGood;noConv;vbf}

RooBifurGaussSignalpT_1_{etaGood;noConv;vbf}

RooRealVarsig_pt0_1_vbf

RooRealVarsig_ptSigmaL_1_vbf

RooRealVarsig_ptSigmaR_1_vbf RooGaussianSignalpT_2_{etaGood;noConv;vbf}

RooRealVarsig_pt0_2_vbf

RooRealVarsig_ptSigma_2_vbf RooGaussianSignalpT_3_{etaGood;noConv;vbf}

RooRealVarsig_pt0_3_vbf

RooRealVarsig_ptSigma_3_vbf RooRealVarsig_ptRelNorm_1_vbf

RooRealVarsig_ptRelNorm_2_vbf RooRealVarn_Background_{etaGood;noConv;vbf}

RooFormulaVarnCat_Signal_{etaGood;noConv;vbf}_reparametrized_hggRooRealVarf_Signal_{etaGood;noConv;vbf}

RooAddPdfsumPdf_{etaMed;noConv;vbf}_reparametrized_hgg RooProdPdfBackground_{etaMed;noConv;vbf} Hfitter::MggBkgPdfBackground_mgg_{etaMed;noConv;vbf}

RooAddPdfBackground_cosThStar_{etaMed;noConv;vbf}

RooGaussianBackground_bump_cosThStar_1_{etaMed;noConv;vbf}

RooRealVarbkg_csth01_{etaMed;vbf}

RooRealVarbkg_csthSigma1_{etaMed;vbf} Hfitter::HftPeggedPolyBackground_poly_cosThStar_{etaMed;noConv;vbf}

RooRealVarbkg_csthPower_{etaMed;vbf}

RooRealVarbkg_csthCoef1_{etaMed;vbf}

RooRealVarbkg_csthCoef2_{etaMed;vbf}

RooRealVarbkg_csthCoef4_{etaMed;vbf}

RooRealVarbkg_csthCoef6_{etaMed;vbf}

RooRealVarbkg_csthRelNorm1_{etaMed;vbf} RooAddPdfBackground_pT_{etaMed;noConv;vbf}

Hfitter::HftPolyExpBackground_pT_2_{etaMed;noConv;vbf}

Hfitter::HftPolyExpBackground_pT_1_{etaMed;noConv;vbf} RooProdPdfSignal_{etaMed;noConv;vbf}RooAddPdfSignal_mgg_{etaMed;noConv;vbf}

RooCBShapeSignal_mgg_Peak_{etaMed;noConv;vbf}

RooFormulaVarmHiggsFormula_{etaMed;noConv;vbf} RooAddPdfSignal_cosThStar_{etaMed;noConv;vbf}RooGaussianSignalcsthstr_1_{etaMed;noConv;vbf}

RooRealVarsig_csthstr0_1_{etaMed;vbf} RooGaussianSignalcsthstr_2_{etaMed;noConv;vbf}

RooRealVarsig_csthstr0_2_{etaMed;vbf} RooRealVarsig_csthstrRelNorm_1_{etaMed;vbf}

RooAddPdfSignal_pT_{etaMed;noConv;vbf}

RooBifurGaussSignalpT_1_{etaMed;noConv;vbf}

RooGaussianSignalpT_2_{etaMed;noConv;vbf}

RooGaussianSignalpT_3_{etaMed;noConv;vbf} RooRealVarn_Background_{etaMed;noConv;vbf}

RooFormulaVarnCat_Signal_{etaMed;noConv;vbf}_reparametrized_hggRooRealVarf_Signal_{etaMed;noConv;vbf}

RooAddPdfsumPdf_{etaBad;noConv;vbf}_reparametrized_hgg RooProdPdfBackground_{etaBad;noConv;vbf} Hfitter::MggBkgPdfBackground_mgg_{etaBad;noConv;vbf}

RooAddPdfBackground_cosThStar_{etaBad;noConv;vbf}

RooGaussianBackground_bump_cosThStar_1_{etaBad;noConv;vbf}

RooRealVarbkg_csth01_{etaBad;vbf}

RooRealVarbkg_csthSigma1_{etaBad;vbf} Hfitter::HftPeggedPolyBackground_poly_cosThStar_{etaBad;noConv;vbf}

RooRealVarbkg_csthPower_{etaBad;vbf}

RooRealVarbkg_csthCoef1_{etaBad;vbf}

RooRealVarbkg_csthCoef2_{etaBad;vbf}

RooRealVarbkg_csthCoef4_{etaBad;vbf}

RooRealVarbkg_csthCoef6_{etaBad;vbf}

RooRealVarbkg_csthRelNorm1_{etaBad;vbf} RooAddPdfBackground_pT_{etaBad;noConv;vbf}

Hfitter::HftPolyExpBackground_pT_2_{etaBad;noConv;vbf}

Hfitter::HftPolyExpBackground_pT_1_{etaBad;noConv;vbf} RooProdPdfSignal_{etaBad;noConv;vbf} RooAddPdfSignal_mgg_{etaBad;noConv;vbf}

RooCBShapeSignal_mgg_Peak_{etaBad;noConv;vbf}

RooFormulaVarmHiggsFormula_{etaBad;noConv;vbf} RooAddPdfSignal_cosThStar_{etaBad;noConv;vbf}RooGaussianSignalcsthstr_1_{etaBad;noConv;vbf}

RooRealVarsig_csthstr0_1_{etaBad;vbf} RooGaussianSignalcsthstr_2_{etaBad;noConv;vbf}

RooRealVarsig_csthstr0_2_{etaBad;vbf} RooRealVarsig_csthstrRelNorm_1_{etaBad;vbf}

RooAddPdfSignal_pT_{etaBad;noConv;vbf}

RooBifurGaussSignalpT_1_{etaBad;noConv;vbf}

RooGaussianSignalpT_2_{etaBad;noConv;vbf}

RooGaussianSignalpT_3_{etaBad;noConv;vbf} RooRealVarn_Background_{etaBad;noConv;vbf}RooFormulaVarnCat_Signal_{etaBad;noConv;vbf}_reparametrized_hgg

RooRealVarf_Signal_{etaBad;noConv;vbf}

RooAddPdfsumPdf_{etaGood;Conv;vbf}_reparametrized_hgg RooProdPdfBackground_{etaGood;Conv;vbf} Hfitter::MggBkgPdfBackground_mgg_{etaGood;Conv;vbf}

RooAddPdfBackground_cosThStar_{etaGood;Conv;vbf}

RooGaussianBackground_bump_cosThStar_1_{etaGood;Conv;vbf} Hfitter::HftPeggedPolyBackground_poly_cosThStar_{etaGood;Conv;vbf}

RooAddPdfBackground_pT_{etaGood;Conv;vbf}

Hfitter::HftPolyExpBackground_pT_2_{etaGood;Conv;vbf}

Hfitter::HftPolyExpBackground_pT_1_{etaGood;Conv;vbf} RooProdPdfSignal_{etaGood;Conv;vbf} RooAddPdfSignal_mgg_{etaGood;Conv;vbf}

RooCBShapeSignal_mgg_Peak_{etaGood;Conv;vbf}

RooFormulaVarmHiggsFormula_{etaGood;Conv;vbf} RooAddPdfSignal_cosThStar_{etaGood;Conv;vbf}RooGaussianSignalcsthstr_1_{etaGood;Conv;vbf}

RooGaussianSignalcsthstr_2_{etaGood;Conv;vbf}

RooAddPdfSignal_pT_{etaGood;Conv;vbf}

RooBifurGaussSignalpT_1_{etaGood;Conv;vbf}

RooGaussianSignalpT_2_{etaGood;Conv;vbf}

RooGaussianSignalpT_3_{etaGood;Conv;vbf} RooRealVarn_Background_{etaGood;Conv;vbf}

RooFormulaVarnCat_Signal_{etaGood;Conv;vbf}_reparametrized_hggRooRealVarf_Signal_{etaGood;Conv;vbf}

RooAddPdfsumPdf_{etaMed;Conv;vbf}_reparametrized_hgg RooProdPdfBackground_{etaMed;Conv;vbf}Hfitter::MggBkgPdfBackground_mgg_{etaMed;Conv;vbf}

RooAddPdfBackground_cosThStar_{etaMed;Conv;vbf}

RooGaussianBackground_bump_cosThStar_1_{etaMed;Conv;vbf}

Hfitter::HftPeggedPolyBackground_poly_cosThStar_{etaMed;Conv;vbf}

RooAddPdfBackground_pT_{etaMed;Conv;vbf}

Hfitter::HftPolyExpBackground_pT_2_{etaMed;Conv;vbf}

Hfitter::HftPolyExpBackground_pT_1_{etaMed;Conv;vbf} RooProdPdfSignal_{etaMed;Conv;vbf}

RooAddPdfSignal_mgg_{etaMed;Conv;vbf}

RooCBShapeSignal_mgg_Peak_{etaMed;Conv;vbf}

RooFormulaVarmHiggsFormula_{etaMed;Conv;vbf} RooAddPdfSignal_cosThStar_{etaMed;Conv;vbf}

RooGaussianSignalcsthstr_1_{etaMed;Conv;vbf}

RooGaussianSignalcsthstr_2_{etaMed;Conv;vbf}

RooAddPdfSignal_pT_{etaMed;Conv;vbf}

RooBifurGaussSignalpT_1_{etaMed;Conv;vbf}

RooGaussianSignalpT_2_{etaMed;Conv;vbf}

RooGaussianSignalpT_3_{etaMed;Conv;vbf} RooRealVarn_Background_{etaMed;Conv;vbf}

RooFormulaVarnCat_Signal_{etaMed;Conv;vbf}_reparametrized_hggRooRealVarf_Signal_{etaMed;Conv;vbf}

RooAddPdfsumPdf_{etaBad;Conv;vbf}_reparametrized_hgg RooProdPdfBackground_{etaBad;Conv;vbf} Hfitter::MggBkgPdfBackground_mgg_{etaBad;Conv;vbf}

RooAddPdfBackground_cosThStar_{etaBad;Conv;vbf}

RooGaussianBackground_bump_cosThStar_1_{etaBad;Conv;vbf}

Hfitter::HftPeggedPolyBackground_poly_cosThStar_{etaBad;Conv;vbf}

RooAddPdfBackground_pT_{etaBad;Conv;vbf}

Hfitter::HftPolyExpBackground_pT_2_{etaBad;Conv;vbf}

Hfitter::HftPolyExpBackground_pT_1_{etaBad;Conv;vbf} RooProdPdfSignal_{etaBad;Conv;vbf}RooAddPdfSignal_mgg_{etaBad;Conv;vbf}

RooCBShapeSignal_mgg_Peak_{etaBad;Conv;vbf}

RooFormulaVarmHiggsFormula_{etaBad;Conv;vbf} RooAddPdfSignal_cosThStar_{etaBad;Conv;vbf}

RooGaussianSignalcsthstr_1_{etaBad;Conv;vbf}

RooGaussianSignalcsthstr_2_{etaBad;Conv;vbf}

RooAddPdfSignal_pT_{etaBad;Conv;vbf}

RooBifurGaussSignalpT_1_{etaBad;Conv;vbf}

RooGaussianSignalpT_2_{etaBad;Conv;vbf}

RooGaussianSignalpT_3_{etaBad;Conv;vbf} RooRealVarn_Background_{etaBad;Conv;vbf}

RooFormulaVarnCat_Signal_{etaBad;Conv;vbf}_reparametrized_hgg

RooRealVarf_Signal_{etaBad;Conv;vbf} RooRealVarbkg_csthPower_{etaGood;jet}

RooRealVarbkg_csthCoef1_{etaGood;jet}

RooRealVarbkg_csthCoef2_{etaGood;jet}

RooRealVarbkg_csthCoef4_{etaGood;jet}

RooRealVarbkg_csthCoef6_{etaGood;jet}

RooRealVarbkg_ptPow2_jet

RooRealVarbkg_ptExp2_jet RooRealVarbkg_ptTail1Norm_jet

RooRealVarsig_csthstr0_1_{etaGood;jet}

RooRealVarsig_csthstr0_2_{etaGood;jet} RooRealVarsig_csthstrRelNorm_1_{etaGood;jet}

RooAddPdfsumPdf_{etaGood;Conv;jet}_reparametrized_hgg RooProdPdfBackground_{etaGood;Conv;jet} Hfitter::MggBkgPdfBackground_mgg_{etaGood;Conv;jet}

RooAddPdfBackground_cosThStar_{etaGood;Conv;jet} RooGaussianBackground_bump_cosThStar_1_{etaGood;Conv;jet}

Hfitter::HftPeggedPolyBackground_poly_cosThStar_{etaGood;Conv;jet}

RooAddPdfBackground_pT_{etaGood;Conv;jet}

Hfitter::HftPolyExpBackground_pT_2_{etaGood;Conv;jet}

Hfitter::HftPolyExpBackground_pT_1_{etaGood;Conv;jet} RooProdPdfSignal_{etaGood;Conv;jet}RooAddPdfSignal_mgg_{etaGood;Conv;jet}

RooCBShapeSignal_mgg_Peak_{etaGood;Conv;jet}

RooFormulaVarmHiggsFormula_{etaGood;Conv;jet} RooAddPdfSignal_cosThStar_{etaGood;Conv;jet}

RooGaussianSignalcsthstr_1_{etaGood;Conv;jet}

RooGaussianSignalcsthstr_2_{etaGood;Conv;jet}

RooAddPdfSignal_pT_{etaGood;Conv;jet}

RooBifurGaussSignalpT_1_{etaGood;Conv;jet}

RooGaussianSignalpT_2_{etaGood;Conv;jet}

RooGaussianSignalpT_3_{etaGood;Conv;jet} RooRealVarn_Background_{etaGood;Conv;jet}

RooFormulaVarnCat_Signal_{etaGood;Conv;jet}_reparametrized_hggRooRealVarf_Signal_{etaGood;Conv;jet}

2.5 2.5

η 1

0 0.5 1 1.5 2 2.5

2 η

0 0.5 1 1.5 2 2.5

1.73 GeV (1) [14.4%]

1.73 GeV (1) [14.4%]

2.21 GeV (2) [13.8%]

2.21 GeV

(2) [13.8%] 2.26 GeV (3) [10.2%]

2.26 GeV (3) [10.2%]

2.58 GeV

(4) [9.4%]

2.58 GeV (4) [9.4%]

3.42 GeV

(5) [13.6%]

3.42 GeV (5) [13.6%]

2.26 GeV (6) [22.7%]

2.26 GeV (6) [22.7%]

3.33 GeV (7) [8.2%]

3.33 GeV

(7) [8.2%]

1.87 GeV (8) [7.8%]

1.87 GeV (8) [7.8%]

η 1

0 0.5 1 1.5 2 2.5

2 η

0 0.5 1 1.5 2 2.5

Pseudorapidity categories (at least one converted photon)

ATLAS detector and physics performance Volume II

Technical Design Report 25 May 1999

680 19 Higgs Bosons

For an integrated luminosity of 100 fb !1 , a Standard Model Higgs boson in the mass range be- tween 105 GeV and 145 GeV can be observed with a significance of more than 5" by using the H # $$ channel alone. Table 19-2 also contains the estimated significances of the H # $$ channel for an integrated luminosity of 30 fb -1 , corresponding to the first three years of LHC operation.

The significances at low luminosity have been evaluated by taking the resulting improvements in mass resolution and background rejection into account. A signal in the $$ channel can only be seen in this case with a significance of % 4" over a narrow mass range between 120 and 130 GeV.

The significances quoted in Table 19-2 are slightly higher than the ones given in the Technical Proposal. The main reason for this is the removal of the so called p T -balance cut, which was ap- plied in order to suppress bremsstrahlung background. Although without this cut the back- ground increases, there is a net gain in the significance. Another reason is the slightly improved mass resolution which is mainly due to a more sophisticated photon energy reconstruction, sep- arating converted and non-converted photons. These gains are somewhat offset by the higher reducible background.

As an example of signal reconstruction above background, Figure 19-4 shows the expected sig- nal from a Higgs boson with m H = 120 GeV for an integrated luminosity of 100 fb -1 . The H # $$

signal is clearly visible above the smooth $$ background, which is dominated by the irreducible continuum of real photon pairs.

19.2.2.2 Associated production: WH, ZH and ttH

The production of the Higgs boson in association with a W or a Z boson or with a tt pair can also be used to search for a low-mass Higgs boson. The production cross-section for the associated production is almost a factor 50 lower than for the direct production, leading to much smaller signal rates. If the associated W/Z boson or one of the top quarks is required to decay leptoni- cally, thereby leading to final states containing one isolated lepton and two isolated photons, the signal-to-background ratio can nevertheless be substantially improved with respect to the direct production. In addition, the vertex position can be unambiguously determined by the lepton charged track, resulting in better mass resolution at high luminosity than for the case of direct H # $$ production.

Figure 19-4 Expected H # $$ signal for m H = 120 GeV and for an integrated luminosity of 100 fb -1 . The signal is shown on top of the irreducible background (left) and after subtraction of this background (right).

10000 12500 15000 17500 20000

105 120 135

m $$ (GeV)

E vents / 2 G eV

0 500 1000 1500

105 120 135

m $$ (GeV)

Signal-background, events / 2 GeV

(19)

Kyle Cranmer (NYU)

Center for Cosmology and Particle Physics

OKC Prospects, Stockholm, Sept. 16, 2010

3-channel top combination

The graph below represents this PDF

‣ where there are several relations between the expected means in the different channels

17

RooSimultaneous simPdf RooProdPdf model_ee

RooGaussian lumiConstraint RooConstV ar 200

RooConstV ar

40 RooRealV ar Lumi RooGaussian alpha_EleEf fConstraint

RooConstV ar 0

RooConstV ar 1 RooRealV ar alpha_EleEf f

RooGaussian

alpha_PDFConstraint

RooRealV ar alpha_PDF RooGaussian alpha_modelConstraint

RooRealV ar alpha_model RooGaussian alpha_jesConstraint RooRealV ar alpha_jes RooGaussian alpha_eesConstraint RooRealV ar alpha_ees

RooGaussian alpha_isrfsrConstraint RooRealV ar alpha_isrfsr

RooGaussian alpha_XsecConstraint RooRealV ar alpha_Xsec RooGaussian alpha_DY_eeConstraint RooRealV ar alpha_DY_ee

RooGaussian alpha_FakeRateConstraint RooRealV ar

alpha_FakeRate

RooPoisson Pois_ee_0

RooRealV ar obsN_0 RooAddition ee_totN_0

RooProduct tbart_ee_overallSyst_x_HistSyst_0 LinInterpV ar tbart_ee_Hist_alpha_0

RooProduct

tbart_ee_SigXsec_x_epsilon

RooRealV ar SigXsecOverSM_comb

LinInterpV ar

tbart_ee_epsilon RooProduct SingleT op_ee_overallSyst_x_HistSyst_0 LinInterpV ar SingleT op_ee_Hist_alpha_0 LinInterpV ar SingleT op_ee_epsilon

RooProduct Diboson_ee_overallSyst_x_HistSyst_0 LinInterpV ar

Diboson_ee_Hist_alpha_0

LinInterpV ar Diboson_ee_epsilon RooProduct Zll_ee_overallSyst_x_HistSyst_0 LinInterpV ar Zll_ee_Hist_alpha_0 LinInterpV ar Zll_ee_epsilon

RooProduct Ztautau_ee_overallSyst_x_HistSyst_0 LinInterpV ar Ztautau_ee_Hist_alpha_0 LinInterpV ar Ztautau_ee_epsilon RooProduct fake_ee_overallSyst_x_Exp_0 RooRealV ar fake_ee_expN_0

LinInterpV ar fake_ee_epsilon

RooProdPdf model_emu

RooGaussian alpha_MuonEf fConstraint

RooRealV ar alpha_MuonEf f RooGaussian alpha_mesConstraint

RooRealV ar

alpha_mes

RooPoisson Pois_emu_0 RooAddition emu_totN_0 RooProduct tbart_emu_overallSyst_x_HistSyst_0 LinInterpV ar tbart_emu_Hist_alpha_0

RooProduct SingleT op_emu_overallSyst_x_HistSyst_0 LinInterpV ar

SingleT op_emu_Hist_alpha_0

LinInterpV ar SingleT op_emu_epsilon RooProduct Diboson_emu_overallSyst_x_HistSyst_0 LinInterpV ar Diboson_emu_Hist_alpha_0

LinInterpV ar Diboson_emu_epsilon RooProduct Ztautau_emu_overallSyst_x_HistSyst_0 LinInterpV ar Ztautau_emu_Hist_alpha_0 LinInterpV ar Ztautau_emu_epsilon

RooProduct fake_emu_overallSyst_x_Exp_0 RooRealV ar fake_emu_expN_0 LinInterpV ar fake_emu_epsilon

RooProdPdf model_mumu

RooGaussian alpha_DY_mumuConstraint

RooRealV ar alpha_DY_mumu RooPoisson Pois_mumu_0 RooAddition mumu_totN_0 RooProduct tbart_mumu_overallSyst_x_HistSyst_0 LinInterpV ar tbart_mumu_Hist_alpha_0

RooProduct SingleT op_mumu_overallSyst_x_HistSyst_0

LinInterpV ar SingleT op_mumu_Hist_alpha_0 LinInterpV ar SingleT op_mumu_epsilon

RooProduct Diboson_mumu_overallSyst_x_HistSyst_0

LinInterpV ar Diboson_mumu_Hist_alpha_0 LinInterpV ar Diboson_mumu_epsilon RooProduct Zll_mumu_overallSyst_x_HistSyst_0 LinInterpV ar Zll_mumu_Hist_alpha_0

LinInterpV ar Zll_mumu_epsilon RooProduct Ztautau_mumu_overallSyst_x_HistSyst_0 LinInterpV ar Ztautau_mumu_Hist_alpha_0 LinInterpV ar Ztautau_mumu_epsilon

RooProduct fake_mumu_overallSyst_x_Exp_0

RooRealV ar fake_mumu_expN_0 LinInterpV ar fake_mumu_epsilon RooCategory channelCat

One may wish to extend the likelihood function in Eq. 15 to include multiple channels (e. g. ee/ µ µ /e µ )

511

or several jet multiplicity bins. Formally, the extension looks very similar for both cases. Let us first

512

consider the case of multiple bins indexed by i. The expectation for the i th bin from the k th signal or

513

background contribution is

514

N ik exp = L ! ik "

j

# ˜ i jk # i jk ( $ j )

# ˜ i jk = ˜ N

exp ik "

j

# i jk ( $ j )

# ˜ i jk . (16)

Note, that we do not add the index to $ j , because we see this as a common source of systematics which

515

is common for the different bins and the different signal and background contributions. The likelihood

516

function is now a product over these bins

517

L( ! sig , L , $ j ) = "

i∈bins

!

Pois(N i obs |N i,tot exp ) × Gaus( ˜ L |L , ! L ) "

j

Gaus( ˜ $ j = 0| $ j , % $ j = 1)

"

. (17)

The likelihood function for multiple channels is similar, with an additional product over the multiple

518

channels. The only subtlety is that k now runs over the set of signal and backgrounds specific to that

519

channel. Similarly, the sources of systematics might also be different for the different channels. Leaving

520

the range of the indices implicit, we arrive at

521

L( ! sig , L , $ j ) = "

l∈{ee,µ µ,eµ}

#

"

i∈bins

!

Pois(N i obs |N i,tot exp ) Gaus( ˜ L |L , ! L ) "

j∈syst

Gaus(0| $ j , 1)

"$

. (18)

5.2 Extracting Measurements from the Profile Likelihood Ratio

522

Armed with the final likelihood function in Eq. 18 and the Asimov dataset, we can now derive the ex-

523

pected uncertainty on the desired cross section measurement. The likelihood function can be maximized

524

to determine the maximum likelihood estimate of all the parameters ˆ ! sig , ˆ L , ˆ $ j . One can then consider

525

the likelihood ratio

526

r( ! sig ) = L( ! sig , ˆ L , ˆ $ j )

L( ˆ ! sig , ˆ L , ˆ $ j ) (19) and the profile likelihood ratio:

527

& ( ! sig ) = L( ! sig , ˆˆ L , ˆˆ $ j )

L( ˆ ! sig , ˆ L , ˆ $ j ) (20) 34

3 observations from data 13 control samples

1 parameter of interest

13 nuisance parameters

References

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