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Nikhef Topical lectures on Statistics (3rd day) 14-16 December 2011 All information (slides, code, exercises) can be found at http://www.nikhef.nl/~ivov/TopicalLectures The Higgs boson 09:00-12:30 Faster-than-light neutrino’s 14:00-17:00 Aart Heijboer and Ivo van Vulpen Simple likelihood plot Can everybody do this ? If you want to reproduce this plot, but cannot please let us know, now or step by later TMath::Poisson( Nevt_bin, alpha ) http://www.nikhef.nl/~ivov/TopicalLectures/SimpleFit/ Di-photon mass Excluded cross-sections 4-lepton mass Significance s Counting events in a mass window Standard Model SM 10 Higgs 5 Data 12 Ok, now what ? Distribution of expected number of events from a large number of LHC’s SM SM+Higgs discovery exclusion data Discovery-aimed: p-value and significance incompatibiliy with the SM-only hypothesis SM 10 Higgs 5 Data 12 Expected significance p-value = 8.35% 1.38 sigma 1) What is the expected significance ? Observed significance p-value = 30.3% 0.5 sigma 2) What is the observed significance ? Discovery-aimed: p-value and significance SM Higgs 10 5 Expected significance p-value = 8.35% 1.38 sigma 3) At what Lumi do you expect to be able to claim a discovery ? 3 times more LUMINOSITY Expected significance SM 30 Higgs 15 Discovery if p-value < 2.87x10-7 p-value = 0.19% 2.9 sigma exected p-value 3.6 sigma observed p-value Exclusion: when/how do you exclude a signal incompatibiliy with the SM+Higgs hypothesis. Excluded if CL < 5% SM 10 Higgs 5 Data 12 Observed CLs+b SM SM+Higgs CLs+b = 18.5% 1) Observed confidence level ? Can we exclude the SM+Higgs hypothesis ? >5% NO Expected CLs+b 2) Expected confidence level ? Did we expect to be able to exclude the SM+Higgs hypothesis ? >5% NO CLs+b = 7.0% Exclusion: What about 1.5 x the Higgs cross-sec ? Expected CLs+b SM 10 Higgs 7.5 Data 12 h /hSM 1.50 CLs+b = 2.0% Observed CLs+b h /hSM 1.50 CLs+b = 6.8% 3) What σh/σhSM can we exclude ? Observed excluded cross-section σh/σhSM , = 1.64 Observed h /hSM to be excluded Excluded cross-sections Expected h /hSM to be excluded We will try to reproduce a few of these numbers Test statistic An (event) variable that tries to separate the 2 hypotheses: 1)The number of events L( 1) LR ˆ) L( 2)Likelihood ratio: L( 1) LR L( 0) used today A) Event counting L(μ) = Used in ATLAS/CMS Π Poisson(n i | SM Higgs) Loop over all bins Probability to observe ni events When you expect λSM+ μλHiggs B) NN output, or complex multi-dimensional fit, … Exercises ATLAS-CONF-2011-161 http://cdsweb.cern.ch/record/1406356/files/ATLASCONF2011161.pdf Mock hγγ analysis and try to reproduce 4 characteristic ATLAS numbers for a single mass (126 GeV), indicated by Quick intro into some of the usefull functions in the code higgs.C In root: .L higgs.C++ higgs() Some crucial functions: init_pdfs ATLAS our imitation Global histograms (TH1F*): pdf_bgr and pdf_sig Some crucial functions: generate_dataset Some crucial functions: compute_test_statistic Some crucial functions: get_test_statistic_distribution See also: plot_with bands (TH1F* h) from utils.h Some additional helper functions that you need (in utils.h) double median (TH1F*) double get_fraction_below(TH1F* h, double x_value) double get_fraction_above(TH1F* h, double x_value) TH1F* read_dataset(Tstring filename)