Statistics in LHC Phenomenology Tilman Plehn MPI f ur Physik & - PowerPoint PPT Presentation

Statistics in LHC Phenomenology Tilman Plehn Searches Neyman–Pearson Higgs to muons SUSY parameters Markov chains Statistics in LHC Phenomenology Tilman Plehn MPI f¨ ur Physik & University of Edinburgh Bonn, 2/2007

Statistics in LHC Outline Phenomenology Tilman Plehn Searches Neyman–Pearson Higgs to muons Maximum signal significance SUSY parameters Markov chains Neyman–Pearson lemma Example: Higgs to muons Supersymmetric parameter space Markov chains

Statistics in LHC Higgs searches — life is tough Phenomenology Tilman Plehn Life at LHC Searches – WBF H → ττ in Standard Model Neyman–Pearson [or MSSM] Higgs to muons – cut analysis promising enough SUSY parameters ⇒ experimentalists at work [for example Atlas–Freiburg–Bonn] Markov chains – neural net better [non-trivially bounded signal regions] – even better with LEP–type event weighting [not just counting experiment] – Higgs discovery channel? ⇒ could we guess such an outcome? [or the opposite] [B. Quayle, ATLAS Higgs meeting, 2003]

Statistics in LHC Neyman–Pearson lemma Phenomenology Tilman Plehn Answer: Neyman–Pearson lemma Searches Neyman–Pearson – correct hypothesis m 1 : Higgs signal Higgs to muons wrong hypothesis m 2 : SM background SUSY parameters – lemma: likelihood ratio p ( d | m 1 ) / p ( d | m 2 ) most powerful estimator Markov chains [lowest probability to mistake right for fluctuation of wrong (type-II error)] – likelihood: for phase–space event p ( d | m ) ∼ |M| 2 [from Monte Carlo] – estimator: plot density with estimator on x axis, cut signal–rich region Application: optimal observables [invite Markus Diehl...] – looking for best way to measure LEP physics – use Neyman–Pearson theorem to construct correlated observables Similar: matrix element method [CDF; DZero] – event likelihood from data and Monte–Carlo [jet–parton identification] – express likelihood of top events as function of m t – maximize probability p ( d | SM , m t ) to measure m t ⇒ likelihood hard to extract from data [single–top]

Statistics in LHC Neyman–Pearson lemma Phenomenology Tilman Plehn Answer: Neyman–Pearson lemma Searches Neyman–Pearson – correct hypothesis m 1 : Higgs signal wrong hypothesis m 2 : SM background Higgs to muons SUSY parameters – lemma: likelihood ratio p ( d | m 1 ) / p ( d | m 2 ) most powerful estimator Markov chains [lowest probability to mistake right for fluctuation of wrong (type-II error)] Optimal significance at parton level [Cranmer, TP] – example: log-likelihood for n -event Poisson statistics [independent channels] Pois ( n | b ) = e − b b n Pois ( n | s + b ) = e − ( s + b ) ( s + b ) n n ! n ! q = log Pois ( n | s + b ) 1 + s 1 + s j „ « „ « X X = − s + n log − → − s j + n j log Pois ( n | b ) b b j j j – independent events with non–trivial distributions j = 1 f ( j ) Pois ( n | s + b ) Q n n 1 + sf ( j ) ! s + b X s q = log = − s + log j = 1 f ( j ) bf ( j ) Pois ( n | b ) Q n j = 1 b b – continuous integration over phase space: s f s → |M s | 2 ! r ) | 2 1 + |M s ( � q ( � r ) = − σ s L + log |M b ( � r ) | 2

Statistics in LHC Neyman–Pearson lemma Phenomenology Tilman Plehn Answer: Neyman–Pearson lemma Searches Neyman–Pearson – correct hypothesis m 1 : Higgs signal wrong hypothesis m 2 : SM background Higgs to muons SUSY parameters – lemma: likelihood ratio p ( d | m 1 ) / p ( d | m 2 ) most powerful estimator Markov chains [lowest probability to mistake right for fluctuation of wrong (type-II error)] Optimal significance at parton level [Cranmer, TP] – from likelihood map q ( � r ) to probability distribution pdf – invert into single–event pdf r d σ b ( � r ) Z ` ´ ρ 1 , b ( q 0 ) = d � δ q ( � r ) − q 0 σ tot , b ρ n , b = ( ρ 1 , b ) n – Fourier–transform and compute n –event pdf: – combine n = 1 , ... into pdf 300 fb -1 ρ b ρ s+b X ρ b ( q ) = Pois ( n | b ) × ρ n , b ( q ) 0.1 n ⇒ integrate to CL b ( q ) = R ∞ dq ′ ρ b ( q ′ ) q 0.05 [5 σ is CLb = 2 . 85 10 − 7] 0 -30 -20 -10 0 10 20 30 q

Statistics in LHC Sub–optimal: detector effects Phenomenology Tilman Plehn Best of all worlds Searches Neyman–Pearson – irreducible & unsmeared: identical signal and background phase space Z Z Higgs to muons d � � � σ tot = dPS M PS d σ PS = r M ( r ) d σ ( r ) SUSY parameters – random numbers � r basis for phase space configurations Markov chains ∆ m width ≪ ∆ m meas ⇒ don’t be ridiculous µµ µµ More realistic – smear observable/random number transfer function W [Gaussian] Z ∞ Z r ⊥ dr ∗ r ) W ( r m , r ∗ d � � � σ tot = dr m M ( r ) d σ ( m ) m −∞ – modified phase–space vector � r = { � r ⊥ , r m } [without back door] – likelihood map over smeared � r ⇒ same procedure as before – complete smearing: replace phase space by set of distributions – lose strict maximum significance claim ⇒ step–by–step into Whizard [Cranmer, TP , Reuter]

Statistics in LHC Example: Higgs to muons Phenomenology Tilman Plehn (min) 1/ σ tot d σ /d ∆ p Weak–boson–fusion Higgs with H → µµ µµ j,j Searches Z QCD Neyman–Pearson – number of signal events small [ σ · BR ∼ 0 . 25 fb ] H Higgs to muons – no distribution with golden cut SUSY parameters ⇒ perfect for multivariate analysis Z ew Markov chains 0 1000 2000 (min) ∆ p µµ j,j [ GeV ] Awful old results [TP , Rainwater] √ σ QCD σ ew L 5 σ [ fb − 1 ] S MH σ H [ fb ] [ fb ] [ fb ] S / B significance △ σ/σ Z Z 14 115 0.25 3.57 0.40 1/9.1 1.7 σ 60 % 2600 14 120 0.22 2.60 0.33 1/7.5 1.8 σ 60 % 2300 14 130 0.17 1.61 0.24 1/6.5 1.7 σ 65 % 2700 14 140 0.10 1.11 0.19 1/7.5 1.2 σ 85 % 4900 200 115 2.57 39.6 5.3 1/10.1 5.3 σ 20 % 270 200 120 2.36 29.2 4.0 1/8.0 5.7 σ 20 % 230 200 130 1.80 18.7 2.7 1/6.9 5.3 σ 20 % 260 200 140 1.14 13.4 2.0 1/7.9 4.0 σ 27 % 500

Statistics in LHC Example: Higgs to muons Phenomenology Tilman Plehn (min) 1/ σ tot d σ /d ∆ p Weak–boson–fusion Higgs with H → µµ µµ j,j Searches Z QCD Neyman–Pearson – number of signal events small [ σ · BR ∼ 0 . 25 fb ] H Higgs to muons – no distribution with golden cut SUSY parameters ⇒ perfect for multivariate analysis Z ew Markov chains 0 1000 2000 (min) ∆ p µµ j,j [ GeV ] Statistical promise – mostly irreducible backgrounds – smearing only relevant for m µµ [mimic by Γ ′ H ?] d σ /dm µµ Z QCD all q 1 – compute likelihood map from matrix elements Z EW → upper limit (target?) on parton–level significance -1 10 H → WBF H → µµ : 3.5 sigma in 300 fb − 1 -2 q > − 1.5 10 [ ∼ 4 . 4 σ with mini-jet veto] – physics: confirm Yukawa coupling -3 10 117 118 119 120 121 122 123 ⇒ maybe, J¨ orn wants to have a look? m µµ [ GeV ]

Statistics in LHC Supersymmetric parameter space Phenomenology Tilman Plehn New physics at the LHC Searches Neyman–Pearson – complex models, including dark matter, flavor physics, low-energy physics,... Higgs to muons – honest parameters: weak-scale Lagrangean SUSY parameters – measurements: masses or edges Markov chains branching fractions cross sections – errors: general correlation, statistics & systematics & theory – problem in grid: huge phase space, local minimum? problem in fit: domain walls, global minimum? [also Fittino: Peter’s talk] First go at problem 10 – ask a friend how SUSY is broken ⇒ mSUGRA 8 – fit m 0 , m 1 / 2 – no problem, include indirect constraints 6 2 (today) – best fit pre-LHC [Ellis, Weinemeyer, Olive, Heiglein] CMSSM, µ > 0 χ 4 tan β = 10, A 0 = 0 ⇒ simple fit tan β = 10, A 0 = +m 1/2 [no theory bias, except they know it is mSUGRA] tan β = 10, A 0 = -m 1/2 2 tan β = 10, A 0 = +2 m 1/2 tan β = 10, A 0 = -2 m 1/2 0 0 200 400 600 800 1000 m 1/2 [GeV]

Statistics in LHC Supersymmetric parameter space Phenomenology Tilman Plehn New physics at the LHC Searches Neyman–Pearson – complex models, including dark matter, flavor physics, low-energy physics,... Higgs to muons – honest parameters: weak-scale Lagrangean SUSY parameters – measurements: masses or edges Markov chains branching fractions cross sections – errors: general correlation, statistics & systematics & theory – problem in grid: huge phase space, local minimum? problem in fit: domain walls, global minimum? [also Fittino: Peter’s talk] First go at problem – ask a friend how SUSY is broken ⇒ mSUGRA – fit m 0 , m 1 / 2 , A 0 , tan β, y t , ... ⇒ best fit to LHC/ILC measurements SPS1a ∆ LHC ∆ LHC ∆ ILC ∆ LHC+ILC masses edges m0 100 3.9 1.2 0.09 0.08 m1 / 2 250 1.7 1.0 0.13 0.11 tan β 10 1.1 0.9 0.12 0.12 A0 -100 33 20 4.8 4.3