(Tentative) Summary of the different methods to measure the spin in H →γγ J. Schaarschmidt, JB. de Vivie (LAL), S. Laplace (LPNHE) HSG1 meeting, September 13 th 2012 1 jeudi 13 septembre 2012
Introduction • Steps to measure spin in H →γγ : • Variable to disentangle the background: diphoton mass . Different methods to exploit it: • 1D fit on mass: • either to obtain S and B yields, for all events or in cos θ * bins • or to compute sWeights for cos θ * sPlots • 2D fit on mass + cos θ * • Angular variable to measure the spin: cos θ * • 3 possible definitions: boost axis, Collins Soper (used here), beam axis • test spin0/spin2 hypotheses using likelihood ratios: • «Tevatron style» L0/L2: test spin0 versus spin2 hypotheses - what has been investigated so far and in this talk • «LHC style» L2/Lmin (see HSG7 meeting on Sept. 12): fit the fraction of spin0/spin2 components - not yet investigated • correlations between mass and cos θ * need to be known for some of the methods 2 jeudi 13 septembre 2012
Samples and technical points • Samples: • Signal (dN/dcos θ * ∼ const): official HSG1 D3PDs • Background: Wisconsin fast sim (thanks to them !) • Spin 2: unofficial D3PDs from J. Albert/F. Bernlochner of Pythia6 Graviton- like m H =125 GeV (thanks to him !) • production modes available : gg (dN/dcos θ * ∼ 1+6cos 2 θ * +cos 4 θ * ) and qq (dN/dcos θ * ∼ 1-cos 4 θ * ) (VBF not treated here) • note: J. Albert and JB de Vivie noticed some weird differences for spin2 between Pythia6 and Pythia8. Anyway, official samples will be made with JHU generator... • Technical points: • workspace similar to official one, but with 2D pdf of mass and cos θ * (so far non-correlated) • here, use Collins-Soper definition of cos θ * (note: this definition does not «suffer» from the asymmetric kinematic cutoffs present in the Boost Axis definition mentionned by Wisconsin in HSG1 Sept. 6) 3 jeudi 13 septembre 2012
cos θ * distributions background 0.04 0.06 Collins-Soper Frame 0.035 Background Background Boost Axis Leading 0.05 Boost Axis Subleading Beam Axis 0.03 0.04 0.025 0.02 0.03 0.015 0.02 Collins-Soper Frame 0.01 Boost Axis 0.01 0.005 Beam Axis 0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 |Cos( *)| Cos( *) � � (plots: M. Kuna) 4 jeudi 13 septembre 2012
cos θ * distributions spin0/spin2 0.04 0.04 0.035 0.035 Spin 2 qq Spin 2 gg 0.03 0.03 0.025 0.025 0.02 0.02 0.015 0.015 Collins-Soper Frame Collins-Soper Frame 0.01 0.01 Boost Axis Boost Axis 0.005 0.005 Beam Axis Beam Axis 0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 |Cos( *)| |Cos( *)| � � 0.04 Spin 0 Signal 0.035 0.03 0.025 0.02 0.015 Collins-Soper Frame Boost Axis 0.01 Beam Axis 0.005 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 |Cos( *)| � (plots: M. Kuna) 5 jeudi 13 septembre 2012
cos θ * distributions 0.05 0.04 0.045 0.035 Collins-Soper Beam Axis 0.04 0.03 0.035 0.025 0.03 0.025 0.02 0.02 0.015 0.015 Background Background 0.01 Spin 0 Signal Spin 0 Signal 0.01 Spin 2 gg Spin 2 gg 0.005 0.005 Spin 2 qq Spin 2 qq 0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 |Cos( *)| � |Cos( *)| � 0.04 Boost Axis 0.035 0.03 0.025 0.02 0.015 Background Spin 0 Signal 0.01 Spin 2 gg Spin 2 qq 0.005 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 |Cos( *)| � (plots: M. Kuna) 6 jeudi 13 septembre 2012
Likelihood ratios definition • cos θ * pdf for signal: sum of spin0, spin2 (gg and qq - vbf not included yet) pdfs with fractions ε 0 , ε 2gg and ε 2qq =(1- ε 0- ε 2gg ) - so far, used ε 2gg =1 for spin2 hypothesis f S e | � 0 , � 2 gg ) = � 0 f S e ) + � S e ) + (1 − � 0 − � 2 gg ) f S c (cos θ ∗ 0 (cos θ ∗ 2 gg (cos θ ∗ 2 qq (cos θ ∗ e ) • the likelihood is given by: − ln L S [+ B ] � n S f S f S n B f B f B � � � � � ��� c, [ m ] ( � 0 , � 2 gg ) = (ˆ n S +ˆ n B ) − ln ˆ c (cos θ ∗ e | � 0 , � 2 gg ) m ( m e | m H ) +ˆ c (cos θ ∗ e ) m ( m e ) e where one includes the background component or not, and the correlations or not (here, without... ) • the likelihood ratios are defined by: «Tevatron way»: fixed fractions «LHC way»: floating fractions L (0 , 1) − 2lnΛ = − 2ln L 2 = − 2ln L (1 , 0) − 2lnΛ = − 2ln L 0 = − 2ln L (0 , 1) L (ˆ � 0 , ˆ � 2 gg ) L 2 L min (Gaussian behavior) ( χ 2 behavior) not investigated yet 7 jeudi 13 septembre 2012
Methods Mass-cos LR for Hypo or Hypothesis testing Signal No No Events used Events used correlation correlation Previous work extraction extraction needed ? binned ? content S+B small M. Fanti, 1 1D fit on mass no(*) no 1D(c)[x1D(m)] S+B Wisconsin mass window S+B full mass 2D fit on J. Albert, 2 yes no 2D(m,c) S+B F. Bernlochner window mass,cos (a bit different) 1D fit on mass 3 S no(*) yes 1D(c) S Wisconsin in cos bins S. Laplace, N. Rhone, 4 S sPlots yes yes 1D(c) S JB de Vivie (*): remaining correlations within a mass or cos θ * bin is a systematic uncertainty → Goal of this presentation: comparison of statistical power of all these methods using inclusive toys 8 jeudi 13 septembre 2012
Statistical interpretation Expected significance of spin0 hypothesis («N( σ )»): asymmetric: usual type II error β symmetric: at q( α = β ) without any a priori belief in spin0/spin2 with prejudice that spin0 is favored drawings: courtesy Jana Probability to exclude spin 2 hypothesis if spin 0 is true («P excl ») Find q for spin2 excl. @ 95%CL ( α 2 =0.05) «How often will we find a value of q that allows to exclude spin 2?»: � q ( α 2 =0 . 05) P (95% CL ) = f S 0 ( q ) dq −∞ 9 jeudi 13 septembre 2012
Method #1 (1D fit on mass) • studies performed by M. Fanti (HSG1 Aug. 9 and Sep. 6) and Wisconsin (Sep. 6) + this study • 1D fit of mass to get S and B, use LR(S+B)(c,m) • advantage: • events outside the signal mass window do not contribute to the spin0/spin2 separation: the LR is effective only in a small mass window, and thus background correlations can be mostly ignored (still need to assess residual effect) • background pdf either from MC (Wisconsin, this study) or data (Fanti: narrow sideband) N( σ ) spin2 L μ P excl L=10.7 fb -1 , μ = 1.85, asym N(toys) = 10k pythia6, 10.7 1.8 57 % this study 1.85 125 GeV 32.1 2.9 94 % pythia6, 5.9 17 % Fanti ∼ 1.53 - 300 GeV 30 39 % JHU, 5.9 0.52 Wisconsin 1.41 - 126.5 GeV 30 1.14 • hard to compare results given the different L/ μ /spin2 samples used and the different ways of quoting results: should agree on benchmark values/stat. interpretation... • still: our result looks way too good ! More almost Gaussian behavior (not exactly so cannot rely on analytical formulae to derive checks needed significances) 10 jeudi 13 septembre 2012
Method #2 (2D fit on mass-cos θ *) • studies performed by J. Albert, F. Bernlochner (HSG1 Apr. 12) + this study • advantage: • make use of full satistical power (cos θ * in fit: improved significance) • drawbacks: • need to correctly modelize mass-cos θ * correlations • hard to fit a 2D pdf in the data directly... here, without correlations L=10.7 fb -1 , μ = 1.85, N( σ ) μ L P excl N(toys) = 2k asym this study 10.7 1.85 1.9 61 % J. Albert, F. «up to up to 90% CL CL exclus exclusion Bernlochner with with 15-25 15-25 fb-1» fb-1» 5% «only» improvement over 1D fit in this ideal case without correlations: not worth the complication of a 2D fit ! 11 jeudi 13 septembre 2012
Method #3 (1D mass fit in bins of cos θ *) • studies performed by Wisconsin (HSG1 Jun. 21) + this study (Jana) • advantages: • obtain the cos θ * distribution of signal events without a priori knowledge • do not need the background cos θ * pdf (only the signal pdf enters the LR) • correlations are less an issue in each cos θ * bins (residual effect to be quantified) toys : • bkg pdfs are obtained from B-only fit of the data (different than slides 7, 8 where MC is used for bkg pdf) • obtain S per cos θ * bin → 10-bins histo • build binned likelihood ratio from histo S+B fit in the More details in Jana’s slides on the sharepoint: data in each https://espace.cern.ch/atlas-phys-higgs-htogamgam/Lists/ cos θ * bin 2012%20HCP/Attachments/6/spin_from_signalyield.pdf 12 jeudi 13 septembre 2012
Method #3 • Results: N( σ ) μ L P excl asym 10.7 1.3 41 % this study 1.85 21.4 1.7 58 % 32.1 2.1 66 % 13 jeudi 13 septembre 2012
Method #4 (sPlots) • Study put on sharepoint in june, based on a simple inclusive toy not using the workspace of the other studies in this talk - no precise significance computation yet • Advantages: • obtain the cos θ * distribution of signal events without a priori knowledge using ALL events (in principle, more powerfull than method #3) • do not need the background cos θ * pdf (only the signal pdf enters the LR) • Drawback: basic sPlots assume no correlations. Extensions exist to handle them, though... • Reminders on sPlots technique: • Likelihood fit on a control variable (m γγ ) • Get covariance matrix from likelihood: • Compute sweights: • Apply sweight to each event to obtain SPlot of test variable (cos θ *) to obtain, on average, the «true» signal distribution... 14 jeudi 13 septembre 2012
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