Bayesian combination of different channels in ttbar single lepton and dilepton analysis

Method

Markov Chain Monte Carlo (MCMC) method is used as implemented in the package Bayesian Analysis Toolkit (BAT-0.3.2) [ Comp. Phys. Comm 180 (2009) 2197-2209 ] to solve the Bayes formula:

the likelihood implicitly depends on systematic uncertainties which affects the parameters. Making explicit such dependence:

where s_i are the different sources of systematics.

• The total likelihood is the product of the likelihoods of the individual channels (e+jets/μ+jets/ee/μμ/eμ);
• The correlations are taken into account during the marginalization to evaluate the posterior of the cross section.

Application

The following set of priors have been used:

• signal cross section: flat prior
• systematic uncertainties
  • data driven systematics: Gamma function priors
  • other systematics: Gaussian priors (truncated if needed in order to avoid negative values)

Input from the lepton+jets analysis

The following values (in standard format) have been used as inputs for the combination program:

Code            Shape           Intr.Unc Description
------          ---------       -------- ------------
LUMI            Gaussian        0.11     Luminosity uncertainty
BTAGEFF         Gaussian        0.10     B-tagging efficiency uncertainty
LTAGEFF         Gaussian        0.50     L-tagging efficiency uncertainty
JES             Gaussian        0.10     Jet Energy Scale uncertainty
JER             Gaussian        0.03     Jet Energy Resolution uncertainty
JRECO           Gaussian        0.10     Jet reconstruction efficiency uncertainty
EID             Gaussian        0.03     Uncertainty on electron ID scale-factors
EES             Gaussian        0.03     Electron Energy Scale uncertainty
EER             Gaussian        0.03     Electron Energy Resolution uncertainty
MID             Gaussian        0.03     Uncertainty on muon ID scale-factors
MES             Gaussian        0.03     Muon Energy Scale uncertainty
MER             Gaussian        0.03     Muon Energy Resolution uncertainty
ISRFSR          Gaussian        0.10     Uncertainty coming from Initial and Final state rediation variation
NLOGEN          Gaussian        0.03     Uncertainty related to the NLO generator (MCatNLO-Powheg)
PSGEN           Gaussian        0.01     Uncertainty related to the Parton Shower (Herwig-Pythia)
PDF             Gaussian        0.03     Parton Distribution Functions uncertainty
OTX             Gaussian        0.01     Uncertainty on Etmiss due to hardware failures 
ZXSEC           Gaussian        0.60     Z+jets cross section uncertainty
STXS            Gaussian        0.10     Single top cross section uncertainty
QCDSTATE        ScaledPoisson   0.54     Stat uncertainty in QCD background for e+jets
QCDSTATM        ScaledPoisson   0.35     Syst uncertainty in QCD background for mu+jets
QCDSYSTE        Gaussian        0.40     Stat uncertainty in QCD background for e+jets
QCDSYSTM        Gaussian        0.40     Syst uncertainty in QCD background for mu+jets
WBGSTATE        ScaledPoisson   0.20     Stat uncertainty in Berend Giele Scaling for e+jets
WBGSTATM        ScaledPoisson   0.22     Stat uncertainty in Berend Giele Scaling for mu+jets
WBGSYST         Gaussian        0.24     Syat uncertainty in Berends Giele Scaling
W2TAGSTAT       ScaledPoisson   0.29     Stat uncertainty in W+2jets tagged
WHFRAC          ScaledPoisson   0.29     Combined unc. from extrapolation ratios, tag fractions (including flavor tag), assumed W+2jet HF composition


Channel: ejets e+jets
Component       Yield   Systematics (name=value)
DATA            17
ttbar           14.53      EID=0.03 BTAGEFF=0.11 LTAGEFF=0.005 JES=0.10 JERUP=0 JERDOWN=0.01 JRECOUP=0 JRECODOWN=0.08 EES=0.017 EER=0 ISRFSRUP=0.06 ISRFSRDOWN=0.11 NLOGEN=0.044 PSGEN=0.011 PDF=0.025 OTX=0.01
QCD             4.78       QCDSYSTE=0.37  QCDSTATE=0.54
W_jets          1.9        WBGSTATE=0.20 WBGSYST=0.25 QCDSYSTE=-0.34 W2TAGSTAT=0.29 WHFRAC=0.29
Z_jets          0.15       ZXSEC=0.60        LUMI=0.11     JES=0.25   LTAGEFF=0.27
SingleTop       0.67       STXS=0.10         LUMI=0.11     JES=0.20   BTAGEFF=0.12

Channel: mjets mu+jets
Component       Yield   Systematics (name=value)
DATA            20
ttbar           14.66      MID=0.023 BTAGEFF=0.11 LTAGEFF=0.005 JES=0.095 JERUP=0 JERDOWN=0.01 JRECOUP=0 JRECODOWN=0.08 MES=0.003 MER=0.005 ISRFSR=0.08  NLOGEN=0.056 PSGEN=0.026 PDF=0.024 OTX=0.01
QCD             0.765      QCDSYSTM=0.56 QCDSTATM=0.35
W_jets          3.2        WBGSTATM=0.22 WBGSYST=0.25 QCDSYSTM=-0.12 W2TAGSTAT=0.29 WHFRAC=0.29
Z_jets          0.08       ZXSEC=0.60        LUMI=0.11     JES=0.25   LTAGEFF=0.28
SingleTop       0.68       STXS=0.10         LUMI=0.11     JES=0.15   BTAGEFF=0.12

Results for the lepton+jets analysis

The following results have been obtained using the values from the above table which refer to an integrated luminosity of 2.9 pb^-1

The following plots show posteriors for some of the most relevant systematics:

Significance evaluation

The significance of the signal has been evaluated looking at the log difference between the maximum and the σ=0 level of the cross section posterior. In case of a normal distribution this difference can be translated in a Z-significance (number of standard deviation distance from mean value) by the following relationship:

The following plot shows such log difference for the above described inputs.

The intercept with the y-axis is ~ 7.3 which, using the above relation, can be converted in a Z-significance of ~ 3.8.

Input from the dilepton analysis

The following values (in standard format) have been used as inputs for the combination program:

Systematics:
JES   Jet energy scale
ELES  Electron energy scale
ELE/MUEFF Electron/muon ID efficiency
EL/MUTRIGEFF Electron/muon trigger efficiency
MURES Muon energy resolution
PDF     PDF uncertainty
ISRFSR   ISR/FSR uncertainty
GENERATOR    Generator uncertainty
PARTSHOW    Parton shower model uncertainty
LUMI Luminosity
*XS   Cross-section
*STAT Statistical error, data driven measurements
*METHOD     Error due to method for data driven measurements


Channel: ee dilepton-e/e
Component    Yield    Systematics (name=value)
ttbar           1.19   JESUP=0.048 JESDOWN=0.058 EESUP=0.02 EESDOWN=0.035 EER=0.008 EID=0.063 ELTRIGEFF=0.008 PDF=0.027 ISRFSRUP=0.027 ISRFSRDOWN=0.027 NLOGENUP=0.0 NLOGENDOWN=0.044 PSGENUP=0.0 PSGENDOWN=0.053 TTSTAT=0.016 PILEUP=0.016 PILEDOWN=0.
DrellYan        0.25    JESUP=0.10 JESDOWN=0.10 DYSTAT=0.38 DYEEMETHOD=0.05 EES=0.58 EER=0.18 ZXSEC=0.01
Ztautau         0.07    ZTTXS=0.465 LUMI=0.11 EID=0.063 ELTRIGEFF=0.008 EESUP=0.105 EESDOWN=0.101 JESUP=0.203 JESDOWN=0.105
Diboson         0.04    JESUP=0.138 JESDOWN=0.105 DBXS=0.312  LUMI=0.11 EID=0.063 ELTRIGEFF=0.008 EESUP=0.06 EESDOWN=0.06 EER=0.029 
SingleTop       0.08    JESUP=0.039 JESDOWN=0.154 EESUP=0.039 EESDOWN=0.039 EERUP=0.115 EERDOWN=0.115 STXS=0.224 LUMI=0.11 EID=0.063 ELTRIGEFF=0.008

Channel: mm dilepton-mu/mu
Component    Yield    Systematics (name=value)
ttbar           1.87   JESUP=0.05 JESDOWN=0.044 MER=0.005 MID=0.024 MES=0.005 MUTRIGEFF=0.008  PDF=0.023 ISRFSRUP=0.016 ISRFSRDOWN=0.016 NLOGENUP=0.07 NLOGENDOWN=0.0 PSGENUP=0.0 PSGENDOWN=0.024 PILEUP=0.0 PILEDOWN=0.0157  TTSTAT=0.014
DrellYan        0.69    JESUP=0.12 JESDOWN=0.12 DYSTAT=0.32 DYMMMETHOD=0.17 MER=0.40 MES=0.03 ZXSEC=0.01 
Ztautau         0.14   ZTTXS=0.437 LUMI=0.11 MID=0.024 MER=.053 MES=0.053 MUTRIGEFF=0.07 JESUP=0.367 JESDOWN=0.053
Diboson         0.07   JESUP=0.190 JESDOWN=0.113 DBXS=0.309 LUMI=0.11 MID=0.024 MUTRIGEFF=0.007 MER=0.01 MES=0.007
SingleTop       0.07   JESUP=0.217  JESDOWN=0.0 STXS=0.232 LUMI=0.11 MID=0.024 MUTRIGEFF=0.008 MER=0.088 MES=0.06

Channel: em dilepton-e/mu
Component    Yield    Systematics (name=value)
ttbar           3.85   JESUP=0.008 JESDOWN=0.016 EID=0.032 ELTRIGEFF=.007 EESUP=0.011 EESDOWN=0.008 MER=0.01 MES=0.002 MID=0.012 MUTRIGEFF=0.0 PDF=0.025 ISRFSRUP=0.019 ISRFSRDOWN=0.019 NLOGENUP=0.009 NLOGENDOWN=0.0 PSGENUP=0.009 PSGENDOWN=0.  PILEUP=0.001 PILEDOWN=0. TTSTAT=0.009
Ztautau         0.13    JESUP=0.054 JESDOWN=0. ZTTXS=0.457 LUMI=0.11 EID=0.032 ELTRIGEFF=0.001 EESUP=0. EESDOWN=0.052 MUTRIGEFF=0.  MID=0.012 MER=0.05 
Diboson         0.15    JESUP=0.067 JESDOWN=0.067 DBXS=0.316   LUMI=0.11 EID=0.032 ELTRIGEFF=0.008 EESUP=0.01 EESDOWN=0.002 EER=0.001 MID=0.012 MUTRIGEFF=0.001 MES=0.005 MER=0.006   
SingleTop       0.22    JESUP=0.01  JESDOWN=0.069  STXS=0.155    LUMI=0.11    EID=0.032 ELTRIGEFF=0.007 EESUP=0.014 EESDOWN=0.014  MID=0.012 MUTRIGEFF=0.001 MER=0.01

Results for the combined lepton+jets and dilepton analysis

The following results have been obtained using the values from the above lepton+jets and dilepton tables which refer to an integrated luminosity of 2.9 pb^-1

Significance evaluation

The significance of the signal has been evaluated as described in the previous section for the lepton+jets combination.

The following plot shows the log difference between the maximum of the cross section posterior pdf and current cross section value.

The intercept with the y-axis is ~ 13.5 which can be converted in a Z-significance of ~ 5.2

Materials

ComBAT The combination tool

http://indico.cern.ch/materialDisplay.py?contribId=6&materialId=slides&confId=80578

-- RiccardoDiSipio - 08-Oct-2010

-- LorenzoBellagamba - 08-Oct-2010