Hard process kinematics (MC level)  

$t\bar{t}$ final states

The final states of the $t\bar{t}$ produced in an event can be grouped by the different decay channels of the W. We group the final states according to the decays of the W into light leptons (e, $\mu$), $\tau$s and quarks (jets). For the $\tau$ special care must be taken to account for the $\tau$ leptonic decays which will yield final states indistinguishable from the ones resulting from W decays into light leptons. From the Particle Data Group we have the following branching ratios:

  • $B(W\rightarrow l\nu_{l})= 0.213 \pm 0.002$
  • $B(W\rightarrow \tau\nu_{\tau})= 0.113 \pm 0.002$
    • $B(\tau\rightarrow l\nu_{l}X)= 0.373 \pm 0.003 $
    • $B(\tau\rightarrow hadrons)=0.627 \pm 0.003 $
  • $B(W\rightarrow q\bar{q}')=0.674 \pm 0.003$
We can easily build the branching ratios for the final states of the $t\bar{t}$ system starting from the W and $\tau$ branching ratios:

$\noindent \{ B(l\nu_{l}) + B(\tau\nu_{\tau})\cdot [B(\tau\rightarrow l\nu_{l}X)+ B(\tau\rightarrow hadrons)]+B(q\bar{q}') \} ^{2}=1$

This yields the following final states:

Expected rates for $t\bar{t}$ final states in the SM
Final state Branching ratio Value Approx.
di-leptonic (DIL)   0.065 $\pm$ 0.002 5/81
$\tau_{had}$ + lepton   0.036 $\pm$ 0.001 3/81
lepton + jets (L+J)   0.344 $\pm$ 0.004 28/81
fully hadronic (HAD)   0.555 $\pm$ 0.005 45/81

Kinematics

For the introductory study of the event kinematics we make use of the All b samples generated with Madgraph + PYTHIA6. We try to extract the signal of the top decay with two leptonic + 2 jets final states. Below you can find several plots related to the kinematics of the top decay products as given by the generator.

10 TeV

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$t\bar{t}$ kinematics at 10 TeV (L=25/pb, $\sigma=317pb$, generator level)

Background modeling (data-driven)

We discuss (at Monte-Carlo level) possible strategies to model background in a data-driven way. Altough background for the dilepton channel is expected to be small ($\approx 10% $) it can increase my mis-reconstruction or mis-selection of the jets coming from each top decay. We call this jet misassignment. The simplest way to test if a jet is coming from a top decay is to correlate it kinematically with a lepton from the same top decay. The plot below shows the distribution for the invariant mass of the jet+lepton pairs in two cases:

  • signal - both jet and lepton come from the same top decay
  • background - the jet comes from radiation (ISR/FSR) or matrix element generated parton

| | |

From the signal distribution one can build the following $\chi^{2}$ to test each jet+lepton:

$\begin{equation*} \chi^{2} = \frac{(\mu-s^{1/2}(j,l))^{2}}{\sigma^{2}}\end{equation*}$

In the equation above $\mu$ is the median and $\sigma$ is the standard deviation of the signal distribution for the invariant mass of the jet+lepton pairs. We test this $\chi^{2}$ using 3 types of pseudo-events (or solution-like events) generated from the MC event (instead of the partons we make use of the matched kT6GenJet):

  • signal events: two jet+lepton pairs correctly assigned
  • 1 misassignment events: one jet+lepton pair correctly assigned and one jet+lepton pair with a jet not matched to the parton from top decay
  • background or 2 misassignment events: two jet+lepton pairs where the jets are not matched to the partons from the top decay
We expect that signal pairs to have lower $\chi^{2}$ with respect to background pairs. The distributions of the $\chi^{2}$ and $\sum\chi^{2}$ are shown below and are in accordance with expectation.

$\chi^[2}$ distributions (generator level)
single $\chi^{2}$ (2 entries/event) sum of single $\chi^{2}$ (1 entry/event)

Background can be modelled as follows:

  • flip method: invert the 3 momentum of the leptons and recompute the $\chi^{2}$
  • swap method: retrieve jets from a different event and recompute the $\chi^{2}$ for the same leptons
Both these procedures should model independently combinatorial background. The plots below show the ratio between the $\chi^{2}$ distribution for background and for these models.

Background models $\chi^{2}$ (generator level)
single $\chi^{2}$ (2 entries/event) sum of single $\chi^{2}$ (1 entry/event)

We see that the flip model does not model so well the background shape for high $\chi^{2}$ values. However we shall use it for comparison and as an estimate for systematic uncertainty on the background model. The background level estimate is done by plotting the $\chi^{2}$ for the data and the $\chi^{2}$ for the background model and finding the ratio between the areas of each one of the distributions, as function of the $\chi^{2}$ cut used to compute the area. We expect that for large $\chi^{2}$ the distribution is background dominated and as so the ratio $A/A_{model}$ yields the background level of the selected events. The procedure is shown in the following table.

Background levels from $\chi^[2}$ assignments with different background models
Flip model Swap model
single $\chi^{2}$ (2 entries/event) sum of single $\chi^{2}$ (1 entry/event) single $\chi^{2}$ (2 entries/event) sum of single $\chi^{2}$ (1 entry/event)

14 TeV

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Top decay kinematics
$p_{T}$ spectrum
top top childs W-boson childs
Angles of emission
wrt to b between lepton and neutrino wrt to W boost, measured in the W CM frame
charged leptons with respect to b
Invariant masses
b and lepton lepton and neutrino  
 

From the $p_{T}$ distributions of the W decay products we conclude that the particles with $I_{Z}=-1/2$ have a slightly harder spectrum than the ones with $I_{Z}=+1/2$ because they are preferentially emitted along the hemisphere where the W is propagating to in the laboratory. This is seen in the distribution of the angle of emission of the lepton and neutrino with respect to the W propagation direction, measured in the CM frame. The angular distribution of the charged leptons is theoretically given by (see R.K. Ellis et. al., QCD and Collider Physics ):

$\frac{1}{N}\frac{dN \left( W \to e\nu \right) }{dcos\theta^{\star}_{e}}= \frac{3}{4\cdot \left(m_{t}^{2}+2M_{W}^{2} \right) } \cdot \left[m_{t}^{2}\sin\theta^{\star}_{e}+M_{W}^{2} \left( 1-\cos\theta^{\star}_{e} \right)^{2} \right] $

Preliminary studies (2007)

  • Preliminary studies (sub-page kept for reference from the prelimiary studies from Nov. - Dec. 2007)

-- PedroSilva - 14 May 2008

Topic attachments
I Attachment History Action Size Date Who Comment
PNGpng BckgLevelsChi2SumwFlipModel.png r1 manage 14.8 K 2008-11-18 - 15:56 PedroSilva  
PNGpng BckgLevelsChi2SumwSwapEvModel.png r1 manage 15.4 K 2008-11-18 - 15:57 PedroSilva  
PNGpng BckgLevelsChi2wFlipModel.png r1 manage 14.5 K 2008-11-18 - 15:57 PedroSilva  
PNGpng BckgLevelsChi2wSwapEvModel.png r1 manage 15.2 K 2008-11-18 - 15:57 PedroSilva  
PNGpng Chi2BckgModels_ratio.png r1 manage 12.9 K 2008-11-18 - 15:58 PedroSilva  
PNGpng Chi2Sum.png r1 manage 17.4 K 2008-11-18 - 15:58 PedroSilva  
PNGpng Chi2Sum_log.png r1 manage 15.4 K 2008-11-18 - 15:58 PedroSilva  
PNGpng JetLeptonKinematics.png r1 manage 15.4 K 2008-11-18 - 16:31 PedroSilva  
PNGpng JetLeptonKinematics_SoverSpB.png r1 manage 14.6 K 2008-11-18 - 16:31 PedroSilva  
PNGpng SingleChi2.png r1 manage 14.2 K 2008-11-18 - 15:59 PedroSilva  
PNGpng SingleChi2_log.png r1 manage 12.6 K 2008-11-18 - 15:59 PedroSilva  
PNGpng SumChi2BckgModels_ratio.png r1 manage 13.5 K 2008-11-18 - 16:08 PedroSilva  
PNGpng dR_bl.png r1 manage 8.4 K 2008-05-29 - 16:00 PedroSilva  
PNGpng deta_bl.png r1 manage 14.2 K 2008-05-29 - 15:59 PedroSilva  
PNGpng dphi_bl.png r1 manage 12.4 K 2008-05-29 - 19:26 PedroSilva  
PNGpng s_bl.png r2 r1 manage 10.2 K 2008-05-29 - 16:00 PedroSilva  
PNGpng s_lnu.png r1 manage 17.4 K 2008-05-14 - 15:56 PedroSilva  
PNGpng theta_lnu.png r1 manage 11.1 K 2008-05-14 - 15:56 PedroSilva  
PNGpng theta_tob.png r1 manage 14.3 K 2008-05-14 - 15:56 PedroSilva  
PNGpng theta_tow_cm.png r1 manage 18.8 K 2008-05-14 - 15:57 PedroSilva  
PNGpng top_childs_pT.png r1 manage 17.0 K 2008-05-14 - 15:57 PedroSilva  
PNGpng top_eta.png r1 manage 19.8 K 2008-11-10 - 11:44 PedroSilva  
PNGpng top_eta_comp.png r1 manage 10.5 K 2008-11-10 - 11:46 PedroSilva  
PNGpng top_pT.png r1 manage 13.5 K 2008-05-14 - 15:57 PedroSilva  
PNGpng top_pt.png r1 manage 16.3 K 2008-11-10 - 11:46 PedroSilva  
PNGpng top_pt_comp.png r1 manage 10.0 K 2008-11-10 - 11:45 PedroSilva  
PNGpng ttbar_deta.png r1 manage 18.5 K 2008-11-10 - 11:47 PedroSilva  
PNGpng ttbar_deta_comp.png r1 manage 11.0 K 2008-11-10 - 11:48 PedroSilva  
PNGpng ttbar_dphi.png r1 manage 16.8 K 2008-11-10 - 11:49 PedroSilva  
PNGpng ttbar_dphi_comp.png r1 manage 8.9 K 2008-11-10 - 11:52 PedroSilva  
PNGpng ttbar_minv.png r1 manage 17.0 K 2008-11-10 - 11:50 PedroSilva  
PNGpng ttbar_minv_comp.png r1 manage 10.4 K 2008-11-10 - 11:52 PedroSilva  
PNGpng w_childs_pT.png r1 manage 16.4 K 2008-05-14 - 15:57 PedroSilva  
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