Covariances for Jets and E T miss





−_π^q2 sinθ _π^q cosθ 0 0 −_sin¹_θ 0

0 0 1





 (5.11)

the parameters needed for the kinematic fitter can be computed for each input track. Note that this approach is valid for electrons and muons only. For other reconstructed objects like jets and missing transverse energy no covariance matrices are provided and must be determined on a statistical basis (cf. Sec.5.2.3).

5.2.3. Covariances for Jets and E

For jets and the missing transverse energy the covariance matrices do not exist for each individual object as in the case of the leptons. Instead, they have to be obtained on a statis-tical basis by comparing the reconstructed objects with their true counterparts in simulated events. One should note that this comparison is the only dependence of the kinematic from simulation; apart from that the KinFitter is completely data-driven. Because the fitter is to be used for the reconstruction of single top or t¯t events here, the covariances are determined

7One should keep in mind that the helix parameters are given with respect to a certain reference point. For vertex-fitted tracks this is either the primary or secondary vertex, for tracks without a vertex constraint this is usually the point of closest approach to the beam-line.

from the cross section weighted sum of Monte Carlo samples for top-pair and single top production.

In order to let the fit operate within the exact phase space model adopted by the analysis, the extraction of the covariances is performed using reconstructed events and physics objects that fulfill the preselection requirements and the corrections described in Chapter4. Events containing at least one good reconstructed jet are used, without applying a b-tag weight selection (PreTag).

To determine the covariance matrices for the reconstructed missing transverse energy, its magnitude and azimuth are compared to that of the neutrino originating from the W boson of the semi-leptonic top quark decay. For the jets such an unique assignment to their true counterparts does not exist. Here, the jets reconstructed at the hadron level are taken for comparison. The matching between the jets at the detector level (dl) and the hadron level (hl) is done by using the parameter

d² = (η_dl−η_hl)² + (ϕ_dl−ϕ_hl)²+ (p_{t, dl}−p_{t, hl})²/p²_{t, hl}. (5.12) The parameter d is computed for each combination of a detector level jet with a hadron level jet and the combination with the smallest value ofd is taken as match. This procedure turned out to be robust and reliable.

Since the detector resolutions are strongly dependent on the transverse momentum and the pseudo-rapidity of the reconstructed objects, the covariance matrices are determined in bins ofptandηof the object in question. The binning is chosen such that sufficient statistics is collected in every bin for a precise determination of each matrix element. The procedure to obtain the average for a matrix element in a certain bin differs for diagonal elements and off-diagonal elements.

For the diagonal elements the residual distributions X_rec−X_true of a variable X – which might bep_t,η,ϕ for the jets, orE_T^miss,ϕfor the missing transverse energy – are fitted with a Gaussian. The square of the resulting width of the residuals gives the average of the wanted variance. An example for this is shown in Fig.5.3a. To determine the off-diagonal elements, the products of the residuals (X_rec−X_true)·(Y_rec−Y_true) of the pair of variables X and Y in question are histogrammed (see Fig. 5.3b). The mean value of the final histogram is a good estimator for the wanted off-diagonal element. The resulting covariance matrix elements in bins ofp_t an ηare collected in Fig. 5.4 for the jets and in Fig.5.5 for the missing transverse energy [PP12].

/ ndf

χ2 165.4 / 77

Constant 0.03602 ± 0.00039 Mean -0.0008207 ± 0.0007222 Sigma 0.08055 ± 0.00061

(GeV)

t, true t, rec - p

p

-0.4 -0.2 0 0.2 0.4

Number of Entries

0 0.01 0.02 0.03 0.04

/ ndf

χ2 165.4 / 77

Constant 0.03602 ± 0.00039 Mean -0.0008207 ± 0.0007222 Sigma 0.08055 ± 0.00061

(a)Determination of jet (∆p_t)²

Mean -0.0004797

) (GeV rad) φtrue

rec- φ

t, true)(

t, rec- p (p

-1 -0.5 0 0.5 1

Number of Entries

0 2 4 6 8 10

12 Mean -0.0004797

(b)Determination of jet∆p_t∆φ

Figure 5.3.: Examples for the determination of the covariance matrix elements. In (a)the deter-mination of a diagonal element is shown. A Gaussian is fitted to the residual distribution of the element in question (here p_tfor jets) and the resulting width squared is taken. For the off-diagonal elements(b)the products of the residuals of both variables (herept andφ) are histogrammed. The mean value of the histogram gives the wanted off-diagonal element.

(GeV)

Figure 5.4.: The histograms show the six covariance matrix elements for the three-momentum vectors of all jets in (p_t, η, ϕ) representation obtained by the statistical method in bins of p_t and η of the jets as described in the text.

(GeV)

Figure 5.5.: The three covariance matrix elements for the missing transverse energy obtained by the statistical method described in the text are histogrammed in bins of p_t and η.

Events

In this analysis the selection of semi-leptonic Wt events containing a high-p_t lepton and jets in the final state is performed with the aid of the kinematic fit technique described in Chap. 5. The top quark decays into a W boson and a quark, therefore the presence of the associated W boson leads to an ambiguity in the event reconstruction since it is not known which of the W bosons decays leptonically into the high-p_t lepton and the neutrino, while the other W decays hadronically into two jets, as depicted in Fig. 6.6 and 6.7. To solve this ambiguity the kinematic fit is run twice for every event: once for testing the hypothesis of the associated W boson decaying hadronically (Sec. 6.3), and a second time for the hypothesis of a leptonic decay of the associated W (Sec. 6.4). The fit with the best χ² value is chosen.

In order to enrich signal-like events a minimum cut on the χ²-probability of the selected fit is imposed. For events containing at least four jets, an additional kinematic fit is performed to test the hypothesis of top-pair production (see Sec. 6.6.1). If the χ²-probability of this fit exceeds a certain threshold, the event will be marked as background and rejected. The final selection criteria are presented in Sec. 6.7, and an immediate visualisation of the analysis workflow is presented in Fig. 6.1.