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\section{Combining cross section measurements for all channels}
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\label{sec:xsec}
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In this section we calculate the cross section for each individual channel and
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combine them together taking into account correlations.
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$\WZ$ cross section is calculated as follows
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\begin{equation}
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\label{eq:xsec}
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\sigma = \frac{N_{obs}-N_{MC}-N_{Z}}{\epsilon \mathcal{L}},
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\end{equation}
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where $N_{obs}$ is the number of observed events, $N_{MC}$
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is the number of background events estimated from Monte Carlo simulation,
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$i.e.$ events from $\Z\gamma$, $\ZZ$, and processes without genuine $\Z$ boson
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in final state. $N_Z$ is the number of instrumental background events from
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processes with genuine $\Z$ boson; these numbers are given in
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Tables~\ref{tab:yields}~and~\ref{tab:yieldsEstimate} for each channel.
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The uncertainty on measured cross section is calculated as given in equation below
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%Eq.~\ref{eq:xsecErr}
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\begin{equation}
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\label{eq:xsecErr}
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\left(\Delta\sigma\right)^2 = \left(\frac{\Delta N_{obs}}{\epsilon\mathcal{L}}\right)^2
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+ \left(\frac{\Delta N_{MC}}{\epsilon \mathcal{L}}\right)^2
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+ \left(\frac{\Delta N_{Z}}{\epsilon \mathcal{L}}\right)^2
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+ \left(\frac{\Delta (\epsilon \mathcal{L})}{\epsilon \mathcal{L}} \frac{N_{WE}}{\epsilon \mathcal{L}}\right)^2 ,
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\end{equation}
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where $\Delta(\epsilon \mathcal{L})/(\epsilon \mathcal{L})$ is
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modeling uncertainty for each channel, given in Table~\ref{tab:FullSys}.
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We normalize cross section to the theoretical cross section, calculated from
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Monte Carlo simulation. The normalized $\WZ$ cross section, $\sigma_n$, is given as
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%\begin{equation}
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\begin{eqnarray}
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\label{eq:normSigma}
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\sigma_n &=&\frac{N_{obs}-N_{MC}-N_{\Z}}{\epsilon\mathcal{L}} \frac{1}{\sigma_{theory}} \nonumber \\
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&=& \frac{N_{obs}-N_{MC}-N_{\Z}}{\epsilon\mathcal{L}} \frac{\epsilon \mathcal{L}}{N_{\WZ}} \nonumber \\
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&=& \frac{N_{obs}-N_{MC}-N_{\Z}}{N_{\WZ}}
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\end{eqnarray}
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%\end{equation}
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%\sigma_{n} = \frac{N_{obs}-N_{MC}-N_{\Z}}{N_{\WZ}}
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where $N_{\WZ}$ is the expected number of signal events, corresponding to a sample of 300 $pb^{-1}$
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integrated luminosity, which is given in the first line of Table~\ref{tab:yields}.
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%Thus, the normalization factors are 7.9, 8.0, 8.9, and 10.1 for $3e$, $2e1\mu$,
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%$2\mu1e$, and $3\mu$, respectively.
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Normalized cross section for each channel is given in
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Table~\ref{tab:fourXsections}.
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\begin{table}[h]
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\begin{center}
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\begin{tabular}{|l|c|} \hline
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Channels & normalized cross section \\ \hline
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$3e$ & 1.00$\pm$0.58 \\
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$2e 1\mu$ & 0.95$\pm$0.45 \\
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$2\mu 1e$ & 0.82$\pm$0.52 \\
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$3\mu$ & 0.98$\pm$0.39 \\ \hline
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\end{tabular}
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\caption{Normalized cross section measured for each channel and corresponding uncertainties.}
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\label{tab:fourXsections}
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\end{center}
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\end{table}
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In order to combine the results for four different channels we use BLUE
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($i.e.$ Best Linear Unbiased Estimate) method~\cite{BLUE}.
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According to this method, final estimation of the cross section is a linear combination
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of cross-sections, measured for each channel separately
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\begin{equation}
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\label{eq:sumXsec}
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\sigma = \sum_{i} \alpha_i\sigma_i
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\end{equation}
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where $\alpha_{i}$ are weighting factors for each measurement, which needs to be determined.
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In the experiment we measure cross section for each channel and calculate uncertainty matrix,
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$\mathbb{E}$. As we consider four channels, $\mathbb{E}$ will be $4 \times $4 symmetric matrix.
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\begin{equation}
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\label{eq:uncertMatrix}
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\mathbb{E} =\left( \begin{array}{cccc}
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\sigma_1^2 & \sigma_{12} & \sigma_{13} & \sigma_{14} \\
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\sigma_{21} & \sigma_2^2 & \sigma_{23} & \sigma_{24} \\
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\sigma_{31} & \sigma_{32} & \sigma_3^2 & \sigma_{34} \\
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\sigma_{41} & \sigma_{42} & \sigma_{43} & \sigma_4^2 \end{array} \right),
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\end{equation}
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%where the diagonal elements correspond to variances of cross-section measurements for channels
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where diagonal elements are the full uncertainties for individual channel squared and
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the off-diagonal elements represent uncertainties of the respective channels from sources of
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correlation, $e.g.$, $\sigma_{12}$ describes uncertainty due to the common sources between the channels $3e$
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and $2e1\mu$ and equals to $\sigma_{1}^{corr} \sigma_{2}^{corr}$, where $\sigma_{1}^{corr}$
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and $\sigma_{2}^{corr}$ are uncertainties on first ($3e$) and second ($2e1\mu$) channels, respectively.
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%==================================================================
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As an example, calculation of $\sigma_{12}=\sigma_1^{corr} \sigma_2^{corr}$ is described below.
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At first we estimate the uncertainty due to modeling sources, common between all channels, which are
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listed in Table~\ref{tab:sys}. These are luminosity, trigger, lepton reconstruction, lepton energy scale,
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PDF, and W transverse mass requirement. Adding these uncertainties together results in 16.2\%.
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Then we list the sources of uncertainty which are common between $3e$ and $2e1\mu$ channels:
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\[ \begin{array}{lc}
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\mbox{Electron charge determination} & 2\%+2\% = 4\% \\
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\mbox{Electron identification} & 4\%+4\% = 8\% \\
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\mbox{Fully correlated modeling} & 16.2\% \\
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\mbox{Background estimated from MC} & \\
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\end{array}\]
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and add these uncertainties as from uncorrelated sources.
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Uncertainty due to electron charge determination, electron identification, and
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fully correlated modeling is
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\begin{equation}
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\label{eq:sigma0corr}
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1.00 \sqrt{4\%^2+8\%^2+16.2\%^2} = 0.185
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\end{equation}
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for $3e$ channel and
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\begin{equation}
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\label{eq:sigma1corr}
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0.95 \sqrt{4\%^2+8\%^2+16.2\%^2} = 0.176
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\end{equation}
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for $2e1\mu$ channel. Hence, the uncertainties for $3e$ and $2e1\mu$ channels due to all
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sources of correlations are
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\begin{subequations}
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\label{eq:covTerms}
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\begin{align}
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\sigma_0^{corr}& = \sqrt{0.185^2 + (\frac{1.3}{7.9})^2} = 0.248 \label{first}\\
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\sigma_1^{corr}& = \sqrt{0.176^2 + (\frac{0.9}{8.0})^2} = 0.209 \label{second}
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\end{align}
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\end{subequations}
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Thus,
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\begin{equation}\label{eq:covTermFin}
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\sigma_{01} = 0.248 * 0.209 = 0.052
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\end{equation}
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Calculating other terms using previously described procedure result in
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\[ \mathbb{E} = \left( \begin{array}{cccc}
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0.335 & 0.052 & 0.095 & 0.041 \\
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0.052 & 0.202 & 0.037 & 0.043 \\
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0.095 & 0.037 & 0.267 & 0.036 \\
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0.041 & 0.043 & 0.036 & 0.155 \end{array} \right) .\]
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In order to determine $\alpha_i$ weighting factors following
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procedure described in paper~\cite{BLUE}
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\begin{equation}
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\label{eq:weights}
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\alpha = \mathbb{E}^{-1}U/(\tilde{U}\mathbb{E}^{-1}U) ,
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\end{equation}
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where $U$ is a four-dimensional vector whose all components are unity.
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Obtained weights are 0.112, 0.281, 0.194, and 0.413 for $3e$, $2e1\mu$, $2\mu1e$, and $3\mu$ channels, respectively.
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Total uncertainty on combined measurement squared is calculated as
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\begin{equation}
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\label{eq:sigmaSquared}
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\sigma^2 = \tilde{\alpha}\mathbb{E}\alpha
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\end{equation}
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Thus, normalized combined cross section is $0.96\pm0.3$.
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Cross sections measured for each channel separately and result of BLUE method is
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shown on Fig.~\ref{fig:xsections}
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\begin{figure}[bt]
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\begin{center}
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\scalebox{0.8}{\includegraphics{figs/combined_xsections.eps}}
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\caption{From bottom to top, the normalized cross section measured for each channel and corresponding uncertainties;
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cross section after combining four measurements using BLUE method. }
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\label{fig:xsections}
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\end{center}
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\end{figure}
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\end{document}
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