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\section{Combine cross section measurements for all channels}
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\label{sec:xsec}
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We calculate $WZ$ cross section using equation given below
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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 number of observed events, $N_{MC}$
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is the number of background events from both non genuine $\Z$
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and genuine $\Z$ physics backgrounds and $N_Z$ is number of instrumental background
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events from 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 error on measured cross section is calculated as given in qeuation 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_{WZ}}{\epsilon \mathcal{L}}\right)^2 ,
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\end{equation}
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where $\frac{\Delta(\epsilon \mathcal{L})}{\epsilon \mathcal{L}}$ represents the
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modeling error for each channel, given in Table~\ref{tab:FullSys}.
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We normalize measured cross section by theoretical cross section. The latter one is obtained by
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using the estimated Monte Carlo number of $WZ$ events corresponding to a sample of 300 $pb^{-1}$
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integrated luminosity, 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 $3e$, $2e1\mu$, $2\mu1e$, and $3\mu$, respectively.
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% Units? Or better, we need to update the note with efficiency*Acceptance numbers.
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%<---- Is unit of 7.9(etc) the same as Luminosity? As I think \epsilon ( or A*\epsilon is the measured in \% )???
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% Remove this example... it does not serve much purpose.
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%As an example, the calculation for 3$e$ channel is provided below.
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%Normalized cross-section and corresponding uncertainty is calculated in Eqs.~\ref{eq:sOverStheory}
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%and ~\ref{eq:deltaSOverStheory}
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%\begin{equation}
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%\label{eq:sOverStheory}
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%\frac{\sigma}{\sigma_theory} = \frac{13.6 - 2.5 - 3.2}{7.9} = 1.0
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%\end{equation}
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%\begin{equation}
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%\label{eq:deltaSOverStheory}
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%\left(\frac{\Delta\sigma}{\sogma_theory}\right)^2 = \frac{13.6 + 1.3^2 + 1.7^2}{7.9^2} + (0.21^2*\frac{7.9}{7.9})^2 = 0.335
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%\frac{\Delta\sigma}{\sogma_theory} = 0.58
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%\end{equation}
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% Remove up to here
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We obtain the cross sections normalized by the theoretical ones as given
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in Table below %~\ref{tab:fourXsections}
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$$
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\begin{tabular}{|l|c|c|} \hline
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Channels & normalized cross section & normalization factor \\ \hline
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$3e$ & 1.00$\pm$0.58 & 7.9 \\
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$2e 1\mu$ & 0.95$\pm$0.45 & 8.0 \\
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$2\mu 1e$ & 0.82$\pm$0.52 & 8.9 \\
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$3\mu$ & 0.98$\pm$0.39 & 10.1 \\ \hline
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\end{tabular}
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\label{tab:fourXsections}
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$$
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1.1 |
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In order to combine the results for four different channles 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, individual estimates, measured for each channel.
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1.1 |
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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 the estimates, which needs to be determined.
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In the experiment we measure cross-section for each channel and calculate matrix of uncertainties,
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$\mathbb{E}$. As we consider four channels, $\mathbb{E}$ will be $4 \times $4 symmetric matrix.
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\[ \mathbb{E} =\left( \begin{array}{cccc}
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\sigma_0^2 & \sigma_{01} & \sigma_{02} & \sigma_{03} \\
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\sigma_{10} & \sigma_1^2 & \sigma_{12} & \sigma_{13} \\
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\sigma_{20} & \sigma_{21} & \sigma_2^2 & \sigma_{23} \\
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\sigma_{30} & \sigma_{31} & \sigma_{32} & \sigma_3^2 \end{array} \right), \]
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1.1 |
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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 the correlated uncertainties of the respective channels,
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$e.g.$, $\sigma_{01}$ describes uncertainty due to the correlation between the channels $3e$ and $2e1\mu$
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and equals to $\sigma_{0}^{corr} \sigma_{1}^{corr}$, where $\sigma_{0}^{corr}$ is uncetrainty
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on zeroth channel ($3e$) due to correlated sources with first channel ($2e1\mu$) and
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$\sigma_{1}^{corr}$ is uncertainty on first channel due to correlated sources woth zeroth channel.
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kkaadze |
1.1 |
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%==================================================================
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As an example, calculation of $\sigma_{01}=\sigma_0^{corr} \sigma_1^{corr}$ is described below.
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At first we estimate the uncertainty which is due to correlated modelling sources, common between all channels,
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listed in Table~\ref{tab:sys}. These sources are Luminosity, trigger, lepton reconstruction, lepton energy scale,
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PDF, W transwers mass requirement. Adding these uncertainties together as uncorrelated results in 16.2\%.
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Then we list the sources of uncertainty which are common between $3e$, $2e1\mu$ channels:
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\[ \begin{array}{lc}
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\mbox{Electron charge determination} & 2*2\% = 4\% \\
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\mbox{Electron identification} & 2*4\% = 8\% \\
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\mbox{fully correlated modeling} & 16.2\% \\
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\mbox{background estimated from MC} & \\
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\end{array}\]
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and add these uncertainties as from uncorrelated sources.
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Uncertainty due to elctron charge identification, electron indetification and
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fully correlated modeling will be
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$\sqrt{4\%^2+8\%^2+16.2\%^2} *1.00 = 0.185$ on $3e$ channel and $\sqrt{4\%^2+8\%^2+16.2\%^2} *0.95 = 0.176$
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on $2e1\mu$ channel, normalized by the corresponding theoretical cross sections.
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Hence, the uncertainties on $3e$ and $2e1\mu$ channels due to all 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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Other terms are also calculated accordingly, which results in following uncertainty matrix
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1.1 |
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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 we follow
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straight forward calculation described in paper ~\cite{BLUE}.
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% I need to correct the syntax in this formula
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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 equal to 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 obtained normalized cross section after combining all four channels is $0.96\pm0.3$.
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Corss sections measured for each channel and result of BLUE method is shown on figure 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{Normalized cross section measured for each channel and corresponding uncertainties. }
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\label{fig:xsections}
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\end{center}
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\end{figure}
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1.1 |
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\end{document}
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