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\section{Combine cross section measurements for all channels} |
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\section{Combining 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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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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|
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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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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 error on measured cross section is calculated as given in qeuation below |
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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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|
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\begin{equation} |
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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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+ \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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|
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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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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 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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|
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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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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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|
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%\begin{equation} |
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%\label{eq:sOverStheory} |
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|
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%\frac{\sigma}{\sigma_theory} = \frac{13.6 - 2.5 - 3.2}{7.9} = 1.0 |
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|
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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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|
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%\begin{equation} |
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%\label{eq:deltaSOverStheory} |
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|
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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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|
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%\end{equation} |
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% Remove up to here |
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%\sigma_{n} = \frac{N_{obs}-N_{MC}-N_{\Z}}{N_{\WZ}} |
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|
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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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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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|
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$$ |
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\begin{tabular}{|l|c|c|} \hline |
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Channels & normalized cross section & normalization factor \\ \hline |
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$3e$ & 1.00$\pm$0.58 & 7.9 \\ |
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$2e 1\mu$ & 0.95$\pm$0.45 & 8.0 \\ |
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$2\mu 1e$ & 0.82$\pm$0.52 & 8.9 \\ |
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$3\mu$ & 0.98$\pm$0.39 & 10.1 \\ \hline |
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\begin{table}[h] |
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\begin{center} |
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\begin{tabular}{|l|c|} \hline |
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Channels & normalized cross section \\ \hline |
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$3e$ & 1.00$\pm$0.58 \\ |
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$2e 1\mu$ & 0.95$\pm$0.45 \\ |
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$2\mu 1e$ & 0.82$\pm$0.52 \\ |
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$3\mu$ & 0.98$\pm$0.39 \\ \hline |
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\end{tabular} |
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\caption{Normalized cross section measured for each channel and corresponding uncertainties.} |
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\label{tab:fourXsections} |
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$$ |
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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 channles we use BLUE |
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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, individual estimates, measured for each channel. |
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of cross-sections, measured for each channel separately |
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|
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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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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 matrix of uncertainties, |
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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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|
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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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|
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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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|
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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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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 $\sigma_{01}=\sigma_0^{corr} \sigma_1^{corr}$ 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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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 $3e$, $2e1\mu$ channels: |
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Then we list the sources of uncertainty which are common between $3e$ and $2e1\mu$ channels: |
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|
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\[ \begin{array}{lc} |
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\mbox{Electron charge determination} & 2*2\% = 4\% \\ |
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\mbox{Electron identification} & 2*4\% = 8\% \\ |
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\mbox{fully correlated modeling} & 16.2\% \\ |
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\mbox{background estimated from MC} & \\ |
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\mbox{Electron charge determination} & 2\%+2\% = 4\% \\ |
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\mbox{Electron identification} & 4\%+4\% = 8\% \\ |
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\mbox{Fully correlated modeling} & 16.2\% \\ |
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\mbox{Background estimated from MC} & \\ |
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\end{array}\] |
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and add these uncertainties as from uncorrelated sources. |
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Uncertainty due to 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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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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\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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Calculating other terms using previously described procedure result in |
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\[ \mathbb{E} = \left( \begin{array}{cccc} |
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0.335 & 0.052 & 0.095 & 0.041 \\ |
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0.095 & 0.037 & 0.267 & 0.036 \\ |
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0.041 & 0.043 & 0.036 & 0.155 \end{array} \right) .\] |
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|
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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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In order to determine $\alpha_i$ weighting factors following |
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procedure described in paper~\cite{BLUE} |
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|
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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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|
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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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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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|
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\begin{equation} |
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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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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{Normalized cross section measured for each channel and corresponding uncertainties. } |
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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} |
