Notes/WZCSA07/xsection.tex

\section{Combining cross section measurements for all channels} 
\label{sec:xsec}

In this section we calculate the cross section for each channel and
provide the method to combine four measurements taking into account the correlation.

$\WZ$ cross section is calculated as follows

\begin{equation}
\label{eq:xsec}
\sigma = \frac{N_{obs}-N_{MC}-N_{Z}}{\epsilon \mathcal{L}},  
\end{equation}

where $N_{obs}$ is the number of observed events, $N_{MC}$
is the number of background events estimated from Monte Carlo simulation,
$i.e.$ events from $\Z\gamma$, $\ZZ$, and processes without genuine $\Z$ boson
in final state. $N_Z$ is the number of instrumental background events from 
processes with genuine $\Z$ boson. These numbers are given in
Tables~\ref{tab:yields}~and~\ref{tab:yieldsEstimate} for each channel.

The uncertainty on measured cross section is calculated as given in equation below
%Eq.~\ref{eq:xsecErr}

\begin{equation}
\label{eq:xsecErr}
\left(\Delta\sigma\right)^2 = \left(\frac{\Delta N_{obs}}{\epsilon\mathcal{L}}\right)^2 
        + \left(\frac{\Delta N_{MC}}{\epsilon \mathcal{L}}\right)^2 
        + \left(\frac{\Delta N_{Z}}{\epsilon \mathcal{L}}\right)^2 
        + \left(\frac{\Delta (\epsilon \mathcal{L})}{\epsilon \mathcal{L}} \frac{N_{WE}}{\epsilon \mathcal{L}}\right)^2 ,
\end{equation} 

where $\frac{\Delta(\epsilon \mathcal{L})}{\epsilon \mathcal{L}}$ is 
modeling uncertainty for each channel, given in Table~\ref{tab:FullSys}.

We normalize cross section to the theoretical cross section, calculated from
Monte Carlo simulation. The normalized $\WZ$ cross section, $\sigma_n$, is given as 

%\begin{equation}
\begin{eqnarray}
\label{eq:normSigma}
\sigma_n &=&\frac{N_{obs}-N_{MC}-N_{\Z}}{\epsilon\mathcal{L}} \frac{1}{\sigma_{theory}} \nonumber \\
        &=& \frac{N_{obs}-N_{MC}-N_{\Z}}{\epsilon\mathcal{L}} \frac{\epsilon \mathcal{L}}{N_{\WZ}} \nonumber \\
        &=& \frac{N_{obs}-N_{MC}-N_{\Z}}{N_{\WZ}}
\end{eqnarray}
%\end{equation}

%\sigma_{n} = \frac{N_{obs}-N_{MC}-N_{\Z}}{N_{\WZ}}

where $N_{\WZ}$ is the expected number of signal events, corresponding to a sample of 300 $pb^{-1}$ 
integrated luminosity, which is given in the first line of Table~\ref{tab:yields}. 
%Thus, the normalization factors are 7.9, 8.0, 8.9, and 10.1 for $3e$, $2e1\mu$, 
%$2\mu1e$, and  $3\mu$, respectively.

Normalized cross section for each channel is given in
Table below %~\ref{tab:fourXsections}  

\begin{table}[h]
  \begin{center}
\begin{tabular}{|l|c|} \hline
Channels          &  normalized cross section \\ \hline
$3e$              &  1.00$\pm$0.58            \\
$2e 1\mu$         &  0.95$\pm$0.45            \\ 
$2\mu 1e$         &  0.82$\pm$0.52            \\
$3\mu$            &  0.98$\pm$0.39            \\ \hline
\end{tabular}
\caption{Normalized cross section measured for each channel and corresponding uncertainties.}   
\label{tab:fourXsections}
\end{center}
\end{table}
  
In  order to combine the results for four different channels we use BLUE 
($i.e.$ Best Linear Unbiased Estimate) method~\cite{BLUE}. 
According to this method, final estimation of the cross section is a linear combination 
of cross-sections, measured for each channel separately 

\begin{equation}
\label{eq:sumXsec}
\sigma = \sum_{i} \alpha_i\sigma_i 
\end{equation}

where $\alpha_{i}$ are weighting factors for each measurement, which needs to be determined. 
 
In the experiment we measure cross section for each channel and calculate uncertainty matrix, 
$\mathbb{E}$. As we consider four channels, $\mathbb{E}$ will be $4 \times $4 symmetric matrix.

\begin{equation}
\label{eq:uncertMatrix}
\mathbb{E} =\left( \begin{array}{cccc}
\sigma_1^2   & \sigma_{12}  & \sigma_{13}  & \sigma_{14}  \\
\sigma_{21}  & \sigma_2^2   & \sigma_{23}  & \sigma_{24}  \\
\sigma_{31}  & \sigma_{32}  & \sigma_3^2   & \sigma_{34}  \\
\sigma_{41}  & \sigma_{42}  & \sigma_{43}  & \sigma_4^2   \end{array} \right),
\end{equation}

%where the diagonal elements correspond to variances of cross-section measurements for channels
where diagonal elements are the full uncertainties for individual channel squared and 
the off-diagonal elements represent uncertainties of the respective channels from sources of
correlation, $e.g.$, $\sigma_{12}$ describes uncertainty due to the common sources between the channels $3e$ 
and $2e1\mu$ and equals to $\sigma_{1}^{corr} \sigma_{2}^{corr}$, where $\sigma_{1}^{corr}$ 
and $\sigma_{2}^{corr}$ are uncertainties on first ($3e$) and second ($2e1\mu$) channels, respectively.  


%==================================================================

As an example, calculation of $\sigma_{12}=\sigma_1^{corr} \sigma_2^{corr}$ is described below.
At first we estimate the uncertainty due to modeling sources, common between all channels, which are
listed in Table~\ref{tab:sys}. These are luminosity, trigger, lepton reconstruction, lepton energy scale, 
PDF, and W transverse mass requirement.  Adding these uncertainties together results in 16.2\%.

Then we list the sources of uncertainty which are common between $3e$ and $2e1\mu$ channels:

\[ \begin{array}{lc}
\mbox{Electron charge determination} & 2\%+2\% = 4\%  \\
\mbox{Electron identification}       & 4\%+4\% = 8\%  \\
\mbox{Fully correlated modeling}     &      16.2\%  \\
\mbox{Background estimated from MC}  &              \\
\end{array}\]

and add these uncertainties as from uncorrelated sources.

Uncertainty due to electron charge determination, electron identification, and 
fully correlated modeling is 
$$\sqrt{4\%^2+8\%^2+16.2\%^2} 1.00 = 0.185$$ 
for $3e$ channel and 
$$\sqrt{4\%^2+8\%^2+16.2\%^2} 0.95 = 0.176$$ 
for $2e1\mu$ channel. Hence, the uncertainties for $3e$ and $2e1\mu$ channels due to all 
sources of correlations are

\begin{subequations}
\label{eq:covTerms}
\begin{align}
\sigma_0^{corr}& = \sqrt{0.185^2 + (\frac{1.3}{7.9})^2} = 0.248 \label{first}\\
\sigma_1^{corr}& = \sqrt{0.176^2 + (\frac{0.9}{8.0})^2} = 0.209 \label{second}
\end{align}
\end{subequations}
Thus, 
\begin{equation}\label{eq:covTermFin}
\sigma_{01} = 0.248 * 0.209 = 0.052
\end{equation}

Calculating other terms using previously described procedure result in

\[ \mathbb{E} = \left( \begin{array}{cccc}
0.335   &    0.052   &    0.095   &    0.041  \\
0.052   &    0.202   &    0.037   &    0.043  \\ 
0.095   &    0.037   &    0.267   &    0.036  \\ 
0.041   &    0.043   &    0.036   &    0.155   \end{array} \right) .\]

In order to determine $\alpha_i$ weighting factors following 
procedure described in paper~\cite{BLUE}
 
\begin{equation}
\label{eq:weights}
\alpha = \mathbb{E}^{-1}U/(\tilde{U}\mathbb{E}^{-1}U) ,
\end{equation}

where $U$ is a four-dimensional vector whose all components are unity.
Obtained weights are 0.112, 0.281, 0.194, and 0.413 for $3e$, $2e1\mu$, $2\mu1e$, and $3\mu$ channels, respectively.
Total uncertainty on combined measurement squared is calculated as

\begin{equation}
\label{eq:sigmaSquared}
\sigma^2 = \tilde{\alpha}\mathbb{E}\alpha
\end{equation}

Thus, normalized combined cross section is $0.96\pm0.3$.
Cross sections measured for each channel separately and result of BLUE method is 
shown on Fig.~\ref{fig:xsections}

\begin{figure}[bt]
  \begin{center}
  \scalebox{0.8}{\includegraphics{figs/combined_xsections.eps}}
  \caption{The normalized cross section measured for each channel and corresponding uncertainties. }
  \label{fig:xsections}
  \end{center}
\end{figure}


\end{document}


Revision:	1.3
Committed:	Fri Sep 5 00:39:57 2008 UTC (16 years, 8 months ago) by kkaadze
Content type:	application/x-tex
Branch:	MAIN
CVS Tags:	Summer08-FinalApproved
Changes since 1.2:	+81 -81 lines
Log Message:	Undate.
#	User	Rev	Content
1	kkaadze	1.3	\section{Combining cross section measurements for all channels}
2	kkaadze	1.1	\label{sec:xsec}
3
4	kkaadze	1.3	In this section we calculate the cross section for each channel and
5			provide the method to combine four measurements taking into account the correlation.
6
7			$\WZ$ cross section is calculated as follows
8	kkaadze	1.1
9			\begin{equation}
10			\label{eq:xsec}
11			\sigma = \frac{N_{obs}-N_{MC}-N_{Z}}{\epsilon \mathcal{L}},
12			\end{equation}
13
14	kkaadze	1.3	where $N_{obs}$ is the number of observed events, $N_{MC}$
15			is the number of background events estimated from Monte Carlo simulation,
16			$i.e.$ events from $\Z\gamma$, $\ZZ$, and processes without genuine $\Z$ boson
17			in final state. $N_Z$ is the number of instrumental background events from
18			processes with genuine $\Z$ boson. These numbers are given in
19			Tables~\ref{tab:yields}~and~\ref{tab:yieldsEstimate} for each channel.
20	kkaadze	1.1
21	kkaadze	1.3	The uncertainty on measured cross section is calculated as given in equation below
22	kkaadze	1.2	%Eq.~\ref{eq:xsecErr}
23	kkaadze	1.1
24			\begin{equation}
25			\label{eq:xsecErr}
26			\left(\Delta\sigma\right)^2 = \left(\frac{\Delta N_{obs}}{\epsilon\mathcal{L}}\right)^2
27			+ \left(\frac{\Delta N_{MC}}{\epsilon \mathcal{L}}\right)^2
28			+ \left(\frac{\Delta N_{Z}}{\epsilon \mathcal{L}}\right)^2
29	kkaadze	1.3	+ \left(\frac{\Delta (\epsilon \mathcal{L})}{\epsilon \mathcal{L}} \frac{N_{WE}}{\epsilon \mathcal{L}}\right)^2 ,
30	kkaadze	1.1	\end{equation}
31
32	kkaadze	1.3	where $\frac{\Delta(\epsilon \mathcal{L})}{\epsilon \mathcal{L}}$ is
33			modeling uncertainty for each channel, given in Table~\ref{tab:FullSys}.
34	kkaadze	1.1
35	kkaadze	1.3	We normalize cross section to the theoretical cross section, calculated from
36			Monte Carlo simulation. The normalized $\WZ$ cross section, $\sigma_n$, is given as
37	kkaadze	1.1
38			%\begin{equation}
39	kkaadze	1.3	\begin{eqnarray}
40			\label{eq:normSigma}
41			\sigma_n &=&\frac{N_{obs}-N_{MC}-N_{\Z}}{\epsilon\mathcal{L}} \frac{1}{\sigma_{theory}} \nonumber \\
42			&=& \frac{N_{obs}-N_{MC}-N_{\Z}}{\epsilon\mathcal{L}} \frac{\epsilon \mathcal{L}}{N_{\WZ}} \nonumber \\
43			&=& \frac{N_{obs}-N_{MC}-N_{\Z}}{N_{\WZ}}
44			\end{eqnarray}
45	kkaadze	1.1	%\end{equation}
46
47	kkaadze	1.3	%\sigma_{n} = \frac{N_{obs}-N_{MC}-N_{\Z}}{N_{\WZ}}
48	kkaadze	1.1
49	kkaadze	1.3	where $N_{\WZ}$ is the expected number of signal events, corresponding to a sample of 300 $pb^{-1}$
50			integrated luminosity, which is given in the first line of Table~\ref{tab:yields}.
51			%Thus, the normalization factors are 7.9, 8.0, 8.9, and 10.1 for $3e$, $2e1\mu$,
52			%$2\mu1e$, and $3\mu$, respectively.
53	kkaadze	1.1
54	kkaadze	1.3	Normalized cross section for each channel is given in
55			Table below %~\ref{tab:fourXsections}
56	kkaadze	1.1
57	kkaadze	1.3	\begin{table}[h]
58			\begin{center}
59			\begin{tabular}{\|l\|c\|} \hline
60			Channels & normalized cross section \\ \hline
61			$3e$ & 1.00$\pm$0.58 \\
62			$2e 1\mu$ & 0.95$\pm$0.45 \\
63			$2\mu 1e$ & 0.82$\pm$0.52 \\
64			$3\mu$ & 0.98$\pm$0.39 \\ \hline
65	kkaadze	1.1	\end{tabular}
66	kkaadze	1.3	\caption{Normalized cross section measured for each channel and corresponding uncertainties.}
67	kkaadze	1.1	\label{tab:fourXsections}
68	kkaadze	1.3	\end{center}
69			\end{table}
70	kkaadze	1.1
71	kkaadze	1.3	In order to combine the results for four different channels we use BLUE
72	kkaadze	1.1	($i.e.$ Best Linear Unbiased Estimate) method~\cite{BLUE}.
73			According to this method, final estimation of the cross section is a linear combination
74	kkaadze	1.3	of cross-sections, measured for each channel separately
75	kkaadze	1.1
76			\begin{equation}
77			\label{eq:sumXsec}
78			\sigma = \sum_{i} \alpha_i\sigma_i
79			\end{equation}
80
81	kkaadze	1.3	where $\alpha_{i}$ are weighting factors for each measurement, which needs to be determined.
82	kkaadze	1.1
83	kkaadze	1.3	In the experiment we measure cross section for each channel and calculate uncertainty matrix,
84	kkaadze	1.2	$\mathbb{E}$. As we consider four channels, $\mathbb{E}$ will be $4 \times $4 symmetric matrix.
85
86	kkaadze	1.3	\begin{equation}
87			\label{eq:uncertMatrix}
88			\mathbb{E} =\left( \begin{array}{cccc}
89			\sigma_1^2 & \sigma_{12} & \sigma_{13} & \sigma_{14} \\
90			\sigma_{21} & \sigma_2^2 & \sigma_{23} & \sigma_{24} \\
91			\sigma_{31} & \sigma_{32} & \sigma_3^2 & \sigma_{34} \\
92			\sigma_{41} & \sigma_{42} & \sigma_{43} & \sigma_4^2 \end{array} \right),
93			\end{equation}
94	kkaadze	1.1
95			%where the diagonal elements correspond to variances of cross-section measurements for channels
96			where diagonal elements are the full uncertainties for individual channel squared and
97	kkaadze	1.3	the off-diagonal elements represent uncertainties of the respective channels from sources of
98			correlation, $e.g.$, $\sigma_{12}$ describes uncertainty due to the common sources between the channels $3e$
99			and $2e1\mu$ and equals to $\sigma_{1}^{corr} \sigma_{2}^{corr}$, where $\sigma_{1}^{corr}$
100			and $\sigma_{2}^{corr}$ are uncertainties on first ($3e$) and second ($2e1\mu$) channels, respectively.
101	kkaadze	1.1
102
103			%==================================================================
104
105	kkaadze	1.3	As an example, calculation of $\sigma_{12}=\sigma_1^{corr} \sigma_2^{corr}$ is described below.
106			At first we estimate the uncertainty due to modeling sources, common between all channels, which are
107			listed in Table~\ref{tab:sys}. These are luminosity, trigger, lepton reconstruction, lepton energy scale,
108			PDF, and W transverse mass requirement. Adding these uncertainties together results in 16.2\%.
109	kkaadze	1.1
110	kkaadze	1.3	Then we list the sources of uncertainty which are common between $3e$ and $2e1\mu$ channels:
111	kkaadze	1.1
112			\[ \begin{array}{lc}
113	kkaadze	1.3	\mbox{Electron charge determination} & 2\%+2\% = 4\% \\
114			\mbox{Electron identification} & 4\%+4\% = 8\% \\
115			\mbox{Fully correlated modeling} & 16.2\% \\
116			\mbox{Background estimated from MC} & \\
117	kkaadze	1.1	\end{array}\]
118
119			and add these uncertainties as from uncorrelated sources.
120
121	kkaadze	1.3	Uncertainty due to electron charge determination, electron identification, and
122			fully correlated modeling is
123			$$\sqrt{4\%^2+8\%^2+16.2\%^2} 1.00 = 0.185$$
124			for $3e$ channel and
125			$$\sqrt{4\%^2+8\%^2+16.2\%^2} 0.95 = 0.176$$
126			for $2e1\mu$ channel. Hence, the uncertainties for $3e$ and $2e1\mu$ channels due to all
127			sources of correlations are
128	kkaadze	1.1
129			\begin{subequations}
130			\label{eq:covTerms}
131			\begin{align}
132			\sigma_0^{corr}& = \sqrt{0.185^2 + (\frac{1.3}{7.9})^2} = 0.248 \label{first}\\
133			\sigma_1^{corr}& = \sqrt{0.176^2 + (\frac{0.9}{8.0})^2} = 0.209 \label{second}
134			\end{align}
135			\end{subequations}
136			Thus,
137			\begin{equation}\label{eq:covTermFin}
138			\sigma_{01} = 0.248 * 0.209 = 0.052
139			\end{equation}
140
141	kkaadze	1.3	Calculating other terms using previously described procedure result in
142	kkaadze	1.1
143			\[ \mathbb{E} = \left( \begin{array}{cccc}
144			0.335 & 0.052 & 0.095 & 0.041 \\
145			0.052 & 0.202 & 0.037 & 0.043 \\
146			0.095 & 0.037 & 0.267 & 0.036 \\
147			0.041 & 0.043 & 0.036 & 0.155 \end{array} \right) .\]
148
149	kkaadze	1.3	In order to determine $\alpha_i$ weighting factors following
150			procedure described in paper~\cite{BLUE}
151	kkaadze	1.1
152			\begin{equation}
153			\label{eq:weights}
154	kkaadze	1.2	\alpha = \mathbb{E}^{-1}U/(\tilde{U}\mathbb{E}^{-1}U) ,
155	kkaadze	1.1	\end{equation}
156
157	kkaadze	1.2	where $U$ is a four-dimensional vector whose all components are unity.
158	kkaadze	1.3	Obtained weights are 0.112, 0.281, 0.194, and 0.413 for $3e$, $2e1\mu$, $2\mu1e$, and $3\mu$ channels, respectively.
159	kkaadze	1.2	Total uncertainty on combined measurement squared is calculated as
160
161			\begin{equation}
162			\label{eq:sigmaSquared}
163			\sigma^2 = \tilde{\alpha}\mathbb{E}\alpha
164			\end{equation}
165
166	kkaadze	1.3	Thus, normalized combined cross section is $0.96\pm0.3$.
167			Cross sections measured for each channel separately and result of BLUE method is
168			shown on Fig.~\ref{fig:xsections}
169	kkaadze	1.2
170			\begin{figure}[bt]
171			\begin{center}
172			\scalebox{0.8}{\includegraphics{figs/combined_xsections.eps}}
173	kkaadze	1.3	\caption{The normalized cross section measured for each channel and corresponding uncertainties. }
174	kkaadze	1.2	\label{fig:xsections}
175			\end{center}
176			\end{figure}
177
178	kkaadze	1.1
179			\end{document}
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