| 1 |
\newpage
|
| 2 |
\appendix
|
| 3 |
\clearpage
|
| 4 |
\section{Signal estimation using $ABCD$ method}
|
| 5 |
|
| 6 |
|
| 7 |
\begin{figure}[hbt]
|
| 8 |
\begin{center}
|
| 9 |
\scalebox{0.4}{\includegraphics{figs/abcd.eps}}
|
| 10 |
\caption{Illustration of ABCD method bin definition for $W \rightarrow \mu\nu$ channels.}
|
| 11 |
\label{fig:ABCDl}
|
| 12 |
\end{center}
|
| 13 |
\end{figure}
|
| 14 |
|
| 15 |
%$ABCD$ method accuracy is dependant on the wea
|
| 16 |
%$ABCD$ method accuracy is better if the method is based on analysis variables which are
|
| 17 |
%weakly correlated, for both signal and background.
|
| 18 |
We estimate \WZ signal yield with final selection cuts for all four analysis channels.
|
| 19 |
Selected events are sorted to non-overlapping bins $A$,$B$,$C$ and $D$ as illustrated in
|
| 20 |
figure \ref{fig:ABCDl}.
|
| 21 |
|
| 22 |
$ABCD$ method accurracy is better if we use variables for which correlation is minimal for both signal and background.
|
| 23 |
\W transverse mass and isolation are usually a suitable combination and were chosen for this analysis.
|
| 24 |
For $W \rightarrow \mu\nu$ channels, $A$, $B$ contain events with
|
| 25 |
muon tracker $P_t$ Isolation $P_t Iso < 2$, while $A$, $C$ contain events with \W
|
| 26 |
transverse mass $ MT_W > 50 GeV$. For $W \rightarrow e\nu$ channels we use {\tt SimpleTight}
|
| 27 |
cut and \W transverse mass cut. All other analysis cuts are applied.
|
| 28 |
|
| 29 |
For the background, assumption of weak correlation is formulated as
|
| 30 |
|
| 31 |
\begin {equation}
|
| 32 |
F_b = \frac{B_A}{B_B} \approx F_b' = \frac{B_C}{B_D}
|
| 33 |
\label{eq_ABCD_bkg}
|
| 34 |
\end {equation}
|
| 35 |
|
| 36 |
$S_i$ and $B_i$ represent signal and background event numbers in $i=A,B,C,D$ bins.
|
| 37 |
For signal we first calculate factors $F_{MT_W}=S_{A+C}/S_{A+B+C+D}$ and
|
| 38 |
$F_{ISO}=S_{A+B}/S_{A+B+C+D}$, defined as fraction of events passing
|
| 39 |
\W transverse mass and the isolation cut respectively.
|
| 40 |
In this analysis they are extracted from signal Monte Carlo sample.
|
| 41 |
From data their values are usually calculated using control samples (templates) $\Z \rightarrow \mumu$
|
| 42 |
or $\Z \rightarrow \epem$ where one lepton simulates neutrino from the \W
|
| 43 |
decay and \Z invariant mass is rescaled to the \W mass.
|
| 44 |
|
| 45 |
Assuming variables' weak correlation for signal, we determine factors $b$, $c$ and $d$ which will be used in calculation of the signal yield in bin $A$
|
| 46 |
|
| 47 |
\begin {eqnarray}
|
| 48 |
b &=& \frac{S_B}{S_A} \approx \frac{S_D}{S_C} \approx \frac{1-F_{MT_W}}{F_{MT_W}} \nonumber \\
|
| 49 |
c &=& \frac{S_C}{S_A} \approx \frac{S_D}{S_B} \approx \frac{1-F_{ISO}}{F_{ISO}} \nonumber \\
|
| 50 |
d &=& \frac{S_D}{S_A} \approx b*c
|
| 51 |
\label{eq_ABCD1}
|
| 52 |
\end {eqnarray}
|
| 53 |
|
| 54 |
This gives a set of equations representing measured sum of signal and background events in each bin, $N_A$, $N_B$, $N_C$ and $N_D$
|
| 55 |
|
| 56 |
\begin{eqnarray}
|
| 57 |
N_A =& S_A + B_A & \quad N_B = b*S_A + B_B \nonumber \\
|
| 58 |
N_C =& c*S_A + B_C & \quad N_D = d*S_A + B_D
|
| 59 |
\label{eq_ABCD2}
|
| 60 |
\end{eqnarray}
|
| 61 |
|
| 62 |
Solving equations \ref{eq_ABCD_bkg}, \ref{eq_ABCD2} we get expression for the number of signal events in bin A, which has all selection cuts applied
|
| 63 |
|
| 64 |
\begin {equation}
|
| 65 |
S_A = \frac{N_A*N_D - N_B*N_C}{N_D+d*N_A - c*N_B - b*N_C}
|
| 66 |
\end {equation}
|
| 67 |
|
| 68 |
Equation \ref{eq_ABCDsigma} is a term for statistical error of signal in bin $A$, $\sigma_{S_A}$. Similarly we determine the background statistical error, $\sigma_{B_A}$, using relation $B_A = N_A - S_A$.
|
| 69 |
In table \ref{tab:ABCD_result} signal and background yields are compared for $ABCD$ method and Monte Carlo.
|
| 70 |
|
| 71 |
|
| 72 |
|
| 73 |
\begin{table}[t]
|
| 74 |
\begin{center}
|
| 75 |
\begin{tabular}{lccccc} \hline \hline
|
| 76 |
& $3e$ & $2e1\mu$ & $1e2\mu$ & $3\mu$\\ \hline
|
| 77 |
%F_{ISO} & $1$ & $2$ & $3$ &
|
| 78 |
%F_{MT_W} & $1$ & $2$ & $3$ &
|
| 79 |
$S_A = N^{\WZ}$ (MC) & 7.9 $\pm$0.2 & 8.0 $\pm$0.3 & 8.9 $\pm$ 0.3 & 10.0 $\pm$ 0.3 \\
|
| 80 |
$S_A = N^{\WZ}$ (ABCD) & 8.6 $\pm$1.7 & 9.0 $\pm$0.3 & 7.9 $\pm$1.6 & 11.3 $\pm$ 0.5 \\ \hline
|
| 81 |
$B_A = N\{ZZ, Zbb, Z\gamma, Z+jets, W+jets, t\bar{t}\}$ (MC) & 6.2 $\pm$0.9 & 1.4 $\pm$0.4 & 4.5 $\pm$0.8 & 1.5 $\pm$ 0.3 \\
|
| 82 |
$B_A = N\{ZZ, Zbb, Z\gamma, Z+jets, W+jets, t\bar{t}\}$ (ABCD) & 5.5 $\pm$2.4 & 0.4 $\pm$0.9 & 5.4 $\pm$2.2 & 0.2 $\pm$ 0.9 \\
|
| 83 |
\hline
|
| 84 |
\end{tabular}
|
| 85 |
\caption{Number of signal and background events for integrated luminosity of 300
|
| 86 |
pb$^{-1}$ calculated using Monte Carlo and ABCD method. Errors are statistical.}
|
| 87 |
\label{tab:ABCD_result}
|
| 88 |
\end{center}
|
| 89 |
\end{table}
|
| 90 |
|
| 91 |
\begin {eqnarray}
|
| 92 |
F &\equiv& N_A*N_D - N_B*N_C \nonumber \\
|
| 93 |
G &\equiv& N_D + d*N_A - b*N_C - c*N_B \nonumber \\
|
| 94 |
\sigma_{S_A}^2 &= \frac{1}{G^4} * & [\
|
| 95 |
( ( N_D*G - F*d) *\sigma_{N_A})^2 \
|
| 96 |
+ ( (-N_C*G + F*c) *\sigma_{N_B})^2 \nonumber \\
|
| 97 |
& &+ ( (-N_B*G + F*b) *\sigma_{N_C})^2 \
|
| 98 |
+ ( ( N_A*G - F ) *\sigma_{N_D})^2 \nonumber \\
|
| 99 |
& &+ ( F*(-N_C+c*N_A) *\sigma_b)^2 \
|
| 100 |
+ ( F*(-N_C+b*N_A) *\sigma_c)^2 \
|
| 101 |
]
|
| 102 |
\label{eq_ABCDsigma}
|
| 103 |
\end {eqnarray}
|
| 104 |
|
| 105 |
|
| 106 |
A Monte Carlo study was done to estimate variable correlation for signal and background,
|
| 107 |
with results shown in table \ref{tab:ABCD_corrres}. Ideally, ratios $(A/B)/(C/D)$ should be $\approx 1$ for the method
|
| 108 |
to produce accurate result. For $\W\rightarrow \mu \nu$ channels there is indication of a stronger correlation, but for
|
| 109 |
all channels there is a significant statistical error, for both signal and background. We conclude that the accurracy of the method is statistically constrained with currently used samples and the extent of correlations is yet inconclusive.
|
| 110 |
|
| 111 |
|
| 112 |
|
| 113 |
\begin{table}[h]
|
| 114 |
\begin{center}
|
| 115 |
\begin{tabular}{lcccc} \hline \hline
|
| 116 |
MC sample & $3e$ & $2e1\mu$ & $1e2\mu$ & $3\mu$\\ \hline
|
| 117 |
%F_{ISO} & $1$ & $2$ & $3$ &
|
| 118 |
%F_{MT_W} & $1$ & $2$ & $3$ &
|
| 119 |
$\WZ$ A/B & 2.2 $\pm$ 0.1 & 1.9 $\pm$ 0.1 & 2.18 $\pm$ 0.09 & 2.0 $\pm$ 0.1 \\
|
| 120 |
$\WZ$ C/D & 1.8 $\pm$ 0.5 & 1.1 $\pm$ 0.3 & 1.5 $\pm$ 0.4 & 2.5 $\pm$ 0.6 \\ \hline
|
| 121 |
$\WZ$ A/B/(C/D) & 1.2 $\pm$ 0.3 & 1.7 $\pm$ 0.5 & 1.5 $\pm$ 0.4 & 0.8 $\pm$ 0.3 \\ \hline
|
| 122 |
\hline
|
| 123 |
$Background$ A/B & 0.25 $\pm$ 0.04 & 0.22 $\pm$ 0.07 & 0.21 $\pm$ 0.05 & 0.4 $\pm$ 0.1 \\
|
| 124 |
$Background$ C/D & 0.23 $\pm$ 0.04 & 0.08 $\pm$ 0.02 & 0.26 $\pm$ 0.06 & 0.08 $\pm$ 0.04 \\ \hline
|
| 125 |
$Background$ A/B/(C/D) & 1.1 $\pm$ 0.3 & 2.7 $\pm$ 1.1 & 0.8 $\pm$ 0.2 & 5.4 $\pm$ 3.0 \\ \hline
|
| 126 |
\hline
|
| 127 |
\end{tabular}
|
| 128 |
\caption{Bin ratios for signal and background Monte Carlo. Errors are statistical.}
|
| 129 |
\label{tab:ABCD_corrres}
|
| 130 |
\end{center}
|
| 131 |
\end{table}
|
| 132 |
|
