| 1 |
\section{Maximum Likelihood (ML) fit}
|
| 2 |
We extract signal yields and lifetimes from the selected samples using an unbinned extended
|
| 3 |
maximum-likelihood fit to the mass $M_{B_s}$ and proper decay length $c t$ of the reconstructed
|
| 4 |
candidates.
|
| 5 |
|
| 6 |
\subsection{Selection}
|
| 7 |
We change some requirements with respect to our cut-and-count analysis in
|
| 8 |
Sec.~\ref{sec:cutcount} on the event variables:
|
| 9 |
we remove the requirement on $\cos\alpha$ and the lifetime significance
|
| 10 |
$ct/\sigma(c t)$ since they present a bias to $c t$. Furthermore, we tighten
|
| 11 |
the requirement on the invariant $K^+ K^-$ mass to lie within 10~MeV
|
| 12 |
of the world average value and the $B_s$ invariant mass between $5.25$ and $5.65$ $GeV/c^2$.
|
| 13 |
This allows us to strongly suppress background events from $B_d\to J/\psi K^*$
|
| 14 |
that have been misreconstructed. The total selection efficiency is $\epsilon = 28.3\%$.
|
| 15 |
The expected composition of the final sample for the fit for an
|
| 16 |
integrated luminosity of 1~pb$^{-1}$ as derived from simulated event
|
| 17 |
samples is listed in Table~\ref{tab:mlpresel}.
|
| 18 |
|
| 19 |
\begin{table}[hpt]
|
| 20 |
\centering
|
| 21 |
\begin{tabular}{|l|c|c|c|c|}
|
| 22 |
\hline
|
| 23 |
& $B_s\rightarrow J/ \psi \phi$ & $b \to X$ & Prompt $J/ \psi$ & $pp\rightarrow \mu \mu X$\\ \hline
|
| 24 |
Total number of events & $970$ & $94 736$ & $920 247$ & $510216$ \\ \hline
|
| 25 |
\verb,L1DoubleMuOpen, & $191$ & $17514$ & $98854$ & $210566$ \\ \hline
|
| 26 |
Pre Kinem. fit & $121$ & $10266$ & $39362$ & $6597$\\
|
| 27 |
After Kinem. fit & $84$ & $1653$ & $3408$ & $1432$ \\
|
| 28 |
\hline
|
| 29 |
Vtx P(KK)$>$2 $\%$ & $76$ & $1008$ & $2406$ & $548$ \\
|
| 30 |
Kaon $p_T$ $>$ 0.7 GeV/c & $64$ & $397$ & $857$ & $243$ \\
|
| 31 |
$\Delta \phi$ mass $<$ 10 MeV & $54$ & $98$ & $213$ & $57$\\ \hline
|
| 32 |
\hline
|
| 33 |
\end{tabular}
|
| 34 |
\caption{\sl Sample composition in 1~pb$^{-1}$ for the Maximum Likelihood
|
| 35 |
fit as predicted from studies of simulated events.}
|
| 36 |
\label{tab:mlpresel}
|
| 37 |
\end{table}
|
| 38 |
|
| 39 |
The event distribution in the two variables $M_{B_s}$ and $ct$
|
| 40 |
for the different samples is presented in Fig.~\ref{fig:mlmb}
|
| 41 |
and Fig.~\ref{fig:mlct}, respectively.
|
| 42 |
%From a Kolmogorov-Smirnov test~\cite{kstest} we determine that
|
| 43 |
%the prompt $B$ and the QCD background are to 90\% compatible
|
| 44 |
%in each of the two variables. Therefore, we merge the two categories
|
| 45 |
%into one prompt background category.
|
| 46 |
|
| 47 |
\begin{figure}[h]
|
| 48 |
\begin{minipage}[b]{0.5\linewidth}
|
| 49 |
\centering
|
| 50 |
\includegraphics[scale=0.4]{figure/BsML.eps}
|
| 51 |
\caption{$B_s$ invariant mass distribution after the "ML" selection
|
| 52 |
for the signal and background categories.}
|
| 53 |
\label{fig:mlmb}
|
| 54 |
\end{minipage}
|
| 55 |
\hspace{1.cm}
|
| 56 |
\begin{minipage}[b]{0.5\linewidth}
|
| 57 |
\centering
|
| 58 |
\includegraphics[scale=0.4]{figure/ctauL.eps}
|
| 59 |
\caption{$B_s$ proper decay length distribution after the "ML"
|
| 60 |
selection for the signal and background categories.}
|
| 61 |
\label{fig:mlct}
|
| 62 |
\end{minipage}
|
| 63 |
\end{figure}
|
| 64 |
|
| 65 |
|
| 66 |
\subsection{Preparation}
|
| 67 |
Since we measure the correlations among the observables to be small in
|
| 68 |
the samples entering the fit, we take the probability density
|
| 69 |
function $\mathcal{P}_{i,c}^j$ for each event $j$ to be a product of the
|
| 70 |
PDFs for the separate observables. For each event hypothesis $i$
|
| 71 |
(signal, backgrounds), we define
|
| 72 |
\begin{equation}
|
| 73 |
\mathcal{P}_{i}^j =\mathcal{P}_i(M_{B_s};\alpha_i)
|
| 74 |
\cdot\mathcal{P}_i(c t,\sigma_{c t};\beta_i)\, ,
|
| 75 |
\end{equation}
|
| 76 |
with shape parameters $\alpha_i$ for $M_B$ and $\beta_i$ for $c t$,
|
| 77 |
evaluated separately for each of the components $i$.
|
| 78 |
The $\sigma_{c t}$ is the error on $c t$ for a given event.
|
| 79 |
We consider three separate components ($i$): signal, B background and prompt $J/ \psi$.
|
| 80 |
The extended likelihood function for the sample is given as:
|
| 81 |
\begin{equation}
|
| 82 |
\mbox{$\cal{L}$} = \exp \left( - \sum_{i} n_{i} \right) \prod_j
|
| 83 |
\left[ \sum_{i} n_{i} P_{i}^{j} \right]
|
| 84 |
\end{equation}
|
| 85 |
The yields $n_i$ and lifetime $\tau_i$ for each sample are then determined by minimizing
|
| 86 |
the quantity $-\ln \cal{L}$~\cite{roofit}.
|
| 87 |
|
| 88 |
% while keeping the PDF parameters $\alpha_i$ and $\beta_i$ fixed.
|
| 89 |
The analysis proceeds with the fit of the entire sample floating yields and the B proper decay
|
| 90 |
length from which we can extract the lifetime.
|
| 91 |
The PDFs are constructed from common functions (Gaussian, exponential, etc) and the parameters
|
| 92 |
are determined on the available data samples.
|
| 93 |
The guiding principle we follow in designing the PDFs is to use the simplest function with the
|
| 94 |
least number of parameters necessary to adequately describe the
|
| 95 |
observed distribution of events for $M_B$ and $c t$ for each component.
|
| 96 |
For the signal $B_s\rightarrow J/ \psi \phi$, we tried to parameterize the $B_s$ invariant mass
|
| 97 |
with two and three Gaussian distributions (Fig.~\ref{fig:sig3G} and Fig.~\ref{fig:sig2G}) but
|
| 98 |
we choose the three-Gaussians parameterization.
|
| 99 |
%We choose the three-Gaussians parameterization and we compare it with the $B_s$ invariant mass
|
| 100 |
%distribution at the generator level plotting the
|
| 101 |
%residual (Fig. 12 and Fig.13).
|
| 102 |
For the proper decay lenght $c t$, generated with value
|
| 103 |
$c\tau = 423$ ps, we fit an exponential convoluted with a gaussian with and without error correction
|
| 104 |
event by event (Fig.~\ref{fig:CtNo} and Fig.~\ref{fig:CtYes}).
|
| 105 |
For the final fit we will use the PDF with error correction event by event. In Table~\ref{tab:pdftable}
|
| 106 |
the summary of the PDFs used for each component.
|
| 107 |
|
| 108 |
\begin{figure}[!h]
|
| 109 |
\vspace{0.5cm}
|
| 110 |
\begin{minipage}[b]{0.5\linewidth}
|
| 111 |
\centering
|
| 112 |
\includegraphics[scale=0.35]{figure/sigBsM3G.eps}
|
| 113 |
\caption{$B_s$ invariant mass signal distribution fitted with three Gaussians.}
|
| 114 |
\label{fig:sig3G}
|
| 115 |
\end{minipage}
|
| 116 |
\hspace{1.cm}
|
| 117 |
\begin{minipage}[b]{0.5\linewidth}
|
| 118 |
\centering
|
| 119 |
\includegraphics[scale=0.35]{figure/sigBsM2G.eps}
|
| 120 |
\caption{$B_s$ invariant mass signal distribution fitted with two Gaussians.}
|
| 121 |
\label{fig:sig2G}
|
| 122 |
\end{minipage}
|
| 123 |
\end{figure}
|
| 124 |
|
| 125 |
\begin{figure}[!h]
|
| 126 |
\begin{minipage}[b]{0.5\linewidth}
|
| 127 |
\centering
|
| 128 |
\includegraphics[scale=0.35]{figure/sigCtau.eps}
|
| 129 |
\caption{$B_s$ proper decay length signal distribution without event by event correction.}
|
| 130 |
\label{fig:CtNo}
|
| 131 |
\end{minipage}
|
| 132 |
\hspace{1.cm}
|
| 133 |
\begin{minipage}[b]{0.5\linewidth}
|
| 134 |
\centering
|
| 135 |
\includegraphics[scale=0.35]{figure/sigCtauErr.eps}
|
| 136 |
\caption{$B_s$ proper decay length signal distribution with event by event correction.}
|
| 137 |
\label{fig:CtYes}
|
| 138 |
\end{minipage}
|
| 139 |
\end{figure}
|
| 140 |
|
| 141 |
\clearpage
|
| 142 |
|
| 143 |
For the B cocktail background we parameterize the $B_s$ invariant mass with a first degree Chebychev polynomial while a double Gaussian convoluted with
|
| 144 |
a double-sided exponential for the proper decay length.
|
| 145 |
|
| 146 |
\begin{figure}[!h]
|
| 147 |
\vspace{0.5cm}
|
| 148 |
\begin{minipage}[b]{0.5\linewidth}
|
| 149 |
\centering
|
| 150 |
\includegraphics[scale=0.35]{figure/BBsM.eps}
|
| 151 |
\caption{$B_s$ invariant mass for B background distribution.}
|
| 152 |
\label{fig:figure1}
|
| 153 |
\end{minipage}
|
| 154 |
\hspace{1.cm}
|
| 155 |
\begin{minipage}[b]{0.5\linewidth}
|
| 156 |
\centering
|
| 157 |
\includegraphics[scale=0.35]{figure/BCtau.eps}
|
| 158 |
\caption{$B_s$ proper decay length for B background distribution.}
|
| 159 |
\label{fig:figure2}
|
| 160 |
\end{minipage}
|
| 161 |
\vspace{0.5cm}
|
| 162 |
\end{figure}
|
| 163 |
|
| 164 |
For the prompt $J/ \psi$ background we parameterize the $B_s$ invariant mass with a first degree Chebychev polynomial while a double Gaussian convoluted with
|
| 165 |
a double-sided exponential for the proper decay length.
|
| 166 |
|
| 167 |
\begin{figure}[!h]
|
| 168 |
\vspace{0.5cm}
|
| 169 |
\begin{minipage}[b]{0.5\linewidth}
|
| 170 |
\centering
|
| 171 |
\includegraphics[scale=0.35]{figure/promptBsM.eps}
|
| 172 |
\caption{$B_s$ invariant mass for prompt background distribution.}
|
| 173 |
\label{fig:figure1}
|
| 174 |
\end{minipage}
|
| 175 |
\hspace{1.cm}
|
| 176 |
\begin{minipage}[b]{0.5\linewidth}
|
| 177 |
\centering
|
| 178 |
\includegraphics[scale=0.35]{figure/promptCtau.eps}
|
| 179 |
\caption{$B_s$ proper decay length for prompt background distribution.}
|
| 180 |
\label{fig:figure2}
|
| 181 |
\end{minipage}
|
| 182 |
\end{figure}
|
| 183 |
|
| 184 |
\begin{table}[htbp]
|
| 185 |
\centering
|
| 186 |
\vspace{0.25cm}
|
| 187 |
\begin{tabular}{ccccc}
|
| 188 |
\hline\hline
|
| 189 |
& \multicolumn{2}{c}{$M_B$} & \multicolumn{2}{c}{ct} \\
|
| 190 |
Components & Function & Parameters & Function & Parameters \\
|
| 191 |
\hline
|
| 192 |
Signal & $G_1+G_2+G_3$ & $\{ \mu_i,\sigma_i \}$ & $G_1 \otimes e^{-ct/ \lambda}$ & $\{ \mu_i,\sigma_i,\lambda \}$ \\
|
| 193 |
B background & Pol$1$ & $\{ \alpha \}$ & $( G_1+G_2 ) \otimes e^{\mp ct_{\pm}/ \lambda_{\pm}}$ & $\{ \mu_i,\sigma_i,\lambda_{\pm} \}$ \\
|
| 194 |
Prompt $J/ \psi$ & Pol$1$ & $\{ \alpha \}$ & $( G_1+G_2 ) \otimes e^{\mp ct_{\pm}/ \lambda_{\pm}}$ & $\{ \mu_i,\sigma_i,\lambda_{\pm} \}$ \\
|
| 195 |
\hline\hline
|
| 196 |
\end{tabular}
|
| 197 |
\caption{Summary of $M_B$ and $ct$ PFDs used in the fit. Where more than one Gaussian (G) is used in
|
| 198 |
a function we denote the separate means and widths with the notation $\mu_i$ and $\sigma_i$, where $i$ is an index that runs over the number
|
| 199 |
of Gaussians (either two or three). For $B$ background and prompt $J/ \psi$ we describe the long tails in the resolution function with two
|
| 200 |
separate exponential functions, one for $ct>0$ and the other for $ct<0$.}
|
| 201 |
\label{tab:pdftable}
|
| 202 |
\end{table}
|
| 203 |
|
| 204 |
|
| 205 |
\clearpage
|
| 206 |
|
| 207 |
\subsection{Fit Validation}
|
| 208 |
|
| 209 |
We have performed a series of detailed studies to demonstrate the accuracy and robustness of our fit strategy.
|
| 210 |
To prove that the fit configuration is unbiased, we have done 100 toy experiments generating the number of
|
| 211 |
expected signal and background yields according to an integrated luminosity of $3$ p$b^{-1}$
|
| 212 |
For each category we examined fitted value for the yields with their errors and the possible bias (``pull'' distributions).
|
| 213 |
We confirm that no biases are observed, and the errors are properly estimated from the fit (see. Table 8).
|
| 214 |
%The mean value for the negative logarithmic value of the likelihood function is $\ln \cal{L} $ $=14765\pm 64$.
|
| 215 |
|
| 216 |
\begin{table}[htbp]
|
| 217 |
\vspace{0.8cm}
|
| 218 |
\centering
|
| 219 |
\begin{tabular}{ccccc}
|
| 220 |
\hline\hline
|
| 221 |
Components & Generated yield & Fitted yield & Pull mean & Pull sigma \\
|
| 222 |
\hline
|
| 223 |
Signal & $161$ & $164\pm16$ & $0.13\pm0.09$ & $0.937\pm0.066$ \\
|
| 224 |
B background & $291$ & $296\pm64$ & $0.09\pm0.10$ & $1.022\pm0.072$ \\
|
| 225 |
Prompt $J/ \psi$ & $636$ & $634\pm64$& $-0.19\pm0.11$ & $1.130\pm0.080$ \\
|
| 226 |
\hline\hline
|
| 227 |
\end{tabular}
|
| 228 |
\caption{Summary table of 100 toy experiments with yields generated from the PDFs.}
|
| 229 |
\end{table}
|
| 230 |
|
| 231 |
\begin{figure}[!h!t]
|
| 232 |
\vspace{0.5cm}
|
| 233 |
\begin{minipage}[b]{0.5\linewidth}
|
| 234 |
\centering
|
| 235 |
\includegraphics[scale=0.37]{figure/toySigN.eps}
|
| 236 |
\caption{Signal yield distribution.}
|
| 237 |
\label{fig:figure1}
|
| 238 |
\end{minipage}
|
| 239 |
\hspace{0.5cm}
|
| 240 |
\begin{minipage}[b]{0.5\linewidth}
|
| 241 |
\centering
|
| 242 |
\includegraphics[scale=0.37]{figure/toySigNerr.eps}
|
| 243 |
\caption{Signal yield error distribution.}
|
| 244 |
\label{fig:figure2}
|
| 245 |
\end{minipage}
|
| 246 |
\end{figure}
|
| 247 |
|
| 248 |
\begin{figure}[!h!t]
|
| 249 |
\vspace{0.5cm}
|
| 250 |
\begin{minipage}[b]{0.5\linewidth}
|
| 251 |
\centering
|
| 252 |
\includegraphics[scale=0.37]{figure/sigPull.eps}
|
| 253 |
\caption{Signal yield distribution.}
|
| 254 |
\label{fig:figure1}
|
| 255 |
\end{minipage}
|
| 256 |
\hspace{0.5cm}
|
| 257 |
\begin{minipage}[b]{0.5\linewidth}
|
| 258 |
\centering
|
| 259 |
\includegraphics[scale=0.37]{figure/toyLnL.eps}
|
| 260 |
\caption{Negative logarithmic likelihood distribution.}
|
| 261 |
\label{fig:figure2}
|
| 262 |
\end{minipage}
|
| 263 |
\end{figure}
|
| 264 |
|
| 265 |
|
| 266 |
|
| 267 |
\clearpage
|
| 268 |
|
| 269 |
\begin{figure}[!h!t]
|
| 270 |
\vspace{0.5cm}
|
| 271 |
\begin{minipage}[b]{0.5\linewidth}
|
| 272 |
\centering
|
| 273 |
\includegraphics[scale=0.37]{figure/toyBbkgN.eps}
|
| 274 |
\caption{B background yield distribution.}
|
| 275 |
\label{fig:figure1}
|
| 276 |
\end{minipage}
|
| 277 |
\hspace{0.5cm}
|
| 278 |
\begin{minipage}[b]{0.5\linewidth}
|
| 279 |
\centering
|
| 280 |
\includegraphics[scale=0.37]{figure/toyBbkgNerr.eps}
|
| 281 |
\caption{B background yield error distribution.}
|
| 282 |
\label{fig:figure2}
|
| 283 |
\end{minipage}
|
| 284 |
\end{figure}
|
| 285 |
|
| 286 |
\begin{figure}[!h!t]
|
| 287 |
\vspace{0.5cm}
|
| 288 |
\begin{minipage}[b]{0.5\linewidth}
|
| 289 |
\centering
|
| 290 |
\includegraphics[scale=0.37]{figure/toyPrN.eps}
|
| 291 |
\caption{Prompt background yield distribution.}
|
| 292 |
\label{fig:figure1}
|
| 293 |
\end{minipage}
|
| 294 |
\hspace{0.5cm}
|
| 295 |
\begin{minipage}[b]{0.5\linewidth}
|
| 296 |
\centering
|
| 297 |
\includegraphics[scale=0.37]{figure/toyPrNerr.eps}
|
| 298 |
\caption{Prompt background yield error distribution.}
|
| 299 |
\label{fig:figure2}
|
| 300 |
\end{minipage}
|
| 301 |
\end{figure}
|
| 302 |
|
| 303 |
\begin{figure}[!h!t]
|
| 304 |
\vspace{0.5cm}
|
| 305 |
\begin{minipage}[b]{0.5\linewidth}
|
| 306 |
\centering
|
| 307 |
\includegraphics[scale=0.37]{figure/BbkgPull.eps}
|
| 308 |
\caption{B background pull distribution.}
|
| 309 |
\label{fig:figure1}
|
| 310 |
\end{minipage}
|
| 311 |
\hspace{0.5cm}
|
| 312 |
\begin{minipage}[b]{0.5\linewidth}
|
| 313 |
\centering
|
| 314 |
\includegraphics[scale=0.37]{figure/promptPull.eps}
|
| 315 |
\caption{Prompt background pull distribution.}
|
| 316 |
\label{fig:figure2}
|
| 317 |
\end{minipage}
|
| 318 |
\end{figure}
|
| 319 |
|
| 320 |
\clearpage
|
| 321 |
|
| 322 |
We repeat the experiment fitting $100$ independent MC cocktail samples with the number of expected events for each category.
|
| 323 |
In the fit we let free to float the yields, the proper decay length for $B_s\rightarrow J/ \psi \phi$, the bias and the scaling factor of
|
| 324 |
the signal resolution function. This test has been performed with statistics for three different integrated
|
| 325 |
luminosity ($3$ p$b^{-1}$, $1$ p$b^{-1}$ and $0.5$ p$b^{-1}$).
|
| 326 |
|
| 327 |
\begin{table}[htbp]
|
| 328 |
\centering
|
| 329 |
\vspace{0.3cm}
|
| 330 |
\begin{tabular}{ccc}
|
| 331 |
\hline\hline
|
| 332 |
Components & Expected & Fit value \\
|
| 333 |
\hline
|
| 334 |
Signal & $161$ & $189\pm19$ \\
|
| 335 |
B background & $291$ & $245\pm63$ \\
|
| 336 |
Prompt $J/ \psi$ & $636$ & $639\pm62$ \\
|
| 337 |
$\lambda$ ($\mu$m) & $423$ & $425\pm38$ \\
|
| 338 |
Bias & & $-1.48\pm0.35$ \\
|
| 339 |
Scaling factor & & $1.08\pm0.28$ \\
|
| 340 |
\hline\hline
|
| 341 |
\end{tabular}
|
| 342 |
\caption{Summary table for experiments with integrated luminosity of $3$ p$b^{-1}$.}
|
| 343 |
\end{table}
|
| 344 |
|
| 345 |
\begin{table}[htbp]
|
| 346 |
\centering
|
| 347 |
\vspace{0.3cm}
|
| 348 |
\begin{tabular}{ccc}
|
| 349 |
\hline\hline
|
| 350 |
Components & Expected & Fit value \\
|
| 351 |
\hline
|
| 352 |
Signal & $54$ & $65\pm11$ \\
|
| 353 |
B background & $97$ & $82\pm36$ \\
|
| 354 |
Prompt $J/ \psi$ & $212$ & $215\pm35$ \\
|
| 355 |
$\lambda$ ($\mu$m) & $423$ & $429\pm56$ \\
|
| 356 |
Bias & & $-1.53\pm0.64$ \\
|
| 357 |
Scaling factor & & $1.25\pm0.46$ \\
|
| 358 |
\hline\hline
|
| 359 |
\end{tabular}
|
| 360 |
\caption{Summary table for experiments with integrated luminosity of $1$ p$b^{-1}$.}
|
| 361 |
\end{table}
|
| 362 |
|
| 363 |
\begin{table}[htbp]
|
| 364 |
\centering
|
| 365 |
\vspace{0.3cm}
|
| 366 |
\begin{tabular}{ccc}
|
| 367 |
\hline\hline
|
| 368 |
Components & Expected & Fit value \\
|
| 369 |
\hline
|
| 370 |
Signal & $27$ & $34\pm7$ \\
|
| 371 |
B background & $49$ & $42\pm26$ \\
|
| 372 |
Prompt $J/ \psi$ & $106$ & $110\pm25$ \\
|
| 373 |
$\lambda$ ($\mu$m) & $423$ & $401\pm80$ \\
|
| 374 |
Bias & & $-1.65\pm1.18$ \\
|
| 375 |
Scaling factor & & $1.72\pm0.81$ \\
|
| 376 |
\hline\hline
|
| 377 |
\end{tabular}
|
| 378 |
\caption{Summary table for experiments with integrated luminosity of $0.5$ p$b^{-1}$.}
|
| 379 |
\end{table}
|
| 380 |
|
| 381 |
It is generally expected that the real backgrounds encountered in collision data could be much higher than predicted by the default (untuned) CMS full simulation samples.
|
| 382 |
|
| 383 |
\clearpage
|
