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\section{Introduction}
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The decay $B_s\rightarrow J/ \psi \phi\rightarrow \mu^+\mu^-K^+K^-$ is of
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particular interest since it allows the study of many properties of the $B^0_s$
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sytem, such as the differences between the widths and the masses of the two
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weak eigenstates, $B_s^H$ and $B_s^L$.
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The measurement of the interference between mixing and decay might reveal $CP$
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violation other than predicted by the Standard Model of Particle Physics
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indicating the presence of new generations of particles and forces between
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them~\cite{physics}. Since $J/\psi \phi$ is a vector-vector final state with
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an unknown mix of $CP$-even and $CP$-odd amplitudes the
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extraction of $CP$ asymmetries requires an angular analysis~\cite{angular}.
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The objective for the Summer'10 conferences is to observe the decay
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statistically significant and if data permits to extract the $B_s$ lifetime.
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These measurements do strongly depend on the secondary vertex reconstruction
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capabilities of the CMS pixel detector and hence serve as benchmark
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in direct comparison to measurements from the Tevatron~\cite{pdg}.
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As in the $B^0_d$ system, the average lifetime of $B_S$ is
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due to the $b$ quark; it is well measured by the CDF~($\tau=1.40\pm 0.14(stat)
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\pm 0.02(syst)\times 10^{-12}$~s)
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and D0 experiments~($\tau=1.444\pm 0.094(stat)\pm 0.020(syst)\times 10^{-12}$~s)
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at the Tevatron of Fermilab.
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\subsection{Strategy}
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We attempt to measure the cross section for the channel, total and
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as function of the transverse momentum, and the $B_s$ lifetime.
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Previous studies~\cite{lotte} indicate that the potential background from
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prompt $J/\psi$ and inclusive $b$ production can be
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identified and separated from the signal using the invariant $J/\psi\phi$
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mass and the proper decay length $ct$ of reconstructed $B_s$ candidates.
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We study these and QCD backgrounds with full detector Monte Carlo simulations
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corresponding to a luminosity at least comparable or exceeding the one
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expected for real data to develop our strategy.
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We propose two approaches: one based on optimized selection criteria
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for event variables (''cut-and-count'') and a two-dimensional maximum likelihood
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fit to the inavriant $J/\psi \phi$ mass and proper decay length $ct$ with relaxed
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requirements on event variables.
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We estimate of the number of $B_s\to J/\psi(\to\mu^+\mu^-) \phi(\to K^+ K^-))$
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events produced in proton-proton collisions at a center-of-mass (CM) energy of 7~TeV
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from the cross section for an integrated luminosity of 1~pb$^{-1}$ the following:
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\begin{eqnarray}
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& & \sigma(B_s\to J/\psi(\to \mu^+\mu^-) \phi(\to K^+ K^-)) = \nonumber \\
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& & \hspace*{4mm} \sigma(pp\to \bar{b}b)\cdot
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2 \cdot B(\bar{b}\to B_s)\cdot \Gamma(B_s\to J/\psi \phi)\cdot \Gamma(J/\psi\to\mu^+\mu^-)\cdot
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\Gamma(\phi\to K^+ K^-)
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\end{eqnarray}
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The value of total cross section for the production of $\bar{b}b$ pairs in proton-proton
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collisions at a CM energy of 7~TeV is expected to be approximately half of the value
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at 14~TeV, namely 250~$\mu$b~\cite{nason,campbell}. With the branching fractions~\cite{pdg}
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listed in Table~\ref{tab:pdgtab} we predict that about 2020 events are produced.
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\begin{table}[ht]
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\begin{center}
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\begin{tabular}{ll}
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\hline
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$B(\bar{b}\to B_s)$ & $(10.7\pm 1.2)$\% \\
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$\Gamma(B_s\to J/\psi \phi)$ & $(1.3\pm 0.4)\cdot 10^{-3}$ \\
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$\Gamma(J/\psi\to \mu^+\mu^-)$ & $(5.93\pm 0.06)$\% \\
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$\Gamma(\phi\to K^+ K^-)$ & $(48.9 \pm 0.5)$\% \\
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\hline
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\end{tabular}
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\label{tab:pdgtab}
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\caption{\sl Cross sections and branching fractions from PDG~\cite{pdg}.}
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\end{center}
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\end{table}
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We establish the efficiency for the trigger and the two reconstruction chains
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in this note. We find the product of both efficiencies to be about 2\% which
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translates into about 40 observed events for each pb$^{-1}$ of integrated luminosity.
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The luminosity is expected to be known to about 10\% which is comparable
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to the statistical error for the first 3~pb$^{-1}$.
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The largest uncertainties in the estimate above of up to 50\% are due to the unknown
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total $pp\to \bar{b}b$ cross section at 7~TeV and the transverse momentum distribution
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of the $b$ quarks~\cite{baines}.
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The blind analysis method has been widely used by the BaBar collaboration and
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experiments at the Tevatron to establish least biased searches, first or
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repeated ones~\cite{aaron}. For an integrated luminosity of 1~pb$^{-1}$
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we approach a possible signal significance of 5$\sigma$, which makes it
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by definition an observation, a first one at CMS, that corroborates this
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approach.
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\subsection{Implementation}
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We establish here the complete analysis with software releases and simulations
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available. We expect that the Summer'10 analysis is likely an update with somewhat
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improved code and newly generated events for signal and background as they
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become available, but methods are the same and efficiencies will not change by much.
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The studies presented here have been performed with analysis software release
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CMSSW 3.3.6~\cite{cmssw}. Simulations of $pp$ hard scatter use PYTHIA
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version 6.416~\cite{pythia}
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\subsection{To Do}
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For the Summer'10 analysis the following issues still need to be addressed:
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\begin{itemize}
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\item Reconstruction efficiencies with new software release CMSSW 3.6.x
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\item QCD background contribution with newly generated samples exceeding the
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expected luminosity in data
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\item
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\end{itemize}
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