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\clearpage
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\section{Pre-Selection of \boldmath{$B_s$} Candidates}
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%The $B_s$ candidates are reconstructed by identifying $J/ \psi$ and
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%$\phi$ candidates.
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% --> at what level in CMSSW; what candidate lists are used
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We start by
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reconstructing $J/ \psi$ candidates from two oppositely charged muons
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that pass the quality requirements
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\verb,TM2DCompatibilityTight,\cite{cmssw}
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and are each associated with a track in the silicon tracker
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(\verb,GlobalMuon, or \verb,TrackerMuon,).
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Furthermore, each pair of muons is required to
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%fit to a common vertex and simultaneously
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to fulfill energy and momentum conservation . %~\cite{kinfit}.
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% with a probability of greater than 1% ???
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We keep all $J/ \psi$ candidates with $p_T>0.5$~GeV/c
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and an invariant mass within $150$~MeV/$c^2$ to the
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world average value~\cite{pdg}.
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Candidate $\phi$ mesons are reconstructed from pairs of oppositely
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charged tracks with $p_T>0.5$ GeV/c and that are selected from a
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sample with the above muon candidate tracks removed. We assume each
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track to be a kaon and calculate the invariant mass of the track pair.
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We keep combinations with an invariant mass within
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$50$~MeV/$c^2$ to the world average~\cite{pdg}.
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We combine a $J/ \psi$ and a $\phi$ candidate to form a
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$B_s$ candidate and require that the invariant mass lies between
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$4.5$~GeV/$c^2$ and $6$~GeV/$c^2$.
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Those candidates are subjected to a kinematic fit~\cite{kinfit}
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that requires momentum and energy conservation in the $B_s$ decay
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and both decay products to originate from a common vertex.
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It does fit the $J/\psi$ decay and
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the dimuon pair is constrained to the nominal $J/ \psi$ mass value.
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We select one $B_s$ candidate per event based on the best vertex
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fit probablility.
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Furthermore, the fit $B_s$ candidate has to lie
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within the mass range $5.2 < M_{B_s} < 5.7$ GeV/$c^2$.
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This selection identifies the correct $B_s\rightarrow J/ \psi \phi$
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candidate in $99 \%$ of all cases as determined from the
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associated 'true' signal decay at generator level (Monte Carlo Truth
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Matching). Fig.~\ref{fig:massBA} shows the invariant $J/\psi \phi$
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mass distribution before and after application of the kinematic fit.
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\begin{figure}[h]
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\vspace{0.5cm}
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\centering
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\includegraphics[scale=0.4]{figure/BsBAfit.eps}
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\caption{$B_s$ invariant mass before (black) and after (red) the
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kinematic fit.}
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\label{fig:massBA}
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\end{figure}
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% The decay $B_d\rightarrow J/ \psi K^*$, $J/\psi\to \mu^+\mu^-$
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% and $K^{*0}\to K^+\pi^-$ with an expected production rate about
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% 7 times larger than our signal channel serves as control channel.
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% The $B_d$ candidates are reconstructed from $J/ \psi$ and $K^*$
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% candidates. The $J/\psi$ candidate reconstruction and selection
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% is the same as for our signal channel.
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% Candidate $K^*$ mesons are reconstructed from pairs of oppositely
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% charged tracks having $p_T>0.5$ GeV/c. We calculate the
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% invariant mass for $K^*$ and we choose the candidates with invariant masses
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% within $200$ MeV/$c^2$ of the world average. $B^0$ candidates, reconstructed
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% by combining a $J/ \psi$ candidate with a $K^*$ candidate are required to have
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% invariant mass between $4.5$ GeV/$c^2$ before performing a kinematic fit using
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% the same method as in the $B_s$ reconstruction. We select one $B^0$ candidate
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% per event based on the best vertex probability and use only B candidates in the
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% mass region $4.9 < M_B < 5.7$ GeV/$c^2$ for final analysis.
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We calculate the two-dimensional proper decay length $c t$ for the
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reconstructed $B$ candidates from the measured distance $L_{xy}$ between
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the production (or primary) and $B$-decay vertices projected onto the
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transverse momentum, and the relativistic boost of the reconstructed
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$B$ meson in the transverse plane the following:
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\begin{equation}
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c t = \frac{M_B}{p^B_T}L_{xy}\, ,
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\end{equation}
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where $M_B$ and $p^B_T$ are the mass and transverse momentum of the B candidate.
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The transverse flight length $L_{xy}$ is the projection of the vector $\vec{s}$
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pointing from the primary to the secondary vertex to the transverse momentum:
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\begin{equation}
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L_{xy} = \frac{\vec{s}\cdot \vec{p}^B_T}{ |\vec{p}^B_T|}
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\end{equation}
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\begin{figure}[h]
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\vspace{0.5cm}
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\centering
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\includegraphics[scale=0.4]{figure/ctauresidual.eps}
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\caption{Proper decay length residual distribution in truth-matched signal events.}
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\label{fig:res}
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\end{figure}
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The generated value of the mean proper decay length $c\tau=423$ $\mu$m.
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The single-Gaussian width of the distribution of the difference in $c t$ between
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reconstructed and generated values as determined by truth-matching
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for signal events (residual distribution)
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is approximately 50~$\mu$m. The distribution is shown in Fig.~\ref{fig:res}.
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