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% 26.02.2007   27.02.2007   28.02.2007
% 01.03.2007   07.03.2007   27.03.2007
% 28.03.2007
%
% The united contribution by N.Nikitin, T.Buanes and K.Toms in the 
% "CERN Yellow Report" 

\begin{document}

\subsection*{The exclusive backgrounds for rare muonic B-decays}

In the current subsection we will give an overview of the backgrounds 
from exclusive $B$-decays with small branching ratios, and exotic decays 
with topologies similar to the signal events (so-called "the exclusive 
backgrounds"). They are not included in the standard Monte Carlo generators 
like PYTHIA or EvtGen\footnote{However these packages have tools to 
include some additional channels, for example in the phase space approach.}, 
thus they will not be included in the samples which are used to estimate 
the combinatorial background. These decays can contribute to the background 
in different ways. The exclusive background can produce from two-body 
hadronic $B$-decays which upon misidentification may give a peak in the 
di-muon invariant mass in the signal region. Then there are three- and 
four-body $B$-decays where a hadron is misidentified yielding a
invariant mass spectrum which due to resolution effects may leak into
the signal region. The last possibility is combinatorial background
with a muon from a rare $B$-decay together with a muon from the semimuonic
decays of the second $b$-quark\footnote{The current opportunity is 
especially significant for $B$-decays with three muons in the final
state. For example $B^+_c\to\mu^+\mu^-\mu^+\nu_{\mu}$.}.

Before applying the signal selection criteria the contribution from 
exclusive backgrounds is much smaller than the contribution from 
combinatorial background. But since the some of the exclusive decays 
have topologies which are similar to the signal, the background rejection
is expected to be less effective on this background. Thus a more thorough
investigation of these modes is necessary. Table \ref{tabl:exclbg}
presents the most important backgrounds from such exclusive decays. 

\begin{table}[th]
\begin{center}
\begin{tabular}{||l|l|c|c|c|c|c||}
\hline
\hline
N  & background Channel & BR Measurement      & \multicolumn{3}{|c|}{Experiment} & Ref. \\
\cline{4-7}
   &            & or SM Predictions &  ATLAS         & LHCb          & CMS     &      \\
\hline
1. & $B^0_{d,s}\to K\pi,\,\pi\pi,\, KK$ & $\sim 2\times 10^{-5}$  & Significant & \multicolumn{2}{|c|}{}  & \cite{pdg06} \\
\cline{1-4} \cline{7-7}
2. & $\Lambda_b\to p\pi^-$ & $< 5\times 10^{-5}$ 90\% C.L. &  \multicolumn{3}{|r|}{Not significant} & \cite{pdg06} \\
   & $\Lambda_b\to p K^- $ & $< 5\times 10^{-5}$ 90\% C.L. &  \multicolumn{3}{|c|}{}   & \cite{pdg06} \\
\hline
3. & $B^0_d\to\pi^-\mu^+\nu_{\mu}$ & $(6.8\pm 0.8)\times 10^{-5}$ & & \multicolumn{2}{|c|}{} & \cite{pdg06}\\
   & $B^0_s\to K^-\mu^+\nu_{\mu}$  & $\sim 7\times 10^{-5}$       & & \multicolumn{2}{|c|}{Not significant} & \cite{ms} \\
\cline{1-3} \cline{7-7}
4. & $B^+\to J/\psi \left( \mu^+\mu^- \right ) K^+$     & $\sim 6\times 10^{-5}$   & Potentially & \multicolumn{2}{|c|}{} & \cite{pdg06} \\
   & $B^+\to J/\psi \left( \mu^+\mu^- \right ) \pi^+$   & $\sim 0.3\times 10^{-5}$ & significant &\multicolumn{2}{|c|}{}  & \cite{pdg06} \\
   & $B_c^+\to J/\psi \left( \mu^+\mu^- \right ) \pi^+$ & $\sim 20\times 10^{-5}$ & & \multicolumn{2}{|c|}{}  & \cite{gklt} \\
\hline
5. & $B\to\left (\pi,\, K,\,\gamma\right )\mu^+\mu^-$         & $10^{-7} - 10^{-8}$    & \multicolumn{3}{|c|}{Not significant} & \cite{pdg06, b2gmm}\\
\hline
6. & $B^+_c\to J/\psi\left (\mu^+\mu^-\right )\mu^+\nu_{\mu}$ & $\sim 30\times 10^{-5}$ &  \multicolumn{2}{|l|}{Potentially} &  & \cite{atlas_note07}\\
   & $B^+_c\to\mu^+\mu^-\mu^+\nu_{\mu}$                       & $< 10\times 10^{-5}$            &  \multicolumn{2}{|l|}{significant} &  & \cite{atlas_note07}\\
\cline{1-3} \cline{5-5} \cline{7-7}
7. & $B^+\to\mu^+\mu^-\mu^+\nu_{\mu}$                         & $\sim 0.1\times 10^{-5}$         &  &  \multicolumn{2}{|c|}{Not significant}  & \cite{atlas_note07}\\
\hline
\hline
\end{tabular}
\end{center}
\caption{The most important backgrounds from rare $B$-decays.} 
\label{tabl:exclbg} 
\end{table}

Let's consider an exclusive background from two-body hadronic decays 
$B^0_{d,s}\to K\pi,\,\pi\pi,\, KK$. If the pions/kaons are misidentified 
as muons these decays will give a peaked background in or near the signal 
region. Since $K$-mesons are heavier than $\mu$, the peak will be shifted 
towards lower masses, and the invariant mass resolution of the detector will
decide whether these channels will contribute to the background or
not. CMS and LHCb have resolution $\sigma^{CMS}_{B\to\mu\mu} = 36$ MeV
\cite{cms_confrep07} and $\sigma^{LHCb}_{B\to\mu\mu} = 18$ MeV
\cite{lhcb_note07}, respectively, which is expected to be good enough to make
this effect negligible. On the fig. \ref{fig:noncombBG} c) is shown the 
invariant misidentifined kaons mass distribution for the decay 
$B^0_s\to K^+K^-$ and dimuon mass distribution for $B^0_s\to\mu^+\mu^-$ 
signal at the CMS detector simulation \cite{cms_note07} confirming the 
above assertion. ATLAS has resolution 
$\sigma^{ATLAS}_{B\to\mu\mu}\approx 80$ MeV \cite{atlas_dc1} which is
probably not good enough to render this background source negligible. 
But for the case of the decay in flight (when both final states hadrons 
decay to muons), the rate can be reduced somewhat by a strict requirement 
on the muon track fit quality and on the matching of inner-detector and 
muon spectrometer track in ATLAS. 

The exclusive background from two-body hadronic $\Lambda_b$-decays are
negligible small for all detectors \cite{lhcb_note07,cms_note07}. For
these decays we can take into account the effective proton identification 
and large mass shift. In addition $\Lambda_b$-barions's production 
cross-section is order times smaller than the production cross-section of 
$B$-mesons at LHC energies. 

The di-muon invariant mass spectrum form the three- and four-body
decays of $B^0_{d,s}$- and $B^{\pm}$-mesons have kinematic end-points 
which are close to the $B_s^0$ signal window, and due to limited resolution 
some events may leak into the window. In particular the channels 
$B^0_d\to\pi^-\mu^+\nu_{\mu}$ and $B^0_s\to K^-\mu^+\nu_{\mu}$ should be 
studied more closely as there is no third charged lepton coming from the 
secondary vertex, and thus the isolation cut will not reduce this background. 
CMS and LHCb have concluded that their contribution is negligible 
because of high mass resolution \cite{cms_note07}, whereas full ATLAS 
simulation results pointed on the small, but non-negligble contribution 
for three-body decays. On The fig. \ref{fig:noncombBG} a) (CMS full
detector simulation results) and b) (ATLAS full detector simulation results) 
we can see the illustration of this assertion for $B^0_s\to K^-\mu^+\nu_{\mu}$ 
decay.

The four-muonic resonant and nonresonat decays of the $B^+_c$-meson can be 
produce the potentially important source of background for all detectors 
\cite{lhcb_note07,cms_note07,atlas_note07} through the mass of $B^+_c$ is 
larger than the masses of $B^0_{d,s}$-mesons. For the LHCb after full detector
simulation and all necessary cuts applied in the mass window $\pm 60$ MeV
nearly $B^0_s$-meson mass this background is approximatly equal to
80 events per one fb$^{-1}$. It is at least two times greater than the 
signal from $B^0_s\to\mu^+\mu^-$ in this area. However this decay produce 
only small contribution to the inclusive background in the signal region 
\cite{lhcb_note07}. For CMS detector because of the kinematical reasons 
these decays do not give the contribution in the signal region 
\cite{cms_note07}. 

% For all channels there is also the possibility of contribution to the
% combinatorial background. This contribution should be treated as any
% other combinatorial background. 

Rare muonic radiative and semi-muonic decays as a background for rare
muonic decays based on the ATLAS detector properties \cite{atlas_note07} 
has also been studied. 
% The ATLAS Electromagnetic Calorimeter detects 
% photons with $p_T(\gamma)>4$ GeV. If $p_T(\gamma)$ is less than this, 
% $B^0_{d,s}\to\mu^+\mu^-\gamma$ decays can produce potential background 
% for signal decays. In order to check this possibility a generator level 
% simulation using the theoretical matrix element \cite{b2gmm} was performed. 
The results of this study using the theoretical matrix element \cite{b2gmm}  
are shown in Figure \ref{fig:noncombBG}d). As can be seen from the figure, 
the background contribution from these channels is small. For rare semi-muonic 
decays we find the same result.


\begin{figure}
\begin{center}
\begin{tabular}{cc}
\mbox{\epsfig{file=bs2kmunu.eps,width=7.cm}} &
\mbox{\epsfig{file=bs2kmunu.atlas.eps,width=7.cm}} \\
\mbox{\epsfig{file=bs2kk.eps,width=7.cm}} &
\mbox{\epsfig{file=b2gmm.eps,width=7.cm}} 
\end{tabular}
\caption{\label{fig:noncombBG} ...
        }
\end{center}
\end{figure}

\begin{thebibliography}{99}

\bibitem{pdg06} PDG06
\bibitem{ms} D. Melikhov, B. Stech, Phys. Rev. D{\bf 62} (2000) 014006.
\bibitem{gklt} S. S. Gershtein, V. V. Kiselev, A. K. Likhoded and 
               A. V. Tkabladze, Phys.Usp.{\bf 38} (1995) pp.1-37.
\bibitem{atlas_dc1} N.Benekos et al. (ATLAS B-Physics Group), ATL-PHYS-2005-002.
\bibitem{lhcb_note07}   LHCb Note in preparation.
\bibitem{cms_confrep07} U.Langenegger, CMS CR 2006/071.
\bibitem{cms_note07}    CMS Note in preparation.
\bibitem{atlas_note07}  ATLAS BG Internal (!) Note.
\bibitem{atlasTDR14}    ATLAS TDR 14, CERN/LHCC/99-14.
\bibitem{b2gmm}    D. Melikhov, N. Nikitin, Phys. Rev. D \textbf{70}, 
                   114028 (2004);
                   D.~I.~Melikhov, N.~V.~Nikitin and K.~S.~Toms, 
                   Phys. At. Nucl. \textbf{68} (2005) 1842.
\end{thebibliography}



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