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\Title{Time integrated ratio of wrong-sign to right-sign $D^0 \rightarrow K\pi$ decays in 2010 data at LHCb}

\bigskip\bigskip

%+\addtocontents{toc}{{\it D. Reggiano}}
%+\label{ReggianoStart}

\begin{raggedright}  

{\it Martin Bessner for the LHCb collaboration}
\bigskip\bigskip
\end{raggedright}


\section{Introduction}

The LHCb experiment~\cite{bib:LHCb} is a single-arm forward spectrometer
well suited to study the 
forward production of heavy quarks ($c,b$) at the LHC.

Mixing in the D meson system is well established~\cite{bib:HFAG}~\cite{bib:pdg10}
and the combination of several
measurements~\cite{bib:d0discover} yields a 
significance for $D^0$ mixing of 10 $\sigma$,
but none of the existing single measurements exceeds
5 $\sigma$ significance.
The time dependent analysis
of {\it wrong-sign} $D^{0} \rightarrow K^{+} \pi^{-}$ decays
provides good sensitivity for mixing.

In 2010, LHCb recorded a data sample corresponding to an integrated luminosity 
of  $\L = (36.4 \pm 3.6) \, \rm{pb}^{-1}$.
These data
are not sufficient to perform a time dependent measurement. However,
the extraction of the time integrated 
ratio of wrong-sign to right-sign $D^0 \rightarrow K\pi$ decays is feasible
and presented here.


\section{Selection}

\label{sec:selection}
For the measurement, $D^0$ mesons from the reconstructed 
decay $D^{*+} \rightarrow D^0 \pi^{+}_{\rm{tag}}$
are used\footnote{Unless otherwise stated, charge conjugate decay modes are implied throughout.}, where the charge of the $\pi_{\rm{tag}}$
allows to determine
the production flavor of the $D^0$-meson.  
For the $D^0 \rightarrow K \pi$ decays,
the charge of the $K$ classifies the decay mode
as right-sign (RS) $D^0 \rightarrow K^- \pi^+$ or
as wrong-sign (WS) $D^0 \rightarrow K^+ \pi^-$ decay.
The wrong-sign decays include contributions from DCS 
decays and from mixing.

The data selection is based on vertex and track
properties, on particle identification in the RICH detectors, and
on kinematics.
WS and RS events are kinematically identical. Therefore, their background shapes are similar.
The main background contribution arises from
adding an unrelated pion
to a correct $D^0$ candidate to form a fake $D^*$ 
(random tag pion ($\pi_{\rm{tag}}$) background).
In total, $28.7\cdot 10^4$ RS and $3.5 \cdot 10^4$ WS $D^0 \rightarrow K \pi$  decay candidates are observed.

For the determination of the signal yields, extended unbinned maximum
likelihood fits within the RooFit toolkit~\cite{bib:roofit} 
of the two dimensional event distribution in the
plane ($m(D^*)-m(D^0), m(D^{0})$) are used.



\section{Results}
\label{sec:wsoverrs}

\subsection{Ratio of wrong-sign to right-sign decays}
\label{subsec:ratio}
The signal yields
from the fits
are taken to compute the 
ratio of WS to RS decays,
$R_{{\rm meas}} = N_{\mathrm{WS}} / N_{\mathrm{RS}} = (0.442 \pm 0.033) \% $.
The error is  statistical only.

In the ratio, asymmetries
of the production rate and detection and trigger efficiencies
cancel to first order and can be neglected.
The relative amount of charm from $B$ decays is assumed to be the same in
WS and RS decays and cancels in the ratio. 

For systematic uncertainties, contributions due to the
 fit model and the choice of
the signal range in $m(D^*)-m(D^0)$
are considered and account for a combined uncertainty of
$\delta R_{{\rm sys.}} = 0.042 \%$.
The uncertainty from double particle  
misidentification is negligible
compared to the other systematic uncertainties.


\subsection{Correction to the wrong-sign to right-sign ratio}
\label{sec:corwsoverrs}
Due to mixing the ratio of WS to RS
decays depends on the $D^0$ decay time.
Therefore, a decay time acceptance function
$\epsilon(t)$ changes the time integrated 
value of the WS to RS ratio.
In order to determine the true ratio
a correction factor has to be applied.

The decay time acceptance is determined from data.
The mixing parameters needed to calculate the correction factor
are taken from HFAG~\cite{bib:HFAG}. 
The correction factor is calculated to be 
$c_{{\rm acc}} = 0.926$ 
and varies with the mixing parameters.
Its uncertainty
is neglected since this is an uncertainty on a correction of $7.4~\%$.


\begin{table}[htb]
\begin{center}
\begin{tabular}{|l|c|}
\hline
& WS/RS of $D\to K\pi$ decays ( \% ) \\ \hline
 $R_{{\rm meas}} $  &   $0.442 \pm 0.033 \; (\rm{stat.})\; \pm 0.042 \; (\rm{sys.})  $  \\ \hline
 $R_{{\rm corr}} $ &  $0.409 \pm 0.031 \; (\rm{stat.})\; \pm 0.039 \; (\rm{sys.})  $ \\ \hline
 $R(\mathrm{PDG})$ & $0.380 \pm 0.018$ \\ \hline
\end{tabular}
\caption{Measured and decay time 
 acceptance corrected ratio of WS to RS
 $D^0 \rightarrow K \pi$ decays.
The last line gives
 the world average.~\cite{bib:pdg10}}
\label{tab:FinalRatioWS2RS}
\end{center}
\end{table}


\section{Summary}
The acceptance corrected value ($R_{{\rm corr}}$) agrees very well with the
world average
%published by PDG~\cite{bib:pdg10}
as shown in 
Table~\ref{tab:FinalRatioWS2RS}.
This demonstrates that 
the signal composition in
data is well understood 
and that this analysis is a 
first step towards a time dependent 
analysis which will allow to extract the mixing parameters $ R_{{\rm D}}, x'^2, y'$.
Due to improvements of the trigger settings,
%and the analysis
the systematic error is expected to reduce significantly with 2011 data.~\cite{bib:lhcbconf}


\begin{thebibliography}{99}

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\bibitem{bib:LHCb} The LHCb collaboration, Journal of Instrumentation (JINST)
  {\bf 3}, S08005 (2008).

\bibitem{bib:HFAG}
http://www.slac.stanford.edu/xorg/hfag/charm/index.html; \newline
D.~Asner {\it et al.} (Heavy Flavor Averaging Group),
[arXiv:1010.1589v2].

\bibitem{bib:pdg10} K. Nakamura {\it et al.} (Particle Data Group), 
J. Phys. G {\bf 37}, 075021 (2010).

\bibitem{bib:d0discover}
B. Aubert et al.  (BABAR Collab.), Phys. Rev.  Lett. {\bf 98}, 211802 (2007);\newline
T. Aaltonen et al.  (CDF Collab.), Phys. Rev. Lett. {\bf 100}, 121802 (2008);\newline
M. Staric et al. (Belle Collab.), Phys. Rev. Lett. {\bf 98}, 211803 (2007).

\bibitem{bib:roofit}
W. Verkerke and D.P. Kirkby, [arXiv:physics/0306116].

\bibitem{bib:lhcbconf}
[LHCb Collaboration], LHCb-CONF-2011-029

\end{thebibliography}
 
\end{document}

