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\Title{Mixing and CP Violation in the B System} 

\bigskip\bigskip

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

\begin{raggedright}  

{\it Sean Benson\index{Benson, S.} on behalf of LHCb collaboration\\
Department of Physics and Astronomy\\
University of Edinburgh\\
EH9 3JZ, United Kingdom}
\bigskip\bigskip
\end{raggedright}

The following sections summarize analyses of the measurement of the \ensuremath{CP}\xspace-violating weak phase $\phi_s$\index{$\phi_s$}\index{CP violation} in the 
\ensuremath{B^0_s \to J/\psi \phi}\xspace~\cite{LHCb-CONF-2012-002} and \ensuremath{B^0_s \to J/\psi \pi^+ \pi^-}\xspace decays~\cite{LHCb:2012ad} (including the resolution
of the ambiguity of $\phi_s$ associated with the sign of the decay width difference in the \ensuremath{B^0_s}\xspace system~\cite{Aaij:2012eq}). Also presented is the
time-integrated analysis of the $\ensuremath{B^0_s}\xspace \to (\phi \to KK) \; (\phi \to KK)$ decay\index{Penguin decays}\index{$B_s^0 \to \phi \phi$} including the \ensuremath{T}\xspace-violating triple product
asymmetries~\cite{Aaij:2012ud}.
All analyses discussed are based on the full 2011 dataset of 1.0~\ensuremath{\mbox{\,fb}^{-1}}\xspace collected
with the LHCb detector at centre-of-mass (COM) energy $\sqrt{s}=7$~TeV.

\section{Direct $\phi_s$ Measurements}
\subsection{The \ensuremath{B^0_s \to J/\psi \phi}\xspace Analysis Method}
\label{sec:jpsiphi}

The \ensuremath{B^0_s \to J/\psi \phi}\xspace decay is selected using a cut based method described in Aaij \textit{et al}. (2011)~\cite{LHCb-PAPER-2011-021}.
This results in $\sim 21200$ signal events with low background.
The decay time resolution of the \ensuremath{B^0_s \to J/\psi \phi}\xspace decay is accounted for in fitting through convolution
of the probably density function (PDF) with a Gaussian function of width $S_{\sigma_t} \cdot \sigma^i_t$,
where $\sigma^i_t$ is the event-by-event decay time resolution of the $i^{th}$ event (determined from 
vertex and decay length uncertainty); $S_{\sigma_t}$ is determined from prompt $\ensuremath{{J\mskip -3mu/\mskip -2mu\psi\mskip 2mu}}\xspace \to \mu^+ \mu^-$ events
to be $1.45 \pm 0.06$, where errors are both systematic (derived from simulation) and statistical.

Decay time acceptance effects due to time-biasing cuts used to select $\ensuremath{{J\mskip -3mu/\mskip -2mu\psi\mskip 2mu}}\xspace \to \mu^+ \mu^-$ events
are determined with the assistance of a prescaled, unbiased trigger. A small drop in acceptance is
also seen at longer lifetimes due to the lower track finding efficiencies associated with tracks from
vertices far from the beam line. A correction on \ensuremath{\Gamma_s}\xspace is found from simulation to be $0.0112 \pm 0.0013$~\ensuremath{{\rm \,ps^{-1}}}\xspace.
Half of this value is applied as a systematic uncertainty.

The efficiency of reconstructing a \ensuremath{B^0_s \to J/\psi \phi}\xspace event also depends on the decay angles in the transversity basis
(described in detail in Reference~\cite{LHCb-PAPER-2011-021}). The correction applied
in the fit is found using Monte Carlo \ensuremath{B^0_s \to J/\psi \phi}\xspace events. The difference in the spectra of kinematic
observables of the tracks in simulated events compared to that observed in the data in addition to the
limited quantity of simulated events are used to determine associated systematic uncertainties.

The sensitivity of the fit to the weak phase $\phi_s$ is greatly enhanced through the ability to 
determine the flavour of the \ensuremath{B^0_s}\xspace meson when it is produced. The methods of determination of the 
flavour and associated uncertainties are described in detail in Reference~\cite{LHCb-PAPER-2011-021}.

\subsection{The $\ensuremath{B^0_s}\xspace \to \ensuremath{{J\mskip -3mu/\mskip -2mu\psi\mskip 2mu}}\xspace \pi \pi$ Analysis Method}
\label{sec:jpsipipi}

The anaysis of the \ensuremath{B^0_s \to J/\psi \pi^+ \pi^-}\xspace decay~\cite{LHCb:2012ad} updates a previous study on the 
$\ensuremath{B^0_s}\xspace \to \ensuremath{{J\mskip -3mu/\mskip -2mu\psi\mskip 2mu}}\xspace f_0(980)$ decay~\cite{LHCb:2011ab} using the
fact that the $775 < m(\pi^+\pi^-) < 1500$~\ensuremath{{\mathrm{\,Me\kern -0.1em V\!/}c^2}}\xspace invariant mass range is 97.5\% \ensuremath{CP}\xspace-odd at 
95\% C.L.~\cite{LHCb:2012ae}.
This then allows for $\phi_s$ to be measured without the need to disentangle \ensuremath{CP}\xspace eigenstates.
As such, a fit to the decay time is sufficient to measure $\phi_s$. The tagging
method and time resolution methods are the same as those used for the \ensuremath{B^0_s \to J/\psi \phi}\xspace decay.

\subsection{Results}
\label{sec:resphis}

Both the \ensuremath{B^0_s \to J/\psi \phi}\xspace and \ensuremath{B^0_s \to J/\psi \pi^+ \pi^-}\xspace analyses utilize unbinned maximum log-likelihood
fitting methods in the measurement of the weak phase $\phi_s$. 
A number of physics parameters are measured at the same time as $\phi_s$ in the analysis of
the \ensuremath{B^0_s \to J/\psi \phi}\xspace decay. These are the decay width (\ensuremath{\Gamma_s}\xspace), the decay width difference between the 
two \ensuremath{B^0_s}\xspace mass eigenstates (\ensuremath{\Delta\Gamma_s}\xspace) and the polarization amplitudes of the P-wave (\ensuremath{|A_0|^2}\xspace, \ensuremath{|A_\parallel|^2}\xspace, \ensuremath{|A_\perp|^2}\xspace) and S-wave (\ensuremath{|A_S|^2}\xspace) contributions
along with corresponding phases\footnote{The convention has been chosen such that
$\ensuremath{\delta_0}\xspace \equiv 0$.} (\ensuremath{\delta_0}\xspace, \ensuremath{\delta_\parallel}\xspace, \ensuremath{\delta_\perp}\xspace, \ensuremath{\delta_S}\xspace) defined at $t=0$. Normalization is
chosen such that $\ensuremath{|A_0|^2}\xspace+\ensuremath{|A_\parallel|^2}\xspace+\ensuremath{|A_\perp|^2}\xspace=1$. In fits the \ensuremath{B^0_s}\xspace oscillation frequency \ensuremath{\Delta m_s}\xspace is constrained within
errors of the LHCb measured value~\cite{LHCb-PAPER-2011-010}.

\begin{table}[t]
\begin{center}
\begin{tabular} {l|ccc}
Parameter                  & Value & Stat. & Syst. \\ 
\hline
$\Gamma_s$ [ps$^{-1}$]        & 0.6580 & 0.0054 & 0.0066\\
$\Delta \Gamma_s$ [ps$^{-1}$] & 0.116 & 0.018 & 0.006\\
$|A_{\perp}(0)|^2$              & 0.246 & 0.010 & 0.013\\
$|A_{0}(0)|^2$                  & 0.523 & 0.007 & 0.024\\
$F_\mathrm{S}$                    & 0.022 & 0.012 & 0.007\\
$\delta_{\perp}$  [rad]         & 2.90  & 0.36  & 0.07\\
$\delta_{\parallel}$  [rad]         & \multicolumn{2}{c}{[2.81, 3.47]}  & 0.13\\
$\delta_s$     [rad]            & 2.90  & 0.36  &  0.08\\
$\phi_s$         [rad]          & -0.001  & 0.101  &  0.027\\ \hline
\end{tabular}
\end{center}
\caption{Results for the physics parameters and their
  statistical and systematic uncertainties. We quote a 68\% C.L. interval for $\delta_{\parallel}$, as described in the text. \label{tab:final+unc.}}
\end{table}

The results of the fit in the \ensuremath{B^0_s \to J/\psi \phi}\xspace decay are given in Table~\ref{tab:final+unc.}. The 68\% C.L. is quoted for \ensuremath{\delta_\parallel}\xspace
as the likelihood is not parabolic about the minimum for this parameter. This is due to the 
central value lying close to the ambiguous solution found through the transformation $\ensuremath{\delta_\parallel}\xspace \to -\ensuremath{\delta_\parallel}\xspace + 2\pi$.

In addition to 
the uncertainties discussed in Section~\ref{sec:jpsiphi}, the only other dominant contribution
is that of direct \ensuremath{CP}\xspace violation (DCPV), which is understood from simplified simulations. 
The uncertainties for tagging calibration, time resolution and \ensuremath{B^0_s}\xspace oscillation frequency are 
included in the fit using Gaussian constraints within their uncertainties.
Studies have shown that these inflate the statistical uncertainty on $\phi_s$ by no more than 5\%.

The result of the measurement of the weak phase $\phi_s$ in the \ensuremath{B^0_s \to J/\psi \pi^+ \pi^-}\xspace decay is found to
be $\phi_s=-0.02\pm0.17\pm0.02$~rad~\cite{LHCb:2012ad}. The systematic uncertainties 
arising from time resolution, time acceptance and tagging are treated in the same way as
in the analysis of the \ensuremath{B^0_s \to J/\psi \phi}\xspace decay.

Both the analysis of the \ensuremath{B^0_s \to J/\psi \phi}\xspace decay and the \ensuremath{B^0_s \to J/\psi \pi^+ \pi^-}\xspace decay contain an ambiguity
in the results associated with the transformations ($\phi_s \leftrightarrow \pi -\phi_s$ ; $\ensuremath{\Delta\Gamma_s}\xspace \leftrightarrow -\ensuremath{\Delta\Gamma_s}\xspace$)
and associated strong phase changes~\cite{Aaij:2012eq}. This ambiguity has been resolved through measuring
the difference in P-wave and S-wave strong phases in different $KK$ invariant mass bins.
Through the separation in to four bins chosen to have roughly equal numbers of events,
a negative trend of strong phase difference is observed with increasing $KK$ invariant mass with significance of 
$4.7\sigma$. This therefore implies that $\ensuremath{\Delta\Gamma_s}\xspace>0$, hence only this result has been quoted
throughout these Proceedings.

\section{Time-integrated Analysis of the \ensuremath{B^0_s \to \phi \phi}\xspace Decay}
\label{sec:phiphi}

The \ensuremath{B^0_s \to \phi \phi}\xspace decay is an example of a flavour changing
neutral current (FCNC) interaction and as such, may only proceed via penguin diagrams in the Standard Model.
A total of $801\pm29$ signal candidates are observed through a cut based selection optimized with the use
of the $_s\mathcal{P}lot$ method~\cite{Pivk:2004ty} to distinguish signal from background. 

The measurement of the polarization amplitudes (\ensuremath{|A_0|^2}\xspace, \ensuremath{|A_\parallel|^2}\xspace, \ensuremath{|A_\perp|^2}\xspace) and strong phase
difference (\ensuremath{\cos\delta_\parallel}\xspace) is performed using a time-integrated, untagged PDF under the assumption
that the time acceptance is uniform and that the \ensuremath{CP}\xspace-violating weak phase is zero.
A maximum log-likelihood fit is then performed to the three helicity angles (see Reference~\cite{Aaij:2012ud}
for more information). The lifetimes of the heavy and light \ensuremath{B^0_s}\xspace mass eigenstates are constrained to be within the errors of the
LHCb measured values~\cite{LHCb:2012ad} taking in to account correlations. S-wave contributions are ignored in the
fit. Data-driven methods indicate the S-wave contribution to be $(1\pm1)\%$, therefore systematic
uncertainties are based on a 2\% S-wave contribution. The angular acceptance is determined from
simulated events. The limited number of simulated events determines the systematic uncertainty due to the angular
acceptance. The time acceptance is understood from Monte Carlo events and simplified simulations
are used to assign a systematic uncertainty from the assumption that it is uniform. The other major source of
systematic uncertainty arises from the background model, where a background histogram from mass sidebands
(defined to be between 60-150~\ensuremath{{\mathrm{\,Me\kern -0.1em V\!/}c^2}}\xspace away from the measured \ensuremath{B^0_s}\xspace mass) is used instead of the nominal flat angular
background.
The polarization amplitudes and strong phase difference are measured to be
\begin{center}
\begin{tabular}{l@{$~=~$}r@{$\,\pm\,$}l@{$\,$(stat) 
      $\pm\,$}l@{$\,$(syst)}l@{\,}l}
$|A_0|^2$ & 0.365 & 0.022 & 0.012  & $\, ,$ \\
$|A_\perp|^2$  & 0.291 & 0.024 & 0.010 & $\, ,$  \\
$|A_\parallel|^2 $ & 0.344 & 0.024 & 0.014& $\, ,$ \\
$\cos(\delta_\parallel)$ &  $-$0.844 &  0.068 & 0.029 & $\, .$
\end{tabular}
\end{center}

Triple product asymmetries are based on T-odd observables U and V (defined in Reference~\cite{Aaij:2012ud}).
Events are separated in to datasets
according to whether $U(V)>0$ and a simultaneous fit is then performed to the obtain the asymmetries (\ensuremath{A_U}\xspace,\ensuremath{A_V}\xspace)
using the $KKKK$ invariant mass as the discriminating observable.

The main systematic uncertainties arise from the choice of signal and background model;
the effect of ignoring the time acceptance and the angular acceptance of the \ensuremath{B^0_s \to \phi \phi}\xspace decay.
The systematic uncertainties on the triple product asymmetries due to acceptance effects 
are estimated using simplified simulation studies (where both the time and angular acceptances
are understood from simulated events). 

Simultaneous fits to the U(V) datasets yield triple product asymmetries of
\begin{center}
\begin{tabular}{l@{$~=~$}r@{$\,\pm\,$}l@{$\,$(stat)
      $\pm\,$}l@{$\,$(syst)}l@{\,}l}
$A_U$ & $-$0.055 & 0.036 & 0.018 & $\, ,$ \\
$A_V$ & 0.010 & 0.036 & 0.018 & $\, .$
\end{tabular}
\end{center}

\section{Summary}
\label{sec:summary}

Direct measurements of the \ensuremath{CP}\xspace-violating weak phase have been measured
using the full 2011 dataset collected with the LHCb dectector at $\sqrt{s}=7$~TeV.
The combination of $\sim 21200$ \ensuremath{B^0_s \to J/\psi \phi}\xspace decays and $\sim 7420$ \ensuremath{B^0_s \to J/\psi \pi^+ \pi^-}\xspace decays yields a measurement
of $\phi_s = -0.002 \pm 0.083 (stat) \pm 0.027 (syst)$~rad. This therefore provides the world's 
most precise measurement of $\phi_s$. Also, it is worth mentioning that we observe the first 
measurement of \ensuremath{\Delta\Gamma_s}\xspace different from zero and have resolved the ambiguity in the $\phi_s - \ensuremath{\Delta\Gamma_s}\xspace$ plane,
i.e. that the heavy \ensuremath{B^0_s}\xspace mass eigenstate lives longer.

We provide the most accurate measurements of the physics parameters in the 
\ensuremath{B^0_s \to \phi \phi}\xspace penguin decay.
%, which are in agreement with those reported from QCD factorization
%methods~\cite{beneke,PhysRevD.80.114026}. 
The most precise measurements of \ensuremath{CP}\xspace violation 
in the \ensuremath{B^0_s \to \phi \phi}\xspace decay through triple product asymmetries is also reported.
% and is found to be in agreement
%with the SM expectation of \ensuremath{CP}\xspace conservation. 

\begin{thebibliography}{10}

\bibitem{LHCb-CONF-2012-002}
LHCb~collaboration
\newblock LHCb-CONF-2012-002 (2012).

\bibitem{LHCb:2012ad}
LHCb~collaboration
\newblock Phys. Lett. B713:378--386 (2012).

\bibitem{Aaij:2012eq}
LHCb~collaboration
\newblock Phys. Rev. Lett. 108:241801 (2012).

\bibitem{Aaij:2012ud}
LHCb~collaboration
\newblock Phys. Lett. B713:369--377 (2012).

\bibitem{LHCb-PAPER-2011-021}
LHCb~collaboration
\newblock Phys. Rev. Lett. 108:101803 (2012).

\bibitem{LHCb:2011ab}
LHCb~collaboration
\newblock Phys. Lett. B707:497--505 (2012).

\bibitem{LHCb:2012ae}
LHCb~collaboration
\newblock arXiv:1204.5643 (2012).

\bibitem{LHCb-PAPER-2011-010}
LHCb~collaboration
\newblock arXiv:1112.4311 (2011).

\bibitem{Pivk:2004ty}
M.~Pivk and F.~R.~Le~Diberder.
\newblock Nucl. Instrum. Meth. A555:356--369 (2005).

%\bibitem{beneke}
%M.~Beneke, J.~Rohrer, and D.~Yang.
%\newblock Nucl. Phys. B774:64--101 (2007).
%
%\bibitem{PhysRevD.80.114026}
%H.~Cheng and C.~Chua.
%\newblock Phys. Rev. D80:114026 (2009).

\end{thebibliography}

\end{document}

