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\newcommand{\dzer}         {D\O}
\newcommand{\dmes}       {\ensuremath{ D}}
\newcommand{\ad}       {\ensuremath{\bar{ D}}}
\newcommand{\dzero}     {\ensuremath{ D^0}}
\newcommand{\adzero}     {\ensuremath{\bar{ D}^0}}
\newcommand{\dst}       {\ensuremath{ D^{*}}}
\newcommand{\adst}       {\ensuremath{\bar{ D}^{*}}}
\newcommand{\dstzero}   {\ensuremath{ D^{*0}}}
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\newcommand{\dstminus}  {\ensuremath{ D^{*-}}}
\newcommand{\dststzero} {\ensuremath{ D^{**0}}}
\newcommand{\adststzero} {\ensuremath{\bar{ D}^{**0}}}
\newcommand{\dstst}     {\ensuremath{ D^{**}}}
\newcommand{\adstst}     {\ensuremath{\bar{ D}^{**}}}
\newcommand{\ds}        {\ensuremath{{D_{s}^{-}}}}
\newcommand{\ads}        {\ensuremath{{D_{s}^{+}}}}
\newcommand{\dmd}       {\ensuremath{{\Delta {m}_{d}}}}
\newcommand{\dms}       {\ensuremath{{\Delta {m}_{s}}}}
\newcommand{\bzerod}    {\ensuremath{ B^{0}_d}}
\newcommand{\bplus}     {\ensuremath{ B^{+}}}
\newcommand{\bs}        {\ensuremath{{B_{s}^{0}}}}
\newcommand{\bsantibs}  {\ensuremath{{B_{s}^{0}-\bar{B_{s}^{0}}}}}
\newcommand{\antibs}        {\ensuremath{{\bar{B_{s}^{0}}}}}
\newcommand{\dstophipi}   {\ensuremath{{D_{s} \rightarrow \phi \pi}}}
\newcommand{\bstodsmunux} {\ensuremath{{B_{s}^{0} \to D_{s}^{-} \mu^+ \nu X}}}
\newcommand{\brbtodstst} {\ensuremath{{BR (\bar{B} \rightarrow D^{**}\ell \bar{\
nu} X)}}}
\newcommand{\bddx}      {\ensuremath{{BR(b\rightarrow D \bar{D}X)}}}
\newcommand{\bdd}       {\ensuremath{{b\rightarrow D \bar{D}X}}}
\newcommand{\bd}        {\ensuremath{{b\rightarrow \bar{D}X}}}
\newcommand{\bccx}      {\ensuremath{{BR(b\rightarrow \psi X)}}}
\newcommand{\bzc}       {\ensuremath{{b \rightarrow 0 \; charm}}}
\newcommand{\bnocharm}  {\ensuremath{{BR(b \rightarrow 0 \; c)}}}
\newcommand{\lnpj}      {\ensuremath{{-ln}(P_j)}}
\newcommand{\phitokk}      {\ensuremath{{\phi \rightarrow  KK}}}
\newcommand{\nc}        {\ensuremath{n_c}}
\newcommand{\ee}        {\ensuremath{{e}^+{e}^-}}
\newcommand{\rphi}      {\ensuremath{{{r}-{\phi}}}}
\newcommand{\EV}        {\ensuremath{{eV}}}
\newcommand{\KV}        {\ensuremath{{keV}}}
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\newcommand{\dedx}      {\ensuremath{{\rm d}E/{\rm d}x}}
\newcommand{\lumi}      {\ensuremath{\mathcal{L}}}
\newcommand{\PM}        {\ensuremath{\pm}}
\newcommand{\bsl}       {\ensuremath{{Br(B \rightarrow \ell \nu X)}}}
\newcommand{\zbb}       {\ensuremath{{Z^0} \rightarrow b \bar{b} \;}}
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\title{$B_{s}$ Mixing at D\O\ }% Version 0.4~\today }

\author{J.~Walder\address{Lancaster
University, Department 
of Physics,  Lancaster, LA1 4YB, UK} on behalf of the D\O\ Collaboration}
       
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%\runtitle{2-column format camera-ready paper in \LaTeX}
%\runauthor{S. Pepping}

\begin{document}

\begin{abstract}
The first direct two-sided bound by a single experiment  on the $\bs$ oscillation
frequency is reported using a sample of semileptonic decays collected 
between 2002 -- 2006 by the Run IIa  D\O\ detector at Fermilab,  corresponding to
approximately $1\,$fb$^{-1}$
of integrated luminosity.  The most probable value of the oscillation
frequency $\Delta m_{s}$ is found from a likelihood scan to be $19$~ps$^{-1}$
and within the range $17 < \Delta m_{s}<21$~ps$^{-1}$ at the $90\%$ C.L. At the
preferred value of $19$~ps$^{-1}$ there is a $2.5\sigma$ deviation from a zero
amplitude hypothesis.
\vspace{1pc}
\end{abstract}

% typeset front matter (including abstract)
\maketitle

%INTRODUCTION
%\section{Introduction\label{intro}}
This proceedings reports on an analysis of Bs mixing recently published
in~\cite{bsmixing}.  The
phenomenon of particle anti-particle oscillations (or mixing)  have
provided
insights into energy scales that had not yet been accessible. For example mixing
in the neutral kaon system led to the prediction of a third flavour
generation~\cite{km}, and oscillations in the $B^{0}_{d}$ system gave
predictions of the top quark mass~\cite{top}.
Measuring the oscillation of the $\bs$ mixing frequency places a 
constraint on the magnitude of the CP violating top quark coupling from the
ratio $|V_{td}/V_{ts}|$ and will perhaps yield a new physics discovery in $b\to
s$ transitions~\cite{bs}. 
Prior to this analysis, and assuming the Standard Model (SM) is correct, global
fits to the unitarity triangle favoured
 $\Delta m_s = 20.9 ^{+4.5}_{-4.2}$~ps$^{-1}$~\cite{ckm_fit}.
This analysis was performed using a data sample of semileptonic
$\bs$ decays collected with the D\O\ detector at Fermilab using $p\bar{p}$  
collisions at $\sqrt{s}=1.96$~TeV corresponding to an integrated
luminosity of approximately $1$~fb$^{-1}$.

%THEORY
%\section{Theory of $\bs$ Mixing}
The $\bs$ system can be described by the matrix evolution equation:
\begin{eqnarray}
i \!\frac{d}{dt} \!
\left( \!\!\begin{array}{c} B^0_s \\ \bar{B}^0_s
       \end{array}  \!\!\right)\!\!
       =\!\!
\left( \!\!\begin{array}{cc}
       M - \frac{i \Gamma}{2} & M_{12} - \frac{i \Gamma_{12}}{2} \\
       M_{12}^* - \frac{i \Gamma_{12}^*}{2} & M - \frac{i \Gamma}{2}
       \end{array} \!\!\right)\!
\left( \!\!\begin{array}{c} B^0_s \\ \bar{B}^0_s
              \end{array} \!\! \right).\nonumber
\label{eqn:mxingmat}
\end{eqnarray}
The two mass eigenstates differ from the flavour eigenstates and are defined
as the eigenvectors of the above matrix.  The heavy~(H) and light~(L) mass 
eigenstates are given by $|B_{s}^{H} \rangle = p | B^0_s \rangle + q |
\bar{B}^0_s\rangle$, $|B_{s}^{L} \rangle = p | B^0_s \rangle - q | \bar{B}^0_s
\rangle$,
%$%\begin{eqnarray}
%|B_{s}^{H} \rangle &=& p | B^0_s \rangle + q | \bar{B}^0_s
%\rangle;%\nonumber\\
%|B_{s}^{L} \rangle &=& p | B^0_s \rangle - q | \bar{B}^0_s \rangle,
%%\label{eqn:blh}\nonumber
%$%\end{eqnarray}
where $|p|^{2}+|q|^{2}=1$.
Denoting $\Delta m_{s} = M_{H} -
M_{L}$, $\Delta \Gamma_{s} = \Gamma_{L}-\Gamma_{H}$, $\Gamma = (\Gamma_{H} +
\Gamma_{L})/2$, the probability for an initial $\bs$ meson at production to
oscillate into a $\antibs$ (or vice-versa) at time $t$  is given by $P^{\rm
osc}= \Gamma{\rm e}^{-\Gamma t}(1-\cos\Delta m_{s}t)/2$, or to not oscillate
with probability $P^{\rm
nosc}= \Gamma{\rm e}^{-\Gamma t}(1+\cos\Delta m_{s}t)/2$, assuming
$\Delta\Gamma_{s}/\Gamma_{s}$ is small and neglecting CP violation.

%\section{The D\O~ Detector}
%Detector
The D\O\ detector is a general purpose spectrometer and
calorimeter~\cite{dzerodet}.
The significant components for this analysis are the muon chambers,
calorimeters and central
tracking region. Enclosed within a 2 T superconducting solenoid is a silicon
micostrip tracker (SMT) and central fiber tracker (CFT) for vertexing and
tracking of charged particles that extends out to a pseudorapidity of $|\eta|=2.0$,
$\eta=-\ln[\tan(\theta/2)]$, where $\theta$ is the polar angle.
The three liquid-argon/uranium calorimeters provide coverage up to
$|\eta|\approx4.0$.  The muon system consists of one tracking layer and
scintillation trigger counters in front of 1.8~T iron toroids with two layers
after the toroids.  Coverage extends to  $|\eta|=2.0$.
%trigger
There a no explicit trigger requirements used in this analysis, however most
events were collected using single-muon triggers.


%SELECTION
%\section{Event Selection}
$\bs$ hadrons are selected using the semileptonic decay\footnote{Charge
conjugate states are implied throughout.}
$\bs \to \mu^{+}\nu\ds X$, where $\ds \to \phi \pi^{-}$,  $\phi \to K^{+}K^{-}$.
The muon required a transverse momentum $\pt(\mu^{+}) > 2$~GeV$/c$, $p(\mu^{+})>3$~GeV$/c$,
and to have a signal in at least two of
layers of the muon system.  All charged tracks in the event are required to have
at least two signals in both the  CFT and SMT and are clustered into jets~\cite{durham}.
The $\ds$ candidate is reconstructed from three charged tracks in the same
jet as the muon.  Two oppositely charged particles with $\pt>0.7$~GeV$/c$ are
assigned the mass of a kaon, and required to have an invariant mass $1.004 < M(K^+ K^-) <
1.034$~GeV$/c^2$, consistent
with a $\phi$ meson.    The third track with charge opposite to that of the
muon, and $\pt>0.5$~GeV$/c$ was assigned the mass of a pion.  The three tracks
are combined to form a common $\ds$ vertex as described in Ref.~\cite{vertex}.
This vertex was required to have a positive displacement relative to the
$p\bar{p}$ collision point (PV), with a significance of at least $4\sigma$.
and $\cos(\alpha)>0.9$, where $\alpha$ is the angle between the $\ds$ momentum
and the direction from the PV to the $\ds$ vertex.  The muon and $\ds$
candidates are required to originate from a common $\bs$ vertex and have an
invariant mass of the $\mu^{+}\ds$ system between 2.6 and 5.4~GeV$/c^2$.

%Likelihood ratio.
A likelihood ratio method~\cite{like_ratio} was utilised to increase the $\bs$ selection
efficiency using the discriminating variables: 
the helicity angle between the
$D_s^-$ and $K^{\pm}$ momenta in the $\phi$ center-of-mass frame;
isolation of the $\mu^+ D_s^-$ system;
 $\chi^2$ of the $D_s^-$ vertex;
invariant masses $M(\mu^+ D_s^-)$ and $M(K^+ K^-)$;
and  transverse momentum $\pt(K^+ K^-)$.
Sideband ($B$) and sideband-subtracted signal ($S$) $M(K^+K^-\pi)$ data
distributions were used to construct the probability distribution functions
({\sl pdf}s) for the discriminants.  The combined likelihood selection
variable was defined to maximise the predicted ratio $S\sqrt{S+B}$.

Following these requirements the number of $\ds$ candidates was 
$N_{\mathrm{tot}}
= 26,\!710 \pm 556 \thinspace \mathrm{(stat)}$, as shown 
in Fig.~\ref{prl_fig1}(a).

\begin{figure}
%\includegraphics[width=0.48\textwidth]{prl_fig1.eps}
\includegraphics[width=\columnwidth]{prl_fig1.eps}
\vspace{-1.2cm}
%\epsfig{file=prl_fig1.eps}
\caption{\label{prl_fig1} 
$(K^+K^-)\pi^-$ invariant mass distribution for
(a) the untagged $B^0_s$ sample, and (b) for candidates
that have been flavour-tagged.  The left and right peaks correspond to
$\mu^+ D^{-}$ and $\mu^+ D^-_{s}$ candidates, respectively. The curve
is a result of fitting a signal plus background model to the data. 
}
\end{figure}

%\section{Initial State Flavour Tagging\label{OST}}
The flavour of the signal $\bs$ meson at production was determined using 
a likelihood ratio method using properties of the opposite-side b-hadron produced
in the event.  
%direct from prl
The performance of the opposite-side
flavour tagger (OST)~\cite{OST} is
characterized by the efficiency
$\epsilon = N_{\mathrm{tag}}/N_{\mathrm{tot}}$, where 
$N_{\mathrm{tag}}$ is the number of tagged $B^0_s$ mesons;
tag purity $\eta_s$, defined as $\eta_s = N_{\mathrm{cor}}/N_{\mathrm{tag}}$, where
$N_{\mathrm{cor}}$ is the number of $B^0_s$ mesons with correct
flavour identification; and the dilution $\mathcal{D}$, related to purity as 
$\mathcal{D} \equiv 2\eta_s -1$.
A reconstructed secondary vertex or lepton $\ell$ (electron or muon) was
defined to be on the opposite side of the $\bs$ meson if 
$\cos\varphi(\vec{p}_{\ell~{\mathrm{or~SV}}}, \vec{p}_B) < 0.8$,
where $\vec{p}_B$ is the reconstructed three-momentum of the 
$B^0_s$ meson, and $\varphi$ is the azimuthal angle about the beam axis.  
%In the construction of the flavour discriminating variables $x_1,
%..., x_n$ for each event,
%an object, either a lepton $\ell$ (electron or muon) or a reconstructed
%secondary vertex (SV), was defined to be on the opposite side from
%the $B^0_s$ meson if
%it satisfied 
%$\cos\varphi(\vec{p}_{\ell~{\mathrm{or~SV}}}, \vec{p}_B) < 0.8$,
%where $\vec{p}_B$ is the reconstructed three-momentum of the 
%$B^0_s$ meson, and $\varphi$ is the azimuthal angle about the beam axis.  
A lepton jet charge was formed
as $Q^{\ell}_J = \sum_i q^i p^i_T/\sum_i p^i_T$, where
all charged particles are summed, including the lepton,
inside a cone
of $\Delta R = \sqrt{(\Delta\varphi)^2 + (\Delta\eta)^2} < 0.5$ centered
on the lepton.
The SV charge
was defined as 
$Q_{\mathrm{SV}} = \sum_i (q^i p_L^i)^{0.6}/\sum_i (p^i_L)^{0.6}$,
where
all charged 
particles associated with the SV are summed,
and $p^i_L$ is the longitudinal momentum of track $i$
with respect to the
direction of the SV momentum. Finally,
event charge is defined as 
$Q_{\mathrm{EV}} =  \sum_i q^i p^i_T/\sum_i p^i_T$,
where the sum is over all tracks with $p_T > 0.5$~GeV$/c$
outside a cone of $\Delta R > 1.5$ centered on the $B^0_s$ direction.
The {\sl pdf} of each discriminating variable was found
for $b$ and $\bar{b}$ quarks
using a large data sample of $B^+ \rightarrow \mu^+ \nu \bar{D}^0$ 
events where the initial state is 
known from the charge of the decay muon.
%For an initial $b$ ($\bar{b}$) quark, the {\sl pdf} for a given 
%variable $x_i$ is denoted 
%$f_i^b(x_i)$ ($f_i^{\bar{b}}(x_i)$),
%and the combined tagging
%variable is defined as
%$d_{\mathrm{tag}} = (1-z)/(1+z)$, 
%where $z=\prod_{i=1}^{n}(f_i^{\bar{b}}(x_i)/f_i^{b}(x_i))$.
%The variable $d_{\mathrm{tag}}$ varies between $-1$ and $1$.
%An event with $d_{\mathrm{tag}} > 0$~$(< 0)$ is tagged as a 
%$b$ ($\bar{b}$) quark.
The likelihood ratio is parameterised to provide an event by event prediction
of b quark.
The OST purity was determined from 
large samples of non-oscillating $B^+ \rightarrow \mu^+ \bar{D}^0 X$ 
 and 
oscillating $B^0_d \rightarrow \mu^+ D^{*-} X$ semileptonic
candidates. 
An average value of
$\epsilon \mathcal{D}^2 =
[2.48 \pm 0.21 \thinspace \mathrm{(stat)} 
          ^{+0.08}_{-0.06} \thinspace \mathrm{(syst)}]\%$ was
obtained~\cite{OST}.
%The estimated event-by-event dilution as a function
%of $|d_{\mathrm{tag}}|$ was determined by 
%measuring $\mathcal{D}$
%%$B^0_d$ oscillations 
%in bins of 
%$|d_{\mathrm{tag}}|$ and parametrizing 
%with a third-order polynomial for
%$|d_{\mathrm{tag}}| < 0.6$. For $|d_{\mathrm{tag}}| > 0.6$,
%$\mathcal{D}$ is fixed to 0.6.
The OST was applied to the $B^0_s \rightarrow \mu^+ D_s^- X$
data sample,
yielding 
$N_{\mathrm{tag}} = 5601 \pm 102 \thinspace \mathrm{(stat)}$ 
candidates having an identified
initial state flavour, as shown in Fig.~\ref{prl_fig1}(b).  The tagging
efficiency was $(20.9\pm0.7)$\%.

%PROPER DECAY TIME
%\section{Proper Decay Time Measurement}
Due to non-reconstructed particles in the event such as the neutrino, the
measured proper
decay length is smeared out and introduces resolution effects.
%After flavour tagging, the proper decay time of candidates is needed;
%however,
%the undetected neutrino and other missing particles in the
%semileptonic $B^0_s$ decay prevent
%a precise determination of the meson's momentum and Lorentz
%boost. This represents an important contribution to the smearing of 
%the proper decay length in semileptonic decays, in addition to the 
%resolution effects. 
A correction factor $K$ was
estimated from a Monte Carlo (MC) simulation by finding
the distribution
of
$K = p_T(\mu^+ D_s^-)/p_T(B)$ for a given decay channel in bins of $M(\mu^+ D_s^-)$.
The proper decay length of each $B^0_s$ meson is then
$c t(B^0_s) = l_M K$, where 
$l_M = M(B^0_s) \cdot (\vec{L}_{T}\cdot
\vec{p}_T(\mu^+ \ds))/(p_{T}(\mu^+ \ds))^{2}$ is the measured
visible proper decay length (VPDL),
$\vec{L}_T$ is the vector from the PV 
to the $B^0_s$ decay vertex in the transverse plane and
$M(B^0_s) = 5.3696$~GeV$/c^2$~\cite{pdg}.

All flavour-tagged events with
$1.72 < M(K^+K^-\pi^-) < 2.22$~GeV$/c^2$ were used in an
unbinned fitting procedure. The likelihood, $\mathcal{L}$, 
for an event to
arise from a specific source in the sample depends 
event-by-event 
on $l_M$,
its uncertainty $\sigma_{l_M}$, the invariant mass
of the candidate $M(K^+K^- \pi^-)$, the predicted
dilution $\mathcal{D}(d_{\mathrm{tag}})$, and
the  selection variable $y_{\mathrm{sel}}$.
%The {\sl pdf}s for $\sigma_{l_M}$, $M(K^+K^- \pi^-)$, $\mathcal{D}(d_{\mathrm{tag}})$
%and $y_{\mathrm{sel}}$
%were determined from data.
Four sources were considered:
the signal $\mu^+ D_s^-(\rightarrow \phi \pi^-)$;
the accompanying peak due to $\mu^+ D^{-} (\rightarrow \phi \pi^-)$;
a small (less than 1\%) reflection  
due to $\mu^+ D^{-} (\rightarrow K^{+} \pi^{-} \pi^-)$, where
the kaon mass is misassigned to one of the pions;
and combinatorial background.
The total fractions of the first two categories were 
determined from the mass fit of Fig.~\ref{prl_fig1}(b).

%Sample composition
The signal sample of $\mu^{+}\ds$ candidates consists mainly of \bs mesons
with some contribution from $B^{0}$ and $B^{+}$ mesons with any $b$-baryon
contribution estimated to be small and is neglected.
The distribution of the VPDl $l$ for non-oscillated and oscillated subsamples
as determined by the OST is modelled for each type of $B$ meson, e.g. for
$\bs$:
%\begin{eqnarray}
%\label{pnososc}
%\lefteqn{p_s^{\mathrm{nos/osc}}(l,K,d_{\mathrm{tag}}) = }  \\
%& & 
%\frac{K}{c \tau_{B_s^0}} \exp(- \frac{Kl}{c \tau_{B_s^0}})
%[1 \pm \mathcal{D}(d_{\mathrm{tag}})
%\cos(\Delta m_s \cdot Kl/c)]/2.  \nonumber
%%\cos(\Delta m_s \cdot Kl/c)]/2. \thinspace \thinspace (1)
%\end{eqnarray}
\begin{eqnarray}
\label{pnososc}
{p_s^{\mathrm{nos/osc}}(l,K,d_{\mathrm{tag}}) }   =
\frac{K}{c \tau_{B_s^0}} \exp(- \frac{Kl}{c \tau_{B_s^0}}) \nonumber \\
\left[ 1 \pm \mathcal{D}(d_{\mathrm{tag}})
\cos(\Delta m_s \cdot Kl/c) \right]/2.  &
\end{eqnarray}
The world averages~\cite{pdg} of $\tau_{B^0_d}$, $\tau_{B^+}$, and
$\Delta m_d$ were used as inputs to the fit.  The lifetime,
$\tau_{B^0_s}$, was allowed to float in the fit. In the amplitude and
likelihood scans described below, $\tau_{B^0_s}$ was fixed to this fitted
value.
%, which agrees with 
%expectations.

The total VPDL
{\sl pdf} for the $\mu^+ D_s^-$ signal is then the sum over all 
decay channels, including branching fractions,
that yield the $D_s^-$ mass peak.
%%The $B^0_s \to \mu^+ D_s^- X$ signal modes (including 
%%$D^{*-}_s$, ${D}_{s0}^{*-}$, and ${D}_{s1}^{'-}$; and
%%$\mu^+$ originating from
%%$\tau^+$ decay)
%%comprise $(85.6 \pm 3.3)$\% of our sample,
%%as determined from a MC
%%simulation.
% which included
%the {\sc PYTHIA} generator v6.2~\cite{pythia} interfaced with the
%{\sc EVTGEN} decay package~\cite{evtgen}, followed by full
%{\sc GEANT} v3.15~\cite{geant}
% modeling of the detector response and event reconstruction.
Backgrounds considered were
decays via 
$B^0_s \to D^+_{(s)} D^-_s X$ and
$\bar{B}^0_d, B^- \to D D^-_s$, followed
by $ D^+_{(s)} \to \mu^+ X$, with 
a real $D_s^-$ reconstructed in the peak and an
associated real $\mu^+$.
Another background taken into account occurs
when the $D_s^-$ meson originates from one $b$ or $c$ quark and the
muon arises from another quark. 
%%This background peaks around the 
%%PV (peaking backgrounds).
%%The uncertainty in each
%%channel covers possible trigger efficiency biases.
%%%
%%Translation from the true VPDL, $l$, to the 
%%measured $l_M$ for a given channel, is achieved
%%by a convolution of the  VPDL detector resolution,
%%of $K$ factors over each normalized distribution, 
%%and by including the reconstruction efficiency as a function of VPDL.
%%MC simulations were used to determined the lifetime-dependent efficiency and
%%VPDL distribution shapes for peaking backgrounds.
%The lifetime-dependent efficiency was
%found for each channel using MC simulations and, as a cross
%check, the efficiency was also determined from the data
%by fixing $\tau_{B^0_s}$ and fitting for the
%functional form of the efficiency.
%The shape of the VPDL distribution for peaking backgrounds
%was found from
%MC simulation, and the fraction from this source was allowed to
%float in the fit.


%%The VPDL uncertainty was determined from the vertex fit using track
%%parameters and their uncertainties. To account for possible mismodeling
%%of these uncertainties, resolution scale factors were introduced as
%%determined by examining the pull distribution of the vertex positions of
%%a sample of $J/\psi \rightarrow \mu^+ \mu^-$ decays. Using these scale
%%factors, the convolving function for the VPDL resolution was the sum of
%%two Gaussians with widths (fractions) of 0.998$\sigma_{l_M}$ (72\%) and
%%1.775$\sigma_{l_M}$ (28\%). 
%A cross check was performed using a MC
%simulation with tracking errors tuned according to the procedure
%described in~\cite{tune}. The 7\% variation of scale factors found in this
%cross check was used to estimate systematic uncertainties due to decay
%length resolution.

Several contributions to the combinatorial backgrounds that have 
different VPDL distributions
were considered.  True
prompt background was modeled with a Gaussian function
with a separate scale factor on the width;
background due to fake vertices around the PV 
was modeled with another Gaussian function;
and long-lived background was modeled with an exponential
function convoluted with the resolution, including
a component oscillating with a frequency 
of $\Delta m_d$. 
% The fractions of these contributions
% and their parameters were determined from the data sample.  
The unbinned fit of the total tagged sample was used 
to determine the various
fractions of signal and backgrounds and the background
VPDL parametrizations.  

\begin{figure}
\includegraphics[width=0.43\textwidth]{prl_fig2.eps}
\vspace{-1.1cm}
\caption{\label{prl_fig2}
Value of $-\Delta\log\mathcal{L}$ as a function of $\Delta m_s$. 
Star symbols do not include systematic uncertainties, and the shaded 
band represents the envelope of all $\log\mathcal{L}$ scan curves
due to different systematic uncertainties.
}
\end{figure}

Figure~\ref{prl_fig2} shows the value of 
$-\Delta\log\mathcal{L}$ as a function of $\Delta m_s$, indicating
a prefered value of 19~ps$^{-1}$, while variation of
$-\log\mathcal{L}$ from the minimum indicates an oscillation
frequency of $17 < \Delta m_s < 21$~ps$^{-1}$ at the 90\% C.L.
The uncertainties are approximately Gaussian inside this interval.
For a true value $\Delta m_s > 22$~ps$^{-1}$ there is insufficient resolution to
measure an oscillation. 
%Using $100$ 
%parametrized MC samples 
From MC samples
with similar statistics,
VPDL resolution, overall tagging performance, and
sample composition of the data
sample, it was determined that for a true value
of $\Delta m_s = 19$~ps$^{-1}$, the probability 
was 15\%
for measuring a value in the range $16 < \Delta m_s < 22$~ps$^{-1}$  
with a $-\Delta \log\mathcal{L}$ lower by at least 1.9 than the
corresponding value at $\Delta m_s = 25$~ps$^{-1}$.

\begin{figure}
\includegraphics[width=0.50\textwidth]{prl_fig3.eps}
\vspace{-1.3cm}
\caption{\label{prl_fig3}
$B^0_s$ oscillation amplitude as a function of oscillation frequency,
$\Delta m_s$.  The solid line shows the $\mathcal{A}=1$ axis for
reference.  The dashed line shows the expected limit including both
statistical and systematic uncertainties.  }
\end{figure}
The amplitude method~\cite{amp_method} was also used.
Equation~\ref{pnososc} was modified to include
the oscillation amplitude $\mathcal{A}$ as an additional coefficient
on the $\cos(\Delta m_s\cdot Kl/c)$ term.
The unbinned fit
was repeated for fixed input values of $\Delta m_s$ and
the fitted value of $\mathcal{A}$ and its uncertainty 
$\sigma_{\mathcal{A}}$ found for each step, as shown in 
Fig.~\ref{prl_fig3}.
At $\Delta m_s = 19$~ps$^{-1}$ the measured data point deviates from
the hypothesis $\mathcal{A}=0$ ($\mathcal{A}=1$) by 2.5 (1.6) standard
deviations, corresponding to a two-sided C.L. of 1\% (10\%), and is in
agreement with the likelihood results. 
In the presence of a signal, however, it is more difficult to define
a confidence interval using the amplitude than by examining the 
$-\Delta\log\mathcal{L}$
curve. Since, on average, these two methods give
the same results, we chose to quantify our $\Delta m_s$ 
interval using the likelihood curve.
A cross-check of the $\bs$ analysis was performed using $B^{0}$ decays and
Figure~\ref{bdamp} shows a peak in the amplitude scan at a value $\Delta
m_{d}\approx 0.5$ps$^{-1}$, compatible with the world average.


\begin{figure}
\includegraphics[width=\columnwidth]{bdamp.eps}
\vspace{-1.1cm}
\caption{\label{bdamp}
$B^{0}_{d}$ oscillation amplitude with statistical uncertainty only for events in
the $D^{-}$mass region in Fig.~\ref{prl_fig1} The red (solid)
line shows the $A$ = 1 axis for reference. The dashed line shows the expected limit including statistical uncertainties
only.
}
\end{figure}


%Systematics
%\section{Systematic Uncertainties}
Systematic uncertainties were addressed by varying inputs within
their range of uncertainties.  Uncertainties included: cut requirements,
{\sl pdf} modelling, $K$-factor distributions, peaking and combinatorial 
backgrounds fractions, and refection contributions.
%Systematic uncertainties were addressed by varying inputs,
%cut requirements,
%branching ratios, and {\sl pdf} modeling. 
%The branching ratios were varied within known uncertainties~\cite{pdg}
%and large variations were taken for those not yet measured.
%The $K$-factor distributions were varied within uncertainties,
%using measured (or smoothed) instead of generated momenta 
%in the MC simulation. The fractions of peaking and combinatorial 
%backgrounds
%were varied within uncertainties.  Uncertainties in the reflection
%contribution were considered. 
The functional form to
determine the dilution $\mathcal{D}(d_{\mathrm{tag}})$ was varied.
The lifetime $\tau_{B^0_s}$ was fixed to its world average
value, and $\Delta \Gamma_s$
was allowed to be non-zero.
The scale factors on the signal and background resolutions 
were varied within uncertainties, and typically generated
the largest systematic uncertainty in the region of interest.
A separate scan of $-\Delta\log\mathcal{L}$ was taken for
each variation, and the envelope of all such curves is indicated
as the band in Fig.~\ref{prl_fig2}.  
The same systematic uncertainties were considered for
the amplitude method using the procedure of Ref.~\cite{amp_method}, 
and, when added in quadrature with the statistical
uncertainties, represent a small effect, as shown in Fig.~\ref{prl_fig3}.
Taking these systematic uncertainties into account,  
we obtain from
the amplitude method an expected limit of $14.1$~ps$^{-1}$ and
an observed lower limit of $\Delta m_s > 14.8$~ps$^{-1}$ at the 95\% C.L.,
consistent with the likelihood scan. 

%\section{Result}
The probability that $B^0_s$-$\bar{B}^0_s$ oscillations with the 
true value of $\Delta m_s > 22$~ps$^{-1}$
 would give a $-\Delta \log\mathcal{L}$ minimum in the
range $16 < \Delta m_s < 22$~ps$^{-1}$ with a depth
of more than 1.7 with respect to the $-\Delta \log\mathcal{L}$ 
value at $\Delta m_s = 25$~ps$^{-1}$, 
corresponding to our observation
including systematic uncertainties, was found to be $(5.0 \pm 0.3)\%$.
This range of $\Delta m_s$ was chosen to encompass the world average
lower limit and the edge of our sensitive region.
This probability was determined by randomly assigning a flavour to each
candidate, effectively simulating a $B^0_s$
oscillation with an infinite frequency.
%To determine this probability, an
%ensemble test using the data sample was performed by randomly
%assigning a flavour to each candidate while retaining all its other
%information, effectively simulating a $B^0_s$
%oscillation with an infinite frequency.  
%Similar probabilities were found using ensembles of parametrized MC
%events.

\begin{figure}
\includegraphics[width=0.48\columnwidth]{rhoeta_withs2abgam_dmsEPS05.eps}
\includegraphics[width=0.48\columnwidth]{rhoeta_withs2abgam_dmsMoriond06.eps}
\vspace{-1.1cm}
\caption{\label{ckm}CKMFitter~\cite{ckmfitter} plots of
$(\bar{\rho},\bar{\eta})$ plane with inputs as of FPCP 2006, excluding (left)
and including (right) the result from this analysis. 
}
\end{figure}

%SUMMARY
%\section{Summary\label{summary}} 
To summarise, a� study of $B^0_s$-$\bar{B}^0_s$ oscillations was
performed using $B^0_s \to \mu^+ D_s^- X$ decays, where $D_s^- \to
\phi \pi^-$ and $\phi \to K^+K^-$, an opposite-side flavour tagging
algorithm, and an unbinned likelihood fit. 
Using the amplitude method an expected limit of $14.1$~ps$^{-1}$ is given and
there is an observed lower limit of $\Delta m_s > 14.8$~ps$^{-1}$ at the 95\%
C.L.
At $\Delta m_s =
19$~ps$^{-1}$, the amplitude method yields a result that
deviates from
the hypothesis $\mathcal{A}=0$ ($\mathcal{A}=1$) by 2.5 (1.6) 
standard deviations, corresponding to a two-sided C.L. of 1\% (10\%).
The likelihood curve is
well behaved near a preferred value of 19~ps$^{-1}$ with a 90\%
C.L. interval of $17 < \Delta m_s < 21$~ps$^{-1}$, assuming Gaussian
uncertainties. 
Ensemble tests indicate that if $\Delta m_s$ lies above the sensitive
region, i.e., above approximately 22~ps$^{-1}$, there is a
$(5.0\pm0.3)$\% probability that it would produce a likelihood minimum
similar to the one observed in the interval $16 < \Delta m_s <
22$~ps$^{-1}$.  This is the first report of a direct two-sided bound 
measured by a single experiment on the 
$B^0_s$ oscillation frequency, and places further constraints on the CKM
unitarity triangle as is shown in Figure~\ref{ckm}.
This result is
consistent with the subsequent observation of oscillations by the CDF
experiment which measures a value
$\Delta m_{s} = 17.31^{+0.33}_{-0.18}({\rm stat}) \pm 0.07 ({\rm
syst})$~\cite{cdfmixing}.


%-------------------------------------------------------------------------%
%BIB
%\vspace{-0.6cm}
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\end{document}
