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%% subsection 3.3 Measuring $\Delta\Gamma_{B_s}$ [slac-pub-7165-0-0-3-3 in slac-pub-7165-0-0-3: slac-pub-7165-0-0-3-4]
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\subsection{\usemenu{slac-pub-7165::context::slac-pub-7165-0-0-3-3}{Measuring $\Delta\Gamma_{B_s}$}}\label{subsection::slac-pub-7165-0-0-3-3}
\label{measure}
We hope to have convinced the reader about the importance of an accurate
measurement of $\Delta\Gamma$.
One method is to substitute $\Gamma_{B_d}$ for the average $B_s$
width $\Gamma_{B_s}$ and to extract $\Delta\Gamma_{B_s}$ from the
time-dependences of untagged flavor specific $B_s$ data samples
\cite{1}. Time-dependent studies of angular distributions of untagged
$\stackrel{(-)}{B}_s \rightarrow J/\psi\phi$ decays allow the extraction
of $\Gamma_L$, and also of $\Gamma_H$ if the CP-odd component
is non-negligible \cite{5,24}.
These and other methods using decay length distributions of fully
reconstructed $B_s$ mesons
are at present statistics limited~\cite{1,5,24}.
As an illustration one may consider the measurement of
\begin{equation}\label{tbsj}
\tau(B_s\to J/\psi \phi)=1.34^{+0.23}_{-0.19}\pm 0.05\,\mbox{ps}
\end{equation}
\noindent recently obtained by the CDF collaboration from a single
lifetime fit of their $\stackrel{(-)}{B}_s \rightarrow J/\psi\phi$
data sample \cite{25}. Next we can write
\begin{equation}\label{glin}
1/\Gamma_L\leq\tau(B_s\to J/\psi\phi),
\end{equation}
\noindent which holds only as an inequality,
because $B_s\to J/\psi\phi$ is not necessarily a pure CP-even final state.
The world average $B_d$ lifetime \cite{26}
\begin{equation}\label{tbdw}
\tau_{B_d}=1.54\pm 0.04\,\mbox{ps}
\end{equation}
\noindent together with the result of section \docLink{slac-pub-7165-0-0-4.tcx}[bsbd]{4}, informs us about
the inverse of the average $B_s$ width $1/\Gamma_{B_s}=\tau_{B_d}$.
We then use
\begin{equation}\label{dggl}
\frac{\Delta\Gamma}{\Gamma}=
2\left(\frac{\Gamma_L}{\Gamma}-1\right)
\end{equation}
\noindent and obtain
\begin{equation}
\left(\frac{\Delta\Gamma}{\Gamma}\right)_{B_s}\geq 0.3\pm 0.4,
\end{equation}
\noindent which is still inconclusive, but can serve to indicate the present
status.
Just establishing a non-vanishing difference in decay length
distributions for partially reconstructed $B_s$ mesons in comparison
to the other $B$ mesons would constitute progress.
The ideal inclusive $b$-hadron data sample should have large statistics
and be highly enriched in $B_s$ decay products originating predominanty
from a single mass eigenstate $B_L$ (or $B_H$). The last requirement
maximizes differentiation between $B_s$ and other $B$-mesons.
The $\phi\phi X$ final state serves as an example \cite{27}.
The probable decay chain is
$B_s\rightarrow D^+_s D^-_s X,$ which is
dominantly CP even \cite{11}.
Both $D_s$'s then decay into $\phi$'s.
While $D_s$ is seen significantly in $\phi$'s,
the $D^+$ is seen in $\phi$'s by
about a factor of 10 less and the $D^0$ even
less than that \cite{28}.
The background due to $B$-meson decays is thus controllable and further
suppressed because $B$'s prefer to be seen as $D^0$ over $D^+$ by a ratio of
2.7~\cite{29}.
If sufficient statistics is available, the $D^\pm_s \phi X$ sample would be
even better.
The inclusive $B_s\rightarrow \phi\ell^+ X$ sample with a high
$P_{T,re\ell}$ lepton, is flavor specific.
Its time dependence is governed by the sum of two exponentials,
${\rm exp}\left(-\Gamma_L t\right)+{\rm exp}\left(-\Gamma_H t\right)$.
Theory predicts $(\Gamma_L+\Gamma_H)/2 = 1/\tau_{B_d}$,
but the observation of the two
exponents requires precise decay length and boost information,
whose accuracy increases the more fully the $B_s$ is reconstructed.
The less reconstructed the $B_s$ data sample, the more important it is
to have a mono-energetic source of $B_s$ mesons. Thus the
more inclusive techniques tend to be more
useful for $e^+ e^- \rightarrow Z^0$
experiments than at hadron accelerators.
Of course, fully reconstructed $B_s$ data samples allow clean
measurements of $\Delta\Gamma_{B_s}$.