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\section{\usemenu{slacpub7165::context::slacpub7165005}{Summary}}\label{section::slacpub7165005}
\label{summary}
In this paper we have analyzed the theoretical prediction
for $\Delta\Gamma_{B_s}$ within the framework of the heavy
quark expansion.
We have calculated the explicit nexttoleading ${\cal O}(1/m_b)$
corrections in the operator product expansion for the
transition matrix element. In addition to the two leading
dimensionsix operators, five new operators of dimension
seven appear at this level. The matrix elements of the latter
operators were evaluated using factorization, which should give
a fair estimate of these subleading corrections.
Their effect on $\Delta\Gamma_{B_s}$,
formally of order ${\cal O}(\Lambda_{QCD}/m_b)$ and
${\cal O}(m_s/m_b)$, turned out to be sizable numerically,
causing a $30\%$ reduction of the leading order prediction.
We performed a numerical investigation of
$\Delta\Gamma_{B_s}$ with emphasis
on theoretical errors, which are presently dominated
by the uncertainties in hadronic matrix elements.
These errors are still rather large and lead to a prediction
of $(\Delta\Gamma/\Gamma)_{B_s}=0.16^{+0.11}_{0.09}$.
However, a systematic improvement of this result is possible,
in particular by progress in lattice QCD. In the future it would
be desireable to measure on the lattice the SP fourfermion operator
along with the VA operator that has received most attention
in the past due to its connection with the mass difference.
Eventually an accuracy
of $10\%$ for $\Delta\Gamma_{B_s}$ should be feasible when the
nexttoleading analysis of shortdistance corrections is also
completed.
The effects of penguin operators and contributions
from CKM suppressed modes have also been considered. They were
shown to give only a few percent relative correction in
$(\Delta\Gamma/\Gamma)_{B_s}$ and are thus negligible in view
of the other uncertainties.
We further studied the $B_sB_d$ lifetime difference and quantified
the expectation $\tau_{B_s}\approx\tau_{B_d}$, estimating
$\tau(B_s)/\tau(B_d)1<1\%$. This result is useful input for
experimental analyses of $\Delta\Gamma_{B_s}$.
To put our theoretical analysis into perspective, we have included
a short discussion of the current experimental situation
concerning $\Delta\Gamma_{B_s}$. Using information on
$\tau(B_s\to J/\psi\phi)$ and $\tau(B_s)=\tau(B_d)$,
we have attempted a preliminary extraction of $\Delta\Gamma_{B_s}$,
obtaining $(\Delta\Gamma/\Gamma)_{B_s}\geq 0.3\pm 0.4$. This is still
inconclusive but can be improved by better statistics in the
future. We have also
proposed an alternative route towards a measurement of
$\Delta\Gamma_{B_s}$ that makes use of the
$\phi\phi X$ and/or $D^\pm_s\phi X$ final states
in $B_s$ decay, which are expected to be dominantly CP even.
The present experimental information may be complemented by the
bound
$(\Delta\Gamma/\Gamma)_{B_s}\leq 2 B(b\to c\bar cs)_{B_s}
\approx 0.44\pm 0.06$.
This bound is not very strong, but it has the advantage of being
valid independently of the heavy quark expansion and it is interesting
for principal reasons.
In addition we have briefly reviewed some phenomenological applications
that could be opened up by further progress on the experimental
as well as the theoretical side. These possibilities include
new methods to study CP violation, complementary information on
$\Delta M_{B_s}$ in case $\Delta\Gamma_{B_s}$ is measured first,
and alternative constraints on $V_{td}/V_{ts}$, especially
for small values of this ratio.
Finally, the theory of inclusive $B$ decays itself can be
expected to profit from a confrontation of the heavy quark expansion
for $\Delta\Gamma_{B_s}$ with experiment.
In this respect $\Delta\Gamma_{B_s}$ provides an important special
case that directly probes ${\cal O}(1/m^3_b)$ contributions.
As we have seen, the topic of $\Delta\Gamma_{B_s}$ touches upon
a rich variety of interesting physics issues and certainly merits
the continued efforts needed to address the problems that are
still unresolved.
\vspace*{0.7cm}
\noindent {\bf Acknowledgements.}
We thank Andrzej Buras for encouragement and useful
conversations. Thanks are also due to Yuval Grossman,
Joe Incandela, Jonathan Lewis, HansG\"unther Moser
and Yossi Nir for discussions.
Fermilab is operated by Universities Research Association, Inc.,
under contract DEAC0276CHO3000 with the United States Department
of Energy.
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