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\section{\usemenu{slacpub7165::context::slacpub7165001}{Introduction}}\label{section::slacpub7165001}
\label{intro}
Mixing phenomena in neutral $B$ meson systems provide an
important testing ground for standard model flavordynamics.
The mass difference between the $B_d$ eigenstates,
$\Delta M_{B_d}$, gave the first evidence for
a large top quark mass and provides a valuable constraint
on $V_{td}$ and the CKM unitarity triangle. A direct measurement of
$\Delta M_{B_s}$, the corresponding quantity for $B_s$ mesons, through
$B_s$$\bar{B}_s$ oscillations, would yield further
information and help to reduce hadronic uncertainties
in the extraction of CKM parameters. Complementary
insight can be gained from the width difference
$\Delta\Gamma_{B_s}$ between the $B_s$ mass eigenstates \cite{1,2}.
This width difference is expected
to be the largest among bottom hadrons \cite{3}, and
it may be large enough to be accessible by
experiment in the near future. The width difference for $B_d$ mesons,
on the other hand, is CKM suppressed and experimentally much harder
to determine.
If $\Delta\Gamma_{B_s}$ is indeed found to be sizable, the observation
of CP violation and the extraction of CKM phases from untagged
$B_s$ data samples can be contemplated \cite{1,4,5}.
This possibility could be important in two respects. First, tagging
any $B_s$ data sample costs in statistics and in purity. Second, the
rapid oscillations dependent on $\Delta M_{B_s} t$ all cancel in
time evolutions of untagged $B_s$ data samples, which are governed
by the two exponentials $\exp(\Gamma_L t)$ and
$\exp(\Gamma_H t)$ alone.
The present article continues previous work
of one of us \cite{1} on the phenomenological potential of
$\Delta\Gamma_{B_s}$,
and focuses on theoretical uncertainties and improvements of
the prediction.
We compute the width difference in the heavy quark
expansion and include explicit $1/m_b$corrections, which
improves over previous estimates of $\Delta \Gamma_{B_s}$ based on a
partonic \cite{6,7,8,9,10} or exclusive \cite{11}
approach and allows us to assess the remaining uncertainties more
reliably. Combined with future measurements of $\Delta \Gamma_{B_s}$
these predictions can be used to derive indirect constraints
on $V_{ts}/V_{td}$ \cite{2} and $\Delta M_{B_s}$.
Nonstandard model sources of CP violation in the $B_s$ system
would reduce $\Delta \Gamma_{B_s}$ compared
to its standard model value, as explained in \cite{12}, so
that a lower bound on the standard model prediction is
especially interesting.
Starting from the flavor eigenstates
$\{B_s\rangle,\, \bar B_s\rangle\}$, $B_s\bar{B_s}$ mixing is
determined by the $2\times 2$ matrix
\begin{equation}
{\bf \cal M} = {\bf M}\frac{i}{2}\,{\bf \Gamma}.
\end{equation}
\noindent with hermitian ${\bf M}$ and ${\bf\Gamma}$.
Due to CPT conservation $M_{11}=M_{22}\equiv M_{B_s}$,
$\Gamma_{11}=\Gamma_{22}\equiv\Gamma_{B_s}$.
We recall that for the $B_s$ system the offdiagonal elements
obey the pattern
\begin{equation}
\left\frac{\Gamma_{12}}{M_{12}}\right\sim {\cal O}\!\left(\frac{m_b^2}
{m_t^2}\right).
\end{equation}
\noindent The mass and lifetime
difference between eigenstates are given by (`H' for `heavy',
`L' for `light')
\begin{equation}\label{delmex}
\Delta M_{B_s} \equiv M_HM_L=2 \,M_{12},\\[0.1cm]
\end{equation}
\begin{equation}\label{delgex}
\Delta \Gamma_{B_s} \equiv \Gamma_L\Gamma_H = \frac{2\,\mbox{Re}\,(
M_{12}^*\Gamma_{12})}{M_{12}}.
\end{equation}
\noindent The corrections to (\docLink{slacpub7165001.tcx}[delmex]{3}) and (\docLink{slacpub7165001.tcx}[delgex]{4})
are extremely suppressed. They enter only at order
$\Gamma_{12}/M_{12}^2$ and vanish altogether in the limit of
exact CP symmetry.
Anticipating the actual hierarchy of eigenvalues,
we have defined both $\Delta M_{B_s}$ and $\Delta\Gamma_{B_s}$
to be positive quantities.
Neglecting CP violating corrections, which are very small in the
standard model (SM), the mass eigenstates are CP eigenstates (up to
corrections of at most $10^{3}$),
and with the phase convention
$CPB_s\rangle=\bar B_s\rangle$ one has
$B_{H/L}\rangle=(B_s\rangle\pm \bar B_s\rangle)/\sqrt{2}$.
Then\footnote{
Subsequently, we present the result of our calculation of $\Gamma_{21}$
as a result for $\Delta\Gamma_{B_s}$ using (\docLink{slacpub7165001.tcx}[delgacp]{5}). If one does
not want to assume standard model CP violation,
(\docLink{slacpub7165001.tcx}[delgacp]{5}) must be generalized to (\docLink{slacpub7165001.tcx}[delgex]{4}),
but our result for $\Gamma_{21}$ is still
valid, provided nonstandard model CP violation modifies only
$M_{12}$, but not $\Gamma_{12}$. Since $\Gamma_{12}$ results
predominantly from tree
decays, this is reasonable to assume.},
using standard CKM phase conventions \cite{13},
\begin{equation}
\label{delgacp}
\Delta \Gamma_{B_s}=2\,\Gamma_{12}=2\,\Gamma_{21}.
\end{equation}
\noindent Note that the lighter state is CP even \cite{1}
and decays more rapidly
than the heavier state. This also follows from the fact that most of the
decay products in the $b\to c\bar{c}s$ transition which are common to
$B_s$ and $\bar{B_s}$ are CP even \cite{11}.
Both the mass and lifetime difference are determined by the familiar
box diagrams that give rise to an effective $\Delta B=2$
Hamiltonian (`$B$' denotes $b$quark number). On distance scales
larger than $1/M_W$, but still smaller than $1/m_b$, this effective
Hamiltonian contains a local $\Delta B=2$ interaction as well as
a bilocal part constructed from two (local) $\Delta B=1$
transitions. The mass difference is given by the real part
of the box diagram and is dominated by the top quark contribution.
For this reason, $M_{12}$
is generated by an interaction that is local already on scales
$x>1/M_W$ and theoretically well under control.
The shortdistance contribution has been calculated to nexttoleading
order in QCD \cite{14}. The longdistance contribution is parametrized
by the matrix element of a single fourquark operator between
$B_s$ and $\bar B_s$ states. Corrections to this result are
suppressed by powers of $m^2_b/M^2_W$
and completely irrelevant for all practical purposes.
The lifetime difference is given by the imaginary part of the
box diagram and determined by real intermediate states,
which correspond to common decay products
of $B_s$ and $\bar{B}_s$, so that only the
bilocal part of the $\Delta B=2$ Hamiltonian can contribute.
The presence of longlived (on hadronic scales) intermediate states
would normally preclude a shortdistance treatment of the lifetime
difference as indeed it does for neutral kaons. But for bottom mesons,
the $b$ quark mass $m_b$ provides an additional shortdistance
scale that leads to a large energy release
(compared to $\Lambda_{QCD}$) into the intermediate states. Thus,
at typical hadronic distances $x > 1/m_b$, the decay is again a
local process. The bilocal $\Delta B=2$ Hamiltonian can be expanded
in inverse powers of the heavy quark mass, schematically:
\begin{equation}
\label{bilocalexpand}
\mbox{Im}\,i\int d^4 x\,T\!\left({\cal O}^{\Delta B=1}(x) {\cal O}^{
\Delta B=1}(0)\right) = \sum_n\frac{C_n}{m_b^n}\,{\cal O}_n^{\Delta B=2}(0)
\end{equation}
\noindent The matrix elements of local $\Delta B=2$ operators
that appear here and in the mass difference are not independent
of $m_b$. Their mass dependence could be made explicit with the
help of Heavy Quark Effective Theory. The difference between the
mass and lifetime difference is that for the lifetime difference
explicit $1/m_b$ corrections arise from the
expansion (\docLink{slacpub7165001.tcx}[bilocalexpand]{6}) even before expanding the matrix
elements of local operators. The heavy quark expansion applies as
well to the diagonal elements
$\Gamma_{ii}\equiv\Gamma_{B_s}\equiv(\Gamma_H+\Gamma_L)/2$
and has been used
to predict the total width of bottom hadrons \cite{3}. A
contribution to $\Gamma_{12}$ requires that the spectator strange
quark and the bottom quark come together within a distance $1/m_b$
in a meson of size $1/\Lambda_{QCD}$. This volume suppression
together with the phase space enhancement, leads to the
estimate
\begin{equation}\label{estimate}
\left\frac{\Gamma_{12}}{\Gamma_{11}}\right\sim 16\pi^2\left(
\frac{\Lambda_{QCD}}{m_b}\right)^3.
\end{equation}
\noindent The application of heavy quark expansions to nonleptonic
decays assumes local duality. The accuracy of this assumption can
not be quantified within the framework itself, at least not to finite order
in the heavy quark expansion. The assumption that the sum over
exclusive modes is accurately described by the heavy quark expansion
might be especially
troubling for $\Delta \Gamma_{B_s}$, since it is saturated by only
a few $D_s^{(*,**)}\bar{D}_s^{(*,**)}$ intermediate states and the energy
release is only slightly larger than one GeV. On the other hand, in the
smallvelocity limit $\Lambda_{QCD}\ll m_b2 m_c\ll m_c$, and the
$N_c\to\infty$limit\footnote{This limit is necessary to justify the
factorization assumption for fourfermion operators.}, local duality
with only a few intermediate states can indeed be verified
explicitly \cite{11}.
This article starts from the hypothesis that duality violations should
be less than $10\%$ for $\Delta \Gamma_{B_s}$.
Aiming at an accuracy of $10\%$, the following corrections
to the leading order result have to be considered:
\renewcommand{\labelenumi}{(\roman{enumi})}
\begin{enumerate}
\item $1/m_b$ corrections from dimension seven operators in
(\docLink{slacpub7165001.tcx}[bilocalexpand]{6}).
\item Deviations from the `vacuum insertion' (`factorization')
assumption for matrix
elements of fourfermion operators.
\item Radiative corrections of order $\alpha_s/\pi$.
\item Penguin and Cabibbosuppressed contributions.
\end{enumerate}
\noindent The major part of this paper is devoted to $1/m_b$ corrections.
We hope to return to radiative corrections in a subsequent publication.
These would bring the shortdistance part of the calculation for
$\Delta\Gamma_{B_s}$ on the same level that has already been
achieved for $\Delta M_{B_s}$.
The result for $\Delta \Gamma_{B_s}$ to nexttoleading order in the
$1/m_b$ expansion is obtained in Sect.~\docLink{slacpub7165002.tcx}[basic]{2}. We use the vacuum
insertion approximation for the dimension seven operators, and express
the result in terms of two nonperturbative parameters that have to
be computed with lattice methods. Sect. \docLink{slacpub7165003.tcx}[pheno]{3} is devoted to the
phenomenology of $\Delta \Gamma_{B_s}$. Numerical results are
discussed in Sect.~\docLink{slacpub7165003.tcx}[numeric]{3.1}, together with the theoretical
uncertainties in $\Delta\Gamma_{B_s}/\Gamma_{B_s}$.
In Sect. \docLink{slacpub7165003.tcx}[limit]{3.2} a generally valid upper bound on
$\Delta \Gamma_{B_s}$ is derived. Sect. \docLink{slacpub7165003.tcx}[measure]{3.3} describes
potential strategies to measure the width difference in experiment.
Some phenomenological applications of such a measurement are
considered in Sect. \docLink{slacpub7165003.tcx}[bbmix]{3.4}.
An issue related to $\Delta \Gamma_{B_s}$ concerns
the total decay rate $\Gamma_{B_s}$ of $B_s$ mesons, averaged
over the longlived and shortlived component. For experimental
investigations of $\Delta\Gamma_{B_s}$ \cite{1} it would be
helpful to know to what extent the average $B_s$ decay rate
$\Gamma_{B_s}$ differs from $\Gamma_{B_d}$. These decay widths are
estimated to coincide to a high accuracy \cite{3}. We
quantify this expectation and detail the contributions that could
give rise to a difference between $\Gamma_{B_s}$ and
$\Gamma_{B_d}$ in Sect.~\docLink{slacpub7165004.tcx}[bsbd]{4}.
A summary is presented in Sect.~\docLink{slacpub7165005.tcx}[summary]{5}.
Penguin and Cabibbosuppressed contributions turn out to shift
$\Delta \Gamma_{B_s}$ by less then $10\%$ and are discussed in
the Appendices, along with a comment
on the lifetime ratio of $B^+$ to $B_d$ mesons.
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