%% slac-pub-7165: page file slac-pub-7165-0-0-3-4.tcx.
%% subsection 3.4 $B_s-\bar B_s$ Mixing and CKM Elements [slac-pub-7165-0-0-3-4 in slac-pub-7165-0-0-3: 6.6\,\mbox{ps}^{-1}\qquad (95\%\ \mbox{C.L.})
\end{equation}
\noindent is significant. It may indicate the need to develop new methods
capable of higher resolving power.
Reliable predictions of $\Delta M_{B_s}$ are therefore important
in order to plan future $B_s$ experiments, in particular if only
lower limits will be available with current vertex techniques.
The most straightforward method makes use of \cite{32}
\begin{equation}
\Delta M_{B_s} =\frac{G^2_F \;M^2_W}{6\pi^2}\eta_B \;\;S_0(x_t )
M_{B_s} B_{B_s} f^2_{B_s} |V_{ts}|^2,
\end{equation}
\noindent where $x_t=m^2_t/M^2_W$.
The current relative uncertainty is about 50\% and is dominated
by the uncertainty in $B_{B_s}$ ($\pm 30\%$),
$f^2_{B_s}$ ($\pm 40\%$), $|V_{ts}|^2$ ($\pm 15\%$) and
$S_0(x_t )$ ($\pm 8\%$). The fractional uncertainty
on $\Delta M_{B_s}$ can be expected to decrease to $\sim 15\%$
by the year 2002,
anticipating improvements in the accuracy of the relevant
parameters $B_{B_s}$ ($\pm 10\%$),
$f^2_{B_s}$ ($\pm 5\%$), $|V_{ts}|^2$ ($\pm 5\%$) and
$S_0(x_t )$ ($\pm 3\%$).
A variant of this method uses the experimental value for
$\Delta M_{B_d}$ and the ratio
\begin{equation}
\frac{(\Delta M)_{B_s}}{(\Delta M)_{B_d}} = \frac{M_{B_s}}{M_{B_d}}
\;\;\frac{B_{B_s}\;f^2_{B_s}}{B_{B_d} \;f^2_{B_d}} \;\;\left |
\frac{V_{ts}}{V_{td}}\right |^2
\end{equation}
\noindent to predict $\Delta M_{B_s}$. This approach will be useful only
if the CKM ratio $|V_{ts}/V_{td}|^2$ is accurately known.
If the first observation of $B_s-\bar B_s$ mixing is a
nonvanishing $\Delta\Gamma_{B_s}$ rather than $\Delta M_{B_s}$,
then a complementary method to predict $\Delta M_{B_s}$
opens up, based on the quantity (see (\docLink{slac-pub-7165-0-0-2.tcx}[dgres]{30}))
\begin{eqnarray}\label{dgdm}
\left(\frac{\Delta\Gamma}{\Delta M}\right)_{B_s} &=&
\frac{\pi}{2} \;\frac{m^2_b}{M^2_W} \;
\left|\frac{V_{cb}V_{cs}}{V_{ts}V_{tb}}\right|^2 \;
\frac{\sqrt{1-4z}}{\eta_B S_0(x_t)} \cdot \nonumber \\
&& \cdot \,\Bigg[\left(2(1-z)K_1+(1-4z)K_2\right)
\left(1+\frac{1}{N_c}\right) \\
&&\hspace*{0.3cm}+\,(1+2z)\left(K_2-K_1\right)
\frac{M^2_{B_s}}{(m_b+m_s)^2}
\left(2-\frac{1}{N_c}\right)\frac{B_S}{B}\,+\, \delta_{1/m}\Bigg].
\nonumber
\end{eqnarray}
\noindent This result is valid to next-to-leading order in the
$1/m_b$ expansion and to leading logarithmic accuracy in QCD.
We have again used factorization for the subleading $1/m_b$
corrections. Note that with the bag parameter $B$ as defined in
(\docLink{slac-pub-7165-0-0-2.tcx}[meqs]{24}), the appropriate QCD correction factor $\eta_B$ is
identical to $C_+(\mu)$ from (\docLink{slac-pub-7165-0-0-2.tcx}[c21pm]{11}) in the leading
logarithmic approximation.
In the ratio $\Delta\Gamma/\Delta M$ the decay constant cancels
and the CKM uncertainty is almost completely removed since
\begin{equation}\label{vctbs}
\left|\frac{V_{cb}V_{cs}}{V_{ts}V_{tb}}\right|^2= 1\pm 0.03.
\end{equation}
At present the accuracy of $\Delta\Gamma/\Delta M$ is still
rather poor, $\Delta\Gamma/\Delta M=(5.6\pm 2.6)\cdot 10^{-3}$.
The breakdown of errors is as follows: $\pm 2.3$ from varying
$B_S/B$ between $0.7$ and $1.3$, ${}^{+1.1}_{-0.7}$ from varying
$\mu$ between $m_b/2$ and $2 m_b$, $\pm 0.4$ from
$m_b=4.8\pm 0.2\,$GeV and $\pm 0.4$ from $m_t=176\pm 9\,$GeV.
The dominant uncertainty is due to $B_S/B$, which has never
been studied before. It is conceivable that a lattice
study could actually calculate $B_S/B$ more
accurately than the bag parameters themselves, because some
systematic uncertainties may be expected to cancel in the ratio.
The quantity $\Delta\Gamma/\Delta M$ might thus be calculable
rather precisely in the future and $\Delta M_{B_s}$ could then
be estimated from the observed $\Delta\Gamma_{B_s}$.
In conjunction with $\Delta M_{B_d}$ this would provide an
alternative way of determining the CKM ratio $|V_{ts}/V_{td}|$,
especially if the latter is around its largest currently
allowed value \cite{2}.
The width difference, and hence its observability increases the
larger $|V_{ts}|\approx|V_{cb}|$ becomes. In contrast, the
ratio
$\Gamma(B\to K^*\gamma)/\Gamma(B\to\{\varrho,\omega\}\gamma)$
is best suited for extracting small $|V_{ts}/V_{td}|$ ratios,
provided the long distance effects can be sufficiently well
understood \cite{33}.
These approaches could complement other methods to determine
$|V_{td}/V_{ts}|$. Such additional possibilities would be
to relate $|V_{ts}|$ to the accurate $|V_{cb}|$ measurements
and to obtain $|V_{td}|$ from $\Delta M_{B_d}$, CKM unitarity
constraints \cite{34},
and in particular $B(K^+\to\pi^+\nu\bar\nu)$ \cite{32,35},
which has the unique advantage of being
exceptionally clean from a theoretical point of view.