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%% subsection 3.1 Numerical Analysis of $(\Delta\Gamma/\Gamma)_{B_s}$ [slac-pub-7165-0-0-3-1 in slac-pub-7165-0-0-3: ^slac-pub-7165-0-0-3 >slac-pub-7165-0-0-3-2]
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\subsection{\usemenu{slac-pub-7165::context::slac-pub-7165-0-0-3-1}{Numerical Analysis of $(\Delta\Gamma/\Gamma)_{B_s}$}}\label{subsection::slac-pub-7165-0-0-3-1}
\label{numeric}
\begin{table}[t]
\addtolength{\arraycolsep}{0.2cm}
\renewcommand{\arraystretch}{1.3}
$$
\begin{array}{|c|c||c|c|c|c|}
\hline
m_b/\mbox{GeV} & \mu & a & b & c &
(\Delta\Gamma/\Gamma)_{B_s} \\
\hline\hline
4.8 & m_b & 0.009 & 0.211 & -0.065 & 0.155 \\ \hline
4.6 & m_b & 0.015 & 0.239 & -0.096 & 0.158 \\ \hline
5.0 & m_b & 0.004 & 0.187 & -0.039 & 0.151 \\ \hline
4.8 & 2 m_b & 0.017 & 0.181 & -0.058 & 0.140 \\ \hline
4.8 & m_b/2 & 0.006 & 0.251 & -0.076 & 0.181 \\ \hline
\end{array}
$$
\caption{\label{table1}
Dependence of $a$, $b$ and $c$ on the $b$-quark mass and renormalization
scale for fixed values of all other short-distance parameters. The last
column gives $(\Delta\Gamma/\Gamma)_{B_s}$ for $B=B_S=1$ (at
the given scale $\mu$), $f_{B_s}=210\,$MeV.}
\end{table}
We first turn to a numerical analysis and discussion of
$(\Delta\Gamma/\Gamma)_{B_s}$ based on (\docLink{slac-pub-7165-0-0-2.tcx}[dgres]{30}). It is
useful to separate the dependence on the long-distance parameters
$f_{B_s}$, $B$ and $B_S$ and write $(\Delta\Gamma/\Gamma)_{B_s}$ as
\begin{equation}
\left(\frac{\Delta\Gamma}{\Gamma}\right)_{B_s} =
\Big[a B + b B_S + c\Big]\left(\frac{f_{B_s}}{210\,\mbox{MeV}}\right)^2,
\end{equation}
\noindent where $c$ incorporates the explicit $1/m_b$ corrections.
To estimate the sensitivity of $(\Delta\Gamma/\Gamma)_{B_s}$ on
the short-distance input parameters, we keep the following parameters
fixed: $m_b-m_c=3.4\,$GeV, $m_s=200\,$MeV, $\Lambda^{(5)}_{LO}=200\,
$MeV. In addition $M_{B_s}=5.37\,$GeV and the semileptonic branching
ratio is $B(B_s\to X e\nu)=10.4\%$. Then $a$, $b$ and $c$ depend only
on $m_b$ and the renormalization scale $\mu$. For some values of
$m_b$ and $\mu$, the coefficients $a$, $b$, $c$ are listed in
Table~\docLink{slac-pub-7165-0-0-3.tcx}[table1]{1}. For a central choice of parameters, which we take
as $m_b=4.8\,$GeV, $\mu=m_b$, $B=B_S=1$ and $f_{B_s}=210\,$MeV, we
obtain
\begin{equation}\label{dgnum1}
\left(\frac{\Delta\Gamma}{\Gamma}\right)_{B_s} = 0.220 - 0.065 = 0.155,
\end{equation}
\noindent where the leading term and the $1/m_b$ correction are
separately displayed. As seen from the Table, the dependence on $m_b$
is weak, but $(\Delta\Gamma/\Gamma)_{B_s}$ increases by almost $20\%$
when the renormalization scale is lowered to $m_b/2$, at fixed $B$ and
$B_S$. These dependences are not specific to the values $B=B_S=1$. The
weak $m_b$ dependence is a somewhat accidental consequence of using the
semileptonic branching ratio to eliminate $V_{cb}$. If instead we
normalize to $\Gamma_{B_s}^{-1}=1.54\,$ps and take $V_{cb}=0.04$,
$(\Delta \Gamma/\Gamma)_{B_s}$ would vary from $0.143$ to $0.166$ under
the same variation of $m_b$ as in the Table. Let us also add the
following more general observations:
(i) The theoretical expression for $\Delta\Gamma_{B_s}$ in
(\docLink{slac-pub-7165-0-0-2.tcx}[dgres]{30}) predicts the sign of this quantity, which a
priori could have either value. $\Delta\Gamma_{B_s}$ is positive
and implies a larger decay rate for the CP even (lighter) state
\cite{10,11}
(see the conventions in the Introduction). The typical magnitude of
$(\Delta\Gamma/\Gamma)_{B_s}$ to leading order in the heavy quark
expansion is about 0.2, larger than other width differences among
bottom hadrons with the possible exception of the case of $\Lambda_b$
(depending on whether theory or present experiments turn out to be right
on $\Lambda_b$).
(ii) The explicit $1/m_b$ corrections are numerically important
and vary strongly with $m_b$. For our central parameter choice
they reduce the leading order prediction by about $30\%$.
Essentially all the various $1/m_b$ correction terms add with
the same sign and make the result somewhat larger than the
natural size of the corrections,
$\Lambda_{QCD}/m_b\approx (M_{B_s}-m_b-m_s)/m_b\approx 8\%$
and $m_s/m_b\approx 4\%$.
(iii) The contribution from the scalar operator $Q_S$ by far
dominates over the contribution from $Q$, because there is a
strong cancellation between terms of different sign in the Wilson
coefficient of the latter operator. This has important
implications for $(\Delta M/\Delta\Gamma)_{B_s}$, which we
discuss below, because hadronic uncertainties cancel only partially
in the ratio $B/B_S$.
(iv) If $B_S=1.3$, a $(\Delta\Gamma/\Gamma)_{B_s}$ of as much as
$0.25$ is not excluded, although this appears unlikely.
On the other hand,
if $B_S<1$, as suggested by the estimate from hybrid logarithms,
and if $f_{B_s}$ turns out to be merely
$180\,$MeV, $(\Delta\Gamma/\Gamma)_{B_s}$ could be as small as
$0.07$, making its experimental detection more difficult.
This discussion shows that to resolve the theoretical uncertainties,
a reliable calculation of $B_S$ is mandatory. Further improvement
then requires a full next-to-leading order calculation of short-distance
corrections.