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\section{\usemenu{slacpub7165::context::slacpub7165002}{$\Delta\Gamma_{B_s}$  Basic Formalism}}\label{section::slacpub7165002}
\label{basic}
The optical theorem relates the total decay width of a particle
to its forward scattering amplitude. The offdiagonal element
$\Gamma_{21}$ of the decay width matrix is given by
\begin{equation}\label{g21t}
\Gamma_{21}=\frac{1}{2M_{B_s}}
\langle\bar B_s{\cal T}B_s\rangle .
\end{equation}
\noindent The normalization of states is
$\langle B_sB_s\rangle=2EV$ (conventional relativistic normalization)
and the transition operator ${\cal T}$ is defined by
\begin{equation}\label{tdef}
{\cal T}=\mbox{Im}\,i\int d^4x\ T\,{\cal H}_{eff}(x){\cal H}_{eff}(0).
\end{equation}
\noindent Here ${\cal H}_{eff}$ is the low energy effective weak
Hamiltonian mediating bottom quark decay. The component that is
relevant for $\Gamma_{21}$ reads explicitly
\begin{equation}\label{heff}
{\cal H}_{eff}=\frac{G_F}{\sqrt{2}}V^*_{cb}V_{cs}
\left(C_1(\mu) (\bar b_ic_j)_{VA}(\bar c_js_i)_{VA}+
C_2(\mu) (\bar b_ic_i)_{VA}(\bar c_js_j)_{VA}\right),
\end{equation}
\noindent where we are neglecting Cabibbo suppressed channels
and the contributions from penguin operators, whose coefficients are
small numerically. These contributions will be considered in the
Appendices. We use the notation $(\bar{q}_1 q_2)_{VA}=\bar q_1
\gamma_\mu (1\gamma_5) q_2$ and similar notation for other
combinations of Dirac matrices. The indices $i,j$ refer to color.
The Wilson coefficient functions $C_{1,2}$
read in the leading logarithmic approximation
\begin{equation}\label{c21pm}
C_{2,1}=\frac{C_+\pm C_}{2}\qquad
C_+(\mu)=\left[\frac{\alpha_s(M_W)}{\alpha_s(\mu)}\right]^{6/23}\quad
C_(\mu)=\left[\frac{\alpha_s(M_W)}{\alpha_s(\mu)}\right]^{12/23}
\end{equation}
\noindent with scale $\mu$ of order $m_b$.
\begin{figure}[t]
\vspace{0cm}
\epsfysize=4.8cm
\epsfxsize=6cm
\centerline{\epsffile{delgfig1.eps}}
\vspace*{1.5cm}
\caption{\label{fig1} Diagram that gives the leading and
nexttoleading in $1/m_b$ terms in the heavy quark expansion of
the forward scattering amplitude.}
\end{figure}
The leading contribution to the $\Delta B=2$ transition operator
is shown in Fig.~\docLink{slacpub7165002.tcx}[fig1]{1},
where the vertices correspond to the
interaction terms in (\docLink{slacpub7165002.tcx}[heff]{10}). The operator product
expansion is constructed using standard methods \cite{3}.
Because of the large momentum flowing through the fermion loop,
it can be contracted to a point. To leading order in $1/m_b$,
the strange momentum can be neglected and the $b$ quark momentum
identified with the meson momentum. The result can be
expressed in terms of two dimension six operators
\vspace*{0.2cm}
\begin{eqnarray}\label{qqs}
Q &=& (\bar b_is_i)_{VA}(\bar b_js_j)_{VA}\\[0.1cm]
Q_S\!\! &=& (\bar b_is_i)_{SP}(\bar b_js_j)_{SP} .
\end{eqnarray}
\noindent The first operator coincides with the single operator that
contributes to the mass difference. The appearance of a second operator
can be traced to the fact that in the calculation of $\Gamma_{21}$
the external $b$ momentum can not be neglected, because its zero
component (in the meson rest frame) provides the large momentum scale.
To include $1/m_b$ corrections, the forward scattering amplitude, evaluated
between onshell quark states, is
expanded in the small strange quark momentum and matched onto operators
with derivatives or with a factor of $m_s$, the strange quark mass, which
we count as $\Lambda_{QCD}$. Operators with additional gluon fields
contribute only to corrections of order $(\Lambda_{QCD}/m_b)^2$ and
need not be considered. It is more direct (and rather trivial at this order)
to use the background field method \cite{15}. Since we do not scale out
the `kinematic' part of order $m_b$ in derivatives acting on $b$ fields,
we do not have immediate power counting. Some operators of higher
dimension in (\docLink{slacpub7165001.tcx}[bilocalexpand]{6}) have to be kept, if they contain
derivatives on $b$ fields, such as $R_2$ below. Using the equations
of motion, we are left with operators with at most one derivative
on $b$ fields and obtain
\begin{eqnarray}\label{tres}
\Gamma_{21} &=& \frac{G^2_F m^2_b}{12\pi (2 M_{B_s})}(V^*_{cb}V_{cs})^2
\sqrt{14z}\cdot \nonumber\\
&&\,\cdot\left[\left((1z)K_1+\frac{1}{2}(14z)K_2\right)\langle Q
\rangle +
(1+2z)\left(K_1K_2\right) \langle Q_S\rangle + \hat{\delta}_{1/m}\right],
\end{eqnarray}
\noindent where $z=m^2_c/m^2_b$ and
\begin{equation}\label{k1k2}
K_1=N_c C^2_1+2C_1 C_2\qquad K_2=C^2_2 .
\end{equation}
\noindent The brackets denote the matrix
element of an operator ${\cal O}$
between a $\bar{B}_s$ and $B_s$ state,
$\langle{\cal O}\rangle\equiv\langle\bar B_s{\cal O}B_s\rangle$.
The $1/m_b$ corrections are summarized in
\begin{eqnarray}\label{oneoverm}
\hat{\delta}_{1/m} &=& (1+2 z)\left[K_1\,(2\langle R_1\rangle
2\langle R_2\rangle) + K_2\,(\langle R_0\rangle 2\langle \tilde{R}_1
\rangle 2\langle \tilde{R}_2\rangle)\right]\nonumber\\
&&\,\,\frac{12 z^2}{14 z}\left[K_1\,(\langle R_2\rangle
+2\langle R_3\rangle) + K_2\,(\langle \tilde{R}_2\rangle
+2\langle \tilde{R}_3\rangle)\right] .
\end{eqnarray}
%&&+ (1+2z)K_2 R_0 2(1+2z)(K_1R_1+K_2\tilde R_1)  \nonumber\\
%&& \left. 2\frac{12z2z^2}{14z}(K_1R_2+K_2\tilde R_2)
% \frac{24z^2}{14z}(K_1R_3+K_2\tilde R_3)\right]
\noindent The subdominant operators are denoted by $R_i$ and
$\tilde{R_i}$ and read ($R_4$ will be needed below)
\begin{eqnarray}
\label{r0qt}
R_0&=&Q_S+\tilde Q_S+\frac{1}{2}Q\qquad
\tilde Q_S=(\bar b_is_j)_{SP}(\bar b_js_i)_{SP}\\
\label{rrt1}
R_1&=&\frac{m_s}{m_b}(\bar b_is_i)_{SP}(\bar b_js_j)_{S+P}\\
\label{rrt2}
R_2&=&\frac{1}{m^2_b}(\bar b_i {\overleftarrow D}_{\!\rho}
\gamma^\mu(1\gamma_5)
D^\rho s_i)( \bar b_j\gamma_\mu(1\gamma_5)s_j)\\
\label{rrt3}
R_3&=&\frac{1}{m^2_b}(\bar b_i{\overleftarrow D}_{\!\rho}
(1\gamma_5)D^\rho s_i) (\bar b_j(1\gamma_5)s_j)\\
\label{rrt4}
R_4&=&\frac{1}{m_b}(\bar b_i(1\gamma_5)iD_\mu s_i)
(\bar b_j\gamma^\mu(1\gamma_5)s_j).
\end{eqnarray}
\noindent The $\tilde{R}_i$ denote the colorrearranged operators
that follow from the expressions for $R_i$ by interchanging $s_i$ and
$s_j$. In deriving (\docLink{slacpub7165002.tcx}[tres]{14}) we omitted total derivative terms,
because fourmomentum is conserved in the
forward scattering amplitude.
The operators $R_i$ and $\tilde{R}_i$ are not all independent at order
$1/m_b$. Relations can be derived by using the equations of motion
and omitting total derivatives. To reduce $R_0$, one can start from
the Fierz identity
\begin{eqnarray}\label{fierz}
(\bar{b}_i\gamma_\mu(1\gamma_5)s_i)(\bar{b}_j\gamma_\nu
(1\gamma_5)s_j) &=& \\[0.1cm]
&&\hspace*{4cm}  (\bar{b}_i\gamma_\mu(1\gamma_5)s_j)
(\bar{b}_j\gamma_\nu(1\gamma_5)s_i)
+ \frac{1}{2}g_{\mu\nu}\,(\bar{b}_i\gamma^\lambda(1\gamma_5)s_j)
(\bar{b}_j\gamma_\lambda(1\gamma_5)s_i)\nonumber
\end{eqnarray}
\noindent and apply derivatives in an appropriate way. Up to
corrections of $1/m_b$ (or less), we find
\begin{eqnarray}
\label{relations}
R_0 &=& 2 \tilde{R_1}R_2+2 R_4\nonumber\\[0.1cm]
\tilde{R}_0 &=& R_0\nonumber\\[0.1cm]
\tilde{R}_2 &=& R_2\\[0.1cm]
\tilde{R}_3 &=& R_3+R_2/2\nonumber\\[0.1cm]
\tilde{R_4} &=& R_4+\tilde{R_1}R_1R_2 .\nonumber
\end{eqnarray}
\noindent The first of these relations shows explicitly that
the matrix element of $R_0$ is $1/m_b$ suppressed compared to
$Q$, which is not directly evident from its definition above.
At this point, we have expressed the $1/m_b$ corrections
to $\Delta \Gamma_{B_s}$ in terms
of five new unknown parameters, in addition to the two nonperturbative
parameters that appear already at leading order, and which also contain
implicit $1/m_b$ corrections. In principle they can all be obtained
within the framework of lattice gauge theory\footnote{The matrix elements
of the subleading operators could be evaluated in the static limit.
However, to consistently include all $1/m_b$ corrections,
$\langle Q\rangle$ and $\langle Q_S\rangle$ must be computed either
in full QCD or in Heavy Quark Effective Theory including $1/m_b$
corrections to the Lagrangian as well as to the effective
theory operators. The
parametrization of $1/m_b$ corrections to $\langle Q\rangle$ has
been analyzed in \cite{16}.}.
Unfortunately, results accurate to $10\%$ are not
yet available, especially not for $\langle Q_S\rangle$ (and all the
subleading operators). We therefore adopt the following strategy: we
parametrize the two operators that appear at leading order. They
can be estimated in vacuum insertion or the large $N_c$ limit, but
should ultimately be computed on the lattice.
The operators $R_i$, $\tilde R_i$, on the other hand, are only of
subleading importance and we shall content ourselves here with the
factorization approximation.
Following standard conventions we express the matrix elements of
$Q$ and $Q_S$ in terms of the corresponding `bag' parameters $B$
and $B_S$
\begin{eqnarray}\label{meqs}
\langle Q\rangle &=& f^2_{B_s}M^2_{B_s}2\left(1+\frac{1}{N_c}\right)B,
\\[0.1cm]
\langle Q_S\rangle\!\! &=& f^2_{B_s}M^2_{B_s}
\frac{M^2_{B_s}}{(m_b+m_s)^2}\left(2\frac{1}{N_c}\right)B_S,
\end{eqnarray}
\noindent where $M_{B_s}$ and $f_{B_s}$ are the mass and decay constant
of the $B_s$ meson and $N_c$ is the number of colors.
The parameters $B$ and $B_S$ are defined such that
$B=B_S=1$ corresponds to the
factorization (or `vacuum insertion') approach, which can provide
a first estimate. Factorization of fourfermion operators
is a controlled approximation
only for large $N_c$ or for a nonrelativistic system.
In the large $N_c$ limit, $B=3/4$ and $B_S=6/5$.
In the sense of these limiting cases, factorization
for realistic $B_s$ mesons
can be expected to yield the correct order of magnitude and, in
particular, the right sign of these matrix elements.
Existing nonperturbative calculations like lattice simulations
for $\langle Q\rangle$, and for its counterpart in the
$K\bar K$ system, are in agreement with this expectation.
Beyond these limits factorization does not reproduce the
correct renormalization scale and scheme dependence,
necessary to cancel the
corresponding, unphysical dependences in the Wilson coefficients.
This raises the additional question, to which we return
below, at what scale factorization should be employed to
estimate the matrix elements.
Without further information a certain variation
of the parameters $B$, $B_S$ should be allowed in performing
a numerical analysis.
Next we consider the subleading operators $R_i$, $\tilde{R_i}$,
where we apply factorization.
Using relations such as ($\alpha,\beta$ refers to
spinor indices, $i,j$ to color as before)
\begin{equation}
\langle\bar{B}_s\bar b_{\alpha i} {\overleftarrow D}_{\!\rho}
D^\rho s_{\beta j}0\rangle = \frac{1}{2} (m_b^2M_{B_s}^2)\,
\langle\bar{B}_s\bar b_{\alpha i} s_{\beta j}0\rangle ,
\end{equation}
\noindent valid to first order in $1/m_b$, all
matrix elements can be expressed in terms of $f_{B_s}$,
$M_{B_s}$ and quark masses. We find
\begin{eqnarray}
\langle R_0\rangle &=& f^2_{B_s}M^2_{B_s}\left(1+\frac{1}{N_c}\right)
\left(1\frac{M^2_{B_s}}{(m_b+m_s)^2}\right)\nonumber\\[0.1cm]
\langle R_1\rangle &=& f^2_{B_s}M^2_{B_s}\frac{m_s}{m_b}
\left(2+\frac{1}{N_c}\right)\nonumber\\[0.1cm]
\langle\tilde R_1\rangle &=& f^2_{B_s}M^2_{B_s}\frac{m_s}{m_b}
\left(1+\frac{2}{N_c}\right)\nonumber\\
\label{mer2}
\langle R_2\rangle &=& f^2_{B_s}M^2_{B_s}
\left(\frac{M^2_{B_s}}{m^2_b}1\right)
\left(1+\frac{1}{N_c}\right)\\[0.1cm]
\langle\tilde R_2\rangle &=& f^2_{B_s}M^2_{B_s}
\left(\frac{M^2_{B_s}}{m^2_b}1\right)
\left(1\frac{1}{N_c}\right)\nonumber\\[0.1cm]
\langle R_3\rangle &=& f^2_{B_s}M^2_{B_s}
\left(\frac{M^2_{B_s}}{m^2_b}1\right)
\left(1+\frac{1}{2N_c}\right)\nonumber\\[0.1cm]
\langle\tilde R_3\rangle &=& f^2_{B_s}M^2_{B_s}
\left(\frac{M^2_{B_s}}{m^2_b}1\right)
\left(\frac{1}{2}+\frac{1}{N_c}\right).\nonumber
\end{eqnarray}
Combining the above results, one can obtain $\Delta\Gamma_{B_s}$
from (\docLink{slacpub7165002.tcx}[tres]{14}). The sensitivity to $V_{cb}$ may be eliminated
by normalizing to the total decay rate $\Gamma_{B_s}$ expressed
in terms of the semileptonic width and branching ratio
\begin{equation}\label{gbsl}
\Gamma_{B_s}=\frac{\Gamma(B_s\to Xe\nu)}{B(B_s\to Xe\nu)}=
\frac{G^2_Fm^5_b}{192\pi^3}V_{cb}^2
\frac{g(z)\,\tilde{\eta}_{QCD}}{B(B_s\to Xe\nu)},
\end{equation}
\begin{equation}\label{gzdef}
g(z)=18z+8z^3z^412z^2\ln z,
\end{equation}
\noindent where
$B(B_s\to Xe\nu)$ is to be taken from experiment\footnote{Since
we show in Sect.~\docLink{slacpub7165004.tcx}[bsbd]{4} that the lifetime difference between
$B_s$ and $B_d$ is tiny, no attention has to be paid to the
flavor content of the $B$ meson.} and $z=m^2_c/m^2_b$ as before.
$\tilde{\eta}_{QCD}$ denotes the oneloop QCD corrections ($m_b$ refers
to the $b$quark pole mass). Their analytic
expression can be found in \cite{17}. At $m_b=4.8\,$GeV,
$m_c=1.4\,$GeV, $\mu=m_b$, and with $\alpha_s(m_b)=0.216$
one has $\tilde{\eta}_{QCD}=0.88$. Since
radiative corrections to $\Delta \Gamma_{B_s}$ are not yet known,
the inclusion of radiative corrections to the semileptonic width
seems somewhat arbitrary. On the other hand, with $V_{cb}=0.04$ and
$\Gamma^{1}_{B_s}=1.54\,$ps, one obtains $m_b\approx 4.8\,$GeV from
(\docLink{slacpub7165002.tcx}[gbsl]{28}), compared to $m_b\approx 4.5\,$GeV without QCD
corrections. We prefer the first value as our central choice for
$m_b$ in the numerical analysis,
but repeat that, in the absence of radiative corrections
to $\Delta \Gamma_{B_s}$, $\tilde{\eta}_{QCD}$ can as well be
considered as a normalization uncertainty that replaces the
normalization uncertainty due to the errors in $V_{cb}$ and
$\Gamma_{B_s}$. Finally one arrives at the following expression:
\begin{eqnarray}\label{dgres}
\frac{\Delta\Gamma_{B_s}}{\Gamma_{B_s}} &=& 16\pi^2
B(B_s\to X e\nu)\frac{\sqrt{14z}}{g(z)\,\tilde\eta_{QCD}}
\frac{f^2_{B_s}M_{B_s}}{m^3_b} \,V^2_{cs} \cdot \nonumber \\
&& \cdot \,\Bigg[\left(2(1z)K_1+(14z)K_2\right)
\left(1+\frac{1}{N_c}\right)B \\
&&\hspace*{0.3cm}+\,(1+2z)\left(K_2K_1\right)
\frac{M^2_{B_s}}{(m_b+m_s)^2}
\left(2\frac{1}{N_c}\right)B_S\,+\, \delta_{1/m} + \delta_{rem}\Bigg].
\nonumber
\end{eqnarray}
\noindent
$\delta_{1/m}$ is related to $\hat{\delta}_{1/m}$, defined in
(\docLink{slacpub7165002.tcx}[oneoverm]{16}), through
\begin{equation}
\hat{\delta}_{1/m} = f_{B_s}^2 M_{B_s}^2 \delta_{1/m} ,
\end{equation}
\noindent and from now on we imply that (\docLink{slacpub7165002.tcx}[mer2]{27}) is used.
We have indicated by $\delta_{rem}$ the contributions from CKMsuppressed
intermediate states $(u\bar{c}, \bar uc, u\bar{u})$ and from penguin
operators in the $\Delta B=1$ effective Hamiltonian, which are
estimated in the Appendices A and B
to be below $\pm 3\%$ and about $5\%$,
respectively, relative to the leading order contribution.
We shall neglect $\delta_{rem}$ in the analysis to follow.
Since $f_{B_s}\sim \Lambda_{QCD}^{3/2}/m_b^{1/2}$, $\Delta\Gamma_{B_s}/
\Gamma_{B_s}\sim 16\pi^2 (\Lambda_{QCD}/m_b)^3$ as in the estimate
(\docLink{slacpub7165001.tcx}[estimate]{7}). Eq.~(\docLink{slacpub7165002.tcx}[dgres]{30}) is valid to leading
(${\cal O}(1/m^3_b)$) and nexttoleading order
(${\cal O}(1/m^4_b)$) in the heavy quark expansion. The most important
neglected terms are radiative corrections of order ${\cal O}(\alpha_s/
m_b^3)$. Implicit here is the assumption that the quantity
$(\Delta\Gamma/\Gamma)_{B_s}$
can indeed be represented to reasonable accuracy by the series in powers
of $\Lambda_{QCD}/m_b$ that is generated by the heavy quark
expansion. As mentioned earlier, this assumption is equivalent to
the assumption of local quark hadron duality.
The leading term in (\docLink{slacpub7165002.tcx}[dgres]{30}), represented by the
contributions proportional to $B$ and $B_S$, agrees with the results
that have been given previously in the literature\footnote{Often
factorization is assumed for the leading order term, so that
$B$ and $B_S$ have
to be set to unity to recover the result.} \cite{6,7,8,9,10}.
Note that we have consistently kept the distinction between quark masses,
arising from the shortdistance loops or the equations of
motion, and the meson mass $M_{B_s}$ from hadronic matrix elements,
since we are aiming at effects beyond leading order in the
heavy quark expansion.
In (\docLink{slacpub7165002.tcx}[dgres]{30}), $K_1,K_2$ and $B,B_S$ should be evaluated at a
scale of order $m_b$. If we wanted to use vacuum insertion to estimate
the bag factors, it is physically clear, especially in the heavy
quark limit $m_b\to \infty$, that vacuum insertion should be applied not
at the scale $m_b$, but at a typical hadronic scale $\mu_h\sim 1\,$GeV.
This still leaves us with an ambiguity as to the choice of $\mu_h$
and in addition with the question, how $B(\mu_h)=B_S(\mu_h)=1$ are
related to $B(m_b)$ and $B_S(m_b)$. This latter question can be
answered in the limit $\mu_h\ll m_b$ and corresponds to
the inclusion of `hybrid logarithms' \cite{18,19}, as done in
\cite{10}. The evolution from $m_b$ to $\mu_h$ is performed in the
leading logarithmic approximation in the static theory and leads
to\footnote{We have checked the calculation of hybrid logarithms and
agree with the findings of \cite{10}.}
\begin{eqnarray}\label{bhlog}
B(m_b) &=& 1\\[0.1cm]
B_S(m_b)\!\!&=& 1\frac{3}{5}\left(1\left[
\frac{\alpha_s(m_b)}{\alpha_s(\mu_h)}\right]^{8/25}\right). \nonumber
\end{eqnarray}
\noindent The first equation in (\docLink{slacpub7165002.tcx}[bhlog]{32}) reflects
the wellknown result that the matrix element of the operator $Q$
has the same leading logarithmic corrections in the static
theory (HQET) as the square of the decay constant,
$f^2_{B_s}$. Taking $\mu_h=0.5,1,2\,$GeV results in
$B_S(m_b)=0.80,0.88,0.94$. (The scale $\mu_h=0.5\,$GeV might
already be too low for a perturbative evolution.)
The $b$quark mass $m_b\approx 4.8\,$GeV is probably not large enough
to make this estimate realistic, even if factorization held at
the scale $\mu_h$. The logarithm $\ln m_b/\mu_h$
is not very large, so that other contributions like
nonlogarithmic ${\cal O}(\alpha_s)$ terms
which are omitted in (\docLink{slacpub7165002.tcx}[bhlog]{32}),
can be expected to be numerically of the same order as the hybrid
logarithms that are retained, especially since summing hybrid
logarithms amounts to a moderate $10\%$ effect (with $\mu_h=1\,$GeV).
The oneloop matching of $Q$ on its counterpart(s) in Heavy
Quark Effective Theory indeed exhibits sizeable cancellations
between logarithms and constants, at least in the particular matching \
scheme considered in \cite{20}. Furthermore, the QCD renormalization
between $m_b$ and $\mu_h$ in (\docLink{slacpub7165002.tcx}[bhlog]{32}) is only valid at leading
order in HQET and neglects $1/m_b$ corrections in the matrix
elements, which is not consistent with our keeping of
explicit $1/m_b$ corrections. On the other hand the $B$ factors
are in principle calculable in
full QCD. In this case they will automatically include $1/m_b$
corrections as well as the hybrid logarithms, among further
important contributions.
For these reasons we prefer to keep the expression for
$(\Delta\Gamma/\Gamma)_{B_s}$ in the form given in (\docLink{slacpub7165002.tcx}[dgres]{30}) and do
not include hybrid renormalization explicitly, with the
understanding that the bag factors will eventually be available
from lattice QCD. In our numerical analysis, we take the conservative,
but perhaps too agnostic attitude that $B_S(m_b)$ could take any
value between $0.7$ and $1.3$, keeping in mind (\docLink{slacpub7165002.tcx}[bhlog]{32}) as
a particular model estimate of $B$ and $B_S$.
The upper end of this range is motivated by the
$N_c\to\infty$ limit, in which $B_S=6/5$.
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