\subsection{Non-Leptonic Decays}
%
%
%
%
In contrast to exclusive nonleptonic processes the $1/m_Q$ 
expansion allows a systematic, QCD-based description of fully 
inclusive nonleptonic decays. In this way one may indeed calculate 
lifetimes and branching fractions. 

We shall first point out some 
points concerning the operator product expansion for inclusive 
nonleptonic decays and then consider a few applications.


\subsubsection{Total Rate}
%
%
%
%
The method of combining the operator product expansion with the 
$1/m_Q$ expansion for the matrix elements may also be applied to 
total rates for inclusive nonleptonic processes. In order to 
discuss the assumption which have to be used we write
\begin{equation}
\Phi (q^2, vq) \int d^4 x \, e^{-iqx}  
            \langle H(v) |T \{ {\cal H}_{eff} (x) 
            {\cal H}_{eff}^\dagger (0) \} | H(v) \rangle .
\end{equation}
which is related to the total rate by 
$\Gamma \propto {\rm Im} \Phi (0,0)$. The resason why the variable 
$q^2$ is introduced is that one may study the analytical structure 
of the correlator of two effective hamiltonians. Similarly to the 
semileptonic case we consider $\Phi$ as a function of $vq$ for fixed 
$q^2$. The analytical structure looke very similar and is depcited  
in fig.\ref{anastruc1}. 

However, there is an important difference between semileptonic decays 
and nonleptonic processes. While in the semileptonic case the integration 
over the neutrino momentum corresponded to a contour integration in 
the complex $vq$ plane, in the nonleptonic case one has to evaluate 
$\Phi$ at a specific point $q^2 = 0$ and $vq = 0$ (equivalent to $q=0$). 
In other words, there is no ``smearing'' in the nonleptonic case and 
hence one has to use local duality. Although this is in general 
a dangerous assumption, in the present case it seems to be justified, 
since $\Phi$ is evaluated at a point far away from the resonance region,
where (for $q^2 = 0$) $vq = (m_Q^2 - m_*^2)/(2 m_Q) = {\cal O} (m_Q)$
\cite{blokMan}. Still it is generally believed that the application of the 
operator product expansion is not as well justified as in the semileptonic
case. 
 
 \begin{figure}
   \epsfysize=7cm
   \centerline{\epsffile{anastruc1.eps}}
   \centerline{\parbox{16cm}{\caption[]{\label{anastruc1}
Sketch of the complex $vq$ plane for $q^2 = 0$ for 
inclusive nonleptonic decays. The thick lines along 
the real axis indicate the branch cuts 
of the function $\Phi$. They correspond to 
$vq < (m_Q^2  - m_*2)/(2m_Q)$ and 
$vq > ... $, 
where $m_*$ is the mass of the lightest hadronic final state which 
may appear in the inclusive decay.  The arrow points at $vq= 0$ which 
is the point to be evaluated for the rate. The resonance region is at 
the right edge of the left-hand cut.
}}}
\end{figure}


\subsubsection{Lifetimes of $B^\pm$, $B^0$ and $\Lambda_b$}
%
%
%
%
The subject of heavy hadron lifetimes is strongly related to nonleptonic 
processes, which have been considered already some time ago 
\cite{rueckl,CabbMaia}; however, the systematic application 
of the $1/m_Q$ 
expansion has turned many assumptions into quantitative arguments. 
As outlined in the last section the $1/m_Q$ expansion allows to calculate
total rates, even for non-leptonic processes, 
and hence a QCD-based calculation 
of lifetimes becomes possible. 

Mainly due to the LEP experiments the data on $b$ hadron lifetimes is 
quite precise. In particular the lifetime ratios are known with precisions
well below ten percent \cite{Joe}
\begin{eqnarray}\label{taudata}
   {\tau(B^-)\over\tau(B_d)} &=& 1.02\pm 0.04 \,, \nonumber\\
   {\tau(B_s)\over\tau(B_d)} &=& 1.01\pm 0.07 \,, \nonumber\\
   {\tau(\Lambda_b)\over\tau(B_d)} &=& 0.78 \pm 0.05 \,.
\end{eqnarray}
where $\tau(B_s)$ is the averaged $B_s$-meson lifetime. 

To leading order in the $1/m_Q$ expansion all lifetime ratios are
unity, since the lifetimes are then determined by the free quark decay. 
If we assume light flavour symmetry for the matrix elements $\lambda_1$ 
and $\lambda_2$, the corrections to the ratio between two meson lifetimes
has to be of order $1/m_Q^3$ and is hence given by dimension six 
operators, which we have not yet discussed in the previous sections. 
Since the value of $\lambda_1$ is different in a meson and a baryon
and $\lambda_2 = 0$ for a $\Lambda_Q$ baryon, we expect
\begin{eqnarray}\label{taucrude}
   {\tau(B^-)\over\tau(B_d)} &=& 1 + O(1/m_b^3) \,,
    \nonumber\\
   {\tau(B_s)\over\tau(B_d)} &=& 1 + O(1/m_b^3) \,,
    \nonumber\\
   {\tau(\Lambda_b)\over\tau(B_d)} &=& 1 + O(1/m_b^2)
\end{eqnarray}
and one already recognizes a potential problem with the lifetime ratio
$\tau(\Lambda_b) / \tau(B_d)$. 

In order to discuss the lifetime ratios for mesons one needs to consider
higher orders in the $1/m_Q$ expansion. Unfortunately, in general the 
number of dimension six operators number of dimension six operators
that appear at order $1/m_Q^3$ is quite large. However, not all of 
these operators contribute to lifetime ratios, since they have to be 
sensitive to the light quark inside the heavy hadron. 

These types of spectator effects have been reconsidered recently 
in \cite{NeuSach} and we shall follow this line of argument.  
Although intuitively these dimension six operators seem to yield 
only small contributions, their effect is enhanced by a phase space 
factor. The non-spectator effects typically yield at the partonic level 
at least a three particle phase space, while the spectator effects 
have to leading order only a two particle final state. This yields 
an enhancement of these contributions by a phase space factor 
$16\pi^2$. Qualitatively, this means that the $1/m_Q$ expansion 
for the lifetime ratios takes the form
\begin{equation}
{\tau(B^-)\over\tau(B_d)} = 1 + \frac{1}{m_b^3} 
                          \left[a_0 +  \frac{1}{m_b} a_1 + \cdots \right]
                          + \frac{16\pi^2}{m_b^3}  
                          \left[b_0 +  \frac{1}{m_b} b_1 + \cdots \right]
\end{equation}
and hence the coefficient $b_0$ is the leading contribution to the 
lifetime ratio. 

We start from the effective hamiltonian for nonleptonic weak decays 
(see also (\ref{Rnonlep}) 
\begin{eqnarray}
   H_{eff} &=& - {4 G_F\over\sqrt{2}}\,V_{cb}\,
    \bigg\{ (C_+ (m_b) + C_- (m_b)) \,\Big[
    \bar d'_L\gamma_\mu u_L\,\bar c_L\gamma^\mu b_L +
    \bar s'_L\gamma_\mu c_L\,\bar c_L\gamma^\mu b_L \Big] \nonumber\\
   &&\phantom{ - {4 G_F\over\sqrt{2}}\,V_{cb}\, }
    \mbox{}+ (C_+ (m_b) - C_- (m_b))\,\Big[
    \bar c_L\gamma_\mu u_L\,\bar d'_L\gamma^\mu b_L +
    \bar c_L\gamma_\mu c_L\,\bar s'_L\gamma^\mu b_L \Big] \nonumber
    + \mbox{h.c.} \,,
\end{eqnarray}
where  $d'=d\,\cos\theta_c + s\,\sin\theta_c$ and $s'=s\,\cos\theta_c
- d\,\sin\theta_c$ are the Cabibbo-rotated down type quarks and
$C_+ (\mu) $ and $C_- (\mu)$ are the Wilson coefficients from 
the running from $M_W$ down to the scale $\mu$ which have been given to 
leading logarithmic sccuracy in (\ref{cpm}).

The correction to the decay rate due to the spectator effects may be 
expressed in terms of four quark operators, which we choose to be 
\begin{eqnarray}\label{4qops}
   O_{V-A}^q &=& \bar b_L\gamma_\mu q_L\,\bar q_L\gamma^\mu b_L
    \,, \nonumber\\
   O_{S-P}^q &=& \bar b_R\,q_L\,\bar q_L\,b_R \,, \nonumber\\
   T_{V-A}^q &=& \bar b_L\gamma_\mu t_a q_L\,\bar q_L\gamma^\mu
    t_a b_L \,, \nonumber\\
   T_{S-P}^q &=& \bar b_R\,t_a q_L\,\bar q_L\,t_a b_R \,,
\end{eqnarray}
where $t_a=\lambda_a/2$ are the generators of colour SU(3). In terms 
of these operators the total decay rate becomes \cite{NeuSach}
\begin{eqnarray}\label{Gspec}
   \Gamma &=& \Gamma_0 + 
    {2 G_F^2 m_b^2\over\pi}\,|V_{cb}|^2\,(1-z)^2\,\Bigg\{
    \bigg( 2 c_1 c_2 + {1\over N_c}\,(c_1^2 + c_2^2) \bigg)\,
    <O_{V-A}^u> + 2 (c_1^2 + c_2^2)\,<T_{V-A}^u> \Bigg\} \nonumber \\
   &-& {2 G_F^2 m_b^2\over 3\pi}\,|V_{cb}|^2\,(1-z)^2\,\Bigg\{
    \bigg( 2 c_1 c_2 + {1\over N_c}\,c_1^2 + N_c\,c_2^2 \bigg)\,
    \bigg[ \bigg( 1 + {z\over 2} \bigg)\,<O_{V-A}^{d'}>
    - (1+2z)\,<O_{S-P}^{d'}> \bigg] \nonumber\\
   &&\phantom{ {2 G_F^2 m_b^2\over\pi}\,|V_{cb}|^2\,(1-z)^2\, }
    \mbox{}+ 2 c_1^2\,\bigg[ \bigg( 1 + {z\over 2} \bigg)\,
    <T_{V-A}^{d'}> - (1+2z)\,<T_{S-P}^{d'}> \bigg] \Bigg\} \nonumber\\
   &-& {2 G_F^2 m_b^2\over 3\pi}\,|V_{cb}|^2\,\sqrt{1-4 z}\,\Bigg\{
    \bigg( 2 c_1 c_2 + {1\over N_c}\,c_1^2 + N_c\,c_2^2 \bigg)\,
    \Big[ (1-z)\,<O_{V-A}^{s'}> - (1+2z)\,<O_{S-P}^{s'}> \Big]
    \nonumber\\
   &&\phantom{ {2 G_F^2 m_b^2\over\pi}\,|V_{cb}|^2\,(1-z)^2\, }
    \mbox{}+ 2 c_1^2\,\Big[ (1-z)\,<T_{V-A}^{s'}> - (1+2z)\,
    <T_{S-P}^{s'}> \Big] \Bigg\} \,,
\end{eqnarray}
where $z=m_c^2/m_b^2$, and $N_c$=3 is the number of colours. 

We shall focus here on the mesonic case, the baryonic case is discussed 
in \cite{NeuSach}. The matrix elements of the four quark operators are
parametrized as 
\begin{eqnarray}\label{Bidef}
   {1\over 2 m_{B_q}}\,\langle B_q|\,O_{V-A}^q\,|B_q\rangle
   &\equiv& {f_{B_q}^2 m_{B_q}\over 8}\,B_1 \,, \nonumber\\
   {1\over 2 m_{B_q}}\,\langle B_q|\,O_{S-P}^q\,|B_q\rangle
   &\equiv& {f_{B_q}^2 m_{B_q}\over 8}\,B_2 \,, \nonumber\\
   {1\over 2 m_{B_q}}\,\langle B_q|\,T_{V-A}^q\,|B_q\rangle
   &\equiv& {f_{B_q}^2 m_{B_q}\over 8}\,\varepsilon_1 \,,
    \nonumber\\
   {1\over 2 m_{B_q}}\,\langle B_q|\,T_{S-P}^q\,|B_q\rangle
   &\equiv& {f_{B_q}^2 m_{B_q}\over 8}\,\varepsilon_2 \,.
\end{eqnarray}
where this parametrization is motivated by factorization, which 
has been briefly discussed in connection with exclusive nonleptonic 
decays. Assuming factorization yields $B_i = 1$ and 
$\varepsilon_i= 0$. Factorization holds exactly in the limit of 
large $N_c$ and hence one expects    
\begin{equation}
   B_i = O(1) \,,\qquad \varepsilon_i=O(1/N_c) \,.
\end{equation}

Following \cite{NeuSach} one may now evaluate the lifetime ratio 
between charged and neutral $B$ mesons in terms of $B_i$ and $
\varepsilon_i$ 
\begin{equation}\label{cidef}
   {\tau(B^-)\over\tau(B_d)} = 1 + k_1 B_1 + k_2 B_2
   + k_3\varepsilon_1 + k_4\varepsilon_2 \,,
\end{equation}
where the coefficients $k_i$ are determined by the mass ratio $z$ 
and the Wilson coefficients $c_1$ and $c_2$.  

\begin{table}[htb]
\centerline{\begin{tabular}{|c|cccc|}\hline
\rule[-0.2cm]{0cm}{0.65cm} $\mu$ & $k_1$ & $k_2$ & $k_3$ & $k_4$ \\
\hline
$m_b/2$ & $+0.044$ & 0.003 & $-0.735$ & 0.201 \\
$m_b$   & $+0.020$ & 0.004 & $-0.697$ & 0.195 \\
$2 m_b$ & $-0.008$ & 0.007 & $-0.665$ & 0.189 \\
\hline
\end{tabular}}
\caption[]{Values of the coefficients $k_i$ for different choices of the 
renormalization scale $\mu$. The value of $z$ is fixed at $z=0.085$.
The table is taken from 
\protect{\cite{NeuSach}}.}
\label{tab1}
\end{table}

In earlier work \cite{bigilife} the assumption of factorization has been 
made and hence the matrix elements of the octet-octet operators 
$\varepsilon_i$ vanish. Using this assumption one obtains for the 
lifetime ratio
\begin{equation}
\frac{\tau (B^-)}{\tau (B_d)} =
1+0.05\cdot \frac{f_B^2}{(200\; \mbox{MeV} )^2}
\end{equation}
which is certainly compatible with data.

On the other hand, the coefficients $k_3$ and $k_4$ of the octett-octett
operators are
larger by a factor five to ten, and if one expects these matrix elements 
to be suppressed by a factor $1/N_C$ it is a possible scenario that 
the nonfactorizable contributions actually dominate the corrections 
to the lifetime ratio.  

The lifetime of the $\Lambda_b$ baryon has also been considered within 
the framework of the $1/m_Q$ expansion. The general conclusion of the 
discussion is that the lifetime ratio $\tau (\Lambda_b)/ \tau (B_d)$
should be slightly smaller than unity, but cannot be as low as the 
measured value for ``reasonable'' values of the parameters. This fact
constitutes a potential problem for the $1/m_Q$ expansion for inclusive
nonleptonic decays, which is still subject of topical research.



\subsubsection{Semileptonic Branching Fraction}
%
%
%
%
Once one is able to calculate lifetimes one may also directly calculate 
the semileptonic branching fraction of $b$ hadrons within the $1/m_Q$ 
expansion. 

The total decay rate is composed of three contributions 
\begin{equation}
\Gamma_{tot} = \Gamma (b \to c \ell \nu) 
+  \Gamma (b \to c \bar{u} d') + \Gamma (b \to c\bar{c}s')
\end{equation}
where the prime indicates the CKM rotated fields 
\begin{eqnarray}
d' &=& V_{ud} d + V_{us} s + V_{ub} b \\
s' &=& V_{cd} d + V_{cs} s + V_{cb} b
\end{eqnarray} 
For further use it is useful to define the ratios 
\begin{equation}
R_{ud} = \frac{\Gamma (b \to c \bar{u} d') }{3\Gamma (b \to c e \nu)}
\quad
R_{cs} = \frac{\Gamma (b \to c \bar{c} s') }{3\Gamma (b \to c e \nu)}
\end{equation}
where the semileptonic rate is given in terms of the semielectronic one 
by
\begin{equation}
\Gamma (b \to c \ell \nu) = 3 f(m_\tau) \Gamma (b \to c e \nu)
\end{equation}
where $f(m_\tau)$ is a known function that 
takes into account phase space effects due to the 
non-negligible $\tau$ mass, $f(m_\tau = 1777 \mbox{ MeV}) = 0.74$. 
These relations allow us to express
the sum of the ratios $R_{ud} + R_{cs}$ in terms of the semileptonic 
branching fraction 
\begin{equation}
R_{ud} + R_{cs} = f(m_\tau) 
\frac{1-Br(b \to c \ell \nu)}{Br(b \to c \ell \nu)}
\end{equation}    
Using the 1996 Particle Data Group data \cite{PDG1996} 
\begin{eqnarray}
Br(B \to X e \nu) &=& (10.4 \pm 0.4) \% \\
Br(B \to X \mu \nu) &=& (10.3 \pm 0.5) \% \\
Br(B \to X \tau \nu) &=& (2.7 \pm 0.4) \% 
\end{eqnarray}
one obtains the result
\begin{equation}
R_{ud} + R_{cs} = 2.37 \pm 0.12
\end{equation}

Up to here we have only used data, and now we discuss the theoretical
predictions for the sum $R_{ud} + R_{cs}$. Due to the mass of the $c$ 
quark one intuitively expects that the calculation of $R_{ud}$ might 
be more reliable than the evaluation of $R_{cs}$, since the latter channel
has two heavy quarks in the final state and one is thus closer to the 
resonance region.

Furthermore, as has been mentioned already several times, the non-perturbative 
corrections are small compared to the perturbative ones, and hence 
the latter will contribute the bulk part of the effect, while the 
non-perturbative corrections are almost negligible. The leading and subleading
perturbative corrections have been available for some time, neglecting the 
masses of the final state quarks \cite{AltarNonlep}. Recently the complete 
calculation of the subleading terms including the masses of the final state 
quarks has been performed \cite{Ball,Ballnew}. However, the mass effects are 
not drastic and do not change the conclusions drawn from the massless case, 
as far as he semileptonic branching fraction is concerned.   

The theoretical expressions for $R_{ud}$ and $R_{cs}$ depend on both quark 
masses $m_b$ and $m_c$. In the following analysis we shall exploit the 
fact that the mass difference  $m_b - m_c$ is quite well known; in the 
heavy mass limit it simply becomes  $<M_B> - <M_D>$, where  $<M_H>$ denotes 
the spin averaged mass on the heavy ground state meson
\begin{equation}
<M_H> = \frac{1}{4}(m_H + 3 m_{H^*}) 
\end{equation}
such that the mass difference is of the order $1/m_Q^2$ with 
\begin{equation}
m_b - m_c = \lambda_1 \left( \frac{1}{2m_b} - \frac{1}{2m_c} \right) 
\end{equation}
Hence a good choice of input parameters is the $b$ quark mass 
$m_b$, the renormalization scale $\mu$ entering the expressions for the 
QCD radiative corrections and the QCD scale parameter 
$\Lambda_{\overline{\mbox{MS}}}^{(5)}$.   

First we shall put in some ``reasonable'' values $\mu = m_b = $ 4.8 GeV 
and $\Lambda_{\overline{\mbox{MS}}}^{(5)} =$ 180 MeV and evaluate from this 
the sum  $R_{ud} + R_{cs}$; we obtain 
\begin{equation}
R_{ud} + R_{cs} = 1.73, 
\end{equation}
while for each individual contribution we have $R_{ud} = 1.27$ and 
$R_{cs} = 0.46$. As we have argued above we expect that $R_{ud}$ may 
be calculated more reliably in the $1/m_Q$ expansion, so we may take 
this calculation and vary the renormalization scale. Since $R_{ud}$
becomes larger as the scale is lowered, we may think of $\mu = m_b / 3$
as a reasonable choice of the renormalization scale, since the energy 
release is shared among the three quarks in the final state. For this
choice one obtains $R_{ud} = 1.52$; using the data we would conclude 
$R_{cs} = 0.85$ which is from the theoretical estimates on the large side. 

Large values like that may be obtained for instance by lowering the 
value of the $b$ quark mass $m_b$; for a value $m_b =$ 4.4 GeV one 
would obtain a $R_{cs}$ in the region as discussed above; however, this 
would imply a value of the $c$ quark mass as low as $m_c = $ 1.1 GeV, 
which is at the lower edge of the commonly accepted values of this 
parameter. 

Another possibility are potentially large QCD radiative corrections appearing 
in the $c\bar c s$ channel. These corrections have been studied previously 
by Ball et al.\cite{Ball,Ballnew}, who find larger QCD radiative corrections 
in this channel due to the long range Coulomb-like interaction between 
the two heavy $c$ quarks in the final state. These corrections shift the result
into the 
right direction, but they do not seem sufficiently large to explain the data.

Finally, all the calculations of the non-leptonic rate are based on the 
concept of parton-hadron duality, which is expected to hold on intuitive grounds; 
however, it may fail for yet unknown reasons, rendering the calculation 
of the non-leptonic inclusive rates useless. On the other hand, it has 
been pointed out \cite{bigilife} that such a failure should show up 
somehow in the $1/m_Q$ expansion, but the $1/m_Q^3$ terms considered so far 
do not give any hint that local duality may fail.    

A large value of $R_{cs}$ has phenomenological implications of its 
own, since it will enhance the number of charm and anticharm quarks produced
in $b$ decays. Defining the number $n_c$ of charm quarks produced per $b$ decay
as (note that it is the sum of $c$ and $\bar c$)
\begin{equation}
n_c = Br(b \to c \ell \nu) + Br (b \to c \bar u d') 
+ 2 Br (b \to c \bar{c} s')
\end{equation}
we may express this quantity in terms of the ratio $R_{cs}$ as
\begin{equation}
n_c = 1 + R_{cs} \frac{Br(b \to c \ell \nu)}{f(m_\tau)}.
\end{equation}
Thus we conclude that for a value of $R_{cs}$ large enough to explain the 
semileptonic branching ratio we have to have  $n_c \sim $ 1.3. 

On the other hand, we may obtain $n_c$ experimentally by summing over the 
various semiinclusive modes. There are experimental data available at 
high and at low energies. At high energies the value of $n_c$ is
\cite{nchigh} $n_c = 1.23 \pm 0.07$ while the average at low energies 
is \cite{nclow} $n_c = 1.12 \pm 0.05$. The weighted average of these 
values is $n_c = 1.16 \pm 0.04$, which is somewhat lower than what one has 
from the semileptonic branching fraction, but this discrepancy is not yet
significant. 

Thus the potential problem is not the semileptonic branching fraction alone, 
but rather the semileptonic branching fraction in combination with the charm 
counting in $b$ decays. Possible resolutions of this problem are that the 
data on charm counting improve and indeed indicate a larger value such as 
$n_c \sim 1.3$. However, if future data confirm a value of $n_c$ as low as 
$n_c = 1.1$ with significantly smaller errors, this would either 
imply a failure of parton-hadron duality even in the ``safe'' channel $c \bar u d$, 
or, if duality were shown to hold in this channel, could indicate a
large branching fraction of $B$ mesons into non-charmed final states, as large  
as 
$$
Br(b \to charmless\,\, final\,\, states) \sim 20 \% .
$$
This is very unlikely 
in the SM, but possible in non-SM scenarios, which enhance the channel 
$b \to s g$. 
