%Title: Heavy quark expansion and inclusive decays
%Author: Thomas Mannel
%Published:     Heavy Flavor Physics. Proceedings of the Seventh International Symposium, University of California, Santa Barbara, 7 - 11 July 1997. Edited by C. Campagnari. Singapore, World Scientific, 1999, p. 61-75.
%====================================================================%
%                  proceedings.tex   24.9.97
%====================================================================%

\documentstyle[sprocl]{article}
%--------------------------------
\makeatletter
\def\fmslash{\@ifnextchar[{\fmsl@sh}{\fmsl@sh[0mu]}}
\def\fmsl@sh[#1]#2{%
  \mathchoice
    {\@fmsl@sh\displaystyle{#1}{#2}}%
    {\@fmsl@sh\textstyle{#1}{#2}}%
    {\@fmsl@sh\scriptstyle{#1}{#2}}%
    {\@fmsl@sh\scriptscriptstyle{#1}{#2}}}
\def\@fmsl@sh#1#2#3{\m@th\ooalign{$\hfil#1\mkern#2/\hfil$\crcr$#1#3$}}
\makeatother
%--------------------------------
%\input{psfig}
\input{epsf}
\bibliographystyle{unsrt}    % for BibTeX - sorted numerical labels by order of
                             % first citation. 

% A useful Journal macro
\def\Journal#1#2#3#4{{#1} {\bf #2}, #3 (#4)}

% Some useful journal names
\def\NCA{\em Nuovo Cimento}
\def\NIM{\em Nucl. Instrum. Methods}
\def\NIMA{{\em Nucl. Instrum. Methods} A}
\def\NPB{{\em Nucl. Phys.} B}
\def\PLB{{\em Phys. Lett.}  B}
\def\PRL{\em Phys. Rev. Lett.}
\def\PRD{{\em Phys. Rev.} D}
\def\ZPC{{\em Z. Phys.} C}
% Some other macros used in the sample text
\def\st{\scriptstyle}
\def\sst{\scriptscriptstyle}
\def\mco{\multicolumn}
\def\epp{\epsilon^{\prime}}
\def\vep{\varepsilon}
\def\ra{\rightarrow}
\def\ppg{\pi^+\pi^-\gamma}
\def\vp{{\bf p}}
\def\ko{K^0}
\def\kb{\bar{K^0}}
\def\al{\alpha}
\def\ab{\bar{\alpha}}
\def\be{\begin{equation}}
\def\ee{\end{equation}}
\def\bea{\begin{eqnarray}}
\def\eea{\end{eqnarray}}
\def\CPbar{\hbox{{\rm CP}\hskip-1.80em{/}}}%temp replacement due to no font


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                %
%    BEGINNING OF TEXT                           %
%                                                %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{document}

\title{HEAVY QUARK EXPANSION AND INCLUSIVE DECAYS}

\author{THOMAS MANNEL}

\address{Institut f\"{u}r Theoretische Teilchenphysik, 
         Universit\"{a}t Karlsruhe, \\
         D--76128 Karlsruhe, Germany}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% You may repeat \author \address as often as necessary      %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\maketitle\abstracts{
Some of the recent developments in the theory and applications of 
the heavy quark expansion for inclusive processes are reviewed. 
The topics concern semileptonic and nonleptonic decays, lifetimes 
and the semileptonic branching fraction.} 
  
\section{Introduction}
%
%
%
%
The heavy mass expansion has become the standard tool in the 
theoretical description of heavy quark processes, since in 
many cases it reduces the model dependence in a significant way
and often allows for purely QCD-based calculations \cite{reviews}. 
The heavy mass limit of QCD may be formulated 
as an effective theory, the 
so called Heavy Quark Effective Theory (HQET)
\cite{HQET}. HQET
exhibits additional symmetries beyond the ones known from QCD
\cite{HQsymmetry},  
and these symmetries restrict the nonperturbative aspects of heavy
quark processes, leading to model independent relations valid in the 
heavy mass limit. Furthermore, the framework of effective theory 
allows us to calculate radiative corrections to the heavy quark 
limit and to access the structure of the recoil (i.e. the $1/m_Q$) 
corrections. 

In order to set up an $1/m_Q$ expansion for inclusive decays HQET is 
combined with the method of the operator product expansion (OPE) 
\cite{inclusive}.The approach applied in this case 
resembles very much the description of deep inelastic scattering (DIS).
While in DIS the large scale is the momentum transfer, one can 
rewrite the expressions in the case of heavy quark decays in such a 
way that the heavy quark mass $m_Q$ is the relevant large scale for 
the (OPE). 

The OPE yields an infinite sum of operators with increasing 
dimension as the first step in obtaining a $1/m_Q$ expansion. 
The dimensions of the operators are compensated by inverse powers 
of a large scale, which is in general of the order of the heavy mass 
scale. The decay probability is then given as forward matrix elements
of these operators between the state of the decaying heavy hadron; 
these matrix elements still have a mass dependence, which then may 
be extracted in terms of a $1/m_Q$ expansion using HQET as for exclusive 
decays. 

In the following we shall first outline the theoretical background  
for inclusive decays and then apply the method to semileptonic and 
nonleptonic processes. 

\section{Theoretical Background}
%
%
%
%
Weak decays of bottom and charm quarks are usually described in terms 
of an effective Hamiltonian ${\cal H}_{eff}$
derived form the full electroweak theory 
in the limit where all relevant scales are much smaller than the 
mass of the weak bosons. Dealing with $\Delta B = \pm 1$ processes
this Hamiltonian contains only a single heavy quark operator. 
Expressed in terms of ${\cal H}_{eff}$ the inclusive decay rate becomes 
\begin{eqnarray} \label{inclusive} 
\Gamma  \propto &&   \sum_X (2 \pi)^4 \delta^4 (P_B - P_X )
| \langle X | {\cal H}_{eff} | B(P_B) \rangle |^2 \\ 
&& = \nonumber 
\int d^4 x \,   \langle B(P_B) |{\cal H}_{eff} (x) 
            {\cal H}_{eff}^\dagger (0) | B(P_B) \rangle 
\\ \nonumber
&& =  2 \mbox{ Im}
\int d^4 x \,   \langle B(P_B) |T \{ {\cal H}_{eff} (x) 
            {\cal H}_{eff}^\dagger (0) \} | B(P_B) \rangle .
\end{eqnarray}
where $| B(P_B) \rangle $ is the decaying heavy meson moving with 
momentum $P_B$ and  $| X \rangle $ is the final state 
with momentum $P_X$, and a sum over $| X \rangle $ is performed 
to obtain the inclusive rate.

The matrix element appearing in (\ref{inclusive}) contains a 
large scale, namely the mass of the heavy quark. The first step 
towards a $1/m_Q$ expansion is to make this large scale explicit. 
This may be achieved by a phase redefinition of the heavy quark 
field removing the large part of the heavy quark momentum $m_Q v$
\begin{equation} \label{phaseredef}
Q_v (x) = e^{im_Q (v \cdot x)} Q (x)     
\end{equation}
This corresponds to the standard splitting of the heavy quark 
momentum $p_Q$ into a large piece and a residual part 
$p_Q = m_Q v +k$; hence the space time dependence of the field 
$Q_v$ is related to the residual momentum $k$ by Fourier 
transformation.  

Inserting this into (\ref{inclusive}) yields
\begin{equation}
\Gamma  \propto 
2 \mbox{ Im} 
\int d^4 x  \, e^{-im_Q vx} \,  
\langle H(v) |T \{ \widetilde{{\cal H}}_{eff} (x) 
   \widetilde{{\cal H}}_{eff} ^\dagger (0) \} | H(v) \rangle 
\end{equation}
where 
\begin{equation} 
\widetilde{{\cal H}}_{eff} ={\cal H}_{eff} ( Q \longrightarrow Q_v)  
\end{equation}
is the effective Hamiltonian with the replacement $Q \to Q_v$.

This relation exhibits the similarity 
between the cross-section calculation in DIS
and the present approach to total rates. In DIS
there appears a large scale, which is the momentum transfer to the 
leptons, while here the mass of the heavy quark appears as a large scale. 

Analogous to DIS, 
the next step is to perform an OPE of the product 
of the two Hamiltonians. After the phase redefinition (\ref{phaseredef})
the remaining matrix element does not involve large momenta of the order 
of the heavy quark mass any more and hence a short-distance expansion  
becomes useful if the mass $m_Q$ is large compared to the scale 
$\bar\Lambda$ determining the matrix element. The OPE 
for the total rate takes the general form
\begin{eqnarray}
&& \int d^4 x  \, e^{im_Q vx} \, 
\langle H(v) |T \{ \widetilde{{\cal H}}_{eff} (x) 
\widetilde{{\cal H}}_{eff}^\dagger (0) \}| H(v) \rangle \\
&& \qquad = \sum_{n=0}^\infty  \left(\frac{1}{2m_Q}\right)^n 
     \hat{C}_{n+3} (\mu) \langle H(v) |{\cal O}_{n+3}| H(v) \rangle_\mu ,
\nonumber 
\end{eqnarray}
where ${\cal O}_n$ are operators of dimension $n$, with their 
matrix elements renormalized at scale 
$\mu$, and $\hat{C}_n$ are the corresponding Wilson coefficients. 
These coefficients encode the short distance physics related to the 
heavy quark mass scale and may be calculated in perturbation theory. 
All long distance contributions connected to the hadronic scale 
$\bar\Lambda$ are contained in the matrix elements of the operators 
${\cal O}_{n+3}$.      

Still the matrix elements of ${\cal O}_{n+3}$ are not independent of the 
heavy quark mass scale, but this mass dependence may be expanded in powers
of $1/m_Q$ by means of HQET. In this way the 
mass dependence of the total decay rate may be accessed completely within 
an expansion in $1/m_Q$.  

The lowest-order term of the OPE are the  
dimension-3 operators. To leading order in the $1/m_Q$ expansion 
only one operator appears, any other operator may be related to 
it up to terms of order $1/m_Q^2$. Usually one chooses 
${\cal O}_3 =  \bar Q_v Q_v$ of which the relevant  
forward matrix element is normalized, since it may be related to 
the conserved current $\bar Q \gamma_\mu Q $.  

Thus the leading contribution can always be evaluated without 
hadronic uncertainties and generically yields the free quark decay rate. 
This has been previously used as a model for inclusive decays, but
now it turns out to be the first term in a systematic $1/m_Q$ expansion 
of total rates. 

Any dimension-four operator may be related to 
the operator 
\begin{equation}
{\cal O}_4  \propto \bar Q_v \Gamma D_\mu Q_v  ,
\end{equation}
the matrix elements of which vanish by the equations of motion. 
This statement is 
equivalent to Luke's theorem \cite{Lu90} derived in the context of 
exclusive decays. 

The first non-trivial non-perturbative contribution comes from
dimension-5 operators and are of order $1/m_Q^2$. Their order of 
magnitude for $B$ decays is small, since $\bar\Lambda^2 /m_Q^2 \sim 
(2-3) \%$. 
 
For mesonic decays there are only the two parameters $\lambda_1$
and $\lambda_2$ defined by the matrix elements between $0^-$ states 
\begin{eqnarray}
2m_B \lambda_1 &=& \langle H (v) | \bar{h}_v  (iD)^2  h_v | H (v) \rangle \\
2m_B \lambda_2 &=& \langle H (v) | \bar{h}_v 
                   \sigma_{\mu \nu} iD^\mu iD^\nu h_v | H (v) \rangle
\end{eqnarray}
The second parameter $\lambda_2$ may be related to the mass splitting 
in the ground state spin symmetry multiplet
\begin{equation} \label{lam2value}
\lambda_2 = 0.12 \mbox{GeV}^2
\end{equation}
We note in passing that the operator defining $\lambda_2$ has 
non-vanishing anomalous dimension and hence $\lambda_2$ in fact 
depends on the scale; the value quoted in  
(\ref{lam2value}) is $\lambda_2 (m_b)$. 

A determination of $\lambda_1$ is much more problematic. For instance, 
$\lambda_1$ may be related to the pole masses of the heavy $b$ and $c$ 
quarks as 
\begin{equation} \label{lam1eq}
m_b - m_c = \bar{M}_B - \bar{M}_D + 
\lambda_1 \left( \frac{1}{2\bar{M}_B} - \frac{1}{2\bar{M}_D} \right) 
+ {\cal O} (1/m_Q^3) 
\end{equation}
where $\bar{M}_{B/D}$ are spin averaged masses of the ground state 
spin symmetry doublet in the $B$ and the $D$ system respectively.  
Relation (\ref{lam1eq}) connects $\lambda_1$ with the quark pole 
masses, which are known to be affected by renormalon ambiguities
\cite{renormalons}; it has been pointed out recently that also
$\lambda_1$ suffers from such a problem \cite{lam1renorm}.  

Once one defines a procedure how to extract a value for $\lambda_1$ 
from experiment or other theoretical considerations one may quote 
various values, which cover quite a large range
\begin{equation}
0.10 \leq - \lambda_1 \leq 0.66
\end{equation}
For details of the various methods we refer the reader to a recent review
\cite{reviews}. In any case, the estimate of the order of magnitude of the 
leading non-perturbative corrections is confirmed by more sophisticated 
methods. 

The higher order terms of the $1/m_Q$ expansion have been investigated 
to some extent. In general, the number of independent operators is 
increasing drastically and a complete analysis becomes difficult. 
However, some of the $1/m_Q^3$ terms are believed to play a role 
in the lifetime patterns of heavy hadrons and we shall return to this 
in section \ref{sec:life}. 

\section{Semileptonic Processes}
%
%
%
%
For semileptonic processes the method of OPE may be applied not only 
to the full $H_{eff}$, but already 
to the product of the two hadronic currents appearing in the 
rate. In this way one may also obtain differential distributions, e.g.
the distribution of the leptonic $q^2$, the energy spectrum of the 
charged lepton as well as the hadronic invariant mass spectrum of
the final state. 

The total rate is obtained by integrating these differential 
distributions, thereby averaging the partonic result for the correlator 
of the two hadronic currents with a weight given by the appropriate 
phase space. Hence in semileptonic processe only the concept of ``global
duality'' is used, i.e. the assumption that sufficiently averaged 
(``smeared'') partonic results equal the corresponding hadronic ones.
For this reason it is generally believed that the results obtained 
for semileptonic processes are on firm theoretical grounds.

In the following we shall quote the results obtained for semileptonic 
processes and discuss the implications for the determination of 
CKM matrix elements.

\subsection{Total Rates and the Determination of $V_{cb}$}
%
%
%
%
Including the first non-trivial non-perturbative corrections 
one obtains for the semileptonic decay $B \to X_c \ell \bar{\nu}_\ell$
to leading order $\alpha_s$ 
\begin{equation} \label{bcsl}
\Gamma(B \to X_c \ell \nu)   
 = \frac{G_F^2 m_b^5}{192\pi^3} |V_{cb}|^2 
      \left[
      \left(1+\frac{\lambda_1}{2m_c^2}\right) z_0 
               \left(\frac{m_c^2}{m_b^2}\right)
      -\frac{9\lambda_2}{2m_c^2} z_2 \left(\frac{m_c^2}{m_b^2}\right)
      \right] , 
\end{equation}
where the two $z_j$ are phase-space functions 
\begin{eqnarray} \label{f1}
z_0(x) &=& 1-8x+8x^3-x^4-12x^2\log x ,
\\  \nonumber
z_2(x) &=& 1-\frac{8}{3}x-8x^2+8x^3+\frac{5}{3}x^4+4x^2\log x  .
\end{eqnarray}
The corresponding expression for the decay $B \to X_u \ell \bar{\nu}_\ell$
is obtained in the limit $m_c \to 0$ and the replacement 
$V_{cb} \to V_{ub}$; it reads
\begin{equation} \label{busl}
\Gamma(B \to X_u \ell \nu) = \frac{G_F^2 m_b^5}{192\pi^3} |V_{ub}|^2
 \left[
      1+\frac{\lambda_1 - 9 \lambda_2}{2m_b^2} 
      \right]   .
\end{equation}
Note that the non-perturbative corrections are indeed small, and 
consequently the main corrections are the radiative corrections 
governed by $\alpha_s (m_Q)$, which are found to be of the order
of 10\% \cite{slqcdcorr}. 

The inclusive decay $B \to X \ell \bar{\nu}_\ell$ is dominated by the 
decays involving charm in the final state and hence the total rate 
for $B \to X \ell \bar{\nu}_\ell$ or more precisely 
$B \to X_c \ell \bar{\nu}_\ell$ may be used to extract $V_{cb}$.
On the first sight this seems to be a bad idea, since superficially
the rate depends on the fifth power of the heavy quark mass which 
is not accurately known, and in addition radiative corrections turned 
out to be large, thereby inducing a sizable scale dependence. 

However, it has been pointed out \cite{vcbincl}, that one may still obtain 
$V_{cb}$ with relatively small theoretical uncertainties, mainly for two 
reasons.

First due to the phase space function appearing in (\ref{bcsl}) the 
dependence on $m_b$ is not really $m_b^5$ but somemwhat milder
\begin{equation}
\Gamma (B \to X_c \ell \bar{\nu}_\ell) \propto 
m_b^5 z_0 \left( \frac{m_c^2}{m_b^2} \right) 
\approx (m_b - m_c)^{3.2} (m_b + m_c)^{1.8}
\end{equation}
where this is valid in the vicinity of a value of the  mass ratio
$m_c / m_b = 0.3$. Using the fact that the difference of the quark 
masses is much better 
known that their individual values (see (\ref{lam1eq}))
one may reduce the uncertainties due  
to the quark mass dependences. 

Second the radiative corrections and the definition of the mass 
are strongly correlated \cite{radcorrel}, 
i.e. the size of the radiative corrections 
depends on the choice of the heavy quark mass. Neglecting non-perturbative
contributions, the radiative 
corrections may in general be written as 
\begin{equation}
\Gamma (B \to X_c \ell \bar{\nu}_\ell) = \frac{G_F^2 m_b^5}{192 \pi^3}
|V_{cb}|^2 z_0 \left( \frac{m_c^2}{m_b^2} \right) 
\left( 1 + a_1 \frac{\alpha_s}{\pi}  + a_2 \left(\frac{\alpha_s}{\pi}\right)^2
+ \cdots \right)
\end{equation}
where the exact value of $a_2$ depends on the renormalization 
scheme used. We shall not go into that much detail here and only give
the orders of magnitude. 

If one chooses e.g.\ to take as the heavy quark mass the pole mass one finds 
$a_1 \approx 1.7$ and $a_2 = {\cal O} (-10)$ and hence large radiative 
corrections where even higher order terms become to be important. 
However, using a running mass (such as the $\overline{MS}$-mass) normalized 
at some small scale $\mu \approx 1$ GeV, the perturbative corrections 
are found to be much better behaved: $a_1 \approx 1$ and 
$a_2 \approx 1$. 

Based on this a fairly reliable and model independent 
extraction of $V_{cb}$ is possible. In fact, it has been argued that 
this method may be at least as reliable as the determination of $V_{cb}$ 
based on exclusive decays. While in exclusive $B \to D^*$ transitions 
the corrections are parametrically of the order $1/m_c^2$ they are in 
the inclusive decay only of order $1/m_b^2$ and hence may be substantially 
smaller. This is, however, compensated by the uncertainty in $m_b$ on which 
the inclusive rate depends, while the leading terms for the exclusive 
rate depends only on the meson masses. 

Two groups have estimated the theoretical uncertainties involved in
the extraction ov $V_{cb}$ from inclusive decays. Bigi, Shifman and Uraltsev
find (see e.g. their review in ref \cite{reviews})
\begin{eqnarray} \label{opti}
|V_{cb}| &=& 0.0419 \left(\frac{Br_{sl}}{10.5 \%} \right)^{1/2} 
 \left(\frac{1.55 \mbox{ ps}}{\tau_B} \right)^{1/2} \\
 && \nonumber \times \left(1 - 
 0.012 \frac{-\lambda_1 - 0.5 \mbox{ GeV}^2}{0.1 \mbox{ GeV}^2} \right) 
 \left(1 \pm 0.015 \pm 0.01 \pm 0.012 \right) 
\end{eqnarray}
where the uncertainties are due to higher orders in perturbation theory, 
to the value of $m_b$ and to orders $1/m_Q^3$ and higher. 

A more pessimistic estimate of the theoretical uncertainties has been given 
by Neubert who assigns a larger uncertainty to the value of the 
$b$ quark mass. He finds \cite{neubertincl}
 \begin{equation} \label{pessi}
|V_{cb}| = (0.040 \pm 0.004) \left(\frac{Br_{sl}}{10.8 \%} \right)^{1/2} 
\end{equation}
where in this uncertainty all sources have been combined. 

Even with the more pessimistic point of view (\ref{pessi}) a model 
independent determination of $V_{cb}$ is feasible from inclusive 
decays, with theoretical uncertainties comparable to the determination 
from $B \to D^* \ell \bar{\nu}_\ell$. The fact that the value of 
$V_{cb}$ extracted in these two ways coincides with good accuracy 
is very reassuring.  

\subsection{Differential Distributions and $V_{ub}$}
%
%
%
%
As pointed out above one may as well calculate differential 
distributions using the OPE and HQET. In this case the relevant 
scale for the OPE is not the heavy quark mass, but rather the 
variable $Q = m_Q v - q$, where $q$ is the momentum transferred
to the leptons. Note that to leading order, $Q^2$ is the invariant
mass of the final state hadrons.

Analogous to DIS the hadronic tensor depends on the 
Lorentz-invariants $\sqrt{Q^2}$ and $vQ$ and one has to distinguish 
various regions of phase space depending on the values of 
these variables. 

In the bulk part of phase space both variables are of order  
$m_Q$ and one may perform the OPE in the way outlined above. 
In this case one obtains the nonperturbative corrections expressed 
in terms of $\lambda_1$ and $\lambda_2$ as for the total rates. 
As an example we quote the result for the lepton energy spectrum for 
$B \to X_u \ell \bar{\nu}_\ell$
\begin{eqnarray} \label{buspec}
\frac{d\Gamma}{dy} &=&  \frac{G_F^2\,|\,V_{ub}|^2\,m_b^5}{192\pi^3}
        \left[ \left( 2y^2 (3-2y)
        + \frac{10y^2}{3} \frac{\lambda_1}{m_b^2}
        + 2y(6+5y) \frac{\lambda_2}{m_b^2} \right) \Theta(1-y) \right. 
\nonumber \\
&&  \vphantom{\frac{G_F^2\,|\,V_{qb}|^2\,m_b^5}{192\pi^3}}
\left. - \frac{\lambda_1 + 33 \lambda_2}{3m_b^2} \delta (1-y)
  -  \frac{\lambda_1}{3m_b^2} \delta ' (1-y) \right]  
\end{eqnarray}
where $y$ is the normalized energy of the charged lepton 
$y = 2E_\ell /m_b$. 

This function is well behaved except close to the endpoint of the
spectrum, where $\delta$-function-like singularities appear. This
problem may be traced back to the fact that close to the endpoint
the OPE breaks down. 

In order to cure this problem we again distinguish two cases. We 
define the endpoint region in which $Q^2$ becomes small although 
the components of $Q$ are still of order $m_Q$: 
$Q^2 = {\cal O} (\Lambda_{QCD} m_Q)$ and $vQ = {\cal O} (m_Q)$.
In this region the OPE needs to be modified and instead of only 
two parameters one has to introduce a ``shape function'' to 
parametrize the nonperturbative input. Without going into 
details, the spectrum for $B \to X_u \ell \bar{\nu}_\ell$ 
is written as
\begin{eqnarray}\label{convol}
   {d \Gamma\over d E_\ell}
   &=& {G_F^2\,|\,V_{qb}|^2\over 12\pi^3}\,E_\ell^2
    \int  dk_+\,f(k_+)\,\Theta\Big[ m_b^* 
    - 2 \,E_\ell \Big] \nonumber\\
   &&\times \Big\{ 3 m_b^{*2}
   - 4 m_b^*\,E_\ell \Big\} \,.
\end{eqnarray}
where the ``shape function'' $f(k_+)$ is given by the formal 
expression
\begin{equation}\label{fdef}
   f(k_+) = \langle B(v)|\,\bar h_v\,\delta(k_+-i D_+)\,h_v\,
   |B(v)\rangle
\end{equation}
and $m^*$ is an ``effective mass'' $m^* = m_b + k_+$.
The first few moments $A_n$ of the structure function 
$$
A_n = \int\mbox{d}k_+\,k_+^n\,f(k_+)
$$
are known: $A_0 = 1$ is the normalization of $f(k_+)$, 
$A_1 = 0$ due to the equations of motion of the heavy quark and 
$A_2 = -\lambda_1 / 3$. 

Closer to the endpoint we enter the resonance region, where 
$Q^2 \sim \Lambda_{QCD}^2$ and $vQ \sim \Lambda_{QCD}$. Here 
the rate is dominated by single states or resonances and the 
appropriate description would be to sum over the lowest few 
resonances.  

The semileptonic $b \to u$ transitions may be used to extract 
$V_{ub}$. However, the total inclusive semileptonic rate is 
dominated by the charm contribution and one has to look at differential 
rates to access $V_{ub}$ by studying phase space regions where 
no charm decays may occur. Traditionally people have used the 
lepton energy looking at large energies beyond the charm endpoint.
Recently a new method has been proposed which uses the hadronic 
invariant mass spectrum, where the part between $m_\pi$ and $m_D$ 
can only be due to $b \to u$ transitions. On the other hand, 
one may calculate this spectrum using the OPE and HQET. It turns 
out \cite{FW,BDU} that for a reasonable ansatz \cite{BDU}
for the shape function the part
of the spectrum which extends beyond $m_D$ is small, of the order
of 20\%. A useful quantity is the normalized integral of the spectrum 
up to a maximal invariant mass $M_{max}$ 
$$
\Phi (M_{max}) = \frac{1}{\Gamma(b \to u \ell \bar{\nu}_\ell)}
                 \int\limits_0^{M_{max}} dM_X \frac{d\Gamma}{dM_X}
$$
In fig.\ref{BDUfig} we plot $\Phi$ as a function of $M_{max}$ 
where the different curves correspond to different values of 
$m_b$. 

\begin{figure}
   \epsfysize=6cm
   \centerline{\epsffile[55 265 545 530]{BDU8.ps}}
   \caption{The integrated hadronic invariant mass spectrum as 
a function on $M_{max}$. The different curves correspond to 
$m_b = 4.72$ GeV (long dashed),  $m_b = 4.82$ GeV (solid), 
$m_b = 4.92$ GeV (short dashed). The ansatz for the shape function 
corresponds to $\lambda_1 = - 0.4$ GeV${}^2$ and $\alpha_s = 0.3$. 
The figure is taken from \protect\cite{BDU}.} 
\label{BDUfig}
\end{figure}

Fig.\ref{BDUfig} indicates that indeed at least 70\% of the hadronic 
invariant mass spectrum lies below $m_D$ and hence method may allow 
for a model independent determination of $V_{ub}$. Note that this 
statement is almost independent of the special ansatz for the 
shape function, since the integrated quantity $\Phi$ depends 
only on the low moments of the shape function, i.e. on 
$\lambda_1$ \cite{FW}.   

\section{Nonleptonic Processes}
%
%
%
%
The method of OPE and HQET may also be used to calculate
rates for purely hadronic processes, in particular lifetimes. 
However, compared to the semileptonic case, where the intergration 
over the leptonic phase space ensured some ``smearing'' of the 
hadronic (or partonic) quantities, the situation for nonleptonic
processes is slightly different. Instead of global duality one 
has to use ``local duality'' which means that partonic and hadronic 
quantities are compared point by point. This is potentially dangerous, 
although in the present application local duality is used far away 
from the resonance region, where it certainly would fail. Not much 
can be said about the reliability of duality in general and local 
duality in particular, so we shall proceed and assume that global as 
well as local duality may be applied. 

\subsection{Lifetimes} \label{sec:life}
%
%
%
%
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{equation}\label{taucrude}
   {\tau(B^-)\over\tau(B_d)} = 1 + O(1/m_b^3) \,,
   {\tau(B_s)\over\tau(B_d)} = 1 + O(1/m_b^3) \,,
   {\tau(\Lambda_b)\over\tau(B_d)} = 1 + O(1/m_b^2)
\end{equation}
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 
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 \cite{early,inclusive}. 

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. 

The contribution $b_0$ can be evaluated in terms of the forward matrix
elements of the dimension-six operators involving the spectator quark.
In total there are four relevant matrix elements which we parametrize
as \cite{NeuSach}
\begin{eqnarray}\label{Bidef}
   {1\over 2 m_{B_q}}\,\langle B_q|
           \bar b_L\gamma_\mu q_L\,\bar q_L\gamma^\mu b_L
                              |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|
                       \bar b_R\,q_L\,\bar q_L\,b_R
                       |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|
          \bar b_L\gamma_\mu t_a q_L\,\bar q_L\gamma^\mu t_a b_L
                       |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|
                       \bar b_R\,t_a q_L\,\bar q_L\,t_a b_R
                       |B_q\rangle
   &\equiv& {f_{B_q}^2 m_{B_q}\over 8}\,\varepsilon_2 \,.
\end{eqnarray}
where $t_a=\lambda_a/2$ are the generators of colour SU(3). Note 
that if we assume factorization we have $B_i = 1$ and 
$\varepsilon_i= 0$. Factorization holds exactly in the limit of 
large $N_c$ and hence one expects $
B_i = {\cal O}(1)$  and $\varepsilon_i=O(1/N_c)$.

Following Ref.\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 $k_i$ can be calculated and depend on the mass ratio 
$m_c / m_b$ and the Wilson coefficients entering the effective 
Hamiltonian. These Wilson coefficients depend on the renormalization 
scale $\mu$ which is of order $m_b$. In table \ref{tab1} we list the 
values for $k_i$ for different scales $\mu$.

\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 $m_c^2 / m_b^2 $ is fixed 
at $0.085$. The table is taken from 
\protect{\cite{NeuSach}}.}
\label{tab1}
\end{table}

It is interesting to note that 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. However, present data is compatible with 
neglecting these contributions. 

A similar analysis maybe performed for the lifetime ratio 
$\tau (\Lambda_b)/ \tau (B_d)$ The result found in \cite{NeuSach}
is
\begin{equation}
\frac{\tau (\Lambda_b)}{\tau (B_d)} = 0.98 + \Delta
\end{equation}
where $\Delta$ is of order $1/m_b^3$ and is most likely only a few 
percent. Given the experimental value which is considerably lower 
the potential problem with the $\Lambda_b$ lifetime still remains. 

\subsection{The Semileptonic Branching Fraction and Charm Counting}
%
%
%
%
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 down quark fields. 
From this one can evaluate the semileptonic branching fraction
$Br_{sl} = \Gamma (B \to X \ell \bar{\nu}_\ell) / \Gamma_{tot}$ 
and the number of charm and anticharm quark created per $B$ decay 
$n_c = 1 + Br(b \to c \bar{c} s) - Br (b \to \mbox{non charm states})$. 
Both quantities may be calculated in terms of a $1/m_Q$ expansion, 
where the nonperturbative contributions
are again parametrized in terms of $\lambda_1$ and $\lambda_2$. However, 
they turn out to be so small that they can safely be neglected. 
Furthermore, the radiative corrections to this quantity have 
been evaluated already up to next-to-leading order \cite{balletal}, 
which introduces a dependence on the renormalization scale $\mu$. 

\begin{figure}
   \epsfysize=6cm
   \centerline{\epsffile{fig_Bslnc.eps}}
   \caption[]{Predictions for $Br_{sl}$ and $n_c$ as a function of 
            $m_c / m_b$ and the renormalization point $\mu$ The data 
            points come form the ``low energy'' and ``high energy''
            measurements (see text). The figure is taken from 
            \protect{\cite{neubertincl}}.}   
\label{bslncfig}
\end{figure}

The naive parton model predicts $Br_{sl} \sim 15\%$ and $n_c \sim 1.2$
Since the nonperturbative corrections are small it was concluded that 
values for $Br_{sl}$ smaller that 12\% are hard to accomodate in theory. 
However, the radiative corrections (the next to leading order analysis, 
including the effect of the charm mass, has been performed in 
\cite{balletal}) lower the value of $Br_{sl}$ significantly. The present 
theoretical situation is shown in fig.\ref{bslncfig} where the shaded 
aerea represents a reasonable variation of $m_c / m_b$ and the 
renormalization point $\mu$.

Since the experimental values obtained from the $\Upsilon (4s)$ 
machines (``low energy data'') and from $Z_0$ decays (``high energy data'')
are slightly different, two experimental data points appear in 
fig.\ref{bslncfig}. The experimental situation is reviewed elsewhere
in these proceedings \cite{lifetimes-sb}.

Given the new experimental and theoretical developments the potential 
problem with the semileptonic branching fraction seems to disappear; 
only the low energy data are still in disagreement with the theoretical 
predictions; however, this discrepancy is not very severe.  

\section{Conclusions}
%
%
%
%
The $1/m_Q$ expansion has become a widely accepted tool also for 
inclusive processes. It relies on the method of OPE, which itself 
is based on the concept of duality. While there is no proof for 
the assumption of duality, at least global duality consitutes 
one of the main ingredients of almost all QCD calculations. 

In semileptonic processes global duality is assumed and hence 
the results for this class of processes seems on firm grounds. 
The total semileptonic rate allows to extract a value of $V_{cb}$ 
with reasonable theoretical acuracy, which is in good agreement 
with the value obtained from exclusive decays. The combination 
ot the two methods has already reduced the relative uncertainty 
in our knowledge of $V_{cb}$ to less than ten percent. 

Differential distributions may be used to extract also $V_{ub}$ 
from inclusive semileptonic decays. In particular, the hadronic 
invariant mass distribution is promising, since only a few tens
of percent of the rate for $B \to X_u \ell \bar{\nu}_\ell$ 
is due to invariant masses larger than $M_D$. Thus a cut on 
the invariant hadronic mass can eleminate the charm background
without affecting the $b \to u$ transitions severely. 
 
The $1/m_Q$ expansion in inclusive nonleptonic processes invokes
the concept of local duality, and the reliability of such 
calculations has been questioned \cite{Blok}. 
Still the pattern of $b$-hadron
lifetimes is reproduced at least qualitatively, although 
at the quantitative level some potential problems, such as 
the $\Lambda_b$ lifetime, remain.    



\section*{Acknowledgments}
I thank the organizers of the conference for the invitation and 
the opportunity to discuss physics in such a beautiful 
environment.

\section*{References}
\begin{thebibliography}{99}
\bibitem{reviews} A subjective selection of recent reviews is: \\
               N.~Isgur and M.~Wise: contribution to {\it Heavy Flavors},
               A.~Buras and M.~Lindner (eds.) 
               (World Scientific, Singapore, 1992);
               M.~Neubert, Phys.\ Rept.\ {\bf 245} (1994) 259; 
               CERN-TH-97-024 (1997), hep-ph/9702375; 
               I. Bigi, M. Shifman and N. Uraltsev, TPI-MINN-97-02-T, 
               (1997), hep-ph/9703290   
               T.~Mannel,  
               J. Phys. {\bf G 21} (1995) 1007; 
               Karlsruhe Preprint TTP97-16, Rep.\ Prog.\ Phys., in print.
\bibitem{HQET} B. Grinstein,
               Nucl. Phys. {\bf B339} (1990) 253; 
               H. Georgi,
               Phys. Lett. {\bf B240} (1990) 447; 
               A. Falk, H. Georgi, B. Grinstein and M. Wise,
               Nucl. Phys. {\bf B343} (1990) 1.               
\bibitem{HQsymmetry}
               N. Isgur and M. Wise,
               Phys. Lett. {\bf B232} (1989) 113 and
               {\bf B237} (1990) 527;
               M.~Voloshin and M.~Shifman,
               Sov. J. Nucl. Phys. {\bf 45} (1987) 292  and
               {\bf 47} (1988) 511. 
\bibitem{inclusive} 
               I. Bigi et al., preprint TPI-MINN-92-67-T, Talk given at 
               {\it Particles \& Fields 92}: 
               7th Meeting of the Division of Particles Fields of
               the APS (DPF 92), Batavia, IL, 10-14 Nov.\ 1992.
               M. Shifman and M. Voloshin,
               Sov. J. Nucl. Phys. {\bf 41} (1985) 120;
               V. Khoze et al.,
               Sov. J. Nucl. Phys. {\bf 46} (1987) 112.
               J.~Chay, H.~Georgi and B.~Grinstein,
               Phys. Lett. {\bf B247} (1990) 399.
               I. Bigi, N. Uraltsev and A.~Vainshtein, 
               Phys.\ Lett.\ {\bf B293} (1992) 430; 
               I. Bigi et al., Minnesota TPI-MINN-92/67-T (1992) and 
               Phys. Rev. Lett. {\bf 71} (1993) 496.
               B.~Blok et al.,
               Phys. Rev. {\bf D49} (1994) 3356.
               A.~Manohar and M.~Wise,
               Phys. Rev. {\bf D49} (1994) 1310.
               T.~Mannel, Nucl.\ Phys. {\bf B423} (1994) 396.
               A.~Falk, M.~Luke and M.~Savage,
               Phys.\ Rev.\ {\bf D49} (1994) 3367.
\bibitem{Lu90}
               M. Luke,
               Phys. Lett. {\bf B252} (1990) 447.
\bibitem{renormalons}
               I. Bigi et al., Phys.\ Rev.\ {\bf D 50} (1994) 2234; 
               M. Beneke and  V. Braun, 
               Nucl.\ Phys.\ {\bf B426} (1994) 301;
\bibitem{lam1renorm}
               M. Neubert,  
               Phys.\ Lett.\ {\bf B393} (1997) 110.
\bibitem{slqcdcorr} 
                    A. Ali and E. Pietarinen, 
                    Nucl. Phys. {\bf B154} (1979) 519;
                    N. Cabibbo, G. Corbo and L. Maiani, 
                    Nucl. Phys. {\bf B155} (1979) 83;
                    G. Altarelli et al., 
                    Nucl. Phys. {\bf B208} (1982) 365;
                    G. Corbo, 
                    Nucl. Phys. {\bf B212} (1983) 99; 
                    M. Jezabek and J. H. K\"uhn, 
                    Nucl. Phys. {\bf B320} (1989) 20.  
               
\bibitem{vcbincl}   I.Bigi and N. Uraltsev
                    Z.\ Phys.\ {\bf C62} (1994) 623.

\bibitem{radcorrel} P. Ball, M. Beneke and V. Braun, 
                    Phys. Rev. {\bf D52} (1995) 3929.
\bibitem{balletal} E. Bagan et al., 
                   Nucl. Phys. {\bf B432} 3 (1994); 
                   Phys.\ Lett.\ {\bf B342} 362 (1995) 
                   [E: Phys.\ Lett.\ {\bf B374} 363 (1996)].
\bibitem{neubertincl} M. Neubert, 
                    CERN-TH-97-019, Talk given at 4th KEK
                    Topical Conference on Flavor Physics, 
                    Tsukuba, Japan, 29-31 Oct 1996, hep-ph/9702310.  
\bibitem{FW} A. Falk, Z. Ligeti and  M. Wise 
             Phys.\ Lett.\ {\bf B406} 225 (1997).
\bibitem{BDU} I. Bigi, R. Dikeman and N. Uraltsev 
              Preprint TPI-MINN-97-21-T, (1997).
\bibitem{early} B. Guberina et al., 
               Phys.\ Lett.\ {\bf B89} 111. (1979), 
               N. Bilic, B. Guberina and J. Trampetic, 
               Nucl. Phys. {\bf B248} 261 (1984), 
               B. Guberina, R. R\"uckl and J. Trampetic,
               Z. Phys.\ {\bf C 33} 297 (1986).
\bibitem{NeuSach} M. Neubert, C. Sachrajda, 
               Nucl.\ Phys.\ {\bf B483} (1997) 339. 
\bibitem{lifetimes-sb} J. Kroll, these proceedings.
\bibitem{Blok} B. Blok, UMN-TH-1452-97, (1997), 
               B. Blok and T. Mannel, 
               Mod.\ Phys.\ Lett.\ {\bf A11} 1263 (1996). 
I
\end{thebibliography}


\end{document}


