%Title: Thirteen Years of Heavy Quark Expansion:   Examples for its Progress and its Problems 
%Author: Thomas Mannel
%Published: <EM> Acta Phys. Polon. </EM> <B> B32 </B> (2001) 1857-1868. 
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\begin{document}
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TTP01-10 \\
4.3.2001
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\begin{center}
{\Large \bf  Thirteen Years of Heavy Quark Expansion: \vspace*{3mm} \\
            Examples for its Progress and its Problems}
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\begin{center}
{\large Thomas Mannel}\vspace*{2mm}\\
     {\sl Institut f\"{u}r Theoretische Teilchenphysik, \\
     Universit\"at Karlsruhe,\\ D -- 76128 Karlsruhe, Germany.}
\end{center}
\vspace{\fill}
\begin{center}
Talk presented at the Epiphany Conference on \\ {\it B Physics and
CP Violation}, \\ Jan.~5.-7., 2001, Cracow, Poland
\end{center}
\vspace{\fill}

\begin{abstract}
\noindent
The heavy mass expansion has become a standard tool that has
significantly evolved since its formulation about thirteen years ago.
Some of the major results of the heavy mass expansion,
namely the determination of $V_{cb}$ (both exclusive and inclusive)
and lifetime calculations, are reviewed and some open problems are
pointed out.
\end{abstract}
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% Acta Physica Polonica titlepage  
\title{Thirteen Years of Heavy Quark Expansion: \\ 
       Examples for its Progress and its Problems
\thanks{Talk presented at the Epiphany Conference on {\it b Physics and
CP Violation}, Jan.~5.-7., 2001, Cracow, Poland}}
\author{Thomas Mannel 
\address{Institut f\"{u}r Theoretische Teilchenphysik, 
     Universit\"at Karlsruhe, \\ D -- 76128 Karlsruhe, Germany 
\and 
     CERN-TH, CH-1211 Geneva 23, Switzerland.}}
\maketitle
\begin{abstract}
The heavy mass expansion has become a standard tool that has
significantly evolved since its formulation about thirteen years ago.
Some of the major results of the heavy mass expansion,
namely the determination of $V_{cb}$ (both exclusive and inclusive)
and lifetime calculations, are reviewed and some open problems are
pointed out.
\end{abstract}
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\section{Introduction}
%
The heavy mass expansion is applied to decay processes of heavy quarks
already for more than thirteen years. After the early work by Shifman,
and Voloshin \cite{Shifman:1988rj}
the symmetries that appear in the heavy mass limit
have been formulated in a clean way by Isgur and Wise 
in two famous papers\cite{Isgur:1989vq,Isgur:1990ed} ,
which are among the most often quoted papers on phenomenological
particle physics of the last decade.
Based on \cite{Shifman:1988rj,Isgur:1989vq,Isgur:1990ed},
one could obtain relations
between form factors which are used (in a refined form) in the context
of the extraction of e.g. $V_{cb}$ from exclusive decays.

This limit of QCD could be formulated as an effective field theory, the
so called Heavy Quark Effective Field Theory (HQET)
\cite{Falk:1990yz,Mannel:1992mc}. It allows us to put the expansion 
in inverse powers of the heavy quark mass $m_Q$ on a field-theoretical
basis and thus to treat the radiative corrections in the heavy mass limit
in a systematic way using the language of Feynman Diagramms.  In
addition this gives us a way to parameterize the sub-leading terms
of the $1/m_Q$ expansion in terms of higher dimensional operators. 

Parallel to this development a description of inclusive decays has been
worked out \cite{Bigi:1993fe,Bigi:1992su,Manohar:1994qn,Mannel:1994su},
including also sub-leading terms in the $1/m_Q$ expansion as well
as radiative corrections. The additional piece of input here is to use
the operator product expansion in a similar way as in deep inelastic
scattering, making use of the fact the $m_Q$ is a scale much larger that
$\Lambda_{QCD}$. Soon it became clear that these two branches
have to merge into a unique description which we now know as the heavy
quark expansion. 

Since then, many detailed calculations have been performed, and many
reviews have been written, such that I can quote only a small
selection of them
\cite{Manohar:2000dt,Isgur:ed.xa,Neubert:1994mb,Bigi:1997fj,Mannel:1997ky}.
In particular, 
the sub-leading (and in some cases even the sub-sub-leading) terms in 
$\alpha_s (m_Q)$ have been computed. Using the data from Cornell, which
became more and more accurate, as well as from the new $B$ factories, we have
even gained some control over the sub-leading terms of the $1/m_Q$ expansion,
which involves certain  non-perturbative parameters.  

The main progress triggered by the discovery of the heavy quark symmetries
and the systematic expansion in powers of $1/m_Q$ 
is in the sector of semi-leptonic processes, exclusive as well as inclusive.
As far as non-leptonic processes are concerned, the heavy quark expansion
can be applied to lifetimes and also to a calculation of the semi-leptonic
branching fraction, and some of the results are briefly outlined
below. However,
up to now no significant progress has been made in the field of exclusive
non-leptonic decays, although here a few new ideas are suggested which
eventually could lead to some progress even for this class of decays
\cite{Beneke:1999br,ctshere}.

In this mini-rewiew I shall pick out a few subjects in which  over the last
years a significant progress has been made. I shall discuss the
exclusive heavy-to-heavy decays and the extraction of $V_{cb}$ in the
next section, in section 3 I shall consider inclusive transitions,
including the determinations of $V_{cb}$ from these processes
as well as a short discussion of the status of the lifetime calculations. 

\section{Exclusive Decays}
%
%
It has become textbook knowledge that 
in the heavy mass limit the heavy ground state hadrons fall into
spin symmetry doublets, corresponding to the two spin directions
of the heavy quark. Furthermore, in the heavy mass limit the heavy
quark becomes a static source of a color field, which is independent
of flavor. These symmetries have simple consequences which can be
expressed in terms of a ``Wigner Eckard Theorem'', very similar to
the case of the well known rotational symmetry. 

This ``Wigner Eckard Theorem'' implies relations between the
form factors for semi-leptonic transitions between heavy mesons,
such as the transitions $B \to D$ and $B \to D^*$. It is in the meantime
common knowledge that these transitions are described by only a single
form factor $\xi$, the so called Isgur-Wise function, which depends on the
scalar product of the four-velocities $v$ and $v'$
of the initial and final state hadrons.

Even more importantly, the heavy quark symmetries also yield an absolute
normalization of the form factors in the heavy quark limit, such that
$\xi (vv' = 1) = 1$. In particular, one obtains an
absolute normalization for some of the form factors relevant in
weak decays. The most prominent example is the decay
$B \to D^* \ell \bar{\nu}$ where one may compute the energy spectrum
$d \Gamma / d (vv')$ of the outgoing $D^*$; an extrapolation to the
normalization point yields
\begin{equation} \label{b2dstar}
\lim_{v \to v'} \frac{1}{\sqrt{(vv')^2-1}} \frac{d \Gamma}{d(vv')} \\ 
= \frac{G_F^2}{4 \pi^3}  |V_{cb}|^2 (m_B - m_{D^*})^2 m_{D^*}^3 
 |\xi_{A1} (vv' = 1)|^2
\end{equation}
where $\xi_{A1}$ is one of the axial-vector
form factors which becomes the Isgur-Wise function in the heavy mass
limit, thus $\xi_{A1} (vv' = 1)=\xi (vv' = 1) = 1$ 

The machinery of HQET allows us to calculate or at least parameterize
the corrections to this relation. The most important property of the
corrections is the absence of terms of the order $1/m_c$\footnote{ 
     The corrections depend in fact on the parameter
     $$
     \Delta = \frac{1}{m_c}-\frac{1}{m_b}
     $$
     so the absence of $1/m_c$ corrections means the that sub-leading
     corrections are of the order $\Delta^2$.}.
This fact is due to a theorem which has been known as the
Ademollo-Gatto Theorem  \cite{Ademollo:1964sr} in the context of current
algebra and has been applied to the case at hand by Luke \cite{Luke:1990eg}.

The radiative corrections to the normalization statement (\ref{b2dstar})
have been computed in the meantime to sub-leading order, i.e.
${\cal O}(\alpha_s^2 (m_Q))$ \cite{Czarnecki:1997hc}, and a complete
BLM resummation of the terms of order $(\beta_0 \alpha_s)^n$  has been
performed in \cite{Uraltsev:1997rd}. The result is that the radiative
corrections are small, they change $\xi_{A1} (1)$ by   
\begin{equation} \label{deltaradcorr}
\Delta_{A1}^{radcorr} = -0.01 \pm 0.01 
\end{equation}
where the conservative estimate of the uncertainty is given
by the size of the ${\cal O}(\alpha_s^2)$ contributions. 

The terms of order $1/m_c^2$ are much harder to discuss, since there is yet
no systematic way to compute them. They involve non-perturbative
matrix elements which either have to be modeled or have to be taken from
future lattice calculations. Various estimates have been discussed over the
last ten years, and it has become clear that early estimates
(e.g. \cite{Falk:1993wt,Mannel:1994kv}) tended to
underestimate their size. The
methods which are used today to estimate the size of these corrections
are based on a sum rule making use of the zero-recoil limit; from
this a typical value is \cite{Bigi:1995ga,Czarnecki:1998wy}
\begin{equation} \label{deltam2}
\Delta_{A1}^{1/m^2} = -0.08 \pm 0.04 
\end{equation}
It is important to notice that the uncertainty quoted in (\ref{deltam2})
is again an estimate which will probably match the taste of most authors
who worked in this field. In other words, the uncertainty is based on 
``common sense'' and, in particular, cannot be interpreted as a
statistical uncertainty with e.g.\ a Gaussian distribution.
The uncertainty quoted in (\ref{deltam2}) contains also an estimate of
the contributions of order $1/m^3$, which, however, are assumed to be 
small.  

A detailed discussion of the central values as well as of the uncertainties
of the normalization  (\ref{b2dstar}) has been performed in the context of
the BaBar workshop \cite{Harrison:1998yr}, where the number 
\begin{equation}
\xi_{A1} (vv' = 1) = 0.913 
\pm 0.007_{pert} \pm 0.024_{1/m^2} \pm 0.011_{1/m^3}
\end{equation}
is given. Although this number represents some consensus among the
theorists who participated in the BaBar workshop, concerns
have been expressed (see e.g. \cite{Bigi:1999dv}),
that the central value as well as the uncertainties should be reconsidered.
At some occasions lower central values as well as
larger errors have been used, e.g.\ in \cite{Bigi:1999dv} a value of
\begin{equation}
\xi_{A1} (vv' = 1) = 0.89 \pm 0.08
\end{equation}
is quoted and the LEP heavy flavor working group used
\begin{equation}
\xi_{A1} (vv' = 1) = 0.89 \pm 0.05
\end{equation}
for their analysis.

It is important to point out once more that all these
values are compatible within the ranges of uncertainties given.
Clearly the issue here is the estimation of the
uncertainties and their interpretation; even an uncertainty as large as
the one quoted in \cite{Bigi:1999dv} cannot be excluded, although it is
considered to be very unlikely. On the other
hand, if the uncertainties are overestimated, 
theorists will be on the safe side,
but on the expense of loosing predictive power. 

In the process of extracting $V_{cb}$ from an extrapolation based on
(\ref{b2dstar}) another potential source of uncertainty is the
formula used for the extrapolation to the point $v=v'$. However,
the slope of the form factor at $v=v'$
can be constrained using dispersion relations \cite{Caprini:1998mu},
which renders this uncertainty negligible.  

Clearly the relevant quantity which is extracted from experiment is the
product $|\xi_{A1} (1) V_{cb}|$. Fortunately the value of $\xi_{A1} (1)$
enters multiplicatively, so one may alway trivially scale out the dependence
on the form factor value. 

The comparison with data and the extraction of $V_{cb}$ will be discussed in
another  contribution to these proceedings \cite{berkhere}



\section{Inclusive Decays}
%
%
%
Inclusive decays can also be described using a $1/m_Q$ expansion. The
main observation is that the heavy quark mass sets a large scale,
the presence of  which can be exploited in a similar way as in deep
inelastic scattering, where the large scale is set by the large
momentum transfer.   

The basic ingredient may be understood very easily.
The inclusive rate is written as
\begin{eqnarray} \label{incansatz}
\Gamma (B \to X) &=& \sum_X  (2\pi)^4 \delta^4(p_B - p_X) 
\langle B | H_I |X\rangle  \langle X | H_I | B\rangle
\\ \nonumber 
&=& \int d^4x \, \langle B | H_I (x) H_I (0) | B\rangle \\
&=& -2 {\rm Im} \int d^4x \, 
\langle B | T [H_I (x) H_I (0)]  | B\rangle
\end{eqnarray}
The effective Hamiltonian $H$ for $\Delta B = \pm1$ contains one
(anti)bottom quark, which can be written as
$$
b(x) = \exp(-im_Q (vx)) \, b_v (x)
$$
where $v$ is the velocity of the decaying $B$ meson. 
This allows us to make the dependence of the matrix element on the
large scale $m_Q$ explicit. i.e. the matrix element involving only the
field $b_v$ may be expanded in inverse powers of $m_Q$.  

Technically this means that one may perform an Operator Product Expansion
(OPE) of the time-ordered product in (\ref{incansatz}) such that
\begin{eqnarray} \label{OPE}
&& \int d^4x \,  T [H_I (x) H_I (0)] =  \\ \nonumber 
&& \qquad \qquad C_0 {\cal O}_0    
 + \frac{1}{m_b} \sum_i C_1^{(i)} {\cal O}_1^{(i)}     
+ \left(\frac{1}{m_b}\right)^2  \sum_i C_2^{(i)} {\cal O}_2^{(i)}  + \cdots
\end{eqnarray}
where ${\cal O}^{(i)}_n$ are local operators of dimension $(mass)^n$
and $C^{(i)}_n$ are coefficients that can be calculated in perturbation
theory.

The rate is obtained by taking the diagonal matrix elements of (\ref{OPE}).
These matrix elements still have a mass dependence (e.g. through the states)
which is expanded in $1/m_Q$ using the techniques of HQET. After this
expansion, the rate is written as
\begin{equation}
\Gamma = \Gamma_0 + \frac{1}{m_Q} \Gamma_1 + \frac{1}{m_Q^2} \Gamma_2
         + \frac{1}{m_Q^3} \Gamma_3 + \cdots
\end{equation}
First few terms of this expansion have been investigated in detail. The
leading term is in general the decay rate of a free quark, since the
matrix element entering this leading term is normalized. Thus one
recovers the parton model result as the leading term of a systematic
expansion. The contribution $\Gamma_1$ vanishes due to a similar argument
as it is used in the derivation of Luke's Theorem, and thus the first
non-trivial corrections to inclusive $b$ hadron decays
are of order $1/m_b^2$. In this order, two parameters appear, which are
defined by the matrix elements
\begin{eqnarray}
2 M_H \lambda_1 &=& \langle H (v) |
                 \bar{Q}_v  (iD)^2  Q_v | H (v) \rangle
\\
6 M_H \lambda_2  &=&
\langle H (v) | \bar{Q}_v (-i)
                      \sigma_{\mu \nu} [iD^\mu , iD^\nu] Q_v 
                                   | H (v) \rangle
\end{eqnarray}
The parameter $\lambda_2$ can be obtained from the splitting within the
spin symmetry doublet for ground state mesons,
$\lambda_2 \approx 0.13 \mbox{ GeV}^2$, while $\lambda_1$ cannot
be easily read off from the spectrum of heavy hadrons. However, it can
be determined from the moments of the hadronic invariant mass spectrum
in inclusive semi-leptonic decays or the moments of the photon-energy
spectrum in $B \to X_s \gamma$. Both determinations yield a consistent
result which is in the range of \cite{Thorndike}
$-0.25 {\rm GeV}^2 \le \lambda_1 \le -0.1 {\rm GeV}^2$.

These values imply that the non-perturbative corrections in inclusive
decays are indeed tiny, namely of the order of
$\lambda_{1,2} / m_b^2 = {\cal O} (1\%)$. However, for the lifetimes these
corrections can be enhanced by large factors of $16 \pi^2$.   
In any case, the non-perturbative corrections are much smaller than
the perturbative ones. 

As far as the parametric dependence of the corrections
on the heavy quark mass is concerned,
inclusive $B$ decays have an advantage over 
exclusive ones, since the corrections are of the
order $1/m_b^2$ as compared to $1/m_c^2$ in exclusive decays.

However, inclusive decays also have disadvantages. In the early days
it was believed that a precise determination of $V_{cb}$ would not
be possible due to the uncertainties induced through the dependence
on the $b$ quark mass. Naively, this parameter enters in the fifth power
such that even a small uncertainty would make the inclusive decays
very uncertain. 

Two facts rescue the situation. Firstly, the phase space
of the quark decay also depends on the $b$ quark (and on the $c$ quark)
mass, in such a way that one may re-express the rate in term of the $b$
quark mass and the difference of the $b$ and the $c$ quark mass, which
is known to a better accuracy, since it may be determined from the
spin averaged $B$ and $D$ meson masses, up to terms of
the order $1/m^2$.

The second fact is the behavior of the radiative corrections. It has
been observed that the radiative corrections are large if the mass in
the partonic calculation is interpreted as the pole
mass \cite{Bigi:1994em}. Schematically one obtains
\begin{equation} \label{bad}
\Gamma \propto m^5_{pole} (1 + a_1 \frac{\alpha_s}{\pi}
       + a_2 \left(\frac{\alpha_s}{\pi}\right)^2 + \cdots
\end{equation}
with 
$a_1 = {\cal O}(1)$ and $ a_2 = {\cal O}(-10)$, 
which means that in this scheme the perturbation series converges
very slowly (if at all). Using instead a short distance definition
$m_{sd}$, one can rearrange the perturbative series   
such that 
\begin{equation} \label{good}
\Gamma \propto m^5_{sd} (1 + b_1 \frac{\alpha_s}{\pi}
       + b_2 \left(\frac{\alpha_s}{\pi}\right)^2 + \cdots
\end{equation}
where now $b_1, b_2 = {\cal O}(1)$.
 
It has been argued in \cite{Bigi:1997fj} that the $\overline{MS}$ mass 
definition, as it is usually used in perturbative calculations,  still does 
not ensure a good convergence of (\ref{good}), since it still has an infrared
sensitivity. According to  \cite{Bigi:1997fj} one should chose a 
short-distance mass that involves a hard cut off, i.e.\ involves powers of the
cut-off; using such a mass, taken at a low scale, yields a good convergence of
(\ref{good}).  

Thus various mass definitions have been proposed to minimize the uncertainties
induced by the ignorance of the heavy quark mass. One way that has been
proposed and which nicely shows this correlation between the radiative
corrections and the quark mass definition is to obtain the quark mass
from heavy quarkonia, i.e. from the spectrum of $\Upsilon$ states. This
calculation involves the application of NRQCD to the energy of the
$\Upsilon (1S)$ state, assuming that possible non-perturbative (which
means in this case non-coulombic) corrections are small. This approach,
suggested by Hoang et al. \cite{Hoang:1999ng,Hoang:1999hm}, 
yields a result of the form 
\begin{equation}
\Gamma  = \frac{G_F^2 |V_{cb}|^2}{192 \pi^3} 
          \left(\frac{m_{\Upsilon(1S)}}{2}\right)^5 0.533  
       (1 - 0.096 - 0.029 + \cdots )
\end{equation}
where now the quark mass is replaced by $m_{\Upsilon(1S)}/2$, which is
known to a very high precision. 
The expansion parameter is now not $\alpha_s$ any more, but the
resulting series seems to converge reasonably well.

Clearly there are also other ways to deal with the interplay of
radiative corrections and the heavy quark mass, but the message of
all this is that the inclusive decays $b \to c \ell \bar{\nu}_\ell$ are
indeed at least competitive compared to the exclusive method described
in the previous section, if not cleaner.

%
\subsection{Determination of $V_{cb}$ from inclusive decays}
%
%
%
After these preliminaries on advantages and disadvantages of the
$1/m_Q$ expansion in inclusive decays  we shall now apply this to the
determination of $V_{cb}$. 
This issue has been investigated in some detail in the context of the
BaBar workshop \cite{Harrison:1998yr}, where
\begin{equation}
|V_{cb}| = \xi_{th}
\left(\frac{Br(\bar{B} \to X_c \ell \bar{\nu})}{10.5\%} \right)^{1/2}
\left(\frac{1.6 {\rm ps}}{\tau_B} \right)^{1/2} 
\end{equation}
with
\begin{equation}
\xi_{th} = 0.0403 (1 \pm 0.030 \pm 0.024 \pm 0.025 \pm 0.012)
\end{equation}
is quoted; the uncertainties are due to higher order radiative
corrections, the mass difference $m_b - m_c$, the $b$ quark mass and the
corrections of order $1/m_b^3$ or higher. 

The latest data on this were given at the ICHEP2000 in
Osaka \cite{GolutvinTalk}, which are
\begin{equation}
V_{cb} = (40.66 \pm 0.36) \times 10^{-3} \times 
(1 \pm 0.015_{pert} \pm 0.010_{m_b} \pm 0.012_{1/m_b^3}) 
\end{equation}

It is noteworthy that for both determinations of $V_{cb}$ the 
uncertainties are already dominated by the theoretical ones. This
relation between the uncertainties will become worse,
since data will improve further
over the next years, while progress on the theoretical side is not
foreseeable.  

Finally one small caveat needs to be mentioned, 
which is the use of duality in inclusive
decays. Clearly the assumption of parton hadron duality is the central
point in any inclusive calculation, and most theorists believe it to be
a safe assumption. I shall return to this point at the end of the
next section. 
This caveat has to be kept in mind when discussing $V_{cb}$ extractions
from inclusive decays, while the exclusive method described in the last
section is safe against possible duality violations.

\subsection{Lifetimes}
%
%
%
Annother inclusive quantity which can be calculated in the framework
of the $1/m_b$ expansion are lifetimes of heavy hadrons. Since in the
$1/m_b$ expansion the leading term is the free quark decay the leading
order prediction for $b$ hadrons is that
\begin{equation}
\tau (B^0) = \tau(B^+) = \tau(B_s) = \tau(\Lambda_b) = \tau(b)
\end{equation}
Furthermore, the first nonperturbative corrections are of order
$\lambda_{1,2}/m_b^2$ and thus very small. In particular, they are
the same for the charged and the neutral $B$ mesons and thus a lifetime
difference can (and will) emerge only at the level of $1/m_b^3$ corrections
and are thus expected to be very small. Clearly, perturbative corrections
to the free quark decay (i.e. the parton model) are short distance
contributions and consequently will also not contribute to the lifetime
difference.  

A lifetime difference between the $\Lambda_b$ and the $B$ meson can
in principle arise already at the level $1/m_b^2$, since the kinetic
energy operator is different for $\Lambda_b$ and $B$; in addition,
$\lambda_2 (\Lambda_b) = 0$ due to heavy quark spin symmetry. The
difference between the kinetic energies can be estimated by comparing
charm and bottom hadrons \cite{Neubert:1997we} 
\begin{eqnarray}
&& [M(\Lambda_b) - M(\Lambda_c)]-[\bar{M}(B) - \bar{M}(D)] \\
\nonumber && \quad =
(\lambda_1 (B) - \lambda_1 (\Lambda_b))
               \left(\frac{1}{m_c} - \frac{1}{m_b} \right)
+{\cal O}(1/m_c^3)
\end{eqnarray}
where $\bar{M}(H) =(M(H_{0^-}) + M(H_{1^-})/4$ is the spin
averaged mass of the mesons. From this estimate one finds that 
the ${\cal O}(1/m_b^2)$ contribution to the lifetime difference
between $\Lambda_b$ and $B$ is as small as 2\%. 

On the other hand, experimentally the lifetime of the $\Lambda_b$ is
significantly smaller than the one of the $B$ meson, by about 20\%
\cite{Groom:2000in}.
In view of the small $1/m_b^2$ contributions, this effect needs
to be explained entirely by $1/m_b^3$ or even higer order terms.

At order $1/m_b^3$ one can indeed identify contributions which can
be large, depending on the values of the unknown hadronic parameters.
These large terms originate from the fact, that the leading free quark
decay is a three particle decay on the partonic level, while the
${\cal O}(1/m_b^3)$ contributions that are sensitive to the spectator
have only a two particle phase space at the partonic level. Thus these
particular contributions receive an enhancement by the phase space
ratio, which introduces a factor of $16\pi^2$. Schematically this is 
\begin{equation}
{\tau(B)\over\tau(\Lambda_b)} = 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}
where $a_i$ and $b_i$ are in general of order one, depending on unknown
hadronic parameters. In \cite{Neubert:1997we} a scan over the parameter
space has been performed with the result that one can marginally fit
the data.

However, the lifetime ratios is still considered to constitute a problem.
If we assume that the charm quark is also heavy, we would need to explain
lifetime ratios such as \cite{Groom:2000in}
\begin{equation}
\frac{\tau(D^+)}{\tau(D^0)} \sim 2.5 \qquad 
\frac{\tau(\Lambda_c)}{\tau(D^+)} \sim 0.2 
\end{equation}
by effects of order $1/m_b^3$. Although these data are consistent with
the naive scaling one infers from the $1/m_b$ expansion, one might feel
uneasy explaining these large effects by third order corrections. 

This has motivated the search for an alternative explanation. First one may
ask the question if the charm quark is indeed heavy enough to be treated in
a $1/m_c$ expansion. However, in applications of HQET to exclusive decays
like the determination of $V_{cb}$ this approximation seems to work quite
well.

Another issue that has to be discussed is again
the assumption of duality in
these processes; this has been a very active field in the last few
years
\cite{Bigi:1999kc,Grinstein:1998xk,Grinstein:1999gc,Lebed:2000gm,%
Bigi:1999qe,Bigi:2001wn}.
Lacking a tool to study duality in four dimensions, most of these studies
were performed in the t'Hooft model, i.e.\ two dimensional QCD. From these
studies it turns out that duality violations are small, at least in these
model studies. Whether this is also the case in four dimensions is still
open; in particular there is no estimate of the uncertainties that have
to be assigned to the fact that duality is used in inclusive heavy quark
processes. 

\section{Conclusions}
%
%
%
%
The heavy quark expansion has become the standard way to describe
heavy quark decays. It has put the theoretical description of these
processes on a firm theoretical basis and makes in many cases models
obsolete. Being a systematic expansion the $1/m_Q$ expansion allows
to at least estimate the size of the higher order corrections. Although
the estimation of higher order terms in most cases depend on model
assumptions (a well known case are the $1/m_Q^2$ corrections in any
of the $V_{cb}$ determinations) the $1/m_Q$ expansion is still preferable
over using a model from the very beginning. 
So the $1/m_Q$ expansion is a substantial progress, in particular in the
light of the forthcoming experimental data, which now can be used to
perform a stringent test of the CKM Sector of the Standard Model. 

Clearly over the time the heavy mass expansion has been explored in
detail and problems showed up, some of which have been solved. One
problem is the obvious question what the meaning of the expansion
parameter is, since many different mass definitions are possible.
This question has been clarified in principle, what remains is to
settle, which definition minimizes the uncertainties.

Another question which is not clarified yet is whether the charm quark
is heavy and can be treated in the $1/m_Q$ expansion. As far as
semi-leptonic processes are concerned the approximation of heavy charm
quark seems reasonable, e.g.\ the inclusive semi-leptonic width of
$D$ mesons comes out satisfactorily. Clearly the assumption of a heavy
charm also enters the determination of $V_{cb}$ via exclusive decays, where it
also seems to work reasonably well. On the other side are the non-leptonic
processes of charm, where the $1/m_Q$ expansion seems to converge badly. From
this point of view one might wonder, why this expansion works so well in the
semi-leptonic sector.
Probably related to this question of the convergence of the heavy
mass expansion  is the $\Lambda_b$ lifetime problem. 

One of the major challenges for the $1/m_Q$ expansion are exclusive
non-leptonic decays. This class of decays is very important for the
determination of the CKM angles $\alpha$, $\beta$ and $\gamma$, which
are related to the CP asymmetries in these processes. Some recent progress
in this direction will be described in another contribution to these
proceedings, however as of now no clear picture (e.g.\ a description in
terms of an operator product expansion) has emerged.

Clearly the heavy mass expansion has become an indispensable tool which
is essential for the test of the CKM sector and the Standard Model CP
violation. In particular, if new physics effects indeed show up in
$B$ decays, it is essential to have a good control over the hadronic
uncertainties, which makes further development of theoretical methods
an important issue. 

\section*{Acknowledgments}
%
I want to thank the organizers of the conference for inviting me and for
giving me the occasion to discuss physics in such a beautiful environment
as the city of Cracow. I also acknowledge support by the German ministry
of education and research bmb+f and by the German science foundation DFG.


\begin{thebibliography}{99}
%
\bibitem{Shifman:1988rj}
M.~A.~Shifman and M.~B.~Voloshin,
%``On Production Of D And D* Mesons In B Meson Decays,''
Sov.\ J.\ Nucl.\ Phys.\ {\bf 47}, 511 (1988).
%%CITATION = SJNCA,47,511;%%
%
\bibitem{Isgur:1989vq}
N.~Isgur and M.~B.~Wise,
%``Weak Decays Of Heavy Mesons In The Static Quark Approximation,''
Phys.\ Lett.\ B {\bf 232}, 113 (1989).
%%CITATION = PHLTA,B232,113;%%
%
\bibitem{Isgur:1990ed}
N.~Isgur and M.~B.~Wise,
%``Weak Transition Form-Factors Between Heavy Mesons,''
Phys.\ Lett.\ B {\bf 237}, 527 (1990).
%%CITATION = PHLTA,B237,527;%%
%
\bibitem{Falk:1990yz}
A.~F.~Falk, H.~Georgi, B.~Grinstein and M.~B.~Wise,
%``Heavy Meson Form-Factors From QCD,''
Nucl.\ Phys.\ B {\bf 343}, 1 (1990).
%%CITATION = NUPHA,B343,1;%%
%

\bibitem{Mannel:1992mc}
T.~Mannel, W.~Roberts and Z.~Ryzak,
%``A Derivation of the heavy quark effective Lagrangian from QCD,''
Nucl.\ Phys.\ B {\bf 368}, 204 (1992).
%%CITATION = NUPHA,B368,204;%%
%
\bibitem{Bigi:1993fe}
I.~I.~Bigi, M.~Shifman, N.~G.~Uraltsev and A.~Vainshtein,
%``QCD predictions for lepton spectra in inclusive heavy flavor decays,''
Phys.\ Rev.\ Lett.\ {\bf 71}, 496 (1993)
[hep-ph/9304225].
%%CITATION = HEP-PH 9304225;%%
%
\bibitem{Bigi:1992su}
I.~I.~Bigi, N.~G.~Uraltsev and A.~I.~Vainshtein,
%``Nonperturbative corrections to inclusive beauty and charm decays: QCD versus phenomenological models,''
Phys.\ Lett.\ B {\bf 293}, 430 (1992)
[hep-ph/9207214].
%%CITATION = HEP-PH 9207214;%%
%
\bibitem{Manohar:1994qn}
A.~V.~Manohar and M.~B.~Wise,
%``Inclusive semileptonic B and polarized Lambda(b) decays from QCD,''
Phys.\ Rev.\ D {\bf 49}, 1310 (1994)
[hep-ph/9308246].
%%CITATION = HEP-PH 9308246;%%
%
\bibitem{Mannel:1994su}
T.~Mannel,
%``Operator product expansion for inclusive semileptonic decays in heavy quark effective field theory,''
Nucl.\ Phys.\ B {\bf 413}, 396 (1994)
[hep-ph/9308262].
%%CITATION = HEP-PH 9308262;%%
%
\bibitem{Manohar:2000dt}
A.~V.~Manohar and M.~B.~Wise,
%``Heavy quark physics,''
{\it  Cambridge Monographs on Particle Physics, Nuclear Physics,
      and Cosmology, Vol. 10}.
%
\bibitem{Isgur:ed.xa}
N.~Isgur and M.~B.~Wise,
%``Heavy Quark Symmetry,''
CEBAF-TH-92-10, appeared in:
Stone, S. (ed.): {\it B physics}, World Scientific 1994,
p. 158-209.
%
\bibitem{Neubert:1994mb}
M.~Neubert,
%``Heavy quark symmetry,''
Phys.\ Rept.\ {\bf 245}, 259 (1994)
[hep-ph/9306320].
%%CITATION = HEP-PH 9306320;%%
%
\bibitem{Bigi:1997fj}
I.~Bigi, M.~Shifman and N.~Uraltsev,
%``Aspects of heavy quark theory,''
Ann.\ Rev.\ Nucl.\ Part.\ Sci.\ {\bf 47}, 591 (1997)
[hep-ph/9703290].
%%CITATION = HEP-PH 9703290;%%
%
\bibitem{Mannel:1997ky}
T.~Mannel,
%``Heavy-quark effective field theory,''
Rept.\ Prog.\ Phys.\ {\bf 60}, 1113 (1997).
%%CITATION = RPPHA,60,1113;%%
%
\bibitem{Beneke:1999br}
M.~Beneke, G.~Buchalla, M.~Neubert and C.~T.~Sachrajda,
%``{QCD} factorization for B --> pi pi decays: Strong phases and CP  violation in the heavy quark limit,''
Phys.\ Rev.\ Lett.\ {\bf 83}, 1914 (1999)
[hep-ph/9905312],
%%CITATION = HEP-PH 9905312;%%
M.~Beneke, G.~Buchalla, M.~Neubert and C.~T.~Sachrajda,
%``QCD factorization for exclusive, non-leptonic B meson decays: General  arguments and the case of heavy-light final states,''
Nucl.\ Phys.\ B {\bf 591}, 313 (2000)
[hep-ph/0006124].
%%CITATION = HEP-PH 0006124;%%
%
\bibitem{ctshere}
C.~Sachrajda, these proceedings
%
\bibitem{Ademollo:1964sr}
M.~Ademollo and R.~Gatto,
%``Nonrenormalization Theorem For The Strangeness Violating Vector Currents,''
Phys.\ Rev.\ Lett.\ {\bf 13}, 264 (1964).
%%CITATION = PRLTA,13,264;%%
%
\bibitem{Luke:1990eg}
M.~E.~Luke,
%``Effects Of Subleading Operators In The Heavy Quark Effective Theory,''
Phys.\ Lett.\ B {\bf 252}, 447 (1990).
%%CITATION = PHLTA,B252,447;%%
%
\bibitem{Czarnecki:1997hc}
A.~Czarnecki and K.~Melnikov,
%``Two-loop QCD corrections to semileptonic b decays at maximal recoil,''
Phys.\ Rev.\ Lett.\ {\bf 78}, 3630 (1997)
[hep-ph/9703291].
%%CITATION = HEP-PH 9703291;%%
%
\bibitem{Uraltsev:1997rd}
N.~Uraltsev,
%``BLM-resummation and OPE in heavy flavor transitions,''
Nucl.\ Phys.\ B {\bf 491}, 303 (1997)
[hep-ph/9610425].
%%CITATION = HEP-PH 9610425;%%
%
\bibitem{Falk:1993wt}
A.~F.~Falk and M.~Neubert,
%``Second order power corrections in the
%  heavy quark effective theory. 1. Formalism and meson form-factors,''
Phys.\ Rev.\ D {\bf 47}, 2965 (1993)
[hep-ph/9209268].
%%CITATION = HEP-PH 9209268;%%
%
\bibitem{Mannel:1994kv}
T.~Mannel,
%``Higher order 1/m corrections at zero recoil,''
Phys.\ Rev.\ D {\bf 50}, 428 (1994)
[hep-ph/9403249].
%%CITATION = HEP-PH 9403249;%%
%
%
\bibitem{Bigi:1995ga}
I.~Bigi, M.~Shifman, N.~G.~Uraltsev and A.~Vainshtein,
%``Sum rules for heavy flavor transitions in the SV limit,''
Phys.\ Rev.\ D {\bf 52}, 196 (1995)
[hep-ph/9405410].
%%CITATION = HEP-PH 9405410;%%
%
\bibitem{Czarnecki:1998wy}
A.~Czarnecki, K.~Melnikov and N.~Uraltsev,
%``Complete O(alpha(s)**2) corrections to zero-recoil sum rules for  B --> D* transitions,''
Phys.\ Rev.\ D {\bf 57}, 1769 (1998)
[hep-ph/9706311].
%%CITATION = HEP-PH 9706311;%%
%
\bibitem{Harrison:1998yr}
P.~F.~Harrison and H.~R.~Quinn  [BABAR Collaboration],
%``The BaBar physics book: Physics at an asymmetric B factory,''
SLAC-R-0504
{\it Papers from Workshop on Physics at an Asymmetric B Factory
(BaBar Collaboration Meeting), Rome, Italy, 11-14 Nov 1996,
Princeton, NJ, 17-20
Mar 1997, Orsay, France, 16-19 Jun 1997 and Pasadena, CA, 22-24 Sep 1997}.
%
\bibitem{Bigi:1999dv}
I.~I.~Bigi,
%``Memo on extracting |V(cb)| and |V(ub)/V(cb)| from semileptonic B  decays,''
hep-ph/9907270.
%%CITATION = HEP-PH 9907270;%%
%
\bibitem{Caprini:1998mu}
I.~Caprini, L.~Lellouch and M.~Neubert,
%``Dispersive bounds on the shape of anti-B --> D(*) l anti-nu form  factors,''
Nucl.\ Phys.\ B {\bf 530}, 153 (1998)
[hep-ph/9712417].
%%CITATION = HEP-PH 9712417;%%
%
\bibitem{berkhere} 
K.~Berkelman, these proceedings.
%
\bibitem{Thorndike}
E.~Thorndike, talk presented at the BCP4 ``Conference on B Physics
and CP Violation'', 18.-23.2.2001, Ise Shima, Japan.
%
\bibitem{Bigi:1994em}
I.~I.~Bigi, M.~A.~Shifman, N.~G.~Uraltsev and A.~I.~Vainshtein,
%``The Pole mass of the heavy quark. Perturbation theory and beyond,''
Phys.\ Rev.\ D {\bf 50}, 2234 (1994)
[hep-ph/9402360].
%%CITATION = HEP-PH 9402360;%%
%
\bibitem{Hoang:1999ng}
A.~H.~Hoang, Z.~Ligeti and A.~V.~Manohar,
%``B decay and the Upsilon mass,''
Phys.\ Rev.\ Lett.\ {\bf 82}, 277 (1999)
[hep-ph/9809423].
%%CITATION = HEP-PH 9809423;%%
%
\bibitem{Hoang:1999hm}
A.~H.~Hoang, Z.~Ligeti and A.~V.~Manohar,
%``B decays in the Upsilon expansion,''
Phys.\ Rev.\ D {\bf 59}, 074017 (1999)
[hep-ph/9811239].
%%CITATION = HEP-PH 9811239;%%
%
\bibitem{GolutvinTalk}
A.~Golutvin, Talk given at {\it 30th International Conference on
High-Energy Physics} (ICHEP 2000), Osaka, Japan, 27 Jul - 2 Aug 2000
%
\bibitem{Neubert:1997we}
M.~Neubert and C.~T.~Sachrajda,
%``Spectator effects in inclusive decays of beauty hadrons,''
Nucl.\ Phys.\ B {\bf 483}, 339 (1997)
[hep-ph/9603202].
%%CITATION = HEP-PH 9603202;%%
%
\bibitem{Groom:2000in}
D.~E.~Groom {\it et al.}  [Particle Data Group Collaboration],
%``Review of particle physics,''
Eur.\ Phys.\ J.\ C {\bf 15}, 1 (2000).
%%CITATION = EPHJA,C15,1;%%
%
\bibitem{Bigi:1999kc}
I.~Bigi, M.~Shifman, N.~Uraltsev and A.~Vainshtein,
%``Heavy flavor decays, OPE and duality in two-dimensional 't Hooft model,''
Phys.\ Rev.\ D {\bf 59}, 054011 (1999)
[hep-ph/9805241].
%%CITATION = HEP-PH 9805241;%%
%
\bibitem{Grinstein:1998xk}
B.~Grinstein and R.~F.~Lebed,
%``Explicit quark-hadron duality in heavy-light meson weak decays in the  't Hooft model,''
Phys.\ Rev.\ D {\bf 57}, 1366 (1998)
[hep-ph/9708396].
%%CITATION = HEP-PH 9708396;%%
%
\bibitem{Grinstein:1999gc}
B.~Grinstein and R.~F.~Lebed,
%``Quark hadron duality in the 't Hooft model for meson weak decays:  Different quark diagram topologies,''
Phys.\ Rev.\ D {\bf 59}, 054022 (1999)
[hep-ph/9805404].
%%CITATION = HEP-PH 9805404;%%
%
\bibitem{Lebed:2000gm}
R.~F.~Lebed and N.~G.~Uraltsev,
%``Precision studies of duality in the 't Hooft model,''
Phys.\ Rev.\ D {\bf 62}, 094011 (2000)
[hep-ph/0006346].
%%CITATION = HEP-PH 0006346;%%
%
\bibitem{Bigi:1999qe}
I.~Bigi and N.~Uraltsev,
%``Pauli interference in the 't Hooft model: Heavy quark expansion and  quark-hadron duality,''
Phys.\ Lett.\ B {\bf 457}, 163 (1999)
[hep-ph/9903258].
%%CITATION = HEP-PH 9903258;%%
%
\bibitem{Bigi:2001wn}
I.~I.~Bigi and N.~G.~Uraltsev,
%``D0 anti-D0 oscillations as a probe of quark-hadron duality,''
Nucl.\ Phys.\ B {\bf 592}, 92 (2001)
[hep-ph/0005089].
%%CITATION = HEP-PH 0005089;%%


\end{thebibliography}
\end{document}


