\section{Inclusive Decays}
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In order to describe inclusive decays in the framework of the $1/m_Q$ 
expansion one combines the method of the operator product expansion 
with heavy quark effective theory \cite{Manifesto}-\cite{FLS93}.
The approach applied here resembles very much the description 
of deep inelastic scattering.  

The first step consists of an operator product expansion 
which yields an infinite sum of operators with increasing dimension. 
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 sections we shall first develop the theoretical 
framework and then consider some applications to various processes.
The method described below also allows us to deal with inclusive 
non-leptonic processes and hence in principle opens the possibility for 
a calculation of lifetimes and branching fractions in the framework of 
the $1/m_Q$ expansion.
Applying this idea to the energy spectra of the charged lepton in 
inclusive semileptonic decays of heavy mesons, the relevant expansion
parameter is not $1/m_Q$, but rather $1/(m_Q - 2E_\ell  )$; the 
denominator is thus the energy release of 
the decay. In almost all phase space the energy release is of 
the order of the heavy mass; it is only in the endpoint region that it
becomes small and hence the expansion breaks down. This problem 
may be fixed by a resummation of terms in the operator product 
expansion, which strongly resembles the summation corresponding to 
leading twist in deep inelastic scattering. Analogously to the 
parton-distribution function, a universal function appears, which 
determines all inclusive heavy-to-light decays.    


\subsection{Operator Product Expansion}
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This section is devoted to a short introduction into the meothod employed 
to set up a $1/m_Q$ expansion for inclusive processes.  
The effective Hamiltonian for a decay of a heavy 
(down-type) quark is in general linear in the decaying heavy 
flavoured quark  
\begin{equation} \label{heff}
{\cal H}_{eff} = \bar{Q} R
\end{equation}
where the operator $R$ describes the decay products. 
In the following we shall consider semileptonic decays, for which  
\begin{equation} \label{rsl}
R_{sl} = \frac{G_F}{\sqrt{2}} V_{Qq} \,\, \gamma_\mu (1-\gamma_5) q 
\,\, (\bar{\nu}_\ell \gamma^\mu (1-\gamma_5) \ell)  , 
\end{equation}
where $q$ is an up-type quark ($c$ or $u$, since we shall consider $b$ 
decays). Similarly, for nonleptonic decays the Cabbibo allowed contribution 
corresponds to  
\begin{eqnarray} \label{Rnonlep}
R_{nl} &=& \frac{G_F}{2\sqrt{2}} V_{Qq} V^*_{q'q''} \left[ (C_+ (m_b) + C_- (m_b))  
         \gamma_\mu (1-\gamma_5) q  \right. (\bar{q}' \gamma^\mu (1-\gamma_5) q'')
       \\ 
       && \qquad \qquad \qquad + (C_+ (m_b) - C_- (m_b)) 
         \gamma_\mu (1-\gamma_5) q''  
       \left. (\bar{q}' \gamma^\mu (1-\gamma_5) q )\right], 
\nonumber 
\end{eqnarray}
where $q'$ ($q''$) is a  down-type (up-type) quark and $V_{Qq}$ the corresponding 
CKM matrix element. The coefficients $C_\pm (m_b)$
are the QCD corrections obtained from the renormalization group running between 
$M_W$ and $m_b$; in leading logarithmic approximation these coefficients are \cite{BurasNLLA}
\begin{equation} \label{cpm}
C_{\pm} (m_b) =[\frac{\alpha _S(M_W^2)}{\alpha _S(m_b^2)}]^{\gamma_{\pm}},
\mbox{ with } \gamma _+=\frac{6}{33-2N_f}=-\frac{1}{2}\gamma _-
\end{equation}
where $\alpha_s (\mu)$ is the one-loop expression (\ref{alpha}) for the running coupling 
coupling constant of QCD.

Finally, for  radiative rare decays we have
\begin{equation}
R_{rare} = \frac{G_F}{\sqrt{2}} V_{tb} V^*_{ts} C_7 (m_b) \frac{e}{16 \pi^2} 
            m_b \sigma_{\mu \nu} (1+\gamma_5) s F^{\mu \nu} 
\end{equation} 
where $C_7 (m_b) $ is again a coefficient obtained from running between $M_W$ 
and $m_b$. Its value is $C_7 (m_b) \sim 0.3$, the corresponding analytical 
expression may be found in \cite{C7}.  

The inclusive decay rate for a heavy hadron $H$ containing the quark $Q$ 
may be related to a forward matrix element by  
\begin{eqnarray} 
&& \Gamma  \propto  \sum_X (2 \pi)^4 \delta^4 (P_B - P_X )
| \langle X | {\cal H}_{eff} | H(v) \rangle |^2 
= \label{inclusive}
\int d^4 x \,   \langle H(v) |{\cal H}_{eff} (x) 
            {\cal H}_{eff}^\dagger (0) | H(v) \rangle 
\\ \nonumber
&&=  2 \mbox{ Im}
\int d^4 x \,   \langle H(v) |T \{ {\cal H}_{eff} (x) 
            {\cal H}_{eff}^\dagger (0) \} | H(v) \rangle .
\end{eqnarray}
where $| X \rangle $ is the final state, which is summed over 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  done by a phase redefinition. This leads to 
\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}
= \bar{Q}_v R \qquad Q_v = e^{-im_Q vx} Q 
\end{equation}
This relation exhibits the similarity 
between the cross-section calculation in deep inelastic scattering 
and the present approach to total rates. In deep inelastic scattering 
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. 

The next step is to perform an operator product expansion of the product 
of the two Hamiltonians. After the phase redefinition 
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 next step is thus to perform an operator-product 
expansion, which has the general form  
$$
\int d^4 x  \, e^{im_Q vx} \, 
\langle H(v) |T \{ \widetilde{{\cal H}}_{eff} (x) 
\widetilde{{\cal H}}_{eff}^\dagger (0) \}| H(v) \rangle
= \sum_{n=0}^\infty  \left(\frac{1}{2m_Q}\right)^n 
     \hat{C}_{n+3} (\mu) \langle H(v) |{\cal O}_{n+3}| H(v) \rangle_\mu ,
$$
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 heavy quark effective theory. This is achived
by expanding the heavy quark fields appearing in the operators ${\cal O}_n$
using (\ref{HarvField}) (or, equivalently, (\ref{FWfield}))
as well as the states by including the corrections to the Lagrangian  
given in (\ref{HarvLag}) (or (\ref{FWlag})) 
as time-ordered products. 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 operator product expansion are the  
dimension-3 operators. Due to Lorentz invariance and parity there 
are only two combinations which may appear, namely   
$\bar  Q_v \fmslash{v} Q_v$ or $\bar Q_v Q_v$. Note that the $Q_v$
operators differ from the full QCD operators only by a phase redefinition, 
and hence $\bar  Q_v \fmslash{v} Q_v = \bar  Q \fmslash{v} Q$ and 
$\bar Q_v Q_v = \bar Q Q$.  The first combination 
is proportional to the $Q$-number current $\bar  Q \gamma_\mu Q$, 
which is normalized even in full QCD,
while the second differs from the first one only by terms of order
$1/m_Q^2$ 
\begin{equation}
\bar Q_v Q_v = v_\mu \bar Q_v \gamma_\mu Q_v 
+ \frac{1}{2 m_Q^2}
\bar{h}_v  \left[ (iD)^2 - (ivD)^2 +
\frac{i}{2} \sigma_{\mu \nu} G^{\mu \nu} \right] h_v  
+ {\cal O} (1/m_Q^3) .
\end{equation}
where $G_{\mu \nu}$ is the gluon field strength.

Thus the matrix elements of the dimension-3 contribution 
is known to be normalized; in the standard normalization of the 
states this implies 
\begin{equation}
\langle H(v) |{\cal O}_3| H(v) \rangle = 
\langle H(v) |\bar Q_v \fmslash{v} H_v | B(v) \rangle = 2 m_H 
\end{equation}
where $m_H$ is the mass of the heavy hadron.
To lowest order in the heavy mass expansion we may furthermore 
replace $m_B = m_Q$ and hence we may evaluate the leading term in 
the $1/m_q$ expansion without any hadronic uncertainty. Generically   
the dimension-3 contribution 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. 

A dimension-four operators contains an additional covariant derivative, 
and thus one has matrix elements of the type 
\begin{equation}
\langle H(v) |{\cal O}_4| H(v) \rangle \propto 
\langle H(v) |\bar Q_v \Gamma D_\mu Q_v | H(v) \rangle = {\cal A}_\Gamma v_\mu  
\end{equation}
Since the equations of motion apply for this tree level matrix element, 
one finds that the constant ${\cal A}_\Gamma$ has to vanish, and thus 
there are no dimension-four contributions. This statement is completely 
equivalent to Luke's theorem \cite{Lu90}, since we are considering 
a forward matrix element, i.e. a matrix element at zero recoil \cite{zerec}.

The first non trivial non-perturbative contribution comes from
dimension-5 operators and are of order $1/m_Q^2$.  
For mesonic decays there are only the two parameters $\lambda_1$
and $\lambda_2$ given in (\ref{lam1}) and (\ref{lam2}), 
which correspond to matrix elements of the subleading terms of 
the Lagrangian. They parametrize the non-perturbative input in the order 
$1/m_Q^2$. For $\Lambda_Q$-type baryons the parameter $\lambda_2$ 
vanishes due to heavy quark spin symmetry, while the kinetic energy parameter 
$\lambda_1$ is nonzero as well. In the framework of the $1/m_Q$ 
expansion this leads to a difference in lifetimes between mesons and
baryons.


\subsection{Semileptonic Decays}
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In the following subsections we shall apply the method outlined above
to inclusive semileptonic processes. Before quoting the results some 
remarks concerning the reliability of the method are in order. 

When considering semileptonic processes the effective hamiltonian may 
be split into a hadronic current and a leptonic one. Hence the operator 
product expansion is in fact an expansion for the two hadronic currents
which takes the form 
\begin{equation} \label{OPEcurr}
H^{\mu \nu} = \int d^4 x  \, e^{i(m_Q v - q)x} \, 
\langle H(v) |T \{\widetilde{J}_{had}^\mu (x) 
\widetilde{J}_{had}^{\dagger \nu} (0) \} | H(v) \rangle
\end{equation}
where the coefficients $A$ depend on $q$, the momentum transferred to 
the leptons. The hadronic tensor $H_{\mu \nu}$ may be expressed in terms 
of scalar functions which will depend on $vq$ and $q^2$. As a function 
of these two variables the analytic structure of $H_{\mu \nu}$ may be 
studied to gain some insight into the reliability of the operator 
product expansion. For fixed $q^2$ we consider $H_{\mu \nu}$ 
as a function of $vq$: $H_{\mu \nu}$ exhibits a cut along the real
axis for values 
$\sqrt{q^2} < vq < (m_Q^2 + q^2 - m_*^2)/(2m_b) $ corresponding 
(for $vq > 0$) to the decay process we are considering. Here 
$m_*$ is the mass of the lowest lying hadronic state that may appear
as a decay product. There are 
also other processes yielding the analytical structure depicted in 
fig.\ref{anastruc}. Calculating the energy spectrum or the total rate
involves an integration over the momentum of the neutrino which corresponds
to an integration over the variable $q$ with a weight given by the 
leptonic tensor. This integration samples the cut of $H_{\mu \nu}$ 
and may be represented by a contour integration along the curve $C$ 
as shown in  fig.\ref{anastruc}. 

Close to the upper edge of the left hand cut $H_{\mu \nu}$ is dominated 
by resonances and hence is not well represented by the partonic expression.
On the other hand, the integration (at least in almost all phase space) 
over $q^2$ induces sufficient ``smearing'' such that the intergals of the 
``true'' (i.e. hadronic) $H_{\mu \nu}$ may be identified with the
corresponding integral of the partonic expression for $H_{\mu \nu}$. 
This assumption, namely that weighted averages of the hadronic 
and partonic expressions are equal, is usually called global duality. 

The notion of global duality has to be contrasted with the notion 
of local duality, where the equality between partonic and hadronic 
quantities is assumed not only in the average, but even for the 
function itself. As we shall see below, local dualtity is needed for 
the description of inclusive nonleptonic processes, while the milder
assumption of global dualitity is needed for semileptonic processes, 
for both the total rate as well as the energy spectrum.    

\begin{figure}
   \epsfysize=7cm
   \centerline{\epsffile{anastruc.eps}}
   \centerline{\parbox{16cm}{\caption[]{\label{anastruc}
Sketch of the complex $vq$ plane for fixed $q^2 > 0$ for 
inclusive semileptonic decays. The thick lines along 
the real axis indicate the branch cuts 
of the hadronic tensor. They correspond to $vq < -\sqrt{q^2}$, 
$\sqrt{q^2} < vq < (m_Q^2 + q^2 - m_*2)/(2m_Q)$ and 
$vq > ... $, 
where $m_*$ is the mass of the lightest hadronic final state which 
may appear in the inclusive decay.  
The countour $C$ is the integration contour corresponding to the integration 
over the neutrino momentum.
}}}
\end{figure}


\subsubsection{Total Rates}
Inserting $R_{sl}$ as given in (\ref{rsl}) one obtains for the total 
inclusive semileptonic decay rate $B \to X_c \ell \nu$
\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) f_1 \left(\frac{m_c}{m_b}\right)
      -\frac{9\lambda_2}{2m_c^2} f_2 \left(\frac{m_c}{m_b}\right)
      \right] , 
\end{equation}
where the two $f_j$ are phase-space functions 
\begin{eqnarray} \label{f1}
f_1(x) &=& 1-8x^2+8x^6-x^8-24x^4\log x ,
\\  \nonumber
f_2(x) &=& 1-\frac{8}{3}x^2-8x^4+8x^6+\frac{5}{3}x^8+8x^4\log x  .
\end{eqnarray}
The result for $B \to X_u \ell \nu_\ell$ is obtained from (\ref{bcsl}) 
as the limit $m_c \to 0$ and the replacement $V_{cb} \to V_{ub}$   
\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}
As discussed above, the leading non-perturbative 
corrections in (\ref{bcsl}) and (\ref{busl}) are parametrized 
by $\lambda_1$ and $\lambda_2$. Estimates for these parameters 
have been discussed in section 2; in order to 
estimate the total effect of the non-perturbative effects we shall 
insert a range of values $-0.3 > \lambda_1 >  -0.6$ GeV${}^2$; from this 
we obtain   
\begin{equation}
\frac{\lambda_1 - 9 \lambda_2}{2m_b^2} \sim -(3 \cdots 4) \%
\end{equation}
This means that the non-perturbative contributions are small, 
in particular compared to the perturbative ones, which have been 
calculated some time ago \cite{QCDradcorr,ACCMM}. For the decay 
$B \to X_u \ell \bar\nu_\ell$ the lowest order QCD corrections 
are given by 
\begin{equation} \label{QCDcorr}
\Gamma (B \to X_u \ell \bar{\nu}_\ell) =  \frac{G_F^2 m_b^5}{192\pi^3} 
       |V_{ub}|^2 \left[ 1 + \frac{2 \alpha}{3 \pi}
                    \left(\frac{25}{4}-\pi^2\right) \right]
= 0.85 |V_{ub}|^2 \Gamma_b,  
\end{equation}
and thus the typical size of QCD radiative corrections is of the order of 
ten to twenty percent. 



\subsection{Energy Spectrum}
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The method of the operator-product expansion may also  be used to 
obtain the non-perturbative corrections to the charged lepton energy 
spectrum \cite{CCGincsl}. In this case the operator product expansion is 
applied not to the full effective Hamiltonian, but rather only to the 
hadronic currents. The rate is written as a product of the hadronic and
leptonic tensor
\begin{equation}
d \Gamma = \frac{G_F^2}{4 m_B} | V_{Qq} |^2 W_{\mu \nu}  
\Lambda^{\mu \nu} d(PS) , 
\end{equation}
where $d(PS)$ is the phase-space differential.  
The short-distance expansion is then performed for 
the two currents appearing in 
the hadronic tensor. Redefining the phase of the heavy-quark fields as in 
(\ref{phaseredef}) one finds that the momentum transfer variable 
relevant for the short-distance expansion is $m_Q v - q$, where 
$q$ is the momentum transfer to the leptons. After integration over the 
momentum of the neutrino the expansion variable is the energy release 
$(m_Q/2) - E_\ell$, where $E_\ell$ is the energy of the charged lepton.  

The structure of the expansion for the spectrum is identical to the one 
of the total rate. The contribution of the dimension-3 operators 
yields the free-quark decay spectrum, there are no contributions from 
dimension-4 operators, and the $1/m_b^2$ corrections are parametrized 
in terms of $\lambda_1$ and $\lambda_2$. Calculating the spectrum for 
$B \to X_c \ell \nu$ yields \cite{Bigiincsl}-\cite{tmincsl}
\begin{eqnarray}
\frac{d \Gamma}{dy} &=& \frac{G_F^2\,|\,V_{cb}|^2\,m_b^5}{192\pi^3}
\Theta(1-y-\rho) y^2 \left[\left\{ 3 (1-\rho) (1-R^2) - 2 y (1-R^3) \right\}
\right. \\ 
&& + \frac{\lambda_1}{[ m_b(1-y)]^2} (3 R^2-4 R^3) 
  - \frac{\lambda_1}{m_b^2(1-y)} (R^2-2 R^3) 
\nonumber \\
&& - \frac{3\lambda_2}{m_b^2(1-y)} (2 R+3 R^2-5 R^3) 
   + \frac{\lambda_1}{3 m_b^2} [ 5y - 2(3-\rho) R^2 + 4 R^3 ] 
\nonumber\\
&& \left. + \frac{\lambda_2}{m_b^2} [ (6+5y) - 12 R
    - (9-5\rho) R^2 + 10 R^3 ] \right] 
    + {\cal O}\left[ (\Lambda/[m_b(1-y)])^3 \right]
\nonumber  
\end{eqnarray}
where we have defined 
\begin{equation}
\rho = \left(\frac{m_c}{m_b}\right)^2 \quad  R = \frac{\rho}{1-y}
\end{equation}
and 
\begin{equation}
y = 2 E_\ell / m_b 
\end{equation}
is the rescaled energy of the charged lepton. 

This expression is somewhat complicated, but it simplifies 
for the decay $B \to X_u \ell \nu$ since then the mass of the quark 
in the final state may be neglected. One finds
\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}

\begin{figure}
   \epsfysize=10cm
   \centerline{\epsffile{mwspec.eps}}
   \caption{The electron spectrum for free quark
$b\rightarrow c$ decay (dashed line), free quark $b\rightarrow u$ decay
(grey line), and  $B\rightarrow X_c e \bar\nu_e$ decay including
$1/m_b^2$ corrections (solid line) with $\lambda_1 = - 0.5$ GeV${}^2$
and $\lambda_2 = 0.12$ GeV${}^2$. The figure is from
\protect\cite{mwincsl}.} 
\label{fig2}
\end{figure}

Figure~\ref{fig2} shows the distributions for inclusive semileptonic 
decays of $B$ mesons. The spectrum close to the endpoint, where the 
lepton energy becomes maximal, exhibits a sharp spike as $y \to y_{max}$. 
In this region we have we have 
\begin{equation}  
\frac{d\Gamma}{dy}  \propto \Theta(1-y-\rho)
\left[ 2 + \frac{\lambda_1}{(m_Q (1-y))^2} 
\left(\frac{\rho}{1-\rho} \right)^2 
\left\{ 3 - 4 \left(\frac{\rho}{1-\rho} \right) \right\}  \right] , 
\end{equation} 
which behaves like 
$\delta$-functions and its derivatives as $\rho \to 0$, which can be 
seen in (\ref{buspec}). This behaviour indicates a breakdown of the operator 
product expansion close to the endpoint, since for the spectra the 
expansion parameter is not $1/m_Q$, but rather $1/(m_Q - qv)$, which 
becomes $1/(m_Q [1-y])$ after the integration over the neutrino momentum. 
In order to obtain a description of the endpoint region, one has to perform
some resummation of the operator product expansion.  
  



\subsubsection{The Endpoint Region}
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Very close to the endpoint of the inclusive semileptonic decay spectra
only a few resonances contribute. In this resonance region one may 
not expect to have a good description of the spectrum using
an approach based on parton-hadron duality; here a sum over a few
resonances will be appropriate. 

In the variable $y$ the size of this resonance region is however of the 
order of $(\bar\Lambda / m_Q)^2$ and thus small. In a larger region of 
the order $\bar\Lambda / m_Q $, which we shall call the endpoint region, 
many resonances contribute and one may hope to describe the spectrum 
in this region using parton-hadron duality. 

In order to obtain an adequate description of this region we consider 
again the correlator of two currents and contract the light quark field
\begin{equation}
   T(q^2,v\cdot q) = \int\mbox{d}^4 x\,e^{i Q\cdot x}\,
   \langle B(v)|\,T\,\Big\{ \bar h_v(x)\,\Gamma_1\,S_q(x,0)\,
   \Gamma_2\,h_v(0) \Big\}\,|B(v)\rangle \,.
\end{equation}
with $Q = q - m_b v$ and $\Gamma_1$ and $\Gamma_2$ two arbitrary Dirac matrices. 
Here $S_q(x,0)$ is the propagator of the $q$-quark in the background
field of the light constituents of the $B$-meson. The Fourier
transform of $S_q(x,0)$ has the form
\begin{eqnarray}\label{propag}
   S_q(Q) &=& {1\over \fmslash{Q} + i\fmslash{D} + i\epsilon}
    \nonumber\\
   &=& (\fmslash{Q} + i\fmslash{D})\,{1\over
    (Q^2 + 2i Q\cdot D - - \fmslash{D}\fmslash{D}
    + i\epsilon)} \,.
\end{eqnarray}
Note that a derivative acting on the rescaled heavy quark field
corresponds to the residual momentum $k=p_b-m_b v$. Since the
residual momentum results from the interactions of the heavy quark
with light degrees of freedom, it is of order $\Lambda_{QCD}$.

We will be interested in the
region where the components of $Q$ are large (of order $m_b$), since
otherwise the operator product expansion breaks down. To leading
order in the heavy mass expansion, we can then neglect the covariant 
derivative in the numerator, as well as the term containing two 
derivatives in the denominator. This gives
\begin{equation}\label{Sendp}
   S_q(Q) = {\fmslash{Q} \over Q^2 + 2i Q\cdot D 
   + i\epsilon} + {\cal O}(\Lambda_{QCD}/m_b) \,.
\end{equation}
The term in the denominator containing the covariant derivative is of
order $\Lambda m_b$. To see whether it is important, we have to
distinguish different kinematic regions. When $Q^2\sim m_b^2$, the
derivative term is suppressed by a factor $\Lambda/m_b$. To leading
order, the propagator corresponds to the propagator of a free quark
with momentum $Q$ and we recover the standard operator product expansion 
as considered above. 

However, close to the endpoint, $Q^2$ is small although the components 
of this vector are large. In the kinematic region 
where $Q^2\sim\Lambda_{QCD} m_b$
(which we shall call endpoint region in the following),  
it is not possible
to neglect the derivative term, since it is of the same order as the 
$Q^2$ term. For even
smaller values $Q^2-m_q^2\sim \Lambda m_q$, the derivative term
becomes the dominant term in the denominator, and the operator
product expansion breaks down. This is the resonance region, where
the invariant mass $Q^2$ of the hadronic final state is close to the
mass-shell of the lowest-lying resonances containing a $q$-quark

It is possible to simplify the result further. To this end, we write
$Q=(v\cdot Q)(n+\delta n)$ and choose the vectors $n$ and $\delta n$
such that $n$ is a null-vector on the forward light cone satisfying
$n^2=0$ and $n\cdot v=1$, and thus $v\cdot\delta n=0$. In the region
where the derivative term in (\ref{Sendp}) is important, i.e.\ for
$v\cdot Q$ of order $m_b$ but $Q^2$ of order $\Lambda m_b$, we have
$n\cdot\delta n\sim \Lambda/m_b$. As long as we restrict ourselves to
the leading term in the large-$m_b$ limit, we can thus neglect
$\delta n$. This leads to the final expression for the effective
background-field propagator:
\begin{equation} \label{final}
   S_q(Q) \simeq \int\mbox{d}k_+\,\delta(k_+-i D_+)\,
   {\rlap{\,/}Q \over Q^2 + 2 k_+\,v\cdot Q 
    + i\epsilon} \,,
\end{equation}
where we define $n\cdot D\equiv D_+$. In light-cone gauge, $n\cdot
A\equiv A_+=0$, the operator $i D_+$ reduces to $i\partial_+$ and
corresponds to the light-cone residual momentum $k_+$ of the heavy
quark inside the $b$-meson. 

The $\delta$-function appearing in (\ref{final}) is still operator 
valued, since it contains a covariant derivative. To calculate the 
rate one has to take a matrix element which defines a distribution 
function of the residual light cone momentum of the heavy quark
\begin{equation}\label{fdef}
   f(k_+) = \langle B(v)|\,\bar h_v\,\delta(k_+-i D_+)\,h_v\,
   |B(v)\rangle
\end{equation}
which corresponds in the language of deep inelastic scattering to the 
leading-twist structure function that determines the
probability to find a $b$-quark with light-cone residual momentum
$k_+$ inside the $B$-meson. We use a mass-independent
normalization of states such that $\langle B(v)|\,\bar h_v\,h_v\,
|B(v)\rangle=1$. It then follows that the structure function is
normalized to unity:
\begin{equation}\label{fnorm}
   \int\mbox{d}k_+\,f(k_+) = 1 \,.
\end{equation}
It will later be convenient to introduce the Fourier transform of the
structure function, which is given by the forward matrix element of a
bilocal operator:\footnote{The definition of a heavy quark structure
function as the Fourier transform of the bilocal operator in
(\ref{bilocal}) was used, in a different context, in
Ref.~\cite{Jaffe}.}
\begin{eqnarray}\label{bilocal}
   \widetilde f(t) &=& \int\mbox{d}k_+\,e^{-i k_+ t}\,f(k_+)
    \nonumber\\
   &=& \langle B(v)|\,\bar h_v(0)\,P\exp\bigg[ -i\int\limits_0^t\!
    \mbox{d}u\,A_+(u n)\bigg]\,h_v(t n)\,|B(v)\rangle \,.
\end{eqnarray}
This function obeys the normalization condition $\widetilde f(0)=1$.
Note that the path-ordered exponential is absent in light-cone gauge.

From the definition of the distribution function $f$ we may infer 
some interesting properties.  The moments of $f(k_+)$ are
given by forward matrix elements of leading-twist, higher-dimension
operators in the heavy quark effective theory. They form a set of
low-energy parameters $A_n$ defined by
\begin{equation}\label{Anrela}
   A_n = \int\mbox{d}k_+\,k_+^n\,f(k_+) = i^n\,\widetilde f^{(n)}(0)
   = \langle B(v)|\,\bar h_v\,(i D_+)^n\,h_v\,|B(v)\rangle \,,
\end{equation}
where $\widetilde f^{(n)}(0)$ is a shorthand notation for the $n$-th
derivative of $\widetilde f(t)$ evaluated at $t=0$. Using the
equation of motion of the heavy quark effective theory, one obtains
for the first three moments
\begin{eqnarray}\label{moments}
   A_1 &=& 0 \,,\qquad A_2 = -{\lambda_1\over 3} \,, \nonumber\\
   A_3 &=& -{v_\nu\over 6}\,\langle B(v)|\,\bar h_v\,\big[
    D_\mu,g_s G^{\mu\nu}\big]\,h_v\,|B(v)\rangle \,,
\end{eqnarray}
where $g_s G^{\mu\nu}=i[D^\mu,D^\nu]$ is the gluon field-strength
tensor, and $\lambda_1$ has been introduced in (\ref{lam1}).

The function $f$ is a universal distribution function, which 
appears in all heavy-to-light inclusive decays; another example 
is the decay $B \to X_s \gamma$ \cite{NeubertBsg,BiMotion}, 
where this function determines 
the photon-energy spectrum in a region of order $1/m_Q$ around
the $K^*$ peak. 

In principle $f$ has to be determined by other methods than the $1/m_Q$
expansion, e.g.\ from lattice calculations or from a model, or it has 
to be determined from experiment by measuring the photon spectrum 
in $B \to X_s \gamma$ or the lepton spectrum in $B \to X_u \ell \bar{\nu}$.
In the context of the model ACCMM model \cite{ACCMM} $f$ has been 
calculated in \cite{BigiACCM}.  

In addition to the moments also the support of 
$f$ is known; it is nonvanishing only for 
$-\infty < k_+ < \bar\Lambda$. A simple one parameter model 
incorporating these properties 
has been suggested in \cite{ManNeu}
\begin{equation} \label{ftoy}
   f(k_+) = {32\over\pi^2\bar\Lambda}\,(1-x)^2 
   \exp\bigg\{ - {4\over\pi}\,(1-x)^2 \bigg\}\,
   \Theta(1-x)  ,
\end{equation}
where $x = k_+ / \bar\Lambda$, and   
the choice $\bar\Lambda = 570$ MeV yields reasonable values 
for the moments. 

The spectrum may be calculated in terms of the distribution function 
$f$ and one finds (assuming the final state quark to be massless)
\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}
Note that
the heavy quark mass $m_b$ does no longer appear explicitly. For
this reason, and in particular when the focus is on the endpoint
region, it would be unnatural to introduce the rescaled lepton energy
$y=2 E_\ell/m_b$. Hence, we will hereafter present our results as
function of the lepton energy $E_\ell$, which is the quantity that is
actually measured in experiments. Note, in particular, that (to the
order we are working) the maximum value of the lepton energy is
correctly reproduced. 

The result (\ref{convol}) represents a resummation of the most
singular contributions in the endpoint region, which correspond to 
a resummation of the highest derivatives of $\delta$-functions as
they appear in (\ref{buspec}), where the explixit caslulation to order
$1/m_Q^2$ has been performed. In order to illustrate the effect 
of the convolution (\ref{convol}) we show 
in fig.~\ref{fig3} the spectrum for 
$B \to X_u \ell \nu_\ell$ using the ansatz (\ref{ftoy}). 

\begin{figure}   
\epsfysize=5cm
   \centerline{\epsffile{mplfig2.eps}}
   \caption{Charged-lepton spectrum in $B\to X_u \ell  \bar\nu$ 
decays. The solid line is (\protect{\ref{convol}}) with the ansatz 
(\protect{\ref{ftoy}}), the dashed
line shows the prediction of the free-quark decay model. The figure 
is from \protect\cite{ManNeu}.}
\label{fig3}
\end{figure}

Including the non-perturbative effects yields a reasonably behaved
spectrum in the endpoint region and the $\delta$-function-like 
singularities have disappeared. Furthermore, the spectrum now extends
beyond the parton model endpoint;   
it is shifted from $E_\ell^{max} = m_Q / 2$ to the physical 
endpoint $E_\ell^{max} = M_H / 2$, since $f$ is non-vanishing for 
positive values of $k_+ < \bar\Lambda = M_H - m_Q$.


\subsubsection{Perturbative Corrections}
%
%
%
%
On top of the large nonperturbative corrections in the endpoint 
region there are also large perturbative corrections which have 
been known for a long time \cite{QCDradcorr}. While the corrections
to the total rate are of he expected size (see \ref{QCDcorr}), 
the endpoint region contains large logarithms, of which the leading
ones are the Sudakov-like double logarithms. Up to terms vanishing 
at the endpoint we have to order $\alpha_2$
\begin{equation} \label{rc}
\frac{d\Gamma}{dy} = \frac{d\Gamma^{(0)}}{dy}
\left[ 1 - \frac{2 \alpha_s}{3 \pi} \left(
\ln ^2 (1-y) + \frac{31}{6} \ln (1-y) + \frac{5}{4} + \pi^2 \right) \right],
\end{equation}
where $y = 2 E_\ell / m_b $ is again the rescaled energy variable and 
$\Gamma^{(0)}$ is the parton model rate which becomes clsoe to the endpoint 
\begin{equation} \label{endpoint}
\frac{1}{\Gamma_b} \frac{d\Gamma^{(0)}}{dy} = 2y^2 (3-2y) \Theta(1-y)
\to 2 \Theta(1-y) 
\end{equation}
The doubly logarithmic terms are believed to exponentiate, yielding 
\begin{equation} \label{rc2}
\frac{d\Gamma}{dy} = \frac{d\Gamma^{(0)}}{dy}
\exp\left[-\frac{2 \alpha_s }{3 \pi} \ln ^2 (1-y)\right]  .
\end{equation}
up to singly logarithmic terms which have been neglected in (\ref{rc2}). 
Note that the exponentiation strongly reduces the rate close to 
the endpoint resulting in a strong modification of the parton model result. 

Since singly logarithmic terms are ommitted, the choice of the 
scale $\mu$ in 
$\alpha_s$ is not obvious and various suggsetions have been made. 
In particular the choice $\mu^2 = m_b^2 (1-y)$ yields a drastic modification
of the spectrum in the endpoint region; using the one loop result for 
$\alpha_s$ we have    
\begin{equation} \label{rc3}
\frac{d\Gamma}{dy} = \frac{d\Gamma^{(0)}}{dy}
\exp\left[-\frac{8}{25} \frac{\ln ^2 (1-y)}
                        {\ln[(m_b^2 / \Lambda_{QCD}^2) (1-y)]} \right] .
\end{equation}
yielding a strong damping in the endpoint region. 

Unlike the nonperturbative corrections the perturbative ones cannot shift 
the endpoint of the spectrom away from the one given by the parton model. 
Still both types, the perturbative and the nonperturbative corrections
are strongly entangled in the endpoint region and a simultaneous 
treatment of both is difficult \cite{sterman}.  

