\section{Exclusive Decays}
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The most straightforward application of heavy quark symmetry are
exclusive decays. In particular the exclusive heavy-to-heavy 
transitions of the form $b \to c$ are strongly restricted by heavy quark 
symmetries and the phenomenological implications are very interesting. 

In this section we shall consider exclusive decays of heavy hadrons, 
focussing mainly on semileptonic decays. Heavy quark symmetries and 
HQET has not (yet?) brought any significant progress in the description 
of exclusive nonleptonic decays and hence only a brief comment is added 
on this kind of decays. 

\subsection{Semileptonic Heavy to Heavy Transitions}
\boldmath 
  \subsubsection{Meson Decays: $B \to D^{(*)} \ell \nu$}
\unboldmath
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The the exclusive semileptonic decays $B \to D^{(*)} \ell \nu$ are the 
``master example'' for the heavy-to-heavy decays which are strongly 
constrained by heavy quark symmetries. The hadronic matrix elements
acquire in the heavy mass limit exactly the form of the Wigner-Eckart
theorem (\ref{wet}) and are thus described interms of a single normalized 
form factor, the Isgur Wise function.

We shall first list the relevant matrix elements and count the form 
factors without assuming heavy quark symmetry. One has
\begin{eqnarray}
 \langle D (v') | \bar{c} \gamma_\mu b | B(v) \rangle  &=&  \sqrt{m_B m_D}
 \left[ \xi_+ (y) (v_\mu + v'_\mu)
     + \xi_- (y) (v_\mu - v'_\mu) \right] \\ 
 \langle D^* (v',\epsilon) | \bar{c} \gamma_\mu b | B(v) \rangle  &=& 
       i \sqrt{m_B m_{D^*}} 
\xi_V (y) \varepsilon_{\mu \alpha \beta \rho} \epsilon^{*\alpha}
                       v^{\prime \beta} v^\rho  \\ 
\langle D^* (v',\epsilon) | \bar{c} \gamma_\mu \gamma_5 b | B(v) \rangle 
   &=& \sqrt{m_B m_{D^*}}
       \left[ \xi_{A1} (y) (vv'+1) \epsilon^*_\mu
      -  \xi_{A2} (y) (\epsilon^* v)  v_\mu \right. \nonumber \\
   && \qquad \qquad \left.   -  \xi_{A2} (y) (\epsilon^* v)  v'_\mu \right] ,
\end{eqnarray} 
where we have defined $y = vv'$. Thus in general there are six form factors,
which in the heavy mass limit for both the $b$ and the $c$ quark may be 
related to the Isgur Wise function as introduced in (\ref{wet})
\begin{equation}
\xi_i (y) = \xi (y) \mbox{ for } i = +,V,A1,A3, \qquad
\xi_i (y) = 0       \mbox{ for } i = -,A2 .
\end{equation} 
In particular, at the non-recoil point $v=v'$ we have due to heavy quark 
symmetry and Luke's theorem
\begin{equation}
\xi_i (1) = 1 + {\cal O} (1/m_Q^2) \mbox{ for } i = +,V,A1,A3, \quad
\xi_i (1) = {\cal O} (1/m_Q)     \mbox{ for } i = -,A2 .
\end{equation} 
Note that the form factors for which there is no normalization at 
$v=v'$ there is also no protection against corrections of lienar 
order in $1/m_Q$.  

The differential rates for the exclusive semiletonic $b \to c$ 
transitions may be expressed in terms of the six form factors as
\begin{eqnarray}
&& \frac{d\Gamma}{dy} (B \to D \ell \nu_\ell)  \label{bd} \\
&& = 
\frac{G_F^2}{48 \pi^3} |V_{cb}|^2 (m_B + m_D)^2 
\left(m_D \sqrt{y^2 -1}\right)^3 
\left| \xi_+ (y) - \frac{m_B - m_D}{m_B + m_D} \xi_- (y) \right|^2 
\nonumber \\
&&\frac{d\Gamma}{dy} (B \to D^* \ell \nu_\ell) \label{bds}   \\
&& = \frac{G_F^2}{48 \pi^3} |V_{cb}|^2 (m_B - m_{D^*})^2 m_{D^*}^2
\left(m_{D^*} \sqrt{y^2 -1}\right) (y+1)^2 |\xi_{A1} (y)|^2 
\sum_{i=0,\pm} |H_i(y)|^2 \nonumber 
\end{eqnarray} 
with 
\begin{eqnarray}
|H_\pm (y)|^2 &=& \frac{m_B^2 - m_{D^*}^2 - 2y m_B m_{D^*}}{(m_B - m_{D^*})^2}
\left[1 \mp \sqrt{\frac{y-1}{y+1}} R_1 (y) \right]^2 \\
|H_0 (y)|^2 &=& \left(1 + \frac{m_B (y-1)}{m_B - m_{D^*}} 
                \left[1-R_2 (y) \right] \right)^2
\end{eqnarray}
where we have defined the form factor ratios 
\begin{equation}
R_1 (y) = \frac{\xi_V (y)}{\xi_{A1} (y)}, \quad 
R_2 (y) = \frac{\xi_{A3} (y) + \frac{m_B}{m_{D^*}} \xi_{A2} (y)}{\xi_{A1} (y)}
\end{equation}
In the heavy mass limit these differential rates depend only on the 
Isgur-Wise function
\begin{eqnarray}
&& \frac{d\Gamma}{dy} (B \to D \ell \nu_\ell)  
\stackrel{m_b, m_c \to \infty}{=} 
\frac{G_F^2}{48 \pi^3} |V_{cb}|^2 (m_B + m_D)^2 
\left(m_D \sqrt{y^2 -1}\right)^3 
| \xi (y)|^2 \\
&& \frac{d\Gamma}{dy} (B \to D \ell \nu_\ell)   
\stackrel{m_b, m_c \to \infty}{=} \\
&&  \frac{G_F^2}{48 \pi^3} |V_{cb}|^2 (m_B - m_{D^*})^2 m_{D^*}^2
\left(m_{D^*} \sqrt{y^2 -1}\right) (y+1)^2 \\ \nonumber 
&& \qquad \left[ 1 + \frac{4y}{y+1}
\frac{m_B^2 - m_{D^*}^2 - 2y m_B m_{D^*}}{(m_B - m_{D^*})^2} \right]
| \xi (y)|^2 \nonumber 
\end{eqnarray} 
These relations allow for a test of heavy quark symmetry, since the 
ratios of the differential rates do not depend on any unknown form factor 
any more. In particular the ratios $R_1$ ($R_2$) measures the ratio of 
the differential transverse (longitudinal) rate and the total differential 
rate. In the heavy mass limit both $R_1$ and $R_2$ are unity; this has to 
be compared to the measurements by CLEO \cite{CLEORi}
\begin{eqnarray}
R_1 &=& 1.24 \pm 0.26 \pm 0.12 \\
R_2 &=& 0.72 \pm 0.18 \pm 0.07 
\end{eqnarray}

\subsubsection{The determination of $V_{cb}$ using HQET}
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From the measured lepton invariant mass spectrum one may determine $V_{cb}$
in a model independent way by extrapolating to the kinematical 
endpoint of maximal momentum transfer to the leptons, corresponding
to the point $v = v'$. At this point heavy quark symmetries determine the 
absolute normalization of some of the form factors and the corrections 
to this normalization have been discussed in section 2. 

The mode $B \to D^* \ell \nu_\ell$ has the advantage of a higher 
branching fraction and hence we shall start the discussion with 
this decay.
The relevant formula may be derived from (\ref{bds}) and reads
\begin{equation}
\lim_{y \to 1} \frac{1}{\sqrt{y^2 - 1}} \frac{d\Gamma}{dy}
(B \to D^* \ell \nu_\ell) = \frac{G_F^2}{4 \pi^3} (m_B - m_{D^*})^2 
m_{D^*}^3 |V_{cb}|^2 |\xi_{A1} (1)|^2 
\end{equation}
The form factor $\xi_{A1}$ is normalized due to heavy quark symmetries
and is hence protected against $1/m_Q$ corrections at $v = v'$ by Lukes 
theorem \cite{Lu90}. Hence we have
\begin{equation}
\xi_{A1} (1) = \eta_A (1+\delta_{1/m^2}) 
\end{equation}
Including QED corrections and the estimate of the $1/m_Q^2$ corrections 
in the way discussed in section 2 one obtains
\begin{equation}
\xi_{A1} (1) = 0.92 \pm 0.03
\end{equation}
From this value one may extract from the extrapolations shown 
in figs.\ref{VCBfig1} and \ref{VCBfig3}
a value for $V_{cb}$
\begin{eqnarray}
|V_{cb}| &=& 0.0362 \pm 0.0019 \pm 0.0020 \pm 0.0014 
             \mbox{ (CLEO)\cite{cleovcb}} \\
|V_{cb}| &=& 0.0345 \pm 0.0025 \pm 0.0027 \pm 0.0015 
             \mbox{ (ALEPH)\cite{ALEPHvcb}} 
\end{eqnarray}
where the last error reflects the theoretical uncertainty of 
the $1/m_Q$ corrections. 
\begin{figure}
   \epsfysize=7cm
   \centerline{\epsffile{fig_CLVcb.eps}}
\caption[]{CLEO data\cite{cleovcb} and the extrapolation 
           used to obtain $V_{cb}$. The figure istaken from the 
           recent review \cite{Neunew}.}
\label{VCBfig1}
\end{figure}

\begin{figure}
   \epsfysize=7cm
   \centerline{\epsffile{fig_ALVcb.ps}}
\caption[]{ALEPH data\cite{ALEPHvcb} and the extrapolation used to 
           obtain $V_{cb}$.}
\label{VCBfig3}
\end{figure}

Recently CLEO has also measured the leptonic invariant mass spectrum 
for the decay $B \to D \ell \nu_\ell$. In a similar way one may perform 
an extrapolation to obtain $V_{cb}$. Here one gets from (\ref{bd})
\begin{eqnarray}
\lim_{y \to 1} && \left(\frac{1}{\sqrt{y^2 - 1}}\right) \frac{d\Gamma}{dy}
(B \to D \ell \nu_\ell) 
\\ \nonumber && = \frac{G_F^2}{48 \pi^3} (m_B + m_D)^2 
m_D^3 |V_{cb}|^2 
\left| \xi_+ (1) - \frac{m_B - m_D}{m_B + m_D} \xi_- (1) \right|^2 
\end{eqnarray}
In this case a form factor enters which is not protected by 
Lukes theorem, $\xi_- (1) = {\cal O} (1/m_Q)$. However, this does not 
spoil the possibility to determine $|V_{cb}|$ from this mode, since the 
$1/m_Q$ corrections are kinematically suppressed by the 
factor $ (m_B - m_D)/(m_B + m_D)$. Here we have 
\begin{equation}
\left| \xi_+ (1) - \frac{m_B - m_D}{m_B + m_D} \xi_- (1) \right| = 
\eta_V (1+ \Delta_{1/m_Q})
\end{equation}
where $\Delta_{1/m_Q}$ are the $1/m_Q$ corrections induced by $\xi_- (1)$.
These corrections have been estimated recently \cite{CapNeu} 
\begin{equation} \label{bdest}
\left| \xi_+ (1) - \frac{m_B - m_D}{m_B + m_D} \xi_- (1) \right| = 
0.98 \pm 0.07
\end{equation}
The data from CLEO \cite{cleovcb1} are shown in fig.\ref{VCBfig2}. 
From the extrapolation 
shown in fig.\ref{VCBfig2} one obtains the value 
\begin{equation}
|V_{cb}| = 0.0353 \pm 0.0046 \pm 0.0044 \pm 0.0025
\end{equation}
where the last error again reflects the theoretical uncertainty. 
Both values of $V_{cb}$ are in very good agreement and are also consistent
with values obtained form other methods, in particular with the 
results from inclusive semileptonic decays \cite{bigiVCB}
\begin{figure}   
\epsfysize=7cm
   \centerline{\epsffile{fig_CL1Vcb.ps}}
\caption[]{Data and the extrapolation used to obtain $V_{cb}$ from 
$B \to D \ell \nu_\ell$. The figure is taken from 
\cite{cleovcb1}.}
\label{VCBfig2}
\end{figure}

%\subsubsection{The slope of the form factors}
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The data on the leptonic invariant mass spectrum extend over the 
whole rage of $y$ and the extrapolation to $y = 1$ has to rely 
on some ansatz for the Isgur Wise function. This may be used in 
turn to extract a value also for the slope of the Isgur Wise function.  

Close to the point $y=1$ one thus parametrizes the Isgur Wise function 
as
\begin{equation}
\xi (y) = 1 - \rho^2 (y-1) + \cdots
\end{equation}
The theoretical predictions\cite{voloshin}
for the slope parameter $\rho^2$ 
depend on matrix elements 
involving excited $D$ mesons which are hard to estimate; thus the
theoretical value is quite uncertain and ranges between 
\begin{equation}
0.5 \le \rho^2 \le 1.1
\end{equation}
Data on the slope parameter come from the LEP experiments as well 
as from ARGUS and CLEO, the results obtained from experiment are 
\begin{equation}
\rho^2 = \left\{ \begin{array}{cc}
                 0.29 \pm 0.21 \pm 0.12 & \mbox{ALEPH\cite{alephrho}} \\
                 0.84 \pm 0.12 \pm 0.08 & \mbox{CLEO\cite{cleorho}} \\
                 1.17 \pm 0.22 \pm 0.06 & \mbox{ARGUS\cite{argusrho}} \\
                 0.81 \pm 0.16 \pm 0.10 & \mbox{DELPHI\cite{delphirho}}
                 \end{array} \right.
\end{equation}
which are compatible with the theoretical expectation.

