\subsection{(Semi)leptonic Heavy to Light transitions}
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Heavy quark symmetries may also be used to restrict the independent
form factors appearing in heavy to light decays \cite{iwlight}. 
For the decays of heavy 
mesons into light $0^-$ and $1^-$ particles heavy quark symmetries restrict 
the number of independent form factors to six, which is just the 
number needed to parametrize the semileptonic decays of this type. 
Furthermore, no absolute normalization of form factors may be obtained from 
heavy quark symmetries in the heavy to light case; only the relative 
normalization of $B$ meson decays heavy to light transitions may be obtained 
from the corresponding $D$ decays.

In general we shall discuss matrix elements of a heavy to light
current which have the following structure
\begin{equation} \label{htlg}
J = \langle  A | \bar{\ell} \Gamma h_v | H (v) \rangle ,
\end{equation}
where $\Gamma$ is an arbitrary Dirac matrix,
$\ell$ is a light quark
($u$, $d$ or $s$) and
$A$ is a state involving only light degrees of freedom.

Spin symmetry implies that the heavy quark index hooks directly 
to the heavy quark index of the Dirac matrix of the current. Thus one
may write for the transition matrix element~(\ref{htlps})
\begin{equation} \label{htltrg}
\langle  A | \bar{\ell} \Gamma h_v | H (v) \rangle =
\mbox{ Tr } \left( {\cal M}_A \Gamma H(v) \right)
\end{equation}
where the matrix $H(v)$ representing the heavy meson has been
given in (\ref{mesonrep0}).
The matrix ${\cal M}_A$ describes the light degrees of freedom
and is the most general matrix which may
be formed from the kinematical variables involved. Furthermore, if
the energies of the particles in the state $A$ are small, i.e.\ of
the order of $\Lambda_{QCD}$, the matrix ${\cal M}_A$ does not depend
on the heavy quark; in particular it does not depend on the heavy mass
$m_H$.
In the following we
shall discuss some examples.


\subsubsection{Heavy Meson Decay Constant}
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The first example is the heavy meson decay constant, where the
state $A$ is simply the vacuum state. The heavy meson decay
constant is defined by
\begin{equation}
\langle 0 | \bar{\ell} \gamma_\mu \gamma_5 h_v | H (v) \rangle =
f_H m_H v_\mu , 
\end{equation}
and since $| A \rangle = | 0 \rangle$ the matrix ${\cal M}_0$ is simply the 
unit matrix times a dimensionful constant\footnote{%
  Note that contributions proportional to
  $\fmslash{v}$ may be eliminated using
  $$
  H(v) \fmslash{v} = - H(v).
  $$ }
and one has, using
(\ref{htltrg})
\begin{equation}
\langle 0 | \bar{\ell} \gamma_\mu \gamma_5 h_v | H (v) \rangle = \kappa
\mbox{ Tr } \left( \gamma \gamma_5 H(v) \right) = 2 \kappa
\sqrt{m_H} v_\mu .
\end{equation}
As discussed above the constant $\kappa$ does not depend on the
heavy mass and thus one infers the well-known
scaling law for the heavy meson
decay constant from the last two equation
\begin{equation} \label{sca}
f_H  \propto \frac{1}{\sqrt{m_H}}
\end{equation}
Including the leading and subleading QCD radiative corrections
one obtains a relation between $f_B$ and $f_D$
\begin{equation}
f_B  = \sqrt{ \frac{m_c}{m_b}}
\left( \frac{\alpha_s (m_b)}{\alpha_s (m_c)} \right)^{-6/25} 
\left[ 1 + 0.894 \frac{\alpha_s (m_c)-\alpha_s (m_b)}{\pi} \right]
f_D \sim 0.69 f_D .
\end{equation}

\boldmath
  \subsubsection{Semileptonic Decays: $B \to (\pi /\rho) \ell \nu$}
\unboldmath
The second example are transitions of a heavy meson
into a light pseudoscalar meson, which we shall denote as $\pi$.
The matrix element corresponding to (\ref{htlg}) is
\begin{equation} \label{htlps}
J_P = \langle \pi(p) | \bar{\ell} \Gamma h_v | H (v) \rangle , 
\end{equation}
where $p$ is the momentum of the light quark,

The Dirac marix ${\cal M}_P$ for the light degrees of freedom
appearing now in (\ref{htltrg})
depends on $p$ and $v$.
It may be expanded in terms of the sixteen independent
Dirac matrices $1$, $\gamma_5$,  $\gamma_\mu$, $\gamma_5 \gamma_\mu$,
and $\sigma_{\mu \nu}$ taking into account that it has to behave
like a pseudoscalar.
The form factors appearing in the decomposition of
${\cal M}_P$
depend on the variable $v \cdot p$, the energy of the light meson
in the rest frame of the heavy one. In order to compare different
heavy to light transition by employing heavy flavor symmetry this
energy must be sufficiently small, since the typical scale for the
light degrees of freedom has to be of the order of $\Lambda_{QCD}$
to apply heavy quark symmetry\footnote{% 
         Note that in this case the variable $v \cdot p$ ranges between
         $0$ and $m_H / 2$ where we have neglected the pion mass. Thus 
         at the upper end of phase space the variable $v \cdot p$ scales
         with the heavy mass and heavy quark symmetries are not applicable
         anymore.}.  
For the case of a light pseudoscalar
meson the most general decomposition of ${\cal M}_P$ is
\begin{equation}
{\cal M}_P = \sqrt{v \cdot p} A (\eta ) \gamma_5
         + \frac{1}{\sqrt{v \cdot p}} B (\eta ) \gamma_5 \fmslash{p} ,
\end{equation}
where we have defined the dimensionless variable
\begin{equation} \label{eta}
\eta = \frac{v \cdot p}{\Lambda_{QCD}} .
\end{equation}

The form factors $A$ and $B$ are universal in the kinematic range
of small energy of the light meson, i.e.\ where the momentum
transfer to the light degrees of freedom is of the order $\Lambda_{QCD}$;
in this region $\eta$ is of order unity. This universality of the
form factors may be used to relate various kinds of heavy to
light transitions, e.g.\ the semileptonic decays like
$D \to \pi e \nu$, $D \to K e \nu$ or $B \to \pi e \nu$
and also the rare decays such as $B \to K \ell^+ \ell^-$
or $B \to \pi \ell^+ \ell^-$, where $\ell$ denotes an electron or a muon.

The relvant hadronic
current for the case of a $0^- \to 0^-$ transition 
may be expressed in terms of two form
factors
\begin{eqnarray} \label{frmfps}
\langle \pi(p) | \bar{\ell} \gamma (1-\gamma_5) h_v | H (v) \rangle
&=& F_1 (v \cdot p ) m_H v_\mu +
    F_2 (v \cdot p ) p_\mu
\\
&=& F_+ (v \cdot p ) (m_H v_\mu + p_\mu ) +
    F_- (v \cdot p ) q_\mu
\nonumber
\end{eqnarray}
where
\begin{equation}
F_\pm (v \cdot p ) = \frac{1}{2} \left( F_1 (v \cdot p ) \pm
                     F_2 (v \cdot p ) \right)
\end{equation}
Inserting this into (\ref{htlps}) one may express $F_\pm$ in terms
of the universal form factors $A$ and $B$
\begin{eqnarray} \label{f1hql}
F_1 (v \cdot p ) &=& F_+ (v \cdot p ) + F_- (v \cdot p )
=   -2 \sqrt{\frac{v \cdot p}{m_H}} A (\eta)
\\  \label{f2hql}
F_2 (v \cdot p ) &=& F_+ (v \cdot p ) - F_- (v \cdot p )
=   -2 \sqrt{\frac{m_H}{v \cdot p}} B (\eta)
\end{eqnarray}
From these relations one may read off the scaling of the form
factors with the heavy mass which was already
derived in \cite{iwlight}. In particular it follows that for small 
$v \cdot p$ $F_1$ is suppressed by one power of the heavy mass
compared to $F_2$. In other words, in the infinite mass limit the 
heavy to light transitions of the form $B \to \pi$ are described in
terms of a single form factor, since $F_+ = - F_-$   

The case of a heavy meson decaying into a light vector meson may be
treated similarly.
The matrix element for the transition of a heavy meson into a light
vector meson (denoted generically as $\rho$ in the following)
is given again by (\ref{htlg}) and is in this case
\begin{equation} \label{htlv}
J_V = \langle \rho(p,\epsilon) | \bar{\ell} \Gamma h_v | H (v) \rangle .
\end{equation}
Using (\ref{htltrg}) one has
\begin{equation} \label{htlvtr}
\langle \rho(p,\epsilon) | \bar{\ell} \Gamma h_v | H (v) \rangle =
\mbox{ Tr } \left( {\cal M}_V \Gamma H(v) \right) ,
\end{equation}
where now the Dirac matrix ${\cal M}_V$ has to be a linear
function of the polarization of the light vector meson.

The most general decomposition
is given in terms of four
dimensionless form factors
\begin{equation} \label{hqlv}
{\cal M}_V = \sqrt{v \cdot p} C(\eta) (v \cdot \epsilon)
    +  \frac{1}{\sqrt{v \cdot p}} D(\eta) (v \cdot \epsilon) \fmslash{p}
 + \sqrt{v \cdot p} E(\eta) \fmslash{\epsilon}
 +  \frac{1}{\sqrt{v \cdot p}} F(\eta) \fmslash{p} \fmslash{\epsilon}
\end{equation}
where the variable $\eta$ has been defined in (\ref{eta}).

Similar to the case of the decays into a light pseudoscalar meson
(\ref{htlvtr}) may be used to relate various exclusive heavy to
light processes in the kinematic range where the energy of the
outgoing vector meson is small. For example,
the semileptonic decays
$D \to \rho e \nu$, $D \to K^* e \nu$ and $B \to \rho e \nu$ are related
among themselves and all of them may be related to the rare heavy
to light decays $B \to K^* \ell^+ \ell^-$
and $B \to \rho \ell^+ \ell^-$
with $\ell = e, \mu$.

\subsubsection{Heavy to light decays of baryons}
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Finally we comment on the heavy to light transitions of baryons. For the 
$\Lambda$-type heavy baryons  spin symmetry relates different
polarizations of the same particle and thus imposes interesting
constraints. Consider for example the matrix element of an operator
$\bar{\ell} \Gamma h_v$ between a heavy $\Lambda_Q$ and a light
spin-1/2 baryon $B_\ell$. It is described by only two form
factors,
\begin{equation}
\langle B_\ell (p) | \bar{\ell} \Gamma h_v | \Lambda_Q  (v) \rangle =
\bar{u}_\ell (p) \{ F_1 (v \cdot p ) + \fmslash{v} F_2 (v \cdot p) \} \Gamma
u_{\Lambda_Q} (v) .
\end{equation}
Thus in this particular case spin symmetry greatly reduces the
number of independent Lorentz-invariant amplitudes which 
describe the heavy to light transitions.

This has some interesting implications for exclusive semileptonic 
$\Lambda_c$ decays. For the case of a left handed current 
$\Gamma = \gamma_\mu (1-\gamma_5)$, the semileptonic decay 
$\Lambda_c \to \Lambda \ell \bar{\nu}_\ell$ is in 
general parametrized in terms  of six form factors
\begin{eqnarray}
\langle \Lambda (p) | \bar{q} \gamma_\mu (1-\gamma_5) c | \Lambda_c (v) \rangle &=&
\bar{u} (p) \left[ f_1 \gamma_\mu
               + i f_2 \sigma_{\mu \nu} q^\nu
               +   f_3 q^\mu  \right] u(p')  \nonumber \\
&+& \bar{u} (p) \left[ g_1 \gamma_\mu
               + i g_2 \sigma_{\mu \nu} q^\nu
               +   g_3 q^\mu  \right] \gamma_5 u(p') ,
\end{eqnarray}
where $p' = m_{\Lambda_c} v$ is the momentum of the $\Lambda_c$ whereas
$q = m_{\Lambda_c} v - p$ is the momentum transfer. From this one
defines the ratio $G_A / G_V$ by
\begin{equation}
\frac{G_A}{G_V} = \frac{g_1 (q^2 = 0)}{f_1 (q^2 = 0)} .
\end{equation}

In the heavy $c$ quark limit one may relate the six form factors
$f_i$ and $g_i$ ($i=1,2,3$) to the two form factors $F_j$
($j=1,2$)
\begin{eqnarray}
f_1 &=& - g_1 = F_1 + \frac{m_\Lambda}{m_{\Lambda_c}} F_2 \\
f_2 &=& f_3 = -g_2 = -g_3 = \frac{1}{m_{\Lambda_c}} F_2
\end{eqnarray}
from which one reads off $G_A / G_V = -1$. This ratio is accessible
by measuring in semileptonic decays $\Lambda_c \to \lambda \ell \bar{\nu}_\ell$ 
the polarization variable $\alpha$
\begin{equation}
\alpha = \frac{2 G_A G_V}{G_A^2 + G_V^2} 
\end{equation}
which is predicted to be $\alpha = -1$ in the heavy $c$ quark limit. 
The subleading corrections to the heavy $c$ quark limit have been 
estimated and found to be small \cite{ML93}
\begin{equation}
\alpha < -0.95 ,
\end{equation}
and recent measurements yield 
\begin{eqnarray} 
\alpha &=& -0.91 \pm 0.49 \quad \mbox{ARGUS \cite{ARGUSalpha}} \\
\alpha &=& -0.89^{+0.17+0.09}_{-0.11-0.05}  \quad \mbox{CLEO\cite{CLEOlamc}}
\end{eqnarray}
and are in satisfactory agreement with the theoretical predictions. 

Recently the CLEO collaboration also measured the ratio of the form factors
$F_1$ and $F_2$, averaged over phase space. Heavy quark symmetries do not 
fix this form factor ratio, at least not for a heavy to light decay, while 
for a heavy to heavy decay the form factor $F_2$ vanishes in the heavy mass 
limit for the final state quark. CLEO measures \cite{CLEOratio}
\begin{equation}
\left\langle \frac{F_2}{F_2} \right\rangle_{\mbox{phase space}} = 
     -0.25 \pm 0.14 \pm 0.08
\end{equation}
which is in good agreement with model estimates \cite{JKlambda}.


