%Title: Exclusive decays of heavy flavours
%Author: Thomas Mannel
%Published: *Acta Phys. Pol.* **B26** (1995) 663-686
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\begin{document}
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\begin{titlepage}
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%{\ulogo U} \hfill
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%{\bf UNIVERSIT\"AT KARLSRUHE} \vspace*{1cm} \\
%{\large \sc Institut f\"ur \\ Theoretische Teilchenphysik }
% \end{flushright}}
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\hskip -3cm
\begin{flushright}
{\bf TTP 95--07} \\
March 1995 \\
hep-ph/yymmnnn
\end{flushright}
\vspace{1cm}
\begin{center}
{\Large\bf EXCLUSIVE DECAYS \vspace*{3mm} \\
OF HEAVY FLAVOURS}
\end{center}
\vspace{0.8cm}
\begin{center}
{\sc Thomas Mannel} \vspace*{2mm} \\
{\sl Institut f\"{u}r Theoretische Teilchenphysik, \\
Universit\"at Karlsruhe \\
D -- 76128 Karlsruhe, Germany.}
\end{center}
\vfill
\begin{abstract}
\noindent
Recent progress in the theoretical description of exclusive heavy flavour
decays is reviewed. After a general discussion of heavy quark symmetries
some applications are studied.
\end{abstract}
\vfill
\begin{center}{\it Contribution to the Crakow Epiphany Conference on Heavy Quarks, \\
in honour of the $60^{th}$ birthday of Kacper Zalewski, \\
January 6, 1995, Crakow, Poland.}
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%{\bf TTP 95--??} \\
%March 1995
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\section{Introduction}
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In the past five years substantial progress has been made in the theoretical
description of systems containing heavy quarks. This progress on the theoretical
side has been accompanied by an enormous improvement of experimental data, which
made the field of heavy quark physics one of the most interesting and prosperous
fields in high energy physics.
The theoretical breakthrough was triggered by the observation that for a heavy
quark one may take advantage of the fact that one may treat such a quark
to first approximation as infinitely heavy \cite{shifman,isgur/wise}.
This infinite mass limit has two
important properties. It may be formulated as an effective field theory, which
is called Heavy Quark Effective Theory (HQET) \cite{HQET}. This
implies that the corrections to this limit may be treated systematically in the
framework of a combined $\alpha_s (m_Q)$ and $\bar\Lambda / m_Q$ expansion,
where $m_Q$ is the mass of the heavy quark and $\bar\Lambda$ is some scale related
to the light degrees of freedom, typically of the order of a few hundred MeV.
The second and even more important point is that the leading term (corresponding
to the infinite mass limit) exhibits two additional symmetries which are not
present in full QCD \cite{isgur/wise}. It was this observation which started the
whole development, since these symmetries of the effective theory allow to
restrict the non-perturbative input needed to describe e.g.\ weak decay matrix
elements. In other words, the hadronic uncertainties may be reduced substantially
and even a few completely model independent statements are possible. The progress
in this field is documented in more or less extensive reviews \cite{reviews}.
Heavy quark symmetries work most efficiently for transitions among heavy quarks;
treating both $b$ and $c$ has heavy the weak transition $b \to c$ are precisely
of this type and thus these weak transitions are strongly constrained by heavy
quark symmetries. In particular, one may even obtain the absolute normalization
of the transition matrix elements at a certain kinematic point, allowing for
a model independent determination of the CKM matrix element $V_{cb}$. If the
final state quark is
light as e.g. in the weak transitions of the type $b \to u$ or $c \to s$ heavy
quark symmetries may still be applied but are less restrictive \cite{iwlight}.
In the present review I try to summarize some of the basics of heavy quark symmetries
and to indicate how to deal with HQET. Numerous papers have appeared since the
pioneering work of Voloshin, Shifman, Eichten, Hill \cite{shifman},
Isgur and Wise \cite{isgur/wise} and it is impossible to
summarize everything in this short review. Heavy quark symmetries lie at the heart
of HQET, and in section 2 we consider these additional symmetries. In section 3
the method of how to calculate corrections in the framework of HQET is outlined.
Section 4 is dedicated to heavy to heavy transitions, where we shall study the
decay $B \to D^{(*)} \ell \bar{\nu}_\ell$ in some detail. Heavy to light decays
will be considered in section 5, where purely leptonic and semileptonic decays
are studied.
\section{Heavy Quark Symmetry}
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The main impact of the heavy quark limit is due to two additional
symmetries which are not present in full QCD; the first is a heavy flavour
symmetry and the second one is the so called spin symmetry.
We shall first study the heavy flavour symmetry. The interaction of
the quarks with the gluons is flavour independent; all flavour dependence
in QCD is only due to the different quark masses. In the $1/m_Q$ expansion
the leading order Lagrangian is mass independent and hence a flavour
symmetry appears relating heavy quarks moving with the same velocity.
For the case of two heavy flavours $b$ and $c$ one has to leading order the
Lagrangian \cite{HQET}
\begin{equation}
{\cal L}_{heavy} = \bar{b}_v (v \cdot D) b_v + \bar{c}_v (v \cdot D) c_v ,
\end{equation}
where $b_v$ ($c_v$) is the field operator $h_v$ for the $b$ ($c$) quark
moving with velocity $v$ and $D = \partial + i g A$ is the QCD covariant
derivative.
This Lagrangian is obviously invariant under the $SU(2)_{HF}$ rotations
\begin{equation}
\left( \begin{array}{c} b_v \\ c_v \end{array} \right) \to
U_v \left( \begin{array}{c} b_v \\ c_v \end{array} \right) \quad
U_v \in SU(2)_{HF} .
\end{equation}
We have put a subscript $v$ for the transformation matrix $U$, since
this symmetry only relates heavy quarks moving with the same velocity.
The second symmetry is the heavy-quark spin symmetry. As is clear form
the Lagrangian in the heavy-mass limit, both spin degrees of freedom
of the heavy quark couple in the same way to the gluons; we may
rewrite the leading-order Lagrangian as
\begin{equation}
{\cal L} = \bar{h}_v^{+s} (iv D) h_v^{+s} + \bar{h}_v^{-s} (iv D) h_v^{-s},
\end{equation}
where $h_v^{\pm s}$ are the projections of the heavy quark field on a
definite spin direction $s$
\begin{equation}
h_v^{\pm s} = \frac{1}{2} (1 \pm \gamma_5 \fmslash{s}) h_v,
\quad s\cdot v = 0 .
\end{equation}
This Lagrangian has a symmetry under the rotations of the heavy quark
spin and hence all the heavy hadron states moving with the velocity $v$
fall into spin-symmetry doublets as $m_Q \to \infty$. In Hilbert space
this symmetry is generated by operators $S_v (\epsilon)$ as
\begin{equation}
[ h_v , S_v (\epsilon) ] = i \fmslash{\epsilon} \fmslash{v} \gamma_5 h_v
\end{equation}
where $\epsilon$ with $\epsilon^2 = -1$ is the rotation axis.
The simplest spin-symmetry doublet in the mesonic case consists of the
pseudoscalar meson $H(v)$ and the corresponding vector meson
$H^* (v,\epsilon)$, since a spin rotation yields
\begin{equation}
\exp\left(iS_v(\epsilon) \frac{\pi}{2} \right) | H (v) \rangle =
(-i) | H^* (v,\epsilon) \rangle ,
\end{equation}
where we have chosen an arbitrary phase to be $(-i)$.
In the heavy-mass limit the spin symmetry partners have to be
degenerate and their splitting has to scale as $1/m_Q$. In other
words, the quantity
\begin{equation}
\lambda_2 = \frac{1}{4} (M_{H^*}^2 - M_H^2)
\end{equation}
has to be the same for all spin symmetry doublets of heavy ground state
mesons. This is well supported by data: For both the $(B,B^*)$ and the
$(D,D^*)$ doublets one finds a value of $\lambda_2 \sim 0.12$ GeV${}^2$.
This shows that
the spin-symmetry partners become degenerate in the infinite mass limit and
the splitting between them scales as $1/m_Q$.
In the infinite mass limit the symmetries imply relations between matrix
elements involving heavy quarks. For a transition between
heavy ground-state mesons $H$ (either pseudoscalar or vector)
with heavy flavour $f$ ($f'$) moving with velocities $v$ ($v'$), one
obtains in the heavy-quark limit
\begin{equation} \label{WET}
\langle H^{(f')} (v') | \bar{h}^{(f')}_{v'} \Gamma h^{(f)}_v
| H^{(f)} (v) \rangle
= \xi (vv') \mbox{ Tr }
\left\{ \overline{{\cal H} (v)} \Gamma {\cal H}(v) \right\} ,
\end{equation}
where $\Gamma$ is some arbitrary Dirac matrix and $H(v)$ are the
representation matrices for the two possibilities of coupling
the heavy quark spin to the spin of the light degrees of freedom,
which are in a spin-1/2 state for ground state mesons
\begin{equation} \label{mesonrep}
{\cal H}(v) = \frac{\sqrt{M_H}}{2} \left\{ \begin{array}{l l}
(1+\fmslash{v}) \gamma_5 & 0^-, \, (\bar{q} Q) \mbox{ meson} \\
(1+\fmslash{v}) \fmslash{\epsilon} & 1^- , \, (\bar{q} Q) \mbox{ meson} \\
& \mbox{with polarization } \epsilon .
\end{array} \right.
\end{equation}
Due to the spin and flavour independence of the heavy mass limit
the Isgur--Wise function $\xi$ is the only nonperturbative information
needed to describe all heavy to heavy transitions within a spin-flavour
symmetry multiplet.
Excited mesons have been studied in \cite{Falk}. They may be classified
by the angular momentum of the light degrees of freedom $j_l$, which
is coupled with the heavy quark spin $S$ to the total angular momentum
$J$ of the meson. Furthermore, the orbital angular momentum $\ell$ determines
the parity $P = (-1)^{\ell+1}$ of the meson. For a given $\ell> 0$ we can have
$j_l = \ell \pm 1/2$ and the coupling of the heavy quark spin yields
two spin symmetry doublets $(J = \ell-1 , J = \ell)$ and
$(J = \ell, J = \ell +1)$. For example, the lowest positive parity
$\ell = 1$ mesons are two spin symmetry doublets $(0^+, 1^+)$ and
$(1^+, 2^+)$. In the $D$ meson system these states have been observed
\cite{CLEOD} and behave as predicted by heavy quark symmetry \cite{EHQexcited}
Similarly as for the mesons heavy-quark symmetries imply that
only one form factor is needed to describe heavy to heavy transitions within
a spin flavour symmetry multiplet; in other words, there is an Isgur Wise
function for each multiplet.
The ground state baryons have been studied in \cite{IWbary,Geobary,MRRbary}.
According to the particle data group they are classified as follows
\begin{eqnarray}
\label{Lam}
\Lambda_h &=& [(qq^\prime)_0 h]_{1/2} \qquad \Xi_h^\prime = [(qs)_0 h]_{1/2} \\
%
\label{Sig}
\Sigma_h &=& [(qq^\prime)_1 h]_{1/2} \qquad \Xi_h = [(qs)_1 h]_{1/2}
\qquad \Omega_h = [(ss)_1 h]_{1/2} \\
%
\label{SigStar}
\Sigma_h^* &=& [(qq^\prime)_1 h]_{3/2} \qquad
\Xi_h^* = [(qs)_1 h]_{3/2} \qquad
\Omega_h^* = [(ss)_1 h]_{3/2}.
\end{eqnarray}
Here, $q, q^\prime$ refer to $u$ and $d$ quarks, $q \neq q^\prime$ for the
$\Lambda_h$, but $q$ may be the same as $q^\prime$ for the $\Sigma_h$ and
$\Sigma_h^*$. The first subscript (0, 1) is the total spin of the light
degrees of freedom, while the second subscript (1/2, 3/2) is the total spin
of the baryon.
Spin symmetry forces these baryons into spin symmetry doublets. For the
$\Lambda$-type baryons (\ref{Lam}) the spin rotations are simply a subset of the Lorentz
transformations, since the light degrees of freedom are in a spin-0 state.
The corresponding spin symmetry doublet is in this case given by the two
polarization directions of the heavy baryon.
From the point of view of heavy quark symmetries the $\Lambda$-type baryons
are the simplest hadrons, although from the quark model
point of view they are composed of three quarks.
The baryons with the light degrees of freedom in a spin one state may be
represented by a pseudovector-spinor object $R^\mu$ with $v_\mu R^\mu=0$
\footnote{One could as well represent the light degrees of freedom by an
antisymmetric tensor instead by a pseudovector; this is a completely
equivalent formulation \cite{MRRbary}.}.
In general $\gamma_\mu R^\mu \neq 0$ because $R^\mu$ contains
spin 1/2 contributions as well as spin 3/2 parts.
In other words, $R^\mu$ contains a Rarita-Schwinger field as well as
a Dirac field. Under Lorentz transformations $R^\mu$ behaves as
\begin{equation}
R^\mu(v) \rightarrow {\Lambda^\mu}_\nu D(\Lambda) R^\nu(\Lambda v),
\end{equation}
where $\Lambda_{\mu \nu}$ and $D(\Lambda)$ are the Lorentz transformations
in the vector and spinor representation respectively, while
under spin rotations we have
\begin{equation} \label{sigspin}
R^\mu(v) \rightarrow - \gamma_5 \fmslash{v} \fmslash{\epsilon} R^\mu(v) .
\end{equation}
The spin-3/2 component of the the pseudovector-spinor object
corresponding to the $\Sigma^*_h$ is projected
out by contracting with $\gamma_\mu$
\begin{equation}
\gamma_\mu R^\mu _{\Sigma_h^*} = 0 .
\end{equation}
The rest of the independent components of $R$ correspond to $\Sigma_h$ baryon:
\begin{equation}
R^\mu_{\Sigma_h} = {1 \over \sqrt{3}} (\gamma^\mu + v^\mu) \gamma_5
u_{\Sigma_h},
\end{equation}
where $u_{\Sigma_h}$ is the Dirac spinor of the $\Sigma_h$ state. Similar
expressions hold for the nonstrange baryons $\Xi^{(*)}_h$ and $\Omega_h^{(*)}$.
The spin rotation (\ref{sigspin}) transform the $\Sigma$-like baryons into
the $\Sigma^*$ states
and vice versa. Thus the spin symmetry doublets for the ground state baryons
are given by the two polarization directions of the baryons in (\ref{Lam}), and
by the two states with corresponding light quark flavour numbers in (\ref{Sig})
and (\ref{SigStar}).
Similar to the case of mesons one may derive a Wigner-Eckart theorem for the
spin symmetry doublets of the baryons
\begin{equation}\label{lambda}
\langle \Lambda_{h}(v) | \bar{h}\Gamma h' | \Lambda_{h'}(v') \rangle = A(v\cdot v')
\bar{u}_{\xi_h}(v) \Gamma u_{\xi_{h'}}(v'),
\end{equation}
where we have allowed for the possibility of two heavy quark flavours $h$
and $h'$. In the same way, one obtains two form factors for the
$\Sigma^{(*)}_h \to \Sigma^{(*)}_{h'}$.
\begin{eqnarray} \label{ss1}
&& \langle \Sigma^{(*)}_h (v) | \bar{h}_v \Gamma h_{v'} |\Sigma^{(*)}_h \rangle \\
&& = \bar{R}^{\mu}_{\Sigma_h^{(*)}} (v) \Gamma {R}^{\nu}_{\Sigma_h^{(*)}}(v')
\, \left[ B(v\cdot v')g_{\mu \nu} + C(v\cdot v')v'_\mu v_\nu \right] .
\nonumber
\end{eqnarray}
Finally, parity does not allow for transitions between $\Lambda$ and
$\Sigma^{(*)}$ type baryons
\begin{equation}\label{sl1}
\langle \Sigma^{(*)}_h(v) | \bar{h}_v \Gamma h_{v'} |\Lambda_h (v) \rangle = 0 ,
\end{equation}
and hence these transitions are not only suppressed by the flavour symmetry
of the light degrees of freedom, but additionally by heavy quark symmetry.
Excited baryons my be studied along the same lines as for the mesons.
The spin symmetry doublets as well as the restrictions on transition matrix
elements have been studied in \cite{Falk}.
Heavy quark symmetries thus lead to a strong reduction of the number of
independent from factors that describe current induced transitions among heavy
hadrons. In addition to that the symmetries even allow us to obtain the
normalization of some of these form factors. Since the currents
\begin{equation}
J^{hh'} = \bar{h}_v \gamma_\mu h^\prime_v = v_\mu \bar{h}_v h^\prime_v
\end{equation}
are the generators of heavy flavour symmetry in the velocity sector $v$,
the normalization of the Wigner-Eckard theorems
(\ref{WET},\ref{lambda},\ref{ss1}) is known at the nonrecoil point $v=v'$.
By standard arguments one obtains for the mesons
\begin{equation} \label{normxi}
\xi (vv' = 1) = 1 ,
\end{equation}
while the corresponding relation for the baryons is
\begin{eqnarray}
A(vv' = 1) &=& \sqrt{m_{\Lambda_h} m_{\Lambda_{h'}}} \label{normA}\\
B(vv' = 1) &=& \sqrt{m_{\Sigma^{(*)}_h} m_{\Sigma^{(*)}_{h'}}} , \label{NormB}
\end{eqnarray}
where the factor involving the squre root of the masses means that the hadron states
in (\ref{lambda}) are normalized relativistically.
\section{Corrections to the Heavy Mass Limit}
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The heavy quark symmetries allow us to obtain relations which hold in the
heavy mass limit, such as the Wigner-Eckart theorems
(\ref{WET},\ref{lambda},\ref{ss1}) and the normalization statements
(\ref{normxi}, \ref{normA}, \ref{NormB}). Corrections due to
finite masses may be calculated systematically using the machinery of
HQET. In this section we consider the strategy of how the
corrections are studied.
In general there are two types of corrections. The short-distance
corrections may be calculated in perturbation theory, based on the
leading order of the $1/m_Q$ expansion of the Lagrangian. The
logarithmic ultraviolet divergences in the effective theory
correspond to logarithmic dependences on the heavy-quark mass $m_Q$
in the full theory, and renormalization group methods may be employed
to perform resummations of these logarithms.
The starting point of a QCD corrections calculation are the Feynman
rules of full QCD and the ones of HQET. In HQET there are only two of
Feynman rules modified compared to full QCD:
\begin{center}
\begin{tabular}{llll}
& full QCD & & HQET \\
Propagator of the heavy quark
& $\displaystyle{\frac{i}{\fmslash{p} - m_Q + i\epsilon}}$ & $\longrightarrow$
& $\displaystyle{\frac{i}{vk + i\epsilon}}, \quad p = mv + k $ \\
Heavy quark gluon vertex
& $ ig \gamma_\mu T^a $ & $\longrightarrow$
& $ ig v_\mu T^a $ \\
& & &
\end{tabular}
\end{center}
For the sake of simplicity and clarity we shall consider the matrix
element of some operator ${\cal O}$ corresponding to some observable
quantity, e.g.\ a current mediating a weak decay. Since HQET is an
effective theory, the machinery of effective theory guarantees the
factorization of long distance effects from the short distance ones,
which are related to the large mass $m_Q$.
Neglecting $1/m_Q$ corrections, this factorization takes the form
\begin{equation} \label{factheo1}
\langle {\cal O} \rangle_{full} =
Z \left( \frac{m_Q}{\mu} \right)
\langle {\cal O} \rangle_{static}(\mu) + {\cal O} (1/m_Q) ,
\end{equation}
where the dependence on $m_Q$ of the coefficient $Z$ is given
as a combined expansion in the coupling strength
$\alpha_s = g^2/(4\pi)$ and logarithms of $m_Q$
\begin{eqnarray} \label{zexp}
Z \left( \frac{m_Q}{\mu} \right) &=& a_{00} \\
&+& a_{11} \left( \alpha_s \ln \left(\frac{m_Q}{\mu} \right) \right)\,
+ a_{10} \alpha_s \nonumber \\
&+& a_{22} \left( \alpha_s \ln \left(\frac{m_Q}{\mu} \right) \right)^2
+ a_{21} \alpha_s
\left( \alpha_s \ln \left(\frac{m_Q}{\mu} \right) \right)
+ a_{20} \alpha_s^2 \nonumber \\
&+& a_{33} \left( \alpha_s \ln \left(\frac{m_Q}{\mu} \right) \right)^3
+ a_{32} \alpha_s
\left( \alpha_s \ln \left(\frac{m_Q}{\mu} \right) \right)^2
+ a_{31} \alpha_s^2
\left( \alpha_s \ln \left(\frac{m_Q}{\mu} \right) \right)
\nonumber \\
&& \hphantom{--------} + a_{30} \alpha_s^3
+ \cdots \nonumber
\end{eqnarray}
This factorization theorem corresponds to the statement that the
ultraviolet divergencies in the effective theory have to match the
logarithmic mass dependences of full QCD. The factorization scale $\mu$ is
an arbitrary parameter, and the physical quantity $\langle {\cal O}
\rangle_{full}$ does not depend on this parameter. However, calculating
the matrix element of this operator in the effective theory and
studying its ultraviolet behaviour allows us to access the mass dependence
of the observable $\langle {\cal O} \rangle$.
The ultraviolet behaviour of the effective theory is investigated by
the renormalization group equations. Differentiating (\ref{factheo1})
with respect to the factorization scale $\mu$ yields the renormalization
group equation
\begin{equation}
\frac{d}{d \ln \mu} \left\{ Z \left( \frac{m_Q}{\mu} \right)
\langle {\cal O} \rangle_{static}(\mu) \right\} = 0
\end{equation}
from which we may obtain an equation which determines the
change of the coefficient $Z$ when the scale is changed
\begin{eqnarray} \label{rge}
&& \left( \frac{d}{d \ln \mu} + \gamma_{\cal O} (\mu) \right)
Z \left( \frac{m_Q}{\mu} \right)= 0
\\
&& \gamma_{\cal O} (\mu) = \frac{d}{d \ln \mu}
\ln ( \langle {\cal O} \rangle_{static}(\mu) )
\nonumber .
\end{eqnarray}
The quantity $\gamma_{\cal O}$ is called the anomalous dimension of the
operator ${\cal O}$ which is universal for all matrix elements of
${\cal O}$, since it is connected with the short distance behaviour of the
insertion of the operator ${\cal O}$.
Eq.(\ref{rge}) describes the renormalization group scaling
in the effective theory. It allows to shift logarithms of the large
mass scale from the matrix element of ${\cal O}$ into the
coefficient $Z$: If the matrix element is renormalized at the large
scale $m_Q$ the logarithms of the type $\ln m_Q$ will apear in the matrix element
of ${\cal O}$ while the coefficient $Z$ at this scale will simply
be
\begin{equation}
Z(1) = a_{00} + a_{10} \alpha_s (m_Q) + a_{20} \alpha_s^2 (m_Q)
+ a_{30} \alpha_s^3 (m_Q) + \cdots
\end{equation}
The renormalization
group equation (\ref{rge}) allows to lower the renormalization point
from $m_Q$ to $\mu$; the matrix element renormalized at $\mu$ will
not contain any logarithms of $M$ any more, they will appear in
the coefficient $Z$ in the way shown in (\ref{zexp}).
In all cases relevant in the present context the matrix elements will
be matrix elements involving
hadronic states, which are in most cases impossible to calculate from
first principles. However, eq.(\ref{rge}) allows to extract the
short distance piece, i.e.\ the logarithms of the large mass $M$
and to separate it into the Wilson coefficients.
The anomalous dimension may be calculated in perturbation theory
in powers of the coupling constant $g$ of the theory.
In general, in a renormalizable theory the coupling constant depends
on the scale $\mu$ at which the theory is renormalized. The scale
dependence of the coupling constant is determined by the
$\beta$ function
\begin{equation} \label{gml}
\frac{d}{d \ln \mu} g (\mu) = \beta (\mu) .
\end{equation}
In a mass independent scheme the renormalization
group functions $\gamma_{\cal O}$ and $\beta$ will depend on the
scale $\mu$ only through their dependence on the coupling constant
\begin{equation}
\beta = \beta(g(\mu)) \qquad
\gamma_{\cal O} = \gamma_{\cal O}( g(\mu)) .
\end{equation}
Hence we may rewrite the renormalization group equation (\ref{rge})
as
\begin{equation} \label{rg1}
\left( \mu \frac{\partial}{\partial \mu}
+ \beta(g) \frac{\partial}{\partial g} +
\gamma_{\cal O}(g) \right) Z \left( \frac{m_Q}{\mu}, g \right) = 0 .
\end{equation}
The renormalization group functions $\beta$ and $\gamma_{\cal O}$
are calculated in perturbation theory; the first term of the $\beta$
function on QCD is obtained from a one-loop calculation and is given by
\begin{equation} \label{beta0}
\beta(g) = - \frac{1}{(4 \pi)^2} \left( 11 - \frac{2}{3} n_f \right)
g^3 + \cdots ,
\end{equation}
where $n_f$ is the number of active flavors, i.e.\ the number
of flavors with a mass less than $m_Q$.
With this input the renormalization group equation may be solved to
yield
\begin{equation} \label{LLA}
Z \left( \frac{m_Q}{\mu} \right) = a_{00} \left(
\frac{\alpha_s (\mu)}{\alpha_s (m_Q)} \right)^
{\displaystyle -\frac{48 \pi^2}{33 - 2 n_f} \gamma_1}
\end{equation}
where $\gamma_1$ is the first coefficient in the perturbative expansion of
the anomalous dimension $\gamma_{\cal O} = \gamma_1 g^2 + \cdots $ and
$\alpha_s (\mu)$ is the one loop expression for the running coupling
constant of QCD
\begin{equation} \label{alpha}
\alpha_s (\mu) = \frac{12 \pi}{(33-2n_f) \ln(\mu^2/\Lambda_{QCD}^2)}
\end{equation}
which is obtained from solving (\ref{gml}) using (\ref{beta0}).
This expression corresponds to a summation of the leading logarithms
$(\alpha_s \ln m_Q)^n$ which is achieved by a one-loop calculation
of the renormalization group functions $beta$ and $\gamma_{\cal Q}$;
in other words, in this way a resummation of the
first column of the expansion (\ref{zexp}) is obtained.
In a similar way one may also resum the second column of (\ref{zexp}),
if the renormalization group functions $\beta$ and $\gamma$ are calculated
to two loops and the finite terms of the one loop expression are included.
Below we present some results up to two loops.
The second type of corrections are the power corrections of order
$1/m_Q^n$, which in
general involve long-distance physics and hence may in general not
be calculated, but have to be parametrized. As an example, consider
a matrix element of a current $\bar{q} \Gamma Q$ mediating
a transition between a heavy meson and some arbitrary state
$| A \rangle$. The full QCD Lagrangian ${\cal L}$ and the fields $Q$
are expanded in terms of a power series in $1/m_Q$ and one obtains
\footnote{It has been pointed out repeatedly \cite{nonunique} that
these expansions
are not unique; one may always shift terms from the fields
into the Lagrangian, which then appear as terms that would
vanish by a naive application of the equations of motion.}
\begin{eqnarray} \label{lexp}
&& {\cal L} = \bar{h}_v (iv D) h_v +
\frac{1}{2m} \bar{h}_v i \fmslash{D} P_- i \fmslash{D} h_v
+ \left( \frac{1}{2m} \right)^2 \bar{h}_v i \fmslash{D} P_-
(-ivD) i \fmslash{D} h_v + \cdots \\
&& Q(x) = e^{-im_Qvx} \left[ 1 + \frac{1}{2m_Q} i \fmslash{D}_\perp +
\left(\frac{1}{2m_Q}\right)^2 (-ivD) P_- i \fmslash{D} + \cdots \right] h_v ,
\end{eqnarray}
where $P_\pm = (1\pm \fmslash{v})/2$ is the projector on the upper/lower
component of the spinors. For the matrix element under consideration
one obtains up to order $1/m_Q$:
\begin{eqnarray} \label{exp}
&& \langle A | \bar{q} \Gamma Q
| M (v) \rangle =
\langle A | \bar{q} \Gamma h_v
| H (v) \rangle \\ \nonumber
&& \qquad + \frac{1}{2m_Q} \langle A | \bar{q} \Gamma
P_- i \fmslash{D} h_v
| H (v) \rangle
-i \int d^4 x \langle A | T \{ L_1 (x) \bar{q} \Gamma h_v \}
| H (v) \rangle
+ {\cal O} (1/m^2)
\end{eqnarray}
where $L_1$ are the first-order corrections to the Lagrangian
as given in (\ref{lexp}). Furthermore, $| M (v) \rangle $ is the
state of the heavy meson in full QCD, including all its mass
dependence, while $| H (v) \rangle$ is the corresponding state in
the infinite mass limit.
Expression (\ref{exp}) displays the generic structure of the
higher-order corrections as they appear in any HQET calculation.
There will be local contributions coming from the expansion of
the full QCD field; these may be interpreted as the corrections to
the currents. The nonlocal contributions, i.e.~the time-ordered
products, are the corresponding corrections to the states and thus
in the r.h.s.\ of (\ref{exp}) only the states of the infinite-mass
limit appear.
Finally we shall review briefly an important result concerning $1/m_Q$
corrections, which is called Lukes theorem. It is a generalization of
the Ademollo Gatto theorem, which states that in the presence of explicit
symmetry breaking the matrix elements of the currents that generate the
symmetry are still normalized up to terms which are second order in the
symmetry breaking interaction.
For the case at hand the relevant symmetry is the heavy flavor symmetry.
This symmetry is an $SU(2)$ symmetry and is generated by three operators
$Q_\pm$ and $Q_3$ with
\begin{eqnarray}
&& Q_+ = \int d^3 x \, \bar{b}_v (x) c_v (x) \quad
Q_- = \int d^3 x \, \bar{c}_v (x) b_v (x) \nonumber \\
&& Q_3 = \int d^3 x \, (\bar{b}_v (x) b_v (x) - \bar{c}_v (x) c_v (x))
\\ \nonumber
&& [Q_+ , Q_- ] = Q_3 \qquad [Q_+ , Q_3 ] = -2 Q_+ \qquad (Q_+)^\dagger = Q_-
\end{eqnarray}
Let us denote the ground state flavour symmetry
multiplet as $| B \rangle$ and $| D \rangle$. Then the operators act in
the following way
\begin{eqnarray}
&& Q_3 | B \rangle = | B \rangle \qquad Q_3 | D \rangle = - | D \rangle \nonumber \\
&& Q_+ | D \rangle = | B \rangle \qquad Q_- | B \rangle = | D \rangle \\
&& Q_+ | B \rangle = Q_- | D \rangle = 0 \nonumber .
\end{eqnarray}
The Hamiltonian of this system has a $1/m_Q$ expansion of the form
\begin{eqnarray}
H &=& H_0^{(b)} + H_0^{(c)} +
\frac{1}{2m_b} H_1^{(b)} + \frac{1}{2m_c} H_1^{(c)} + \cdots \\
\nonumber
&=& H_0^{(b)} + H_0^{(c)} +
\frac{1}{2} \left(\frac{1}{2m_b}+\frac{1}{2m_c}\right)
(H_1^{(b)}+H_1^{(c)}) \\
\nonumber
&& \hphantom{H_0^{(b)} + H_0^{(c)}}
+ \frac{1}{2} \left(\frac{1}{2m_b}-\frac{1}{2m_c}\right)
(H_1^{(b)}-H_1^{(c)}) + \cdots \\
&=& H_{symm} + H_{break} \nonumber .
\end{eqnarray}
In the second equation, the first line is still symmetric under heavy flavour
$SU(2)$ while the term in the second line does not commute any more with $Q_\pm$,
but it still commutes with $Q_3$. In other words, to order $1/m_Q$ we still
have common eigenstates of $H$ and $Q_3$, which we shall denote as
$\tilde{| B \rangle}$ and $\tilde{| D \rangle}$. Sandwiching the
commutation relation we get
\begin{eqnarray}
1&=& \tilde{\langle B |} Q_3 \tilde{| B \rangle} =
\tilde{\langle B |} [Q_+ , Q_- ] \tilde{| B \rangle}
\\ \nonumber
&=& \sum_n \left[ \tilde{\langle B |} Q_+ \tilde{| n \rangle}
\tilde{\langle n |} Q_- \tilde{| B \rangle} -
\tilde{\langle B |} Q_- \tilde{| n \rangle}
\tilde{\langle n |} Q_+ \tilde{| B \rangle}\right] \\
\nonumber
&=& \sum_n \left[ |\tilde{\langle B |} Q_+ \tilde{| n \rangle}|^2 -
|\tilde{\langle B |} Q_- \tilde{| n \rangle}|^2 \right]
\end{eqnarray}
where $\tilde{| n \rangle}$ form a complete set of states of the Hamiltonian
$H_{symm} + H_{break}$. The matrix elements may be written as
\begin{equation}
\tilde{\langle B |} Q_\pm \tilde{| n \rangle} =
\frac{1}{E_B - E_n} \tilde{\langle B |} [H_{break}, Q_\pm ] \tilde{| n \rangle}
\end{equation}
where $E_B$ and $E_n$ are the energies of the states $\tilde{| B \rangle}$
and $\tilde{| n \rangle}$ repectively. In the case
$\tilde{| n \rangle} = \tilde{| D \rangle}$ the matrix element will
be of order unity, since both the numerator as well as the energy difference in the
denominator are of the order of the symmetry breaking. For all other states the
energy difference in the denomiantor is nonvanishing in the symmetry limit,
and hence this difference is of order unity; thus the matrix element for these
states will be of the order of the symmetry breaking. From this we conclude
\begin{equation}
\tilde{\langle B |} Q_+ \tilde{| D \rangle} =
1 + {\cal O}\left[\left(\frac{1}{2m_b}-\frac{1}{2m_c}\right)^2\right] .
\end{equation}
In particular, the weak transition currents at the nonrecoil point $v=v'$
are proportional to these symmetry generators and hence we may conclude that
for some of these matrix elements we only have corrections of the order
$1/m_Q^2$.
\section{Heavy to Heavy Transitions}
%
%
%
%
For the case of a heavy to heavy transition the
Wigner Eckart theorem (\ref{WET}) implies that there is only a single
form factor which describe the weak decays of heavy
hadrons; furthermore, the heavy mass limit yields the normalization
of this form factor at the kinematic point $v = v'$.
Treating both the $b$ and the $c$ quark as heavy, the
semileptonic decays $B \to D^{(*)} \ell \nu$ are the phenomenologically
relevant examples. The matrix elements for these transitions are in general
parametrized in terms of six form factors
\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 $y = vv'$. Due to the Wigner Eckart theorem
(\ref{WET}) these six from factors are related to the Isgur Wise
function by
\begin{equation}
\xi_i (y) = \xi (y) \mbox{ for } i = +,V,A1,A3, \qquad
\xi_i (y) = 0 \mbox{ for } i = -,A2 .
\end{equation}
Since heavy quark symmetries also yield the normalization of the
Isgur Wise function, we know the absolute value of the differential
rate at the point $v = v'$ in terms of the meson masses and $V_{cb}$.
Hence we may use this to extract $V_{cb}$ from these decays in a model
independent way
by extrapolating
the lepton spectrum to the kinematic endpoint $v=v'$.
Using the mode $B \to D^{(*)} \ell \nu$ one obtains the relation
\begin{equation} \label{extra}
\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} (1)|^2 ,
\end{equation}
where $\xi_{A1}$ is equal to the Isgur Wise function in the
heavy mass limit, and hence $\xi_{A1} (1) = 1$.
Corrections to this relation have been calculated along the lines
outlined above in leading and subleading order. A complete discussion
may be found in more extensive review articles (see e.g. Neubert's review
\cite{reviews}), including reference to the original papers. Here we only
state the final result
\begin{eqnarray}
&& \xi_{A1} (1) = x^{6/25} \left[ 1 + 1.561 \frac{\alpha_s (m_c) - \alpha_s (m_b)}{\pi}
- \frac{8 \alpha_s (m_c)}{3 \pi} \right.
\\ \nonumber
&& \quad + z \, \left\{\frac{25}{54} - \frac{14}{27} x^{-9/25}
+ \frac{1}{18} x^{-12/25} + \frac{8}{25} \ln x
\right\} %\\ \nonumber
\left. - \frac{\alpha_s (\bar{m})}{\pi} \frac{z^2}{1-z} \ln z \right]
+ \, \delta_{1/m^2} ,
\label{msquared}
\end{eqnarray}
where we use the abbreviations
$$
x = \frac{\alpha_s (m_c)}{ \alpha_s (m_b)}, \quad z = \frac{m_c}{m_b}
$$
and $\bar{m}$ is a scale somewhere between $m_b$ and $m_c$.
Up to the term $\delta_{1/m^2}$ all these contributions may be calculated
perturbatively, including the dependence on $z$. The quantity
$\delta_{1/m^2}$ parametrizes the nonperturbative contributions, which enter
here at order $1/m^2$. These corrections may be expressed in terms of the
kinetic energy $\lambda_1$, the chromomagnetic moment $\lambda_2$, which are
given in terms of matrix elements of higher-order terms of the Lagrangian
\begin{eqnarray}
\lambda_1 &=& \frac{\langle H (v) | \bar{h}_v (iD)^2 h_v | H (v) \rangle}
{2 M_H}
\label{lam1} \\
\lambda_2 &=& \frac{\langle H (v) | \bar{h}_v \sigma_{\mu \nu} iD^\mu iD^\nu h_v
| H (v) \rangle}
{2 M_H} ,
\label{lam2}
\end{eqnarray}
where the normalization of the states is chosen to be
$\langle H (v) | \bar{h}_v h_v | H (v) \rangle = 2 M_H$, where $M_H$ is the
mass of the heavy meson in the static limit.
In terms of these parameters $\delta_{1/m^2}$ may be written as
\begin{eqnarray}
\delta_{1/m^2} &=& - \left(\frac{1}{2m_c} \right)^2
\frac{1}{2} \left( -\lambda_1 + \lambda_2 \vphantom{\int} \right.
\\ \nonumber &+& \left.
(-i)^2 \frac{1}{2\sqrt{M_B M_D}} \int d^4 x \,d^4y \,
\langle B^*(v,\epsilon) | T \left[ {\cal L}_b^{(1)} (x) \bar{b}_v c_v
{\cal L}_c^{(1)} (y) \right]
| D^*(v,\epsilon) \rangle \right) \\
&+& \vphantom{\int}{\cal O} (1/m_c^3,1/m_b^2, 1/(m_c m_b) ) , \nonumber
\end{eqnarray}
where ${\cal L}_Q^{(1)}$ is the first order Lagangian for the quark $Q$ as given
in (\ref{lexp}) and $M_B$ ($M_D$) are the masses of the $B$ ($D$) meson in the
heavy quark limit.
Here we display only the largest contribution of order $1/m_c^2$; the complete
expression, including the $1/m_b^2$ and $1/(m_c m_b)$ terms, may be found
in \cite{FN92,zerec}.
The matrix elements $\lambda_1$, $\lambda_2$ and the one of the time-ordered
product have to be estimated in a model or need to be taken from data. The parameter
$\lambda_2$ has been discussed above and is given by the mass splitting
between the ground state spin symmetry partners. The kinetic energy $\lambda_1$
is currently subject of intensive discussions; it may not be read off from
the hadron spectrum and thus it is not easy to access. It is not yet determined
from data and only theoretical estimates exist; from its definition one
is led to assume
$\lambda_1 < 0$; a more restrictive inequality
\begin{equation}
-\lambda_1 > 3 \lambda_2
\end{equation}
has been derived in a quantum mechanical framework in \cite{BiMotion}
and using heavy-flavour sum rules \cite{Bisumrule}.
Furthermore, there exists also a sum rule estimate \cite{BBsumrule}
for this parameter
\begin{equation}
\lambda_1 = - 0.52 \pm 0.12 \mbox{ GeV}^2 .
\end{equation}
Similarly it is not easy to obtain information on the matrix element involving
the time-ordered product, and thus the corrections of order $1/m^2$ will finally
limit our ability to determine the CKM matrix element $V_{cb}$ in a model independent
way, at least along the lines as described above.
Various estimates for $\delta_{1/m^2}$ have been given in the literature. The
first estimate of this correction has been given in \cite{FN92} using the GISW
model \cite{GISW}, which is based on a wave function for the light quark. In this
work $ \delta_{m^2} = -2\% \ldots -3\% $ has been obtained. Another estimate with
weaker assumptions yields $\delta_{m^2} = 0 \ldots -5\%$ \cite{zerec}, but both
estimates have been critizised recently as being too small. Based on heavy flavour
sum rules it has been argued in \cite{SU94} that the $1/m^2$ corrections can be
quite large $\delta_{m^2} = 0\% \ldots -8\% $ \cite{SU94}. These various estimates
indicate the size of the theoretical error involved in the determination of $V_{cb}$
from the exclusive channel $B \to D^* \ell \bar{\nu}_\ell$.
This result has been used to extract $V_{cb}$ from data. In figure 1
the latest data \cite{CLEOvcb} are shown.
From this fit one obtains \cite{CLEOvcb}
\begin{equation}
|V_{cb}| = 0.0362 \pm 0.0019 \pm 0.0020 \pm 0.0014,
\end{equation}
where $\xi(1) = 0.97 \pm 0.04$ has been used. The third error in $|V_{cb}|$
is due to the theoretical uncertainties, which by now almost match the
experimental ones.
%\begin{figure}[p]
% \epsfysize=14cm
%\begin{center}
% \leavevmode
% \epsffile[70 290 520 690]{vcbfig.ps}
%\vspace*{14cm}
% \caption{
%Latest data \protect\cite{CLEOvcb} for the product $|V_{cb}| \xi (vv')$
%as a function of $y = vv'$.}
%\label{fig1}
%\end{center}
%\end{figure}
Similar statements can be made for the semileptonic decays of heavy baryons.
The decay $\Lambda_b \to \Lambda_c \ell \bar{\nu}$ is again parametrized by
a single form factor, and this even remains true for the $1/m_Q$ correction
terms \cite{GGWbary,GCbary}. However, at present data on these decays are
still sparse, although first measurements have been performed \cite{LEPbary}.
The $\Sigma$-type baryons will in general decay either strongly or
electromagnetically and hence a weak decay will be completely unobservable.
The only candidate where a weak decay may be observable is the $\Omega_Q$
baryon, since a strong decay would require the emission of a kaon and this
may be suppressed due to the too small phase space.
\section{Heavy to Light Transitions}
%
%
%
%
%
Heavy quark symmetries may also be used to restrict the independent
form factors appearing in heavy to light decays. 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 the
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{mesonrep}).
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.
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 on 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}
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
any more.}.
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 like $B \to K \ell^+ \ell^-$
or $B \to \pi \ell^+ \ell^-$ where $\ell$ denotes an electron or a muon.
As an example we give the relations between exclusive semileptonic
heavy to light decays. The relvant hadronic
current for this case 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}.
This may be used to normalize the semileptonic $B$ decays into light mesons
relative to the semileptonic $D$ decays. One obtains
\begin{equation} \label{bdphql}
F_\pm^B (v \cdot p ) =
\frac{1}{2} \left(\sqrt{\frac{m_D}{m_B}}
\pm \sqrt{\frac{m_B}{m_D}} \right)
F_+^D (v \cdot p )
+ \frac{1}{2} \left(\sqrt{\frac{m_D}{m_B}}
\mp \sqrt{\frac{m_B}{m_D}} \right)
F_-^D (v \cdot p )
\end{equation}
Note that $F_+$ for the $B$ decay is expressed in terms of
$F_+$ {\it and} $F_-$ for the $D$ decays. In the limit of vanishing
fermion masses only $F_+$ contributes, which means that the $F_-$
contribution to the rate is of the
order of $m_{lepton} / m_H$. Thus it will be extremely difficult to
determine experimentally.
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 $\ell = e, \mu$.
Finally we comment on the heavy to light transitions of baryons. For the
$\Lambda$-type heavy baryons (\ref{Lam}) 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 acessible
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}.
\section{Conclusions}
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The field of heavy quark physics has gone through a remarkable development
over the last few years due to new theoretical ideas as well as to a
major improvement of data. In particular the progress in the technology
of detectors (e.g.\ silicon vertex detectors) opened the possibility to
study $b$ physics even at machines which originally were not designed
for this kind of research. In this way also the high energy colliders
(in particular LEP and TEVATRON) could contribute substantially in this
area, since they allow to measure states (such as the $B_s$ and the $b$
flavoured baryons) which lie above the threshold of the
$\Upsilon (4s)$-$B$-factories.
From the theoretical side the heavy quark limit and HQET brought an
important success, since it provides a model independent and QCD based
framework for the description of processes involving heavy quarks.
As far as exclusive heavy to heavy decays are concerned, the additional
symmetries of the heavy mass limit restrict the number of
nonperturbative functions in a model independent way; furthermore,
heavy quark symmetries fix the absolute normalization of some of the
transition amplitudes at the point of maximum momnetum transfer.
In heavy to light
decays heavy quark symmetries do not work as efficiently; in this case
only the relative normalization of $B$ decays versus the corresponding
$D$ decays may be obtained.
Corrections may be studied systematically HQET. As
in any effective field theory this framework allows a clean separation between
short distance effects, connected with the large mass $m_Q$, and the
long distance pieces, which are related to small hadronic scales of order
$\Lambda_{QCD}$. The short distance part may be calculated in renormalization
group improved perturbation theory, while the long distance piece needs to
be parametrized, but is restricted by heavy quark symmetries.
The phenomenological impact of the heavy quark limit is tremendous. Its
main field of application are the semileptonic decays of $b$ and $c$ hadrons,
where the hadronic matrix elements are studied in the heavy mass expansion.
In particular for $b \to c$ decays, when both quarks are treated as heavy,
one has many model independent statements concerning the decay rates; in
addition, also the absolute normalization of the matrix elements is known,
allowing us the extraction of $V_{cb}$ without strong model dependences.
Heavy quark symmetries also relate exclusive semileptonic transitions with
the exclusive rare decays, which are based on $b \to s \gamma$ or
$b \to s \ell^+ \ell^-$ decays. These are, however, of the heavy to
light type and thus are not as strongly restricted as the heavy to heavy
ones.
HQET does not yet have much to say about exclusive nonleptonic decays; even
for the decays $B \to D^{(*)} D^{(*)}_s$, which involves three heavy quarks,
heavy quark symmetries are not sufficient to yield useful relations between
the decay rates \cite{MRRnonlep}. Of course, with additional assumptions
such as factorization one can go ahead and relate the
nonleptonic decays to the semileptonic ones; however, this is a very strong
assumption and it is not clear in what sense factorization is an
approximation. On the other side, the data of the nonleptonic $B$ decays
supports factorization, and first attempts to understand this from QCD and
HQET have been untertaken \cite{DGnonlep}; however, the problem of the
exclusive nonleptonic decays still needs clarification and hopefully the
heavy mass limit will also be useful here.
\section*{Acknowledgements}
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It was great pleasure and honour for me to be invited and to talk at the
symposium celebrating Kacper Zalewski's $60^{th}$ birthday.
I want to thank the organizers of the conference for the invitation and their
hospitality during my stay in Crakow.
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\newpage
\section*{Figure Captions}
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\noindent
{\bf Figure 1:}
Latest data \protect\cite{CLEOvcb} for the product $|V_{cb}| \xi (vv')$
as a function of $y = vv'$.
\end{document}