\subsection{Corrections to the Symmetry Limit}
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Heavy quark symmetries hold in the limit in which the heavy quark mass is 
considered to be infinitely heavy. However, such a limit is in general only useful 
if one has some way to at least estimate the corrections to the limiting 
behaviour. 

In the case at hand, there are two sources of corrections, namely QCD radiative
corrections, characterized by the small parameter $\alpha_s (m_Q)$ and power 
corrections governed by the small parameter $\Lambda_{QCD} / m_Q$. We shall 
start our discussion with the perturbatively calculable QCD radiative corrections. 

\subsubsection{QCD Radiative Corrections}
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The effective theory approach allows us to separate long and short distance 
contributions, and the scale where this separation takes place is taken to be 
the renormalization scale $\mu$. Since we are dealing with QCD, the coupling 
constant becomes smaller as the scale becomes larger, and thus we may calculate 
the short distance contributions perturbatively, as long as 
$\mu \gg \Lambda_{QCD}$. 

The usual procedure is to use renormalization group methods to improve perturbation 
theory. The coupling constant depends on the scale logarithmically, and hence 
$\alpha_s (m_Q)$ contains nonanalytic dependencies on $1/m_Q$. Renormalization 
group improvement means that one may resum these logarithms using as an input the 
perturbatively calculated renormalization group functions.

In order to exhibit this formalism we shall consider a simple example, which 
we will study in some detail, namely a heavy-to-light current matrix element
of the form 
\begin{equation} \label{expr}
\langle A | J | M (v) \rangle = 
\langle A | \bar{q} \Gamma Q 
              | M (v) \rangle = Z \left( \frac{m_Q}{\mu} \right)  
\langle A | \bar{q} \Gamma h_v 
              | H (v) \rangle |_\mu 
   + {\cal O} (1/m_Q)  
\end{equation}
where the left hand side is the matrix element in full QCD (depending 
on the full heavy quark field $Q$ and on the full pseudoscalar meson state
$ | M (v) \rangle$) while the right hand side is the static limit containing the 
static heavy quark field and the state in the static limit $| H (v) \rangle$. 

The renormalization constant $Z$ contains the short distance contribution of 
the matrix element and may be calculated perturbatively. It is a dimensionless
quantity and hence it depends (for constant coupling $\alpha_s$) on the ratio  
$m_Q / \mu$. Its perturbative expansion takes the form
\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}
where $\alpha_s = g^2/(4\pi)$ and the coefficients $a_{ij}$ have to be 
calculated using the Feynman rules of HQET. 

Before discussing these Feynman rules and a sample calculation, we shall 
continue the introduction to the renormalization group methods. 
 
The factorization theorem of the long and short distance contributions
implies 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 A | \bar{q} \Gamma Q | M (v) \rangle $
does not depend on this parameter. However, calculating
the matrix element of this operator in the effective theory and
studying its ultraviolet behavior allows us to access the mass dependence
of the matrix element $\langle A | \bar{q} \Gamma Q | M (v) \rangle $.

The ultraviolet behavior of the effective theory is investigated by
the renormalization group equation (RGE), which is obtained in the usual way 
from differentiating (\ref{expr}) with respect to the factorization 
scale $\mu$. The left hand side is independent of the arbitrary renormalization 
point $\mu$, and one may obtain an equation for the renormalization point 
dependence of $Z$  
\begin{equation} \label{rge}
\left( \mu \frac{\partial}{\partial \mu}
+ \beta(g) \frac{\partial}{\partial g} +
\gamma_J (g) \right)  Z \left( \frac{m_Q}{\mu}, g \right) = 0 .
\end{equation}
where $\gamma_J (g)$ is the anomalous dimension of the current $J$ 
which is related to the ultraviolet behavior of the matrix elements of 
$J$. The function $\beta(g)$ defines the running of the coupling constant 
\begin{equation} \label{gml}
\frac{d}{d \ln \mu} g (\mu) = \beta (\mu) .
\end{equation}
Both functions $\gamma_J (g)$ and $\beta(g)$ are calculable
in perturbation theory using a loopwise expansion, where the first term 
of the $\beta$ function of QCD is well known
\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 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_J = \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 $\gamma_J$;  
in other words, in this way a resummation of the
first column of the expansion (\ref{zexp}) is achieved.

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 non-logarithmic terms of the one loop 
expression are included.

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 $J$ 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 $J$ 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 $m_Q$ 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 $m_Q$
and to separate it into the Wilson coefficients.

Finally, the case we have considered as an example is indeed very simple; 
in general all operators of a given dimension may mix under renormalization, 
i.e. instead of a simple anomalous dimension a matrix of anomalous dimensions
may occur and the renormalization group equation \ref{rge} becomes a 
system of differential equations.  

In order to perform a calculation in HQET one has to derive the Feynman
rules of the effective theory. The dimension four piece of the Lagrangian
is unique and one may use the leading piece of 
(\ref{HarvLag}) or (\ref{FWlag}) to define 
a renormalizable field theory, and one may read off the 
Feynman rules for this theory in the usual way. For a static heavy quark 
moving with velocity $v$ one obtains

\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}
while all other rules remain unchanged. 

After having obtained the Feynman rules, one may proceed to calculate the 
anomalous dimensions by extracting the divergent pieces from loop diagrams.
We shall stick to our example of the heavy to light current and give an outline 
how such a calculation is performed. 

\begin{figure}
   \epsfysize=5cm
   \centerline{\epsffile{hqself.ps}}
   \centerline{\parbox{16cm}{\caption{\label{QCDfig1}
The self energy of a heavy quark. The double line represents the propagator of 
the static quark with velocity $v$ and residual momentum $k$ or $k-q$ respectively. 
}}}
\end{figure}

We first consider the self energy of a heavy quark which we shall need to 
obtain the wave function renormalization of the heavy quark. 
The loop diagram, which is depicted in fig.\ref{QCDfig1}, 
yields a divergent result which we shall regularize in dimensional regularization.
As usual these divergencies are compensated by counterterms in the Lagrangian 
and in the case of the self energy this amounts to add acounterterm of the 
same form as the leading order Lagrangian. Once the divergent part of the 
self energy diagram is compensated one has to define the finite part of the 
diagram, in other words, one may still perform finite renormalizations 
corresponding to a change of the renormalization point $\mu$. Hence 
a counter term is added as
\begin{equation}
{\cal L} = \bar{h}_v^{bare} (ivD) h_v^{bare} 
         = \bar{h}_v (ivD) h_v - (1-Z_h) \bar{h}_v (ivD) h_v 
         = Z_h \bar{h}_v (ivD) h_v
\end{equation}
which corresponds to a replacement of the bare field by the renormalized one 
\begin{equation}   
h_v^{bare} = \sqrt{Z_h (\mu)} h_v|_\mu
\end{equation}
where we have explicitly displayed the $\mu$ dependence and $h_v|_\mu$ is 
the heavy quark field renormalized at the point $\mu$. 

In dimensional regularization, 
the dependence on the renormalization scale $\mu$ enters through 
the fact that in this regularization scheme one has to introduce some scale 
parameter which keeps the coupling dimensionless; this scale parameter is to be 
identified with the renormalization scale $\mu$. 

The one-loop integral one has to perform in order to calculate $Z_h$ is
\begin{eqnarray}
i \Sigma_H (k) &=& (-ig)^2 (\mu)^\epsilon 
                   \int \frac{d^{4-\epsilon} q}{(2 \pi)^{4-\epsilon}}
                   v_\lambda T^a \frac{i}{v(k+q)} v^\lambda \frac{-i}{q^2} T^a \\
\nonumber      &=& -\frac{4}{3} g^2 (\mu)^\epsilon 
                   \int \frac{d^{4-\epsilon} q}{(2 \pi)^{4-\epsilon}}
                   \frac{1}{q^2 \, v(k+q)}  
\end{eqnarray}
Since we only want to extract the wave function renormalization we only need to 
evaluate the divergent part of the integral, i.e.\ the contribution proportional 
to the pole in $\epsilon$. One obtains 
\begin{equation}
\Sigma_H (k) = \frac{16}{3} (vk) \frac{g^2}{16 \pi^2} 
               \left( \frac{1}{\epsilon} + \ln(\mu) \right)
             + {\cal O} (1) 
             = (vk) (Z_h - 1) 
\end{equation}
where we have neglected finite terms as $\epsilon \to 0$ and $\mu \to 0$. 
Obviously the divergence may be compensated by defining a the wave function
renormalization in such a way that the pole in $\epsilon$ is removed. However, 
one may still add a finite renormalization on top of the infinite one corresponding
to a choice of the renormalization point $\mu$. One could choose the scale $m_Q$ 
as a reference point and thus the logarithm appearing here corresponds to 
the logarithms in (\ref{zexp}) and the renormalization constant $Z_h$ becomes
\begin{equation}
\sqrt{Z_h} = 1 -  \frac{8}{3} \frac{g^2}{16 \pi^2} 
             \left( \frac{1}{\epsilon} + \ln(\mu) \right)
\end{equation}

\begin{figure}
   \epsfysize=5cm
   \centerline{\epsffile{qself.ps}}
   \centerline{\parbox{16cm}{\caption{\label{QCDfig2}
The self energy of a light quark. 
}}}
\end{figure}

Similarly one may as well calculate the self energy of a quark in full QCD
as depicted in fig.\ref{QCDfig2}. 
The wave function renormalization of a dynamic quark comes out to be 
\begin{equation}
\sqrt{Z_q} = 1 -  \frac{4}{3} \frac{g^2}{16 \pi^2} 
             \left( \frac{1}{\epsilon} + \ln (\mu) \right)
\end{equation}

As a side remark we note that the result for the wave function renormalization 
of the heavy quark may be related to finite terms contained in the renormalized 
self energy of a dynamic quark. Keeping the mass of the quark one finds in 
the finite renormalized self energy close to mass shell terms of the form 
\begin{equation}
\Sigma_{full}^{ren} (p) = - \frac{16}{3} \frac{g^2}{16 \pi^2}
                          (\fmslash{p} - m_Q) \frac{p^2 - m_Q^2}{m_Q^2}
                          \ln\left(\frac{m_Q^2}{m_Q^2 - p^2}\right) 
        + \cdots 
\end{equation}
where the dots denote nonlogarithmic terms. Splitting the momentum of 
the heavy quark as $p = m_Q v + k$ and projecting on the large components
one finds 
\begin{equation}
\Sigma_{full}^{ren} (p) = - \frac{16}{3} \frac{g^2}{16 \pi^2}
                          P_+ (vk)
                          \ln\left(\frac{m_Q^2}{vk}\right)  
        + \cdots 
\end{equation}
which contains a term which diverges logarithmically as $m_Q \to \infty$.
The coefficient of this logarithmic divergence is equal to the divergent 
contribution as calculated in the effective theory, showing that the 
ultraviolett behaviour of the effective theory is indeed related to the 
mass dependences in the full theory.

Our goal is to perform a calculation of the one-loop anomalous dimension of 
heavy quark currents \cite{qcdcorr}. 
It is well known that in full QCD the vector and the 
axial vector currents have vanishing anomalous dimension, since both  
are conserved in the massless limit. On the other hand, the rsult for these 
currents will exhibit logarithmic dependences on the quark masses, which 
become logarithmic divergencies in the heavy mass limit. Thus we expect 
nontrivial anomalous dimension once one of the heavy quarks becomes static.
In other words, we shall now evaluate the anomalous dimension of the 
heavy-light current 
\begin{equation}
J = \bar{q} \Gamma h_v
\end{equation}
The one loop diagram that needs to be evaluated is depicted in 
fig.\ref{QCDfig3},
where the circle-cross vertex represents the Dirac matrix $\Gamma$. 
The result is again divergent and this divergence needs to be compensated  
by an appropriate counterterm. This counterterm takes the form
\begin{equation}
\mbox{counterterm for } J = \left[\frac{\sqrt{Z_h Z_q}}{Z_J} - 1\right] J
= [(\sqrt{Z_h} - 1) + (\sqrt{Z_q} - 1) - (Z_J - 1)] J + {\cal O} (g^4)
\end{equation}
where the appropriate renormalization factors for the external lines have been
taken into account. 

\begin{figure}
   \epsfysize=5cm
   \centerline{\epsffile{hlcurr.ps}}
   \centerline{\parbox{16cm}{\caption{\label{QCDfig3}
The one loop correction to the heavy to light current operators. The 
double line represents a static heavy quark moving with velocity $v$.
}}}
\end{figure}

Inorder to evaluate the divergent part one may neglect all external momenta 
and the necessary integral $I$ becomes
\begin{eqnarray}
I &=& (-ig)^2 (\mu)^\epsilon 
                   \int \frac{d^{4-\epsilon} q}{(2 \pi)^{4-\epsilon}}
                   \gamma_\lambda T^a \frac{i\fmslash{q}}{q^2} \Gamma  
                   \frac{i}{vq} v^\lambda T^a \frac{-i}{q^2}  \\
\nonumber      &=& -\frac{8}{3} \frac{g^2}{16 \pi^2} 
                   \left(\frac{1}{\epsilon} + \ln \mu \right)  
\end{eqnarray}
where we have again dropped all finite terms as $\epsilon \to 0$ and 
$\mu \to 0$. From this result we obtain for the renormalization constant
$Z_J$
\begin{equation}
Z_J = 1+ \frac{12}{3} \frac{g^2}{16 \pi^2} 
                   \left(\frac{1}{\epsilon} + \ln \mu \right)
\end{equation}
From the renormalization constant $Z_J$ we evaluate the anomalous dimension 
as 
\begin{equation}
\gamma_J = \mu \frac{\partial}{\partial \mu} \ln Z_J = 
           \frac{12}{3} \frac{g^2}{16 \pi^2} + {\cal O} (g^4) 
\end{equation}
which yields for the renormalization group improved result for $Z_J$
\begin{equation} \label{LLAh2l}
Z_J = \left( 
\frac{\alpha_s (\mu)}{\alpha_s (m_Q)} \right)^
          {\displaystyle -\frac{6}{33 - 2 n_f}}
\end{equation}

In a similar fashion one may evaluate the counter term for the heavy-heavy 
current 
\begin{equation}
{\cal J}(v,v') = \bar{h}_{v'} \Gamma h_v 
\end{equation}
and derive from this the anomalous dimension. Here a specific feature 
of HQET enters the game, namely the anomlous dimension depends on the velocities
of the heavy quarks. At the first sight this seems to be strange, since in 
general the anomalous dimensions do not depend on the kinematic variables of 
the matrix element. However, in HQET the velocity of the heavy quark is
not a kinematic quantity rather it is an external parameter, while the real 
kinematic quantities are the residual momenta of which the anomalous dimension 
is independent.

\begin{figure}
   \epsfysize=5cm
   \centerline{\epsffile{hhcurr.ps}}
   \centerline{\parbox{16cm}{\caption{\label{QCDfig4}
The one loop correction to the heavy to heavy current operators. The 
double lines represent static heavy quarks moving with velocity $v$
and $v'$ respectively.
}}}
\end{figure}

For the one loop correction to the heavy-heavy current one has to evaluate 
the diagram shown in fig.\ref{QCDfig4}. The counter term reads
\begin{equation}
\mbox{counterterm for } {\cal J} = \left[\frac{Z_h}{Z_J} - 1\right] J
= [(Z_h - 1) - (Z_{\cal J} - 1)] {\cal J} + {\cal O} (g^4)
\end{equation}
and the renormalization constant is found to be
\begin{equation}
Z_{\cal J} = 1 -  \frac{16}{3} \frac{g^2}{16 \pi^2} [(vv') r(vv') - 1]   
                   \left(\frac{1}{\epsilon} + \ln \mu \right)
\end{equation}
where we have defined a function 
\begin{equation}
r(x) = \frac{1}{\sqrt{x^2 - 1}} \ln [\sqrt{x^2 - 1} + x]
\end{equation}
From the renormalization constant we derive the anomlous dimension to be 
\begin{equation}
\gamma_{\cal J} =  -  \frac{16}{3} \frac{g^2}{16 \pi^2} [(vv') r(vv') - 1]
+ {\cal O} (g^4)
\end{equation}
which vanishes as $vv' \to 1$. This is in fact a result which holds to all orders
in perturbation theory since the heavy-heavy current for equal velocities is 
the generator of heavy flavour symmetry and hence a conserved current, which 
has to have vanishing anomalous dimension in all order of perturbation theory.

Inserting this result into the formula for the resummation of the leading 
logarithms one has
\begin{equation}
Z_{\cal J} =  \left( 
\frac{\alpha_s (\mu)}{\alpha_s (m_Q)} \right)^
          {\displaystyle -\frac{8}{33 - 2 n_f} [vv' r(vv') - 1]}
\end{equation}

As an application we consider the left handed current 
\begin{equation}
L_\nu = \bar{c} \gamma_\nu (1 - \gamma_5) b
\end{equation}
and start at some large scale as e.g. the scale of the weak bosons $M_W$. 
Since in full QCD this current does not have an anomalous dimension, there will 
not be large logarithms of the form $\ln(M_W^2 / m_b^2)$. Scaling down using the 
renormalization group one arrives at the scale of the $b$ quark mass. 
At that point we match to an effective theory in which the $b$ quark becomes 
a static quark, where tree level matching is sufficient as long as we stick to 
the one-loop result for the renormalization group functions. Hence we find for 
some scale $\mu$ below $m_b$ 
\begin{equation}
\bar{c} \gamma_\nu (1 - \gamma_5) b = 
          \left(\frac{\alpha_s (\mu)}{\alpha_s (m_b)} \right)^
          {\displaystyle -\frac{6}{25}} 
(\bar{c} \gamma_\nu (1 - \gamma_5) b_v)|_\mu 
\end{equation}
The charm quark mass is still a perturbative scale and hence we may scale 
down further using the perturbatively calculated renormalization group functions.
At the charm quark mass scale we replace also the charm quark by a static quark
and scale down further to some scale $\mu$ below $m_c$ which is assumed to be still
large enough to use perturbation theory. Thus one has
\begin{eqnarray}
&& \langle D(p'=M_D v') | \bar{c} \gamma_\nu (1 - \gamma_5) b | B(p=M_B v) \rangle 
\\ \nonumber
&& = \left(\frac{\alpha_s (m_c)}{\alpha_s (m_b)} \right)^
          {\displaystyle -\frac{6}{25}}
   \left( \frac{\alpha_s (\mu)}{\alpha_s (m_c)} \right)^
          {\displaystyle -\frac{8}{33 - 2 n_f} [vv' r(vv') - 1]} 
\\ \nonumber && \qquad \qquad \times
\langle D(p'=M_D v')|\bar{c}_{v'} \gamma_\nu (1 - \gamma_5) b_v  
         | B(p=M_B v) \rangle_\mu
\end{eqnarray}
where we have sandwiched the current now between states to indicate that 
the velocities entering the heavy quark operators are the velocities 
of the initial and final state particles. 

Using the renomalization group we have thus achieved a resummation of 
logarithms. The scaling between $m_b$ and $m_c$ has moved logarithms of the 
form $\ln (m_b / m_c)$ from the matrix element into the renormalization 
(or Wilson coefficient) $Z_j$, 
while the scaling between $m_c$ and $\mu$ has moved the logarithms of the 
form $\ln (m_c / \mu)$ into the renormalization constant $Z_{\cal J}$. 
In fact, for the example discussed above the result is known including
the two loop anomlaous dimension \cite{twoloop}

The procedure described here has the disadvantage that numerically it 
is useful only in the limit $m_b \gg m_c$ such that $\ln (m_b / m_c) \gg 1$
is a large logarithm and terms of order 
$(m_c / m_b)^n  \alpha_s (m_b) \ln (m_b / m_c))$ may be neglected.
In real life we have $m_b /m_c \sim 3$ and one may 
think of simply performing the one loop calculation including the masses
in full QCD and use this to match directly QCD to a theory with two 
static quarks, where heavy quark symmetries hold. In this way one 
obtains all terms of order $(m_c / m_b)^n  \alpha_s  \ln (m_b / m_c))$. 
However, the price to pay is a scale ambiguity in the scale of $\alpha_s$
which has to be taken at some scale $\bar{m}$ between $m_b$ and $m_c$. 
Numerically this is not a problem since $\alpha_s$ does not run much between 
$m_c$ and $m_b$. 

Recently the latter procedure has been applied at the two loop level 
\cite{Czarn} such that the terms of order 
$(m_b / m_c)^n  \alpha_s^2 (\bar{m})  \ln (m_b / m_c))$ are now known. 
The usual way to parametrize the corrections is by introducing 
corrections factors 
\begin{equation}
V_\mu \to \eta_V V_\mu , \qquad  A_\mu \to \eta_A A_\mu
\end{equation}
for the $b\to c$ vector and axial vector current respectively. The 
coefficients $\eta_V$ and $\eta_A$ 
are known up to terms of order $\alpha_s^3 (\bar{m})$. Taking 
the numbers of \cite{Czarn} one has 
\begin{eqnarray}
\eta_V &=& 1 + 0.018 + 0.004 + {\cal O}(\alpha_s^3 (\bar{m})) 
        =  1.022 \pm 0.004 \\
\eta_A &=& 1 - 0.033 - 0.007 + {\cal O}(\alpha_s^3 (\bar{m})) 
        =  0.960 \pm 0.007
\end{eqnarray}
The uncertainty given here is the uncertainty due to the terms of order
$\alpha_s^3 (\bar{m})$ and is conservatively estimated by the size of 
the calculated $\alpha_s^2 (\bar{m})$ corrections. 

Given the fact that the calculated QCD corrections are already below the
level of $1 \%$ one needs to worry also about QED corrections. In fact, 
the QED corrections to processes of this type have been calculated 
already some time ago in the context of $\mu$ decay\cite{sirlin}. 
Similar to the 
QCD corrections they may be factorized and in principle one could apply 
also renormalization group methods. However, the running of the QED
coupling is negligible at this level and the QED corrections factor 
is given by
\begin{equation}
\eta_{A/V} \to  \eta_{A/V} \left[1 + \frac{\alpha_{QED}}{\pi}
                           \ln \frac{M_Z}{m_b} \right] 
           \sim 1.013 \,\, \eta_{A/V}
\end{equation}
Hence short distance QED corrections enhance the matrix elements by 
$1.3 \%$.
 


