\subsubsection{Power Corrections}
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While the QCD radiative corrections are related to logarithmic dependences
on the heavy mass, one also expects power corrections which are connected 
with inverse powers of the heavy mass. As we shall see, the numerator is 
some scale related to the light degrees of freedom which is thus of order
$\Lambda_{QCD}$. This type of corrections involves long distance contributions
and hence is generically nonperturbative. Consequently, at higher orders 
in $1/m_Q$ new, nonperturbative parameters or even form factors enter the 
game, which cannot be calculated in the framework of HQET and which are 
in general not related to the nonperturbative input used in the static limit. 

As an example we consider again the matrix element of a heavy-light current 
$\bar{q} \Gamma Q$ mediating
a transition between a heavy meson and some arbitrary state 
$| A \rangle$ involving only light degrees of freedom.
The full QCD Lagrangian ${\cal L}$ and the fields $Q$ 
are expanded in terms of a power series in $1/m_Q$ in the way described in 
section 2, and the matrix element under consideration 
up to order $1/m_Q$ takes the form:
\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{HarvLag}) or (\ref{FWlag}). 
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 
power 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.  

However, it has already been discussed in section 2
that local terms may be shifted form the field and vice versa by 
appropriate field redefinitions. Hence the distinction between corrections 
to the states (as being the time-ordered products with the corrections to the 
Lagrangian) is not unique. 

Another example for higher order terms is the mass of the heavy hadron,
for which also a $1/m_Q$ expansion exists. In the infinite mass limit this mass is given in terms
of the quark mass plus some ``binding energy'' 
$\bar{\Lambda}$. The corrections are of order $1/m_Q$
and are given by the matrix elements of the leading 
correction term of the Lagrangian. One obtains 
\begin{equation} \label{massrel}
m_H = m_Q \left(1 + \frac{\bar\Lambda}{m_Q} + \frac{1}{2 m_Q^2} 
      \left(\lambda_1 + d_H \lambda_2 \right)
+ {\cal O} (1/m_Q^3) \right)  
\end{equation}
where $d_H = 3$ for the $0^-$ and $d_H = -1$ for the $1^-$ meson. Up to 
this order no nonlocal terms appear; such terms show up the first time 
at order $1/m_Q^3$. 
The parameters $\bar\Lambda$, $\lambda_1$ and $\lambda_2$ correspond to 
matrix elements involving higher order terms that appear in the effective
theory Lagrangian 
\begin{eqnarray} 
\bar{\Lambda} &=& 
\frac{\langle 0 | q \stackrel{\longleftarrow}{ivD} \gamma_5 h_v | H (v) \rangle}
                       {\langle 0 | q \gamma_5 h_v | H (v) \rangle}
\label{Lambar} \\ 
\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 = 2 (m_Q + \bar\Lambda) $. 
These parameters may be interpreted as the binding energy of the heavy meson in 
the infinite mass limit ($\bar\Lambda$), the expectation value of 
the kinetic energy  of the heavy quark ($\lambda_1$) and its energy 
due to the chromomagnetic moment of the heavy quark ($\lambda_2$)
inside the heavy meson. The latter two parameters play an important role since 
they parametrize the non-perturbative input needed in the subleading order of the 
$1/m_Q$ expansion.   

As has been discussed already in connections with the residual mass term 
the matrix elements $\bar\Lambda$, $\lambda_1$ and $\lambda_2$ depend on the 
renormalization point and/or on the definition of the mass which is used as an 
expansion parameter. Another issue are the renormalon ambiguities which 
again are related to the definition of the mass. We shall not discuss this
issue here, rather refer the reader to the literature \cite{renormalons}.  

The only parameter which is easy to access is
$\lambda_2$, since it is related to the mass splitting between $H(v)$ 
and $H^* (v , \epsilon)$. From the $B$-meson system we obtain
\begin{equation}
\lambda_2 (m_b) = \frac{1}{4} (M_{H^*} - M_H) = 0.12 \mbox{ GeV}^2; 
\end{equation}
from the charm system the same value is obtained.  This shows that indeed
the spin-symmetry partners are degenerate in the infinite mass limit and 
the splitting between them scales as $1/m_Q$. 

The other parameters appearing in (\ref{massrel}) are not simply 
related to the hadron spectrum.
Using the pole mass for $m_Q$ in (\ref{massrel}), QCD sum rules yield for
a value of $\bar\Lambda = 570 \pm 70$ MeV \cite{reviews}. More 
problematic is the parameter $\lambda_1$; 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}
This value is compatible with the bounds; however, it is unexpectedly large, 
since it corresponds to a rms-momentum of the heavy quark inside the meson 
of 
\begin{equation}
\sqrt{\langle \vec{p}^2 \rangle} \sim 720 \mbox{ MeV} 
\end{equation}
which is large compared to the naive guess of $-\lambda_1 \sim (\Lambda_{QCD})^2$ 
This is the reason why also smaller values of $\lambda_1$ have been used 
in the literature. 

Recently there has been an attempt \cite{GKLW96} to extract 
$\bar\Lambda$ and $\lambda_1$ from the shape of the lepton energy spectrum 
in inclusive semileptonic 
$B$ decays (see sections 4.3 and 4.4). The values obtained from this analysis are
$\bar\Lambda = 0.39 \pm 0.11$ GeV and $-\lambda_1 = 0.19 \pm 0.10$ GeV${}^2$, 
where the $\overline{\rm MS}$ definition of the mass has been used. 
The uncertainties
quoted are only the $1\sigma$ statistical ones; the systematical uncertainties 
of this approach are difficult to estimate.  

Finally we shall review an important results concerning $1/m_Q$ 
corrections, one of which is called Luke's theorem \cite{Lu90}. It is a 
generalization of 
the Ademollo Gatto theorem \cite{AdemolloGatto}, 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 and 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
$1/m_Q^2$. 

Reparametrization invariance also puts restrictions on the higher order 
corrections, since it relates different orders of the $1/m_Q$ expansion. 
In particular for the correction terms to the Lagrangian reparametrization 
invariance yields restrictions, i.e. it relates different oreders. 

 