\section{Theoretical Background}
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In this first part of the review the theoretical background of Heavy 
Quark Effective Theory (HQET) is explained. The first section deals with 
the Lagrangian aof HQET and the corresponding expansion of the fields. 
The second section deals with one of the major achievements of the 
infinite mass limit, namely the additional symmetries that appear 
as $m_Q \to \infty$. Finally, in the third section the corrections to the 
limiting behaviour are discussed and some sample calculations are 
considered.

\boldmath
\subsection{$1/m_Q$ Expansion of QCD}
\unboldmath
  \subsubsection{Lagrangian and fields} 
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Different approaches have been suggested to perform the infinite mass 
limit of QCD. Here we shall discuss only two of them in some detail. 
The first one \cite{MRRderivation}
is the one which is along the lines of the effective 
theory discussion of the introduction; here the starting point will 
be the generating functional of Greens functions of full QCD, which 
will be represented as a functional integral. After identifying the 
heavy degrees of freedom these may be integrated out explicitly, 
leading to a non-local action, which in the second step is expanded 
into local terms, which define the Lagrangian of the effective theory.

The second way \cite{MainzHQET}
is along the lines of the textbook way to obtain the 
non-relativistic limit of the Dirac equation by transforming away 
the couplings to the small components of the spinor. These 
transformations, the so-called Foldy Wouthuysen transformations, 
yield a different Lagrangian and also a different expansion for the 
fields. However, it became clear that these two approaches are 
completely equivalent, since they yield the same physical amplitudes, 
and which representation should be used depends on the problem under
consideretion. 
    
The first way to formulate the heavy mass limit is by integrating 
out heavy degrees of freedom from the functional integral of QCD 
Greens functions \cite{MRRderivation}. This 
integration may in fact be done explicitly, since for the case at 
hand it amounts to a Gaussian functional integration. Formulating 
the heavy mass limit in this way clearly exhibits that it 
corresponds to an effective theory in the usual sense. 
We start from the generating functional of the QCD Greens functions
\begin{equation} \label{gf}
Z(\eta,\bar{\eta},\lambda) = \int [d Q][d \bar{Q}]
                                  [d \phi_\lambda ]
\exp \biggl\{ i S + i S_\lambda
     + i \int d^4 x \, (\bar{\eta} Q + \bar{Q} \eta
                        + \phi_\lambda \lambda )
     \biggr\},
\end{equation}
where $\phi_\lambda = q,\, A_\mu^a$ denotes the light degrees of freedom
(light quarks $q$ and gluons $A_\mu$) with the action $S_\lambda$, while 
$S$ denotes the piece of the action for the heavy quark $Q$ including its coupling
to the gluons 
\begin{equation} \label{act}
S = \int d^4 x \, \bar{Q} (i \fmslash{D} - m_Q ) Q.
\end{equation}
where 
\begin{equation}
D_\mu = \partial_\mu + i g A_\mu, \quad
\end{equation}
is the covariant derivative of QCD. We have introduced source terms 
$\eta$ for the heavy quark and $\lambda$ for the light degrees of freedom. 

We shall consider hadrons containing a single heavy quark, and we assume 
that this heavy hadron moves with a velocity $v$
\begin{equation}
v = \frac{p_{\mbox{hadron}}}{m_{\mbox{hadron}}}, \quad v^2 = 1, \quad v_0 > 0
\end{equation}
This velocity vector may be used to 
split the heavy quark field $Q$  into an ``upper'' component $\phi$ and a
``lower'' one $\chi$
\begin{eqnarray} \label{splitu}
\phi_v = \frac{1}{2} (1+\fmslash{v}) Q, && \fmslash{v} \phi_v = \phi_v \, , \\
\chi_v = \frac{1}{2} (1-\fmslash{v}) Q, && \fmslash{v} \chi_v = -\chi_v \, , 
\label{splitl}
\end{eqnarray}
and to define a decomposition of the covariant derivative into a
``longitudinal'' and a ``transverse'' ($\perp$)
part
\begin{equation} \label{splitd}
D_\mu = v_\mu (v \cdot D) + D_\mu^{\perp} \, , \quad
\left\{ \fmslash{D}^{\perp} \, , \, \fmslash{v} \right\} = 0.
\end{equation}
Using (\ref{splitu}-\ref{splitd}) the action
(\ref{act}) of the heavy quark field takes the
form
\begin{equation} \label{actul}
S = \int d^4 x \, \biggl[
    \bar{\phi}_v \{ i (v \cdot D) - m_Q \} \phi_v -
    \bar{\chi}_v \{ i (v \cdot D) + m_Q \} \chi_v 
+ \bar{\phi}_v i \fmslash{D}^{\perp} \chi_v
    + \bar{\chi}_v i \fmslash{D}^{\perp} \phi_v \biggr].
\end{equation}
The heavy quark in the meson is very close to being on shell, and thus
the space time dependence of the heavy quark field is mainly that
of a free particle moving with velocity $v$. This suggests a
reparametrization of the fields by removing
the space time dependence of a solution of the free Dirac equation. 
We shall chose the ``particle-type'' parametrization corresponding to 
the ``positive energy solution'' of the Dirac equation 
\begin{equation} \label{phaseredef}
\phi_v = e^{-im_Q (v \cdot x)} h_v \, , \qquad
\chi_v = e^{-im_Q (v \cdot x)} H_v,
\end{equation}
such that the space time dependence of the remaining fields $h_v$ and 
$H_v$ is determined by the residual momentum $k = p - m_Q v$, which is 
due to binding effects of the heavy quark inside the heavy hadron, and 
which is a ``small'' quantity of order $\Lambda_{QCD}$.  

Expressed in these fields the action of the heavy quark becomes 
\begin{equation} \label{acth}
S = \int d^4 x \, \left[
    \bar{h}_v i (v \cdot D) h_v -
    \bar{H}_v \{ i (v \cdot D) + 2m_Q \} H_v 
    + 
      \bar{h}_v i \fmslash{D}^{\perp} H_v
    + \bar{H}_v i \fmslash{D}^{\perp} h_v \right],
\end{equation}

The term containing the sources is also rewritten in terms of the
fields $h_v$ and $H_v$
\begin{equation}
 \int d^4 x \, (\bar{\eta} \psi + \bar{\psi} \eta)
 =  \int d^4 x \, (\bar{\rho}_v h_v + \bar{h}_v \rho_v
             + \bar{R}_v H_v + \bar{H}_v R_v ),
\end{equation}
where $\rho_v$ and $R_v$ are now source terms for the upper component 
field $h_v$ and the lower componenent part $H_v$ respectively. 

In terms of the new variables the generating functional reads
\begin{eqnarray} \label{Genfh}
&& Z ( \rho_v ,  \bar{\rho}_v , R_v , \bar{R}_v, \lambda )
=  \int [d h_v][d \bar{h}_v][d H_v][d \bar{H}_v]
        [d \phi_\lambda ] \\
&&   \exp \left\{ i S +  S_\lambda+
    i \int d^4 x \, (\bar{\rho}_v h_v + \bar{h}_v \rho_v
             + \bar{R}_v H_v + \bar{H}_v R_v
             + \phi_\lambda \lambda )
          \right\},
\nonumber
\end{eqnarray}
where the action $S$ for the heavy quark is given in eq.(\ref{acth}).

From (\ref{acth}) it is obvious that the 
heavy degree of freedom is the lower component field $H_v$, since it has
a mass term $2m_Q$, while the upper component field $h_v$ is a massless 
field describing the static heavy quark. In the heavy mass limit only 
the Greens functions involving the field $h_v$ have to be calculated, 
and hence we integrate over $H_v$ in the functional integral 
(\ref{Genfh}) with the sources of the lower component field $R_v$ and 
$\bar{R}_v$ set to zero. This can be done explicitly, since it is a Gaussian 
integration 
\begin{equation}
Z ( \rho_v ,  \bar{\rho}_v , \lambda )
=
     \int [d h_v][d \bar{h}_v]
          [d \lambda ] \Delta 
   \exp \left\{ i S + S_\lambda +
     i \int d^4 x \, (\bar{\rho}_v^+ h_v^+ + \bar{h}_v^+ \rho_v^+
                      + \phi_\lambda \lambda )
       \right\},
\end{equation}
where now the action functional for the heavy quark becomes a non-local object 
\begin{equation}  \label{actnl}
S = \int d^4 x \,  \left[
    \bar{h}_v^+ i (v \cdot D) h_v^+
    - \bar{h}_v^+  \fmslash{D}^{\perp}
    \left( \frac{1}{ i (v \cdot D) + 2m_Q - i \epsilon} \right)
    \fmslash{D}^{\perp} h_v^+ \right].
\end{equation}
This Gaussian integration corresponds to the replacement 
\begin{equation}
H_v = \left( \frac{1}{2m_Q + ivD} \right) i \fmslash{D}_\perp h_v
\end{equation}
for the lower component field.
Furthermore, the Gaussian integration yields a determinant 
$\Delta$. In the full theory one may also perform this Gaussian 
integration, and the determinant obtained contains all the closed 
loops of quarks. After renormalization of the full theory their 
contribution starts at order $1/m^2$ with an Uehling potential like term.
In the effective theory one may take the determinant $\Delta$ to be a 
constant, if the terms of order $1/m_Q^2$ and higher comming from the 
closed heavy quark loops are included by matching to the full theory. 
Since we shall discuss only the leading term of the $1/m_Q$ expansion 
in this section, we may drop the determinant in what follows.    

The non-locality of the action functional is connected to the large scale 
set by the heavy quark mass, and the non-local terms may be expanded in 
terms of an infinite series of local operators, which come with increasing
powers of $1/m_Q$. In the context of a field theory this corresponds 
to a short distance expansion and hence these operators have to be 
renormalized. The tree level relations may be read off from the geometric 
series expansion of the non-local term in (\ref{actnl}). In this way we 
obtain the expansion of the field and the Lagrangian 
\begin{eqnarray} \label{HarvField}
Q(x) &=& e^{-im_Qvx} \left[ 1 + 
         \left( \frac{1}{2m_Q + ivD} \right) i \fmslash{D}_\perp \right] h_v \\ 
     &=& e^{-im_Qvx} \left[ 1 + \frac{1}{2m_Q} \fmslash{D}_\perp +  \nonumber 
     \left(\frac{1}{2m_Q}\right)^2 (-ivD) \fmslash{D}_\perp + \cdots \right] h_v \\  
\label{HarvLag}
{\cal L} &=& \bar{h}_v (iv D) h_v + 
\bar{h}_v i \fmslash{D}_\perp \left( \frac{1}{2m_Q + ivD} \right) 
          i \fmslash{D}_\perp h_v \\
         &=& \bar{h}_v (iv D) h_v + \frac{1}{2m_Q} \bar{h}_v 
             (i\fmslash{D}_\perp)^2 i h_v
          +  \left( \frac{1}{2m_Q}\right)^2 \bar{h}_v 
                            (i\fmslash{D}_\perp) (-ivD) 
         \nonumber  (i\fmslash{D}_\perp)  h_v + \cdots
\end{eqnarray}

A Greens function with an operator insertion is treated in a similar way; 
the net effect of this is that the heavy quark fields in the operator 
insertion 
are replaced by the expansion (\ref{HarvField}). 
  

The second way we shall consider here is the traditional one which is based 
on the Foldy Wouthuysen transformations \cite{MainzHQET}. 
The basic idea is to find a transformation
\begin{equation}
Q(x) = e^{-iS} Q^\prime (x) ,
\end{equation}
which is unitary in spinor space and which decouples the large and small 
component fields in the Lagrangian. Defining the large and small component 
fields in the same way as above, one writes
\begin{eqnarray} \label{FWfields}
h^{(+)}(x) &=& \frac{1+\fmslash{v}}{2} e^{\textstyle im_Qvx} 
               \left(\cdots
               e^{\textstyle -\frac{1}{2m_Q^3}{\cal O}_3} 
               e^{\textstyle -\frac{1}{2m_Q^2}{\cal O}_2}
               e^{\textstyle -\frac{1}{2m_Q}{\cal O}_1}\right) Q(x) \\
h^{(-)}(x) &=& \frac{1-\fmslash{v}}{2}e^{\textstyle -im_Qvx}
               \left(\cdots
               e^{\textstyle -\frac{1}{2m_Q^3}{\cal O}_3} 
               e^{\textstyle -\frac{1}{2m_Q^2}{\cal O}_2}
               e^{\textstyle -\frac{1}{2m_Q}{\cal O}_1}\right) Q(x)
\nonumber
\end{eqnarray}
and determines the operators ${\cal O}_i$ in such a way that (to given order
in $1/m_Q$ no coupling terms between $h^{(+)}$ and $h^{(-)}$ appear 
in the action (\ref{act}). This may be done recursively and one obtains 
for the first few operators
\begin{eqnarray}
{\cal O}_1 &=& i\fmslash{D}_{\perp} \\
{\cal O}_2 &=& \frac{1}{2}(i\fmslash{D}_{\perp}\fmslash{v} (ivD) +
                            \fmslash{v} (ivD) \fmslash{D}{\perp}) \\
{\cal O}_3 &=& \frac{1}{3}(i\fmslash{D}{\perp})^3 +
  \frac{1}{4}(i\fmslash{D}_{\perp})(ivD)^2 + 
  \frac{1}{2}(ivD) (i\fmslash{D}_{\perp}) (ivD) 
  \frac{1}{4} (ivD)^2 (i\fmslash{D}_{\perp})) 
\end{eqnarray}
Expanding (\ref{FWfields}) in a power series in $1/m_Q$ one finds for the
fields
\begin{equation} \label{FWfield}
Q(x) =  e^{-im_Q v\cdot x}\left[ 1 +\frac{1}{2m_Q}(i\fmslash{D}_{\perp}) +
\frac{1}{4m_Q^2}\left( v\cdot D \fmslash{D}_{\perp} - \frac{1}{2}
\fmslash{D}_{\perp}^2 \right) + \cdots \right] h_v (x) 
\end{equation}
where $h_v (x) = h^{(+)} (x) + h^{(-)} (x)$. 

The diagonalized Lagrangian reads
\begin{equation} \label{FWlag}
{\cal L} = \bar h_v (iv \cdot D) h_v 
              + \tilde{K}_1 + \tilde{M}_1 + \tilde{E}_1 
              + \tilde{K}_2 + \tilde{M}_1 + \tilde{E}_1 
              + \cdots 
\end{equation}
where we have defines the abbreviations
\begin{eqnarray}
\tilde{K}_1 &=& \bar{h}_v \frac{(iD)^2}{2m_Q} h_v, \quad 
\tilde{M}_1 = 
\frac{(-i)}{2m_Q}\bar{h}_v \sigma_{\mu \nu}(iD^\mu)(iD^\nu) h_v, \quad
\tilde{E}_1 = \bar{h}_v \frac{(ivD)^2}{2m_Q} h_v \\
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\tilde{K}_2 &=& \frac{1}{8m_Q^2}
   \bar{h}_v [(iD^\mu), [(-ivD),(iD_\mu)]] h_v \\ \nonumber 
\tilde{M}_2 &=& \frac{(-i)}{8m_Q^2}
   \bar{h}_v \sigma_{\mu \nu}\{ (iD^\mu), [ (-ivD),(iD^\nu)]\} h_v, \quad
\tilde{E}_2 = \bar{h}_v \frac{(ivD)^3}{2m_Q} h_v 
\end{eqnarray}
The Lagrangian as well as the fields have a different expansion compared 
to the one given in (\ref{HarvField}) and \ref{HarvLag}; 
the differences starting at order 
$1/m_Q^2$. In particular, the Lagrangian 
(\ref{FWlag}) does not exhibit any explicit time derivative 
$iv\partial$; the term $ivD$ appears only in commutators where it 
becomes an electric field or derivatives of it. 

On the first sight it seems that one obtains indeed a different 
answer, depending on how the heavy mass limit is performed. However,
as far as physical observables are concerned, all approaches yield the
same result. Hence it is a matter of convenience which approach is 
preferable. 

The way terms may be reshuffled in order to obtain the same result 
for physical observables comes from the fact, that the corrections 
to the static limit originate from two sources. The first type of 
$1/m_Q$ corrections are due to the expansion 
of the field $Q$, yielding a series of local operators, since the 
original quark current $\bar{q} \Gamma Q$ was also a local operator.
The second type originates from the corrections to the Lagrangian which 
are treated as perturbations and hence appear as time-ordered products 
of the current with the corrections terms of the Lagrangian. In the 
approach where the small component field is integrated out the higher 
order term in the $1/m_Q$ expansion of the Lagrangian contain explicit 
factors of $(ivD)$ corresponding in the rest frame to time derivatives. 

Inserting such a time derivative into a time-ordered product one 
obtains a local term, since
\begin{equation}
\int d^4 x \langle T \left[ 
     \bar{h}_v (x) {\cal O} (ivD) h_v(x) \bar{h}_v (0) \Gamma q(0)
           \right] \rangle 
 \propto \langle 
     \bar{h}_v (o) {\cal O} P_+ \Gamma q(0)
            \rangle 
\end{equation}
where ${\cal O}$ is some arbitrary combination of Dirac matrices and 
derivatives and $P_+$ is the projection on the upper component. 
The (nonlocal) contraction of $h_v(x) \bar{h}_v (0)$ is proportional 
to $1/(ivD)$ which gets cancelled by the corresponding factor in the 
numerator, resulting in a local contribution. 

In this way certain terms of the Lagrangian may be rehuffled into the 
fields and vice versa. Formally this may be obtained by a field 
redefinition, since $(ivD)$ is contained in the leading order term of the 
Lagrangian. In other words, all approaches yield at least on shell 
the same results and they are equivalent in the sense of field 
redefinitions.






