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\begin{document}

\draft
 
{\tighten
\preprint{\vbox{\hbox{TECHNION 96-15}
                \hbox{MZ-TH/96-17} }}
 
\title{Spectator Effects in the Heavy Quark Effective Theory}
 
\author{B. Blok$^1$, J.G. K\"orner$^2$, D. Pirjol$^1$ and J.C. Rojas$^2$\footnote{On leave
of absence from the Universidad de Santiago de Chile, Avenida Bernardo O'Higgins 3363,
Santiago, Chile}}

\address{$^1$Department of Physics, Technion-Israel Institute of Technology, Haifa 32000, Israel\\
         $^2$Johannes-Gutenberg Universit\"at, Institut f\"ur Physik (ThEP), Staudinger Weg 7,
         55099 Mainz, Germany}

\maketitle

\begin{abstract}%
We present a complete analysis of the Heavy Quark Effective Theory Lagrangian at order $1/m^2$
in the leading logarithmic approximation, including effects induced by spectator quarks.
At this order new correction terms appear in the effective Lagrangian, as
four-quark operators containing both heavy and light quark fields. We compute the coefficients
of these operators to one-loop order and in the leading-logarithmic approximation. Two of them 
break the heavy
quark spin symmetry and we estimate their contribution to the hyperfine splitting of the heavy
mesons in the factorization approximation. We find that they make a positive contribution to
the hyperfine splitting of about 10\% of the measured splitting in the charm case and of up
to 5\% in the bottom case.

\end{abstract}

}%end tighten

\newpage

\section{Introduction}

The heavy mass expansion established itself as a valuable tool for the study of hadrons containing
one heavy quark \cite{IW,Ne,Sh}. This expansion is formulated most naturally in terms of an effective
theory, the
heavy quark effective theory (HQET), which is an approximation to QCD with one heavy quark.
The corrections to this approximation are controlled by the small parameter $\Lambda_{QCD}/m$, where
$\Lambda_{QCD}\simeq 300$ MeV is a typical scale of low-energy QCD and $m$ is the heavy quark mass.

The applications of the heavy quark effective theory to physical problems presented so far
include correction terms of order $1/m^2$ \cite{FN,Man,BSUV} and in some recent instances even
$1/m^3$ \cite{BDS,GK}. 
We investigate in this paper the effects induced by spectator quarks in the HQET.
They appear first at order $1/m^2$ and are associated
in an effective theory language
with 4--quark operators containing both heavy- and light-quark fields.
These operators mix under renormalization with the other operators of dimension six already present
in the HQET Lagrangian at tree level and their inclusion changes the form of the renormalized
Lagrangian.

We define the operators which appear in Section II and compute their mixing under renormalization.
The coefficients of these operators can be obtained from a one-loop matching calculation, which
is presented in Section III, together with a leading-log renormalization-group improvement.
As an application we estimate in Section IV their total contribution to the
hyperfine splitting of heavy-light mesons in the framework of the factorization approximation.

\section{Dimension-6 Operators}

The Lagrangian of the heavy quark effective theory is written as an expansion in the inverse
heavy quark mass \cite{IW,Ne,Sh}
\begin{equation}\label{1}
{\cal L}_{HQET} = \bar h(iv\cdot D)h + \frac{1}{2m}{\cal L}_1 + \frac{1}{(2m)^2}{\cal L}_2 + \cdots
\end{equation}
The heavy quark field $h$ annihilates static heavy quarks moving with a fixed velocity $v$ and
satisfies
the condition $\frac12(1+\vsl)h = h$. The finite-mass effects appear as interaction terms in
the Lagrangian suppressed by powers of $1/m$. The first two correction terms have the well-known
expressions \cite{L,EH,FGL,KT,BKP}
\begin{eqnarray}\label{2}
{\cal L}_1 &=& c_k(\mu) \bar h(iD)^2h + c_m(\mu) \bar h\frac{g}{2}\sigma_{\mu\nu}F^{\mu\nu}h
+ c_e(\mu)\bar h(iv\cdot D)^2h\\
{\cal L}_2 &=& c_D(\mu) {\cal O}_D + c_{SO}(\mu) {\cal O}_{SO}\,.\label{3}
\end{eqnarray}
The coefficients of the operators in (\ref{2}) are known in the leading
logarithmic approximation \cite{EH,FGL}
\begin{eqnarray}\label{6}
c_k(\mu) = 1\,,\qquad c_m(\mu) = -\left(\frac{\alpha_s(\mu)}{\alpha_s(m)}\right)^{-3/\beta_0}
\,,\qquad c_e(\mu) = 2-3\left(\frac{\alpha_s(\mu)}{\alpha_s(m)}\right)^{-8/(3\beta_0)}
\end{eqnarray}
where $\beta_0=11-\frac23n_f$ is the one-loop coefficient of the beta function for $n_f$ active
quarks. The coefficient of the last operator in (\ref{2}) is gauge-dependent; the value
quoted above corresponds to the Feynman gauge.
%will not be needed in the following
%as this operator vanishes by the equation of motion of the heavy quark field $iv\cdot Dh=0$.

To order $1/m^2$ there are two local operators contributing at tree-level, the Darwin term and the
spin-orbit interaction energy respectively
\begin{eqnarray}\label{4}
{\cal O}_D &=& \frac{g}{2}(D^\mu F_{\mu\nu}^a)(\bar hv_\nu t^ah)\\\label{5}
{\cal O}_{SO} &=& ig(\bar h\sigma_{\alpha\nu}v_\mu F^{\mu\nu}D^\alpha h) +
\frac{ig}{2} (D_\alpha F^a_{\mu\nu})(\bar h\sigma^{\alpha\nu} v^\mu t^ah)\,.
\end{eqnarray}
The coefficients $c_D(\mu)$ and $c_{SO}(\mu)$ were calculated in \cite{BKP} (see also \cite{CKO,BO})
using a different operator basis and in the absence of light quarks. We reconsider here their
calculation in a basis more suited for our present purposes, with a different result for $c_D(\mu)$.

In addition to the operators shown in (\ref{3}), there are three other types of dimension-6
operators which must be added to ${\cal L}_2$:
\begin{enumerate}
\item[a)] the gluonic operator
\begin{equation}\label{7}
{\cal O}_{gl} = \frac14 gf_{abc}F_{\mu\nu}^aF_{\nu\lambda}^bF_{\lambda\mu}^c\,.
\end{equation}
Its coefficient has been computed many times in the past (see e.g. \cite{Shifman,BKP,CS}).
To one-loop order ${\cal O}_{gl}$ does not mix with other dimension 6 operators and
will be neglected in the following.

\item[b)] Operators which vanish by the equation of motion of the heavy quark field $iv\cdot Dh=0$.
Even though their expectation values vanish, they can contribute when considering mass corrections 
to the matrix elements of currents. A discussion of their renormalization is given in Appendix
A.

\item[c)] 4-quark operators built out of heavy-light and light-light quark fields.
It is these operators which will be the main point of interest of this paper.
\end{enumerate}
  There are four independent heavy-light 4-quark operators, which can be conveniently chosen as
follows
\begin{eqnarray}\label{8}
{\cal O}_1^{hl} &=& \frac{g^2}{2}\sum_q(\bar q\gamma_\mu t^aq)(\bar h\gamma^\mu t^ah)\\\label{9}
{\cal O}_2^{hl} &=& \frac{g^2}{2}\sum_q(\bar q\gamma_\mu\gamma_5t^aq)(\bar h\gamma^\mu\gamma_5t^ah)\\
{\cal O}_3^{hl} &=& \frac{g^2}{2}\sum_q(\bar q\gamma_\mu q)(\bar h\gamma^\mu h)\\
{\cal O}_4^{hl} &=& \frac{g^2}{2}\sum_q(\bar q\gamma_\mu\gamma_5 q)(\bar h\gamma^\mu\gamma_5 h)\,.
\label{11}
\end{eqnarray}
The summation extends over all dynamic quarks, e.g. u,d,s,c for the case of a heavy bottom quark.

The structure of the possible 4-quark operators containing only light quark fields is more
complicated. In the general case of more than one light flavor there are eight such operators.
A detailed discussion of their renormalization properties can be found in \cite{SVZ,JK,BB}.
Here we consider for the purpose of illustration only the simpler case with $n_f=1$, when
there are only four such operators. They can be chosen as in (\ref{8}-\ref{11}):
\begin{eqnarray}\label{12}
{\cal O}_1^{ll} &=& \frac{g^2}{2}(\bar q\gamma_\mu t^aq)(\bar q\gamma^\mu t^aq)\\
{\cal O}_2^{ll} &=& \frac{g^2}{2}(\bar q\gamma_\mu\gamma_5t^aq)(\bar q\gamma^\mu\gamma_5t^aq)\\
{\cal O}_3^{ll} &=& \frac{g^2}{2}(\bar q\gamma_\mu q)(\bar q\gamma^\mu q)\\
{\cal O}_4^{ll} &=& \frac{g^2}{2}(\bar q\gamma_\mu\gamma_5 q)(\bar q\gamma^\mu\gamma_5 q)\,.
\label{15}
\end{eqnarray}
These operators are first induced to two-loop order through mixing with the gluonic operator
(\ref{7}).

The complete basis of the dimension-6 operators includes also three nonlocal operators consisting
of time-ordered products of dimension-5 operators
\begin{eqnarray}\label{16}
{\cal O}_{kk} &=& \frac{i}{2}\int\mbox{d}^4x T[\bar h(iD)^2h](x)\,[\bar h(iD)^2h](0)\\
{\cal O}_{km} &=& i\int\mbox{d}^4x T[\bar h(iD)^2h](x)\,[\bar h\frac{g}{2}\sigma\cdot Fh](0)\\
{\cal O}_{mm} &=& \frac{i}{2}\int\mbox{d}^4x T[\bar h\frac{g}{2}\sigma\cdot Fh](x)
\,[\bar h\frac{g}{2}\sigma\cdot Fh](0)\,.
\label{18}
\end{eqnarray}

We will use a compact vector notation for the thirteen operators
(\ref{4},\ref{5},\ref{8}-\ref{18}), defined as follows
\begin{equation}\label{19}
\hat {\cal O} = \left( \begin{array}{c}
{\cal H}\\
{\cal O}^{hl}\\
{\cal O}^{ll}\end{array} \right)\,,\qquad{\rm with}\,\,
{\cal H} = \left(\begin{array}{c}
{\cal O}_D\\
{\cal O}_{SO}\\
{\cal O}_{kk}\\
{\cal O}_{km}\\
{\cal O}_{mm}\end{array}\right)\,,\,\,\,
\hat {\cal O}^{hl} = \left(\begin{array}{c}
{\cal O}_1^{hl}\\
{\cal O}_2^{hl}\\
{\cal O}_3^{hl}\\
{\cal O}_4^{hl}\end{array}\right)\,,\,\,\,
\hat {\cal O}^{ll} = \left(\begin{array}{c}
{\cal O}_1^{ll}\\
{\cal O}_2^{ll}\\
{\cal O}_3^{ll}\\
{\cal O}_4^{ll}\end{array}\right)\,\,\,\,.
\end{equation}
An analogous notation will be used for the coefficients of these operators. Requiring that the total
renormalized Lagrangian ${\cal L}_2$ be scale-independent gives a renormalization-group equation
for the coefficients $\hat c(\mu)$
\begin{equation}\label{20}
\mu\frac{\mbox{d}}{\mbox{d}\mu}\hat c - \hat \gamma^T\hat c = 0\,.
\end{equation}
The anomalous dimension matrix can be written as
\begin{eqnarray}\label{21}
\hat\gamma = \left( \begin{array}{ccc}
A & B & 0 \\
C & D & 0 \\
0 & E & F \end{array}\right)\,.
\end{eqnarray}
To one-loop order the blocks in this matrix take the values
\begin{mathletters}
\label{22}
\begin{equation}\label{22a}
A = \frac{g^2}{(4\pi)^2}\left( \begin{array}{ccccc}
4 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 \\
-\frac{146}{9} & 0 & 0 & 0 & 0 \\
0 & -12 & 0 & 6 & 0 \\
-10 & 0 & 0 & 0 & 12 \end{array}\right)\,,
\end{equation}
\begin{equation}\label{22b}
B = \frac{g^2}{(4\pi)^2}\left( \begin{array}{cccc}
9 & 0 & 0 & 0 \\
0 & -5 & 0 & -\frac83 \\
-18 & 0 & 0 & 0 \\
0 & -\frac{40}{3} & 0 & -\frac{64}{9} \\
9 & -\frac53 & 0 & -\frac89 \end{array}\right)\,,
\end{equation}
\begin{equation}\label{22c}
C = \frac{g^2}{(4\pi)^2}\left( \begin{array}{ccccc}
-\frac{n_f}{3} & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 \end{array}\right)\,,
\end{equation}
\begin{equation}\label{22d}
D = \frac{g^2}{(4\pi)^2}\mbox{diag}(13, 13-\frac43 n_f,
22 - \frac43 n_f, 22-\frac43 n_f)\,,
\end{equation}
\begin{equation}\label{22e}
E = \frac{g^2}{(4\pi)^2}\left( \begin{array}{cccc}
\frac{10}{9} & 0 & 0 & 0 \\
-\frac29 & 0 & 0 & 0 \\
\frac43 & 0 & 0 & 0 \\
\frac43 & 0 & 0 & 0  \end{array}\right)\,,
\end{equation}
\begin{equation}\label{22f}
F = \frac{g^2}{(4\pi)^2}\left( \begin{array}{cccc}
-\frac{137}{9} - \frac43 n_f & 5 & 0 & \frac83 \\
\frac{41}{9} & 13-\frac43 n_f & \frac83 & 0 \\
\frac83 & 12 & 22-\frac43 n_f & 0 \\
\frac{44}{3} & 0 & 0 & 22-\frac43 n_f \end{array}\right)\,.
\end{equation}
\end{mathletters}
The blocks $D$ and $F$ contain contributions which take into account the running of the $g$ factors
contained in the definition of the four-quark operators.
We have implicitly used the fact established in \cite{BKP} that the form of ${\cal O}_{SO}$ as
given in (\ref{5}) is preserved under renormalization. We agree with \cite{BO} on the renormalization
of the Darwin term in $A$ in the absence of the mixing with the four-quark operators.
The lower diagonal block $F$ has been calculated previously in \cite{CS}.

\section{Matching and running}

The coefficients of the operators ${\cal H}$ (see (\ref{19})) are given, at the matching scale
$\mu=m$, by their tree-level values $c_{{\cal H}} =$ column$(1,1,1,-1,1)$.
The coefficients of the non-local operators (\ref{16}-\ref{18}) are given simply by the products
of the respective local dimension-5 operators
\begin{eqnarray}\label{ckk}
c_{kk}(\mu) &=& 1\\
c_{km}(\mu) &=& -\left(\frac{\alpha_s(\mu)}{\alpha_s(m)}\right)^{-\frac{3}{\beta_0}}\\
c_{mm}(\mu) &=& \left(\frac{\alpha_s(\mu)}{\alpha_s(m)}\right)^{-\frac{6}{\beta_0}}\,.\label{cmm}
\end{eqnarray}
Integration of the
RG-equation (\ref{20}) is complicated by the mixing between ${\cal O}_D$ and ${\cal O}_1^{hl}$.
The exact solution for $c_D(\mu)$ and $c_1^{hl}(\mu)$ in the leading-log approximation has the form
\begin{mathletters}
\label{23}
\begin{eqnarray}\label{23a}
c_D(\mu) &=& \frac{\lambda_1-13}{\sqrt{81-12n_f}}\left\{
\left(1-\frac{2}{\lambda_1}\left(\frac{190}{9}-\lambda_2\right)-\frac{\lambda_2-3}{\lambda_1-12}
\right)z^{-\lambda_1/(2\beta_0)} + \frac{\lambda_2-3}{\lambda_1-12}z^{-6/\beta_0}\right.\\
& &\left.+\, \frac{2}{\lambda_1}\left(\frac{190}{9}-\lambda_2\right)\right\} -
(\lambda_1\leftrightarrow\lambda_2)\nonumber\\
\label{23b}
c_1^{hl}(\mu)  &=& \frac{9}{\sqrt{81-12n_f}}\left\{
\left(1-\frac{2}{\lambda_1}\left(\frac{190}{9}-\lambda_2\right)-\frac{\lambda_2-3}{\lambda_1-12}
\right)z^{-\lambda_1/(2\beta_0)} + \frac{\lambda_2-3}{\lambda_1-12}z^{-6/\beta_0}\right.\\
& &\left.+\, \frac{2}{\lambda_1}\left(\frac{190}{9}-\lambda_2\right)\right\} -
(\lambda_1\leftrightarrow\lambda_2)\,.\nonumber
\end{eqnarray}
Here we have denoted $z=(\alpha_s(\mu)/\alpha_s(m))$ and 
\begin{equation}
\lambda_{1,2} = \frac12(17 \pm \sqrt{81-12n_f})
\end{equation}
\end{mathletters}
are the eigenvalues of the 2$\times$2 mixing matrix of ${\cal O}_D$ and ${\cal O}_1^{hl}$.

For practical purposes it is sufficient to use a truncated form for these coefficients, which is
obtained when one sets $n_f=0$ in (\ref{22c}). For this case one obtains
\begin{eqnarray}\label{tr1}
c_D(\mu) &=& -\frac54\left(\frac{\alpha_s(\mu)}{\alpha_s(m)}\right)^{-6/\beta_0} - \frac{65}{36}
\left(\frac{\alpha_s(\mu)}{\alpha_s(m)}\right)^{-2/\beta_0} + \frac{73}{18}\\\label{tr2}
c_1^{hl}(\mu)  &=& \frac94\left(\frac{\alpha_s(\mu)}{\alpha_s(m)}\right)^{-6/\beta_0}
+ \frac{65}{36}\left(\frac{\alpha_s(\mu)}{\alpha_s(m)}\right)^{-2/\beta_0}
-\frac{308}{117}\left(\frac{\alpha_s(\mu)}{\alpha_s(m)}\right)^{-13/2\beta_0} - \frac{37}{26}\,.
\end{eqnarray}
When expanded in powers of $\alpha_s(m)\ln(\mu/m)$, these truncated forms differ from the
exact ones (\ref{23}) only starting at three-loop order for $c_D$ and at four-loop order for
$c_1^{hl}$. The numerical error commited when using them can be consequently expected to be
very small.

The coefficient of the local spin symmetry breaking operator ${\cal O}_{SO}$ (\ref{5}) is
\cite{BKP}
\begin{eqnarray}\label{cSO}
c_{SO}(\mu) = 2\left(\frac{\alpha_s(\mu)}{\alpha_s(m)}\right)^{-3/\beta_0} - 1\,.
\end{eqnarray}
This coefficient has been also obtained without an explicit calculation in  \cite{CKO}
from reparametrization invariance arguments \cite{LM}. However, it does not seem possible to
constrain the coefficient of the Darwin term $c_D$ by using similar arguments, as suggested
in \cite{CKO}. A discussion of the constraints which can be obtained with the help of
reparametrization invariance arguments on the renormalized $1/m^2$ HQET Lagrangian is
given in Appendix B.

Finally, the coefficients of the other heavy-light four-quark operators (\ref{9}-\ref{11}) are
found to be
\begin{eqnarray}\label{c2hl}
c_2^{hl}(\mu) &=& -\frac{10}{21-4n_f}z^{-3/\beta_0} + \frac{5}{3-4n_f}z^{-6/\beta_0}
-\frac{15}{39-4n_f}\\
& & +\, \frac{20(99+96n_f-16n_f^2)}{(4n_f-39)(4n_f-21)(4n_f-3)}
z^{-(13-\frac43 n_f)/(2\beta_0)}\nonumber\\
c_3^{hl}(\mu) &=& 0\\\label{c4hl}
c_4^{hl}(\mu) &=& -\frac{4}{3(12-n_f)}z^{-3/\beta_0} + \frac{4}{3(15-2n_f)}z^{-6/\beta_0}
-\frac{4}{33-2n_f}\\
& & +\, \frac{4(-639+156n_f-8n_f^2)}{3(n_f-12)(2n_f-33)(2n_f-15)}
z^{-(22-\frac43 n_f)/(2\beta_0)}\,.\nonumber
\end{eqnarray}

A full one-loop determination of the coefficients $c^{hl}_i(\mu)$
requires an explicit matching calculation. This involves computing the box diagrams for
heavy-light quark scattering in QCD shown in Fig.1. When expanded in powers of $1/m$ the total result
for these diagrams is
\begin{eqnarray}\label{24}
I_{QCD} &=& \frac{ig^4}{(4\pi)^2}\left\{-\frac{3}{2\lambda^2}[\gamma_\mu t^a]_{\beta\alpha}
[\gamma^\mu t^a]_{\delta\gamma} + \frac{1}{m}\left[
-\frac{3\pi}{4\lambda}[\gamma_\mu t^a]_{\beta\alpha}[\gamma^\mu t^a]_{\delta\gamma} +
\frac{5\pi}{9\lambda}[\gamma_\mu\gamma_5t^a]_{\beta\alpha}[\gamma^\mu\gamma_5t^a]_{\delta\gamma} 
\nonumber\right.\right.\\
& &\hspace{-1cm}\left.\left.\,+
\frac{8\pi}{27\lambda}[\gamma_\mu\gamma_5]_{\beta\alpha}[\gamma^\mu\gamma_5]_{\delta\gamma}
\right]
+\frac{1}{m^2}\left[\frac94[\gamma_\mu t^a]_{\beta\alpha}[\gamma^\mu t^a]_{\delta\gamma} +
\frac{5}{12}\left(\ln\frac{\lambda^2}{m^2}-2\right)
[\gamma_\mu\gamma_5t^a]_{\beta\alpha}[\gamma^\mu\gamma_5t^a]_{\delta\gamma}
\right.\right.\nonumber\\
& &\hspace{-1cm}\left.\left.\,+
\frac29\left(\ln\frac{\lambda^2}{m^2}-2\right)
[\gamma_\mu\gamma_5]_{\beta\alpha}[\gamma^\mu\gamma_5]_{\delta\gamma}\right] + {\cal O}(1/m^3)
\right\}
\end{eqnarray}
To simplify the calculation we have taken a massless light quark scattering in the forward direction.
The infrared singularities have been regulated with a finite gluon mass $\lambda$.

A computation of the same diagrams using the effective theory Feynman rules for the heavy line
with insertions of all $1/m$ corrections up to second order gives, in the $\overline{MS}$ scheme,
\begin{eqnarray}\label{25}
I_{HQET} &=& \frac{ig^4}{(4\pi)^2}\left\{-\frac{3}{2\lambda^2}[\gamma_\mu t^a]_{\beta\alpha}
[\gamma^\mu t^a]_{\delta\gamma} + \frac{1}{m}\left[
-\frac{3\pi}{4\lambda}[\gamma_\mu t^a]_{\beta\alpha}[\gamma^\mu t^a]_{\delta\gamma} +
\frac{5\pi}{9\lambda}[\gamma_\mu\gamma_5t^a]_{\beta\alpha}[\gamma^\mu\gamma_5t^a]_{\delta\gamma} 
\nonumber\right.\right.\\
& &\hspace{-1cm}\left.\left.\,+
\frac{8\pi}{27\lambda}[\gamma_\mu\gamma_5]_{\beta\alpha}[\gamma^\mu\gamma_5]_{\delta\gamma}
\right]
+\frac{1}{m^2}\left[
\frac{5}{12}\left(\ln\frac{\lambda^2}{\mu^2}+2\right)
[\gamma_\mu\gamma_5t^a]_{\beta\alpha}[\gamma^\mu\gamma_5t^a]_{\delta\gamma}
\right.\right.\nonumber\\
& &\hspace{-1cm}\left.\left.\,+
\frac29\left(\ln\frac{\lambda^2}{\mu^2}+2\right)
[\gamma_\mu\gamma_5]_{\beta\alpha}[\gamma^\mu\gamma_5]_{\delta\gamma}\right] + {\cal O}(1/m^3)
\right\}
\end{eqnarray}

The first two terms in the $1/m$ expansion agree identically, as does the logarithmic dependence
on the infrared regulator $\lambda$ in the $1/m^2$ part. Imposing equality of these two expressions
requires adding the four-quark operators (\ref{8}-\ref{11}) to the effective theory with
coefficients
\begin{eqnarray}\label{26}
c_1^{hl}(\mu) &=& \frac{18g^2}{(4\pi)^2}\\
c_2^{hl}(\mu) &=& \frac{g^2}{(4\pi)^2}\left(\frac{10}{3}\log\frac{\mu^2}{m^2} -
\frac{40}{3}\right)\\
c_3^{hl}(\mu) &=& 0\\\label{29}
c_4^{hl}(\mu) &=& \frac{g^2}{(4\pi)^2}\left(\frac{16}{9}\log\frac{\mu^2}{m^2} -
\frac{64}{9}\right)\,.
\end{eqnarray}
The logarithmic terms agree, as they should, with the leading terms obtained by expanding the
LLA sums (\ref{23b},\ref{c2hl}-\ref{c4hl}).

It is important to point out that the constant terms in the full one-loop expressions for these
coefficients are scheme-dependent. It is only after renormalization-group improvement that the
${\cal O}(\alpha_s)$ terms become well-defined. In this process the scheme-dependence of the constant
terms in (\ref{26}-\ref{29}) is cancelled by the scheme-dependence of a two-loop anomalous dimension
\cite{BFRS}.
However, there are reasons to expect the partial results (\ref{26}-\ref{29}) to give at least a
qualitative estimate of the full ${\cal O}(\alpha_s)$ correction, based on empirical evidence
\cite{BG} that the contribution of the two-loop anomalous dimension is often much smaller than the
constant term in the one-loop expression of the coefficient.

On the other hand, the logarithmic terms in (\ref{26}-\ref{29}) and the LLA sums
(\ref{23b},\ref{c2hl}-\ref{c4hl})
are scheme-independent. However, the use of the leading-log coefficients is only then likely to
give a
meaningful result, when the logarithmic terms in (\ref{26}-\ref{29}) dominate clearly over the
constant terms. When this is not the case, then the use of the full one-loop results
(\ref{26}-\ref{29}),
with all the above limitations, is likely to be a better approximation to the true value of the
coefficient. Such a situation will be encountered in the case of the charm quark to be discussed
below.

\section{Application}

The importance of considering the four-quark operators (\ref{8}-\ref{11}) becomes immediately
apparent
if one notes that the equation of motion for the gluon field gives ${\cal O}_D =  {\cal O}_1^{hl}$.
The coefficients of these two operators (\ref{23a}) and (\ref{23b}) get contributions of the
same order
in $\alpha_s$ (except to ${\cal O}(\alpha_s^0)$, where only $c_D$ is nonvanishing), so that
neglecting 
${\cal O}_1^{hl}$ will lead to an incomplete result.
Adding the contribution of  ${\cal O}_1^{hl}$  together with that of the Darwin term
gives, in the leading logarithmic
approximation, the following ``effective'' coefficient for the spin symmetry conserving operator
at order $1/m^2$ (we use here the truncated forms (\ref{tr1},\ref{tr2}) for these coefficients)
\begin{equation}\label{34}
c_D = \left(\frac{\alpha_s(\mu)}{\alpha_s(m)}\right)^{-6/\beta_0} -
\frac{308}{117}\left[\left(\frac{\alpha_s(\mu)}{\alpha_s(m)}\right)^{-13/(2\beta_0)} - 1
\right]\,.
\end{equation}
At one-loop order the leading log following from this expression coincides with the one which would
be obtained from (\ref{23a}), but at higher orders they are different.

More interesting from a practical point of view is the fact that the operators ${\cal O}_2^{hl}$ and
${\cal O}_4^{hl}$ break the heavy quark spin symmetry and thus contribute to the hyperfine
splittings of the heavy hadrons. The remainder of this Section is dedicated to a discussion of
these effects.

We first briefly review the derivation of the mass formula for the mass of a heavy hadron, including
the contributions of
the operators (\ref{8}-\ref{11}). The mass of a heavy hadron can be written as an expansion in
inverse powers of the heavy quark mass as \cite{Man,BSUV}
\begin{eqnarray}\label{mass}
m_{B^{(*)}} =m_b+\bar\Lambda  + \langle\mbox{T} {\cal H}_{int}\exp \left(-i\int\mbox{d}^4x
{\cal H}_{int}(x)\right)\rangle
\end{eqnarray}
with ${\cal H}_{int}=-{\cal L}_{int} = -\frac{1}{2m_b}{\cal L}_1 -\cdots$ the interaction
Hamiltonian.
The expectation value is
taken between eigenstates of the leading order HQET Lagrangian  which have the mass
$m_b+\bar\Lambda$.
This gives, including terms up to order $1/m_b^2$,
\begin{eqnarray}\label{ass}
m_{B^{(*)}} &=& m_b+\bar\Lambda - \frac{1}{2m_b}\langle {\cal L}_1\rangle - 
\frac{1}{(2m_b)^2}\left(\langle {\cal L}_2\rangle + 2(c_k(\mu))^2\langle {\cal O}_{kk}\rangle
\right.\\
&+&\left.\, c_k(\mu)c_m(\mu)\langle {\cal O}_{km}\rangle + 2(c_m(\mu))^2\langle {\cal O}_{mm}\rangle
+ \sum_{i=1}^4c_i^{hl}(\mu)\langle {\cal O}_i^{hl}\rangle\right) + \cdots\,.
\nonumber
\end{eqnarray}
The expectation values appearing in this formula can be parametrized as
\begin{eqnarray}\label{para}
\langle {\cal L}_1\rangle &=& -c_k(\mu)\mu_\pi^2 + c_m(\mu)\frac{d_H}{3}\mu_G^2\\
\langle {\cal L}_2\rangle &=& c_D(\mu)\mu_D^3 + \frac{d_H}{3} c_{SO}(\mu)\mu_{SO}^3\\
\langle {\cal O}_{kk}\rangle &=& \mu_{\pi\pi}^3\\
\langle {\cal O}_{km}\rangle &=& \frac{d_H}{3}\mu_{\pi G}^3\\
\langle {\cal O}_{mm}\rangle &=& \mu_{GG}^3\,,
\end{eqnarray}
with $d_H=3$ for a pseudoscalar meson and --1 for a vector meson.

The matrix elements of the four-quark operators (\ref{8}-\ref{11}) can be estimated with the
help of the factorization approximation \cite{BSUV2}. An alternative model-independent determination
of ${\cal O}_1^{hl}$ (based on a method proposed in \cite{C}) has been given in \cite{CP}.
This is done by first applying a Fierz transformation to bring the operator into the form
$(\bar h\Gamma' q)(\bar q\Gamma' h)$, after which the vacuum state is inserted between
the two currents.
Only color singlet currents will give a nonvanishing contribution. In this way we obtain
\begin{eqnarray}
& &\frac{1}{2m_B}\langle\bar B(v)|\frac{g^2}{2}\sum_q (\bar q\gamma_\mu\gamma_5 t^aq)(\bar h
\gamma^\mu\gamma_5 t^ah)|\bar B(v)\rangle = -\frac{2\pi\alpha_s}{3}f_B^2 m_B\\
& &\frac{1}{2m_B}\langle\bar B(v)|\frac{g^2}{2}\sum_q (\bar q\gamma_\mu\gamma_5q)(\bar h
\gamma^\mu\gamma_5 h)|\bar B(v)\rangle = -\frac{\pi\alpha_s}{2}f_B^2 m_B\\
& &\frac{1}{2m_B}\langle\bar B^*(v,\varepsilon)|\frac{g^2}{2}\sum_q (\bar q\gamma_\mu\gamma_5 t^aq)
(\bar h\gamma^\mu\gamma_5 t^ah)|\bar B^*(v,\varepsilon)\rangle = \frac{2\pi\alpha_s}{9}f_B^2 m_B\\
& &\frac{1}{2m_B}\langle\bar B^*(v,\varepsilon)|\frac{g^2}{2}\sum_q (\bar q\gamma_\mu\gamma_5q)
(\bar h\gamma^\mu\gamma_5 h)|\bar B^*(v,\varepsilon)\rangle = \frac{\pi\alpha_s}{6}f_B^2 m_B\,.
\end{eqnarray}

Here we have used the following relations which are valid in the heavy mass limit
\begin{eqnarray}
& &\langle\bar B(v)|\bar h\gamma_\mu\gamma_5q|0\rangle = if_Bm_B v_\mu\\
& &\langle\bar B^*(v,\varepsilon)|\bar h\gamma_\mu q|0\rangle = f_{B}m_{B} \varepsilon_\mu^*\,.
\end{eqnarray}

Combining these relations one obtains the following expression for the hyperfine splitting
\begin{eqnarray}
m_{B^*} - m_B &=& \frac{1}{2m_b}c_m(\mu)\frac{4\mu_G^2}{3} + \frac{4}{3(2m_b)^2}\left(
c_{SO}(\mu)\mu_{SO}^3 + c_k(\mu)c_m(\mu)\mu_{\pi G}^3\right.\\
& &\left.\, - c_2^{hl}(\mu)\frac{2\pi\alpha_s}{3}
f_B^2 m_B -  c_4^{hl}(\mu)\frac{\pi\alpha_s}{2}f_B^2 m_B\right) + {\cal O}(1/m_b^3)\,.\nonumber
\end{eqnarray}

\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline
%& & & &\\[-0.3cm]
%\rule{0pt}{11.5pt}
& \multicolumn{2}{|c|}{LLA}
& \multicolumn{2}{c|}{One-loop}\\
%[0.4cm]
\cline{2-5}
 & $\mu=0.5$ GeV
 & $\mu=1$ GeV
 & $\mu=0.5$ GeV
 & $\mu=1$ GeV\\[0.1cm]
\hline
\hline
& & & &\\[-0.3cm]
$c_2^{hl}(\mu/m_c)$ & -0.268 & -0.074 & -0.653 & -0.503 \\[0.4cm]
\hline
& & & &\\[-0.3cm]
$c_4^{hl}(\mu/m_c)$ & -0.117 & -0.038 & -0.348 & -0.268 \\[0.4cm]
\hline
& & & &\\[-0.3cm]
$c_2^{hl}(\mu/m_b)$ & -0.401 & -0.232 & -0.577 & -0.483 \\[0.4cm]
\hline
& & & &\\[-0.3cm]
$c_4^{hl}(\mu/m_b)$ & -0.156 & -0.105 & -0.308 & -0.268 \\[0.4cm]
\hline
\end{tabular}
\end{center}
\begin{quote} {\bf Table 1.}
The coefficients of the spin-symmetry violating four-quark operators for the charm and bottom case
at two values of the factorization scale $\mu=0.5$ GeV and $\mu=1$ GeV.
\end{quote}

This relation can be written only in terms of observable quantities by expressing the quark mass
in terms of the pseudoscalar meson mass and multiplying with $m_B + m_{B^*}$
\begin{eqnarray}\label{hyper}
m_{B^*}^2 - m_B^2 &=& c_m(\mu)\frac{4\mu_G^2}{3} + \frac{4}{3(2m_B)}\left(
2\bar\Lambda \mu_G^2 + c_{SO}(\mu)\mu_{SO}^3 + c_k(\mu)c_m(\mu)\mu_{\pi G}^3\right.\\
& &\left.\, - c_2^{hl}(\mu)\frac{2\pi\alpha_s}{3}
f_B^2 m_B -  c_4^{hl}(\mu)\frac{\pi\alpha_s}{2}f_B^2 m_B\right) + {\cal O}(1/m_B^2)\,.\nonumber
\end{eqnarray}

In Table 1 we present values for the coefficients of the two four-quark operators which contribute
to the hyperfine splittings. They are given in both the leading-log approximation and the
``full'' one-loop form at two different values of the factorization scale $\mu=0.5$ and 1 GeV.
In computing these values we used $\Lambda_{QCD}=250$ MeV and $m_c=1.39$ GeV, $m_b=4.8$ GeV. For the
bottom case we neglected the change in the running of $\alpha_s$ across the charm threshold, which
gives a negligible error.

We will assume, as discussed in Sect.III, that the contribution of the two-loop anomalous dimension 
to the finite part of the coefficients $c_2^{hl}$ and $c_4^{hl}$ can be neglected. Judging from the
one-loop expression, this would imply that the leading-log approximation is a very poor one in the
charm case, where the logarithmic term accounts for at most 30\% of the total correction.
The situation
is slightly better in the bottom case, where its contribution is enhanced to about 50\%.

Nevertheless, for purposes of illustration we show in Table 2 the combined contributions of
these two operators
to the hyperfine splitting of the D and B mesons using for their coefficients both approximation
methods.
For the reasons discussed above we tend to
prefer the full one-loop results over those obtained from the leading log approximation.
In computing the matrix elements of the operators in Table 2 we have used the
following values $f_D=170$ MeV, $f_B=180$ MeV, $m_D=1.97$ GeV, $m_B=5.28$ GeV.

\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
& & & & & &\\[-0.3cm]
%\rule{0pt}{11.5pt}
$\mu_{fact}$ & $\langle {\cal O}_2^{hl}\rangle_{D^*-D}$
& $\langle {\cal O}_4^{hl}\rangle_{D^*-D}$
& $\langle {\cal O}_2^{hl}\rangle_{B^*-B}$
& $\langle {\cal O}_4^{hl}\rangle_{B^*-B}$
& $\Sigma_D$ (MeV) & $\Sigma_B$ (MeV)\\[0.3cm]
\hline
\hline
& & & & & & \\[-0.3cm]
0.5 GeV & (543 MeV)$^3$ & (493 MeV)$^3$ & (783 MeV)$^3$ & (712 MeV)$^3$ 
& 7.4(LLA)& 2.7(LLA)\\
& & & & & 18.9(1-l.) & 4.2(1-l.)\\[0.3cm]
\hline
& & & & & & \\[-0.3cm]
1.0 GeV & (431 MeV)$^3$ & (392 MeV)$^3$ & (622 MeV)$^3$ & (565 MeV)$^3$ 
& 1.1(LLA) & 0.8(LLA)\\
& & & & & 7.3(1-l.) & 1.8(1-l.)\\[0.3cm]
\hline
\end{tabular}
\end{center}
\begin{quote} {\bf Table 2.}
Matrix elements of the four-quark operators ${\cal O}_2^{hl}$ and ${\cal O}_4^{hl}$ in the
factorization approximation for two different values of the factorization scale $\mu=0.5$ and
1 GeV. In the last two columns the total contribution of these two operators to the
hyperfine splittings of the D and B mesons is shown in both the leading-log approximation and using
the one-loop result (1-l.) for their coefficients.
\end{quote}

Next we turn to a numerical discussion of the relation (\ref{hyper}) for the hyperfine splittings. In
writing (\ref{hyper}) we have neglected two types of corrections: a) electromagnetic effects
\cite{RW}
and b) SU(3) violation \cite{RS,DiB}, both of which appear already at order $1/m_b$. The former can
be neglected by limiting oneself to non-strange heavy mesons. From the
measured hypefine splittings $\Delta_{D^0}=D^{*0}-D^0 = 142.12 \pm 0.05$ MeV, 
$\Delta_{D^+}=D^{*+}-D^+ = 140.64 \pm 0.1$ MeV, 
$\Delta_{B}=B^{*}-B = 46.2 \pm 0.85$ MeV, one can see that the electromagnetic correction in the
charm case is less than 2 MeV, whereas in the bottom case it is expected to be under 1 MeV \cite{RW}.

The relation (\ref{hyper}) contains the as yet unknown parameters $\mu_{SO}^3$ and $\mu_{\pi G}^3$.
The first parameter vanishes exactly for $S$-wave mesons in potential models as the corresponding
operator
(\ref{5}) corresponds to the spin-orbit interaction energy of the heavy quark \cite{BSUV}. The second
parameter is not easy to estimate in a model-independent way. The contributions of the four-quark
operators are shown in the last two columns of Table 2. Their total contribution is positive and
amounts to about 10-20 MeV in the charm case and 1-5 MeV in the bottom case. 

Comparing with the measured hyperfine splittings one can see that a positive $1/m_B$ correction to
(\ref{hyper}) agrees with the sign of the correction to the heavy mass scaling law.
Unfortunately, the lack of information on the precise values of $\mu_{SO}^3$ and
$\mu_{\pi G}^3$ prevents us from making a more quantitative analysis of the deviations from scaling. 

Finally we note the relevance of these corrections for the model-independent determination of the
matrix element $\mu_G^2$ from the measured hyperfine splittings in the B system. From (\ref{hyper})
we obtain 0.358 GeV$^2$ = $\mu_G^2 + 0.026 + 0.021$ + terms proportional to $\mu_{SO}^3/m_B$ and
$\mu_{\pi G}^3/m_B$. The two numbers on the r.h.s. correspond to $\bar\Lambda\mu_G^2$ and
respectively to the four-quark operators' contribution. In this estimate we used $\bar\Lambda=400$
MeV and $\mu_G^2$ =0.35 GeV$^2$. This shows that the theoretical error of this determination
coming from the $1/m_B$ terms in (\ref{hyper}) is conceivably as high as 13\%. 

\acknowledgements
D.P. is grateful to A. Czarnecki, C.D. Lu and D.X. Zhang for discussions and help and acknowledges
support of the Deutsche Forschungsgemeinschaft (DFG) while part of this work was done.
J.G.K. acknowledges partial support from BMFT, FRG under contract 06MZ566. J.C.R. acknowledges
the support of the Alexander von Humboldt Foundation.

\newpage
\appendix
\section*{A}

A complete study of the renormalized HQET Lagrangian at order $1/m^2$ requires also the
computation of the coefficients of operators which vanish by the equation of motion of the
heavy quark field. There are altogether six such operators: three local operators which will be
chosen as in \cite{BKP}
\begin{eqnarray}\label{B1}
{\cal O}_8 &=& \frac{ig}{2}\bar h\sigma_{\mu\nu}F^{\mu\nu}(v\cdot D)h +
\frac{ig}{2}\bar h(v\cdot D)\sigma_{\mu\nu}F^{\mu\nu}h\\
{\cal O}_9 &=& i\bar hD^2(v\cdot D)h + i\bar h(v\cdot D)D^2h\\\label{B3}
{\cal O}_{10} &=& \bar h(iv\cdot D)^3h\,,
\end{eqnarray}
and three nonlocal operators consisting of time-ordered products of dimension 5 operators
\begin{eqnarray}\label{B4}
{\cal O}_{ee} &=& \frac{i}{2}\int\mbox{d}^4x\,\mbox{T}[\bar h(iv\cdot D)^2h](x)
[\bar h(iv\cdot D)^2h](0)\\
{\cal O}_{ke} &=& i\int\mbox{d}^4x\,\mbox{T}[\bar h(iD)^2h](x)
[\bar h(iv\cdot D)^2h](0)\\
{\cal O}_{me} &=& i\int\mbox{d}^4x\,\mbox{T}[\bar h\frac{g}{2}\sigma\cdot Fh](x)
[\bar h(iv\cdot D)^2h](0)\,.
\end{eqnarray}
They will be combined together in a vector ${\cal H}_e$ defined as
\begin{eqnarray}\label{B7}
{\cal H}_e = \mbox{column}({\cal O}_8\,, {\cal O}_9\,, {\cal O}_{10}\,,
{\cal O}_{ee}\,, {\cal O}_{ke}\,, {\cal O}_{me})\,.
\end{eqnarray}

These operators mix only with the operators in ${\cal H}$ (\ref{19}). The renormalized operators
${\cal H}$ and ${\cal H}_e$ satisfy a renormalization group equation which can be written in
matrix form as
\begin{eqnarray}\label{B8}
\mu\frac{\mbox{d}}{\mbox{d}\mu} 
\left( \begin{array}{c}
{\cal H}\\
{\cal H}_e\end{array} \right) +
\left( \begin{array}{cc}
A & G\\
0 & H\end{array} \right)
\left( \begin{array}{c}
{\cal H}\\
{\cal H}_e\end{array} \right) = 0\,.
\end{eqnarray}
The block $A$ has been given in (\ref{22a}), and the remaining ones are given by
\begin{eqnarray}
G = \frac{g^2}{(4\pi)^2}
\left( \begin{array}{cccccc}
0 & 0 & -\frac{16}{3} & 0 & 0 & 0 \\
3 & 0 & 0 & 0 & 0 & 0 \\
0 & \frac{32}{3} & \frac{128}{3} & 0 & -\frac{32}{3} & 0 \\
-\frac{14}{3} & 0 & 0 & 0 & 0 & -\frac{32}{3} \\
9 & 0 & -\frac{32}{3} & 0 & 0 & 0 \end{array} \right)\,,\\
H = \frac{g^2}{(4\pi)^2}
\left( \begin{array}{cccccc}
12 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & -\frac{32}{3} & 0 & 0 & 0 \\
0 & 0 & \frac{16}{3} & 0 & 0 & 0 \\
0 & 0 & \frac{16}{3} & \frac{32}{3} & 0 & 0 \\
0 & -\frac{16}{3} & -32 & -\frac{64}{3} & \frac{16}{3} & 0 \\
-\frac23 & 0 & 0 & 0 & 0 & \frac{34}{3}\end{array} \right)\,.
\end{eqnarray}

At the scale $\mu=m$ the coefficients of the operators in (\ref{B7}) are given by
\begin{equation}
c_e = \mbox{column}(0\,,0\,,0\,,1\,,-1\,,1)\,.
\end{equation}
Integrating the RG equation (\ref{B8}) with this initial condition one obtains the following
expressions for the coefficients of the three local operators in (\ref{B7}) in the leading log
approximation
\begin{eqnarray}\label{B12}
c_8(\mu) &=& -2z^{-3/\beta_0} + 3z^{-17/(3\beta_0)} - \frac54 z^{-6/\beta_0} + \frac14
- \frac{9}{2\beta_0}z^{-6/\beta_0}\ln z\\\label{B13}
c_9(\mu) &=& 3z^{-8/(3\beta_0)} - 3\\
c_{10}(\mu) &=& -\frac35 z^{-6/\beta_0} - \frac{65}{9} z^{-2/\beta_0} + 9 z^{-16/(3\beta_0)}
+ \frac{23}{30} z^{-8/(3\beta_0)} - \frac{35}{18}\,.\label{B14}
\end{eqnarray}
We denoted here $z=(\alpha_s(\mu)/\alpha_s(m))$. In deriving (\ref{B14}) we used the truncated
expression (\ref{tr1}) for the coefficient of the Darwin term.
It is important to emphasize that these coefficients
are (just as $c_e(\mu)$ in (\ref{4})) gauge dependent. Their expressions given above correspond to
the Feynman gauge.

\newpage
\section*{B}

The matching relation connecting the heavy quark field $Q$ in QCD with the effective theory field
$h$ is \cite{KT,BKP}
\begin{eqnarray}\label{A1}
Q(x) = \Omega\exp(-imv\cdot x\vsl) h(x)\,,
\end{eqnarray}
with
\begin{eqnarray}\label{A2}
\Omega = \exp\left(\frac{i\Dslp}{2m}\right)\exp\left(\frac{1}{4m^2}\left[
(i\Dsl_\parallel)(i\Dslp) + (i\Dslp)(i\Dsl_\parallel)\right]\right)\cdots
\end{eqnarray}
and $\Dsl_\parallel = v\cdot D\vsl$, $\Dslp=\Dsl-\Dsl_\parallel$. The field $h$ contains
both ``upper'' and ``lower'' components $h_\pm$, satisfying $\vsl h_\pm = \pm h_\pm$. One has
\begin{eqnarray}
\exp(-imv\cdot x\vsl)h = e^{-imv\cdot x}\frac{1+\vsl}{2}h + e^{imv\cdot x}\frac{1-\vsl}{2}h
= e^{-imv\cdot x}h_+ + e^{imv\cdot x}h_-\,.
\end{eqnarray}
It can be shown \cite{KT} that the field transformation (\ref{A2}) decouples the two
components $h_+$ and $h_-$. The HQET Lagrangian (\ref{1}-\ref{3}) used in the main text refers
only to the $h_+$ part.
% (the subscript is suppressed for convenience whenever no confusion can arise).

The most general form for the $1/m^2$ term in the HQET Lagrangian includes, in addition to the
operators introduced so far, also the operators
\begin{eqnarray}\label{A4}
{\cal O}_8 &=& \frac{ig}{2}\bar h\sigma_{\mu\nu}F^{\mu\nu}(v\cdot D)h +
\frac{ig}{2}\bar h(v\cdot D)\sigma_{\mu\nu}F^{\mu\nu}h\\
{\cal O}_9 &=& i\bar hD^2(v\cdot D)h + i\bar h(v\cdot D)D^2h\\\label{A6}
{\cal O}_{10} &=& \bar h(iv\cdot D)^3h\,.
\end{eqnarray}
Their renormalization properties are studied in the Appendix A.

The purpose of this Appendix is to show that requiring the invariance of the HQET Lagrangian 
(\ref{1}) under a small change in the velocity $v$ (the so-called reparametrization invariance
\cite{LM}) fixes the coefficients $c_{SO}(\mu)$ \cite{CKO} and $c_9(\mu)$. However, no constraint
for $c_D(\mu)$ is obtained in this way, in contrast to \cite{CKO}.

We start by computing the change in the $h_+$ field under an infinitesimal change of the velocity
parameter $v\to v'=v+\Delta v$. This can be obtained from (\ref{A1}) by taking into account the
invariance of the QCD field under this transformation $\delta Q=0$ as
\begin{eqnarray}\label{A7}
\delta h_+ = \delta\left[ \frac{1+\vsl}{2}\exp(imv\cdot x)\Omega^{-1}\right]\Omega
\exp(-imv\cdot x\vsl)h\,.
\end{eqnarray}
Explicitly, to first order in $\Delta v$ and up to second order in $1/m$,
\begin{eqnarray}\label{A8}
\delta h_+ = \Delta v^\mu\left\{ imx_\mu + \frac12 \gamma_\mu + \frac{i}{4m}
\left[\gamma_\mu \Dslp + D_\mu\right] + {\cal O}(1/m^2)\right\} h_+\,.
\end{eqnarray}
Note that no negative component field $h_-$ is introduced by this transformation.

The variation of the HQET Lagrangian (\ref{1}-\ref{3}) is obtained by the variation of the
effective field (\ref{A8}) and the variation of the $v$-dependent operators, including the
operators (\ref{A4}-\ref{A6}). After some algebra one obtains
\begin{eqnarray}\label{A9}
\delta {\cal L} &=& (1-c_k(\mu)) \bar h_+i\Delta v\cdot Dh_+
-\frac{1}{m}\left\{-(c_e+c_9-1)\bar h_+(\Delta v\cdot D)(v\cdot D)h_+\right.\\
&-&\left.\,\frac14\left(c_e+1+c_9+c_m+\frac{c_{SO}+1}{2}\right)
\bar h_+ig\gamma\cdot \Delta v\gamma^\nu v^\mu F_{\mu\nu} h_+\right.\nonumber\\
&-&\left.\,\frac14\left(c_e+1+c_9-c_m-\frac{c_{SO}+1}{2}\right)
\bar h_+ig\gamma^\nu\gamma\cdot \Delta v v^\mu F_{\mu\nu} h_+{\cal O}(1/m^2)\right\}\nonumber\,.
\end{eqnarray}
This will vanish for any $\Delta v$ provided that the following identities hold
\begin{eqnarray}\label{A10}
c_k(\mu) &=& 1\\\label{A11}
c_9(\mu) &=& -1 - c_e(\mu)\\\label{A12}
c_{SO}(\mu) &=& -1 -2 c_m(\mu)\,.
\end{eqnarray}

The first constraint has been given in \cite{LM} and is perhaps the best known application of
the reparametrization invariance principle. The constraint (\ref{A12}) has been presented in
\cite{CKO} and its prediction for $c_{SO}$ agrees with the explicit calculation in \cite{BKP}
(see (\ref{cSO})). The relation (\ref{A11}) together with (\ref{6}) predicts the following value
for $c_9(\mu)$
\begin{eqnarray}
c_9(\mu) = 3\left[\left(\frac{\alpha_s(\mu)}{\alpha_s(m)}\right)^{-8/(3\beta_0)} - 1\right]\,.
\end{eqnarray}
This agrees with the expression of this coefficient obtained by direct computation in leading
log approximation (\ref{B13}).

\newpage
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\end{references}

\begin{figure}
\centerline{\epsfxsize=10truecm \epsfbox{Fig1.eps} }
\caption[1]{Diagrams contributing to heavy-light quark scattering which are needed for the matching
conditions (35-38).}
\end{figure}




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