%Title: Non-Abelian Dipole Radiation and the Heavy Quark Expansion
%Author: Andrzej Czarnecki, Kirill Melnikov, and Nikolai Uraltsev
%Published: <EM> Phys. Rev. Lett. </EM> <B> 80 </B> (1998) 3189. 
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%\documentstyle[tighten,aps]{revtex}

\begin{document}
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\preprint{TTP97-33, hep-ph/9708372}

\title{Non-Abelian Dipole Radiation and the Heavy Quark Expansion}
\author{Andrzej Czarnecki\thanks{Present address:
Physics Department,
Brookhaven National Laboratory,  Upton, New York 11973.}
and Kirill Melnikov}
\address{Institut f\"ur Theoretische Teilchenphysik, \\
Universit\"at Karlsruhe,
D-76128 Karlsruhe, Germany}
\author{Nikolai~Uraltsev\thanks{Permanent address:
Petersburg Nuclear Physics Institute,
Gatchina, St.~Petersburg 188350, Russia}}
\address{Dept. of Physics, University of Notre Dame du Lac,\\
Notre Dame, IN 46556, USA}
\maketitle


\begin{abstract}
Dipole radiation in QCD is derived to the second order in
$\alpha_s$. A power-like evolution of the spin-singlet heavy quark
operators is obtained to the same accuracy. We give an ${\cal
O}(\alpha_s^2)$ relation between the short-distance low-scale running
heavy quark mass and the $\overline{\rm MS}$ mass.  We discuss the
properties of an effective QCD coupling $\alpha_s^{(d)}(E)$ which
governs the dipole radiation. We argue that this coupling is
advantageous for describing heavy quark physics.
\end{abstract}

\vspace*{3mm}
Theoretical description of heavy flavor decays  benefits from a strong
hierarchy between the mass  of the decaying quark  and the typical scale
of the strong interactions, $m_{b(c)} \gg \Lambda_{\rm QCD}$.  The
current level of precision requires an accurate treatment of
nonperturbative effects even in beauty decays.  A consistent genuinely
QCD-based framework for this  is provided by the heavy quark expansion
(HQE) which combines the Wilson operator product expansion  (OPE)  with
the nonrelativistic expansion. It allows a  simultaneous precise
description  of both perturbative and nonperturbative effects.

An important class of applications of the HQE are semileptonic weak
transitions between $b$ and $c$ quarks,  in which  both initial and
final quarks are heavy. The most informative predictions can be made for
the transition amplitudes in the so-called small velocity (SV) limit
\cite{SV}, when the velocities of  heavy hadrons in initial and final
states  are small. The heavy quark in this case plays a role of a slowly
moving source of the color Coulomb field which affects the light degrees
of freedom in the hadrons. This physical picture  is formalized in the
Wilsonian approach by integrating out high-momentum degrees of freedom
of full QCD and resorting to an effective low-energy theory.

A peculiarity of the effective theory for heavy flavor hadrons is its
essentially Minkowskian nature.  Any process in which the velocity of
the heavy quark $Q$ changes involves gluon radiation with energy and
momentum in the whole range up to the quark masses.  On the other
hand, when the energy loss is small compared to $m_Q$, the radiation
off the heavy colored particles is almost a classical effect and
therefore a universal process-independent description is possible.  In
the small velocity limit it is the familiar dipole radiation, with some
peculiarities due to the non-Abelian nature of QCD.

Even though the dipole radiation has the most obvious manifestation in
the SV processes, it is relevant in  many other situations,
 from the zero-recoil transitions to inclusive decay widths.  The
reason is that the OPE ensures certain universality of the effects of
soft physics originating at the momentum scale well below $m_Q$ for
all genuinely short-distance observables.

In this letter we derive the non-Abelian dipole radiation to second
order in $\alpha_s$ and use it to obtain the power evolution of a number
of spin-singlet heavy quark operators to order $\alpha_s^2$. In
particular, a gauge-invariant relation between a short-distance
low-scale running heavy quark mass and the $\overline{\rm MS}$ mass
$\bar{m}_Q(m_Q)$ is given, which becomes a practical necessity since
the two-loop accuracy in the processes with heavy quarks is becoming
the state of the art.

The radiation by a  heavy charged particle when its velocity changes
is   well known  from classical electrodynamics. It is
obtained from the Li{\'e}nard--Wiechert retarded
potentials $A_\mu(\vec r,\,t)$ at $\vec{r} \rightarrow \infty$ and has the
form
\begin{equation}
\frac{1}{\omega}\frac{{\rm d}I(\omega)}{{\rm d}\omega} \;=\;
\frac{\alpha}{\pi}
\left(\frac{1}{|\vec v\,|}\ln{\frac{1+|\vec v\,|}{1-|\vec v\,|} } -2
\right) \frac{1}{\omega}
\:=\: \frac{2}{3}\,\frac{\alpha}{\pi}
\frac{\vec{v}^{\,2}}{\omega} \;+\;{\cal O}(\vec{v}^{\,4})\;,
\label{5}
\end{equation}
where $\omega$ is frequency, $I(\omega)$ intensity,
and $\alpha$ is the fine structure constant.
We will be interested only in
the dipole term $\propto \vec{v}^{\,2}$ in QCD, and
do not consider multipole radiation proportional to higher powers
of $\vec v$.

In QED the same relation holds, with $1/\omega\; {\rm d}I/{\rm
d}\omega$ giving the probability of radiating soft photon(s) with
energy $\omega$. Moreover, there are no higher-order corrections in
$\alpha_{\rm em}$ provided $\omega \ll m$.  The effects of the photon
interaction with vacuum fluctuations are also suppressed, by powers of
$\omega/ m_e$, and the soft radiation in QED is governed by
$\alpha(0)$.

Due to gluon self-interaction the dipole radiation in non-Abelian
theory is different.  Our main interest lies in the domain where
$\omega$ is large compared to $\Lambda_{\rm QCD}$ (the nonperturbative
multipole expansion of the color-singlet systems was discussed in
\cite{mult}). In this regime perturbative calculations can be
performed.


Below we introduce some standard notations used for heavy quarks.  Let
us consider a process of scattering of a color-singlet ``weak''
current $J$ with momentum $q$ on a heavy quark $Q$ in the SV
kinematics.  For simplicity, the initial quark is assumed to rest.
The initial $Q$ and final state $\tilde Q$ quarks can have arbitrary
masses; however, both masses must be large, so that the
nonrelativistic expansion can be applied.  The SV limit $\vec{v}=
{\vec q}/m_{\tilde Q}$, $|\vec{v}\,| \ll 1$, is kept by
adjusting $\vec{q}$ appropriately.  For simplicity we consider the
case of equal masses, $m_{\tilde Q}=m_Q$, although nothing depends on
this assumption.  The current $J$ must have a non-vanishing tree level
nonrelativistic limit (e.g., $J_0^\dagger =\bar{Q} \gamma_0 \tilde Q$
or $J_S^\dagger =\bar{Q} \tilde Q$), otherwise it can be arbitrary.

 Inclusive processes of the scattering of heavy quarks are described
by an appropriate structure function $W(q_0,\,\vec{q}\,)$ which is a
sum of all transition probabilities induced by $J$ into final states
with momentum $\vec{q}$ and energy $m_Q +q_0$.  The optical theorem
relates it to the discontinuity of the forward transition amplitude
$T(q_0,\,\vec{q}\,)$ at physical values of $q_0\,$:
\begin{equation}
T(q_0,\,\vec{q}\,)\;=\; \frac{i}{2m_Q}
\int\, d^4 x\: {\rm e}\,^{-iqx}\;
\langle {Q} | {T\,J(x) J^\dagger (0)} |{Q} \rangle \;,\;\;
W(q_0,\,\vec{q}\,) = 2\: \mbox {Im}\: T(q_0,\,\vec{q}\,)\;.
\label{9}
\end{equation}

In the heavy quark limit the spin degrees of freedom become
irrelevant, and we assume averaging over spin states (also color, for
perturbative calculations).  The structure functions have a threshold
$
q_0^{\rm min}\;= \; \sqrt{\vec{q}^{\,2}+m_{Q}^2}-m_Q \simeq
{m_{Q} \vec{v}^{\,2}}/{2}
$
corresponding to elastic transitions.
The variable $ \omega \;=\; q_0-q_0^{\rm min}
$
measures the hadron excitation energy in the final state.

The nonrelativistic expansion and the perturbative treatment are both
justified if $\Lambda_{\rm QCD} \ll \omega \ll m_Q$, and this
hierarchy is assumed in what follows. The nonperturbative aspects will
be addressed later. In perturbative calculations we can use quark
states instead of actual heavy hadrons, as was tacitly assumed above.
The structure function $W$ takes the following form in the heavy quark
limit:
\begin{equation}
W(\omega,\,\vec{v})\;=\; N\,\delta(\omega) +
\,\frac{2\vec{v}^{\,2}}{3}\, \frac{d(\omega)}{\omega} + {\cal
O}\left(\vec{v}^{\,4}\right)\;.
\label{14}
\end{equation}
At $\vec v =0$ only the elastic peak is present.  The dipole radiation is
described by $d(\omega)$ and $N$ is a (velocity-dependent)
factor depending on the current. 

Motivated by the dipole radiation in QED, we define the dipole coupling
$\alpha_s^{(d)}$ by projecting Eq. (\ref {14}) on its second term:
\begin{equation}
C_F \frac{\alpha_s^{(d)}(\omega)}{\pi \omega}\;=\;
\lim_{\vec v \rightarrow 0} \;
\lim_{m_Q \rightarrow \infty}\,\frac{3}{2\vec{v}^{\,2}}
\frac{W(\omega,\,\vec{v})}{\int_0^{\omega}\,
W(\omega',\,\vec{v})\,{\rm d}\omega'}
\;.
\label{16}
\end{equation}
Here $\omega$ is assumed to be positive.  The denominator in the last
ratio eliminates the overall normalization of the effective
nonrelativistic current.  In this form the heavy quark limit
$m_Q\rightarrow \infty$ yields a finite result. The normalization
integral includes the elastic peak, which makes it infrared safe at
arbitrary $\vec v$.  On the other hand, the exact upper limit does not
matter in the SV kinematics, since it affects the ratio only by
corrections $\sim \vec{v}^{\,2}$.  In this way, the inelastic SV
structure function determines the effective QCD running
coupling~\footnote{S. Brodsky suggested to call it ``radiation
charge''. A more precise term could be also a ``dipole radiation
charge'', to distinguish from soft radiation in the ultrarelativistic
case.}  driving the dipole radiation of gluons. It is a dimensionless
function of the ratio $\Lambda_{\rm QCD}/\omega$.

The OPE and factorization of the infrared effects ensures that
$\alpha_s^{(d)}(\omega)$ is universal: it does not depend on the
choice of the weak current $J$ probing the heavy quark structure, or
on the ratio of the quark masses. Also, there is no dependence on the
type of the initial heavy hadron as soon as the onset of the
quark-hadron duality is passed.  This property always holds
perturbatively.  The coupling $\alpha_s^{(d)}$ obeys the usual
renormalization group (RG) equation with the standard first two terms
in the $\beta$-function.  Because of its physical definition, the
evolution of $\alpha_s^{(d)}$ is properly defined when heavy flavor
thresholds are passed.

In practice one needs to know the relation between
$\alpha_s^{(d)}$ and the standard
reference coupling $\alpha_s^{\overline{\rm MS}}$. We
calculated it to the second order in
$\alpha _s^2$ using the technique of Ref.~\cite{cmtech}:
\begin{equation}
\frac{\alpha_s^{(d)}(\mu)}{\pi} =
\frac{\alpha_s^{\overline{\rm MS}}(\mu)}{\pi} +
\left[\left(\frac{5}{3} - \ln{2}\right)\frac{\beta_0}{2}
- C_A \left(\frac{\pi^2}{6}-\frac{13}{12} \right)\right]
\left(\frac{\alpha _s}{\pi}\right)^2\;,
\label{eq:zwiaz}
\end{equation}
where $C_A=N_c$ for the ${\rm SU}(N_c)$ gauge group and
$\beta_0=\frac{11}{3}C_A-\frac{2}{3}n_f$.  The term $\sim C_A$ in 
eq.~(\ref{eq:zwiaz})
represents the so-called genuine (non-BLM) second-order effect; it has
purely non-Abelian origin.  The BLM-type terms \cite{BLM} which
account for the effects of running of the QCD coupling in the
leading-order calculations are readily computed to any order in this
case (the results of \cite {blmope} can be directly used for such
calculation).  The term $\beta_0^2 \alpha _s^3$ and the expression for
the case of massive flavors are given in \cite{hep}.  Even in the
Abelian theory with light flavors $\alpha_s^{(d)}(\mu)$ deviates from
the $V$-scheme coupling $\alpha_s^{(V)}(\mu)$ \cite{BLM}, their
commensurate scales are somewhat different. This is a purely kinematic
effect; $\alpha_s^{(d)}(\mu)$ incorporates integration over the
invariant mass of the light fermion pair in the interval from $0$ to
$\mu$, while in $\alpha_s^{(V)}$ the fermion loop enters at a fixed
spacelike momentum.

A similar effective
coupling for emission of soft gluons in the
ultrarelativistic case $|\vec{v}|\to 1$ has been also
calculated to two loops
\cite{v1};
its exact definition, however, is less direct.
It differs from $\alpha_s^{(d)}$ in the
second-order terms, although the difference is small numerically.

A construction of the HQE in QCD requires a proper definition of the
basic objects of the effective theory, in the first place the heavy
quark mass, kinetic operator and all other composite operators.  A
consistent approach is possible if an explicit separation scale $\mu$
between the low- and high-momentum domains is introduced.  All
operators then depend on $\mu$.  A normalization-point--independent
pole mass of the heavy quark, although appearing in purely
perturbative calculations at a given order, cannot be completely
defined when nonperturbative effects are addressed. This applies also
to the parameter $\overline{\Lambda}$ measuring the difference between
the heavy hadron mass $M_{H_Q}$ and $m_Q$ in the heavy quark
limit. Instead, one must use in the OPE the short-distance mass
$m_Q(\mu)$ and, correspondingly, $\overline{\Lambda}(\mu)= \lim
_{m_Q\rightarrow \infty} M_{H_Q}-m_Q(\mu)$ with $\Lambda_{\rm QCD} \ll
\mu \ll m_Q$ \cite{pole}. The low-scale short-distance mass can be
defined in different ways. We recall, however, that using the standard
${\overline {\rm MS}}$ mass if $\mu \ll m_Q$ is unnatural and should
be avoided \cite{rev}.


A physically appropriate gauge-invariant scheme suggested in
\cite{blmope,rev,five} is based on the SV sum rules relating the
moments of the SV structure functions to the local heavy-quark
operators \cite{optical}.  The normalization point is introduced as an
upper cutoff in the integral over excitation energy.  For example, for
a heavy hadron $H_Q$ one defines
\begin{eqnarray}
\overline{\Lambda}(\mu)\; &\equiv & \; \lim_{m_Q \rightarrow \infty}\;
\left[ M_{H_Q}-m_Q(\mu)\right]\, =
\, \lim_{\vec v \rightarrow 0}\;
\lim_{m_Q \rightarrow \infty}\, \frac{2}{\vec v^{\,2}}
\frac{\int_0^\mu \; {\omega\, W(\omega,\,\vec v\,)\:{\rm d}\omega}}
{\int_0^\mu \;{W(\omega,\, \vec v\,)\:{\rm d}\omega}} \;\;,
\nonumber \\
\mu_\pi^2(\mu)\;&\equiv & \;
\frac{\langle {H_Q}|{\bar Q (i\vec{D}\,)^2 Q} |{H_Q} \rangle _\mu}
{\langle {H_Q} | {\bar Q Q} |{H_Q} \rangle _\mu}
\,=\,
\lim_{\vec v \rightarrow 0} \; \lim_{m_Q \rightarrow \infty}\,
\frac{3}{\vec{v}^{\,2}}
\frac{\int_0^{\mu}\,\omega^2\,  W(\omega,\,\vec{v})\:{\rm d}\omega}
{\int_0^{\mu}\:W(\omega,\,\vec{v})\:{\rm d}\omega}
\;,
\label{28}
\end{eqnarray}
etc.
These operator relations suggest that the
perturbative evolution is described by one and the same function,
$\alpha_s^{(d)}(\mu)$ of Eq.~(\ref{16}):
\begin{equation}
\frac{{\rm d}\overline{\Lambda}(\mu)}{{\rm d}\mu}\;=\;\frac {4}{3} \,
C_F \frac {\alpha_s^{(d)}
(\mu)}{\pi}\;,\qquad
\frac{{\rm d}\mu_\pi^2(\mu)}{{\rm d}\mu^2}\;=\;
C_F \frac{\alpha_s^{(d)}(\mu)}{\pi}\;.
\label{30}
\end{equation}
The same $\alpha _s^2$ running of $\mu_\pi^2$
was obtained in \cite{xi} from the zero-recoil transitions.

A  similar $\mu$-dependence holds for the renormalized slope of the
Isgur-Wise function when it is defined in the same physical way
\cite{rev} based on the Bjorken sum rule:
\begin{equation}
\mu\frac{{\rm d}
\left [\varrho^2(\mu)-\frac{1}{4} \right ]}{{\rm d}\mu}\;=\;
\frac{4}{3}\,
\frac{2\alpha_s^{(d)}(\mu)}{3\pi}\;.
\label{30a}
\end{equation}
Due to short-distance effects, the observable transition form factors
do not have a literal heavy quark limit at $\vec v \ne 0$ but vanish,
and require factoring out the perturbative suppression due to gluon
radiation, to yield the effective $\mu$-dependent IW function
$\xi(v^2;\,\mu)$.  In analogy with Eqs.~(\ref{28}), it is convenient
to define $\xi(v^2;\,\mu)$ for small $v$ as
$$
\xi(v^2;\,\mu)\; = \;\lim_{m_Q\rightarrow \infty}
\frac{F(v)} {\left[\frac{1}{2\pi}\int_0^\mu\; W(\omega,\, \vec
v\,)\:{\rm d}\omega \right]^{1/2} }\;,\qquad
\varrho^2(\mu) =
\left. -2\frac{{\rm d}\xi(v^2;\,\mu)}{{\rm d} v^{\,2}}\right|_{v=0}\;,
$$
where $F$ is the form factor, say,  of
the vector current $\bar Q \gamma_0 Q$
and $W$ is the
corresponding structure function. This is the scheme suggested in
\cite{rev} for which Eq.~(\ref{30a}) holds.

The RG equations (\ref {30}) are proportional to $\alpha_s^{(d)}$.
Hence, the OPE shows that the first-order running is not renormalized
in the Abelian theory without light fermions to {\it all orders} in
the coupling \cite{blmope}, since in such a theory $\alpha_s^{(d)}$
identically coincides with $\alpha (0)$.

According to \cite{five}, one can calculate perturbatively a physical
(gauge-invariant) low-scale running mass $m_Q(\mu)$ with $\mu \ll m_Q$
by subtracting, order by order, the infrared part given by the SV sum
rules, from the pole mass $m_Q^{\rm pole}$:
$$
m_Q(\mu)=\left[m_Q^{\rm pole}\right]_{\rm pert} -
\left[ \overline{\Lambda} (\mu)\right]_{\rm pert} - \frac{1}{2m_Q(\mu)}
\left[\mu_\pi^2(\mu) \right]_{\rm pert}\;.
$$
This mass determines the nonrelativistic kinetic energy term in the
renormalized heavy quark Hamiltonian.  The two-loop relation between
this $m_Q(\mu)$ and the $\overline{\rm MS}$ mass is
\begin{eqnarray}
\label{mass}
m_Q(\mu) = {\bar m} \Bigg \{1&+&\frac {4}{3}\frac {\alpha _s (\bar m)}{\pi}
\left (1 - \frac {4}{3} \frac {\mu}{\bar m} - \frac {\mu^2}{2\bar m^2}
\right)
\\
+ \left (\frac {\alpha _s (\bar m)}{\pi} \right )^2 \Bigg [
K &-&\frac {8}{3} +  \frac {\mu}{\bar m} \left (\frac {8\beta _0}{9} X_1
+ \frac {8\pi^2}{9} - \frac {52}{9} \right )
+
\frac {\mu ^2}{\bar m ^2} \left (\frac {\beta _0}{3}X_2
+ \frac {\pi^2}{3} - \frac {23}{18} \right )
\Bigg ] \Bigg \}
\nonumber
\end{eqnarray}
where
$$
K = \frac{\beta_0}{2}\left(\frac {\pi^2}{6} +\frac {71}{48} \right)\,+\,
\frac {665}{144} + \frac {\pi^2}{18}\left(2\ln{2} - \frac{19}{2}\right)
-\frac {1}{6} \zeta (3),
$$
$$
X_1 = \ln{\frac {2\mu}{\bar m}} - \frac {8}{3},\;\;\;
X_2 = \ln{\frac {2\mu}{\bar m}} - \frac {13}{6}
$$
and $\bar m = \bar m(\bar m)$ is the $\overline{\rm MS}$ mass
normalized at the scale $\bar m$. The ${\cal O}(\alpha _s ^2)$
relation between the pole mass and the $\overline{\rm MS}$ mass
\cite{Broadhurst} was used here.  We neglected small terms $\sim
\mu^3/m_Q^2$ which can be incorporated in the same way, if
necessary. The next-order BLM correction to Eq.~(\ref{mass}) is given
in \cite{hep}.

The nonperturbative effects in the dipole radiation are suppressed by
powers of $\Lambda_{\rm QCD}/\omega$.  In particular, the actual
structure functions in Eqs.~(\ref{9}) refer to a heavy hadron ($B$
meson, $\Lambda_b$ etc.) but not an isolated heavy quark. Likewise,
the complete effective coupling $\alpha_s^{(d)}$ defined in this way,
as any other effective charge \cite{grunberg}, acquires
power-suppressed nonperturbative contributions; they differ, for
example, in $B$ and $\Lambda_b$ even in the heavy quark limit.  One
can quantify the nonperturbative effects in $\alpha_s^{(d)}(\omega)$
by constructing its $1/\omega$ expansion; the methods used in the OPE
are applicable here (see \cite {hep} for more details).  The
nonperturbative effects in $\alpha_s^{(d)}$ are expressed through the
expectation values of local heavy quark operators $\bar Q O_k Q$.

The first nontrivial operator is the kinetic operator $\bar Q (i\vec
D\,)^2 Q$, which, by dimension counting, would lead to $1/\omega^2$
effects. However, it does not contribute to $\alpha_s^{(d)}$. Indeed,
we can consider the initial heavy quark moving with the spacelike
momentum $|\vec{p}| \ll \omega$. The structure function for such a
state is directly related to the one at rest, and is modified only in
terms $\sim \vec{p}^{\,2} \vec{v}^{\,4}$ which can be neglected.
Since the kinetic expectation value for such a state is non-zero, the
corresponding coefficient function must vanish.


There is one spin-singlet, so-called Darwin operator
$O_D= -i\bar Q D_k D_0 D_k Q$ yielding $1/\omega^3$-suppressed
terms (for simplicity, we consider the case of a $B$ meson). The OPE
relates its effect \cite{hep} to the logarithmic anomalous dimension
$\hat\gamma_D$ of $O_D$:
$$ \frac{\delta_{\rm
D}\alpha_s^{(d)}(\omega)} {\alpha_s^{(d)}(\omega)} =
\frac{3\pi\hat\gamma_D(\omega)}{8\alpha_s^{(d)}(\omega)}
\frac{\rho_D^3(\omega)}{\omega^3}
\simeq
-\left(\frac{\alpha _s(\mu)}{\alpha 
_s(\omega)}\right)^{-\frac{13}{2\beta_0}}
\frac{39\rho_D^3(\mu)}{32\omega^3} \,\approx \,
-\left(\frac{0.55 \,\mbox{GeV}}{\omega}\right)^3
,
$$
where the sum rule for the Darwin term
\cite{rev,motion,pirjol} was used, and
$$ \mu\frac{\rm d}{{\rm d} \mu} \rho_D^3(\mu)
= \hat\gamma_D(\mu) \rho_D^3(\mu),~~\hat\gamma_D(\mu) \simeq -
\frac{13\alpha_s^{(d)}(\mu)}{4\pi},~~
$$
$$
\rho_D^3(\mu)\;=\; \frac{\langle {B} |{O_D} |{B} \rangle
_\mu}{2M_B}\simeq \frac{2\pi\alpha _s}{9} \tilde f_B^2 M_B \approx
0.1\,\mbox{GeV}^3\; \mbox{ at } \; \alpha _s(\mu)\simeq 1\;.
$$
We also employed the factorization estimate of $\rho_D^3$
\cite{motion}; the anomalous dimensions were calculated in
\cite{vslog}.
We see that  the nonperturbative effects are expected to
die out quickly at large energy,
$\; \delta_{\rm np}\alpha_s^{(d)}(\omega)\; \sim \;
\left({\Lambda_{\rm QCD}}/{\omega}\right)^k, \;\;k\ge 3\:.
$

As an effective QCD coupling, $\alpha_s^{(d)}$ has an additional
advantage: it directly measures the strength of the interaction,
whereas such popular couplings as $\alpha _s^{(D)}$ or $\alpha
_s^{(R)}$ determine the {\em deviation} of the corresponding
observables, Adler $D$-function $D(Q^2)$ or $R(e^+e^- \to
\mbox{hadrons})$, from their tree-level value.  From this perspective,
$\alpha_s^{(d)}$ is closer to the so-called $V$-scheme coupling
$\alpha _s^{(V)}$ related to the heavy quark potential $V(R)$
\cite{appel,BLM,peter}.  An exact definition of $\alpha _s^{(V)}$ in
higher orders is not evident at the moment, however.  Emergence of
terms $\alpha _s^k\ln{\alpha _s}$ with $k\ge 4$ in its relation to
$\alpha_s^{\overline{\rm MS}}$ \cite{appel} may indicate that $\alpha
_s^{(V)}$ is not a genuinely short-distance quantity.

We defined $\alpha_s^{(d)}$ directly in Minkowski space.  Since the
OPE is applicable here, this is not too important.  However, the
nonperturbative terms related to the asymptotic nature of the power
expansion of the OPE which are exponentially suppressed in the
Euclidean domain, may yield an oscillating component which is only
power-suppressed in $\alpha_s^{(d)}$ (for a recent discussion, see
\cite {inst}).  The coupling $\alpha_s^{(d)}$ may also show some
(smeared) resonance structure at the heavy flavor thresholds.  To
eliminate these features one can easily construct the Euclidean
counterpart of $\alpha_s^{(d)}$.

The coupling $\alpha_s^{\overline{\rm MS}}$ proved to be indispensable
for complicated multi-loop calculations. However, it is unphysical in
some respects and its use as an expansion parameter sometimes obscures
the perturbative expansion \cite{stan}. At the moment the complete
${\cal O}(\alpha _s^2)$ relation between $\alpha_s^{\overline{\rm
MS}}$ and $\alpha_s^{(d)}$ is sufficient.  A calculation of order 
$\alpha _s^3$ seems feasible as well.

Using $\alpha_s^{(d)}$ is advantageous in heavy quark decays and may
improve the accuracy of the perturbative estimates.  Since the
velocity of the charm hadrons is typically small in $B$ decays,
$\alpha_s^{(d)}$ directly determines the gluon bremsstrahlung and thus
sums the lowest-momentum component of the corrections, where the
effects of higher orders in a differently defined $\alpha_s$ could be
large.

The value of $\alpha_s^{(d)}(E)$ can in principle be fitted from
double-differential distributions in $b\rightarrow c\, \ell\nu$
decays, using the definition (\ref{16}) for any measured combination
of the actual semileptonic structure functions of $B$ meson.  However,
the corrections due to finite $m_Q$ and $\vec{v}$ must be accounted
for.  The dependence of the average invariant mass of hadrons in the
final state on $q^2$, $E_\ell$ or other suitable kinematical
constraint on the bremsstrahlung energy is expected to be sensitive to
$\alpha_s^{(d)}$.  Other possibilities to measure it exist in the
near-threshold production of $b\bar{b}$ in ${\rm e^+e^-}$ annihilation
or in top quark physics.  All they require, however, additional
investigation.\vspace*{.15cm}

To summarize, we have derived the non-Abelian dipole radiation by a
nonrelativistic color particle to the second order in the strong
coupling.  By virtue of the SV sum rules this determines the two-loop
power mixing of the number of spin-singlet effective local heavy quark
operators. We have given the two-loop relation between the
gauge-invariant short-distance low-scale heavy quark mass $m_Q(\mu)$
suitable for the Wilson OPE and $m_Q(m_Q)$ in the $\overline{\rm MS}$
scheme. We have discussed the properties of the effective dipole
radiation coupling $\alpha_s^{(d)}$ which is useful for heavy quark
physics. A more detailed discussion can be found in \cite{hep}.

\vspace*{.2cm}
{\bf Acknowledgments:} \hspace{.4em}N.U. is grateful
to I.~Bigi, S.~Brodsky, Yu.~Dokshitzer, A.~Mueller, M.~Shifman,
A.~Vainshtein and M.~Voloshin for discussions.  This work was
supported in part by NSF under the grant number PHY 92-13313, by BMBF
under grant number BMBF-057KA92P and by Graduiertenkolleg
``Teilchenphysik'' at the University of Karlsruhe.

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