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\begin{center}
{\large\sc Gluon condensate term in heavy quark mass
}\\
\vspace*{4mm}
V.V.Kiselev 
\\
\vspace*{3mm}
{\it Russian State Research Center "Institute for High Energy Physics",} \\
{\it Protvino, Moscow Region,  142284,  Russia. }\\
{\it E-mail: kiselev@mx.ihep.su, Fax: +7337}
\end{center}

\begin{abstract}
We investigate a connection between a renormalon ambiguity of heavy quark mass
and the gluon condensate contribution into the quark dispersion law related
with a virtuality, being a displacement of heavy quark from the perturbative
mass-shell, which happens inside a hadron.
\end{abstract}

\vspace*{1cm}
PACS numbers: 12.39.Hg, 12.38.-t

\newpage
Operator Product Expansion is among the most powerful tools in the heavy quark
physics. In this respect it is usually applied in the form of series in the
inverse heavy quark mass, determining the characteristic energy scale, say, in
sum rules or for decays etc. \cite{1}. It is well recognized that the Wilson
coefficients standing in front of quark-gluon operators can contain the
uncertainty caused by the factorization of perturbative contribution and the
nonperturbative matrix elements of composite operators. In this case the
restriction on internal virtualities in Feynman diagrams has to be introduced
to control the dependence on an ``infrared'' energy scale $\lambda$. Usually,
the gluon propagator is modified by replacement: $1/k^2\to 1/(k^2-\lambda_g^2)$
or the cut off the gluon momenta is performed as $k^2>\lambda^2$ \cite{2}. The
calculation results depend on these parameters. Say, a peculiar behaviour at
$\lambda_g^2\to 0$ appears in physical quantities. For example, a perturbative
correlator of two heavy quark currents acquires a power correction like
$\lambda^4/m^4$, where $m$ is the heavy quark mass \cite{3}. Physically, it
means that the OPE expansion can be valid if we sum the perturbative and
nonperturbative parts with the vacuum expectation of gluon operator which has
the same low energy scale dependence: the gluon condensate $\sim \lambda^4$.
Then the $\lambda$-dependent term can be eaten due to the appropriate
definition of OPE with the condensates. Another case takes place for the
uncertainty in the heavy quark mass, where the perturbative calculation of
self-energy with the gluon virtuality cut off leads to the linear term in
$\lambda$. However, there is no appropriate operator whose vacuum expectation
is proportional to the first power of low energy scale \cite{1}. It was shown
that the mentioned uncertainty proportional to the powers of factorization
scale $\lambda$ can be related with the perturbative summation of higher order
diagrams, which in the limit of infinitely large number of flavors has the
divergency of series in $\beta_0\alpha_s$, where $\beta_0$ denotes the first
coefficient of Gell-Mann--Low function in QCD. The Borel transform of such
series has some peculiar points, which provide the uncertainty in the inverse
transformation. This uncertainty, related with the divergency of perturbative
series is called the renormalon \cite{4}, since the physical contents of such
fact is clarified by the representation, where the series are combined in the
running of QCD coupling constant dependent of the gluon virtuality. The
coupling has the singularity, which is the indication of confinement. In this
way, the uncertainty in powers of $\Lambda_{QCD}$ appears again.
These facts imply that the OPE for fixed values of physical quantities (say,
partial widths or coupling constants in the sum rules) in terms of perturbative
heavy quark mass results in the heavy quark mas, whose value extracted form the
data, strongly depends on the order of calculation in $\alpha_s$-series
\cite{1}: the mass value is significantly changed from order to order.

Thus, the heavy quark quantities have the renormalon uncertainties connected
to the infrared confinement in QCD. Some of them can be eaten by the
appropriate definition of OPE with condensates. The heavy quark mass is of a
special interest, since its infrared uncertainty cannot be
straightforwardly adopted by the vacuum expectation of an operator with the
dimension 1 in the energy scale.

In present paper we evaluate the gluon condensate contribution to the
dispersion law of heavy quark. We find that the corresponding operator is
divided by the third power of quark virtuality, which results in the
appropriate dimension of term in the heavy quark mass. We discuss how this fact
can be used to cancel the infrared uncertainty of mass.

We perform the calculation of diagram shown in Fig.1 in the technique of
fixed-point gauge \cite{5} with the NRQCD propagators of heavy quarks \cite{6}.

\setlength{\unitlength}{1mm}
\begin{figure}[th]
\begin{center}
\begin{picture}(100,40)
\put(5,0){
\epsfxsize=8cm \epsfbox{1.ps}}
\put(7,9){$p$}
\put(80,9){$0$}
\put(43,2){$p+k_1$}
\put(60,2){$p+k_1+k_2$}
\put(33,40){$k_1$}
\put(54,40){$k_2$}
\end{picture}
\end{center}
\caption{The diagram with the gluon condensate contribution to the two-point
effective action of heavy quark.}
\end{figure}

The covariant form of two-point heavy quark effective action $\bar h_v \Gamma
h_v$ can be represented as
\begin{eqnarray}
\Gamma &=& p\cdot v - \frac{(p\cdot v)^2-p^2}{2 m}+\nonumber \\
&&\frac{\pi^2}{24}\left\langle
\frac{\alpha_s}{\pi} G^2_{\mu\nu}\right \rangle \left [ \frac{(p\cdot
v)^2-p^2}{m^2}\frac{1}{\left(p\cdot v - \frac{(p\cdot v)^2-p^2}{2
m}\right)^3}+\frac{1}{m} \frac{1}{\left(p\cdot v - \frac{(p\cdot v)^2-p^2}{2
m}\right)^2}\right],
\label{main}
\end{eqnarray}
where $v$ denotes the four-velocity of hadron containing the heavy quark.
The validity of (\ref{main}) holds under the certain condition on the region of
kinematical variables: the gluon condensate term in the dispersion law of quark
is less than the leading contribution. 

In the rest frame of hadron $v=(1,{\bf 0})$ we have
$$
p\cdot v - \frac{(p\cdot v)^2-p^2}{2 m} = p_0 -\frac{{\bf p}^2}{2 m} = 
\Delta E,
$$
where $\Delta E$ denotes a heavy quark virtuality inside the hadron. The
perturbative mass-shell is defined by the following expression:
$$
\Delta E = 0.
$$
It is quite clear that the confined quark cannot reach the mass-shell
and there is a minimal displacement from the surface of free quark motion,
which is the nonperturbative quantity. So, we suppose that
$$
\Delta E \sim \Lambda_{QCD}.
$$
In what follows we apply the model with the quark dispersion law determined by
the form dictated by the account of gluon condensate in (\ref{main}):
\begin{equation}
p_0 = \omega_0+\frac{{\bf p}^2}{2 \tilde m},
\end{equation}
where again $\omega_0\sim \Lambda_{QCD}$ and $\tilde m$ denotes the effective
heavy quark mass, which differs from the perturbative pole mass due to the
contribution of gluon condensate. In the nonrelativistic rest frame we
have\footnote{In NRQCD, where $|{\bf p}|/m < 1$, the gluon condensate
correction to the heavy quark action $\Gamma$ tends to zero at large
virtualities $Q=\Delta E$ as $O(1/Q^2)$ and $O(1/Q^3)$ for the static and
dynamic terms, respectively. However, the correction remains small even at
lower scales.}
\begin{equation}
\Gamma = p_0 - \frac{{\bf p}^2}{2 m}+\frac{\pi^2}{24}\left\langle
\frac{\alpha_s}{\pi} G^2_{\mu\nu}\right \rangle \left [ \frac{{\bf
p}^2}{m^2\Delta E^3}+\frac{1}{m \Delta E^2}\right].
\label{main2}
\end{equation}
Then, we can derive that
\begin{equation}
\tilde m = m + \frac{\pi^2}{12}\left\langle
\frac{\alpha_s}{\pi} G^2_{\mu\nu}\right \rangle \frac{1}{\Delta E^3}.
\label{mm}
\end{equation}
Eq.(\ref{mm}) shows that at $\langle \frac{\alpha_s}{\pi} G^2_{\mu\nu}
\rangle\sim \Lambda_{QCD}^4$ the contribution of gluon condensate to the heavy
quark mass is about $\Lambda_{QCD}$, i.e. it is linear in the infrared scale of
energy, when the operator determining this term is the fourth power of scale.

Note, that the second term independent of ${\bf p}^2$ in gluon condensate
contribution shown in (\ref{main2}) results in the correction to the static
energy of heavy quark, so that
\begin{equation}
\delta \omega_0 = \frac{\pi^2}{24}\left\langle
\frac{\alpha_s}{\pi} G^2_{\mu\nu}\right \rangle \frac{1}{m \Delta E^2}.
\label{small}
\end{equation}
Furthermore, the gluon condensate contributes to $\omega_0$ in two ways: the
first one is explicitly given by (\ref{small}), the second is related with the
redefinition of heavy quark mass ($m\to \tilde m$). Indeed, in this case we
have to redefine the ``large'' momentum of heavy quark by the substitution for
$m v$ by $\tilde m v$ and so on, which means that the resulting change of
static energy is given by
$$
\Delta \omega_0 = \tilde m - m +\delta\omega_0 \sim
\Lambda_{QCD}\left(1+\ae \frac{\Lambda_{QCD}}{2m}\right), \;\;\; \ae\sim 1.
$$
Then, we can see that after the account for the gluon condensate the
displacement of static energy can be basically adopted in the mass $\tilde m$.

Furthermore, we can write down the following relations for the perturbative
dependence of heavy quark quantities on the scale $\lambda$:
\begin{equation}
\frac{d m^{\rm pert}}{d\lambda} = \frac{d \omega_0}{d\lambda} = \frac{\Delta
E}{d\lambda},
\label{pert}
\end{equation}
where in the second equality we neglect the dynamical term and remain the
static energy. Then the linear dependence on $\lambda$ in $m$ appears in to
ways: the first is the direct calculation of self-energy diagram for the heavy
quark, which results in
$$
\frac{d m^{(1)}}{d\lambda} = C_m \alpha_s \lambda,
$$
and the second is contributing from the gluon condensate term due to the
$\Delta E$ dependence according to (\ref{mm}) and (\ref{pert}) (the vacuum
condensate of gluon operator has the higher power: $\lambda^4$), so that 
$$
\frac{d m^{(2)}}{d\lambda} = - \frac{\pi^2}{4}\left\langle
\frac{\alpha_s}{\pi} G^2_{\mu\nu}\right \rangle \frac{1}{\Delta E^4}\;
C_m \alpha_s \lambda.
$$
Then, we see that at $\Delta E \approx \omega_0$ the heavy quark mass can be
physically independent on the introduction of factorization scale $\lambda$,
i.e. $\frac{d m}{d\lambda}=\frac{d m^{(1)}}{d\lambda}+\frac{d
m^{(2)}}{d\lambda}=0$, if
$$
\omega_0^4 = \frac{\pi^2}{4}\left\langle
\frac{\alpha_s}{\pi} G^2_{\mu\nu}\right \rangle.
$$
At $\langle \frac{\alpha_s}{\pi} G^2_{\mu\nu} \rangle\approx (0.37\;{\rm
GeV})^4$ \cite{7} the evaluation gives
$$
\omega_0\approx 0.46\;{\rm GeV.}
$$
Neglecting the dynamical term in the heavy quark virtuality we obtain the
following estimate of displacement for the heavy quark mass due to the gluon
condensate:
\begin{equation}
\Delta m \approx \frac{1}{3} \omega_0 \approx 0.16\; {\rm GeV,}
\label{strict}
\end{equation}
which can serve as the constrain of maximal value.

There is a small difference between the nonperturbative contributions to the
mass of heavy quark bound inside a heavy quarkonium $Q_1\bar Q_2$ and a heavy
meson $Q\bar q$ containing a single heavy quark. It is caused by the deviation
of $\Delta E$ from $\omega_0$. Indeed, in the heavy quarkonium we have the
following phenomenological regularity extracted from the approximately
identical arrangement of excitation levels in the systems with the various
quark contents: the kinetic energy of quarks is the flavor-independent value,
$$
T = \frac{\langle{\bf p}^2\rangle}{2 m_1}+\frac{\langle{\bf p}^2\rangle}{2 m_2}
\approx 0.4\;\; {\rm GeV.}
$$
So, in the system with the hidden flavor $Q_1=Q_2$ we can put
$\frac{\langle{\bf p}^2\rangle}{2 m} \approx 0.2\;\; {\rm GeV}$, which can
increase the minimal virtuality of heavy quark, and
$$
\Delta E\approx \omega_0+0.2\;\; {\rm GeV}\approx 0.66\;{\rm GeV}.
$$
At the chosen value of gluon condensate we find
$$
\Delta m =\tilde m-m \approx 0.05\;\;{\rm GeV},
$$
in the heavy quarkonium.

In the heavy meson with the single heavy quark, the appropriate parameter is
the average square of heavy quark momentum inside the hadron: $\mu_\pi^2=
\langle {\bf p}^2\rangle\sim 0.3\;\;{\rm GeV}^2$. Then, the flavor-dependent
contribution to $\Delta E$ results in
$$
\Delta E_c\approx 0.56\;\;{\rm GeV},\;\;\;\;
\Delta E_b\approx 0.49\;\;{\rm GeV},
$$
for the $c$ and $b$ quarks, respectively. Numerically we find
$$
\Delta m_c \approx 0.09\;\;{\rm GeV},\;\;\;\;
\Delta m_b \approx 0.13\;\;{\rm GeV}.
$$
We see that because the displacement of charmed quark from the mass-shell can
be greater than that of beauty quark, the nonperturbative contributions to
their masses are different, though we expect that the difference is not far
away from the accuracy of estimates. Of course, the increase of virtuality
$\Delta E$ due to the kinetic term is a phenomenological assumption, and the
most strictly justified estimate is (\ref{strict}).

Thus, the main statement on the nonperturbative displacement of heavy quark
masses remains the following: it is about the confinement scale. However, we
can get some definite estimates for these values.

To conclude, we have shown that the Operator Product Expansion including the
gluon condensate results in the following dispersion law for the heavy quark:
$$
p_0({\bf p}) = \omega_0 + \frac{{\bf p}^2}{2 m},
$$
where the correction to the heavy quark mass is given by 
$$
\Delta m = \frac{\pi^2}{12}\left\langle
\frac{\alpha_s}{\pi} G^2_{\mu\nu}\right \rangle \frac{1}{\omega_0^3},
$$
and the infrared ambiguity in the mass caused by the corresponding renormalon,
can be cancelled at
$$
\omega_0^4 = \frac{\pi^2}{4}\left\langle
\frac{\alpha_s}{\pi} G^2_{\mu\nu}\right \rangle.
$$
Of course, the conclusion is drawn to the given, linear order in $\alpha_s$,
and the well known divergency of heavy quark pole mass with the increase of
$\alpha_s$-order probably can be removed, if the higher order corrections to
the Wilson coefficient of gluon condensate as well as the higher condensates
will be included into the consideration in the same manner.

The author expresses the gratitude to A.L.Kataev for fruitful
discussions and valuable remarks. 

This work is in part supported by the Russian Foundation for Basic Research,
grants 99-02-16558 and 96-15-96575.

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