%Title: Corrections of order $\beta_0^3\alpha_s^3$ to the energy levels and  wave function 
%Author: Matthias Steinhauser
\documentclass[slac_one]{revtex4}
\usepackage{graphicx}
\usepackage{fancyhdr}
\pagestyle{fancy}
\fancyhead{} % clear all fields
\fancyfoot{} % clear all fields
%\fancyhead[C]{\it {2005 International Linear Collider Workshop -
%Stanford U.S.A.}}
\fancyfoot[LE,LO]{0403}
\renewcommand{\headrulewidth}{0pt}
\renewcommand{\footrulewidth}{0pt}
\renewcommand{\sfdefault}{phv}
\renewcommand{\baselinestretch}{1.2}
\setlength{\textheight}{235mm}
\setlength{\textwidth}{178mm}%{170mm}
\setlength{\topmargin}{-20mm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newcommand{\grtsim}{\mbox{\raisebox{-3pt}{$\stackrel{>}{\sim}$}}}
\newcommand{\lessim}{\mbox{\raisebox{-3pt}{$\stackrel{<}{\sim}$}}}

\newcommand{\note}[1]{{\tiny (note)}\marginpar {\scriptsize #1}}

\newcommand{\lmu}[1]{L_{#1}}
\newcommand{\sss}{S(S+1)}
\newcommand{\LrmuEuler}{L_r}
\newcommand{\Las}{L_{\alpha_s}}
\newcommand{\LE}[1]{L^E_{#1}}
\newcommand{\PolyEuler}[1]{P_{#1}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\begin{document}

%- {{{ Header:

%Title of paper
\title{{\small{2005 International Linear Collider Workshop - Stanford,
U.S.A.
\hfill TTP/05-09, SFB/CPP-05-28
}}\\ %% Please keep this conference title here
\vspace{12pt}
Corrections of order $\beta_0^3\alpha_s^3$ to the energy levels and
wave function} %% Paper title goes here

% Repeat the \author .. \affiliation  etc. as needed
%
% \affiliation command applies to all authors since the last
% \affiliation command. The \affiliation command should follow the
% other information

\author{M. Steinhauser}
\affiliation{Institut f{\"u}r Theoretische Teilchenphysik,
  Universit{\"a}t Karlsruhe, 76128 Karlsruhe, Germany}

\begin{abstract}

  One of the most important tasks of a future linear collider is the
  precise measurement of the threshold production cross section of top
  quark pairs. In order to match the expected experimental precision
  it is necessary to perform theoretical predictions which include
  corrections up to third order in perturbation theory. In this contribution
  we report about the corrections of order $\beta_0^3\alpha_s^3$ to the
  heavy quark spectrum and wave function. Physical applications both for
  the top and bottom quark system are discussed.

\end{abstract}

%\maketitle must follow title, authors, abstract
\maketitle

\thispagestyle{fancy}

% body of paper here - Use proper section commands
% References should be done using the \cite, \ref, and \label commands
% Put \label in argument of \section for cross-referencing
%\section{\label{}}

%- }}}
%- {{{ Motivation:

\section{Motivation}

A future linear collider running at a center-of-mass energy close to
the production threshold of a top quark pair allows for a precise 
determination of the mass and width of the top
quark, the strong coupling and --- in case the Higgs boson is not too
heavy --- also of the top quark Yukawa coupling.
In order to match the expected experimental
precision~\cite{Martinez:2002st}  
it is important to compute higher-order quantum corrections to this process.
In this context we refer to the 
Refs.~\cite{Hoa_etal,KniPenSmiSte,PenSte,KPSS2,Hoa}.


The ground state energy to the heavy quarkonium system 
has been computed in Ref.~\cite{KniPenSmiSte,PenSte}
through ${\cal O}(\alpha_s^5m_q)$ including the third-order correction
to the Coulomb approximation. The result has been used to
extract $m_b$ from the $\Upsilon(1S)$ meson mass and to derive a
formula which relates the top quark mass to the maximum of the cross
section below the nominal threshold. The properties of the
excited states are more sensitive to the nonperturbative phenomena, and
the corresponding perturbative estimates cannot be used, {\it e.g.}, for
the accurate determination of the heavy-quark mass by direct comparison to
the meson masses.  However, they have to be taken into account in the
framework of the nonrelativistic sum rules~\cite{NSVZ} which is based on
the concept of quark-hadron duality and keeps the nonperturbative
effects under control.
For the practical analysis only a few states
with small principal quantum numbers $n$ and zero orbital momentum $l$
are of interest.

A further interesting aspect of investigating the excited states
with reliable perturbative results at hand is the possibility
to test the effects and
structure of the nonperturbative QCD vacuum.

In this contribution we report about the corrections of order
$\beta_0^3\alpha_s^3$ to the energy levels for $n=1,2$ and 
$3$~\cite{PSS,BKS}.
In combination with Ref.~\cite{PenSte} this
leads to complete third-order results for the energy levels.

As far as the wave function at the origin is concerned 
a complete result is only
available through ${\cal O}(\alpha_s^2)$ \cite{PenPiv1,MelYel1}.  The
${\cal O}(\alpha_s^2)$ correction has turned out to be so sizeable that
the application of perturbation theory for a system of two heavy
quarks seemed to be questionable ---  even for top quarks.
Thus it is indispensable to gain full control over the
next order. Logarithmically enhanced contributions of order
${\cal O}(\alpha_s^3\ln^2\alpha_s)$ and
${\cal O}(\alpha_s^3\ln\alpha_s)$
have been computed in Refs.~\cite{KniPen2,ManSte}
and~\cite{KPSS2,Hoa}, respectively.
In Refs.~\cite{PSS,BKS} the next step has been taken and 
contributions of order $\beta_0^3\alpha_s^3$ have been evaluated.
Due to the large numerical value of $\beta_0$ these corrections are
supposed to be numerically dominant.

%- }}}
%- {{{ Energy levels and wave functions to ${\cal O}(\beta_0^3\alpha_s^3)$:

\section{Energy levels and wave functions to ${\cal O}(\beta_0^3\alpha_s^3)$}
\label{sec1}

In the framework of nonrelativistic effective theory
\cite{CasLep,BBL,PinSot1,BenSmi} the corrections to the heavy quarkonium
parameters are obtained by evaluating the corrections to the Green
function of the effective Schr\"odinger equation \cite{KniPenSmiSte}.
The $\beta_0^3$ part of the third-order contribution, 
which is the main purpose of this contribution,
results from the 
leading renormalization group running of the static potential, $V_C(r)$.
The coordinate space representation of $V_C$ can, e.g., be found in
Ref.~\cite{PSS}. At $n^{\rm th}$ order it contains $\beta_0^i$
terms ($i=1,\ldots,n$)
which are accompanied by logarithms $\ln(\mu r)$ raised to power
$1,\ldots,i$.
At third order in perturbation theory 
one has to consider single iterations of the $\beta_0^3$ term,
double iterations of the $\beta_0^2$ and $\beta_0$ term and triple
iterations of the first-order corrections proportional to $\beta_0$.
For the practical computation we use the method elaborated in
Refs.~\cite{KPP,PenPiv1,PenPiv2}.  In this way we obtain the corrections
to the energy levels and wave function at the origin in the form of
multiple harmonic sums.  For general $n$ the result is rather
cumbersome. However, for a specific $n$ the summation can be performed
analytically.

As already mentioned in the Introduction, for practical purpose
in the context of sum rules it is certainly sufficient to 
have results for $n=1,~2,~3$ and $l=0$. For this reason,
in Ref.~\cite{PSS} the analytical results have been presented
for these values.
In Ref.~\cite{BKS} a more general formula has been derived, which however
still contains unevaluated sums. After specifying $n$ the same results
as in Ref.~\cite{PSS} are obtained.

Let us for completeness present the new results.
For vanishing angular momentum the perturbative part of the energy level
with principal quantum number $n$ can be written as
\begin{eqnarray}
  E_n^{\rm p.t.}&=&E^C_n+\delta E^{(1)}_n+\delta E^{(2)}_n+\delta
  E^{(3)}_n
  +\ldots\,,
  \label{enseries}
\end{eqnarray}
where $\delta E^{(k)}_n$ stands for corrections of order $\alpha_s^k$ and
the leading order Coulomb energy is given by
$E^C_n=-{C_F^2\alpha_s^2m_q/(4n^2)}$.
For the ${\cal O}(\beta_0^3\alpha_s^3)$ term we obtain
\begin{eqnarray}
  \delta^{(3)}_{\beta_0^3}  E_1&=&
  E^C_1\left({\beta_0\alpha_s\over \pi}\right)^3
  \Bigg[32 \lmu{1}^3 + 40 \lmu{1}^2 + \left({16\pi^2\over 3}+64\zeta(3)
    \right)\lmu{1}
%    \nonumber\\&&
    -8+{4\pi^2}+{2\pi^4\over 45}
    +64\zeta(3)-{8\pi^2\zeta(3)}+96\zeta(5)\Bigg]\,, 
  \nonumber\\
  \delta^{(3)}_{\beta_0^3}  E_2
  &=&E^C_2\left({\beta_0\alpha_s\over \pi}\right)^3
  \Bigg[32 \lmu{2}^3+{88}\lmu{2}^2+\left({32}+{16\pi^2\over 3}
    +128\zeta(3)\right)\lmu{2}
    -{102}+{52\pi^2\over 3}
    +{4\pi^4\over 45}+112\zeta(3)
    \nonumber\\&&
    -{32\pi^2\zeta(3)}+384\zeta(5)\Bigg]\,, 
  \nonumber\\
  \delta^{(3)}_{\beta_0^3}  E_3&=&
  E^C_3\left({\beta_0\alpha_s\over \pi}\right)^3
  \Bigg[32 \lmu{3}^3+{120}\lmu{3}^2
    +\left({136\over 3}+{16\pi^2\over 3}
    +192\zeta(3)\right)\lmu{3}
    -{9514\over 27}    
    +{427\pi^2\over 9}+{2\pi^4\over 15}+{140\zeta(3)}
    \nonumber\\&&
    -{72\pi^2\zeta(3)}
    +864\zeta(5)\Bigg]\,,
\label{Ebt3}
\end{eqnarray}
where $\lmu{n}= \ln(n\mu/(C_F\alpha_s(\mu)m_q))$ 
and $\zeta(i)$ is Riemann's $\zeta$ function.
The remaining contributions to $\delta E_n^{(3)}$ can be found in
Ref.~\cite{PSS}. 


The perturbative expansion for the wave
function can be written as follows
\begin{eqnarray}
  |\psi_n(0)|^2&=&|\psi^C_n(0)|^2\left(1+\delta^{(1)}\psi_n
  +\delta^{(2)}\psi_n+\delta^{(3)}\psi_n+\ldots\right)\,,
\end{eqnarray}
where $|\psi^C_n(0)|^2={C_F^3\alpha_s^3 m_q^3/(8\pi n^3)}$
is  the leading order Coulomb value. Our result for the
${\cal O}(\beta_0^3\alpha_s^3)$ term reads
\begin{eqnarray}
  \delta^{(3)}_{\beta_0^3} \psi_1&=&\left({\beta_0\alpha_s\over
    \pi}\right)^3
  \left[80\lmu{1}^3+\left({52}-{80\pi^2\over 3}\right)\lmu{1}^2
    +\left(-40-{6\pi^2}+{10\pi^4\over 9}
    +200\zeta(3)\right)\lmu{1}\right.
    \nonumber\\
    &&
    -{20}+{22\pi^2\over 3}
    -{7\pi^4\over 5}+{4\pi^6\over 105}
    +112\zeta(3)-{12\pi^2\zeta(3)}-16\zeta(3)^2-40\zeta(5)\Bigg]
    \,, 
    \nonumber\\
    \delta^{(3)}_{\beta_0^3} \psi_2&=&\left({\beta_0\alpha_s\over
      \pi}\right)^3 
    \left[80\lmu{2}^3+\left({332}-{160\pi^2\over 3}\right)\lmu{2}^2
      +\left({308}-{266\pi^2\over 3}+{40\pi^4\over 9}
      +400\zeta(3)\right)\lmu{2}\right.
      \nonumber\\
      &&
      -{361}+{73\pi^2\over 3}
      -{26\pi^4\over 45}+{32\pi^6\over 105}
      +496\zeta(3)-{48\pi^2\zeta(3)}-128\zeta(3)^2-160\zeta(5)\Bigg]
      \,, 
      \nonumber\\
      \delta^{(3)}_{\beta_0^3} \psi_3&=&
      \left({\beta_0\alpha_s\over \pi}\right)^3 
      \left[80\lmu{3}^3+\left({612}-80\pi^2\right)\lmu{3}^2
        +\left({2893\over 3}-{228\pi^2}+{10\pi^4}
        +600\zeta(3)\right)\lmu{3}\right.
        \nonumber\\
        &&
        -{100679\over 54}+{183\pi^2\over 2}
        +{52\pi^4\over 15}+{36\pi^6\over 35}
        +{1374\zeta(3)}
	-{108\pi^2\zeta(3)}
	-432\zeta(3)^2
        \nonumber\\
        &&-360\zeta(5)\Bigg]
        \,.
\end{eqnarray}

In the next Section we discuss the numerical impact of the new terms
to the system of two heavy quarks at threshold.

%- }}}
%- {{{ Phenomenological applications:

\section{Phenomenological applications}
\label{sec3}

%- {{{ Excited states of bottomonium and charmonium

\subsection{Excited states of bottomonium}
The mass of the $\Upsilon(nS)$ meson can be decomposed into perturbative
and nonperturbative contributions
\begin{eqnarray}
  M_{\Upsilon(nS)} &=&
  2m_b + E_n^{\rm p.t.}+ \delta^{\rm n.p.}E_n\,.
  \label{mqq}
\end{eqnarray}
As discussed in the previous Section, the perturbative 
contribution $E_n^{\rm p.t.}$ is know up to ${\cal O}(m_q\alpha_s^5)$.
For $n=1$ the phenomenological application of the 
result to the $\Upsilon(1S)$ meson mass 
has been discussed in Ref.~\cite{PenSte} and a competitive extraction
of the bottom quark mass has been obtained.
It is thus tempting to investigate Eq.~(\ref{mqq}) also for excited
states. In this context it is convenient to consider the ratio
\begin{eqnarray}
  \rho_n&=&{E_n-E_1\over 2m_b+E_1}\,,
  \label{rho}
\end{eqnarray}
which depends only logarithmically on the bottom quark mass
and does not suffer from renormalon contributions.

Including successively higher orders one obtains for $\mu=\mu_s=2.1$~GeV
\begin{eqnarray}
 10^2 \times\rho^{\rm p.t.}_2 &=& 1.49\left(
  1 +0.79_{\rm NLO} + 1.18_{\rm NNLO} + 1.21_{\rm N^3LO}+\ldots
  \right)
  \,,
  \nonumber\\
 10^2 \times\rho^{\rm p.t.}_3  &=& 1.77\left(
  1 + 0.92_{\rm NLO} + 1.37_{\rm NNLO} + 1.55_{\rm N^3LO}+\ldots
  \right)
  \,.
  \label{eq::rho}
\end{eqnarray}
In Tab.~\ref{tab1} the sum of the individual orders is compared
to the experimental
result given by $\rho^{\rm exp}_n =(M_{\Upsilon(nS)}-M_{\Upsilon(1S)})/
M_{\Upsilon(1S)}$.
It is worth mentioning that although the convergence of the series 
is not good, the N$^3$LO perturbative result is in impressive
agreement with the experimental values both for $n=2$ and $n=3$.
We would like to emphasize the role of the perturbative corrections
necessary to bring theory and experiment into agreement. 
From this observation one can conclude that the magnitude of the
nonperturbative effects for the excited state, $\delta^{\rm n.p.}E_2$,
is of the same size as $\delta^{\rm n.p.}E_1$.
The latter was estimated to $\approx60$~MeV in Ref.~\cite{PenSte}.
Similar conclusion has been made in Ref.~\cite{BVS} in a somewhat
different framework.

\begin{table}[b]
\begin{center}
\renewcommand{\arraystretch}{1.2}
\begin{tabular}{|l||c|c|}
  \hline
  & $\Upsilon(2S)$ & $\Upsilon(3S)$ \\
  \hline
  $ 10^2 \times\rho^{\rm p.t.}_n$ 
  & $6.2^{+1.7}_{-1.2}$ & $8.6^{+2.4}_{-1.8}$ \\
  $ 10^2 \times\rho^{\rm exp}_n$  
  & $5.95$& $9.46$\\
  \hline
\end{tabular}
\caption{\label{tab1}
  Perturbative versus experimental results for the parameter $\rho_n$
  as defined in Eq.~(\ref{rho}).
  The theoretical uncertainty corresponds to
  $\alpha_s(M_Z)=0.118\pm 0.003$.}
\end{center}
\end{table}

%- }}}
%- {{{ $\Upsilon$ leptonic width

\subsection{$\Upsilon(1S)$ leptonic width}

It is tempting to use the new corrections to the wave function in
order to predict the decay rate of the $\Upsilon(1S)$ meson into
leptons.
In the nonrelativistic effective theory the leading order approximation
reads
\begin{eqnarray}
  \Gamma^{\rm LO}(\Upsilon(1S)\to l^+l^-) &\equiv&
  \Gamma_1^{\rm LO} 
  \,\,=\,\,
  \frac{4\pi N_cQ_b^2\alpha^2|\psi_1^C(0)|^2}{3m_b^2}
  \,,
\end{eqnarray}
with $N_c=3$ and $Q_b=-1/3$. Combining the known
perturbative results up to ${\cal O}(\alpha_s^3\ln\alpha_s)$ (see
Ref.~\cite{KPSS2}) with the ${\cal O}(\beta_0^3\alpha_s^3)$ contribution
discussed in Section~\ref{sec1} we find
\begin{eqnarray}
  \Gamma_1 &\approx& \Gamma_1^{\rm LO}(1 -0.445_{\rm NLO}
  +1.75_{\rm NNLO}-1.67_{\rm N^3LO^\prime}+\ldots)
  \,,
  \label{gamnum}
\end{eqnarray}
where the prime indicates that the N$^3$LO corrections are not complete.
Though the perturbative corrections are huge, the rapid growth of the
perturbative coefficients stops at NNLO if we assume that the
$\beta_0^3$ term sets the scale of the nonlogarithmic third-order
contribution.

In Fig.~\ref{fig::mu}(a), the width is plotted as a
function of $\mu$ including the LO, NLO, NNLO and N$^3$LO$'$ approximations
along with the experimental value.  
As one can see the available ${\cal O}(\alpha_s^3)$ 
terms stabilize the series and significantly 
reduce the scale
dependence.  At the scale $\mu'\approx 2.7$~GeV, which is close
to the physically motivated scale $\mu_s$, the N$^3$LO$'$ corrections
vanish and at the scale $\mu''\approx 3.1$~GeV the result becomes
independent of $\mu$; {\it i.e.}, the N$^3$LO$'$ curve shows a local
maximum. In the whole range of $\mu$ between approximately 
2~GeV and 5~GeV the result
for the width agrees with the experimental value within the error bar
due to the uncertainty of the strong coupling constant. This may
signal that the missing perturbative corrections are rather moderate.
Furthermore, this result constitutes a significant improvement as
compared to the NLL approximation discussed in
Ref.~\cite{Pineda:2003be}.

For a definite conclusion, however,
one has to wait until the third-order corrections are completed.  The
potentially most important part to be computed is
the ultrasoft contribution which includes $\alpha_s(\mu)$ normalized at
relatively low ultrasoft scale $\mu_{us}\sim \alpha_s^2m_q$.  Currently
only a partial result for this contribution exists \cite{KniPen1}.

\begin{figure*}[t]
  \centering
  \begin{tabular}{cc}
    \includegraphics[width=80mm]{bot_mu.eps}
    &
    \includegraphics[width=80mm]{top_mu.eps}
    \\
    (a) & (b)
  \end{tabular}
  \caption{\label{fig::mu} (a) $\Gamma_1$ normalized to 
    $\hat{\Gamma}_1\equiv\Gamma_1^{\rm LO}\big|_{\alpha_s\to \alpha_s(\mu_s)}$ 
    as a function of $\mu$ at LO (dotted), NLO (dashed), NNLO
    (dotted-dashed) and N$^3$LO$'$ (full line).  The horizontal line
    corresponds to the experimental value $\Gamma^{\rm
      exp}(\Upsilon(1S)\to e^+e^-)=1.31$~keV.
    For the N$^3$LO$'$ result, the band
    reflects the errors due to $\alpha_s(M_Z)=0.118\pm 0.003$.  
    (b) The analog plot for $R_1$ with 
    $\hat{R}_1\equiv R_1^{\rm LO}\big|_{\alpha_s\to \alpha_s(\mu_s)}$.}
\end{figure*}

%- }}}
%- {{{ Top quark threshold production.

\subsection{Top quark threshold production}

In contrast to the bottom system 
the nonperturbative effects in the case of the top quark are negligible.
However, due to the relatively large top quark width, $\Gamma_t$,
its effect has to be taken into account properly~\cite{FadKho}
since the Coulomb-like resonances below threshold are smeared out.
Actually, the cross section only shows a small bump which is
essentially the remnant of the ground state pole.
The higher poles and
continuum, however, affect the position of the resonance peak and move
it to higher energy. The value of the normalized cross section
$R=\sigma(e^+e^-\to t\bar t)/\sigma(e^+e^-\to\mu^+\mu^-)$ at the
resonance energy is dominated by the contribution from the {\it  would-be} 
toponium ground state which in the leading approximation
reads $R_1^{\rm LO}=6\pi
N_cQ_t^2|\psi_1^C(0)|^2/\left(m_t^2\Gamma_t\right)$, where $Q_t=2/3$.
Numerically we find
\begin{eqnarray}
  R_1 &\approx& R_1^{\rm LO}(1 -0.243_{\rm NLO}
  +0.435_{\rm NNLO}-0.268_{\rm N^3LO^\prime}+\ldots)
  \,.
  \label{rnum}
\end{eqnarray}
The new third-order corrections proportional to $\beta_0^3$ amount to
approximately $-7\%$ of the LO approximation at the soft scale which is
the same order of magnitude as the ${\cal O}(\alpha_s^3)$ linear
logarithmic term.  The available N$^3$LO terms improve the stability of
the result with respect to the scale variation as can be seen in
Fig.~\ref{fig::mu}(b).  The absence of a rapid growth of the
coefficients along with the alternating-sign character of the series and
the weak scale dependence suggest that the missing perturbative
corrections are moderate and most likely are in the few-percent range.  It
is interesting to note that the perturbative contributions of different
orders, which are relatively large when taken separately, cancel in the
sum to give only a few percent variation of the leading order result.
  
%- }}}

%- }}}
%- {{{ Acknowledgments:

% If you have acknowledgments, this puts in the proper section head.
\begin{acknowledgments}
I would like to thank Alexander Penin and Vladimir Smirnov for the 
pleasant and fruitful collaboration.
This work was supported by the ``Sonderforschungsbereich/Transregio~9''
and by the ``Impuls- und Vernetzungsfonds'' of the
Helmholtz Association, contract number VH-NG-008.
\end{acknowledgments}

%- }}}
%- {{{ References

\begin{thebibliography}{99}

%\include{refs}

\bibitem{Martinez:2002st}
M.~Martinez and R.~Miquel,
%``Multi-parameter fits to the t anti-t threshold observables at a future  e+ e-
%linear collider,''
Eur.\ Phys.\ J.\ C {\bf 27} (2003) 49.
%[arXiv:hep-ph/0207315].
%%CITATION = HEP-PH 0207315;%%

\bibitem{Hoa_etal}
A. H. Hoang, M. Beneke, K. Melnikov, T. Nagano, A. Ota, A. A. Penin,
A. A. Pivovarov, A. Signer, V. A. Smirnov, Y. Sumino, T. Teubner,
O. Yakovlev, and A. Yelkhovsky,
Eur.\ Phys.\ J. direct C { 3} (2000) 1.

\bibitem{KniPenSmiSte} 
B.A. Kniehl, A.A. Penin, 
V.A. Smirnov, and M. Steinhauser, Nucl.\ Phys.\ {\bf B635} (2002) 357.

\bibitem{PenSte} A.A. Penin and M. Steinhauser,
Phys.\ Lett.\ B {538} (2002) 335.

\bibitem{KPSS2} B.A. Kniehl, A.A. Penin, V.A. Smirnov, and M. Steinhauser,
Phys.\ Rev.\ Lett.\ {90}  (2003) 212001;
Erratum {\it ibid.}\ {91} (2003) 139903.

\bibitem{Hoa} A.H. Hoang, Phys.\ Rev.\ D {69} (2004) 034009.

\bibitem{NSVZ} V.A. Novikov, L.B. Okun, M.A. Shifman, 
A.I. Vainshtein, M.B. Voloshin, and
V.I. Zakharov, Phys.\ Rev.\ Lett.\ { 38} (1977) 626;
Phys.\ Rev.\ Lett.\ { 38} (1977) 791, Erratum;
Phys.\ Rep.\ C { 41} (1978) 1.

\bibitem{PSS} A.A.~Penin, V.A.~Smirnov, and M. Steinhauser,
  Nucl.\ Phys.\ B {\bf 716} (2005) 303.
%%%  [arXiv:hep-ph/0501042].

\bibitem{BKS}
  M.~Beneke, Y.~Kiyo, and K.~Schuller,
  Nucl.\ Phys.\ B {\bf 714} (2005) 67.


\bibitem{PenPiv1} A.A. Penin and A.A. Pivovarov,
Phys.\ Lett.\ B { 435} (1998) 413;
Nucl.\ Phys.\ B { 549} (1999) 217.

\bibitem{MelYel1} K. Melnikov and A. Yelkhovsky,
Phys.\ Rev.\ D { 59} (1999) 114009.

\bibitem{KniPen2} B.A. Kniehl and A.A. Penin,
Nucl.\ Phys.\ {B 577} (2000) 197.
\bibitem{ManSte}
A.V. Manohar and I.W. Stewart,
Phys.\ Rev.\ D {63} (2001) 054004.

\bibitem{PinSot1} A. Pineda and J. Soto,
Nucl.\ Phys.\ Proc.\ Suppl. {64}  (1998) 428;\\
N. Brambilla, A. Pineda, J. Soto, and A. Vairo,
Nucl.\ Phys.\ {B 566} (2000) 275.

\bibitem{CasLep} W.E. Caswell and G.P. Lepage,
Phys.\ Lett.\ B {167} (1986) 437.

\bibitem{BBL} G.T. Bodwin, E. Braaten, and G.P. Lepage,
Phys.\ Rev.\ D {51} (1995) 1125; Erratum {\it ibid.}\ {55} (1997) 5853.

\bibitem{BenSmi} M. Beneke and V.A. Smirnov,
Nucl.\ Phys.\ {B 522} (1998) 321.

\bibitem{KPP} J.H. K\"uhn, A.A. Penin, and A.A. Pivovarov,
Nucl.\ Phys.\ {B 534} (1998) 356.

\bibitem{PenPiv2} A.A. Penin and A.A. Pivovarov,
Nucl.\ Phys.\ {B 550} (1999) 375;
Yad.\ Fiz.\ {64} (2001) 323
[Phys.\ Atom.\ Nucl.\ {64} (2001) 275].

\bibitem{BVS}  N. Brambilla, Y. Sumino, and  A. Vairo,
Phys.\ Lett.\ {B513} (2001) 381.

\bibitem{Pineda:2003be}
A.~Pineda,
Acta Phys.\ Polon.\ B {34} (2003) 5295.

\bibitem{KniPen1} B.A. Kniehl and A.A. Penin,
Nucl.\ Phys.\ {B 563} (1999) 200.

\bibitem{FadKho} V.S. Fadin and V.A. Khoze,
Pis'ma Zh.\ Eksp.\ Teor.\ Fiz.\ { 46} (1987) 417
[JETP Lett.\ { 46} (1987) 525].



\end{thebibliography}

%- }}}

\end{document}
