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\journal{Nuc. Phys. (Proc. Suppl.)}

\begin{document}

\begin{frontmatter}

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\title{Heavy-quark masses from low-energy moments of heavy-quark currents}

%% use optional labels to link authors explicitly to addresses:
 \author{Peter Marquard}
  \address{Institut f\"ur Theoretische Teilchenphysik\\ Karlsruher Institut
    f\"ur Technologie (KIT) \\ 76131 Karlsruhe }
%\cortext[cor1]{Speaker}
\ead{peter.marquard@kit.edu}


\begin{abstract}
%% Text of abstract
\noindent
We present the newest results for the determination of heavy-quark
masses from low-energy moments of the vacuum polarization function. The
results presented here update an earlier determination of the quark
masses and incorporate new theoretical calculations of the low-energy
moments of the vacuum polarization and new experimental data at the
bottom threshold. 

\end{abstract}

\begin{keyword}
heavy-quark masses\sep sum rules
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\end{frontmatter}

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\section{Introduction}
The precise determination of charm and bottom quark masses has always been an 
important task both for theory and experiment. The most precise values have
been obtained \cite{Chetyrkin:2009fv} from an analysis of the ITEP sum rules
\cite{Novikov:1977dq} 
(for reviews see Refs.~\cite{Reinders:1984sr,Colangelo:2000dp,Ioffe:2005ym}), 
combining data for the heavy-quark production cross section in
electron-positron collision with dispersion relations and a four-loop
evaluation of the vacuum polarization induced by the heavy quark
current. In this contribution,
we summarize the most recent progress, which includes data recently
published by the BABAR collaboration \cite{:2008hx} and new perturbative 
results.

\section{Analytic Results}
The determination of the heavy quark masses in \cite{Chetyrkin:2009fv}
follows closely Refs.~\cite{Kuhn:2007vp,Kuhn:2001dm,Kuhn:2002zr}.  It is
based on the direct comparison of the theoretical and experimental
evaluations of the contributions to the derivatives of the polarization
function $\Pi_Q(q^2)$, the former evaluated in perturbative QCD, the
latter through moments of the measured cross section for heavy-quark
production in electron-positron annihilation. Using dispersion
relations, the moments of $R_Q$~\footnote{For the precise definition of
  $R_Q$, in particular the treatment of gluon splitting into $Q\bar Q$,
  the subtraction of singlet contributions, and the role of
  nonperturbative terms in the case of charm quarks we refer to
  Ref.~\cite{Kuhn:2007vp}.},
\begin{equation}
  {\cal M}_n \equiv \int \frac{{\rm d}s}{s^{n+1}} R_Q(s)
  \,,
  \label{eq:Mexp}
\end{equation}
can be related to the derivatives of the vacuum polarization function at $q^2=0$,
\begin{equation}
  {\cal M}_n = \frac{12\pi^2}{n!}
  \left(\frac{{\rm d}}{{\rm d}q^2}\right)^n
  \Pi_Q(q^2)\Bigg|_{q^2=0}
  \,.
\label{eq:Mtheo}
\end{equation}
 In its domain of
analyticity $\Pi_Q(q^2)$ can be cast into the form
\begin{equation}
  \Pi_Q(q^2) = Q_Q^2 \frac{3}{16\pi^2} \sum_{n\ge0}
                       \bar{C}_n z^n
  \,,
  \label{eq:pimom}
\end{equation}
with $z=q^2/(4m_Q^2)$. Here  $m_Q=m_Q(\mu)$ is the heavy quark
mass with charge $Q_Q$ in the $\overline{\rm{MS}}$ scheme at the scale
$\mu$. The coefficients 
$\bar C_n$ depend on $\alpha_s$ and 
on the heavy quark mass through logarithms of the form $l_{m_Q} = \ln
(m_Q^2(\mu)/\mu^2)$. Equating theoretically calculated and
experimentally measured moments, the heavy quark mass is given by 
\begin{equation}
  m_Q(\mu) = \frac{1}{2}
  \left( \frac{9 Q_Q^2\bar{C}_n}{4 {\cal M}_n^{\rm exp}}\right)^{1/(2n)} 
  \,.
  \label{eq:m_Q}
\end{equation}
As a perturbative series the coefficients $\bar{C}_n$ can be written as
\begin{equation}
  \begin{split}
    \bar{C}_n =&\, \bar{C}_n^{(0)} + \frac{\alpha_s(\mu)}{\pi} \left(
      \bar{C}_n^{(10)} + \bar{C}_n^{(11)}l_{m_Q} \right)
    \\&
    +\left(\frac{\alpha_s(\mu)}{\pi}\right)^2 \left( \bar{C}_n^{(20)}
      +\bar{C}_n^{(21)}l_{m_Q}
      + \bar{C}_n^{(22)}l_{m_Q}^2 \right)
    \\&
    + \left(\frac{\alpha_s(\mu)}{\pi}\right)^3 \left( \bar{C}_n^{(30)} 
      + \bar{C}_n^{(31)}l_{m_Q} 
      + \bar{C}_n^{(32)}l_{m_Q}^2 
    \right.\\&\left.\mbox{}\qquad
      + \bar{C}_n^{(33)}l_{m_Q}^3 \right)
    + \ldots \,.
  \end{split}
% \nonumber\\
  \label{eq:cn}
\end{equation}
The terms of order $\alpha_s^2$ were evaluated up to $n=8$ in
Refs.~\cite{Chetyrkin:1995ii,Chetyrkin:1996cf,Chetyrkin:1997mb} and even
up to n=30 in Refs.~\cite{Boughezal:2006uu,Maier:2007yn}. The four-loop
contributions to $\bar C_0$ and $\bar C_1$ were calculated in
Refs.~\cite{Chetyrkin:2006xg,Boughezal:2006px}.  For the higher moments
the analysis of \cite{Kuhn:2007vp} was based on estimates for $\bar
C_n^{(30)}$ with $n=2,~3,~4$, which lead to an additional uncertainty in
the mass determination. Recently, the exact results for the
second~\cite{Maier:2008he} and third~\cite{Maier:2009fz} moments were
obtained. Combining these coefficients with additional information on
the threshold and the high-energy behaviour and using the analyticity of
$\Pi_Q(q^2)$ and Pad\'e approximations, fairly precise numerical results
were obtained~\cite{Kiyo:2009gb} for the higher coefficients up to
$n=10$. (For an earlier analysis along similar lines see
Ref.~\cite{Hoang:2008qy}.) For the lowest four moments the four-loop
coefficients $\bar C_n^{(30)}$ are listed in Tab.~\ref{tab:tabc} both
for the charm and the bottom quark. Using a different approach based on
threshold resummation the low-energy expansion coefficients have also
been estimated in \cite{Greynat:2010kx}. All determinations agree within
the error, with \cite{Kiyo:2009gb} giving the smallest errors. A
comparison between the works is shown in Tab. \ref{tab:compC}.  It
should be emphasized that these results are well within the estimates
used in the analysis of \cite{Kuhn:2007vp}.
\begin{table}
  \centering
{\scalefont{0.9}  \begin{tabular}{ccccc}
    \hline
    &$\bar{C}^{(30)}_1$&$\bar{C}^{(30)}_2$&$\bar{C}^{(30)}_3$&$\bar{C}^{(30)}_4$\\\hline
    charm&-5.6404&-3.4937&-2.8395&-3.349(11)\\
    bottom&-7.7624&-2.6438&-1.1745&-1.386(10)\\\hline
  \end{tabular}}
  \caption{Lowest four expansion coefficients $\bar C_n^{(30)}$ for charm and bottom
    quarks. The first three coefficients are known analytically, the
    next is known with good accuracy from a Pad\'e approximation \cite{Kiyo:2009gb}.}\label{tab:tabc}
\end{table}
\begin{table}
  \centering
{\scalefont{0.9}  $\begin{array}{llll}
    \hline
%    & \bar{C}^{(3)}_3
& \bar{C}^{(3)}_4 & \bar{C}^{(3)}_5 & \bar{C}^{(3)}_6 %&\bar{C}^{(3)}_7 & K^{(3)}_0 
\\\hline
    \text{Hoang et al}
%& -\text{3.28}\pm\text{0.57}
& -\text{4.2}\pm\text{1.2}  & -\text{5.0}\pm\text{1.7}
    & -\text{5.3}\pm\text{2.0} %& -\text{5.2}\pm\text{2.3} & -10\pm11
    \\
    \text{Greynat, Peris}
%&-\text{2.840}\text{ (exakt)}
& -\text{3.6}\pm\text{0.5}  & -\text{4.4}\pm\text{1.2}
    & -\text{4.7}\pm\text{1.8} %& -\text{4.6}\pm\text{2.3} &\phantom{-}18\pm87
    \\
\text{Kiyo et al}
%&-\text{2.840}\text{ (exakt)}
& -\text{3.349}(11)    &-\text{3.737}(32)
    & -\text{3.735}(61) %& -\text{3.39}(10)  & \phantom{-}17(11)
\\\hline
  \end{array}$ }
  \caption{Comparison of different estimates for the low-energy expansion coefficients $\bar C_n^{(30)}$. }
  \label{tab:compC}
\end{table}
The
impact of these new results on the quark mass determination will be  
shown below. 

\section{Bottom Production Close to Threshold}
\label{sec:bot_prod}


The determination of the bottom quark mass, as performed in
\cite{Kuhn:2007vp,Kuhn:2001dm} relies heavily on the precise measurement of 
 $R=\sigma(e^+e^-\to {\rm hadrons})/\sigma_{\text{pt}}$
 (with $\sigma_{\text{pt}} =  \frac{4\pi\alpha^2}{3s}$),
which enters the moments as defined above. 
Specifically, it is the contribution from the heavy quark current
denoted as $R_b$ with the light-quark contribution
subtracted.
It is convenient to split the integration region 
into three pieces: The lowest region covering the narrow 
resonances, an intermediate ``threshold'' region between 10.62~GeV
and 11.24~GeV, and the perturbative region above 11.24~GeV, where the
measurement is replaced by the perturbative QCD prediction. The choice
of 11.24~GeV corresponds to the upper end of the energy range covered by a
CLEO measurement 
more than 20 years ago \cite{Besson:1984bd}. It also coincides approximately
with the 
energy reach of a recent BABAR measurement \cite{:2008hx}. In the analysis of \cite{Kuhn:2007vp},
$\Upsilon(4S)$ with its mass
$M_{\Upsilon(4S)}=10.5794(12)$~GeV and width $\Gamma_{\Upsilon(4S)}=20.5$~MeV
has been considered together 
with the three lower, narrow resonances and thus the continuum part of the
bottom cross section was taken from 10.62 GeV upwards. Until recently the 
only measurement in the threshold region has been the one from the CLEO
collaboration, which quotes a systematic error of about 6\%. No radiative
corrections had been applied. In Ref.~\cite{Kuhn:2007vp} it has been argued, that a
normalization factor 1/1.28 is necessary to reconcile these data
with more recent and more precise CLEO results below the $\Upsilon(4S)$-resonance 
and with perturbative QCD at the high end. These ``rescaled'' data
were the basis of the subsequent extraction of the bottom quark mass.
However, in view of these uncertainties an overall systematic error of
10\% was attributed to the contribution of the moments from this region.
Thus, although this contribution to the moments is relatively small, its
impact on the error was larger or equal than the one from the other 
two regions combined.


Recently a measurement of $R_b$ in the energy region between 10.54~GeV and
11.20~GeV was performed by the BABAR collaboration
with significantly improved statistics and with a correlated systematic
error between 2.5\% and 3\% \cite{:2008hx}. This allows an independent
determination of the contribution to the moments with significantly
reduced systematic error.
However, no radiative corrections were applied to the published data and the
 radiative tails of the four lower $\Upsilon$ resonances were included in the
 quantity denoted $R_b$. Therefore, the data has to be corrected for
 these effects. The details of this procedure can be found in \cite{Chetyrkin:2009fv}.

% In a first step we subtract the radiative
% tail of the $\Upsilon(1S)$, $\Upsilon(2S)$ and $\Upsilon(3S)$
% resonances, which is explicitly given in Ref.~\cite{:2008hx}.
% Subsequently we subtract the radiative tail of the $\Upsilon(4S)$ resonance. For the resonance
% shape we use a Breit-Wigner function with an electronic width of $\Gamma_{ee}(\Upsilon(4S))=0.272$~keV 
% and an energy-independent total width of
% $\Gamma_{\text{tot}}(\Upsilon(4S))=20.5$~MeV \cite{PDG}. For the radiator
% function  
% $G(z)$ we take the functional dependence as used in \cite{Chetyrkin:1996tz}, based on 
% the resummed NNLO result of \cite{Jadach:1988gb}:
% \begin{equation}
% G(z) = \beta (1-z)^{\beta-1}\,{\rm e}^{\delta_{\text{yfs}}}\,F\,\left(
%  \delta_C^{V+S} + \delta_C^H \right)\,,
% \label{eqGc}
% \end{equation}
% with
% \begin{align}
% \beta =& \frac{2\alpha}{\pi}(L-1)\,,\qquad L = \ln\frac{s}{m_e^2}\,,
% \qquad F = \frac{{\rm e}^{-\beta\gamma_E}}{\Gamma(1+\beta)}\,,\nonumber\\
% \delta_{\text{yfs}} =& \frac{\alpha}{\pi} \left( \frac{L}{2} - 1 + 2\zeta(2) 
%  \right)\,,\\
% \delta_c^{V+S} =& 1+\frac{\alpha}{\pi}(L-1)+\frac{1}{2}
%  \left(\frac{\alpha}{\pi}\right)^2 L^2\,,\nonumber\\
% \delta_C^H =& -\frac{1-z^2}{2}+\frac{\alpha}{\pi}L\left[-\frac{1}{4}
%  \left(1+3z^2\right)\ln z -1+z\right]\,.\nonumber
% \end{align}

% The remainder $\hat{\sigma}$ corresponds to the continuum cross section
% distorted by initial-state radiation (ISR) and modified by vacuum polarization. It
% is related to $\sigma$, the cross section without ISR, through
% \begin{equation}
% \hat\sigma(s) = \int_{z_0}^1 {\rm d}z\,G(z)\,\sigma(sz)\,,
% \label{eqisrint}
% \end{equation}
% %hat sigma = G** sigma
% where the lower bound of the integration is given by $z_0 = (10.62
% \text{GeV})^2/s$ corresponding to the point where the continuum cross
% section (after subtraction of the $\Upsilon(4s)$ resonance) vanishes.

% Given $\hat \sigma$, we can solve for $\sigma$ in an iterative way as
% follows: Let us define $\delta G(z) \equiv G(z) -\delta(1-z)$ and
% evaluate a successive series of approximations,
% \begin{equation}
%   \label{eq:sigma_i}
%   \sigma_i = \sigma_{0} -\int_{z_0}^{1}{\rm d}z 
%   \delta G(z) \sigma_{i-1}(sz)\,,
% \end{equation}
% using as starting point $\sigma_0=\hat{\sigma}$.
% % In practice it is sufficient to terminate with $\sigma_2$, corresponding
% % to two iterations. The effect of a third iteration is below XX \%. 
% The difference between $\sigma_i$ and $\sigma$ can be estimated by
% evaluating Eq. (\ref{eqisrint}) with $\sigma_i$ in place of $\sigma$.
% After five iterations the resulting function differs from $\hat{\sigma}$
% by less than $0.5\%$.

% Finally, the effect of the vacuum
% polarization must be taken into account and the result is normalised
% relative to the point cross section,
% \begin{equation}
%   \label{eq:R_b}
%   R_b=\sigma \frac{3s}{4\pi\alpha^2(s)}\,.
% \end{equation}

% The integration
% region in Eq.~\eqref{eqisrint} covers the energy interval between 10.62~GeV
% and 11.24~GeV, whence a constant value $(\alpha/\alpha(s))^2=0.93$ 
% has been adopted. 

% In Fig. \ref{fig:R_b} we show the BABAR  data \cite{:2008hx} (after
% subtraction of the radiative tails of $\Upsilon(1S)$ to
% $\Upsilon(4S)$), together with $R_b$ after deconvolution of ISR 
% and correcting for the running of $\alpha(s)$. We also
% show the CLEO data \cite{Besson:1984bd} after the aforementioned rescaling.

% \begin{figure}
%   \centering
%   \includegraphics[width=\linewidth]{figures/cleobabardat.eps}
%   \caption[]{Comparison of rescaled CLEO data for $R_b$ with BABAR data
% %    before (\includegraphics{sym_BaBar_orig}) and after
% %    (\includegraphics{sym_BaBar}) deconvolution.} 
%     before and after deconvolution. The black bar on the right corresponds to
%     the theory prediction~\cite{Harlander:2002ur}.} 
%   \label{fig:R_b}
% \end{figure}

% It is now straightforward to evaluate the contribution to the moments.
% The result is listed in Tab.~\ref{tab:Mn5} and compared to our earlier analysis
% based on the CLEO result~\cite{Besson:1984bd}. The error of this new result is  
% dominated by the correlated systematic error of the BABAR measurement
% which amounts to about 3.5\%. In addition we use a 2\% error for the uncertainty
% from the details of the matching between the tail of $\Upsilon(4S)$ and
% the continuum around $\sqrt{s} = 10.62~\text{GeV}$, which we add in quadrature. 

% As is evident from Tab.~\ref{tab:Mn5}, the agreement between old and new
% result  
% is remarkable giving additional confidence in the procedure used in
% Ref.~\cite{Kuhn:2007vp}.
% The new experimental input and the new information on the
% coefficients $\bar C^{30}_n$  lead to a significant
% reduction of the error on $m_b$, as shown below.

% \begin{table}[t]
% \begin{center}
% {
% {\scalefont{0.9}
% \begin{tabular}{l|llll}
% $n$                                                 
% & 1           & 2           & 3           & 4 \\\hline
% ${\cal M}_{n,\  {\rm old}}^{\rm dat}\times 10^{(2n+1)}$  
% & $0.296(32)$ & $0.249(27)$ & $0.209(22)$ & $0.175(19)$\\ 
% ${\cal M}_{n,\  {\rm new}}^{\rm dat}\times 10^{(2n+1)}$  
% & $0.287(12)$ & $0.240(10)$ & $0.200(8)$ & $0.168(7)$\\ 
% \hline
% ${\cal M}_{n,\  {\rm new}}^{\rm exp}\times 10^{(2n+1)}$  
% & $4.592(31)$ & $2.872(28)$ & $2.362(26)$ & $2.170(26)$\\ 
% %${\cal M}_{n,\  {\rm new}}^{\rm thr}\times 10^{(2n+1)}$  &&&&
%   \end{tabular}
% }
% }
% \caption{\label{tab:Mn5}Moments in $(\text{GeV})^{-2n}$ for the bottom quark
%   system from the 
%   threshold region $\langle 10.62$ GeV, $11.24\ \text{GeV}\rangle$
%   from Ref.~\cite{Kuhn:2007vp} (old) and this letter (new).
%   Also the new total experimental moments are given.
% }
% \end{center}
% \end{table}



%----------------------------------------------------------------------------------------------------


\section{Quark Masses}
\label{sec:mass}

%The same approach is also applicable for the case of the bottom quark. 
Using the new moments with their significantly reduced experimental
error, one obtains the results 
for the bottom quark mass at the scale $\mu=10\,$GeV as listed in
Tab.~\ref{tab::mb}. In comparison with the previous determination a minute 
upwards shift of 1~MeV (resulting from an upward shift of +3~MeV from
the new data and a downward shift of $-$2~MeV from the new theory input) 
and a reduction of both experimental and theory error is observed. 
The three results based on $n=1$, $2$ and $3$ are of comparable
precision. The relative size of the contribution from the continuum above 11.24
GeV which is modelled by perturbative QCD decreases for the higher moments $n=2$
and 3. On the other hand the theory uncertainty, exemplified by the $\mu$
dependence is still acceptable. We therefore adopt the result from $n=2$ (which
is roughly between the $n=1$ and $n=3$ values and which exhibits the smallest 
error) as our final result
\begin{align}
m_b(10~\text{GeV}) =&\,3610(16)~\text{MeV}\,, \notag\\
m_b(m_b) =&\,4163(16)~\text{MeV}\,.
\label{eq:mb}
\end{align}
These values are well consistent with the previous determination \cite{Kuhn:2007vp}
$m_b(10~\text{GeV})=3609(25)$~MeV and $m_b(m_b)=
4164(25)$~MeV. Recently, the bottom-quark mass was also obtained from
lattice calculations \cite{McNeile:2010ji} and ratios of moments
\cite{Narison:2010cg}. 

\begin{table}[t]
\begin{center}
{

\begin{tabular}{l|l|rrr|l|l}
\hline
$n$ & $m_b(10~\text{GeV})$ & 
exp & $\alpha_s$ & $\mu$ &
total &$m_b(m_b)$
\\
\hline
        1&  3597&  14&  7&   2&  16&  4151 \\
        2&  3610&  10&  12&  3&  16&  4163 \\
        3&  3619&  8&  14&   6&  18&  4172 \\
        4&  3631&  6&  15&  20&  26&  4183 \\
%%%%%
\hline
\end{tabular}
}
\caption{\label{tab::mb}Results for $m_b(10~\text{GeV})$ and $m_b(m_b)$ 
in MeV
  obtained from Eq.~\eqref{eq:m_Q}.
  The errors are from experiment, $\alpha_s$ and the variation of $\mu$.
}
\end{center}
\end{table}

In the absence of new data the analysis of $m_c$ will be based on the moments
listed in Tab.~6 of Ref.~\cite{Kuhn:2007vp}. As emphasized in \cite{Kuhn:2001dm,Kuhn:2007vp} it is convenient
to  consider as primary quantity the running quark mass at scale 3~GeV. 
This is the natural
scale for the sum rule (corresponding roughly to the charm
threshold) and, as a consequence of the smaller strong
coupling  constant, the perturbative series exhibits a more stable behaviour.

If not stated otherwise, all input parameters and assumptions are
identical to those of Ref.~\cite{Kuhn:2007vp}.
In particular we adopt $\alpha_s(M_Z)=0.1189$.
% Compared to \cite{Kuhn:2007vp}, the shift induced by the 
%analytic results for $\bar{C}_n^{(30)}$ amounts to 3, 4 and 8~MeV for
%n=2, 3 and 4 respectively.
The results for all four moments are nicely consistent, and the three lowest
moments exhibit comparable errors. %, ranging  between 13~MeV and 17~MeV. 
Note, that the
relative composition of the experimental input varies strongly from low to
high moments: For $n=1$ the contributions from narrow resonances and continuum 
are roughly comparable, for $n=3$ the continuum contribution amounts to 
about 10\%. 
Furthermore, the experimental contribution to the error decreases with
increasing $n$%from 9~MeV to 5~MeV
, the $\mu$-dependence, reflecting the theory
uncertainly, increases.% from 2~MeV to 7~MeV. 
Despite the significant
differences in the composition of the errors, the results are perfectly
consistent. Since the result 
from $n=1$ has the smallest dependence on the strong coupling and the smallest
total error we take as our final value 
\begin{equation}
\label{eq:mc_3}
m_c(3~\text{GeV}) = 986(13)~\text{MeV}\,,
\end{equation}
and consider its  consistency with $n=2$, $3$ and $4$ as additional
confirmation.


Transforming this to the scale-invariant mass $m_c(m_c)$~\cite{Chetyrkin:2000yt}, including the
four-loop coefficients  of the renormalization group functions one
finds 
%~\footnote{In Ref.~\cite{Kuhn:2007vp} an imprecise routine has been
%  used to compute the scale-invariant charm quark mass with the result
%  $m_c(m_c)=1286$~MeV for the central value. In this letter we solve the
%  coupled system of differential equations numerically which has also been
%  done in Ref.~\cite{Allison:2008xk} where $m_c(m_c)=1268$~MeV has been
%  obtained using $\alpha_s(M_Z)=0.1176$.}
$m_c(m_c)=1279(13)~\text{MeV}$. Let us recall at this point that a recent
lattice determination, 
combining a lattice simulation for the data for the pseudoscalar correlator
with the perturbative  three- and four-loop result
\cite{Chetyrkin:1997mb,Sturm:2008eb,Maier:2009fz} has led to
$m_c(3~\text{GeV})= 986(10)$~MeV \cite{Allison:2008xk} in remarkable
agreement with most recent analysis \cite{Chetyrkin:2009fv}.

\section{Summary}
\label{sec:sum}

Based on new four-loop results for the higher derivatives 
of the vacuum polarization function and new BABAR data 
for bottom quark production in the threshold region, a 
reanalysis of the charm- and bottom-quark mass determination 
has been performed. The new data, a posteriori, give 
additional support to the analysis of CLEO data 
presented in Ref.~\cite{Besson:1984bd} and, furthermore, lead to a 
significant reduction of the experimental error. 
The new theory results for the higher moments lead to a 
further reduction of the theory uncertainty and, 
equally important, demonstrate the consistency between the    
analysis based on different moments. The final results,
$m_c(3 {\rm GeV})= 0.986(13)~{\rm GeV}$ and 
$m_b(10{\rm GeV}) =3.610(16)~{\rm GeV}$ are consistent with the earlier 
determination in Ref.~\cite{Kuhn:2007vp} and, together with
Ref.~\cite{Allison:2008xk}, constitute the most precise
determination of charm- and bottom-quark masses to date.

\section*{Acknowledgements}
The work was done in collaboration with K.G. Chetyrkin, Y. Kiyo, J.H. K\"uhn, A. Maier,
P. Maierh\"ofer, A.V. Smirnov, M. Steinhauser and C. Sturm and supported
by the Deutsche Forschungsgemeinschaft through the SFB/TR-9
``Computational Particle Physics''.

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