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\begin{document}

\markboth{Johann H. K\"UHN}
{Heavy Quark Masses}

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\title{PRECISE HEAVY QUARK MASSES
\footnote{Presented at the Workshop on ''Determination of the
  Fundamental Parameters of QCD'', Singapore, March 18-21, 2013.} 
}

\author{\footnotesize JOHANN H. K\"UHN
%\footnote{
%Typeset names in 8 pt Times Roman, uppercase. Use the footnote to 
%indicate the present or permanent address of the author.}}
}

\address{Institut f\"ur Theoretische Teilchenphysik, Karlsruhe Institut f\"ur Technologie,\\
76128 Karlsruhe, Germany\\
Johann.Kuehn@KIT.edu
%University Department, University Name, Address\\
%City, State ZIP/Zone,
%Country\footnote{State completely without abbreviations, the 
%affiliation and mailing address, including country and e-mail address. 
%Typeset in 8 pt Times Italic.}\\
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}
%\author{SECOND AUTHOR}
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%\address{Group, Laboratory, Address\\
%City, State ZIP/Zone, Country
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\maketitle

\pub{Received (Day Month Year)}{Revised (Day Month Year)}

\begin{abstract}
Recent theoretical and experimental improvements in the determination of
charmed- and bottom-quark masses are discussed. The final results, 
$m_c(3\,{\rm GeV})=986(13)\,{\rm MeV}$ and 
$m_b(m_b)=4163(16)\,{\rm MeV}$, are among the most precise
determinations of these two fundamental parameter. A critical analysis
of the theoretical and experimental uncertainties is presented and
possibilities for further improvements of the experimental input are
discussed.


\keywords{Perturbative Quantum Chromodynamics, Charmed Quarks, Bottom Quarks} 

\end{abstract}

\ccode{PACS Nos.: 12.38.Bx, 14.65.Dw, 14.65.Fy}


\section{Introduction}
The most striking deficiency of the Standard Model (SM) of particle
physics is its large number of presently uncalculable ``fundamental'' 
parameters.	The gauge sector requires three constants only,
which in the context of a Grand Unified Theory (GUT) might even be
reduced to one universal gauge coupling. The quartic Higgs-boson
self-coupling, together with  the quadratic terms is sufficient for 
the description of spontaneous symmetry breaking and, as a consequence, 
the generation of gauge boson and fermion masses. A proliferation of 
parameters  is observed in the fermion mass matrix, in other words in 
the Yukawa couplings which parametrize the strength of the interactions 
between fermions and the Higgs boson. The determination of this mass 
matrix, in turn, is equivalent to the determination of fermion masses 
and mixing angles. A large number of detailed experiments is devoted 
to the  measurement of the Cabbibo-Kobayashi-Maskawa angles in the 
quark sector and the corresponding angles characteristic for the 
lepton mixing. The precise determination of  the quark masses
is of similar importance. In contrast to lepton masses the determination 
of quarks masses requires not only experimental but also considerable 
theoretical effort, as we will see below.  

To date no convincing framework exists which would allow to predict
these parameters from first principles, and at best 
one might hope to find relations between quark and lepton masses 
within the third family. Despite this lack of predictive power there 
is strong interest in precise quark mass values which are decisive for  
numerous phenomenological issues.
The $u$-, $d$- and $s$-quark masses are responsible for the explicit
breaking of chiral symmetry and, as a consequence, for the non-vanishing 
masses of the pions, interpreted as pseudo-Goldstone bosons. Furthermore, 
they must be included in any analysis of pion and kaon scattering
performed in the framework of  chiral perturbation theory  ($\chi$PT),
(For a detailed discussion and further references see
Ref.~\refcite{Leutwyler}.)
 which leads to excellent predictions for a large variety of phenomena.
For the strange quark with its mass of around 100~MeV the applicability 
of $\chi$PT is at the borderline. Mass determinations based on sum 
rules for various correlators 
\cite{Jamin:2001zr,Baikov:2005rw,Chetyrkin:2005kn}
 or on the analysis of strangeness violating tau-lepton 
decays\cite{Chetyrkin:1998ej,Pich:1999hc,Baikov:2004tk}
are based on perturbative QCD (pQCD), 
(despite the fact that they employ relatively low scales of 
order 1.5 to 2~GeV) and are in reasonable agreement with   
$\chi$PT and with lattice results\cite{Colangelo:2010et,latticetalk}.

The following discussion  will be focused on charm and bottom quarks. 
With mass values around one and four GeV, respectively, these are 
sufficiently heavy to allow for a perturbative treatment, if suitable 
correlators are analyzed in well chosen momentum regions. Furthermore, 
for the observables considered in this paper, perturbation theory is 
known to sufficiently high order, such that the theory error is well 
under control.

The precise determination of $m_c$ and $m_b$ is motivated by
phenomenological and conceptual considerations. Let us just recall 
that the  charmed quark mass not only governs the properties of the 
charmonium states (See for example Ref.~\refcite{Kniehl:2003ap}.), 
it is also a crucial ingredient in
predictions\cite{Buras:2005gr,Brod:2008ss}
 for the decay $K\to\pi\nu\bar\nu  $. Both charm and bottom quark masses
are required for predictions of $\Gamma(B\to X_b\ell\nu)$ and thus 
crucial for the determination of the CKM element $V_{cb}$. 
The bottom quark mass, finally, is required to predict rates, photon- 
and lepton-spectra in $B$-meson decays, it governs the masses of 
bottonium states (see e.g.\ Ref.~\refcite{Penin:2005eu}) and 
determines the rate of the dominant decay mode 
of the Higgs boson, $\Gamma(H\to b\bar b)$, and hence all experimentally 
observable branching ratios. Last not least, once the masses of top and
bottom quark can be related e.g.\ in the framework of a Grand Unified
Theory, the relative errors $\delta m_b/m_b$ and $\delta m_t/m_t$ (with 
both $\overline{MS}$ masses evaluated at the scale $m_t$) should be comparable. 
As we will see below, for  
$m_b(m_t)=2.701\pm0.022\,{\rm GeV}$ and
$m_t=161.47\pm 0.85\,{\rm GeV}$, (based on a pole mass of
$M_t=173.20\pm0.87\,{\rm GeV}$ as determined by a 
Tevatron\footnote{ This value is in very good agreement 
with the recent ATLAS and CMS 
results\cite{Yazgan} of $173.2\pm0.6\pm0.8\,{\rm GeV}$ and 
$173.2\pm0.6\pm0.8\,{\rm GeV}$ respectively.}
analysis\cite{Petrillo} and $\alpha_s=0.1189(20)$)
this requirement is indeed fulfilled.




\section{Quark Mass Definitions}

Masses of stable elementary particles are obviously determined by
kinematic measurements, and the same is true for unstable particles, 
as long as their decay rate is far smaller than the particle mass.  
This ``pole mass'' definition $M_{pole}$ is directly related to the location 
of the pole of the corresponding propagator. In practical 
calculations (For an overview see e.g. Ref~\refcite{Chetyrkin:1996ia}.) 
involving fundamental fermions (both quarks and leptons) it is often 
useful to use the $\overline{MS}$ mass $m(\mu)$ in intermediate steps.
In the case of leptons, however, the pole mass remains the fundamental
observable that can be measured with arbitrary precision, at least in
principle. The situation is different for quarks which cannot be
produced or studied in isolation. Correspondingly the pole mass cannot be
determined through kinematic measurements with arbitrary precision and
uncertainties of order $\Lambda_{QCD}$ necessarily remain. In
perturbative calculations this is reflected in relatively large higher
order corrections, indicating the appearance of of a divergent series
related to the presence of ''renomalons''\cite{Beneke:1998ui}. Nevertheless, 
in some cases (e.g.\ the determination of the top quark mass from its
decay products) it is convenient or even unavoidable to use the pole
mass as primary quantity.

The $\overline{MS}$-mass as second and for many cases most convenient
choice is directly related to the mass parameter in the Lagrangian. 
In perturbation theory a regularisation and renormalisation prescription
is required to arrive at an ultraviolet finite result, with
dimensional regularisation and minimal subtraction as convenient
conventions. To complete the prescription a choice for the
renormalisation scale $\mu$ has to be adopted, and the $\mu$-dependence 
of the $\overline{MS}$-mass  $m(\mu)$ is governed by the mass anomalous
dimension, 
\begin{equation}
\mu^2\frac{d}{d\mu^2} \overline{m}(\mu)
 =  \overline{m}(\mu)\gamma_m(\alpha_s) \equiv
-\overline{m}\sum_{i\geq0}\gamma_m^{{i}}
\left(\frac{\alpha_s}{\pi}\right)^{i+1},
\label{anom-mass-def}
\end{equation}
with $\gamma^i$ known\cite{Chetyrkin:1997dh,Vermaseren:1997fq} to $i=3$,
corresponding to four-loop precision\footnote{Recently\cite{Kostia} 
even the five-loop result has become available.}.
Two-, three- and partial four-loop results
\cite{Gray:1990yh,Fleischer:1998dw,Chetyrkin:1999qi,Melnikov:2000qh,Lee:2013sx}
 are available for the
conversion\footnote{These relations, together with the solution of 
Eq.~\ref{anom-mass-def} and issues of ``matching'' at flavour thresholds
are conveniently encoded in the
{\tt Mathematica} and {\tt C++} programs 
{\tt  RunDec}\cite{Chetyrkin:2000yt} 
and CRunDec\cite{Schmidt:2012az}.}
between $M_{pole}$ and $m(\mu)$. Various other mass definitions,
particularly suited for specific applications, have been suggested in
the literature. Examples are ``kinetic mass''\cite{Bigi:1996si}, 
employed e.g.\ in weak decays of $B$ mesons,
(Note that the kinetic mass has not yet been related
to the $\overline{MS}$-mass in ${\cal O}(\alpha_s^3)$ accuracy.) 
 ``potential subtracted mass''\cite{Beneke:1998rk} 
and ``$1S$-mass''\cite{Hoang:1998ng}, used in
connection with quarkonium physics.  The following
discussion will be restricted to the $\overline{MS}$-mass which, as 
stated above, is closely related to the Lagrangian and is used, at
least in intermediate steps, in most higher order calculations. Last not
least it is best suited for the mass determination based on moments of 
the $R$-ratio as discussed in the following.

\section{Correlators, Moments and Quark Masses}

The idea behind this approach relies on the observation that the
correlator of the electromagnetic current $j_\mu$
\begin{equation}
(-q^2g_{\mu\nu} + q_\mu q_\nu)\Pi(q^2)\equiv i\int {\rm d}xe^{iqx}\langle
T j_\mu(x)j_\nu(0)\rangle
\end{equation}
can be evaluated perturbatively in the deep-Euklidean region $q^2\ll 0$.
As a consequence of the large quark mass the lowest pole of $\Pi(q^2)$
is located at $q^2=M_{J/\psi}^2$, the branching cut even starts at 
$(2M_D)^2\approx (3.739\,{\rm GeV})^2$. The perturbative evaluation is,
therefore, possible even at  $q^2=0$ which is already sufficiently far
away from the physical, the threshold region. The same statements apply,
a forteriori, to bottom quark production. However, some caveats must be
considered. If perturbative QCD (pQCD) could be applied to   
$\Pi(q^2)$ at $q^2$ and all its derivatives, this would correspond to
a perturbative description of $\Pi(q^2)$ for arbitrary $q^2$.
It has therefore been argued that pQCD should be applicable only to the
lowest terms of the Taylor series
\begin{equation}
  \label{eq:Pi}
  \Pi_Q(q^2)\equiv Q_Q^2\frac{3}{16\pi^2} \sum_{n\geq0}\bar{C}_nz^n
  \,,
\end{equation}
with $z=q^2/4m_Q^2(\mu)$ and $m_Q(\mu)$ being the $\overline{MS}$ mass
at scale $\mu$. As stated before, the coefficients $\overline{C}_n$ can
be calculated in pQCD and cast into the following generic form
\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}
with $\alpha_s\equiv \alpha_s(\mu)$, $\ell_{m_Q}\equiv \ln m_Q^2/\mu^2$
and $\bar C_n^{(ij)}$ being pure numbers. The coefficients
$\bar C_n^{(ij)}$ with $j=0$
are obtained from increasingly complex
calculations, those with $n\ge 1$ can be reconstructed from
the lower ones, employing the renormalisation group with the
coefficients of the anomalous mass dimension $\gamma_m$ and the 
$\beta$ function to the appropriate order. The two-loop result 
$\overline{C}_n^{(10)}$ can be obtained directly from the Taylor
expansion of the QED polarisation function, evaluated by Kallen and
Sabry long time ago. The three-loop result has been calculated in
Ref.~\refcite{Chetyrkin:1995ii,Chetyrkin:1996cf}
up to $n=4$, (later\cite{Chetyrkin:1997mb} up to $n=8$) in terms of rational and
transcendental numbers with the help of the FORM program
MATAD\cite{Steinhauser:2000ry}, using a
recursive algorithm.
(More recently the results even up to $n=30$ have become available%
\cite{Boughezal:2006uu,Maier:2007yn}.) 
The four-loop coefficients up to $n=3$ for vector, axial vector, scalar
and pseudoscalar correlators were obtained%
\cite{Chetyrkin:2006xg,Boughezal:2006px,Maier:2008he,Maier:2009fz,Sturm:2008eb}
using Integration-by-Parts identities\cite{Chetyrkin:1981qh} in 
combination with Laporta's algorithm\cite{Laporta:1996mq,Laporta:2001dd} 
which leads to an algebraic reduction of
$C^{40}_n$ to a small set of ``master integrals''. These have
been evaluated first
numerically with high precision\cite{Schroder:2005va,Chetyrkin:2006dh}, 
subsequently 
in a series of papers in terms of rational and transcendental numbers. 
(A complete list of references can be found in Ref.~\refcite{Maier:2009fz}.) 
Numerical results for moments with higher $n$
(with an estimated precision better than one percent for $n=4$) are based on the
approximate reconstruction of the full function $\Pi(q^2)$, using
information at $q^2=0$, the high energy behaviour and the threshold
region combined with interpolations based on Pad\`e approximations\cite{Kiyo:2009gb}.
(For an analysis along similar lines see Ref.~\refcite{Hoang:2008qy}.)

Exploiting the analyticity of $\Pi(q^2)$ around $q^2=0$ and using dispersion
relations, the derivatives at $q^2=0$ can be expressed as weighted
integrals over the imaginary part of $\Pi(q^2)$, which in turn is given
by the cross section for electron-positron annihilation into
hadrons. Let us denote the normalised cross section for heavy quark
production as $R_Q(s)\equiv \sigma_Q(s)/\sigma_{\rm point}(s)$. The
moments of $R_Q$, defined as
\begin{equation}
  {\cal M}^{\rm exp}_n \equiv \int \frac{{\rm d}s}{s^{n+1}} R_Q(s)
  \,,
  \label{eq:Mexp}
\end{equation}
can be directly related to the perturbatively calculated Taylor
coefficients. In total one thus obtains the $\overline{MS}$ quark mass in
terms of experimentally weighted integrals of $R_Q$ and the perturbatively
calculable coefficients $\bar C_n$,
\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}
This strategy has been suggested originally in 
Ref.~\refcite{Shifman:1978bx} and applied to a
precise charm and bottom mass determination in Ref.~\refcite{Kuhn:2001dm}
once the
three-loop results had become available. A significantly improved reanalysis
based on four-loop moments and with new data has been performed
in Ref.~\refcite{Kuhn:2007vp}, additional
updates and improvements from new data and the precis evaluation of the 
higher moments can be 
found in Refs.~\refcite{Chetyrkin:2009fv,Chetyrkin:2010ic},
which are the basis of the subsequent discussion.
For the extraction of $R_Q$ from the data the issue of singlet
contributions and secondary radiation of heavy quarks has been discussed
in some detail in Ref.~\refcite{Kuhn:2007vp}. 
Furthermore, the potential influence of a
non-vanishing gluon condensate 
$\langle\frac{\alpha_s}{\pi}GG\rangle=0.006\pm0.012\,{\rm GeV}^4$ 
has been analysed in  Refs.~\refcite{Kuhn:2007vp,Chetyrkin:2010ic},
which, for moments with $n\le3$, leads to shifts of one to
two MeV and an associated uncertainty of three MeV at most.

\section{Results}

Let us now present the experimental results for the moments, first for
charm, later for bottom. For charm the integration region is split into
one covering the narrow resonances $J/\psi$ and $\psi'$, the ``threshold
region'' between $2 m_D$ and 4.8~GeV and the perturbative continuum
above. 
\begin{table}
\tbl{Comparison of the theory predictions for 
  $R(s)$ with the experimental results at a few selected 
  values for $\sqrt{s}$.}
%\centering
{\begin{tabular}{c|llllll}
  \hline
%%%%%
$\sqrt{s}$(GeV) & 2.00 & 3.65 & 3.732 & 4.80 & 9.00 & 10.52 \\
$R^{\rm th}(s)$ &2.209(91)&2.161(18)&2.160(17)&3.764(64)&3.564(17)&3.548(12)\\
$R^{\rm exp}(s)$&2.18(7)(18)&2.157(35)(86)&2.156(86)(86)&3.66(14)(19)&3.62(7)(14)&3.56(1)(7)
\\
 Experiment &   BESS & BESS & BESS  &   BESS & MD-1  & CLEO \\
 \hline
%%%%%
\end{tabular}
\label{tab:Rcomparison}
}
\end{table}
Note that we assume the validity of pQCD in this region with high
precision, an assumption that is well consistent with present
measurements (see Table~\ref{tab:Rcomparison}) 
but for the moment remains an assumption, to be
verified e.g. by future BESS experiments.

  \begin{table}[t]
\tbl{Experimental moments
  in $(\mbox{GeV})^{-2n}$ as defined in
  Eq.~(\ref{eq:Mexp}), separated according to the contributions from
  the narrow resonances,
  the charm threshold region and the continuum region
  above $\sqrt{s}=4.8$~GeV. In the last column the NLO contribution from the
  gluon condensate is shown.} 
%    \centering
{
\begin{tabular}{l|lll|l||l}
%%%%%
\hline
$n$ & ${\cal M}_n^{\rm res}$
& ${\cal M}_n^{\rm thresh}$
& ${\cal M}_n^{\rm cont}$
& ${\cal M}_n^{\rm exp}$
& ${\cal M}_n^{\rm np}$(NLO)
\\
& $\times 10^{(n-1)}$
& $\times 10^{(n-1)}$
& $\times 10^{(n-1)}$
& $\times 10^{(n-1)}$
& $\times 10^{(n-1)}$
\\
\hline
$1$&$  0.1201(25)$ &$  0.0318(15)$ &$  0.0646(11)$ &$  0.2166(31)$
 &$-0.0002(5)$ \\
$2$&$  0.1176(25)$ &$  0.0178(8)$ &$  0.0144(3)$ &$  0.1497(27)$
 &$-0.0005(10)$ \\
$3$&$  0.1169(26)$ &$  0.0101(5)$ &$  0.0042(1)$ &$  0.1312(27)$
 &$-0.0008(16)$ \\
$4$&$  0.1177(27)$ &$  0.0058(3)$ &$  0.0014(0)$ &$  0.1249(27)$
 &$-0.0013(25)$ \\
\hline
\end{tabular}
    \label{tab:contglu}
}
  \end{table}

  \begin{table}[t]
  \tbl{Results for $m_c(3\, \text{GeV} )$ in $\text{GeV}$. The errors
      are from experiment, $\alpha_s$, variation of $\mu$ and the
     gluon condensate.}
%    \centering
%\scalefont{1}{
{
\begin{tabular}{l|l|llll||l}
%%%%%
\hline
$n$ & $m_c(3~\mbox{GeV}) $ & exp & $\alpha_s$ & $\mu$ 
& ${\rm np}_{\rm NLO}$ & total\\
 \hline
1& 0.986& 0.009&  0.009&  0.002& 0.001& 0.013\\
2& 0.975& 0.006&  0.014&  0.005& 0.002& 0.016\\
3& 0.975& 0.005&  0.015&  0.007& 0.003& 0.017\\
4& 0.999& 0.003&  0.009&  0.031& 0.003& 0.032\\
\hline
\end{tabular}
%}
    \label{tab:charmTab}}
  \end{table}

The results for the moments from one to four and the error budget are 
listed in Table~\ref{tab:contglu}, 
those for the quark mass in Table~\ref{tab:charmTab}.
The moment with $n=1$ is most robust from the theory side, as
evident from the relatively smaller coefficient in the perturbative
series. 
This argument can be made more quantitatively by rewriting eq.~(\ref{eq:m_Q}) 
in the form
\begin{eqnarray}
m_c &=& 
\frac{1}{2}
\bigg( \frac{9 Q_c^2}{4}\frac{\bar{C}_n^{\rm Born}}{{\cal M}_n^{\rm exp}}\bigg)^\frac{1}{2n} 
(1 + r^{(1)}_n \alpha_s + r^{(2)}_n\alpha_s^2 + r^{(3)}_n \alpha_s^3) 
\nonumber\\
&\propto&
1 - 
\left(\begin{array}{c}0.328\\0.524\\0.618\\0.662\end{array}\right)\alpha_s - 
\left(\begin{array}{c}0.306\\0.409\\0.510\\0.575\end{array}\right)\alpha_s^2 -
\left(\begin{array}{c}0.262\\0.230\\0.299\\0.396\end{array}\right)\alpha_s^3,
\label{mc} 
\end{eqnarray}
where the entries correspond to the moments with $n=1$, 2, 3 and 4. 
Note, that the coefficients are decreasing with increasing order of
$\alpha_s$. Estimating
the relative error through $r_n^{\rm max}(\alpha_s(3\;{\rm GeV}))^4$ leads to
1.4 /  2.3 / 2.7 / 2.9  per mille and thus to an estimate even
smaller than the one based on the $\mu$ dependence 
listed as theory error in the fifth
column of Table~\ref{tab:charmTab}. Also the contribution from the gluon condensate is
smallest for the lowest values of $n$.
The advantage of $n=2$ and even more so of $n=3$ lies in the
reduced contribution from the continuum region. Note that all four
results are mutually consistent. In view of the smallest sensitivity to
$\alpha_s$ and the choice of the renormalisation scale $\mu$ we adopt
the value $m_c(3\,{\rm GeV})=986\pm13\,{\rm MeV}$ as derived from $n=1$ as
our final result. This  is in very good agreement with the result
 $m_c(3\,{\rm GeV})=987\pm9\,{\rm MeV}$ obtained in 
Ref.~\refcite{Bodenstein:2011ma},
which is based on a different integration kernel.\footnote{For more
details, references and a closely related discussion of $m_b$ see
Ref.~\refcite{Bodenstein}.}

Tables~\ref{tab:contglu} and \ref{tab:charmTab} and also
illustrate the path to a further reduction of the error. Let us restrict
the discussion to $n=1$,2 and 3. For $n=1$ important contributions arise
from all three regions. Improved determinations of $\Gamma_e(J/\psi)$
would reduce the errors of all three moments. Improved measurements of
$R_Q$ in the threshold region and at 4.8~GeV would have a strong impact
on $n=1$ and strengthen our confidence in the validity of pQCD close to,
but above 4.8~GeV. Another interesting option would be a simultaneous fit
to all three moments, taking the proper experimental correlations into
account.

The theory error is presently deduced from a variation of $\mu$
between two and
four GeV. Taking, as an alternative, the last calculated perturbative
coefficient would lead to a comparable, somewhat smaller error.
An important contribution to the error budget arises from the
uncertainty in the strong coupling, where $\alpha_s(M_Z)=0.1189\pm0.002$
has been taken from Ref.~\refcite{Bethke:2006ac}. Taking as an alternative
$\alpha_s(M_Z)=0.1184\pm0.0007$ as obtained from a more recent
compilation\cite{Bethke:2012jm} would obviously lead to a significant 
reduction of $\delta m$. 
\begin{figure}[t]
\begin{center}
\begin{tabular}{c}
\hspace{-1.cm}
\leavevmode
\epsfxsize=1.0\textwidth
\epsffile{mc3cmp.eps}
\\
\hspace{-1.cm}
\leavevmode
\epsfxsize=1.0\textwidth
\epsffile{mbcmp.eps}
\end{tabular}
\end{center}
\vspace*{-2em}
\caption{
Comparison of recent determinations of $m_c(3\;{\rm GeV})$ and $m_b(m_b)$.
\protect\label{fig::mc_compare}
}{}
\end{figure}
A comparison of selected $m_c$ determinations is shown in 
Fig.~\ref{fig::mc_compare}. The
excellent agreement between lattice\cite{Allison:2008xk,McNeile:2010ji}
and perturbative QCD results is particularly encouraging.

Similar statements can be made for the determination of the bottom quark
mass. A recent measurement of the cross section in the threshold
region between 10.6~GeV and 11.2~GeV was employed in 
Ref.~\refcite{Chetyrkin:2009fv} and has lead to a significant reduction
of the experimental error on $m_b$. Still, additional measurements in
the region around and above 11~GeV would be welcome in order to confirm
the validity of perturbative QCD relatively close to threshold. 
The result for the second moment has been adopted as our final answer
\begin{equation}
m_b(10\,{\rm GeV}) = 3610(16)\,{\rm MeV}
\end{equation}  
and corresponds to $m_b(m_b)=3610(16)\,{\rm MeV}$. 
Our results are in excellent agreement with a completely independent lattice
determination, as shown in Fig.~\ref{fig::mc_compare}.













\section*{Acknowledgments}

I would like to thank K.~G.~Chetyrkin, A.~Maier, P.~Maierhofer,
P.~Marquard, M.~Steinhauser and C.~Sturm for a pleasant collaboration on
precise quark mass determinations, and K. K. Phua, Harald Fritzsch and
C. Dominguez for  the invitation to the  
{\em International Workshop on Determination of
  the Fundamental Parameters of QCD} and for financial support through
Nanyang Technological University. This work was supported by the
Deutsche Forschungsgemeinschaft through the \
SFB/TR-9 {\em Computational   Particle Physics}.



\begin{thebibliography}{0}
\bibitem{Leutwyler} Heinrich Leutwyler, these proceedings.
%\cite{Jamin:2001zr}
\bibitem{Jamin:2001zr} 
  M.~Jamin, J.~A.~Oller and A.~Pich,
  %``Light quark masses from scalar sum rules,''
  Eur.\ Phys.\ J.\ C {\bf 24}, 237 (2002)
  [hep-ph/0110194].
  %%CITATION = HEP-PH/0110194;%%
  %90 citations counted in INSPIRE as of 06 May 2013

%\cite{Baikov:2005rw}
\bibitem{Baikov:2005rw}
  P.~A.~Baikov, K.~G.~Chetyrkin and J.~H.~Kuhn,
  %``Scalar correlator at O(alpha(s)**4), Higgs decay into b-quarks and bounds on the light quark masses,''
  Phys.\ Rev.\ Lett.\  {\bf 96}, 012003 (2006)
  [hep-ph/0511063].
  %%CITATION = HEP-PH/0511063;%%
  %80 citations counted in INSPIRE as of 06 May 2013
%\cite{Chetyrkin:2005kn}
\bibitem{Chetyrkin:2005kn} 
  K.~G.~Chetyrkin and A.~Khodjamirian,
  %``Strange quark mass from pseudoscalar sum rule with O(alpha(s)**4) accuracy,''
  Eur.\ Phys.\ J.\ C {\bf 46}, 721 (2006)
  [hep-ph/0512295].
  %%CITATION = HEP-PH/0512295;%%
  %36 citations counted in INSPIRE as of 06 May 2013

%\cite{Chetyrkin:1998ej}
\bibitem{Chetyrkin:1998ej} 
  K.~G.~Chetyrkin, J.~H.~Kuhn and A.~A.~Pivovarov,
  %``Determining the strange quark mass in Cabibbo suppressed tau lepton decays,''
  Nucl.\ Phys.\ B {\bf 533}, 473 (1998)
  [hep-ph/9805335].
  %%CITATION = HEP-PH/9805335;%%
  %96 citations counted in INSPIRE as of 06 May 2013
%\cite{Pich:1999hc}
\bibitem{Pich:1999hc} 
  A.~Pich and J.~Prades,
  %``Strange quark mass determination from Cabibbo suppressed tau decays,''
  JHEP {\bf 9910}, 004 (1999)
  [hep-ph/9909244].
  %%CITATION = HEP-PH/9909244;%%
  %95 citations counted in INSPIRE as of 06 May 2013
%\cite{Baikov:2004tk}
\bibitem{Baikov:2004tk} 
  P.~A.~Baikov, K.~G.~Chetyrkin and J.~H.~Kuhn,
  %``Strange quark mass from tau lepton decays with O(alpha(s)**3) accuracy,''
  Phys.\ Rev.\ Lett.\  {\bf 95}, 012003 (2005)
  [hep-ph/0412350].
  %%CITATION = HEP-PH/0412350;%%
  %63 citations counted in INSPIRE as of 06 May 2013
%\cite{Colangelo:2010et}
\bibitem{Colangelo:2010et} 
  G.~Colangelo, S.~Durr, A.~Juttner, L.~Lellouch, H.~Leutwyler, V.~Lubicz, S.~Necco and C.~T.~Sachrajda {\it et al.},
  %``Review of lattice results concerning low energy particle physics,''
  Eur.\ Phys.\ J.\ C {\bf 71}, 1695 (2011)
  [arXiv:1011.4408 [hep-lat]].
  %%CITATION = ARXIV:1011.4408;%%
  %173 citations counted in INSPIRE as of 06 May 2013
\bibitem{latticetalk} C.~McNeile, these proceedings.
%\cite{Kniehl:2003ap}
\bibitem{Kniehl:2003ap}
  B.~A.~Kniehl, A.~A.~Penin, A.~Pineda, V.~A.~Smirnov and M.~Steinhauser,
  %``M(eta(b)) and alpha(s) from nonrelativistic renormalization group,''
  Phys.\ Rev.\ Lett.\  {\bf 92} (2004) 242001
   [Erratum-ibid.\  {\bf 104} (2010) 199901]
  [hep-ph/0312086].
  %%CITATION = HEP-PH/0312086;%%
  %105 citations counted in INSPIRE as of 02 May 2013
\bibitem{Buras:2005gr}
   A.~J.~Buras, M.~Gorbahn, U.~Haisch and U.~Nierste,
   %``The Rare decay K+ ---> pi+ nu anti-nu at the next-to-next-to-leading 
order in QCD,''
   Phys.\ Rev.\ Lett.\  {\bf 95} (2005) 261805
   [hep-ph/0508165].
   %%CITATION = HEP-PH/0508165;%%
\bibitem{Brod:2008ss}
   J.~Brod and M.~Gorbahn,
   %``Electroweak Corrections to the Charm Quark Contribution to K+ ---> 
pi+ nu anti-nu,''
   Phys.\ Rev.\ D {\bf 78} (2008) 034006
   [arXiv:0805.4119 [hep-ph]].
   %%CITATION = ARXIV:0805.4119;%%
%\cite{Penin:2005eu}
\bibitem{Penin:2005eu} 
  A.~A.~Penin, V.~A.~Smirnov and M.~Steinhauser,
  %``Heavy quarkonium spectrum and production/annihilation rates to order beta**3(0) alpha**3(s),''
  Nucl.\ Phys.\ B {\bf 716}, 303 (2005)
  [hep-ph/0501042].
  %%CITATION = HEP-PH/0501042;%%
  %40 citations counted in INSPIRE as of 15 May 2013

\bibitem{Petrillo} Gianluca Petrillo, Rencontre de Modriond, March 13th,
  2013.
\bibitem{Yazgan} Efe Yazgan, Rencontre de Modriond, March 13th, 2013.
%\cite{Chetyrkin:1996ia}
\bibitem{Chetyrkin:1996ia} 
  K.~G.~Chetyrkin, J.~H.~Kuhn and A.~Kwiatkowski,
  %``QCD corrections to the $e^{+} e^{-}$ cross-section and the $Z$ boson decay rate: Concepts and results,''
  Phys.\ Rept.\  {\bf 277}, 189 (1996).
  %%CITATION = PRPLC,277,189;%%
  %168 citations counted in INSPIRE as of 06 May 2013
%\cite{Beneke:1998ui}
\bibitem{Beneke:1998ui} 
  M.~Beneke,
  %``Renormalons,''
  Phys.\ Rept.\  {\bf 317}, 1 (1999)
  [hep-ph/9807443], and references therein.
  %%CITATION = HEP-PH/9807443;%%
  %434 citations counted in INSPIRE as of 06 May 2013
%\cite{Chetyrkin:1997dh}{Vermaseren:1997fq}
\bibitem{Chetyrkin:1997dh} 
  K.~G.~Chetyrkin,
  %``Quark mass anomalous dimension to O (alpha-s**4),''
  Phys.\ Lett.\ B {\bf 404}, 161 (1997)
  [hep-ph/9703278].
  %%CITATION = HEP-PH/9703278;%%
  %224 citations counted in INSPIRE as of 10 May 2013
%\cite{Vermaseren:1997fq}
\bibitem{Vermaseren:1997fq} 
  J.~A.~M.~Vermaseren, S.~A.~Larin and T.~van Ritbergen,
  %``The four loop quark mass anomalous dimension and the invariant quark mass,''
  Phys.\ Lett.\ B {\bf 405}, 327 (1997)
  [hep-ph/9703284].
  %%CITATION = HEP-PH/9703284;%%
  %307 citations counted in INSPIRE as of 10 May 2013
\bibitem{Kostia}
P.~Baikov K.~Chetyrkin, Talk presented by K. Chetyrkin at
ACAT 2013,16-21 May 2013, Beijing, China. 
%\cite{Gray:1990yh}
\bibitem{Gray:1990yh} 
  N.~Gray, D.~J.~Broadhurst, W.~Grafe and K.~Schilcher,
  %``THREE LOOP RELATION OF QUARK (modified) MS AND POLE MASSES,''
  Z.\ Phys.\ C {\bf 48}, 673 (1990).
  %%CITATION = ZEPYA,C48,673;%%
  %486 citations counted in INSPIRE as of 29 Apr 2013
%\cite{Fleischer:1998dw}
\bibitem{Fleischer:1998dw} 
  J.~Fleischer, F.~Jegerlehner, O.~V.~Tarasov and O.~L.~Veretin,
  %``Two loop QCD corrections of the massive fermion propagator,''
  Nucl.\ Phys.\ B {\bf 539}, 671 (1999)
  [Erratum-ibid.\ B {\bf 571}, 511 (2000)]
  [hep-ph/9803493].
  %%CITATION = HEP-PH/9803493;%%
  %86 citations counted in INSPIRE as of 29 Apr 2013
%\cite{Chetyrkin:1999qi}
\bibitem{Chetyrkin:1999qi} 
  K.~G.~Chetyrkin and M.~Steinhauser,
  %``The Relation between the MS-bar and the on-shell quark mass at order alpha(s)**3,''
  Nucl.\ Phys.\ B {\bf 573}, 617 (2000)
  [hep-ph/9911434].
  %%CITATION = HEP-PH/9911434;%%
  %178 citations counted in INSPIRE as of 29 Apr 2013
%\cite{Melnikov:2000qh}
\bibitem{Melnikov:2000qh} 
  K.~Melnikov and T.~v.~Ritbergen,
  %``The Three loop relation between the MS-bar and the pole quark masses,''
  Phys.\ Lett.\ B {\bf 482}, 99 (2000)
  [hep-ph/9912391].
  %%CITATION = HEP-PH/9912391;%%
  %241 citations counted in INSPIRE as of 29 Apr 2013
%\cite{Lee:2013sx}
\bibitem{Lee:2013sx} 
  R.~Lee, P.~Marquard, A.~V.~Smirnov, V.~A.~Smirnov and M.~Steinhauser,
  %``Four-loop corrections with two closed fermion loops to fermion self energies and the lepton anomalous magnetic moment,''
  JHEP {\bf 1303}, 162 (2013)
  [arXiv:1301.6481 [hep-ph]].
  %%CITATION = ARXIV:1301.6481;%%

%\cite{Chetyrkin:2000yt}
\bibitem{Chetyrkin:2000yt} 
  K.~G.~Chetyrkin, J.~H.~Kuhn and M.~Steinhauser,
  %``RunDec: A Mathematica package for running and decoupling of the strong coupling and quark masses,''
  Comput.\ Phys.\ Commun.\  {\bf 133}, 43 (2000)
  [hep-ph/0004189].
  %%CITATION = HEP-PH/0004189;%%
%\cite{Schmidt:2012az}
\bibitem{Schmidt:2012az} 
  B.~Schmidt and M.~Steinhauser,
  %``CRunDec: a C++ package for running and decoupling of the strong coupling and quark masses,''
  Comput.\ Phys.\ Commun.\  {\bf 183}, 1845 (2012)
  [arXiv:1201.6149 [hep-ph]].
  %%CITATION = ARXIV:1201.6149;%%
  %2 citations counted in INSPIRE as of 29 Apr 2013
%\cite{Bigi:1996si}
\bibitem{Bigi:1996si} 
  I.~I.~Y.~Bigi, M.~A.~Shifman, N.~Uraltsev and A.~I.~Vainshtein,
  %``High power n of m(b) in beauty widths and n=5 ---> infinity limit,''
  Phys.\ Rev.\ D {\bf 56}, 4017 (1997)
  [hep-ph/9704245].
  %%CITATION = HEP-PH/9704245;%%
  %114 citations counted in INSPIRE as of 11 May 2013
%\cite{Beneke:1998rk}
\bibitem{Beneke:1998rk} 
  M.~Beneke,
  %``A Quark mass definition adequate for threshold problems,''
  Phys.\ Lett.\ B {\bf 434}, 115 (1998)
  [hep-ph/9804241].
  %%CITATION = HEP-PH/9804241;%%
  %200 citations counted in INSPIRE as of 11 May 2013
%\cite{Hoang:1998ng}
\bibitem{Hoang:1998ng} 
  A.~H.~Hoang, Z.~Ligeti and A.~V.~Manohar,
  %``B decay and the Upsilon mass,''
  Phys.\ Rev.\ Lett.\  {\bf 82}, 277 (1999)
  [hep-ph/9809423].
  %%CITATION = HEP-PH/9809423;%%
  %169 citations counted in INSPIRE as of 11 May 2013
%\cite{Chetyrkin:1995ii}{Chetyrkin:1996cf}{Chetyrkin:1997mb}
\bibitem{Chetyrkin:1995ii} 
  K.~G.~Chetyrkin, J.~H.~Kuhn and M.~Steinhauser,
  %``Heavy quark vacuum polarization to three loops,''
  Phys.\ Lett.\ B {\bf 371}, 93 (1996)
  [hep-ph/9511430].
  %%CITATION = HEP-PH/9511430;%%
  %110 citations counted in INSPIRE as of 11 May 2013
%\cite{Chetyrkin:1996cf}
\bibitem{Chetyrkin:1996cf} 
  K.~G.~Chetyrkin, J.~H.~Kuhn and M.~Steinhauser,
  %``Three loop polarization function and O (alpha-s**2) corrections to the production of heavy quarks,''
  Nucl.\ Phys.\ B {\bf 482}, 213 (1996)
  [hep-ph/9606230].
  %%CITATION = HEP-PH/9606230;%%
  %213 citations counted in INSPIRE as of 11 May 2013
%\cite{Chetyrkin:1997mb}
\bibitem{Chetyrkin:1997mb} 
  K.~G.~Chetyrkin, J.~H.~Kuhn and M.~Steinhauser,
  %``Heavy quark current correlators to O (alpha-s**2),''
  Nucl.\ Phys.\ B {\bf 505}, 40 (1997)
  [hep-ph/9705254].
  %%CITATION = HEP-PH/9705254;%%
  %117 citations counted in INSPIRE as of 11 May 2013
%\cite{Steinhauser:2000ry}
\bibitem{Steinhauser:2000ry} 
  M.~Steinhauser,
  %``MATAD: A Program package for the computation of MAssive TADpoles,''
  Comput.\ Phys.\ Commun.\  {\bf 134}, 335 (2001)
  [hep-ph/0009029].
  %%CITATION = HEP-PH/0009029;%%
  %130 citations counted in INSPIRE as of 30 Apr 2013

%\cite{Boughezal:2006uu}
\bibitem{Boughezal:2006uu} 
  R.~Boughezal, M.~Czakon and T.~Schutzmeier,
  %``Four-Loop Tadpoles: Applications in QCD,''
  Nucl.\ Phys.\ Proc.\ Suppl.\  {\bf 160}, 160 (2006)
  [hep-ph/0607141].
  %%CITATION = HEP-PH/0607141;%%
  %34 citations counted in INSPIRE as of 30 Apr 2013
%\cite{Maier:2007yn}
\bibitem{Maier:2007yn} 
  A.~Maier, P.~Maierhofer and P.~Marquard,
  %``Higher Moments of Heavy Quark Correlators in the Low Energy Limit at O(alpha**2(s)),''
  Nucl.\ Phys.\ B {\bf 797}, 218 (2008)
  [arXiv:0711.2636 [hep-ph]].
  %%CITATION = ARXIV:0711.2636;%%
  %33 citations counted in INSPIRE as of 30 Apr 2013

%\cite{Chetyrkin:2006xg}
\bibitem{Chetyrkin:2006xg} 
  K.~G.~Chetyrkin, J.~H.~Kuhn and C.~Sturm,
  %``Four-loop moments of the heavy quark vacuum polarization function in perturbative QCD,''
  Eur.\ Phys.\ J.\ C {\bf 48}, 107 (2006)
  [hep-ph/0604234].
  %%CITATION = HEP-PH/0604234;%%
  %63 citations counted in INSPIRE as of 30 Apr 2013
%\cite{Boughezal:2006px}
\bibitem{Boughezal:2006px} 
  R.~Boughezal, M.~Czakon and T.~Schutzmeier,
  %``Charm and bottom quark masses from perturbative QCD,''
  Phys.\ Rev.\ D {\bf 74}, 074006 (2006)
  [hep-ph/0605023].
  %%CITATION = HEP-PH/0605023;%%
  %77 citations counted in INSPIRE as of 30 Apr 2013
%\cite{Maier:2008he}
\bibitem{Maier:2008he} 
  A.~Maier, P.~Maierhofer and P.~Marquard,
  %``The Second physical moment of the heavy quark vector correlator at O(alpha**3(s)),''
  Phys.\ Lett.\ B {\bf 669}, 88 (2008)
  [arXiv:0806.3405 [hep-ph]].
  %%CITATION = ARXIV:0806.3405;%%
  %33 citations counted in INSPIRE as of 30 Apr 2013
%\cite{Maier:2009fz}
\bibitem{Maier:2009fz} 
  A.~Maier, P.~Maierhofer, P.~Marquard and A.~V.~Smirnov,
  %``Low energy moments of heavy quark current correlators at four loops,''
  Nucl.\ Phys.\ B {\bf 824}, 1 (2010)
  [arXiv:0907.2117 [hep-ph]].
  %%CITATION = ARXIV:0907.2117;%%
  %21 citations counted in INSPIRE as of 30 Apr 2013
%\cite{Sturm:2008eb}
\bibitem{Sturm:2008eb} 
  C.~Sturm,
  %``Moments of Heavy Quark Current Correlators at Four-Loop Order in Perturbative QCD,''
  JHEP {\bf 0809}, 075 (2008)
  [arXiv:0805.3358 [hep-ph]].
  %%CITATION = ARXIV:0805.3358;%%
  %19 citations counted in INSPIRE as of 12 May 2013
%\cite{Chetyrkin:1981qh}
\bibitem{Chetyrkin:1981qh} 
  K.~G.~Chetyrkin and F.~V.~Tkachov,
  %``Integration by Parts: The Algorithm to Calculate beta Functions in 4 Loops,''
  Nucl.\ Phys.\ B {\bf 192}, 159 (1981).
  %%CITATION = NUPHA,B192,159;%%
  %839 citations counted in INSPIRE as of 30 Apr 2013
%\cite{Laporta:1996mq}
\bibitem{Laporta:1996mq} 
  S.~Laporta and E.~Remiddi,
  %``The Analytical value of the electron (g-2) at order alpha**3 in QED,''
  Phys.\ Lett.\ B {\bf 379}, 283 (1996)
  [hep-ph/9602417].
  %%CITATION = HEP-PH/9602417;%%
  %256 citations counted in INSPIRE as of 30 Apr 2013
%\cite{Laporta:2001dd}
\bibitem{Laporta:2001dd} 
  S.~Laporta,
  %``High precision calculation of multiloop Feynman integrals by difference equations,''
  Int.\ J.\ Mod.\ Phys.\ A {\bf 15}, 5087 (2000)
  [hep-ph/0102033].
  %%CITATION = HEP-PH/0102033;%%
  %392 citations counted in INSPIRE as of 30 Apr 2013
%\cite{Laporta:1996mq,Laporta:2001dd}


%\cite{Schroder:2005va}
\bibitem{Schroder:2005va} 
  Y.~Schroder and A.~Vuorinen,
  %``High-precision epsilon expansions of single-mass-scale four-loop vacuum bubbles,''
  JHEP {\bf 0506}, 051 (2005)
  [hep-ph/0503209].
  %%CITATION = HEP-PH/0503209;%%
  %48 citations counted in INSPIRE as of 11 May 2013
%==========================masters========================
%\cite{Chetyrkin:2006dh}
\bibitem{Chetyrkin:2006dh} 
  K.~G.~Chetyrkin, M.~Faisst, C.~Sturm and M.~Tentyukov,
  %``epsilon-finite basis of master integrals for the integration-by-parts method,''
  Nucl.\ Phys.\ B {\bf 742}, 208 (2006)
  [hep-ph/0601165].
  %%CITATION = HEP-PH/0601165;%%
  %35 citations counted in INSPIRE as of 11 May 2013



%=============================masters end =====================
%\cite{Kiyo:2009gb}
\bibitem{Kiyo:2009gb} 
  Y.~Kiyo, A.~Maier, P.~Maierhofer and P.~Marquard,
  %``Reconstruction of heavy quark current correlators at O(alpha(s)**3),''
  Nucl.\ Phys.\ B {\bf 823}, 269 (2009)
  [arXiv:0907.2120 [hep-ph]].
  %%CITATION = ARXIV:0907.2120;%%
  %24 citations counted in INSPIRE as of 30 Apr 2013
%\cite{Hoang:2008qy}
\bibitem{Hoang:2008qy} 
  A.~H.~Hoang, V.~Mateu and S.~Mohammad Zebarjad,
  %``Heavy Quark Vacuum Polarization Function at O(alpha**2(s)) O(alpha**3(s)),''
  Nucl.\ Phys.\ B {\bf 813}, 349 (2009)
  [arXiv:0807.4173 [hep-ph]].
  %%CITATION = ARXIV:0807.4173;%%
  %24 citations counted in INSPIRE as of 30 Apr 2013
%\cite{Shifman:1978bx}
\bibitem{Shifman:1978bx} 
  M.~A.~Shifman, A.~I.~Vainshtein and V.~I.~Zakharov,
  %``QCD and Resonance Physics. Sum Rules,''
  Nucl.\ Phys.\ B {\bf 147}, 385 (1979).
  %%CITATION = NUPHA,B147,385;%%
  %4064 citations counted in INSPIRE as of 15 May 2013
%\cite{Kuhn:2001dm}
\bibitem{Kuhn:2001dm} 
  J.~H.~Kuhn and M.~Steinhauser,
  %``Determination of alpha(s) and heavy quark masses from recent measurements of R(s),''
  Nucl.\ Phys.\ B {\bf 619}, 588 (2001)
  [Erratum-ibid.\ B {\bf 640}, 415 (2002)]
  [hep-ph/0109084].
  %%CITATION = HEP-PH/0109084;%%
  %129 citations counted in INSPIRE as of 15 May 2013
%\cite{Kuhn:2007vp}
\bibitem{Kuhn:2007vp} 
  J.~H.~Kuhn, M.~Steinhauser and C.~Sturm,
  %``Heavy Quark Masses from Sum Rules in Four-Loop Approximation,''
  Nucl.\ Phys.\ B {\bf 778}, 192 (2007)
  [hep-ph/0702103 [HEP-PH]].
  %%CITATION = HEP-PH/0702103;%%
  %113 citations counted in INSPIRE as of 15 May 2013
%\cite{Chetyrkin:2009fv}%\cite{Chetyrkin:2010ic}
\bibitem{Chetyrkin:2009fv} 
  K.~G.~Chetyrkin, J.~H.~Kuhn, A.~Maier, P.~Maierhofer, P.~Marquard, M.~Steinhauser and C.~Sturm,
  %``Charm and Bottom Quark Masses: An Update,''
  Phys.\ Rev.\ D {\bf 80}, 074010 (2009)
  [arXiv:0907.2110 [hep-ph]].
  %%CITATION = ARXIV:0907.2110;%%
  %103 citations counted in INSPIRE as of 15 May 2013
%\cite{Chetyrkin:2010ic}
\bibitem{Chetyrkin:2010ic} 
  K.~Chetyrkin, J.~H.~Kuhn, A.~Maier, P.~Maierhofer, P.~Marquard, M.~Steinhauser and C.~Sturm,
  %``Precise Charm- and Bottom-Quark Masses: Theoretical and Experimental Uncertainties,''
  Theor.\ Math.\ Phys.\  {\bf 170}, 217 (2012)
  [arXiv:1010.6157 [hep-ph]].
  %%CITATION = ARXIV:1010.6157;%%
  %15 citations counted in INSPIRE as of 15 May 2013
%\cite{Bodenstein:2011ma}
\bibitem{Bodenstein:2011ma} 
  S.~Bodenstein, J.~Bordes, C.~A.~Dominguez, J.~Penarrocha and K.~Schilcher,
  %``QCD sum rule determination of the charm-quark mass,''
  Phys.\ Rev.\ D {\bf 83}, 074014 (2011)
  [arXiv:1102.3835 [hep-ph]].
  %%CITATION = ARXIV:1102.3835;%%
  %7 citations counted in INSPIRE as of 15 May 2013

%\cite{Bethke:2006ac}
\bibitem{Bethke:2006ac} 
  S.~Bethke,
  %``Experimental tests of asymptotic freedom,''
  Prog.\ Part.\ Nucl.\ Phys.\  {\bf 58}, 351 (2007)
  [hep-ex/0606035].
  %%CITATION = HEP-EX/0606035;%%
  %182 citations counted in INSPIRE as of 30 Apr 2013
\bibitem{Bodenstein} S.~Bodenstein, these proceedings.
%\cite{Bethke:2012jm}
\bibitem{Bethke:2012jm} 
  S.~Bethke,
  %``World Summary of $\alpha_s$ (2012),''
  Nucl.\ Phys.\ Proc.\ Suppl.\  {\bf 234}, 229 (2013)
  [arXiv:1210.0325 [hep-ex]].
  %%CITATION = ARXIV:1210.0325;%%
  %10 citations counted in INSPIRE as of 30 Apr 2013
%\cite{Allison:2008xk}{McNeile:2010ji}
\bibitem{Allison:2008xk} 
  I.~Allison {\it et al.}  [HPQCD Collaboration],
  %``High-Precision Charm-Quark Mass from Current-Current Correlators in Lattice and Continuum QCD,''
  Phys.\ Rev.\ D {\bf 78}, 054513 (2008)
  [arXiv:0805.2999 [hep-lat]].
  %%CITATION = ARXIV:0805.2999;%%
  %104 citations counted in INSPIRE as of 11 May 2013
%\cite{McNeile:2010ji}
\bibitem{McNeile:2010ji} 
  C.~McNeile, C.~T.~H.~Davies, E.~Follana, K.~Hornbostel and G.~P.~Lepage,
  %``High-Precision c and b Masses, and QCD Coupling from Current-Current Correlators in Lattice and Continuum QCD,''
  Phys.\ Rev.\ D {\bf 82}, 034512 (2010)
  [arXiv:1004.4285 [hep-lat]].
  %%CITATION = ARXIV:1004.4285;%%
  %96 citations counted in INSPIRE as of 11 May 2013



\end{thebibliography}

\end{document}


