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\begin{center}
{\large \bf 
Precise Charm and Bottom Quark Masses\footnote{Presented at the Workshop
{\em Constraining the hadronic contributions to the muon's anomalous
  magnetic moment $g-2$ and $\alpha_{\rm em}(M_Z)$}, Trento, April 10-12, 2013}
\\ }
\vspace{5mm} J.H. K\"uhn\footnote{Johann.Kuehn@KIT.edu} 

\vspace{5mm}
Institut f\"ur Theoretische Teilchenphysik, Karlsruhe Institut f\"ur Technologie,\\
76128 Karlsruhe, Germany
\end{center}
\vspace{5mm}

{\bf 1. Method\\}

Exploiting the analyticity of the vacuum polarization function
$\Pi(q^2)$ 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.~\cite{Shifman:1978bx} and applied to a
precise charm and bottom mass determination in Ref.~\cite{Kuhn:2001dm}
once the
three-loop results had become available. A significantly improved reanalysis
based on four-loop moments as obtained in Refs.~%
\cite{Chetyrkin:2006xg,Boughezal:2006px,Maier:2008he,Maier:2009fz,Sturm:2008eb,%
Kiyo:2009gb}
and with new data has been performed
in Ref.~\cite{Kuhn:2007vp}, additional
updates and improvements from new data and the precise analytical evaluation of the 
perturbative moments can be 
found in Refs.~\cite{Chetyrkin:2009fv,Chetyrkin:2010ic}.
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.~\cite{Kuhn:2007vp}. 
Furthermore, the potential influence of a
non-vanishing gluon condensate 
$\langle\frac{\alpha_s}{\pi}GG\rangle=0.006\pm0.012\,{\rm GeV}^4$ 
on the charm mass determination
has been analysed \cite{Kuhn:2007vp,Chetyrkin:2010ic}
and found to be small.\\

{\bf 2. 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. Note that we assume the validity of pQCD in this region with high
precision, an assumption that is well consistent with present
measurements but for the moment remains an assumption, to be
verified e.g. by future BESS experiments (for charm) and Belle (for bottom).

  \begin{table}[t]
{
\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}
\caption{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.} 
} 
\label{tab:contglu}
  \end{table}

  \begin{table}[t]
%    \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}
%}
  \caption{Results for $m_c(3\, {\rm GeV} )$ in ${\rm GeV}$. The errors
      are from experiment, $\alpha_s$, variation of $\mu$ and the
     gluon condensate.}
    \label{tab:charmTab}}
\  \end{table}
The results for the moments from one to four and the error budget are 
listed in Table~1, 
those for the quark mass in Table~2.
The moment with $n=1$ is most robust from the theory side, as
evident from the relatively smaller coefficient in the perturbative
series.  
In view of the smallest sensitivity to
$\alpha_s$ and to the choice of the renormalisation scale $\mu$ we adopt
the value as derived from $n=1$ as our final result:
\begin{equation}
m_c(3\,{\rm GeV})=986(13)\,{\rm MeV}. 
\end{equation}
 

Tables~1 and 2 also
illustrate the path to a further reduction of the error. 
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.

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.~\cite{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 $m_b(m_b)=3610(16)\,{\rm MeV}$. 

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\end{thebibliography}


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