%Title: Precise Charm- and Bottom-Quark Masses: Recent Developments
%Author: J. H. K&uuml;hn
%Published:     FLAVIAnet Topical Workshop Kazimierz, Poland, 23-27 July 2009. To be published in the proceedings.

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\begin{document}
\title{Precise Charm- and Bottom-Quark Masses:\\ Recent Developments}
\author{J. H. K\"uhn
\address{Karlsruhe Institute of Technology}
}
\thanks{FLAVIAnet Topical Workshop 2009}
\maketitle
\begin{abstract}
Recent theoretical and experimental improvements in the determination of
charm- and bottom-quark masses are discussed. The final results,
$m_c(3\,\text{GeV})=986(13)\,$MeV and $m_b(m_b)=4163(16)\,$MeV represent
the presently most precise determinations of these two fundamental Standard Model
parameters.
\end{abstract}
%\PACS{}

The past years have witnessed significant improvement in the
determination of charm- and bottom-quark masses as a consequence of
improvements in experimental techniques as well as theoretical
calculations. Both masses are critical ingredients in the evaluation of
various observables, e.g. the masses of charm and bottom mesons and the
corresponding quarkonia, the rates for weak decays of $B$ mesons and the
closely related CKM matrix elements $|V_{cb}|^2$ and $|V_{ub}|^2$. Rare
kaon decays and the $b\to s\gamma$ transition are critically affected by
contributions from virtual charmed quarks. If the Higgs boson is as
light as expected from electroweak precision measurements, its decay is
dominated by bottom quarks with a rate proportional $m_b^2$.
From the more theoretical side it is clear that $m_b$ is a decisive
input for any attempt of Yukawa unification, in the framework of
$SU(5)$, relating $\tau$ and bottom Yukawa couplings, or $SO(10)$,
relating those of $\tau$, bottom and top quarks. Requiring comparable
relative precision of top and bottom masses, $\delta m_b/ m_b \sim
\delta m_t/ m_t$, an error $\delta m_b$ around or below 25 MeV is
required to fully exploit the current precision of $m_t$ with $\delta
m_t \approx 1 {\rm GeV}$, not to speak of the potential precision of a
future linear collider.

As stated above, quark mass determinations can be based on a variety of
observations and theoretical calculations. The one presently most precise
 follows an idea advocated by the ITEP group more than
thirty years ago \cite{Shifman:1978bx}, and has gained renewed interest after significant
advances in higher order perturbative calculations discussed in this
paper have been achieved.  It exploits the fact that the vacuum polarization function
$\Pi(q^2)$ and its derivatives, evaluated at $q^2=0$, can be considered
short distance quantities with an inverse scale characterized by the
distance between the reference point $q^2=0$ and the location of the
threshold $q^2=(3{\rm GeV)}^2 $ and $q^2=(10{\rm GeV})^2$ for charm and
bottom respectively. This idea has been taken up in \cite{Kuhn:2001dm} after the
first three-loop evaluation of the moments became available \cite{Chetyrkin:1995ii,Chetyrkin:1996cf,Chetyrkin:1997mb} and
has been further improved in \cite{Kuhn:2007vp} using four-loop results
\cite{Chetyrkin:2006xg,Boughezal:2006px} for the
lowest moment. An analysis which is based on the most recent theoretical
\cite{Maier:2008he,Maier:2009fz,Kiyo:2009gb} and experimental progress has been performed in \cite{Chetyrkin:2009fv}
and will be reviewed in the following.

Let us recall some basic notation and definitions. The vacuum
polarization $\Pi_Q(q^2)$ induced by a heavy quark $Q$ with charge
$Q_Q$ (ignoring in this short note the so-called singlet contributions),
is an analytic function with poles and a branch cut at and above
$q^2=M_{J/\psi}^2$. Its Taylor coefficients $\bar C_n$, defined through
\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}
can be evaluated in pQCD
\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
     + \bar{C}_n^{(33)}l_{m_Q}^3 \right) +
    \ldots \,.
  \end{split}
  \label{eq:c_exp}
\end{equation}
Here $z\equiv q^2/4m_Q^2$, where $m_Q=m_Q(\mu)$ is the running
$\overline{\text{MS}}$ mass at scale $\mu$. Using a once-subtracted dispersion
relation
\begin{equation}
  \label{eq:disp_rel}
  \Pi_Q(q^2)=\frac{1}{12\pi^2}\int^\infty_0 ds \frac{R(s)}{s(s-q^2)}
\end{equation}
(with $R_Q$ denoting the familiar $R$-ratio for the production of heavy quarks),
the Taylor coefficients can be expressed through moments of $R_Q$.
Equating perturbatively calculated and experimentally measured moments,
\begin{equation}
  \label{eq:M_exp}
  {\cal M}_n^{\text{exp}}=\int \frac{ds}{s^{n+1}}R_Q(s)
\end{equation}
leads to an ($n$-dependent) determination of the quark mass
\begin{equation}
  \label{eq:m_Q}
  m_Q=\frac{1}{2}\left(\frac{9Q_Q^2}{4}\frac{C_n}{{\cal M}_n^{\text{exp}}}\right)\,.
\end{equation}
The consistency of the results for different $n$
  and their stabilization with increasing orders in perturbation theory
  gives confidence in their reliability.

  Significant progress has been made in the perturbative evaluation of
  the moments since the first analysis of the ITEP group. The ${\cal
    O}(\alpha_s^2)$ contribution (three loops) has been evaluated more
  than 13 years ago \cite{Chetyrkin:1995ii,Chetyrkin:1996cf,Chetyrkin:1997mb}, as far as the terms up to $n=8$ are
  concerned, recently even up to $n=30$ \cite{Boughezal:2006uu,Maier:2007yn}.  About ten years later the
  lowest two moments ($n=0,1$) of the vector correlator were evaluated
  in ${\cal O}(\alpha_s^3)$, i. e. in four-loop approximation
  \cite{Chetyrkin:2006xg,Boughezal:2006px}. 
  The corresponding two lowest moments for the pseudoscalar
  correlator were obtained in \cite{Sturm:2008eb} in order to derive the charmed
  quark mass from lattice simulations \cite{Allison:2008xk}.  In \cite{Maier:2008he,Maier:2009fz} the second and
  third moments were evaluated for vector, axial and pseudoscalar
  correlators. Combining, finally, these results with information about
  the threshold and high-energy behaviour in the form of a Pad\'e
  approximation, the full $q^2$/dependence of all four correlators was
  reconstructed and the next moments, from four up to ten, were obtained
  with adequate accuracy. A list of the relevant coefficients is shown in Table~\ref{tab:2} (for
an earlier, less precise analysis, see Ref. \cite{Hoang:2008qy}).

\begin{table}
  \centering
  $\begin{array}{|c|c|c|c|}
\hline
                    &   n_l=3     & n_l=4       & n_l=5       \\\hline
\bar{C}^{(3),v}_1    & -5.6404     & -7.7624     & -9.6923       \\\hline
\bar{C}^{(3),v}_2    & -3.4937     & -2.6438     & -1.8258       \\\hline
\bar{C}^{(3),v}_3    & -2.8395     & -1.1745     & 0.4113       \\\hline
\bar{C}^{(3),v}_4    & -3.349(11)  & -1.386(10)  & 0.471(9)    \\\hline
\bar{C}^{(3),v}_5    & -3.737(32)  & -1.754(32)  & 0.104(27)   \\\hline
\bar{C}^{(3),v}_6    & -3.735(61)  & -1.910(63)  & -0.228(54)  \\\hline
\bar{C}^{(3),v}_7    & -3.39(10)   & -1.85(10)   & -0.46(9)    \\\hline
\bar{C}^{(3),v}_8    & -2.85(13)   & -1.67(14)   & -0.66(12)   \\\hline
\bar{C}^{(3),v}_9    & -2.22(17)   & -1.47(18)   & -0.91(16)   \\\hline
\bar{C}^{(3),v}_{10} & -1.65(20)   & -1.37(22)   & -1.30(19)   \\\hline
  \end{array}$

\caption{Expansion coefficients from the reconstructed vector 
polarization function for different numbers of light
    quarks in the $\overline{\text{MS}}$ scheme. $C^{(3),v}_{1-3}$
are exact.}
  \label{tab:2}
\end{table}

Most of the experimental input had already been compiled and exploited
in \cite{Kuhn:2007vp}, where it is described in more detail. However, until recently
the only measurement of the cross section above but still close
$B$-meson threshold was performed by the CLEO collaboration more than
twenty years ago \cite{Besson:1984bd}. Its large systematic uncertainty was responsible for a
sizable fraction of the final error on $m_b$.  This measurement has been
recently superseded by a measurement of BABAR \cite{:2008hx} with a
systematic error between 2 and 3\%. In \cite{Chetyrkin:2009fv} the radiative
corrections were unfolded and used to obtain a significantly improved
determination of the moments.
\begin{figure}
  \centering
  \includegraphics[width=\linewidth]{cleobabardat}
  \caption{Comparison of rescaled CLEO data for $R_b$ with BABAR data
    before and after deconvolution. The black bar on the right corresponds to
    the theory prediction~\cite{Harlander:2002ur}.}
  \label{fig:R_b}
\end{figure}
\begin{table}
  \centering
    \begin{tabular}{l|r@{}l|rrrr|c}
\hline
$n$ & \multicolumn{2}{|c|}{$m_c(3~\text{GeV})$} & 
exp & $\alpha_s$ & $\mu$ & np & 
total
\\
\hline
%%%%%
       1&   &986&  9&   9&   2&  1 & 13 \\
       2&   &976&  6&  14&   5&  0 & 16 \\
       3&   &978&  5&  15&   7&  2 & 17 \\
       4&  1&004&  3&   9&  31&  7 & 33 \\
%%%%%
\hline
\end{tabular}

\caption{\label{tab:mc1}Results for $m_c(3~\text{GeV})$ in MeV
  obtained from Eq.~\eqref{eq:m_Q}. The errors are from experiment,
  $\alpha_s$, variation of $\mu$
  and the gluon condensate.
}
\end{table}
\begin{table}
\centering
\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{table}
The final results for $m_c (3 {\rm GeV})$ and $m_b (10 {\rm GeV})$ are
listed in Tables \ref{tab:mc1} and \ref{tab::mb}. Despite the significant
differences in the composition of the errors, the results for different
values of $n$ are perfectly
consistent. For charm the result 
from $n=1$ has the smallest dependence on the strong coupling and the smallest
total error, which 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 \cite{Chetyrkin:2009fv}
$m_c(m_c)=1279(13)~\text{MeV}$. Let us recall at this point that a recent
study \cite{Allison:2008xk}, 
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 in remarkable
agreement with \cite{Kuhn:2007vp,Chetyrkin:2009fv}.  

The treatment of the bottom quark case proceeds along
similar lines. However, in order to suppress the theoretically evaluated
input above 11.2 GeV (which corresponds to roughly 60\% for the lowest,
40\% for the second and 26\% for the third moment), the result from the
second moment has been adopted as our final result,
\begin{equation}
m_b(10~\text{GeV}) =3610(16)~\text{MeV},
\label{eq:m_b}
\end{equation}
corresponding to $m_b(m_b)=4163(16){\rm MeV}$. The explicit $\alpha_s$
dependence of $m_c$ and $m_b$ can be found in
\cite{Chetyrkin:2009fv}. When considering the ratio of charm and bottom
quark masses, part of the  $\alpha_s$ and of the $\mu$ dependence cancels
\begin{equation}
  \label{eq:m_ratio}
  \frac{m_c(3~\text{GeV})}{m_b(10~\text{GeV})}=
  0.2732 -\frac{\alpha_s-0.1189}{0.002}\cdot 0.0014 \pm 0.0028 
  \,,
\end{equation}
which might be a useful input in ongoing analyses of bottom decays.

The results presented in \cite{Chetyrkin:2009fv} constitute the most precise values for the
charm- and bottom-quark masses available to date. Nevertheless it is
tempting to point to the dominant errors and thus identify potential
improvements.  In the case of the charmed quark the error is dominated
by the parametric uncertainty in the strong coupling
$\alpha_s(M_Z)=0.1185\pm0.002$. Experimental and theoretical errors are
comparable, the former being dominated by the electronic width of the
narrow resonances. In principle this error could be further reduced by
the high luminosity measurements at BESS III. A further reduction of the
(already tiny) theory error, e.~g.~through a five-loop calculation
looks difficult. Further confidence in our result can be obtained from
the comparison with the forementioned lattice evaluation.

Improvements in the bottom quark mass determination could originate
from the experimental input, e.\ g.\ through an improved
determination of the electronic widths of the narrow $\Upsilon$
resonances or through a second, independent measurement of the $R$ ratio
in the region from the $\Upsilon(4S)$ up to 11.2 GeV. As shown in Fig. \ref{fig:R_b},
there is a slight mismatch between the theory prediction above 11.2 GeV
and the data in the region below with their systematic error of less
than 3\%.

{\it To summarize:}
Charm and bottom quark mass determinations have made significant progress
during the past years. A further reduction of the theoretical and experimental
error seems difficult at present. However, independent experimental results on
the $R$ ratio would help to further consolidate the present situation.
The confirmation 
by a recent  lattice analysis with similarly small uncertainty gives 
additional confidence in the result for $m_c$.


{\it Acknowledgements:} I would like to thank K. Chetyrkin, A. Maier,
P.~Maier\-h\"ofer, P. Marquard, M. Steinhauser and C. Sturm for their collaboration and
A. Maier for help in preparing this manuscript.

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