%Title: Determination of $\alpha_s$ and the charm and bottom quark masses from recent measurements of $R(s)$ 
%Author: J.H. K\"uhn and M. Steinhauser
%Published: <EM> Proceedings of ICHEP 2002, Elsevier Science B.V., Bentvelsen et al. Eds.  </EM> <B>  </B> (2002) 539-541. 
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\begin{document}    

\title{
\vskip-1.6cm{\baselineskip14pt
\centerline{\normalsize TTP02--27 \hfill}
}
\vskip1.0cm
Determination of $\alpha_s$ and the charm and bottom quark masses
from recent measurements of $R(s)$
}

\author{
{J.H. K\"uhn}$^{a}$
and
{M. Steinhauser}$^b$
  \\[3em]
  {\normalsize ${}^a$ Institut f\"ur Theoretische Teilchenphysik,}
  {\normalsize Universit\"at Karlsruhe, D-76128 Karlsruhe, Germany}
  \\[.5em]
  {\normalsize ${}^b$ II. Institut f\"ur Theoretische Physik,}
  {\normalsize Universit\"at Hamburg, D-22761 Hamburg, Germany}
}

\begin{abstract}
Recent experimental data for the total cross
section $\sigma(e^+e^-\to\mbox{hadrons})$ are compared
with the up-to-date
theoretical prediction of perturbative QCD for those energies where
perturbation theory is reliable. This allows for a 
determination of the strong coupling
$\alpha_s^{(4)}(5~\mbox{GeV})=0.235\pm0.047$.
The precise data from the charm threshold region,
when combined with the recent evaluation of moments with three loop
accuracy, lead to a direct determination of the short distance
$\overline{\rm MS}$ charm quark mass $m_c(m_c)=1.304(27)$.
Applying the
same approach to the bottom quark we obtain $m_b(m_b)=4.191(51)$~GeV. 
\end{abstract}

\date{}
\maketitle

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\section{Introduction}

During the past years new and more precise data for 
$\sigma(e^+e^-\to\mbox{hadrons})$ 
have become available in the low energy region between 2 and 10~GeV. 
At the same time increasingly precise calculations have been
performed in the framework of perturbative QCD (pQCD), 
both for the cross section as a
function of the center-of-mass energy $\sqrt{s}$,
including quark mass effects, and for its moments which
allow for a precise determination of the quark mass.
A fresh look at the determination of $\alpha_s$ and
the evaluation of the charm quark mass with the help of sum rules is
thus an obvious task.
We will concentrate on low moments
as suggested by the ITEP group long ago~\cite{NSVZ}.
This is a natural route to determine directly a short distance
mass, say $m_c(m_c)$ or even $m_c(3~\mbox{GeV})$ in the 
$\overline{\rm MS}$ scheme~\cite{NSVZ,ShiVaiZak79}.
In the present work we concentrate on the mass in the 
$\overline{\rm MS}$ scheme, $m_c(\mu)$, and adopt $\mu=3$~GeV as our
default value,
a scale characteristic for the present problem and sufficiently high
to ensure convergence of the perturbative series.
As already discussed in~\cite{NSVZ}, fixed order pQCD is adequate, if
only low moments are used in the analysis.

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

\section{\label{sec:R}$\alpha_s$ from the continuum region}

As stated in the Introduction, we distinguish
two energy regions: first,
the continuum region where pQCD and local duality
are expected to give reliable predictions for the hadronic cross
section and, second, the charm threshold region starting
from the $D$ meson threshold at $3.73$~GeV up to approximately
$5$~GeV, where the cross section exhibits rapid variations,
plus the $J/\Psi$ and $\Psi^\prime$ resonances.
The former will be mainly sensitive to the value of
$\alpha_s$. The latter will be used to evaluate moments and
to determine the charm quark mass.

The most recent theoretical results for $R(s)$ based on NNLO pQCD,
including quark mass effects, were employed in~\cite{KueSte01}.
Assigning
a fully correlated systematic error of $4.3$\% to the BES 
data~\cite{Zhao_priv} they find
\begin{eqnarray}
  \alpha_s^{(3)}(3~\mbox{GeV}) &=&
  0.369^{+0.047}_{-0.046}{}^{+0.123}_{-0.130}
  \,,
  \label{eq:asBES_1}
\end{eqnarray}
from the BES data below $3.73$~GeV and 
\begin{eqnarray}
  \alpha_s^{(4)}(4.8~\mbox{GeV}) &=& 
  0.183^{+0.059}_{-0.064}{}^{+0.053}_{-0.057}
  \,,
  \label{eq:asBES_2}
\end{eqnarray}
from the point at $4.8$~GeV.
The uncertainties refer to the
uncorrelated (statistical) and correlated errors, respectively. 
The same analysis is applicable to the
MD-1 results.
From their average value
$R((8.9~\mbox{GeV})^2)=3.578\pm0.021\pm0.140$
as quoted in their summary
one obtains
\begin{eqnarray}
  \alpha_s^{(4)}(8.9~\mbox{GeV}) &=&
  0.193^{+0.017}_{-0.017}{}^{+0.127}_{-0.107}
  \,,
  \label{eq:asMD1_2}
\end{eqnarray}
From the CLEO value $R((10.52~\mbox{GeV})^2)=3.56\pm0.01\pm0.07$ one
obtains
\begin{eqnarray}
  \alpha_s^{(4)}(10.52~\mbox{GeV}) &=&
  0.186^{+0.008}_{-0.008}{}^{+0.061}_{-0.057}
  \,.
  \label{eq:asCLEO}
\end{eqnarray}

Combining the results 
of Eqs.~(\ref{eq:asBES_1}),~(\ref{eq:asBES_2}),~(\ref{eq:asMD1_2})
and~(\ref{eq:asCLEO}) by assuming uncorrelated systematic
errors and evolving them to a common scale of $5$~GeV leads to
\begin{eqnarray}
  \alpha_s^{(4)}(5~\mbox{GeV}) &=&
  0.235^{+0.047}_{-0.047}
  \,.
\end{eqnarray}  
Evolving this value up to $M_Z$
\begin{eqnarray}
  \alpha_s^{(5)}(M_Z) &=&
  0.124^{+0.011}_{-0.014}
  \,,
\end{eqnarray}  
good agreement with other determinations~\cite{pdg00}
is observed.


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

\section{\label{sec:charm}Charm 
  quark mass determination from the threshold region}

The optimal method for the determination of the charm quark
mass is based on the direct comparison of theoretical and experimental
moments of the charm quark contribution to the photon polarization
function.
In the limit of small momentum the latter can be cast 
into the form~\cite{CheKueSte97}
\begin{eqnarray}
  \Pi_c(q^2) &=& Q_c^2 \frac{3}{16\pi^2} \sum_{n\ge0}
                       \bar{C}_n z^n
  \,,
  \label{eq:pimom}
\end{eqnarray}
with 
$z=q^2/(4m_c^2)$ where $m_c=m_c(\mu)$ is the $\overline{\rm MS}$ 
charm quark mass at the scale $\mu$.
The perturbative series for the coefficients $\bar{C}_n$ 
up to $n=8$ is known
analytically~\cite{CheKueSte96,CheKueSte97} up to order $\alpha_s^2$.
We define the moments through
\begin{eqnarray}
  {\cal M}_n^{\rm exp} &=& \int \frac{{\rm d}s}{s^{n+1}} R_c(s)
  \nonumber\\
  &=&
  {\cal M}_n^{\rm th} \,\,=\,\, 
  \frac{9}{4}Q_c^2
  \left(\frac{1}{4 m_c^2}\right)^n \bar{C}_n
  \,.
  \label{eq:Mth}
\end{eqnarray}
The charm quark mass can be obtained from
\begin{eqnarray}
  m_c(\mu) &=& \frac{1}{2} 
  \left(\frac{\bar{C}_n}{{\cal M}_n^{\rm exp}}\right)^{1/(2n)} 
  \,.
  \label{eq:mc1}
\end{eqnarray}
The analysis of the experimental moments with $n=1,\ldots,4$ leads to
the results displayed in Fig.~\ref{fig:mom}.


The moment with $n=1$ is evidently least sensitive to non-perturbative
contributions from condensates, to the Coulombic higher order effects, the
variation of $\mu$ and the parametric $\alpha_s$ dependence.
Hence 
\begin{eqnarray}
  m_c(m_c) &=& 1.304(27)~\mbox{GeV}
  \,.
  \label{eq:mcfinal}
\end{eqnarray}
is adopted as the final result. Using the two- and three-loop 
relation~\cite{GraBroGraSch90,CheSte99,MelRit99}
between the pole- and the $\overline{\rm MS}$-mass this corresponds to
\begin{eqnarray}
  M_c^{\rm(2-loop)} &=& 1.514(34)~\mbox{GeV}
  \,,
  \nonumber\\
  M_c^{\rm(3-loop)} &=& 1.691(35)~\mbox{GeV}
  \,.
\end{eqnarray}

\begin{figure}[t]
  \begin{center}
    \begin{tabular}{c}
      \leavevmode
      \epsfxsize=6.9cm
      \epsffile[40 250 550 580]{mcmc.ps}
    \end{tabular}
  \end{center}
  \vspace{-3.em}
  \caption{\label{fig:mom}$m_c(m_c)$ for $n=1,2,3$ and $4$.
  For each value of $n$ the results from left to right correspond
  the inclusion of terms of order $\alpha_s^0$, $\alpha_s^1$ and
  $\alpha_s^2$ in the coefficients $\bar{C}_n$.
  Note, that for $n=3$ and $n=4$ the errors can not be determined
  with the help of Eq.~(\ref{eq:mc1})
  in those cases where only the two-loop corrections of order $\alpha_s$ are
  included into the coefficients $\bar{C}_n$ as the equation cannot be
  solved for $m_c(3~\mbox{GeV})$.
  }
\end{figure}

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

\section{\label{sec:bottom}The bottom quark mass}

The same approach is also applicable to the determination of
$m_b$.
Again a significant improvement of the stability of the prediction 
by inclusion of the NNLO terms is observed.
As final result one finds
\begin{eqnarray}
  m_b(m_b) &=& 4.191(51)~\mbox{GeV}
  \,,
  \label{eq:mbmb}
\end{eqnarray}
well consistent with evaluations based on the analysis of bottonium
and high spectral
moments~\cite{PenPiv1,MelYel,BenSig,Hoang:2000fm,Pineda:2001zq,PinSte02}.
The $\overline{\rm MS}$ result of Eq.~(\ref{eq:mbmb}) corresponds to
a pole mass of~\cite{GraBroGraSch90,CheSte99,MelRit99}
\begin{eqnarray}
  M_b^{\rm(2-loop)} &=& 4.651(57)~\mbox{GeV}
  \,,
  \nonumber\\
  M_b^{\rm(3-loop)} &=& 4.819(57)~\mbox{GeV}
  \,,
  \label{eq:Mb}
\end{eqnarray}
using ${\cal O}(\alpha_s^2)$ and ${\cal O}(\alpha_s^3)$ accuracy,
respectively. 

\section{Perspectives~\cite{KueSte02}}

Precise measurements of the $R$-ratio in the charm- and bottom-threshold
region are well within reach at CLEO. The values of $\alpha_s$ as derived from
present measurements in Section~\ref{sec:R} and the impact of a reduction of
the relative error down to 1\% are demonstrated in
Fig.~\ref{fig::asrun}~\cite{KueSte02}.

\begin{figure}[t]
  \begin{center}
    \begin{tabular}{c}
      \leavevmode
      \epsfxsize=6.9cm
      \epsffile[40 250 550 580]{asrun.ps}
    \end{tabular}
  \end{center}
  \vspace{-3.em}
  \caption{\label{fig::asrun}$\alpha_s$ as a function of $\sqrt{s}$.
    Results from $\tau$ and
    $Z$ decays and those extracted in Section~\ref{sec:R} are shown.
    The two error bars on the data points indicate the statistical
    (inner) and systematical (outer) uncertainty.
    For the combined result (indicated by a star)
    at $\sqrt{s}=5.0$~GeV only the 
    error after adding the statistical and systematical uncertainty in
    quadrature is shown.
    For illustration $\delta\alpha_s$
    reduced to the error as expected from $\delta R/R=1\%$ are also shown
    (slightly displaced in order to make the presentation more visible).
          }
\end{figure}

The quark mass determinations are at present strongly affected by the
relatively large error in the electronic widths of the narrow charmonium and
bottomium resonances and by the poorly measured threshold region. More precise
measurements might well lead to quark mass determinations with $\delta
m_Q\approx 15$~MeV and 30~MeV for charm and bottom, respectively.


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\begin{thebibliography}{99}

\def\ap#1#2#3{  {Ann.\ Phys.\ B }{\bf #1} (#2) #3}
\def\app#1#2#3{  {Act.\ Phys.\ Pol.\ B }{\bf #1} (#2) #3}
\def\cpc#1#2#3{  {Comp.\ Phys.\ Commun.\ }{\bf #1} (#2) #3}
\def\cmp#1#2#3{  {Comm.\ Math.\ Phys.\ }{\bf#1} (#2) #3}
\def\epjc#1#2#3{ {Eur.\ Phys.\ J.\ C }{\bf #1} (#2) #3}
\def\fortp#1#2#3{{Fortschr.\ Phys.\ }{\bf#1} (#2) #3}
\def\jcp#1#2#3{  {J.\ Comp.\ Phys.\ }{\bf#1} (#2) #3}
\def\nima#1#2#3{ {Nucl.\ Inst.\ Meth.\ A }{\bf #1} (#2) #3}
\def\npb#1#2#3{  {Nucl.\ Phys.\ B }{\bf #1} (#2) #3}
\def\nca#1#2#3{  {Nuovo Cim.\ A }{\bf #1} (#2) #3}
\def\plb#1#2#3{  {Phys.\ Lett.\ B }{\bf #1} (#2) #3}
\def\prc#1#2#3{  {Phys.\ Reports }{\bf #1} (#2) #3}
\def\prd#1#2#3{  {Phys.\ Rev.\ D }{\bf #1} (#2) #3}
\def\pR#1#2#3{   {Phys.\ Rev.\ }{\bf #1} (#2) #3}
\def\prl#1#2#3{  {Phys.\ Rev.\ Lett.\ }{\bf #1} (#2) #3}
\def\pr#1#2#3{   {Phys.\ Reports }{\bf #1} (#2) #3}
\def\ppnp#1#2#3{ {Prog.\ Part.\ Nucl.\ Phys.\ }{\bf #1} (#2) #3}
\def\sovnp#1#2#3{{Sov.\ J.\ Nucl.\ Phys.\ }{\bf #1} (#2) #3}
\def\tmf#1#2#3{  {Teor.\ Mat.\ Fiz.\ }{\bf #1} (#2) #3}
\def\yadfiz#1#2#3{{Yad.\ Fiz.\ }{\bf #1} (#2) #3}
\def\zpc#1#2#3{  {Z.\ Phys.\ C }{\bf #1} (#2) #3}
\def\ibid#1#2#3{ {ibid.\ }{\bf #1} (#2) #3}

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\end{thebibliography}

\end{document}

