%Title: Quark masses from low-energy moments of heavy-quark current correlators
%Author: P. Marquard
%Published:     <EM>PoS(EPS-HEP 2009)204</EM>

\documentclass{PoS}
\usepackage{amsmath}

\title{Quark masses from low-energy moments of heavy-quark current correlators}

\ShortTitle{Quark masses from low-energy moments of heavy-quark current correlators}

\author{\speaker{Peter Marquard}\thanks{In collaboration with 
K.~Chetyrkin, Y.~Kiyo, J.H.~K\"uhn, A.~Maier, P.~Maierh\"ofer,
A.V.~Smirnov, M.~Steinhauser, C.~Sturm}\\
        Institut f\"ur Theoretische Teilchnphysik \\
        Universit\"at Karlsruhe, KIT\\
        E-mail: \email{peter.marquard@kit.edu}}


      \abstract{We present an update of the determination of charm- and
        bottom-quark masses from the low-energy expansion of the vacuum
        polarization function in combination with experimental data for
        $R(s)$. This analysis leads to the most precise determination of
        the masses of the charm and bottom quarks:
        $m_c(3\,\mbox{GeV}) = 986(13)\, \mbox{MeV}$, $m_b(m_b) =
        4163(16)\,\mbox{MeV}$. }

\FullConference{European Physical Society Europhysics Conference on High Energy Physics,
EPS-HEP 2009,\\
		 July 16 - 22 2009\\
		 Krakow, Poland}

\begin{document}

\section{Introduction}
\label{sec:introduction}
The precise determination of the masses of the charm and bottom quark
are of utmost importance since many observables in flavor
physics are highly sensitive to them. Important examples are the
semi-leptonic decay rates of $B_d$ and $B_c$ mesons
\begin{align}
  \label{eq:1}
  &\Gamma(B_d\to \ell \nu K)  \propto m_b^5,  &\Gamma(B_c\to \ell \nu K)  \propto m_b^5 f(m_c^2 / m_b^2) .
\end{align}
Another example is the decay rate of the Higgs boson into bottom
quarks where the dominant uncertainty stems from the dependence on the
bottom-quark mass.
\begin{equation}
  \label{eq:2}
  \Gamma(H\to b\bar b) = \frac{G_F M_H}{4 \sqrt{2} \pi}
  m_b^2(1+5.6667 a_s + \cdots - 825.7 a_s^4 ) , \quad a_s=\alpha_s / \pi .
\end{equation}

One possible and maybe the most precise way to determine the masses of
the charm and the bottom quark are low-energy sum rules. Here one
compares weighted integrals ${\cal M}_n^{exp}$ of the experimentally
measured R-ratio
\begin{align}
  \label{eq:3}
&{\cal M}_n = \int ds \frac{R(s)}{s^{n+1}},   &  R(s) = \frac{\sigma(e^+ e^- \to hadrons)}{e^+ e^-  \to \mu^+ \mu^-}
\end{align}
on the one hand and coefficients $C_n^{th}$ of the low-energy
expansion of the vacuum polarization function
\begin{align}
  \label{eq:4}
&\Pi^{\mu\nu}(q^2) = (q^2 g^{\mu\nu} - q^\mu q^ \nu) \Pi(q^2), 
& \Pi(q^2) = Q_q^2 \frac{3}{16 \pi^2 }\sum C_n^{\mathrm th}  z^n , && z = \frac{q^2 }{ 4 m_q^2}
\end{align}
on the other hand. Relating Eq. (\ref{eq:3}) and Eq. (\ref{eq:4}) by
means of a dispersion relation the quark mass is given by
\begin{equation}
  \label{eq:5}
  m_q = \frac{1}{2} \left ( \frac{9 Q_q^2 C_n^{\mathrm th}}{{4 \cal M}_n} \right )^{\frac{1}{2n}}.
\end{equation}
In the following we will discuss how to calculate the low-energy
expansion of the vacuum polarization function and use it in combination
with experimental data for the R-ratio in order to determine the values of the
quark masses.
\section{Low-energy expansion of the vacuum polarization function and reconstruction of $\bf R(s)$ using Pad\'e approximations}
\label{sec:low-energy-expansion}
The low-energy expansion of the vacuum polarization has been computed up
to four-loops in perturbative QCD
\cite{Chetyrkin:2006xg,Boughezal:2006px,Maier:2008he,Sturm:2008eb,Maier:2009}.
While at four-loops it was only possible to calculate the first three
terms, at three-loop level as many as 30 terms are available
\cite{Maier:2007yn,Boughezal:2006uu}. In the four-loop case 
the limiting factor is the factorial growth of the number
of integrals appearing in the calculation. The details of the
calculation are beyond the scope of this article, so we only give the
numerical results in Table~\ref{tab:moments}.  

Besides the terms in the low-energy expansion discussed above much more
information is available for the behavior of the vacuum polarization
function. There are two more distinct energy regions, namely the
euclidean high-energy and the threshold region. In both regions some
terms in the corresponding expansions are known. Combining all the
available information one can try to reconstruct the full energy
dependence of the vacuum polarization function
\cite{Hoang:2008qy,Kiyo:2009}. This approach relies on the construction
of Pad\'e approximants, which are a generalization of the well known
Taylor series. Once the full energy dependence of the vacuum
polarization function is known it can be used to obtain precise predictions
for the coefficients of the low-energy expansion. These predictions are
given in Table \ref{tab:moments}. Examples for the form of vacuum
polarization function are shown in Fig. \ref{fig:pade}. There we show its
behavior below threshold, while above threshold we show $R(s)$ which
can be related to the vacuum polarization function using $R(s) =
12 \pi \mathrm{Im} (\Pi(s + i\epsilon))$.
\begin{table}
  \centering
\fontsize{9}{12}\selectfont
  \begin{tabular}{cc}
  $\begin{array}{l|c|c|c}
\hline
                    &   n_l=3     & n_l=4       & n_l=5       \\\hline\hline
{C}^{(3)}_1    & -5.6404     & -7.7624     & -9.6923       \\
{C}^{(3)}_2    & -3.4937     & -2.6438     & -1.8258       \\
{C}^{(3)}_3    & -2.8395     & -1.1745     & 0.4113       \\
{C}^{(3)}_4    & -3.349(11)  & -1.386(10)  & 0.471(9)    \\
{C}^{(3)}_5    & -3.737(32)  & -1.754(32)  & 0.104(27)   \\
{C}^{(3)}_6    & -3.735(61)  & -1.910(63)  & -0.228(54)  \\
{C}^{(3)}_7    & -3.39(10)   & -1.85(10)   & -0.46(9)    \\
{C}^{(3)}_8    & -2.85(13)   & -1.67(14)   & -0.66(12)   \\
{C}^{(3)}_9    & -2.22(17)   & -1.47(18)   & -0.91(16)   \\
{C}^{(3)}_{10} & -1.65(20)   & -1.37(22)   & -1.30(19)    \\\hline
  \end{array}$
\end{tabular}
\caption{Low-energy expansion coefficients of the vector current
  correlator in four-loop approximation. The heavy quark mass is
  renormalized in the $\overline {\mbox{MS}}$ scheme. The number of
  light quarks is given by $n_l$. The first three coefficients have
  been calculated directly while the others are obtained from the
  reconstructed form of the vacuum polarization function.}
\label{tab:moments}
\end{table}
\begin{figure}
  \centering
  \includegraphics[scale=0.60]{V3charmle_stddev.eps}\hfill
  \includegraphics[scale=0.60]{V3charmR.eps}
  \caption{Pad\'e approximations of the vacuum polarization function
    below threshold (left) and the $R$-ratio above threshold (right). In
    addition, the expansions about $q^2=0$, $q^2=4 m^2$ and $q^2\to
    \infty$ are shown. The $R$-ratio is plotted against the velocity of
    the heavy quarks. }
  \label{fig:pade}
\end{figure}
\section{Determination of the Quark Masses}
\label{sec:masses}
The main experimental ingredient is the measurement of $R(s)$ at the
threshold region. In case of the charm quark the available data is in
good agreement with the predictions of perturbative QCD which is valid
above threshold. The analysis leads to a value of $m_c(3\,\mbox{GeV}) =
986(13)\, 
\mbox{MeV}$, which together with the analysis in \cite{Allison:2008xk} based on
lattice simulations constitutes the most precise determination of the
charm-quark mass. In the case of the bottom quark, the situation was
less clear for quite some time since the CLEO data from 1985 was clearly
in disagreement with the predictions from perturbative QCD and was
therefore rescaled to match these predictions. The situation has greatly
been improved with new data from the BABAR collaboration (cp.
Fig. \ref{fig:rs}). Incorporating the new data in the analysis for the
bottom-quark mass and using the improved predictions for the theoretical
low-energy parameters shown above one obtains a value of $m_b(m_b) = 4163(16)\,\mbox{MeV}$ for the
mass of the bottom quark. The full details of the analysis can be found in
\cite{Chetyrkin:2009fv}.
\begin{figure}
  \centering
  \includegraphics[scale=0.4]{cleobabardat.eps}
  \caption{$R(s)$ below the $b \bar b$ threshold. In the figure the
    rescaled CLEO data is shown in comparison with the new data from the
  BABAR collaboration.}
  \label{fig:rs}
\end{figure}

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\end{document}


