% Please use the skeleton file you have received in the 
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\documentclass{PoS}

\usepackage[]{graphicx}
\graphicspath{{figures/}}
\usepackage{amsmath}
\DeclareMathOperator{\Li}{Li}
\DeclareMathOperator{\Res}{Res}

\title{From low-energy moments of $\Pi(q^2)$ to $R(s)$}

\ShortTitle{From low-energy moments of $\Pi(q^2)$ to $R(s)$}

\author{\speaker{Peter Marquard}\thanks{In collaboration with Y.~Kiyo,
    A.~Maier, P.~Maierh\"ofer}\\
        Karlsruhe Institute of Technology\\
        E-mail: \email{peter.marquard@kit.edu}}

%\author{Another Author\\
%        Affiliation\\
%        E-mail: \email{...}}

      \abstract{ We construct approximate formulas for the ${\cal
          O}(\alpha_s^3)$ QCD contributions to the vector current
        correlator, which is valid for arbitrary values of momenta and
        masses. The derivation is based on conformal mapping and the
        Pad\'e approximation procedure and incorporates known expansions
        in the low energy, threshold and high energy regions. We use our
        results to estimate additional terms in these expansions.  }

\FullConference{RADCOR 2009 - 9th International Symposium on Radiative Corrections (Applications of Quantum Field Theory to Phenomenology) ,\\
		 October 25 - 30 2009\\
		 Ascona, Switzerland}

\begin{document}

\section{Introduction}
\label{sec:intro}
Correlators of heavy quark currents in different kinematical regions are of interest for a number of
 phenomenological applications. These two-point functions only depend on
 two scales, namely the square of the external four-momentum $q^2$ and
 the heavy quark mass $m$. Many of the applications focus on one of
 three distinct kinematical regions: The low energy region with $q^2
 \approx 0$, the quark pair production threshold at $q^2=4m^2$ and the
 (euclidean) high energy region $-q^2 \to \infty$. 

Moments in the low energy expansion can be used for
 precise extractions of charm and bottom quark masses via sum rules
 \cite{Novikov:1977dq} (for reviews see \cite{Reinders:1984sr}),
 whereas threshold and high energy expansions are directly related to
 production cross sections for $t \bar{t}$, 
charmed hadrons or bottom hadrons in the respective energy regions.

The aforementioned expansions for the correlators in the three regions
are known relatively well: In the low energy region at ${\cal O}
(\alpha_s^2)$ the leading eight coefficients were computed more than ten
years ago \cite{Chetyrkin:1995ii}, and
as of today as many as $30$ moments are known
\cite{Boughezal:2006uu}.  At ${\cal O}(\alpha_s^3)$,
however, only the first three physical moments of the vector correlator
are available \cite{Sturm:2008eb}.

Threshold expansions for correlators are expansions in the small heavy
quark velocity $v=\sqrt{1-4m^2/q^2} \ll 1$. Currently all of the
necessary machinery for NNLO threshold expansion is known (see for
instance Ref. \cite{Hoang:2000yr} and references therein). This means,
that all terms of order $(\alpha_s^n/v^{n-1})\cdot\{1,\,v,\,v^2\}$ are
in principle known. Explicit expansions for vector
correlators can be derived from Refs.
\cite{Penin:1998ik}.

For high energies the leading seven coefficients in the expansion are
known at ${\cal O}(\alpha_s^2)$ for the vector 
current \cite{Chetyrkin:1997qi}.  At order $\alpha_s^3$ the
first two terms in the high energy expansion of the vector correlator
have been published in Refs. \cite{Baikov:2004ku}.  More
information is available for the absorptive parts of the correlators,
which correspond to the logarithmic terms in the high energy
expansions. Here the first three coefficients are known for the vector
 correlator
\cite{Gorishnii:1991vf}.

Still, it would be desirable to have results
for the correlators which are valid for arbitrary energies in addition
to these expansions. One obvious
benefit would be the possibility to expand such a result in a
kinematical region of interest in order to obtain even more coefficients in
the expansion. It would also be possible to predict values of cross sections for
intermediate regions between threshold and high energies, where the mere
expansions may not be very accurate anymore. Last but not least, the
full energy dependence is essential for those QCD sum rule approaches to
quark mass determination which use either the Borel transformation of
the correlator \cite{Novikov:1977dq} or so-called contour-improved
perturbation theory \cite{Pivovarov:1991bi}.

Unfortunately, analytical results which are valid for arbitrary energies
only exist up to ${\cal O}(\alpha_s)$ \cite{Kallen:1955fb}. Still, it is
possible to reconstruct the full energy dependence approximately from
the known expansions at higher orders. Using a seminumerical method
based on Pad\'e approximations \cite{Baker:1975}, the correlator of the 
vector current was reconstructed at ${\cal O}(\alpha_s^2)$
\cite{Chetyrkin:1995ii}. Moreover, it
was demonstrated in \cite{Hoang:2008qy} that in spite of the rather low
amount of information available at ${\cal O}(\alpha_s^3)$ it is still
viable to reconstruct the vector correlator and predict expansion
coefficients with decent accuracy.

In this work we use the Pad\'e approximation method to reconstruct the
vector heavy quark current correlators at ${\cal O}(\alpha_s^3)$ and
derive approximations to previously unknown expansion coefficients in
the low energy, threshold and high energy regions.

% This paper is structured in the following way:
% In Section \ref{sec:meth} we set up our conventions and explain the
% general approximation method. Section \ref{sec:calc} contains details of
% the application of the method to the different correlators and the
% choice of physically meaningful approximants. The results for the
% reconstructed correlators and the estimates for new expansion
% coefficients can be found in Section \ref{sec:res}. In Section
% \ref{sec:conc} we summarise the work and give our conclusions.

\section{Calculation}
\label{sec:meth}


\subsection{Polarisation functions}
\label{sec:polf}

It is convenient to explicitly extract the Lorentz structure of the
heavy quark vector current correlator and define the polarisation function
$\Pi^v(q^2)$:
\begin{equation}
  \label{eq:1}
   (-q^2 g_{\mu\nu} + q_\mu q_\nu) \Pi^v (q^2) = i\int dx e^{iqx} \langle 0 |Tj_\mu^v(x)
   j_\nu^v(0)|0\rangle , \quad j_\mu^v = \bar\psi \gamma_\mu \psi \,.
\end{equation}
  We do not take
into account singlet contributions originating from diagrams with
massless cuts  and choose the normalisation
$\Pi^v(0)=0 $.
The perturbative expansions of the polarisation function reads
\begin{equation}
  \label{eq:3}
  \Pi^v = \Pi^{(0),v} + C_F
  \Pi^{(1),v} \frac{\alpha_s}{\pi}+ 
  \Pi^{(2),v} \left(\frac{\alpha_s}{\pi}\right)^2+  \Pi^{(3),v} \left(\frac{\alpha_s}{\pi}\right)^3+ \dots\,,
\end{equation}
where $C_F=\frac{4}{3}$ is the quadratic Casimir operator for the adjoint representation.
A natural variable to describe the behaviour of $\Pi^v$ is given by $  z
= \frac{q^2}{4m^2}$ 
where $m$ denotes the heavy quark mass defined in the on-shell scheme.

% For the construction of approximants we need expansions in the low
% energy, threshold and euclidean high energy regions, which correspond to
% $z=0,\,1\text{ and } -\infty$ respectively. Around $z=0$ the expansion reads
% \begin{equation}
%   \label{eq:5}
%   \Pi^{(i),v}(z) = \frac{3}{16\pi^2}\sum_{n=1}^\infty C^{(i),v}_n z^n\,.
% \end{equation}
% The renormalisation scale $\mu$ is set to $m$, so that the coefficients
% $C^{(i),v}_n$ are simply real numbers. 

% Around $z=1$ we have
% \begin{equation}
%   \label{eq:6}
%   \Pi^{(i),v}(z) = \sum_{k=k_0}^\infty \sum_{l\geq 0}
%   K^{(i),v}_{\frac{k}{2}\,l}
%   (1-z)^\frac{k}{2} \log^l (1-z)\,.
% \end{equation}
% The lower bound $k_0$ of the sum is $3-i$ for the axial-vector and
% scalar correlators and $1-i$ for the vector and pseudo-scalar
% correlators. In the na\"ive Taylor series we omit
% the second index, i.e. $ K^{(i),v}_\frac{k}{2} \equiv K^{(i),v}_{\frac{k}{2}\,0}$
% %The terms with negative powers of $(1-z)$ are due to the Coulomb
% %singularity present in the vector and pseudo-scalar correlator. 

% Finally, for $z \to -\infty$ we define
% \begin{equation}
%   \label{eq:7}
%   \Pi^{(i),v}(z) = \sum_{n=0}^\infty \sum_{l\geq 0}
%   D^{(i),v}_{n\,l} \left(\frac{1}{z}\right)^n \log^l(-4z)\,,
% %  \Pi^{(i),v}(z) = \frac{3}{16\pi^2}\sum_{n=0}^\infty \sum_{l\geq 0}
% %  D^{(i),v}_{n\,l} \left(\frac{1}{z}\right)^n \log^l\left(-\frac{4}{z}\right)\,,
% \end{equation}
% where we again use the convention $ D^{(i),v}_n \equiv  D^{(i),v}_{n\,0}$.

\subsection{Pad\'e approximation}
\label{sec:pade1}
The Pad\'e approximant $p_{n,m}(x)$ to a function $f$ is defined as
\begin{equation}
  \label{eq:8}
  p_{n,m}(x) =  \frac{\sum_{i=0}^n a_i x^i}{1+\sum_{i=1}^m b_i x^i}\,.
\end{equation}

A na\"ive application of the Pad\'e approximation method will, however, fail
for the functions $\Pi^{(i),v}(z)$ 
%in the perturbative
%expansions of the polarisation functions $\Pi^{v}(z)$ 
because contrary to the Pad\'e approximants (Eq. (\ref{eq:8})) they are
not meromorphic everywhere in the complex plane. There are two major
aspects of this problem, which have to be considered:
%\begin{itemize}
%\item 
% Aside from power-like terms, there are also logarithmic
%   contributions in the threshold and high energy expansions of $\Pi^{(i),v}(z)$.
   The functions $\Pi^{(i),v}(z)$ diverge logarithmically for $z \to -\infty$. There
  are also logarithmic contributions at threshold.
%\item 
There is a branch cut along the real axis starting from $z=1$.
%\end{itemize}
This behaviour can obviously not be reproduced accurately by a Pad\'e approximation.

The first problem related to the appearance of logarithms can be cured
by splitting $\Pi^{(i),v}(z)$ into two parts,
\begin{equation}
  \label{eq:10}
  \Pi^{(i),v}(z) = \Pi^{(i),v}_{reg}(z) + \Pi^{(i),v}_{log}(z)\,,
\end{equation}
so that $\Pi^{(i),v}_{log}(z)$ is a suitable function containing
all known logarithmic contributions to $\Pi^{(i),v}(z)$.  
In this way the problem reduces to finding an approximation to
$\Pi^{(i),v}_{reg}(z)$. 
% Obviously 
% $\Pi^{(i),v}_{log}(z)$ is not uniquely defined (for example, an
% arbitrary non-logarithmic term can be added to it)
%  and different
% prescriptions for its construction will lead to different reconstructed
% polarisation functions in the end.

% In the case of the vector and axial-vector polarisation functions, 
% the functions $\Pi^{(i),v}(z)$ with $i>1$ also show the well-known Coulomb
% singularity at threshold in addition to the logarithmic behaviour. Being
% proportional to $\left(1-z\right)^{(1-i)/2}$ this singularity is, however,
%  meromorphic and can be  incorporated into the construction of
%  the Pad\'e approximants. 


The second problem related to the branch cut is treated in a different
way: We map the complex plane (including its cut) onto the unit circle
in such a way that the branch cut is mapped onto the perimeter. This can
be achieved via the conformal transformation
$  z \to \frac{4\omega}{(1+\omega)^2}$. 
The functions $\Pi^{(i),v}_{reg}(\omega)$ are now suitable for the
Pad\'e approximation procedure.

% \begin{figure}
% $\begin{array}{ccc}
% \includegraphics*[width =.99\linewidth]{cont.1.eps}
% \end{array}$
% \caption{Conformal mapping of the complex plane onto the unit
%   circle. The points $z=0$ and $z=1$ correspond to $\omega=0$ and
%   $\omega=1$, respectively. $z\pm \infty$ goes to $\omega=-1$. The
%   branch cut starting at $z=1$ is mapped onto the perimeter of the circle. } 
% \label{fig:1}
% \end{figure}


% In the next step we  fix the coefficients $a_i,\,b_i$ in
% Eq. \eqref{eq:8}. From the expansions in the low energy, threshold and
% euclidean high energy regions we know the values of the $\Pi^{(i),v}(z)$ and
% some of their derivatives for $z=0$, $z=1$, and $z \to
% - \infty$. While the information from the first two regions can be used
% directly to fix the derivatives of $\Pi^{(i),v}_{reg}(\omega)$ at
% $\omega=0$ and $\omega=1$, respectively, the expansion around $z \to-
% \infty$ is quite different from the expansion around $\omega=-1$.

% In order to also incorporate the information
% from the high energy region one can use the following auxiliary function
% \cite{Chetyrkin:1998ix}
% \begin{equation}
%   \label{eq:12}
%   P_n(\omega) = 
%   \frac{(4\omega)^{n-1}}{(1+\omega)^{2n}}\left(\Pi^{(i),v}_{reg}(\omega)-\sum_{j=0}^{n-1}\frac{(1+\omega)^{2j}}{(4\omega)^j} \frac{1}{j!} \left(\frac{d}{d(1/z)}\right)^j \Pi^{(i),v}_{reg}(z)\Big|_{z=-\infty}\right)\,,
% \end{equation}
% where $n$ is the highest known power of $1/z$ in the high
% energy expansion. $P_n(-1)$ corresponds to the coefficient of $1/z^n$,
% while all other terms in the expansion of $\Pi^{(i),v}_{reg}(z)$
% around $z\to - \infty$ together with the terms from the low energy expansion
% determine the behaviour of $P_n(\omega)$ around $\omega=0$. If the
% expansion around $z=0$ is available up to $z^m$ the values 
% $P_n(0), P_n'(0), \dots,P_n^{(n+m-1)}(0)$ together with $P_n(-1)$
% can be used for the construction of Pad\'e approximants. Additional
% knowledge about the $l$ leading terms in the threshold expansion
% translates into $P_n(1), P_n'(1), P_n''(1),\dots,P_n^{(l-1)}(1)$.

% Note that the construction Eq. \eqref{eq:12} in general produces
% half-integer powers of $1/z$ in the high energy expansion of
% the reconstructed function $\Pi^{(i),v}_{reg}(z)$. Since only terms
% with integer powers of $1/z$ may appear in the expansion, we explicitely
% require that the terms with half-integer powers vanish.
% These additional constraints do not have the form of Eq. \eqref{eq:9}. 
% As a result, the solution to the system of equations
% which determines the coefficients of the Pad\'e approximant is, in general, no longer
% unique.


% The Coulomb singularity proportional to $\left(1-z\right)^{(1-i)/2}$, which
% appears in the vector and the axial-vector correlator, can easily be
% incorporated by defining $P_n(\omega)$ with an additional factor
% $(1-\omega)^{1-i}$ on the right hand side of Eq. \eqref{eq:12} so
% that the limit $\omega \to 1$ becomes regular. 
\subsection{Subtractions}
\label{sec:subtractions}

As explained in Section \ref{sec:pade1}, the first step consists of
absorbing the logarithmic contributions in the high energy and threshold
expansions into a function $\Pi^{(3),v}_{log}(z)$. This function must be
chosen carefully in order not to introduce undesired additional
singularities in $\Pi^{(3),v}_{reg}(z)$. It is very convenient to use
lower order analytical results $\Pi^{(0),v}$ and $\Pi^{(1),v}$ as
auxiliary functions \cite{Hoang:2008qy}. $\Pi^{(0),v}$ is
given by
\begin{equation}
  \label{eq:13}
  \Pi^{(0),v}(z)=\frac{3}{16 \pi^2}\left(\frac{20}{9}+\frac{4}{3z}-\frac{4(1-z)(1+2z)}{3z}G(z)\right)\,,
\end{equation}
with
\begin{align}
  \label{eq:14}
G(z) =& \frac{1}{2z}\log\left(u\right)/\sqrt{1-\frac{1}{z}}\,,
\hspace{1cm} u = (\sqrt{1-\frac{1}{z}}-1)/(\sqrt{1-\frac{1}{z}}+1)\,. 
\end{align}
The expansions of the vector  correlator (with $n_l=3$) around
threshold and for $q^2\to \infty$ are given by 
\begin{align}
  \Pi^{(3),v} (z) =& 2.63641/(1 - z) 
  + ( - 25.2331 - 7.75157 \log(1 - z))/\sqrt{1 - z}
- 11.0654 \log(1 - z) \notag \\ &+ 1.42833 \log^2(1 - z)  
- 0.421875 \log^3(1 - z) +  K_0  +\mathcal{O}(\sqrt{1-z})\label{eq:4} 
\end{align}
and 
\begin{align}
&\Pi^{(3),v}(z)=-6.172 - 0.070 \log(-4 z) + 0.121 \log^2(-4 z) -
 0.037 \log^3(-4 z)  + 1/z (-4.333 \notag \\ &- 3.756 \log(-4 z) +
   2.118 \log^2(-4 z) -
   0.319 \log^3(-4 z)) + 1/z^2( D_2  -5.130 \log(-4 z) +   \notag \\&
0.318 \log^2(-4 z) + 0.401 \log^3(-4 z) -
   0.079 \log^4(-4 z))+\mathcal{O}(1/z^3) \, \label{eq:2},
 \end{align}
respectively. The
construction of $\Pi^{(3),v}_{log}(z)$ is now based on the expansions of
$\Pi^{(1),v}$ and $G(z)$ in the threshold and high-energy region:
\begin{equation}
  \label{eq:16}
  G(z) = \frac{\pi}{2}\frac{1}{\sqrt{1-z}}+{\cal O}((1-z)^0)\,, \quad 
\Pi^{(1),v}(z)=-\frac{3}{16}\log\left(1-z\right)+\text{const} +{\cal O}(\sqrt{1-z}) \,,
\end{equation}
\begin{equation}
  \label{eq:19}
  G(z) = \frac{-\log(-4z)}{2z}+{\cal O}\left(\frac{1}{z^2}\right)\,.
\end{equation}
Since these expansions have the desired behavior in the interesting
regions we make the ansatz
\begin{equation}
  \label{eq:20}
  \Pi^{(3),v}_{log}(z) = \sum_{i>0,j} k_{ij} \Pi^{(1),v}(z)^i
  G(z)^j+\sum_{m,n} d_{mn} \big(z\,G(z)\big)^m \left(1-\frac{1}{z}\right)^{\left \lceil
      \frac{m}{2}\right \rceil} \frac{1}{z^n}\,,
\end{equation}
where $ \lceil\dots\rceil$ means rounding up to the next integer
number. The coefficients $k_{ij}$ and $d_{mn}$ are chosen for
$\Pi^{(3),v}_{log}(z)$ to reproduce the known behavior in the threshold
and high-energy region given by Eqs. (\ref{eq:4}) and (\ref{eq:2}).

The choice of $\Pi^{(3),v}_{log}(z)$ is of course not unique. A perfect
reconstruction of the polarisation function would clearly not depend on
the specific choice, so variations of $\Pi^{(3),v}_{log}(z)$ can be
used to estimate the quality of the approximation procedure. Following
Ref. \cite{Hoang:2008qy}, we introduce additional parameters $a_i$ and $b_i$ for this purpose.
We modify the ansatz
Eq. \eqref{eq:20} for $\Pi^{(3),v}_{log}(z)$ in the following way:
In the first sum, we multiply the summands with  $i=3,\,j=0$  and
$i=j=1$ (which roughly correspond to terms proportional to  $\log^3
(1-z)$ and $\log(1-z)/\sqrt{1-z}$) by factors  $a_1+1/z$ and $a_2+1/z$,
respectively. In the second sum all summands corresponding to the two highest powers
of logarithms are multiplied by factors
$1+1/(b_1z)$ for $m=3$ and $1+1/(b_2z)$ for $m=4$.

Except for the conditions $a_i \neq -1$ and $b_i \neq 0$ the values of
the parameters are in principle arbitrary. We vary them independently
with
% \begin{equation}
%   \label{eq:21}
$ a_i \in  \{-1 \pm 1,\, -1\pm 4,\, -1 \pm 16,\, -1 \pm 64\}\,,\, 
  b_i \in  \{\pm1,\, \pm 4,\, \pm 16,\, \pm64\}\,.$
% \end{equation}

% \subsection{Pad\'e approximation}
% \label{sec:pade2}

In the next step we determine the coefficients of the Pad\'e approximants
from the expansions in the low energy, threshold, and euclidean high
energy region. The low energy expansion for the vector current and
$n_l=3$ reads
\begin{equation}
  \label{eq:22}
  \Pi^{(3),v}(z)= 6.95649 z + 7.2478 z^2 + 7.31855 z^3 +\mathcal{O}(z^4)
  \, .
\end{equation}
The expansions around threshold and for high energies are listed in
Eqs. (\ref{eq:4}) and (\ref{eq:2}), respectively.

Following Ref. \cite{Hoang:2008qy}, we additionally require that terms proportional to
$z^{-\frac{3}{2}}$ and $z^{-\frac{5}{2}}$ are absent in the high energy
expansion. 

% \subsection{Discussion of approximants}
% \label{sec:disc-appr}

% It turns out that some of the approximants have poles inside the unit
% circle which translate to unphysical poles inside the complex $z$-plane for the
% reconstructed polarisation functions. For this reason we immediately discard
% approximants which have poles for $|\omega|<1$.

% There is an additional subtlety for poles which do not lie inside, but
% close to the unit circle. As pointed out in Ref. \cite{Hoang:2008qy}, these poles can have a
% huge numerical effect on the behaviour of the approximant on the perimeter of the
% circle. More specifically, they can lead to unphysical peaks in the
% cross sections derived from the imaginary parts of the polarisation
% functions above threshold. 

% To avoid such resonances we discard approximants 
% that have pronounced maxima on the perimeter of the circle, i.e.
% \begin{equation}
%   \label{eq:24}
%   \max \bigg| p_{n,m}(\omega)\big|_{|\omega|=1}\bigg|>\rho\,,
% \end{equation}
% where the value of
% $\rho$ is chosen heuristically using the following two criteria:
% first, the real and imaginary parts of the polarisation functions should
% show no significant additional peaks; second, an adequate number of
% ${\cal O}(1000)$ to ${\cal O}(10000)$ ``good'' approximants should remain.
% In practice, we choose $\rho=3$ for the vector, scalar and pseudo-scalar
% correlators and $\rho=1.2$ for the axial-vector correlator. 

\subsection{Results}
\label{sec:res}

From the Pad\'e approximants we reconstruct the polarisation
function. Its imaginary part corresponding to hadron
production cross section is plotted above the charm threshold
(i.e. with $n_l=3$) in
Fig. \ref{fig:2}. 
\begin{figure}
  \centering
    \includegraphics[width=.47\linewidth]{V3charmR.eps}
    \includegraphics[width=.47\linewidth]{cv4_distr}
    \caption{Left: Imaginary part of the four loop contributions to the
      polarisation
      function above the charm threshold. The plots show $v R^{(3),v}=v
      12\pi \text{Im}(\Pi^{(3),v})$ as functions of
      $v=\sqrt{1-1/z}$. The solid black line is the mean from all
      approximants, the area covered by three standard deviations is
      shown by bands. The dashed lines show the expansions in the
      threshold and high energy regions. Right: Distribution of the
      values of $C_4^{(3),v}$ in on-shell scheme from different Pad\'e
      approximants to the charm vector correlator.}
   \label{fig:2}
\end{figure}
The reconstructed functions can be expanded again to obtain additional
low energy, threshold and high energy coefficients. We find that the
values of the coefficients are strongly peaked around the mean value
(for an example, see Fig. \ref{fig:2}). As a consequence we give our
errors in terms of standard deviations.  As expected
%from Fig. \ref{fig:2}, 
the error is a lot
smaller for the low energy coefficients in comparison to the coefficients
in the threshold and high energy regions. 

% Comparing our  predictions for the vector correlator with the previous results \cite{Hoang:2008qy} (see Table \ref{tab:6}), we find indications for a
% different sign of the threshold constant $K^{(3),v}_0$, but good agreement for all low
% energy moments. Furthermore, the errors
% are decreased notably, depending on the coefficient by about one order
% of magnitude. This is mainly due to the additional information from
%  $C^{(3),v}_3$, whereas $D^{(3),v}_0$ and $D^{(3),v}_1$ seem to play
%  only a minor role. 
% The impact of additional information on
% the quality of the predictions from the Pad\'e approximants can also be
% seen in Table \ref{tab:7}, where we compare results for the pseudo-scalar
% correlator with three and four low energy coefficients as input.

\begin{table}
  \centering
$\begin{array}{|c||c|c|c|c|c|c|c|c|c|c|c|c|}
                    & 
C^{(3),v}_4          &
C^{(3),v}_5          &
C^{(3),v}_6          &
C^{(3),v}_7          &
C^{(3),v}_8          &
K^{(3),v}_0          &
D^{(3),v}_2          \\\hline\hline
  n_l=3     & 
 383.073(11) &
 378.688(32) &
 373.536(61) &
 368.23(9)   &
 363.03(13)  &
   17(11)    &
 2.0(42)     \\
n_l=4       & 
 339.913(10) &
 338.233(32) &
 335.320(63) &
 331.90(10)  &
 328.33(14)  &
  17(29)     &
 1.2(83)     \\
n_l=5       &
 298.576(9) &
 299.433(27)&
 298.622(54)&
 296.99(9)  &
 294.94(12) &
  16(10)    &
 1.4(21)    
\end{array}
$
\caption{Expansion coefficients from the reconstructed  polarisation functions for different numbers of light
  quarks in the on shell scheme. $C^{(3),v}_{1-3}$  are known exactly. The errors always apply to the last digits, i.e. $2.0(42)$
  means an error of $4.2$.}
  \label{tab:111}
\end{table}
\section{Conclusion}
\label{sec:conc}
We have used the Pad\'e approximation method to reconstruct the full
energy dependence of the heavy quark correlator of the vector current at
order $\alpha_s^3$. As input we have used information from expansions in
the low energy, threshold and high energy regions.  Expanding the
reconstructed correlators, we have obtained predictions for additional
coefficients in these expansions.  We find that these predictions are
fairly accurate for low energy coefficients but less precise for the
coefficients in the threshold and high energy expansions.


\begin{thebibliography}{99}
\providecommand{\url}[1]{\texttt{#1}}
\providecommand{\urlprefix}{URL }
\providecommand{\eprint}[2][]{\url{#2}}

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V.~A. Novikov, et~al.
\newblock \emph{Phys. Rept.}, 41:1--133, 1978.
\vspace{-1ex}
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\end{thebibliography}

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