%Title: Four-loop moments of the heavy quark vacuum polarization function in perturbative QCD
%Author: K.G. Chetyrkin, J.H. K\"uhn and C. Sturm
%Published: * Eur. Phys. J. * ** C48 ** (2006) 107-110.
%hep-ph/0604234
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\newcommand{\M}[3]{{\mathrm{T}}^{}_{#2#3}}
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\begin{document}
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\noindent
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\begin{center}
\begin{Large}
\begin{bf}
Four-loop moments of the heavy quark vacuum polarization
function in perturbative QCD
\end{bf}
\end{Large}
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\vspace{0.8cm}
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\begin{large}
K.~G.~Chetyrkin$\rm \, ^{a,}$
\footnote{On leave from Institute for Nuclear Research of the
Russian Academy of Sciences, Moscow, 117312, Russia.},
J.~H.~K\"uhn$\rm \, ^{a \,}$
{\normalsize and } C.~Sturm$\rm \, ^{b \,}$
\end{large}
\vskip .7cm
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{\small {\em
$\rm ^a$ Institut f{\"u}r Theoretische Teilchenphysik,
Universit{\"a}t Karlsruhe,
D-76128 Karlsruhe,
Germany}}
\vskip .3cm
{\small {\em
$\rm ^b$ Dipartimento di Fisica Teorica,
Universit{\`a} di Torino, I-10125 Torino, Italy
{\rm{\&}}
INFN, Sezione di Torino, Italy}}
\vspace{0.8cm}
{\bf Abstract}
\end{center}
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\begin{quotation}
\noindent
New results at four-loop order in perturbative QCD for the first two Taylor
coefficients of the heavy quark vacuum polarization function are presented.
They can be used to perform a precise determination of the charm- and
bottom-quark mass. Implications for the value of the quark masses are
briefly discussed.
\vspace{1pc}
\end{quotation}
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\section{Introduction}
\label{Intro}
Two-point correlators are of central importance for many theoretical
and phenomenological investigations in Quantum Field Theory. As
a consequence they are studied in great detail in the framework
of perturbative calculations. Due to simple kinematics (only one
external momentum) even multi-loop calculations can be performed. The
results for all physically interesting diagonal and non-diagonal
correlators including {\em full} quark mass dependence
are available up to $\mathcal{O}\left(\alpha_s^2\right)$
\cite{Chetyrkin:1996cf,Chetyrkin:1998ix,Chetyrkin:1997mb}.\\
%%%%%
At four-loop order the two-point correlators can be considered in two
limits: In the high energy limit massless propagators need to be
calculated, in the low energy limit vacuum integrals
(``tadpole diagrams'')
arise. The evaluation of the latter in three-loop approximation
has been pioneered in ref. \cite{Broadhurst:1992fi} and automated in
ref. \cite{Steinhauser:2000ry}.\\
%%%%%
Recently first results for physical quantities, which are related to
four-loop tadpole diagrams, have been obtained. The four-loop matching
condition for the strong coupling constant $\alpha_s$ at a heavy
quark threshold has been calculated in ref.~\cite{Chetyrkin:2005ia,Schroder:2005hy}.
The four-loop QCD contribution to the electroweak $\rho$-parameter
induced by the singlet diagrams of the Z-boson self-energy has been
computed in ref.~\cite{Schroder:2005db}.\\
%%%%
The detailed knowledge of the heavy quark correlator is important for the
precise determination of heavy quark masses with the help of QCD sum rules. As
known from ref.~\cite{Kuhn:2001dm,Kuhn:2002zr}, the determination of the
charm- and bottom-quark mass is further improved, if the four-loop
corrections, hence $\Order(\als^3)$, for the lowest Taylor coefficients of the
vacuum polarization function are available. The subset of four-loop
contributions to the lowest two moments, which involve two internal loops from
massive and massless fermions coupled to gluons, hence of
$\Order(\als^3\*\nf^2)$, has already been calculated in
ref.~\cite{Chetyrkin:2004fq}. The symbol $\nf$ denotes the number of active
quark-flavors, contributing through fermion-loops inserted into gluon lines.
The terms being proportional to $\als^j\*\nl^{j-1}$ are even known to all
orders $j$ in perturbative QCD~\cite{Grozin:2004ez}. The symbol $\nl$ denotes
the number of light quarks, considered as massless. In this paper the complete
four-loop contributions originating from non-singlet diagrams for the first
two Taylor coefficients are presented. Singlet contributions have been
studied in
ref.~\cite{Groote:2001vr,Portoles:2001yu,Portoles:2002rt}.\\
%%%%
In general, the tadpole diagrams encountered in these calculations contain
both massive and massless lines. As is well-known, the computation of the
four-loop $\beta$-function can be reduced to the evaluation of four-loop
tadpoles composed of { completely massive} propagators only. Calculations for
this case have been performed in \cite{vanRitbergen:1997va,Czakon:2004bu,Kajantie:2003ax}.
%%%%
The outline of this paper is as follows. In section
\ref{GeneralNotations} we briefly introduce the notation and discuss
generalities. In section \ref{Calculations} we discuss the reduction to
master integrals, describe the solution of the linear system of
equations, give the result for the lowest two moments and discuss the impact
on the quark mass determination.
Our conclusions and a brief summary are given in section
\ref{DiscussConclude}.
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\section{Notation and Generalities}
\label{GeneralNotations}
The correlator $\Pi^{\mu\nu}(q)$ of two currents is defined as
\begin{equation}
\Pi^{\mu\nu}(q,j)=i\*\int dx\,e^{iqx}\langle 0|Tj^\mu(x) j^\nu(0)|0
\rangle\,,
\label{correl}
\end{equation}
with the current
$j^{\mu}(x)=\overline{\Psi}(x)\gamma^\mu\Psi(x)$ being composed out of the
heavy quark fields $\Psi(x)$. The function $\Pi^{\mu\nu}(q)$ is conveniently
written in the form:
\begin{equation}
\label{vacpol}
\Pi^{\mu\nu}(q)=\left(-q^2\*g^{\mu\nu}+q^{\mu}\*q^{\nu}\right)\*\Pi(q^2)\,.
\end{equation}
The vanishing of the longitudinal contribution, as well as the confirmation of
$\Pi^{\mu}_{\phantom{\mu}\mu}(q^2=0)=0$, has been used as a check of the
calculation. The function $\Pi(q^2)$ is of phenomenological interest, because
it can be related to the ratio
$R(s)=\sigma(e^+\*e^-\rightarrow\mbox{hadrons})/\sigma(e^+\*e^-\rightarrow\mu^+\mu^-)$
with the help of dispersion relations:
\begin{equation}
\Pi(q^2)={1\over12\*\pi^2}\int_{0}^{\infty}\!ds\,{R(s)\over (s-q^2)}\;\;\mbox{mod subtr.}
\label{PIR}
\end{equation}
Performing the derivative of eq.~(\ref{PIR}) with respect to $q^2$ one obtains
on the one hand ``experimental'' moments
\begin{equation}
\mathcal{M}^{\mbox{\footnotesize{exp}}}_n=\int\!ds\,{R(s)\over s^{n+1}}\,,
\end{equation}
which can be evaluated from the $R$-ratio. On the other hand
one can define ``theoretical'' moments
\begin{equation}
\mathcal{M}^{\mbox{\footnotesize{th}}}_n=
Q_{q}^2\*{9\over4}\*\left({1\over4\*\overline{m}_q^2}\right)^n\*\overline{C}_n\,,
\end{equation}
which are related to the Taylor coefficients
$\overline{C}_n$ of the vacuum polarization function:
\begin{equation}
\overline{\Pi}(q^2) = {3\*Q_q^2\over16\*\pi^2}\*\sum_{n\ge0}
\overline{C}_n\*\overline{z}^n\,,
\label{PiExp}
\end{equation}
with $\overline{z}=q^2/(4\*\overline{m}^2)$. Symbols carrying a bar denote
that renormalization has been performed in $\overline{\mbox{MS}}$-scheme. The
Taylor expansion in $q^2$ around $q^2=0$ leads to massive tadpole integrals.
The first and higher derivatives can be used for a precise determination of the
charm- and bottom-quark mass. However, also the lowest expansion coefficient
$\overline{C}_0$ has an interesting physical meaning, since it relates the
coupling of the electromagnetic interaction in
different renormalization schemes. \\
%%%%
Sum rules as tool for the determination of the charm- and bottom-quark
mass have been suggested since long in ref.~\cite{Novikov:1977dq}. This
method has then been applied later also to the determination of the
bottom-quark mass~\cite{Reinders:1984sr}. One of the most precise
determinations of the charm- and bottom-quark mass being based on sum
rules in connection with the calculation of three-loop moments in
perturbative QCD has been performed in ref.~\cite{Kuhn:2001dm}, with the
values of $\overline{m}_c(\overline{m}_c)=1.304(27)$~GeV and
$\overline{m}_b(\overline{m}_b)=4.191(51)$~GeV as results for the charm-
and the bottom-quark mass.
%%%%
It is convenient to define the expansion of the Taylor coefficients
$\overline{C}_n$ of the vacuum polarization function in the strong
coupling constant $\als$ as
\begin{equation}
\Cb_n=\Cb_n^{(0)}
+\left({\als\over\pi}\right)^1\*\Cb_n^{(1)}
+\left({\als\over\pi}\right)^2\*\Cb_n^{(2)}
+\left({\als\over\pi}\right)^3\*\Cb_n^{(3)}+\dots\,.
\label{CbarExp}
\end{equation}
Due to the distinct mass hierarchy of the quarks in the Standard-Model, one
can consider for the above moments one species of quarks as massive ($\nh=1$)
and all lighter ones as massless. The number of active quarks $\nf$ is then
decomposed according to $\nf=\nl+\nh$. For convenience the symbol $\nh$ is
kept explicitly in the following.
%%
%%
%%
%%
\section{Calculations and Results}
\label{Calculations}
%%%%%
The Feynman-diagrams have been generated with the help of the program
{\tt{QGRAF}} \cite{Nogueira:1991ex}. After performing the expansion in
the external momentum $q$ all integrals can be expressed in terms of 55
independent vacuum topologies with additional increased powers of
propagators and additional irreducible scalar products.\\
%%%%
In order to reduce this host of integrals to a small set of master integrals
the traditional Integration-by-parts(IBP) method has been used in combination
with Laporta's algorithm \cite{Laporta:1996mq,Laporta:2001dd}. The resulting
system of linear equations has been solved with a {\tt{FORM3}}
\cite{Vermaseren:2000nd,Vermaseren:2002rp,Tentyukov:2006ys} based program, in
which partially also ideas described in
ref.~\cite{Laporta:2001dd,Mastrolia:2000va,Schroder:2002re} have been
implemented. The rational functions in the space-time dimension $d$, which
arise in this procedure, have been simplified with the program
{\tt{FERMAT}}~\cite{Lewis}. Masking of large integral coefficients has been
implemented, a strategy also adopted in the program
{\tt{AIR}}~\cite{Anastasiou:2004vj}. In order to achieve the reduction to 13
master integrals more than 31 million IBP-equations have been generated and
solved. This leads to integral-tables with solutions for around five million
integrals. Furthermore all symmetries of the 55 independent topologies have
been taken into account in an automated way, by reshuffling the powers of the
propagators of a given topology in a unique way. Taking into account
symmetries is important in order to keep
the size of the integral-tables under control.\\
%%%%
The first four master integrals shown in fig.~\ref{masters:simple} can be
calculated completely analytically in terms of $\Gamma$-functions. The fifth
integral ($T_{52}$) can be obtained from results of
ref.~\cite{Broadhurst:1992fi,Broadhurst:1996az}.
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\begin{figure}[!ht]
%%%%%%%%%%%%%%%%%%%%
\begin{center}
\begin{minipage}[b]{2cm}
\begin{center}
\includegraphics[height=1.5cm,bb=126 332 460 665]{Topo40401-Dash}
$\M{}{4}{1}$
% 40401
\end{center}
\end{minipage}
%
%%%%%%%%%%%%%%%%%%%%
%
\begin{minipage}[b]{2cm}
\begin{center}
\includegraphics[height=1.5cm,bb=170 320 415 666]{Topo40523-Dash}
\\
$\M{}{5}{1}$
% 40523
\end{center}
\end{minipage}
%
%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%
%
\begin{minipage}[b]{2cm}
\begin{center}
\includegraphics[height=1.5cm,bb=126 320 460 678]{Topo40504-Dash}
$\M{}{5}{3}$
% 40504
\end{center}
\end{minipage}
%
%%%%%%%%%%%%%%%%%%%%
%
\begin{minipage}[b]{2cm}
\begin{center}
\includegraphics[height=1.5cm,bb=126 320 460 678]{Topo40618-Dash}
$\M{}{6}{3}$
% 40618
\end{center}
\end{minipage}
%
%%%%%%%%%%%%%%%%%%%%
%
\begin{minipage}[b]{2cm}
\begin{center}
\includegraphics[height=1.5cm,bb=170 320 415 666]{Topo40524-Dash}
\\
$\M{}{5}{2}$
% 40524
\end{center}
\end{minipage}
%
\end{center}
\caption{Factorized or analytically known master integrals. The solid (dashed) lines
denote massive (massless) propagators.
\label{masters:simple}}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\noindent
The remaining eight master integrals (fig.~\ref{MasterTopologies}) have been
determined with high precision numerics with the difference equation method
\cite{Laporta:2001dd} in ref.~\cite{Schroder:2005va}. Independently they have
been determined in ref.~\cite{Chetyrkin:2006dh} by constructing an
$\vep$-finite basis. Some of these master integrals have also been calculated
in
ref.~\cite{Laporta:2002pg,Chetyrkin:2004fq,Kniehl:2005yc,Schroder:2005db}.\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%
\begin{figure}[!ht]
\begin{center}
\begin{minipage}[b]{2cm}
\begin{center}
\includegraphics[height=1.5cm,bb=126 320 460 678]{Topo40505-Dash}
$\M{}{5}{4}$
% 40505
\end{center}
\end{minipage}
%
%%%%%%%%%%%%%%%%%%%%
\begin{minipage}[b]{2cm}
\begin{center}
\includegraphics[height=1.5cm,bb=126 320 460 678]{Topo406112-Dash}
$\M{}{6}{2}$
% 406112
\end{center}
\end{minipage}
%
%%%%%%%%%%%%%%%%%%%%
%
\begin{minipage}[b]{2cm}
\begin{center}
\includegraphics[height=1.5cm,bb=126 320 460 678]{Topo406111-Dash}
$\M{}{6}{1}$
% 406111
\end{center}
\end{minipage}
%
%
%
\begin{minipage}[b]{2cm}
\begin{center}
\includegraphics[height=1.5cm,bb=126 332 460 678]{Topo40602-Dash}
$\M{}{6}{4}$
% 40602
\end{center}
\end{minipage}
%
\vspace{5mm}
%%%%%%%%%%%%%%%%%%%%
%
\begin{minipage}[b]{2cm}
\begin{center}
\includegraphics[height=1.5cm,bb=126 332 460 678]{Topo407118-Dash}
$\M{}{7}{1}$
% 407118
\end{center}
\end{minipage}
%
%%%%%%%%%%%%%%%%%%%%
%
\begin{minipage}[b]{2cm}
\begin{center}
\includegraphics[height=1.5cm,bb=126 332 460 666]{Topo407117-Dash}
$\M{}{7}{2}$
% 407117
\end{center}
\end{minipage}
%
%%%%%%%%%%%%%%%%%%%%
%
\begin{minipage}[b]{2cm}
\begin{center}
\includegraphics[height=1.5cm,bb=126 332 460 666]{Topo40802-Dash}
$\M{}{8}{1}$
% 40802
\end{center}
\end{minipage}
%
%%%%%%%%%%%%%%%%%%%%
%
\begin{minipage}[b]{2cm}
\begin{center}
\includegraphics[height=1.5cm,bb=126 320 460 678]{Topo40903-Dash}
$\M{}{9}{1}$
% 40903
\end{center}
\end{minipage}
\end{center}
\caption{Master integrals where only a few terms of their
$\vep$-expansion are known analytically. The solid (dashed) lines
denote massive (massless) propagators.
\label{MasterTopologies}}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%
Inserting the master integrals and performing renormalization of the strong
coupling constant $\als$, the external current and the mass
$m=\overline{m}(\mu)$ leads to the following result ($\mu=\overline{m}$):
\begin{eqnarray}
\label{C0}
\overline{C}_0^{(3)} &=&
\nl\*\nh\*\Bigg(
- {2\over9}\*a_4
+ {7043\over34992}
- {1\over108}\*\Log{2}{4}
+ {\pi^2\over108}\*\Log{2}{2}
+ {49\over12960}\*\pi^4
\nonumber\\&&\qquad\quad
- {127\over324}\*\z3
\Bigg)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \nh^2\*\left(
{610843\over2449440}
- {661\over2835}\*\z3
\right)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \nl^2\*\left(
{17897\over69984}
- {31\over162}\*\z3
\right)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\nonumber\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
&+&\nl\*\Bigg(
- {50\over81}\*a_4
- {71629\over46656}
- {25\over972}\*\Log{2}{4}
+ {25\over972}\*\Log{2}{2}\*\pi^2
+ {8533\over116640}\*\pi^4
\nonumber\\&&\qquad\quad
- {21343\over3888}\*\z3
\Bigg)
\nonumber\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
&+&\nh\*\Bigg(
- {28364\over405}\*a_4
- {83433703\over8164800}
- {7091\over2430}\*\Log{2}{4}
+ {7091\over2430}\*\Log{2}{2}\*\pi^2
\nonumber\\
&&\qquad
+ {14873\over18225}\*\pi^4
- {14509529\over340200}\*\z3
+ {5\over3}\*\z5
\Bigg)
\nonumber\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
&+& {64\over3}\*a_5
+ {12007\over243}\*a_4
- {8572423579\over604661760}
+ {12007\over5832}\*\Log{2}{4}
- {8\over45}\*\Log{2}{5}
\nonumber\\
&-& {12007\over5832}\*\Log{2}{2}\*\pi^2
+ {8\over27}\*\Log{2}{3}\*\pi^2
- {1074967\over699840}\*\pi^4
+ {34\over135}\*\log{2}\*\pi^4
+ {\pi^6\over486}
\nonumber\\
&+& {53452189\over349920}\*\z3
- {28\over243}\*\z3^2
- {37651\over648}\*\z5
- {1\over432}\*T_{54,3}
+ {7\over1296}\*T_{62,2}\,,\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{C1}
\overline{C}_1^{(3)} &=&
\nl\*\nh\*\Bigg(
-{116\over243}\*a_4
+ {262877\over787320}
- {29\over1458}\*\Log{2}{4}
+ {29\over1458}\*\Log{2}{2}\*\pi^2
+ {1421\over174960}\*\pi^4
\nonumber\\&&\qquad
- {38909\over58320}\*\z3
\Bigg)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \nh^2\*\left(
{163868\over295245}
- {3287\over7290}\*\z3
\right)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \nl^2\*\left(
{42173\over98415}
- {112\over405}\*\z3
\right)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\nonumber\\
&+&\nh\*\Bigg(
- {1394804\over8505}\*a_4
- {27670774337\over1414551600}
- {348701\over51030}\*\Log{2}{4}
\nonumber\\&&\qquad
+ {348701\over51030}\*\Log{2}{2}\*\pi^2
+ {1447057\over765450}\*\pi^4
- {95617883401\over943034400}\*\z3
+ {128\over27}\*\z5
\Bigg)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\nonumber\\
&+&\nl\*\Bigg(
- {4793\over7290}\*a_4
- {9338899\over2099520}
- {4793\over174960}\*\Log{2}{4}
+ {4793\over174960}\*\Log{2}{2}\*\pi^2
\nonumber\\&&\qquad
+ {372689\over839808}\*\pi^4
- {48350497\over1399680}\*\z3
\Bigg)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\nonumber\\
&-&{127168\over1215}\*a_5
- {22152385\over61236}\*a_4
+ {237787820456749\over380936908800}
- {22152385\over1469664}\*\Log{2}{4}
\nonumber\\
&+&{15896\over18225}\*\Log{2}{5}
+ {22152385\over1469664}\*\Log{2}{2}\*\pi^2
- {15896\over10935}\*\Log{2}{3}\*\pi^2
\nonumber\\
&-&{29962031\over176359680}\*\pi^4
- {67558\over54675}\*\log{2}\*\pi^4
+ {60701\over1071630}\*\pi^6
+ {282830677079\over881798400}\*\z3
\nonumber\\
&-&{242804\over76545}\*\z3^2
- {653339\over2430}\*\z5
- {5849\over272160}\*T_{54,3}
+ {60701\over408240}\*T_{62,2}\,,
\end{eqnarray}
where Riemann's zeta-function $\zeta_n$ and the polylogarithm-function
$\mbox{Li}_n(1/2)$ are defined by
\begin{equation}
\zeta_n=\sum_{k=1}^{\infty}{1\over k^n}\quad\mbox{and}\quad
a_n=\mbox{Li}_n(1/2)=\sum_{k=1}^{\infty}{1\over2^{k}k^{n}}\,.
\label{zateunda}
\end{equation}
The two numerical constants that appear in eq.~(\ref{C0}) and eq.~(\ref{C1})
were obtained in eq.~(19) and eq.~(20) of ref.~\cite{Chetyrkin:2006dh}:
\begin{equation}
T_{54,3}=-8445.8046390310298\dots \;\;\mbox{and}\;\;
T_{62,2}=-4553.4004372195263\dots\,.
\end{equation}
Numerically the coefficients $\overline{C}_0$ and $\overline{C}_1$ are given
by:
\begin{eqnarray}
\label{C0Numerik}
\overline{C}_0 &=& \left({\alpha_s\over\pi}\right) \* 1.4444
+ \left({\alpha_s\over\pi}\right)^2\*
\left(1.5863 + 0.1387\*\nh + 0.3714\*\nl \right)
\nonumber\\
&+& \left({\alpha_s\over\pi}\right)^3\*
( 0.0252\*\nh\*\nl + 0.0257\*\nl^2 - 0.0309\*\nh^2
- 3.3426\*\nh - 1.2112\*\nl
\nonumber\\
&&\qquad\quad+1.4186 )+\dots\\
\mbox{and}&&\nonumber\\
\label{C1Numerik}
\overline{C}_1 &=&1.0667
+ \left({\alpha_s\over\pi}\right) \* 2.5547
+ \left({\alpha_s\over\pi}\right)^2\*
\left(0.2461 + 0.2637\*\nh + 0.6623\*\nl \right)
\nonumber\\
&+& \left({\alpha_s\over\pi}\right)^3\*
( 0.1658\*\nh\*\nl + 0.0961\*\nl^2 + 0.0130\*\nh^2
- 6.4188\*\nh - 2.9605\*\nl
\nonumber\\
&&\qquad\quad+
8.2846)+\dots\,.
\end{eqnarray}
The coefficient $\overline{C}_1$ can be used to evaluate the influence of the
four-loop correction on the value of the charm- and bottom-quark mass given in
section~\ref{GeneralNotations}. The $\overline{\mbox{MS}}$-charm-quark mass is
reduced by about 2 MeV, for bottom-quarks we find a reduction by about 4 MeV.
%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Summary and Conclusion}
\label{DiscussConclude}
The calculation of Taylor expansion coefficients of the vacuum
polarization function is important for a precise determination of the
charm- and bottom-quark mass. Within this work we have presented a new
result at four-loop order in perturbative QCD for the first two
coefficients of the Taylor expansion. For the computation the
traditional IBP-method in combination with Laporta's algorithm has been
used in order to reduce all appearing integrals on a small set of master
integrals. The new four-loop contributions decrease the value of the
$\overline{\mbox{MS}}$-charm- (bottom-)quark mass by around 2~MeV
(4~MeV). With the knowledge of the four-loop contributions the
theoretical uncertainty is well under control in the view of the current
and foreseeable precision of experimental data.\\
\noindent
{\bf Acknowledgments:}\\
%%%%
The authors would like to thank M.~Faisst for numerous discussions and for
providing numerical cross checks of partial results. The work was supported by
the Deutsche Forschungsgemeinschaft through the SFB/TR-9 ``Computational
Particle Physics''. The work of C.S. was also partially supported by MIUR
under contract 2001023713$\_$006.
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