%Title: Recent Results on Four-loop Tadpoles
%Author: K.G. Chetyrkin, J.H. K\"uhn and C. Sturm
%Published: <EM> Nucl. Phys. (Proc. Suppl.)  </EM> <B> B160 </B> (2006) 230-234. <EM>Proceedings of the 8th Zeuthen Workshop on Elementary Particle Theory, Loops and Legs in Quantum Field Theory</EM>, 23-28 April 2006, Eisenach, Germany. 
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% declarations for front matter
\title{Recent Results on  Four-loop Tadpoles}

\author{K.~G.~Chetyrkin%
        \address[KAUNI]{Institut f{\"u}r Theoretische Teilchenphysik,
	                Universit\"at Karlsruhe, 
                        D-76128 Karlsruhe, 
                        Germany}%
        \thanks{On leave from Institute for Nuclear Research of the 
     	        Russian Academy of Sciences, Moscow, 117312, Russia.},
        J.~H.~K{\"u}hn\addressmark[KAUNI]
        and
       \underline{C.~Sturm}\address{Dipartimento di Fisica Teorica, 
                         Universit{\`a} di Torino, 
                         I-10125 Torino, Italy %
                        \& INFN, Sezione di Torino}
        }
       
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}

\begin{abstract}
The experimental measurable $R$-ratio can be used to perform a precise
determination of the charm- and bottom-quark mass with the help of
theoretical computable expansion coefficients of the vacuum
polarization function.  The four-loop contributions in perturbative QCD
to the first two expansion coefficients of the vacuum polarization
functions are presented.
\vspace{1pc}
\end{abstract}

\maketitle

\section{Introduction}
Due to the progress in the last years in calculational  techniques a number
 of physical quantities related to  the computation of four-loop
 tadpole diagrams have been evaluated recently. For example, the
 matching condition for the strong coupling constant $\alpha_s$ at a
 heavy quark threshold  has been determined in
 ref.~\cite{Schroder:2005hy,Chetyrkin:2005ia} to four-loop order in the modified minimal
 subtraction scheme.  This  requires
 the calculation of tadpole diagrams with massive and massless
 propagators. The calculation of the four-loop $\beta$-function, which
 in contrast requires the evaluation of completely massive tadpole
 integrals, has been performed previously in
 ref.~\cite{vanRitbergen:1997va,Kajantie:2003ax,Czakon:2004bu}.\\
%%
Moments of the cross section for electron-positron annihilation
into hadrons are  also related to  tadpole  diagrams.
In this context the correlator of two electromagnetic currents
$j^\mu(x)=\overline{\Psi}(x)\gamma^\mu\Psi(x)$ of a heavy quark with
mass $m_q$ is important. It is connected to the vacuum polarization
function $\Pi(q^2)$ through:
\begin{eqnarray*}
(q^{\mu}\*q^{\nu}-q^2\*g^{\mu\nu})\*\Pi(q^2)&\!\!\!\!\!=&\!\!\!\!\!
i\!\int\!\!dx e^{iqx}\langle0|Tj^{\mu}(x)j^{\nu}(0)|0\rangle,
%\nonumber\\
\end{eqnarray*}
where $q^\mu$ is the external momentum. On the one hand the vacuum
polarization function can be linked to moments~$\mathcal{M}_n$ with the
help of dispersion relations:
\begin{equation}
\mathcal{M}_n^{\mbox{\scriptsize{exp}}}=\!\!\int\!ds{R(s)\over 
s^{n+1}}={12\pi^2\over n!}\left({d\over dq^2}\right)^n \left.\Pi(q^2)\right|_{q^2=0}\!,
\label{moments}
\end{equation}
$ \mathcal{M}_n^{\mbox{\scriptsize{exp}}}$
depends on the experimental measurable $R$-ratio
$R(s)=\sigma(e^+e^-\rightarrow \mbox{hadrons})/\sigma(e^+e^-\rightarrow
\mu^+\mu^-)$. On the other hand the derivatives of the vacuum
polarization function can be calculated in perturbative QCD. 
Let us define
\begin{equation}
\overline{\Pi}(q^2)
%\Big|_{\mbox{\scriptsize{non-singlet}}}
={3\*Q_q^2\over16\pi^2}\sum_{n\ge0}\overline{C}_n\*z^n\,,
\label{expansion}
\end{equation}
where $z=q^2/(4\*\overline{m}_q^2)$ and $Q_q$ is the electric charge
of the heavy quark. For  quantities with a  bar renormalization is
performed in the $\overline{\mbox{MS}}$-scheme. The expansion in
equation~(\ref{expansion}) leads to tadpole integrals. From the first
and the dhigher derivatives of the vacuum polarization function one can
extract the heavy quark mass by combining
equation~(\ref{moments})~and~(\ref{expansion}):
\begin{equation}
\overline{m}_q(\mu)={1\over2}\left({\*9\*Q_q^2\*\overline{C}_n\over4\*\mathcal{M}_n^{\mbox{\scriptsize{exp}}}}\right)^{1/(2\*n)}\,.
\label{masses}
\end{equation}
%%
This method has been originally used to determine the charm- and later
the bottom-quark mass~\cite{Novikov:1977dq,Reinders:1984sr}. Taking into
account moments up to three-loop
order~\cite{Chetyrkin:1996cf,Chetyrkin:1997mb}  precise results for
both quark masses have been obtained~\cite{Kuhn:2001dm,Kuhn:2002zr}. 
In  view of the present and
foreseeable precision of experimental data it is important to know the
four-loop contributions~\cite{Chetyrkin:2006xg} whose evaluation is
subject of this work.
%%

This article is structured as follows: In section~\ref{Calculation} we
 outline  the computation of the four-loop contributions to the
vacuum polarization function in the low energy limit. We discuss the
calculational methods, the determination of the master integrals and
finally present the result. In section~\ref{Pheno} we examine the
phenomenological implications of this result and in
section~\ref{SumConc} we close with a brief summary and conclusions.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Outline of the calculation}
\label{Calculation}
\subsection*{Calculation methods}
In a first step about 700 Feynman diagrams were generated using the
program {\tt{QGRAF}}~\cite{Nogueira:1991ex} and classified according to
the number of inserted closed fermion loops. In four-loop
approximation one thus obtains three classes of diagrams for the
considered problem, shown on fig.~\ref{Decomposition}(a)-(c). Within
this work we consider the calculation of the non-singlet diagrams at
four-loop order. Contributions of singlet type diagrams like the
representative in fig.~\ref{Decomposition}(d) were studied in
ref.~\cite{Groote:2001vr,Portoles:2001yu,Portoles:2002rt}.
\begin{figure}[!ht]
\vspace{-0.6cm}
\begin{center}
\begin{minipage}{2.8cm}
\begin{center}
\includegraphics[width=2.8cm,bb=194 558 387 685]{nf2}
(a)
\end{center}
\end{minipage}
\hspace{0.6cm}
\begin{minipage}{2.8cm}
\begin{center}
\includegraphics[width=2.8cm,bb=194 562 387 685]{nf1}
(b)
\end{center}
\end{minipage}\\[0.1cm]
%\hspace*{0.6cm}
\begin{minipage}{2.8cm}
\begin{center}
\includegraphics[width=2.6cm,bb=232 454 424 573]{NSingulett}
(c)
\end{center}
\end{minipage}
\hspace{0.6cm}
\begin{minipage}{2.8cm}
\begin{center}
\includegraphics[width=2.6cm,bb=231 452 423 574]{Singulett}
(d)
\end{center}
\end{minipage}
\end{center}
\vspace{-0.9cm}
\caption{The first diagram (a) belongs  to the $\nf^2$-contribution,
         the second diagram (b) to the $\nf^1$-contribution, the
         third diagram (c) contributes to the $\nf^0$-contribution,
         which does not involve any insertion of closed fermion
         loops. The fourth diagram (d) is a singlet-diagram. The
         wavy lines denote the external vector current and the spiral
         lines represent gluons. \label{Decomposition}}
\end{figure}\\
%%
The multitude of tadpole integrals have been reduced to a small set
of master integrals with the traditional Integration-by-parts (IBP)
method in combination with the Laporta-algorithm
\cite{Laporta:2001dd,Laporta:1996mq}. The generation and solution of the
linear system of equations has been implemented in a program
\cite{Sturm:2005} based on
{\tt{FORM}}\cite{Tentyukov:2006ys,Vermaseren:2002rp,Vermaseren:2000nd}
and {\tt{FERMAT}}\cite{Lewis}, which is used to simplify the rational
functions in the space time dimension $d$. Furthermore, all symmetries of
the Feynman integrals have been taken into account in an automated
way. This  helps to reduce the number of initial
IBP-identities, which need to be generated and solved. Likewise, the
consideration of symmetries is important to keep the size of the
integral tables containing the solutions of the linear system of
equations under control. All integrals can thus be mapped 
on a small set of 13 master integrals.

\subsection*{Master integrals}
The 13 master integrals have been evaluated in
ref.~\cite{Schroder:2005va} with the help of the difference equation
method~\cite{Laporta:1996mq,Laporta:2001dd,Laporta:2002pg}. Independently
they have been calculated in ref.~\cite{Chetyrkin:2006dh} by
constructing an $\varepsilon$-finite basis of master integrals.\\
%%
While solving the system of IBP-identities a division by
$\varepsilon=(4-d)/2$ can appear. The coefficient functions of
the master integrals can contain so called spurious poles. Master
integrals with spurious poles in their  coefficient functions need to
be evaluated deeper in the $\varepsilon$-expansion. In general each
additional order in $\varepsilon$ in which a master integral has to be
evaluated is increasingly  cumbersome, in particular,  if numerical methods
need to be applied. For this reason it is useful to select a set of
master integrals whose coefficient functions are finite in the limit
$\varepsilon\rightarrow0$ by exploiting the freedom in the choice of the
master integrals. This defines an $\varepsilon$-finite basis.    
 Trivial master integrals, which can be completely
expressed in terms of $\Gamma$-function,  are not considered in this 
construction. These  are shown in
fig.~\ref{Analy}.
\begin{figure}[!hb]
\begin{center}
\begin{minipage}{2.8cm}
\begin{center}
\includegraphics[width=1.8cm,bb=128 305 475 677]{T63}\\
$T_{63}$
\end{center}
\end{minipage}
\begin{minipage}{2.8cm}
\begin{center}
\includegraphics[width=1.8cm,bb=128 305 475 677]{T53}\\
$T_{53}$
\end{center}
\end{minipage}\\[0.cm]
\begin{minipage}{2.8cm}
\begin{center}
\includegraphics[width=1.8cm,bb=166 335 403 668]{T51}\\
$T_{51}$
\end{center}
\end{minipage}
\begin{minipage}{2.8cm}
\begin{center}
\includegraphics[width=2.2cm,bb=128 317 475 664]{T41}\\[0cm]
$T_{41}$
\end{center}
\end{minipage}
\end{center}
\vspace{-0.7cm}
\caption{Master integrals which are known completely analytically. The
  solid (dashed) lines denote massive (massless) propagators.\label{Analy}}
\end{figure}\\
%%
For the remaining nine master integrals an $\varepsilon$-finite basis
has been constructed in ref.~\cite{Chetyrkin:2006dh}. The master
integrals are shown in fig.~\ref{epfinite}. The $\varepsilon$-finite
master integrals have been calculated semi-numerically in
ref.~\cite{Chetyrkin:2006dh} with the help of Pad{\'e}
approximations~\cite{Fleischer:1994ef,Baikov:1995ui,Faisst:2004kz,Faisst:2005}. The
result is in perfect agreement with ref.~\cite{Schroder:2005va}.
\begin{figure}[!hb]
\begin{minipage}{2.4cm}
\begin{center}
\includegraphics[width=1.8cm,bb=111 347 445 681]{T81f}\\[0cm]
$T_{81}^f$
\end{center}
\end{minipage}
\begin{minipage}{2.4cm}
\begin{center}
\includegraphics[width=1.8cm,bb=111 347 445 681]{T72f}\\[0cm]
$T_{72}^f$
\end{center}
\end{minipage}
\begin{minipage}{2.4cm}
\begin{center}
\includegraphics[width=1.8cm,bb=111 347 445 681]{T71f}\\[0cm]
$T_{71}^f$
\end{center}
\end{minipage}\\[0.1cm]
\begin{minipage}{2.4cm}
\begin{center}
\includegraphics[width=1.8cm,bb=111 347 445 681]{T64f}\\[0cm]
$T_{64}^f$
\end{center}
\end{minipage}
\begin{minipage}{2.4cm}
\begin{center}
\includegraphics[width=1.8cm,bb=111 347 445 681]{T61f}\\[0cm]
$T_{61}^f$
\end{center}
\end{minipage}
\begin{minipage}{2.4cm}
\begin{center}
\includegraphics[width=1.8cm,bb=110 353 439 682]{T52f}\\[0cm]
$T_{52}^f$
\end{center}
\end{minipage}\\[0.1cm]
\begin{minipage}{2.4cm}
\begin{center}
\includegraphics[angle=90,width=1.8cm,bb=126 305 477 677]{T62f}\\[0cm]
$T_{62}^f$
\end{center}
\end{minipage}
\begin{minipage}{2.4cm}
\begin{center}
\includegraphics[height=1.8cm,bb=128 305 475 677]{T91f}\\[0cm]
$T_{91}^f$
\end{center}
\end{minipage}
\hspace{0.1cm}
\begin{minipage}{2.3cm}
\begin{center}
\includegraphics[height=1.8cm,bb=121 353 537 682]{T54f}\\[0cm]
\hspace{-0.5cm}$T_{54}^f$
\end{center}
\end{minipage}
\caption{Master integrals being member of the $\varepsilon$-finite
basis. The solid (dashed) lines denote massive (massless)
propagators. One master integral ($T_{54}^f$) has a scalar
product.\label{epfinite}}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection*{Result}
If ones inserts the results for the master integrals into the 
solution provided by  the Laporat-algorithm 
and performs renormalization in the
$\overline{\mbox{MS}}$-scheme one obtains for the first two expansion
coefficients $\overline{C}_0$ and $\overline{C}_1$ 
(adopting  the renormalization scale $\mu=m$):
\begin{eqnarray}
\label{C0}
\overline{C}_0&=&
%1.33333333333333\*l_\mu
%\nonumber\\
%%%%%%%%%%%%%%%%%%%%%%%%%%
+\left({\alpha_s\over\pi}\right)\phantom{^1}
 1.44444444444444
\nonumber\\&&
+\left({\alpha_s\over\pi}\right)^2\big(
 1.72492329642573
\nonumber\\&&\qquad\qquad
+0.37139917695473\*\nl\big)
\nonumber\\&&
+\left({\alpha_s\over\pi}\right)^3\big(
-1.95490692882665
\nonumber\\&&\qquad\qquad
-1.18600984426309\*\nl
\nonumber\\&&\qquad\qquad
+0.02570664656045\*\nl^2\big)\,,
%\nonumber
\end{eqnarray}
\begin{eqnarray}
\label{C1}
%%%%%%%%%%%%%%%%%%%%%%%%%%
\overline{C}_1&=&
%\phantom{\left({\alpha_s\over\pi}\right)^0}
+1.06666666666667
\nonumber\\&&
+\left({\alpha_s\over\pi}\right)\phantom{^1}
 2.55473251028807
\nonumber\\&&
+\left({\alpha_s\over\pi}\right)^2\big(
 0.50988169828765
\nonumber\\&&\qquad\qquad
+0.66227709190672\*\nl\big)
\nonumber\\&&
+\left({\alpha_s\over\pi}\right)^3\big(
 1.87882284654395
\nonumber\\&&\qquad\qquad
-2.79472185790743\*\nl
\nonumber\\&&\qquad\qquad
+0.09610140043295\*\nl^2\big)\,.
\end{eqnarray}
The symbol $\nl$ denotes the number of light active quarks which are
considered as massless. The $\nf^2$-contribution\cite{Chetyrkin:2004fq}
and $\nf^1$-contribution are known completely analytically. The terms 
proportional to $\alpha_s^k\*\nl^{k-1}$ are known to all orders
$k$ in~\cite{Grozin:2004ez}.
The results for $\overline{C}_0$ and $\overline{C}_1$ in the analytical form
are given in~\cite{Chetyrkin:2006xg}.   
\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Phenomenological implications}
\label{Pheno}
In this section we discuss the influence of the newly computed
correction on the charm- and bottom-quark masses as well as its effect
on the (unphysical) renormalization scale dependence of the
scale-invariant quark masses. In our analysis we will closely follow
\cite{Kuhn:2001dm}. In particular, we will take  from
\cite{Kuhn:2001dm} the value of the first ``experimental'' moment as
defined in eq.~(\ref{moments}).  The latest experimental information on
$R(s)$ which appeared after publication of \cite{Kuhn:2001dm} will be
taken into account in a future study.
%%
 
Using relation~(\ref{masses}) for the first moment at three- and
four-loop approximations one can assess the influence of the new
four-loop order on the values of the charm- and bottom-quark mass.
\ice{
 with
the help of the equation:
\begin{equation}
\overline{m}_q(\mu) = \frac{1}{2}\, 
\left(
\frac{9}{4} \,  Q_q^2\,
\frac{
\overline{C}_1(\alpha_s(\mu),\overline{m}_q(\mu),\mu)
}{\mathcal{M}_1^{\mbox{\scriptsize{exp}}}}
\right)^{1/2}
\label{mass_eq}
{}.
{}
\end{equation}
%%
}

Let us first summarize the current status of the charm- and bottom-quark
masses as obtained in \cite{Kuhn:2001dm} from the first moment
evaluated to order $\alpha_s^2$:
\begin{eqnarray}
%\overline{m}_c(3\GeV) &=& 1.0270 \pm 0.0022 \GeV
\overline{m}_c(3\GeV) &=& 1.027 \pm 0.002 \GeV
{},
\label{mc3:3l:KS}
\\
%\overline{m}_b(10\GeV) &=& 3.6652 \pm 0.0050 \GeV
\overline{m}_b(10\GeV) &=& 3.665 \pm 0.005 \GeV
{}.
\label{mb10:3l:KS}
\end{eqnarray}
Note that here and below we display only the uncertainties coming from
the variation of the renormalization scale $\mu$ in the region $ \mu =
10 \pm 5 \GeV$ for the bottom-quark and $ \mu = 3 \pm 1 \GeV$ for the
charm-quark respectively. The transformation of the results to the scale
invariant mass $\overline{m}_q(\overline{m}_q)$ using the {\em
three-loop coefficients} of the renormalization group (RG) functions
leads to
\begin{eqnarray}
%\overline{m}_c(\overline{m}_c) &=& 1.3042 \pm 0.0020 \GeV
\overline{m}_c(\overline{m}_c) &=& 1.304 \pm 0.002 \GeV
{},
\label{mcmc:3l:KS}
\\
%\overline{m}_b(\overline{m}_b) &=& 4.2047 \pm 0.0048 \GeV
\overline{m}_b(\overline{m}_b) &=& 4.205 \pm 0.005 \GeV
{}.
\label{mbmb:3l:KS}
\end{eqnarray}
%%

The inclusion of the four-loop contribution to the function
$\overline{C}_1$ leads for the case of the charm-quark to the following
modification of eq.~(\ref{mc3:3l:KS}):
\begin{equation}
%\overline{m}_c(3\GeV) = 1.02267 {}^{+0.00005}_{-0.00045} \GeV
\overline{m}_c(3\GeV) = 1.023 {}^{+0.00005}_{-0.0005} \GeV
\label{mc3:4l}
{},
\end{equation}
which corresponds to the following result for the scale-invariant mass:
\begin{equation}
%\overline{m}_c(\overline{m}_c) = 1.31049 {}^{+0.00004}_{-0.00041} \GeV
\overline{m}_c(\overline{m}_c) = 1.310 {}^{+0.00004}_{-0.0004} \GeV
\label{mcmc:4l}
{}.
\end{equation}
%%

Note that for both equations above we have used the evolution with
four-loop accuracy as required by the self-consistency
considerations. If
%, like K\"uhn and Steinhauser ({\em and like Czakon
%and Co}) 
we would use the RG equation only with the three-loop accuracy
this would not only lead to somewhat larger errors (by around 20-25\%)
but also to a {\em significantly smaller} value of
$\overline{m}_c(\overline{m}_c) = 
%1.30049 
1.300 \GeV$ .
%%

For the case of the bottom-quark our results read:
\begin{equation}
%\overline{m}_b(10\GeV) =  3.66478 {}^{+0.00023}_{-0.00109} \GeV
\overline{m}_b(10\GeV) =  3.665 {}^{+0.0002}_{-0.0011} \GeV
\label{mb10:4l}
{},
\end{equation}
which corresponds to the  following result for the scale-invariant mass:
\begin{equation}
%\overline{m}_b(\overline{m}_b) = 4.20992 {}^{+0.00022}_{-0.00106} \GeV
\overline{m}_b(\overline{m}_b) = 4.210 {}^{+0.0002}_{-0.0011} \GeV
\label{mbmb:4l}
{}.
\end{equation}
Again the equations above correspond to the use of the four-loop evolution
equation. Due to the larger scale involved the use of the three-loop
running would move the $\overline{m}_b(\overline{m}_b)$ by about
five~MeV down (leaving it practically unchanged from the result of
\cite{Kuhn:2001dm} within our accuracy) and again would lead to a slight
increase of the error.
%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Summary and conclusion}
\label{SumConc}

The calculation of the expansion coefficients of the vacuum polarization
function allows a precise determination of the charm- and bottom-quark
masses. The reduction to master integrals of the first two expansion
coefficients $\overline{C}_0$ and $\overline{C}_1$ at four-loop order in
perturbative QCD has been completed using the IBP-method in combination
with the Laporta-Algorithm. Furthermore all master integrals were
obtained with two independent methods: the difference equation method
and the method of the $\varepsilon$-finite basis. The influence of the
new four-loop contributions on the values of the charm- and bottom-quark
mass is small: the newly computed four-loop term moves up the
charm-quark mass by 6~MeV and the bottom-quark mass  by 5~MeV.\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\noindent
{\bf Acknowledgments:}\\ 
%%%%
The authors are grateful to M.~Faisst for numerous discussions and
for providing numerical cross checks of partial results. 
We thank M.~Czakon  for an interesting discussion. 
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.\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\noindent
{\bf Note added.}
\\
\noindent
The results of our calculations as expressed in eq. (\ref{C1})
have been confirmed in the work \cite{Boughezal:2006px}, where also a
strong reduction of the theoretical error due to the unphysical scale
dependence of $\overline{m}_q(\overline{m}_q)$ has been found.
Furthermore, the complete four-loop QCD correction to the electroweak
$\rho$ parameter were calculated in ~\cite{Chetyrkin:2006bj} and
confirmed in \cite{Boughezal:2006xk} (note that the calculation of the
so-called singlet contribution to the full result was performed in an
earlier work \cite{Schroder:2005db}).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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