%Title: Multiloop Tadpoles
%Author: M. Faisst, K.G. Chetyrkin, Johann H. K�hn
%Published: <EM> Nucl.Phys.Proc.Suppl. </EM> <B> 135 </B> (2004) 307-310. 
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\title{Multiloop Tadpoles}

\author{M.~Faisst\address[ttp]{Institut f\"ur Theoretische
    Teilchenphysik, \\ Universit\"at Karlsruhe, D-76128 Karlsruhe,
    Germany}, K.G.~Chetyrkin\addressmark, and
    Johann~H.~K\"uhn\addressmark}
       
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\runtitle{Multiloop Tadpoles}
\runauthor{M.~Faisst}

\begin{document}

\begin{abstract}
We present a semi-numerical approach which allows the calculation of
four-loop tadpole diagrams. The method is introduced by presenting the
results for a scalar four-loop diagram. Finally, the application
of the method to the calculation of a physical quantity like the four-loop
vacuum polarization in QCD is discussed.
\vspace{1pc}
\end{abstract}

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\maketitle

\section{Introduction}
The great success of modern high precision phenomenology would not have
been possible without the development of a variety of calculational
methods for the evaluation of Feynman diagrams. A basic ingredient
which enters at many quite different stages are the vacuum diagrams,
also called tadpole diagrams. They appear as factorizible parts of
complicated Feynman diagrams or directly build up the diagrammatic
representation of a quantity of physical interest like, for example, the
low momentum expansion of the vacuum polarization. There are also
expansion techniques \cite{asyexpansion} which
transform complicated diagrams into a combination of simpler massless
diagrams and tadpole diagrams. Since the latter manifest 
themselves in a variety of applications, the detailed control over this
important building block is mandatory. Furthermore, given 
the increasing resolution of modern experiments, one is forced to also
push ahead the efforts on the theory side, calculating higher
orders in perturbation theory to be able to keep the theoretical
uncertainties below the experimental precision. 

The results of the evaluation of a complete set of master diagrams for
tadpoles up to the three-loop order, including an algorithm to reduce an
arbitrary vacuum diagram to this set of master integrals,
are available \cite{tad3l}. This algorithm is implemented in the 
computer algebra package {\tt MATAD} \cite{matad} written in {\tt
  FORM} \cite{FORM}. The obvious next step is the
treatment of four-loop tadpole diagrams. In many applications the method 
of Laporta \cite{laporta:alg} is at present a highly useful approach
for the treatment of multiloop diagrams. In contrast, we followed a
different strategy which was already successfully applied
in \cite{pade:work}. In the next sections we will use the calculation of
a scalar tadpole as an example. Then we will indicate 
how to apply the same technique to the vacuum polarization in QCD.

\section{Method\label{sec:method}}

The idea of this approach is the reduction of the number of analytical
integrations by one, having then to deal with three-loop diagrams. The
remaining one-dimensional integration is performed numerically. The
integrand is obtained by cutting one line of the four-loop tadpole, thus
obtaining a three-loop self-energy diagram. Since there is no
other four-momentum involved, the self-energy diagram only depends
on the momentum $p^2$ flowing through the cut line. Thus one is
able to immediately perform the angular part of the remaining loop
integration and is left with a one-dimensional integration over the
modulus of the momentum $p$. Following this procedure, the calculation of
the four-loop tadpole is transformed to the problem of evaluating a three-loop
self-energy.

\begin{table*}[ht]
\centering
\begin{tabular}{lc|c|c|c|c|c|c}
\multirow{2}{1cm}{\normalsize $\times 10^{-5}$}
& & \multicolumn{6}{c}{high energy} \\
\multicolumn{2}{c|}{} & 3 & 4 & 5 & 6 & 7 & 8 \\ \hline
\multirow{3}{1.3cm}{low energy}
& 3 &  -191.42436 &  -6.01348 &  \phantom{-}0.44044 &
\phantom{-}0.09165 & 0.00151 & -0.01738 \\ \cline{2-8}
& 4 &    \phantom{00}-5.15001 &   \phantom{-}1.94591 & -0.10397 &
-0.00344 & 0.00052 & \phantom{-}0.00029 \\ \cline{2-8} 
& 5 &   \phantom{0}-16.06059 &   \phantom{-}0.74717 &
\phantom{-}0.01367 & -0.00101 & 0.00388 & -0.00025 \\
\end{tabular}

\caption{\label{tab:cut1}Difference between the known result $1.34894
  80217 09708 \dots$ by Laporta and our method taking cut 1. The
  dependence on the number of low- and high-energy input terms is
  given. Note that each value has to be multiplied by a factor of
  $10^{-5}$.}
%\end{table*}

\phantom{h}

%\begin{table*}[h!]
\begin{tabular}{lc|c|c|c|c|c|c}
\multirow{2}{1cm}{\normalsize $\times 10^{-5}$}
& & \multicolumn{6}{c}{high energy} \\
\multicolumn{2}{c|}{} & 3 & 4 & 5 & 6 & 7 & 8 \\ \hline
\multirow{3}{1.3cm}{low energy}
& 3 & -579.78184 &  8.92004 & -0.08593 &  \phantom{-}0.01259 &
\phantom{-}0.00456 & \phantom{-}0.00010 \\ \cline{2-8} 
& 4 & -186.75432 &  1.21461 & -0.04923 & -0.00242 & -0.00008 & -0.00004
\\ \cline{2-8} 
& 5 & \phantom{0}-41.00154  &  0.53958 & -0.10894 &  \phantom{-}0.00332
&  \phantom{-}0.00009 & -0.00002 \\ 
\end{tabular}
\caption{\label{tab:cut2}Difference between the known result $1.34894
  80217 09708 \dots$ by Laporta and our method taking cut 2. The
  dependence on the number of low- and high-energy input terms is
  given. Note that each value has to be multiplied by a factor of
  $10^{-5}$.}
\end{table*}
These three-loop diagrams can be treated with the help of asymptotic
expansions. Specifically we obtain the self-energy in the low-
and high-energy limit. These two limits can be combined in a
Pad\/e-approximation, a rational function with the same low- and
high-energy expansion as the exact self-energy. It had been shown in
\cite{pade:work} and \cite{vacpol3l} that this way of reconstructing the
general dependence on the external momentum leads to a very good
agreement with the full analytical result.

\section{A Scalar Four-Loop Tadpole\label{sec:scalar}}
In the following we will consider the diagram shown in figure
\ref{fig:tad4l}, 
\begin{figure}[h]
  \centering
  \includegraphics[bb=271 555 330 586,height=1.5cm]{tad4l.eps}
  \caption{Scalar four-loop diagram with all lines massive. The bold lines
    indicated by the numbers represent two possible cuts of the
    diagram, leading to different three-loop self-energy diagrams. 
  \label{fig:tad4l}}
\end{figure}
a scalar tadpole diagram with all lines representing a massive
propagator. Two independent cuts are taken at the bold lines indicated
by number $1.$ and $2.$ In this way we obtain two different three-loop
self-energy diagrams which have to be evaluated according to section
\ref{sec:method}. The calculation of a physical quantity involves a
large number of diagrams and the set-up thus makes use of several
computer algebra packages. The program {\tt EXP} \cite{EXP} extracts the
subdiagrams needed for the asymptotic expansions. After the  
application of the expansion procedures, the resulting integrals are
tadpoles with propagators of one mass scale in combination with massless
propagator type integrals. Their computation can be performed using the
program packages {\tt MATAD} and {\tt MINCER} \cite{MINCER}. We obtain
the low- and high-energy expansion of the diagrams and construct the
Pad\/e approximation. After performing the final numerical integration, we
find the results summarized in tables \ref{tab:cut1} and
\ref{tab:cut2}. The results are compared to the value $1.34894 80217 09708
\dots$ given by Laporta in \cite{Laporta:tad}. The numbers (multiplied
by $10^{-5}$) in the tables \ref{tab:cut1} and \ref{tab:cut2} give the
difference between the result by Laporta and our method in its
dependence on the number of low- and high-energy input terms. One
observes that these values converge quite fast towards the precisely
known result. Already three terms in both the low- and the high-energy
expansion lead to an agreement better than $1\%$. This also shows that
the final result does not depend on the line where the cut 
is performed, although the speed of convergence differs for the two
cases. Given these results, we conclude that this semi-numerical method
works very well and can be applied to the calculation of a generic set
of tadpole diagrams.

\section{Vacuum Polarization}

At present our major goal is the calculation of the low energy expansion
of the four-loop vacuum polarization in the framework of QCD. The Taylor
expansion of the relevant photon self-energy leads to a series in the 
external momentum $q^2$, times a set of four-loop tadpole diagrams. On the
four-loop level this amounts to about 700 diagrams. Facing this number of 
diagrams, one has to use automatized methods. We use the program 
{\tt QGRAF} \cite{QGRAF} to generate the diagrams. In figure
\ref{fig:method1}
\begin{figure}[h]
  \centering
  \parbox{2.2cm}{\includegraphics[height=2cm]{method1.eps}} $\rightarrow$
  \parbox{3cm}{\includegraphics[height=2cm]{method2.eps}}
  \caption{\label{fig:method1}General idea how diagrams are generated}
\end{figure}
the implementation is indicated. The cut is always located next to the
same external vertex, thus generating vertex type diagrams shown on the
right side of figure \ref{fig:method1}. By using {\tt EXP} to treat the
output of {\tt QGRAF}, it is straight forward to implement the expansion
with respect to the external momentum $q$ and the low- and high energy
expansions with respect to the loop momentum $p$.

The following steps can, in principle, be performed in the same way as
described in the previous section for the scalar example. In contrast to
the scalar diagram in figure \ref{fig:tad4l}, where we are dealing with 
finite contributions at all stages, the sum of the whole set of four-loop
QCD diagrams obtained as described still exhibits some
divergencies. These divergencies cancel after the final integration and
the full renormalization. The same approach as presented in section
\ref{sec:scalar} would lead to a numerical cancellation of these
divergencies. But as the sum of all QCD diagrams $\Pi_n$ on the right
hand side in figure \ref{fig:method1} has the same divergence structure
as the quark self-energy in QCD, one could remove these by multiplying
with the quark field renormalization $Z_2$ in the appropriate way:
\begin{equation}\label{eq:div1}
\begin{split}
\overline{\Pi}_{3} = \bigg[& \frac 1{Z_2} \bigg(
\parbox{1cm}{\includegraphics[width=1cm]{vertex0.eps}} +
\parbox{1cm}{\includegraphics[width=1cm]{vertex1.eps}}
+\dots+\\
&\parbox{1cm}{\includegraphics[width=1cm]{vertex2.eps}}
 +\dots+
\parbox{1cm}{\includegraphics[width=1cm]{vertex3.eps}}
 +\dots \, \bigg) \bigg] \bigg|_{\tiny \alpha_s^3}.
\end{split}
\end{equation}
The low- and high-energy expansion of the $ \alpha_s^3$ contribution
$\overline{\Pi}_{3}$ is now finite and we are able to apply the procedure
described in section \ref{sec:scalar} to get a finite numerical four-loop
result. It is left to correct for the multiplication with $Z_2$ in
equation (\ref{eq:div1}). This is achieved by adding the analytically
known low-energy expansion up to the three-loop or the contributions up to
order $\alpha_s^2$ respectively with the appropriate combination of
$Z_2$ shown in equation
\begin{equation}\label{eq:div2}
\begin{split}
\Pi_{3}=Z_2 &\bigg( \int dp \, \overline{\Pi}_{3}(p^2)
 + \big[ \frac 1{Z_2} \big( 
\parbox{0.7cm}{\includegraphics[width=0.7cm]{vac1l1.eps}} \\
&+\parbox{0.7cm}{\includegraphics[width=0.7cm]{vac2l1.eps}}
 +\dots+
\parbox{0.7cm}{\includegraphics[width=0.7cm]{vac3l2.eps}}
 +\dots\big) \big] \big|_{\leq \alpha_s^2} \bigg).
\end{split}
\end{equation}
To obtain a more compact expression this can be rewritten as
follows: 
\begin{equation}\label{eq:div3}
\begin{split}
\Pi_{3}= & \int dp \, \overline{\Pi}_{3}(p^2)
 + \bigg[ \bigg(1- \frac 1{Z_2}\bigg) \bigg( 
\parbox{0.7cm}{\includegraphics[width=0.7cm]{vac1l1.eps}}\phantom{.} \\
&+\parbox{0.7cm}{\includegraphics[width=0.7cm]{vac2l1.eps}}
 +\dots+
\parbox{0.7cm}{\includegraphics[width=0.7cm]{vac3l2.eps}}
 +\dots\bigg)
\bigg] \bigg|_{\alpha_s^3}.
\end{split}
\end{equation}


\section{Status}

At present we have recalculated the first physically interesting moment
of the vacuum polarization up to the three-loop level. By taking five
terms of the low- and high-energy expansions respectively, we reproduce
the known results \cite{vacpol3l} to an accuracy at the per-mill
level. The evaluation of the four-loop diagrams is in progress. Given
the same number of terms in the expansions, we expect a four-loop result
of an accuracy in the order of $1\%$.

\section{Summary}

We have described a semi-numerical method for the calculation of four-loop
tadpole diagrams. It has been shown that this framework leads to good
agreement with already available results. The calculation of the four-loop
vacuum polarization is under way.

  This work was supported by the {\it Graduiertenkolleg
``Hochenergiephysik und Teil\-chen\-astrophysik''}, by BMBF under grant 
No. 05HT9VKB0, and the SFB/TR 9 (Computational Particle Physics).

\def\plb#1#2#3{{\it Phys.~Lett.~}{\bf B #1} (#2) #3}
\def\jcp#1#2#3{{\it J.~Comp.~Phys.~}{\bf #1} (#2) #3}

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%%CITATION = HEP-PH 0009029;%%

\bibitem{FORM}
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\end{thebibliography}
\end{document}

