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\section{Reduction technique for integrals with internal self energies}
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\label{sec:reduct-techn-integr}

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\subsection{Algorithm}
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The required CPU time for the reduction of Feynman integrals of a given topology to master integrals using Laporta's algorithm strongly depends on the powers of the propagators and scalar products. For a fixed depth of a Taylor expansion in an external momentum of a Feynman amplitude the maximal number of propagator powers will appear on those integrals which contain the maximal number of self energy insertions. In the case of $4$-loop tadpoles we have at most $3$ self energy subgraphs which lead to a sum of propagator powers raised by $2$ compared to the generic case without self energies. Such integrals are thus particularly difficult for Laporta's algorithm. On the other hand the presence of self energy insertions can be exploited to perform the reduction to master integrals in two less expensive steps, first, the reduction of the self energy subgraphs and, second, the reduction of the remaining integral. In the second step Intgration-by-Parts relations are constructed in which the self energy master integrals are treated as objects depending only on their external momentum $q$. In the following the procedure is illustrated for an arbitrary integral which contains a self energy insertion:
%
\begin{align} \label{initial-se-topo}
  T~=~~\GenericSeTopo~~~~=~~\int d^dq R^{\mu_1\dots \mu_n}(q,m) S_{\mu_1\dots \mu_n}(q,m) P_{m_0}^{a_0}(q^2)\ ,
\end{align}
%
where $S_{\mu_1\dots \mu_n}(q,m)$ is a rank $n$ tensor self energy integral, $R^{\mu_1\dots \mu_n}(q,m)$ is the rest graph, and $P_{m_0}^{a_0}(q^2) = (q^2-m_0^2)^{-a_0}$ is the connecting propagator with mass $m_0$. All masses are assumed to be either zero or $m$. $S_{\mu_1\dots \mu_n}(q,m)$ is expressed in terms of a complete set of Lorentz structures $\{\Pi^p_{\mu_1\dots \mu_n}\}$ of rank $n$ consisting of the momentum $q$ and the metric $g^{\mu\nu}$, and the (scalar) self energy master integrals $S_z(q^2,m)$:
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\begin{align} \label{se-tensor-to-scalar}
  S_{\mu_1\dots \mu_n}(q,m) = \sum_p \Pi^p_{\mu_1\dots \mu_n} \sum_z c_z^p(d,q^2,m) S_z(q^2,m) \ .
\end{align}
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The coefficients $c_z^p(d,q^2,m)$ are rational functions of the space-time dimension $d$, the momentum $q^2$, and the mass $m$. Inserting this decomposition into (\ref{initial-se-topo}) gives
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\begin{align}
  T &= \int d^dq \sum_p R^{\mu_1\dots \mu_n}(q,m) \Pi^p_{\mu_1\dots \mu_n}(q) P_{m_0}^{a_0}(q^2) \sum_z c_z^p(d,q^2,m) S_z(q^2,m).
\end{align}
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The rest graph tensor is contracted with the Lorentz structures to
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\begin{align}
  R^{\mu_1\dots \mu_n}(q,m) \Pi^p_{\mu_1\dots \mu_n}(q) = R^p(q^2,m).
\end{align}
%
From the structure of the IBP identites we know, that the $q^2$ dependence of the denominators of the coefficients $c_z^p(d,q^2,m)$ factorizes into propagator-like objects $P_k(q^2,m)=(q^2-k^2m^2)^{-1}$, where $k$ takes values from a finite set of integers depending on the type of the self energy. Partial fractioning leads to
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\begin{align}\label{coeffpartfrac}
  P_{m_0}^{a_0}(q^2) c_z^p(d,q^2,m) = \sum_k \sum_{r_k} \tilde{c}_z^{p,k,r_k}(d) P_k^{r_k}(q^2,m),
\end{align}
%
where $P_{m_0}(q^2)$ was written as $P_k(q^2,m)$ with $m_0=km$. By
convention we choose $r_k\ge 0$ for $k\ne 0$ so that the decomposition
(\ref{coeffpartfrac}) is unique. After applying these transformations the integral $T$ takes the form
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\begin{align}
  T = \int d^dq \sum_p R^p(q^2,m) \sum_z \sum_k \sum_{r_k} \tilde{c}_z^{p,k,r_k}(d) P_k^{r_k}(q^2,m) S_z(q^2,m),
\end{align}
%
which can be written more conveniently as
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\begin{align}\label{T-to-se-topo}
  T = \sum_k \sum_z \sum_p \sum_{r_k} \tilde{c}_z^{p,k,r_k}(d) T^k_z(\rho^p,r_k,m)
\end{align}
%
with
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\begin{align}\label{reducedtopoTkz}
  T^k_z(\rho^p,r_k,m) = \int d^dq\;R^p(q^2,m) P_k^{r_k}(q^2,m) S_z(q^2,m).
\end{align}
%
$\rho^p$ denotes the set of propagator powers of the rest graph. That means the initial integral $T$ is expressed as a linear combination of integrals of the type $T^k_z(\rho^p,r_k,m)$ in which the self energy insertions appear only as master integrals and all cross talking momenta between the self energy and the rest graph are removed. Figure \ref{t-to-se-topo-fugure} illustrates eq. (\ref{reducedtopoTkz}) for a four-loop vacuum diagram.

\begin{figure}[t]
  \begin{align*}
    \TopoC{SF}~~~&=~~\sum_{b_i,r_0}\tilde{c}_1^{b_i,0,r_0}(d) \TopoD{SF}{0} ~+~ \sum_{b_i,r_2}\tilde{c}_1^{b_i,2,r_2}(d) \TopoD{WF}{2}\\[3mm]
                &+~~\sum_{b_i,r_0}\tilde{c}_2^{b_i,0,r_0}(d) \TopoE{TF}{0} ~+~ \sum_{b_i,r_2}\tilde{c}_2^{b_i,2,r_2}(d) \TopoE{UF}{2}
  \end{align*}
  \caption{The figure above shows an example of eq. (\ref{T-to-se-topo}) for a four-loop tadpole. Note that on the right hand side besides massless (dashed) lines and lines with mass $m$ (solid) also a line with mass $2m$ (double line) appears which is not present in the initial integral. The next step would be to repeat the procedure for the two-loop self energy.}
  \label{t-to-se-topo-fugure}
\end{figure}

The next step is to construct Integration-by-Parts identities for the integrals $T^k_z$ in which the self energy insertions are treated as objects depending only on their external momenta. The identities have the form
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\begin{align}\label{ibp-se}
  0 & = \int d^dq\,\frac{\partial}{\partial k_\mu}\ell_\mu R^p(q^2,m) P_k^{r_k}(q^2,m) S_z(q^2,m)\\
    & = \delta_{k\ell}\,d\,T^k_z(\rho^p,r_k,m) + \int d^dq\,\Big[\ell_\mu\frac{\partial}{\partial k_\mu}\Big]_I R^p(q^2,m) P_k^{r_k}(q^2,m) S_z(q^2,m)\nonumber\\
    & = \delta_{k\ell}\,d\,T^k_z(\rho^p,r_k,m)\nonumber\\
    & + \int d^dq \left(\Big[\ell_\mu\frac{\partial}{\partial k_\mu}\Big]_I R^p(q^2,m) \right) P_k^{r_k}(q^2,m) S_z(q^2,m)\nonumber\\
    & + \int d^dq R^p(q^2,m) \left(\ell_\mu\frac{\partial}{\partial k_\mu} P_k^{r_k}(q^2,m) \right) S_z(q^2,m)\nonumber\\
    & + \int d^dq R^p(q^2,m) P_k^{r_k}(q^2,m) \left(\ell_\mu\frac{\partial}{\partial k_\mu} S_z(q^2,m) \right),\nonumber
\end{align}
%
where $k$ is a loop momentum (i.\,e. either $q$ or a loop momentum of $R^p$) and $\ell$ is a loop momentum or, if present, an external momentum. The notation $\ell_\mu[\partial/\partial k_\mu]_I R^p(q^2,m)$ means that the derivative acts on the integrand of $R^p(q^2,m)$. This part and the contribution $\ell_\mu\partial/\partial k_\mu P_k^{r_k}(q^2,m)$ are treated like in traditional IBP. The difference is the treatment of $\partial/\partial k_\mu S_z(q^2,m)$ in the case $k=q$ where the derivative acts on the self energy. The derivative is explicitly performed and the resulting integrals are reduced to the self energy master integrals $S_z$. The coefficients appearing in this reduction require partial fractioning in $q^2$ again like in (\ref{coeffpartfrac}). Afterwards all terms in (\ref{ibp-se}) are expressed by $T^k_z$ and the system of equations is solved by the Laporta algorithm.

In the case of four-loop tadpoles with a one-loop self energy insertion the restgraph is a two-loop self energy. Therefore the procedure above is applied to the two-loop self energy in a second pass. The topologies which appear in our calculation are depicted in Fig. (\ref{se-topos}).

\begin{figure}[t]
  \begin{center}
    $\TopoB{SA}{(0,1,2)}~~\TopoB{SB}{(0,1,3)}~~\TopoB{SC}{(0,1,2)}~~\TopoB{SD}{(0,2)}~~\TopoB{SE}{(0,2)}~~\TopoB{SF}{(0,2)}~~\TopoB{SG}{(0,2)}$
  \end{center}
  \caption{All four-loop tadpoles with self energy insertions which appear in our calculation can be mapped to these seven topologies. Dotted lines are massless and solid lines carry the mass $m$. The wavy propagator which connects the self energies appears with the masses $km$ where $k$ is an element of the list below the corresponding diagram.}
  \label{se-topos}
\end{figure}

Let us add a few remarks at this point. The number of integers on which a topology $T^k_z$ depends is given by the number of propagators of the initial topology $T$ minus the number of propagators of the self energy subgraphs and the cross talking scalar products. In the case of four-loop tadpoles only one propagator power of initially ten (eight propagators and two irreducible scalar products) survives this procedure. Therefore the combinatorics of the IBP system for the $T^k_z$ is much better behaved than for the initial topology. There is a price to pay for this simplification, of course. The reduction of subgraphs and the partial fractioning of propagators which carry the same momentum but different masses leads to a system of IBP relations which couples different topologies. Reducing a single propagator power from a self energy can produce two powers on an external leg, therefore in the worst case the total number of propagator powers might be doubled. If the rest graph is sufficiently simple these issues are easy to deal with. Futhermore, the tensor reduction of large powers of cross-talking scalar products will produce huge intermediate expressions and many terms in eq. (\ref{T-to-se-topo}). These expressions simplify to the full reduction to master integrals when the solution of the IBP system (\ref{ibp-se}) is inserted. The procedure was implemented in a \texttt{Mathematica} program.


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\subsection{Implementation in {\tt FIRE}}
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Additionally to the stand-alone implementation mentioned above, for
one-loop self energy insertions the procedure was also realized with the
help of the {\tt FIRE}  algorithm \cite{Smirnov:2008iw}.

Instead of the tensor reduction (\ref{se-tensor-to-scalar}) the
so-called \textit{region-bases} feature of \texttt{FIRE} is used. The
tensor structure arises from scalar products of loop-momenta between the
self energy and the rest graph. The \texttt{SBases} program
\cite{Smirnov:2006tz} can construct a basis of recursion relations to
reduce the power of these cross-talking scalar products to zero without
caring about the rest of the intgeral. Such a basis is called a
\textit{region-basis}. In the second step the scalar self energy
integrals are reduced to master integrals. This can either be done by a
region-basis, or with the help of precalculated reduction tables in
combination with the \textit{Rules} feature of \texttt{FIRE}.

In the case of one-loop self energies with one mass (either massive-massive or massive-massless) two master integrals appear, the massive tadpole and the self energy with both propagator powers equal to one. Integrals which contain the tadpole are treated like usual integrals with one loop less than the initial integral, because the tadpole factorizes from the rest of the intgeral. The treatment of integrals with the self energy master insertion is a bit more complicated. One has to introduce a new object in Feynman diagrams. We call this a \textit{heavy dot} which can be either present (integral has a self energy master insertion), or not (integral has a tadpole insertion) and is therefore represented by an index which is either zero or one.

\begin{center}
  $\TopoA{SFo}\quad\longrightarrow\quad\FireSeTopo{1}\quad$and$\quad\FireSeTopo{0}$
\end{center}

Note that we also have to introduce additional lines which carry masses depending on the self energy type. The massive-massive self energy needs a massless and a double-massive line, the massive-massless self energy needs a massless and a usual massive line that is external with respect to the self energy (one of these is naturally present in the initial diagram). These lines are needed to absorb the momentum dependence of the coefficients from the reduction of the self energies to master integrals. The partial fractioning of these two lines is done by a recursion relation which is added to the region basis. The IPBs for the remaining integrals are constructed as in eq. (\ref{ibp-se}) where the creation or destruction of a heavy dot is done by a shift operator for the corresponding index, just like for usual propagators. The reduction procedure aims on reducing the heavy dot index to zero if possible. The {\tt FIRE} program has a setting allowing to use the heavy dots and can create SBases for integrals with heavy dots present. However, the IBP generator coming with \texttt{FIRE} at the moment is not able to construct such IBP relation. As it has been explained above, the procedure can be repeated if a second self energy is present and one can again consider a region-basis corresponding to the second self energy subdiagram, insert rules reducing it to masters and finally result in a problem with two heavy dots.


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\subsection{Top level reduction}
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Although the approach from above is not applicable to general Feynman
integrals, in combination with \texttt{FIRE} for the top level reduction
it can be used for the calculation of arbitrary integrals. In the
highest sectors the SBases are used to reduce the integrals until enough
lines are contracted so that only integrals with internal self energies
are left. From experience we know that the SBases algorithm often fails
in sectors where self energy insertions are present. At this point a
program that implements the above procedure is plugged in with the rules
framework of \texttt{FIRE} to automatically finish the reduction. This
way we do not need SBases for the lower sectors for which in some cases
we could not find a basis. In combination with the top level reduction
single integrals can be calculated if needed or for checks. However, at
present this approach is not suited for large scale calculations. The
self energy formalism itself is very efficient and can be used to
significantly reduce the effort to invest in Laporta algorithm.



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