%Title: Differential Equations for Two-Loop Four-Point Functions
%Author: T.Gehrmann, E. Remiddi
%Published: * Nucl. Phys. * ** B580 ** (2000) 485-518.
%hep-ph/9912329
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\begin{document}
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\begin{titlepage}
\renewcommand{\thefootnote}{\fnsymbol{footnote}}
\vspace*{-1cm}
\begin{flushright}
TTP99--49\\
(hep-ph/9912329)\\
December 1999 \\
\end{flushright}
\vskip 3.5cm
\begin{center}
{\Large\bf Differential Equations for Two-Loop Four-Point Functions}
\vskip 1.cm
{\large T.~Gehrmann} and {\large E.~Remiddi}\footnote{Supported by
Alexander-von-Humboldt
Stiftung, permanent address: Dipartimento di Fisica,
Universit\`{a} di Bologna, I-40126 Bologna, Italy}
\vskip .7cm
{\it Institut f\"ur Theoretische Teilchenphysik,
Universit\"at Karlsruhe, D-76128 Karlsruhe, Germany}
\end{center}
\vskip 2.6cm
\begin{abstract}
At variance with fully inclusive quantities, which have been computed
already at the two- or three-loop level, most exclusive observables are
still known only at one-loop, as further progress was hampered
so far by the greater computational problems encountered in the study
of multi-leg amplitudes beyond one loop.
We show in this paper how the use of tools already employed in inclusive
calculations can be suitably extended to the
computation of loop integrals appearing in the virtual corrections to
exclusive observables, namely two-loop four-point
functions with massless propagators and up to one off-shell leg.
We find that multi-leg integrals, in addition to integration-by-parts
identities, obey also identities resulting from Lorentz-invariance. The
combined set of these identities can be used to reduce the large number of
integrals appearing in an actual calculation to a small number of
master integrals.
We then write down explicitly the differential equations in the external
invariants fulfilled by these master integrals,
and point out that the equations can be used
as an efficient method of evaluating the master integrals themselves.
We outline strategies for the solution of the differential equations,
and demonstrate the application of the method on several examples.
\end{abstract}
\vfill
\end{titlepage}
\newpage
\renewcommand{\theequation}{\mbox{\arabic{section}.\arabic{equation}}}
\section{Introduction}
\setcounter{equation}{0}
\setcounter{footnote}{0}
\renewcommand{\thefootnote}{\arabic{footnote}}
Perturbative corrections to many inclusive quantities have been computed
to the two- and three-loop level in past years. From the technical
point of view, these inclusive calculations correspond to the computation of
multi-loop two-point functions, for which many elaborate
calculational tools have been developed. In contrast,
corrections to exclusive
observables, such as jet production rates, could up to now only be
computed at the one-loop level. These calculations require the
computation of multi-leg amplitudes to the required number of loops,
which beyond the one-loop level
turn out to be a calculational challenge obstructing further progress.
Despite considerable progress made in recent times, many of the
two-loop integrals relevant for the calculation of jet observables
beyond next-to-leading order are still unknown. One particular class of
yet unknown
integrals appearing in the two-loop corrections to three jet production
in electron-positron collisions, to two-plus-one jet production in
electron-proton collisions and to vector boson plus jet production in
proton-proton collisions are two-loop four-point functions with massless
internal propagators and one external leg off shell.
Taking these two-loop four-point
integrals as an example, we elaborate in this paper several
techniques to compute multi-leg amplitudes beyond one loop. We
demonstrate how integration-by-parts identities (already known to be a very
valuable tool in inclusive calculations) and identities following from
Lorentz-invariance (which are non trivial only for integrals depending
on at least
two independent external momenta) can be used to reduce the large number
of different integrals appearing in an actual calculation to a small
number of master integrals. This reduction can be carried out
mechanically (by means of a small chain of
computer programs), without
explicit reference to the actual structure of the integrals under
consideration and can also be used for the reduction of
tensor integrals beyond one loop.
The master integrals themselves, however, can not be computed from
these identities. We derive differential equations in the external
momenta for them. Solving these
differential equations, it is possible to compute the master integrals
without explicitly carrying out any loop integration, so that this
technique appears to be a valuable alternative to conventional
approaches for the computation of multi-loop integrals.
The plan of the paper is as follows. In Section~\ref{sec:master}
we review the derivation of the integration by parts (IBP) identities
and introduce the Lorentz invariance (LI) identities.
In Section~\ref{sec:diffeq} the differential equations for the master
amplitudes are obtained.
The practical application of these tools is outlined in detail
in Section~\ref{sec:oneloop} on a self-contained rederivation of the
one-loop massless box integral with one off shell
leg.
Section~\ref{sec:twoloop} contains examples of
massless two-loop four-point functions with one off-shell
leg, evaluated for arbitrary space-time dimension.
We show which of these functions can be reduced to simpler
functions and which are genuine master integrals, and compute some of
the master integrals by solving the corresponding differential
equations. Finally, Section~\ref{sec:conc} contains
conclusions and an outlook on
potential future applications of the tools developed here. The
higher transcendental functions appearing in our results for the
one and two loop
integrals are summarized in an Appendix, where we also discuss how these
functions can be expanded around the physical number of space-time
dimensions.
\section{Reduction to Master Integrals}
\label{sec:master}
\setcounter{equation}{0}
Any scalar massless two-loop integral can be brought into the form
\begin{equation}
I(p_1,\ldots,p_n) = \int \frac{\d^d k}{(2\pi)^d}\frac{\d^d l}{(2\pi)^d}
\frac{1}{D_1^{m_1}\ldots D_{t}^{m_t}} S_1^{n_1}
\ldots S_q^{n_q} \; ,
\label{eq:generic}
\end{equation}
where the $D_i$ are massless scalar propagators, depending on $k$, $l$ and the
external momenta $p_1,\ldots,p_n$ while $S_i$ are scalar products
of a loop momentum with an external momentum or of the two loop
momenta. The topology (interconnection of
propagators and external momenta) of the integral is uniquely
determined by specifying the set $(D_1,\ldots,D_t)$
of $t$ different propagators in the graph. The integral itself is then
specified by the powers $m_i$ of all propagators and by the selection
$(S_1,\ldots,S_q)$ of scalar products and their powers $(n_1,\ldots,n_q)$ .
(all the $m_i$ are positive integers greater or equal to 1, while the
$n_i$ are greater or equal to 0).
Integrals of the same topology with the same dimension $r=\sum_i m_i$
of the denominator and same total number $s=\sum_i n_i$ of scalar products
are denoted as a class of integrals $I_{t,r,s}$.
Any four-point function depends on three linearly independent
external momenta, $p_1$, $p_2$ and $p_3$. At the two-loop level, one can
combine the two loop momenta $k$ and $l$ and these external momenta
to form 9 different scalar products involving $k$
or $l$. As the propagators present in the graph are (linearly independent)
combinations of scalar products, only $9-t$
different scalar products can appear explicitly in an integral with $t$
different propagators. Since a two-loop four-point function can have at
most seven different propagators, as can be found by considering the
insertion of a propagator into a one-loop four-point function, one has
in general $t\leq 7$, while the
minimum number of massless propagators in a two-loop graph is $t=3$,
corresponding to a two-point function.
The number of different two-loop four-point
integrals for given $t$ (number of different propagators), $r$
(sum of powers of all propagators) and $s$ (sum of powers of all
scalar products) can be computed from simple combinatorics:
\begin{equation}
N(I_{t,r,s}) = { r-1 \choose r-t} {8-t+s \choose s} \; .
\label{eq:intnum}
\end{equation}
The number $N(I_{t,r,s})$ of the integrals grows quickly as $r, s$
increase, but the integrals are related among themselves
by various identities.
One class of identities follows from the fact that the integral over the
total derivative with respect to any loop momentum vanishes in
dimensional regularization
\begin{equation}
\int \frac{\d^d k}{(2\pi)^d} \frac{\partial}{\partial k^{\mu}}
J(k,\ldots) = 0,
\end{equation}
where $J$ is any combination of propagators, scalar products
and loop momentum vectors. $J$ can be a vector or tensor of any rank.
The resulting identities~\cite{chet} are called integration-by-parts (IBP)
identities and can for two-loop integrals be cast into the form
\begin{eqnarray}
\int \frac{\d^d k}{(2\pi)^d} \frac{\d^d l}{(2\pi)^d}
\frac{\partial}{\partial k^{\mu}} v^{\mu} f(k,l,p_i) & = & 0, \nonumber \\
\int \frac{\d^d k}{(2\pi)^d} \frac{\d^d l}{(2\pi)^d}
\frac{\partial}{\partial l^{\mu}} v^{\mu} f(k,l,p_i) & = & 0,
\end{eqnarray}
where the integrand $f(k,l,p_i)$ is a scalar function, containing
propagators and
scalar products and $v_{\mu}$ can be any external or loop momentum vector.
As a consequence, one obtains for a graph with $m$ loops and $n$
independent external momenta a total number of $N_{{\rm IBP}} = m(n+m)$.
For a two-loop four-point function, this results in ten IBP identities for
each integrand.
In addition to the IBP identities, one can also exploit the fact that
all integrals under consideration are Lorentz scalars (or, perhaps
more precisely, ``$d$-rotational'' scalars) , which are
invariant under a Lorentz (or $d$-rotational) transformation of the
external momenta. In
order to derive the resulting Lorentz invarianve (LI) identities, we
consider an infinitesimal Lorentz transformation
\begin{equation}
p^{\mu} \to p^{\mu} + \delta p^{\mu} =
p^{\mu} + \delta \epsilon^{\mu}_{\nu} p^{\nu} \qquad
\mbox{with} \qquad \delta \epsilon^{\mu}_{\nu} = - \delta
\epsilon^{\nu}_{\mu}\;,
\end{equation}
which should not change the scalar Feynman integral
\begin{equation}
I(p_1+\delta p_1,
\ldots , p_n+\delta p_n)=
I(p_1,\ldots,p_n) \; .
\end{equation}
Expanding
\begin{equation}
I(p_1+\delta p_1,
\ldots , p_n+\delta p_n) =
I(p_1,\ldots,p_n) + \delta p_1^{\mu}
\frac{\partial}{\partial p_1^{\mu}}
I(p_1,\ldots,p_n) + \ldots + \delta p_n^{\mu}
\frac{\partial}{\partial p_n^{\mu}}
I(p_1,\ldots,p_n)\; ,
\end{equation}
one arrives at
\begin{equation}
\delta \epsilon^{\mu}_{\nu} \left( p_1^{\nu}\frac{\partial}{\partial
p_1^{\mu}} + \ldots + p_n^{\nu}\frac{\partial}{\partial
p_n^{\mu}} \right) I(p_1,\ldots,p_n) = 0\; .
\end{equation}
Since $\delta \epsilon^{\mu}_{\nu}$ has six independent components, the
above equation contains up to six LI identities. These are however not
always linearly independent. To determine the maximum number of linearly
independent identities, one uses the antisymmetry of $\delta
\epsilon^{\mu}_{\nu}$ to obtain
\begin{equation}
\left(p_1^{\nu}\frac{\partial}{\partial
p_{1\mu}} - p_1^{\mu}\frac{\partial}{\partial
p_{1\nu}} + \ldots + p_n^{\nu}\frac{\partial}{\partial
p_{n\mu}} - p_n^{\mu}\frac{\partial}{\partial
p_{n\nu}}\right) I(p_1,\ldots,p_n) = 0 \;.
\label{eq:li}
\end{equation}
This equation can be contracted with all possible antisymmetric
combinations of $p_{i\mu}p_{j\nu}$ to yield LI identities for $I$.
For a three-point vertex, two of the external momenta are linearly
independent ($n=2$), and only one antisymmetric combination of them can
be constructed, resulting in one LI identity ($N_{{\rm LI}}=1$).
A four-point function
depends on three external momenta ($n=3$), allowing to construct three
linearly independent antisymmetric combinations, which yield three
LI identities ($N_{{\rm LI}}=3$).
The full potential of the LI identities can only be
exploited for integrals involving five or more external legs, which
allow to construct six linearly independent antisymmetric combinations
of external momenta, thus projecting out all six LI identities
($N_{{\rm LI}}=6$).
Since $I$ is a scalar, it can not depend on the momenta $p_i$ itself,
but only on scalar products $s_{ij} = 2 p_i\cdot p_j$ of the external
momenta. Replacing
\begin{equation}
\frac{\partial}{\partial p_{i\mu}} = \sum_j 2\left(p_{i\mu}+p_{j\mu}\right)
\frac{\partial}{\partial s_{ij}}\;,
\end{equation}
one finds that (\ref{eq:li}) becomes a trivial identity, independent of
$I$. However, the derivatives in (\ref{eq:li}) can be interchanged with
the loop integrations in $I$, such that they do not act anymore on the
integral $I$, but on the integrand of $I$. After this interchange,
(\ref{eq:li}) becomes a non-trivial relation between different
integrals.
In the case of two-loop four-point functions, one has a total of 13
equations (10 IBP + 3 LI) for each integrand corresponding to an
integral of class $I_{t,r,s}$, relating integrals of the same topology
with up to $s+1$ scalar products and $r+1$ denominators, plus integrals
of simpler topologies ({\it i.e.}~with a smaller number of different
denominators).
The 13 identities obtained starting from an integral $I_{t,r,s}$ do contain
integrals of the following types:
\begin{itemize}
\item $I_{t,r,s}$: the integral itself.
\item $I_{t-1,r,s}$: simpler topology.
\item $I_{t,r+1,s}, I_{t,r+1,s+1}$ : same topology, more complicated than
$I_{t,r,s}$.
\item $I_{t,r-1,s}, I_{t,r-1,s-1}$: same topology, simpler than
$I_{t,r,s}$.
\end{itemize}
Quite in general, single identities of the above kind can be used
to obtain the reduction of $I_{t,r+1,s+1}$ or $I_{t,r+1,s}$ integrals
in terms of $I_{t,r,s}$ and simpler integrals - rather than to
get information on the $I_{t,r,s}$ themselves.
Integrations-by-parts identities are widely applied in multi-loop
calculations of
inclusive quantities (see e.g.~\cite{krev} for a review), which are related to
two-point functions. In these calculations,
only a relatively small number of different topologies has to be
considered, but the
integrals appearing in the calculation can bear large powers
of propagators and
scalar products, arising for example from expansions in masses or
momenta. In these
calculations, it is desirable to have reduction formulae for
arbitrary powers of propagators and
scalar products. These can be obtained from IBP identities derived
for an integral with arbitrary powers (to be treated symbolically)
of scalar products and propagators; the derivation
of these symbolic reduction formulae requires a lot of ingenuity,
based on the direct inspection of the explicit form of the IBP identities
for each considered topology, and could not be carried
out mechanically.
For loop integrals with a large number of external legs, IBP identities
are needed for a large number of different topologies, but in general
for relatively small powers of
propagators and scalar products. In this case, it would therefore
be desirable to
have a mechanical procedure for solving, for any given topology,
IBP and LI identities for integrals with fixed
powers of the propagators and scalar products.
If one considers the set of all the identities obtained starting from
the integrand of all the $N(I_{t,r,s})$ integrals of class $I_{t,r,s}$,
one obtains
$(N_{{\rm IBP}}+ N_{{\rm LI}}) N(I_{t,r,s})$ identities
which contain $N(I_{t,r+1,s+1})+N(I_{t,r+1,s})$
integrals of more complicated structure. From (\ref{eq:intnum}) it can
be read off that with increasing $r$ and $s$
the number of identities grows faster than the number
of new unknown integrals\footnote{The importance of this fact was
first pointed out by S.~Laporta and exploited in~\cite{laporta}.}.
As a consequence, if for a given $t$-topology one considers the set of
all the possible equations obtained by considering all the integrands up to
certain values $r^*, s^*$ of $r, s$, for large enough $r^*, s^*$
the resulting system of equations is overconstrained and can be used for
expressing the more complicated integrals, with greater values of $r, s$
in terms of simpler ones, with smaller values of $r, s$. (Let us observe
that, the system being overconstrained, the equations cannot be all
independent; it is not {\it a priori} known how many equations are in
fact linearly independent and, correspondingly, how many integrals
of the topology under consideration
will remain after reduction).
The required values $r^*$ and $s^*$ for $r$ and $s$
can be found by counting the number of accumulated equations (equations
for all integrals with $r\leq r^*$ and $s\leq s^*$) and comparing them
with the number of accumulated unknown integrals, with
$(r\leq r^* + 1,s \leq s^* + 1)$, but excluding
$(r=t,s=s^* + 1)$. As an example, we list in Table~\ref{tab:unknown}
the number of equations and unknowns for two-loop
four-point functions with
seven denominators $t=7$. It can be seen that a complete reduction
requires at least one of the combinations ($r^*,s^*$): (7,2); (8,1);
(9,0).
\begin{table}[h]
\fbox{$t=7$} \\
\parbox{7.5cm}{\rule[-4mm]{0cm}{1cm}different $I_{t,r,s}$}
\parbox{7.5cm}{accumulated \parbox{4cm}{equations\\unknowns}}\\
\parbox{7.5cm}{
\begin{tabular}{|r||r|r|r|r|r|}\hline
\backslashbox{$r$}{$s$}
& 0 & 1 & 2 & 3 & 4 \\ \hline\hline
7 \hspace{0.3cm} & 1 & 2 & 3 & 4 & 5 \\ \hline
8 \hspace{0.3cm} & 7 & 14 & 21 & 28 & 35 \\ \hline
9 \hspace{0.3cm} & 28 & 56 & 84 & 112 & 140 \\ \hline
10 \hspace{0.3cm} & \phantom{3}84 & 168 & 252 & 336 & 420\\ \hline
\end{tabular}
}
\parbox{7.5cm}{
\begin{tabular}{|r||r|r|r|r|}\hline
\backslashbox{$r$}{$s$}
& 0 & 1 & 2 & 3 \\ \hline\hline
& 13 & 39 & 78 & 130 \\
\raisebox{1.5ex}[-1.5ex]{7}\hspace{0.3cm} & 22 & 45 & 76 & 115 \\ \hline
& 104 & 312 & 624 & 1040 \\
\raisebox{1.5ex}[-1.5ex]{8}\hspace{0.3cm} & 106 & 213 & 354 & 535 \\ \hline
& 468 & 1404 & 2808 & 4680 \\
\raisebox{1.5ex}[-1.5ex]{9}\hspace{0.3cm} & 358 & 717 & 1196 & 1795 \\ \hline
\end{tabular}
}
\caption{Comparison of the number of (IBP and LI) identities
to the number of new unknowns (different integrals with
$t=7$) appearing in these equations for a two
loop box integral with $t=7$ internal propagators. The identities for
$I_{t,r,s}$ contain at most $I_{t,r+1,s+1}$. It can be seen that for
growing $r$ and $s$, the number of equations, upper number in each
box, exceeds the number of unknowns, given by the lower number.}
\label{tab:unknown}
\end{table}
The above table illustrates that typically hundereds of equations have to be
solved in order to obtain a reduction towards simpler integrals. The
task is performed automatically (and independently of the topology!)
by a computer program invoking repeatedly the
computer algebra packages FORM~\cite{form} and MAPLE~\cite{maple}.
For any given four-point two-loop topology,
this procedure can result either in a reduction
towards a small number (typically one or two) integrals of the
topology under consideration and integrals of simpler
topology (less different denominators), or even in a complete
reduction of all integrals of the topology under consideration
towards integrals with simpler topology.
Left-over integrals of the topology under consideration are called
irreducible master integrals or just
master integrals. If a topology turns out to contain irreducible master
integrals, one is in principle free to choose which integrals are taken
as master integrals, as long as the chosen integrals are not related by
the IBP and LI identities. In our reduction, we choose
$I_{t,t,0}$ for
topologies with one master integral and $I_{t,t,0}$ together
with the required number of integrals of type
$I_{t,t+1,0}$ for topologies with more than
one master integral.
As a final point, it is worth noting that the procedure described above can
be used to reduce a tensor integral towards scalar integrals. All
integrals appearing in the projection of an arbitrary tensorial integral
onto a tensor basis will take the form (\ref{eq:generic}), i.e.~they can
be classified as $I_{t,r,s}$ and reduced to master integrals.
In the following section, we will demonstrate that the established reduction
of all the integrals to a few master integrals can be used also to write
differential equations in the external invariants $s_{ij}$ for the master
integrals themselves, and then how the differential equations can be used
to compute these master integrals.
\section{Differential Equations for Master Integrals}
\label{sec:diffeq}
\setcounter{equation}{0}
The IBP and LI identities discussed in the previous section allow to
express integrals of the form (\ref{eq:generic}) as a linear
combination of a few master integrals, {\it i.e.} integrals which are
not further reducible, but have
to be computed by some different method.
For the case of massless two-loop
four-point functions, several techniques have been proposed in the
literature, such as for example the application of a Mellin-Barnes
transformation to all propagators~\cite{smirnov,tausk}
or the negative dimension
approach~\cite{glover}. Both techniques rely on an explicit integration
over the loop momenta, with differences mainly in the representation
used for the propagators.
So far, these techniques were only applied to a
limited number of master integrals: Smrinov~\cite{smirnov}
has recently used the
Mellin-Barnes method to compute the planar double box integral
for the case of all external legs on shell (massless case);
the same method has been applied
by Tausk~\cite{tausk} for the computation of the non-planar on-shell
double box integral;
the negative dimension
approach has been applied by Anastasiou, Glover and
Oleari~\cite{glover} to compute the class of two-loop box integrals
which correspond to a one-loop bubble insertion in one of the
propagators of the one-loop box. A general method for the computation of
the master integrals appearing in two-loop four-point functions
has however not yet been found. So far, it has also not even
been clear
(apart from the planar double box topology, where Smirnov and Veretin
have recently demonstrated the existence of two master
integrals~\cite{smir2}) how many are the master integrals for a given
topology. Solving the identities discussed in the previous section, we
are now able to identify the irreducible master integrals. A list of
reducible two-loop four-point topologies will be given in
Section~\ref{sec:twoloop}.
A method for the computation of master integrals avoiding the explicit
integration over the loop momenta is to derive differential equations in
internal propagator masses or in external momenta for the master integral,
and to solve these with appropriate boundary conditions.
This method has first been suggested by Kotikov~\cite{kotikov} to relate
loop integrals with internal masses to massless loop integrals.
It has been elaborated in detail and generalized to differential
equations in external momenta in~\cite{remiddi}; first
applications were presented in~\cite{appl}.
In the case of four-point functions with one external off shell leg
and no internal masses, one has three independent
invariants, resulting in three differential equations.
The derivatives in the invariants $s_{ij}=(p_i+p_j)^2$
can be expressed by derivatives in the external momenta:
\begin{eqnarray}
\sab \frac{\partial}{\partial \sab} & = & \frac{1}{2} \left( +
p_1^{\mu} \frac{\partial}{\partial p_1^{\mu}} +
p_2^{\mu} \frac{\partial}{\partial p_2^{\mu}} -
p_3^{\mu} \frac{\partial}{\partial p_3^{\mu}}\right) \nonumber \\
\sac \frac{\partial}{\partial \sac} & = & \frac{1}{2} \left( +
p_1^{\mu} \frac{\partial}{\partial p_1^{\mu}} -
p_2^{\mu} \frac{\partial}{\partial p_2^{\mu}} +
p_3^{\mu} \frac{\partial}{\partial p_3^{\mu}}\right)
\label{eq:derivatives} \\
\sbc \frac{\partial}{\partial \sbc} & = & \frac{1}{2} \left( -
p_1^{\mu} \frac{\partial}{\partial p_1^{\mu}} +
p_2^{\mu} \frac{\partial}{\partial p_2^{\mu}} +
p_3^{\mu} \frac{\partial}{\partial p_3^{\mu}}\right) \nonumber
\end{eqnarray}
The combinations of derivatives and momenta appearing on the right hand
side of (\ref{eq:derivatives}) are obviously linearly independent from
the combinations appearing in the LI identities (\ref{eq:li}) (which
vanish identically when acting on a function depending on the scalars
$\sab,\sac,\sbc$).
The three derivatives of an integral $I_{t,r,s}(\sab,\sac,\sbc,d)$
are not linearly independent, but related due to the properties of $I$
under rescaling of all external momenta:
\begin{equation}
I_{t,r,s}(\sab,\sac,\sbc,d) = \lambda^{-\alpha(d,r,s)}
I_{t,r,s}(\lambda^2\sab,\lambda^2\sac,\lambda^2\sbc,d)\; ,
\end{equation}
where $\alpha(d,r,s)$ is the mass dimension of the integral. For a $m$-loop
integral in $d$ space-time dimensions
with $r$ powers of denominators and $s$ scalar products, one
finds $\alpha(d,r,s) = m d + 2s -2r$. The above equation yields the
rescaling relation
\begin{equation}
\left[ - \frac{\alpha}{2} +
\sab \frac{\partial}{\partial\sab} + \sac
\frac{\partial}{\partial\sac} +
\sbc \frac{\partial}{\partial\sbc} \right] I_{t,r,s}(\sab,\sac,\sbc,d)
=0 \; .
\label{eq:rescale}
\end{equation}
Let us now first consider the case of a topology with only one master
integral, which is chosen to be $I_{t,t,0}$, defined as
\begin{equation}
I_{t,t,0}(\sij,s_{jk},s_{ki},d) =
\int \frac{\d^d k}{(2\pi)^d}\frac{\d^d l}{(2\pi)^d}
f_{t,t,0}(k,l,p_i) \; ,
\label{eq:tt0}
\end{equation}
where $ f_{t,t,0}(k,l,p_i) $ is a suitable integrand of the form
appearing in (\ref{eq:generic}).
It is evident that acting with the two sides of (\ref{eq:derivatives})
in the two sides of (\ref{eq:tt0}) will express the scalar
derivatives of $I_{t,t,0}$ as a combination of
integrals of the same type as appearing in the IBP and LI identities for
$I_{t,t,0}$, including integrals of type $I_{t,t+1,1}$ and
$I_{t,t+1,0}$.
These can all be reduced to $I_{t,t,0}$ and to integrals of simpler topology
by applying the IBP and LI identities. This reduction results in
differential equations for $I_{t,t,0}$ of the form:
\begin{eqnarray}
\sij \frac{\partial}{\partial \sij} I_{t,t,0}(\sij,s_{jk},s_{ki},d) & = &
A(\sij,s_{jk},s_{ki},d)
I_{t,t,0}(\sij,s_{jk},s_{ki},d) \nonumber \\
& & +
F(\sij,s_{jk},s_{ki},d,I_{t-1,r,s}(\sij,s_{jk},s_{ki},d))\;,
\label{eq:master1}
\end{eqnarray}
where $\sij$, $s_{jk}$, $s_{ki}$ are the three invariants,
$A(\sij,s_{jk},s_{ki},d)$ turns out to be a rational function of the
invariants and of $d$,
$F(I_{t-1,r,s})$ is a linear combination (with coefficients
depending on $\sij,s_{jk},s_{ki}$ and $d$)
of integrals of type
$I_{t-1,r,s}$, containing only topologies simpler than $I_{t,t,0}$, but
potentially with high powers of some denominators and scalar products.
These $I_{t-1,r,s}$, which refer to a simpler topology and therefore can be
considered as known
in a bottom-up approach, play the role of an inhomogeneous term in the
equation; one can then look for the proper solution
of (\ref{eq:master1}) in a straightforward way.
The master integral
$I_{t,t,0}(\sij,s_{jk},s_{ki},d)$ can indeed be obtained
by matching the general
solution of (\ref{eq:master1}) to an appropriate boundary
condition. Quite in general, finding a boundary condition is
a simpler problem than evaluating the whole
integral, since it depends on a smaller number of kinematical
variables. In some cases, the boundary condition can even be
determined from the differential equation itself:
for $\sij=0$, (\ref{eq:master1}) yields, if $A(0,s_{jk},s_{ki},d)\neq 0$,
\begin{equation}
I_{t,t,0}(0,s_{jk},s_{ki},d) = -
\left[A(0,s_{jk},s_{ki},d)\right]^{-1}
F(0,s_{jk},s_{ki},d,I_{t-1,r,s}(0,s_{jk},s_{ki},d))\; .
\end{equation}
For $A$ vanishing at $\sij=0$ one can consider $I_{t,t,0}$ in the limit
where one of the external momenta vanishes, corresponding to the
vanishing of both invariants involving this momentum, e.g.~$\sij=0$ and
$s_{ki}=0$ for $p^{\mu}_i=0$. In this case,
$I_{t,t,0}$ reduces to a three-point vertex function with one off shell
external leg.
All these functions, which might be determined by iterating the procedure
just described for the considered 4-point function, have actually
already been computed at the two-loop
level in~\cite{kl} using IBP identities to reduce all possible
topologies to a few master integrals, which, in this case, can be
computed straightforwardly using Feynman parameters
(cf.~Section~\ref{sec:twoloop}). Starting from the
boundary condition in $\sij=s_{ki}=0$, one can determine
$I_{t,t,0}(0,s_{jk},s_{ki},d)$ by solving the differential equation in
$s_{ki}$ -- this provides the desired boundary condition in
$\sij=0$.
For topologies with more than one master integral, (\ref{eq:master1})
will be replaced by a system of coupled, linear, first order
differential equations for all master integrals of the topology under
consideration. The determination of the master integrals from these
equations follows the same lines as discussed above, with the only
difference that the general solution for the system of coupled
differential equations is harder to obtain than for a single
equation. In case the coupled equations can not be decoupled by
an appropriate choice of variables,
several mathematical techniques can be employed
here~\cite{kamke}: the system of $n$ coupled first order
differential equations can for
example be rewritten into one $n$-th order differential equation, which
is then solved with standard methods. In some cases, the system can also
be transformed into a form which is known to be solved by
generalized hypergeometric series~\cite{kamke,bateman,grad,exton}.
It is clear from the above discussion, that the determination of a
master integral of a certain topology with $t$ different denominators
requires the knowledge of all the integrals appearing in the inhomogeneous
term. These integrals are subtopologies of the topology of the
integral under
consideration, and contain at most $t-1$ different denominators.
The determination of master integrals has therefore to proceed
bottom-up from
simpler topologies with less different denominators towards more complicated
topologies with more different denominators. For the case under special
consideration, massless
two-loop four-point functions with up to one external leg
off-shell, this implies that one has to progress from the simplest
master integrals with $t=3$ (off-shell two-point function) to construct
all master integrals up to $t=7$.
\section{A Paedagogical Example: the One-Loop Four-Point Function}
\label{sec:oneloop}
\setcounter{equation}{0}
To illustrate how the method explained above works in practice, we
present in this section a detailed and self-contained derivation of the
one-loop four-point function
\begin{equation}
\boxLO = \int \frac{\d^d k}{(2\pi)^d} \frac{1}{k^2 (k-p_2)^2
(k-p_2-p_3)^2 (k-p_1-p_2-p_3)^2}\; .
\label{eq:boxLOdef}
\end{equation}
The topology of this integral is given by the set of its
four propagators, it is the only $t=4$ topology at one loop. In the
notation introduced above, (\ref{eq:boxLOdef}) corresponds to
an integral $I_{4,4,0}$, the simplest integral of this topology with
all propagators appearing in first power and with no scalar products.
For the one-loop four-point function, one has
four independent scalar products involving $k_{\mu}$ ($k^2$ and
$k_{\mu}p_i^{\mu}$) and four linear independent denominators. This
implies that any scalar product involving $k_{\mu}$ can be rewritten as
linear combination of propagators and invariants $s_{ij} \equiv 2 p_{i \mu}
p_j^{\mu}$.
Consequently,
no integrals with scalar products in the numerator can appear for this
topology.
The reduction of all four integrals with one squared propagator ($I_{4,5,0}$)
can be carried out by considering the IBP identities for
$I_{4,4,0}$:
\begin{equation}
\int \frac{\d^d k}{(2\pi)^d} \frac{\partial}{\partial k^{\mu}}
\frac{v^{\mu}}{k^2 (k-p_2)^2
(k-p_2-p_3)^2 (k-p_1-p_2-p_3)^2} = 0 \;,
\label{eq:IBPLO}
\end{equation}
where $v^{\mu}$ can be the loop momentum
$k^{\mu}$ or any of the external momenta
$p_i^{\mu}$, thus yielding four identities. It turns out that
LI identities and IBP identities for integrals with higher powers of the
propagators do not contain additional information which would allow
a reduction of $I_{4,4,0}$. The integral (\ref{eq:boxLOdef}) is therefore
a master integral. It is the only master integral for this topology.
To proceed towards the differential equations in the invariants $\sab$,
$\sac$ and $\sbc$ for this
master integral, let us consider the derivatives in the external
momenta:
\begin{eqnarray}
p_1^{\mu} \frac{\partial}{\partial p_1^{\mu}} \boxLO & = &
\int \frac{\d^d k}{(2\pi)^d} \frac{1}{k^2 (k-p_2)^2
(k-p_2-p_3)^2 (k-p_1-p_2-p_3)^2} \nonumber \\
& &
\left( \frac{2p_1^{\mu}
(k-p_1-p_2-p_3)_{\mu}}{(k-p_1-p_2-p_3)^2}\right)\;, \\
p_2^{\mu} \frac{\partial}{\partial p_2^{\mu}} \boxLO & = &
\int \frac{\d^d k}{(2\pi)^d} \frac{1}{k^2 (k-p_2)^2
(k-p_2-p_3)^2 (k-p_1-p_2-p_3)^2}\nonumber \\
& &
\left(\frac{2p_2^{\mu}
(k-p_1-p_2-p_3)_{\mu}}{(k-p_1-p_2-p_3)^2} + \frac{2p_2^{\mu}
(k-p_2-p_3)_{\mu}}{(k-p_2-p_3)^2} + \frac{2p_2^{\mu}
(k-p_2)_{\mu}}{(k-p_2)^2}\right)\;, \\
p_3^{\mu} \frac{\partial}{\partial p_3^{\mu}} \boxLO & = &
\int \frac{\d^d k}{(2\pi)^d} \frac{1}{k^2 (k-p_2)^2
(k-p_2-p_3)^2 (k-p_1-p_2-p_3)^2} \nonumber \\
& &
\left(\frac{2p_3^{\mu}
(k-p_1-p_2-p_3)_{\mu}}{(k-p_1-p_2-p_3)^2} + \frac{2p_3^{\mu}
(k-p_2-p_3)_{\mu}}{(k-p_2-p_3)^2} \right)\; .
\end{eqnarray}
The right hand sides of the above equations contain
terms with
\begin{itemize}
\item[{(i)}] four different propagators with one squared propagator, no
scalar product
\item[{(ii)}] four different propagators, no squared propagator, no
scalar product
\item[{(iii)}] three different propagators, one squared propagator.
\end{itemize}
The terms of type (i) can now be reduced to type (ii) and (iii) by using
the integration-by-parts identities (\ref{eq:IBPLO}).
One obtains:
\begin{eqnarray}
p_1^{\mu} \frac{\partial}{\partial p_1^{\mu}} \boxLO & = & - \boxLO
+ \boxLOpamb\; , \nonumber \\
p_2^{\mu} \frac{\partial}{\partial p_2^{\mu}} \boxLO & = & - \boxLO
+ \boxLOpdmc\; , \nonumber \\
p_3^{\mu} \frac{\partial}{\partial p_3^{\mu}} \boxLO & = & (d-6) \boxLO
- \boxLOpamb - \boxLOpdmc\; , \nonumber
\end{eqnarray}
where $(\bullet)$ denotes a squared propagator
and $(\times)$ stands for a pinched (cancelled) propagator.
A check on these equations is provided by the rescaling
relation (\ref{eq:rescale})
\begin{equation}
\left[ 4-\frac{d}{2} + \sab \frac{\partial}{\partial\sab} + \sac
\frac{\partial}{\partial\sac} +
\sbc \frac{\partial}{\partial\sbc} \right] \boxLO = 0\; ,
\end{equation}
which is related to the above equations by
\begin{equation}
\sab \frac{\partial}{\partial\sab} + \sac
\frac{\partial}{\partial\sac} +
\sbc \frac{\partial}{\partial\sbc} = \frac{1}{2} \sum_{i=1}^3 p_i^{\mu}
\frac{\partial}{\partial p_i^{\mu}} \; .
\end{equation}
Inserting the derivatives obtained above, one finds that the rescaling
relation is indeed fulfilled.
The three-propagator terms appearing in the above equations can be further
reduced by using the IBP identities for the corresponding vertex;
the vertex amplitudes, in turn, can all be expressed in terms of bubble
integrals, and one finally finds
\begin{equation}
\boxLOpamb = \frac{d-3}{p_2\cdot (p_1+p_3)} \left[
\frac{1}{(p_1+p_2+p_3)^2} \bubbleLO{p_{123}} - \frac{1}{(p_1+p_3)^2}
\bubbleLO{p_{13}} \right]\; .
\end{equation}
A similar identity is obtained by exchanging $p_1
\leftrightarrow p_2$.
The differential equations for the one-loop box integral then follow
from (\ref{eq:derivatives}).
The set of differential equations reads:
\begin{eqnarray}
\sab \frac{\partial}{\partial \sab}
\boxLO
& = & - \frac{d-4}{2} \boxLO \nonumber \\
& & + \frac{2(d-3)}{\sab+\sac}\left[\frac{1}{\sabc} \bubbleLO{p_{123}}
-\frac{1}{\sbc} \bubbleLO{p_{23}} \right] \nonumber \\
& & + \frac{2(d-3)}{\sab+\sbc}\left[\frac{1}{\sabc} \bubbleLO{p_{123}}
-\frac{1}{\sac} \bubbleLO{p_{13}} \right]\;, \label{eq:boxLOsab}\\
\sac \frac{\partial}{\partial \sac}
\boxLO
& = & \frac{d-6}{2} \boxLO \nonumber \\
& & - \frac{2(d-3)}{\sab+\sac}\left[\frac{1}{\sabc} \bubbleLO{p_{123}}
-\frac{1}{\sbc} \bubbleLO{p_{23}} \right]\;, \label{eq:boxLOsac}\\
\sbc \frac{\partial}{\partial \sbc}
\boxLO
& = & \frac{d-6}{2} \boxLO \nonumber \\
& & - \frac{2(d-3)}{\sab+\sbc}\left[\frac{1}{\sabc} \bubbleLO{p_{123}}
-\frac{1}{\sac} \bubbleLO{p_{13}} \right]\;, \label{eq:boxLOsbc}
\end{eqnarray}
where $\sabc = \sab+\sac + \sbc$. The one-loop bubble diagrams in the
inhomogenous term yield:
\begin{equation}
\bubbleLO{p} =
\left[\frac{(4\pi)^{\frac{4-d}{2}}}{16\pi^2}\frac{ \Gamma (3-d/2)
\Gamma^2 (d/2-1)}{ \Gamma (d-3)} \right] \; \frac{-2i}{(d-4)(d-3)}
\left( -p^2\right)^{\frac{d-4}{2}} \equiv A_{2,LO}
\left( -p^2\right)^{\frac{d-4}{2}} \;.
\end{equation}
The boundary conditions in $\sij=0$ can be readily read off from the
above:
\begin{eqnarray}
\boxLO(\sab=0) & = & \frac{4(d-3)}{(d-4)}\frac{1}{\sac\sbc}\Bigg[
\bubbleLO{p_{123}} \nonumber \\
& & \hspace{1.3cm}
- \bubbleLO{p_{13}} -
\bubbleLO{p_{23}} \Bigg]\;, \label{eq:boundLOsab} \\
\boxLO(\sac=0) & = & \frac{4(d-3)}{(d-6)}\frac{1}{\sab}\Bigg[\frac{1}{\sabc}
\bubbleLO{p_{123}} - \frac{1}{\sbc} \bubbleLO{p_{23}} \Bigg]\;, \\
\boxLO(\sbc=0) & = & \frac{4(d-3)}{(d-6)}\frac{1}{\sab}\Bigg[\frac{1}{\sabc}
\bubbleLO{p_{123}} - \frac{1}{\sac} \bubbleLO{p_{13}} \Bigg] \;.
\end{eqnarray}
The result for the one-loop box integral can in principle
be obtained by integrating
any of the differential equations
(\ref{eq:boxLOsab})--(\ref{eq:boxLOsbc}). In practice,
it turns out to be more appropriate to introduce a new set of
variables, namely $\sac$, $\sbc$ and $\sabc=\sab+\sac+\sbc$, corresponding
to the arguments appearing in the two-point functions in the
inhomogeneous terms. This transformation yields a differential equation
in $\sabc$, which will be used for integration. Note that this
transformation also modifies the differential equations in $\sac$ and
$\sbc$. The differential equation in $\sabc$ reads:
\begin{eqnarray}
\lefteqn{\frac{\partial}{\partial \sabc}
\boxLO
+ \frac{d-4}{2(\sabc-\sac-\sbc)} \boxLO = } \nonumber \\
& & + \frac{2(d-3)}{(\sabc-\sbc)(\sabc-\sac-\sbc)}
\left[\frac{1}{\sabc} \bubbleLO{p_{123}}
-\frac{1}{\sbc} \bubbleLO{p_{23}} \right] \nonumber \\
& & + \frac{2(d-3)}{(\sabc-\sac)(\sabc-\sac-\sbc)}
\left[\frac{1}{\sabc} \bubbleLO{p_{123}}
-\frac{1}{\sac} \bubbleLO{p_{13}} \right] \label{eq:boxLOq}\; .
\label{eq:boxLOmaster}
\end{eqnarray}
The boundary condition in $\sabc=0$ can not be trivially determined from
this equation, reflecting the fact that the massless one-loop
box integral with all external legs on shell is not reducible to simpler
subtopologies by IBP identities. The boundary condition in
$\sabc=-\sac-\sbc$ can however be determined from (\ref{eq:boundLOsab}).
Equation~(\ref{eq:boxLOmaster}) is a linear, inhomogeneous first order
differential equation of the form
\begin{displaymath}
\frac{\partial y(x)}{\partial x} + f(x) y(x) = g(x),
\end{displaymath}
which can be solved by
introducing an integrating factor (see for instance~\cite{bronstein} or
any standard book on differential equations)
\begin{displaymath}
M(x) = e^{\int f(x) \d x},
\end{displaymath}
such that $y(x)=1/M(x)$ solves the homogenous differential equation
($g(x)=0$). This
yields the general solution of the inhomogenous equation as
\begin{displaymath}
y(x) = \frac{1}{M(x)} \left( \int g(x) M(x) \d x + C\right),
\end{displaymath}
where the integration constant $C$ can be adjusted to match the boundary
conditions.
For~(\ref{eq:boxLOmaster}), we have at once the integrating factor
\begin{equation}
M(\sabc) = (\sac + \sbc - \sabc )^{\frac{d-4}{2}}\; .
\label{eq:intfac}
\end{equation}
This factor is not unambigous, since
\begin{equation}
M'(\sabc) = (\sabc -\sac - \sbc)^{\frac{d-4}{2}}\; .
\label{eq:intfacal}
\end{equation}
would also be a vaild integrating factor. We select
(\ref{eq:intfac}) by requiring a real integrating factor
in the Euclidian region:
$-\sabc \ge -\sac - \sbc \ge 0$. The final result for the
box integral does
not depend on the selection of the integrating factor; using
(\ref{eq:intfacal}), one must however be more careful in applying
analytic continuation formulae and in multiplying non-integer powers of
the invariants.
With the integrating factor (\ref{eq:intfac}), the one-loop box integral
reads:
\begin{eqnarray}
\boxLO (\sabc,\sac,\sbc) &=&
2(d-3) A_{2,LO} \left( \sac + \sbc - \sabc
\right)^{2-\frac{d}{2}}
\int^{\sabc}\hspace{-0.25cm} \d \sabcp \left( \sac + \sbc - \sabcp
\right)^{\frac{d}{2}-3}\nonumber \\
& & \hspace{-0.9cm}
\Bigg[ \frac{\left(-\sac\right)^{\frac{d}{2}-3}}{ \sac - \sabcp }
+\frac{\left(-\sbc\right)^{\frac{d}{2}-3}}{\sbc - \sabcp}
- \frac{2\sabcp - \sac - \sbc}{(\sac - \sabcp)(\sbc - \sabcp) }
\left(-\sabcp\right)^{\frac{d}{2}-3} \Bigg].
\label{eq:boxLOintform}
\end{eqnarray}
From a computational point of view, the evaluation of the box amplitude
of (\ref{eq:boxLOdef}) has been reduced to the one dimensional integration
corresponding to the solution of (\ref{eq:boxLOmaster}).
The lower boundary of the integral is independent of $\sabc$ and can be
adjusted arbitrarily. The first two terms in (\ref{eq:boxLOintform})
can be easily integrated by shifting the integration variable to
$\sabcp - \sac - \sbc$, which is then integrated between $0$ and
$\sabc - \sac - \sbc$. To integrate the last term, one introduces a new
variable $\sabcp (\sabcp - \sac - \sbc)$, which is integrated between
$0$ and $\sabc (\sabc - \sac - \sbc)$. The resulting integrals yield
can be identified as integral representation of the hypergeometric function
$\,_2F_1$.
With this choice of variables and
boundaries, no constant term is required to match the boundary
conditions. The result for the one-loop box integral then reads:
\begin{eqnarray}
\lefteqn{\boxLO = - \frac{4(d-3)}{d-4} A_{2,LO}
\frac{1}{\sac\sbc}} \nonumber \\
& & \hspace{0.46cm}
\Bigg[ \left(\frac{\sac \sbc}{\sac - \sabc}\right)^{\frac{d}{2}-2}
\,_2F_1\left(2-d/2, 2-d/2; 3-d/2; \frac{\sabc - \sac - \sbc}{\sabc -\sac}
\right)\nonumber \\
& & \hspace{0.4cm}
+ \left(\frac{\sac \sbc}{\sbc - \sabc}\right)^{\frac{d}{2}-2}
\,_2F_1\left(2-d/2, 2-d/2; 3-d/2; \frac{\sabc - \sac - \sbc}{\sabc -\sbc}
\right) \nonumber \\
& & \hspace{0.4cm} - \left( \frac{-\sabc \sac \sbc}{(\sac - \sabc)
(\sbc - \sabc)}
\right)^{\frac{d}{2}-2} \,_2F_1\left(2-d/2, 2-d/2; 3-d/2;
\frac{\sabc (\sabc - \sac - \sbc)}{(\sabc -\sac)(\sabc - \sbc)}\right) \Bigg]
\;.
\nonumber \\
\end{eqnarray}
This expression can be safely continued from the Euclidian to the
Minkowskian region. The arguments of the hypergeometric functions are
ratios of invariants, they are not changed by the analytic
continuation. The non-integer powers of invariants appearing as
coefficients acquire imaginary parts, their signs are uniquely
determined by the convention $-p^2 = -p^2 - i0$. The above equation
reproduces the well-known result from the literature, e.g.~\cite{ert}.
When the box integral is expressed in the above form,
where no expansion around $d=4$ has yet
been performed, analytic continuations, e.g.~to the on-shell case $\sabc=0$ or
to collinear and soft limits $s_{ij}=0$, can be made with ease.
\section{Results on Two-Loop Four-Point Functions}
\label{sec:twoloop}
\setcounter{equation}{0}
In the following, we shall outline how the techniques derived in
Sections \ref{sec:master} and \ref{sec:diffeq}
can be applied to the computation of two-loop integrals
appearing in amplitudes for the decay of one massive into
three massless particles: two-loop four-point functions with one off
shell leg. The main purpose of this section is to illustrate
applications of the tools developed above to non-trivial problems;
the list of integrals given here is far from complete. We provide a
comprehensive list of master integrals only up to $t=5$, for
$t=6$ and $t=7$ only reducible integrals are quoted.
A prefactor common to all massles scalar integrals is
\begin{equation}
S_d = \left[(4\pi)^{\frac{4-d}{2}}\frac{ \Gamma (3-d/2)
\Gamma^2 (d/2-1)}{ \Gamma (d-3)} \right] \; ,
\end{equation}
which is also appearing in all counterterms in the $\overline{{\rm
MS}}$--scheme.
In the following, the notation of external momenta is a follows:
$p_i$ denotes an on-shell momentum, $p_{ij(k)}$ denotes an off-shell
momentum, being the sum of two (three) on-shell momenta $p_i$, $p_j$(,
$p_k$) with $s_{ij(k)}=(p_{ij(k)})^2$. $p$ is an arbitrary momentum.
\subsection{$t=3$}
For two-loop integrals with $t=3$, only one topology exists: the
two-loop vacuum bubble. The
corresponding integral fulfils a homogeneous differential equation,
which can not be used to infer any boundary condition. The integral can
however be computed using Feynman parameters:
\begin{equation}
\bubbleNLO{p} = \left(\frac{S_d}{16 \pi^2}\right)^2
\frac{\Gamma(3-d)\Gamma^2(d-3)}{\Gamma^2(3-d/2)\Gamma(d/2-1)\Gamma(3d/2-3)}
\left( -p^2 \right)^{d-3} = A_3 \left( -p^2 \right)^{d-3}\;.
\end{equation}
\subsection{$t=4$}
Several different two-loop topologies exist for $t=4$. Two types of
two-point functions are encountered. The first can be reduced to
\begin{equation}
\bubblexNLO{p}= \frac{3d-8}{d-4} \frac{1}{p^2} \bubbleNLO{p}
\end{equation}
using IBP identities. The second is the product of two one-loop
bubble integrals and yields
\begin{equation}
\doublebubbleNLO{p} = \left( A_{2,LO} \right)^2 (-p^2)^{d-4}\; ,
\end{equation}
which generalizes trivially to a three-point function
\begin{equation}
\doublebubblexNLO{p_{123}}{p_{12}}{p_3}
= \left( A_{2,LO} \right)^2
(-\sabc)^{\frac{d}{2}-2}(-\sab)^{\frac{d}{2}-2}\; .
\label{eq:doublebubblex}
\end{equation}
Only one of the master integrals at $t=4$ fulfils a homogeneous
differential equation: a two-loop vertex integral with one
off-shell leg. This integral can also be computed using Feynman parameters:
\begin{equation}
\triangleNLO{p_{12}}{p_1}{p_2}
= \left(\frac{S_d}{16 \pi^2}\right)^2 \frac{-2 \Gamma^2(d-3)
\Gamma(d-4) \Gamma(3-d)}{\Gamma(3-d/2)\Gamma^2(d/2-1)\Gamma(3d/2-4)}
\left(-s_{12}\right)^{d-4} = A_4 \left(-s_{12}\right)^{d-4}\; .
\end{equation}
The other vertex integral topology with one off shell leg can be reduced
using IBP identities:
\begin{equation}
\trianglexNLO{p_{12}}{p_1}{p_2} = -\frac{3d-8}{d-4} \frac{1}{s_{12}}
\bubbleNLO{p_{12}} \; .
\end{equation}
Among the three vertex integrals with two off shell legs, only one can
be reduced using IBP and LI identities:
\begin{equation}
\trianglexNLO{p_{123}}{p_{12}}{p_3} = \frac{3d-8}{d-4} \frac{1}{s_{123}
- s_{12}} \left[ \bubbleNLO{p_{12}} - \bubbleNLO{p_{123}} \right] \;.
\end{equation}
The two remaining ones are master integrals, fulfilling inhomogeneous
differential equations. For a vertex $p_{123} \to p_{12} + p_3$, the
appropriate variables for the differential equations are $s_{123}$ and
$s_{12}$. To illustrate the structure of the differential equations, we
quote them for one of the master integrals:
\begin{eqnarray}
s_{123} \frac{\partial}{\partial s_{123}}
\triangleNLO{p_{123}}{p_{12}}{p_3} & = & \frac{d-4}{2}\; \frac{2\sabc -
\sab}{\sabc - \sab} \triangleNLO{p_{123}}{p_{12}}{p_3} \nonumber \\
& &-
\frac{3d-8}{2}\; \frac{1}{\sabc - \sab} \bubbleNLO{p_{12}}\;, \nonumber \\
\sab \frac{\partial}{\partial \sab} \triangleNLO{p_{123}}{p_{12}}{p_3}
& = & - \frac{d-4}{2}\; \frac{\sab}{\sabc - \sab}
\triangleNLO{p_{123}}{p_{12}}{p_3} \nonumber \\
&& +
\frac{3d-8}{2}\; \frac{1}{\sabc - \sab} \bubbleNLO{p_{12}}\;.
\end{eqnarray}
The boundary conditions for $\sabc=0$ or $\sab=0$ are obtained directly
from the vertex integrals with one off shell leg quoted above. Using
these, one finds
\begin{equation}
\!\triangleNLO{p_{123}}{p_{12}}{p_3}\! = A_4 \left( \sab -
\sabc\right)^{\frac{d}{2}-2} \left(-\sabc\right)^{\frac{d}{2}-2}
- \frac{3d-8}{2(d-3)} A_3 \frac{\left(-\sab\right)^{d-3}}{-\sabc}
\,_2F_1\left(\frac{d}{2}-1, 1; d-2; \frac{\sab}{\sabc}\right).
\end{equation}
The second master integral can be obtained from this by analytic
continuation of the hypergeometric function:
\begin{equation}
\trianglexNLO{p_{123}}{p_3}{p_{12}} = -\frac{3d-8}{d-4} A_3
\left( -\sab \right)^{\frac{d}{2}-2} \left(-\sabc\right)^{\frac{d}{2}-2}
\,_2F_1\left(\frac{d}{2}-1, 2-\frac{d}{2}; 3-\frac{d}{2};
\frac{\sabc-\sab}{\sabc}\right)\; .
\end{equation}
Vertex integrals with three off shell legs can not appear as
subtopologies in two-loop four-point functions with one off shell leg.
\subsection{$t=5$}
The two-loop two-point function with $t=5$ is a well known
example~\cite{chet,krev} for the application of IBP identities:
\begin{equation}
\bubblecrossNLO{p} =
\frac{2(3d-8)(3d-10)}{(d-4)^2} \frac{1}{\left(p^2\right)^2}
\bubbleNLO{p}- \frac{2(d-3)}{d-4} \frac{1}{p^2} \doublebubbleNLO{p}\; .
\end{equation}
The four different $t=5$
three-point functions with one off-shell leg can also
be reduced by using IBP and LI identities:
\begin{eqnarray}
\triangleaNLO{p_{12}}{p_1}{p_2} &= & \frac{(d-3)(3d-8)(3d-10)}{(d-4)^3}
\;\frac{1}{(\sab)^2} \bubbleNLO{p_{12}}\;,\\
\trianglebNLO{p_{12}}{p_1}{p_2} &= & \frac{(d-3)(3d-10)}{(d-4)^2}
\; \frac{1}{\sab} \triangleNLO{p_{12}}{p_1}{p_2} \nonumber \\
&& -\frac{(d-3)(3d-8)(3d-10)}{(d-4)^3}
\;\frac{1}{(\sab)^2} \bubbleNLO{p_{12}}\;, \\
\trianglecNLO{p_{12}}{p_1}{p_2} &= & -\frac{(3d-8)(3d-10)}{(d-4)^2}
\;\frac{1}{(\sab)^2} \bubbleNLO{p_{12}}\;,\\
\triangledNLO{p_{12}}{p_1}{p_2} &= & -\frac{3d-10}{2(d-4)}
\; \frac{1}{\sab} \triangleNLO{p_{12}}{p_1}{p_2}\;.
\end{eqnarray}
By applying IBP and LI identities, it is likewise possible to reduce
all but one $t=5$ three-point functions with two off-shell legs:
\begin{eqnarray}
\triangleaNLO{p_{123}}{p_{12}}{p_3} &= & - \frac{(d-3)(3d-10)}{(d-4)^2}
\; \frac{1}{\sabc - \sab} \trianglexNLO{p_{123}}{p_3}{p_{12}} \nonumber\\
&& + \frac{(d-3)(3d-8)(3d-10)}{(d-4)^3}
\;\frac{1}{\sab(\sabc-\sab)} \bubbleNLO{p_{12}}\;, \\
\trianglebNLO{p_{123}}{p_3}{p_{12}} &= &
\frac{(d-3)(3d-10)}{(d-4)^2}
\; \frac{1}{\sabc - \sab} \triangleNLO{p_{123}}{p_{12}}{p_3} \nonumber\\
&& - \frac{(d-3)(3d-8)(3d-10)}{(d-4)^3}
\;\frac{1}{\sabc(\sabc-\sab)} \bubbleNLO{p_{123}}\;, \\
\trianglecNLO{p_{123}}{p_{12}}{p_3} &= & \frac{(3d-8)(3d-10)}{(d-4)^2}
\; \frac{1}{\sab(\sabc-\sab)} \bubbleNLO{p_{12}}
\nonumber \\
&& - \frac{(3d-8)(3d-10)}{(d-4)^2}\;\frac{1}{\sabc(\sabc-\sab)}
\bubbleNLO{p_{123}}\;, \\
\trianglecNLO{p_{123}}{p_3}{p_{12}} &= & \frac{3d-10}{2(d-4)}
\; \frac{1}{\sabc-\sab} \trianglexNLO{p_{123}}{p_3}{p_{12}} \nonumber\\
&& - \frac{(3d-8)(3d-10)}{2(d-4)^2}\;\frac{1}{\sabc(\sabc-\sab)}
\bubbleNLO{p_{123}}\;, \\
\triangledNLO{p_{123}}{p_{12}}{p_3} &= & - \frac{3d-10}{2(d-4)}
\; \frac{1}{\sabc-\sab} \triangleNLO{p_{123}}{p_{12}}{p_3}\nonumber\\
&&+ \frac{(3d-8)(3d-10)}{2(d-4)^2}\;\frac{1}{\sab(\sabc-\sab)}
\bubbleNLO{p_{12}}\; .
\end{eqnarray}
The remaining three-point function is a master integral, which can be
found by solving the corresponding differential equations:
\begin{eqnarray}
\trianglebNLO{p_{123}}{p_{12}}{p_3} &= &\frac{2(d-3)}{d-4}\,
\left(A_{2,LO}\right)^2\;
\left(-\sab\right)^{d-4} \frac{1}{-\sabc}
\,_2F_1\left(
1-\frac{d}{2},d-3; d-2; \frac{\sabc-\sab}{\sabc} \right) \nonumber \\
& & - \frac{(3d-8)(3d-10)}{(d-4)^2} \, A_3 \nonumber \\
& &
\Bigg[
\frac{1}{\sab}\, \left(-\sabc\right)^{d-4}
\,_3F_2\left( \frac{d}{2}-1,1,d-3; 3-\frac{d}{2},d-2;
\frac{\sab-\sabc}{\sab} \right) \nonumber \\
&& + \frac{1}{\sabc}\, \left(-\sab\right)^{d-4}
\,_3F_2\left( \frac{d}{2}-1,1,d-3; 3-\frac{d}{2},d-2;
\frac{\sabc-\sab}{\sabc} \right)\Bigg] \; .
\end{eqnarray}
For $t=5$, one finds four different topologies for two-loop four-point
functions. These are all master integrals obeying inhomogeneous
differential equations in the external invariants. Solving these
equations and matching the boundary conditions, all master integrals
can be determined.
\begin{eqnarray}
\boxbubbleaNLO{p_{123}}{p_1}{p_2}{p_3} & = & \frac{3d-10}{d-4}
\, A_4 \, \frac{1}{\sbc}\left(-\sac\right)^{d-4}
\left(\frac{\sab+\sac}{\sbc}\right)^{3-d}\nonumber \\
&& \hspace{1.4cm}
\,_2F_1\left(d-3,\frac{d}{2}-2;\frac{d}{2}-1;\frac{\sab}{\sab+\sac}\right)
\nonumber\\
&& -\frac{3d-10}{2(d-4)}\, A_4 \, \frac{1}{\sab}
\left( -\sab-\sbc\right)^{d-4}
\left(\frac{\sab+\sac}{\sab}\right)^{\frac{d}{2}-3}\nonumber \\
&& \hspace{1.4cm}
\,_2F_1\left(3-\frac{d}{2},4-d;5-d;\frac{\sac\sbc}{(\sab+\sac)(\sab+\sbc)}
\right)\nonumber \\
&&\hspace{-1cm}
-\frac{(3d-8)(3d-10)}{4(d-3)(d-4)} \,A_3\,
\frac{1}{\sab\sabc}
\left(-\sac\right)^{d-3}
\left(\frac{\sab+\sbc}{\sabc}\right)^{\frac{d}{2}-2}
\left(\frac{\sab+\sac}{\sab}\right)^{\frac{d}{2}-3} \nonumber\\
&& \hspace{-1cm}
\,_2F_1\left(\frac{d}{2}-1,1;d-2;\frac{\sac}{\sabc}\right)
\,_2F_1\left(3-\frac{d}{2},4-d;5-d;\frac{\sac\sbc}{(\sab+\sac)(\sab+\sbc)
}\right)\nonumber \\
&&\hspace{-1cm}-\frac{(3d-8)(3d-10)}{2(d-4)^2}\, A_3 \, \frac{1}{-\sab-\sbc}
\left(-\sac\right)^{d-4}
\left(\frac{\sab+\sbc}{\sab+\sac}\right)^{\frac{d}{2}-1}
\left(\frac{\sabc}{\sab}\right)^{\frac{d}{2}-2}\nonumber \\
&& F_1\left(4-d,2-\frac{d}{2},2-\frac{d}{2},5-d,\frac{\sbc}{\sab+\sbc},
\frac{\sbc}{\sabc}\right) \;,
\\
\boxbubblebNLO{p_{123}}{p_1}{p_2}{p_3} & = &
\frac{(3d-10)(3d-8)}{(d-4)(d-6)}\, A_3\, \frac{1}{\sab+\sbc} \bigg[
\left(-\sac\right)^{d-4}
\,_2F_1\left(1,1;4-\frac{d}{2};\frac{\sbc}{\sab+\sbc} \right) \nonumber \\
&& \hspace{0.5cm} - \left(-\sabc\right)^{d-4}
F_1\left(1,1,4-d,4-\frac{d}{2};\frac{\sbc}{\sab+\sbc},
\frac{\sbc}{\sabc}\right) \bigg] \nonumber\\
&& - \frac{2(3d-8)(3d-10)}{(d-4)(d-6)}\, A_3\, \left(-\sabc\right)^{d-5}
\,_2F_1\left(1,2-\frac{d}{2};\frac{d}{2}-1;\frac{\sab}{\sab+\sac}\right)
\nonumber \\
&& \hspace{-0.7cm} \left[ \,_2F_1\left(1,5-d;4-\frac{d}{2}; \frac{\sbc}{\sabc}\right) +
\frac{d-6}{d-4}
\,_2F_1\left(1,5-d;3-\frac{d}{2}; \frac{\sab+\sac}{\sabc}\right) \right]
\!\! ,
\\
\boxxaNLO{p_{123}}{p_1}{p_2}{p_3} & = & \frac{(3d-8)(3d-10)}{(d-4)^2}\;
A_3 \;\frac{1}{\sac\sbc} \;\Bigg[ \nonumber \\
&& \hspace{-2cm} -\left(\frac{\sac\sbc}{-(\sab+\sbc)}\right)^{d-3}
\,_2F_1\left( d-3,d-3;d-2;\frac{\sab}{\sab+\sbc}\right) \nonumber \\
&& \hspace{-2cm}-\left(\frac{\sac\sbc}{-(\sab+\sac)}\right)^{d-3}
\,_2F_1\left( d-3,d-3;d-2;\frac{\sab}{\sab+\sac}\right) \\
&&\hspace{-2cm}
+\left(\frac{-\sabc\sac\sbc}{(\sab+\sbc)(\sab+\sac)}\right)^{d-3}
\,_2F_1\left(
d-3,d-3;d-2;\frac{\sab\sabc}{(\sab+\sac)(\sab+\sbc)}\right)\Bigg] \;,
\nonumber\\
\boxxbNLO{p_{123}}{p_1}{p_2}{p_3} & = & - A_4 \frac{(d-3)(3d-10)}{(d-4)^2}
\left( -\sabc \right)^{d-4} \left(-\sac-\sbc\right)^{3-d} \nonumber \\
&& \Bigg[ \left(-\sbc\right)^{d-4} F_1\left(4-d,d-3,2-\frac{d}{2},5-d,
\frac{\sac}{\sac+\sbc},\frac{\sac}{\sabc}\right)\nonumber \\
&& +\left(-\sac\right)^{d-4} F_1\left(4-d,d-3,2-\frac{d}{2},5-d,
\frac{\sbc}{\sac+\sbc},\frac{\sbc}{\sabc}\right)\Bigg]\nonumber \\
&& -A_3\frac{(3d-8)(3d-10)}{2(d-4)^2} \frac{1}{-\sabc} \nonumber \\
&& \Bigg[ \left(-\sac\right)^{d-4} S_1\left(\frac{d}{2}-1,1,1,d-2,5-d,
\frac{\sbc}{\sabc},\frac{\sac}{\sabc}\right)\nonumber \\
&&
+\left(-\sbc\right)^{d-4} S_1\left(\frac{d}{2}-1,1,1,d-2,5-d,
\frac{\sac}{\sabc},\frac{\sbc}{\sabc}\right)\Bigg]\;.
\end{eqnarray}
The first two integrals are one-loop bubble insertions into the one-loop
box and have already been computed for arbitrary powers of the propagators
in~\cite{glover}. The last two integrals were, to our knowledge, not
known up to now.
In the reduction of integrals of the last topology, one finds two master
integrals, whose differential equations decouple in the variable
$\Delta\equiv\sac-\sbc$. The second master integral can be found
by rearranging one of the differential equations:
\begin{eqnarray}
\lefteqn{\boxxbdotNLO{p_{123}}{p_1}{p_2}{p_3} =}\nonumber \\
&& - \frac{4\sab\sabc}{\sac\sbc}
\frac{\partial}{\partial\sab} \boxxbNLO{p_{123}}{p_1}{p_2}{p_3}\nonumber \\
&& + (d-4)\frac{3\sab+\sabc}{\sac\sbc} \boxxbNLO{p_{123}}{p_1}{p_2}{p_3}
\nonumber \\
&& -\frac{(d-3)(3d-10)}{d-4}\frac{1}{\sac\sbc} \left(
\triangleNLO{p_{123}}{p_{13}}{p_2} + \triangleNLO{p_{123}}{p_{23}}{p_1}\right)
\nonumber \\
&& +\frac{2(d-3)(3d-8)(3d-10)}{(d-4)^2}\frac{1}{\sac\sbc}\left(
\frac{1}{\sac} \bubbleNLO{p_{13}} + \frac{1}{\sbc}\bubbleNLO{p_{23}}
\right)\; .
\end{eqnarray}
Finally, products of one-loop vertex with one-loop bubble integrals also
yield topologies with $t=5$. These can all be reduced to
(\ref{eq:doublebubblex}) and are not quoted explicitly.
The complete
list of integrals at $t=5$ which were derived in this section can now be
used to compute all integrals at $t=6$ and $t=7$ which can be reduced
using IBP and LI identities. The results of this reduction are
summarized in the following.
\subsection{$t=6$}
Two-loop integrals with $t=6$ must be three- or four-point functions.
Since we are concerned with subgraphs that can appear in the reduction of
four-point functions with one off shell leg, we need to consider
three-point functions with up to two off shell legs. For general
three-point functions at $t=6$, one finds three distinct topologies:
two planar and one crossed arrangement of the loop momenta. The crossed graphs
correspond to master integrals, while the planar graphs are reducible,
as first pointed out in~\cite{kl}, where the three-point integral
with one off shell leg was computed. We reproduce these results:
\begin{eqnarray}
\triaplanNLO{p_{12}}{p_1}{p_2} &= & \frac{3(d-3)(3d-10)}{(d-4)^2}
\frac{1}{\sab^2} \triangleNLO{p_{12}}{p_1}{p_2} \nonumber \\
&& + \frac{4(d-3)^2}{(d-4)^2} \frac{1}{\sab^2} \doublebubbleNLO{p_{12}}
\nonumber \\
&&-\frac{6(d-3)(3d-8)(3d-10)}{(d-4)^3} \frac{1}{\sab^3} \bubbleNLO{p_{12}}
\; ,\\
\triaplanxNLO{p_{12}}{p_1}{p_2} &= & -\frac{3(d-3)(3d-10)}{2(d-4)(d-5)}
\frac{1}{\sab^2} \triangleNLO{p_{12}}{p_1}{p_2} \nonumber \\
&&+ \frac{3(d-3)(3d-8)(3d-10)}{(d-4)^2(d-5)}
\frac{1}{\sab^3} \bubbleNLO{p_{12}}\; .
\end{eqnarray}
The results for two off shell legs read:
\begin{eqnarray}
\triaplanNLO{p_{123}}{p_1}{p_{23}} &= & \frac{1}{\sabc}
\trianglebNLO{p_{123}}{p_{23}}{p_1} + \frac{(d-3)(3d-10)}{(d-4)^2}
\frac{1}{(\sab+\sac)\sabc} \trianglexNLO{p_{123}}{p_1}{p_{23}} \nonumber \\
&&+\frac{2(d-3)(3d-10)}{(d-4)^2} \frac{1}{(\sab+\sac)\sabc}
\triangleNLO{p_{123}}{p_1}{p_{23}}\nonumber \\
&& -\frac{4(d-3)^2}{(d-4)^2} \frac{1}{(\sab+\sac)\sabc}
\doublebubblexNLO{p_{123}}{p_{23}}{p_1}\nonumber \\
&& +\frac{4(d-3)^2}{(d-4)^2} \frac{1}{(\sab+\sac)\sabc}\; ,
\doublebubbleNLO{p_{123}}\nonumber \\
&& +\frac{(d-3)(3d-8)(3d-10)}{(d-4)^3}
\frac{1}{\sbc(\sab+\sac)\sabc} \bubbleNLO{p_{23}} \nonumber \\
&& -\frac{4(d-3)(3d-8)(3d-10)}{(d-4)^3}
\frac{1}{(\sab+\sac)\sabc^2}
\bubbleNLO{p_{123}}\; ,\\
\triaplanxNLO{p_{123}}{p_{23}}{p_1} &= &-\frac{3(d-3)(3d-10)}{2(d-4)(d-5)}
\frac{1}{(\sab+\sac)^2} \left(\triangleNLO{p_{123}}{p_{23}}{p_1}
+\!\!\trianglexNLO{p_{123}}{p_{23}}{p_1}\right) \nonumber \\
&& +\frac{3(d-3)(3d-8)(3d-10)}{2(d-4)^2(d-5)}\frac{1}{\sbc(\sab+\sac)^2}
\bubbleNLO{p_{23}} \nonumber \\
&& +\frac{3(d-3)(3d-8)(3d-10)}{2(d-4)^2(d-5)}\frac{1}{\sabc(\sab+\sac)^2}
\bubbleNLO{p_{123}} \; .
\end{eqnarray}
For $t=6$, both
one-loop bubble insertions on propagators of the one-loop box can be reduced:
\begin{eqnarray}
\boxbubbleapNLO{p_{123}}{p_1}{p_2}{p_3} & = & -\frac{3(\sab+\sac)}{\sac\sbc}
\boxbubbleaNLO{p_{123}}{p_1}{p_2}{p_3} \nonumber \\
&& + \frac{3(3d-10)}{2(d-4)} \frac{1}{\sac\sbc}\left(
\triangleNLO{p_{23}}{p_2}{p_3}- \triangleNLO{p_{123}}{p_2}{p_{13}}
\right)\nonumber \\
&& -\frac{3(3d-8)(3d-10)}{2(d-4)^2} \frac{1}{\sac^2\sbc}
\bubbleNLO{p_{13}}\; ,\\
\boxbubblebpNLO{p_{123}}{p_1}{p_2}{p_3} & = &
\frac{3}{\sac} \boxbubblebNLO{p_{123}}{p_1}{p_2}{p_3}
+ \frac{3(3d-10)}{2(d-4)} \frac{1}{\sac\sbc}
\trianglexNLO{p_{123}}{p_1}{p_{23}} \nonumber \\
&& +\frac{3(3d-8)(3d-10)}{2(d-4)^2} \frac{1}{\sac\sbc\sabc}
\bubbleNLO{p_{123}}\;.
\end{eqnarray}
Among the remaining four planar diagrams at $t=6$, three are reducible to
simpler subtopologies:
\begin{eqnarray}
\boxxamNLO{p_{123}}{p_1}{p_2}{p_3} & = & -3 \frac{\sab}{\sac\sbc}
\boxxaNLO{p_{123}}{p_1}{p_2}{p_3} \nonumber \\
&& -\frac{6(d-3)}{d-4}\frac{1}{\sac}
\boxbubblebNLO{p_{123}}{p_1}{p_2}{p_3} \nonumber \\
&& -\frac{3(d-3)(3d-10)}{(d-4)^2} \frac{1}{\sac\sbc}
\trianglexNLO{p_{123}}{p_1}{p_{23}} \nonumber \\
&& -\frac{3(d-3)(3d-8)(3d-10)}{(d-4)^3} \nonumber \\
&& \hspace{1cm} \left(\frac{1}{\sac^2\sbc}
\bubbleNLO{p_{13}}+ \frac{1}{\sac\sbc^2} \bubbleNLO{p_{23}}\right)\; , \\
\boxxapNLO{p_{123}}{p_1}{p_2}{p_3} & = & -3 \frac{\sab}{\sac\sbc}
\boxxaNLO{p_{123}}{p_1}{p_2}{p_3} \nonumber \\
&& +\frac{6(d-3)}{d-4} \frac{\sab+\sac}{\sac\sbc}
\boxbubbleaNLO{p_{123}}{p_1}{p_2}{p_3} \nonumber \\
&& -\frac{3(d-3)(3d-10)}{(d-4)^2} \frac{1}{\sac\sbc}
\triangleNLO{p_{23}}{p_2}{p_3} \nonumber \\
&& +\frac{3(d-3)(3d-10)}{(d-4)^2} \frac{1}{\sac\sbc}
\triangleNLO{p_{123}}{p_2}{p_{13}} \nonumber \\
&& -\frac{3(d-3)(3d-8)(3d-10)}{(d-4)^3} \nonumber \\
&& \hspace{1cm} \left(\frac{1}{\sac\sbc^2}
\bubbleNLO{p_{23}} - \frac{1}{\sac\sbc\sabc} \bubbleNLO{p_{123}}\right)\; , \\
\boxxbmNLO{p_{123}}{p_1}{p_2}{p_3} & = & -\frac{1}{d-4}
\boxxbdotNLO{p_{123}}{p_1}{p_2}{p_3}\nonumber\\
&& + \frac{6(d-3)}{d-4} \frac{\sab+\sac}{\sac\sbc}
\boxbubbleaNLO{p_{123}}{p_1}{p_2}{p_3} \nonumber \\
&& - \frac{3(d-3)(3d-10)}{(d-4)^2} \frac{1}{\sac\sbc}
\triangleNLO{p_{23}}{p_2}{p_3} \nonumber \\
&& +\frac{3(d-3)(3d-10)}{(d-4)^2} \frac{1}{\sac\sbc}
\triangleNLO{p_{123}}{p_2}{p_{13}} \nonumber \\
&& +\frac{3(d-3)(3d-8)(3d-10)}{(d-4)^3} \frac{1}{\sac^2\sbc}
\bubbleNLO{p_{13}}\; .
\end{eqnarray}
One of the two remaining non-planar diagrams is also reducible, the other
non-planar topology contains two master integrals. The reducible
integral reads:
\begin{eqnarray}
\boxxbmcrossNLO{p_{123}}{p_1}{p_2}{p_3} & = &
\hspace{0.28cm}
\frac{3(d-4)}{2d-9}\, \frac{\sab}{\sac\sbc}\boxxaNLO{p_{123}}{p_1}{p_2}{p_3}
\nonumber \\
&&+\frac{3(d-4)}{2d-9}\, \frac{\sbc}{\sab\sac}\boxxaNLO{p_{123}}{p_3}{p_2}{p_1}
\nonumber \\
&&+\frac{3(d-4)}{2d-9}\, \frac{\sac}{\sab\sbc}\boxxaNLO{p_{123}}{p_1}{p_3}{p_2}
\nonumber \\
&&+ \frac{3(d-3)(3d-8)(3d-10)}{(d-4)^2(2d-9)}\,
\frac{\sac+\sbc}{\sab^2\sac\sbc}\bubbleNLO{p_{12}} \nonumber \\
&&+ \frac{3(d-3)(3d-8)(3d-10)}{(d-4)^2(2d-9)}\,
\frac{\sab+\sbc}{\sab\sac^2\sbc}\bubbleNLO{p_{13}} \nonumber \\
&&+ \frac{3(d-3)(3d-8)(3d-10)}{(d-4)^2(2d-9)}\,
\frac{\sab+\sac}{\sab\sac\sbc^2}\bubbleNLO{p_{23}} \nonumber \\
&&- \frac{3(d-3)(3d-8)(3d-10)}{(d-4)^2(2d-9)}\,
\frac{1}{\sab\sac\sbc}\bubbleNLO{p_{123}}\; .
\end{eqnarray}
\subsection{$t=7$}
At $t=7$ different one finds six different topologies. Three of them
are triangle insertions to the one-loop box. These three integrals
are all reducible, two of them contain only master integrals up to $t=5$ in
their reduction:
\begin{eqnarray}
\lefteqn{\boxtriaaNLO{p_{123}}{p_1}{p_2}{p_3} =}\nonumber \\
&& -\frac{6(d-3)(3d-14)}{(d-4)(d-6)}\, \frac{1}{\sbc^2}\,
\boxbubblebNLO{p_{123}}{p_2}{p_1}{p_3}\nonumber \\
&& + \frac{3(3d-14)}{(d-6)} \, \frac{\sab^2}{\sac^2\sbc^2}\,
\boxxaNLO{p_{123}}{p_1}{p_2}{p_3} \nonumber \\
&& -\frac{6(d-3)(3d-14)}{(d-4)(d-6)}\, \frac{(\sab+\sac)^2}{\sac^2\sbc^2}\,
\boxbubbleaNLO{p_{123}}{p_1}{p_2}{p_3} \nonumber \\
&& - \frac{3(d-3)(3d-10)(3d-14)}{2(d-4)^2(d-5)(d-6)} \, \frac{(2d-10)\sab +
(3d-14) \sbc}{\sac\sbc^2(\sab+\sbc)} \trianglexNLO{p_{123}}{p_2}{p_{13}}
\nonumber \\
&& + \frac{3(d-3)(3d-10)(3d-14)}{2(d-4)^2(d-5)(d-6)} \, \frac{(2d-10)\sab +
(3d-14) \sac}{\sac^2\sbc^2} \triangleNLO{p_{23}}{p_2}{p_3} \nonumber\\
&& - \frac{3(d-3)(3d-10)(3d-14)}{2(d-4)^2(d-5)(d-6)} \, \frac{(2d-10)
\sab\sabc +(3d-14) \sac\sbc }{\sac^2\sbc^2(\sab+\sbc)}
\triangleNLO{p_{123}}{p_2}{p_{13}} \nonumber \\
&& - \frac{3(d-3)(3d-8)(3d-10)(3d-14)}{2(d-4)^3(d-5)(d-6)} \,
\frac{(2d-10)\sab + (d-6)\sbc}{\sac^2\sbc^2(\sab+\sbc)}
\bubbleNLO{p_{13}}\nonumber\\
&& + \frac{3(d-3)(3d-8)(3d-10)(3d-14)}{(d-4)^3(d-5)(d-6)}\,
\frac{(d-5)\sab - (d-4)\sac}{ \sac^2\sbc^3} \, \bubbleNLO{p_{23}}\nonumber\\
&& + \frac{3(d-3)(3d-8)(3d-10)(3d-14)}{2(d-4)^3(d-5)(d-6)}\,
\frac{(2d-10)\sab\sabc + (d-6)
\sac\sbc}{\sac^2\sbc^2\sabc(\sab+\sbc)}
\bubbleNLO{p_{123}}\; ,\\
\lefteqn{\boxtriabNLO{p_{123}}{p_1}{p_2}{p_3} =}\nonumber \\
&& \frac{2(2d-9)}{(d-4)(d-6)}\,\frac{2\sab+\sac+\sbc}{\sac\sbc}
\boxxbdotNLO{p_{123}}{p_1}{p_2}{p_3}\nonumber \\
&& -\frac{3(d-4)}{d-6}\, \frac{(\sac+\sbc)^2}{\sac^2\sbc^2}
\boxxbNLO{p_{123}}{p_1}{p_2}{p_3}\nonumber \\
&& -\frac{6(d-3)(3d-14)}{(d-4)(d-6)}\, \frac{(\sab+\sbc)^2}{\sac^2\sbc^2}
\boxbubbleaNLO{p_{123}}{p_2}{p_1}{p_3} \nonumber\\
&& -\frac{6(d-3)(3d-14)}{(d-4)(d-6)}\, \frac{(\sab+\sac)^2}{\sac^2\sbc^2}
\boxbubbleaNLO{p_{123}}{p_1}{p_2}{p_3} \nonumber\\
&& +\frac{3(d-3)(3d-10)(3d-14)}{2(d-4)^2(d-5)(d-6)}\,
\frac{(2d-10)\sab+(3d-14)\sbc}{\sac^2\sbc^2} \triangleNLO{p_{13}}{p_1}{p_3}
\nonumber \\
&& +\frac{3(d-3)(3d-10)(3d-14)}{2(d-4)^2(d-5)(d-6)}\,
\frac{(2d-10)\sab+(3d-14)\sac}{\sac^2\sbc^2} \triangleNLO{p_{23}}{p_2}{p_3}
\nonumber \\
&& -\frac{3(d-3)(3d-10)}{(d-4)^2(d-6)}\, \frac{(3d-14)\sab + (4d-18)\sac
-(d-4)\sbc }{\sac^2\sbc^2} \triangleNLO{p_{123}}{p_{13}}{p_2}
\nonumber \\
&& -\frac{3(d-3)(3d-10)}{(d-4)^2(d-6)}\, \frac{(3d-14)\sab -(d-4)\sac
+ (4d-18)\sbc }{\sac^2\sbc^2} \triangleNLO{p_{123}}{p_{23}}{p_1}
\nonumber \\
&& -\frac{3(d-3)(3d-8)(3d-10)}{(d-4)^3(d-5)(d-6)}
\frac{(d-5)(3d-14)(\sab+\sac)+(d-4)^2\sbc}{\sac^3\sbc^2}
\bubbleNLO{p_{13}}\nonumber \\
&& -\frac{3(d-3)(3d-8)(3d-10)}{(d-4)^3(d-5)(d-6)}
\frac{(d-5)(3d-14)(\sab+\sbc)+(d-4)^2\sac}{\sac^2\sbc^3}
\bubbleNLO{p_{23}}\;.
\end{eqnarray}
The remaining three topologies are the double box and two different
momentum arrangements of the crossed box. These topologies contain
each two master integrals.
\section{Conclusions and Outlook}
\label{sec:conc}
\setcounter{equation}{0}
Progress in the computation of exclusive observables, such as for
example jet production rates, beyond the next-to-leding
order has up to now been hampered mainly by difficulties in
the calculation of virtual two-loop integrals with more than two external
legs. In contrast to this, many inclusive observables (which correspond
from the calculational point of view to two-point functions) are
known to next-to-next-to-leading order and even beyond. These
higher order calculations relied on a variety of elaborate technical tools
for the computation of the virtual integrals. In this paper, we outline
how techniques known from multi-loop calculations of two-point integrals
can be modified and extended towards the computation of integrals with a
larger number of external legs.
We demonstrate that
the large number of different two-loop integrals appearing in an actual
calculation can be reduced to a small number of scalar
master integrals by using
the well-known integration-by-parts identities~\cite{chet} together with
identities following from Lorentz-invariance which are unique to
multi-leg integrals. As a by-product of this reduction, one is also able to
reduce two-loop integrals with tensorial structure to scalar integrals.
In contrast to two-point integrals, where only a few
topologically different
graphs can appear with potentially large powers of propagators and
scalar products,
one finds that the reduction of three- and four-point integrals gives
rise to a large number of topologically different graphs, which appear
however only with small powers of propagators
and scalar products. The reduction of two-point functions usually
proceeds via solving manually
the integration-by-parts identities for arbitrary
powers of propagators and denominators in a given graph topology;
this procedure seems to be not practicable for multi-leg integrals.
To
accomplish the reduction
of these, we developed an algebraic programme which automatically
derives and solves the integration-by-parts and Lorentz-invariance
identities for a given graph up to some pre-selected fixed number of powers in
denominators and scalar products independent of the topology.
To compute the scalar master integrals, we derive differential equations
in the external momenta~\cite{remiddi}
for them; the boundary conditions of these
differential equations correspond to simpler integrals, where for example
one of the external momenta vanishes. These differential equations can be
solved (for arbitrary space-time dimensions)
by employing standard mathematical methods. We observe that
the differential equations for the master integrals we considered up to now
are solved by generalized hypergeometric functions. We illustrate the
application of this method in detail on the example of
the one-loop four-point function with one off shell leg.
Using the differential equation method,
we provide a complete list of all master integrals with up to $t=5$
different denominators that can appear in the reduction of two-loop
four-point functions with one off shell leg. We also list
all reducible integrals with $t=6$ and $t=7$ different propagators.
The computaion of the master integrals with $t=6$ and $t=7$ is still an
outstanding task.
The differential equations for these outstanding master integrals are of
similar structure as the differential equations for master integrals
with a smaller number of different propagators. The main problem towards
a complete computation of these integrals is at present the integration
of the inhomogeneous term, containing itself already hypergeometric
functions arising from the subtopologies.
It is worthwhile to point out similarities and differences between the
differential equation method employed in this paper and other
methods employed for similar calculations in the
literature. Both the negative dimension approach of~\cite{glover} and
the Mellin-Barnes transformation method employed in~\cite{smirnov,tausk}
rely on choosing a particular assignment of momentum vectors to the
internal loop propagators. After this assignment, a representation of
the propagators in terms of a multiple sum (negative dimension approach)
or an integral transformation (Mellin-Barnes method) is employed,
such that the integral over the loop momentum can be carried out explicitly.
The final result for the integral is then retrieved by resummation of
a multiple sum or by an inverse integral transformation. Both methods,
when employed for arbitrary space-time dimension, give rise
to generalized hypergeometric functions, which can be represented as
multiple sums as well as inverse Mellin-Barnes integrals~\cite{bateman}.
In the differential equation method, one assigns momentum vectors to
the loop propagators only for the sake of deriving the
differential equations and the IBP and LI identities.
After applying these identities to simplify the differential equations,
one obtaines a relation between the derivative
(with respect to an external momentum) of a
master integral, the master integral itself and other master integrals
with simpler topology, independent of the parametrization chosen for the
internal propagators. These differential equations can then be solved
analytically by
integration; the resulting integrals correspond to the integral
representations of generalized hypergeometric functions~\cite{bateman}.
Using the differential equation method, one can therefore circumvent
the explicit loop momentum integration needed in the other methods and
one arrives at a representation of the hypergeometric functions, which is
presumably more transparent than a multiple sum or an inverse integral
transformation. In the integral representation, it is in particular
straightforward to identify linear combinations of different
hypergeometric functions, which are difficult to disentagle in the
other representations. At present, it should however not be claimed that any
of the methods is superior, since none of them could yet be
employed to compute all outstanding two-loop four-point master integrals.
As a final point, we note that the methods derived in this paper
contain a
high level of redundancy, which allows for a number of non-trivial
checks on the results obtained with them. The automatic reduction to
master integrals using integration-by-parts and Lorentz-invariance
identities corresponds to the solution of a linear system of equations
containing more identities than unknowns. The existence of a solution to
this system provides therefore already a check on the self-consistency of
the identities. In computing
the master integrals from differential equations, one integrates one
of the three differential equations in the external invariants, such that
the result can be checked by inserting it in the remaining two differential
equations.
In short, this paper demonstrates how techniques developed for
multi-loop calculation of two-point functions can be extended towards
integrals with a larger number of external legs. As a first example of
the application of these tools in practice, we computed some
up to now unknown
two-loop four-point functions, relevant for jet calculus beyond the
next-to-leading order. Using these tools as alternative or complement to
already existing techniques, one should be able to proceed towards
complete two-loop calculations of exclusive quantities.
\section*{Acknowledgements}
We are grateful to Jos Vermaseren for his assistance in the use of the
algebraic program FORM.
One of the authors (E.R.) wants to thank the Alexander-von-Humboldt
Stiftung for supporting his stay at the Institut f\"ur Theoretische
Teilchenphysik of the University of Karlsruhe.
The research work presented in this paper was supported in part by
the DFG (Forschergruppe ``Quantenfeldtheorie, Computeralgebra und
Monte-Carlo Simulation'', contract KU 502/8-2).
\begin{appendix}
\renewcommand{\theequation}{\mbox{\Alph{section}.\arabic{equation}}}
\section{Special Functions}
\setcounter{equation}{0}
This appendix summarizes the series and integral representations
of the hypergeometric functions appearing in the master integrals.
The properties of these functions, in particular their
region of analyticity, their analytic continuation as well as reduction
formulae, can be found in the
literature~\cite{glover,bateman,grad,exton}.
Hypergeometric functions
are sums with coefficients formed from Pochhammer symbols
\begin{equation}
\left( a\right)_n \equiv \frac{\Gamma(a+n)}{\Gamma(a)}.
\end{equation}
The hypergeometric functions of a single variable are given by:
\begin{eqnarray}
\;_2F_1 \left(a,b;c;z\right) & = & \sum_{n=0}^{\infty}
\frac{(a)_n(b)_n}{(c)_n} \frac{z^n}{n!}\; ,\\
\;_3F_2 \left(a,b_1,b_2;c_1,c_2;z\right) & = & \sum_{n=0}^{\infty}
\frac{(a)_n(b_1)_n(b_2)_n}
{(c_1)_n(c_2)_n} \frac{z^n}{n!}\;.
\end{eqnarray}
Two types of hypergeometric functions of two variables also appear in
our results:
\begin{eqnarray}
F_1\left(a,b_1,b_2;c;z_1,z_2\right) &=& \sum_{m,n=0}^{\infty}
\frac{(a)_{m+n}(b_1)_m(b_2)_n}
{(c)_{m+n}} \frac{z_1^m}{m!} \frac{z_2^n}{n!}\; ,\\
S_1\left(a_1,a_2,b;c,d;z_1,z_2\right) &=& \sum_{m,n=0}^{\infty}
\frac{(a_1)_{m+n}(a_2)_{m+n}(b)_m}
{(c)_{m+n}(d)_m} \frac{z_1^m}{m!} \frac{z_2^n}{n!}\; .
\end{eqnarray}
These functions have the following integral representations:
\begin{eqnarray}
\;_2F_1 \left(a,b;c;z\right) & = & \frac{\Gamma(c)}{\Gamma(b)\Gamma(c-b)}
\int_0^1 \d t\; t^{b-1} (1-t)^{c-b-1} (1-tz)^{-a} \nonumber \\
&& \hspace{2cm} \mbox{Re}(b) > 0, \qquad \mbox{Re}(c-b)>0 \\
\;_3F_2 \left(a,b_1,b_2;c_1,c_2;z\right) & = &
\frac{\Gamma(c_1)\Gamma(c_2)}{\Gamma(b_1)\Gamma(c_1-b_1)\Gamma(b_2)
\Gamma(c_2-b_2)}\nonumber \\
&& \int_0^1 \d t_1 \int_0^1 \d t_2\;
t_1^{b_1-1} t_2^{b_2-1} (1-t_1)^{c_1-b_1-1}(1-t_2)^{c_2-b_2-1}
(1-t_1t_2z)^{-a}\nonumber \\
&& \hspace{-0.7cm}
\mbox{Re}(b_1) > 0, \qquad \mbox{Re}(c_1-b_1)>0, \qquad
\mbox{Re}(b_2) > 0, \qquad \mbox{Re}(c_2-b_2)>0 \\
F_1\left(a,b_1,b_2;c;z_1,z_2\right)&=&\frac{\Gamma(c)}{\Gamma(a)\Gamma(c-a)}
\int_0^1 \d t\; t^{a-1}(1-t)^{c-a-1} (1-tz_1)^{-b_1}(1-tz_2)^{-b_2}
\nonumber \\
&& \hspace{2cm} \mbox{Re}(a) > 0, \qquad \mbox{Re}(c-a)>0 \\
S_1\left(a_1,a_2,b;c,d;z_1,z_2\right) &=&
\frac{\Gamma(c)\Gamma(d)}{\Gamma(a_1)\Gamma(c-a_1)\Gamma(b)\Gamma(d-b)}
\nonumber \\
&& \int_0^1\d t_1\int_0^1\d t_2 t_1^{a_1-1} t_2^{b-1} (1-t_1)^{c-a_1-1}
(1-t_2)^{d-b-1} \left(1-t_1t_2z_1-t_1z_2\right)^{-a_2} \nonumber\\
&&\mbox{Re}(a_1)>0, \quad \mbox{Re}(c-a_1)>0,\quad
\mbox{Re}(b)>0, \quad \mbox{Re}(d-b)>0
\end{eqnarray}
\section{Expansion of Hypergeometric Functions}
\setcounter{equation}{0}
To separate divergent and finite parts
of the loop integrals derived in this paper, one has to expand them
around the physical number of space-time dimensions in the parameter
$\epsilon=(4-d)/2$. We demonstrate in this appendix, that this expansion
can, at least for the hypergeometric functions in one variable, carried out
in a mechanical way, giving rise to harmonic polylogarithms (HPL),
a generalization of Nielsen's polylogarithms~\cite{nielsen}
introduced in~\cite{hpl}.
Expanding the integral representation of $\,_2F_1$ in $\epsilon$
yields simple powers of $(t,1-t,1-tz)$ times the product of some
number of $(\ln t, \ln (1-t), \ln(1-tz))$. The powers of
$(t,1-t,1-tz)$ can be integrated by parts until one obtains
non-trivial integrals
\begin{displaymath}
\int_0^1 \d t \left(\frac{1}{t},\frac{1}{1-t},\frac{1}{1-tz}\right)
\ln^{n_1}t \ln^{n_2}(1-t) \ln^{n_3}(1-tz)\;.
\end{displaymath}
All these integrals are combinations of harmonic polylogarithms
$H(\vec{a};z)$, where $\vec{a}$ is a vector of indices with
$w=n_1+n_2+n_3+1$ components. $w$ is called the weight of the
harmonic polylogartihm.
The proof by induction in $w$ is
trivial, once the HPL formalism~\cite{hpl} is recalled:
\begin{enumerate}
\item
Definition of the three HPLs at $w=1$:
\begin{eqnarray}
H(1;z) & \equiv & -\ln (1-z)\; ,\nonumber \\
H(0;z) & \equiv & \ln z\; ,\nonumber \\
H(-1;z) & \equiv & \ln (1+z)
\label{eq:levelone}
\end{eqnarray}
and the three fractions
\begin{eqnarray}
f(1;z) & \equiv & \frac{1}{1-z} \;, \nonumber \\
f(0;z) & \equiv & \frac{1}{z} \;, \nonumber \\
f(-1;z) & \equiv & \frac{1}{1+z} \; ,
\end{eqnarray}
such that
\begin{equation}
\frac{\partial}{\partial z} H(a;z) = f(a;z)\qquad \mbox{with}\quad
a=+1,0,-1\;.
\end{equation}
\item For $w>1$:
\begin{eqnarray}
H(0,\ldots,0;z) & \equiv & \frac{1}{w!} \ln^w z\; ,\\
H(a,\vec{b};z) & \equiv & \int_0^z \d x f(a;x) H(\vec{b};x)\; ,
\end{eqnarray}
which results in
\begin{equation}
\frac{\partial}{\partial z} H(a,\vec{b};z) = f(a;z) H(\vec{b};z)\;.
\end{equation}
This last relation is a convenient tool for verifying identities among
different HPLs. Such identities can be verified by first checking a
special point (typically $z=0$)
and subsequently checking the derivatives. If agreement in
the derivatives is not obvious, this procedure can be repeated until one
arrives at relations involving only HPLs with $w=1$.
\item
The HPLs fulfil an algebra (see Section 3 of~\cite{hpl}), such that
a product of two HPLs (with weights $w_1$ and $w_2$)
of the same argument $z$ is a combination of HPLs of argument
$z$ with weight $w=w_1+w_2$.
\end{enumerate}
Using these properties of the HPL, one can show that
the integrals appearing in the $\epsilon$-expansion of the
hypergeometric function can be reexpressed as
\begin{equation}
\int_0^1 \d t \left(\frac{1}{t},\frac{1}{1-t},\frac{1}{1-tz}\right)
\ln^{n_1}t \ln^{n_2}(1-t) \ln^{n_3}(1-tz) \to
\int_0^1 \d t \left(\frac{1}{t},\frac{1}{1-t},\frac{1}{t-1/z}\right)
H(\vec{a},t) H( \vec{b},zt)\;.
\label{eq:NTint}
\end{equation}
Following the argumentation of Section 7 of~\cite{hpl}, one can
show that the integral on the right hand side of the above equation
yields a linear combination of
HPLs of weight $w=w_a+w_b+1$. The proof goes via induction in $w_b$.
For $w_a=w_b=0$ one has $ H(\vec{a};t) H( \vec{b};zt)=1$.
The $t$-integral in (\ref{eq:NTint}) yields then a combination
of HPL of weight $w=1$
(\ref{eq:levelone}). Likewise, for $w_b=0$ the right hand side of
(\ref{eq:NTint}) will yield a linear combination of HPLs
of weight $w=w_a+1$ and of argument $z$, as proven in Section 7 of~\cite{hpl}.
Considering
\begin{equation}
\frac{\partial}{\partial z}
\int_0^1 \d t \left(\frac{1}{t},\frac{1}{1-t},\frac{1}{t-1/z}\right)
H(\vec{a};t) H(B, \vec{b};zt)\;,
\label{eq:induct}
\end{equation}
we observe that
\begin{eqnarray}
\frac{\partial}{\partial z} \frac{\d t}{t-1/z} &=& \frac{1}{z^2}
\, \frac{1}{t-1/z}\, \d t\,
\frac{\partial}{\partial t} + \mbox{boundary terms} \;,
\nonumber \\
\frac{\partial}{\partial z} H(B,\vec{b};zt) &=& t\,f(B;zt)\,H(\vec{b};zt)
\nonumber \;.
\end{eqnarray}
Making these replacements in (\ref{eq:induct}) and applying
partial fractioning to all denominators, we are left with
\begin{equation}
\left(\frac{1}{z},\frac{1}{1-z},\frac{1}{1+z}\right)
\int_0^1 \d t \left(\frac{1}{t},\frac{1}{1-t},\frac{1}{t-1/z}\right)
H(\vec{a};t) H(\vec{b};zt)\;,
\end{equation}
which is a combination of HPLs with argument $z$ and
weight $w=w_a+w_b+1$ multiplied with
$(1/z,1/(1-z),1/(1+z))$. Integrating
(\ref{eq:induct}) over $z$ will thus yield a combination of HPLs with
argument $z$ and weight $w+1$, which completes the proof by induction.
The $\epsilon$-expansion of $\,_3F_2$, corresponding
to a double integral in $t_1$ and $t_2$, is obtained by carrying out the
procedure described here twice, again resulting in a combination
of HPLs.
A systematic $\epsilon$-expansion of $F_1$ and $S_1$, which are functions
two variables $z_1$ and $z_2$, will in general go beyond the harmonic
polylogarithms in one variable.
\end{appendix}
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\end{document}