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### Description

A key ingredient in an automated evaluation of two-loop multileg processes is a

fast and numerically stable evaluation of scalar Feynman integrals. In this respect, the calculation of two-loop three- and four-point functions in the general complex mass case so far relies on multidimensional numerical integration through sector decomposition whereby a reliable result has a high computing cost, whereas the derivation of a fully analytic result remains beyond reach. It would therefore be useful to perform part of the Feynman

parameter integrations analytically in a systematic way to let only a reduced

number of integrations to be performed numerically. Such a working program has been initiated for the calculation of massive two-loop $N$-point functions

using analytically computed building blocks. This approach is based on the implementation of two-loop scalar $N$-point functions in four dimensions $^{(2)}I_{N}^{4}$ as double integrals in the form:

$$
^{(2)}I_{N}^{4}
\sim
\sum \int_{0}^{1} d \rho \int_{0}^{1} d \xi \, P(\rho,\xi) \;
^{(1)}\widetilde{I}_{N+1}^{4}(\rho,\xi)
$$
where the building blocks $^{(1)}\widetilde{I}*{N+1}^{4}(\rho,\xi)$ involved in the integrands are similar to "generalised" one-loop $(N+1)$-point Feynman-type integrals, and where $P(\rho,\xi)$ are weighting functions. The $^{(1)}\widetilde{I}*{N+1}^{4}(\rho,\xi)$ are "generalised" in the sense that the integration domain spanned by the Feynman parameters defining them is no longer the usual simplex ${ 0 \leq z_{j} \leq 1, j = 1,\cdots,N+1; \sum_{j=1}^{N+1} z_{j}=1}$ at work for the one-loop $(N+1)$-point function, but another domain (e.g. a cylinder with triangular basis) which depends on the topology of the two-loop $N$-point function considered. The generalisation concerns also the underlying kinematics, which, besides external momenta, depends on two extra Feynman parameters $\rho$ and $\xi$. The parameter space spanned by this kinematics is larger than the one spanned in one-loop $(N+1)$-particle processes at colliders.The only two remaining integrations over $\rho,\xi$ to be performed numerically represent a substantial gain w.r.t. a fully numerical integration of the many Feynman parameter two-loop integrals.
As a first step in this direction, the method developped has been successfully applied to the usual one-loop four-point function for arbitrary masses and kinematics as a ``proof of concept'', showing its ability to circumvent the subtleties of the various analytic continuations in the kinematical variables in a systematic way, in a series ofthree articles. The target work, namely its practical implementation to compute the building blocks $^{(1)}\widetilde{I}_{N+1}^{4}(\rho,\xi)$ is to be elaborated and presented

in a future series of articles.