On hyperlogarithms and Feynman integrals with divergences and many scales

Hyperlogarithms provide a tool to carry out Feynman integrals in Schwinger parameters. So far, this method has been applied successfully mostly to finite single-scale processes. However, it can be employed in more general situations. We give examples of integrations of three- and four-point integrals in Schwinger parameters with non-trivial kinematic dependence, including setups with off-shell external momenta and differently massive internal propagators. The full set of Feynman graphs admissible to parametric integration is not yet understood and we discuss some counterexamples to the crucial property of linear reducibility. In special cases we observe how a change of variables can restore this prerequisite for direct integration and thereby enlarge the set of accessible graphs. Working in dimensional regularization, we furthermore clarify how a simple application of partial integration can be used to convert divergent parametric integrands to convergent ones. In contrast to the subtraction of counterterms, this scheme is ideally suited for our method of integration.