Padé approach to top-quark mass effects in gluon fusion amplitudes

Gluon fusion processes like single and double Higgs production exhibit slow convergence and pose severe computational challenges. We show how the top-quark mass dependence of the virtual amplitudes can be reconstructed with a conformal mapping and Pad\'e approximants based on the expansion in the inverse top-quark mass and the non-analytic terms in the expansion around the top threshold. The method is then applied at two- and three-loop order.