Validity of Padé approximations in vacuum polarization at three- and four-loop order.

The heavy-quark contribution to the polarization function Π at higher perturbative orders is presently only known approximately. We scrutinize the accuracy of state-of-the-art approximations at three- and four-loop order. At three loops, we present for the first time a result with arbitrary numerical precision for general kinematics and compare to the best Padé estimate. At four loops, we calculate the fourth (inverse) moment of the nonsinglet heavy-quark vacuum polarization in order to test the prediction for this moment based on the Padé approximation