The vacuum polarization due to a virtual heavy quark pair and
specifically the coefficients of its Taylor expansion in the external
momentum are closely related to moments of the cross section for
quark-antiquark pair production in electron-positron
annihilation. Relating measurement and theoretically calculated Taylor
coefficients, an accurate value for charm- and bottom-quark mass can be
derived, once corrections from perturbative QCD are sufficiently well
under control. Up to three-loop order these have been evaluated
previously. We now present a subset of four-loop contributions to the
lowest two moments, namely those from diagrams which involve two
internal loops from massive and massless fermions coupled to virtual
gluons, hence of order alphas^3 nf^2. The calculation demonstrates the
applicability of Laporta's algorithm to four-loop vacuum diagrams with
both massive and massless propagators and should be considered a first
step towards the full evaluation of the order alphas^3 contribution.
