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We evaluate, analytically, a specific class of eighth--order and tenth--orderQED contributions to the anomalous magnetic moment of the muon. They aregenerated by Feynman diagrams involving lowest order vacuum polarizationinsertions of leptons $l=e,\mu$, and $\tau$. The results are given in the formof analytic expansions in terms of the mass ratios $m_e/m_\mu$ and$m_\mu/m_\tau$. We compute as many terms as required by the error induced bythe present experimental uncertainty on the lepton masses. We show how theMellin--Barnes integral representation of Feynman parametric integrals allowsfor an easy analytic evaluation of as many terms as wanted in these expansionsand how its underlying algebraic structure generalizes the standardrenormalization group properties. We also discuss the generalization of thistechnique to the case where two independent mass ratios appear. Comparison withprevious numerical and analytic evaluations made in the literature, wheneverpertinent, are also made.

Muon Anomaly from Lepton Vacuum Polarization and The Mellin--Barnes Representation