Three-loop QED contributions to the $g-2$ of charged leptons with two internal fermion loops and a class of Kampé de Fériet series
Résumé
The three-loop QED mass-dependent contributions to the g-2 of each of the charged leptons with two internal closed fermion loops, sometimes called A3(6)(m1m2,m1m3) in the g-2 literature, is revisited using the Mellin-Barnes (MB) representation technique. Results for the muon and τ lepton anomalous magnetic moments A3,μ(6) and A3,τ(6), which were known as series expansions in the lepton mass ratios up to the first few terms only, are extended to their exact expressions. The contribution to the anomalous magnetic moment of the electron A3,e(6) is also explicitly given in closed form. In addition to this, we show that the different series representations derived from the MB representation collectively converge for all possible values of the masses. Such unexpected behavior is related to the fact that these series bring into play double hypergeometric series that belong to a class of Kampé de Fériet series which we prove to have the same simple convergence and analytic continuation properties as the Appell F1 double hypergeometric series.