Analytical calculations of fifth-order electromagnetic corrections to the anomalous magnetic moment of leptons within the Mellin-Barnes representation

We investigate the explicit, analytical expressions for the fifth-order electromagnetic corrections in the fine structure constant α to the anomalous magnetic moment of leptons aL (L = e, μ, τ) from diagrams with insertions of the vacuum polarization operator consisting of pure lepton loops. Our approach is based on the consecutive application of dispersion relations for the polarization operator and the Mellin – Barnes transform for the propagators of massive particles. Exact analytical expressions for the corrections to aL from vacuum polarization by four identical loops are obtained. Asymptotic expansions are found in the limit of both small and large values of the lepton mass ratio  (r = mℓ /mL), r≪ 1 and r→∞ The resulting expansions are compared with the corresponding expressions given in the literature.