Analytical Calculations of Contributions to the Anomalous Magnetic Moments of Leptons from Vacuum Polarization by Lepton Loops within the Mellin–Barnes Representation

Abstract—Explicit analytical expressions have been derived for electromagnetic corrections to the anomalous magnetic moments aL of leptons (L = e, μ, and τ) of the tenth order of the radiative corrections from Feynman diagrams with insertions of the polarization operator with four lepton loops, where one is formed by leptons of the same kind as initial the lepton L, and three others are formed by leptons of type $$\ell \ne L$$ . The studies are based on the consecutive application of dispersion relations for the polarization operator and Mellin–Barnes transform for the propagators of massive particles. The exact formulas of expansion in the limit of low and high values of the ratios of lepton masses r = $${{{{m}_{\ell }}} \mathord{\left/ {\vphantom {{{{m}_{\ell }}} {{{m}_{L}}}}} \right. \kern-0em} {{{m}_{L}}}}$$ , r → 0 and r → ∞, and compared with the corresponding asymptotic expressions of aL, available in the literature.