Twelfth Order Corrections to Lepton Anomalies Due to Vacuum Polarization Insertions by Lepton Loops within the Mellin–Barnes Representation

Explicit analytical expressions are obtained for the electromagnetic corrections to the lepton anomalous magnetic moments $${{a}_{L}}$$ ( $$L = e, \\mu $$ , and τ) in the twelfth order of expansion with respect to the electric charge e . The Feynman diagram with insertions of the vacuum polarization operator consisting of five identical lepton loops formed by leptons of type $$\\ell $$ different from the initial lepton L ( $$\\ell $$ ≠ L ) is considered. The approach is based on the consecutive application of dispersion relations for the vacuum polarization operator and on the Mellin–Barnes transform for the propagators of massive particles. Exact expressions are expanded into a series with respect to the lepton mass ratio $$r = {{{{m}_{\\ell }}} \\mathord{\\left/ {\\vphantom {{{{m}_{\\ell }}} {{{m}_{L}}}}} \\right. \\kern-0em} {{{m}_{L}}}}$$ , and detailed investigation of the asymptotic expressions in the regions of both small ( $$r \\to 0$$ ) and large ( $$r \\to \\infty $$ ) values of r is performed. It is shown that for real leptons (electron, muon, and τ lepton), calculations can be performed using simpler approximate formulas with consideration of the number of terms depending on the required accuracy, instead of using exact expressions.