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We investigate the radiative quantum electrodynamic (QED) corrections to thelepton ($L=e, ~\mu $ and $\tau$) anomalous magnetic moment due to thecontributions of diagrams with insertions of the photon vacuum polarisationoperator consisting solely of four closed lepton ($l=e, ~\mu $ and $\tau$)loops. Moreover, we focus on specific operators with two loops formed byleptons $L$ of the same type as the external one, the other two formed byleptons $\ell$ different from $L$. The approach is essentially based on theemployment of the Mellin-Barnes representation of the $x$-parametrization ofthe corresponding Feynman diagrams. This allows one to obtain, for the firsttime, exact analytical expressions for the radiative corrections of the tenthorder w.r.t. the electromagnetic coupling constant $e$. Analytically, theradiative corrections are expressed in terms of the ratio $r=m_\ell/m_L$ of theinternal to external lepton masses. The dependence on $r$ is investigatednumerically in the whole interval of $r$, $0 < r < \infty$. To make comparisonswith earlier published results possible, our exact analytical expressions areexpanded about $r=0$ and $r \to \infty$ and compared with the correspondingasymptotic expansions known in the literature.

Analytical calculations of the tenth order QED radiative corrections to lepton anomalies within the Mellin-Barnes representation