Radiative QED corrections to the lepton anomaly from four-loop vacuum polarization insertions with three identical leptons

Analytical expressions for the tenth order electromagnetic corrections to the lepton ($L=e,\mu$ and $\tau$) anomaly $a_L$ are derived explicitly for a class of Feynman diagrams with insertions of the vacuum polarization operator consisting of four closed lepton loops. We consider a particular case when one loop is formed by the lepton $L$ of the same kind as the one under consideration, the other three loops being formed by leptons $\mathrm{\ell }\mathrm{\ne }L$. The method is based on the consecutive application of dispersion relations for the polarization operator and the Mellin–Barnes transform for the propagators of massive particles. The result is expressed in terms of the mass ratio $r=m_{\mathrm{\ell }}\mathrm{/}{m}_L$. We investigate the behaviour of the exact analytical expressions as $r\to 0$ and $r\to \mathrm{\infty }$ and compare them with the corresponding asymptotic expansions known in the literature.