High precise measurements of vector and axial-vector spectral functions in hadronic decays of the lepton stimulate the further theoretical researches of various forms of perturbative expansions and nonperturbative effects. Advantages and self-consistency of the the Shirkov-Solovtsov analytic approach in describing the hadronic decays the lepton are demonstrated.

Analytical expressions for the tenth order electromagnetic corrections to the lepton (L = e, µ and
τ ) anomaly aL 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 ℓ ≠ 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 = mℓ/mL. We investigate
the behaviour of the exact analytical expressions as r → 0 and r → ∞ and compare them with the
corresponding asymptotic expansions known in the literature.