Behavior of the mass operator in a superstrong magnetic field: Summation of the perturbation theory diagrams

An expression is obtained for the radiative correction to the electron mass in a superstrong magnetic field in the approximation of large logarithms ($\ln\xi\gg 1$, $\xi=B/B_0$, $B_0=m^2/e=4,41\cdot 10^{13}$ Gs); the correction has the form $m\{\exp((\alpha/4\pi)\ln^2\xi)-1\}$. This result shows that the correction reaches values of the order of the mass only in anomalously large fields $B\sim 10^{15} B_0$.