Electron Self-Energy in Strong Magnetic Field: Summation of Double Logarithmic Terms

We study the electron self--energy in a strong magnetic field when the parameter \eta\equiv (\alpha/2\pi) \ln^2 (eB/m^2_0) \sim 1 and explore the transition between the perturbative regime \eta<<1 and the nonperturbative massless QED regime \eta>>1 where chiral symmetry is broken spontaneously and electrons acquire the dynamically induced mass.