Atomic levels in superstrong magnetic fields and $D=2$ QED of massive electrons: screening

The photon polarization operator in superstrong magnetic fields induces the dynamical photon “mass” which leads to screening of Coulomb potential at small distances $z\ll1/m$, $m$ is the mass of an electron. We demonstrate that this behaviour is qualitatively different from the case of $D=2$ QED, where the same formula for a polarization operator leads to screening at large distances as well. Because of screening the ground state energy of the hydrogen atom at the magnetic fields $B\gg m^2/e^3$ has the finite value $E_0=-me^4/2\ln^2(1/e^6)$.