Strong fields and QED as function of the g-factor

Precision QED experiments (muon $g-2$ and Lamb shift) require understanding of QED with arbitrary gyromagnetic ratio $g>2$. We will first show that the need to have a renormalizable theory requires for $g>2$ reformulation in terms of Klein-Gordon-Pauli (KGP) equation. Using KGP, we obtain the nonperturbative effective action of QED within Schwinger proper time method in arbitrarily strong quasi-constant external electromagnetic fields as a function of $g$. The expression is divergent for $|g|>2$, given the magnetic instability of the vacuum due to the lowest Landau orbit eigenenergy having an indefinite value in strong magnetic fields. The spectrum of Landau eigenvalues for KGP in a magnetic field is an exact periodic function of $g$, no states are disappearing from the spectrum. This periodicity allows to establish a generalized form of the effective action valid for all $g$. We show the presence of a cusp at the periodic points $g=\ldots-6,-2,2,6\ldots$. Consequently, the QED beta function and parts of light-by-light scattering differ from perturbative computation near to $g=2$ and an asymptotically free domain of $g$ for QED arises. We further show that only for $g=(2N+1)$ there is exact correspondence of a field-dependent quasi-temperature and the Unruh Temperature.