Singular properties of QED vacuum response to applied quasi-constant electromagnetic fields

Abstract: Employing the Bogoliubov coefficient summation method and introducing the gyromagnetic ratio $g\neq 2$ we derive an explicit functional form of $\mathfrak{Im}V^\mathrm{EHS}_g$, the imaginary part of Euler-Heisenberg-Schwinger (EHS) type effective action. We show that $\mathfrak{Im}V^\mathrm{EHS}_g$ is periodic in $g$ for any (quasi-)constant electromagnetic field configuration, and equal to the imaginary part obtained using a periodic in $g$ Ramanujan integrand in the proper time representation of $V^\mathrm{EHS}_g$. This validates the Ramanujan representation of $V^\mathrm{EHS}_g$ for both real and imaginary parts and allows writing the effective action in a suitably modified Schwinger proper time format. As a function of the ratio $b/a$ between ${\mathcal{B}} \to b$ and ${\mathcal{E}}\to a$ covariant generalizations of EM fields, we explore the singular properties of $\mathfrak{Im}V^\mathrm{EHS}_g$ at $g=2\pm 4k, k=0,\pm1,\pm2\ldots$ involving the pseudoscalar $ab\equiv \vec{\mathcal{E}}\cdot\vec{\mathcal{B}}$ in perturbative and nonperturbative behavior. We study the $e^-e^+$-decay vacuum instability, incorporating the physical value of $g-2$ vertex diagrams when summing infinite irreducible loops. We obtain an effective expansion parameter $\chi_b=\alpha b/2a$ ($\alpha=e^2/4\pi$), characterizing the onset of nonperturbative in $g-2$ suppression of vacuum instability. We demonstrate the $\chi_b$ domains for which perturbative expansion in $\alpha$ breaks down: The EM vacuum subject to critical electric field strength is stabilized in magnetic-dominated \lq magnetar\rq\ environments. Considering separately the case of ${\mathcal{E}}$ and ${\mathcal{B}}$ fields, we generalize to all $g$ the temperature representation of the $V^\mathrm{EHS}_g$ effective action.