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

Employing the Bogoliubov coefficient summation method and introducing the gyromagnetic ratio $g\mathrm{\ne }2$ we derive an explicit functional form of $\mathfrak{I}\mathfrak{m}{V}_g^{\mathrm{E}\mathrm{H}\mathrm{S}}$, the imaginary part of Euler-Heisenberg-Schwinger (EHS) type effective action. We show that $\mathfrak{I}\mathfrak{m}{V}_g^{\mathrm{E}\mathrm{H}\mathrm{S}}$ 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 $V_g^{\mathrm{E}\mathrm{H}\mathrm{S}}$. This validates the Ramanujan representation of $V_g^{\mathrm{E}\mathrm{H}\mathrm{S}}$ 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 $b\mathrm{/}a$ between $\mathcal{B}\to b$ and $\mathcal{E}\to a$ covariant generalizations of EM fields, we explore the singular properties of $\mathfrak{I}\mathfrak{m}{V}_g^{\mathrm{E}\mathrm{H}\mathrm{S}}$ at $g=2\pm 4k,k=0,\pm 1,\pm 2\dots$ 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 ${\chi }_b=\alpha b\mathrm{/}2a$ ($\alpha =e^2\mathrm{/}4\pi$), 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 $V_g^{\mathrm{E}\mathrm{H}\mathrm{S}}$ effective action.