outlook in Sec. \[seq:Con+Out\].

Calculation {#sec:calculation}
===========

Starting point of our calculation is the one-loop effective Lagrangian in constant external electromagnetic fields (“Heisenberg-Euler effective Lagrangian”) [@Heisenberg:1935qt]. It can be compactly represented as [@Schwinger:1951nm] (cf. also [@Dittrich:2000zu; @Jentschura:2001qr]), $${\cal L}=\frac{e^2}{8\pi^2}\int_{-i\eta}^{\infty-i\eta}\frac{{\rm d}s}{s}\,{\rm e}^{-i(m^2-i\epsilon)s}\left[ab\coth(eas)\cot(ebs)-\frac{a^2-b^2}{3}
 -\frac{1}{(es)^2}\right], \label{eq:effL}$$ with $\{\epsilon,\eta\}\to0^+$, elementary charge $e$ and electron mass $m$. The secular invariants $$a=(\sqrt{{\cal F}^2+{\cal G}^2}-{\cal F})^{1/2}\,, \quad b=(\sqrt{{\cal F}^2+{\cal G}^2}+{\cal F})^{1/2}\,,$$ are made up of the gauge and Lorentz invariants of the electromagnetic field, $${\cal F}=\frac{1}{4}F_{\mu\nu}F^{\mu\nu}=\frac{1}{2}(\vec{B}^2-\vec{E}^2)\,, \quad {\cal G}=\frac{1}{4}F_{\mu\nu}{}^*F^{\mu\nu}=-\vec{E}\cdot\vec{B}\,. \label{eq:invs}$$ Here $^*F^{\mu\nu}=\frac{1}{2}\epsilon^{\mu\nu\alpha\beta}F_{\alpha\beta}$ denotes the dual field strength tensor, and $\epsilon^{\mu\nu\alpha\beta}$ is the totally antisymmetric tensor; $\epsilon^{0123}=1$. Our metric convention is $g_{\mu \nu}=\mathrm{diag}(-1,+1,+1,+1)$, and we use $c=\hbar=1$. For completeness note that $ab=\sqrt{{\cal G}^2}$, $a^2-b^2=-2{\cal F}$ and $a^2+b^2=2\sqrt{{\cal F}^2+{\cal G}^2}$.

Strictly speaking, the Heisenberg-Euler Lagrangian  describes the effective nonlinear interactions between constant electromagnetic fields mediated by electron-positron fluctuations in the vacuum. The typical spatial (temporal) extents to be probed by these fluctuations are of the order of the Compton wavelength (time) $\lambda_c=\tau_c=1/m$, with $\lambda_c=3.86\cdot10^{-13}{\rm m}$ and $\tau_c=1.29\cdot10^{-21}{\rm s}$. Hence, [Eq. ]{} can also be adopted for inhomogeneous electromagnetic fields whose typical spatial (temporal) variation is on scales much larger than the Compton wavelength (time), i.e., for [*soft*]{} electromagnetic fields 