that may locally be approximated by a constant. Many electromagnetic fields available in the laboratory are compatible with this requirement. Within the above restrictions, [Eq. ]{} can serve as a starting point to study the effective interaction between dynamical photons and inhomogeneous background electromagnetic fields.

For this purpose it is convenient to decompose the electromagnetic field strength tensor $F^{\mu\nu}$ introduced above as $F^{\mu\nu}\to F^{\mu\nu}(x)+f^{\mu\nu}(x)$ into the field strength tensor of the background field $F^{\mu\nu}(x)$ and the photon field strength tensor $f^{\mu\nu}(x)$ [@BialynickaBirula:1970vy]. To linear order in $f\equiv f^{\mu\nu}$, the Lagrangian can then be compactly written as $${\cal L}=f^{\mu\nu}(x)\frac{\partial{\cal L}}{\partial F^{\mu\nu}}(x)+{\cal O}(f^2)\,. \label{eq:Seff}$$ Here we neglected higher-order terms with two or more photons.

Equation  is straightforwardly differentiated with respect to $F^{\mu\nu}$, yielding $$\frac{\partial{\cal L}}{\partial F^{\mu\nu}}
 =\frac{1}{2}\frac{1}{a^2+b^2}\left[\left(b\frac{\partial{\cal L}}{\partial b}-a\frac{\partial{\cal L}}{\partial a}\right)F_{\mu\nu}
 +{\cal G}\left(\frac{1}{b}\frac{\partial{\cal L}}{\partial b}+\frac{1}{a}\frac{\partial{\cal L}}{\partial a}\right){}^*F_{\mu\nu}\right].$$ In particular at leading order in a double expansion of the integrand in [Eq. ]{} in terms of $a$ and $b$ the propertime integral can be performed easily, resulting in $${\cal L}=\frac{e^2}{8\pi^2}\frac{1}{45}\frac{e^2}{m^4}\Bigl[(a^2+b^2)^2+3(ab)^2+ {\cal O}(\varepsilon^6)\Bigr], \label{eq:effLpert}$$ with ${\cal O}(a)\sim{\cal O}(b)\sim{\cal O}(\varepsilon)$; cf. also [@Dunne:2004nc] providing the weak field expansion coefficients of the Heisenberg-Euler effective Lagrangian explicitly to all orders. From [Eq. ]{} we obtain the compact 