Fractional Effective Action at strong electromagnetic fields

In 1936, Weisskopf showed that for vanishing electric or magnetic fields the strong-field behavior of the one loop Euler-Heisenberg effective Lagrangian of quantum electro dynamics (QED) is logarithmic. Here we generalize this result for different limits of the Lorentz invariants $\vec{E}^2-\vec{B}^2$ and $\vec{B}\cdot\vec{E}$. The logarithmic dependence can be interpreted as a lowest-order manifestation of an anomalous power behavior of the effective Lagrangian of QED, with critical exponents $\delta=e^2/(12\pi)$ for spinor QED, and $\delta_S=\delta/4$ for scalar QED.