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 $\delta =e^2\mathrm{/}(12\pi )$ for spinor QED, and ${\delta }_S=\delta \mathrm{/}4$ for scalar QED.