Magnetic Catalysis in Quantum Electrodynamics

We derive the (Wilsonian) low energy effective Lagrangian for Quantum Electrodynamics under external constant magnetic field by integrating out all electrons except those in the lowest Landau level. We find the one-loop effective Lagrangian contains a marginal four-Fermi interaction with anomalous dimension, $\frac{(\ln 2{)}^2}{2{\pi }^2}{e}^4$. Renormalization group analysis shows that the four-Fermi interaction will break chiral symmetry in QED if the external magnetic field is extremely strong, $B>{10}^{42}$ gauss, or if the Landau gap, $\sqrt{\mathrm{\mid }eB\mathrm{\mid }}>6.5\times {10}^{10}$ GeV.