RG analysis of magnetic catalysis in dynamical symmetry breaking

We perform the renormalization group analysis on dynamical symmetry breaking under a strong external magnetic field studied recently by Gusynin, Miransky, and Shovkovy. We find that any attractive four-Fermi interaction becomes strong at low energy, thus leading to dynamical symmetry breaking. When the four-Fermi interaction is absent, the $\beta$ function for the electromagnetic coupling vanishes in the leading order in $\frac{1}{N}$. By solving the Schwinger-Dyson equation for the fermion propagator, we show that in the $\frac{1}{N}$ expansion, for any electromagnetic coupling, dynamical symmetry breaking occurs due to the presence of a Landau energy gap by the external magnetic field.