Large N dynamics in QED in a magnetic field

The expression for the dynamical mass of fermions in QED in a magnetic field is obtained for a large number of the fermion flavor N in the framework of $1/N$ expansion. The existence of a threshold value $N_{\mathrm{thr}},$ dividing the theories with essentially different dynamics, is established. For the number of flavors $N\ll N_{\mathrm{thr}},$ the dynamical mass is very sensitive to the value of the coupling constant ${\alpha }_b,$ related to the magnetic scale $\mu =\sqrt{|\mathrm{eB}|}.$ For N of the order of $N_{\mathrm{thr}}$ or larger, a dynamics similar to that in the Nambu–Jona-Lasinio model with a cutoff of the order of $\sqrt{|\mathrm{eB}|}$ and the dimensional coupling constant $G\sim 1/\left(N|\mathrm{eB}|\right)$ takes place. In this case, the value of the dynamical mass is essentially ${\alpha }_b$ independent (the dynamics with an infrared stable fixed point). The value of $N_{\mathrm{thr}}$ separates a weak coupling dynamics (with ${\tilde{\alpha }}_b\equiv N{\alpha }_b\ll 1)$ from a strong coupling one (with ${\tilde{\alpha }}_b\gtrsim 1)$ and is of the order of $1/{\alpha }_b.$