The gauge independence of the dynamical fermion mass generated through chiral symmetry breaking in QED in a strong, constant external magnetic field is critically examined. We show that the bare vertex approximation, in which the vertex corrections are ignored, is a consistent truncation of the Schwinger–Dyson equations in the lowest Landau level approximation. The dynamical fermion mass, obtained as the solution of the truncated Schwinger–Dyson equations evaluated on the fermion mass shell, is shown to be manifestly gauge independent. By establishing a direct correspondence between the truncated Schwinger–Dyson equations and the 2PI (two-particle-irreducible) effective action truncated at the lowest nontrivial order in the loop expansion as well as in the 1/Nf expansion (Nf is the number of fermion flavors), we argue that in a strong magnetic field the dynamical fermion mass can be reliably calculated in the bare vertex approximation.
