Dynamical mass generation in QED with magnetic fields: arbitrary field strength and coupling constant

We study the dynamical generation of masses for fundamental fermions in quenched quantum electrodynamics, in the presence of magnetic fields of arbitrary strength, by solving the Schwinger-Dyson equation (SDE) for the fermion self-energy in the rainbow approximation. We employ the Ritus eigenfunction formalism which provides a neat solution to the technical problem of summing over all Landau levels. It is well known that magnetic fields catalyze the generation of fermion mass m for arbitrarily small values of electromagnetic coupling \alpha. For intense fields it is also well known that m \propto \sqrt eB. Our approach allows us to span all regimes of parameters \alpha and eB. We find that m \propto \sqrt eB provided \alpha is small. However, when \alpha increases beyond the critical value \alpha_c which marks the onslaught of dynamical fermion masses in vacuum, we find m \propto \Lambda, the cut-off required to regularize the ultraviolet divergences. Our method permits us to verify the results available in literature for the limiting cases of eB and \alpha. We also point out the relevance of our work for possible physical applications.