Critical coupling constants for relativistic wave equations and vacuum breakdown in quantum electrodynamics

We study the strong-coupling limit of some relativistic wave equations describing bound states of oppositely charged fermions or bosons 1 and 2, of arbitrary mass. Using both numerical and analytic momentum-space methods we find the value ${\gamma }_{\max }$ of γ=-$e_1$ $e_2$/4π for which the lowest-lying bound state disappears from the spectrum, as well as the smaller value ${\gamma }_{\mathrm{dec}}$ for which 2 becomes unstable to the decay into the composite (1,2) system and the antiparticle 1¯. We also consider the limit $m_2$→∞ and discuss the connection of our results with the so-called breakdown of the vacuum in quantum electrodynamics for a sufficiently strong external field.