An estimate is made of the effect of expanding about the full nucleon and meson propagators as well as the mean meson fields in certain model renormalizable relativistic field theories of nuclear matter. Meson masses are determined self-consistently by satisfying, at zero momentum only, the Schwinger-Dyson equations which result when the energy functional is truncated at the two-loop level. The tachyonic pion problem of the standard mean field expansion is eliminated but the opposite problem of a far too massive pion in normal nuclear matter arises. NUCLEAR STRUCTURE Relativistic mean field theories of nuclear matter, tachyonic pions. Approximate solutions of Schwinger-Dyson equations.