  A nonperturbative Schwinger-Dyson analysis of mass generation is presented
for a non-Hermitian PT-symmetric field theory in four dimensions of an axion
coupled to a Dirac fermion.The model is motivated by phenomenological
considerations.The axion has a quartic self-coupling $\lambda$ and a Yukawa
coupling $g$ to the fermion. The Schwinger-Dyson equations are derived for the
model with generic couplings. In the non-Hermitian case there is an additional
nonperturbative contribution to the scalar mass. In a simplified rainbow
analysis the solutions for the SD equations, are given for different regimes of
the couplings $g$ and $\lambda$.
