  We consider the renormalisation properties of non-Hermitian Yukawa theories
involving a pseudoscalar (axion) field at or near $4$ dimensions. The
non-Hermiticity is \cPT-symmetric where $\mathcal P$ is a linear idempotent
operator (such as parity) and $\mathcal T$ is an anti-linear idempotent
operator (such as time-reversal). The coupling constants of the Yukawa and
quartic scalar coupling terms reflect this non-Hermiticity. The path integral
representing the field theory is used to discuss the Feynman rules associated
with the field theory. The fixed point structure associated with the
renormalisation group has \cPT- symmetric and Hermitian fixed points. At two
loops in the massless theory, we demonstrate the flow from Hermitian to
non-Hermitian fixed points. From the one-loop renormalisation of a massive
Yukawa theory, a self-consistent Nambu-Jona Lasinio gap equation is established
and its real solutions are discussed.
