PT symmetric fermionic field theories with axions: Renormalization and dynamical mass generation

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.