  The renormalisation group flow of a Hermitian field theory is shown to have
trajectories which lead to a non-Hermitian Parity-Time ($\mathcal{PT}$)
symmetric field theory for an axion coupled to a fermion in spacetime
dimensions $D=4-\epsilon$, where $\epsilon >0 $. In this renormalisable field
theory, the Dirac fermion field has a Yukawa coupling $g$ to a pseudoscalar
(axion) field and there is quartic pseudoscalar self-coupling $u$. The
robustness of this finding is established by considering flows between
$\epsilon$ dpependent Wilson-Fisher fixed points and also by working to
\emph{three loops} in the Yukawa coupling and to \emph{two loops} in the
quartic scalar coupling. The flows in the neighbourhood of the non-trivial
fixed points are calculated using perturbative analysis, together with the
$\epsilon$ expansion. The global flow pattern indicates flows from positive $u$
to negative $u$; there are no flows between real and imaginary $g$. Using
summation techniques we demonstrate a possible non-perturbative
$\mathcal{PT}$-symmetric saddle point for $D=3$.
