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The renormalisation group flow of a Hermitian field theory is shown to havetrajectories which lead to a non-Hermitian Parity-Time ($\mathcal{PT}$)symmetric field theory for an axion coupled to a fermion in spacetimedimensions $D=4-\epsilon$, where $\epsilon >0 $. In this renormalisable fieldtheory, the Dirac fermion field has a Yukawa coupling $g$ to a pseudoscalar(axion) field and there is quartic pseudoscalar self-coupling $u$. Therobustness 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 thequartic scalar coupling. The flows in the neighbourhood of the non-trivialfixed 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 $g$. Usingsummation techniques we demonstrate a possible non-perturbative$\mathcal{PT}$-symmetric saddle point for $D=3$.

physical review. d/physical review. d.

Renormalisation group flows connecting a $4-\epsilon$ dimensional  Hermitian field theory to a $\mathcal{PT}$-symmetric theory for a fermion  coupled to an axion