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

The renormalisation group flow of a Hermitian field theory is shown to have trajectories which lead to a non-Hermitian Parity-Time ($\mathcal{P}\mathcal{T}$) 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 $\mathcal{P}\mathcal{T}$-symmetric saddle point for $D=3$.