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

## TL;DR

This paper demonstrates that renormalisation group flows in a Hermitian fermion-axion field theory can lead to non-Hermitian, $ ext{PT}$-symmetric theories in $4- ext{epsilon}$ dimensions, with detailed perturbative analysis up to three loops.

## Contribution

It shows the existence of $ ext{PT}$-symmetric non-Hermitian theories emerging from Hermitian theories via RG flows in $4- ext{epsilon}$ dimensions, including multi-loop calculations and non-perturbative considerations.

## Key findings

- RG flows connect Hermitian and $ ext{PT}$-symmetric theories.
- Flows from positive to negative pseudoscalar self-coupling $u$.
- Possible non-perturbative $ ext{PT}$-symmetric saddle point in 3D.

## Abstract

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 $D=3$.