Wilsonian approach to the interaction ${\phi }^2(i\phi {)}^{\epsilon }$

We study the renormalisation of the non-Hermitian $\mathcal{P}\mathcal{T}$-symmetric scalar field theory with the interaction ${\phi }^2(i\phi {)}^{\epsilon }$ using the Wilsonian approach and without any expansion in $\epsilon$. Specifically, we solve the Wetterich equation in the local potential approximation, both in the ultraviolet regime and with the loop expansion. We calculate the scale-dependent effective potential and its infrared limit. The theory is found to be renormalisable at the one-loop level only for integer values of $\epsilon$, a result which is not yet established within the $\epsilon$-expansion. Particular attention is therefore paid to the two interesting cases $\epsilon =1,2$, and the one-loop beta functions for the coupling associated with the interaction $i{\phi }^3$ and $-{\phi }^4$ are computed. It is found that the $-{\phi }^4$ theory has asymptotic freedom in four-dimensional spacetime. Some general properties for the Euclidean partition function and $n$-point functions are also derived.