Instantons, analytic continuation, and PT -symmetric field theory

Ordinary Hermitian $\ensuremath{\lambda}{\ensuremath{\phi}}^{4}$ theory is known to exist in $d<4$ dimensions when $\ensuremath{\lambda}>0$. For negative values of the coupling, it has been suggested that a physical meaningful definition of the interacting theory can be given in terms of $\mathcal{P}T$-symmetric field theory. In this work, we critically reexamine the relation between analytically continued Hermitian field theory with quartic interaction and $\mathcal{P}T$-symmetric field theory, including $O(N)$ models. We find that in general $\mathcal{P}T$-symmetric field theory does not correspond to the analytic continuation of the Hermitian theory to negative coupling, except at high temperature where the instanton contribution present in the analytically continued theory can be neglected.