Can negative bare couplings make sense? The $\vecφ^4$ theory at large $N$

Abstract: Scalar $\lambda\phi^4$ theory in 3+1D, for a positive coupling constant $\lambda>0$, is known to have no interacting continuum limit, which is referred to as quantum triviality. However, it has been recently argued that the theory in 3+1D with an $N$-component scalar $\vec{\phi}$ and a $(\vec{\phi}\cdot\vec{\phi})^{\,2}=\vec{\phi}^{\,4}$ interaction term does have an interacting continuum limit at large $N$. It has been suggested that this continuum limit has a negative (bare) coupling constant and exhibits asymptotic freedom, similar to the $\mathcal{P}\mathcal{T}$-symmetric $-g\phi^4$ field theory. In this paper I study the $\vec{\phi}^{\,4}$ theory in 3+1D at large $N$ with a negative coupling constant $-g<0$, and with the scalar field taking values in a $\mathcal{P}\mathcal{T}$-symmetric complex domain. The theory is non-trivial, has asymptotic freedom, and has a Landau pole in the IR, and I demonstrate that the thermal partition function matches that of the positive-coupling $\lambda>0$ theory when the Landau poles of the two theories (in the $\lambda>0$ case a pole in the UV) are identified with one another. Thus the $\vec{\phi}^{\,4}$ theory at large $N$ appears to have a negative bare coupling constant; the coupling only becomes positive in the IR, which in the context of other $\mathcal{P}\mathcal{T}$-symmetric and large-$N$ quantum field theories I argue is perfectly acceptable.