Can negative bare couplings make sense? The $\vec{\phi}^4$ 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. The spirit of renormalization is that observables do not depend on the renormalization scale. Here we see even if the coupling is taken negative above the scale of the Landau pole, thermodynamic observables are unaffected. 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.