Comment on the cosmological constant for $\lambda \phi^4$ theory in $d$ spacetime dimensions

In a recent article we showed that the analog of the cosmological constant in two spacetime dimensions for a wide variety of integrable quantum field theories has the form $\rho_{\rm vac} = - m^2 /2 \mathfrak{g}$ where $m$ is a physical mass and $\mathfrak{g}$ is a generalized coupling, where in the free field limit $\mathfrak{g} \to 0$, $\rho_{\rm vac}$ diverges. We speculated that in four spacetime dimensions $\rho_{\rm vac}$ takes a similar form $\rho_{\rm vac} = - m^4/2 \mathfrak{g}$, but did not support this idea in any specific model. In this article we study this problem for $\lambda \phi^4$ theory in $d$ spacetime dimensions. We show how to obtain the exact $\rho_{\rm vac}$ for the sinh-Gordon theory in the weak coupling limit by using a saddle point approximation. This calculation indicates that the cosmological constant can be well-defined, positive or negative, without spontaneous symmetry breaking. We also show that $\rho_{\rm vac}$ satisfies a Callan-Symanzik type of renormalization group equation. For the most interesting case physically, $\rho_{\rm vac}$ is positive and can arise from a marginally relevant negative coupling $\mathfrak{g}$ and the cosmological constant flows to zero at low energies.