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 ${\rho }_{\mathrm{v}\mathrm{a}\mathrm{c}}=-m^2\mathrm{/}2\mathfrak{g}$ where $m$ is a physical mass and $\mathfrak{g}$ is a generalized coupling, where in the free field limit $\mathfrak{g}\to 0$, ${\rho }_{\mathrm{v}\mathrm{a}\mathrm{c}}$ diverges. We speculated that in four spacetime dimensions ${\rho }_{\mathrm{v}\mathrm{a}\mathrm{c}}$ takes a similar form ${\rho }_{\mathrm{v}\mathrm{a}\mathrm{c}}=-m^4\mathrm{/}2\mathfrak{g}$, 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 ${\rho }_{\mathrm{v}\mathrm{a}\mathrm{c}}$ 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 ${\rho }_{\mathrm{v}\mathrm{a}\mathrm{c}}$ satisfies a Callan-Symanzik type of renormalization group equation. For the most interesting case physically, ${\rho }_{\mathrm{v}\mathrm{a}\mathrm{c}}$ is positive and can arise from a marginally relevant negative coupling $\mathfrak{g}$ and the cosmological constant flows to zero at low energies.