  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.
