Scalar field theory in 3+1 dimensions

We study the existence of a stable ground state for the most general renormalizable single scalar field theory in four dimensions within a variational approach. In its regularized version we find a theory with an energy density which is not bound from below but with a metastable local minimum such that when the cutoff is removed the theory is interacting, finite, and possesses a stable ground state. In fact, we find that this theory is not stable unless it is symmetric. This generalizes Stevenson’s recent results on even ${\phi }^4$ theory.