Renormalization group domains of the scalar Hamiltonian . ∗

Using the local potential approximation of the exact renormalization group (RG) equation , we show the various domains of values of the parameters of the O(1)-symmetric scalar hamiltonian. In three dimensions, in addition to the usual critical surface S c (attraction domain of the Wilson-Fisher fixed point), we explicitly show the existence of a first-order phase transition domain S f separated from S c by the tricritical surface S t (attraction domain of the Gaussian fixed point). S f and S c are two distinct domains of repulsion for the Gaussian fixed point, but S f is not the basin of attraction of a fixed point. S f is characterized by an endless renormalized trajectory lying entirely in the domain of negative values of the ϕ 4-coupling. This renormalized trajectory exists also in four dimensions making the Gaussian fixed point ultraviolet stable (and the ϕ 4 4 renormalized field theory asymptotically free but with a wrong sign of the perfect action). We also show that very retarded classical-to-Ising crossover may exist in three dimensions (in fact below four dimensions). This could be an explanation of the unexpected classical critical behavior observed in some ionic systems.