Non-trivial continuum limit of a PHI44 model with negative coupling constant

We present a rigorous, non-perturbative construction of a non-trivial euclidean phi44 field theory with negative coupling constant, as a scaling limit of stable lattice theories. In the present paper the usual propagator is replaced by a hierarchical one for pedagogical reasons. The resulting model has similar renormalization problems as the standard phi44 theory. Our non-perturbative construction has the standard renormalized perturbation expansion as an asymptotic expansion, and is asymptotically free. The analysis can be extended to the conventional model with the use of a rigorous block-spin formulation to yield a non-trivial continuum theory with similar properties, but possibly violating reflection positivity in a non-perturbative way. (orig.)