Fixed-Point Structure of Scalar Fields

We search for alternatives to the trivial ${\phi }^4$ field theory by considering nonpolynomial potentials. Such theories are renormalizable when the natural cutoff dependences of the coupling constants are taken into account. We find a continuum of fixed points, which includes the well-known Gaussian fixed point. The fixed-point density has a maximum at a location corresponding to a theory with a Higgs boson mass of approximately 2700 GeV. The Gaussian fixed point is UV stable in some directions in the extended parameter space. Along such directions we obtain nontrivial asymptotically free theories.