  We search for alternatives to the trivial $\phi^4$ field theory by including
arbitrary powers of the self-coupling. Such theories are renormalizable when
the natural cutoff dependencies 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 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 non-trivial asymptotically free
theories.
