  A scalar field theory with 4-derivative kinetic terms and 4-derivative cubic
and quartic couplings is presented as a proxy for quantum quadratic gravity
(QQG). The scalar theory is renormalizable and asymptotically free and the
remaining key issue is unitarity, or more precisely positivity, just as it is
in QQG. We have extended calculations for the optical theorem and for a
differential cross section, both in the high energy limit, to show how
positivity constrains the theory. The results also show how it is that
differential cross sections can have good high energy behavior. Finally we use
the scalar theory to extend the Stuckelberg theory of a massive U(1) gauge
boson to a renormalizable theory of a self-interacting gauge boson.
