  We search for novel Lorentz- and CPT-violating field theories, beyond those
contained in the superficially renormalizable standard model extension. We find
a new class of scalar field self-interactions which are nonpolynomial in form,
involving arbitrarily high powers of the field. Many of these interactions
correspond to nontrivial asymptotically free theories. These theories are
stable if rotation invariance remains unbroken. These results indicate that
certain forms of Lorentz violation, if they exist, may naturally be quite
strong.
