Calculability conditions are discussed for local gauge theories with Higgs-type symmetry breaking. We focus on the naturalness of $\mu e$ universality, the naturalness of the Cabibbo angle $\theta$, the naturalness of $\mathrm{CP}$-violating phases, and the naturalness of the nonleptonic $\Delta I=\frac{1}{2}$ rule. In this context we examine many published gauge models and construct others to illuminate the questions at hand. We note that naturalness of $\mu e$ universality for charged currents does not necessarily imply universality for neutral currents (natural "restricted" universality), and we emphasize the need for ${\nu }_e$-beam experiments. For SU(2) × U(1) and SU(2) × U(1) × U(1) we give first examples of how a nontrivial natural $\theta$ can appear. Models with $\mathrm{CP}$ violation are classified as to whether their $\mathrm{CP}$-violating phases are natural or not. For O(4) × U(1) we give a first example in which all the above naturalness criteria can be implemented. Here the natural $\mu e$ universality is necessarily restricted. The principal tool used in these investigations is the strict renormalizability relative to a gauge group enlarged by discrete symmetries, and the union of representations reducible under the gauge group to irreducible ones under the enlarged group. To implement this program, it is sometimes necessary to introduce Higgs couplings involving right-handed neutrinos; here the zero neutrino mass is associated with a discrete symmetry which remains unbroken upon spontaneous breakdown. We also find that strict renormalizability can lead to mass relations between fermions. In O(4) × U(1) models, such mass relations as well as right-handed neutrinos are necessary ingredients. Furthermore, for these models the spontaneity of $\mathrm{CP}$ violation acquired an operational significance, namely, as a discrete symmetry necessary (but not sufficient) to give a $\mathrm{CP}$-violating phase a natural value (90°). While the models we discuss are rather cumbersome, particularly due to the complexity of the symmetry-breaking mechanism, we expect that the tools we have developed may well have wider applicability.

• Received 7 February 1974

DOI:https://doi.org/10.1103/PhysRevD.10.539