Custodial symmetry, Georgi-Machacek model, and other scalar extensions

In an SU(2) gauge theory, if the gauge bosons turn out to be degenerate after spontaneous symmetry breaking, obviously these mass terms are invariant under a global SU(2) symmetry that is unbroken. The pure gauge terms are also invariant under this symmetry. This symmetry is called the {\em custodial symmetry} (CS). In $\rm SU(2)\times U(1)$ gauge theories, CS implies a mass relation between the $W$ and the $Z$ bosons. The Standard Model (SM), as well as various extensions of it in the scalar sector, possess such a symmetry. In this paper, we critically examine the notion of CS and show that there may be three different classes of CS, depending on the gauge couplings and self-couplings of the scalars. Among old models that preserve CS, we discuss the Two-Higgs Doublet Model and the one doublet plus two triplet model by Georgi and Machacek. We show that for two-triplet extensions, the Georgi-Machacek model is not the most general possibility with CS. Rather, we find, as the most general extension, a new model with more parameters and hence a richer phenomenology. Some of the consequences of this new model have also been discussed.