Low-energy effective theory, unitarity, and nondecoupling behavior in a model with heavy Higgs-triplet fields

We discuss the properties of a model incorporating both a scalar electroweak Higgs doublet and an electroweak Higgs triplet. We construct the low-energy effective theory for the light Higgs doublet in the limit of small (but nonzero) deviations in the ρ parameter from one, a limit in which the triplet states become heavy. For Δρ>0, perturbative unitarity of $WW$ scattering breaks down at a scale inversely proportional to the renormalized vacuum expectation value of the triplet field (or, equivalently, inversely proportional to the square root of Δρ). This result imposes an upper limit on the mass scale of the heavy triplet bosons in a perturbative theory; we show that this upper bound is consistent with dimensional analysis in the low-energy effective theory. Recent articles have shown that the triplet bosons do not decouple, in the sense that deviations in the ρ parameter from one do not necessarily vanish at one-loop in the limit of large triplet mass. We clarify that, despite the nondecoupling behavior of the Higgs triplet, this model does not violate the decoupling theorem since it incorporates a large dimensionful coupling. Nonetheless, we show that if the triplet-Higgs boson masses are of order the grand unified theory scale, perturbative consistency of the theory requires the (properly renormalized) Higgs-triplet vacuum expectation value to be so small as to be irrelevant for electroweak phenomenology.