Models in which the Higgs sector is extended by a single electroweak scalar multiplet X can possess an accidental global U(1) symmetry at the renormalizable level if X has isospin T≥2. We show that all such models with an accidental U(1) symmetry are excluded by the interplay of the cosmological relic density of the lightest (neutral) component of X and its direct-detection cross section via Z exchange. The sole exception is the $T=2$ multiplet, whose lightest member decays on a few-day to few-year time scale via a Planck-suppressed dimension-5 operator.
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Earl et al. (2013) studied this question.
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