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The most general renormalizable interaction between the Higgs sector and a new gauge-singlet scalar $S$ is governed by two interaction terms: cubic and quartic.
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D. Curtin et al., Exotic decays of the 125 GeV Higgs boson , Phys. Rev. D90
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I. Brivio and M. Trott, The Standard Model as an Effective Field Theory , Phys. Rept. 793
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