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In Higgs physics of the Standard Model, there is only one Higgs. The Higgs belong to a Higgs doublet. After electroweak symmmetry breaking, there is only one remaining Higgs, and 3 Goldstone bosons.

My question is then, why do we call the Higgs doublet with the word "doublet" while there is only one Higgs ? I would think that it should be called like Higgs isospin doublet ? What is "doublet" used for : for the Higgs or for the isospin ?

If I would like to express a model with several Higgs, should I say "multiple Higgs", or "Higgs belonging to a multiplet", or "Higgs belonging to a Higgs multiplet" ?. I ask this, because if I use "belonging to a multiplet", it seems that it would be wrong because the case of only one Higgs would belong already to a doublet of Higgs.

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