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To my understanding, the Higgs field condensed at the same time as the electroweak symmetry was broken. The fields' doublet components merged, while the singlet components became the Higgs boson and the photon.

Was this merging inevitable simply because the fields had a similar symmetry? This seems to be implied in the answer to the following question, quote below: Why some particles interact with the Higgs field and others don't?

The gauge invariance in fact dictates that the Higgs field has to interact with other fields, the gauge fields. Because the Higgs field carries no color charges like quarks, it's neutral under $SU(3)_{QCD}$ which means that it doesn't have any interactions with gluons.

However, the Higgs field has nonzero charges under the electroweak $SU(2)_W$ because the Higgs field is a doublet

To rephrase my question, is the electroweak charge a direct result of the symmetry group $SU(2)$, and not a specific property of the electroweak field itself?

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The Higgs field gives mass to the W and Z bosons which comprise the "weak force". If those bosons didn't have mass, they would act like photons, so the weak force would be like electromagnetism.

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