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If one has a gauge theory with a specific symmetry group and we add to it a scalar field with a non-zero VEV, how do we know in general to which symmetry group will the original symmetry be spontaneously broken?

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It is broken to whichever subgroup of the original group that leaves the VEV unchanged.

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@mike stone 's answer is impeccable, but requires emphasizing that the answer therefore depends on the representation of the Higgs multiplet under that group.

For most of the groups used in physics, this was done in a legendary paper Group theory of the spontaneously broken gauge symmetries, Ling-Fong Li (1974), Phys Rev D9 1723, routinely used as a "phonebook" if you don't recall the standard answers.

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