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edited Sep 26, 2021 at 19:38

Whenever a theory has a local gauge symmetry, it automatically also has a global symmetry. After all you can just look at the constant gauge transformations. So the answer to this part is yes, but the last comment (i.e. "and it has global symmetry $G_{SM}$") is redundant.
In theories with spontaneous symmetry breaking, we start with a Lagrangian that describes physics at a high energy scale and has a certain gauge symmetry and then discover that at lower energy scales the resulting effective field theory does no longer posses this symmetry. The low energy field theory really has a smaller gauge group (after fixing unitary gauge) so the answer is No.
What do you mean by "Goldstone mechanism"? There is the Goldstone theorem which basically says that massless Goldstone bosons arise when symmetries are spontaneously broken and the Higgs mechanism in which these Goldstone bosons are "eaten up" to give mass to the gauge bosons.

Whenever a theory has a local gauge symmetry, it automatically also has a global symmetry. After all you can just look at the constant gauge transformations. So the answer to this part is yes, but the last comment (i.e. "and it has global symmetry $G_{SM}$") is redundant.
In theories with spontaneous symmetry breaking, we start with a Lagrangian that describes physics at a high energy scale and has a certain gauge symmetry and then discover that at lower energy scales the resulting effective field theory does no longer posses this symmetry. The low energy field theory really has a smaller gauge group so the answer is No.
What do you mean by "Goldstone mechanism"? There is the Goldstone theorem which basically says that massless Goldstone bosons arise when symmetries are spontaneously broken and the Higgs mechanism in which these Goldstone bosons are "eaten up" to give mass to the gauge bosons.

Whenever a theory has a local gauge symmetry, it automatically also has a global symmetry. After all you can just look at the constant gauge transformations. So the answer to this part is yes, but the last comment (i.e. "and it has global symmetry $G_{SM}$") is redundant.
In theories with spontaneous symmetry breaking, we start with a Lagrangian that describes physics at a high energy scale and has a certain gauge symmetry and then discover that at lower energy scales the resulting effective field theory does no longer posses this symmetry. The low energy field theory really has a smaller gauge group (after fixing unitary gauge) so the answer is No.
What do you mean by "Goldstone mechanism"? There is the Goldstone theorem which basically says that massless Goldstone bosons arise when symmetries are spontaneously broken and the Higgs mechanism in which these Goldstone bosons are "eaten up" to give mass to the gauge bosons.

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created Sep 26, 2021 at 17:56

Whenever a theory has a local gauge symmetry, it automatically also has a global symmetry. After all you can just look at the constant gauge transformations. So the answer to this part is yes, but the last comment (i.e. "and it has global symmetry $G_{SM}$") is redundant.
In theories with spontaneous symmetry breaking, we start with a Lagrangian that describes physics at a high energy scale and has a certain gauge symmetry and then discover that at lower energy scales the resulting effective field theory does no longer posses this symmetry. The low energy field theory really has a smaller gauge group so the answer is No.
What do you mean by "Goldstone mechanism"? There is the Goldstone theorem which basically says that massless Goldstone bosons arise when symmetries are spontaneously broken and the Higgs mechanism in which these Goldstone bosons are "eaten up" to give mass to the gauge bosons.