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I’m trying to understand the assumptions under which Goldstone theorem is valid. Is it just a Lagrangian that depends on scalar fields and has no gauge potentials? Or is it also true for more general Lagrangians?

Given a Lagragian $L$ which contains no gauge bosons and which is invariant under the group $G$ but the ground state is only invariant under the subgroup $H$, then there exists dim G- dim H Goldstone bosons (massless and spinless).

You also need Lorentz invariance. Check this out https://arxiv.org/pdf/1207.0457.pdf

It also doesn't have to be spinless. Its quantum numbers are the same as the conserved charge it is spontaneously breaking. One well known example with spin is Goldstinos.

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