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If symmetry breaking, e.g. with a potential $V=-\mu^2\phi^2 + \lambda \phi^4 $ occurs at a certain energy scale, and I now evolve to another scale via the Callan-Symanzik equations, does that change the behavior of the symmetry breaking?

If I understood it right effective $\phi^6,\phi^8,...$ interactions will start to shoot up. Will I have to look at their coefficients too and consider symmetry breaking every time I change the renormalization scale?

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Contrary to what the language implies, spontaneous symmetry breaking in usual vacuum QFT is not something that "occurs at a scale". Rather, the symmetry breaking scale is the scale below one may not pretend that the symmetry is whole (and e.g. the gauge bosons massless), see this answer. Speaking intuitively, the symmetry breaking scale is the scale below which the terms violating the symmetry become non-neglegible.

There is another notion of symmetry breaking in the language of phase transition where it actually takes place, but there are no phases for a vacuum field theory - these you get only for a thermal theory with variable temperature.

The rewriting of the action of a spontaneously broken symmetry in terms of fluctuations around the VEV ($\phi_0 = \langle \phi_0 \rangle + \phi$) takes place "before" renormalization. Although the VEV of the field may in principle acquire corrections by renormalization like any other parameter in the Lagrangian, it is not the case that you renormalize the theory and then go through the procedure of symmetry breaking again. In fact, the corrections to the VEV may play an important role in the low energy effective theory. Furthermore, renormalization must respect the structure of the theory in the sense that the Ward-Takahashi identities of the (broken) gauge symmetry (called Slavnov-Taylor identities in the non-Abelian case) must be preserved. This imposes severe restrictions on the RG flow of the theory compared to non-symmetric theories.

For an explicit discussion of this and the example of a $\mathrm{U}(1)$-Higgs model, see "Spontaneous symmetry breaking with Wilson renormalization group" by Bonini and D'Attanasio.

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