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Since you start with a group $G$ which is a symmetry of the theory (ie of Lagrangian) there will be some generators for $G$. Call them $T_1,\ldots,T_N$. In the original Lagrangian, (before the Higgs gets a vev) you can do a transformation with any of the $T_i$'s and the Lagrangian will not change. After the field gets a vev, you can try the same and you will ...

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From your comments, it sounds like you're looking for Deconfinement. At higher energies the quarks and antiquarks are no longer bound, they are asymptotically free, and thus chiral symmetry is restored.

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To study Higgs mechanism, we can use a Lagrangian of the form: \begin{align} \mathcal{L}=(D_{\mu}\phi)^2-\frac{1}{4} F_{\mu\nu}F^{\mu\nu}-V(|\phi|) \end{align} Where: \begin{align} V(|\phi|)&=-2v^2|\phi|^2+|\phi|^4 \\ &=(|\phi|^2-v^2)^2-v^4 \\ D_\mu \phi&=\partial_{\mu}\phi+\mathrm{i}e A_{\mu} \phi \\ F_{\mu\nu}&=\partial_{\mu} ...

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I don't think it's proper to say that the spontaneous breaking of anything is attributed to the Higgs mechanism. It's a postulate of the theory that the potential takes a certain shape, and that shape leads to a symmetry, and that the symmetry is spontaneously broken. The Higgs mechanism is a consequence of that. The symmetry that is broken is the local ...

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You will get a better answer than this but, assuming all you want is a rough analogy compared to your chair idea, here goes. For a local gauge symmetry analogy, forget the chair. Imagine you are building a house on rough surface, with the ground sloping all over the place. Obviously you want the floor to be level, no matter how uneven the ground is. So ...

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From the previous comments, I'm quite sure $W_\mu$ transforms with a phase-factor. $$W_\mu \rightarrow e^{i\theta}W_\mu$$ therefore mixing the charged components of the $W$ field.

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