I have a basic question about spontaneous symmetry breaking for theory with one complex scalar field $\phi= \phi_1 +i \phi_2$ with Lagrangian
$$ \mathcal{L}=\partial _\mu \phi ^*\partial ^\mu \phi +\mu ^2\phi ^*\phi - \frac{\lambda}{2}(\phi ^*\phi )^2 $$
with potential $V(\phi )=-\mu ^2|\phi |^2+\frac{\lambda}{2}|\phi|^4$. Standard analysis of this potential gives that the set of minima of this potential is circle parametrized by $\phi_{\theta} =\frac{\mu}{\sqrt{\lambda}}e^{i\theta}$ for $\theta \in \mathbb{R}$. If we continue the discussion as in every standard paper / script (eg. this and this) dealing with this ussue we can write the complex field $\phi$ as
$$ \phi = (\phi_0 + \eta) \cdot e^{i \zeta} $$
where $\phi_0$ is an arbitrary choosen minimum of $V$ and $\eta$ and $\zeta$ are two real fields, while $\eta$ is the massive field and the $\zeta$ massless field (the "Goldstone boson").
Let's talk about the vacuum expectation values (VEV). We have $\langle 0|\phi |0\rangle \neq 0 $ but $\langle 0|\eta |0\rangle = \langle 0|\zeta |0\rangle =0$ for the 'new' field and my question is just what is the importance of VEV in the maschinery of spontaneous symmetry breaking? What is physically so striking that the former field has nonzero VEV, while the vev of the two new fields is zero?
The only point which seems reasonable dealing with vev in this setting is that accurately using perturbation theory it is often a neccessary assumption that the field has vev zero. Here we obtained two new field with this property, so from this point of view this spontaneous symmetry breaking seems to provide a transformation making the theory better approachable by perturbation theory techniques.
Now the question is if this is the only and the main reason why the vev is claimed to be so important for considerations on the spontaneous symmetry breaking or are there more reasons?
In addition I saw in some books on this topic remarks without deeper explanations that fields with nonzero vev are is some sense 'not physical ' and 'bad behaved'. But I nowhere found explanations explaning what is going on this 'slogan'. In light of it it seems that this procedure gives us new fields which have zero vev, so these might be easier to treat in some sense and they might be 'more physical'. Could anybody explain to me what this sloppy (at least for me) slogan on fields with nonvanishing vev means?
