It seems that there are two definitions of the Higgs mode; one primarily used in particle theory, and one used in condensed matter. In the former, we can consider a simple Lagrangian of the form
$$ \mathcal{L}=\partial_\mu \phi^* \partial^\mu \phi -\mu^2 |\phi|^2-\lambda |\phi|^4 $$
Imposing that the above remain invariant under a local gauge transformation
$$\phi\rightarrow e^{i\alpha(x)}\phi$$
We can then expand about the vacuum in terms of fields $\eta(x)$ and $\xi(x)$, with the former representing the component of the $\phi$ field perpendicular to the potential's minimum and the latter the $\phi$ component tangent to the minimum. A global symmetry will yield a mass for $\eta$ and not for $\xi$. However, given a local symmetry as written above, we can take a gauge such that the Lagrangian yields two massive particles, a vector gauge boson, and a massive scalar.
My confusion stems from the discussion of the Higgs amplitude mode in Pekker and Varma's paper on the subject. Namely, they (as well as many other condensed matter theorists) state that, given the order parameter
$$\psi(r,t)=|\psi(r,\,t)|e^{i\theta(r,\,t)}$$
That fluctuations of the phase $\theta$ are Goldstone modes and fluctuations of the amplitude are Higgs modes. Now, Pekker and Varma make the point that this is not "truly" a Higgs mode:
One might well ask when it is appropriate to call an amplitude mode in condensed matter physics a Higgs mode. Higgs was dealing with a locally gauge-invariant problem. Of the condensed matter problems discussed here, only the superconductors are locally gauge invariant, the cold bosons in a lattice are only globally gauge invariant, and the antiferromagnets are locally anisotropic. So it might be appropriate to refer to only the amplitude mode(s) in superconductors as Higgs mode(s).
However, the Higgs mechanism in high energy is primarily defined as emerging from "spontaneously broken" local gauge symmetries. We might have amplitude fluctuations (parameterized by $\eta$) for global symmetries, but this does not lead to a Higgs mechanism. Therefore, my question is how, exactly, is the Higgs "amplitude" mode the same thing as the "true" Higgs boson of particle theory? To me, it seems like fluctuations of the amplitude of the order parameter are just one small piece of the Higgs, while the condensed matter community adopts this as the fundamental definition. Would it be correct to say that, on the high-energy side, "pure" amplitude modes only emerge via "spontaneously broken" local gauge symmetries, and therefore any "pure" amplitude mode can be seen as an analogous Higgs mechanism in condensed matter, even in the absence of "breaking" a local gauge? Any clarification would be appreciated.
