Understanding the Plasmon dispersion relation to first approximation

Plasmons are quantized oscillations of the charge density in solids, and are found in basically all conducting materials in nature.

As I am playing around with an electron microscope I refurbished, I revisited a very old problem: the dispersion relation of plasmons in materials.

For light in vacuum, the dispersion relation is required for propagation at $c$, and you have $omega=ck$ as a strict relation. For acoustic phonons, you similarly have a linear dispersion relation via Goldstones theorem (with the usual subtleties).

The plasmon however, is always gapped/massive, and I would assume then that Goldstone’s theorem and its extensions would have nothing to say about the plasmon dispersion relation.

It is still possible to estimate the dispersion relation within the self-consistent field approximation/RPA (basically valid in the large electron density limit). In this approximation you generally get a dispersion relation for the plasmon of the following form (Section 15 of Quantum Theory of Many-Particle Systems by Fetter and Walecka):

$$omega(q) = omega_0left(1+frac{9}{10}left(frac{q}{q_{mathrm{TF}}}right)^2 + O(q^4)right)=omega_0left(1+a q^2 + O(q^4)right)$$

Here $q_mathrm{TF}$ is the Thomas-Fermi wavelength, which is simply
$q_mathrm{TF} = sqrt{frac{6pi n e^2}{E_f}}$.
Where $n$ is the electron density, and $E_f$ is the Fermi energy, which in 3D is proportional to $n^{2/3}$. The plasmon frequency is given by $omega_0=sqrt{frac{4pi n e^2}{m}}$

We can rewrite this expression in terms of the Fermi energy and plasma frequency as well

$$omega(q)-omega_0 propto frac{E_F}{omega_0} q^2 propto n^{1/6}q^2$$

Although the calculation for this dispersion term is very simple and straightforward, I am struggling to understand why I would expect

1. To lowest order, $q^2$ dispersion for the plasmon
   and
2. A coefficient of the form $a=frac{omega_0}{q_{mathrm{TF}}^2}$

There are basically 3 contributions to calculating $1=V,mathrm{Re}left[Piright]$, which gives the plasmon energy: the Coulomb interaction ($q^{-2}$ in 3D), the phase space over which $Pi$ is integrated (a mix of $omega$ and $q$, and how the energy of electrons disperse $q^2$. Which ones are giving what contributions?

Naively if I think of the $q^2$ from coming due to purely kinetic energy contributions, you would expect the coefficient $apropto n$, but instead we have $apropto n^{1/6}$. Additionally, the classical solution (parallel plate capacitor essentially) predicts no dispersion, so the effect should be due to the existence of the Fermi surface and its dispersion.