= Solution
A neutral <superfluid> can select a phase of its complex <order parameter>, breaking a global $U(1)$ symmetry. A spatially uniform change of that phase costs no <energy>. Slowly varying changes therefore produce a gapless <Goldstone mode>. For a dilute weakly repulsive <Bose gas>, the <Bogoliubov quasiparticle dispersion> is
$$
E_k=\sqrt{\epsilon_k(\epsilon_k+2g_Bn_0)},\qquad
\epsilon_k=\frac{\hbar^2k^2}{2m_B},
$$
so $E_k\simeq\hbar c_sk$ at long wavelengths, with $c_s=\sqrt{g_Bn_0/m_B}$. Density and phase fluctuations are canonically coupled components of this sound branch, rather than two independent positive-frequency modes. Superfluid helium and dilute atomic condensates supply neutral examples, although the dilute-gas formula is not quantitatively valid for strongly interacting liquid helium.
In a <superconductor>, the pair field is charged and its phase couples to the electromagnetic <gauge field>. Only combinations such as $\nabla\theta-q\mathbf A/\hbar$ are physically meaningful. The <Anderson-Higgs mechanism in a superconductor> produces electromagnetic screening and removes the neutral sound mode as an independent gapless charged-density oscillation. In a homogeneous three-dimensional continuum, long-range Coulomb <forces> give a longitudinal plasma gap, approximately $\omega^2=\omega_p^2+c_s^2k^2$ at small $k$. The static transverse response gives the <Meissner effect> and
$$
\lambda_L^{-2}=\mu_0\frac{n_se^2}{m},
$$
when $n_s$ counts superconducting <electrons> of effective <mass> $m$. Counting pairs instead uses density $n_s/2$, charge $-2e$ and <mass> $2m$, giving the same stiffness. The condensate amplitude may also have a gapped <Higgs mode>, but that amplitude oscillation is distinct from the electromagnetic Higgs mechanism. Local gauge transformations remain a redundancy; this is not spontaneous breaking of an observable local symmetry in the same sense as the neutral global phase symmetry.
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