Well, Goldstone's 1961 sombrero potential model amply illustrates the basics. Let me vulgarize them.
In O(2) language, cavalierly about normalizations,
he thinks of a complex scalar field theory, whose real and imaginary components resolve to a two real scalar d.o.f. system, with prototype potential
$$
\frac{\lambda}{4} (A^2 + B^2 +\epsilon v^2)^2 .
$$
The O(2) isorotation symmetry sends A to $A\cos \theta- B \sin \theta$; and B to the orthogonal combination. The conserved current is $J_\mu = A\partial _\mu B - B\partial_\mu A $, i.e. $\partial \cdot J=0$, and
$
i\theta [Q, A]=\delta A ~,
$
and likewise for B, so then
$$
\delta A =- \theta B, ~~~~\delta B=\theta A ~.
$$
Slide the parameter ε from 1 (generic for non vanishing positive) ; to 0; to -1 (generic for non vanishing negative), and monitor the qualitative fate of the two fields along these three cases.
For ε=1, A and B are twins. They have the same mass, $\sqrt{\lambda} v$, as the minimum of the potential is at $\langle A\rangle =\langle B\rangle = 0$, and so
$\langle \delta A\rangle= \langle \delta B\rangle= 0 $, that is, $Q|0\rangle=0$, so the vacuum is invariant under isorotation, $e^{i\theta Q}|0\rangle=|0\rangle$. The potential looks like so:
Decreasing ε to 0, the mass of A and B decreases to 0, but they remain twins, and their rotation into each other is still linear, and the vacuum is still symmetric—all the above relations (except for the vanished mass) are the same as above.
As soon as ε goes negative, something catastrophically, qualitatively different happens: SSB. Take ε to be -1 for simplicity. Now the quartic potential morphs into the iconic Goldstone sombrero, and the minimum is this entire flat circle on the A-B plane.
The symmetry slides you around that degenerate bottom (orbit) without resistance.
A choice is thus forced for the ground state: suppose you pick, arbitrarily, $\langle B\rangle=0$ and $\langle A\rangle =v$. Since you are interested in excitations around this vacuum, change to variables of convenience, $A\equiv v+h$, so h is the excitation around this vacuum with $\langle h\rangle=0$.
The quartic potential now expands to
$$
\frac{\lambda}{4} \left ( B^4+ h^4 +2h^2B^2 +4vh(B^2+h^2) +4v^2 h^2 \right).
$$
There is no mass term for B, but a hefty mass for h (formerly known as A ; you could take this mass to infinity, if inclined, by sending λ there), similar to the mass it used to have before the SSB.
- The crucial part is $\langle B \rangle= \langle h \rangle= \langle \delta A \rangle=0$ , but $\langle \delta B \rangle= \theta v$ , non-vanishing : the hallmark of the Goldstone boson, since $\delta B= \theta v+ \theta h$ ; call v the order parameter. This shift nonlinearity in the goldston's transform precludes the existence of a mass term for it, as that term would not be invariant under the symmetry, still all-powerful, but a bit hidden (whom are we fooling? This is dubbed the Nambu-Goldstone realization).
The current $J_\mu = v\partial_\mu B+ h\partial _\mu B - B\partial_\mu h $ , is, of course, still conserved, but, check that now $Q|0\rangle\neq 0 $ : the symmetry shifts the vacuum around the bottom of the sombrero, exciting sloshing Bs out of it—it taps the bowl of jello with a spoon. $|B(p=0)\rangle$ is degenerate with $|0\rangle$, as $\langle B(p)| J_\mu (x)|0\rangle\sim e^{ip\cdot x} v p_\mu$ .
Check that $[Q,H]=0$ , so $H (Q|0\rangle)= Q(H|0\rangle)= QE_0|0\rangle= E_0(Q|0\rangle)$ .
By contrast, oscillations of the massive h (the σ, or "Higgs") correspond to rolling up and down the walls of the valley of the sombrero, transversely to the valley axis.
- The takeaway: As ε slides from 1 to 0 to -1, the mass of the "higgs", A/h, decreases to 0 and then back to above its former value; by contrast, the mass of B decreases to 0, and stays there: but, suddenly, for ε<0, it morphs into a goldston.
