= Solution
{c}
The <Goldstone theorem> states that spontaneous breaking of a continuous internal <global symmetry in field theory> in a local relativistic <quantum field theory> produces a zero-<mass> <Goldstone boson> for each independent broken generator, under the usual positive-state-space and conserved-current hypotheses. For $G\to H$ there are $\dim G-\dim H$ symmetry-required <Goldstone bosons>. This count assumes a Lorentz-invariant <vacuum state>; nonrelativistic counting can differ.
At the classical level, invariance of the <scalar potential> gives $\partial_iV\,(T^a\phi)_i=0$. Differentiating at a stationary <vacuum expectation value> $\phi_0$ yields
$$
\boxed{(M^2)_{ij}(T^a\phi_0)_j=0,\qquad (M^2)_{ij}=\left.\partial_i\partial_jV\right|_{\phi_0}.}
$$
These are the <Goldstone directions in the scalar mass matrix>. Broken generators have $T^a\phi_0\ne0$, so their independent directions have zero quadratic <mass> with a canonical <kinetic term>. The theorem guarantees these modes; it does not exclude further accidental zero-<mass> modes. A local <gauge symmetry> is a redundancy and does not meet the physical global-symmetry hypothesis: the <Higgs mechanism> can instead supply longitudinal <particle polarizations> of massive <gauge bosons>.
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