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We do the exact same thing in the standard model when we break the electroweak symmetry. There are 3 broken generators $\delta_-/2, \sigma_1/2, \sigma_2/2$, and a leftover preserved $U(1)_{EM}$ generator, $\delta_+/2$, where
$$
\delta_\pm = \frac{1}{2}(\mathbf{1}\pm \sigma_3)
$$
are linear combinations of the EW gauge group generators. You can verify that ...
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Just to augment @ChiralAnomaly's answer with some algebra, we can compute a squared mass$$\frac{\partial^2V}{\partial\phi\partial\phi^\ast}=4\lambda\phi^\ast\phi-\mu^2$$for each solution of $\frac{\partial V}{\partial\phi}=0$. If $\mu^2\ge0$, $|\phi|^2=\frac{\mu^2}{2\lambda}$; if $\mu^2<0$, $\phi=0$. In both cases$$\frac{\partial^2V}{\partial\phi\partial\...
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People often use the name "mass term" for a quadratic term in the lagrangian. That careless habit comes from free field theories, where it really is a mass term. But in general, the theory's physical predictions are determined by the whole theory, not just by one term in the lagrangian.
The mass of the Higgs particle is real and positive. It's not ...
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This is a verbose placeholder for an answer, as Weinberg himself in his QTFvII, Ch 19.5, p 195 et seq, beats the SU(2)×SU(2) σ-model of Gell-Mann and Levy to a pulp. For simplicity, you may eliminate the σ, and thus move on an O(4)/O(3) hypersphere parameterized by three projective coordinates, his Goldstone πs, or ζs, a manifold manifestly isospin ...
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