Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 305 / 3 / b / iii

For ${\mu }^2>0$, the minima satisfy ${\Phi }^{†}\Phi =v^2/2$ with $v^2={\mu }^2/\lambda$. A global $SU(3)$ rotation can chooseThe transformations acting as $SU(2)$ on the first two entries leave this vector fixed, so the symmetry-breaking pattern isThere are $8-3=5$ broken generators. If the symmetry were global, the Goldstone theorem would therefore give five massless scalar modes, exactly the five tangent directions of the vacuum manifold $SU(3)/SU(2)$.