Past exam of the advanced quantum field theory for particle physics course of the University of Oxford / 2021 / 3 / a / i

Put $r^2={\Phi }^T\Phi$. The potential has its minimum atChoose the vacuum expectation value ${\Phi }_0=(0,0,v{)}^T$. Since the weak hypercharge is zero, $Q=T^3$ on this real scalar triplet, and $T^3{\Phi }_0=0$. Neither $T^1$ nor $T^2$ annihilates ${\Phi }_0$. The stabilizer subgroup is consequentlyIn particular, spontaneous symmetry breaking has not reduced the two neutral gauge generators to a single one: $Y$ annihilates every component and remains unbroken independently of $T^3$.

The scalar mass matrix isWith canonical kinetic normalization $\frac{1}{2}(\partial h{)}^2$, the radial Higgs boson therefore hasThe two tangential zero eigenvalues are Goldstone modes, absorbed by the two massive gauge bosons through the Higgs mechanism.