 Solution

Assume $\lambda >0$ so that the Higgs potential is bounded below, and work at tree level in four spacetime dimensions with metric $(+---)$. For ${\mu }^2>0$, the minimum is $\Phi =0$. The quadratic terms describe three massless gauge bosons and two complex scalars of mass $\sqrt{{\mu }^2}$, equivalently four real scalar degrees of freedom. These are the perturbative particle masses about this vacuum, without a claim about the nonperturbative asymptotic spectrum of an interacting non-Abelian theory.

Expanding the gauge covariant derivative with $T_j={\tau }_j/2$ gives the interaction termsIn the second term, the antisymmetric part of ${\tau }_j{\tau }_k$ vanishes against the symmetric gauge-field product. The corresponding Feynman diagram vertices have one vector and a scalar–antiscalar pair, two vectors and a scalar–antiscalar pair, or four scalar legs with two particles and two antiparticles. The first vertex carries a momentum difference from its derivative. The Pauli matrix in it determines which doublet components couple; the two-vector scalar vertex is proportional to ${\delta }_{jk}$. There is no cubic scalar interaction about the zero-field vacuum.