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

This cannot be the sole symmetry-breaking sector of the Standard Model. Its gauge-boson mass matrix has two zero eigenvalues, so besides the electromagnetic photon it predicts an extra massless neutral gauge boson. Every linear combination of $W^3$ and $B$, including the combination normally called $Z$, is massless. The observed massive Z boson is a concrete contradiction. Choosing $g,g^{\prime },m,\lambda$ differently cannot change these two zero eigenvalues.

A real scalar triplet can occur as an additional sector, but a further vacuum expectation value, for example that of a Higgs doublet, is then necessary to obtain the observed neutral gauge boson mass pattern. This does not rescue the triplet-only model posed here.