Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 305 / 2 / b / Solution

Write $H=2^{-1/2}{e}^{-i{\xi }^a{T}^a}(0,v+h{)}^{\mathsf{T}}$. Differentiating the exponential to first order in the fields, or using its Maurer-Cartan form exactly, shows that the angular fields enter $D_{\mu }H$ throughwith higher-order commutator terms dictated by the same group-valued combination. A local $SU(2{)}_L$ transformation can set ${\xi }^a=0$. In this unitary gauge,The three Goldstone modes have become the longitudinal polarizations of the three massive electroweak gauge bosons, which is the Higgs mechanism.