Write $H=2^{-1/2}e^{-i\xi^aT^a}(0,v+h)^{\mathsf T}$. Differentiating the exponential to first order in the fields, or using its Maurer-Cartan form exactly, shows that the angular fields enter $D_\mu H$ through
$$
\partial_\mu\xi^aT^a+gW_\mu^aT^a+\frac{g'}2B_\mu
$$
with 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>,
$$
H=\frac1{\sqrt2}\begin{pmatrix}0\\v+h\end{pmatrix}.
$$
The three Goldstone modes have become the longitudinal polarizations of the three massive electroweak gauge bosons, which is the <Higgs mechanism>.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-305/2/b/solution