Past exam of the theoretical physics I course of the University of Cambridge / 2017 / 3 / c / Solution

Write $\varphi =(v+h/\sqrt{2})e^{i\vartheta }$. The covariant kinetic term containsA local U(1) gauge symmetry can remove $\vartheta$ in unitary gauge. The angular fluctuation no longer appears as an independent physical massless Goldstone boson; its degree of freedom supplies the longitudinal polarization of the massive vector field. At quadratic order,The factor of two follows from the convention that $v$ is the complex field's vacuum value. This Higgs mechanism changes a two-polarization massless electromagnetic field into a three-polarization massive Proca field; the remaining radial scalar keeps $m_h^2=2\lambda v^2$. Static fields in the homogeneous broken phase are screened on a scale of order $m_A^{-1}$ in these units. The gauge symmetry is a redundancy of the description; the physical statement is the changed spectrum and disappearance of an independent massless angular mode.