Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 305 / 1 / i / Solution

Use unitary gauge and write $\varphi =v+h/\sqrt{2}$. The quadratic Lagrangian containsThus the radial Higgs excitation is a spin-zero particle with $m_h^2=2\lambda v^2$, while the gauge boson is a massive spin-one particle with $m_A^2=2e^2{v}^2$. The phase of $\varphi$ is the Goldstone boson eaten by the Higgs mechanism to supply the longitudinal polarization of $A_{\mu }$.