Past exam of the advanced quantum field theory for particle physics course of the University of Oxford / 2025 / 3 / c / Solution

Unitary gauge fixes the scalar phase to zero. Locally choose $\alpha =-\pi /f$ in part b; then ${\pi }^{\prime }=0$ and $A_{\mu }^{\prime }=A_{\mu }+{\partial }_{\mu }\pi /(ef)$. The remaining fields are the radial Higgs mode and a massive vector. Its quadratic Lagrangian density is $-F^2/4+m_A^2{A}^2/2$ and its gauge-boson propagator isFor $k^2\ne 0$, introduce the transverse and longitudinal orthogonal projectors $P_{\mu \nu }^T=g_{\mu \nu }-k_{\mu }{k}_{\nu }/k^2$ and $P_{\mu \nu }^L=k_{\mu }{k}_{\nu }/k^2$. Omitting the infinitesimal in this algebraic decomposition,At large momentum in Euclidean space, the transverse coefficient falls as $k^{-2}$ but the longitudinal coefficient approaches a constant of order $m_A^{-2}$. Thus power counting applied directly to unitary gauge Feynman diagrams suggests increasingly severe ultraviolet divergences. This hides the renormalizability of physical scattering amplitudes; it does not disprove it. A finite-$\xi$ R-xi gauge exposes the better propagator scaling while preserving the fields needed to track the cancellations.