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

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 $A_{\mu }$.

Varying $A_{\nu }$ gives the Maxwell equationsOn the vacuum manifold, choose unitary gauge so that $\varphi =v$. Then $\mathbf{J}=-2e^2{v}^2\mathbf{A}$ and the London equation isFor a static configuration, $\nabla \times \mathbf{B}=\mathbf{J}$ and $\nabla \cdot \mathbf{B}=0$, soThe magnetic field therefore decays exponentially inside the condensate. This is the Meissner effect, with penetration depth equal to the inverse gauge-boson mass.

Writing $\Phi =({\varphi }_1,{\varphi }_2{)}^T$, the scalar sector has a global $SU(2)$ flavour symmetry in addition to the gauged common $U(1)$ phase. A vacuum can be chosen as ${\Phi }_0=(0,v{)}^T$. A diagonal combination of the gauge phase and the $SU(2)$ rotation generated by ${\sigma }_3$ leaves it invariant.

The radial scalar has $m_h^2=2\lambda v^2$, the gauge boson has $m_A^2=2e^2{v}^2$, and two real scalars remain massless. The four real scalar directions consist of one radial mode and three angular modes; one angular mode is eaten, leaving the two Goldstone bosons expected from the two broken physical global generators. The vacuum set before quotienting is $S^3$, and the manifold of gauge-inequivalent vacua is the complex projective line