Mexican-hat potential, spontaneous symmetry breaking, and massless Goldstone bosons
The Mexican-hat potential V(φ) = μ²|φ|² + λ|φ|⁴ undergoes spontaneous symmetry breaking when μ² < 0. The vacuum manifold is a circle of degenerate minima at |φ| = v = √(−μ²/2λ). Goldstone's theorem (1961): for each broken continuous symmetry generator, there is a massless boson. Here U(1) has 1 generator → 1 massless Goldstone mode (angular fluctuations) and 1 massive Higgs mode (radial). External field h explicitly breaks symmetry, generating a pseudo-Goldstone mass m_π² ∝ h/v (Gell-Mann–Oakes–Renner relation in QCD).