Spontaneous Symmetry Breaking & Goldstone Theorem

Mexican hat potential V(φ) = -μ²|φ|² + λ|φ|⁴ · Massive radial (Higgs) mode · Massless angular (Goldstone) mode

μ² (mass²): 1.0 λ (coupling): 0.5 Field point φ: 0.3

Goldstone theorem: When a continuous global symmetry is spontaneously broken, there appears one massless scalar boson (Goldstone boson) per broken generator. V(φ) = -μ²|φ|² + λ|φ|⁴ has U(1) symmetry. Minimum at |φ|=v=√(μ²/2λ). Radial fluctuations: mass² = 4μ² (Higgs mode). Angular fluctuations: mass² = 0 (Goldstone mode, flat direction). Dispersion: ω²=k²c² (Goldstone, acoustic), ω²=m²+k²c² (massive).