Higgs Mechanism & Mexican Hat Potential
Spontaneous symmetry breaking, order parameter, and Goldstone mode
Potential Parameters
μ² (mass parameter)
-1.0
λ (self-coupling)
1.0
Field angle φ
0°
⚡ Broken Symmetry Phase
Vacuum: |φ| = v = √(-μ²/λ)
Higgs mass: mH = √(2|μ²|)
Goldstone mode: massless
U(1) symmetry spontaneously broken
Animate Ball Rolling
V(φ) = μ²|φ|² + λ|φ|⁴
Mexican hat shape when μ² < 0:
• Higgs boson = radial oscillation
• Goldstone = angular (massless)
• In gauge theories: Goldstone eaten by gauge boson → massive vector (W, Z)