Chiral Symmetry Breaking

The Mexican hat potential V(σ,π) = λ(σ²+π²−v²)² governs chiral symmetry. At low temperature the quark condensate ⟨ψ̄ψ⟩ ≠ 0 breaks the symmetry; Nambu-Goldstone pions emerge.

Mexican Hat Potential + Order Parameter Trajectory

Parameters

V(σ,π) = λ(σ²+π²−v²)²
v² = v₀²(1 − T/T_c)
m_σ² = 8λv², m_π = 0

Order Parameter

v² (min radius²)1.000
⟨ψ̄ψ⟩ ~ v1.000
σ mass m_σ---
π mass m_π---
Phase---
Physics note:
QCD chiral transition T_c ≈ 150 MeV
Pions are pseudo-Goldstone bosons
(light because m_quark ≠ 0)
Nambu & Goldstone (1960, 1961)