Chiral Symmetry

Chirality as a quantum number

For a Dirac fermion ψ, define the chirality projectors P_L = (1 − γ⁵)/2 and P_R = (1 + γ⁵)/2. Then ψ_L = P_L ψ and ψ_R = P_R ψ are independent Weyl spinors. In the massless limit they decouple completely:

L_kinetic = i psi-bar gamma^mu d_mu psi
         = i psi_L-bar gamma^mu d_mu psi_L + i psi_R-bar gamma^mu d_mu psi_R

You can multiply ψ_L by one phase and ψ_R by a different phase. The theory does not care. That extra freedom is U(1)_L × U(1)_R chiral symmetry; with N_f flavors it enlarges to SU(N_f)_L × SU(N_f)_R. A Dirac mass term m ψ-bar ψ = m (ψ_L-bar ψ_R + ψ_R-bar ψ_L) explicitly mixes the two chiralities and breaks the symmetry.

Chiral symmetry of QCD

The up and down quark masses (2.2 and 4.7 MeV) are tiny compared to the QCD scale Λ_QCD ≈ 250 MeV. In the massless approximation QCD has:

Symmetry	Fate	Consequence
U(1)_V (baryon number)	Exact	Quarks survive in pairs of quarks (mesons) or triples (baryons)
SU(N_f)_V (isospin / flavor)	Exact in chiral limit	Approximate flavor multiplets: proton/neutron, π±°
SU(N_f)_A (axial isospin)	Spontaneously broken	N_f² − 1 Goldstone bosons: pions, kaons, η
U(1)_A (axial baryon)	Broken by anomaly	η′(958) is heavy, not a Goldstone

The quark condensate

The QCD vacuum is not empty. It contains a chirally non-invariant expectation value:

<0 | q-bar q | 0>  ≈  -(250 MeV)³  ≈  -1.6 × 10⁻² GeV³

q-bar q = q_L-bar q_R + q_R-bar q_L pairs a left-handed quark with a right-handed antiquark. The condensate transforms as a bifundamental of SU(N_f)_L × SU(N_f)_R, so a nonzero VEV picks out a diagonal SU(N_f)_V subgroup that leaves it invariant, and breaks the orthogonal axial directions. The vacuum chooses one direction in chiral space; the axial generators rotate it — and Goldstone bosons appear.

Worked example: the pion mass

For two flavors (u, d), the Gell-Mann-Oakes-Renner relation gives the leading-order pion mass:

m_pi^2 = -(m_u + m_d) <q-bar q> / f_pi^2

Plug in numbers:

• m_u + m_d ≈ 7 MeV (MS-bar at 2 GeV)
• ⟨q-bar q⟩ ≈ −(272 MeV)³ per flavor
• f_π = 92.4 MeV

Result: m_π² ≈ 7 × 10−³ × (0.272)³ / (0.0924)² GeV² ≈ 0.0173 GeV², so m_π ≈ 132 MeV. Experimental: m_π± = 139.6 MeV. The small discrepancy is higher-order chiral perturbation theory. The pion is not exactly massless because the chiral symmetry is not exact — quark masses break it explicitly.

How many Goldstone bosons?

N_f	Broken generators	Goldstone bosons	Identification
1 (u only)	1 (U(1)_A, but anomalous)	0 (anomaly)	No Goldstone; η′ heavy
2 (u, d)	3	3	π±, π° (140 MeV)
3 (u, d, s)	8	8	π (140), K (495), η (548)
4 (+c)	15	~3-8 useful	c too heavy; chiral PT breaks down

Two-flavor chiral symmetry is the best approximation: pions are by far the lightest hadrons. Three-flavor is decent: the SU(3) octet structure shows up clearly in the pseudoscalar mass formulas (Gell-Mann-Okubo: 4 m_K² ≈ m_π² + 3 m_η², satisfied to about 5%).

JavaScript — chiral projection and the GMOR pion mass

// Chiral projectors P_L, P_R for a 4-component Dirac spinor
// In the Weyl/chiral basis, gamma_5 = diag(-1, -1, 1, 1)
function projL(psi) {  // (1 - gamma_5)/2 picks upper two components
  return [psi[0], psi[1], 0, 0];
}
function projR(psi) {  // (1 + gamma_5)/2 picks lower two components
  return [0, 0, psi[2], psi[3]];
}

// Mass term mixes them: m * (psi_L-bar psi_R + psi_R-bar psi_L)
// Without mass, the two halves never interact -- chiral symmetry is exact.

// Gell-Mann-Oakes-Renner pion mass
const m_u = 2.2e-3;      // GeV
const m_d = 4.7e-3;      // GeV
const condensate = -Math.pow(0.272, 3); // GeV^3, per flavor, MS-bar 2 GeV
const f_pi = 0.0924;     // GeV

const m_pi_sq = -(m_u + m_d) * condensate / (f_pi * f_pi);
const m_pi    = Math.sqrt(m_pi_sq);
console.log('m_pi (GMOR) =', (m_pi * 1000).toFixed(1) + ' MeV');
// ~132 MeV; experimental 139.6 MeV (charged pion)

// Strange-flavor kaon
const m_s = 0.095;
const m_K_sq = -(m_u + m_s) * condensate / (f_pi * f_pi);
console.log('m_K (GMOR) =', (Math.sqrt(m_K_sq) * 1000).toFixed(0) + ' MeV');
// ~480 MeV; experimental 494 MeV. Decent for leading order.

// Chiral symmetry breaking scale (cutoff of chi-PT)
const Lambda_chi = 4 * Math.PI * f_pi;
console.log('Lambda_chi =', (Lambda_chi * 1000).toFixed(0) + ' MeV');
// ~1162 MeV -- below this, pion EFT is valid

Where chiral symmetry matters

• Low-energy QCD. Pion physics, kaon physics, η-η′ mixing, baryon chiral PT.
• Lattice QCD. Domain-wall and overlap fermions preserve chiral symmetry exactly on the lattice; crucial for matrix elements.
• Heavy-ion collisions. Quark-gluon plasma transition at T_c ≈ 155 MeV is a chiral-symmetry restoration crossover.
• Neutron-star cores. At baryon density ρ > few ρ_0, chiral symmetry may partially restore; color-superconducting phases possible.
• Electroweak symmetry breaking analog. Same mathematical structure as Higgs mechanism (technicolor models exploit this).
• Anomaly physics. π° → γγ rate is fixed by the axial anomaly; this is one of QCD's sharpest tests.

Common mistakes

• Conflating chirality with helicity. For massless particles they coincide. For massive particles helicity is frame-dependent; chirality is frame-independent.
• Calling the chiral symmetry exact. Quark masses break it explicitly. The pion has a (small) nonzero mass.
• Forgetting the anomaly. U(1)_A is broken at the quantum level, not just by spontaneous breaking. The η′ is therefore heavy.
• Missing the difference between vector and axial breaking. SU(N_f)_V (isospin) is approximately preserved; only SU(N_f)_A is spontaneously broken.
• Treating the condensate as a quark-antiquark molecule. It is a vacuum expectation value of a composite operator, not a literal bound state.
• Using chiral PT above 4πf_π. The expansion fails around 1.2 GeV; do not try to predict the rho meson with leading chiral PT.