chiralsymmetry

Chiral symmetry is a property of certain quantum field theories in which the left-handed and right-handed components of fermion fields can be rotated independently without changing the action. For massless Dirac fermions, the Lagrangian is invariant under global SU(N)L × SU(N)R transformations, where the left- and right-handed parts of the fermion field transform separately. The vector subgroup with L = R corresponds to SU(N)V. In practical terms, chiral symmetry distinguishes transformations that act differently on left- and right-handed states.

In quantum chromodynamics (QCD), chiral symmetry is an approximate global symmetry of the light-quark sector when quark masses are small. In the limit m_u, m_d, m_s → 0, the theory possesses SU(3)L × SU(3)R chiral symmetry, which is spontaneously broken to SU(3)V by the quark condensate ⟨ψ̄ψ⟩. This breaking gives rise to eight pseudo-Goldstone bosons (the pions, kaons, and eta). Real quark masses explicitly break the symmetry, imparting small masses to these particles.

In the Standard Model, chiral symmetry is explicitly broken by fermion mass terms and by Yukawa couplings to the Higgs, while the weak interactions are chiral, coupling only to left-handed fermions. The Higgs mechanism provides mass through gauge-invariant Yukawa couplings, so chiral symmetry is not an exact symmetry of the observed spectrum, though the approximate chiral symmetry of QCD remains a useful organizing principle at low energies.

Chiral anomalies, notably the axial U(1)A anomaly in QCD, break certain chiral currents at the quantum level and influence the symmetry structure of gauge theories. Overall, chiral symmetry remains a central organizing principle in particle physics, even when it is explicitly or anomalously broken.