Chiral symmetry and its breaking

In addition to fundamental symmetries playing a crucial role for establishing the Standard Model of fundamental interactions, approximate symmetries provide essential insight into the respective phenomena and shed light on the underlying physics. Here we give a brief pedagogical introduction to chiral symmetry as an approximate but still rather accurate symmetry of strong interactions and its spontaneous breaking in the vacuum of Quantum Chromodynamics. Special attention is paid to a microscopic picture of this phenomenon and understanding a dual nature of the chiral pion that is the Goldstone boson related to spontaneous breaking of chiral symmetry and the lowest pseudoscalar quark-antiquark state in the spectrum of hadrons simultaneously.

💡 Research Summary

The paper provides a pedagogical yet technically detailed overview of chiral symmetry in the context of strong interactions and its spontaneous breaking in the QCD vacuum. It begins by recalling the Dirac formalism in the Weyl (chiral) representation, where the four‑component spinor ψ is split into two independent two‑component spinors ξ and η. In the massless limit the right‑handed (ψ_R) and left‑handed (ψ_L) components decouple, a fact encoded by the projection operators P_R = (1+γ⁵)/2 and P_L = (1–γ⁵)/2. The authors emphasize that the mass term m₀ ψ̄ψ mixes these chiralities, destroying the separate conservation of right‑ and left‑handed currents.

Next, the paper derives the Noether currents associated with global U(1) phase rotations (the vector current j^μ = ψ̄γ^μψ) and axial rotations (the axial current j⁵^μ = ψ̄γ^μγ⁵ψ). For massless fermions both currents are classically conserved, leading to a U(1)_V × U(1)A symmetry. However, when electromagnetic interactions are included, the Adler–Bell–Jackiw anomaly adds a term proportional to F{μν}\tilde F^{μν}, breaking the axial symmetry at the quantum level.

Extending to N_f ≥ 2 flavors, the authors introduce the non‑abelian chiral symmetry SU(N_f)_L × SU(N_f)R. The corresponding vector and axial currents j_a^μ = ψ̄γ^μ t_a ψ and j{5a}^μ = ψ̄γ^μγ⁵ t_a ψ are conserved in the chiral limit, and the associated charges generate two commuting SU(N_f) algebras. In real QCD the up, down, and strange quarks are light enough that this symmetry is an excellent approximation.

The core of the manuscript is the discussion of spontaneous chiral symmetry breaking (SBCS). Using BCS‑type reasoning and effective four‑fermion models such as the Nambu–Jona‑Lasinio (NJL) and its generalized version (GNJL), the authors show how a non‑zero quark condensate ⟨ψ̄ψ⟩ emerges in the vacuum. This condensate dynamically generates a momentum‑dependent constituent quark mass M(p) even when the bare mass vanishes, thereby breaking the axial symmetry and giving rise to Goldstone bosons.

The “chiral pion” is examined in depth. As a Goldstone boson of the breaking SU(2)_A → U(1), the pion appears as one of three massless excitations in the chiral limit. Simultaneously, within quark‑model frameworks it is the lowest‑lying pseudoscalar q q̄ bound state. The authors reconcile these pictures by invoking the partially conserved axial current (PCAC) and the Gell‑Mann–Oakes–Renner relation m_π² f_π² = −(m_u + m_d)⟨ψ̄ψ⟩, which quantifies how explicit quark masses give the pion a small but finite mass.

Various realizations of chiral symmetry—linear versus non‑linear sigma models, the role of the axial anomaly, and the impact of explicit symmetry‑breaking terms—are compared, highlighting how each captures different aspects of low‑energy hadron phenomenology. The paper concludes that chiral symmetry, though approximate, is indispensable for understanding the spectrum and interactions of light hadrons, and that the dual nature of the pion exemplifies the deep connection between symmetry principles, vacuum structure, and observable particle properties. Future directions suggested include lattice QCD studies of the condensate, exploration of color‑superconducting phases at high density, and higher‑order chiral perturbation theory calculations.