This paper is a guided review of chiral symmetry in quantum chromodynamics (QCD) and what happens to that symmetry when matter gets hot or dense. The author starts from a basic puzzle: pions, the lightest particles that feel the strong force, behave like “pseudoscalars” and play a key role in why nuclear matter has the densities and energies we see. Those facts are tied to the deeper structure of the QCD vacuum, the lowest-energy state of the theory.

The review explains chirality — a property of quarks that for massless particles matches the notion of right- or left-handed motion — by reworking the Dirac equation. For massless quarks, chirality is a good label and matches helicity (spin direction relative to motion). For massive quarks the two chiralities mix. The paper points out a useful analogy: this mixing is mathematically like the way electrons pair up in superconductors. That analogy motivates the idea that quark masses and related phenomena might come from spontaneous symmetry breaking, not just from numbers put into the equations by hand.

Because the up, down, and strange quarks are much lighter than other quarks, chiral symmetry is a good approximate symmetry for them. The paper lists current estimates of those masses, for example mu(2 GeV) = 2.16 ± 0.07 MeV, md(2 GeV) = 4.70 ± 0.07 MeV, and ms(2 GeV) = 93.5 ± 0.8 MeV, and explains how the symmetry is broken spontaneously in the QCD vacuum. That breaking produces Nambu–Goldstone bosons, whose lightest members are the pions. The author also discusses the U(1)A anomaly — a specific quantum effect — and how it affects particle masses such as the η′ meson.

To make contact with data and intuition, the review surveys several effective models. These include Nambu–Jona-Lasinio (NJL) type chiral quark models, a three-flavor linear sigma model with a determinant term used to explain the η′ mass term, and a parity-doublet model for nucleons. The parity-doublet approach is being used to build equations of state for nuclear and neutron-star matter that include the possibility that chiral symmetry is (partly) restored in dense conditions. The paper also gives an intuitive account, based on the Hellmann–Feynman theorem, of how the chiral condensate (a measure of symmetry breaking) can be reduced when temperature or density change.