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This paper is a clear, lecture-style review of what theoretical physicists can say exactly about an imaginary version of quantum chromodynamics (QCD) in which one or more quark masses are set to zero. The author focuses on global symmetries of the theory — in particular scale (conformal) symmetry and chiral symmetry (the symmetry between left- and right-handed quarks) — and on how those symmetries are modified by quantum effects. The review assumes the CP-violating theta parameter is zero and aims to be pedagogical rather than to present new numerical results.
At the classical level, massless QCD has no built‑in scale: scaling all distances and fields leaves the action unchanged. Quantum mechanics changes that. The review explains dimensional transmutation — the quantum generation of a physical energy scale — and how the strong coupling runs with energy. In the leading approximation the strong coupling αs(E) behaves like 2π/(b0 ln(E/ΛQCD)), where b0 = (11 Nc − 2 Nf)/3 (Nc is the number of colors, Nf the number of light quark flavors) and ΛQCD is the emergent scale (about 200 MeV). That breakdown of classical scale invariance shows up as a nonzero trace of the quantum energy–momentum tensor, a statement known as the trace or conformal anomaly (written in the paper as an operator identity relating the trace to the beta function and the gluon field strength).
The review also covers chiral symmetries that are exact when quark masses vanish. It discusses the different forms these symmetries take, the well‑known chiral anomaly (a quantum effect that breaks some classically conserved chiral currents), and the spontaneous breaking of non‑singlet chiral symmetry that is central to low‑energy hadron physics. The author describes how these ideas are organized into an effective chiral Lagrangian, the low‑energy theory that captures the physics of the lightest hadrons in the massless‑quark limit.
A significant part of the paper treats how to define and study these symmetry questions on the lattice, the standard regulator used to make the path integral well defined. The lattice introduces an ultraviolet cutoff and hence participates in dimensional transmutation. The review explains practical and conceptual lattice issues: a naive lattice fermion formulation runs into Nielsen–Ninomiya’s no‑go theorem (which forbids a straightforward lattice version of a single chiral fermion), and ways around it such as the Ginsparg–Wilson construction are discussed. Related topics included are rigorous inequalities like the Vafa–Witten theorem, the spectral density of the Euclidean Dirac operator, and how the anomaly appears in the infrared.
Why this matters: studying the massless‑quark limit sharpens our understanding of the role of symmetry in strong interactions. Real QCD has very light up and down quarks (roughly 3 and 5 MeV) and a moderately light strange quark (about 100 MeV), so the massless idealization is useful for understanding approximate symmetries in nature. Important caveats noted in the review are that the low‑energy region of QCD is nonperturbative (so the simple running formulas lose accuracy there), some statements remain beyond rigorous proof (for example, a full mathematical proof of confinement is still an open problem mentioned in the text), and the paper studies an idealized “imaginary world” that helps clarify formal points but is not a literal description of the real world.