The classical QED Lagrangian with massless fermions has two conserved currents: the vector current j^mu = psi-bar gamma^mu psi and the axial current j^mu_5 = psi-bar gamma^mu gamma_5 psi. What happens to the axial current at the quantum level?
ABoth currents remain conserved
BThe axial current acquires a non-zero divergence proportional to F_{mu nu} F-tilde^{mu nu} — the chiral anomaly — due to the triangle diagram where two photons couple to the axial current through a fermion loop
CThe vector current becomes anomalous instead
DBoth currents are broken by quantum effects
The Adler-Bell-Jackiw (ABJ) anomaly states that partial_mu j^mu_5 = (e^2)/(16 pi^2) F_{mu nu} F-tilde^{mu nu}, where F-tilde is the dual field strength. This is computed from the triangle diagram with one axial vertex and two vector vertices. The anomaly is exact — it receives no higher-order corrections (Adler-Bardeen theorem). The vector current remains conserved (no anomaly), which is essential for electric charge conservation. The anomaly has real physical consequences: it explains the decay rate of pi^0 -> gamma gamma, which would be zero without the anomaly.
Question 2 Multiple Choice
If the gauge symmetry of a theory is anomalous (i.e., the gauge current has an anomaly), the theory is inconsistent and must be discarded. Why is a gauge anomaly more dangerous than a global anomaly?
BBecause a gauge anomaly breaks the Ward identities that ensure unitarity and renormalizability — without gauge invariance at the quantum level, negative-norm states (ghosts) do not decouple, probability is not conserved, and the theory makes no sense
CBecause gauge anomalies produce infinite cross sections
DBecause gauge anomalies violate energy conservation
Gauge invariance is not merely a convenient symmetry — it is essential for the consistency of the theory. It ensures that unphysical polarization states of gauge bosons decouple (unitarity), that divergences can be systematically removed (renormalizability), and that the number of physical degrees of freedom is correct. If quantum corrections break gauge invariance (a gauge anomaly), all of this fails. This is why anomaly cancellation is a non-negotiable consistency condition on the particle content of any gauge theory. A global anomaly (breaking a global symmetry) is physically interesting but not fatal.
Question 3 True / False
In the Standard Model, the anomalies from quarks and leptons within each generation cancel exactly. This cancellation is a coincidence with no deeper explanation.
TTrue
FFalse
Answer: False
Anomaly cancellation in the Standard Model requires specific relationships among the hypercharges and representations of quarks and leptons. Within each generation, the sum of certain products of charges (the anomaly coefficients for SU(3)^2 U(1), SU(2)^2 U(1), U(1)^3, and gravitational anomalies) vanishes. This cancellation appears highly non-trivial when the particle content is taken as given, but it follows automatically if the Standard Model is embedded in a grand unified theory (like SU(5) or SO(10)), where quarks and leptons live in unified representations. Anomaly cancellation is therefore evidence for a deeper structure.
Question 4 Short Answer
Explain how the chiral anomaly resolves the puzzle of neutral pion decay (pi^0 -> gamma gamma) and why this decay would be forbidden without it.
Think about your answer, then reveal below.
Model answer: The pion is a pseudo-Goldstone boson of the spontaneously broken chiral symmetry of QCD. Its coupling to photons occurs through the axial current: the matrix element involves <0|j^mu_5|pi^0> coupled to two photons. If the axial current were exactly conserved (no anomaly), partial_mu j^mu_5 = 0, the coupling to the two-photon state would vanish (by taking the divergence of the amplitude and using current conservation), and the pi^0 would not decay to two photons. The chiral anomaly gives partial_mu j^mu_5 = (e^2 N_c)/(16 pi^2) F_{mu nu} F-tilde^{mu nu}, which provides the coupling. The predicted decay rate, proportional to N_c^2 alpha^2 m_pi^3/(f_pi^2), agrees with experiment when N_c = 3 colors. This was one of the first confirmations that quarks come in three colors.
This is a remarkable story: a formal mathematical anomaly in the quantum theory resolves a physical puzzle and simultaneously provides evidence for the color degree of freedom. It shows that anomalies are not defects but essential features of the quantum theory with observable consequences.