Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 305 / 2 / i / Solution

For $q=(u,d{)}^T$, massless two-flavour Quantum chromodynamics hasClassically its continuous global symmetry is $SU(2{)}_L\times SU(2{)}_R\times U(1{)}_V\times U(1{)}_A$, up to finite central quotients. The Adler-Bell-Jackiw anomaly destroys the continuous axial $U(1{)}_A$ in the quantum theory, leaving $SU(2{)}_L\times SU(2{)}_R\times U(1{)}_V$, together with an anomaly-preserved discrete axial subgroup.