Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 305 / 4 / c / i / Solution

Introduce the linear sigma model fieldA suitable renormalizable potential iswith a small symmetry-breaking source proportional to the quark-mass matrix when desired. A vacuum $⟨\Sigma ⟩=vI$ is fixed precisely by $L=R$, so the unbroken group is the diagonal isospin $SU(2{)}_V$. The three broken axial generators produce the pion triplet ${\pi }^{\pm },{\pi }^0$, which become pseudo-Goldstone bosons when $m_u,m_d$ are restored.