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

The condensate transforms as $⟨{\bar{q}}_L{q}_R⟩\mapsto L⟨{\bar{q}}_L{q}_R⟩R^{†}$, so its value proportional to the identity produces chiral symmetry breakingThe vacuum manifold is the coset $(SU(2{)}_L\times SU(2{)}_R)/SU(2{)}_V\cong SU(2)$ and has three Goldstone modes. A field $U(x)\in SU(2)$ transforms as $U\mapsto LU{R}^{†}$. Lorentz invariance and chiral symmetry leave, at two-derivative order, one invariant metric on this coset, soSince the Lagrangian density has dimension four and $U$ is dimensionless, the pion decay constant $f_{\pi }$ has mass dimension one.