Gauge Invariance, Chiral Symmetry, and the Decayη→π0γγ

We consider a closed-fermion-loop model for the decay $\eta \to {\pi }^0\gamma \gamma$. We incorporate chiral symmetry and gauge invariance and derive an effective Lagrangian for the above decay using a method first introduced by Schwinger. The effective Lagrangian is derived by treating the mesons as soft. The effect of symmetry breaking is also considered. We come to the surprising conclusion that in the limit of exact ${\mathrm{SU}}_2$ × ${\mathrm{SU}}_2$ or ${\mathrm{SU}}_3$ × ${\mathrm{SU}}_3$ the decay is strictly forbidden. The symmetry-breaking limit gives an extremely suppressed width for the decay $\eta \to {\pi }^0\gamma \gamma$.