Gauge invariance, chiral symmetry, and the decay eta → π0γγ

We consider a closed-fermion-loop model for the decay η→π⁰γγ. 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 SU₂ × SU₂ or SU₃ × SU₃ the decay is strictly forbidden. The symmetry-breaking limit gives an extremely suppressed width for the decay η→π⁰γγ.