Electromagnetic Meson Decays from an Effective Lagrangian with Chiral Symmetry

A nonet-symmetrical and PCAC-violating term is introduced into a Lagrangian with $S{U}_3\otimes S{U}_3$ chiral symmetry and PCAC (hypothesis of partially conserved axial-vector current). Breaking of the nonet symmetry of this term is achieved through the renormalization of the fields. This renormalization is necessary because of the requirement that the kinetic-energy terms of the Lagrangian be correctly normalized after the removal of the tadpole terms. Using as input the decay rate for ${\pi }^0\to 2\gamma$, we predict $\Gamma (\eta \to 2\gamma )=0.41\mathrm{}\pm 0.08$ keV and $\Gamma (X^0\to 2\gamma )=6\mathrm{}\pm 1$ keV. The result for $\Gamma (\eta \to 2\gamma )$ agrees with experiment within 2 standard deviations. It is argued that errors of about 20% should be expected in our calculations, mainly because of the neglect of the effects due to the finite width of the various particles.