Chiral Sum-Rules for ${\cal L}^{WZ}_{(6)}$ Parameters and Application to $\pi^0,\eta,\eta'$ Decays

The chiral expansion of the low energy processes $\pi^0\to\gamma\gamma$ and $\eta\to\gamma\gamma$ is reconsidered with particular emphasis on the question of the evaluation of the two low-energy parameters from ${\cal L}^{WZ}_{(6)}$ which are involved at chiral order six. It is shown how extensive use of sum-rules and saturation with resonances as well as constraints from asymptotic QCD effectively determine their values. Predictions for the widths are presented for both standard and non-standard values of the quark mass ratio $m_s/{\hat m}$. A precise relation is established between the usual phenomenological $\eta-\eta'$ mixing parameters and those of the chiral expansion. The large size of the chiral correction to the $\eta$ decay can be understood on the basis of a simple counting rule: $O(1/N_c)\sim\ O(m_q)$. It is shown how this counting rule eventually allows one to include the $\eta'$ into the effective lagrangian in a consistent and systematic way.