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

Abstract

The chiral expansion of the low energy processes $π^0\toγγ$ and $η\toγγ$ 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 $η-η'$ mixing parameters and those of the chiral expansion. The large size of the chiral correction to the $η$ 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 $η'$ into the effective lagrangian in a consistent and systematic way.