Off-shell Behavior of the $\pi\!-\!\eta$ Mixing Amplitude

We extend a recent calculation of the momentum dependence of the $\rho-\omega$ mixing amplitude to the pseudoscalar sector. The $\pi\!-\!\eta$ mixing amplitude is calculated in a hadronic model where the mixing is driven by the neutron-proton mass difference. Closed-form analytic expressions are presented in terms of a few nucleon-meson parameters. The observed momentum dependence of the mixing amplitude is strong enough as to question earlier calculations of charge-symmetry-breaking observables based on the on-shell assumption. The momentum dependence of the $\pi\!-\!\eta$ amplitude is, however, practically identical to the one recently predicted for $\rho-\omega$ mixing. Hence, in this model, the ratio of pseudoscalar to vector mixing amplitudes is, to a good approximation, a constant solely determined from nucleon-meson coupling constants. Furthermore, by selecting these parameters in accordance with charge-symmetry-conserving data and SU(3)-flavor symmetry, we reproduce the momentum dependence of the $\pi\!-\!\eta$ mixing amplitude predicted from chiral perturbation theory. Alternatively, one can use chiral-perturbation-theory results to set stringent limits on the value of the $NN\eta$ coupling constant.