Anomalous ω-Z-γ vertex from hidden local symmetry

We formulate the general form of the ω-Z-γ vertex in the framework based on the hidden local symmetry, which arises from the gauge-invariant terms for intrinsic parity-odd part of the effective action. Those terms are given as the homogeneous part of the general solution (having free parameters) to the Wess-Zumino anomaly equation and hence are not determined by the anomaly, in sharp contrast to the Harvey-Hill-Hill (HHH) action where the relevant vertex is claimed to be uniquely determined by the anomaly. We show that, even in the framework that HHH was based on, the ω-Z-γ vertex is actually not determined by the anomaly but by the homogeneous (anomaly-free) part of the general solution to the Wess-Zumino anomaly equation having free parameters in the same way as in the hidden local symmetry formulation: The HHH action is just a particular choice of the free parameters in the general solution. We further show that the ω-Z-γ vertex related to the neutrino (ν)-nucleon (N) scattering cross section σ(νN→νN(N')γ) is determined not by the anomaly but by the anomaly-free part of the general solution having free parameters. Nevertheless, we find that the cross section σ(νN→νN(N')γ) is related through the Ward-Takahashi identity to Γ(ω→π0γ) which has the same parameter dependence as that of σ(νN→νN(N')γ) and hence the ratio σ(νN→νN(N')γ)/Γ(ω→π0γ) is fixed independently of these free parameters. Other set of the free parameters of the general solution can be fixed to make the best fit of the ω→π0l+l- process, which substantially differs from the HHH action. This gives a prediction of the cross section σ(νN→νN(N')γ*(l+l-)) to be tested at ν-N collision experiments in the future.