About the inclusion of an infinite number of resonances in anomalous decays

The extracted value for the $g^{eff}_{omega rhopi}$ effective coupling from experimental data, considering only the $rho$ meson, resumes not only the $rho$ meson effect but also all its additional radial excitation modes. By explicitly adding the radial excitations of the $rho$ meson, considering a particular form of the spectrum and relations among the couplings, we identify the single $g_{omega rho pi}$ and the $rho$ radial excitations effect in the $omega rightarrow pi^0 gamma$ decay. We obtain that the individual coupling is in the range $g_{omegarhopi}= 8.2 - 8.6 {text GeV}^{-1}$, which is about 40% smaller than the effective $g^{eff}_{omegarhopi}$. We verify the consistency with the chiral approach in the $pi^0 rightarrow gammagamma$ and $gamma^* rightarrow 3pi$ processes. Besides the model dependence, our description succeeds in exhibiting how each contribution came into the game. In particular, we show that for the $gamma^* rightarrow 3 pi$ decay, the usual relation $mathcal{A}^{VMD}_{gamma3pi}=(3/2)mathcal{A}^{WZW}_{gamma3pi}$, encodes all the vector contributions and not only the $rho$ meson one. In addition, we find that there is an almost exact (accidental) cancelation between the radial excitations and the contact term contributions.