Paper detail

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

The extracted value for the $g^{eff}_{ωρπ}$ effective coupling from experimental data, considering only the $ρ$ meson, resumes not only the $ρ$ meson effect but also all its additional radial excitation modes. By explicitly adding the radial excitations of the $ρ$ meson, considering a particular form of the spectrum and relations among the couplings, we identify the single $g_{ωρπ}$ and the $ρ$ radial excitations effect in the $ω\rightarrow π^0 γ$ decay. We obtain that the individual coupling is in the range $g_{ωρπ}= 8.2 - 8.6 {\text GeV}^{-1}$, which is about 40% smaller than the effective $g^{eff}_{ωρπ}$. We verify the consistency with the chiral approach in the $π^0 \rightarrow γγ$ and $γ^* \rightarrow 3π$ processes. Besides the model dependence, our description succeeds in exhibiting how each contribution came into the game. In particular, we show that for the $γ^* \rightarrow 3 π$ decay, the usual relation $\mathcal{A}^{VMD}_{\gamma3π}=(3/2)\mathcal{A}^{WZW}_{\gamma3π}$, encodes all the vector contributions and not only the $ρ$ meson one. In addition, we find that there is an almost exact (accidental) cancelation between the radial excitations and the contact term contributions.