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

Abstract

The usual procedure to extract the $g^{eff}_{\omega\rho\pi}$ effective coupling from experimental data resumes not only the $\rho$ meson effect but also all its additional radial excitation modes. In this work, we consider a particular form of the spectrum and relations among the couplings to explicitly add the radial excitations of the $\rho$ meson from the $\omega \rightarrow \pi^0 \gamma$ and $\omega \rightarrow 3\pi$ decays. This allows to identify the single \linebreak $g_{\omega\rho\pi}= 8.3 \pm 0.8$~GeV$^{-1}$, which is about 40% smaller than the effective $g^{eff}_{\omega\rho\pi}$. We verify the consistency with the chiral approach in the $\pi^0 \rightarrow \gamma\gamma$ and $\gamma^* \rightarrow 3\pi$ processes, exhibiting the role of each contribution. In particular, we show that for the $\gamma^* \rightarrow 3 \pi$ decay, the usual relation $\mathcal{A}^{VMD}_{\gamma3\pi}=(3/2)\mathcal{A}^{WZW}_{\gamma3\pi}$, encodes all the vector contributions and not only the $\rho$ meson one. In addition, we find that there is a strong (accidental) cancelation between the radial excitations and the contact term contributions.