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

Abstract

The extracted value for the $g_{\omega \rho \pi }^{eff}$ 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 $\omega \to {\pi }^0\gamma$ decay. We obtain that the individual coupling is in the range $g_{\omega \rho \pi }=8.2-8.6{\text{G}eV}^{-1}$, which is about 40% smaller than the effective $g_{\omega \rho \pi }^{eff}$. We verify the consistency with the chiral approach in the ${\pi }^0\to \gamma \gamma$ and ${\gamma }^{\ast }\to 3\pi$ processes. Besides the model dependence, our description succeeds in exhibiting how each contribution came into the game. In particular, we show that for the ${\gamma }^{\ast }\to 3\pi$ decay, the usual relation ${\mathcal{A}}_{\gamma 3\pi }^{VMD}=(3/2){\mathcal{A}}_{\gamma 3\pi }^{WZW}$, 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.

Comments: Discussion extended and several references added. Results remain unchanged. To appear in Rev. Mex. Fis