Identify hadron anomalous couplings at colliders

We investigate the identification of the Wess-Zumino-Witten (WZW) Lagrangian at colliders such as BESIII and the Super-$\tau$-Charm Facility. Our analysis concentrates on the radiative decays of $\eta$ and $\eta'$ mesons, including $\eta^{(\prime)} \to \gamma\gamma$, $\eta^{(\prime)} \to \gamma\ell^+ \ell^-$, $\eta^{(\prime)} \to \pi^+\pi^-\gamma$, and $\eta^{(\prime)} \to \pi^+\pi^-\ell^+\ell^-$, as well as semileptonic kaon decays such as $K^+ \to \pi^+\pi^- e^+ \nu_e$. Employing the hidden local symmetry framework to incorporate vector meson contributions, we compute the decay amplitudes and form factors. For the decay $\eta \to \pi^+\pi^-\gamma$, the box anomaly dominates, and we find that the anomalous coupling can be experimentally determined to percent-level precision at BESIII. In contrast, vector meson contributions are significant in the decay $\eta' \to \pi^+\pi^-\gamma$. Using experimental data for $\eta' \to \pi^+\pi^-\gamma$, we obtain ${\cal B}_{\mathrm{box}}^{\text{exp}} = (1.70 \pm 0.05)\%$, which is approximately ten times larger than previously expected experimentally. We observe good agreement between our calculated anomalous couplings and experimental results. In kaon decays, WZW terms uniquely contribute to the form factor $H$, which can be extracted from parity-conserving decay distributions. While predictions at the chiral point closely match experimental values (e.g., $H^+ = -2.31$ versus $-2.27 \pm 0.10$), we find that intermediate vector meson states introduce substantial corrections, potentially as large as 25%. We strongly advocate for revisiting these experiments to achieve improved precision in form factor extractions.