$φ(2170)$ decaying to $φη$ and $φη^\prime$

Abstract: In this work we calculate the decay widths of $\phi(2170)$ to $\phi\eta$ and $\phi\eta<sup>\prime$ by considering $\phi(2170)$ as a $\phi K\bar K$ state, with $K\bar K$ clustering as $f_0(980)$. These decay widths have been recently determined by the BESIII, BaBar and Belle collaborations with the aim of unraveling the nature of $\phi(2170)$. By analyzing the data on the cross sections of the process $e<sup>+e<sup>-\to\phi\eta$, two solutions were found by the BESIII collaboration with the same $\chi<sup>2/n.d.f$, mass and width for $\phi(2170)$. However, different values for $$\mathcal{B}<sup>{\phi(2170)}<em>{\phi\eta}\Gamma^{\phi(2170)}</em>{e<sup>+e<sup>-}$$ were obtained: $0.24<sup>{+0.12}_{-0.07}$ eV (solution I) and $10.11<sup>{+3.87}_{-3.13}$ eV (solution II). In case of $$\mathcal{B}<sup>{\phi(2170)}<em>{\phi\eta^{\prime}\Gamma^{\phi(2170)}}</em>{e<sup>+e<sup>-}$$, a value of $7.1\pm 0.7\pm 0.7$ was determined by the BESIII collaboration from fits to the data on the cross section of $e<sup>+e<sup>-\to\phi\eta<sup>\prime$. The Belle collaboration has also determined $$\mathcal{B}<sup>{\phi(2170)}<em>{\phi\eta}\Gamma^{\phi(2170)}</em>{e<sup>+e<sup>-}$$ more recently, although the statistical significance related to the signal of $\phi(2170)$ in the data is low. We calculate the decay widths of $\phi(2170)$ to $\phi\eta$ and $\phi\eta<sup>\prime$, and compare their ratio $$\Gamma^{{\phi(2170)}_{\phi} \eta}/\Gamma^{{\phi(2170)}_{\phi} \eta<sup>\prime}$$ with the one determined by using the above mentioned experimental data. Considering the theoretical and experimental uncertainties, the lower limit of our theoretical result ($\simeq 2.6$) is close to the upper value obtained ($\simeq 2$) by using the solution II of BESIII for $$\mathcal{B}<sup>{\phi(2170)}<em>{\phi\eta}\Gamma^{\phi(2170)}</em>{e<sup>+e<sup>-}$$, as well as to the upper limit found ($\simeq 2.8$) when considering the solutions III and IV of the Belle collaboration for $$\mathcal{B}<sup>{\phi(2170)}<em>{\phi\eta}\Gamma^{\phi(2170)}</em>{e<sup>+e<sup>-}$$.