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

Abstract

In this work we calculate the decay widths of $ϕ(2170)$ to $ϕη$ and $ϕη^\prime$ by considering $ϕ(2170)$ as a $ϕK\bar K$ 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 $ϕ(2170)$. By analyzing the data on the cross sections of the process $e^+e^-\toϕη$, two solutions were found by the BESIII collaboration with the same $χ^2/n.d.f$, mass and width for $ϕ(2170)$. However, different values for $\mathcal{B}^{ϕ(2170)}_{ϕη}Γ^{ϕ(2170)}_{e^+e^-}$ were obtained: $0.24^{+0.12}_{-0.07}$ eV (solution I) and $10.11^{+3.87}_{-3.13}$ eV (solution II). In case of $\mathcal{B}^{ϕ(2170)}_{ϕη^\prime}Γ^{ϕ(2170)}_{e^+e^-}$, 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 $e^+e^-\toϕη^\prime$. The Belle collaboration has also determined $\mathcal{B}^{ϕ(2170)}_{ϕη}Γ^{ϕ(2170)}_{e^+e^-}$ more recently, although the statistical significance related to the signal of $ϕ(2170)$ in the data is low. We calculate the decay widths of $ϕ(2170)$ to $ϕη$ and $ϕη^\prime$, and compare their ratio $Γ^{ϕ(2170)}_{ϕη}/Γ^{ϕ(2170)}_{ϕη^\prime}$ 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 $\mathcal{B}^{ϕ(2170)}_{ϕη}Γ^{ϕ(2170)}_{e^+e^-}$, as well as to the upper limit found ($\simeq 2.8$) when considering the solutions III and IV of the Belle collaboration for $\mathcal{B}^{ϕ(2170)}_{ϕη}Γ^{ϕ(2170)}_{e^+e^-}$.