There are attempts in the literature to theoretically explain the large breaking of isotopic invariance in the decay $\eta (1405)\to f_0(980){\pi }^0\to 3\pi$ by the mechanism containing the logarithmic (triangle) singularity, i.e., as being due to the transition $\eta (1405)\to (K^{*}\overline{K}+{\overline{K}}^{*}K)\to (K^{+}{K}^{-}+K^0{\overline{K}}^0){\pi }^0\to f_0(980){\pi }^0\to 3\pi$. The corresponding calculations were fulfilled for a hypothetic case of the stable $K^{*}$ meson. Here, we show that the account of the finite width of the $K^{*}$ (${\mathrm{\Gamma }}_{K^{*}\to K\pi }\approx 50\mathrm{MeV}$) smoothes the logarithmic singularities in the amplitude and results in the suppression of the calculated decay width $\eta (1405)\to f_0(980){\pi }^0\to 3\pi$ by the factor of 6–8 as compared with the case of ${\mathrm{\Gamma }}_{K^{*}\to K\pi }=0$. We also analyze the difficulties related with the assumption of the dominance of the $\eta (1405)\to (K^{*}\overline{K}+{\overline{K}}^{*}K)\to K\overline{K}\pi$ decay mechanism and discuss the possible dynamics of the decay $\eta (1405)\to \eta \pi \pi$. The decisive improvement of the experimental data on the $K\overline{K}$, $K\pi$, $\eta \pi$, and $\pi \pi$ mass spectra in the decay of the resonance structure $\eta (1405/1475)$ to $K\overline{K}\pi$ and $\eta \pi \pi$, and on the shape of the resonance peaks themselves in the $K\overline{K}\pi$ and $\eta \pi \pi$ decay channels is necessary for further establishing the $\eta (1405)\to 3\pi$ decay mechanism.

• Received 14 April 2015

DOI:https://doi.org/10.1103/PhysRevD.92.036003