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Isospin breaking decay $η(1405) \to f_0(980)π^0 \to 3π$

There are attempts in the literature to theoretically explain the large breaking of isotopic invariance in the decay $η(1405)$\,$\to$\,$f_0(980)π^0$\,$\to$\,$ 3π$ by the mechanism containing the logarithmic (triangle) singularity, i.e., as being due to the transition $η(1405)\to(K^*\bar K+\bar K^*K)\to(K^+K^-+K^0\bar K^0)π^0\to f_0(980)π^0\to3π$. 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 $K^*$ ($Γ_{K^*\to Kπ}\approx50$ MeV) smoothes the logarithmic singularities in the amplitude and results in the suppression of the calculated decay width $η(1405)\to f_0(980) π^0\to3π$ by the factor of $6-8$ as compared with the case of $Γ_{K^*\to Kπ}$\,=\,0. We also analyze the difficulties related with the assumption of the dominance of the $η(1405)\to(K^*\bar K+\bar K^*K)\to K\bar Kπ$ decay mechanism and discuss the possible dynamics of the decay $η(1405)\toηππ$. The decisive improvement of the experimental data on the $K\bar K$, $ Kπ$, $ηπ$, and $ππ$ mass spectra in the decay of the resonance structure $η(1405/1475)$ to $K\bar Kπ$ and $ηππ$, and on the shape of the resonance peaks themselves in the $K\bar Kπ$ and $ηππ$ decay channels is necessary for the further establishing the $η(1405)\to3 π$ decay mechanism.