Isospin breaking decay η(1405) → f0(980)π0 → 3π

There are attempts in the literature to theoretically explain the large breaking of isotopic invariance in the decay eta(1405) -> f(0)(980)pi(0) -> 3 pi by the mechanism containing the logarithmic ( triangle) singularity, i. e., as being due to the transition eta(1405) -> (K*(K) over bar + (K) over bar *K) -> (K+ K- + K-0 (K) over bar (0))pi(0) -> f(0)(980)pi(0) -> 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* (Gamma(K* -> K pi) approximate to 50 MeV) smoothes the logarithmic singularities in the amplitude and results in the suppression of the calculated decay width eta(1405) -> f(0)(980)pi(0) -> 3 pi by the factor of 6- 8 as compared with the case of Gamma(K* -> K pi) = 0. We also analyze the difficulties related with the assumption of the dominance of the eta(1405) -> (K*(K) over bar + (K) over bar *K) -> (K (K) over bar pi decay mechanism and discuss the possible dynamics of the decay eta(1405) -> eta pi pi. The decisive improvement of the experimental data on the K (K) over bar, K pi, eta pi, and pi pi mass spectra in the decay of the resonance structure eta(1405/1475) to K (K) over bar pi and eta pi pi, and on the shape of the resonance peaks themselves in the K (K) over bar pi and eta pi pi decay channels is necessary for further establishing the eta(1405) -> 3 pi decay mechanism.