Isotopic symmetry breaking in the $\eta(1405)\to f_0 (980)\pi^0\to\pi^+\pi^-\pi^0$. Decay through a $K\bar K$ Loop diagram and the role of anomalous Landau thresholds

Anomalous isotopic symmetry breaking in the $\eta(1405)\to f_0 (980)\pi^0\to\pi^+\pi^-\pi^0$ decay through a mechanism featuring anomalous Landau thresholds in the form of logarithmic triangle singularities, i.e., through the $\eta(1405)\to (K^*\bar K+\bar K^*K)\to(K^+K^-+K^0\bar K^0)\pi^0\to f_0(980)\pi^0 \to\pi^+\pi^- \pi^0$ transition, has been analyzed. It has been shown that this effect can be correctly quantified only by taking into account the nonzero $K^*$ width. Different scales of isotopic symmetry breaking associated with the $K^+-K^0$ mass difference are compared.