Final State Interactions and Khuri-Treiman Equations in $\eta\to 3\pi$ decays

Using extended Khuri-Treiman equations, we evaluate the final state interactions due to two-pion rescatterings to the decays $\eta\to \pi^0 \pi^+ \pi^-$ and $\eta\to \pi^0 \pi^0 \pi^0$. As subtraction to the dispersion relation we take the one-loop chiral perturbation theory result of Gasser and Leutwyler. The calculated corrections are moderate and amount to about $14\%$ in the amplitude at the center of the decay region. A careful analysis of the errors inherent to our approach is given. As a consequence, the experimental rate of the decay can only be reproduced if the double quark mass ratio $Q^{-2}\equiv\frac{m_{d}-m_{u}} {m_{s}-{\hat m}} \cdot\frac{m_{d}+m_{u}}{m_{s}+{\hat m}}$ is increased from the usual value of $1/(24.1)^2$ to $1/(22.4 \pm 0.9)^2$. We have also calculated the ratio of the rates of the two decays and various Dalitz Plot parameters. In particular, the linear slope $a$ in the charged decay is different from the one-loop value and agrees better with experiment.