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Recent experiments on $\eta\to 3\pi$ decays have provided an extremelyprecise knowledge of the amplitudes across the Dalitz region which representstringent constraints on theoretical descriptions. We reconsider an approach inwhich the low-energy chiral expansion is assumed to be optimally convergent inan unphysical region surrounding the Adler zero, and the amplitude in thephysical region is uniquely deduced by an analyticity-based extrapolation usingthe Khuri-Treiman dispersive formalism. We present an extension of the usualformalism which implements the leading inelastic effects from the $K\bar{K}$channel in the final-state $\pi\pi$ interaction as well as in the initial-state$\eta\pi$ interaction. The constructed amplitude has an enlarged region ofvalidity and accounts in a realistic way for the influence of the two lightscalar resonances $f_0(980)$ and $a_0(980)$ in the dispersive integrals. It isshown that the effect of these resonances in the low energy region of the $\eta\to 3\pi$ decay is not negligible, in particular for the $3\pi^0$ mode, andimproves the description of the energy variation across the Dalitz plot. Someremarks are made on the scale dependence and the value of the double quark massratio $Q$.

Extended chiral Khuri-Treiman formalism for $\eta\to 3\pi$ and the role of the $a_0(980)$, $f_0(980)$ resonances