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

Recent experiments on $\eta\to 3\pi$ decays have provided an extremely precise knowledge of the amplitudes across the Dalitz region which represent stringent constraints on theoretical descriptions. We reconsider an approach in which the low-energy chiral expansion is assumed to be optimally convergent in an unphysical region surrounding the Adler zero, and the amplitude in the physical region is uniquely deduced by an analyticity-based extrapolation using the Khuri-Treiman dispersive formalism. We present an extension of the usual formalism 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 of validity and accounts in a realistic way for the influence of the two light scalar resonances $f_0(980)$ and $a_0(980)$ in the dispersive integrals. It is shown 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, and improves the description of the energy variation across the Dalitz plot. Some remarks are made on the scale dependence and the value of the double quark mass ratio $Q$.