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Analyticity of $ηπ$ isospin-violating form factors and the $τ\toηπν$ second-class decay

We consider the evaluation of the $ηπ$ isospin-violating vector and scalar form factors relying on a systematic application of analyticity and unitarity, combined with chiral expansion results. It is argued that the usual analyticity properties do hold (i.e. no anomalous thresholds are present) in spite of the instability of the $η$ meson in QCD. Unitarity relates the vector form factor to the $ηπ\to ππ$ amplitude: we exploit progress in formulating and solving the Khuri-Treiman equations for $η\to 3π$ and in experimental measurements of the Dalitz plot parameters to evaluate the shape of the $ρ$-meson peak. Observing this peak in the energy distribution of the $τ\to ηπν$ decay would be a background-free signature of a second-class amplitude. The scalar form factor is also estimated from a phase dispersive representation using a plausible model for the $ηπ$ elastic scattering $S$-wave phase shift and a sum rule constraint in the inelastic region. We indicate how a possibly exotic nature of the $a_0(980)$ scalar meson manifests itself in a dispersive approach. A remark is finally made on a second-class amplitude in the $τ\toππν$ decay.