Form factors of the isovector scalar current and the $etapi$ scattering phase shifts

A model for S-wave $etapi$ scattering is proposed which could be realistic in an energy range from threshold up to above one GeV, where inelasticity is dominated by the $Kbar{K}$ channel. The $T$-matrix, satisfying two-channel unitarity, is given in a form which matches the chiral expansion results at order $p^4$ exactly for the $etapitoetapi$, $etapito Kbar{K}$ amplitudes and approximately for $Kbar{K}to Kbar{K}$. It contains six phenomenological parameters. Asymptotic conditions are imposed which ensure a minimal solution of the Muskhelishvili-Omn`es problem, thus allowing to compute the $etapi$ and $Kbar{K}$ form factor matrix elements of the $I=1$ scalar current from the $T$-matrix. The phenomenological parameters are determined such as to reproduce the experimental properties of the $a_0(980)$, $a_0(1450)$ resonances, as well as the chiral results of the $etapi$ and $Kbar{K}$ scalar radii which are predicted to be remarkably small at $O(p^4)$. This $T$-matrix model could be used for a unified treatment of the $etapi$ final-state interaction problem in processes such as $etato eta pipi$, $phitoetapigamma$, or the $etapi$ initial-state interaction in $etato3pi$.

Analyticity of $etapi$ isospin-violating form factors and the $tautoetapi u$ second-class decay

We consider the evaluation of the $etapi$ 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 $eta$ meson in QCD. Unitarity relates the vector form factor to the $etapi to pipi$ amplitude: we exploit progress in formulating and solving the Khuri-Treiman equations for $etato 3pi$ and in experimental measurements of the Dalitz plot parameters to evaluate the shape of the $rho$-meson peak. Observing this peak in the energy distribution of the $tauto eta pi u$ 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 $etapi$ 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 $tautopipi u$ decay.