Form factors of the isovector scalar current and the $\eta \pi$ scattering phase shifts

A model for S-wave $\eta \pi$ scattering is proposed which could be realistic in an energy range from threshold up to above one GeV, where inelasticity is dominated by the $K\bar{K}$ 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 $\eta \pi \to \eta \pi$, $\eta \pi \to K\bar{K}$ amplitudes and approximately for $K\bar{K}\to K\bar{K}$. 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 $\eta \pi$ and $K\bar{K}$ 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 $\eta \pi$ and $K\bar{K}$ 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 $\eta \pi$ final-state interaction problem in processes such as ${\eta }^{\mathrm{\prime }}\to \eta \pi \pi$, $\varphi \to \eta \pi \gamma$, or the $\eta \pi$ initial-state interaction in $\eta \to 3\pi$.