$\eta(1295)\to3\pi$ decays

The radially excited pseudoscalar state $\eta(1295)$ is still poorly understood [as well as its $SU(3)$ partners $\pi(1300)$ and $K(1460 )$] and we want to attract attention experimenters to its comprehensive study. As for the main three-body decay $\eta(1295)\to \eta\pi\pi$, here the main interest is the measurements of the shapes of the $\eta\pi$ and $\pi\pi$ mass spectra in which the contributions from the $a_0(980)$ resonance and the large $\pi\pi$ $S$ wave ($\sigma$) should manifest themselves. To describe these mass spectra, we propose to use a simple isobar model with a single fitting parameter characterizing the relative intensity of the $a_0 (980)$ and $\sigma$ state production. Our main goal is to discuss the dynamics of the isospin-breaking decays $\eta(1295)\to\pi^+ \pi^-\pi^0$ and $\eta(1295)\to3\pi^0$, whose experimental studies could continue the impressive story of isospin violations in the decays of light isoscalar mesons $\omega\to2\pi$, $\eta\to3 \pi$, $\eta'\to3\pi$, $\eta(1405)\to3\pi$, and $f_1(1285)\to3\pi$. We estimate the widths of the isospin-breaking decays $\eta(1295)\to 3\pi$ produced via the mixing of the $\pi^0-\eta$ and $a^0_0(980) -f_0(980)$ states and also due to the $\pi^0(1300)-\eta(1295)$ mixing. Processes that have the potential for detecting $\eta(1295) \to3\pi$ decays are discussed.