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$η(1295)\to3π$ decays

The radially excited pseudoscalar state $η(1295)$ is still poorly understood [as well as its $SU(3)$ partners $π(1300)$ and $K(1460 )$] and we want to attract attention experimenters to its comprehensive study. As for the main three-body decay $η(1295)\to ηππ$, here the main interest is the measurements of the shapes of the $ηπ$ and $ππ$ mass spectra in which the contributions from the $a_0(980)$ resonance and the large $ππ$ $S$ wave ($σ$) 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 $σ$ state production. Our main goal is to discuss the dynamics of the isospin-breaking decays $η(1295)\toπ^+ π^-π^0$ and $η(1295)\to3π^0$, whose experimental studies could continue the impressive story of isospin violations in the decays of light isoscalar mesons $ω\to2π$, $η\to3 π$, $η'\to3π$, $η(1405)\to3π$, and $f_1(1285)\to3π$. We estimate the widths of the isospin-breaking decays $η(1295)\to 3π$ produced via the mixing of the $π^0-η$ and $a^0_0(980) -f_0(980)$ states and also due to the $π^0(1300)-η(1295)$ mixing. Processes that have the potential for detecting $η(1295) \to3π$ decays are discussed.