Inspired by the event accumulation around 2.6 GeV in the $eta^primepi^+pi^-$ invariant mass spectrum of $J/psito gamma eta^primepi^+pi^-$, which was reported by the BESIII Collaboration, we carry out the study of the mass spectrum and decay behavior
of four radial excitations in the pseudoscalar meson family, which include $eta^{(prime)}(6S)$ and $eta^{(prime)}(7S)$. Combining with these analysis, we present the calculation of the reactions induced by a pion or kaon on the proton target which are relevant to these four discussed states. According to this information, we give concrete experimental suggestion of searching for them, which will become a new task for future experiments.
The unflavored light meson families, namely $omega_2$, $rho_2$, and $phi_2$, are studied systematically by investigating the spectrum and the two-body strong decays allowed by Okubo-Zweig-Iizuka rule. Including the four experimentally observed states
and other predicted states, phenomenological analysis of the partial decay widths can verify the corresponding assignments of these states into the families. Moreover, we provide typical branching ratios of the dominant decay channels, especially for missing ground states, which is helpful to search for or confirm them and explore more properties of these families at experiment.
