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Inspired by the event accumulation around 2.6 GeV in the$\eta^\prime\pi^+\pi^-$ invariant mass spectrum of $J/\psi\to \gamma\eta^\prime\pi^+\pi^-$, which was reported by the BESIII Collaboration, wecarry out the study of the mass spectrum and decay behavior of four radialexcitations 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 theproton target which are relevant to these four discussed states. According tothis information, we give concrete experimental suggestion of searching forthem, which will become a new task for future experiments.

Potential higher radial excitations in the light pseudoscalar meson family