Observation of $η_c(2S) \to 3(π^+π^-)$ and measurements of $χ_{cJ} \to 3(π^+π^-)$ in $ψ(3686)$ radiative transitions

Abstract: The hadronic decay $\eta_c(2S) \to 3(\pi<sup>+\pi<sup>-)$ is observed with a statistical significance of 9.3 standard deviations using $(448.1\pm2.9)\times10<sup>6$ $\psi(3686)$ events collected by the BESIII detector at the BEPCII collider. The measured mass and width of $\eta_c(2S)$ are $(3643.4 \pm 2.3 (\rm stat.) \pm 4.4 (\rm syst.))$ MeV/$c<sup>2$ and $(19.8 \pm 3.9 (\rm stat.) \pm 3.1 (\rm syst.))$ MeV, respectively, which are consistent with the world average values within two standard deviations. The product branching fraction $$\mathcal{B}[\psi(3686)\to \gamma \eta_c(2S)]\times\mathcal{B}[\eta_c(2S)\to3(\pi<sup>+\pi<sup>-)]$$ is measured to be $$(9.2 \pm 1.0 (\rm stat.) \pm 0.9 (\rm syst.))\times10<sup>{-6}$$. Using $$\mathcal{B}[\psi(3686)\to \gamma \eta_c(2S)]=(7.0<sup>{+3.4}_{-2.5})\times10<sup>{-4}$$, we obtain $$\mathcal{B}[\eta_c(2S) \to 3(\pi^{+\pi^-)]} = (1.31 \pm 0.15 (\rm stat.) \pm 0.13 (\rm syst.)(^{{+0.64}_{-0.47})} (\rm extr))\times10<sup>{-2}$$, where the third uncertainty is from $\mathcal{B}[\psi(3686) \to \gamma \eta_c(2S)]$. We also measure the $\chi_{cJ} \to 3(\pi<sup>+\pi<sup>-)$ ($J=0, 1, 2$) decays via $\psi(3686) \to \gamma \chi_{cJ}$ transitions. The branching fractions are $$\mathcal{B}[\chi_{c0} \to 3(\pi^{+\pi^-)]} = (2.080\pm0.006 (\rm stat.)\pm0.068 (\rm syst.))\times10<sup>{-2}$$, $$\mathcal{B}[\chi_{c1} \to 3(\pi^{+\pi^-)]} = (1.092\pm0.004 (\rm stat.)\pm0.035 (\rm syst.))\times10<sup>{-2}$$, and $$\mathcal{B}[\chi_{c2} \to 3(\pi^{+\pi^-)]} = (1.565\pm0.005 (\rm stat.)\pm0.048 (\rm syst.))\times10<sup>{-2}$$.