Search for $η_c(2S)\to 2(π^+π^-)$ and improved measurement of $χ_{cJ}\to 2(π^+π^-)$
We search for the hadronic decay $η_c(2S)\to 2(π^+π^-)$ in the $ψ(3686)\toγη_c(2S)$ radiative decay using $(27.12\pm 0.14)\times 10^8$ $ψ(3686)$ events collected by the BESIII detector at the BEPCII collider. No significant signal is found, and the upper limit of $\mathcal{B}[ψ(3686)\toγη_c(2S)]\mathcal{B}[η_c(2S)\to 2(π^+π^-)]$ is determined to be $0.78\times 10^{-6}$ at the 90\% confidence level. Using $ψ(3686)\toγχ_{cJ}$ transitions, we also measure the branching fractions of $\mathcal{B}[χ_{cJ(J=0,1,2)}\to 2(π^+π^-)]$, which are $\mathcal{B}[χ_{c0}\to 2(π^+π^-)]=(2.127\pm 0.002~(\mathrm{stat.})\pm 0.101~(\mathrm{syst.}))$\%, $\mathcal{B}[χ_{c1}\to 2(π^+π^-)]=(0.685\pm 0.001~(\mathrm{stat.})\pm 0.031~\mathrm{syst.}))$\%, and $\mathcal{B}[χ_{c2}\to 2(π^+π^-)]=(1.153\pm 0.001~(\mathrm{stat.})\pm 0.063~(\mathrm{syst.}))$\%.