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Based on $448.1 \times 10^6$ $\psi(3686)$ events collected with the BESIIIdetector, the decays $\psi(3686)\to\gamma\chi_{cJ}, \chi_{cJ} \to\gamma\gamma~(J=0, 1, 2)$ are studied. The decay branching fractions of$\chi_{c0,c2} \to \gamma\gamma$ are measured to be$\mathcal{B}(\chi_{c0}\to\gamma\gamma) = (1.93 \pm 0.08 \pm 0.05 \pm0.05)\times 10^{-4}$ and $\mathcal{B}(\chi_{c2}\to\gamma\gamma) = (3.10 \pm0.09 \pm 0.07 \pm 0.11)\times 10^{-4} $, which correspond to two-photon decaywidths of $\Gamma_{\gamma\gamma}(\chi_{c0}) = 2.03 \pm 0.08 \pm 0.06 \pm 0.13~\rm{keV}$ and $\Gamma_{\gamma\gamma}(\chi_{c2}) = 0.60 \pm 0.02 \pm 0.01 \pm0.04 ~\rm{keV}$ with a ratio of$\mathcal{R}=\Gamma_{\gamma\gamma}(\chi_{c2})/\Gamma_{\gamma\gamma}(\chi_{c0})=0.295 \pm 0.014 \pm 0.007 \pm 0.027$, where the uncertainties are statistical,systematic and associated with the uncertainties of$\mathcal{B}(\psi(3686)\to\gamma\chi_{c0,c2})$ and the total widths$\Gamma(\chi_{c0,c2})$, respectively. For the forbidden decay of$\chi_{c1}\to\gamma\gamma$, no signal is observed, and an upper limit on thetwo-photon width is obtained to be $\Gamma_{\gamma\gamma}(\chi_{c1})<5.3~\rm{eV}$ at the 90\% confidence level. The ratio of the two-photon widthsbetween helicity-zero and helicity-two components in the decay$\chi_{c2}\to\gamma\gamma$ is also measured to be $f_{0/2} =\Gamma^{\lambda=0}_{\gamma\gamma}(\chi_{c2})/\Gamma^{\lambda=2}_{\gamma\gamma}(\chi_{c2})= (0.0 \pm 0.6 \pm 1.2)\times 10^{-2}$, where the uncertainties are statisticaland systematic, respectively.

Improved measurements of two-photon widths of the $\chi_{cJ}$ states and helicity analysis for $\chi_{c2}\to\gamma\gamma$