Two-photon widths of the ${\chi }_{c0,2}$ states and helicity analysis for ${\chi }_{c2}\to \gamma \gamma$

Based on a data sample of $106\times {10}^6$ ${\psi }^{\prime }$ events collected with the BESIII detector, the decays ${\psi }^{\prime }\to \gamma {\chi }_{c0,2}$, ${\chi }_{c0,2}\to \gamma \gamma$ are studied to determine the two-photon widths of the ${\chi }_{c0,2}$ states. The two-photon decay branching fractions are determined to be $\mathcal{B}({\chi }_{c0}\to \gamma \gamma )=(2.24\pm 0.19\pm 0.12\pm 0.08)\times {10}^{-4}$ and $\mathcal{B}({\chi }_{c2}\to \gamma \gamma )=(3.21\pm 0.18\pm 0.17\pm 0.13)\times {10}^{-4}$. From these, the two-photon widths are determined to be ${\Gamma }_{\gamma \gamma }({\chi }_{c0})=(2.33\pm 0.20\pm 0.13\pm 0.17)\mathrm{keV}$, ${\Gamma }_{\gamma \gamma }({\chi }_{c2})=(0.63\pm 0.04\pm 0.04\pm 0.04)\mathrm{keV}$, and $\mathcal{R}={\Gamma }_{\gamma \gamma }({\chi }_{c2})/{\Gamma }_{\gamma \gamma }({\chi }_{c0})=0.271\pm 0.029\pm 0.013\pm 0.027$, where the uncertainties are statistical, systematic, and those from the PDG $\mathcal{B}({\psi }^{\prime }\to \gamma {\chi }_{c0,2})$ and $\Gamma ({\chi }_{c0,2})$ errors, respectively. The ratio of the two-photon widths for helicity $\lambda =0$ and helicity $\lambda =2$ components in the decay ${\chi }_{c2}\to \gamma \gamma$ is measured for the first time to be $f_{0/2}={\Gamma }_{\gamma \gamma }^{\lambda =0}({\chi }_{c2})/{\Gamma }_{\gamma \gamma }^{\lambda =2}({\chi }_{c2})=0.00\pm 0.02\pm 0.02$.

Figure 1

Upper plot: The fitted $E_{{\gamma }_1}$ spectrum for the ${\psi }^{\prime }$ data sample. The expected positions of $E_{{\gamma }_1}$ from ${\chi }_{c0}$, ${\chi }_{c1}$, ${\chi }_{c2}$ are indicated by arrows. The dashed curve shows the background line shape fixed to the shape in Fig. 2. Lower plot: The number of standard deviations, $\chi$, of data points from the fitted curves.

Figure 2

The background $E_{{\gamma }_1}$ spectrum. The points are from the off-${\psi }^{\prime }$ data. The curve is from a fit to the $\psi (3770)$ data.

Figure 3

The $E_{{\gamma }_1}$ spectrum for the radiative photon in the samples ${\psi }^{\prime }\to {\gamma }_1{\chi }_{c0,2}$, ${\chi }_{c0,2}\to K^{+}{K}^{-}$.

Figure 4

Distributions of $\cos {\theta }_1$, $\cos {\theta }_2$ and ${\varphi }_2$ for the ${\psi }^{\prime }\to \gamma {\chi }_{c2}$, ${\chi }_{c2}\to \gamma \gamma$, where the dots with error bars are background-subtracted data and the histograms are the fitted results.