The production of charmonium states from two-photon collisions has been measured and searched for by the Belle Collaboration at the KEKB $e^{+}e^{-}$ collider. In this analysis, the process $\gamma \gamma \rightarrow X \rightarrow J/\psi\gamma$, $J/\psi \rightarrow \ell^{+} \ell^{-} (\ell = e \: {\rm or} \: \mu)$ is analyzed by using a 971.1 fb$^{-1}$ Belle data sample, where $X$ is a charmonium state. Previously, Belle and CLEO collaborations have reported measurements of $\gamma \gamma \rightarrow \chi_{c2}(1P) \rightarrow J/\psi\gamma$, $J/\psi \rightarrow \ell^{+} \ell^{-}$ using 32.6 fb$^{-1}$ and 14.4 fb$^{-1}$ data samples, respectively. By analyzing an about 30 times larger data sample, we perform a more precise measurement of the $\chi_{c2}(1P)$ and search for other charmonium(-like) states. From the Monte Carlo samples the detection efficiency is estimated to be 7-15% under some conditions. Also shown are the results of a feasibility study using a signal MC sample from which the expected accuracy for a measurement of the $\chi_{c2}(1P)$ is evaluated.
We obtain $\Gamma_{\gamma\gamma}(\chi_{c2}(1P))$$\mathcal{B}(\chi_{c2}(1P)$$\rightarrow$$J/\psi\gamma)$$\mathcal{B}(J/\psi$$\rightarrow$$\ell^{+} \ell^{-})$=13.5$\pm$0.2 (stat.)$\pm$0.5(syst.) eV. It turns out that the statistical error will be 1.5%. This result is 6.5 times more precise than the previous one.