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The process $e^+e^- \to \gamma\chi_{cJ}$ ($J$=1, 2) is studied via initialstate radiation using 980 fb$^{-1}$ of data at and around the $\Upsilon(nS)$($n$=1, 2, 3, 4, 5) resonances collected with the Belle detector at the KEKBasymmetric-energy $e^+e^-$ collider. No significant signal is observed exceptfrom $\psi(2S)$ decays. Upper limits on the cross sections between$\sqrt{s}=3.80$ and $5.56~{\rm GeV}$ are determined at the 90% credibilitylevel, which range from few pb to a few tens of pb. We also set upper limits onthe decay rate of the vector charmonium [$\psi(4040$), $\psi(4160)$, and$\psi(4415)$] and charmoniumlike [$Y(4260)$, $Y(4360)$, and $Y(4660)$] statesto $\gamma\chi_{cJ}$.

Measurement of $e^+e^- \to \gamma\chi_{cJ}$ via initial state radiation at Belle