Measurement of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msup><mml:mi>e</mml:mi><mml:mo>+</mml:mo></mml:msup><mml:msup><mml:mi>e</mml:mi><mml:mo>−</mml:mo></mml:msup><mml:mo stretchy="false">→</mml:mo><mml:mi>γ</mml:mi><mml:msub><mml:mi>χ</mml:mi><mml:mrow><mml:mi>c</mml:mi><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>c</mml:mi><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mi>c</mml:mi><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math> cross sections at center-of…

The $e^+e^-\to\gamma\chi_{cJ}$ ($J=0,1,2$) processes are studied at center-of-mass energies ranging from 3.773 to 4.600 GeV, using a total integrated luminosity of 19.3 fb$^{-1}$ $e^+e^-$ annihilation data accumulated with the BESIII detector BEPCII. We observe for first time $e^+e^-\to\gamma \chi_{c1,c2}$ signals $\sqrt{s}=$ 4.180 GeV statistical significances 7.6$\sigma$ and 6.0$\sigma$, respectively. production cross section each energy is also measured. find that line shape \chi_{c1}$ can be described conventional charmonium states $\psi(3686)$, $\psi(3770)$, $\psi(4040)$, $\psi(4160)$. Compared this, \chi_{c2}$ channel, one more additional resonance added describe shape. Its mass width measured $M=4371.7\pm7.5\pm1.8$ MeV/$c^2$ $\Gamma^{tot}=51.1\pm17.6\pm1.9$ MeV, where uncertainties second systematic. significance this estimated 5.8$\sigma$, its parameters agree $Y(4360)$ previously reported in $e^+e^-\to\pi^+\pi^-\psi(3686)$, $Y(4390)$ $e^+e^-\to\pi^+\pi^-h_c$ within uncertainties. No significant signal \chi_{c0}$ process observed, upper limits Born sections $\sigma_{B}(e^+e^-\to\gamma \chi_{c0})$ 90\% confidence level reported.