Study of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mi>χ</mml:mi><mml:mrow><mml:mi>c</mml:mi><mml:mi>J</mml:mi></mml:mrow></mml:msub></mml:math>radiative decays into a vector meson

The decays ${\ensuremath{\chi}}_{cJ}\ensuremath{\rightarrow}\ensuremath{\gamma}V$ ($V=\ensuremath{\phi}$, ${\ensuremath{\rho}}^{0}$, $\ensuremath{\omega}$) are studied with a sample of radiative ${\ensuremath{\psi}}^{\ensuremath{'}}\ensuremath{\rightarrow}\ensuremath{\gamma}{\ensuremath{\chi}}_{cJ}$ events in a sample of $(1.06\ifmmode\pm\else\textpm\fi{}0.04)\ifmmode\times\else\texttimes\fi{}{10}^{8}{\ensuremath{\psi}}^{\ensuremath{'}}$ events collected with the BESIII detector. The branching fractions are determined to be: $\mathcal{B}({\ensuremath{\chi}}_{c1}\ensuremath{\rightarrow}\ensuremath{\gamma}\ensuremath{\phi})=(25.8\ifmmode\pm\else\textpm\fi{}5.2\ifmmode\pm\else\textpm\fi{}2.3)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}$, $\mathcal{B}({\ensuremath{\chi}}_{c1}\ensuremath{\rightarrow}\ensuremath{\gamma}{\ensuremath{\rho}}^{0})=(228\ifmmode\pm\else\textpm\fi{}13\ifmmode\pm\else\textpm\fi{}22)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}$, and $\mathcal{B}({\ensuremath{\chi}}_{c1}\ensuremath{\rightarrow}\ensuremath{\gamma}\ensuremath{\omega})=(69.7\ifmmode\pm\else\textpm\fi{}7.2\ifmmode\pm\else\textpm\fi{}6.6)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}$. The decay ${\ensuremath{\chi}}_{c1}\ensuremath{\rightarrow}\ensuremath{\gamma}\ensuremath{\phi}$ is observed for the first time. Upper limits at the 90% confidence level on the branching fractions for ${\ensuremath{\chi}}_{c0}$ and ${\ensuremath{\chi}}_{c2}$ decays into these final states are determined. In addition, the fractions of the transverse polarization component of the vector meson in ${\ensuremath{\chi}}_{c1}\ensuremath{\rightarrow}\ensuremath{\gamma}V$ decays are measured to be ${0.29}_{\ensuremath{-}0.12\ensuremath{-}0.09}^{+0.13+0.10}$ for ${\ensuremath{\chi}}_{c1}\ensuremath{\rightarrow}\ensuremath{\gamma}\ensuremath{\phi}$, $0.158\ifmmode\pm\else\textpm\fi{}{0.034}_{\ensuremath{-}0.014}^{+0.015}$ for ${\ensuremath{\chi}}_{c1}\ensuremath{\rightarrow}\ensuremath{\gamma}{\ensuremath{\rho}}^{0}$, and ${0.247}_{\ensuremath{-}0.087\ensuremath{-}0.026}^{+0.090+0.044}$ for ${\ensuremath{\chi}}_{c1}\ensuremath{\rightarrow}\ensuremath{\gamma}\ensuremath{\omega}$, respectively. The first errors are statistical and the second ones are systematic.