Radiative decays $\chi_{cJ}\rightarrow V\gamma $ within the QCD factorisation framework

We present a discussion of radiative decays $\chi_{cJ}\rightarrow\gamma \rho (\omega, \phi)$. The decay amplitudes are computed within the QCD factorisation framework. NRQCD has been used in the heavy meson sector as well as a collinear expansion in order to describe the overlap with light mesons in the final state. The colour-singlet contributions to all helicity amplitudes have been computed using the light-cone distribution amplitudes of twist-2 and twist-3. All obtained expressions are well defined at least in the leading-order approximation. The colour-octet contributions have also been studied in the Coulomb limit in order to exhibit their scaling behaviour. In order to understand the relevance of the different contributions we perform numerical estimates using the colour-singlet contributions. For the $\chi_{c1}\rightarrow \gamma V_\perp$ decays, to vector mesons with transverse polarisation, we find that the colour-singlet contribution potentially allows for a reliable description. On the other hand, for the $\chi_{c1}\rightarrow \gamma V_\Vert$ decays, to vector mesons with longitudinal polarisation, our finding indicate that the colour-octet mechanism may be important for a good description. We expect that more accurate measurements of the decay $\chi_{c2}\rightarrow\gamma V_{\perp,\Vert}$ can help to better understand the mechanism of radiative decays.