Polarization study of the P-wave charmonium radiative decay into a light vector meson at $e^{+}{e}^{-}$ collider experiment

In this work, a formalism is presented for the helicity amplitude analysis of the decays $\psi (2S)\to {\gamma }_1{\chi }_{cJ},{\chi }_{cJ}\to {\gamma }_{2}V(V={\rho }^0,\phi ,\omega )$~(the subscript 1,2 is used to distinguish the two radiative photons), and the polarization expressions of the P-wave charmonia ${\chi }_{cJ}$ and the vector mesons ${\rho }^0,\phi ,\omega$ for experimental measurements at Electron-Positron Collider. In addition, we derive the formulae of the angular distributions of the ${\chi }_{c1,2}\to \gamma V$ to extract the degree of transverse polarization $P_T$ of $e^{+}{e}^{-}$ pairs with symmetric beam energy as well as the ratios of two helicity amplitudes $x$ (in ${\chi }_{c1}$ decays) and $x,y$ (in ${\chi }_{c2}$ decays) representing the relative magnitudes of transverse to longitudinal polarization amplitude, and validate it by performing the Monte Carlo simulation. Finally, the statistical sensitivity of $P_T$, $x$ and $y$ are estimated based on the large $\psi (2S)$ data samples collected at the current and proposed future $e^{+}{e}^{-}$ collider experiment.