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.