Spin correlations in polarizations of P-wave charmonia $\chi_{cJ}$ and impact on $J/\psi$ polarization

Based on a general form of the effective vertex functions for the decays of P-wave charmonia $\chicj$, angular distribution formulas for the subsequent decays $\chicj\rightarrow \jpsi \gamma$ and $\jpsi \to \mu^+\mu^-$ are derived. The formulas are the same as those obtained in a different approach in the literature. Our formulas are expressed in a more general form, including parity violation effects and the full angular dependence of $\jpsi$ and muon in the cascade decay $\chicj\to\jpsi\gamma\to\mu^+\mu^-\gamma$. The $\chicj$ polarization observables are expressed in terms of rational functions of the spin density matrix elements of $\chicj$ production. Generalized rotation-invariant relations for arbitrary integer-spin particles are also derived and their expressions in terms of observable angular distribution parameters are given in the $\chi_{c1}$ and $\chi_{c2}$. To complement our previous direct-$\jpsi$ polarization result, we also discuss the impact on the observable prompt-$\jpsi$ polarization. As an illustrative application of our angular distribution formulas, we present the angular distributions in terms of the tree-level spin density matrix elements of $\chi_{c1}$ and $\chi_{c2}$ production in several different frames at the Large Hadron Collider. Moreover, a reweighting method is also proposed to determine the entire set of the production spin density matrix elements of the $\chi_{c2}$, some of which disappear or are suppressed for vanishing higher-order multipole effects making the complete extraction difficult experimentally.