Spin correlations in polarizations of P-wave charmonia χcJ and impact on J/ψ polarization

Based on a general form of the effective vertex functions for the decays of P-wave charmonia chi(cJ), angular distribution formulas for the subsequent decays chi(cJ) -> J/psi gamma and J/psi -> 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 J/psi and muon in the cascade decay chi(cJ) -> J/psi gamma -> mu(+)mu(-)gamma. The chi(cJ) polarization observables are expressed in terms of rational functions of the spin density matrix elements of chi(cJ) 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-J/psi. polarization result, we also discuss the impact on the observable prompt-J/psi 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.