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

Abstract

Based on a general form of the effective vertex functions for the decays of P-wave charmonia $\text{chicj}$, angular distribution formulas for the subsequent decays $\text{chicj}\to \text{jpsi}\gamma$ and $\text{jpsi}\to {\mu }^{+}{\mu }^{-}$ 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 $\text{jpsi}$ and muon in the cascade decay $\text{chicj}\to \text{jpsi}\gamma \to {\mu }^{+}{\mu }^{-}\gamma$. The $\text{chicj}$ polarization observables are expressed in terms of rational functions of the spin density matrix elements of $\text{chicj}$ 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-$\text{jpsi}$ polarization result, we also discuss the impact on the observable prompt-$\text{jpsi}$ 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.

Comments: Version published in PRD, 23 pages, 18 figures