Polarization transfer in $\psi'\to\psi\pi\pi$: a complete spin density matrix analysis framework

A theoretical framework based on the Spin Density Matrix (SDM) formalism is developed to describe polarization transfer in the decay chain $e^+e^- \rightarrow \psi^\prime \rightarrow \psi\pi\pi$. Explicit relations connecting the SDMs of $\psi^\prime$ and $\psi$ are derived, generalizing Cahn's analysis into a complete SDM treatment. For the dominant $S$-wave $\pi\pi$ emission, the SDM is shown to be perfectly preserved, $\rho_\psi = \rho_{\psi^\prime}$, rendering the $\psi$ an ideal probe of the initial polarization state. Deviations arising from $D$-wave contributions are quantified, and a self-consistency experimental test is proposed that simultaneously validates the framework and constrains partial wave amplitudes. This formalism provides a consistent basis for extracting $\psi$ polarization and for amplitude analyses of subsequent $\psi$ decays in a continuum-background-free environment. The framework extends to other hadronic transitions, including $\psi' \to h_c\pi^0$ in charmonium and $\Upsilon(nS) \to \Upsilon(mS)\pi\pi$ in bottomonium, as well as to electroweak processes such as $e^+e^- \to Z^\ast \to ZH$, where the same angular-momentum structure governs polarization transfer -- offering a unified probe of dynamics from charmonium to the Higgs sector.