Polarization transfer in $ψ'\toψππ$: 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 $e^+e^- \rightarrow ψ^\prime \rightarrow ψππ$. Explicit relations connecting the SDMs of $ψ^\prime$ and $ψ$ are derived, generalizing Cahn's analysis into a complete SDM treatment. For the dominant $S$-wave $ππ$ emission, the SDM is shown to be perfectly preserved, $ρ_ψ= ρ_{ψ^\prime}$, rendering the $ψ$ 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 $ψ$ polarization and for amplitude analyses of subsequent $ψ$ decays in a continuum-background-free environment. The framework extends to other hadronic transitions, including $ψ' \to h_cπ^0$ in charmonium and $Υ(nS) \to Υ(mS)ππ$ 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.