Polarization transfer in $\ddot{I}{ˆ}^{\prime }\to \ddot{I}ˆ\ddot{I}€\ddot{I}€$: a complete spin density matrix analysis framework

Abstract

A theoretical framework based on the Spin Density Matrix (SDM) formalism is developed to describe polarization transfer in the decay chain $e^{+}{e}^{-}\to \ddot{I}{ˆ}^{\prime }\to \ddot{I}ˆ\ddot{I}€\ddot{I}€$. Explicit relations connecting the SDMs of $\ddot{I}{ˆ}^{\prime }$ and $\ddot{I}ˆ$ are derived, generalizing Cahn's analysis into a complete SDM treatment. For the dominant $S$-wave $\ddot{I}€\ddot{I}€$ emission, the SDM is shown to be perfectly preserved, $\ddot{I}{}_{\ddot{I}}ˆ=\ddot{I}{}_{\ddot{I}{ˆ}^{\prime }}$, rendering the $\ddot{I}ˆ$ 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 $\ddot{I}ˆ$ polarization and for amplitude analyses of subsequent $\ddot{I}ˆ$ decays in a continuum-background-free environment. The framework extends to other hadronic transitions, including $\ddot{I}{ˆ}^{\prime }\to h_c\ddot{I}{€}^0$ in charmonium and $\hat{I}\yen (nS)\to \hat{I}\yen (mS)\ddot{I}€\ddot{I}€$ 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.

13 pages, 5 figures