Polarization transfer in $ψ'\toψππ$: 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 $e^+e^- \rightarrow ψ^\prime \rightarrow ψππ$. 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.

Figures (5)

• Figure 1: Schematic diagram for the transition $\psi^\prime \to \psi\pi\pi$. The quantum numbers $(\ell, L)$ denote the orbital angular momentum of the $\pi\pi$ system and the relative motion between $(\pi\pi)$ and $\psi$, respectively. Parity and charge conjugation conservation restrict $\ell$ and $L$ to even values, with $\ell=0,2$ and $L=0,2$ being dominant. Blobs represent non-perturbative hadronization processes; wavy lines denote soft gluon exchanges mediating the transition.
• Figure 2: Two-dimensional dilepton angular distribution $W(\theta,\phi)$ from Eq. \ref{['eq:WthetaPhi']} for different polarization states. (a) Full transverse polarization: $\lambda_\theta=1, \lambda_\phi=\lambda_{\theta\phi}=0$. (b) Mixed polarization: $\lambda_\theta=0.5, \lambda_\phi=0.3, \lambda_{\theta\phi}=0.1$. (c) Nearly isotropic: $\lambda_\theta=0.1, \lambda_\phi=0.05, \lambda_{\theta\phi}=0.02$. The distributions demonstrate how SDM elements shape experimentally observable patterns.
• Figure 3: Observed Fisher information for (left) $\lambda_\theta$, (middle) $\lambda_\phi$, and (right) $\lambda_{\theta\phi}$ on the $(\cos\theta, \phi)$ plane. High Fisher information (bright regions) indicates phase space regions that provide maximum statistical sensitivity. For $\lambda_\theta$, sensitivity peaks near the beam axis ($|\cos\theta| \to 1$). For $\lambda_\phi$ and $\lambda_{\theta\phi}$, maximum sensitivity occurs at intermediate polar angles with specific azimuthal structure.
• Figure 4: Three-path self-consistent test of polarization transfer. Path A measures the parent SDM $\rho'_{\text{(expt)}}$ from $\psi' \to \mu^+\mu^-$. Path B extracts the transition matrix $T$ from PWA of $\psi' \to \psi\pi\pi$. These are combined to calculate $\rho_{\text{(calc)}}$. Path C directly measures the daughter SDM $\rho_{\text{(expt)}}$ from $\psi \to \mu^+\mu^-$ in the decay chain. Consistency between $\rho_{\text{(calc)}}$ and $\rho_{\text{(expt)}}$ validates the entire framework and constrains partial wave amplitudes.
• Figure 5: Schematic for $\psi' \to h_c\pi^0$. Only $L=0,2$ orbital angular momenta are allowed.