The leptonic widths of high $\psi$-resonances in unitary coupled-channel model

Understand

The leptonic widths of high $\psi$-resonances are calculated in a coupled-channel model with unitary inelasticity, where analytical expressions for mixing angles between $(n+1)\,^3S_1$ and $n\,^3D_1$ states and probabilities $Z_i$ of the $c\bar c$ component are derived.

• Since these factors depend on energy (mass), different values of mixing angles $\theta(\psi(4040))=27.7^\circ$ and $\theta(\psi(4160))=29.5^\circ$, $Z_1\,(\psi(4040))=0.76$, and $Z_2\,(\psi(4160))=0.62$ are obtained.
• It gives the leptonic widths $\Gamma_{ee}(\psi(4040))=Z_1\, 1.17=0.89$~keV, $\Gamma_{ee}(\psi(4160))=Z_2\, 0.76=0.47$~keV in good agreement with experiment.
• For $\psi(4415)$ the leptonic width $\Gamma_{ee}(\psi(4415))=~0.55$~keV is calculated, while for the missing resonance $\psi(4510)$ we predict $M(\psi(4500))=(4515\pm 5)$~MeV and $\Gamma_{ee}(\psi(4510)) \cong 0.50$~keV.