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

## TL;DR

This paper calculates the leptonic widths of high $\psi$-resonances using a coupled-channel model with inelasticity, deriving analytical expressions for mixing angles and probabilities, and provides predictions consistent with experimental data.

## Contribution

It introduces a unitary coupled-channel model with analytical expressions for mixing angles and probabilities, enabling accurate calculation of leptonic widths of high $\psi$-resonances.

## Key findings

- Calculated leptonic widths for $\psi(4040)$ and $\psi(4160)$ match experimental values.
- Predicted leptonic width and mass for the missing $\psi(4510)$ resonance.
- Derived energy-dependent mixing angles and $car c$ component probabilities.

## Abstract

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.