Leptonic widths of high ψ -resonances in a unitary coupled-channel model

Abstract

The leptonic widths of high $\ensuremath{\psi}$-resonances are calculated in a coupled-channel model with unitary inelasticity, where analytical expressions for the mixing angles between $(n+1){^{3}S}_{1}$ and $n{^{3}D}_{1}$ states and probabilities ${Z}_{i}$ of the $c\overline{c}$ component are derived. These factors depend on energy (mass) and can be different for $\ensuremath{\psi}(4040)$ and $\ensuremath{\psi}(4160)$. However, our calculations give a small difference between the mixing angles, $\ensuremath{\theta}(\ensuremath{\psi}(4040))=(2{8}_{\ensuremath{-}2}^{+1})\ifmmode^\circ\else\textdegree\fi{}$ and $\ensuremath{\theta}(\ensuremath{\psi}(4160))=(2{9}_{\ensuremath{-}3}^{+2})\ifmmode^\circ\else\textdegree\fi{}$, and $\ensuremath{\sim}10%$ difference between the probabilities ${Z}_{1}(\ensuremath{\psi}(4040))=0.8{5}_{\ensuremath{-}0.02}^{+0.05}$ and ${Z}_{2}(\ensuremath{\psi}(4160))=0.79\ifmmode\pm\else\textpm\fi{}0.01$. It provides the leptonic widths ${\mathrm{\ensuremath{\Gamma}}}_{ee}(\ensuremath{\psi}(4040))=\phantom{\rule{0ex}{0ex}}(1.0\ifmmode\pm\else\textpm\fi{}0.1)\text{ }\text{ }\mathrm{keV}$ and ${\mathrm{\ensuremath{\Gamma}}}_{ee}(\ensuremath{\psi}(4160))=(0.62\ifmmode\pm\else\textpm\fi{}0.0.07)\text{ }\text{ }\mathrm{keV}$ in agreement with experiment; for $\ensuremath{\psi}(4415)$ the value ${\mathrm{\ensuremath{\Gamma}}}_{ee}(\ensuremath{\psi}(4415))=(0.66\ifmmode\pm\else\textpm\fi{}0.06)\text{ }\text{ }\mathrm{keV}$ is obtained, while for the missing resonance $\ensuremath{\psi}(4510)$, we predict its mass, $M(\ensuremath{\psi}(4500))=(4512\ifmmode\pm\else\textpm\fi{}2)\text{ }\text{ }\mathrm{MeV}$, and ${\mathrm{\ensuremath{\Gamma}}}_{ee}(\ensuremath{\psi}(4510))=(0.68\ifmmode\pm\else\textpm\fi{}0.14)\text{ }\text{ }\mathrm{keV}$.