$\mathcal{R}(3780)$ Resonance Interpreted as the $1^3{D}_1$-Wave Dominant State of Charmonium from Precise Measurements of the Cross Section of $e^{+}{e}^{-}\to$ Hadrons

Abstract

We report the precise measurements of the cross section of $e^{+}{e}^{-}\to$ hadrons at center-of-mass energies from 3.645 to 3.871 GeV. We thereby perform the most precise study of the cross sections and find a complex system composed of three resonances of $\mathcal{R}(3760)$, $\mathcal{R}(3780)$, and $\mathcal{R}(3810)$. For the first time, we measure the $\mathcal{R}(3810)$ electronic width to be $(19.4\pm 7.4\pm 12.1)$ eV. For the $\mathcal{R}(3760)$ resonance, we measure the mass to be $(3751.9\pm 3.8\pm 2.8)$ MeV/$c^2$, the total width to be $(32.8\pm 5.8\pm 8.7)$ MeV, and the electronic width to be $(184\pm 75\pm 86)$ eV. For the $\mathcal{R}(3780)$ resonance, we measure its mass to be $(3778.7\pm 0.5\pm 0.3)$ MeV/$c^2$, total width to be $(20.3\pm 0.8\pm 1.7)$ MeV, and electronic width to be $(265\pm 67\pm 83)$ eV. Forty-seven years ago, the $\ddot{I}ˆ(3770)$ resonance was discovered, and was subsequently interpreted as the $1^3{D}_1$-wave dominant state of charmonium. However, our analysis of the total-hadron cross sections indicates that the $\ddot{I}ˆ(3770)$ is not a single state, but a complex system composed of the $\mathcal{R}(3760)$, $\mathcal{R}(3780)$, and $\mathcal{R}(3810)$ resonances. Among these, we interpret the $\mathcal{R}(3780)$ is a resonance dominated by the $1^3{D}_1$ charmonium state.