$\mathcal R(3780)$ Resonance Interpreted as the $1^3D_1$-Wave Dominant State of Charmonium from Precise Measurements of the Cross Section of $e^+e^-\rightarrow$ Hadrons

Abstract

We report the precise measurements of the cross section of $e^+e^-\rightarrow$ 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 $\psi(3770)$ resonance was discovered, and was subsequently interpreted as the $1^3D_1$-wave dominant state of charmonium. However, our analysis of the total-hadron cross sections indicates that the $\psi(3770)$ 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 $1^3D_1$ charmonium state.