Observation of ${e}^{+}{e}^{-}\to {\pi }^{+}{\pi }^{-}\psi \left(3770\right)$ and ${D}_{1}\left(2420{\right)}^{0}{\overline{D}}^{0}+c.c.$

Several intermediate states of the reaction channels ${e}^{+}{e}^{-}\to {\pi }^{+}{\pi }^{-}{D}^{0}{\overline{D}}^{0}$ and ${e}^{+}{e}^{-}\to {\pi }^{+}{\pi }^{-}{D}^{+}{D}^{-}$ are studied using the data samples collected with the BESIII detector at center-of-mass energies above 4.08 GeV. For the first time in this final state, a $\psi \left(3770\right)$ signal is seen in the $D\overline{D}$ invariant mass spectrum, with a statistical significance of $5.2\sigma $ at $\sqrt{s}=4.42\text{}\text{}\mathrm{GeV}$. There is also evidence for this resonance at $\sqrt{s}=4.26$ and 4.36 GeV with statistical significance of $3.2\sigma $ and $3.3\sigma $, respectively. In addition, the Born cross section of ${e}^{+}{e}^{-}\to {\pi }^{+}{\pi }^{-}\psi \left(3770\right)$ is measured. The proposed heavy-quark-spin-symmetry partner of the $X\left(3872\right)$, the state ${X}_{2}\left(4013\right)$, is also searched for in the $D\overline{D}$ invariant mass spectra. No obvious signal is found. The upper limit of the Born cross section of the process ${e}^{+}{e}^{-}\to {\rho }^{0}{X}_{2}\left(4013\right)$ combined with the branching fraction is measured. Also, the processes ${e}^{+}{e}^{-}\to {D}_{1}\left(2420\right)\overline{D}+c.c.$ are investigated. The neutral mode with ${D}_{1}\left(2420{\right)}^{0}\to {D}^{0}{\pi }^{+}{\pi }^{-}$ is reported with statistical significance of $7.4\sigma $ at $\sqrt{s}=4.42\text{}\text{}\mathrm{GeV}$ for the first time, and evidence with statistical significance of $3.2\sigma $ and $3.3\sigma $ at $\sqrt{s}=4.36$ and 4.60 GeV is seen, respectively. No evident signal for the process ${e}^{+}{e}^{-}\to {D}_{1}\left(2420{\right)}^{0}{\overline{D}}^{0}+c.c.,{D}_{1}\left(2420{\right)}^{0}\to {D}^{*+}{\pi }^{-}$ is reported. Evidence for ${e}^{+}{e}^{-}\to {D}_{1}\left(2420{\right)}^{+}{D}^{-}+c.c.,{D}_{1}\left(2420{\right)}^{+}\to {D}^{+}{\pi }^{+}{\pi }^{-}$ is reported with statistical significance of $3.1\sigma $ and $3.0\sigma $ at $\sqrt{s}=4.36$ and 4.42 GeV, respectively.