Observation of $e^{+}{e}^{-}\to {\pi }^{+}{\pi }^{-}\psi (3770)$ and $D_1(2420{)}^0{\bar{D}}^0+c\mathrm{.}c\mathrm{.}$

Several intermediate states of the reaction channels $e^{+}e^{-} \rightarrow \pi^{+} \pi^{-} D^{0} \bar{D}^{0}$and$e^{+}e^{-} \rightarrow \pi^{+} \pi^{-} D^{+}D^{-}$ 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 $\psi (3770)$ signal is seen in the $D\bar{D}$ invariant mass spectrum, with a statistical significance of $5.2\sigma$ at $\sqrt{s}=4.42$ 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 $e^{+}{e}^{-}\to {\pi }^{+}{\pi }^{-}\psi (3770)$ is measured. The proposed heavy-quark-spin-symmetry partner of the $X(3872)$, the state $X_2(4013)$, is also searched for in the $D\bar{D}$ invariant mass spectra. No obvious signal is found. The upper limit of the Born cross section of the process $e^{+}{e}^{-}\to {\rho }^0{X}_2(4013)$ combined with the branching fraction is measured. Also, the processes $e^{+}e^{-}\to D_{1}(2420)\bar{D} + c.c.$areinvestigated\mathrm{.}Theneutralmodewith$D_{1}(2420)^{0}\to D^{0}\pi^{+}\pi^{-}$isreportedwithstatisticalsignificanceof7.4$\sigma$at$\sqrt{s} = 4.42$ 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 $e^{+}e^{-}\to D_{1}(2420)^{0}\bar{D}^{0} + c.c., D_{1}(2420)^{0}\to D^{*+}\pi^{-}$ is reported. Evidence for $e^{+}e^{-}\to D_{1}(2420)^{+}D^{-} +~c.c., D_{1}(2420)^{+}\to D^{+}\pi^{+}\pi^{-}$ is reported with statistical significance of 3.1$\sigma$ and 3.0$\sigma$ at $\sqrt{s}$ = 4.36 and 4.42 GeV, respectively.