Can the π⁺ χ_{c 1} resonance structures be D^* over(D, ̄)^* and D₁ over(D, ̄) molecules?

We use QCD sum rules to study the recently observed resonance-like structures in the π⁺ χ_{c 1} mass distribution, Z₁⁺ (4050) and Z₂⁺ (4250), considered as D^{* +} over(D, ̄)^{* 0} and D₁⁺ over(D, ̄)⁰ + D⁺ over(D, ̄)₁⁰ molecules with the quantum number J^P = 0⁺ and J^P = 1^-, respectively. We consider the contributions of condensates up to dimension eight and work at leading order in α_s. We obtain m_{D* D*} = (4.15 ± 0.12) GeV, around 100 MeV above the D^* D^* threshold, and m_{D1 D} = (4.19 ± 0.22) GeV, around 100 MeV below the D₁ D threshold. We conclude that the D^{* +} over(D, ̄)^{* 0} state is probably a virtual state that is not related with the Z₁⁺ (4050) resonance-like structure. In the case of the D₁ D molecular state, considering the errors, its mass is consistent with both Z₁⁺ (4050) and Z₂⁺ (4250) resonance-like structures. Therefore, we conclude that no definite conclusion can be drawn for this state from the present analysis.