Possible \(D_{1}D_{1}\), \(D_{1} \bar{D}_{1}\), \(B_{1}B_{1}\) and \(B_{1} \bar{B}_{1}\) molecular states and the recoil corrections

Recoil correction appears at \(O\big (\frac{1}{M}\big )\), which turns out to be very essential for the hadronic molecules with heavy flavor. In the past, we always thought that the recoil corrections were unfavorable to the formation of the molecular states, but our research reveals its importance to form the di-hadron bound states. In some cases, we are unable to find the bound states without considering the recoil corrections. Under SU(2) chiral symmetry, we have studied the \(D_1 D_1\), \(D_1 \bar{D}_1\), \(B_1 B_1\) and \(B_1 \bar{B}_1\) systems in the framework of the One-Boson-Exchange (OBE) model with the treatments of the S–D wave mixing effect and the recoil corrections. Our results indicate that the recoil corrections cannot be ignored in some cases. For both the \(D_1 D_1\) and the \(B_1 B_1\) systems with \(I(J^P)=1(2^+)\), although the bound-state solution can be found without considering the recoil corrections, the inclusion of such corrections significantly enhances the credibility of the molecular picture, as evidenced by smaller cutoff parameters and larger binding energies in the refined results. In contrast, for the \(D_1\bar{D}_1\) system with \(I(J^P)=0(0^+)\) and \(I(J^P)=0(1^+)\), the recoil corrections markedly decrease the binding energy of these channels. Notably, for the \(B_1 B_1\) system with \(I(J^P)=1(0^+)\), the bound-state solution exists only when the recoil corrections are taken into account. Besides, di-hadron systems formed by \(B_{1}\) mesons are more likely to form bound states than those formed by \(D_{1}\) mesons within the same \(I(J^P)\).