Abstract

In this work, we systematically study the $D{\overline{D}}^{*}/B{\overline{B}}^{*}$ and $D{D}^{*}/\overline{B}{\overline{B}}^{*}$ systems with the Bethe-Salpeter equation in the ladder and instantaneous approximations for the kernel. By solving the Bethe-Salpeter equation numerically with the kernel containing the direct and crossed one-particle exchange diagrams and introducing three different form factors (monopole, dipole, and exponential form factors) at the vertices, we find only the isoscalar $D{\overline{D}}^{*}/B{\overline{B}}^{*}$ and $D{D}^{*}/\overline{B}{\overline{B}}^{*}$ systems can exist as bound states. This indicate that the $X(3872)$ and ${T}_{cc}^{+}$ could be accommodated as ${I}^{G}({J}^{PC})={0}^{+}({1}^{++})$ $D{\overline{D}}^{*}$ and $(I){J}^{P}=(0){1}^{+}\text{ }\text{ }D{D}^{*}$ bound states while the bound-state explanations for ${Z}_{b}(10610)$ and ${Z}_{c}(3900)$ are excluded.

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