We study the possible bound states of the $D_1D$ system in the Bethe-Salpeter (BS) formalism in the ladder and instantaneous approximations. By solving the BS equation numerically with the kernel containing one-particle exchange diagrams and introducing three different form factors (monopole, dipole, and exponential form factors) at the vertices, we investigate whether the isoscalar and isovector $D_1D$ bound states may exist, respectively. We find that $Y(4260)$ could be accommodated as a $D_1D$ molecule, whereas the interpretation of $Z_2^+(4250)$ as a $D_1D$ molecule is disfavored. The bottom analog of $Y(4260)$ may exist but that of $Z_2^+(4250)$ does not.
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