Studying the $D_{1}D$ molecule in the Bethe-Salpeter equation approach

We study the possible bound states of the $D_{1}D$ 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 $D_{1}D$ bound states may exist, respectively. We find that $Y(4260)$ could be accommodated as a $D_{1}D$ molecule, whereas the interpretation of $Z_2^{+}(4250)$ as a $D_{1}D$ molecule is disfavored. The bottom analog of $Y(4260)$ may exist but that of $Z_2^{+}(4250)$ does not.

Figure 1

The direct-channel (a) and cross-channel (b) Feynman diagrams for the $D_{1}D$ system at the tree level.

Figure 2

Dependence of the cutoff ${\mathrm{\Lambda }}_M$ (a), ${\mathrm{\Lambda }}_D$ (b), and ${\mathrm{\Lambda }}_E$ (c) of $Y(4260)$ without considering $\sigma$ meson exchanged between the constituents on parameters ${\zeta }_1$ and ${\mu }_1$.

Figure 3

Numerical results of the wave function $\tilde{f}(|p_t|)$ for $Y(4260)$ without considering $\sigma$ meson contribution in the $D_1\overline{D}$ molecular picture with (a) the monopole form factor, (b) the dipole form factor, and (c) the exponential form factor for the parameters $\mu =0.1{\mathrm{GeV}}^{-1}$ and $\zeta =0.1$.

Figure 4

Dependence of the cutoff ${\mathrm{\Lambda }}_M$ (a), ${\mathrm{\Lambda }}_D$ (b), and ${\mathrm{\Lambda }}_E$ (c) for the $I=0$ $D_{1}D$ bound state without considering $\sigma$ meson contribution on the parameters ${\zeta }_1$ and ${\mu }_1$ with $E_b=-60\mathrm{MeV}$.

Figure 5

Dependence of the cutoff ${\mathrm{\Lambda }}_M$ (a), ${\mathrm{\Lambda }}_D$ (b), and ${\mathrm{\Lambda }}_E$ (c) for the $I=0$ $B_{1}B$ bound state without considering $\sigma$ meson contribution on the parameters ${\zeta }_1$ and ${\mu }_1$ with $E_b=-60\mathrm{MeV}$.

Figure 6

Numerical results of the wave function $\tilde{f}(|p_t|)$ for $Y(4260)$ in the $D_1\overline{D}$ molecular picture with (a) the monopole form factor, (b) the dipole form factor, and (c) the exponential form factor for the parameters $\mu =0.1{\mathrm{GeV}}^{-1}$ and $\zeta =0.1$.