Compositeness of $T_{cc}$ and $X\left(3872\right)$ by considering decay and coupled-channels effects

The compositeness of weakly bound states is discussed using the effective field theory from the viewpoint of the low-energy universality. We introduce a model with coupling of the single-channel scattering to the bare state, and study the compositeness of the bound state by varying the bare state energy. In contrast to the naive expectation that the near-threshold states are dominated by the molecular structure, we demonstrate that a noncomposite state can always be realized even with a small binding energy. At the same time, however, it is shown that a fine tuning is necessary to obtain the noncomposite weakly bound state. In other words, the probability of finding a model with the composite dominant state becomes larger with the decrease of the binding energy, in accordance with the low-energy universality. For the application to exotic hadrons, we then discuss the modification of the compositeness due to decay and coupled-channels effects. We quantitatively show that these contributions suppress the compositeness, because of the increase of the fraction of other components. Finally, as examples of near-threshold exotic hadrons, the structures of $T_{cc}$ and $X\left(3872\right)$ are studied by evaluating the compositeness. We find the importance of the coupled-channels and decay contributions for the structures of $T_{cc}$ and $X\left(3872\right)$, respectively.

Figure 1

Schematic illustrations of the $T_{cc}$ system (left) and $X\left(3872\right)$ system (right).

Figure 2

The compositeness $X$ as a function of the normalized bare state energy ${\nu }_0/E_{\mathrm{typ}}$ with the binding energies $B=E_{\mathrm{typ}}$ (solid line), $B=0.01E_{\mathrm{typ}}$ (dashed line), and $B=B_{\mathrm{cr}}=0.243E_{\mathrm{typ}}$ (dotted line).

Figure 3

The illustration of the definitions of ${\nu }_c$ and $P_{\mathrm{comp}}$ with the bound state with $B=E_{\mathrm{typ}}$. $P_{\mathrm{comp}}$ is the fraction of the shaded region to the entire horizontal axis.

Figure 4

The fraction of the composite dominant region $P_{\mathrm{comp}}$ as a function of normalized binding energy $B/E_{\mathrm{typ}}$.

Figure 5

The compositeness obtained from the model calculation (13) (solid line) and from the central value of the weak-binding relation (21) (dashed line) with the fixed binding energy $B=E_{\mathrm{typ}}$ [panel (a)] and $B=0.01E_{\mathrm{typ}}$ [panel (b)].

Figure 6

The compositeness $X$ as a function of the normalized bare state energy ${\nu }_0/E_{\mathrm{typ}}$ for ${\lambda }_0=0$ (solid line), ${\lambda }_0=-\left|{\lambda }_0^{\mathrm{cr}}\right|$ (dotted line), and ${\lambda }_0=\left|{\lambda }_0^{\mathrm{cr}}\right|$ (dashed line) with $B=0.01E_{\mathrm{typ}}$.

Figure 7

The compositeness $X$ as a function of the normalized coupling constant of the four-point interaction ${\lambda }_0/\left|{\lambda }_0^{\mathrm{cr}}\right|$ for $B=E_{\mathrm{typ}}$ (solid line) and $B=0.01E_{\mathrm{typ}}$(dashed line). The bare state energy ${\nu }_0$ is fixed as ${\nu }_0=0.5E_{\mathrm{typ}}$.

Figure 8

The fraction of the composite dominant region $P_{\mathrm{comp}}$ as a function of normalized bare state energy ${\nu }_0/E_{\mathrm{typ}}$. The solid line stands for $P_{\mathrm{comp}}$ for ${\lambda }_0=0$, the dashed line for repulsive ${\lambda }_0=\left|{\lambda }_0^{\mathrm{cr}}\right|$, and the dotted line for attractive ${\lambda }_0=-\left|{\lambda }_0^{\mathrm{cr}}\right|$.

Figure 9

The compositeness $\tilde{X}$ as a function of the normalized bare state energy ${\nu }_0/E_{\mathrm{typ}}$ for $-B\le {\nu }_0\le E_{\mathrm{typ}}$. The solid (dashed) lines represent the results with $\mathrm{\Gamma }\ne 0$ ($\mathrm{\Gamma }=0$). Panel (a) corresponds to the case with $(B,\mathrm{\Gamma }/2)=(0.01E_{\mathrm{typ}},0.1E_{\mathrm{typ}})$, (b) to $(B,\mathrm{\Gamma }/2)=(0.01E_{\mathrm{typ}},E_{\mathrm{typ}})$, (c) to $(B,\mathrm{\Gamma }/2)=(E_{\mathrm{typ}},0.1E_{\mathrm{typ}})$, and (d) to $(B,\mathrm{\Gamma }/2)=(E_{\mathrm{typ}},E_{\mathrm{typ}})$.

Figure 10

The fraction of the composite dominant region $P_{\mathrm{comp}}$ as a function of the normalized binding energy $B/E_{\mathrm{typ}}$. The solid, dashed, and dotted lines stand for $\mathrm{\Gamma }=0, \mathrm{\Gamma }/2=0.1E_{\mathrm{typ}}$, and $\mathrm{\Gamma }/2=E_{\mathrm{typ}}$, respectively.

Figure 11

The compositeness as a function of the normalized bare state energy ${\nu }_0/E_{\mathrm{typ}}$ for $-B\le {\nu }_0\le E_{\mathrm{typ}}$ at fixed binding energy and the threshold energy differences $(B,\mathrm{\Delta }\omega )=(0.01E_{\mathrm{typ}},0.01E_{\mathrm{typ}})$ [panel (a)], $(B,\mathrm{\Delta }\omega )=(0.01E_{\mathrm{typ}},E_{\mathrm{typ}})$ [panel (b)], $(B,\mathrm{\Delta }\omega )=(E_{\mathrm{typ}},0.01E_{\mathrm{typ}})$ [panel (c)], and $(B,\mathrm{\Delta }\omega )=(E_{\mathrm{typ}},E_{\mathrm{typ}})$ [panel (d)]. The solid lines represent $X_1+X_2$, the dotted $\text{lines}{X}_1$, and the dashed lines the compositeness in the the single-channel case.

Figure 12

The fraction of the threshold channel composite dominant region $P_{\mathrm{comp}}$ as a function of the normalized binding energy $B/E_{\mathrm{typ}}$ for fixed threshold energy difference. The solid line represents the single-channel case, the dotted line $\mathrm{\Delta }\omega =10E_{\mathrm{typ}}$, and the dashed line $\mathrm{\Delta }\omega =E_{\mathrm{typ}}$.

Figure 13

The compositeness $\tilde{X}$ as a function of the bare state energy ${\nu }_0$. Panel (a) [(b)] shows the result of $T_{cc}$ [$X\left(3872\right)$]. The solid lines stand for the sum of the compositenesses of threshold and coupled channels, ${\tilde{X}}_1+{\tilde{X}}_2$, the dotted lines show ${\tilde{X}}_1$, and the dashed lines show ${\tilde{X}}_1+{\tilde{X}}_2$ with setting $\mathrm{\Gamma }=0$. The cutoff is fixed as $\mathrm{\Lambda }=140$ MeV.

Figure 14

Same as Fig. 13, but the cutoff is fixed as $\mathrm{\Lambda }=770$ MeV.