Static tetraquark and pentaquark potentials

We evaluate the static $qq\overline{q}\overline{q}$ and $qqqq\overline{q}$ potentials in the quenched theory at $\beta =5.8$ and $\beta =6.0$ on a lattice of size ${16}^3\times 32$. We compare the static potentials to the sum of two meson potentials for the tetraquark system and to the sum of the baryonic and mesonic potentials for the pentaquark state, as well as, with the confining potential obtained in the strong coupling expansion.

Figure 1

The Wilson loop for the $qq\overline{q}\overline{q}$ system.

Figure 2

The Wilson loop for the $qqqq\overline{q}$ system.

Figure 3

The tetraquark minimal length. The angles at the junctions are $120°$.

Figure 4

The geometry used for the computation of the tetraquark static potential where we have taken the baryonic junction at the origin and the antibaryonic junction at $\mathbf{y}=(R_1,0,0)$. The quarks are located at positions ${\mathbf{r}}_1=(0,R_2/2,0),{\mathbf{r}}_2=(0,-R_2/2,0)$ and the antiquarks at ${\mathbf{r}}_3=(R_1,R_2/2,0)$ and ${\mathbf{r}}_4=(R_1,-R_2/2,0)$.

Figure 5

The pentaquark minimal length. The angles at the junctions are all $120°$.

Figure 6

Plateau values for the tetraquark potential. Top for the case where $R_1/a=2$ and $R_2/a=8$; middle for $R_1/a=4$ and $R_2/a=4$; bottom for $R_1/a=8$ and $R_2/a=2$. The data obtained using variational method $1$ are denoted by the circles, using variational method $2$ by the filled triangles and using Eq. (13) by the crosses. The circles and filled triangles are shifted horizontally for clarity.

Figure 7

The tetraquark static potential at $\beta =6.0$ (crosses) and $\beta =5.8$ (filled triangles) for $R_2=0.6\mathrm{f}\mathrm{m}$ as a function of the minimal length in physical units. We have applied a constant shift to the data at $\beta =5.8$. The jackknife errors are smaller than the size of the symbols.

Figure 8

The ratio $-\log [W(t)/W(t-1)]$ for the tetraquark static potential versus time in lattice units. Upper graph for $R_1=0.2\mathrm{f}\mathrm{m}$ and $R_2=0.6\mathrm{f}\mathrm{m}$ at $\beta =6.0$. Lower graph for $R_1=1.2\mathrm{f}\mathrm{m}$ and $R_2/a=0.6\mathrm{f}\mathrm{m}$ at $\beta =5.8$. Horizontal lines show the value obtained from fitting the lattice data to a constant with the solid line showing the value that we choose. The initial time used in the fit is shown by the position where the line begins. The dotted curve shows the result of fitting to Eq. (18).

Figure 9

The tetraquark static potential at $\beta =6.0$ versus the minimal length. Data obtained for $R_2=0.2$, 0.4, 0.6 and 0.8 fm are denoted with crosses, filled triangles, filled squares and filled circles, respectively.

Figure 10

The tetraquark static potential versus $R_1$ for $R_2=0.2\mathrm{f}\mathrm{m}$ (crosses), $0.4\mathrm{f}\mathrm{m}$ (filled triangles), $0.6\mathrm{f}\mathrm{m}$ (filled squares) and 0.8 fm (filled circles). The solid line is twice the $q\overline{q}$ potential.

Figure 11

The tetraquark static potential at $\beta =5.8$ for $R_2=0.3\mathrm{f}\mathrm{m}$ compared with $V_{\mathrm{m}\mathrm{i}\mathrm{n}}^{4q}$ (solid line) and with $2V_{q\overline{q}}(R_1)$ (dashed line).

Figure 12

The tetraquark static potential at $\beta =6.0$ for $R_2=0.8\mathrm{f}\mathrm{m}$ compared with $V_{\mathrm{m}\mathrm{i}\mathrm{n}}^{4q}$ (solid line) and with $2V_{q\overline{q}}(R_1)$ (dashed line).

Figure 13

The geometry used for the computation of the pentaquark static potential where we have taken the baryonic junctions at $\mathbf{x}=(R_1,0,0)$ and at $\mathbf{z}=(0,R_1,0)$ and the antibaryonic junction at $\mathbf{y}=(0,0,0)$. The quarks are located at positions ${\mathbf{r}}_1=(R_1,0,R_2/2),{\mathbf{r}}_2=(R_1,0,-R_2/2)$, ${\mathbf{r}}_3=(0,R_1,R_2/2)$ and ${\mathbf{r}}_4=(0,R_1,-R_2/2)$ and the antiquark at ${\mathbf{r}}_5=(0,0,R_1)$.

Figure 14

Plateau values for the pentaquark potential. Top for the case where $R_1/a=1$ and $R_2/a=8$; middle for $R_1/a=4$ and $R_2/a=2$; bottom for $R_1/a=8$ and $R_2/a=2$. The rest of the notation is as in Fig. 6.

Figure 15

The pentaquark static potential at $\beta =6.0$ (crosses) and $\beta =5.8$ (filled triangles) for $R_2=0.6\mathrm{f}\mathrm{m}$ as a function of the minimal length. The data at $\beta =5.8$ are shifted by a constant.

Figure 16

The pentaquark static potential at $\beta =6.0$ versus the minimal length. The notation is the same as that used in Fig. 9.

Figure 17

The pentaquark static potential versus $R_1$ for $R_2=0.2\mathrm{f}\mathrm{m}$ (crosses), $0.4\mathrm{f}\mathrm{m}$ (filled triangles), $0.6\mathrm{f}\mathrm{m}$ (filled squares) and $0.8\mathrm{f}\mathrm{m}$ (filled circles). The baryonic potential shifted by a constant is shown by the open squares.

Figure 18

The pentaquark static potential at $\beta =5.8$ for $R_2=0.3\mathrm{f}\mathrm{m}$ compared with $V_{\mathrm{m}\mathrm{i}\mathrm{n}}^{5q}$ (solid line) and with the sum of the baryonic and mesonic potentials (dashed line).

Figure 19

The pentaquark static potential at $\beta =6.0$ for $R_2=0.8\mathrm{f}\mathrm{m}$ compared with $V_{\mathrm{m}\mathrm{i}\mathrm{n}}^{5q}$ (solid line) and with the sum of the baryonic and mesonic potentials (dashed line).