Lattice QCD study of four-quark components of the isosinglet scalar mesons: Significance of disconnected diagrams

We study the possible significance of four-quark states in the isosinglet scalar mesons ($J^{PC}=0^{++}$, $I=0$) by performing two-flavor full lattice QCD simulations on an $8^3\times 16$ lattice using the improved gauge action and the clover-improved Wilson quark action. In particular, we evaluate the propagators of molecular and tetraquark operators together with singly disconnected diagrams. In the computation of the singly disconnected diagrams we employ the $Z_2$-noise method with the truncated eigenmode approach. We show that the quark loops given by the disconnected diagrams play an essential role in propagators of tetraquark and molecular operators.

Figure 1

The diagrams for the propagator of the molecular operator $G^{\text{molec}}(t)$.

Figure 2

The diagrams for the propagator of the tetraquark operator $G^{\text{tetra}}(t)$.

Figure 3

The propagators of the molecule and its components at $\kappa =0.146$, 0.147, and 0.148. The solid diamonds represent the propagators of the molecule. The open circles, squares, and triangles represent the components of the molecular operators $2D(t)$, $C(t)$, and $-6A(t)$ in Eq. (6), respectively. The $D(t)$ and $A(t)$ diagrams are dominant in the molecule. The plots of $2D(t)$ and $-6A(t)$ are shifted to $t/a\pm 0.2$ for visibility.

Figure 4

The propagators of the tetraquark operator and its components at $\kappa =0.146$, 0.147, and 0.148. The solid squares represent the propagators of tetraquark. The open circles and triangles represent $4D^{\prime }(t)$ and $-4A^{\prime }(t)$, respectively, which are plotted at $t/a\pm 0.2$ for visibility. The singly disconnected diagrams $A^{\prime }(t)$ are dominant in the tetraquark.

Figure 5

The effective masses of the molecule (open triangles) and tetraquark (open squares) without the singly disconnected diagrams at $\kappa =0.146$, 0.147, and 0.148. The data are plotted at $t/a\pm 0.2$ for visibility.

Figure 6

The effective masses of the molecule (solid triangles) and tetraquark (solid squares) with the singly disconnected diagrams at $\kappa =0.146$, 0.147, and 0.148. The data are plotted at $t/a\pm 0.2$ for visibility.

Figure 7

The quark mass dependence of the square of the $\pi$ meson mass (solid circles), double the $\pi$ meson mass (open triangles), the $\rho$ meson mass (open circles), the mass of the molecule (diamonds), and the mass of the tetraquark (open squares). We plot the masses of the molecule both with (solid diamonds) and without (open diamonds) the singly disconnected diagram. The chiral limit is given by ${\kappa }_c=0.152(6)$.