Cluster reducibility of multiquark operators

It is shown that the multiquark gauge-invariant operators can, in general, be decomposed into combinations of products of ordinary hadronic operators, exhibiting their cluster reducibility. The latter property inhibits the formation of completely compact multiquark bound states. Multiquark operators still play a crucial role in the description of exotic states in regions of configuration space where the hadronic clusters are close to each other. Our proof gives a foundation for a unified viewpoint, where the multiquark-type and the molecular-type approaches play complementary roles, at the gauge-invariant nonlocal operator level.

Figure 1

Examples of line $C_{yx}$.

Figure 2

Pictorial representation of mesonic and baryonic gauge-invariant operators, with phase factors running along straight lines; in the baryonic case, $\epsilon$ is the Levi-Civita symbol, put here as a reminder of the completely antisymmetric structure of the junction vertex of the three phase-factor lines.

Figure 3

Pictorial representation of (a) tetraquark, (b) pentaquark, and (c) hexaquark operators. Hexaquark operators with three quark and three antiquark fields are omitted.

Figure 4

Group product law of phase factors: the product of the two phase factors along the lines $C_{yx}$ and $C_{zy}$, respectively, is equal to the phase factor along the composite line $C_{zyx}$.

Figure 5

Decomposition of the tetraquark operator into a combination of products of mesonic operators.

Figure 6

Decomposition of the pentaquark operator into a combination of products of mesonic and baryonic operators; the ellipsis indicates other types of the link, as in Fig. 5.

Figure 7

Decomposition of the hexaquark operator into a combination of products of baryonic operators; the ellipsis designates other types of the link, as in Fig. 5.

Figure 8

Mesonic and baryonic hybrid operators; $G$ is the gluon field strength.

Figure 9

Mesonic and baryonic operators in the $SU(N_c)$ case; $\epsilon$ is the Levi-Civita symbol in $N_c$ dimensions.

Figure 10

Tetraquark operators in the $SU(N_c)$ case, where two extreme cases are shown. The first diagram contains $(N_c-1)$ quarks and $(N_c-1)$ antiquarks, with a single link between the two subsystems. The last diagram contains two quarks and two antiquarks, with $(N_c-2)$ links between the subsystems.

Figure 11

Pentaquark operators in the $SU(N_c)$ case, with two extreme cases shown. The first diagram contains $2(N_c-1)$ quarks and $(N_c-2)$ antiquarks. The last diagram contains $(N_c-1)+2$ quarks and one antiquark.

Figure 12

Hexaquark operators in the $SU(N_c)$ case, with two extreme cases shown. The first diagram contains $N_c(N_c-1)$ quarks; at the central junction point, apart from the bifurcation into the two horizontal lines, there are bifurcations into $(N_c-2)$ lines, which in turn bifurcate each into $(N_c-1)$ lines. The last diagram contains $2(N_c-1)+2$ quarks.