Covariant equations for the tetraquark and more

We derive covariant equations for a system of two quarks and two antiquarks where the effect of quark-antiquark annihilation is taken into account. In our approach, only pairwise interactions are retained, while all possibilities of overcounting are excluded by (i) keeping terms in the kernel that are consistent with a meson-meson and diquark-antidiquark substructure, and (ii) introducing four-body equations with a novel structure that specifically avoids the generation of overcounted terms. The resulting tetraquark bound state equations are given for the case of general two-body interactions, and for the specific case of separable interactions that lead to a description of the tetraquark in terms of meson-meson and diquark-antidiquark degrees of freedom where the effects of quark-antiquark annihilation are included. The inclusion of $2q2\overline{q}$- and $q\overline{q}$-channel coupling in our approach enables a wide variety of applications of our equations to other processes within the $2q2\overline{q}$ system, and to other two-particle plus two-antiparticle systems.

Figure 1

Examples of terms involving $q\overline{q}$ annihilation contributing to the tetraquark amplitude within a model involving only meson, diquark and antidiquark constituents: (a) $MM$ scattering, (b) $D\overline{D}$ scattering, (c) $D\overline{D}\leftarrow MM$ transition. Such terms are not taken into account when the covariant four-body equations of KK [3] are used to describe the $2q2\overline{q}$ system.

Figure 2

Structure of the four-body kernel $K_2$ where only two-body forces are included. Each of the summed diagrams corresponds to the term $K_{a{a}^{\prime }}$ of Eq. (4), where index $a$ ($a^{\prime }$) is given by the numerical labels of the two top (bottom) quark ($q$) or antiquark ($\overline{q}$) lines. Note that the precise mathematical meaning of each diagram is given by Eq. (10).

Figure 3

Disconnected part of the $q\overline{q}$ $t$ matrix $X_a$. Left arrows indicate particles, as labeled, while right arrows indicate the corresponding antiparticles.

Figure 4

The amplitude $A_{23}$ (disconnected part of the $q\overline{q}$ $t$ matrix $X_{23}$). With the initial (final) quark assigned momentum label $p_2$ ($k_2$), and corresponding antiquark assigned momentum $-p_3$ ($-k_3$), so that $p_2-p_3=0$, the expression for $A_{23}$ is given as in Eq. (14).

Figure 5

Example of overcounting resulting from a replacement $T_a\to X_a=T_a+A_a$ in the KK equations: (a) amplitude $T_{14}{A}_{23}$ contained in the inhomogeneous term $X_{14,23}$ of Eq. (20), (b) amplitude $A_{23}{T}_{12}{T}_{34}{A}_{23}$ contained in the second iteration term $X_{14,23}{G}_0{X}_{12,34}{G}_0{X}_{13,23}$. The amplitude in (b) is already contained in the amplitude in (a).

Figure 6

Example of overcounting inherent in Eq. (20). Illustrated is amplitude $A_{13}{A}_{14}$, which is generated in the first iteration of Eq. (20), but that is also obtained from $A_{13}$ when the initial state antiquark labels 3 and 4 are interchanged through antisymmetrization.

Figure 7

Diagrammatic representation of the kernels of Eqs. (50): (a) kernels with $q\overline{q}$ intermediate states, as given explicitly in Eq. (54), (b) kernels with $2q2\overline{q}$ intermediate states, as given explicitly by Eq. (50b).

Figure 8

Form of the interaction between quark 1 and antiquark 4 inside the amplitude $A_{23}{T}_{12}{T}_{34}{A}_{23}$.