Color SU(3) symmetry, confinement, stability, and clustering in the q(2)(q)over-bar(2) system
We examine the assumptions underlying the (model-dependent) predictions of q(2)(q) over bar (2) or tetraquark spectra. The models implemented so far have used only two-body interactions proportional to the color charges; that assumption is the source of many serious shortcomings. We extend the analysis to three- and four-body interactions based on color SU(3) algebra, while including all relevant information one has about three-quark forces from lattice QCD. Thus we find that (quasi)stable tetraquarks are not necessarily a consequence of color SU(3) dynamics, let alone of QCD. We make this statement and the conditions under which it holds more precise in the text. In the process we are led to a set of sufficient conditions for a mathematical description of the hadronic world as we know it, i.e., of baryons and q (q) over bar mesons, without going into the question of tetraquark existence. These conditions are as follows. (1) Stability: All the (colorless and colored) states energies mu...st be bounded from below. (2) Confinement: A color singlet q (q) over bar potential energy must (infinitely) rise with the separation distance. (3) Color ordering: Colored states must be heavier than color-neutral ones. (4) Clustering: Any multiquark color-singlet state Hamiltonian must turn into a sum of three-quark (baryons) and quark-antiquark (mesons) cluster Hamiltonians, in the limit of asymptotically large separations. We discuss the consistency of these four requirements with color SU(3) symmetry and with each other.