Abstract
Due to the cluster reducibility of multiquark operators, a strong interplay exists in tetraquarks between the compact structures, resulting from the direct confining forces acting on quarks and gluons, and the molecular structure, dominated by the mesonic clusters. This issue is studied within an effective field theory approach, where the compact tetraquark is treated as an elementary particle. The key ingredient of the analysis is provided by the primary coupling constant of the compact tetraquark to the two mesonic clusters, considered here in the framework of a scalar interaction. Under the influence of this coupling, an initially formed compact tetraquark bound state evolves towards a new structure, where a molecular configuration is also present. In the strong-coupling limit, the evolution may end with a shallow bound state of the molecular type. The strong-coupling regime is also favored by the large Nc properties of QCD. The interplay between compact and molecular structures may provide a natural explanation of the existence of many shallow bound states.
Highlights
The theoretical issue faced by exotic hadrons, called “multiquark states”, is whether they are formed like ordinary hadrons, by means of the confining forces that act on the quarks and gluons, or whether they are formed like molecular states, by means of the effective forces that act on ordinary hadrons [16]. (The term “molecule” refers here to the color-neutral character of hadrons, in analogy with the molecules formed by atoms [30,31])
Molecular-type states [30,31,36,37], called “hadronic molecules”, are studied by means of effective field theories, based on approximate symmetry properties and nonrelativistic approximation [38–45]. The reason these two competing alternatives are arising is related to the fact that the multiquark operators that generate multiquark states are not color-irreducible, in contrast to the ordinary hadron case, in the sense that they are decomposable along combinations of clusters of ordinary hadron operators [16,46]
The formation of compact tetraquarks as definite stable bound states remains a matter of debate. This is related to the “cluster reducibility” problem, in the sense that the multiquark operators that create tetraquarks are reducible to a combination of mesonic clusters, and the compact tetraquark state would rapidly dislocate into them and would be transformed into a molecular-type object [16,46,87,88]
Summary
The experimental discoveries over the last two decades of new particle candidates, corresponding to “exotic hadrons” [1–12], not fulfilling the scheme of the standard quark model [13–16], has given rise to thorough theoretical investigations for the understanding of the nature and structure of these states; recent review articles can be found in [17–29]. The diquark being considered in particular in its color-antisymmetric representation (ignoring here spin degrees of freedom) within a very small volume (pointlike or almost pointlike approximation), one always has tetraquark (or multiquark) bound states [68–74]; this is due to the fact that, in that approximation, all forces acting on the various small volumes (or points) are of the attractive confining types This is not the case of the molecular scheme, where the occurrence of a bound state depends on the strength of the attractive forces. In the strong coupling limit, the shift may even transform the compact tetraquark into a shallow bound state, typical of loosely bound hadronic molecules This phenomenon is best represented by means of the “elementariness” parameter Z, introduced by Weinberg [36], which measures the probability of a bound state to be considered as elementary, and the complementary quantity, (1 − Z ), representing its “compositeness”. A few detailed analytic expressions, approximating energy eigenvalues, are gathered in the Appendix
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