Zooming in on Multiquark Hadrons within QCD Sum-Rule Approaches

Abstract

Aiming at self-consistent descriptions of multiquark hadrons (such as tetraquarks, pentaquarks, hexaquarks) by means of QCD sum rules, we note that the totality of contributions to two-point or three-point correlation functions that involve, respectively, either two or just a single operator capable of interpolating the particular multiquark under study can be straightforwardly disentangled into two disjoint classes defined by unambiguously identifiable members. The first is formed by so-called multiquark-phile contributions which indeed might support multiquarks. In the case of flavour-exotic tetraquarks, by definition composed of four (anti–) quarks of mutually different flavours, a tetraquark-phile contribution has to exhibit two or more gluon exchanges of appropriate topology. The second consists of contributions evidently not bearing any relation to multiquarks; these must be discarded when studying multiquarks by QCD sum rules. The first class only should enter the “multiquark-adequate” QCD sum rules for exotic hadrons.

Tetraquark-Adequate QCD Sum Rules

“…Insights of this kind prove to be of utmost importance for the exploitation of such correlation functions in Sections 3 and 4. For the two types ( 8) and ( 9) of flavour-cryptoexotic four-point correlation functions, some tetraquark-phile contributions of lowest order O(α 2 s ) illustrating their conceivable topology are exemplified, for both varieties of flavour-preserving correlation functions (8), by Figure 1 and, for all flavour-rearranging correlation functions (9), by Figure 2. 5) of the flavour-preserving type (8), specified in Definition 2: exemplary contributions [2,7] established to be tetraquark-phile (according to Proposition 1), of the lowest perturbative order capable of satisfying the preconditions of Proposition 1, that is, of the order O(α 2 s ).…”

“…The decisive move towards a discussion of the flavour-cryptoexotic tetraquarks (2) that complies with their exotic nature is the identification of those contributions to the four-point correlation functions (5) that belong to the tetraquark-phile type demanded by Proposition 1. For the case of the three-flavour tetraquarks (3) adopted for illustration, the crucial four-point correlation functions are precisely those of the form (8) and (9). The systematic and thorough scrutiny for their tetraquark-phile contributions may be accomplished, order by order in the (perturbative) expansion of the four-point correlation functions (5) with respect to the strong coupling (1), by application of the (analytic) tool provided by the Landau equations [11].…”

“…In contrast to this, in the class of tetraquark mesons that exhibit the maximum number of four mutually different open quark flavours (reviewed in Ref [9](5), on the one hand, and related flavour-rearranging correlation functions(5), on the other hand, the large-N c behaviour of the respective tetraquark-phile contributions differs by one order of N c .…”

“…For any flavour-rearranging correlation function ( 9), the case-by-case judgement might turn out to be in order. The actual feasibility of any such analysis has been claimed and a conceivable route briefly indicated, for the flavour-preserving quark distributions (8), in Reference [24] but, for the flavour-rearranging quark distribution (9), in Reference [12].…”

“…Figure 2.Flavour-cryptoexotic four-current correlation function(5) of the flavour-rearranging type(9), specified in Definition 2: exemplary contributions[2,7] established to be tetraquark-phile (according to Proposition 1), of the lowest perturbative order capable of satisfying the preconditions of Proposition 1, that is, of the order O(α 2 s ). This order emerges from two internal exchanges of gluons, depicted in form of curly lines.…”