Mission Target: Exotic Multiquark Hadrons -- Sharpened Blades

Motivated by recent experimental progress in establishing the likely existence of (variants of) exotic hadrons, predicted to be formed by the strong interactions, various proposed concepts and ideas are compiled in an attempt to draft a coherent picture of the achievable improvement in the theoretical interpretation of exotic hadrons in terms of the underlying quantum field theory of strong interactions.


Significance of Fundamental Diverseness of Ordinary Hadrons and Multiquark States
Within the framework of (relativistic) quantum field theories, all strong interactions are described -at fundamental level -by quantum chromodynamics (QCD), a renormalizable gauge theory, invariant under local transformations forming a representation of the compact non-Abelian Lie group SU (3). Two sorts of particles constitute the (basic) dynamical degrees of freedom of QCD: massless vector gauge bosons labelled gluons, transforming (inevitably) according to the eight-dimensional adjoint representation 8 of SU (3), and spin- 1 2 fermions q a , labelled quarks, each distinguished from all others by some quark flavour degree of freedom a ∈ {u, d, s, c, b, t(, . . .?)} (1) and transforming according to the three-dimensional fundamental representation 3 of SU (3). The (few) fundamental parameters characterizing QCD are the masses m a of the quarks q a as well as the strong coupling g s , frequently adopted in form of a strong fine-structure coupling This designation as quantum chromodynamics derives from the fact that the quark and gluon degree of freedom affected by their gauge-group transformation is referred to as their colour. Among others, QCD features the phenomenon of colour confinement: not the (coloured) quarks and gluons but exclusively their colour-singlet hadron bound states [1] invariant under the action of the QCD gauge group are, in form of isolated states, experimentally observable. Closer inspection reveals that the hadron states have to be divided into two disjoint categories: • Conventional (ordinary) hadrons include all mesons that consist of only a pair of quark and antiquark, as well as all baryons that consist of three quarks or of three antiquarks. • Exotic hadrons are characterized by non-conventional quark and/or gluon compositions comprising multiquark states (tetraquarks, pentaquarks, hexaquarks, heptaquarks, etc.), "hybrid" quark-gluon bound states, or pure-gluon bound states (nick)named glueballs.
There is a (crucial) fundamental difference between conventional hadrons and exotic hadrons, based on a (more or less) trivial observation: any colour-singlet multiquark arrangement of a number of quarks and/or antiquarks may be decomposed (in one or more ways) into a set of states that are also colour singlets but consist of lesser numbers of quarks and/or antiquarks. Therefore, an (initially) tightly bound, "compact" multiquark hadron may reconfigure to molecular-type clusters of (ultimately) conventional hadrons, loosely bound by some residual arXiv:2302.14439v3 [hep-ph] 8 May 2023 forces [2,3]. In view of this, trustworthy attempts to describe exotic hadrons should (struggle to) take into account, too, the potential mixing of these two "phases" of multiquark hadrons.

Tetraquark Mesons -the Example of Multiquark Exotic Hadron States Par Excellence
All tetraquark mesons T are bound states of two antiquarks q a , q c and two quarks q b , q d , henceforth calling the masses of the four (anti-) quarks constituting such state m a , m b , m c , m d . On group-theoretical grounds, the presence of these mesons in the hadron spectrum without coming into conflict with confinement of colour is rendered possible by the appearance of two SU(3) singlet representations 1 in the (appropriate) tensor product of two fundamental SU(3) representations 3 and two (complex-conjugate) fundamental SU(3) representations 3 [24,36], as this product's decomposition into irreducible SU(3) representations 1, 8, 10, 10, 27 reveals: As far as its flavour degrees of freedom are concerned, the four quark constituents of any tetraquark state (3) may contribute, at most, four different quark flavours and, trivially, carry at least one, the same for all the four (anti-) quarks. Owing to such simultaneous involvement of both quarks and antiquarks, however, the latters' hadron bound states need not feature all of the available quark flavours. Table 1 presents the listing [21] of conceivable quark-flavour arrangements in the tetraquark state (3), with respect to both the number of different flavours a = b = c = d provided by two quarks and two antiquarks as well as the number of flavours exhibited by the related hadron, which might differ from the former number either because of mutual flavour-antiflavour compensations or because of quark-flavour double occurrences. 3 q a q b q c q b 4 q a q b q a q c 4 q a q b q b q c 2 q a q b q c q c 2 2 q a q b q a q b 4 q a q a q a q b 2 q a q a q b q a 2 q a q b q b q a 0 q a q a q b q b 0 1 q a q a q a q a 0 Needless to say, at least from the experimental point of view it may be more satisfactory if the exotic nature of a (suspected) multiquark is established already by its observed content of quark flavours. The corresponding species of multiquarks may be told apart by relying on For the quark-flavour arrangements of tetraquarks, Table 1 offers several options to meet the requirement of being considered flavour-exotic: of course, there can exist merely one flavour arrangement that incorporates four mutually different quark flavours; however, there exist a few self-evident options for flavour-exotic tetraquarks to comprise not more than two or three different quark flavours by involving one or even two double appearances of a given flavour.
Quite recently, various candidates for tetraquark states that are manifestly flavour-exotic by exhibiting (in accordance with Definition 1) four open quark flavours have been observed by experiment. Regarding the flavour compositions of these candidates, there are states each encompassing exactly one of all four lightest quarks [6,7,10,11] and "doubly flavoured" ones containing only three different flavours but one of these twice [8,9] (see summary of Table 2).

Correlation Functions of Hadron Interpolating Operators: Application to Multiquarks
For descriptions of hadronic states in terms of QCD, a pivotal contact point between the realm of QCD and the realm of hadrons is established by the concept of hadron interpolating operators. For a fixed hadron H under consideration, its -not necessarily unique -hadron interpolating operator, generically called O, is a gauge-invariant local operator composed of the QCD dynamical degrees of freedom, the quark and gluon field operators, that betrays its nonzero overlap with the hadron |H by the nonvanishing matrix element emerging from its getting sandwiched between the hadronic state |H and the QCD vacuum |0 : 0|O|H = 0. In all subsequent implementations of hadron interpolating operators, features such as parity or spin degrees of freedom can be safely ignored; they get therefore notationally suppressed.
For a conventional meson consisting of a quark of flavour b and an antiquark of flavour a, the most evident option for its interpolating operator is the quark-antiquark bilinear current For exotic hadrons belonging to the subset of tetraquark mesons characterized in Equation (3), the search for appropriate tetraquark interpolating operators, specifically named θ, is greatly facilitated by the observation [40] that (by means of suitable Fierz transformations [41]) every colour-singlet operator that is composed of two quarks and two antiquarks can be expressed by a linear combination of only two different products of colour-singlet conventional-meson interpolating operators of quark-bilinear-current shape (5). Thus, this "operator basis" reads Moreover, taking into account some useful identities recalled, for instance, by Equations (32) and (36) of Reference [26] or Equations (1) and (2) of Reference [37] may be regarded either as a kind of shortcut to or as explicit verification of these findings. The tetraquark interpolating operators (6) will provide some kind of playground for (most of) the ensuing considerations. That pleasing observation [40] points out a promising route how to reasonably proceed. Namely, the enabled basic two-current structure (6) of the tetraquark interpolating operators θ suggests to start (envisaged) analyses of tetraquarks from correlation functions -in general, defined by vacuum expectation values of time-ordered products, symbolized by T, of chosen field operators -of four quark-bilinear operators (5). If tolerated by the involved dynamics, in appropriate four-point correlation functions of such kind tetraquark states should become manifest by their contributions in form of intermediate-state poles. Momentarily focusing to only essential aspects, all these four-current correlation functions are of the general structure Upon application of well-understood procedures, the correlation functions (7) entail also the amplitudes encoding scatterings of two conventional mesons into two conventional mesons. Because of the two-current structure (6), contact with tetraquark states, in form of correlation functions involving tetraquark interpolating operators θ, can be established by identification or contraction of configuration-space coordinates of proper quark-bilinear currents j, forming • twice configuration-space contracted two-point correlation functions of two operators (6) • once contracted three-point correlation functions of one operator (6) and two operators (5) An immediate implication of the mere conceptual nature of unconventional multiquark states is, as already stressed in Section 1, their potential to undergo clustering without getting into conflict with colour confinement [2]. For the correlation-function underpinned analyses of tetraquark properties, this finding should be regarded as a strong hint that, presumably or even very likely, not all QCD-level contributions to some correlation function are, in general, of relevance for such formation of a tetraquark pole. It appears opportune to distinguish any contribution that may play a rôle in tetraquark studies even by nomenclature; this is done in Definition 2. A QCD contribution to a correlation function (7) is termed tetraquark-phile [18,23] if it is (potentially) capable of supporting the formation of a tetraquark-related intermediate-state pole.
As a guidance through the process of filtering all of the QCD-level contributions as implicitly requested by Definition 2, a self-evident, easy to implement criterion may be devised [17,19]: For a given four-point correlation function (7) with external momenta in initial state p 1 , p 2 and external momenta in final state q 1 , q 2 , considered as a function of the Mandelstam variable a QCD-level contribution is supposed to be tetraquark-phile if it exhibits a nonpolynomial dependence on s and if it develops an intermediate-state four-quark-related branch cut starting at the branch point For any contribution to a correlation function, the capability of supporting the formation of a tetraquark pole by satisfying all requirements in Proposition 1 may be straightforwardly and unambiguously decided by consulting the related Landau equations [42]: the existence of an appropriate solution to (the relevant set of) those Landau equations indicates the presence of an expected branch cut. References [19,26,37] show some examples worked out in all details. As announced in Section 1, the benefit of implementing such programme is exemplified for the meanwhile even experimentally observed [6,7,10,11] subset of all those flavour-exotic tetraquarks that exhibit not less than (the feasible maximum of) four unequal quark flavours: At least for the case of the definitely flavour-exotic tetraquarks (12), there exist two definitely distinguishable quark-flavour distributions in (from the point of view of intermediate states) incoming and outgoing states of a correlation function (7): its quark-flavour arrangements in initial and final state might be either identical or different. These two possibilities got names:
For the two categories of correlation functions (7), it is straightforward yet worthwhile (since instructive) to investigate their contributions of lowest orders to the perturbative expansions in powers of the strong fine-structure constant (2). Representative examples of contributions are given, for flavour-preserving cases, in Figures 1 and 2 (7) of flavour-rearranging type (left) and (right) contraction (9) to a correlation function of one tetraquark interpolating operator (6) and two quark-bilinear currents (5) [37]: typical contribution of lowest tetraquark-phile perturbative order O(α 2 s ).
• For flavour-preserving correlation functions, the line of argument proves to be, more or less, evident. All the contributions of the type of Figure 1(a) or of the type of Figure 1(b), involving at most one gluon exchange, are doubtlessly disconnected. The contributions that involve a single gluon exchange between their two (otherwise disconnected) quark loops vanish identically, due to the vanishing of the sum over colour degrees of freedom of each of the two quark loops. Phrased slightly more technically, this can be traced back to the tracelessness of all generators of a special unitary group, governing the couplings of quarks and gluons. Consequently, exclusively contributions that involve, at least, two gluon exchanges of an appropriate topology may be viewed as tetraquark-phile. These insights get, of course, corroborated by identifying these tetraquark-phile contributions according to Proposition 1 by explicit inspection [17] by way of their Landau equations. Replacing any double contraction (8) in Figure 1 by a single contraction (9) confirms the tetraquark-phile nature of contributions of the type of Figure 2 or related higher orders.
• For flavour-rearranging correlation functions, a simple optical guidance in this analysis is, beyond doubt, hardly imaginable: already the lowest-order contributions turn out to be connected. Rather, one has to gladly accept any assistance offered by that tool called Landau equations. For the three lowest-order contributions exemplified in Figure 3, the usage of this formalism is demonstrated, in full detail, in Appendix A of Reference [19], in the Appendix of Reference [37], as well as in Section 4 of Reference [26]. For this kind of analysis, it might prove advantageous to recast the encountered plots into box shape, by "unfolding" all these plots [15,19,26,37]. These efforts' outcome is that contributions of the type of Figure 3(a) or of the type of Figure 3(b), being characterized by no or only one internal gluon exchange, do not incorporate the requested four-quark singularities. Involvement of this feature starts not before the level of two gluon exchanges of suitable positioning, which then holds, of course, also for the single contractions (9) in Figure 4.
As an overall summary of the two classes of definitely flavour-exotic correlation functions (7) identified by Definition 4, the systematic scrutiny of their lowest-order contributions betrays that tetraquark-phile contributions (an essential ingredient, since providing the singularities that, upon summation, may support the development of intermediate-state tetraquark poles) will not emerge before the next-to-next-to-lowest order in a series expansion in powers of the strong fine-structure constant (2), that is, in terms of α s , have to be at least of the order O(α 2 s ).

Number of Colour Degrees of Freedom, Unfixed: Large-N c Limit and 1/N c Expansion
Quite generally, first insights, even if only of qualitative nature, may be gained from the reduction of the complexity of QCD, enacted by the increase of the number of colour degrees of freedom and, in parallel, the decrease of the strength of the strong-interaction coupling g s . In some more detail, that simplification of QCD [12,13] proceeds along the following moves: • Generalize QCD to the gauge theories invariant under a non-Abelian Lie group SU(N c ). The dynamical degrees of freedom of each of the latter quantum field theories hence are its gauge bosons, still retaining their designation as gluons and transforming according to the (N 2 c − 1)-dimensional, adjoint representation of SU(N c ), and its fermionic quarks that transform according to the N c -dimensional, fundamental representation of SU(N c ). • Allow the number of colour degrees of freedom, N c , to increase from N c = 3 to infinity: • For the strong coupling strength g s , demand the related decrease, with rising N c , to zero: Clearly, for the strong fine-structure coupling α s this requirement implies the behaviour Therefore, in the large-N c limit, the product N c α s approaches a meaningful finite value.
Only by establishing a careful balance between the growth of N c and the vanishing of α s , the latter requirement allows for both reasonable generalization of QCD to its large-N c limit and exploitation of any corresponding 1/N c expansion, that is, the expansion in powers of 1/N c . According to the above characterization of large-N c QCD, for each QCD contribution to a correlation function its behaviour in the large-N c limit gets determined by two ingredients: • the number of closed loops of the colour degrees of freedom carried by quarks or gluons, • the number of either the strong couplings (14) or the strong fine-structure constants (15).
Keeping this in mind, the large-N c behaviour of arbitrary correlation functions will be found. In particular, for the tetraquark-phile (and therefore tetraquark-pole relevant) contributions, indicated by the subscript "tp", to definitely flavour-exotic correlation functions (7), one gets • for any flavour-preserving contribution of the type employed by Figure 1(c) or Figure 2, • for each flavour-rearranging contribution of the kind adopted by Figure 3(c) or Figure 4, This general discrepancy between the large-N c behaviour of the flavour-preserving and of the flavour-rearranging four-point correlation functions expressed, for all contributions of any tetraquark-phile type, by Equations (16) and (17), on the one hand, and by Equation (18), on the other hand, has a startling or even disturbing implication for the spectra of tetraquark mesons to be expected in the large-N c limit. In the scattering of a pair of conventional mesons, a tetraquark T betrays its existence by contributing in form of an intermediate-state pole. Its couplings to conventional mesons are governed by transition amplitudes A(T ←→ M ab M cd ). Given the discrepancy between those classes of contributions for large N c , consistency in the large-N c limit turns out [17,19] to impose constraints on any involved transition amplitudes. The QCD predictions for the large-N c behaviour of the correlation functions introduced in Section 3 cannot be matched, at hadron level, by the presence of merely a single tetraquark state [22]. Rather, fulfillment of the large-N c behaviour requested by Equations (16), (17) and (18) by the tetraquark-pole contributions necessitates the pairwise occurrence of tetraquarks, that is to say, of a minimum of two (corresponding) tetraquarks [17,19]. The two tetraquarks, generically denoted by T A and T B , have to exhibit unequal N c dependences of their transition amplitudes to the two possible quark-flavour divisions among the two conventional mesons in initial and final states; their dominant decay channels, however, exhibit the same large-N c behaviour. Thus, in the large-N c limit their total decay widths, Γ, behave in a similar fashion, and the large-N c interrelationships of the four involved transition amplitudes are of the kind Table 3 compares several available expectations for the large-N c dependence of the total decay rates Γ of definitely exotic and cryptoexotic tetraquarks, indicating a few discrepancies likely resulting from differences in underlying assumptions or contributions considered as crucial.  [17,19]

Multiquark-Adequate QCD Sum Rules Recognizing "Peculiarities" of Exotic Hadrons
From a mainly theoretical point of view, the description of any hadronic bound states of the fundamental degrees of freedom of QCD in a thoroughly analytical fashion appears to be most favourable; a promising approach complying with this intention, well-grounded in the framework of relativistic quantum field theories, is realized by the QCD sum rule formalism.
In the version originally devised by Shifman, Vainshtein, Zakharov [27] and others [28], a QCD sum rule embodies an analytical relationship between, on the one hand, properties of the hadron state (formed by the strong interactions) in the focus of one's current interest and, on the other hand, the (few) basic parameters of their underlying quantum field theory, QCD. In principle, every routine derivation of a QCD sum rule follows meanwhile well-established procedures [29]. The starting point of the construction of a QCD sum rule is the evaluation of an appropriate correlation function -which clearly has to involve an operator interpolating the hadron under investigation -in parallel both at the phenomenological hadron level and at the fundamental QCD level, followed (of course) by equating both evaluations' outcomes: • In the course of QCD-level evaluation, Wilson's operator product expansion [30] (enabling conversion of a nonlocal product of operators into a series of local operators) is invoked to separate nonperturbative and (to some extent calculable) perturbative contributions.

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The perturbative contributions, identical to the lowest term of this operator product expansion, can be inferred in form of a series in powers of the strong coupling (2).

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The nonperturbative contributions involve, apart from derivable prefactors, vacuum condensates, i.e., the vacuum expectation values of products of quark and/or gluon field operators, which may be interpreted as a kind of effective parameters of QCD.
• In the course of hadron-level evaluation, the insertion of a complete set of hadron states guarantees that the hadron under study shows up by way of its intermediate-state pole.
By application of dispersion relations (and, if necessary, a sufficient number of subtractions), both perturbative QCD-level evaluation and hadron-level evaluation can be reexpressed (for the sake of convenience) in the form of dispersion integrals of appropriate spectral densities. The predictive value and therefore usefulness of the QCD-hadron relations constructed in this manner is perceptibly increased by taking consecutively both the following measures: 1.
Subject both sides of such a relation to a Borel transformation to another variable called Borel parameter τ. This results in the entire removal of any subtraction term introduced and the suppression of the hadron-level contributions above the hadronic ground state. Under a Borel transformation, all vacuum condensates in the nonperturbative QCD-level contributions get multiplied by powers of τ. So, these terms are called power corrections.

2.
Rely on the assumption of quark-hadron duality, which postulates a (needless to stress, approximately realized) cancellation of all perturbative QCD-level contributions above suitably defined effective thresholds, s eff , against all higher hadron-level contributions, consisting of hadron excitations and hadron continuum. In implementing this concept, the problem of pinning down the nature of s eff may be dealt with in two different ways: • Without knowing better, just a guessed fixed value of the parameter s eff is adopted: • In contrast, slipping in limited information about a targeted hadron state opens the possibility [31][32][33][34] to work out the expected s eff dependence on Borel parameters τ: The roadmap for the construction of QCD sum rules sketched above has originally been drafted for analyses of conventional hadrons. Its unreflected application (in unchanged form) also to multiquark states seems, in view of the far-reaching discrepancies between the exotic and the conventional categories of hadrons, to be either too optimistic or a little bit too naïve. Rather, one should be open for (potentially favourable) modifications of the customary QCD sum-rule approach, modifications that might be capable of improving the achieved accuracy of the predictions of QCD sum rules for the class of multiquark exotic hadrons. In particular, upon performing necessary evaluations of correlation functions at QCD level one might find advantageous to take into account the QCD contributions' feature of being tetraquark-phile, in Definition 2 implied to be desirable and by Proposition 1 given its precise meaning, or not. With respect to the power corrections, in any QCD sum-rule derivation indispensable for its QCD-level evaluation, the problem of whether a given nonperturbative vacuum-condensate contribution is tetraquark-phile or not may be analyzed along the lines indicated in Section 3 (as has been demonstrated at the example of definitely flavour-exotic tetraquarks [25,35,37]).
Targeting definitely flavour-exotic tetraquarks (12), the versions of correlation functions (7) indicated in Definition 4 have to be discriminated and hence subjected to separate treatment.

•
In the flavour-preserving case, one has to start from the four-point correlation functions Applying the traditional QCD sum-rule manipulations to twofold contractions (8) of the correlation functions (25) yields as outcome of this enterprise a relationship, depicted in Figure 5, that incorporates a (vast) multitude of QCD-level and hadron-level quantities.   However, a more in-depth analysis [35] reveals that, already on diagrammatic grounds, this conglomerate decomposes, in fact, into two QCD sum rules for conventional mesons ( Figure 6) and one further QCD sum rule that, potentially, supports the development of tetraquark poles and rightly deserves the label of being "tetraquark-adequate" (Figure 7). In the course of its QCD-level evaluation, this latter QCD sum rule receives, exclusively, tetraquark-phile contributions, in the sense of Proposition 1; all the perturbative among these enter in form of dispersion integrals of tetraquark-adequate spectral densities, ρ p . An analogous reflection for single contractions (9) of the correlation functions (25) leads to similar QCD sum-rule findings, all perturbative tetraquark-phile QCD contributions being encoded, in dispersive formulation, in tetraquark-adequate spectral densities ∆ p . +...  • In the flavour-rearranging case, one has to deal with the four-point correlation function Here, irrespective of (ultimately necessary) spatial contractions (8) and (9) of four-point correlation functions (7), the analysis is unfortunately not thus straightforward as in the flavour-preserving case: Within QCD-level evaluation, all tetraquark-phile contributions (defined by requiring them to satisfy the constraint formulated in Proposition 1) may be identified, case by case, by inspection of the solutions of the relevant Landau equations. Within hadron-level evaluation, that QCD-level characteristic of being tetraquark-phile or not is mirrored by the ability of any contributions at hadron level to accommodate, in their s channel, two-meson intermediate states or not, in addition to a possible presence of tetraquark intermediate-state poles [37]. Hardly surprisingly, these insights translate the outcome of the QCD sum-rule formalism based on the correlation function (26) into a quark-hadron relation of (expected) two-component structure symbolically shown in Figure 8. All perturbative tetraquark-phile QCD-level contributions find their way into a tetraquark-adequate QCD sum rule arising from a precursor as in Figure 8(b) by spectral densities ρ r in the double-contractions case (8) and ∆ r in the single-contraction case (9).
In terms of these hadronic properties, all effective-threshold improved multiquark-adequate QCD sum rules resulting from (once or twice) contracted four-point correlation functions (7) assume, for the example of definitely flavour-exotic tetraquarks, symbolically the form [35,37] ( The general lesson to be learned from the above for both perturbative and nonperturbative QCD contributions to QCD sum-rule approaches applied to any type of multiquark hadrons: paying attention to deploy exclusively spectral densities and power corrections computed in multiquark-phile manner should avoid or, at least, diminish the "contamination" of inferred QCD sum-rule predictions by input not related at all to the multiquark hadrons under study.

Summary, Conclusion and Outlook -Multiquark-Instigated Theoretical Adaptations
The multiquark states among the conceivable exotic hadrons feature a characteristic not shared by conventional hadrons, namely, cluster reducibility [2,39], that is to say, their ability to fragment into colour-singlet bound states of lesser numbers of constituents, eventually into a set of conventional hadrons. A promising implication for various theoretical approaches to multiquarks is the advantage gained by pertinent modification of one's favoured formalism.
Here, such improvements have been illustrated for the set of flavour-exotic tetraquarks. An analogous contemplation can be (and has been) done for the class of flavour-cryptoexotic tetraquarks [17][18][19][20][21]26]. It goes without saying that there one gets confronted with additional complications: the potential mixing of these tetraquark states with conventional mesons that carry precisely the quantum numbers of those tetraquarks. Mutatis mutandis, these findings should be straightforwardly transferable to any other multiquark states, such as the likewise established [4] pentaquark baryons. The numerical impact of proposed changes may only be quantified by confronting (definite) multiquark predictions with experimental counterparts.