Tetraquarks in large-N_c QCD
aa r X i v : . [ h e p - ph ] F e b Tetraquarks in large- N c QCD
Wolfgang Lucha a , Dmitri Melikhov b,c,d , Hagop Sazdjian e, ∗ a Institute for High Energy Physics, Austrian Academy of Sciences, Nikolsdorfergasse 18,A-1050 Vienna, Austria b D. V. Skobeltsyn Institute of Nuclear Physics, M. V. Lomonosov Moscow StateUniversity, 119991 Moscow, Russia c Joint Institute for Nuclear Research, 141980 Dubna, Russia d Faculty of Physics, University of Vienna, Boltzmanngasse 5, A-1090 Vienna, Austria e Universit´e Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France
Abstract
The generalization of the color gauge group SU(3) to SU( N c ), with N c takingarbitrarily large values, as had been proposed and developed by ’t Hooft, hasallowed for a decisive progress in the understanding of many qualitative, aswell as quantitative, aspects of QCD in its nonperturbative regime. In par-ticular, the notion of valence quarks receives there a precise meaning. Thepresent work reviews the various aspects of the extension of this approach tothe case of tetraquark states, which are a category of the general class of ex-otic states, also called multiquark states, whose internal valence-quark struc-ture does not match with that of ordinary hadrons, and which have received,in recent years, many experimental confirmations. The primary question ofdescribing, or probing, on theoretical grounds, multiquark states is first ex-amined. The signature of such states inside Feynman diagrams in relationwith their singularities is highlighted. The main mechanisms of formation oftetraquark states, provided by the diquark model and the molecular scheme,are considered together with their specific implications. The properties oftetraquark states at large N c are analyzed through the Feynman diagramsthat describe two-meson scattering amplitudes. It turns out that, in thatlimit, the possible formation of tetraquark states is mainly due to the mutual ∗ Corresponding author
Email addresses: [email protected] (Wolfgang Lucha), [email protected] (Dmitri Melikhov), [email protected] (HagopSazdjian)
Preprint submitted to Progress in Particle and Nuclear Physics February 5, 2021 nteractions of their internal mesonic clusters. These essentially arise from thequark-rearrangement, or quark-interchange, mechanism. In coupled-channelmeson-meson scattering amplitudes, one may expect the occurrence of twoindependent tetraquark states, each having priviledged couplings with thetwo mesons of their dominant channel. The question of the energy balanceof various schemes in the static limit is also analyzed. The clarification ofthe mechanisms that are at work in the formation of tetraquarks is the mainoutcome from the large- N c approach to this problem. Keywords:
QCD, tetraquarks, large N c , multiquark operators Contents1 Introduction 32 Large- N c limit 8 N c ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.3 Multilocal operators . . . . . . . . . . . . . . . . . . . . . . . 39 N c N c ? . . . . . . . . . . . . . . . . 632 Molecular states 71
The possibility of the existence of exotic hadrons, containing more va-lence quarks and antiquarks than a quark-antiquark pair for mesons andthree quarks for baryons, had been considered since the early days of thequark model [1, 2]. The lack of experimental data about these hypotheticalobjects during the following decades has pushed them for a long time into amarginal situation. However, with the advent of Quantum Chromodynamics(QCD) as a theory of strong interactions, with the fundamental propertiesof asymptotic freedom and confinement [3–5], the road was open for detailedtheoretical investigations on the subject [6, 7].Progress has been achieved for the last two decades when several exper-iments have been able to detect, with sufficient precision, many candidatesto represent exotic hadrons, the latter not being matched with the quantumnumbers or the constituent content of the usual quark model [8–17]. Thesediscoveries stimulated in turn vast theoretical investigations in order to un-derstand and interpret the detailed dynamics that give rise to their existence.General accounts of them can be found in recent review articles [18–28].While in the framework of QCD theory one explains in a satisfactoryway the spectrum and transition properties of ordinary hadrons [29–39], theproblem of exotic hadrons comparatively encounters additional difficulties.In QCD, which is a non-Abelian gauge theory, observable quantities must becolor gauge invariant. This naturally explains why quarks and gluons, thebuilding blocks of the theory, are not individually observed in nature; onlycolor-singlet objects, actually represented by the ordinary hadrons, whichare bound states of quarks and gluons, are detected as free particles. This3roperty is commonly depicted by the term “confinement of quarks and glu-ons”. It should be emphasized, however, that the term confinement has here astronger meaning than the terms “color screening”, also used in the literature.The bound states of quarks and gluons do not resemble the positronium- orhydrogen-like bound states of Quantum Electrodynamics (QED). Hadrons,which, in principle, are infinite in number, have masses squared that in-crease linearly with respect to their spin, lying along “Regge trajectories”.This is due to the fact that the potential energy of the binding forces in-creases with respect to the mutual distances between quarks and/or gluons[40, 41]. However, the confining forces are only operative between coloredobjects or within color-singlet objects. There are no confining or long-rangevan der Waals forces that would operate between hadrons. The latter mutu-ally interact by means of short-range forces, represented by contact terms ormeson-exchange terms.The color gauge invariance principle should also apply to the constitutionof exotic hadrons, also called “multiquark states”. However, one realizes herethat color-singlet multiquark states can easily be generated by products orcombinations of products of free or interacting ordinary hadron states [42].Since ordinary hadrons mutually interact by means of short-range forces,one should not find in that case spectra similar to that of the confininginteractions found earlier in QCD. Rather, one would have loosely boundstates, or corresponding resonances, typical of molecular or nuclear states,whose illustrated representative is the deuteron [43–46].The existence of molecular-type exotic hadrons does not a priori excludethe formation of exotic hadrons by means of confining forces acting directlyon the quarks and gluons. In such a case, one would expect to obtain boundstates of compact size, due to the confining nature of all the operative forces.Such states are also designated as “compact exotic hadrons”, in contradis-tinction to the “molecular states”.The eventual existence of compact multiquark states would implicitlymean that they are color or cluster irreducible, in the sense that they cannotbe decomposed as combinations of products of simpler color gauge invari-ant states, which are the usual hadronic states. This basic property doesnot seem, however, realizable. When the multiquark generating operators orsources are expressed in local form, they can always be reexpressed, by meansof Fierz rearrangements, as combinations of products of ordinary hadroniccurrents [42]. More generally, even when the multiquark operators take amultilocal form with the aid of gauge links, the color or cluster reducibility4henomenon continues to occur [47]. This means that, on formal grounds,compact multiquark states are not the most favored outcome of the exotichadron construction. To ensure their existence on theoretical grounds, oneshould be able to find underlying dynamical mechanisms that ensure theirstability against the natural forces of dislocation, represented by the forma-tion of internal hadronic clusters.The primary mechanism that might operate for the formation of compactmultiquark states is the “diquark” one [7, 48], which has been first utilizedin the study of baryon spectroscopy [49–51]. It is based on the observationthat in SU(3), the color gauge group underlying QCD, the fundamental rep-resentation to which the quarks are belonging is the triplet ; therefore, twoquarks belong either to the antitriplet representation , which is antisym-metric, or to the sextet representation , which is symmetric. The forcesthat act between the quarks are generally attractive in the representationand will have the tendancy to form bound states of diquarks, which, in thecase of compact sizes, will behave as nearly pointlike antiquarks, thus in turnbeing attracted by a third quark.The idea of the diquark mechanism has naturally been extended to thecase of exotic hadrons [52–55], giving rise to detailed investigations [56–64].This model, associated with approximate flavor symmetry and flavor inde-pendence of the confining forces, generally predicts, in the case of existenceof exotic hadrons, several flavor multiplets of such states.On the other hand, the effective forces that operate for the formation ofmolecular-type states are more dependent on the quark flavors and hencemay not predict as many flavor multiplets as the diquark mechanism. One ofthe salient features of the molecular picture is its tendancy to predict boundstates lying near the two-hadron threshold.The difficulties encountered in clearly predicting the domains or condi-tions of existence of each category of exotic hadrons, compact or molecular,are intimately related to the fact that QCD theory is characterized by hav-ing a nonperturbative regime at large distances, which is not yet analyticallysolved. Efficient tools are provided through the recognition of approximatesymmetries and the use of related effective field theories, among which onemay quote chiral perturbation theory [65–68], heavy-quark effective theo-ries [69–73] and effective theories involving nucleons [74–81]. Other analyticapproaches are forced to hinge on simplifying models, which in turn are sub-jected to theoretical debates.For the time being, lattice theory remains one of the most powerful tools5or the solution of QCD in its nonperturbative regime. It is essentially basedon a numerical approach, discretizing the continuum of spacetime over aeuclidean finite volume lattice [40, 82]. Lattice theory has also made decisiveprogress in the calculation of scattering amplitudes [83, 84]. Concerning theexotic hadrons, it has already provided positive results, mainly in sectorscontaining two heavy quarks [85–95]. However, lattice theory results are notyet sufficiently precise to allow making a distinction between compact andmolecular structures; this requires the calculation of appropriate form factorsthat might give additional information about the size of the observed states.Among the approaches for the analysis of the nonperturbative proper-ties of QCD, the large- N c limiting method, originally introduced by ’t Hooft[96–98], has been proven to be one of the most fruitful ones. It consists ingeneralizing the color gauge group SU(3) of QCD to the more general caseof SU( N c ), with N c considered as a parameter. It turns out that the large- N c limit of the theory, associated with a scaling of the coupling constantas g = O (1 / √ N c ), has more simplifying features than in the finite- N c case.The color-singlet parts of Feynman diagrams can then be classified accord-ing to their topological properties: it is the “planar” diagrams that are thedominant ones, while the other types of diagram can systematically be clas-sified, according to their more complicated topology, within a perturbativeexpansion in 1 /N c , as providing nonleading contributions.This approach does not solve the theory, but allows, assuming that thelarge- N c limit is a smooth one, a better understanding of some of its salientfeatures. In this limit, the spectrum of the theory is essentially made of an in-finite tower of free stable mesons, their mutual interactions occurring throughnonleading effects in 1 /N c [99, 100]. This clearly shows that, in the hadronicworld, the ρ -meson, for instance, is as elementary as the pion and could notbe considered as a composite object of two pions [42, 43]. Another outcomeis a natural explanation of the OZI (Okubo, Zweig, Iizuka) rule [2, 101, 102],which asserts that leading strong interaction processes involving hadrons arethose that have nonzero connecting quark lines between the initial and fi-nal states. Processes not satisfying this rule occur in nondominant ordersof the 1 /N c expansion and are naturally subleading [99, 100]. The large- N c limit brings also a natural explanation of the absence, at leading order, ofthe quark-antiquark sea inside hadrons and a theoretical support to Reggephenomenology, in which, in first approximation, hadronic processes are welldescribed by tree diagrams with hadron exchanges [103].The large- N c approach to QCD has received much attention during the6ast decades in many phenomenological calculations related to hadronicphysics. Its main virtue is to provide a theoretical basis for qualitative sim-plifications and for the understanding of the data.The large- N c limiting procedure has also been, over the last two decades, adecisive tool for new investigations in the search for possibly existing dualityrelations between gauge and string theories [104–109].The purpose of the present article is to present a review of the mainproperties of the large- N c approach, with emphasis put on its applications toexotic hadrons. Rather than focusing on particular candidates or particulardata, our aim is to introduce the general aspects of the method, which mightbe applicable to a wide variety of situations.The large- N c analysis plays a crucial role in the recognition of those QCDFeynman diagrams that might contribute to the formation of exotic hadrons.It is in the large- N c limit that the counting of the quark content of a hadronicstate takes a systematic mathematical meaning, associated with the order ofexpansion with respect to 1 /N c . However, a straightforward transposition tothe case of exotic hadrons of results known from the large- N c approach toordinary hadrons might lead, in some cases, to wrong predictions. It is herethat the analysis of the singularities of Feynman diagrams with respect tothe quark content becomes of primary importance. This is usually done withthe help of the Landau equations [110, 111].In summary, the large- N c approach may be considered as one of the basictools for a systematic investigation of the nonperturbative regime of QCD,with the objective of gaining complementary information with respect toother well-established approaches.The paper is organized as follows. In Sec. 2, we present the general as-pects of the large- N c approach. Section 3 is devoted to a review of the variousdescriptions of exotic states by means of interpolating currents or multilocaloperators. In Sec. 4, emphasis is put on the singularities of Feynman dia-grams and the Landau equations for the recognition of a possible presenceof tetraquark states. Section 5 studies the properties of tetraquarks throughthe Feynman diagrams of meson-meson scattering amplitudes in terms ofquark and gluon lines. Various cases of quark flavor contents are considered.In Sec. 6, some salient features of the molecular scheme, in relation witheffective theories, are reviewed. In Sec. 7, the question of the reducibility ofmultiquark operators is considered and the energy balance of various config-urations is studied in the static limit. The notion of geometric partitioningis introduced. A summary and concluding remarks follow in Sec. 8.7 . Large- N c limit Quantum Chromodynamics is a non-Abelian gauge theory with the colorgauge group SU(3), with three quark fields ψ a ( a = 1 , , , three antiquark fields ψ b ( b = 1 , , ¯3 and eight gluon fields A Bµ ( B = 1 , . . . , [3–5]. The primary,CP conserving, Lagrangian density, written in matrix form in color spaceand with N f different quark flavors, reads L = −
12 tr c F µν F µν + N f X j =1 ψ j ( iD µ γ µ − m j ) ψ j , (1)where D is the covariant derivative, D µ = 1 ∂ µ − igT B A Bµ , F is the fieldstrength, F µν = ( i/g )[ D µ , D ν ] = T B F Bµν , g is the coupling constant and T B are the generators of the gauge group in the fundamental representation,with normalization tr( T A T B ) = (1 / δ AB . The quantization of the theoryrequires the introduction into the previous Lagrangian density of a gauge-fixing term together with a part containing auxiliary anticommuting scalarfields, the so-called Faddeev-Popov ghosts [111–115].QCD has the property of asymptotic freedom, which asserts that thetheory becomes almost free at short distances, or at high energies, while itbecomes unstable at large distances, or at low energies. This is interpreted asthe sign of a new regime, characterized by the confinement of the fundamentalparticles of the theory. One striking feature of the theory is that the couplingconstant, which appears in the primary Lagrangian density (1), is not afree parameter: it is absorbed into the definition of the mass scale, usuallydenoted by Λ QCD [116, 117], a phenomenon called “dimensional (or mass)transmutation”. Actually, the mass of the proton is mainly determined byΛ
QCD and not by the masses of the quarks that enter into its constitution;even if the quarks were massless, the proton would continue having a massof the order of its physical mass. This is in contrast to the behavior in theelectroweak sector of the Standard Model, where the Higgs mechanism isat the origin of the mass scales. Therefore, the masses of the light quarks u, d, s do not play a major role in the theory and could, in many cases, betaken as zero. 8he presence of a free parameter in a theory allows one to search forapproximate solutions for some particular values of the parameter and thento apply perturbation theory around those values [118]. QED and the short-distance regime of QCD provide particular examples of this procedure, theexpansion being realized around the free theory. ’t Hooft observed that QCDpossesses a hidden free parameter, represented by the dimension of the colorgauge group SU(3), provided one considers it as part of the general class ofnon-Abelian gauge theories SU( N c ), with the particular physical value N c = 3of the parameter N c [96]. He studied the limit of large values of N c , with thequark fields belonging to the fundamental representation, which is of dimen-sion N c , and the gluon fields belonging to the adjoint representation, whichis of dimension ( N c − β -function, which displays the implicit mass-scale dependence of the couplingconstant [4, 5, 119, 120] and which is a gauge and renormalization-groupindependent quantity up to two loops. At one-loop order it reads β ( g ) ≡ µ ∂g∂µ = − π (cid:16) N c − N f (cid:17) g , (2)where µ is the mass scale at which renormalization has been defined. Asymp-totic freedom is realized for a negative value of β ; this is indeed the case withthe physical values N c = 3 and N f = 6. Taking now large values of N c , whilekeeping N f fixed, one notices that the negativity of β is strengthened. Toensure, however, a smooth limit, so that Λ QCD remains independent of N c ,one should admit that at the same time the coupling constant g scales with N c like N − / c ; the product g N c then remains constant with respect to N c : g N c ≡ λ = O ( N c ) . (3)The corresponding β -function is, for large N c , β ( λ ) ≡ µ ∂λ∂µ = − π λ + O (cid:16) N c (cid:17) . (4)A similar conclusion is also obtained at the two-loop level.Generally, the quark flavor number, N f , manifests itself through quark-loop contributions. Equation (4) shows that in the large- N c limit, with fixed N f , the quark loop contributions are expected to be of nonleading order. This9s one of the important qualitative simplifications that occur on practicalgrounds in the large- N c limit.Other types of large- N c limits have also been considered in the past andpresently for various purposes. The simultaneous limits of large values of N c and N f , with N c /N f fixed, has been considered by Veneziano [121]; itis evident, from the previous observation, that in that case the quark loopscontinue contributing to leading order. In another limiting procedure, oneassumes that the quark fields belong to the second-rank antisymmetric tensorrepresentation [122–125]; for N c = 3, this representation is equivalent tothe antitriplet one and therefore the physical content of actual QCD is notmodified. For general N c , the dimension of that representation is N c ( N c − / N c ;this prevents the quark loops from disappearing from the leading order.The above variants of the large- N c limit have their own phenomenologicaladvantages. We shall stick, however, in the present review, to the moretraditional scheme developed by ’t Hooft, because of its greater simplicity.Reviews about the large- N c limit can be found in [21, 126–134]. To study in more detail the properties of the theory in the large- N c limit,it is advantageous to use a color two-index notation for the gluon fields [96].Since they belong to the adjoint representation and the latter is contained inthe direct product of the fundamental and antifundamental representations,one may represent the gluon fields with the notation A ab,µ , its relationshipwith the conventional notation being the following: A ab,µ = ( A Bµ T B ) ab , A a † b,µ = A ba,µ , A aa,µ = 0 , a, b = 1 , . . . , N c , (5)the third equation being a consequence of the property of the T ’s traceless-ness. A similar notation can also be adopted for the ghost fields; however,for the simplicity of presentation, we shall not explicitly write down ghostfields in the remaining part of the paper, nor shall we draw the correspond-ing Feynman diagrams; their presence does not modify the main qualitativefeatures that are drawn from the gluon fields.With the above convention, the color contents of the quark and gluonpropagators are h ψ ai,α ( x ) ψ b,j,β ( y ) i = δ ij δ ab S αβ ( x − y ) , (6)10here i and j are flavor indices, α and β spinor indices, and S is the color-independent Dirac field propagator, h A ab,µ ( x ) A cd,ν ( y ) i = (cid:16) δ ad δ cb − N c δ ab δ cd (cid:17) D µν ( x − y ) , (7)where D is the color-independent part of the gluon propagator. The termproportional to 1 /N c in the last equation ensures the traceless property ofthe gluon field. However, because of the factor 1 /N c , it could be neglected inleading-order calculations; this amounts to replacing the gauge group SU( N c )by U( N c ) and the ( N c −
1) gluon fields by N c ones. The corresponding ap-proximation is of order 1 /N c (cf. Ref. [126], Appendix C). This considerablysimplifies the diagrammatic representation of the gluon propagator: as far asthe color indices are concerned, the gluon propagates as a quark-antiquarkpair, which suggests a double-line representation for the gluon propagator.Figure 1 depicts, in two columns, the correspondence between the conven-tional and the double-line representations. a ¯ aa b ¯ b ¯ a a ¯ aa ¯ a ¯ b b Figure 1: The quark and gluon propagators in the conventional and the double-line rep-resentations, first and second columns, respectively. Lower indices are distinguished bybars.
Concerning the interaction parts, they have structures of the followingtypes: ψ a γ µ A ab,µ ψ b , A ab,µ A bc,ν ∂ µ A c,νa , A ab,µ A bc,ν A c,µd A d,νa , quark flavor beingconserved. One notices that a lower color index is always contracted withthe upper index of a neighboring field and this ensures the continuity oflines arriving at a vertex and departing from it. The corresponding vertexdiagrams are presented in Fig. 2 in both representations.The double-line representation allows a better control of the color flowinside Feynman diagrams. To have a first glance of it, we consider the twolowest-order contributions to the gluon self-energy, represented by a quarkloop (Fig. 3a) and a gluon loop (Fig. 3b), respectively. Each color loop pro-duces a factor N c . The external gluon field color indices being fixed, diagram11 ¯ b b ¯ cc ¯ dd ¯ a b ¯ ba ¯ ad ¯ d c ¯ ca ¯ b b ¯ cc ¯ a b ¯ ba ¯ ac ¯ ca ¯ b b ¯ a ¯ b ba ¯ a Figure 2: The quark-quark-gluon, three-gluon and four-gluon vertices in the conventionaland the double-line representations, first and second columns, respectively. Lower indicesare distinguished by bars. g ; taking into account thelarge- N c behavior of the latter [Eq. (3)], one finds that the large- N c behaviorof diagram (a) is O ( N − c ). On the other hand, diagram (b) contains oneinternal color loop, providing an additional factor N c with respect to theprevious diagram. Therefore, the large- N c behavior of diagram (b) is O ( N c ).Thus, among the two diagrams of Fig. 3, it is diagram (b) which contributesto the leading-order behavior. (a) O ( N − c ) (b) O ( N c ) Figure 3: The lowest-order gluon self-energy contributions: (a) quark loop; (b) gluon loop.The corresponding orders in large- N c behavior are also indicated. The origin of the difference of contributions between the two diagrams caneasily be understood: it is related to the distinct representations of the colorgroup to which the quark and the gluon fields belong: one has N c quarksin the fundamental representation against ( N c −
1) gluons in the adjointrepresentation. At leading orders, internal gluon lines produce N c times morecontributions than quark lines, a feature that the double-line representationdisplays explicitly through the supplementary color loops. This means that,at leading orders, the quark loops, which actually are in N f duplicates, canbe neglected altogether, except when they appear as contractions of externalquark lines. This is one of the main advantages of the large- N c limitingprocedure adopted by ’t Hooft.As a second example of the large- N c counting rules, we consider, still inthe gluon self-energy part, one-gluon exchange diagrams containing either aquark loop or a gluon loop (Fig. 4). Diagram (a) contains two color loops,producing a factor N c , together with six vertices, producing a factor N − c [Eq. (3)]. Its global behavior is therefore O ( N − c ). Diagram (b) has oneadditional color loop and thus its behavior is O ( N c ). One again verifies thegeneral property of the nonleading character of the internal quark loops.This second example outlines some other general features of the large- N c a) O ( N − c ) (b) O ( N c ) Figure 4: One-gluon exchange inside the gluon self-energy diagram: (a) including a quarkloop; (b) including a gluon loop. The corresponding orders in large- N c behavior are alsoindicated. behavior, that we would like to emphasize. The leading N c -behavior of Fig. 4is the same as in Fig. 3. This means that the large- N c behavior does not followthe usual perturbative expansion in the coupling constant. The number ofcolor loops may balance the occurrence of new vertices. The example of Fig. 4may easily be generalized to more complicated types of diagram, where onefinds again the same leading large- N c behavior. The common feature of allthese diagrams is that they can be mapped on a plane, and more generally ona two-dimensional surface, without allowing crossings of color lines. For thisreason, they are called planar diagrams , which can be considered as belongingto a particular topological class in color space. It is to be emphasized thatthey are infinite in number.On the other hand, the nonleading diagrams of Figs. 3 and 4, which con-tain the quark loops, could also be considered as planar. They, however,display an additional topological property, which is associated with the no-tion of a hole . Comparing diagrams (a) and (b) of both figures, one maydistinguish figures (a) from figures (b) by the occurrence in (a) of a holein place of the internal color loop that exists inside the gluon loop in (b).Therefore, diagrams (a) above can be considered as planar, but with onehole inside the plane. It is evident that each occurrence of a hole produces afactor N − c in the large- N c counting rules.The diagrams that do not fulfill the planarity condition are called non-planar . They occur when, after their projection on a plane, some of the colorlines cross each other. An example of such a case is provided by the two-gluoncrossed-exchange diagram between two quark lines. Figure 5 displays a fewFeynman diagrams occurring in the perturbative expansion of the two-point14unction of the color-singlet bilinear current j ¯ kℓ , h j ¯ kℓ ( x ) j † ¯ kℓ ( y ) i , defined as j ¯ kℓ = ψ a,k ψ aℓ , (8)where a is a color index and k and ℓ are fixed flavor indices; Dirac matricesand spinor indices have been omitted, as not being of primary importance inthe present evaluation. Diagrams (a), (b) and (c) are planar and provide theleading large- N c behavior. Diagram (d), representing the two-gluon crossed-ladder diagram, is nonplanar. One observes that it contains only one colorloop, against the three color loops of the two-gluon ladder diagram (c). Itslarge- N c behavior is therefore O ( N − c ), against the O ( N c ) behavior of thethree planar diagrams (a), (b) and (c). (a) O ( N ) (b) O ( N )(c) O ( N ) (d) O ( N − ) Figure 5: A few Feynman diagrams involved in the perturbative expansion of the correla-tion function h j ( x ) j † ( y ) i . (a) A quark loop; (b) one-gluon exchange; (c) two-gluon ladderdiagram; (d) two-gluon crossed-ladder diagram. The corresponding orders in large- N c behavior are also indicated. Nonplanar diagrams can be characterized by a specific topological invari-ant, the number of handles . Keeping in Fig. 5d one of the gluon propagatorsin space with respect to the projection plane, one observes that it plays therole of a handle of a two-dimensional surface. From the previous analysis,one deduces that each handle introduces a factor N − c with respect to theplanar diagram contribution. 15he diagrams of Fig. 5 are examples of vacuum-to-vacuum diagrams cor-responding to the connected part of correlation functions of gauge invariantlocal currents, each made of a quark and an antiquark field. Vacuum-to-vacuum diagrams can also be generated by gluon field currents, made ofbilinear functions of gluon field strengths. An example is the current G [ µν ][ ησ ] = F Aµν F Aησ = 2 F ab,µν F ba,ησ . (9)The leading-order behavior of the corresponding two-point function is pro-vided by the planar diagrams, two of which are represented in Fig. 6. Their G G (a) O ( N c ) GG (b) O ( N c ) Figure 6: Vacuum-to-vacuum planar diagrams made of gluon lines, corresponding to thetwo-point function of the gluonic current G [Eq. (9)]. large- N c behavior is O ( N c ). Comparing this with the leading-order behaviorof the quark bilinear case [Fig. 5], we observe that the latter can be deducedfrom the former by considering the external boundary quark loop as a holeinside the planar diagram topology.We thus arrive at a general formula about the leading-order behavior of adiagram, characterized by the number of two topological invariants in colorspace, the hole, made of a color quark loop, and the handle, made of a gluonpropagator. Designating by B the number of holes and by H the number ofhandles, the power of the large- N c behavior is N − B − Hc ≡ N χc , (10)where χ is called the Euler characteristic (cf. Ref. [126], Appendix A).Planar diagrams without holes or with a number of holes determinedby the external boundary color quark loops will provide the leading-orderbehavior at large N c . The inclusion of handles and additional holes con-tributes to nonleading terms in 1 /N c . Within this approach, 1 /N c appearsas the effective perturbative expansion parameter of the theory. On the other16and, planar diagrams are infinite in number; this means that, even with theplanarity approximation, one is not yet able to solve in a simple way thetheory. Nevertheless, one hopes that the 1 /N c expansion will provide manyqualitative simplifications and an improvement in the understanding of thedynamics of the theory. Properties of physical states can be investigated by considering corre-lation functions of gauge-invariant local currents, having the same quantumnumbers. For mesons, the natural candidates are the quark bilinear currents,as defined in Eq. (8).Considering the two-point function of such a current (generally, its con-nected part), typical Feynman diagrams of its perturbative expansion in thecoupling constant have been presented in Fig. 5, where the first three, (a),(b) and (c), correspond to planar diagrams with leading-order behaviors in N c . The most salient feature of these, and of all planar diagrams, is the factthat they contain only two quark lines (or propagators). A larger number ofquark lines can appear only in nonleading diagrams, containing an additionalnumber of holes. States characterized by a single quark-antiquark pair, as aleading descriptive element, correspond to the ordinary meson states. There-fore, the two-point function of the current, saturated by a complete set ofhadronic intermediate states, reduces, at leading order in N c , to a sum ofmeson poles: Z d xe ip.x h j ( x ) j † (0) i = X n iF n p − M n + iǫ = O ( N c ) , (11)where F n is defined as the matrix element of j between vacuum and themeson state | n i : h | j | n i = F n . (12)The number of meson states must be infinite. This is dictated by theasymptotic behavior of the left-hand side: because of asymptotic freedom,its high-energy behavior is known and contains logarithmic factors, whichcannot be reproduced by a finite number of terms in the sum (11) [99].This also entails a generic behavior at large N c for each term of the seriesin connection with the behavior of the left-hand side ( O ( N c )). The mostnatural solution is that the meson masses (for finite n ) remain finite at large17 c , while the couplings F n increase like N / c : M n = O ( N c ) , F n = O ( N / c ) . (13)From the complete decomposition of the two-point function into a seriesof poles [Eq. (11)], one also deduces that the meson states are stable at large N c . If the mesons were unstable, they would have finite widths, manifested asfinite imaginary parts in the pole terms, which, in turn, would imply, throughthe unitarity property of the theory, the existence of unitarity cuts and theappearance of many-particle states; these would be manifested through theexistence of more than two quark lines in the leading-order diagrams, whichis not the case.In obtaining Eq. (11), we have assumed that all planar diagrams contain-ing two quark lines are perturbative representatives of single meson states.Since planar diagrams contain, in general, many gluon lines, the questionarises as to whether such diagrams may also contain independent glueballstates, which might be formed as gauge-invariant bound states of severalgluon fields. This question is best analyzed through the study of the sin-gularities and the imaginary part of the corresponding diagrams, by cuttingthem with a vertical line. An example of this is presented in Fig. 7. One ob- O ( N ) a ¯ ab ¯ bc ¯ c Figure 7: A planar diagram, contributing to the two-point function of the current j , withtwo gluon propagators submitted, together with the quark and antiquark propagators, tothe vertical cut (dashed). The color indices of the various lines on the right of the verticalcut are explicitly indicated. Lower indices are distinguished by bars. serves that each gluon propagator, cut by the vertical line, is connected, withits color indices, to the neighboring propagator, and nowhere a color-singletgluonic cluster emerges. The set of the above gluon propagators belongs tothe adjoint representation of the gauge group; on the other hand, the set18f the quark and antiquark propagators belongs also to the adjoint repre-sentation. It is the connection of the two sets that produces a color-singletrepresentation. Therefore, the corresponding state is color-irreducible, in thesense that it is not decomposable into the product of other color-singlet rep-resentations. Hence, no independent glueball state may be generated fromsuch a diagram. This property is very general for the two-point function andmay be verified on more complicated planar diagrams.Equation (11) can be diagrammatically described by representing themeson propagators by straight line segments and displaying the large- N c behavior of the couplings [Eqs. (12) and (13)] (cf. Fig. 8, where the connectedpart of the two-point function is schematically represented in the form h jj i c ). h jj i c = P n N / c N / c F n F n Figure 8: Equation (11) in diagrammatic form.
The study of the correlation functions of the currents j can be continuedwith the case of the three-point function h jjj i , where j is a generic current,such that connections between three j s are possible with quark lines. Thesimplest planar diagram, for the connected part, is presented in Fig. 9. O ( N c ) Figure 9: A planar diagram for the three-point function h jjj i c ; here, the arrowed linesindicate quarks, whereas the dashed straight line indicates the cut. Cutting the diagram with a straight line in any direction and position,one meets always a pair of quark-antiquark lines, which means that the sin-19ularities of the diagram are located at meson poles . These may be threeor two in number. The first category involves, as a residue, the (amputated)vertex function of three meson sources, providing the three-meson couplingconstant. The second category provides the coupling of a current j to twomesons. The corresponding equation is represented diagrammatically, to-gether with the relevant large- N c behaviors, in Fig. 10. h jjj i c = P N / c N / c N / c N − / c + P N c N / c N / c Figure 10: Diagrammatic representation of the three-point function h jjj i c in terms ofmeson propagators and couplings, at leading order of N c ; here, the unarrowed lines indicatemesons. The large- N c behaviors of the various couplings are explicitly indicated. One finds that the three-meson couplings behave, at large N c , as N − / c .This also determines the decay amplitude of mesons into two mesons, whichvanishes at large N c ; the mesons are thus stable in this limit, a result thatconfirms the stability property already deduced from the decomposition ofthe two-point function [Eq. (11)]. The second type of diagram in Fig. 10determines the coupling of the current j to two mesons, or, equivalently, totwo pairs of quarks and antiquarks. Its behavior is N c , a factor 1 /N / c lessthan the coupling F n [Eqs. (12) and (13)]. If one interprets these couplingsas probability amplitudes of creating, by the current j , from the vacuumquark-antiquark pairs, one deduces that at large N c mesons are made of onepair of quark-antiquark, while sea quarks, represented by additional quark-antiquark pairs, occur only as nonleading effects. This fact is a phenomenolo-gically confirmed property, which is explained here in a natural way throughthe 1 /N c expansion method. The choice of the form and geometry of the cuts allows a selection of particular so-lutions of the Landau equations [110]; more generally, if p , q and r are the momentaassociated with the external currents, then the singularities occur, according to the Lan-dau equations, in the variables p , q and r .
20e next consider four-point functions h jjjj i of generic currents j , suchthat connections between neighboring currents can be realized with quarklines. The simplest planar diagram, for the connected part, is presented inFig. 11. O ( N c ) Figure 11: A planar diagram for the four-point function.
The singularities are again located at meson poles. The decompositionof the connected part of the four-point function in terms of meson propaga-tors and couplings is diagrammatically presented, together with the relevantlarge- N c behaviors of the couplings, in Fig. 12. One finds that the four-mesoncouplings are of order N − c , smaller by a factor 1 /N / c than the three-mesoncouplings. This, in turn, entails that the decay amplitudes of mesons intothree mesons are also of order N − c . The couplings of the currents j to three-meson states, or, equivalently, to three pairs of quarks and antiquarks, areof order N − / c .Factoring out, in the above decomposition, the four meson propagators,together with their couplings to the external currents, one obtains the scat-tering amplitude of two mesons into two mesons, which is of order N − c : T ( M M → M M ) = O ( N − c ) . (14)It is worthwile noticing that, at that leading order, the scattering amplitude isexpressed as a series of tree diagrams involving the infinite number of mesonpropagators and mutual couplings. This is in qualitative accordance withRegge phenomenology, where the dominant contributions come from treediagrams of hadron exchanges and couplings [103], even though the Reggebehavior itself cannot be demonstrated by the sole large- N c limit.The above procedure can be continued to higher numbers of currents.One particular outcome, as was already evident, is the relative decrease of21 jjjj i c = P N / c N / c N / c N / c N − c + P N / c N / c N / c N / c N − / c N − / c + P N / c N / c N / c N / c N − / c N − / c + P N − / c N / c N / c N / c Figure 12: Diagrammatic representation of the four-point function h jjjj i c in terms ofmeson propagators and couplings, at leading order of N c . The large- N c behaviors of thevarious couplings are explicitly indicated. the order in N c -behavior of the meson couplings when the number of mesonsincreases. Thus, the n -meson couplings behave as N − n/ c . One consequenceof this property is that when a meson may decay into many mesons, it prefer-entially decays first into two mesons (or, possibly, into three mesons if thereis a selection rule), which in turn decay into two or three mesons, and soforth. This is also a phenomenologically confirmed fact.Another phenomenon occurring in hadron physics is related to the so-called OZI rule (after Okubo, Zweig and Iizuka) [2, 101, 102], which stipulatesthat, in the case of three light quarks, mesons generally are members of nonetsof the flavor group U(3), rather than of separate octets and singlets of thegroup SU(3) (best illustrated by the ϕ − ω mixing) . A violation of therule concerns processes where the quark lines are completely disconnectedbetween the initial and final states; such processes should be negligible. Anexample is illustrated by the two-point function h j ¯ kk ( x ) j † ¯ ℓℓ (0) i , where k and ℓ The pseudoscalar mesons are an exception, due to the chiral anomaly problem [67, 135];however, the anomaly vanishes at large N c . N c is O ( N c ), smaller by a factor1 /N c with respect to the behavior of the planar diagrams of Fig. 5. Onceagain, the large- N c limit provides a natural explanation of this qualitativeeffect, widely verified by experimental data. j ¯ kk j † ¯ ℓℓ O ( N ) Figure 13: OZI-rule violating diagram: Two gluon propagators joining two quark loopswith completely different flavor content. The order in large- N c behavior is to be comparedwith the leading-order behaviors of Fig. 5. To summarize, the large- N c limiting procedure leads, according to theorders of N c , to a hierarchical classification of the various processes that occurin QCD in its nonperturbative regime, providing a qualitative understandingof many typical phenomena that characterize the strong-interaction physicsof hadrons. As far as the meson sector is concerned, at leading order in N c , QCD reduces to a theory of an infinite number of free stable mesons,made of a quark-antiquark pair and of gluons, whose masses squared areexpected to lie along Regge trajectories. At this level, all mesons are on equalfooting; their differences arise only from their specific quantum numbers.The interactions among these mesons, which are responsible for their strongdecays and nontrivial scattering processes, appear at nonleading orders of N c .From this point of view, strong-interaction physics of mesons correspondsto a weakly interacting effective field theory, with an expansion parametergiven by 1 /N c , dominated by tree diagrams of meson exchanges and contactterms, as compared to the underlying strongly interacting theory, which isresponsible for the confinement of quarks and gluons.Studies of the influence of the large- N c limit on meson properties can befound in Refs. [67, 136–142]. 23 .4. Baryons One would like to complete the large- N c approach by extending it to thephysics of baryons. However, here, the method applied to the case of mesonsturns out to be inapplicable.The main reason of that difficulty is related to the description itself ofbaryonic states at large N c . While for mesons, the change of the gaugegroup from SU(3) to SU( N c ) did not need any change in their description,characterized by their couplings to the local bilinear currents (8), baryons,and, more precisely, the currents to which they may couple preferentially,require a change of description. In SU(3), baryonic states are coupled tocurrents that are trilinear in quark fields and completely antisymmetric incolor indices to ensure gauge invariance. A typical such current is: j (3) B ( x ) = 13! ǫ abc ψ a ( x ) ψ b ( x ) ψ c ( x ) , (15)where ǫ is the Levi-Civita symbol (a completely antisymmetric tensor) and,for simplicity, we have considered quarks with the same flavor and omittedthe spin indices. In passing to SU( N c ), one has to generalize the abovedefinition by using the Levi-Civita tensor in N c dimensions, involving N c indices, which, in turn, requires the use of N c quark fields. The baryoniccurrrent then becomes: j ( N c ) B = 1 N c ! ǫ a a ··· a Nc ψ a ψ a · · · ψ a Nc . (16)Considering now the two-point function of this current, one can try to evalu-ate the N c dependence of the corresponding Feynman diagrams, as was donein Fig. 5 for the mesonic currents. Typical diagrams are presented in Fig. 14.Taking into account the normalization factor included in the definitionof the current (16), the class of diagrams not containing gluon propagators[Fig. 14a] behaves as O ( N c ), which fixes the normalization for the other typesof diagrams. A diagram containing one gluon propagator [Fig. 14b], joiningtwo quark propagators, contains a damping factor 1 /N c coming from the cou-pling constant squared [Eq. (3)]. However, there are N c ( N c − / ∼ N c / O ( N c /N c ) = O ( N c ). A diagram containing two gluon propagators [Fig. 14c],joining different quark lines, contains the damping factor 1 /N c coming from24 a) O ( N c ) (b) O ( N c ) (c) O ( N c ) Figure 14: Typical Feynman diagrams of the two-point function of the baryonic current(16): (a) without gluon propagators; (b) with one gluon propagator (sample); (c) withtwo gluon propagators (sample). The order in large- N c behavior of the contributions ofall diagrams of each category is also indicated. the coupling constants at the vertices; this is to be multiplied by the totalnumber of such diagrams, which is of the order of N c ; the total contributionof this category of diagrams is therefore O ( N c ). We observe that the pertur-bative expansion of the two-point function introduces at each order of theexpansion a new factor N c , which makes the corresponding series formallydivergent at large N c . Contrary to the case of the two-point functions ofthe mesonic currents, it is not possible here to group, in a stable way, ele-ments of the perturbative series into topological classes having well-defined N c behaviors.Another complication arises from the fact that, inside baryons, the recog-nition of the color topological categories of diagrams is less evident thanfor mesons; the reason for this is related to the fact that, in baryons, allquark colors flow in the same direction, while in mesons the color of the anti-quark flows in the opposite direction to that of the quark. We illustrate thisphenomenon by focusing on the first gluon-exchange diagrams between twoquark lines. They are presented, in the double-line representation, in Fig. 15.Figure 15a corresponds to the one-gluon exchange diagram. We observethat, because of the similar directions of the quark color flows, the two colorlines of the gluon propagator cross each other. As it stands, this diagramcannot be factorized in a plane, as in the planar-diagram case, into separatecolor flows without crossing. One could, of course, unfold one quark line, byinversing its drawing and flow, to make the diagram planar, however, thisoperation may, in general, be forbidden by the rest of the bigger diagraminto which the above diagram is embedded. The behavior of the diagram,with fixed external color indices, is O ( N − c ). Figure 15b corresponds to the25 a) O ( N − c ) (b) O ( N − c ) (c) O ( N − c ) Figure 15: Gluon exchanges between two quark lines in the double-line representation: (a)one-gluon exchange; (b) two-gluon ladder exchange; (c) two-gluon crossed-ladder exchange.The order in large- N c behavior of each diagram, with external color indices fixed, is alsoindicated. ladder exchange of two gluon propagators. Here also, we meet the previousphenomenon. Contrary to the meson case, the diagram does not have anycolor loop and behaves at large N c , with fixed external color indices, like O ( N − c ), a factor 1 /N c less than in the case of mesons. Figure 15c correspondsto the crossed-ladder exchange of two gluon propagators. Here, the diagramcontains an internal color loop and behaves like O ( N − c ). It thus appearsas the partner of the one-gluon exchange diagram for summation purposes.This property remains true for n -gluon exchange diagrams where each gluonline is crossed by the other ( n −
1) gluon lines.The above results seem to prevent the consideration, in a simple way, ofthe large- N c limit in the sector of baryons, at least in a way that is parallelto that of mesons.A way out of this difficulty was proposed by Witten [99]. He noticed thatwhereas the perturbative interaction between two quarks is small, of theorder of 1 /N c , it is the big number of quarks inside the baryons and the sumof the mutual interactions that are at the origin of the divergence. In such acase, diagrammatic considerations, which focus on the mutual interactions ofa few neighboring quarks, are of little help. Every quark experiences a globalstrong force representing an average form of the sum of the forces exerted bythe other quarks. One is then in a situation where a self-consistent mean-field approximation can be used. The problem simplifies further in the case ofheavy quarks, where nonrelativistic theory applies in the form of the Hartreeequations. Witten showed that these equations can be consistently solved,yielding a coherent description of the baryonic sector in the large- N c limit.The main property that characterizes these equations is that they globallyscale as N c at large N c ; this is due to the fact that each of their components –26he total kinetic energy, the total potential energy and the mass of the baryon– has the same N c -behavior. Therefore, N c factors out of the equations,leaving N c -independent equations, which ensure the stability of the resultunder perturbations with respect to 1 /N c . In particular, the size and shapeof the baryons turn out to be independent of N c . The method is applied tothe case of the ground state, as well as to the excited states, of baryons.The above procedure is also applied to the study of processes like baryon-baryon, baryon-antibaryon and baryon-meson scatterings. All these pro-cesses have the property of leading to equations that globally scale like N c ,which is then factoured out.The properties and dynamics of the baryons are thus mainly described,at large N c , by semi-classical equations, rather than by microscopic quantumequations.How to interpret the dissymmetry that emerges, at large N c , betweenthe mesonic and baryonic sectors? In this respect, Witten has made the fol-lowing crucial observation [99]: The mesonic sector of QCD is described bya weakly interacting effective field theory, whose interaction scale is of theorder of 1 /N c . On the other hand, weakly coupled field theories often de-velop nonperturbative solutions, like solitons or monopoles [143–146], whosemass scale is governed by the inverse of the weak coupling. This is, in par-ticular, the case for some electroweak theories with spontaneously brokensymmetry, characterized, say, by a coupling constant squared α , which pos-sess a magnetic monopole type solution, whose mass is of the order of 1 /α [143–145]. The structure of the monopole is determined, for small α , bysolving classical equations, from which α drops out, the size and shape ofthe monopole then becoming independent of α . Similarly, the mass of thebaryons in QCD is of the order of the inverse of the coupling 1 /N c , i.e., of theorder of 1 / (1 /N c ) = N c . Therefore, baryons can be considered as the QCDanalogs of the solitons or magnetic monopoles, while mesons and glueballsare the analogs of ordinary particles.More detailed investigations in the baryonic sector can be found in Refs.[128, 147–156]. We discuss, in this subsection, the generalization of the Standard Model(SM) to the case when the color subgroup of the SM symmetry group becomesSU( N c ) instead of SU(3). 27e first briefly recall the quantum numbers of the quark and lepton fieldsin the SM (see, e.g., [111, 113–115]). The SM of elementary particles is agauge theory based on the spontaneously broken SU L (2) × U Y (1) × SU( N c )symmetry, where the SU L (2) × U Y (1) sector describes electroweak (EW)interactions of the fundamental fermion fields of the SM, quarks and leptons.The SM contains three generations of fermion matter fields, quarks andleptons. In each generation, left-handed fermions compose doublets withrespect to the SU L (2) group, whereas the right-handed matter fields areSU L (2) singlets. For instance, in the first generation, the SM contains twoweak doublets q L = (cid:18) u L d L (cid:19) , l L = (cid:18) ν L e L (cid:19) , (17)and four right-handed SU L (2) singlets (if one includes the Dirac right-handedneutrino field in the set of the SM fields): u R , d R , e R , ν R . (18)The quark and lepton doublets have the same SU L (2) charges: +1 / − / L (2) charges. The U Y (1) quantumnumbers – the weak hypercharges Y – for each SU L (2) multiplet are inde-pendent from each other (the upper and the lower components of any doublethave the same Y ). In the SM one has Y lL = − , Y eR = − , Y νR = 0 ,Y qL = , Y uR = , Y dR = − . (19)The electric charge is related to the SU L (2) and U Y (1) quantum numbers bythe Gell-Mann–Nishijima relation Q = I + Y / , (20)where I is the eigenvalue of the third component of the weak isospin . Forthe left-handed doublets, I = ± /
2, for the right-handed singlets, I = 0.The SM is free from the chiral (axial) anomaly, since the quark-loopcontribution to the anomaly cancels against the lepton-loop contribution.This happens since quark and lepton charges satisfy the relation X leptons Q l + X quarks Q q = 0 . (21)28uark fields belong to the fundamental representation of the SU(3) colorgroup, so summation over quarks includes summation over color indices run-ning from 1 to 3. Notice that the left-handed fermion fields and the right-handed fermion fields satisfy Eq. (21) separately.When one generalizes the color group SU(3) to SU( N c ), the lepton quan-tum numbers remain unchanged, as leptons do not interact with the gluons,but the EW, i.e., SU L (2) × U(1), quantum numbers of the quark fields shouldbe changed. To obtain the electric charge and weak hypercharge of the quarksfor arbitrary N c , the following constraints are imposed: (i) The left-handedquark fields (cid:18) u L d L (cid:19) , (cid:18) c L s L (cid:19) and (cid:18) t L b L (cid:19) remain weak isospin doublets with I = ± /
2, while the right-handed quark fields remain SU L (2) singlets. Elec-troweak quantum numbers of all quarks satisfy the Gell-Mann–Nishijima re-lation (20). (ii) The SU L (2) × U(1) × SU( N c ) Standard Model should be freeof axial anomalies. Quark fields are in the fundamental representation of the SU ( N c ) color group, so there are N c different quark colors. The anomalycancellation condition requires that the sum of all electric charges (of leptonsand quarks) vanishes.In calculating the sum of charges in (21) for the left-handed particles,the terms ± / N c and thus one comes to the following relation Y lL + N c Y qL = 0 . (22)Taking into account that the hypercharge for the left-handed doublet remainsthe same as in the SM, Y lL = −
1, one obtains Y qL = N c . Proceeding in thesame way for the right-handed particles, one finds Y lL = − , Y eR = − , Y νR = 0 , (23) Y qL = N c , Y uR = 1 + N c , Y dR = − N c . (24)The quark electric charges therefore become Q u,c,t = 12 + 12 N c , Q d,s,b = −
12 + 12 N c . (25)The electric charges of mesons are not changed compared to the case N c = 3, since they contain one quark and one antiquark, whose Y -termscancel each other. Masses of mesons, as quark-antiquark composites, remainfinite, O ( N c ), at large N c . 29aryons are bound states of N c quarks. Their electric charges changecompared to the SU(3) case and generally increase with N c : for odd N c theircharges are integers, while for even N c they are half-integers.For the classification of baryons, one first uses strong isospin ; this ap-proximate symmetry of strong interactions is related to the smallness of themasses of the u and d quarks compared to Λ QCD and to the values of the QCDvacuum condensates. One may impose the conditions that, in the SU( N c )theory, u and d quarks still form a doublet with I = +1 / − /
2, re-spectively, while all other quarks are strong isosinglets , and that all quarkssatisfy the Gell-Mann–Nishijima relation (20) for strong hypercharges of thequarks. For the light quarks u , d and s , one finds [157]: Y u = Y d = 1 N c , Y s = − N c . (26)Baryons in large N c have masses of O ( N c ). One then is entitled to use spin-flavor symmetry [150] and to classify baryons into spin-flavor representationsat leading order in 1 /N c . In general, the latter are decomposed into distinctflavor multiplets with increasing spins. It is then possible to assign the fa-miliar baryons into representations where they keep the same values of spin,isospin, hypercharge and electric charge as in the SU(3) case. For instance,the proton is composed of ( N c + 1) / u quarks and ( N c − / d quarks andone may verify that its electric charge is +1. The reader may consult Ref.[157] for a more detailed account of the classification scheme of baryons. Let us briefly comment on the possible magnitude of the corrective factorsin the 1 /N c expansion method in phenomelogical applications, where one hasto use the physical value 3 of N c .The qualitative successes that the 1 /N c expansion method has obtained inQCD in the understanding of the hierarchy of various processes and phenom-ena brings an indirect or implicit justification of the validity of the method,which hinges, like other perturbative methods, on the smallness of the ex-pansion parameter 1 /N c . When the corrective factors to the leading termsare of order 1 /N c , the latter become, for N c = 3, of the order of 1/10, whichis indeed a small quantity. For corrections of order 1 /N c (not to be con-founded with leading terms of order 1 /N c ), one has a quantity of the orderof 1/3, which is not fairly small. Generally, corrections of order 1 /N c come30rom internal quark loops, which, furthermore, are proportional to the fla-vor number N f . Considering only the case of three light quarks ( u, d, s ), N f = 3, one would have N f /N c = 1, which is not a perturbative parameter.Actually, it becomes important here to know about the size of the coefficientthat accompanies the factor N f /N c . A hint about the latter coefficient isprovided by the expression of the beta function, Eq. (2). Comparing bothterms of the right-hand side of Eq. (2), one deduces that, for N f = 3 and N c = 3, the corrective term coming from the quark loops with respect to theleading term, is approximately equal to 0.18, which is a small quantity thatmight represent an acceptable value for an expansion procedure. In reality,on phenomenological grounds, the corresponding corrective factors, exam-ples of which are provided by the quark-sea contributions inside hadrons (inthe nonperturbative regime) and by the OZI-rule violations, are even muchsmaller, being of the order of a few percent.Therefore, one may consider the perturbative expansion, in terms of 1 /N c ,at least for the first corrective terms, as a phenomenologically well-establishedprocedure.In conclusion, the large- N c limit of QCD, proposed by ’t Hooft and com-pleted by Witten’s proposal about baryons, leads to a consistent and simpli-fied picture of the hadronic world and of its strong interaction dynamics. In this subsection, we briefly outline the role of the large- N c limit inthe correspondence established between string theories and quantum fieldtheories, mostly known as the AdS/CFT correspondence [104–109]. Thissubject being out of the scope of the present review, we only focus here onthe philosophy that has guided the related investigations. The interestedreader is invited to consult the quoted references.String theory has been present in hadronic physics from the early daysof the discoveries of hadron resonances. The existence of a large numberof hadron resonances, lying along Regge trajectories, was very suggestive ofstring theory spectra. Later, the advent of the quark model and of QCDintroduced the concept of confinement of quarks and gluons, a property thatis also shared by strings, at the endpoints of which are attached quarks.It was noticed by ’t Hooft, on the basis of the diagrammatic expansion inthe large- N c limit, that the resemblance between QCD theory in its nonper-turbative regime and string theory is much enforced [96] in that limit. Thespectrum of mesons is then very similar to that of free strings, whose coupling31onstant would be of the order of 1 /N c . This would confirm the existence ofthe duality relation existing between the two theories. The string picture isthen expected to be induced in QCD by the chromoelectric flux tubes and bythe Wilson lines that ensure gauge invariance of multilocal operators. How-ever, this string theory seems to be an effective theory [158, 159]. Flat-spacestring theories are consistently formulated in ten dimensions and the scalesthat they involve are of the order of the Planck scale, rather than of thehadronic scale [160–163].On the other hand, the analysis applied to the large- N c limit of QCD israther general and could be applied to other gauge theories with differenttypes of gauge symmetries. Duality relations could thus be searched for ina wider area of theories, general gauge field theories on the one hand andgeneral string theories on the other. The basic idea is that some physicaltheories might have two equivalent descriptions, each with different variables,such that the strong coupling regime of one of them corresponds to the weak-coupling regime of the other. The corresponding investigations have beenbased on searches for theories having common global-symmetry properties,which survive the changes of descriptions.From the latter point of view, it turns out that the conformal invarianceis the simplest symmetry that solves the problem. Thus, N = 4 super-Yang-Mills (SYM) theories, in four dimensions, are found to be dual to type-IIBstring theory in Anti-de Sitter (AdS) space, in five dimensions; the fifthdimension of the latter space is necessary for a consistent introduction ofgravity. This means that, in the duality relation, the gauge theory appearsas a boundary value of the string theory. To ensure complete consistency ofthe relation, one must start, as emphasized before, from a ten-dimensionalspace. The gauge theory then corresponds to the partially compactifiedtheory through a six-dimensional manifold M , while the string theory iscompactified through a five-dimensional sphere S , leading to the followingcorrespondence of spaces: R , × M ⇐⇒ AdS × S .As in other cases, the large- N c limit simplifies the above relationship andmakes the duality relation more transparent. In particular, it reduces thecurvature of the Anti-de Sitter space and constrains the gravity theory to itsweak coupling regime, therefore allowing calculations of the gauge theory inits strong coupling regime.In spite of this theoretical progress, the above duality relation does notdirectly apply to QCD theory, for several reasons. First, the latter theoryis not conformally invariant; scale symmetry is broken by quantization and32he bound-state spectrum displays there a mass gap with towers of discretemasses. Second, QCD is not supersymmetric. Third, quarks belong to thefundamental representation of the color group, while in N = 4 SYM fermionsbelong, in four (identical) copies, to the (sole) adjoint representation with thesix-fold presence of scalar partners. For these reasons, the treatment of QCDneeds more elaborate pathways to establish the bridge with string theory[106]. The AdS/QCD correspondence remains, for the moment, at the levelof phenomenological approaches or of model building [164–169].
3. How to describe multiquark states?
Properties of physical states are usually probed in quantum field theoryby the study of Green’s functions or correlation functions, using interpolat-ing currents having nonvanishing couplings to them. This was the case formesons [Eq. (8)] and baryons [Eq. (15)]. In the case of bound states, moredetailed informations are obtained from the solution of bound-state equa-tions, which generally require the use of multilocal operators as interpolatingprobes.The problem is similar, in principle, in the case of exotic hadrons ormultiquark states. One has to find appropriate interpolating currents toextract from correlation functions their specific properties. Here, however,additional complications arise. First, because of the increasing number ofquarks in multiquark states, the number of the corresponding interpolatingcurrents also increases and several combinations of them may be as goodcandidates as the individual ones. An optimal choice, for practical purposes,would be the one that would provide the strongest coupling to the state understudy. However, the physical properties of the states are independent of theinitial choice of interpolating currents, provided the latter have nonvanishingcouplings to them. Second, in varying N c , the definition of the multiquarkstate itself may change. An example of this phenomenon has been met withthe ordinary baryons, for which the interpolating current had to be modified[Eq. (16)]. This phenomenon is rather general for multiquark states, in whichcase the large- N c generalization of the interpolating currents is no longerunique: one has to deal with different schemes of well-known multiquarkstates of the case N c = 3, such as tetraquarks, pentaquarks and hexaquarks.We shall review, in this section, the various possibilities that one meetsfor the choice of interpolating currents and operators for the study of theproperties of multiquark states. We shall first consider the case of the gauge33roup SU(3) and then its generalization to SU( N c ). For simplicity, we shallignore spin/Dirac indices and concentrate on color and flavor indices. Theinclusion of spin/Dirac indices can be done with the incorporation of appro-priate Dirac matrices, taking into account the total spin and parity of thestates. We first consider the case of tetraquarks, which are mesons expected tobe represented by two pairs of valence quarks and antiquarks . To avoidmixing problems with ordinary meson states, we shall consider four differentquark flavors, referred to by indices i, j, k, ℓ ; color indices will be designatedby a, b, c, . . . .Since the tetraquark is a color-singlet state, one has to find interpolatingcurrents that are globally color singlets. As mentioned in the Introduction,an evident choice is the product of two mesonic color-singlet currents of thetype of Eq. (8). Designating by T (1 , the tetraquark current, one has twodifferent choices: T (1 , ij, ¯ kℓ ( x ) = j ¯ ij ( x ) j ¯ kℓ ( x ) = (cid:16) ψ a,i ψ aj (cid:17) ( x ) (cid:16) ψ b,k ψ bℓ (cid:17) ( x ) , (27) T (1 , iℓ, ¯ kj ( x ) = j ¯ iℓ ( x ) j ¯ kj ( x ) = (cid:16) ψ a,i ψ aℓ (cid:17) ( x ) (cid:16) ψ b,k ψ bj (cid:17) ( x ) . (28)Another choice corresponds to the “diquark” combinations, by groupingthe two quarks and the two antiquarks into antisymmetric or symmetric rep-resentations. In the first case, one obtains with the two quarks the antitripletrepresentation, ¯3 , and with the two antiquarks the triplet representation, ;the two may then be combined into the singlet representation. In the sec-ond case, the two quarks are in the sextet representation, , and the twoantiquarks in the antisextet representation, ¯6 , which also can be combinedto yield the singlet representation. Designating by T ( ∓ , ∓ ) the correspondingcurrents, one has T ( − , − )¯ i ¯ k,jℓ ( x ) = 12 ǫ abc (cid:16) ψ a,i ψ b,k (cid:17) ( x ) ǫ dec (cid:16) ψ dj ψ eℓ (cid:17) ( x ) , (29) T (+ , +)¯ i ¯ k,jℓ ( x ) = 14 (cid:16) ψ a,i ψ b,k + ψ b,i ψ a,k (cid:17) ( x ) (cid:16) ψ aj ψ bℓ + ψ bj ψ aℓ (cid:17) ( x ) , (30) For brevity, we shall often refer to them as four-quark states. ǫ is the Levi-Civita tensor, already introduced in Eq. (15). The twocurrents T ( ∓ , ∓ ) are not independent of the two former currents T (1 , . Byusing in Eq. (29) the relation ǫ abc ǫ dec = δ ad δ be − δ ae δ bd , (31)and grouping in Eq. (30) the quark fields in bilinear current forms, one finds T ( − , − )¯ i ¯ k,jℓ = −
12 ( T (1 , ij, ¯ kℓ + T (1 , iℓ, ¯ kj ) , T (+ , +)¯ i ¯ k,jℓ = −
12 ( T (1 , ij, ¯ kℓ − T (1 , iℓ, ¯ kj ) . (32)Clearly, one can also reexpress the currents T (1 , as combinations of thecurrents T ( ∓ , ∓ ) .Finally, one can also choose tetraquark currents made of products ofbilinear currents in the octet representation, : T (8 , ij, ¯ kℓ ( x ) = (cid:16) ψ a,i ( T A ) ab ψ bj (cid:17) ( x ) (cid:16) ψ c,k ( T A ) cd ψ dℓ (cid:17) ( x ) , (33) T (8 , iℓ, ¯ kj ( x ) = (cid:16) ψ a,i ( T A ) ab ψ bℓ (cid:17) ( x ) (cid:16) ψ c,k ( T A ) cd ψ dj (cid:17) ( x ) , (34)where the T A s are the generators of SU(3) in the fundamental representation.By using the relation( T A ) ab ( T A ) cd = 12 (cid:16) δ ad δ cb − N c δ ab δ cd (cid:17) , (35)with N c = 3, one can reexpress these currents in terms of the currents T (1 , : T (8 , ij, ¯ kℓ = − (cid:16) T (1 , ij, ¯ kℓ + T (1 , iℓ, ¯ kj (cid:17) , T (8 , iℓ, ¯ kj = − (cid:16) T (1 , ij, ¯ kℓ + 13 T (1 , iℓ, ¯ kj (cid:17) . (36)Therefore, only two currents are independent for the probe of tetraquarkswith four different quark flavors. Their specific choice is a matter of taste orpractical usefulness and does not prejudge in any way the physical structureof the tetraquark. It is the calculation of their couplings to the latter whichultimately may provide the physical information.The above procedure of construction of currents can readily be generalizedto other multiquark states. We briefly sketch some of them.Pentaquarks are expected to be dominated by four valence quarks andone valence antiquark. We consider the case of four different flavors for thequarks, the antiquark having one of these flavors. A pentaquark current is35ost easily constructed as a product of a bilinear mesonic current [Eq. (8)]and of a trilinear baryonic current [Eq. (15)], an example of which is P (1 , ij,ikℓ ( x ) = j ¯ ij ( x ) j B,ikℓ ( x ) = (cid:16) ψ a,i ψ aj (cid:17) ( x ) 13! ǫ bcd (cid:16) ψ bi ψ ck ψ dℓ (cid:17) ( x ) . (37)Other currents commonly used are based on the diquark antisymmetricrepresentation: P (¯3 , − , − )¯ i,ij,kℓ ( x ) = 14 ǫ abc ψ a,i ( x ) ǫ bde (cid:16) ψ di ψ ej (cid:17) ( x ) ǫ cd ′ e ′ (cid:16) ψ d ′ k ψ e ′ ℓ (cid:17) ( x ) . (38)Hexaquarks are dominated by six valence quarks (or by three quarks andthree antiquarks, a case that we omit below). Their currents can commonlybe represented as products of two baryonic currents or products of threeantisymmetric diquark currents (here considered with four different quarkflavors): H (1 , ijk,ijℓ ( x ) = j B,ijk ( x ) j B,ijℓ ( x )= 13! ǫ abc (cid:16) ψ ai ψ bj ψ ck (cid:17) ( x ) 13! ǫ a ′ b ′ c ′ (cid:16) ψ a ′ i ψ b ′ j ψ c ′ ℓ (cid:17) ( x ) , (39) H ( − , − , − ) ij,ik,jℓ ( x ) = 18 ǫ abc ǫ aa a (cid:16) ψ a i ψ a j (cid:17) ( x ) ǫ bb b (cid:16) ψ b i ψ b k (cid:17) ( x ) ǫ cc c (cid:16) ψ c j ψ c ℓ (cid:17) ( x )(40)(no summation over repeated flavor indices). N c ) In passing to SU( N c ), nontrivial modifications occur in the definitionsof the interpolating currents that we met in the case of SU(3). This is re-lated to the fact that these currents are generally defined as products ofirreducible tensors and with the increase of N c the number of such tensorsincreases in turn. Adopting the notation of irreducible representations basedon Young tableaux (see, e.g., Ref. [170]), [ ℓ , ℓ , · · · ], where the nonincreas-ing integers ℓ i ( i = 1 , , . . . ) denote the number of boxes in each column,the defining fundamental representation to which belongs the quark field issimply [1], while the antifundamental representation to which belongs theantiquark field is [ N c − N c = 3, the latter reduces to [2], which36mplies that a two-index antisymmetric tensor of quark fields belongs to theantitriplet representation ¯3 . When N c >
3, one has ( N c −
2) distinct anti-symmetric irreducible representations, which generalize the case of N c = 3.The two-index antisymmetric representation is then [2], while the two-indexsymmetric representation is [1 , N c − J ) quarkfields, or ( N c − J ) antiquark fields, where J = 0 , , , . . . , ( N c − ǫ tensor, which now contains N c indices: j ( J, − ) a a ··· a J = ǫ a a ··· a J b b ··· b Nc − J ψ b ψ b · · · ψ b Nc − J , (41) j a a ··· a J ( J, − ) = ǫ a a ··· a J b b ··· b Nc − J ψ b ψ b · · · ψ b Nc − J , (42)flavor indices being ignored. The case J = 0 reproduces, up to a multiplica-tive constant, the baryonic color singlet current (16).The equivalent forms of representation (29) are then T ( J, − ) = 1( N c − J )! j a a ··· a J ( J, − ) j ( J, − ) a a ··· a J , J = 1 , , . . . , ( N c − . (43)The choice J = ( N c −
2) reproduces the antisymmetric representation ofEq. (32). The choice J = 1 corresponds to the grouping of the quark fieldsinto the antifundamental representation.The above currents do not exhaust all the possibilities of constructing in-terpolating currents. One still has the possibility of incorporating symmetricrepresentations, as in Eq. (30), which we omit here for simplicity.Other types of representation are the products of ordinary mesonic cur-rents, like in Eqs. (27) and (28). Using in Eqs. (43) contractions of the two ǫ tensors, as in Eq. (31), which are present in j and j , one can reexpressthe antisymmetric-type tensor currents as combinations of products of suchmesonic bilinear currents, as in Eq. (32). (In the case of one quark flavor,these reduce to powers of a single current.) The same result is also obtainedwith the symmetric representations. We notice, in particular, that the sim-plest quadrilinear currents of the types of Eqs. (27) and (28), met in theSU(3) case, may continue playing the role of interpolating currents in theSU( N c ) case, corresponding to the choice J = N c − J close to 1, the tetraquark will have a solitonic structure, as in thecase of baryons .The fact that all tetraquark currents can be reexpressed as combinationsof products of ordinary mesonic currents is an indication that they are color-reducible, unlike the currents of ordinary mesons and baryons. This hasthe consequence, that, at large N c , the leading behavior of their correlationfunctions is given by that of products of correlation functions of ordinarymesonic currents, representing disconnected propagation of free mesons andnot of tetraquarks [42, 99, 126]. We consider here, as an example, the case ofthe two-point function of the current (27). At large N c , it behaves at leadingorder as h T (1 , ij, ¯ kℓ ( x ) T (1 , † ¯ ij, ¯ kℓ (0) i = N c →∞ h j ¯ ij ( x ) j † ¯ ij (0) ih j ¯ kℓ ( x ) j † ¯ kℓ (0) i = O ( N c ) , (44)where the N c -behavior is obtained from Fig. 5a. This means that the searchfor tetraquark states in correlation functions has to go beyond the leadingorder [171].For pentaquarks, the currents constructed as products of a mesonic and abaryonic current are still valid, provided one uses for the latter its expressionof SU( N c ) [Eqs. (16) and (41)]: P (1 , ( x ) = j ( x ) j (0 , − ) ( x ) . (45)Generalizations of the antisymmetric tensor currents (38) are: P ( J,K, − ) = j a ··· a J ,b ··· b K ( J + K, − ) j ( J, − ) a a ··· a J j ( K, − ) b b ··· b K ,J, K = 1 , , . . . , ( N c − , ( J + K ) ≤ ( N c − . (46)For hexaquarks, the analogs of representations (39) are H (1 , ( x ) = j (0 , − ) ( x ) j (0 , − ) ( x ) . (47) For simplicity of language, we shall continue using for the exotic states the same namesas in SU(3), independently of their internal structure. H (1 , ,..., , − ) = ǫ a a ··· a Nc j (1 , − ) a j (1 , − ) a · · · j (1 , − ) a Nc , (48) H (1 , ,N c − , − ) = ǫ a a ··· a Nc j (1 , − ) a j (1 , − ) a j ( N c − , − ) a ··· a Nc . (49)Like the tetraquark currents, pentaquark and hexaquark currents are de-composable along combinations of products of ordinary hadronic currents;their two-point functions satisfy properties similar to that of Eq. (44).Graphical representations of the currents introduced in the present sub-section will be presented in Sec. 3.3, in the more general case of multilocaloperators.Multiquark-state currents of the types introduced above have been con-sidered and studied in Refs. [172, 173]. The description of multiquark states may also necessitate in some in-stances the use of more general probes than local currents. Bound-stateequations require the use of multilocal fields. Lattice gauge theory, whichworks in a discretized spacetime, is another instance where the theory is for-mulated from the start by means of such operators. It is therefore necessaryto find the corresponding generalizations of the various currents that we metin our previous study. We shall focus our attention on representations thatpreserve the gauge invariance of the theory.Gauge-invariant operators are constructed by using path-ordered gluon-field phase factors [174–177], also called gauge links or Wilson lines, havingthe form U ab ( C yx ) = (cid:16) P eig Z C yx dz µ T B A Bµ ( z ) (cid:17) ab , (50)where C yx is an oriented curve going from x to y and P represents the path-ordering of the gluon fields according to their position on the line C yx ; theintegration goes from x to y along that line. The phase factors U ( C yx ) arethe color parallel transporters along the lines C yx [178].The color-trace operation on a phase factor taken along a closed contour C xx defines a gauge-invariant operator, called the Wilson loop. Its vacuumexpectation value plays an important role in defining gauge-invariant staticpotential energies [40, 41, 82]. 39auge-invariant operators coupling to mesons and baryons (here for thegroup SU(3)) are M = q a ( y ) U ab ( C yx ) q b ( x ) , (51) B = 13! ǫ abc U ad ( C xy ) q d ( y ) U be ( C xt ) q e ( t ) U cf ( C xz ) q f ( z ) , (52)where flavor and spin indices have been omitted. A pictorial representationof them, with phase factor lines chosen along straight line segments, is givenin Fig. 16. Meson and baryon local currents [Eqs. (8) and (15), respectively]are obtained (up to the defining multiplicative constants) by concentratingthe quark and antiquark coordinates at single points and by shrinking in thelatter expressions the phase factors to 1. q ( y ) q ( x ) (a) q ( y ) q ( t ) q ( z ) ǫx (b) Figure 16: Pictorial representation of the gauge-invariant meson (a) and baryon (b) oper-ators; in the baryon case, ǫ is the completely antisymmetric Levi-Civita tensor, indicatingthe antisymmetric property of the three-line vertex. As a general remark, let us emphasize that physical properties of statesshould be independent of the shape of the lines C yx , provided they are contin-uous and smoothly varied upon deformations from straight lines. The latterare generally chosen for their simplicity and for their adequacy in lattice cal-culations [40]. The above property can be verified in the case of bound-stateenergies, which are obtained from the behavior of Wilson-loop vacuum aver-ages at large time separations. In QCD, the Wilson-loop vacuum average isexpected to satisfy in that limit the area law and, more generally, the mini-mal surface property [40, 179, 180]. Smooth deformations of the phase-factor In this and the following subsections, for ease of pictorial representations, the quarkfields are designated by the notation q , rather than ψ . ( u ) q ( v ) q ( u ) q ( v ) ǫ ǫx y (a) q ( v ) q ( v ) q ( v ) q ( v ) q ( u ) ǫ ǫǫx yz (b) q ( v ) q ( v ) q ( v ) q ( v ) q ( v ) q ( v ) ǫ ǫǫx yzt ǫ (c) Figure 17: Pictorial representation of the gauge-invariant (a) tetraquark, (b) pentaquark,and (c) hexaquark operators. lines inside the bound-state definition do not change its energy, but only af-fect the expression of the wave function (cf. Sec. 7.2 below and Ref. [180],Appendix A).Similar constructions can also be applied to the multiquark states. Theywere promoted in the past by Rossi and Veneziano [181, 182] and are called“string-junction” or “ Y -shaped-junction” type representations. They aremainly considered for the antisymmetric representations, typical of the di-quark picture [52–54], in which the interquark forces are expected to beattractive, leading to the emergence of multiquark bound states. Pictorialrepresentations of these are given in Fig. 17. The local multiquark antisym-metric representation currents [Eqs. (29), (38) and (40)] are obtained (up tothe defining multiplicative constants) by concentrating quark and antiquarkcoordinates at single points and by shrinking the phase factors to 1.Generalizations of the previous operators to the SU( N c ) case are straight-forward, following the constructions of the corresponding local currents [Eqs.(8), (16), (43), (46), (48) and (49)]. The case of mesons and baryons isdisplayed in Fig. 18. Tetraquark, pentaquark and hexaquark operators aregraphically represented in Figs. 19, 20 and 21.41 ( N c ) Figure 18: Meson and baryon operators in the SU( N c ) case. ǫ ǫ ( N c − ) ( ( N c − ) ) · · · ǫǫ ( N c − ) Figure 19: Tetraquark operators in the SU( N c ) case. The two extreme cases, with J = 1and J = ( N c − N c −
1) quarks and ( N c −
1) antiquarks, with a single link between the two string junctions.The last diagram contains two quarks and two antiquarks, with ( N c −
2) links betweenthe string junctions. ǫ ǫǫ ( N c − ) ( ( N c − ) ) ( N c − ) · · · ǫ ǫǫ ( N c − ) ( ( N c − ( Figure 20: Pentaquark operators in the SU( N c ) case. Two extreme cases, with J = 1, K = 1 and J = 1, K = ( N c − N c −
1) quarks and ( N c −
2) antiquarks. The last diagram contains( N c −
1) + 2 quarks and one antiquark. ǫǫ ǫǫ ( N c − ) ( ( N c − ) ) ( N c − ) ) ( N c − ) ( ( N c ) · · · ǫ ǫǫ ǫ ( N c − ) ( ( N c − ) ( N c − ( Figure 21: Hexaquark operators in the SU( N c ) case. Two extreme cases, correspondingto Eqs. (48) and (49), are displayed. The first diagram contains N c ( N c −
1) quarks. Thelast diagram contains 2( N c −
1) + 2 quarks.
In summary, multiquark states can generally be described, or theoreti-cally probed, by several operators, each highlighting a particular aspect ofthe state under study. In passing to large N c , the number of these opera-tors increases and the structure of the multiquark states may become morecomplicated. One however hopes that only few of them will represent thedominant representative scheme, which might correspond to the outcome ofmore dynamical investigations. 43 . Singularities of Feynman diagrams connected with multiquarkstates While the large- N c limit approach is a method aiming to explore theproperties of the theory in its nonperturbative regime, here, for QCD, in itsconfining regime, it still hinges, as we have seen in Sec. 2, on the analysis ofFeynman diagrams, which are representative of the perturbative regime ofthe theory. Although this might seem contradictory, it should be emphasizedthat one is not considering a single or a finite number of Feynman diagrams,but rather classes of Feynman diagrams which are distinguished by theirtopological properties in color space. Thus, in Fig. 5, the diagrams (a),(b) and (c) are parts of the same class of planar diagrams, depicting thetwo-point correlation function of meson currents, having the same large- N c behavior. This class contains an infinite number of other diagrams, involvingmany-gluon exchanges, but the same number (viz., two) of quark lines. Thelarge- N c approach assumes that the infinite sum of diagrams contained inthis topological class produces the bound states of mesons and maintains theconfining property of the theory [96, 99, 126].Each Feynman diagram participating in the above summation process,though not explicitly displaying confining properties or bound-state attri-butes, should carry a minimum amount of common qualitative features withthe other diagrams in order to produce at the end the desired nonperturbativeeffects. In the example of the two-point function of the meson currents givenabove, it is the number of quark lines which is common to all the summeddiagrams. It is this number that allows the introduction of the notion ofvalence quarks. This is then manifested in each Feynman diagram throughthe singularity structure in momentum space, represented by a discontinuityin the total invariant mass squared, the so-called Mandelstam s -variable,starting from the two-quark threshold and going to infinity. The summationof diagrams transforms this singularity into a series of meson poles.The same procedure also applies to the two-point functions of the baryoniccurrents. For N c = 3, it is the Feynman diagrams with three quark lineswhich should be representative of the leading valence-quark structure. Whenthe large- N c limit is taken, the number of valence quarks for baryons increaseswith N c , however, the singularity content of each diagram should remain, inthat a threshold singularity in the s -variable should be present and shouldlead after summation to a series of baryon poles.In passing to the case of exotic (multiquark) states, one expects a gen-44ralization of the above phenomenon. An exotic state with a number A ofvalence quarks should be represented, at leading order in N c , by diagramscontaining a number A of quark lines. Here, however, two kinds of difficul-ties emerge, which were not present in the case of ordinary hadrons. First,Feynman diagrams with A quark lines may contain color-singlet disconnectedpieces; this is a consequence of the fact that the multiquark currents are gen-erally expressible as combinations of products of ordinary currents (cf. Sec. 3,Eqs. (27), (28), (37), (39) and (44)). Such diagrams, which represent propa-gation of free particles, cannot participate in the formation of bound statesand hence should not be taken into account. Second, there are still connectedFeynman diagrams, having A quark lines, which do not possess singularitiesin the s -variable. Their singularities concern the u - or t -variables and there-fore cannot not participate in the multiquark pole production process andshould not be considered.In summary, the counting of quark lines in a given Feynman diagramis no longer sufficient for its consideration in the formation process of themultiquark bound state. A more precise criterion, based on the analysisof the singularity structure in the s -variable, is necessary. This criterion isprovided by the Landau equations [110, 111], which allow one to analyze inmore detail the singularity properties of Feynman diagrams. We shall brieflysketch below the Landau equations and shall consider a few typical exampleswhich will be helpful in the analyses of multiquark-state properties.A generic expression of a Feynman diagram is I ( p ) = Z L Y ℓ =1 d k ℓ (2 π ) I Y i =1 q i − m i + iǫ ) , (53)where p represents a collection of external momenta and q i ( I in number) arelinear functions of the p s and of the loop variables k ℓ ( L in number).The Landau equations are λ i ( q i − m i ) = 0 , i = 1 , . . . , I, (54) I X i =1 λ i q i · ∂q i ∂k ℓ = 0 , ℓ = 1 , . . . , L, (55)where the λ s are Lagrange multipliers to be determined. Some of the param-eters λ may vanish or may be compatible with vanishing values.45e are mainly interested in the location of the singularities producedby the quark propagators. Gluons being massless, the singularities of theirpropagators generally start at the same positions as those produced by thequark propagators. We therefore shall not consider, in general, gluon propa-gators in the Landau equations as independent sources of singularities; thisis realized by putting, from the start, the corresponding λ s equal to zero.However, gluon lines may participate in the production of quark singularitiesthrough the momentum they carry.Since multiquark states are expected to decay into ordinary hadrons, orto have couplings with them in the case of bound states, it is easier to studytheir properties through the scattering amplitudes of ordinary hadrons andthe corresponding Feynman diagrams.We consider, for definiteness, the case of tetraquarks made of four quarkswith different flavors; the quarks will be designated by indices 1 and 3 andthe antiquarks by indices 2 and 4. As we have seen in Sec. 3.2, such adescription still remains valid for general N c , although in the latter case otherrepresentations also emerge (cf. Fig. 19). Within the present representation,the tetraquark may couple to two mesons and therefore may be probed in two-meson scattering processes. Using the bilinear currents defined in Eq. (8),one may consider, in momentum space, Fourier transforms of the correlationfunctionsΓ D ≡ h j ( x ) j ( y ) j † ( z ) j † (0) i , Γ D ≡ h j ( x ) j ( y ) j † ( z ) j † (0) i , (56)Γ R ≡ h j ( x ) j ( y ) j † ( z ) j † (0) i , Γ R ≡ h j ( x ) j ( y ) j † ( z ) j † (0) i , (57)where the subscript D refers to direct-channel processes and the subscript R to quark recombination or rearrangement channel processes.We first consider two typical diagrams of the direct channel 1 process– a disconnected diagram and a connected one – represented in Figs. 22aand b, respectively. (The diagram with one-gluon exchange between the twodisconnected color-singlet diagrams is zero.) The total conserved momentumof the two-meson processes is designated by P , with the usual definition s = P .The Landau equations of the disconnected diagram are themselves sep-arable into two independent subsets, leading to the physical singularities P = ( m + m ) and P = ( m + m ) , which do not involve s . Thesesingularities refer to the internal structure of each meson, which propagatesfreely and independently from the other. It is evident that this diagram can-46 P P P P − kkP − k ′ k ′ j j † j j † (a) O ( N c ) P P ′ P P ′ P − kk − k ′′ k ′ + k ′′ P − k ′ k ′′ j j † j j † (b) O ( N c ) Figure 22: Disconnected (a) and connected (b) diagrams in the direct channel M M → M M of the meson-meson scattering amplitude. not be involved in the formation of a bound state or of a resonance. One canalso consider all other planar diagrams which include the gluon exchangesinside each quark loop, associated with the above disconnected diagram, stillfinding the same singularities as above, confirming the fact that gluon prop-agators generally do not modify the location of singularities found with thesole quark propagators.For the connected diagram of Fig. 22, it is sufficient to consider a verticalcut passing between the two gluon lines. The Landau equations are then: λ (( P − k ) − m ) = 0 , λ (( k − k ′′ ) − m ) = 0 ,λ (( k ′ + k ′′ ) − m ) = 0 , λ (( P − k ′ ) − m ) = 0 , (58) − λ ( P − k ) + λ ( k − k ′′ ) = 0 , − λ ( k − k ′′ ) + λ ( k ′ + k ′′ ) = 0 ,λ ( k ′ + k ′′ ) − λ ( P − k ′ ) = 0 . (59)The following definitions hold: P = P + P = P ′ + P ′ , s = P ,t = ( P − P ′ ) , u = ( P − P ′ ) . (60)The system of equations (58) and (59) can be solved, leading to the physicalsingularity at s = ( P i =1 m i ) . The fact that the four quark masses arepresent means that we have four-quark intermediate states, which, togetherwith the contributions of other diagrams involving more gluon lines, willgenerate two-interacting-meson states and possibly tetraquark states.47e next consider, in Fig. 23, two diagrams of the recombination channel1. Here, the following definitions hold: P = ( P + P ) = ( P ′ + P ′ ) , s = P ,t = ( P − P ′ ) , u = ( P − P ′ ) . (61) j j † j j † P P ′ P P ′ P − kk P − P ′ + kP ′ − k (a) O ( N c ) j j † j j † P P ′ P P ′ P − k ′ k P − P ′ + k ′ P − P ′ + k P ′ − kk ′ − k (b) O ( N c ) Figure 23: Diagrams in the quark-rearrangement channel M M → M M of themeson-meson scattering amplitude, not concerned, at large N c , with tetraquark states. The Landau equations of diagram (a) have several subsets of physicalsolutions: t = ( m + m ) , u = ( m + m ) , P = ( m + m ) , etc., butno singularities in s are found. The t - and u -channel singularities will besaturated, with similar diagrams involving gluon lines, by one-meson states.The singularities in the external momenta squared are those that are presentin the external meson propagators. The Landau equations of diagram (b) alsolead to the same sets of singularities as diagram (a). Therefore, the diagramsof Fig. 23, which apparently display four quark lines, do not participatein the formation of two-meson interacting systems, nor to possibly existingtetraquark states. Nevertheless, they produce, in the mesonic world, contact-type interactions, as well as one-meson exchange diagrams.One may have another view of the preceding results, by referring to thetopological properties of the diagrams in color space. The latter are planardiagrams and can be unfolded, as suggested in Ref. [183], to make explicitthe color flow. The unfolded plane corresponds now to the ( u, t ) plane (Fig.24). The t - and u -channel singularities are obtained by cutting the boxdiagrams by horizontal and vertical lines, respectively. It is evident, here,that the corresponding singularities are two-quark singularities, typical ofone-meson states. The s -channel singularities are obtained by cutting the48ox diagrams by oblique and curved lines passing through the four quarkpropagators. However, when the diagram is color-planar, as is the case here,the cuts produce disconnected singularities, concentrated at opposite cornersand corresponding to radiative corrections of external-meson propagators orof current vertices. s -channel singularities may arise only when the diagramis color-nonplanar. u tj j † j † j (a) O ( N c ) j j † j † j (b) O ( N c ) Figure 24: Diagrams of Fig. 23 in unfolded form.
A typical color-nonplanar diagram, which contributes to the s -channelsingularities, is presented in Fig. 25, together with its unfolded form, wherethe nonplanarity is manifest. j j † j j † P P P P k ′ k P − k k − k ′′ k ′ + k ′′ P − k ′ k ′′ (a) O ( N − c ) j j † j † j (b) O ( N − c ) Figure 25: (a) Diagram in the quark-rearrangement channel M M → M M of themeson-meson scattering amplitude, participating, at large N c , in the formation of possibletetraquark states. (b) The same diagram in the unfolded ( u, t ) plane, displaying its color-nonplanar character. The efficient Landau equations are obtained by cutting the diagram (a)of Fig. 25 by a vertical line passing between the two gluon lines. One nowfinds a physical singularity in the s -channel at the position s = ( P i =1 m i ) ,49s in the case of Fig. 22b. Therefore, this diagram, together with otherdiagrams of the same color-topological class, will contribute to the formationof two-meson interacting states and eventually to that of tetraquark states.It is worthwhile noticing the difference of behavior in N c of the diagram(b) of Fig. 22 and the diagram (a) of Fig. 25 – O ( N c ) for the first and O ( N − c )for the second – contributing to the formation of tetraquarks in the directand recombination channels, respectively. This outlines the color-topologicaldifference that exists between them: the former is planar, while the latter isnonplanar, a feature that may have consequences for the various couplingsof tetraquarks to two-meson states.Other examples or details of Landau equations can be found in Ref. [184].
5. Tetraquarks at large N c This section is devoted to a detailed study of some of the properties oftetraquarks at large N c . We have seen, in Secs. 3.2 and 3.3, that tetraquarksat large N c may be described by ( N c −
2) inequivalent classes of operatorshaving different numbers of valence quarks [Eq. (43)], generically each having( N c − J ) valence quarks and ( N c − J ) valence antiquarks, where J takes valuesfrom 1 to ( N c − J = ( N c − N c ) does not change theirdescription. For the representations with 1 ≤ J ≤ ( N c − J close to 1, the latter become similar to thedescription found for baryons: the number of valence quarks and antiquarksgrows with N c and presumably also the mass of the tetraquarks. Here, thetetraquark becomes rather a solitonic object, requiring a different type oftreatment. It is not clear, for the time being, which representation providesthe most faithful description in the limit N c = 3. Since, however, the repre-sentation with J = ( N c −
2) does not require any modification of treatment,it remains the most practical one from the mathematical viewpoint. This iswhy we shall concentrate in the present section and review on this represen-tation. The reader may consult Refs. [172, 173] for a detailed account of theproperties of tetraquarks in higher representations.50ne particular feature of tetraquark and multiquark currents, met inSec. 3, is their decomposition property into combinations of products of or-dinary mesonic or baryonic currents, reflecting their color reducibility. Atlarge N c , two-point functions of tetraquark currents are dominated by thecontributions of their disconnected parts [Eq. (44)]. This fact has led Wittenand Coleman to conclude that multiquark states should not exist in QCD, atleast as confined states [99, 126]. Actually, as was emphasized by Weinberg[171], the situation may be more complex. Dynamical effects may still beat work, preventing the multiquark states from being dissociated into theirelementary mesonic or baryonic clusters. In such a case, the multiquark statewill appear as a pole, or a narrow resonance, in nonleading terms of the 1 /N c expansion. Contrary to the case of ordinary mesons and baryons, in order todetect the conditions in which multiquark states may appear, one has to go,in the 1 /N c expansion of correlation functions of currents, beyond the leadingorders. Studies in this line of approach can be found in Refs. [183–189].Since the tetraquark state can couple, in the present representation, totwo mesonic currents, it can naturally be probed in meson-meson scatteringamplitudes, appearing as a possible pole or a resonance. We therefore shallstudy the tetraquark properties at large N c through the N c -leading or sub-leading typical Feynman diagrams that may contribute to its emergence. Weshall first concentrate, in the following, on the case of fully exotic tetraquarks,containing four different quark flavors. This has the advantage of excludingmixings with ordinary mesons, which often may prevent one from drawinga clear conclusion. The case of cryptoexotic tetraquarks will be consideredafterwards. Most of the material needed for this study has been already introducedin Sec. 4. We consider two pairs of quarks and antiquarks, with four differentflavors, which we distinguish by the labels 1 and 3 for the quarks and 2 and4 for the antiquarks. We then consider the four-current correlation functions(56) and (57), describing two direct channels and two quark-recombinationchannels, designated by D D R R
2, respectively.We first consider the direct channels. The corresponding leading discon-nected and connected diagrams, for channel D
1, have been given in Fig. 22.It is understood that each such diagram is accompanied by an infinite numberof other diagrams with many-gluon exchanges belonging to the same color-topology class (here, planar). It is only the connected part of the correlation51unction that may provide information about the corresponding scatteringamplitude. To isolate the latter, one has to factorize in the connected part ofthe correlation function the external meson propagators, together with themeson couplings [Eqs. (12) and (13)]. These diagrams, as has been shown byEqs. (58)–(60), have four-quark singularities in the s -channel and hence mayparticipate in the formation, as intermediate states, of two-meson states, aswell as of possible tetraquark states, the latter henceforth being designatedby T .One then obtains the leading large- N c behaviors for the two-meson scat-tering amplitudes in channels D D A ( M M → M M ) ∼ A ( M M → M M ) = O ( N − c ) , (62) A ( M M → M M → M M ) ∼ A ( M M → M M → M M )= O ( N − c ) , (63) A ( M M → T → M M ) ∼ A ( M M → T → M M )= O ( N − c ) . (64)We next consider the recombination channels (57). Typical N c -leadingand -subleading diagrams have been shown in Figs. 23 and 25. Only diagramsof the type of Fig. 25 do have s -channel four-quark singularities and hencemay participate in the formation of two-meson and tetraquark states. Onthe other hand, the N c -leading diagrams, such as those of Fig. 23, contributeto parts of the scattering amplitude that do not have s -channel singularities.One obtains the following large- N c behaviors of the scattering amplitudes inchannels R R A ( M M → M M ) ∼ A ( M M → M M ) = O ( N − c ) , (65) A ( M M → M M → M M ) ∼ A ( M M → M M → M M )= O ( N − c ) , (66) A ( M M → T → M M ) ∼ A ( M M → T → M M )= O ( N − c ) . (67)One can analyze Eqs. (62) and (65) in terms of effective meson vertices.Four-meson vertices of the direct type appear as being of order N − c , while52hose of the recombination type of order N − c (Figs. 26a and 27a): g ( M M M M ) ∼ g ( M M M M ) = O ( N − c ) , (68) g ( M M M M ) = O ( N − ) . (69)Four-meson contact terms are also accompanied by glueball-exchange andone-meson-exchange terms (Figs. 26b and 27b). M M M M N − c (a) O ( N − c ) M M M M G N − c N − c (b) O ( N − c ) Figure 26: (a) Four-meson vertex in the direct channel D D M M M M N − c (a) O ( N − c ) M M M M M N − / c N − / c (b) O ( N − c ) M M M M M N − / c N − / c (c) O ( N − c ) Figure 27: (a) Four-meson vertex in the recombination channel R R The determination of the behaviors of four-meson vertices, including con-tact terms and meson exchanges, allows us to evaluate the contributions oftwo-meson intermediate states in the above processes. They are summarizedin Fig. 28, where we have kept, for simplicity, only contact-type interac-tions. They consistently reproduce the behaviors expected from Eqs. (63)and (66). Similar conclusions could also be obtained from the glueball- andone-meson-exchange diagrams.The validity of the behaviors displayed in Fig. 28 can also be verifiedon individual Feynman diagrams with gluon exchanges, using the Landau53 − c N − c M M M M M M (a) O ( N − c ) N − c N − c M M M M M M (b) O ( N − c ) N − c N − c M M M M M M N − c M M (c) O ( N − c ) Figure 28: Leading-order contributions of two-meson intermediate states to the direct, (a)and (b), and recombination, (c), channels. equations and recognizing the type of intermediate state that can be ob-tained from the summation, with respect to multigluon exchanges, of suchtypes of diagrams. Two examples are displayed in Fig. 29. In diagram (a),contributing to the direct channel D
1, the intermediate state is manifestlythe two-meson state M M , while the external mesons are M M . Thiscorresponds to Fig. 28a. In diagram (b) of Fig. 29, contributing to the re-combination channel R
1, the intermediate state is composed of M M onthe right and of M M on the left. This corresponds to Fig. 28c. (Also,one should not forget that in the right and left corners of the diagrams ofFig. 29, near the vertices of the currents j , one still has planar multigluonexchanges, whose infinite sum reconstitutes the external mesons.)The properties of possibly existing tetraquark states can be extractedfrom Eqs. (64) and (67). One observes that a single tetraquark alone cannotsatisfy these two equations. At least two different tetraquarks, which wedesignate by T A and T B , are needed to fulfill the conditions imposed bythese equations. The results for tetraquark to two-meson-state transitionamplitudes are the following (see Fig. 30): A ( T A → M M ) ∼ O ( N − c ) , A ( T A → M M ) ∼ O ( N − c ) , (70) A ( T B → M M ) ∼ O ( N − c ) , A ( T B → M M ) ∼ O ( N − c ) . (71)If the tetraquarks lie above the two-meson thresholds, the decay widths54 j j † j j † (a) O ( N c ) 1 43 2 j j † j j † (b) O ( N − c ) Figure 29: (a) Feynman diagram contributing to the reconstruction of the two-mesonintermediate state M M in the direct channel D R N − c N − c M M M M T A (a) O ( N − c ) N − c N − c M M M M T B (c) O ( N − c ) N − c N − c M M M M T A (b) O ( N − c ) N − c N − c M M M M T B (d) O ( N − ) Figure 30: Leading-order contributions of tetraquarks T A and T B to the direct (a,c) andrecombination (b,d) channels. T A ) ∼ Γ( T B ) = O ( N − c ) , (72)which are smaller than those of the ordinary mesons [Γ = O ( N − c )] by onepower of N c .To have an insight into the internal structure of the two tetraquark can-didates, one can transcribe the informations about the four-meson couplingscoming from Eqs. (68)–(69) into an effective Lagrangian, expressed throughquark color-singlet bilinears: L eff , int = − λ N c [( q q )( q q )( q q )( q q ) + ( q q )( q q )( q q )( q q )] − λ N c [( q q )( q q )( q q )( q q ) + ( q q )( q q )( q q )( q q )] , (73)where we have explicitly factored out the N c -dependence of the couplingconstants. One then deduces from Eqs. (70) and (71) that the tetraquarkfields T A and T B should have, at leading order at large N c , the followingstructure in terms of the quark color-singlet bilinears: T A ∼ ( q q )( q q ) , T B ∼ ( q q )( q q ) . (74)This result favors a color singlet-singlet structure of the tetraquarks inthe exotic case. We notice that, according to Eqs. (70) and (71), the maindecay channels of the tetraquarks are not of the dissociative type, but ratherof the quark rearrangement type.It is worth emphasizing here that the two-meson contributions, foundin Eqs. (63) and (66) [Fig. 28], saturate by themselves, at large N c , thesingularity structure emerging from the Feynman diagrams. Contrary tothe ordinary-meson case, they are in competition with the contributions ofpossibly existing tetraquarks. Therefore, the presence of the latter does notappear as mandatory for the saturation of the large- N c equations. The resultsfound above about the tetraquark couplings to two mesons and about theirdecay widths have, therefore, the meaning of upper bounds. Eventually, onemight encounter an intermediate situation, where one of the tetraquarks, T B ,say, is absent from the spectrum for some dynamical reason. In that case,one tetraquark, T A , would exist and, if it lies above the two-meson threshold,it would be observed through its preferred decay channel [Eq. (70)].56 .2. Cryptoexotic states We next consider cryptoexotic channels, involving three different quarkflavors, designated by 1, 2 and 3. As in Eqs. (56) and (57), we considercorrelation functions describing two direct and two recombination channels:Γ D ≡ h j ( x ) j ( y ) j † ( z ) j † (0) i , Γ D ≡ h j ( x ) j ( y ) j † ( z ) j † (0) i , (75)Γ R ≡ h j ( x ) j ( y ) j † ( z ) j † (0) i , Γ R ≡ h j ( x ) j ( y ) j † ( z ) j † (0) i . (76)Leading and subleading diagrams of the direct channel D (a) O ( N c ) j j † j j † j j † j j † (b) O ( N c ) j j † j j † (c) O ( N c ) (d) O ( N c ) j j † j j † Figure 31: Leading- and subleading-order diagrams of the direct channel D Diagram (b) of Fig. 31 corresponds to the leading-order contribution tothe meson-meson scattering amplitude. It has only a two-quark singularityin the s -channel, which represents the contribution of a single-meson inter-mediate state.Diagram (c) represents contributions from radiative corrections to theprevious diagram. In the space of meson states, the first part of the inter-mediate states contributes to the formation of a single-meson state, whichthen emits two virtual mesons, or a tetraquark, and reabsorbs them later.This diagram may also describe a mixing between a single-meson state anda tetraquark state, having the same quantum numbers.Diagram (d) represents a direct contribution of two-meson states and/orof a tetraquark state.For the direct channel D
2, the structure of the diagrams is similar to thatof Fig. 22 and is represented in Fig. 32.For the recombination channel R s -channel singularities [cf. Fig. 23a and related comment], while diagrams (b)57 a) O ( N c ) j j † j j † (b) O ( N c ) j j † j j † Figure 32: Leading- and subleading-order diagrams of the direct channel D and (c) receive contributions from four-quark intermediate states in the s -channel. Similar conclusions also hold for the recombination channel R j j † j j † (a) O ( N c ) j j † j j † (b) O ( N c ) j j † j j † (c) O ( N − c ) Figure 33: Leading- and typical subleading-order diagrams of the recombination channel R R From the previous results, one may obtain information about the variousscattering and transition amplitudes and effective meson couplings, as inEqs. (62)–(69). The effective meson-meson interactions at the vertex levelare summarized in Fig. 34.The contributions of two-meson intermediate states are represented gra-phically in Fig. 35.The tetraquark contributions are extracted in the same way as for theexotic channels. Because of the presence of the additional diagram (b) ofFig. 33, they are of the same order in all four channels and hence a singletetraquark T might accommodate all the corresponding constraints. How-ever, a more detailed analysis of the mechanism of formation of two-meson58 − / c N − / c M M M M M (a) O ( N − c ) M M M M M N − / c N − / c (b) O ( N − c ) M M M M N − c (c) O ( N − c ) M M M M N − c (d) O ( N − c ) M M M M N − c (e) O ( N − c ) Figure 34: Tree-level vertex diagrams with meson propagators in the direct channel D D R u -channel analogof diagram (b) not drawn.) N − c N − c M M M M M M (a) O ( N − c ) N − c N − c M M M M M M (c) O ( N − c ) N − c N − c M M M M M M (b) O ( N − c ) N − c N − c M M M M M M (d) O ( N − c ) Figure 35: Two-meson intermediate-state contributions to the direct channel D D R T A and T B , whose field structures, in terms ofvalence quarks and antiquarks, are T A ∼ ( q q )( q q ) , T B ∼ ( q q )( q q ) . (77)This conclusion is based on calculations similar to those presented for theexotic case in Sec. 5.4, where, now, the cryptoexotic case gives rise to addi-tional diagrams. The results are graphically summarized in Fig. 36. Actually,diagrams (a) and (d) of that figure may not exist, but could be representedby the mixing-mechanism diagrams (b) of Figs. 38 and 37, respectively (seebelow). This issue depends more sensitively on the formation mechanism ofthe tetraquarks. N − c N − c M M M M T A (a) O ( N − c ) N − c N − c M M M M T A (c) O ( N − c ) N − c N − c M M M M T B (b) O ( N − c ) N − c N − c M M M M T A (d) O ( N − c ) Figure 36: Tetraquark-state contributions to the direct channel D
1, (a) and (b), the directchannel D
2, (c), and the recombination channel R
1, (d).
The decay widths of the tetraquarks into two mesons are again of order N − [Eq. (72)].Diagram (b) of Fig. 33 may also describe mixings of two-meson or tetraquarkstates with a single-meson state that appears in the left part of the diagram.Figure 37 graphically describes this phenomenon.Mixings of a single-meson state with two-meson and tetraquark statesalso exist in the direct channel D
1, as was previously mentioned, emergingfrom diagrams of the type of Fig. 31c. Since in the quark loop the quark and60 − / c N − / c N − c M M M M M M M (a) O ( N − c ) N − / c N − / c N − c M M M T A M M (b) O ( N − c ) Figure 37: Mixings, in the recombination channel R
1, of a single-meson state with two-meson (a) and tetraquark (b) states. the antiquark can have any flavor, the resulting two-meson and tetraquarkstates may belong to another class of cryptoexotic states. Figure 38 graph-ically describes this phenomenon. Cryptoexotic tetraquarks may therefore N − / c N − / c N − / c N − / c M M M M M M M M (a) O ( N − c ) N − / c N − / c N − / c N − / c M M M T A M M M (b) O ( N − c ) Figure 38: Mixings, in the direct channel D
1, of a single-meson state with two-meson (a)and tetraquark (b) states. decay into two mesons either through a direct coupling or through a mixingwith single-meson states. In both cases, the transition amplitude is of order O ( N − c ).Cryptoexotic channels with two quark flavors can be treated in the sameway as before. Here, the correlation functions to be considered areΓ D ≡ h j ( x ) j ( y ) j † ( z ) j † (0) i , Γ D ≡ h j ( x ) j ( y ) j † ( z ) j † (0) i , (78)Γ R ≡ h j ( x ) j ( y ) j † ( z ) j † (0) i , Γ R ≡ h j ( x ) j ( y ) j † ( z ) j † (0) i . (79)Most of the leading and subleading diagrams are similar to those found inthe three-flavor case. In addition, one finds, in the direct channel D
1, anni-hilation-type diagrams involving at least two gluon lines, which produce, asintermediate states in the s -channel, glueballs (cf. Figs. 39 and 13). Mixingsof tetraquarks with glueball states are of subleading order. Therefore, themain conclusions about tetraquark decay widths and two-meson intermediatestates remain unchanged. 61 a) O ( N c ) j j j † j † (b) O ( N c ) j j j † j † Figure 39: (a) Glueball appearance in the intermediate states of the s -channel of thescattering process M M → M M . (b) Similar phenomenon in the scattering process M M → M M . This diagram also contributes in the t -channel of the scatteringprocess M M → M M (cf. Fig. 26b). We now consider the case of an open flavor, where two quark fields havethe same flavor. The corresponding four-point correlation function isΓ ≡ h j ( x ) j ( y ) j † ( z ) j † (0) i . (80)Here, the direct and the recombination channels are identical, with the com-mon scattering process M M → M M . The corresponding leading andmain subleading diagrams are represented in Fig. 40. (a) O ( N c ) j j † j j † j j † j j † (b) O ( N c ) (c) O ( N c ) j j † j j † Figure 40: Leading and subleading diagrams in the open-flavor channel of the correlationfunction (80).
The effective meson-meson interaction vertex, the two-meson intermediate-state contribution and the tetraquark state are graphically represented inFig. 41. The decay width of the tetraquark into two mesons is of order N − c .62he tetraquark state couples to the local current ( q q )( q q ), which shouldbe antisymmetrized with respect to the quark field q , taking into accountits spin degrees of freedom. M M M M N − c (a) O ( N − c ) N − c N − c M M M M M M (b) O ( N − c ) N − c N − c M M M M T (c) O ( N − c ) Figure 41: Four-meson effective vertex (a), two-meson intermediate states (b), andtetraquark intermediate state (c), corresponding to the open-flavor scattering process M M → M M . (Meson-exchange diagrams, as in Fig. 27, are not represented.) N c ? The fact that, in the exotic and cryptoexotic channels, the possibly exist-ing tetraquarks should have, at N c -leading order of the connected diagrams,an internal structure made of two mesonic clusters [Eqs. (74) and (77)] re-quires further clarification for the understanding of such a result. The mainobservation is that these solutions do not correspond to the expected diquark-antidiquark structure, which would result from a confining mechanism of thefour-constituent system.The reason for this can be traced back to the different behaviors of quark-antiquark and diquark systems at large N c . In quark-antiquark systems,ladder-type gluon-exchange diagrams are planar and therefore the infinitenumber of such diagrams contributes with equal power of N c to the wholesum (cf. Fig. 5). This is a necessary condition to possibly form, at that orderof N c , a bound-state pole in the corresponding scattering amplitude.This is not the case for diquark (or antidiquark) systems. Ladder-typediagrams are damped by factors of N − c at each inclusion of a gluon line63cf. Fig. 15). Here, it is the completely crossed diagrams which play therole of planar diagrams. However, the sum of such diagrams is of no helpfor the formation of bound states, since completely crossed diagrams do nothave s -channel singularities and are parts of the definition of the irreduciblekernel of any integral or bound-state equation. The formation of a boundstate (or of a quasi-bound state) of a diquark necessitates the summation ofladder diagrams. One cannot obtain, with such type of summation, a boundstate which would be stable with respect to a given order of N c .The above features can also be formulated in terms of the quark-antiquarkand quark-quark scattering amplitudes, designated by T , the first being inthe color-singlet representation and the second in the antisymmetric rep-resentation. Designating by K the one-gluon-exchange kernel, wherein thecoupling constant has been redefined according to ’t Hooft’s limit, Eq. (3),and from which color indices have been either factorized or summed togetherwith nearby other color-tensor contributions, the two scattering amplitudessatisfy, after summation of all ladder diagrams, the integral equations T qq = 1 N c K + K ∗ G ∗ T qq , (81) T qq = 1 N c K + 1 N c K ∗ G ∗ T qq , (82)where G represents the two quark propagators, the star operation takesinto account the eventual integrations with respect to the momenta, and theleading behaviors in N c have been factorized.In the quark-antiquark case, Eq. (81), a rescaling of T in the form T /N c removes the factor 1 /N c from the equation and transforms the latter intoan N c -independent equation. If the latter equation is assumed, with anappropriate form of the gluon propagator, to be valid in the confining regime,then the resulting meson bound states will have, at leading order of N c , N c -independent masses, thus confirming the soft behavior of the latter undervariations of N c from infinity down to N c = 3.This is not the case for the diquark system. The previous rescaling of T does not remove the 1 /N c dependence of the kernel of the integral equation(82). To have a qualitative understanding of the consequences of the 1 /N c dependence of the kernel, which we assume being transmitted to the confininginteraction, we consider the illustrative example of two heavy quarks, withreduced mass µ , satisfying the Schr¨odinger equation with a linearly confining64otential: h E − p µ − N c σr i φ = 0 , (83)where σ is the string tension and the 1 /N c dependence of the potential hasbeen explicitly factorized. This equation leads to the formation of confinedbound states, with bound state energies and mean spatial sizes scaling withrespect to N c in the following way: E ∼ (cid:16) N c √ σ µ (cid:17) / √ σN c , h r i ∼ (cid:16) N c √ σ µ (cid:17) / √ σ . (84)At large N c , the bound-state energies decrease and the diquark masses tendto the two-quark mass threshold. On the other hand, the mean spatial sizesof the diquarks increase, although weakly, with N c and the diquarks ceaseto be compact objects, in contradiction with the initial objective of findingdiquark and antidiquark systems with compact sizes. In this situation, thesystem might easily switch to the N c -dominant configuration made of twomesonic clusters.The above results can also be understood with the aid of the quadraticCasimirs of the various representations. Assuming that the confining in-teraction kernel has the same color-representation property as the gluonpropagator, one can evaluate the relative strengths of the various two-bodypotentials (for more details, cf. Ref. [21], Appendix B). For the quark-antiquark system in the singlet representation, the potential is proportionalto − ( N c − / (2 N c ), where the minus sign reflects the attractive natureof the potential. For the same system in the adjoint representation, it isproportional to +1 / (2 N c ). For the diquark system in the antisymmetric rep-resentation [2] (Sec. 3.2), it is proportional to − ( N c + 1) / (2 N c ), while inthe symmetric representation [1 , N c − / (2 N c ).We observe that, at large N c , the ratio between the diquark potential inthe antisymmetric representation and the quark-antiquark potential in thesinglet representation decreases like 1 / ( N c − N c = 3, however, the latter ratio is only 1/2, which might make possible theformation of diquark systems inside tetraquarks.To remedy the above difficulties of the diquark scheme, several dynamicalmechanisms have been advocated, either at the experimental production level[60, 190], or at the inner interaction level [191].65n theoretical grounds, an alternative viewpoint has been advocated inRef. [187]. It was argued that the planar diagrams of the type of Figs. 22band 29a do not represent, in spite of gluon exchanges and s -channel cuts,genuine interactions between mesons, but rather depict different ways of rep-resenting color or momentum flows in a system of two noninteracting mesons,and therefore tetraquark formation graphs should begin from nonplanar di-agrams. This argument, if accepted, brings the contribution of the direct-channel D D N − c , instead of N − c .However, one still has a discrepancy with the recombination channels, whichremain at order N − c . To remain at the end with one type of tetraquark, inthe diquark-antidiquark antisymmetric representation, one is obliged to im-pose an additional selection rule, according to which tetraquarks may appearonly in direct or recombination channels.The main point of the above argument, the noninteracting feature of thetwo mesons, does not seem, however, well-founded. Diagrams of the typesof Figs. 22b and 29a are representatives of an infinite set of planar diagramscontaining at the right and left corners of the quark loops (near the currents j ) planar gluons exchanged between antiquark 2 and quark 1, and betweenantiquark 4 and quark 3, which means that the initial and final parts ofthe diagrams already contain mesons M and M ; any gluon exchangedbetween these mesons represents a genuine interaction and not merely acolor- or momentum-flow artifact. It is only these kinds of diagram that candescribe two-meson-loop formation, as depicted in Fig. 28, which is of order N − c . We shall describe below in more detail the mechanism of the two-mesoninteraction. The question as to whether the planar diagrams may produce bythemselves tetraquark poles is more involved and requires further analysis,which we shall also present hereafter.We come back to the N c -dominant structure of the four-quark system,made of two mesonic clusters [Eqs. (74) and (77)]. The principal questionthat remains to be answered is whether such solutions are compatible withthe formation of tetraquarks. Considering, for definiteness, the direct chan-nel D T A ∼ ( q q )( q q ), which is gen-erated by means of sums, with respect to ladder-gluon lines, of diagramsof the type of Fig. 29a. The gluon lines between quark 1 and antiquark 4generate the scattering amplitude T . Similarly, the gluon lines betweenantiquark 2 and quark 3 generate the scattering amplitude T . The twoscattering amplitudes are disconnected from each other. Nevertheless, they66re embedded into the structure of the meson-meson scattering amplitude T [(21)(43) , (21)(43)] ≡ T ( M M → M M ) and are subjected to additionalloop integrations, providing a connected structure. A typical contributionis graphically represented in Fig. 42. Other contributions involve either T alone or T alone. φ † φ φ † φ T T O ( N − c ) Figure 42: Typical contribution of the quark-antiquark scattering amplitudes T and T to the meson-meson scattering amplitude of the direct channel D φ and φ are the wave functions of the external mesons.Other contributions involve either T alone or T alone. At leading order in N c , the scattering amplitudes T qq are saturated by aninfinite sum of stable mesons [Sec. 2.3], which we label by a global increasingindex n , the value n = 0 corresponding to the ground state. They have thenthe following structure: T qq ( P qq , . . . ) = 1 N c K − iN c ∞ X n =0 φ qq,n φ † qq,n P qq − P n , (85)where P qq is the total momentum of the quark-antiquark system and P n isthe mass squared of the n th meson, with the φ qq,n s being the correspondingwave functions . The one-gluon-exchange contribution (or its equivalent onein a confining scheme) remains outside the sum, since it cannot contribute The wave functions φ are equal, up to normalization factors, to the Bethe–Salpeterwave functions multiplied by the inverse of the two quark propagators. s -channel singularities of the object of Fig. 42 are governedby those coming from the product T T . Each of these scattering ampli-tudes has meson poles as singularities and hence behaves as a sum of effectivemeson propagators. The s -channel singularities that result from the integra-tions are therefore those of two-meson scattering amplitudes. They do notinvolve, however, any pole-type singularity which might signal the possiblepresence of a tetraquark state. It might, however, happen that the infinitesum of the two-meson contributions, which all have the same type of singu-larities but are located at different positions, produces a pole-type singularitythrough a divergence occurring in the vicinity of some particular point. Sucha possibility, which closely depends on the behaviors of the various overlap-pings of wave functions, is the only one which might exist within the classof N c -leading diagrams. It would no longer be the result of purely confininginteractions, since the latter have been absorbed by the meson formations,but rather would be the result of a residual effect of them, coming from theexistence of the tower of an infinite number of meson states. The possibilityof such a mechanism needs, however, further detailed investigations.In case the above mechanism does not produce a bound state, a secondpossibility of producing a tetraquark pole remains: an iteration mechanismof diagrams of the type of Fig. 42. The iteration kernel might be a kernel ofthe recombination channel, an example of which is presented in Fig. 43.The structure of the first iteration, in parallel to a corresponding Feyn-man diagram, is presented in Fig. 44. An infinite series of such iterations,which act in convolution in momentum space, might generate, after summa-tion, a tetraquark pole in the scattering amplitude. However, each iterationintroduces here a factor of N − c , which means that the effective interactionkernel of the possibly resulting integral equation is of order N − c and hencevanishes when N c tends to infinity. Since the confining interactions are ex-plicitly absent between the meson clusters, one faces here two possibilities.In the first one, the resulting interaction is of the residual long-range type(van der Waals, or another hidden mechanism; cf., for an example of this,Sec. 7). If it is globally attractive, then a bound state will exist, approachingthe lowest two-meson threshold when N c goes to infinity. However, the exis-tence of such types of forces has not been observed up to now in the domain68 † φ † φ φ T T T T O ( N − c ) Figure 43: Contributions of the quark-antiquark scattering amplitudes T and T , on theone hand, and of T and T , on the other, to the meson-meson scattering amplitude ofthe recombination channel R N − c . j j † j j † (a) O ( N − c ) φ † φ † φ φ T T T T T T O ( N − c ) Figure 44: Iteration of the scattering amplitudes T and T with insertions of the scat-tering amplitudes T and T . Each insertion introduces a factor of N − c . (a) TypicalFeynman diagram of the first iteration. (b) Generation, after summation of all gluon lineswith the same planarity, of one of the corresponding diagrams of the scattering amplitude T ( M M → M M ). To be compared with Figs. 29a and 42.
69f light quarks. Whether they exist and are operative with heavy quarksremains a key issue of this mechanism.In the second possibility, one rather is in the presence of effective short-range molecular-type interactions between mesons. The properties of thesetypes of interaction will be considered in Sec. 6. The main property that con-cerns us here is that when the strength of the interaction tends to zero, andmore generally becomes lower than a critical strength, no bound states exist,but resonances might appear; in the strength-vanishing limit, the resonancemasses are pushed towards infinity. Therefore, when N c tends to infinity, notetraquark bound states or low-mass resonances are expected to occur. Itis only for finite (possibly large) values of N c that tetraquark bound statesmight appear in the spectrum, in the vicinity of the two-meson threshold.The detailed prediction of the above possibilities depends on the relatedintegral equation that governs the pole production. The exact expression ofthe latter takes, however, a rather intricate form, since it actually involves,because of the infinite number of meson states, an infinite number of coupledequations, with off-mass-shell scattering amplitudes. This problem is not yetanalyzed in the literature.Bound-state equations for four-quark states, based on the Bethe–Salpeterand Dyson–Schwinger equations framework, have been considered in Refs.[192–195], where a detailed study of the various mechanisms of formation oftetraquark states has been undertaken. The results confirm the dominantrole of meson clusters inside the tetraquark bound states.The same mechanism as described above could also be applied to the otherdirect channel, D
2, of the meson-meson scattering amplitude [Eq. (56)] byexchanging the roles of the antiquarks 2 and 4. Here, however, the quarkflavors and masses having been changed, the structure of the kernel of theiteration is different from that of channel D D N c analysis of Feynman diagrams, seems to be dominated by the meson-meson interactions through a collaborative iteration of direct and recombina-tion sectors . From that viewpoint, we are rather close to a molecular-typestructure, in which the role of confining interactions has been absorbed bythe formation of mesons. However, residual long-range type interactions maystill survive and contribute in a more specific way.We shall come back, through another perspective, to the comparisonof the diquark-formation and the mesonic-cluster-formation mechanisms inSec. 7.
6. Molecular states
Molecular structure of exotic hadrons had been considered since the earlydays of the charmonium discovery [44–46]. Taking the analogy of the forma-tion of atomic molecules or of nuclei, it is natural to consider the possibilityof the existence of bound states or resonances resulting from the direct inter-action of ordinary hadrons. Since the latter mutually interact by means ofshort-range forces, generated by meson exchanges, one is entitled to use effec-tive field theories [65–81], adapted to the range of energies and masses thatare involved [197–214]. In particular, the presence, in the exotic hadrons, ofheavy quarks allows the use, at least partially, of a nonrelativistic formalism,which considerably facilitates the analysis of the problem.
The main idea of the formulation of effective field theories is that thedescription of the dynamics of physical systems depends on the energy scaleor the distance scale at which one evaluates physical observables. At low en-ergies, or at large distances, the dynamics should be insensitive to the detailsof the dynamics at high energies or at short distances. It is then advanta-geous to integrate out the degrees of freedom that describe the short-distance Quark recombination or interchange channels in meson-meson scattering have beenconsidered in Ref. [196]. M , interacting by means of aYukawa-type coupling with a scalar field of mass m and coupling constant g . The effective theory is obtained by integrating out the scalar field andkeeping only the fermion field. In the latter theory, the leading-order inter-action is represented by a four-fermion contact term, with coupling constant h . Nonleading interactions are represented by higher-dimensional operators.72atching conditions are implemented by considering the elastic (off-energy)two-fermion scattering amplitude. In the full theory, the most importantcontributions come from the series of ladder diagrams, while in the effectivetheory, the equivalent contributions come from the series of chains of bubblediagrams, generated by the four-fermion contact term. These diagrams arerepresented in Fig. 45, where the main parameters of the two theories arealso displayed. This model has also direct connection with the evaluationof the nucleon-nucleon scattering amplitude at low energies, considered in[75–77]. h M + + + · · · (b) gg Mm + + + · · · (a) Figure 45: (a) The series of ladder diagrams in the full theory. (b) The series of chains ofbubble diagrams in the effective theory. The masses and coupling constants are displayed.
In the full theory, the bound-state problem is governed by the Schr¨odingerequation with the (attractive) Yukawa potential. Bound states exist onlywhen the coupling constant (squared) is greater than some critical value: g ≥ g . = 1 . × (cid:16) πmM (cid:17) . (86)At the critical value, the bound state appears at the two-fermion threshold(zero binding energy) and when g is gradually increased, the binding energyincreases and the bound state goes down in the potential well. For higher val-ues of g , new bound states may appear, representing excited states. When g < g . , the bound states disappear, having been transformed into reso-nances. For values of g approaching zero, the lowest-energy resonance hasan energy (more precisely its real part) that increases up to infinity.Nonrelativistic or semirelativistic theories are generally expanded in in-verse powers of the heavy-fermion mass M . A kinematic quantity that isadequate for such expansions is the ratio of the c.m. momentum Q of the73ermions to the mass M , Q/M . Introducing the velocity v of the fermionsthrough the relation Q = M v , the above expansion is therefore an expansionwith respect to v ; small velocities ensure rapid convergence of the corre-sponding series. In some cases, infrared singularities of loop diagrams arerepresented by negative powers of v and demand a separate treatment orisolation of such terms.Considering now the effective theory, the matching condition with the fulltheory, up to some order in the loop counting, implies a redefinition of theeffective coupling h . The latter then takes the following form: h = ∞ X n =0 h ( n ) , (87)where h ( n ) represents the contribution coming from the n -loop calculation.The calculations, up to two loops, provide the following expansion of h : h = h (0) h (cid:16) g M πm (cid:17) + ln (cid:16) (cid:17) (cid:16) g M πm (cid:17) + · · · i , h (0) = g m . (88)The chain diagrams of Fig. 45b can be summed using the full h as theeffective coupling. The bubble diagram is ultraviolet divergent and requiresrenormalization. After the integration of the temporal component of the loopvariable is done, the three-dimensional part is regularized by dimensionalregularization, in which case no subtraction is needed. The series is simply ageometric series and is easily summed. One finds for the off-energy scatteringamplitude A = h hM / ( − E ) / / (4 π ) , (89)where E is the total (nonrelativistic) energy of the two fermions. For h negative, there is a bound state with energy (cf. also [75]) E = − π h M . (90)(The binding energy is B = − E .) When | h | → ∞ , the bound-state energytends to zero and the bound state approaches the threshold. We have seenthat in the full theory this situation occurs when g tends to g . from above[Eq. (86)]. This means that the above limit of h occurs for a finite value of g , which is g . ; therefore, the series (87) diverges for that value of g , which74ignals the fact that the correspondence between the full and the effectivetheories is no longer perturbative. In this domain of the coupling constant,the higher-dimensional operators, which have been neglected in the aboveevaluations, become relevant to all orders and may signal the breakdownof the effective theory. On the other hand, when h tends to zero in itsnegative domain, the bound-state energy tends to −∞ and the bound statedisappears from the bottom of the energy domain. In the full theory, thishappens when g tends to + ∞ . However, when g gradually increases, newbound states appear in the spectrum of the full theory, representing excitedstates; these are not reproduced in the effective theory. In the domain ofpositive values of h , bound states do not exist. This, therefore, correspondsto the domain 0 ≤ g ≤ g . ; here, the full theory displays resonances, but inthe effective theory they are absent. (The effective theory might display aresonance in the case of derivative-type couplings, a situation that occurs inchiral perturbation theory (cf. Sec. 6.3).) A perturbative matching betweenthe full and the effective theories occurs only in the weak-coupling regime ofthe former, i.e., g ≃
0, which entails h ≃ h > g . h g Figure 46: A schematic behavior of the effective-theory coupling constant h with respectto variations of the full-theory coupling constant squared g . is a consequence of the fact that the bubble diagram is ultraviolet diver-75ent, while the box diagram in the full theory is finite and therefore reflectssmoothly the variations of the coupling constant. In x -space, the bound-state equation of the effective theory is governed by the three-dimensional δ -function potential, which requires renormalization of the coupling constant[216].In spite of the breakdown of perturbation theory in the region g ∼ g . and the lack of explicit correspondence between g and h , it is possible todeduce further information through parameters that have direct connectionwith experimental measurements. These are the S -wave scattering length a and the effective range r e . When the bound state approaches the threshold,the scattering length increases and tends to ∞ at threshold, while the ef-fective range remains finite. It is then shown that h is related to the exactscattering length of the Yukawa theory [75, 77]: h = − πM a, (91)whereas higher-dimensional derivative terms are suppressed by powers of(4 πr e /a ). Therefore, the bound-state energy (90) remains a valid result forlarge values of the scattering length. However, the radius of convergence ofthe effective theory is much reduced and is given by values of the momentum k of the order of p / ( ar e ) and not by r − e ∼ m .The above considerations allow us to foresee the implications of the large- N c limit on the possibility of formation of bound states. Here, the full theoryis QCD, while the effective theory is a meson theory, where mesons mutuallyinteract either by meson exchanges, characterized by some effective genericcoupling g , or by contact terms, characterized by an effective coupling h .We have seen in Sec. 2.3 that three-meson couplings and four-meson cou-plings generally scale like 1 /N / c and 1 /N c , respectively (cf. Figs. 10 and12). Therefore, in the large- N c limit, the meson-interaction couplings tendto zero and one reaches the situation where the mesons become free noninter-acting particles. For large finite values of N c , the effective theory is then inthe weak-coupling regime, which corresponds to the phase g ≃ h ≃ h >
0. In this situation, one does not expect to find bound states. Rather,one should have resonances located far from the two-meson thresholds.When N c is decreased down to finite values, close to the physical value3, two types of evolution might be expected, depending on the quark massesthat are involved and also on the detailed quantum numbers of the systemthat is considered. For light quarks, the evolution probably remains in the76hase g ≤ g . , h >
0, in which case the lowest-mass resonance approachesthe two-meson threshold, but still remains sufficiently far from it. This iscorroborated by results obtained within the framework of chiral perturba-tion theory (cf. Sec. 6.3). For systems involving heavy quarks, it seems thatthe evolution reaches the critical region of the vicinity of the two-mesonthreshold, characterized by g ∼ g . and | h | ∼ ∞ , where a bound state ora resonance might appear. This is corroborated by the experimental obser-vations of exotic hadrons involving heavy quarks. The above descriptions,while remaining at the level of observations, require, however, more dynam-ical justifications at the QCD level. The description of the internal structure of a bound state depends uponthe scale at which the latter is probed. The mean size of the bound stateis one of the criteria that can be used to distinguish two situations: (i) alarge size, characterizing a loosely bound state, in which one may distinguishthe existence of two or several clusters; (ii) a compact size, characterizing anundecomposable or elementary object. It is evident, however, that probingthe latter object with higher precision, one may discover that, in turn, it isalso decomposable into more elementary clusters. The most natural exampleof this situation comes from the nuclei, which, in first approximation, canbe described as made of nucleons, considered as pointlike objects. However,observing the nucleons at shorter distances, one realizes that the latter arethemselves made of more elementary particles, which are the quarks and thegluons.The same criterion also applies to the case of exotic hadrons. Molecular-type exotic hadrons would be described as composed of clusters of ordi-nary hadrons, interacting by means of effective forces, while compact exotichadrons would be described by the direct interactions of quarks and glu-ons, without leading to the appearance of hadronic clusters. The solution ofbound-state equations and comparison of their predictions with experimentaldata would be sufficient to settle the question of the nature of a bound state;however, in many cases, the interactions that are at work have a nonpertur-bative character and do not allow for a deductive solution of the problem.In this case, the knowledge of external criteria, related to experimental data,could bring a complementary view to the efforts of understanding the prob-lem. 77n this respect, Weinberg proposed in the past, in the framework of nonrel-ativistic mechanics, an independent criterion for the probe of compositenessof the deuteron, whose binding energy is much smaller than the scale of thestrong interactions that govern nuclear physics [43, 217]. Defining Z as theprobability of having the deuteron as an elementary particle (0 ≤ Z ≤ − Z ).The vicinity of the deuteron state to the two-nucleon threshold allows one touse, for the scattering phase shift δ of the two nucleons, the effective rangeexpansion in terms of the scattering length a and the effective-range r e k cot δ ≃ − a + 12 r e k , (92)and to relate the latter quantities to the binding energy and the parameter Z . Weinberg finds a = [2(1 − Z ) / (2 − Z )] R + O ( m − π ) , r e = [ − Z/ (1 − Z )] R + O ( m − π ) ,R = (2 µB ) − / , (93)where B is the deuteron binding energy, B = 2 .
22 MeV, µ the proton-neutronreduced mass, m π the pion mass, and R represents the deuteron radius. If thedeuteron is composite, Z ≃ a takes its maximum value, while r e ≃
0; inthe opposite case, if the deuteron is elementary, Z ≃ a ≃ r e → −∞ .Experimental data, a = +5 .
41 fm and r e = 1 .
75 fm [43, 218], clearly favorthe composite nature of the deuteron, made of a proton and a neutron.Notice that, for Z ≃
0, the relationship between the binding energy and thescattering length of Eq. (93) reduces to Eq. (90), using (91).Weinberg’s criterion has been investigated by many authors and extendedto a wider range of applicability, such as to resonances, multi-channel pro-cesses and relativistic cases [219–226]. Oller has proposed a new criterionfor compositeness, based on the use of the number operators of free particles[227]. The compositeness criterion thus brings a complementary constraintfor the analysis of the internal structure of bound states and resonances.
Molecular systems made of light quarks ( u, d, s ) need the use of a rel-ativistic formalism. Since the hadronic clusters interact by means of short-range forces and the light quarks have a relativistic motion, one guesses thatthe mean forces experienced by the clusters are weaker than in the case of78eavy quarks, which tend to stabilize the general motion. Therefore, theeffective theory is expected here to be in its weak-coupling regime, whereno bound states can be produced. Rather, one expects the appearance ofresonances, generally located, mostly for the case of the quarks u and d , farfrom the two-hadron threshold.The light-quark effective field theory which describes QCD at low energiesis chiral perturbation theory (ChPT) [65–68]. The case of SU(2) L × SU(2) R chiral symmetry, involving pions, has been widely studied in the literatureand precise experimental tests have confirmed its validity [228, 229].The domain of validity of ChPT, concerning the meson momenta, ex-tends over a few hundreds of MeV. Since the perturbative expansion is donepolynomially (up to logarithms) in the momenta, it is not expected to findpoles with the first few terms and to be able to probe directly the propertiesof resonances. To extend the domain of predictivity of ChPT, it has beencombined with dispersion relations, using analyticity and crossing-symmetryproperties [228, 229]. Extending the above setup to the complex plane, ithas been shown that the ππ scattering amplitude in its partial S wave, withisospin 0, possesses a pole in the second Riemann sheet at the complex massvalue M = (441 − i f (500) /σ meson, whose existence had been controversial for several decades,but which later had re-emerged through new experimental results as a wideresonance. Confirmation of the above result has been obtained in [231–233].It had also been suggested that properties of resonances could be moredirectly probed by using elastic unitarity in its full form, by means of the in-verse amplitude method [234–238]. A thorough study of the scalar resonance f (500) /σ , based on the combined frameworks of ChPT, dispersion relationsand unitarization, has been undertaken by Pel´aez et al. [239–241]. Theiranalysis also leads to a pole position in the second Riemann sheet of thecomplex plane, at the mass value M = (449 − i S -wave isospin-0 ππ scattering amplitude in itsleading order, O ( p ), of ChPT. The latter reads t I =0 ℓ =0 ( s ) ≡ t ( s ) = 2 s − M π πF π , (94)where s is the Mandelstam variable and F π is the pion decay constant, definedas in Eq. (12) with an axial-vector current ( F π ≃ . M π is the79ion mass. Notice that in ChPT the meson-meson interactions begin withderivative couplings, this is why an s -dependence appears at leading order.The elastic unitarity condition readsIm t ( s ) = σ ( s ) | t ( s ) | , σ ( s ) = (cid:16) − M π /s (cid:17) / . (95)This shows that the imaginary part of the scattering amplitude is of higherorder, O ( p ), than the real part. In terms of the inverse amplitude it takesthe form Im 1 t ( s ) = − Im t ( s ) | t ( s ) | = − σ ( s ) . (96)Therefore, the imaginary part of the inverse of the amplitude is explicitlyknown and reduces to a kinematic factor. This allows one to complete ex-pression (94), by incorporating in it information (96), and considering (94)as the real part of the amplitude: t ( s ) = Re t ( s )1 − iσ ( s )Re t ( s ) . (97)In searching for poles in the second Riemann sheet, one considers thecomplex conjugate of the amplitude of the first Riemann sheet. Expression(97) becomes t II ( s ) = Re t ( s )1 + iσ II ( s )Re t ( s ) . (98)Notice that this equation is the analog of Eq. (89), obtained by summing aseries of bubble diagrams. In the lower s -plane, σ ( s ) undergoes the change σ II ( s ) = − ( σ ( s ∗ )) ∗ . The pole position is obtained from the zero of the de-nominator, σ ( s σ )(2 s σ − M π ) = i πF π , (99)which gives √ s σ = (493 − i N c behavior of the various contributions. At large N c , the pion mass remains unaffected at leading order, M π = O ( N c ), while80he decay constant scales like N / c , F π = O ( N / c ) [Eq. (13)]. Taking intoaccount these behaviors, one finds that Eq. (99) reduces to √ s σ = (1 − i ) √ πF π = O ( N / c ) , (100)which shows that the mass and width of the f (500) /σ meson increase like p N c at large N c . This is in accordance with the weak-coupling regime ofmolecular effective theories that we have met in Sec. 6.1. This shows thatthe f (500) /σ meson is mainly made of two pions, rather than of a pair ofquark and antiquark, in which case the mass should remain stable underchanges of N c and the width would decrease [239, 240] (cf. also [247, 248]).Nevertheless, because of mixing possibilities, the f (500) /σ seems to have asmall component of qq state.The detailed analysis applied to the case of f (500) /σ has also been ap-plied to the case of the ρ meson, which appears as a resonance in the P wave.Here, however, the behaviors of the mass and the width under variations of N c confirm the fact that the ρ meson is mainly made of a qq pair [239, 240].In conclusion, the molecular scheme, considered as an effective theory,provides a systematic tool of investigation of the properties of many exotic-type states, either in the domain of heavy quarks, or that of light quarks.
7. The cluster reducibility problem
One salient feature of the multiquark currents, met in Secs. 3.1 and 3.2, istheir decomposition property into combinations of products of meson and/orbaryon currents, typical examples of which are Eqs. (32) and (36). Thissuggests that multiquark states are not color-irreducible, unlike ordinaryhadrons, and, therefore, could not be put on the same footing as the lat-ter states. The consequences of this fact are easily conceivable. If, within amultiquark state, clusters of ordinary hadrons may be formed, and since themutual interactions of the latter are not confining, the multiquark state willhave the tendancy to be dissociated into its hadronic components or to betransformed into a loosely bound state of hadrons.One might still think that the above color-reducibility property concernsonly couplings to local operators which involve a few moments of the corre-sponding bound state wave function. Actually, the property is very generaland concerns also the couplings to multilocal operators [47].81 .1. Cluster reducibility of multilocal operators
We shall briefly sketch, in this subsection, the case of multilocal operators.The proof of cluster reducibility of multilocal operators is based on twoproperties of the gauge links (50): they are elements of the gauge groupSU(3) (and, more generally, of SU( N c )), and have a determinant equal to 1.Hence, they satisfy the group composition law U ab ( C zy ) U bc ( C yx ) = U ac ( C zyx ) , (101)where C zyx is the line composed of the union of the two lines C zy and C yx , witha junction point at y . The expression of the determinant of U is (cf. Ref. [127],Appendix C)det( U ( C yx )) = 1 = 13! ǫ a a a ǫ b b b U a b ( C yx ) U a b ( C yx ) U a b ( C yx ) . (102)Considering, for definiteness, the tetraquark operator of Fig. 17a, multiplyingit with the determinant of the gauge link of the line C yx and using thecontraction property of two ǫ tensors, ǫ c c c ǫ d d d = δ c d δ c d δ c d + X k i ( − P δ c k d δ c k d δ c k d , (103)where the sum runs over all permutations of the indices k i , with the signof the parity of the permutation represented by ( − P , one ends up with adecomposition of the operator into a sum of products of two meson opera-tors of the type (51), where the gauge link lines have a polygonal structure.Further simplification occurs, using the backtracking condition [130] U ab ( C yx ) U bc ( C xy ) = δ ac , (104)which expresses the unitarity property of the parallel transport operation,leading to the decomposition displayed graphically in Fig. 47, which is themultilocal form of the first of Eqs. (32). The meson operators that appearon the right-hand side of the equation come out with polygonal phase-factorlines.The same procedure can be applied to the cases of pentaquark and hex-aquark operators. The pentaquark operator of Fig. 17b is decomposed into acombination of products of a mesonic and a baryonic operator (cf. Fig. 48).The hexaquark operator of Fig. 17c is decomposed into a combination ofproducts of two baryonic operators (cf. Fig. 49).82 qq qǫ ǫ = + Figure 47: Decomposition of the tetraquark operator into a combination of products oftwo meson operators. q qq qqǫ ǫǫ = ǫ + · · · Figure 48: Decomposition of the pentaquark operator into a combination of products of ameson and a baryon operator. The ellipsis indicates the remaining three other products. q qq qq qǫ ǫǫ ǫ = ǫ ǫ + · · · Figure 49: Decomposition of the hexaquark operator into a combination of products oftwo baryon operators. The ellipsis indicates the remaining other products. N c ) case [47], by appropriately generalizing the ǫ -tensor properties [127].Thus, for tetraquarks, the first operator in Fig. 19 decomposes into ( N c − N c −
1) mesonic operators. The last operator in that figuredecomposes into combinations of two mesonic operators and of Wilson loops,according to the types of paths used between the two junction points. (In thecase of straight lines, the Wilson loops disappear.) Similar decompositionsoccur for the other operators (not drawn) of that figure denoted by the ellipsesand corresponding to intermediate-type representations.For pentaquarks, the first operator in Fig. 20 decomposes into a combi-nation of products of ( N c −
2) mesonic operators and one baryonic operator.The last operator of the figure decomposes into a combination of productsof one mesonic operator and one baryonic operator, and possibly of Wilsonloops.For hexaquarks, the first operator in Fig. 21 decomposes into a combi-nation of ( N c −
1) baryonic operators, while the last operator of that figuredecomposes into a combination of products of two baryonic operators, andpossibly of Wilson loops.
The main question that remains to be clarified concerning the clusterreducibility property of multiquark operators is whether it survives in thequantized theory. This is not a trivial question, because the equivalencerelations, which were established on formal grounds between Y -shaped oper-ators and combinations of products of ordinary hadronic operators, involveproducts of some phase factors along the same lines, but belonging to differ-ent global paths (cf. Figs. 47, 48 and 49, right-hand sides of the equalities).The disentanglement of such phase factors, in order to introduce them insidedifferent hadronic states, corresponding to the clusters, might not be a neu-tral operation and might imply energy loss or gain. We shall show below, onsimple examples, that this is indeed the case and that the result depends onthe geometrical properties, in coordinate space, of the representations thatare considered, favoring only one of the two sides of the equivalence relations.General energy considerations are much more transparent in the caseof static quarks, corresponding to the infinite mass limit of heavy quarks,kinematic effects of the motion of quarks being then suppressed; actually,the latter, for finite masses, do not affect the main property of confinementof the theory and introduce only nonleading terms in the confining regime.84herefore, the static limit is expected to provide, in the confining regime,the main qualitative properties that are searched for.To extract the interaction energy properties of a static system, one gener-ally considers correlation functions of appropriate operators having couplingsto such systems. The calculation involves, among others, the propagators ofthe static quarks in the presence of gluon fields, which are essentially propor-tional to the gluon-field path-ordered phase factors along the time direction[41, 82, 115]. Considering in the correlation functions gauge-invariant mul-tilocal operators, which involve in their definitions gluon-field phase factors,one ends up with the vacuum expectation value of an expression involvingthe color trace of phase factors along a closed contour (or traces along closedcontours in the case of non-connected operators), which defines the vacuumaverage of the Wilson loop. On the other hand, inserting in the correlationfunction a complete set of intermediate hadronic states and taking the to-tal evolution time T to infinity, one selects the ground-state hadron, whichyields a factor e − iET , where E is the corresponding interaction energy, thequark-mass contributions having been factorized. Then the comparison ofthis term with the Wilson-loop contribution allows for the determination of E . We first consider the simplest example, corresponding to a mesonic oper-ator [Eq. (51)] with a phase factor along a straight-line segment [Fig. 16a] oflength R , taken in the three-dimensional space orthogonal to the time axis; R is equal to the distance between the quark and the antiquark, consideredat equal times. The resulting Wilson-loop contour, for an evolution of thesystem during a time T , is a rectangle of length T and width R , representedin Fig. 50. T R
Figure 50: Wilson-loop contour resulting from the time evolution of a meson containing astatic quark-antiquark pair.
The evaluation of the vacuum average of Wilson loops, in the absence of85 completely analytic solution of QCD in its confining regime, can be numer-ically done in lattice gauge theory, where one works in Euclidean spacetime.Furthermore, making there the strong-coupling expansion leads to analyticpredictions, which are generally confirmed by experimental data [40, 82, 115].Generally, in the evaluation of the energy balance, one uses the fact that, inEuclidean space, the generating functional of Green’s functions is dominatedby field configurations which minimize the energy.At leading order of the strong-coupling expansion, the Wilson-loop vac-uum average is obtained by paving the minimal area enclosed by the contourwith the lattice plaquettes. In the case of the rectangle above, the area issimply the product RT ; the Wilson-loop vacuum average is then propor-tional to the factor e − σRT , where σ is the “string tension”, here defined as σ = a ln( g ), where a is the lattice spacing and g the QCD coupling constant.Upon comparing this factor with e − ET (the Euclidean version of e − iET ), oneobtains E = σR, (105)which means that the static quark-antiquark interaction energy increaseslinearly with the separation distance. This result analytically establishesthe confinement of quarks. It has also been confirmed by direct numericalevaluations in lattice theory [249]. Corrections, coming from finite mass andspin effects, have been evaluated in Ref. [250].We next examine the question of the possible influence on the energy ofthe state coming from a deformation of the phase-factor line in the definitionof the mesonic operator (51). We choose, as a simple example, a rectangu-lar line in position space, orthogonal to the time direction (Fig. 51a). TheWilson-loop contour, generated by this operator during a time evolution T , isrepresented in Fig. 51b by the dashed oriented line. The area mapped by thelattice plaquettes in the strong-coupling approximation is equal to R ( T +2 d ).It is only the factor multiplying the variable T that contributes to the inter-action energy; from this, one deduces the same relation as in Eq. (105), whichshows that the phase-factor line deformation in the mesonic operator doesnot modify the energy of the state. The resulting change in the Wilson-loopvacuum average comes from the factor e − σ dR , which is absorbed in the me-son wave-function expressions. Also, line deformations having componentson the time axis, with finite size ∆ t , say, cannot modify the energy of thestate, since the size ∆ t becomes negligible in front of T , when the lattergoes to infinity. 86 ( y ) q ( x ) Rd (a) TRd (b)
Figure 51: (a) A mesonic operator with a rectangular phase-factor line; (b) the Wilson-loopcontour, generated after a time evolution T , represented by the dashed oriented line. It is worth noticing that, in the continuum theory, it is expected that,for large contours, the Wilson-loop vacuum average is saturated by the min-imal surface enclosed by the contour [40, 179, 180]. The minimal surfacecorresponding to the contour of Fig. 51b does not coincide with the unionof the three rectangular areas delineated by the contour and paved by theplaquettes of the lattice. According to the defining equation of the minimalsurfaces, it lies outside these areas [180]. However, when the limit T → ∞ is taken, it shrinks to the above areas and thus provides the same result asthat obtained on the lattice.We now consider the problem of the tetraquark operator in the diquark-antidiquark antisymmetric representation together with its reduction intotwo mesonic operators, as represented in Fig. 47. As a consequence of theindependence of the energy of a system of the types of phase-factor lines,as shown above, one can deform, in the mesonic operators of the right-handside of the equality in Fig. 47, the phase-factor lines to transform them intostraight line segments joining the quark to the antiquark. In the config-uration adopted in that figure, the first diagram of the two-meson systeminvolves smaller lengths for the distances between the quark and the anti-quark inside the mesons with respect to the second diagram. According toEq. (105), the energy of the first diagram being thus smaller than that of thesecond diagram, one can drop for the present study the contribution of thelatter, which will give negligible contributions compared to the first one inthe Wilson-loop evaluations. (We recall that the quarks are static.) Since weare interested in qualitative aspects, we make further simplifications in thegeometric configurations of each representation. We choose equal distances ℓ between the quark and the antiquark in each meson. The two mesonic87perators are placed in the same plane, along parallel directions, the twoquarks and the two antiquarks being aligned along vertical lines, separatedby a distance d (see Fig. 52a, top). The same configuration of the quarksand the antiquarks is also chosen for the tetraquark operator (Fig. 52b, top). d T ℓd T ℓ d q q ℓ q q d = ⇒ (a) qq qqd ℓ = ⇒ (b) Figure 52: (a) Two mesonic operators and the Wilson loop contours generated after atime evolution T . (b) Tetraquark operator in the diquark-antidiquark antisymmetric rep-resentation and the Wilson-loop contour generated after a time evolution T , representedby the dashed oriented lines. The evolution of the two systems during a time T generates Wilson loopswhose contours are represented in Fig. 52. The system of two mesons gener-ates two independent factorized Wilson loops, each with a rectangular con-88our. (In the continuum theory, factorization of Wilson loops occurs at lead-ing order of large N c [100]; nonfactorizable contributions are of the OZI-ruleviolating type, cf. Fig. 13.) The total area enclosed by the contours is 2 ℓT .The tetraquark operator generates a single Wilson loop, whose contour iscomposed of a central rectangle of width ℓ and of four wings, each havinga width equal to d/
2. The total area enclosed in the contour is ( ℓ + 2 d ) T .One then obtains for the two-meson system and for the tetraquark systemthe following interaction energies: E . = σ (2 ℓ ) , E tetrq . = σ ( ℓ + 2 d ) . (106)The two energies are not equal. The system which will dominate in thegenerating functional of Green’s functions is the one that has the smallestenergy. A fair dominance of the tetraquark system thus requires σ ( ℓ + 2 d ) ≪ σ (2 ℓ ) = ⇒ d ≪ ℓ/ . (107)The meaning of Eq. (107) is that, when the distance between the twomesons is much smaller compared to their mean size, the tetraquark opera-tor may be considered as the representative of the system. More generally,one might also have situations where the two mesons are overlapping eachother. Therefore, condition (107) depicts situations where the two quark-antiquark pairs are located in a small volume, in which the two mesons arenot sufficiently separated from each other. When a clear separation of thetwo mesons is realized, such that d ≫ ℓ/
2, then the latter system becomesthe dominant one.The analysis presented above can easily be extended to other multiquarksystems, like pentaquarks and hexaquarks, and also to the case of the gen-eral group SU( N c ), with similar conclusions: the multiquark operator, con-structed in the diquark-type antisymmetric representation (or, equivalently,in the string-junction-type or Y -shaped-type representations) is representa-tive of the system only when all quarks and antiquarks are positioned in asmall volume of space, where the mesonic or baryonic clusters are overlap-ping each other or are very close to each other; outside such a volume, themesonic and baryonic clusters become more faithful representatives of thesystem under study.The strong-coupling approximation in lattice gauge theories for four-quark systems has been first considered by Dosch [251]. The results pre-sented above have been confirmed by direct numerical calculations on thelattice [89, 252–255]. 89n conclusion, the cluster separability property of multiquark operators,obtained on formal grounds, has a weaker significance when energy balanceis considered. Although the string-junction-type representation does not sur-vive in all space, it may still dominate in small volumes, from which it mayinfluence, by continuity on the frontier of the volume, the properties of thesystem in the external volume. We shall study, in the next subsection, inmore detail, the contributions of each type of description to the interactionenergy of the system. A general feature of the static interaction energies is that they continuerepresenting the dragging guide of the system under consideration even whenthe constituents are in motion, after, of course, taking into account the kine-matic modifications. We shall now consider the case of moving quarks andantiquarks in the nonrelativistic approximation, corresponding, in practice,to heavy quarks and antiquarks. This generalization is sufficient to deducethe essential qualitative aspects of the problem.We denote henceforth the interaction energies by V . The system that isconsidered is the tetraquark system in its Y -shaped representation (Fig. 53a)and in its mesonic-cluster-type representations (Figs. 53b and c), as deducedfrom the cluster reducibility relation of Fig. 47. The two quarks are desig-nated by 1 and 3, the antiquarks by 2 and 4, and the two junction points ofthe Y -shaped representation by k and ℓ . The quarks and antiquarks beingin motion, the lengths of the various segments of Fig. 53 are now variablesof the problem. On the other hand, the positions of the junction points k and ℓ are not predetermined; they are obtained after a minimization of theinteraction energy V of the Y -shaped representation with respect to thesepoints. The latter are called “Steiner points” in the literature. In general, fora configuration of the type of Fig. 53a, the point k corresponds to the posi-tion from which the pairs of points (1 , , ℓ ) and ( ℓ,
1) are seen under 120 ◦ and similarly for the point ℓ . In principle, the minimization program shouldbe applied continuously for every configuration of the quark and antiquarkpositions; this, however, is a lengthy time consuming task and generally oneis satisfied with simple geometric configurations, which are proved as intro-ducing only tiny quantitative errors.The interaction or potential energy of the two-mesonic clusters is com-posed of the contributions of Figs. 53b and c. According to Eq. (105), they90
14 3 ℓ k (a) (b) (c)
Figure 53: The tetraquark system in (a) the Y -shaped representation, (b) and (c) two-mesonic cluster representations. are, respectively, V , = σ ( r + r ) , V , = σ ( r + r ) , (108)where we have designated by r ij the three-dimensional distance between po-sitions i and j .According to the positions of the quarks and antiquarks, the interactionenergy that prevails is the one that is minimal: V . = min( V , , V , ) , (109)which can also be rewritten as V . = σ ( r + r ) θ (( r + r ) − ( r + r ))+ σ ( r + r ) θ (( r + r ) − ( r + r )) . (110)On the other hand, the Y -shaped potential takes the form V Y = σ ( r k + r k + r ℓ + r ℓ + r kℓ ) . (111)The final potential is then the minimum of V . and V Y : V tetrq . = min( V Y , V . ) , (112)which is explicitly dependent on the positions of the quarks and the an-tiquarks. The Y -shaped potential will dominate in small volumes, wherethe quarks and the antiquarks are close to each other, while the two-mesonpotential will dominate when the two mesonic clusters are well separated.91he two-meson potential (110), which is composed of the contributionsof two different clusters, exclusive to each other, is based on the quark re-arrangement mechanism when two quarks or two antiquarks come close toeach other. This potential is known in the literature under the name of “flip-flop” [256–268]. One of its features is that it is free of long-range van derWaals-type forces, which unavoidably occur in additive quark models withconfining potentials [269–273]. Actually, Ref. [257] adopted it just on thebasis of the latter property. It also introduced the concept of “configuration-space partitioning”, due to the appearance of different potentials accordingto the positions of the quarks and the antiquarks. However, one should beaware that potential (110) is not a mere model proposition, but is the out-come of QCD lattice calculations in the strong-coupling limit, and is verifiedin full numerical calculations [89, 251–255]. In the continuum theory, it hassupport from Wilson-loop calculations in the large- N c limit [179, 180].In calculations with potential (112), one is interested in the possible ex-istence of tetraquark bound states, which would be stable under strong in-teractions and would thus provide a clear experimental signal. This wouldhappen if the bound state is located below the two-meson thresholds. Nar-row resonance-type states, which would be located above two-meson thresh-olds, might also give signals about the existence of quasi-stable tetraquarkstates. Detailed calculations have been done in Refs. [262–268]. In partic-ular, Ref. [262] provides instructive details about the contributions of thevarious forces. Ignoring spin degrees of freedom, systems made of quarkswith two different flavors, of the types QQ ¯ q ¯ q and QqQq , have been consid-ered. It turns out that, for the existence of a bound state, the Y -shapedpotential plays a minor role; the main role is played by the flip-flop poten-tial. Thus the system QQ ¯ q ¯ q has always a bound state for any value of theratio of the quark masses, with a binding energy rather small compared tothe strong-interaction energy scale. The system QqQq has bound states onlyfor comparable masses of the two quarks.It may seem puzzling how the flip-flop potential, where the two clustersdo not directly interact, can produce a bound state. Actually, the interactionis hidden in the quark rearrangement mechanism; furthermore, the smallnessof the binding energy is not due to the existence of a small parameter, butrather due to the smallness of the overlapping region of the two domains ofexistence of each component of the total potential (110).The existence of a stable tetraquark bound state with the structure QQ ¯ q ¯ q had also been predicted by Manohar and Wise [274] on the basis of the heavy-92uark limit. In this limit, the interaction between the two heavy quarksis well described by the short-range component of the confining potentialand, because of its attractive nature, it produces a deeply bound diquarkstate, which behaves like an almost pointlike color-antitriplet heavy anti-quark, which then forms with the two light antiquarks an antibaryon-likebound state. This mechanism has also been advocated in recent studies, to-gether with the heavy-quark symmetry, to predict, on quantitative grounds,the tetraquark bound-state masses [275–277].Although the predictions about the existence of tetraquark bound statesof the type QQ ¯ q ¯ q seem to be similar in the two approaches based on thegeometric partitioning, on one hand, and on the heavy-quark symmetry, onthe other, the lines of approach do not seem to have common features. Inthe heavy-quark symmetry approach, the heavy diquark system is reducedto a pointlike antiquark, while in the geometric partitioning approach it isjust the contrary that is used, that is, the cluster reducibility of the sys-tem into mesonic clusters and their mutual interaction through the quarkrearrangement mechanism.Coming back to the geometric partitioning idea, it seems to provide a re-fined analysis of the conditions under which the reducibility of the multiquarkoperators and states occurs. The string-junction or Y -shaped junction-typedescription of multiquark states seems to survive only in small volumes ofspace, leaving the rest of space to the description based on the mesonic orbaryonic clusters, which continue interacting by means of the quark exchangemechanism, producing, in turn, weakly bound multiquark states.Nevertheless, the qualitative, as well as quantitative, conclusions reachedup to now by means of the latter descriptions cannot be considered as defini-tive. The main reason of this is related to the fact that a complete descriptionof a multiquark state necessitates, even in the nonrelativistic limit, the use ofa multichannel interacting system, relating the independent basis states, thenumber of which is not limited to one. Taking the example of the tetraquark,we have seen, in Sec. 3.1, that there are two independent basis states or sec-tors, which could be taken to be the sectors (21)(43) and (41)(23) in theircolor-singlet-singlet representation, respectively, or the sector (21)(43) in itssinglet-singlet and octet-octet representations, or the diquark-antidiquarksectors in their antisymmetric and symmetric representations. Usually, thediquark-antidiquark sector in its symmetric representation, as well as theoctet-octet sector, are discarded on the basis that their internal interactionis repulsive and could not lead to the formation of diquark or antidiquark or93uark-antiquark intermediate bound states. However, one should also takeinto account the fact that the mutual interaction between these clusters toform a color-singlet state is still attractive with a strength at least twicegreater than the conventional quark-antiquark interaction forming a colorsinglet. The existence of such forces might still substantially modify thepredictions obtained up to now. Therefore, complementary studies are stillneeded in this approach to reach a definitive conclusion.As a last remark, geometric partitioning, which has been formulated ina nonrelativistic framework, cannot be considered, in general, as an instan-taneous phenomenon. It is the result of a transition process from an ener-getically favorable configuration to another one. This transition involves thequark rearrangement or interchange mechanism. Therefore, it might be thatgeometric partitioning is actually a simplified description of the more com-plicated quark rearrangement mechanism, which involves, in its generality,many Feynman diagrams.
8. Summary and concluding remarks
The extension of the color gauge group SU(3) to SU( N c ), with large N c ,as had been proposed by ’t Hooft, has been revealed to form an efficient toolfor the investigation of the nonperturbative regime of QCD. Without solvingthe theory, it has clarified many theoretical questions that had been raisedwith the emergence of QCD, some of which having already appeared withthe early days of the quark model.It is in the large- N c limit that the notion of confinement takes its ideal-ized formulation. In this limit, quark-pair creation being suppressed, mesonsappear as made of a pair of quark and antiquark, therefore providing to thenotion of valence quarks a precise meaning. The 1 /N c expansion method,starting from leading terms, provides a systematic tool for a qualitative un-derstanding of the order of magnitude of physical processes and observables.However, in this limit, baryons undergo a huge transition, since the num-ber of their constituents increases with N c and they tend to have a solitonicstructure, necessitating a treatment different from that of mesons.It is then natural to apply the large- N c analysis to the case of exotichadrons, which, in the language of valence quarks, are states containingmore quarks and antiquarks than the ordinary hadrons. Many newly exper-imentally discovered or observed particles fall into this category, since their94uantum numbers or decay modes do not fit into the scheme of the ordinaryquark model.The main theoretical question that arises at this level is whether suchstates are color irreducible, like ordinary hadrons. The latter, at the valence-quark level, cannot be decomposed into simpler color-singlet states. Theanswer, for the exotic states, is negative. They are decomposable into com-binations of products of ordinary hadrons. This property is true not only forlocal interpolating currents, but also for multilocal operators involving gaugelinks. This means that exotic states are not natural extensions of ordinaryhadrons and could not be solutions resulting from fully confining forces, witha spectroscopy made of towers of bound states with increasing masses. Sucha situation might be reached only with the existence of hidden fine-tuningmechanisms that could favor the confining forces to take place, without beingdestabilized by cluster decomposition.In passing to the gauge group SU( N c ), new technical complications arise.It turns out that exotic states can be probed or described by several inequiv-alent representations, each containing different numbers of valence quarks.Thus, tetraquarks, which in the case of SU(3) are described as made of twopairs of valence quarks and antiquarks, have now ( N c −
2) different represen-tations, a generic representation having J pairs of quarks and antiquarks, J taking values from 2 to ( N c − N c , the diagonal channels (or direct channels) aredominant, while the off-diagonal channels (or quark-rearrangement or quark-interchanging channels) are subdominant. This has as a consequence that ifthere are tetraquark states, they are two in number, each one correspondingto the diagonal-channel solution, and each of them having a dominant cou-pling with the two mesons of that channel, of the order of 1 /N c . In case of95 possibility of decay, the corresponding decay width would be of the orderof 1 /N c . This solution does not favor the diquark scheme, which, because ofthe confinement constraint, is built on a single antisymmetric representationand thus predicts a single tetraquark, having equal couplings with the twomesons of each channel (spin quantum numbers having been ignored, as notbeing essential for these analyses).It should be stressed that, contrary to the case of ordinary hadrons, thelarge- N c analysis does not imply the existence of tetraquarks. These arein competition with contributions of two-meson intermediate states, whichconsistently can saturate the various equations. Tetraquarks, if they exist,are additonal contributions to the intermediate states. Therefore, predictionsobtained about tetraquarks at large N c should be considered as upper boundsfor the related quantities.The large- N c analysis also allows the study of the formation mechanism oftetraquark states. At leading order, it is the two-meson clusters that providethe main contributions and these are expected not to mutually interact bymeans of confining forces. In that scheme, the main formation mechanismis generated by the internal contributions of quark-rearrangement (quark-interchange) processes. These sum up and might produce tetraquark polesin the meson-meson scattering amplitudes. There are two possible interpre-tations of the global outcome of this mechanism: (i) The whole summation isreduced to an effective meson-meson interaction, producing molecular-typetetraquarks. (ii) The interaction between the meson clusters, even if notconfining, is of the residual type of confining interactions, not reducible tomeson exchanges or contact terms. Only a more detailed investigation of thecorresponding dynamics might provide a clarification of that issue.For cryptoexotic states, with three or two quark flavors, mixing diagrams,involving ordinary mesons as intermediate states, complicate the extractionof the tetraquark properties. Nevertheless, the existence of four-channel pro-cesses and the fact that the quark-rearrangement mechanism is still at work,imply, in general, the possibility of the existence of two different tetraquarks,eventually having priviledged couplings with the mesons of the diagonal-typechannels.Finally, the idea of geometric partitioning, which takes into account theenergy balance of meson-cluster- and string-junction-type configurations, hasprovided further clarification about the dominant configurations which mightproduce tetraquark states. Except in small volumes, where the four quarksare located, most of space is dominated by two-meson-cluster configurations.96he transition from one of these configurations to the other implies againthe quark-rearrangement mechanism.The mechanism of formation of tetraquarks is not yet fully understood,due to the complexity of experimental data and the lack of explicit theoret-ical solutions. However, the large- N c approach provides a complementaryview to the problem by establishing a hierarchy among the various types ofcontributions. The general outcome that emerges from that approach is thatthe tetraquark formation is mainly dominated by the quark-rearrangementmechanism, which operates at the internal level of the processes.The problem of exotic hadrons still remains a challenge for all theoreticalapproaches. Acknowledgements
D. M. acknowledges support from the Austrian Science Fund (FWF),Grant No. P29028. H. S. acknowledges support from the EU research andinnovation programme Horizon 2020, under Grant agreement No. 824093.D. M. and H. S. are grateful for support under joint CNRS/RFBR GrantNo. PRC Russia/19-52-15022. The figures (except Fig. 46) were drawn withthe aid of the package Axodraw2 [278].
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