DSF/13(2006) LAPTH-1150/06 LPSC-06-37 hep-ph/0608001 Chromomagnetism, flavour symmetry breaking and S-wave tetraquarks F. Buccella,1,∗ H. Høgaasen,2,† J.-M. Richard,3,‡ and P. Sorba4,§ 1Universit`a di Napoli Federico II, Dipartimento di Scienze Fisiche and INFN, Sezione di Napoli, Complesso Universitario di Monte Sant’ Angelo, via Cintia I-80126 Napoli, Italy 2Department of Physics, University of Oslo, Box 1048 NO-0316 Oslo, Norway 3Laboratoire de Physique Subatomique et Cosmologie, Universit´e Joseph Fourier–IN2P3-CNRS 53, avenue des Martyrs, 38026 Grenoble cedex, France 4Laboratoire d’Annecy-le-Vieux de Physique Th´eorique (LAPTH) 9, chemin de Bellevue, B.P. 110, 74941 Annecy-le-Vieux Cedex, France (Dated: February 2, 2008) The chromomagnetic interaction, with full account for flavour-symmetry breaking, is applied to S-wave configurations containing two quarks and two antiquarks. Phenomenological implications are discussed for light, charmed, charmed and strange, hidden-charm and double-charm mesons, 7 and extendedto their analogues with beauty. 0 0 PACSnumbers: 12.39.-x,12.39.Mk,12.40.Yx 2 n a J I. INTRODUCTION 0 2 Thequestionoftheexistenceofmultiquarkhadronsbeyondordinarymesonsandbaryonshasbeenaddressed 4 since the beginning of the quark model. It has been particularly discussed recently with the firm or tentative v discovery of new hadron states in a variety of experiments. For a review of recent results, see, e.g., Refs. [1, 2, 1 3, 4]. 0 Different mechanismshavebeen proposedto formstable ormetastable multiquarksin the groundstate. The 0 mostnaturalmechanism,especiallyforstatesclosetoahadron–hadronthreshold,isprovidedbynuclearforces, 8 0 extrapolated from the nucleon–nucleon interaction, and acting between any pair of hadrons containing light 6 quarks. This led several authors to predict the existence of DD∗ and D∗D(+c.c.) molecules [5, 6, 7, 8, 9]. 0 According to these authors (see, also, [10, 11, 12]), the latter configurationis perhaps seen in the X(3872) [13], / though other interpretations have been proposed for this narrow meson resonance with hidden charm [14, 15]. h p Stable or metastable multicharmed dibaryons are also predicted in this nuclear-physics type approach [16]. - Flavour independence is a key property of QCD, at least in the heavy-quark limit. Quarks are coupled p to the gluon field through their colour, not their mass, and this induces a static interquark potential which e h is independent of the flavour content, in the same way as the same Coulomb interaction is kept acting on : antiprotons, kaons, muons and electrons when exotic atoms and molecules are studied [17]. The mechanism v by which the hydrogen molecule is more deeply bound than the positronium molecule remains valid, mutatis i X mutandis, in hadron physics with flavour independence and favoursthe binding of (QQq¯q¯) below the threshold r of two heavy-flavouredmesons, when the quark-mass ratio Q/q increases [18, 19, 20, 21, 22, 23, 24, 25]. a The best known mechanism for multiquark binding is based on spin-dependent forces. In the late 70’s, Jaffe [26,27]proposeda(q2q¯2)pictureofsomescalarmesons,asasolutiontothepuzzleoftheirlowmass,decayand production properties, and abundance. He also discovered that the colour–spin operator entering the widely- accepted models sometimes provides multiquark states with a coherentattractionwhich is larger than the sum of the attractive terms in the decay products, hence favouring the formation of bound states. An example is the so-called H dibaryon [28], with spin S = 0 and quark content (ssuudd), tentatively below any threshold ∗Electronicaddress: [email protected] †Electronicaddress: [email protected] ‡Electronicaddress: [email protected] §Electronicaddress: [email protected] 2 made of two lightbaryons. This predictionstimulated anintense experimentalwork,whichdid not leadto any positive evidence, see, e.g., [29]. This model for the H also provoked much theoretical activity. New configurations were found, in which the chromomagnetic effects are favourable, such as the 1987-vintage pentaquark [30, 31], P = (Qq¯4). A compre- hensive systematics of multiquark configurations with favourable chromomagnetic effects can be found, e.g., in [32, 33, 34] and the references therein. Intheearlydaysofmultiquarkinvestigations,chromomagneticeffectswerealsointensivelyusedinanattempt to explain the narrow hadronic resonances which were observed at that time [35]. Models were proposed for these hadrons, with two clusters of complementary colour separated by an orbital momentum, to prevent the decay into colour singlets without pair creation and thereby give possible long lived states [36, 37, 38, 39]. The effective mass of each cluster was computed from the chromomagnetic interaction and effective quark masses. Themoststudiedstateswerethetetraquarkscalled“baryonium”andthepentaquarkscalled“mesobaryonium”. In the limit of exact SU(N) flavour-symmetry,the chromomagnetic model reads F H = m C λ˜ .λ˜ σ .σ , (1) i i j i j − i i<j X X where σ is the spin operator and λ˜ the colour operator acting on the ith quark, and each effective mass i i m includes the constituent quark mass and its chromoelectric energy (binding effect). There is already an i abundant literature on how to estimate the expectation value of the chromomagnetic operator for multiquark configurations,inparticularusingsomepowerfulgroup-theoreticaltechniques. TheHamiltonian(1)isexpressed in terms of Casimir operators of the spin, colour and spin–colour groups. When the overall strength factor C is replaced by a coupling C which depends on the quark flavour, an explicit basis is required to estimate the ij eigenstates of H. NotethattheroleofSU(3) symmetrybreakinghasalreadybeenanalysedintheliterature,see,inparticular, F [40, 41, 42, 43] for the H and the P. It often happens that the corrections weaken or even spoil the binding predicted in the SU(3) limit. F Inthispaper,adetailedformalismispresentedtofullyaccountforflavour-symmetrybreakinginthechromo- magneticinteraction,andanapplicationisgiventothesectorofsystemsmadeoftwoquarksandtwoantiquarks in a relative S-wave, i.e., scalar (JP = 0+), axial (1+) and tensor (2+) mesons. The question then is how to extrapolate the strength of the chromomagnetic interaction from the meson or baryon sector to the case of multiquark configurations. There has been several investigations of multiquark states using the remarkable know-how of few-body physics. The strategy here consists in writing down an explicit Hamiltonian with kinetic energy operator, spin-independent confining forces and spin-dependent terms, tuning the parameters to reproduce some known mesons and baryons, and solve the multiquark problem. This involves an extrapolation of the linear quark– antiquark potential toward multiquark states and an ad-hoc regularisation of the contact interaction, which then can be treated beyond first order. The present approach is somewhat complementary. The role of the chromomagnetic interaction is analysed from the point of view of the symmetry properties, to deduce patterns shared by a whole class of models. The study is restricted to the chromomagnetic model, though it has been challenged recently by models where the hyperfine splittings of hadrons is described by instanton-induced forces or spin-flavour terms. The multiquark sector in these models is reviewed by Stancu [44] or Sakai et al. [43]. It is well known that a colour singlet configuration with two quarks and two antiquarks has at least one component which is a product of two colour singlets. Hence most states are very broad, since unstable against spontaneous dissociation, and give only indirect signatures. However,in rare circumstances, the dissociation is kinematically suppressed, resulting into a remarkably small width. This is the scenario proposed recently for the X(3872) [15]. Theapplicationswillbefocusedonfour-quarkmesonswithspinS =0,1or2,andvariousflavourcontent. As alreadymentioned,therearepromisingpossibilitiesintheexoticsectorwithtwoheavyquarks,especially(bcq¯q¯), butthesestateshavenotyetbeenexperimentallysearchedfor. However,thereareindicationsofsupernumerary statesinthecharmoniumspectrum[4,13,45,46]. Thesingle-charmstates(cqq¯q¯)andanalogueswithstrangeness werepredictedmanyyearsago[47],andtherecentfindingsintheD spectrummightrevealsomeofthesesates. s Thehottestsectoristheoneofscalarmesons. RecentexperimentsatLEARandatB-factorieshaveconfirmed the years of data taking and analysis: there are far too many scalar mesons below 2 GeV for the only qq¯(q 3 denotes u or d) and ss¯states, even including the radial excitations. The fashion evolved from the multiquarks of Jaffe to glueballs and hybrids, but seemingly tends again toward multiquarks. It is hardly possible to propose an ultimate solution to this problem. It appears clearly from the detailed phenomenological analyses [48, 49, 50, 51, 52] that states with different quark and gluon content are abundantly mixed and acquire an appreciable mass shift due to their coupling to the real or virtual decay channels. Nevertheless, such a mixing shouldoperatebetweenproperlyidentifiedbarestates,andsomeclarificationwillbesuggestedinthefour-quark sector which is a key ingredient of the mixing scheme. This paper is organised as follows: in Sec. II, the most general chromomagnetic Hamiltonian is presented and diagonalised for systems of two quarks and two antiquarks. The application to various flavour sectors is presented in Sec. III, before the conclusions in Sec. IV. II. THE CHROMOMAGNETIC HAMILTONIAN A. General considerations The interaction Hamiltonian acting on the colour and spin degrees of freedom, and generalising (1), is H = m +H , H = C λ˜ .λ˜ σ .σ . (2) i CM CM ij i j i j − i i,j X X It is inspired by one-gluon-exchange [53], in which case C contains a factor α /(m m ), where α is the ij s i j s coupling constant of QCD and m the mass of the ith quark, and also the probability of finding the quarks i (or antiquarks) i and j at the same location. The above model is more general. The coefficients C which ij presumablyincorporatenon-perturbativeQCDcontributions,dependonthequarkmassesandontheproperties of the spatial wave function, as in the one-gluon-exchange model. The solution of the eigenvalue problem for the Hamiltonian (2) is of interestnot only for spectroscopy,but in all circumstances where two quarks and two antiquarks are in a relative S-wave, for instance when studying the violation of the OZI rule [54]. For a quark–antiquark meson, λ˜ .λ˜ = 16/3 and σ .σ = +1 for spin S = 1 and 3 for S = 0. The 1 2 1 2 Hamiltonian(2)accountsnaturallyh forthieob−servedhyperfinesplittingssuchasJ/ψ η or−D∗ D . Thisleads − c s− s to the strength parameters shown in Table I. As the spin-singlet state of bottomonium and the spin-triplet state of (bc¯) are not yet known experimentally, and in these sectors, the data have been replaced by model calculations [55]. TABLE I: Valuesof Cq′q¯(in MeV) estimated from meson masses n s c b n¯ 29.8 s¯ 18.4 8.6 c¯ 6.6 6.7 5.5 ¯b 2.1 2.2 6.8a 4.1a aThisisextractedfromoneofthemodelcalculations compiledinRef.[55] For ordinary baryons, the colour operator λ˜ .λ˜ = 8/3 is the same for all pairs and factors out. For spin i j S =3/2,H =8(C +C +C )/3pushesup∆,Σ∗−,etc. ForspinS =1/2(qqq′)baryonswithtwoidentical CM 12 23 31 quarks, H = 8/3(C 4C ) is attractive. In the general case (q q q ) of spin 1/2 such as Λ or Σ with CM 12 13 1 2 3 0 breaking of isospin symm−etry, or Ξ+(csu), a basis c [(q q ) q ] , [(q q ) q ] , (3) 1 2 1 3 1/2 1 2 0 3 1/2 canbe chosen,withsymmetric orantisymmetriccouplingofthe firsttwoquarks(the index, hereandinsimilar further states, denotes the value of the spin) in which the chromomagnetic interaction reads 8 C 2C 2C √3(C C ) HCM = 3 1√23−(C 13−C )23 233C− 13 . (4) (cid:20) 23− 13 − 12 (cid:21) 4 The N ∆ system gives access to C . Then the Λ Σ Σ∗ multiplet gives C and another value of C qq qs qq close to th−e previous one. Then Ξ, Ξ∗ and Ω− depen−d on−m +4C /3 and m +8C /3 and, to the extent s ss s ss { } that these parameters do not change much from Ξ to Ω, C can be obtained. The value shown for C is from ss cc model calculations of double-charm baryons [56]. The values of the strength factors C are displayed in Table ij II. TABLE II:Approximatevalues of Cii′ (in MeV) estimated from baryon masses n s c n [19−20] s [12−14] [5−10] c 4 5 5b bThisisextractedfromoneofthemodelcalculations inRef.[56] For tetraquarks and higher multiquark states, there is the known complication that an overall colour singlet canbe builtfromseveralmannersofarranginginternalcolour. Thesecolourstatesusuallycanmixandonehas to diagonalisethe interactionHamiltonian. In the case of tetraquarks,the mostnatural basis is constructedby coupling the quarks q and q in colour ¯3 or 6 and spin s=0 or 1, to the extent allowedby the Pauliprinciple, 1 2 and similarly for the antiquarks. However, for studying the decay properties, it is convenient to translate the state content in the basis [(q q¯ )c(q q¯)c] or [(q q¯ )c(q q¯ )c]. Here, and in the rest of this article, the upper 1 3 2 1 4 2 3 index c denotes the colour of the cluster. It runs over c = 1 and c = 8 in this decomposition. The relevant crossing matrices should be derived with care, as some errorsand misprints occurredin the early literature. In particular,theorderadoptedforcouplingq andq ,forinstance,resultsintophasefactorsthatdonotinfluence 1 2 the physicscontent,butshouldbe treatedconsistentlythroughoutthecalculation. The resultspresentedbelow have been checked in particular against [57] in the limit of isospin symmetry, and [58]. B. Group theoretical considerations The operator = λ˜ .λ˜ σ .σ can be elegantly expressed in terms of the Casimir operators of the spin i j i j O − SU(2) , colour SU(3) and spin–colour SU(6) groups, as stressed in [27, 39, 59] for special configurations or s c cs P more general cases. For an N-constituent system consisting of n quarks and n¯ = N n antiquarks, with the same strength C = C in the quark sector, C = C in the antiquark sector, and C− = C′ for all quark–antiquark pairs, it ij ij ij can be shown that 8 8 2H = C C (Q) C (Q) C (Q) 16n C C (Q) C (Q) C (Q) 16n¯ CM 6 3 2 6 3 2 − − − 3 − − − − 3 − (cid:20) (cid:21) (cid:20) (cid:21) 8 8 8 +C′ C (T) C (Q) C (Q) C (T)+C (Q)+C (Q C (T)+ C (Q)+ C (Q) , (5) 6 6 6 3 3 3 2 2 2 − − − − 3 3 3 (cid:20) (cid:21) whereC ,C andC aretheCasimiroperatorsofSU(2) ,SU(3) andSU(6) ,respectively,forthequark(Q)or 2 3 6 s c cs antiquark (Q) sector or the whole system (T). The normalisation adopted here is such that C = S(S +1) 2 for a spin S, and C (3) = 16/3 and C (6) = 70/3 for the lowest representations. If it is further assumed that 3 6 C =C =C′, the well-known formula [27] 1 4 1 8 8 =8N + C (T) C (T) C (T)+C (Q)+ C (Q) C (Q)+C (Q)+ C (Q) C (Q) , (6) 6 2 3 3 2 6 3 2 6 O 2 − 3 − 2 3 − 3 − is recovered. Itispossibletomakesomegeneralconsiderationsontheeigenvaluesofthechromomagneticinteractionforthe scalar, axial and tensor tetraquarks. Consider first the flavour-symmetry limit, which is a good approximation for the states built from light (q = u, d) quarks and antiquarks. In this limit, the matrix representation H CM simplifies to two 2 2 matrices for the scalars, two 2 2 and two 1 1 for the axials and two 1 1 for the × × × × 5 tensors. Theinteractionbetweenthequarksandtheantiquarks,whichdependsstronglyontheSU(6) Casimir cs operators of the tetraquark, has a tendency to give eigenstates which approximately belong to the irreducible representations of that algebra. This observation has also interesting consequences for the decay properties of tetraquarks. In fact, many years ago, Jaffe [26, 27] stressed that all the multiquarks have “open door” channels, that is to say, can decay into two colour singlets by simple rearrangementof the constituents, see, also, Refs. [60, 61]. Only phase space can possibly block this spontaneous dissociation. More recently, this property has been related [62] to the transformation properties of the multiquark states withrespecttoSU(6) . Since thepseudoscalar(π,K,η,η′)andthevector(ρ,K∗,ω,φ)mesonstransformasa cs singletanda35,respectively,the“opendoor”pseudoscalar–pseudoscalar(PP)channelswillbeSU(6) singlets cs andthe pseudoscalar–vector(PV)channelswillbe 35-pletsofthesamealgebra. The‘opendoor”vector–vector (VV) channels will be found for the states transforming in a representation contained in the product of two 35 representations (1, 35, 189, 280, 280 and 405). Thescalarstatesbuiltfromlightquarksbelongtotherepresentations1+405ofSU(6) forthecaseofisospin cs I =0 and to the representations 1+189 for I =0,1,2. Indeed, the quarks symmetric (resp. antisymmetric) in colour–spinbelongtothe(6 6) =21(resp.(6 6) =15representationsofSU(6) . Fromthedecomposition S A cs × × of the SU(6) representations with respect to SU(3) SU(2) cs c s × 21=(6,3)+(3,1) , 15=(6,1)+(3,3) , (7) and the SU(6) products of representations cs 21 21=1+35+405 , 15 15=1+35+189 . (8) × × it is readily seen that two (1,1) singlets of SU(3) SU(2) come from the 21 21 and 15 15 products, and c s × × × also that the 35 representation does not contain any (1,1) singlet of SU(3) SU(2) . c s × In order to apply Eqs. (6) and (5) to these states, the following SU(6) Clebsch–Gordan coefficients are necessary 6 1 1 = 21;(6,3) 21;(6,3) + 21;(3,1) 21;(3,1) , | i 7| i √7 r (cid:12) (cid:11) (cid:12) (cid:11)(cid:12) (cid:11) 1 (cid:12) 6(cid:12) (cid:12) 405 = 21;(6,3) 21;(6,3) 21;(3,1) 21;(3,1) , | i √7| i − 7 r (cid:12) (cid:11) (cid:12) (cid:11)(cid:12) (cid:11) 2 (cid:12) 3(cid:12) (cid:12) 1 = 15;(6,1) 15;(6,1) + 15;(3,3) 15;(3,1) , | i 5| i 5 r r 3 (cid:12)(cid:12) (cid:11) 2(cid:12)(cid:12) (cid:11)(cid:12)(cid:12) (cid:11) 189 = 15;(6,1) 15;(6,1) 15;(3,3) 15;(3,1) . (9) | i 5| i − 5 r r Now, the SU(6)cs Casimir dependence of the(cid:12)(cid:12) chromom(cid:11) agnetic(cid:12)(cid:12) contribu(cid:11)t(cid:12)(cid:12)ion to th(cid:11)e mass of the tetraquarks showninEq.(9)impliesthatthelighteststatesareapproximatelysinglets,whiletheheavierstatestransforming approximatelyasthe405orthe189representation,havelargecouplingtoVVandsmallcouplingtoPPchannels. As for the axial sector, the lightest state will be a I = 0 transforming as a 35, followed by two I = 1 states andaI =0, 1, 2clustertransforminginthe sameway,whilethe heavieststatesarethe twoI =1transforming approximatelyas280+280. Due toparityconservation,a1+ statecannotdecayinto twopseudoscalarmesons, the heavieststatesareexpectedto havea smallamplitude to PVandmaylie below the thresholdforVV. Note that the four 35 may be too light to decay into PV. Finally,thetensorstates,whichhaveS-waveamplitudesintoVV,maybeunderthresholdforthatfinalstate. When states with one or more strange constituents are considered, the chromomagnetic interaction involve different gyromagnetic factors and short-range correlations. These symmetry-breaking effects mix states with different SU(6) transformation properties for the qq and q¯q¯pairs, but many of the qualitative features of the cs symmetrylimitremain,bothforthehierarchyofmassesanddecaypatterns. However,fordetailedphenomeno- logical applications, it is desirable to have explicit estimates of the eigenstates of H , and for this purpose, CM instead of using a basis of SU(2) , SU(3) and SU(6) representations, it is preferable to couple explicitly the s c cs quarks in states of given spin and colour, and similarly for the antiquarks. This new basis turns out also more convenienttoimposethe constraintsdue toPauliprinciple. The calculationsarenowcarriedoutinsomedetail for the scalar, axial and tensor configurations. 6 C. Scalar tetraquarks Considerfirstthe caseoftotalspinS =0. Inthe [(q q )(q¯ q¯ )]basis,the diquarkandthe antidiquarkshould 1 2 3 4 bear conjugate colour, (¯3,3) or (6,¯6), and the same spin 0 or 1. The Hamiltonian (1) acts on the four states: φ =(q q )6 (q¯ q¯ )¯6 , φ =(q q )3 (q¯ q¯ )3 , 1 1 2 1⊗ 3 4 1 2 1 2 0⊗ 3 4 0 φ =(q q )6 (q¯ q¯ )¯6 , φ =(q q )3 (q¯ q¯ )3 . (10) 3 1 2 0⊗ 3 4 0 4 1 2 1⊗ 3 4 1 The colour-magnetic interaction in this basis reads A A H = 1 2 , (11) CM − B1 B2 (cid:20) (cid:21) with 2 2 submatrices × 4 20 (C +C )+ (C +C +C +C ) 2√6(C +C +C +C ) A = 3 34 12 3 14 13 23 24 14 13 23 24 , 1 " 2√6(C14+C13+C23+C24) 8(C34+C12) # A =B† = 2 (C C +C C ) 5 2√6 , (12) 2 1 √3 13− 14 24− 23 0 2 (cid:20) (cid:21) 4(C +C ) 2√6(C +C +C +C ) 34 12 14 13 23 24 B2 = 2√6(C− +C +C +C ) 8(C +C C C C C ) . " 14 13 23 24 −3 34 12− 14− 13− 23− 24 # In the states φ and φ , the quarks are symmetric in colour–spin and belong to the (6 6) =21 dimensional 1 2 S × representation of SU(6) , and the antiquarks belong to a 21 representation. In φ and φ , the quarks are cs 3 4 coupled antisymmetrically in a (6 6) =15 representation, and the antiquarks in a 15. If only three flavours A are involved, φ and φ fall into t×he ¯3 3 = 1+8 representations of SU(3) , which is called a nonet in the 1 2 F familiar notation of this symmetry grou×p, and φ and φ fall into the 6 ¯6=1+8+27 representations. 3 4 × If the two quarks q and q or the two antiquarks q¯ and q¯ , are identical, the states φ and φ are excluded 1 2 3 4 1 2 by the Pauli principle, and in the space spanned by φ and φ , the Hamiltonian H is expressedby the 2 2 3 4 CM × matrix B . 2 − In the limit where one antiquark,sayq¯ , is veryheavy anddecouples,i.e., C =C =0, the problemreduces 4 i4 4i to the previously discussed [63, 64] chromomagnetic problem of a spin 1/2, colour triplet (qqq¯) triquark. It always contains a colour singlet qq¯pair, leading to superallowed decays, if kinematically permitted. Intheflavour-symmetrylimit,withthefurtherassumptionthatthequark–quarkandquark–antiquarkcolour– spin interaction strengths are equal, H reduces to CM 88/3 8√6 0 0 8√6 16 0 0 HCM =−C 0 0 8 8√6, (13) −  0 0 8√6 16/3     with eigenvalues 43.3656C and 1.9678C in the nonet subspace spanned by φ and φ , and 19.3656C and 1 2 − − − +22.0322C in the 36-plet spanned by φ and φ , which separates out exactly. The lightest state in the nonet 3 4 and the lightest one in the 36-plet are split by 24C, i.e., about 400 MeV, exceeding twice the mass difference between strange and non-strange quarks. This led one to predict that the flavour nonet and 36-plet are well separated. Itwillbe shownlaterthat this isnotanylongerthe case,ifflavoursymmetryis brokenalsoinH CM (and not only in the constituent masses). From Eqs. (11-12), it is seen that for the separation of the 36-plet from the nonet to remain, with a block- diagonal form for H , it suffices that C = C and C = C , or C = C and C = C , i.e., both CM 13 14 23 24 13 23 14 24 quarks have the same coupling to each antiquarks, or vice-versa. It also persists that the lowest eigenvalue is found in the nonet subspace spanned by φ and φ . 1 2 7 E (GeV) [qs][q¯s¯] qq q¯q¯ qq q¯q¯ 1.0-{ }{ } { }{ } [qs][q¯s¯] [qq][q¯q¯] [qq][q¯q¯] 0.5- 0- with SU(N) broken SU(N) F F FIG. 1: Spectrum of light multiquark scalars, with a shift ms−mq added for each strange quark or antiquark, and the chromomagnetic term calculated either in the SU(N)F limit (left) or with SU(N)F breaking (right). In the labels, [qq] denotes thesymmetric spin-colour coupling, and {qq} the antisymmetric one, and q=u, d. Asanillustration,themassspectrumoflight0+mesonsisshowninFig.1withandwithoutflavoursymmetry breaking in H , with realistic values for the strength factors C . CM ij For completeness, the crossing matrix is provided between the basis (10) where quarks and antiquarks are paired, and the basis α =(q q¯ )1 (q q¯ )1 , α =(q q¯ )1 (q q¯ )1 , 1 1 3 0⊗ 2 4 0 2 1 3 1⊗ 2 4 1 α =(q q¯ )8 (q q¯ )8 , α =(q q¯ )8 (q q¯ )8 , (14) 3 1 3 0⊗ 2 4 0 4 1 3 1⊗ 2 4 1 with quark–antiquark coupling, it is 3√2 √3 √6 3 1 √6 3 3√2 √3 6−3 √6 √3 −3√2 . (15) − − √3 3√2 3 √6   − −   D. Axial tetraquarks ThecasewherethetotalspinisS =1issomewhatmorecomplicatedthanthespinS =0caseastherecoupling to spin 1 can be done in severalways. The colour-magneticHamiltonian now acts overa six-dimensional space with basis ψ =(q q )6 (q¯ q¯ )¯6 , ψ =(q q )¯3 (q¯ q¯ )3 , 1 1 2 1⊗ 3 4 1 2 1 2 1⊗ 3 4 1 ψ =(q q )¯3 (q¯ q¯ )3 , ψ =(q q )6 (q¯ q¯ )¯6 , (16) 3 1 2 0⊗ 3 4 1 4 1 2 1⊗ 3 4 0 ψ =(q q )¯3 (q¯ q¯ )3 , ψ =(q q )6 (q¯ q¯ )¯6 . 5 1 2 1⊗ 3 4 0 6 1 2 0⊗ 3 4 1 Ifthe twoquarks(antiquarks)areidenticalinflavour,ψ , ψ andψ (ψ , ψ andψ ) areexcludedbythe Pauli 1 3 4 1 5 6 principle. This is the case in particular for the manifestly exotic states. 8 The colour-magnetic Hamiltonian can be written in terms of 2 2 blocks as × A A A 1 2 3 H = B B B , (17) CM 1 2 3 −"C C C # 1 2 3 with 2 2(C +C )+5(C +C +C +C ) 3√2(C C +C C ) A1 = 3 343√21(2C13 C1144+C1234 C2233) 24 4(C34+C12)1+3−2(C1144+C2143−+C2233+C24) , (cid:20) − − − (cid:21) 2 6(C C +C C ) 5√2(C C +C C ) A2 =B1† = 3 2√2(C1313−+C1414 C2324− C2423) − 6(C1313+−C1414 C2423−C232)4 , (cid:20) − − − − − (cid:21) 2 6(C +C C C ) 5√2(C +C C C ) A3 =C1† = 3 −2√2(C1313 C1414−+C2423− C2324) 6(C1313 C1414+−C2324−C242)3 , (cid:20)− − − − − (cid:21) 2 4(3C C ) 3√2(C +C +C +C ) B2 = 3 3√2(C +1C2−+3C4 +C ) − 123C 146C 23 24 , (18) 13 14 23 24 12 34 (cid:20)− − (cid:21) 2 2(C C +C C ) 0 B3 =C2† = 3 − 13− 140 24− 23 5(C13 C14+C24 C23) , (cid:20) − − − (cid:21) 2 4C +12C 3√2(C +C +C +C ) C3 = 3 3√2(C−14+12C13+C3243+C24) − 146C121+32C3423 24 . (cid:20)− − (cid:21) In the limit of flavoursymmetry whereC =C, i,j, the eigenstates of H have well defined transformation ij CM ∀ properties under the relevant flavour-symmetry group, and the colour-magnetic Hamiltonian H reduces to CM the well-known matrix 6 0 0 0 0 0 0 0 0 0 0 0 8C 0 0 2 3√2 0 0  − . (19) − 3 0 0 3√2 1 0 0  0 0 −0 −0 2 3√2  −  0 0 0 0 3√2 1   − −    witheigenvalues 16C,0, 40C/3,32C/3, 40C/3,32C/3. The correspondingflavourmultiplets are9and36 − − − for the first eigenvalues, 18=10+8 for the next two ones, and 18=10+8 for the last two ones Moreover,for the interesting case (QQud) case where the two heavy quarksQ are identicalandthe two light antiquarks obey isospin symmetry, H also takes the block-diagonalform CM C +C +10C 0 0 0 0 0 34 12 14 0 2C 2C +4C 0 0 0 0 34 12 14 4 0 − − 0 2C +6C 6√2C 0 0  34 12 14 − − − 3 0 0 6√2C14 C12 3C34 0 0   − −   0 0 0 0 2C12+6C34 6√2C14   − −   0 0 0 0 6√2C 3C +C   − 14 − 12 34   (20) Note that, contrary to what happens for the spin S = 0 case, the lowest eigenvalue of the colourmagnetic HamiltoniansurvivesthePauliprinciple,i.e.,remainswhenthebasisstatesψ ,ψ andψ areremoved,atleast 1 3 4 for all the physically acceptable values of the parameters (see next section). The crossing matrix from the basis (16) to the basis β =(q q¯ )1 (q q¯ )1 , β =(q q¯ )1 (q q¯ )1 , 1 1 3 0⊗ 2 4 1 2 1 3 1⊗ 2 4 0 β =(q q¯ )1 (q q¯ )1 , β =(q q¯ )8 (q q¯ )8 , (21) 3 1 3 1⊗ 2 4 1 4 1 3 0⊗ 2 4 1 β =(q q¯ )8 (q q¯ )8 , β =(q q¯ )8 (q q¯ )8 , 5 1 3 1⊗ 2 4 0 6 1 3 1⊗ 2 4 1 9 is 2 √2 1 √2 1 √2 − − 2 √2 1 √2 1 √2 1  0 0 √−2 2 √2 −2  . (22) 2√3√2 2 √2 1 √2 1  √2 −2 −√2 −1 √2 1     0 −0 2 √2 − 2 √−2   − −    E. Tensor tetraquarks The surveyis ended by the caseof spinS =2. In the diquark–antidiquarkcoupling scheme, the chromomag- netic Hamiltonian H , written in the basis CM ξ =(q q )6 (q¯ q¯ )¯6 , ξ =(q q )¯3 (q¯ q¯ )¯3 , (23) 1 1 2 1⊗ 3 4 1 2 1 2 1⊗ 3 4 1 reads 2 2(C +C ) 5(C +C +C +C ) 3√2(C +C C C ) 12 34 − 13 24 14 23 − 13 24− 23− 14 . (24) − 3(cid:20) −3√2(C13 +C24−C23−C14) −4(C12+C34)−2(C13+C24+C23+C14) (cid:21) With two quarks identical in flavour,the state ξ is excluded by the Pauli principle. 1 As all spins are aligned, the crossing matrix from the basis (24) to the basis γ =(q q¯ )1 (q q¯ )¯1 , γ =(q q¯ )8 (q q¯ )¯8 , (25) 1 1 3 1⊗ 2 4 1 2 1 3 1⊗ 2 4 1 reduces to the standard crossing matrix of colour 1 √2 1 . (26) √3 1 √2 (cid:20) − (cid:21) III. APPLICATION TO TETRAQUARKS This section is devoted to consequences of the chromomagnetic interaction applied to four-quark states for the various flavour configurations. A. Adjusting the parameters ThestrengthparametersC forquark–antiquarkpairscanbeextractedfromordinarymesons,andaregiven ij in Table I. They can be considered as upper bounds, as the two-body correlations are stronger in mesons than in tetraquarks. The quark–quark analogues, deduced from the baryon spectrum, are listed in Table II. These parameters can be used to extrapolate the model from ordinary hadrons to multiquarks. A tempting alternative strategy consists of extracting the parameters from states which are assumed to be dominantly tetraquarks [65] and to apply the model to predict new tetraquark states. However, the observed states verylikely resultfromanintricatemixing of four-quark,two-quark,hybridandgluoniumstates,andthe fit can be biased if this mixing is ignored. There is no way to determine the effective masses unambiguously, as they incorporate binding effects which depend on the environment. In particular, the values of m extracted from baryons are usually higher than i those from mesons. This is, indeed, a general property that baryons are heavier, per quark, than mesons, for instance (Ω)/3> (φ)/2. Theinequality(qqq)/3>(qq¯)/2canbederivedinalargeclassofmodelsinspired by QCDM[66]; in thiMs review article, and refs. therein, it is also reminded that (qq¯)+(QQ¯) 2(Qq¯), hence ≤ masses deduced from hidden flavour are found lighter than from open flavour. In a multiquark such as (cc¯qq¯), a compromise has be found, as in Ref. [15]. 10 The chromomagnetic model will thus be used mainly to predict the ordering of the various spin and flavour configurations. Estimating the absolute masses would require a more careful treatment of the chromoelectric effects. B. Light mesons This is the most delicate sector. Experimentally, states are often broad and overlapping. Theoretically, the quark–antiquark spectrum is not as easily described as in the case of heavy quarks, and states with exotic internal structure are thus harder to single out. Moreover,mixing of configurations is more appreciable in this sector. Justtoillustratethecomplexity,thediagramswithinternalqq¯ ss¯transitionthroughanintermediate gluon (see Fig. 2) mix ten tetraquark states with I = 0, and six with↔I = 1 [57]. Hence great care is required when discussing this sector. q q q s q¯ s¯ q¯ q¯ FIG. 2: Example of mixing through internal annihilation There are many scalar mesons below about 2 GeV, and different scenarios have been proposed for their assignment, see, e.g., [51, 52] and references therein. For instance, Klempt et al. [67], using a relativistic quarkmodelandaninstanton-inducedinteraction,proposedthatthemainlyquark–antiquarkmultipletincludes f0(980)andf0(1500). Alternatively,itistemptingtoassignf0(1370)asbeingmainlyqq¯,inits3P configuration, 0 whichisexpectedtolieslightlybelowits3P (perhapsmixedwith1P )and3P partners,inanalogywithwhat 1 1 2 is observed in the case of charmonium. This is the point of view adopted, e.g., in [48, 49], where the f0(1500) is tentatively identified with the lowest gluonium state, the second qq¯ state with isospin I = 0 being slightly higher. The expert view point of the latest issue of the review of particles properties [52] suggests to organize thescalarsinalow-lyingnonetconsistingoff0(600),f0(980),a(980)andK∗(800)=κ,asecondmultipletbeing 0 madeoff0(1370),f0(1500),a(1450)andK∗(1430),withthecaveatthatthef0(1500)iscopiouslymixedwiththe 0 f0(1710)tosharetheirqq¯,ss¯andgluoniumcontent. ThisisnottoofarfromtherecentanalysisbyNarison[68]. If it is assumed that C = C , C = C = 0.625C , and C = C = C2 /C , and if the values qq qq¯ qs qs¯ qq ss ss¯ qs qq C =19.2MeV,m =320MeVandm =445MeVareadopted,thetwoabovenonetscomewithmasses(439, qq q s 722,980)MeV and (1242,1376,1512)MeV, respectively, in our simple chromomagnetic model. The agreement with experimental masses is perhaps too good, as mixing with other configurations is expected to shift these results. Anyhow, these parameters also give the remaining tetraquark spectrum. In particular, for the Y = 2 axialsconsideredin[65] with(qqs¯s¯) content,a firststateis found at1310MeVanda heavieroneat1620MeV, while forthe 27 (I =1)Y =2, the mass is predicted to be about1540MeV. States with both openandhidden strangenessare obtainedat 1510and 1870Mev (0+), 1500,1640and1760MeV (1+) and 1810 MeV (2+), and an axial state with a double hidden strangeness is predicted at 1780 MeV. Moreover, the φω resonance found at BES II [69] at 1812MeV could be identified with the multiquark scalar with hidden strangeness. A difficulty with this description of tetraquarks, or at least, with this choice of parameters, is the prediction of a scalar multiplet with I = 0, 1, 2 at about 800MeV, without experimental evidence for a I = 1 resonance nor for a I = 2 one in that region.1 Also, a puzzlingly light I = 0 state would be predicted with this set of parameters. 1 Note that the repulsive character of the I =2ππ phase-shift inthis regionis not a definite obstacle, as well the pentaquark is