Flavor Physics and CP Violation Conference, Victoria BC, 2019 1

Deciphering the XYZ States

M. B. VoloshinWilliam I. Fine Theoretical Physics Institute, University of Minnesota, Minneapolis, MN 55455, USA

School of Physics and Astronomy, University of Minnesota, Minneapolis, MN 55455, USA andInstitute of Theoretical and Experimental Physics, Moscow, 117218, Russia

I give a brief account of current topics in description of exotic multiquark mesonic resonanceswith hidden heavy (c or b) flavor, the so-called XYZ states, in terms of hadronic molecules andhadro-quarkonium systems. Also discussed are the recently observed hidden-charm pentaquarksincluding additional ways of producing them in experiments.

I. INTRODUCTION

Experiment has revealed a number of states withhidden charm and hidden bottom that do not fit thestandard quark model template of mesons consistingof a quark and antiquark and baryons consisting ofthree quarks. Thus far about a dozen or more mesonsabove the open charm threshold require presence in-side them of a light quark-antiquark pair along witha hidden charm cc̄ quark pair. Two isotopic dou-blets of mesons with a heavier bb̄ quark pair witha similar four-quark structure have been observed.More recently, sightings of pentaquarks consisting ofthree light quarks and a cc̄ pair have been reported.This family of hadronic states, referred to as exotic,and expanding from the first observation [1] of thecharmonium-like peak X(3872) up to the most re-cently reported [2] narrow pentaquarks, conspicuouslychallenges the theory for explaining their ‘internalworkings’ and possibly predicting their yet unknownproperties and new states of similar nature.

Detailed reviews of the experimental status of thenew states and of various theoretical models can befound in a number of very recent papers [3–6]. Here Ipresent a brief account of the theoretical situation as Isee it. The existing schemes for description of the ex-otic states are based on the idea that the complicatedmultiquark dynamics splits into simpler few-body cor-relations that may be tractable theoretically. The dis-cussed types of such two-body correlations are shownin Figure 1. Although all of the shown configurationsare likely present, to an extent, in the XYZ mesons,as will be discussed, there are good reasons to believe(or at least a strong hope) that the configurations oftwo first types, molecules and hadro-quarkonium, aredominant in some of the observed exotic states. Onthe contrary, it can be argued that the third struc-ture in Figure 1 corresponding to significant correla-tions within diquarks and antidiquarks does not havea justification within QCD dynamics. As to the mosttheoretically ‘unpleasant’ fourth structure, where allinteractions are of similar strength and no few-bodycorrelations can be considered as more important thanother, one can only hope that it is not very importantin at least some of the exotic hadrons. The study ofbaryonic exotic states, the pentaquarks, is still at an

early stage, so that only preliminary remarks can bemade at present regarding the internal dynamics ofthese baryons.

In what follows I discuss the configurations of Fig-ure 1 within specific mesonic XYZ states, and thenbriefly discuss the hidden-charm pentaquarks.

Q

Q̄

q̄

q

B(∗)

B̄(∗)

ρ, ω, π, . . .Molecule :

q

(QQ̄)

q̄

Hadro− quarkonium :

(Qq)

(Q̄q̄)

Diquarkonium :

Qq̄qQ̄

A mess :

FIG. 1: Types of internal dynamics of exotic heavymesons.

II. MOLECULES

The notion of hadronic molecules made of heavy-light hadrons goes back to the following simple con-sideration [7]. The interaction between such hadrons,mediated by exchange of light quarks and antiquarksdoes not depend on the mass of the heavy quark in thelimit, where that mass is large. On the other hand thekinetic energy is inversely proportional to the heavymass. Thus for the states of heavy hadron pairs, where

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2 Flavor Physics and CP Violation Conference, Victoria BC, 2019

the light-quark exchange results in an attraction thereshould be bound or/and resonance levels for a suffi-ciently heavy heavy quark. In such a near thresholdstate, the motion of the constituents proceeds at longdistances (much like the motion of the proton and neu-tron in deuteron), where the heavy hadrons largelyretain their individual structure. The relevant sizeof such states can be estimated in terms of the bind-ing/excitation energy δ (relative to the threshold) andthe mass M of the constituents as

r ∼ 1/√M |δ| ≈ (1) 4.5 fm√

1 MeV|δ| charmonium− like

2.8 fm√

1 MeV|δ| bottomonium− like ,

so that the notion of molecules applies in a narrowmass band around the threshold with δ not exceeding10-20 MeV for charmonium-like states and not ex-ceeding several MeV for the bottomonium-like ones.

It was however not known a priori whether charmedor bottom quarks would be ‘sufficiently heavy’ un-til in 2003 the Belle experiment reported [1] obser-vation of a charmonium-like peak X(3872) in theJ/ψπ+π− channel. The peak is very narrow [8],Γ < 1.2 MeV, and the mass of the peak practically co-incides with the threshold for D∗0D̄0 charmed mesonpair: M(X)−M(D∗0)−M(D0) = −0.01±0.18 MeV.According to the present understanding (see e.g. inRef. [4]) the X(3872) peak is dominantly a shallowbound or a virtual JPC = 1++ state of an S-wavepair of neutral charmed mesons D∗0D̄0+D̄∗0D0. Thisstructure successfully explains the isospin violation oforder one that is evident from simultaneous existence(with a comparable rate) of the decays to the finalstates J/ψρ0, J/ψω [8, 9] and χc1π

0 [10].A very clean example of molecular states is

presented by the bottomonium-like Zb(10610) andZb(10650) resonances [12], whose masses are withinfew MeV from respectively the B∗B̄ and B∗B̄∗ thresh-olds. The understanding that these these resonancesare bound or virtual states made of the correspondingheavy meson pair is strongly supported by an anal-ysis of the behavior of heavy quark spin in produc-tion and decay of these resonances [11]. Namely thestrength of the interaction depending on the spin ofa heavy quark in QCD is inversely proportional tothe quark mass. For this reason there is an approxi-mate Heavy Quark Spin Symmetry (HQSS) for heavyquarks which symmetry is exact in the limit of infi-nite quark mass, and is a very good approximationfor soft processes involving bottom quarks. An illus-tration of the quality of this approximation can befound e.g. in the relative rate of HQSS suppressed andallowed transitions between the bottomonium states:

Γ[Υ(2S) → Υ(1S)η]/Γ[Υ(2S) → Υ(1S)ππ] ∼ 10−3.In a widely separated meson-antimeson pair the spinof the b quark (antiquark) is fully correlated with thespin of the light antiquark (quark) that is contained inthe corresponding pseudoscalar or vector B̄ (B) me-son. For this reason the spins of the b and b̄ are notcorrelated with each other and the bb̄ pair is in a mixedspin state:

B∗B̄ − B̄∗B ∼ 0−H ⊗ 1−L + 1−H ⊗ 0−L

B∗B̄∗ ∼ 0−H ⊗ 1−L − 1−H ⊗ 0−L , (2)

where SH (SL) stands for the total spin of the heavy(light) quark-antiquark pair. If the Zb resonances con-tain widely separated heavy meson-antimeson pairs,one would expect that the spin structures (2) shouldbe to some accuracy retained in the two molecularstates. It is exactly this behavior that has been ob-served in experiment [12] with the Zb resonances de-caying with comparable rates to the ortho- (SH = 1)and the para- (SH = 0) states of bottomonium withemission of a pion: Zb → Υ(nS)π and Zb → hb(kP )π.Moreover the relative signs of the transition ampli-tudes implied by this picture [11] are in a remarkableagreement with the observed relative phase of the twoZb resonance contribution to the processes Υ(5S) →Zbπ → Υ(nS)ππ and Υ(5S)→ Zbπ → hb(kP )ππ.

The spin structure (2) of free meson pairs is pre-served withing the Zb resonances inasmuch as the de-pendence of the interaction through light degrees offreedom on the light spin SL is not essential. In par-ticular, such dependence would induce transitions be-tween the two states of meson pairs in Eq.(2) andgive rise to decay of the heavier Zb resonance to thelighter meson pair, Zb(10650)→ B∗B̄+c.c., which de-cay is perfectly allowed and can only be suppresseddue to the spin orthogonality in Eq.(2). The data [13]show no indication of such decay, so that possibly theinteraction within the heavy meson-antimeson pairsindeed does not depend on the spin of light quarksimplying an existence of a certain ‘accidental’ LightQuark Spin Symmetry (LQSS). Such symmetry is notexpected within QCD. Moreover it is explicitly bro-ken by pion exchange. The observed suppression ofa spin-dependent interaction then can be described interms of a form factor suppression [14] of the pion ex-change or as an effect of a contact term [15] effectivelycanceling the pion contribution.

If the light-quark interaction between the heavy me-son can indeed be approximated as spin-idependent,one should then expect [16] existence of near-thresholdresonances in the S-wave states of the meson pairs re-lated to the Zb(10610) and Zb(10650) by the quarkspin symmetry. There are four such states:

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Flavor Physics and CP Violation Conference, Victoria BC, 2019 3

Wb2 : 1−(2+) :(1−H ⊗ 1−L

)∣∣J=2

, B∗B̄∗ ; (3)

Wb1 : 1−(1+) :(1−H ⊗ 1−L

)∣∣J=1

, B∗B̄ + B̄∗B;

W ′b0 : 1−(0+) :

√3

2

(0−H ⊗ 0−L

)+

1

2

(1−H ⊗ 1−L

)∣∣J=0

, B∗B̄∗ ;

Wb0 : 1−(0+) :1

2

(0−H ⊗ 0−L

)−√

3

2

(1−H ⊗ 1−L

)∣∣J=0

, BB̄ ,

where the quantum numbers IG(JP ) and the corre-sponding meson-antimeson thresholds are indicated.Due to their negative G parity the WbJ states can-not be produced by a single pion emission from abottomonium-like resonance produced in e+e− annihi-lation. They can be produced by emission of a photonin similar processes, however the rate is possibly verysmall. The lowest mass resonance Wb0 may be acces-sible in a two pion process Υ(6S)→ Wb0ππ, howeverthe rate of this process is hard to predict. Finally,the most favorable setting for a search in the processe+e− → WbJρ requires c.m. energy in excess of 11.4- 11.5 GeV.

An application of the same considerations tocharmonium-like molecules made of charmed mesonshas some peculiarity due to apparently weaker con-straints from the HQSS and also due to a signifi-cant isotopic breaking near threshold arising from themass difference between the neutral and charged D(∗)

mesons. For instance, the X(3872) peak has the JPC

quantum numbers similar to the neutral component ofthe Wb1 isotopic triplet. However it is not clear whatis the significance of the strong isospin breaking inthe masses of the charmed mesons for the emergenceof the D∗0D̄0 threshold peak. For this reason it wouldbe troublesome to predict a full structure of S wavethreshold molecules made of charmed mesons. TheZc(3900) and Zc(4020) appear to be hidden-charmanalogs of the bottomonium-like Zb resonances. How-ever the heavy charmed quark spin properties are notas clear cut. The Zc(3900) resonance has been ob-served in the channel J/ψπ [17] but not (yet?) inhcπ [18]. For the peak Zc(4020) the situation is re-versed: the decays to hcπ are ‘seen’ but those to J/ψπare ‘not seen’. It thus appears that the mixing ofthe ortho- and para- spin states of the cc̄ pair is lessstraightforward in this case than for the Zb resonancesalthough some hints at decays of Zc(3900) to theparacharmonium states, hπ and ηcρ are reported [19].Furthermore, the heavier resonance Zc(4020) appearsnot to decay into the lighter meson pair D∗D̄+c.c.much in the same manner as its bottomonium-likecounterpart Zb(10650).

III. HADRO-QUARQONIUM

Some manifestly exotic hidden-charm resonanceshave masses that are not particularly close to anytwo-body thresholds, so that it would be trouble-some to interpret them as molecular states. Thesestates include Z(4430)± [8] decaying to ψ(2S)π±,Zc(4100)± [20] and Zc(4200)± [21] in the respectivechannels ηcπ

± and J/ψπ±, and a pair of resonancesZc(4050)± and Zc(4250)± observed [22] in the decaychannel χc1π

±. (A subsequent search [23] for the lat-ter two peaks however turned to be unsuccessful, nei-ther any other confirmation was reported in over adecade, so that the experimental status of these twostates is not quite clear.) The decay channels withcharmonium are essentially the only experimentallyobserved decay modes, while less, or very little, isknown about decays of these peaks into final stateswith open charm. This implies that the latter statesdo not entirely dominate the decay modes of the ex-otic hidden-charmed resonances. (As an example ofsuch saturation of the width by open charm channelsthe resonance ψ(3770) can be mentioned.)

Such behavior of the exotic resonances strongly sug-gests [24, 25] that their structure is dominantly de-scribed by the hadro-charmonium picture shown asthe second type of configuration in the Figure 1. Theheavy cc̄ pair is a state of charmonium embedded inan excited light-quark matter due to a two (or multi-) gluon interaction. The binding of charmonium inlight-quark matter has been considered long ago [26–28] in terms of interaction of charmonium inside nu-clei. The hadro-charmonium resonances are differentin that instead of the nucleus the compact cc̄ state isinside a spatially large light-matter excitation. In thispicture the decays into charmonium and light mesonsare due to a de-excitation of the light matter withthe cc̄ state remaining. The decay width is then setby the scale typical for excited light-quark resonancesand should thus be broad - in tens or hudreds of MeV.It is thus expected that such decays mostly proceedinto the charmonium state that is already contained‘inside’, in agreement with the observed appearance ofparticular cc̄ in the decay products (e.g. ψ(2S) ratherthan J/ψ in the decays of ψ(4330)±). Naturally, onecan expect that ‘other’ cc̄ states should be present

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among the decay products at a sub-dominant leveldue to a ‘deformation’ of charmonium by the binding.

The tendency of hadro-quarkonium to not decayoverwhelmingly into states with open heavy flavor canbe semi-quantitatively argued in the limit of largeheavy quark mass [29]. Indeed, such decay requiresa reconnection of the dominant bindings within thefour-quark complex, as illustrated in Figure 2. In the

� �� �Q QQ̄ Q̄

� �� �q̄q̄

qq

�

Hadro-Quarkonium D D̄

FIG. 2: Reconnection of binding in hadro-charmonium de-cay into open heavy flavor

limit of large heavy quark mass MQ one can consideran effective potential between the heavy quark andantiquark in terms of Born-Oppenheimer approxima-tion, where the potential is the energy of the systemas a function of distance between static Q and Q̄.Clearly, at short distances there is a Coulomb-like po-tential well, and at long distances the energy goes toa constant corresponding to widely separated heavymesons. At intermediate distances however, the en-ergy is increased due to the gluonic bindings beingout of the minimal energy state, as shown in Figure 3.The discussed decay into open heavy flavor channel

�

VQQ̄

r

�tunneling

ΛQCD

FIG. 3: Behavior of a Born-Oppenheimer type potentialbetween heavy quarks

can then be considered as tunneling through the po-tential barrier. Given that the parameters of the po-tential are determined by ΛQCD one can expect thedependence of the tunneling rate on the mass MQ as

Γ(open flavor) ∼ exp

(−C

√MQ/ΛQCD

), (4)

where C is a numerical constant.The dominant QCD interaction giving rise to the

analog of van der Waals force between a compactquarkonium and light matter is the chromoelectricdipole E1 that does not depend on the heavy quarkspin. Thus in the hadro-charmonium picture one

can expect that the exotic states appear in multi-plets whose components are related by HQSS in thesame way as the states of quarkonium are related bythis symmetry. It is very likely that an example ofsuch multiplet is provided by the Zc(4100) resonancedecaying into ηcπ and the Zc(4200) observed in theJ/ψπ± channel. The quantum numbers JP of theseresonances are not yet well known, but the data arequite compatible with them being 0+ and 1+ respec-tively. One can thus suggest [30] that these are hadro-charmonium resonances made by embedding in the Swave the ηc or J/ψ charmonium into the same excitedpion-like hadronic state. Clearly the mass differencebetween the resonances agrees (within expeted accu-racy) with that between J/ψ and ηc, and the observedtotal widths do not contradict within the errors to theexpectation that they should be equal. Furthermore,the rate of production in B decays (also within errors)agrees with the expected relation

B[B0 → Zc(4100)−K+]

B[B0 → Zc(4200)−K+]≈ B[B0 → ηcπ

−K+]

B[B0 → J/ψπ−K+]

∣∣∣∣ .(5)

The leading HQSS breaking interaction in QCD is wellknown, so that one can also predict a relation betweensub-dominant heavy-spin violating decays [30]

Γ[Zc(4100)→ J/ψρ] ≈ 3 Γ[Zc(4200)→ ηcρ] , (6)

with the branching fraction for each of these processesexpected in the ballpark from several percent to a fewtens percent relative to the observed decays to ηcπand J/ψπ.

If the Zc(4100) and Zc(4200) are identified as the 1Sstates of charmonium ‘stuck’ in an excited pion, thenit is possible that there are similar states with an ex-cited Kaon instead of a pion, that are about 150 MeVheavier: Zcs(4250) and Zcs(4350) [31]. These strangehadro-charmonia should be SU(3) flavor symmetrypartners of the non-strange ones, in the same manneras the excited resonances K(1460) and π(1300) arepartners. It scan be expected that the flavor SU(3)symmetry should be applicable to hadro-quarkoniumwith about the same accuracy as to ordinary lighthadrons. This expected behavior is quite differentfrom that for molecular states. Indeed, in the lat-ter systems the SU(3) breaking by the strange quarkmass is a large effect in comparison with the bind-ing/excitation energy of a molecular state. Even theisotopic mass differences can be of importance in themolecules, as is the case for X(3872). The strangehadro-charmonium resonances, decaying to ηcK andJ/ψK should be observable in the decays of thestrange Bs mesons, Bs → ZcsK, at the same rate [31](corresponding to the branching fraction ∼ 10−5) asthe non-strange ones, B → ZcK.

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Flavor Physics and CP Violation Conference, Victoria BC, 2019 5

IV. HIDDEN-CHARM PENTAQUARKS

Some time ago the LHCb experiment has re-ported [32, 33] an observation of resonant structures inthe hidden-charm pentaquark channel J/ψp producedin decays of the b hyperon Λb → J/ψpK−. The initialobservation indicated a broad, Γ = 205±18±86 MeV,structure Pc(4380) and a narrower peak Pc(4450)with Γ = 39 ± 5 ± 19 MeV. Very recently a re-fined picture was presented [2] with the peak near4450 MeV resolved into two narrower ones, Pc(4440)with Γ = 20.6 ± 4.9+8.7

−10.1 MeV and Pc(4457) with

Γ = 6.4 ± 2.0+5.7−1.9 MeV, and an additional observed

narrow peak Pc(4312) with Γ = 9.8± 2.7+3.7−4.5 MeV.

The models of the internal dynamics of the bar-ionic pentaquark states essentially follow the samelines as in the previous discussion of the mesonic four-quark systems. In particular, the resonances withmass in the vicinity of a threshold for a charmed hy-peron and a charmed (anti)meson can be tested forbeing of the molecular kind. The newly reportedthree narrow resonances tantalizingly suggest [2] sucha structure composed of Σc and D̄(∗) bound in the S-wave. Namely, the thresholds for Σ+

c D̄0 and Σ+

c D̄∗0

are higher than the central values of the masses ofPc(4312) and Pc(4457) by respectively about 5 and 2MeV, while the Pc(4440) could have binding energyof about 20 MeV due to the spin-spin interaction ofΣc and D̄∗ through the light degrees of freedom. Inthis case the quantum numbers JP of the three statesshould be 1/2−, 1/2− and 3/2−, which assignmentdoes not contradict the data, but is not yet estab-lished either. The isotopic spin of a baryonic hidden-charm state Xcc̄ produced in the decays Λb → X+

cc̄K−

due to the underlying quark process b → cc̄s is nec-essarily equal to 1/2. However the isospin violationby the mass differences in the Σc and D(∗) isotopicmultiplets can be enhanced in the pentaquarks due tocloseness of the threshold [34]. In the molecular modelone should expect a strong decay of pentaquarks intothe final state(s) with Λc instead of Σc: Pc → ΛcD̄

(∗).Indeed, there is approximately 160 - 300 MeV of en-ergy available for such decays, and there appears tobe no principle forbidding them to proceed from amolecular state due to the ΣcD̄

(∗) → ΛcD̄(∗) scatter-

ing. The experimental status of these decays is notclear at present. However if they are not found, themolecular model may have difficulty explaining theirsuppression. An alternative model for pentaquarks ,based on the hadro-quarkonium picture and free fromthis potential difficulty is developed in Refs. [35–37].

Further studies of hidden-charm pentaquarks wouldbe greatly facilitated if additional to the productionat LHC ways of experimentation with them could befound. One such suggested [38–40] alternative sourceof pentaquarks decaying to J/ψp is their formationin the s channel by photon beam on hydrogen tar-

get, γp→ Pc. First data from the GlueX experimentsearching for this process have just appeared [41] withno evidence yet of pentaquark resonance(s), and set-ting a model-dependent upper limit of approximately2% on the branching fraction B(Pc → J/ψp). Giventhat this is just a first measurement and the study iscurrently in flux, it appears to be premature to drawfrom the reported data any far reaching conclusions onthe existence and the properties of the pentaquarks.

Another possible source of hidden-charm pen-taquarks can be provided by antiproton - deuteriumcollisions [42] and can be studied e.g. in the PANDAexperiment [43] at FAIR. The mechanism of formationin the s channel of a resonance coupled to a charmo-nium state and a nucleon Pc → (cc̄) + N is shown inFigure 4.

FIG. 4: The graph for the hidden-charm pentaquark for-mation in p̄ − d collision. Dashed lines denote nucleons,the solid lines, as marked, are for the deuteron (d), thecharmonium state (cc̄) and the pentaquark Pc.

The dominant part of the wave function of the nu-cleons inside the deuteron, treated as a loose boundstate, can be effective only if the kinematical con-straints in the graph do not require the relative mo-mentum of the neutron and the proton in the triangleto be large in comparison with the inverse nucleonsize. Considering the process in the rest frame of thedeutron (which frame coincides with the lab frame ina realistic experiment, e.g. in PANDA), one readilyfinds that both nucleons in the triangle can be on-shelland simultaneously at rest if the mass M of the pen-taquark is related to the mass m of the charmoniumstate and the nucleon mass µ as M = M0(m) with

M20 (m) = 2m2 + µ2 . (7)

(The small binding energy ε = −2.22 MeV in thedeuteron is obviously neglected in this expression.)In particular, for the (cc̄) charmonium mass of J/ψand ηc the special value of the pentaquark mass isestimated as respectively M0(mJ/ψ) = 4.48 GeV andM0(mηc) = 4.33 GeV. It can be readily noted thatthe former of these values is quite close to the mea-sured mass of Pc(4440) and Pc(4457), while the lat-ter is close to the mass of Pc(4312). It can be notedin connection with this kinematical observation that

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6 Flavor Physics and CP Violation Conference, Victoria BC, 2019

unlike mesonic states the same pentaquark resonancecan generally couple to both J/ψN and ηcNchannels.The low-momentum wave function for the motion in-side the deuteron is applicable in a range of the pen-taquark mass around the special value (7) so that infact all the so far reported Pc states can be studiedin both decay channels. The resulting expected crosssection at the maximum of the Breit-Wigner peak fora pentaquark can be estimated in terms of the branch-ing fraction B[Pc → (cc̄) + n] as [42]

σ(p̄+ d→ Pc) ∼

10−33 cm2

{Γ[(cc̄)→ pp̄]

1 keV

}B[Pc → (cc̄) + n] ,(8)

This estimate clearly appears to favor studies in theηc+n channel due to a much larger than for J/ψ decaywidth into pp̄: Γ(ηc → pp̄) ∼ 50 keV.

V. REMARKS ON DI-DIQUARKS

It is mentioned in the introduction that the thirdtype of configuration in Figure 1 with dominant cor-relations being within diquarks does not have a justi-fication within QCD. Here I would like to somewhatexpand on this remark. The usual argument in fa-vor of dominant configurations with color antisym-metric diquark pairs is that there is an attraction inthe antisymmetric state and a repulsion in the sym-metric one (a detailed discussion of the di-diquarkmodel can be found in the review [44]). This argu-ment is based on considering a one-gluon exchangebetween the quarks. Following this argumentationand assuming that the one-gluon exchange providesa relevant guidance, it is helpful to consider in fullthe one-gluon exchange potential in a system of twoquarks and two antiquarks constrained by the condi-tion that the system is an overall color singlet. De-note the constituents of the system as q1q̄2q3q̄4, wherethe subscripts stand for the variables, e.g. the co-ordinates ~ri, of the quarks and the antiquarks (theodd are for the quarks and the even for the anti-quarks). There are two orthogonal color configura-tions corresponding to the overall color neutrality:with color-symmetric diquarks u = {q1q3}{q̄2q̄4} and

with color-antisymmetric w = [q1q3][q̄2q̄4], where thecurly (straight) braces stand for the color symmetriza-tion (antisymmetrization). The one-gluon exchangein fact mixes these two configurations, so that thepotential V in the space of (w, u) has the form of amatrix [45] written in terms of the Coulomb factorscij = αs/|~ri − ~rj |:

V = −1

4

(N2

c−1Nc

r + Nc+1Nc

t√N2c − 1 s√

N2c − 1 s

N2c−1Nc

r − Nc−1Nc

t

),

(9)with Nc being the number of colors and the nota-tions are used: r = c12 + c34 + c14 + c23, s =c12+c34−c14−c23, t = 2c13+2c24−c12−c14−c23−c34.The attraction (repulsion) within the antisymmetric(symmetric) diquark is described by the term t whilethe term s describes the mixing between these con-figurations. One can readily see that at large num-ber of colors the t term is by the factor N−1

c smallerthan the s term (and these terms are comparable atNc = 3). Thus the mixing is parametrically more im-portant at large Nc than the difference between the at-traction and repulsion (or at least equally importantat Nc = 3) and the configurations with either spe-cific symmetry cannot be dominant. This conclusionapplies if there are no other parameters that wouldcompensate for the color suppression. The only situa-tion where such ‘overriding’ parameter is present is ina system with a double-heavy diquark, QQq̄q̄, wherethe quarks Q are very heavy in comparison with eitherthe masses of the antiquarks q̄, or with ΛQCD if the an-tiquarks are light. In this case the Coulomb attractionwithin the color antisymmetric QQ pair is sufficientlyenhanced by the mass of Q in order to overcome themixing [45]. In the hidden-charm or hidden-bottommultiquark systems there is no such parameter andthere is no grounds whatsoever to consider the modelwith dominantly color-antisymmetric diquarks.

Acknowledgments

This work is supported in part by U.S. Departmentof Energy Grant No. de-sc0011842.

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