Pentaquark implications for exotic mesons

PHYSICAL REVIEW D 71, 014017 (2005)

Pentaquark implications for exotic mesons

T. J. Burns* and F. E. Close†

Rudolph Peierls Centre for Theoretical Physics, University of Oxford, 1 Keble Road, Oxford OX1 3NP, United Kingdom

J. J. Dudek‡

Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606, USA(Received 30 November 2004; published 18 January 2005)

*Email add†Email add‡Email add

1550-7998=20

If the exotic baryon ��1540� is a correlated udud �s with JP � 1�

2 , then there should exist an exoticmeson #� with strangeness �2, JP � 1�, mass �1:6 GeV, decaying to K�K0 with widthO�10–100� MeV. The 1�1400; 1600� may be broad members of 10� 10 in such a picture, and it ispossible that vector mesons in the 1.4–1.7 GeV mass range are related to this multiplet. We suggest thatdynamical correlations may limit the spectrum of JP�C� such that the 1�� states are isolated.

DOI: 10.1103/PhysRevD.71.014017 PACS numbers: 12.40.Yx

I. INTRODUCTION

If the narrow �� seen in nK� and pK0 is confirmed as apentaquark, then correlations among quarks in strong QCDplay an essential role. There is considerable literaturerecognizing that a ud pair in color �3 with net spin 0 feels

a strong attraction [1,2]. We denote this ��ud�3c0 �, the sub-

script denoting its spin, the superscript the color, and ( )denoting the quasiparticle. Reference [3] considers thefollowing subcluster for the pentaquark: ��ud0��ud0� �swith a P wave between the assumed bosonic �ud� correla-tions. By contrast Ref. [4] assumes that the ��ud0� seed isattracted in a P wave to a strongly bound ‘‘triquark’’��ud6c1 �s�. These models assign the �� to 10 of flavor.

An essential feature of these dynamics is that in theS wave the chromomagnetic repulsion of like flavors de-stabilizes the configuration such that decay to meson �baryon in the S wave has such a large width that they areeffectively nonexistent [5]. It is in the P wave that poten-tially interesting pentaquarks emerge.

Mixing between the two configurations ��ud�3c0 ��s and

��ud6c1 �s� has been shown to lead to an eigenstate of lowmass, which may be identified with the ��1540� [6–8].Further, this mixing potentially stabilizes the ud �s configu-ration, underpinning the metastability of the �� [8].

The point of departure for this paper is to note that ifeither of these correlations is realized empirically for theP wave, then on model independent grounds one can re-

place ��ud�3c0 � by �q, which implies the existence of 10 and

10 exotic mesons. A specific example is the analog of�� ud �s��ud� ! #� �ud �s��s. While many of thesestates may be broad and unmeasurable, we shall suggestthat if the mixing that lowers and stabilizes the �� con-figuration in Ref. [8] applies, then there should be anobservable #� with JP � 1� and strangeness �2, together

ress: [email protected]: [email protected]: [email protected]

05=71(1)=014017(9)$23.00 014017

with other JP�C� � 1�� states with rather characteristicsignatures. If the �� should survive high statistics data andwith a width of �1 MeV, then the observation or otherwiseof such mesons may help to discriminate among models forthe dynamical origin of that metastability. Recent pro-posals [8] to explain the anomalously narrow width ofthe �� ought to carry over to the meson world, althoughdue to greater phase space we expect that the meson analog#� will have a width of the order of 10–100 MeV. Analogswith JP � 0�, 2�, 3� also arise but are expected to bebroad and unobservable.

The possibility that the 1�1400� and 1�1600� [9]could belong to these multiplets is discussed. In addition,the pattern of vector mesons in the 1.4–1.7 GeV massregion [9] may also have some overlap with these ideas.

II. EXOTIC MESONS

The idea that mesons beyond q �q exist is not new. Jaffe[10] noted that the attractive forces alluded to the abovelead to a low lying qq �q �q S-wave nonet. There is goodevidence that the scalar mesons below 1 GeVare intimatelyrelated to such a picture [11] and the idea of these ascorrelated diquarks has been resurrected by Maiani et al.[12]. Models of multiquark mesons typically predict alarge number of states in various flavor multiplets andspin states, including exotics whose flavor or JPC cannotbe constructed from q �q. Any model of exotics must facethe fact that experimentally there is at most only a handfulof such candidates. The full set of flavor representationsfrom 3 � 3 � �3 � �3 � ��3� 6� � �3� �6� is

(i) �3

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� 3 � 8 1, Jaffe’s original nonet,
(ii) � 6 � 3� ��3 � �6� � 10 8a 10 8b � 18 18,
the decuplets into which Ref. [13] assigns the1�1400� state [9], and accompanying octets, and

(iii) 6
� �6 � 27� 8� 1. All of these states were included in the original study ofqq �q �q in the S wave [10]. Within the dynamical assump-tions made there, states other than the nonet (i) werepredominantly predicted to be very broad and effectively
2005 The American Physical Society

T. J. BURNS, F. E. CLOSE, AND J. J. DUDEK PHYSICAL REVIEW D 71, 014017 (2005)

unobservable. With L � 1 in the system there are manymultiplets with negative parity, J � 0, 1, 2, 3 and C � �.Either all of the states are broad and unobservable or someorganizing principle is required if one wishes to identifyone or two states as members, such as in [13], and explainaway the remainder. We examine the qq �q �q systems in L �1 in light of the recent interest in diquark correlations,focusing on the supposed dominance of ‘‘good’’ diquarksover ‘‘bad’’ [5]. We shall see that the correlation oftriquark-antiquark (and charge conjugate) leads to a lim-ited spectrum of JP�C� states and the possibility that themasses of the 1�� multiplets are lowered to mass scalesakin to those of the1�1400; 1600� claimed in� and�0

respectively1 [9].

III. THE #�: AN EXOTIC MESON ANALOGOF THE ��

If the �� is confirmed as a resonant state with ��1 MeV, then the stability of the correlations, at least in theP wave, raises interesting questions for the existence ofobservable meson analogs in representations 18 and 18. Inthe (ud �Q) correlation, at least, the same dynamics that leadto �� in 10 imply a 10 of mesons which will include a #�

with strangeness �2, and its 10 antiparticle with strange-ness �2. In a later section we will show that a dynamicalpicture in which a triquark-antiquark (and charge conju-gate) are in L � 1 suggests that the 1�� multiplet lieslowest.

Neither of the correlations of [3] or [4] alone deriveseither the low mass or width of the ��1540� readily [15].The triquark correlation in Ref. [4] has the �ud in the �ud�s�in the configuration �ud61, chosen for its maximal attrac-tion. However, it has been widely noted [6–8] that mixingwith the �ud�30 via either one-gluon exchange, instantonforces, or the effect of virtual KN loops leads to oneeigenstate that is lower in energy than either of the un-mixed states. That this could cause a decoupling of the�� ! KN, along the lines suggested in Ref. [16], ispossible but has not been demonstrated (and this mecha-nism has problems if virtual K�N loops are included).Reference [8] argues that including both gluon exchangeand instanton forces in the mixing analysis leads to effec-tive masses for the light eigenstates �ud �s� 750 MeV and�ud � 450 MeV. The reason for �ud�s being some350 MeV more bound than the lightest uds � state isthat the gluon and instanton forces are twice as attractivein the q �q channel than in the qq case. Further, asm�ud �s� &

m�K� �m�d=u�, for constituent quark mass m�q�, thepentaquark cannot dissociate into the Ku�d�. It is proposed[8] that rearrangement is suppressed in L � 1 with theresult that the �� is metastable.

1We refer the reader to [14] for an analysis of the experimentaldata that disputes a resonant interpretation for the 1�1400�.

014017

The reduced mass of the 750 MeV triquark and450 MeV diquark is 280 MeV. The similarity of this tothat of the �, which is �m��=4 � 255 MeV, suggests asimilar price for L excitation in the two systems. Usingm�f2�1525�=f1�1420��m�� 400–500 MeV as ameasure of the orbital excitation energy,2 the mass scalefor �� is �1600–1700 MeV. Spin-orbit splitting mightreduce this to 1540 MeV [18].

But now consider the �ud accompanying the ud �s in theP wave: we can replace this by any of �u, �d, or �s to form ameson. Were we to do so for any combination qiqj �qk and�ql, we would have a 10 and 10 of mesons with a P waveinternally. The1�1400; 1600� could be members of such amultiplet (this was originally proposed on symmetrygrounds in Ref. [13]); however, an inescapable conse-quence of such a proposal is that there exists an exoticud�s �s meson with strangeness �2. With m��s� �m�ud wepredict this to be at �1600 MeV. Assuming that the��1540� is narrow due to a mixing between �ud6c1 �s and�ud3c0 �s such that the low mass eigenstate decouples fromKN, then the #� should exist with a ‘‘normal’’ width. Witha mass even at 1700 MeV the phase space ratio for # !KK and �� ! KN is �16. The KK� channel is open; thephase space enhancement in this case is �5 but the spincounting will elevate this so that we may expect a similarbranching ratio to that of the KK mode. The three bodyKK mode will also contribute, but uncorrelated threebody modes are not expected to dominate over two bodyones. The net result is that we expect ��#�� & 100 MeV,such that the #� should be detectable (likewise for theequivalent strangeness �2 member of the 10).

Surprisingly it is not immediately possible to excludesuch an exotic state in K�K0 if its mass is �1:6–1:7 GeV,and a dedicated search is suggested in, e.g., K�N !K�K0� [19]. If the dynamics of Ref. [8] underpins theformation of triquarks, other exotic members of the mul-tiplets are probably broad and will be difficult to observe,as discussed in the next section.

IV. OTHER MEMBERS OF THE MESON 10 10

A unified convention for constructing the symmetrystates for multiquarks is given in Table 1 of [20], repro-duced here as Table I. This gives the combinations of threelabels for the symmetric and mixed (�) states; the mixed( ) and antisymmetric follow trivially. The labels A, B, andC are defined as A �ud � �s; B �ds � �u; C

�su � �d, where �ud �ud� du�=��2

petc., and the sign

of the antisymmetric combination is important.In the meson case it is useful to adopt the ordering

A1B2 ��qq �q�, C3 �q. Triquarks �qq �q are in flavor �3 �

2Note that this is more reasonable than the 207 MeV claimedby Karliner and Lipkin [4] on the basis of an analogy to the Dsspectrum; see [17] for a critique.

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TABLE I. Pentaquark wave functions where ABC are defined in the text. Note that consistency requires the meson octet to be definedwith each q �q positive except for � � �u �d; �K0 � �s �d, and then 0 � �u �u� d �d�=

��2

p. In this convention �8 � �2s�s� u �u�

d �d�=��6

p.

10 85

�� AAAp ��ACA� CAA� AAC�=

��3

p��ACA� CAA� 2AAC�=

��6

p

n �ABA� BAA� AAB�=��3

p�ABA� BAA� 2AAB�=

��6

p

�� CAC� ACC� CCA�=��3

p�CAC� ACC� 2CCA�=

��6

p

�0 ��ABC� BAC� ACB� CAB� BCA� CBA�=��6

p��ABC� BAC� ACB� CAB� 2BCA� 2CBA�=

��12

p

�0 ��ABC� ACB� BAC� CAB�=2�� BAB� ABB� BBA�=

��3

p�BAB� ABB� 2BBA�=

��6

p

�� CCC�0 �CBC� BCC� CCB�=

��3

p�CBC� BCC� 2CBB�=

��6

p

�� CBB� BCB� BBC�=��3

p��CBB� BCB� 2BBC�=

��6

p

�� BBB

PENTAQUARK IMPLICATIONS FOR EXOTIC MESONS PHYSICAL REVIEW D 71, 014017 (2005)

�3 � �6 3. To make a 10 8 of mesons (or pentaquarks)the triquark must be in flavor �6, which is composed of thefollowing members:

AA � �ud �s;

fABg � ��ud �u� �ds�s�=��2

p;

fCAg � ��ud �d� �su�s�=��2

p;

BB � �ds �u;

fBCg � ��ds �d� �su �u�=��2

p;

CC � �su �d:

The nonexotic combinations in which qiqj �qj are inflavor 3 allow the possibility of qj �qj ! gluons and hencemixing with conventional hadrons; such combinationshave no advantages in forming metastable states and com-prise the P-wave excited version of Jaffe’s nonet [1]. Thecombinations fABg, fBCg, and fCAg are at least stableagainst qj �qj ! gluons [by SU�3�F] although in the mecha-nism of Kochelev et al. [8], it is only the AA triquark that isfully stable: all others are unstable against decay into or�s.

Using the conventions of [20], a full set of mesonrepresentations can be constructed:

#� � AAA;

K�

10� ��ACA� CAA� AAC�=

��3

p;

K�8b � ��ACA� CAA� 2AAC�=

��6

p;

�

10� �CAC� ACC� CCA�=

��3

p;

014017

�8b � �CAC� ACC� 2CCA�=

��6

p;

“K�” � CCC:

The other charge combinations follow by acting on thesewith I� accordingly. Charge conjugate analogs of thesecorrespond to a 10 8.

Reference [4] considered only the exotic states at thecorner of the pentaquark 10: AAA, BBB, and CCC. Thedynamics of Ref. [8], discussed earlier, suggest that theconfigurations BBB and CCC are energetically disfavoredand unstable. For instance, the CC triquark ��su �d� isenergetically disfavored, due to the presence of �su inplace of �ud, for which the instanton forces are lessattractive, and unstable because m��su �d�>m�� m�s�enables decay, likewise for the BB triquark. One conse-quence could appear to be that only the �� will be narrowin such a dynamics: the ��;�� contain ��su �d� or ��ds �u�which are unstable against decay into � s, while theother states mix with 8.

The absence of prominent signals other than the �� inthe 10, in particular, the �5 [21], may thus be explained: ifsuch correlations occur, then they create only metastableconfigurations if either �ud and/or ��ud �s� are involved.We argue that the same situation arises in the 10 and 10 ofmesons: the only exotic combinations with a chance ofsuppressed widths are the #� and #�, containing thetriquarks AA and �A �A respectively.

The exotic strangeness �1 ‘‘K�’’ ��su �d� �d and ‘‘K��’’��ds �u� �u contain triquarks that are unstable against emission in the model of Ref. [8]. Thus we expect thattheir widths will be at least 300 MeV. Identifying suchstates will be a challenge. The mass gap between the #�

and the ‘‘K�;0;�;��’’ states will be �2m�s� d� �m��us � �ud�. As m��us � �ud� is likely to be at leastas large as m�s� d� the spread is likely to be only�100 MeV.

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The remaining combinations occur in nonexotic multip-lets and in general will mix. Consider, for example, theJP � 1� states 0 and 0

1. Depending on the mixing anglebetween the 10 and 8 basis states, the mass eigenstates canbe degenerate or separated by up to 2�m�s� �m�d��. In thislatter case the physical states are the ideal mass eigenstatesqq �q �q and qs �q �s . This small mass range for the statessuggests that the hidden flavor mass basis is more repre-sentative than an SU�3�F multiplet basis. Thus we need tocount the number of states. For the nonstrange unchargedstates with I � 1 and I � 0 there are six permutations ofthe distinct ABC labels. In the SU�3�F symmetry basisthese correspond to

10:I � 1; 8�:�I � 1� � �I � 0�;

8 :�I � 1� � �I � 0�; 1:I � 0:

The latter trio correspond to the familiar ;!8;!1 combi-nations in the case of 1� and as such are indistinguishablefrom radially excited q �q nonets. The former would corre-spond to a pair of s and a single !8 and hence would benovel. Thus counting the population of I � 1 and I � 0vector mesons within an energy range of �300 MeV canhint at which underlying multiplets are present.

While the �1700�, !�1650�, ��1680� form a candidatenonet, their masses are somewhat unnatural. The �1450�and !�1420� are missing a partner to complete the set.Depending on whether this is I � 0 or I � 1 could benovel. The K��1410� appears to be anomalously low inmass for q �q systems but fits naturally into the 10 configu-ration. The widths of most of these states are hundreds ofMeV, in accordance with the expected widths of a triquark-quark configuration.

The general feature is that for a given JP�C� six non-strange uncharged members are expected within a fewhundred MeV. If any of the plethora of observed states[9] is associated with these, such that their widths of

014017

�300 MeV give a scale for their (in)stability, then a #�

seems an unavoidable consequence.

V. DYNAMICS AND JPC IN L � 1 qq q q MESONS

An L � 1 qq �q �q system has a variety of JPC combina-tions. The relative masses and potential stability of thesecan depend upon the correlations of strong QCD. We nowinvestigate the different dynamical arrangements for anL � 1 qq �q �q system, distinguished by the configurationof the orbital angular momentum. We will see that dynam-ics may favor a triquark configuration for the 18 18 ofmesons, and that such a configuration has a limited spec-trum of JPC states. The same dynamical assumptions,namely, the prevalence of good diquarks over bad, explainsthe absence of higher representations such as 27.

In the diquark-diquark correlation, the qq and �q �q sys-tems are each in L � 0 with L � 1 between them. Forquark pairs which are symmetric in space, the remainingdegrees of freedom must be antisymmetric. The allowedcombinations are as follows, for total quark spin S, with f g

and � denoting flavor 6 (6) and 3 (3), respectively, andsuperscripts and subscripts denoting color and spin:

10� 8 10� 8

�i� fqqg�31� �q �q30 �qq�30f �q �qg31 S � 1; JP � 0�; 1�; 2�;

�ii� fqqg60� �q �q�61 �qq61f �q �qg�60 S � 1; JP � 0�; 1�; 2�:

A different set of configurations arises if the L � 1 isbetween a pair of quarks or antiquarks, and the concept of a‘‘diquark’’ dissolves. The quark pairs are now spatiallyantisymmetric, so that to satisfy the Pauli principle, thesame flavor and color correlations as above will have spinflipped from 0 (antisymmetric) to 1 (symmetric), or viceversa. The resulting combinations are, where q j q denotesa pair of quarks in L � 1,

10� 8 10� 8

�iii� fq j qg�30� �q �q30 �qq�30f �q j �qg30 S � 0; JP � 1�;

�iv� fq j qg61� �q �q�61 �qq61f �q j �qg�61 S � 0; JP � 1�;

S � 1; JP � 0�; 1�; 2�;

S � 2; JP � 1�; 2�; 3�;

�v� fqqg60� �q j �q�60 �q j q�31f �q �qg31 S � 0; JP � 1�;

�vi� fqqg�31� �q j �q31 �q j q60f �q �qg�60 S � 0; JP � 1�;

S � 1; JP � 0�; 1�; 2�;

S � 2; JP � 1�; 2�; 3�:

The first thing to notice is that the two different pictures have different J couplings. If the diquark-diquark picturedescribes the 1�1400; 1600� mesons, then S � 1 only, and so we would also expect 0�� and 2�� partners at comparablemass, shifted by spin-orbit splittings. Conversely, the latter picture allows for S � 0, 1 or 2. If we supposed that the

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dynamics were such that the S � 0 state was favored (andwe will argue that this could be so), then the apparentabsence of 0�� and 2�� siblings to the 1 mesons isnatural.

In the diquark-diquark picture we see that a meson in 18or 18 is made of a good and a bad diquark: in Jaffe’soriginal paper [1], the absence of S-wave mesons in thisflavor representation was due to these repulsive color-magnetic interactions. By contrast, we see that in thesecond picture, in which the L � 1 is ‘‘within’’ a diquark,the spatial antisymmetry annuls these repulsive forces,turning a bad diquark of a given flavor (color-spin sym-metric) into a good diquark (color-spin symmetric), as inconfigurations (iii) and (iv) [or conversely turns a gooddiquark into a bad diquark in configurations (v) and (vi)].However, once orbital angular momentum separatesquarks, the short-range hyperfine interaction is heavilysuppressed, so it is better to describe these L � 1 diquarksas neither good nor bad, but ‘‘null.’’

What then can we say about the dynamics of a �qq�f �q j �qg system? On the one hand, we could consider thissystem a direct mirror image of the Jaffe-Wilczek correla-tion [3] for the ��,

�ud j �ud�s! �q j �q�qq:

On the other hand, the ��qq�30 �q� and ��qq61 �q� systems inS � 1=2 are very light due to attractive color-magnetic andinstanton forces [7,8], so it would be natural to consider theconfigurations (iii) and (iv) as tightly bound S-wave qq �qtriquarks in a relative P wave with a �q, (along with theircharge conjugates).3 Each of these systems has a gooddiquark, and the presence of a �q in an S wave with thesediquarks lowers the energy further.

References [6,7] noted that the color-magnetic interac-tion mixes the S � 1=2 �qq�30 �q and �qq61 �q triquarks [con-figurations (iii) and (iv)] to give the lowest eigenvalue�21:88C in the case of full SU�3�F symmetry [7], a con-siderably stronger attraction than the diquark-diquark sys-tem, for which the lightest configuration has an energy shiftof �16=3C. Reference [8] noted that instanton forcescause the same mixing and lower the energy of the triquarkcorrelation further.

Note also that the mixing of the (iii) and (iv) configura-tions, with their downward energy shifts, occurs only in theS � 0, JP � 1� state. Thus if this correlation is dominant,the lowest lying multiplets can be those with JP � 1�, allother JP states being higher in energy. In any event, thisdynamical picture limits the spectrum of JPC that could beexpected, so that this picture is less readily falsified by thedearth of experimental evidence of mesons in 10 or 10 than

3Provided the triquark is compact spatially, the differencebetween a P wave between the antiquarks and a P wave betweenthe free antiquark and the triquark can be neglected.

014017

the diquark-diquark picture in which there are necessarily0� and 2� partners.

The same dynamics suggests that higher representationssuch as �6 � 6 � 27 8 1 are not dynamically favored.fqqgf �q �qg systems are made of pairs of bad diquarks, andeven moving to the triquark picture cannot annul theserepulsive forces.

Thus we suggest that although tetraquark states in gen-eral will tend to be broad, the JP � 1� 10 and 10, possiblymixed with 8, may have the best chance of being observ-able. There are subtleties involved with forming the C �

� eigenstates associated with 18 18; these are discussedin the Appendix.

The phenomenology of the controversial 1�1400� and1�1600� is consistent with the triquark correlation. Thedynamically preferred arrangement in triquark languagehas S � 0, which fits neatly with the absence of clear 0�

and 2� partners to the 1�1400� and 1�1600�. It is plau-sible that the mixed �qq�30 �q and �qq61 �q system could be theonly stable triquark correlation, and that states which donot benefit from this mixing (S � 1; 2) are not dynamicallypreferred.

Note that our conclusion that the 1� 10 lies lowestassumes Fermi-Dirac symmetry arguments apply to the �qin the triquark and the free �q even after cluster decom-position. If one were to give up the symmetry entirely andtreat the triquark and antiquark as independent fermions,such that there is no simple Pauli principle in practice, thenwe could expect 0� multiplets also to be low lying. TheS-wave decay 0� ! K �K is likely to be broad unlessm�0�� & 1:2 GeV, in which case an interesting questionof metastability could arise [22]. This extreme situationcould also lead to ‘‘forbidden’’ S-wave configurations suchthat 10 could occur in 0� and the ��1540� could also beS wave with negative parity [22]. Such a state does notarise within our assumptions, as in the S wave they mirrorJaffe’s model, which has 10 only in 1� and is expected tobe broad [10]. We do not consider the 0� ud�s �s further inthis paper.

VI. JPC � 1�� MULTIPLETS IN CORRELATEDQUARK MODELS

In the search for evidence of gluonic degrees of freedomin strong QCD, attention has focused, in particular, on theprediction of exotic quantum numbers such as JPC � 1��,which are forbidden for q �q in a potential but allowed if thegluonic degrees of freedom are excited. The masses of thelightest such hybrids are predicted from lattice QCD to beabove 1.8 GeV [23]. Thus the appearance of 1�1600� !�0 is intriguing as this was long ago suggested as aselection rule by Lipkin (see the citation to unpublishedremarks by Lipkin in [24]) and then in a modern context in[25] where the decay of a hybrid 1�� was predicted to have�0 >�. A problem is that such a state could also occur

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from qq �q �q and the mass pattern of the other members ofthe nonet would need to be identified in order to distinguishthis from a canonical q �qg hybrid. In this context there isalso reported a companion state 1�1400� which is seen in� but not �0. If such a decay conserves flavor, and if� �8, then such a state cannot belong to an 8. Motivatedby this state, Ref. [13] proposed that 1�1400� belong to a10 10 qq �q �q multiplet, not a hybrid q �qg state in 8.

As shown in the Appendix, we can take a linear combi-nation of �

10 and �

10to give an IG�JPC� � 1��1�� state

overlapping to ��8, and likewise combinations of �8a

and �8b

overlapping to ��1. Since the combinationsCAC and CCA have two strange masses, while ACC haszero strange masses, we can express the SU�3�F symmetrybasis states X and Y for the 1��1�� states as

X � ��10 �

10�=

��2

p� �0�8

� ��2

pqs �q �s�qq �q �q�=

��3

p;

Y � ��8a �

8b�=

��2

p� �0�1

� ��2

pqs �q �s�2qq �q �q�=

��6

p;

where qs �q �s and qq �q �q denote the (normalized) parts of thewave function with two and zero strange quarks, respec-tively. The orthogonal linear combinations give anIG�JPC� � 1��1�� state overlapping to meson statesKK and . For Ref. [13], who assume � �8 and �0 �1, it is states X, Y respectively that decay to � and �0.However, the physical � and �0 are not pure 8 and 1. Onecould choose the mixing of the decuplet and octet states Xand Y to enforce decays to the � and �0 respectively;this would require the mixing angle of X; Y to be the sameas the �� 0 mixing angle. The eigenstates mixed in thisway would then correspond to 1�1400� ! � and1�1600� ! �0. Conversely, mass eigenstates that areideal

XH � qq �q �q; XL � qs �q �s

will decay to �n and �s respectively.In order to go from mass eigenstates to symmetry basis

states, it is necessary to have a mixing amplitude A�XH !XL>m�qq �q� �m�qs�s�. The stability of qs �s in contrast toqq �q seems to argue against that, but a definite answer isbeyond our ability to determine without further assump-tions. If 1�1400; 1600� are identified with these statesthen the quark mass eigenstates appear to be nearly real-ized: the masses based on the simplest flavor counting areconsistent with �

1 �1400� and �1 �1600� as the (domi-

nantly) qq �q �q and qs �q �s states. This agrees with our earlierprediction that the ud �s �s state #� has mass �1600 MeV,which in turn is consistent with the ��1540�. A pair of states with similar masses to those of the two 1s arerequired. There are known problems with identifying the �1460; 1600� as simply q �q states [26] and the existence of

014017

�1250� remains uncertain. If the latter exists as a q �qcandidate, then the other pair may be related to the qq �q �qand qs �q �s states. Conversely, if the �1460� is the lightestsuch resonance, then the K��1410� mass is more in tunewith the pattern of interest here than it would be for a q �qnonet. A test will be the presence or absence of isoscalarpartners to these states. A 10 or 10 have no such I � 0, �1,or ! states whereas the 8 does; a canonical nonet wouldhave the ;!;� analogs.

In any case, the �300 MeV width of the 1�1400� and1�1600� is consistent with a triquark dynamics. The wavefunction of a state, in any decuplet-octet mixing sce-nario, is composed of triquarks that can decay to � s,� q, or �s � q (or charge conjugates, for the antitri-quark component).

In [13] it is shown that if the IG�JPC� � 1��1��

1�1400� ! � is a member of a 10 10 then theremust also be an IG�JPC� � 1��1�� partner state ! and K �K. Using symmetry arguments, Ref. [13] showsthat this ‘‘supermultiplet’’ of 10 10 decaying to twopseudoscalar mesons (PP) ought to be accompanied byanother supermultiplet decaying to a pseudoscalar andvector meson (PV) with a 1��1�� ! ! and a1��1�� ! ;K� �K. Note, however, that the 1�� in thePV supermultiplet discussed in Ref. [13] must be accom-panied by siblings 0�� and 2��. By visualizing the systemas a triquark-antiquark in a P wave, we can immediatelyunderstand the origin of the two supermultiplets and theirJPC quantum numbers.

In normal q �q mesons, q and �q have opposite parity, sothat P-wave states have P � �. Conversely, in thetriquark-antiquark picture, the triquark and antiquarkhave the same parity, so that P-wave states have P � �.In the Appendix we show that the wave function of a qq �q �qmeson has an extra degree of freedom compared to a q �qmeson, manifested in the freedom to take 10� 10 versus10� 10 (or equivalently 8a � 8b versus 8a � 8b), givingC � � and C � � for each J. So if we take the q �qspectrum of states,

S � 0; J � 1�� S � 1; J � 0��; 1��; 2��; (1)

flip the parity, and take C � �, we acquire precisely thespectrum of [13]:

S � 0; J � 1�� S � 1; J � 0��; 1��; 2��; (2)

S � 0; J � 1�� S � 1; J � 0��; 1��; 2��: (3)

In the work of Chung et al. [13] there is no dynamicalpicture distinguishing the two supermultiplets, and henceno suggestion as to why the PV supermultiplet has not beenexperimentally observed. We show in the Appendix thatthere are different dynamics underpinning the supermul-tiplets. The PP supermultiplet, to which the observed1 !� belongs, has S � 0, and we saw earlier that this spinconfiguration is the only one in which mixing from one

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gluon exchange and instanton forces allows a light andmetastable triquark. On the contrary, the PV multiplet hasS � 1, a configuration in which mixing cannot occur,resulting in heavier and possibly unstable triquarks. Thismight account for the experimental elusiveness of the PVsupermultiplet.

Earlier we noted that the diquark-diquark picture canonly have S � 1, J � 0�, 1�, 2�. In the Appendix weshow that for the diquark-diquark configuration the super-multiplet label . is given by . � ��1�l, so that such asystem can exist in only one supermultiplet (. � �1 in theP wave). Because of angular momentum conservation inthe decays of the 0� and 2�, this supermultiplet can onlybe PV. Thus, provided it is valid to treat diquarks aseffective bosons, the 1�� 1�1400� ! � and1�1600� ! �0 cannot be in the diquark-diquarkarrangement.

VII. CONCLUSION

In conclusion, the confirmation of a narrow ��1540�and the absence of other narrow members of a 10 can beexplained by correlations that suggest there should be a #�

with canonical width & 100 MeV together with a familyof broad partners. This particular dynamics is exceptionalas received wisdom has been that all these states should‘‘fall apart.’’ If the �� is an artifact, or if its narrow widthis due to some mechanism other than the mixing amongcorrelations as discussed here, then the 1�� tetraquarkmesons will be all broad as in [5,10]. In any event, if theexistence of a resonant �� with narrow width survivesfurther scrutiny, then a 10 of mesons with moderate widths,of which the # may have a canonical width, meritsinvestigation.

Models which consider four-quark mesons produce aconsiderable multiplicity of flavor and spin states. Usinga triquark-quark correlation for tetraquarks there can be anexception to this rule, with only a reduced set of statesappearing and the possibility that the states with JP � 1�

in the 10 and 10 may be observable. It seems possible toassociate certain otherwise peculiar states in the mesonspectrum with those predicted here.

This model is trivially falsifiable by comparison of itsprediction of flavor exotic states with experiment. We havenoted that while the �1700�, !�1650�, and ��1680� forma candidate nonet, their masses are somewhat unnatural.The �1450� and !�1420� are missing a partner to com-plete the set and determine whether they are in a nonet or10. The K��1410� appears to be anomalously low in massfor q �q systems but fits naturally into the 10 configuration.The widths of most of these states are also consistent withthis picture.

The general feature is that for a given JP�C� six non-strange uncharged members are expected within a fewhundred MeV. If any of the plethora of observed states

014017

[9] is associated with these, such that their widths of�300 MeV give a scale for their (in)stability, then a #�

with a canonical width seems an unavoidable consequence.

ACKNOWLEDGMENTS

This work is supported, in part, by grants from theParticle Physics and Astronomy Research Council, theEU-TMR program ‘‘Euridice’’ HPRN-CT-2002-00311, aClarendon Fund Bursary, and by the U.S. Department ofEnergy under Contract No. DE-AC05-84ER40150. Wethank H. J. Lipkin for comments about clusterdecomposition.

APPENDIX

To obtain charge conjugation eigenstates for tetraquarkmesons we need to write their wave functions in qq �q �q and�q �q qq form. We demonstrate the procedure for the non-strange neutral members of the 10 and 10, noting that thesame analyses can be applied to the 8a and 8b and that wecan easily generalize to the I3 � �1 members with theusual G parity operator.

From the wave functions given earlier , we find

010 � 6 � 3 � ��fdsg� �d �s � fsug��s �u � fudg� �u �d�=

��3

p;

(A1)

~ 010 � 3 � 6 � �� d �sfdsg � � �s �ufsug � � �u �dfudg�=

��3

p;

(A2)

010

� �3 � �6 � ��dsf �d �sg � �suf�s �ug � �udf �u �dg�=��3

p;

(A3)

~ 010

� �6 � �3 � ��f �d �sg�ds � f �s �ug�su � f �u �dg�ud�=��3

p:

(A4)

Notice C010 � ~0

10and C0

10� ~0

10, so if we denote the

10 and 10 wave functions

/�10; .� � 010 � . ~0

10; /�10; .� � 010� . ~0

10;

(A5)

then our . is precisely that defined by Eq. (2) in [13]:

C/�10; .� � ./�10; .�; C/�10; .� � ./�10; .�:(A6)

We have the freedom to choose the phase between the 10and 10 , which we denote by a, so that the full wavefunctions of a four-quark 10 10 have 2 degrees of free-dom, a and . ,

0�.; a� � 010 � . ~0

10 � a�010� . ~0

10�: (A7)

We see that C0�.; a� � .a0�.; a� and the doubling ofstates follows: in the . � �1 multiplet we have a 1��

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�.a � �� and 1�� .a � ��, and in the . � �1multiplet we have a 1�� .a � �� and 1�� .a ��. Looking at the qq �q �q part of the wave functiononly, we see that under interchange of quarks 2 $ 3 linearcombinations of 0

10 and 010

correspond to the flavor wave

functions of [13] in q �qq �q form: the 1�� state is 010 �

010

�0�8, and the 1�� state is 010 � 0

10��

13

q�� K0K0 � �K�K��; our � is defined as

�u �d [20], hence the phase difference compared to [13].Likewise for linear combinations of the wave functions 8aand 8b.

Let us first look at the diquark-diquark correlation,treating the diquarks as effective bosons. Denoting fqqgby b and �qq by B, in shorthand notation we can write0

10 � b �B, ~010 �

�Bb, 010

� B �b, ~010

� �bB, so that thefull wave function can be expressed

0�.; a� � b1 �B2 � . �B1b2 � a�B1�b2 � . �b1B2�; (A8)

where the labels 1 and 2 denote the spin and space degreesof freedom of the bosons. C0�.; a� � .a0�.; a� in thisshorthand notation, as expected. Interchanging the spaceand spin labels in the wave function brings a factor ��1�l

0�.; a� � ��1�l�b2 �B1 � . �B2b1 � a�B2�b1 � . �b2B1�;

(A9)

and since bosons are commuting variables (�b1; �B2 � 0),

0�.; a� � .��1�l0�.; a�; (A10)

so in the diquark-diquark picture . � ��1�l.In the triquark-antiquark configuration there are two

spin 1=2 fermions in L � 1. Denoting the triquarks fABg,

014017

fCAg, and fBCg by D, U, and S,

010 � ��u �U� d �D� s �S�=

��3

p� q �Q (A11)

and analogously for Eqs. (A2)–(A4). The full wave func-tion is

0�.; a� � q1 �Q2 � . �Q1q2 � a�Q1 �q2 � . �q1Q2�; (A12)

� ��1�s�1��1�l�q2 �Q1 � . �Q2q1 � a�Q2 �q1 � . �q2Q1�

(A13)

� ��1�l�s.01�.; a�: (A14)

Thus in the triquark-antiquark picture . � ��1�l�s.In the work of Chung et al. [13], the quantum number .

distinguishes the two supermultiplets in the 10 10 sys-tem, being labeled by . � �1 (PP) and . � �1 (PV), orvice versa (the overall phase of . is not important). At firstglance, it appears as though the total spin S of the quarksdistinguishes the two supermultiplets: the S � 0 statesbeing PP and the S � 1 states being PV, but some cautionis needed. By angular momentum conservation for P-wavedecays, 0�, 2� ! PV but not ! PP, so we can immediatelyassign the 0� and 2� states to the PV multiplet. Since wecan have both 1� ! PV and ! PP in the P wave, there isnothing, a priori, that tells us to which supermultiplet a 1�

must belong. However, we have shown that for a P-wavetriquark-antiquark system the supermultiplet label . isgiven by . � ��1�s�1. We can confirm, then, that the totalspin S of the quarks determines precisely to which super-multiplet a state belongs. The states with S � 0 belong tothe PP supermultiplet (. � �1), and those with S � 1 tothe PV supermultiplet (. � �1).

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