Meson-Meson molecules and compact four-quark states

0.07 MB·English

by  J. Vijande

Meson-Meson molecules and compact four-quark states
J. Vijande∗ and A. Valcarce†
∗DepartamentodeFísicaAtómica,MolecularyNuclear,UniversidaddeValencia(UV)andIFIC(UV-CSIC),
Valencia,Spain.
†DepartamentodeFísicaFundamental,UniversidaddeSalamanca,Salamanca,Spain.
0
1
0 Abstract. Thephysics of charm hasbecome one of thebest laboratories exposing thelimitationsof thenaive constituent
2 quarkmodelandalsogivinghintsintoamorematuredescriptionofmesonspectroscopy,beyondthesimplequark–antiquark
configurations. In this talk we review some recent studies of multiquark components in the charm sector and discuss in
n
particularexoticandnon-exoticfour-quarksystems.
a
J Keywords: Hadonspectra,exoticstates
PACS: 14.40.Nd,14.40.Lb,14.40.-n
7
2

Morethanthirtyyearsaftertheso-calledNovemberrevolution[1],heavyhadronspectroscopyremainsachallenge.
]
h Theformerlycomfortableworldofheavymesonsisshakenbynewresults[2].Thisstartedin2003withthediscovery
p of the D∗ (2317) and D (2460) mesons in the open-charmsector. These positive-parity states have masses lighter
s0 s1
- thanexpectedfromquarkmodels,andalsosmallerwidths.Outofthemanyproposedexplanations,theunquenching
p
e ofthenaivequarkmodelhasbeensuccessful[3].Whena(qq¯)pairoccursinaP-wavebutcancoupletohadronpairs
h inS-wave,thelatterconfigurationdistortsthe(qq¯)picture.Therefore,the0+ and1+ (cs¯)statespredictedabovethe
[ DK(D∗K) thresholdscoupleto the continuum.Thismixesmeson–mesoncomponentsin the wave function,an idea

advocatedlongagotoexplainthespectrumandpropertiesoflight-scalarmesons[4].
1
v Thispossibilityof(cs¯nn¯)(nstandsforalightquark)componentsinD∗s hasopenthediscussionaboutthepresence
6 ofcompact(cc¯nn¯)four-quarkstatesinthecharmoniumspectroscopy.Somestatesrecentlyfoundinthehidden-charm
7 sector mayfit in the simple quark-modeldescriptionas (cc¯) pairs(e.g.,X(3940),Y(3940),andZ(3940)asradially
8 excited c , c , and c ), but others appear to be more elusive, in particular X(3872), Z(4430)+, and Y(4260).
c0 c1 c2
4
The debate on the nature of these states is open, with special emphasis on the X(3872). Since it was first reported
.
1 by Belle in 2003 [5], it has gradually become the flagship of the new armada of states whose properties make
0 their identification as traditional (cc¯) states unlikely. An average mass of 3871.2±0.5MeV and a narrow width
0 of less than 2.3MeV have been reported for the X(3872). Note the vicinity of this state to the D0D∗0 threshold,
1
M(D0D∗0) = 3871.2±1.2 MeV. With respect to the X(3872) quantum numbers, although some caution is still
:
v requireduntilbetterstatistic isobtained[6],anisoscalarJPC =1++ state seemstobethebestcandidatetodescribe
Xi thepropertiesoftheX(3872).
Anotherhotsector,atleastfortheorists,includesthe(ccn¯n¯)states,whicharemanifestlyexoticwithcharm2and
r
a baryonnumber0.Shouldtheyliebelowthethresholdfordissociationintotwoordinaryhadrons,theywouldbenarrow
andshowupclearlyintheexperimentalspectrum.Therearealreadyestimatesoftheproductionratesindicatingthey
couldbeproducedanddetectedatpresent(andfuture)experimentalfacilities[7].Thestabilityofsuch(QQq¯q¯)states
has been discussed since the early 80s [8], and there is a consensus that stability is reached when the mass ratio
M(Q)/m(q) becomes large enough. See, e.g., [9] for Refs. This effect is also found in QCD sum rules [10]. This
improvedbindingwhenM/mincreasesisduetothesamemechanismbywhichthehydrogenmolecule(p,p,e−,e−)
ismuchmoreboundthanthepositroniummolecule(e+,e+,e−,e−).WhatmattersisnottheCoulombcharacterofthe
potential,butitspropertytoremainidenticalwhenthemasseschange.Inquarkphysics,thispropertyisnamedflavour
independence.Itisreasonablywellsatisfied,withdeparturesmainlyduetospin-dependentcorrections.
Thequestioniswhetherstabilityisalreadypossiblefor(ccn¯n¯)orrequiresheavierquarks.InRef.[9], amarginal
bindingwasfoundforaspecificpotentialforwhichearlierstudiesfoundnobinding.Thisillustrateshowdifficultare
suchfour-bodycalculations.
Besidestryingtounravelthepossibleexistenceofbound(ccn¯n¯)and(cc¯nn¯)statesoneshouldaspiretounderstand
whether it is possible to differentiate between compact and molecular states. A molecular state may be understood
asa four-quarkstate containinga singlephysicaltwo-mesoncomponent,i.e., a uniquesinglet-singletcomponentin
the colour wave function with well-defined spin and isospin quantum numbers. One could expect these states not
beingdeeplyboundandthereforehavingasizeoftheorderofthetwo-mesonsystem,i.e.,D ∼1.Oppositetothat,
R
a compact state may be characterized by its involved structure on the colour space, its wave function containing
differentsinglet-singletcomponentswithnonnegligibleprobabilities.Onewouldexpectsuchstateswouldbesmaller
thantypicaltwo-mesonsystems,i.e.,D <1.LetusnoticethatwhileD >1butfinitewouldcorrespondtoameson-
R R
mesonmoleculeD K−→→¥ ¥ wouldrepresentanunboundthreshold.Thus,dealingwithfour-quarkstatesanimportant
R
questioniswhetherweareinfrontofacolorlessmeson-mesonmoleculeoracompactstate,i.e.,asystemwithtwo-
bodycoloredcomponents.Whilethefirststructurewouldbenaturalinthenaivequarkmodel,thesecondonewould
openanewareaonthehadronspectroscopy.
Therearethreedifferentwaysofcouplingtwoquarksandtwoantiquarksintoacolorlessstate:
[(q q )(q¯ q¯ )] ≡ {|3¯ 3 i,|6 6¯ i}≡{|3¯3i12,|66¯i12} (1)
1 2 3 4 12 34 12 34 c c
[(q q¯ )(q q¯ )] ≡ {|1 1 i,|8 8 i}≡{|11i ,|88i } (2)
1 3 2 4 13 24 13 24 c c
[(q q¯ )(q q¯ )] ≡ {|1 1 i,|8 8 i}≡{|1′1′i ,|8′8′i }, (3)
1 4 2 3 14 23 14 23 c c
beingthethreeofthemorthonormalbasis.Eachcouplingschemeallowstodefineacolorbasiswherethefour–quark
problemcan be solved.Thefirst basis, Eq. (1),beingthe mostsuitable one to dealwith the Pauliprincipleis made
entirelyofvectorscontaininghidden–colorcomponents.Theothertwo,Eqs.(2)and(3),arehybridbasiscontaining
singlet–singlet(physical)andoctet–octet(hidden–color)vectors.
Toevaluatetheprobabilityofphysicalchannels(singlet-singletcolorstates)oneneedstoexpandanyhidden-color
vectorofthefour-quarkstatecolorbasisintermsofsinglet-singletcolorvectors.Givenageneralfour-quarkstatethis
requirestomixtermsfromtwodifferentcouplings,2and3.In [11]thetwoHermitianoperatorsthatarewell-defined
projectorsonthetwophysicalsinglet-singletcolorstateswerederived,
1
P|11ic = (cid:0)PQˆ+QˆP(cid:1)2(1−|ch11|1′1′ic|2)
1
P|1′1′ic = (cid:0)PˆQ+QPˆ(cid:1)2(1−|ch11|1′1′ic|2), (4)
whereP,Q,Pˆ,andQˆ aretheprojectorsoverthebasisvectors(2)and(3),
P = |11i h11|
cc
Q = |88i h88| , (5)
cc
and
Pˆ = |1′1′(cid:11)cc(cid:10)1′1′|
Qˆ = |8′8′(cid:11)cc(cid:10)8′8′| . (6)
Byusingthemandtheformalismof [11],thefour-quarknature(unbound,molecularorcompact)canbeexplored.
Such a formalism can be applied to any four-quark state, however, it becomes much simpler when distinguishable
quarksare present. This would be, for example,the case of the (nQn¯Q¯) system, where the Pauli principle does not
apply.Inthissystemthebases(2)and(3)aredistinguishableduetotheflavorpart,theycorrespondto[(nc¯)(cn¯)]and
[(nn¯)(cc¯)],andthereforetheyareorthogonal.Thismakesthattheprobabilityofaphysicalchannelcanbeevaluated
intheusualwayfororthogonalbasis[12].Thenon-orthogonalbasesformalismisrequiredforthosecaseswherethe
Pauli Principle applies either for the quarksor the antiquarks pairs. Relevant expressions can be found in [11]. We
show in Table 1 some examples of results obtained for heavy-lighttetraquarks. One can see how independently of
theirbindingenergy,allofthempresentasizableoctet-octetcomponentwhenthewavefunctionisexpressedinthe2
coupling.Letusfirstofallconcentrateonthetwounboundstates,D >0,onewithS=0andonewithS=1,given
E
inTable1.Theoctet-octetcomponentofbasis(2)canbeexpandedintermsofthevectorsofbasis(3)asexplainedin
theprevioussection.Then,theprobabilitiesareconcentratedintoasinglephysicalchannel,MMorMM∗[MMstands
fortwoidenticalpseudoscalarD(B)mesonsandMM∗forapseudoscalarD(B)mesontogetherwithitscorresponding
vector excitation, D∗ (B∗)]. In other words, the octet-octet component of the basis (2) or (3) is a consequence of
havingidenticalquarksandantiquarks.Thus,four-quarkunboundstatesarerepresentedbytwoisolatedmesons.This
1
0.8
M
M0.6
P
0.4
0.2
-180 -150 -120 -90 -60 -30 0
D E (MeV)
FIGURE1. PMMasafunctionofD E.
conclusionis strengthenedwhenstudyingtherootmeansquareradii,leadingtoa picturewherethetwoquarksand
thetwoantiquarksarefaraway,hx2i1/2≫1fmandhy2i1/2≫1fm,whereasthequark-antiquarkpairsarelocatedat
atypicaldistanceforameson,hz2i1/2≤1fm.LetusnowturntotheboundstatesshowninTable1,D <0,onein
E
thecharmsectorandtwointhebottomone.Incontrasttotheresultsobtainedforunboundstates,whentheoctet-octet
componentofbasis(2)isexpandedintermsofthevectorsofbasis(3),oneobtainsapicturewheretheprobabilities
inallallowedphysicalchannelsarerelevant.Itisclearthattheboundstate mustbegeneratedbyaninteractionthat
itisnotpresentintheasymptoticchannel,sequesteringprobabilityfromasinglesinglet-singletcolorvectorfromthe
interactionbetweencoloroctets.Suchsystemsareclearexamplesofcompactfour-quarkstates,inotherwords,they
cannotbeexpressedintermsofasinglephysicalchannel.
Wehavestudiedthedependenceoftheprobabilityofaphysicalchannelonthebindingenergy.Forthispurposewe
haveconsideredthesimplestsystemfromthenumericalpointofview,the(S,I)=(0,1)ccn¯n¯state.Unfortunately,this
stateisunboundforanyreasonablesetofparameters.Therefore,webinditbymultiplyingtheinteractionbetweenthe
lightquarksbyafudgefactor.Suchamodificationdoesnotaffectthetwo-mesonthresholdwhileitdecreasesthemass
ofthefour-quarkstate.TheresultsareillustratedinFigure1,showinghowintheD →0limit,thefour-quarkwave
E
functionisalmostapuresinglephysicalchannel.Closetothislimitonewouldfindwhatcouldbedefinedasmolecular
states.Whentheprobabilityconcentratesintoasinglephysicalchannel(P →1)thesystemgetslargerthantwo
M1M2
isolated mesons [11]. One can identify the subsystems responsible for increasing the size of the four-quark state.
Quark-quark (hx2i1/2) and antiquark-antiquark(hy2i1/2) distances grow rapidly while the quark-antiquark distance
(hz2i1/2)remainsalmostconstant.Thisreinforcesourpreviousresult,pointingtothe appearanceof two-meson-like
structureswheneverthebindingenergygoestozero.
Inanotherrecentinvestigation,thefour-bodySchrödingerequationhasbeensolvedaccuratelyusingthehyperspher-
ical harmonic(HH) formalism[12], with two standardquarkmodelscontaininga linear confinementsupplemented
byaFermi–Breitone-gluonexchangeinteraction(BCN),andalsobosonexchangesbetweenthelightquarks(CQC).
The modelparameterswere tunedin the meson and baryonspectra. The results are givenin Table 2, indicatingthe
quantumnumbersof the state studied, the maximumvalue of the grand angularmomentumused in the HH expan-
sion,K ,andtheenergydifferencebetweenthemassofthefour-quarkstate,E ,andthatofthelowesttwo-meson
max 4q
thresholdcalculatedwiththesamepotentialmodel,D .Forthe(ccn¯n¯)systemwehavealsocalculatedtheradiusof
E
thefour-quarkstate,R ,anditsratiotothesumoftheradiiofthelowesttwo-mesonthreshold,D .
4q R
As can be seen in Table 2 (left), in the case of the (cc¯nn¯) there appear no compact bound states for any set of
quantumnumbers,includingthesuggestedassignmentfortheX(3872).Independentlyofthequark–quarkinteraction
TABLE1. Heavy-lightfour-quarkstatepropertiesforselected
quantum numbers. All states have positive parity and total or-
bitalangular momentumL=0.EnergiesaregiveninMeV.The
notation M1M2 |ℓ stands for mesons M1 and M2 with a rela-
tive orbital angular momentum ℓ. P[|3¯3i12(|66¯i12)] stands for
c c
the probability of the 33¯(6¯6) components given in Equation (1)
andP[|11i (|88i )]forthe11(88)components giveninEqua-
c c
tion(2).PMM,PMM∗,andPM∗M∗ havebeen calculatedfollowing
theformalismof[11],andtheyrepresenttheprobabilityoffind-
ingtwo-pseudoscalar(PMM),apseudoscalarandavector(PMM∗)
ortwovector(PM∗M∗)mesons
(S,I) (0,1) (1,1) (1,0) (1,0) (0,0)
Flavor ccn¯n¯ ccn¯n¯ ccn¯n¯ bbn¯n¯ bbn¯n¯
Energy 3877 3952 3861 10395 10948
Threshold DD|S DD∗|S DD∗|S BB∗|S B1B|P
D E +5 +15 −76 −217 −153
P[|3¯3i12] 0.333 0.333 0.881 0.974 0.981
. c
P[|66¯i12] 0.667 0.667 0.119 0.026 0.019
c
P[|11i ] 0.556 0.556 0.374 0.342 0.340
c
P[|88i ] 0.444 0.444 0.626 0.658 0.660
c
PMM 1.000 − − − 0.254
PMM∗ − 1.000 0.505 0.531 −
PM∗M∗ 0.000 0.000 0.495 0.469 0.746
TABLE2. (cc¯nn¯)and(ccn¯n¯)results.
(cc¯nn¯) (ccn¯n¯)
CQC BCN CQC
JPC(Kmax) E4q D E E4q D E JP(Kmax) E4q D E R4q D R
0++(24) 3779 +34 3249 +75 0+(28) 4441 +15 0.624 >1
0+−(22) 4224 +64 3778 +140 1+(24) 3861 −76 0.367 0.808
1++(20) 3786 +41 3808 +153 I=0 2+(30) 4526 +27 0.987 >1
1+−(22) 3728 +45 3319 +86 0−(21) 3996 +59 0.739 >1
2++(26) 3774 +29 3897 +23 1−(21) 3938 +66 0.726 >1
2+−(28) 4214 +54 4328 +32 2−(21) 4052 +50 0.817 >1
1−+(19) 3829 +84 3331 +157 0+(28) 3905 +50 0.817 >1
1−−(19) 3969 +97 3732 +94 1+(24) 3972 +33 0.752 >1
0−+(17) 3839 +94 3760 +105 I=1 2+(30) 4025 +22 0.879 >1
0−−(17) 3791 +108 3405 +172 0−(21) 4004 +67 0.814 >1
2−+(21) 3820 +75 3929 +55 1−(21) 4427 +1 0.516 0.876
2−−(21) 4054 +52 4092 +52 2−(21) 4461 −38 0.465 0.766
andthequantumnumbersconsidered,thesystemevolvestoawellseparatedtwo-mesonstate.Thisisclearlyseenin
the energy,approachingthe thresholdmade of two free mesons, and also in the probabilitiesof the differentcolour
componentsof the wave function and in the radius [12]. Thus, in any manner one can claim for the existence of a
compactboundstateforthe(cc¯nn¯)system.
AcompletelydifferentbehaviourisobservedinTable2(right).Here,therearesomeparticularquantumnumbers
where the energy is quickly stabilized below the theoretical threshold. Of particular interest is the 1+ ccn¯n¯ state,
whoseexistencewaspredictedmorethantwentyyearsago[13].Thereisaremarkableagreementontheexistenceof
anisoscalarJP=1+ ccn¯n¯boundstateusingbothBCNandCQCmodels,ifnotinitsproperties.FortheCQCmodel
thepredictedbindingenergyislarge,−76MeV,D <1,andaveryinvolvedstructureofitswavefunction(theDD∗
R
componentofitswavefunctiononlyaccountsforthe50%ofthetotalprobability)whatwouldfitintocompactstate.
Opposite to that, the BCN modelpredictsa rather small binding,−7 MeV, and D is largerthan 1, althoughfinite.
R
Thisstatewouldnaturallycorrespondtoameson-mesonmolecule.
ConcerningtheothertwostatesthatarebelowthresholdinTable2(right)amorecarefulanalysisisrequired.Two-
mesonthresholdsmustbedeterminedassumingquantumnumberconservationwithinexactlythesameschemeusedin
thefour–quarkcalculation.Dealingwithstronglyinteractingparticles,thetwo-mesonstatesshouldhavewelldefined
total angular momentum, parity, and a properly symmetrized wave function if two identical mesons are considered
(coupledscheme). When noncentralforcesare nottaken into account,orbitalangularmomentumand totalspin are
alsogoodquantumnumbers(uncoupledscheme).Wewouldliketoemphasizethatalthoughweusecentralforcesin
ourcalculationthe coupledscheme is the relevantonefor observations,since a small non-centralcomponentin the
potentialisenoughtoproduceasizeableeffectonthewidthofastate.Thesestatearebelowthethresholdsgivenby
theuncoupledschemebutabovetheonesgivenwithinthecoupledschemewhatdiscardthesequantumnumbersas
promisingcandidatesforbeingobservedexperimentally.
BindingincreasesforlargerM/m,butinthe(bbn¯n¯)sector,thereisnoproliferationofboundstates.Wehavestudied
allgroundstatesof(bbn¯n¯)usingthesameinteractingpotentialsasinthedouble-charmcase.Onlyfourboundstates
havebeenfound,withquantumnumbersJP(I)=1+(0),0+(0),3−(1),and1−(0).Thefirstthreeonescorrespondto
compactstates.
To conclude, let us stress again the important difference between the two physical systems which have been
considered. While for the (cc¯nn¯), there are two allowed physical decay channels, (cc¯)+(nn¯) and (cn¯)+(c¯n), for
the(ccn¯n¯)onlyonephysicalsystemcontainsthepossiblefinalstates,(cn¯)+(cn¯).Therefore,a(cc¯nn¯)four-quarkstate
willhardlypresentboundstates,becausethesystemwillreorderitselftobecomethelightesttwo-mesonstate,either
(cc¯)+(nn¯)or(cn¯)+(c¯n).Inotherwords,iftheattractionisprovidedbytheinteractionbetweenparticlesiand j,it
doesalsocontributetotheasymptotictwo-mesonstate.Thisdoesnothappenforthe(ccn¯n¯)iftheinteractionbetween,
forexample,thetwoquarksisstronglyattractive.Inthiscasethereisnoasymptotictwo-mesonstateincludingsuch
attraction,andthereforethesystemmightbind.
Onceallpossible(ccn¯n¯),(bbn¯n¯)and(cc¯nn¯)quantumnumbershavebeenexhaustedveryfewalternativesremain.
Ifadditionalboundfour-quarkstatesorhigherconfigurationareexperimentallyfound,thenothermechanismsshould
beatwork,forinstancebasedondiquarks[4,14,15].
This work has been partially funded by the Spanish Ministerio de Educación y Ciencia and EU FEDER under
Contract No. FPA2007-65748, by Junta de Castilla y León under Contract No. SA016A17, and by the Spanish
Consolider-Ingenio2010ProgramCPAN(CSD2007-00042).
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