We study charmonium ($c\bar c n\bar n$), bottomonium ($b\bar b n\bar n$) and
exotic ($cc\bar n\bar n$ and $bb\bar n\bar n$) four-quark states by means of a
standard non-relativistic quark potential model. We look for possible bound
states. Among them we are able to distinguish between meson-meson molecules and
compact four-quark states.Comment: 4 pages, 1 figure, invited parallel talk at the 5th International
  Conference on Quarks and Nuclear Physics (QNP09), Beijing, September 21-26,
  2009.To be published in Chinese Physics 

A Valcarce

J Vijande

Vijande J

Chinese Physics C

English

arXiv

We study charmonium (c (c) over bar cn (n) over bar), bottomonium (b (b) over barn (n) over bar) and exotic (ccnn and bbnn) four-quark states by means of a standard non-relativistic quark potential model We look for possible bound states Among them we are able to distinguish between meson-meson molecules and compact four-quark state

Vijande Asenjo, Javier

Valcarce Mejía, Alfredo

Repositori d'Objectes Digitals per a l'Ensenyament la Recerca i la Cultura

arXiv:0912.3080v1  [hep-ph]  16 Dec 2009CPC(HEP & NP), 2009, 33(X): 1—4 Chinese Physics C Vol. 33, No. X, Xxx, 2009Molecular and compact four-quark states *A. Valcarce1;1) J. Vijande1;2)1 Departamento de F´ısica Fundamental, Universidad de Salamanca,Salamanca, Spain2 Departamento de F´ısica Ato´mica, Molecular y Nuclear, Universidad de Valencia (UV) and IFIC (UV-CSIC), Valencia, SpainAbstract We study charmonium (cc¯nn¯), bottomonium (bb¯nn¯) and exotic (ccn¯n¯ and bbn¯n¯) four-quark statesby means of a standard non-relativistic quark potential model. We look for possible bound states. Amongthem we are able to distinguish between meson-meson molecules and compact four-quark states.Key words tetraquarks, quark-models, charmonium, exoticsPACS 14.40.Nd,14.40.Lb,14.40.-n1 IntroductionThe discoveries on several fronts of unusual char-monium states like X(3872) and Y (4260) and open-charmmesons with unexpected masses likeD∗sJ(2317)and D∗0(2308), have re-invigorated the study of thehadron spectra. Their anomalous nature has trig-gered several interpretations, among them, the exis-tence of compact four-quark states or meson-mesonmolecules. This challenging situation resembles thelong-standing problem of the light-scalar mesons,where it has been suggested that some resonancesmay not be ordinary qq¯ states, though there is lit-tle agreement on what they actually are [1]. In thiscase, four-quark states have been justified to coexistwith qq¯ states because they can couple to JPC =0++without orbital excitation [2].Four-quark systems present a richer color struc-ture than standard baryons or mesons. Although thecolor wave function for ordinary mesons and baryonsleads to a single vector, working with four–quarkstates there are different vectors driving to a singletcolor state out of colorless or colored quark-antiquarktwo-body components. Thus, in dealing with four–quark states an important question is whether weare in front of a colorless meson–meson molecule ora compact state (i.e., a system with two-body col-ored components). Whereas the first structure wouldbe natural in the naive quark model, the second onewould open a new area in hadron spectroscopy.We have derived the necessary formalism to eval-uate the physical channels probability(singlet–singletcolor states) in an arbitrary four–quark wave func-tion. For this purpose we have expanded any hidden–color vector of the four–quark state color basis, i.e.,vectors with non–singlet internal color couplings, interms of singlet–singlet color vectors. Such a proce-dure gives rise to a wave function expanded in termsof color singlet-singlet nonorthogonal vectors, wherethe determination of the probability of physical chan-nels becomes cumbersome. However, it allows to dif-ferentiate among unbound, compact and molecularfour-quark states.2 FormalismThere are three different ways of coupling twoquarks and two antiquarks into a colorless state:[(q1q2)(q¯3 q¯4)] ≡ {|3¯12334〉, |6126¯34〉}≡ {|3¯3〉12c , |66¯〉12c } (1a)[(q1q¯3)(q2 q¯4)] ≡ {|113124〉, |813824〉}≡ {|11〉c, |88〉c} (1b)[(q1q¯4)(q2 q¯3)] ≡ {|114123〉, |814823〉}≡ {|1′1′〉c, |8′8′〉c} , (1c)Received 21 October 2009∗ This work has been partially funded by Ministerio de Ciencia y Tecnolog´ıa under Contract No. FPA2007-65748, by EUFEDER, and by Junta de Castilla y Leo´n under Contracts No. SA016A17 and GR121)E-mail: valcarce@usal.es2)E-mail: javier.vijande@uv.esc©2009Chinese Physical Society and the Institute of High Energy Physics of the ChineseAcademy of Sciences and the Instituteof Modern Physics of the Chinese Academy of Sciences and IOP Publishing LtdNo. X A. Valcarce et al: Molecular and compact four-quark states 2each forms an orthonormal basis. Each couplingscheme allows us to define a color basis where thefour–quark problem can be solved. The first basis,Eq. (1a), being the most suitable one to deal with thePauli principle is made entirely of vectors containinghidden–color components. The other two, Eqs. (1b)and (1c), are hybrid bases containing singlet–singlet(physical) and octet–octet (hidden–color) vectors.An arbitrary four-quark wave function can beexpressed in terms of physical components, singlet-singlet color states, by means of an infinite expansionthan can be resumed as [3],|Ψ〉 =12(PQˆ+QˆP) 11−cos2α|Ψ〉+12(PˆQ+QPˆ) 11−cos2α|Ψ〉 . (2)Thus, one obtains two hermitian operators that arewell–defined projectors on the two physical singlet–singlet color statesP |11〉c=(PQˆ+QˆP) 12(1−cos2α)P |1′1′〉c=(PˆQ+QPˆ) 12(1−cos2α). (3)Thus, given an arbitrary state |Ψ〉 its projection ona particular subspace E is given by |Ψ〉|E = PE |Ψ〉.Then, the probability of finding such an state on thissubspace isE |〈Ψ|Ψ〉|E = 〈Ψ|P†EPE |Ψ〉= 〈Ψ|P2E|Ψ〉= 〈Ψ|PE|Ψ〉 .(4)Therefore, once the projection operators have beenconstructed [Eq. (3)], the probabilities for findingsinglet–singlet components are given by,P |Ψ〉([11]) =〈Ψ | P |11〉c|Ψ〉P |Ψ〉([1′1′]) =〈Ψ | P |1′1′〉c|Ψ〉. (5)Using Eq. (3) it can be easily checked that P |Ψ〉([11])+P |Ψ〉([1′1′])= 1, whereP |Ψ〉([11]) =12(1−cos2α)[〈Ψ |PQˆ |Ψ〉+〈Ψ | QˆP |Ψ〉]P |Ψ〉([1′1′]) =12(1−cos2α)[〈Ψ | PˆQ |Ψ〉+〈Ψ |QPˆ |Ψ〉]. (6)3 ResultsThe stability of a four–quark state can be ana-lyzed in terms of ∆E, the energy difference betweenits mass and that of the lowest two-meson threshold,∆E =E4q−E(M1,M2) , (7)where E4q stands for the four–quark energy andE(M1,M2) for the energy of the two-meson threshold.Thus, ∆E < 0 indicates all fall-apart decays are for-bidden, and therefore one has a proper bound state.∆E ≥ 0 will indicate that the four–quark solution cor-responds to an unbound threshold (two free mesons).Thus, an energy above the threshold would simplymean that the system is unbound within our varia-tional approximation, suggesting that the minimumof the Hamiltonian is at the two-meson threshold.Another helpful tool analyzing the structure of afour–quark state is the value of the root mean squareradii: 〈x2〉1/2, 〈y2〉1/2, and 〈z2〉1/2. Bound four–quarkstates can be distinguished from two free mesons bymeans of their root mean square radiusRMS4q(2q) =(∑4(2)i=1mi〈(ri−R)2〉∑4(2)i=1 mi)1/2, (8)and in particular, their corresponding ratio,∆R=RMS4qRMSM1 +RMSM2, (9)where RMSM1+RMSM2 stands for the sum of the radiiof the mesons corresponding to the lowest threshold.We show in Table 1 the results obtained for severaldifferent four–quark states in the bottom and charmsectors. One can see how independently of their bind-ing energy, all of them present a sizable octet-octetcomponent when the wave function is expressed inthe (1b) coupling. Let us first of all concentrate onthe two unbound states, ∆E > 0, one with ST =0 andone with ST = 1, given in Table 1. The octet-octetcomponent of basis (1b) can be expanded in terms ofthe vectors of basis (1c) as explained in the previoussection. Thus, once expressions (6) are consideredone finds that the probabilities are concentrated intoa single physical channel, MM or MM∗. In otherNo. X A. Valcarce et al: Molecular and compact four-quark states 3words, the octet-octet component of the basis (1b)or (1c) is a consequence of having identical quarksand antiquarks. Thus, four-quark unbound states arerepresented by two isolated mesons. This conclusionis strengthened when studying the root mean squareradii, leading to a picture where the two quarks andthe two antiquarks are far away, 〈x2〉1/2  1 fm and〈y2〉1/2 1 fm, whereas the quark-antiquark pairs arelocated at a typical distance for a meson, 〈z2〉1/2 ≤ 1fm.Table 1. Four–quark state properties for selected quantum numbers. All states have positive parity and totalorbital angular momentum L= 0. Energies are given in MeV and distances in fm. The notation M1M2 |`stands for mesons M1 and M2 with a relative orbital angular momentum `. P [|3¯3〉12c (|66¯〉12c )] stands for theprobability of the 33¯(6¯6) components given in Eq. (1a) and P [ | 11〉c ( | 88〉c)] for the 11(88) components givenin Eq. (1b). PMM , PMM∗ , and PM∗M∗ stand for the probability of two pseudoscalar, pseudoscalar-vector ortwo vector mesons.(ST , 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 10948Threshold DD |S DD∗ |S DD∗ |S BB∗ |S B1B |P∆E +5 +15 −76 −217 −153P [|3¯3〉12c ] 0.333 0.333 0.881 0.974 0.981P [|66¯〉12c ] 0.667 0.667 0.119 0.026 0.019P [ | 11〉c] 0.556 0.556 0.374 0.342 0.340P [ | 88〉c] 0.444 0.444 0.626 0.658 0.660PMM 1.000 − − − 0.254PMM∗ − 1.000 0.505 0.531 −PM∗M∗ 0.000 0.000 0.495 0.469 0.746〈x2〉1/2 60.988 13.804 0.787 0.684 0.740〈y2〉1/2 60.988 13.687 0.590 0.336 0.542〈z2〉1/2 0.433 0.617 0.515 0.503 0.763RMS4q 30.492 6.856 0.363 0.217 0.330∆R 69.300 11.640 0.799 0.700 0.885Let us now turn to the bound states shown in Ta-ble 1, ∆E < 0, one in the charm sector and two inthe bottom one. In contrast to the results obtainedfor unbound states, when the octet-octet componentof basis (1b) is expanded in terms of the vectors ofbasis (1c), equations (6) indicate a picture where theprobabilities in all allowed physical channels are rel-evant. It is clear that the bound state must be gen-erated by an interaction that it is not present in theasymptotic channel, sequestering probability from asingle singlet–singlet color vector from the interactionbetween color octets. Such systems are clear exam-ples of compact four–quark states, in other words,they cannot be expressed in terms of a single physicalchannel. Moreover, as can be seen in Table 1, theirtypical sizes point to compact objects 20% smallerthan a standard two–meson system.We have studied the dependence of the probabil-ity of a physical channel on the binding energy. Forthis purpose we have considered the simplest systemfrom the numerical point of view, the (ST ,I) = (0,1)ccn¯n¯ state. Unfortunately, this state is unboundfor any reasonable set of parameters. Therefore, webind it by multiplying the interaction between thelight quarks by a fudge factor. Such a modificationdoes not affect the two–meson threshold while it de-creases the mass of the four–quark state. The re-sults are illustrated in Fig. 1, showing how in the∆E→ 0 limit, the four–quark wave function is almosta pure single physical channel. Close to this limitone would find what could be defined as molecularstates. Moreover, the size of the four–quark stateincreases when ∆E → 0. When the probability con-centrates into a single physical channel (PMM → 1)the size of the system gets larger than the sum of twoisolated mesons [3]. We have identified the subsys-tems responsible for increasing the size of the four–quark state. Quark-quark (〈x2〉1/2) and antiquark-antiquark (〈y2〉1/2) distances grow rapidly while thequark–antiquark distance (〈z2〉1/2) remains almostconstant. This reinforces our previous result, point-ing to the appearance of two-meson-like structureswhenever the binding energy goes to zero.No. X A. Valcarce et al: Molecular and compact four-quark states 4-180 -150 -120 -90 -60 -30 0∆E (MeV)0.20.40.60.81PMMFig. 1. PMM as a function of ∆E.Although the present analysis has been performedby means of a particular quark interacting poten-tial [4], the conclusions derived are independent ofthe quark-quark interaction used. They mainly relyon using the same hamiltonian to describe tensorsof different order, two and four-quark componentsin the present case. When dealing with a completebasis, any four-quark deeply bound state has to becompact. Only slightly bound systems could be con-sidered as molecular. Unbound states correspond toa two-meson system. A similar situation would befound in the two baryon system, the deuteron couldbe considered as a molecular-like state with a smallpercentage of its wave function on the ∆∆ chan-nel, whereas the H−dibaryon would be a compactsix–quark state. When working with central forces,the only way of getting a bound system is to havea strong interaction between the constituents thatare far apart in the asymptotic limit (quarks or an-tiquarks in the present case). In this case the short-range interaction will capture part of the probabilityof a two-meson threshold to form a bound state. Thiscan be reinterpreted as an infinite sum over physicalstates. This is why the analysis performed here is soimportant before any conclusion can be made con-cerning the existence of compact four–quark statesbeyond simple molecular structures.If the prescription of using the same hamiltonianto describe all tensors in the Fock space is relaxed,new scenarios may appear. Among them, the inclu-sion of many–body forces is particularly relevant. InRef. [5] the stability of QQn¯n¯ and QQ¯nn¯ systems wasanalyzed in a simple string model considering only amultiquark confining interaction given by the mini-mum of a flip-flop or a butterfly potential in an at-tempt to discern whether confining interactions notfactorizable as two–body potentials would influencethe stability of four–quark states. The ground stateof systems made of two quarks and two antiquarksof equal masses was found to be below the dissoci-ation threshold. While for the cryptoexotic QQ¯nn¯the binding decreases when increasing the mass ratiomQ/mn, for the flavor exotic QQn¯n¯ the effect of masssymmetry breaking is opposite. Others scenariosmay emerge if different many–body forces, like many–body color interactions [6] or ’t Hooft instanton–basedthree-body interactions [7], are considered.4 ConclusionsWe have discussed the formalism to express thewave function of a general four–quark state in termsof physical channels, i.e., those constructed by usingcolor singlet–singlet states. We have studied charmo-nium (cc¯nn¯), bottomonium (bb¯nn¯) and exotic (ccn¯n¯and bbn¯n¯) four-quark states by means of a standardnon-relativistic quark potential model. We look forpossible bound states. Among them we are ableto distinguish between meson-meson molecules andcompact four-quark states. The importance of per-forming a complete analysis of the system, energy andwave function, in the vicinity of a two-meson thresh-old has been emphasized.References1 Amsler C and Tornqvist N A, Phys. Rep., 2004, 389: 61.2 Jaffe R L, Phys. Rep., 2005, 409: 1.3 Vijande J and Valcarce A, Phys. Rev. C, 2005, 72: 025206.4 Vijande J, Ferna´ndez F, and Valcarce A, J. Phys. G, 2005,31: 481.5 Vijande J, Valcarce A, and Richard J M, Phys. Rev. D,2007, 76: 114013; Ay C, Richard J M, and Rubinstein JH, Phys. Lett. B, 2009, 674: 227.6 Dmitrasˇinovic´ V, Phys. Lett. B, 2001, 499: 135; Phys. Rev.D, 2003, 67: 114007.7 ’t Hooft G, Phys. Rev. D, 1976, 14: 3432.

Molecular and compact four-quark states

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5th International Conference on Quarks and Nuclear Physics (QNP09). Inst High Energy Phys Chinese Acad Sci, Beijing, PEOPLES R CHINA, SEP 21-25, 2009We study charmonium (c (c) over bar cn (n) over bar), bottomonium (b (b) over barn (n) over bar) and exotic (ccnn and bbnn) four-quark states by means of a standard non-relativistic quark potential model We look for possible bound states Among them we are able to distinguish between meson-meson molecules and compact four-quark states.Peer reviewe

Vijande, Javier

Valcarce, A.

Digital.CSIC

http://dx.doi.org/10.1088/1674-1137/34/9/026

Molecular and compact four-quark states

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Repositori d'Objectes Digitals per a l'Ensenyament la Recerca i la Cultura

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