Study ofqqqcc¯ five quark system withthree kinds ofquark-quark hyperfine interaction S. G. Yuan1,2,4, K. W. Wei3, J. He1,5, H. S. Xu1,2, B. S. Zou2,3 1. Instituteof Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China 2. TheoreticalPhysicsCenterforScienceFacilities,ChineseAcademyofSciences, Beijing100049, China 3. Instituteof HighEnergy Physics, Chinese Academy of Sciences, Beijing100049, China 4. Graduate University of Chinese Academy of Sciences, Beijing 100049, China 5. ResearchCenterforHadronandCSRPhysics,InstituteofModernPhysicsofCASandLanzhouUniversity,Lanzhou730000,China (Dated:January5,2012) Thelow-lyingenergyspectraoffivequarksystemsuudcc¯ (I=1/2,S=0)andudscc¯ (I=0,S= 1)areinves- − tigatedwiththreekindsofschematicinteractions: thechromomagnetic interaction, theflavor-spindependent interactionandtheinstanton-inducedinteraction. Inallthethreemodels,thelowestfivequarkstate(uudcc¯or udscc¯)hasanorbitalangularmomentum L = 0andthespin-parity JP = 1/2 ; themassofthelowestudscc¯ − stateisheavierthanthelowestuudcc¯state. 2 1 PACSnumbers:12.39.Jh,14.20.Pt 0 2 n I. INTRODUCTION theqqqcc¯consistedofthecoloredquarkclusterqqqcandc¯. a The five quark configuration qqqqs¯ and qqqqc¯ with ex- J otic quantum numbers have been extensively studied in the Theconventionalpictureoftheprotonandthecorrespond- 4 chiralquarkmodel[14–17],colormagneticinteractionmodel ing excitedstates are a boundstate of threelightquarksuud [18,19]andinstanton-inducedinteractionmodel[20]. Inthis in constituent quark model (CQM). Recently, an new mea- ] work, we study the mass spectra of the hidden charm sys- h surement about parity-violating electron scattering (PVES) tems uudcc¯ and udscc¯ with three types of hyperfine interac- t in JLab affords new information about the contributions of - tions, color-magnetic interaction (CM) based on one-gluon l strange quarks to the charge and magnetization distributions c exchange,chiralinteraction(FS)basedonmesonexchange, u of the proton, which provides a direct evidence of the pres- and instanton-induced interaction (Inst.) based on the non- n ence of the multiquark components in the proton [1]. The perturbativeQCDvacuumstructure. [ importanceoftheseaquarksintheprotonisalsofoundinthe measurementofthed¯/u¯ asymmetryinthenucleon[2]. Thispaperisorganizedasfollows. InSectionII,weshow 1 the wave functionsof five quarkstates and Hamiltoniansfor v Theoretically, the systematic investigationof baryonmass thethreetypesofinteractions.InSectionIII,themassspectra 7 spectraanddecaypropertiesinCQM showslargedeviations inthepositiveandnegativesectorsarepresented. Thepaper 0 of theoreticalvaluesfrom the experimentaldata [3], such as endswithabriefsummary. 8 the large Nη decay branch ratio of N (1535) and the strong ∗ 0 coupling of Λ(1405) to the K¯N. Riska and co-authors sug- . 1 gested that the mixtures of three-quarkcomponentsqqq and II. THEWAVEFUNCTIONANDHAMILTONIAN 0 the multiquark components qqqqq¯ reduce these discrepan- 2 cies [4–6]. In a recent unquenched quark model, by taking 1 As dealing with the conventional three quark model we into account the effects of multiquark components via 3P : 0 need the wave functionsand Hamiltonian to study the spec- v paircreationmechanism,itisalsoveryencouragingtounder- Xi standtheprotonspinproblemandflavorasymmetry[7]. The trum. qqqqq¯componentscouldalsobeintheformofmeson-baryon r a configurations,suchasN∗(1535)asaKΣboundstate[8]and A. Wavefunctionsoffivequarksystems Λ(1405)asaK¯N boundstate[9]. In the early 1980s, Brodsky et al. proposed that there are Beforegoingtohidden-charmfivequarkuudcc¯ andudscc¯ non-negligibleintrinsicuudcc¯components( 1%)inthepro- ∼ systemswithisospinandstrangenessas(I,S) = (1/2,0)and ton [10]. Later the study of Shuryak and Zhitnitsky show a (I,S) = (0, 1),respectively,wefirstconsiderthefourquark significant charm componentin η′ also [11]. It is natural to − subsystem, which can be coupled to an antiquark to form expectthehighexcitedbaryonscontainalargehiddencharm a hidden-charm five quark system. We use the eigenvalue five quark components too. Recently, some narrow hidden method as given in Ref.[21] to derive the udsc wave func- charm N and Λ resonanceswere predictedto be dynami- c∗c¯ ∗cc¯ tions of the flavor symmetry [211] , [22] , [4] , [31] and cally generatedin the PBand VBchannelswith mass above F F F F [1111] ,whichcorrespondtotheSU(4)flavorrepresentation 4GeVandwidthsmallerthan100MeV[12,13]. Theseres- F 15,20,35,45,and1,respectively. Fortheseflavormultiplets onances,ifobserved,definitelycannotbeaccommodatedinto combinedwith c¯, the followingdecompositionof the SU(4) the frame of conventional qqq quark models. A interesting representationintoSU(3)representationscanbefound, questioniswhetherthesedynamicallygeneratedN andΛ c∗c¯ ∗cc¯ resonancescanbedistinguishedfrompenta-quarkconfigura- 15 4 = 80+10+80+10+31+61+151 (1) tionstates[4–6]. Todistinguishthetwohadronstructurepic- × tures,itshouldbeworthwhiletoexplorethemassspectrumof + 31+32+62+31+3 1 − 2 20 4 = 80+100+31+61 (2) 1 5 × + 151+31+151+6−1+80+60 + 2Xi<j(C[ri−rj]2+V0)+Hhyp. (7) 35 4 = 100+350+241+61+152+32 (3) where m denotes the constituent masses of quarks u,d,s,c × i + 34+15−1+100+61+32+83+13+13 (andtheantiquarkc¯),andP~cmandMarethetotalmomentum andtotalmass 5 m ofthefivequarksystem. C andV are 45 4 = 80+100+270+241+61+151 (4) Pi=1 i 0 × constants. As pointed out by Glozman and Riska [23], one + 31+61+152+32+62+32+83+13 maytreattheheavy-lightquarkmassdifferencebyincluding + 15−1+100+80+61+31+32, aflavordependentperturbationtermH0′′, 1 4 = 10+31 (5) 5 ~p2 P~2 where×the upper indexes denote the charm number. The H′ = X(mi+ 2mi )− 10cmm i=1 decompositionnotationsinRef.[22]areadopted. Inthecur- 5 rentworkweonlyconsiderthehidden-charmfivequarksys- 1 tem,whichmeanscharmnumberC=0.Thestateslyinginthe + 2X(C[ri−rj]2+V0)+H0′′ +Hhyp. (8) octet 80 and singlet 10 of the qqqcc¯ states carry the isospin i<j and strangeness as (I,S)=(1/2,0) and (I,S)=(0, 1) for the withmdenotingtheu,d,squarkmass. TheHamiltonianmay − octet, and (I,S)=(0, 1) for the singlet, respectively, which be rewritten as a sum of 4 separated hamiltonians in Jacobi − arethestatesweneed. Theoctetcanbederivedfrom[211] , coordinates.TheperturbationtermH′′ hasthefollowingform F 0 [22] and[31] . Thesingletcanbederivedfrom[211] and F F F [b1in1e1d1]wFi.thOannlytiqcuaanrk[4c¯],Fwshyimchmdeotreysfnoortmcodnetcauinlptlheetwisohsepnincoamnd- H0′′ = −X4 (1− mm){2~pmi2 − 5m(3mmcP~+2cm2m )}δic i=1 c c strangenessquantumnumberswewant. Theuudcwavefunc- m ~p2 tion can be constructed directly by replacement rules men- (1 ) 5 δ , (9) 5c¯ tionedinRef.[23]. Theexplicitformofuudcandudscwave − − mc¯ {2m} functionsarerelegatedtoAppendixA.Thephaseconvention where the Kronecker symbol δ means that the flavor- ic issameasinRefs.[5,6]. dependentterm is nonzerowhen theith quarkoffourquarks Thegeneralexpressionintheflavor-spincouplingscheme ischarmquark.Ifthecenter-of-masstermisdropped,thema- forthesefivequarkwavefunctionsisconstructedas trix element of perturbation term on the harmonic oscillator ψ(i)(J,Jz) = X X stateinthenegativeparitysector(L=0)willbe Ca,b[,1c4,d],e,fLz,Sz,szC[CFS(i)]e C[FS(i)]c hH0′′i[4]X[1111]CFS[211]C[31]FS =−43δ, (10) [X(i)]f[CFS(i)]e [C(i)]d[FS(i)]c [F]a[S(i)]b whereδ = (1 m/m )ω withthe oscillatorfrequencyω = · [X(i)]f,Lz[F(i)]a,Tz[S(i)]b,SzψC[211]d √5C/m. For o−ther stcates5considered in this work the ma5trix (S,S ,L,L J˜,J˜)(J˜,J˜,1/2,s J,J ) elementscanbealsowrittenassuchsimpleform. z z z z z z · | | · ξ¯szϕ(rc¯)ψ¯Cϕ¯. (6) quTarhkestienrmtheHhhaydprorenflse.cItns tthheishwyoprekrfiwneecinotnesriadcetriotnhrbeeettwyepeens whereJ˜isthetotalangularmomentumoffourquarkandS the ofthehyperfineinteractions,i.e.,flavor-spininteraction(FS) totalspinoffourquark,iisthenumberoftheqqqcc¯configura- based on meson exchange, color-magnetic interaction (CM) tioninbothpositiveandnegativeparitysectors,whichwillbe basedonone-gluonexchange,andinstanton-inducedinterac- givenexplicitlylater. ψ¯C,ϕ¯ andξ¯ representthecolor,flavor sz tion(inst.) basedonthenon-perturbativeQCDvacuumstruc- andspinorwavefunctionsoftheantiquark,respectively.ϕ(r ) c¯ ture. representsthespacewavefunctionforantiquark.Thesymbols The flavor-spin dependent interaction reproduceswell the C[.] are S Clebsch-Gordan coefficients for the indicated [..][...] 4 light-quark baryon spectrum, especially the correct ordering color-flavor-spin([CFS]),colorψ¯C,flavor-spin([FS]),flavor of positive and negative parity states in all the considered ([F]),spin([S]),andorbital([X])wavefunctionsoftheqqqc spectrum[25]. Theflavordependentinteractionhasbeenex- system. tendedtoheavybaryonssectorinRef.[23]. GiventhatSU(4) flavorsymmetryisbrokenmainlythroughthequarkmassdif- ferences,thehyperfineHamiltoniancanbewrittenasthefol- B. Hamiltonians lowingform[23,26] Toinvestigatethemassspectrumofthefivequarksystem, 4 m2 14 the non-relativisticharmonic oscillator Hamiltonian is intro- HFS =−CχXmm X~λiF ·~λFjσ~i·σ~j, (11) i,j i j F=1 ducedasinthelightflavorcase[24]: where σ and λF are Pauli spin matrices and Gell-Mann 5 ~p2 P~2 i i H = X(mi+ i ) cm SU(4)F flavor matrices, respectively,andCχ a constantphe- i=1 2mi − 2M nomenologically20∼30MeV.Inthechiralquarkmodel[25], 3 onlythehyperfineinteractionsbetweenquarksareconsidered the C parameters of Ref. [29], determined by a fit to the i,j whiletheinteractionsbetweenthequarksandtheheavyanti- charmed ground states. For the Inst. model, the parameters quarkc¯areneglected. are determined by a fit to the splittings between the baryon Thechromomagneticinteraction,whichhaveachievedcon- groundstatesN(938),∆(1232),Λ(1116),Σ0(1193),Ω(1672), siderable empirical success in describing the splitting in Λc(2286),Σc(2455),Ξ0c(2471),Ξ′c0(2578)andΞ∗c0(2645).The baryonspectra[27],areintensivelyusedinthestudyofmul- fityieldsaratioofabout / 2/3,whichisthesame ns nn W W ≃ tiquarkconfigurations[22,28–30]. Acommonlyusedhyper- asinRef.[34]. TheparameterV foreachmodelisadjusted 0 fineinteractionisasthefollowing[29], to reproducethe mass of N (1535)as the lowest JP = 1/2 ∗ − N resonanceofpenta-quarknature. HCM =−XCi,j~λci ·~λcjσ~i·σ~j, (12) ∗ i,j TABLE I: The parameters (in the unit of MeV) for three kinds of where σ is the Pauli spin matrice, λc is the Gell-Mann i i hyperfineinteractions. SU(3) colormatrices,andC thecolormagneticinteraction C i,j CM[29] C 20 C 14 C 4 C 5 strength.Thequark-antiquarkstrengthfactorsarefixedbythe qq qs qc sc C 6.6 C 6.7 C 5.5 V -208 hyperfinesplittings of the mesons. For an antiquarkthe fol- qc¯ sc¯ cc¯ 0 FS [24] C 21 V -269 lowingreplacementshouldbeapplied[31]:λ~c λ~c∗. χ 0 The instanton induced interaction, introd→uce−d first by Inst. Wnn 315 Wns 200 Wnc 70 Wsc 52 V -213 0 t Hooft[32] for[ud]-quarksandthenextendedto threefla- ′ vorcase [33] andfourflavorcase [34], isalsoquitesuccess- With all these Hamiltonianparametersfixed andthe wave ful in generating the hyperfine structure of the baryon spec- functionsoffivequarksystemoutlinedintheSect.II,thema- trum. The nonrelativistic limit of the unregularized quark- trixelementsofHamiltonianforvariousfive-quarkstatescan quark t Hooftinteractionhastheform[34–37], ′ becalculated. H = 4 (nn)+ (ns) For the uudcc¯ and udscc¯ systems, the lowest states Inst − PDS=0⊗(cid:2)Wnn PFA Wns PFA are expected to have all five quark in the spatial ground + (nc)+ (sc) Wnc PFA Wsc PFA (cid:3)⊗PC3¯ state of [4]X configuration and hence negative parity. For 2 (nn)+ (ns) the construction of color-flavor-spin wave-functions, the − PDS=1⊗(cid:2)Wnn PFA Wns PFA convenient coupling schemes for the FS and CM mod- + (nc)+ (sc) , (13) Wnc PFA Wsc PFA (cid:3)⊗PC6 els are different, i.e., [1111]CFS[211]C[f]FS[f]F[f]S and whereWf1f2 istheradialmatrixelementofthecontactinter- [c1o1n1fi1g]uCrFaSti[ofn]sF[ffo]rCSth[2e1u1u]Cd[cfc¯]Sa,nrdesupdescctic¯veslyys.tTemhesfloafvospr-ascpiianl actionbetweena quarkpair withflavors f and f , (f f ) 1 2 F 1 2 the projector onto flavor-antisymmetric quark pairsP; A and ground state [4]X for the FS and CM models are listed in PC3¯ Table II, where the configurations 1′ > and 3′ > are only theprojectorsontocolorantitripletandcolorsextetpairs, | | PC6 for the udscc¯ system. For the udscc¯ system, the [211]′F and respectively; and theprojectorsontoantisymmet- PDS=0 PDS=1 u c u s ricspin-singletandsymmetricspin-tripletstates,respectively. d d For a three quark system, only two quarks qq in a spin sin- [211] correspondtotheWeylTableaus s and c ,re- glet state with the flavor antisymmetry can interact through F spectively. the instantoninducedinteraction. Here, we phenomenologi- callyconsidertheinstanton-inducedinteractionofthencand scquarkpairs,althoughsomeauthors[38,39]assumethatthe TABLEII: The flavor-spinconfigurations for the uudcc¯ and udscc¯ heavyflavor decoupleswhen the quarkgets heavierthan the systemsofspacialgroundstate[4]X fortheFS andCMmodels. Λ . FS model CMmodel QCD |1′> [31]FS[211]′F[22]S [211]′F[31]CS[211]C[22]S III. MASSSPECTRAOFuudcc¯ANDudscc¯SYSTEMS |31′>> [[3311]]FS[[221111]]′F[[2321]]S [[221111]]′F[[3311]]CS[[221111]]C[[2321]]S | FS F S F CS C S 2> [31] [31] [22] [31] [211] [211] [22] | FS F S F CS C S Inthissection,wepresentthenumericalresultsforthelow- 3> [31] [211] [31] [211] [31] [211] [31] | FS F S F CS C S lying spectra of the five quark systems of uudcc¯ and udscc¯ 4> [31] [22] [31] [22] [22] [211] [31] | FS F S F CS C S with the hyperfine interaction given by the color-magnetic 5> [31] [31] [31] [31] [211] [211] [31] | FS F S F CS C S interaction , the flavor-spin interaction, and the instanton- 6> [31] [31] [4] [31] [211] [211] [4] | FS F S F CS C S inducedinteraction,respectively. Forthekineticpartandthe confinement potential part of the Hamiltonian, we take the The correspondingseven udscc¯ wave functionswith spin- parameters of Refs. [23, 24], i.e., m = m = 340 MeV, parity 1/2 are 1,1/2 , 1,1/2 , 2,1/2 , 3,1/2 , u d − ′ − − − ′ − m = 460 MeV, m = 1652 MeV and C = m ω2/5 with 3,1/2 , 4,1/2 |, and i5,1|/2 . i Th|e fiveiwa|ve funci- s c u 5 | −i | −i | −i ω =228MeV. tionswithspin-parity3/2 are 3,3/2 , 3,3/2 , 4,3/2 , 5 − ′ − − − | i | i | i All other parameters for three different hyperfine interac- 5,3/2 and 6,3/2 . The one wave function with spin- − − tions are listed in Table I. For the FS model, theC param- p| arity 5i/2 is|6,5/2i. They form three subspace of JP = χ − − | i eter is taken from Ref. [24]. For the CM model, we take 1/2 ,3/2 and5/2 ,respectively. − − − 4 The energies for these different configurations have been TABLEV:Themixingcoefficientsofthestateswithspin-parity1/2 calculated with three kinds of hyperfineinteractions and are − undertheCMinteractionincludingtheqq¯interaction. listedinTableIII. udscc¯ 1 > 1> 2> 3 > 3> 4> 5> ′ ′ | | | | | | | 4273 -0.54 0.06 -0.02 0.84 -0.05 -0.01 0.01 TABLEIII:Energies(inunitofMeV)oftheudscc¯anduudcc¯system 4377 -0.05 0.61 0.08 -0.12 -0.77 -0.15 -0.11 ofthespacialgroundstatewiththreekindsofhyperfineinteractions fordifferentflavor-spinconfigurations. 4453 0.83 -0.03 0.10 0.52 -0.15 -0.09 0.03 4469 -0.07 -0.17 -0.20 -0.05 -0.11 -0.95 -0.09 CM FS Inst 4494 -0.02 0.46 0.64 -0.02 0.40 -0.30 0.36 conf. udscc¯ uudcc¯ udscc¯ uudcc¯ udscc¯ uudcc¯ Jp 4576 0.14 0.61 -0.55 0.06 0.45 -0.03 -0.31 |1′> 4404 −− 4169 −− 4211 −− 12− 4649 0.03 0.08 -0.48 -0.02 -0.11 0.02 0.87 |31′>> 444344238520 43−−7−−2 444111665996 40−−1−−7 444222228227 41−−2−−5 12321−−− u4u2d6c7c¯ 0|1.6>1 0|2.1>1 -0|3.7>7 -0|4.0>3 -0|5.1>2 |3> 4441 4333 4200 4059 4322 4167 21− 4363 0.31 0.37 0.24 0.82 0.17 | 4538 4430 4200 4059 4322 4167 23− 4377 0.36 0.57 0.34 -0.56 0.34 2 2> 4552 4436 4182 4052 4347 4195 1− 4471 0.63 -0.57 0.45 -0.05 -0.26 |4> 4471 4368 4229 4096 4360 4202 21− 4541 0.07 -0.44 -0.15 0.03 0.88 | 2 4572 4468 4229 4096 4360 4202 3− 2 |5> 4617 4508 4258 4133 4386 4237 12− TABLE VI: The mixing coefficients of the states with spin-parity 4585 4477 4258 4133 4386 4237 32− 1/2−undertheFS interaction. |6> 4629 4526 4362 4236 4461 4322 32− udscc¯ 1′> 1> 2> 3′ > 3> 4> 5> 4719 4616 4362 4236 4461 4322 52− 4084 -|0.03 -0|.75 -0|.66 | 0 | 0 | 0 | 0 4154 0 0 0 0.39 -0.70 -0.58 0.12 Forsubspacesof JP = 1/2 and3/2 , somenon-diagonal − − 4160 0.95 -0.22 0.21 0 0 0 0 matrixelementsofHamiltoniansarenotzeroandleadtothe 4171 0 0 0 0.92 0.35 0.18 -0.06 mixtureoftheconfigurationswiththesamespin-parity.After 4253 0 0 0 -0.03 0.42 -0.35 0.84 considering the configuration mixing, the eigenvaluesof the 4263 -0.29 -0.62 0.73 0 0 0 0 Hamiltoniansofthefivequarkudscc¯anduudcc¯systemsinthe 4278 0 0 0 0.07 -0.46 0.71 0.53 spatialgroundstatearelistedinTableIV.Thecorresponding mixingcoefficientsofthestateswithspin-parity1/2 forthree uudcc¯ 1> 2> 3> 4> 5> − | | | | | differentmodelsarelistedinTablesV-VII.Thespinsymme- 3933 0.76 0.65 0 0 0 try [4] is orthogonalto the spin symmetry[31] and [22] . 4013 0 0 -0.78 -0.60 0.17 S S S Thereisnomixingbetweentheconfiguration[31]FS[31]F[4]S 4119 0 0 0.52 -0.47 0.71 andother7configurations. 4136 0.64 -0.76 0 0 0 4156 0 0 0.35 -0.65 -0.68 TABLEIV:Energies(inunitofMeV)theudscc¯anduudcc¯systems inthespatialgroundstateunderthreekindsofhyperfineinteractions (i.e.,withconfigurationmixingconsidered). For the lowest spatial excited states, one quark should be in p-wave,whichresultsinapositiveparityforthefivequark CM FS Inst. system.Fortheudscc¯system,therearethirtyfourwavefunc- JP udscc¯ uudcc¯ udscc¯ uudcc¯ udscc¯ uudcc¯ tions with spin-parity 1/2+ and 3/2+, twenty two with 5/2+ 12− 4273 4267 4084 3933 4209 4114 and four with 7/2+. Similarly, there are too many states for 21− 4377 4363 4154 4013 4216 4131 uudcc¯ system. Here, ten of all states with spin-parity 1/2+, 21− 4453 4377 4160 4119 4277 4204 fiveloweststateswithspin-parity3/2+,fiveloweststateswith 1− 4469 4471 4171 4136 4295 4207 5/2+, and all the states with spin-parity 7/2+ are listed Ta- 2 1− 4494 4541 4253 4156 4360 4272 bleVIIIintermsoftheenergy. 2 1− 4576 4263 4362 Whileinthenegativeparitysectortherearethreesubspaces 2 21− 4649 4278 4416 for 1/2−, 3/2− and 5/2−, respectively, for the positive par- 3− 4431 4389 4184 4013 4216 4131 ity sector, thereare foursubspacesfor 1/2+, 3/2+, 5/2+ and 23− 4503 4445 4171 4119 4295 4204 7/2+, respectively. Intheprocessofthecalculation,we take 2 theL-S couplingschemewithstandardClebsch-Gordancoef- 3− 4549 4476 4263 4136 4362 4272 2 ficientsoftheangularmomentum[40].Fortheflavor-spinand 3− 4577 4526 4278 4236 4416 4322 2 instanton-inducedinteractions, due to the ignoringof quark- 23− 4629 4362 4461 antiquark interaction, the 1/2+ and 3/2+ states of the same 52− 4719 4616 4362 4236 4461 4322 configuration [f]FS[f]F[f]S degenerate. In the CM model, the two states of the same configuration but different four 5 ofconfigurationsofcertainstate,whichwillresultindifferent TABLE VII: The mixing coefficients of the states with spin-parity patternsof the electromagneticandstrongdecays. Themix- 1/2 undertheInst.interaction. − ingeffecthasbeenexploredinlightquarksector,suchasthe udscc¯ 1 > 1> 2> 3 > 3> 4> 5> ′ ′ decayofnucleonresonancesN (1440)[6]andN (1535)[41]. | | | | | | | ∗ ∗ 4209 0.99 -0.07 0.08 0 0 0 0 For the udscc¯ system, in the CM model without qq¯ inter- 4216 0 0 0 0.97 0.12 0.02 0.19 action,theSU(3)flavorsingletwithhiddencharm,whichhas 4277 -0.04 -0.94 -0.35 0 0 0 0 4295 0 0 0 -0.19 0.86 -0.21 0.42 four quark configuration [211]F′[31]CS[211]C[22]S, is dom- inant in the lowest energy state, with a small admixture of 4360 -0.10 -0.34 0.93 0 0 0 0 [211] [31] [211] [22] . The mixing of the two configu- F CS C S 4362 0 0 0 -0.07 0.13 0.97 0.19 rations is due to the flavor dependence of the C . After i,j 4416 0 0 0 -0.10 -0.47 -0.12 0.87 considering the qq¯ interaction in CM model, the configura- uudcc¯ |1> |2> |3> |4> |5> tion [211]F′[31]CS[211]C[31]S (∼ 72%) becomes the domi- 4089 0.94 0.35 0 0 0 nant wave function component, with a strong admixture of 4096 0 0 0.86 -0.20 0.47 [211]F′[31]CS[211]C[22]S ( 27%),asshowninTableV.The ∼ 4157 0 0 0.11 0.97 0.20 qq¯ interactionleadstoa furthermixingofthetwo spinsym- metry configurations of [22] and [31] , besides the flavor 4175 -0.35 0.94 0 0 0 S S symmetrybreakingeffects. IntheFSmodel,theloweststate 4242 0 0 -0.50 -0.12 0.86 has a dominant four-quark configuration [31] [211] [22] FS F S ( 42%), with a strong admixtures of [31] [31] [22] FS F S ∼ TABLE VIII: Energies (in unit of MeV) of positive parity (L=1) and [31]FS[211]′F[22]S, as shown in Table VII. In the Inst model, the lowest state predominantly has the configuration qqqcc¯ states with quantum numbers of N - and Λ -resonances un- derthreekindsofinteraction,withconfigu∗rationmi∗xingconsidered. [211]C[31]FS[211]F′[22]S,whichisthesameastheCMcase withoutqq¯ interaction. CM FS Inst. JP udscc¯ uudcc¯ udscc¯ uudcc¯ udscc¯ uudcc¯ For the uudcc¯ system, there is no hidden charm SU(3) 1+ 4622 4456 4291 4138 4487 4396 flavor singlet state. In the CM model after taking into ac- 21+ 4636 4480 4297 4140 4501 4426 count the qq¯ interaction, the lowest energy state is mainly 21+ 4645 4557 4363 4238 4520 4426 the admixture of [211] [31] [211] [31] ( 67%) and 21+ 4658 4581 4439 4320 4540 4470 [211] [31] [211] [22]F( 2C7S%), asCshowSn i∼n Table V. In 21+ 4690 4593 4439 4367 4557 4482 the FFS moCdSel, theCloweSst∼state is the four-quark configura- 21+ 4696 4632 4467 4377 4587 4490 tion[31] [211] [22] ( 52%), with a strongadmixtureof 21+ 4714 4654 4469 4404 4590 4517 [31] [31FS] [22]F( 4S2%∼). In the present Inst. model, as- 21+ 4728 4676 4486 4489 4614 4518 sumiFnSgpheFnomeSno∼logicallythatthe tHooft’sforcealsoop- 21+ 4737 4714 4492 4508 4616 4549 erates between a light and a charm q′uark, the configuration 21+ 4766 4720 4510 4515 4626 4566 [211] [31] [211] [22] should be the lowest, as the spin 23+ 4623 4457 4291 4138 4487 4396 [22] FandflCaSvor[2C11] Scontainmoreantisymmetrizedquark 23+ 4638 4515 4297 4140 4501 4426 pairsS. In the Inst modFel,if it is assumedthatthelightquark 23+ 4680 4561 4363 4238 4520 4426 andcharmquarkdecouples,the[211] [31] [211] [22] and 23+ 4692 4582 4439 4320 4540 4470 [31] [31] [211] [22] states degenFerateCaSnd shoCuld bSe the 23+ 4695 4625 4439 4367 4557 4482 loweFst. CS C S 2 5+ 4705 4539 4297 4140 4501 4426 25+ 4719 4649 4439 4320 4540 4470 If the the flavor SU(3) symmetry is restored and the 25+ 4773 4689 4467 4367 4587 4482 light quark and charm quark decouples, the udscc¯ is 25+ 4793 4696 4486 4404 4615 4490 lower than the uudcc¯. For the positive parity udscc¯ 25+ 4821 4710 4492 4515 4632 4517 states, under the CM interaction with the qq¯ interac- 2 7+ 4945 4841 4638 4508 4698 4566 tion, the lowest state has predominantly the four-quark 27+ 4955 4862 4671 4551 4712 4634 configuration [31]F[31]CS[211]C[31]S, with a strong ad- 27+ 4974 4919 4705 4587 4765 4669 mixture of the configurations [22]F[31]CS[211]C[22]S and 27+ 5010 4759 4797 [31]F[31]CS[211]C[22]S. In the FS model, the lowest 2 positive parity state has predominantly the configuration [4] [22] [22] . The Inst model predicts that the lowest FS C S state is the configuration [1111] [31] [211] [22] , which quarkangularmomentum J˜haveasmallsplittingmagnitude F CS C S can form the SU(3) flavor singlet state when combined with ofseveralMeVasshowninTableVIII.Hereonlythemasses theantiquark. of severallower energystates, whichare moreinteresting to us,arelistedinTableVIII. Differenthyperfineinteractionspredictdifferentconfigura- The non-zero off-diagonal matrix elements introduce the tionsforthelowestfivequarkstates,whichwillresultindif- mixtureoftheconfigurationswiththesamequantumnumber. ferent decay patterns and can be checked by future experi- The different hyperfine interactions give different admixture ments. 6 IV. SUMMARYANDDISCUSSIONS picturewith threekindsofthe residualinteractions,thelow- est udscc¯ system is heavier than the uudcc¯ system. So the In this work we have estimated the low-lying energy lev- meson-baryonpictureandthepenta-quarkpicturegivediffer- els of the five quark systems uudcc¯ and udscc¯ with the hid- entpredictiononthemassorderofthesuper-heavyN∗andΛ∗ den charm by using the three kinds of hyperfine interac- withhiddencharm. tions. The hidden charm states are obtained by diagonaliz- In theCM model, the lowest 1/2− and 3/2− states, corre- ingthehyperfineinteractionsineachsubspacewiththesame spondingtothesamefour-quarkconfiguration,aresplitbythe spin-parity. For the colormagnetic interaction, flavor-spin- quark-antiquarkinteraction. Andthe3/2− state ofthe udscc¯ dependent interaction and Inst.-induced interaction, all the and uudcc¯ system is about150 MeV heavier than the corre- models predict that the lowest states of the five quark sys- sponding1/2− state. In the FS and Inst. models, dueto the temsudscc¯anduudcc¯havethespin-parity1/2 . Theabsolute lackofthequark-antiquarkinteraction,thetwostatesdegen- − value of the negative hyperfine energy for the configuration erate. [4] [22] [22] inthepositiveparitysectorislargerthanthe In addition, we have also discussed the admixture pattern FS F S case of the [31]FS[211]F′[22]S in the negative parity sector. of the configurations with the same quantum numbers. The But this difference cannot overcome the orbital excited en- quarkmassdifferenceandquark-anti-quarkinteractionarethe ergyoftheP-wavefivequarksystem. Thisisincontrastwith two sources of generating the configuration mixing, and the the situation in the light flavor sector with the chiral hyper- latter more importantfor the configurationmixing and mass fine interaction [24], due to the fact that the hyperfine split- splitting of penta-quark states. Since various configurations ting depends on the quark masses and gets weak for heavy willresultindifferentelectromagneticandthestrongdecays, quarks. In addition, for the flavor-spin interaction, the low- the studyof the decay propertiesmay providea goodtest of est uudcc¯ state has negative parity, which is opposite to the themodels. lowest positive parity state of uuddc¯ system containing only Experimental observation of the super-heavy N and Λ ∗ ∗ oneheavyantiquark[15]. Thefourquarksuuddwithcolored with hidden charm and their decay propertiesfrom pp¯ reac- quarkclusterconfiguration[31]X[4]FS[22]F[22]S arestrongly tionatPANDA andep reactionatJLab12GeV upgradeare attractive due to the diquark structure [ud][ud]c¯. However, of great interests for our understanding dynamics of strong thecquarkinthediquarks[ud][uc]withthesameflavor-spin interaction. symmetry reducesto a large extent the hyperfineinteraction energy. The instanton-induced interaction only operates on thecolorsextetandantitripletdiquark,andthusfavorsaswell Acknowledgements the similar diquark structure. The P-wave diquark-triquark structure [ud][udc¯] is discussed under the colormagnetic in- teraction [19] and is almost as low as the [ud][ud]c¯ [42]. It This work is supported by the National Natural Science wouldbeofintereststostudytheconfigurationsof[ud][ucc¯] Foundation of China (Grant Nos. 10875133, 10821063, and[ud][scc¯]. 10905077, 10925526, 11035006 and 11147197), the Min- The coupled-channelunitary approach [12] predicted that istry of Education of China (the project sponsored by SRF the boundstate D¯ Λ is 30 50 MeV lowerthanthe bound forROCS,SEMunderGrantNo. HGJO90402)andChinese s c stateD¯Σ . Inthechiralquark∼model[13],thereonlyexiststhe Academy of Sciences (the Special Foundation of President c boundstateD¯Σ . Inthepresentmodel,forthecolored-cluster underGrantNo. YZ080425). c [1] A.Achaetal.,Phys.Rev.Lett.98,032301(2007). [9] E.OsetandA.Ramos,Nucl.Phys.A635,99(1998). [2] G.T. Garvey and J.-C. Peng, Progr. Part. Nucl. Phys. 47, 203 [10] S.J.Brodsky,P.Hoyer,C.PetersonandN.Sakat,PhysLettB (2001). 93,451(1980). [3] S.CapstickandW.Roberts, Prog. Part.Nucl. Phys.45, S241 [11] E. V. Shuryak and A. R. Zhitnitsky, Phys. Rev. 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[22] = [uduc+cudu+ducu+ucud [37] JishnuDey,MiraDey,andPeterVolkovitsky,Phys.Lett.B261, | F2i √8 493(1991). duuc ucdu cuud udcu], (A8) [38] Gerard ′tHooft,arXiv:hep-th/9903189v3. − − − − [39] SachikoTakeuchi,Nucl.Phys.A642,543(1998). Theexplicitformsoftheflavorsymmetry[31] F [40] K.Nakamuraetal.(ParticleDataGroup),J.Phys.G37,075021 (2010). 1 [41] C.S.AnandB.S.Zou,Eur.Phys.J.A39,195(2009). |[31]F1i = √18[2uucd+2cuud+2ucud−cudu−ucdu [42] KingmanCheung,Phys.Rev.D69,094029(2004). [43] C.S.An,B.Saghai,S.G.YuanandJunHe,Phys.Rev.C81, ducu udcu dcuu cduu], (A9) − − − − 045203(2010). 1 [31] = [6uudc 3duuc 3uduc 4dcuu 4cduu | F2i 12 − − − − AppendixA:Thewavefunctionsforfourquarksubsystem + 5cudu+5ucdu+2uucd cuud ducu − − ucud udcu], (A10) 1. Flavorandspincouplings − − Take the decomposition of the flavor-spin configuration 1 [31] = [ 3duuc+3uduc 3ducu+3udcu [31]FS[211]F[22]S asanexample, | F3i √48 − − 1 2dcuu+2cduu cudu+ucdu cuud [31]FS1 = [211] F1[22] S1+ [211] F2[22] S2 , − − − | i √2{| i | i | i | i } + ucud]. (A11) (A1) 1 3. Theflavorwavefunctionoffourquarksubsystemudsc [31] = √2[211] [22] + [211] [22] FS2 F3 S2 F2 S2 | i 2{− | i | i | i | i [211] [22] (A2) Theexplicitformsoftheflavorsymmetry[31] : − | iF1| iS1} F 1 1 |[31]FS3i = 2{[211]iF1|[22]iS2+|[211]iF2|[22]iS1 |[31]F1i = √12[ucsd−cdsu+uscd−dcsu+sucd + [211] F3[22] S1 (A3) sdcu+cusd dscu+scud scdu | i | i } − − − 8 + csud csdu], (A12) ucsd sucd scdu csud dscu (A17) − − − − − − } Theexplicitformsoftheflavorsymmetry[211]′ 1 F [31] = 3(usdc sduc+ucds cdus | F2i √96{ − − 1 + sudc dsuc+cuds dcus) [211]′ = cdsu ucsd+ucds cdus − − | F1i 2√3{ − − + 2(scdu scud+csdu csud) − − + cuds scdu+dcsu cusd + ucsd cdsu+uscd dcsu+sucd − − − − + dscu csdu+csud dcus sdcu+cusd dscu , (A13) − − } − − } (A18) 1 |[31]F3i = √32{2(udsc−dusc+udcs−ducs) [211]′ = 1 2(udcs sdcu+dscu ducs+sucd uscd) | F2i 6{ − − − + sduc+usdc+cdsu+ucsd+cdus + cdus cdsu+ucds ucsd+scud + ucds+uscd+sdcu dcsu sucd sudc − − − − − scdu+dcsu cuds+cusd cuds cusd dsuc dscu dcus (A14) − csud+csdu−dcus − − − − − } − − } (A19) Theexplicitformsoftheflavorsymmetry[211] : F 1 [211] = 3(sudc sduc+usdc dsuc 1 | F1i 4√6{ − − |[211]′F3i = 6√2{3(udsc+sudc+dsuc−dusc + sdcu sucd+dscu uscd) − − sduc usdc)+cdsu+ucsd+udcs+sucd + 2(csud csdu+scud scdu) − − − − + scdu+cuds+csud+dscu+dcus ducs + cusd cdsu+cdus cuds − − − cdus ucds dcsu uscd + dcus ucds+ucsd dcsu , (A15) − − − − − − } scud sdcu cusd csdu − − − − } (A20) 1 [211] = 6(udsc dusc) F2 | i 12√2{ − + 5(dcsu cdsu+cusd ucsd) 4. Thewavefunctionofspinsymmetryoffourquark − − subsystem + 4(scdu csdu+csud scud) − − + 3(sduc dsuc+usdc sudc) − − Thewavefunctionsforspinsymmetry[22]S, + 2(ducs udcs)+cuds ucds+sucd − − sdcu+dscu cdus+dcus uscd ,(A16) [22] = 1 2 +2 − − − } S1 | i √12{ |↑↑↓↓i |↓↓↑↑i−|↓↑↑↓i−|↑↓↑↓i , (A21) 1 − |↓↑↓↑i−|↑↓↓↑i} [211] = 2(udcs+cuds+dcus ducs 1 | F3i 6{ − [22] = + .(A22) S2 cdus ucds)+dcsu+uscd+scud | i 2{|↑↓↑↓i |↓↑↓↑i−|↓↑↑↓i−|↑↓↓↑i} − − + sdcu+cusd+csdu cdsu MorecanbefoundinRef.[43]. −