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Exploring open-charm decay mode $\Lambda_c\bar{\Lambda}_c$ of charmonium-like state $Y(4630)$ PDF

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Preview Exploring open-charm decay mode $\Lambda_c\bar{\Lambda}_c$ of charmonium-like state $Y(4630)$

Exploringopen-charmdecaymodeΛ Λ¯ ofcharmonium-likestateY(4630) c c Xuewen Liu1,∗ Hong-Wei Ke2,† Xiang Liu3,4,‡ and Xue-Qian Li1§ 1School of Physics, Nankai University, Tianjin 300071, China 2School of Science, Tianjin University, Tianjin 300072, China 3School of Physical Science and Technology, Lanzhou University, Lanzhou 730000, China 4ResearchCenterforHadronandCSRPhysics,LanzhouUniversity&InstituteofModernPhysicsofCAS,Lanzhou730000,China ThenewlyobservedX,Y,ZexoticstatesaredefinitelynotinthestandardQQ¯(cid:48)structures,thustheirexistence composes a challenge to our understanding on the fundamental principles of hadron physics. Therefore the studies on their decay patterns which are determined by the non-perturbative QCD will definitely shed light ontheconcernedphysics. Generallythefour-quarkstatesmightbeinamolecularstateortetraquarkortheir mixture.Inthiswork,weadoptthesuggestionthatY(4630)isacharmonium-liketetraquarkmadeofadiquark and an anti-diquark. If it is true, its favorable decay mode should be Y(4630) decaying into an open-charm baryonpair, sincesuchatransitionoccursviastronginteractionandissuper-OZI-allowed. Inthiswork, we 6 calculatethedecaywidthofY(4630) → Λ Λ¯ intheframeworkofthequarkpaircreation(QPC)model. Our 1 c c numerical results on the partial width computed in the tetraquark configuration coincide with the Belle data 0 withinacertainerrortolerance. 2 p PACSnumbers:14.40.Rt,13.30.Eg,13.25.Jx,12.38.Lg e S I. INTRODUCTION in principle can be depicted by hadronic loops even though 7 the propagators in the loops do not correspond to real color- 2 singlet particles (see in text), so they suffer from a loop sup- In2007,theBellecollaborationreportedthata JPC = 1−− ] resonance peak Y(4630) with mass M = 4634+9 MeV and pression. Even though the most promising tetraquark candi- h −11 date Z(4430)+ decays into the [cq¯][c¯q] mode ψ(2S)π+ [10– p widthΓ = 92+−4312 MeVappearedattheinvariantmassspectra 13] with a broad width Γ = 172±13 MeV, this case is very p- ofthee+e− →Λ+cΛ−c channel[1]. different from Y(4630). Since its mass is below the Λ Λ¯ c c e Besides an interpretation that the observed Y(4630) is the thresholditwouldoverwhelminglydecayintoopen-charmed h 53S1 charmoniumstate[2,3],therearemanyalternativesug- mesons. For Y(4630) case, as its mass is above the double- [ gestions for the observed peak, for example, Y(4630) was baryon threshold, the strong decay of such tetraquark state 2 considered to be induced by a threshold effect instead of be- is OZI-super allowed. Therefore, following the suggestions v ingagenuineresonance[4], thenitwasalsointerpretedasa given by other groups here we will assume that the decay 2 molecular state made of ψ(2S) and f0(980) by another theo- modeY(4630) → ΛcΛ¯c wouldbedominant,namelythispar- 6 retical physics group [5]. Among those proposals, the sug- tialwidthcouldbeatthesameorderofthetotalwidth. 7 gestion that Y(4630) is a tetraquark state is more favorable 0 [6, 7]. In Ref. [6], the Y(4630) is identified as the ground A tetraquark is assumed to be made of the diquark- 0 state with its orbital angular momentum L = 1. It is noted antidiquark[cq][c¯q¯],whereqisalightquarkeitheruordand . 1 that the mass and width of Y(4630) are consistent within er- [cq] resides in a color anti-triplet whereas [c¯q¯] is in a color 0 rorswiththosefortheY(4660)state(M =4652±10±8MeV, triplet (in later calculations we do not distinguish between 6 Γ = 68±11±1MeV), which is found in the invariant mass u and d at all). In this work, we suppose that Y(4630) is a 1 spectrumofψ(2S)π+π− bytheBellecollaboration[8,9]. By tetraquarkinthedynamicpicturesuggestedbyBrodskyetal. : v analyzing the Λ Λ¯ and ψ(2S)π+π− spectra, Cotugno et al. [14].Inthetetraquarkadiquarkandananti-diquarkarebound c c Xi suggested that the Y(4630) and Y(4660) could be the same togetherviatheQCDconfinement,butareseparatedbyasub- tetraquarkstate,andisthefirstradialexcitationoftheY(4360) stantialdistanceoncetheyarecreated. ThustheY(4630)state r a withL=1[7]. can be considered as a two-body meson-like state. The pic- tureweadoptinthisworkisslightlydifferentfromthatpro- In fact, Y(4630) as a [cq] [c¯q¯] tetraquark would more 3¯ 3 posedbyMaianietal. [6,15],wheretheauthorsstudiedthe likely decay into charmed baryon-pair [5, 7], and the ratio BR(Y → Λ Λ¯ )/BR(Y → ψ(2S)π+π−) = 25±7[7]suggests tetraquarkstatesbymeansoftheirspinstructureofaHamil- c c thatthedoublebaryondecaymodeΛ Λ¯ isstronglypreferred. tonianformalism[16],infact,thetwopicturesareinprinciple c c consistent. Underthisassignment,westudythestrongdecay However,theredefinitelymayexistotherdecaymodesbe- ofY(4630)bycomputingthewidthofY(4630)→Λ Λ¯ inthe sideoftheΛ Λ¯ pair,suchasDD¯,DD¯∗,D∗D¯∗,J/ψη,ψ(2S)η, c c c c quarkpaircreation(QPC)model. Thecorrespondingreaction etc. Such processes occur via color rearrangements which mechanism is that first the diquark-antidiquark bound state is dissociated into a “free” diquark-antidiquark system and a lightquark-antiquarkpairiscreatedfromthevacuum,thenthe quark and anti-quark would join the diquark and antidiquark ∗Electronicaddress:[email protected] respectively to constitute a baryon-anti-baryon pair. Indeed, †Electronicaddress:[email protected] ‡Electronicaddress:[email protected] thisassociationcanbeviewedasthatduetosoftgluonemis- §Electronicaddress:[email protected] sion a light-quark pair is created and the soft gluons tear off 2 the diquark-antidiquark bound state, then by absorbing light isexpressedas quark and antiquark respectively they transit into color sin- g4l6e3t0bMaryeVon,ss.otMhaotraeosvueprp,rMesΛsico+niMndΛ¯ucciesdobnylymsalticghhitnlygdbieffloewr- T = −3γ(cid:88)(cid:104)1m;1 −m|00(cid:105) (cid:90) dk5 dk6δ3(k5+k6) entmomentaasappearingatsimilarhadronicprocesses,does m (cid:32) (cid:33) k −k not exist. Surely the whole dissociation process is governed ×Y 5 6 χ56 ϕ56 ω56 d†(k )b† (k ), (1) by non-perturbative QCD, so that one needs to introduce a 1m 2 1,−m 0 0 5i 5 6j 6 fewphenomenologicalfactorswhichcanonlybeobtainedby fittingavailabledata. whereiand jaretheSU(3)-colorindicesofthecreatedquark √ The paper is organized as follows: after this introduction, and anti-quark. ϕ56 = (uu¯ + dd¯ + ss¯)/ 3 and ω56 = δ 0 0 ij wecalculatetherateofY(4630)decayingintotheΛcΛ¯c pair are for flavor and color singlets, respectively. χ516,−m is a spin in section. IIA&IIB and perform a numerical analysis in triplet.Heretheindices5and6distinguishbetweenthequark Sec. IIC. The other decays of Y(4630) are discussed in Sec. and antiquark respectively as shown in Fig. 1. Y (k) ≡ (cid:96)m III.Sec. IVisdevotedtooursummary. |k|(cid:96)Y (θ ,φ ) denotes the (cid:96)th solid harmonic polynomial. γ (cid:96)m k k isadimensionlessconstantforthestrengthofquarkpaircre- ationfromvacuumandisfixedbyfittingdata. II. THEY(4630)→Λ Λ¯ STRONGDECAY c c In this work, we use the two-body wave function for the q1 diquark-antidiquark bound system Y(4630), since the con- c2 Λc stituents (diquark and antidiquark) are treated as two point- q1 q5 likecolorsources. Inthisstructure,thediquark Qqofcolor- c2 q5 anti-tripletinthetetraquarkisinanalogtoaheavyQ¯ residing Y(4630) 0++q¯6 inacommonmesonQQ¯ whileQ¯q¯issimilartoQbythesame q¯3 q¯6 colorconfiguration. c¯4 q¯3 Λ¯c ThespinwavefunctionsofaY(JPC =1−−)statewithL=1 c¯4 in the basis of |Sqc,Sq¯c¯,Stotal,L(cid:105)J=1 can be assigned in four FIG. 1: The QPC mechanism for decay Y(4630) → Λ +Λ¯ , we distinctstatesas[6] c c label the quark c and antiquark c¯ with subscripts 2 and 4, as well understood.qstandsforthelightquarku/d. Y = |0,0,0,1(cid:105) , 1 √ 1 Y = 1/ 2(|1,0,1,1(cid:105) +|0,1,1,1(cid:105) ), 2 1 1 Y = |1,1,0,1(cid:105) , 3 1 Y = |1,1,2,1(cid:105) . 4 1 In the dynamical picture of tetraquark, the (anti)diquark is considered to be a point-like color source, then the two- In the following, we present all the details of calculating bodywavefunction(meson-like)shouldbeagoodapproxima- Y(4630)→ΛcΛ¯cintheQPCmodel. tiontodescribetheinnerstructureofY(4630). Includingthe color(ω[12][34]),spin(χ[12][34]),flavor(ϕ[12][34])andthespatial Y Y Y (Ψ (p ,p ))parts,thewavefunctioniswrittenas nYLYMLY 1 2 A. ImplementationintheQPCmodel (cid:12) (cid:69) culTahteetQhePrCatemsoodfeOl [k1u7b–o2-3Z]wheaigs-bIiezeunkaw(iOdeZlIy)aapllpolwieeddtsotrocanlg- =(cid:12)(cid:12)Y((cid:112)nY22ESY+1L(cid:88)Y JYM(cid:10)JYL)(KMY)S M |J M (cid:11) decays[24–37],whichobviouslycomposethedominantcon- Y Y LY Y SY Y JY tributionstothetotalwidthsoftheconcernedhadrons. (cid:90) MLY,MSY As indicated in the introduction, we suppose Y(4630) as × dp1dp2δ3(KY−p1−p2)ΨnYLYMLY (p1,p2) a tetraquark in the diquark-antidiquark structure, thus in our case,thedecayofY(4630)isadissociationprocesswherethe ×χSYMSYϕ[Y12][34]ω[Y12][34]|[q1 q2](p1)[q¯3q¯4](p2)(cid:105),(2) diquark and antidiquark bound state is loosened by a quark- antiquark pair which is created in vacuum. Concretely, the whereweusethe(super)subscript1∼4tomarkthe(anti)quark quark and antiquark of the pair excited out from the vacuum in the tetraquark as clearly shown in Fig 1. K is the Y would join the diquark and antidiquark respectively to com- 3-momentum of Y(4630), p is the 3-momentum of the 1(2) pose a Λ Λ¯ pair, and the process is graphically shown in (anti)diquark. S = S + S is the total spin. J = Fig.1. c c L +S denotesYthetot[aql1qa2n]gula[rq¯3mq¯4o]mentumofY(4630).Y Y Y The quantum number of the created quark pair is 0++ Wealsoconsiderthediquark-quarkpicture[38–42]forthe [17, 18]. In the non-relativistic limit, the transition operator Λ baryon in where the internal degrees of freedom of the c 3 diquarkareneglectedasinthetetraquarks,thenwehave parts of the two-body wave functions of Y(4630). Their ex- (cid:90) plicitformsarecollectedintheappendixB.Thewavefunction (cid:12)(cid:12)(cid:12)Λc(MSΛc)(KΛc)(cid:69)= (cid:112)2EΛc dp1dk3δ3(cid:0)KΛc −p1−k3(cid:1) ofWΛcitwhitlhlebetrcaonnsistiidoenreadmipnlitthuedenegxitvseencitnionE.q. (6), the matrix ××Ψ|[Λqc(qp1],(kp3))χq12,(MkSΛc)(cid:105)ϕ,[Λ1c2]3ω[Λ1c2]3 (3) eMleMmJeYnMtJΛccManJΛ¯cbeasrewritten in terms of the helicity amplitude 1 2 1 3 3 where the (super)subscripts in the expressions correspond (cid:104)ΛcΛ¯c|T|Y(4630)(cid:105)=δ3(KΛc +KΛ¯c −KY)MMJΛcMJΛ¯c. (8) 3to-mthoemecnotnusmtituoefntΛq,uaprk(kan)disththeed3iq-muaormk,enatnudmKoΛfcthise tdhie- ThedecaywidthofY(4630)→ΛcΛ¯cisthen c 1 3 tqJhuPear=skp(iqn21u+pararoknj)de.cLtiTo=nhes0t,qatuseoa.nwtuemonnluymubseersMoSfΛcΛ(=c MareJΛck)ntoowlnabaesl ΓY =π2M|KY2| 2JY1+1 MJMΛ(cid:88)c,MJMΛ¯c (cid:12)(cid:12)(cid:12)(cid:12)MMJΛcMJΛ¯c(cid:12)(cid:12)(cid:12)(cid:12)2, Thewavefunctionsrespectthenormalizationconditions where|K|,asaforementioned,isthe3-momentumofthefinal (cid:104)Y(K )|Y(K(cid:48))(cid:105) = 2E δ3(K −K(cid:48)), (4) statesinthecenterofmassframe. Y Y Y Y Y (cid:104)Λc(KΛc)|Λc(K(cid:48)Λc)(cid:105) = 2EΛc δ3(KΛc −K(cid:48)Λc). (5) B. Baryonwavefunction For Y(4630) → Λ + Λ¯ process, the transition hadronic c c matrixelementiswrittenas ThecharmedbaryonΛc isconsideredasthe[cq]-qpicture inourscenario,thenatwobodywavefunction,whichcanbe (cid:104)ΛcΛ¯c|S|Y(4630)(cid:105) = I−i2πδ(Ef −Ei)(cid:104)ΛcΛ¯c|T|Y(4630)(cid:105). gained by solving the Schro¨dinger equation, could be a rea- I−nKthec=enKte.rTohfetnh,ewmeahsasvferameofY(4630),KY =0andKΛc = sonFaobrleouarppcroonxcirmetaeticoanl.culation, we employ a non-relativistic Λ¯ c Cornell-likepotentialwheretheconcernedfreeparametersare (cid:113) (cid:104)ΛcΛ¯c|T|Y(4630)(cid:105)=−3γ 8EYEΛcEΛ¯c fiinxgedthbeySficthtirno¨gditnhgeemraesqsusaptieocntrwaeofocbhtaairnmtehdebwarayvoenfsu.nBcytiosnolovf- (cid:88) (cid:88) × (cid:104)1m;1 −m|00(cid:105) Λ . ThegeneralHamiltonianofadiquark-quarksystem(i.e. c MLY,MSY,m MSΛc,MSΛ¯c atwobodysystem)canbewrittenas ×(cid:104)s m ;s m |1 −m(cid:105)(cid:104)L M S M |J M (cid:105) 5 5 6 6 Y LY Y SY Y JY p2 p2 ×(cid:104)S12MS12S34MS34|SYMSY(cid:105)(cid:104)S12MS12S5MS5|SΛcMSΛc(cid:105) H = 2m[cq] +m[cq]+ 2mq +mq+V(r), (9) ×(cid:104)SΛcMSΛc00|JΛcMJΛc(cid:105)(cid:104)S34MS34S6MS6|SΛ¯cMSΛ¯c(cid:105) [cq] q ×(cid:104)SΛ¯cMSΛ¯c00|JΛ¯cMJΛ¯c(cid:105)(cid:104)ϕ[Λ1c2]5ϕ[Λ¯3c4]6|ϕ[Y12][34]ϕ506(cid:105) wherethem[cq](p[cq])andmq(pq)arethemasses(3-momenta) ofthediquark[cq]andquarkqrespectively. ×(cid:104)ω[Λ1c2]5ω[Λ¯3c4]6|ω[Y12][34]ω506(cid:105)IMLY,m(K). (6) It is worth of pointing out that in literature, the diquark- quarkstructuremightbedifferent,namelythetwolightquarks The expressions of Eq. (6) for Y states are explicitly make a light diquark and the heavy quark stands as a color 1,2,3,4 writtenoutintermsofIMLY,m(K)aslistedintheAppendixA. source. Insteadthebaryonstillmightbein[Qq]3¯q3 structure The spatial integral IMLY,m(K) manifests an overlap between [43],especiallyinourcasethediquark(anti-diquark)doesnot thespacialpartsoftheinitialstate(includingthecreatedlight havetimetorecombineintoQ3[qq]3¯ bycolorrearrangement, quarkpair)andthefinalstate,andisexpressedas namelytheoriginaldiquarkstructurewouldremaintomakea (cid:90) colorsingletbaryonbyabsorbingalightquark. Theinterac- IMLY,m(K)= dp1dp2dk5dk6 tionpotentialis ××δδ33((pk15++pk26))δΨ3∗Λ(K(pΛ1c,−k5p)1Ψ−∗Λ¯k(p5)2δ,3k(6K)Λ¯c −p2−k6) V(r)=−43αrs +brκ+c, (10) c c p −p (cid:16)k −k (cid:17) ×Ψ ( 1 2)Y 5 6 where −4/3 is the color factor specific to 3-3¯ attraction, b is (cid:90)nYLYMLY 2 1m 2 the string tension and c is a global zero-point energy. Here = dpΨ∗ (p−µK)Ψ∗ (−p+νK) wetake the brκ +c partas theconfinement whichis slightly Λc Λ¯c different form the usual Cornell br + c potential. α is the (cid:16) (cid:17) s ×Ψ (p)Y p−K , (7) phenomenologicalstrongcouplingconstant. nYLYMLY 1m In this work, since only the wave function of Λ+(2286) c whereµ=m /(m +m )andν=m /(m +m ). Fol- which is in S-wave is needed, the hyperfine interactions in- [cq] [cq] q [c¯q¯] [c¯q¯] q¯ lowing the literature in this field, we employ the simple har- cludingthespin-spininteraction,thespin-orbitinteractionand monicoscillator(SHO)wavefunctionstostandforthespacial thecolortensorinteraction[30]arenotincluded. 4 With the diquark mass m[cq] =1.86 GeV which is calcu- nr=1 state nr=2 state 250 140 lated by the QCD sum rules [44] and the light constituent Y quark mass mq =0.33 GeV, the parameters are fixed to be: 1 120 αs = 0.45,b = 0.135 GeV2,κ = 0.84,c = 0.333 GeV. Here, 200 100 Belle as theoretical inputs, we ignore possible inaccuracies of the Y parameters. V)150 2 V) 80 Y1 e e The fitted spectra are presented in Table. I, and a compar- M( Belle M( 60 ison with the experimental data and other theoretical predic- ΓY 100 ΓY tfiuonncstiionnliotferΛatu(2re28a6re)iaslspololitsteteddininFtihge.2ta.ble. Theradialwave 50 40 Y2 Y4 c Y Y4 20 Y 3 3 0 0 TABLEI:Thefittedspectraofcharmedbaryonswithdifferentquan- 1 2 3 4 5 6 1 2 3 4 5 6 tumnumbers,includingacomparisonwiththeexperimentaldataand RY(GeV-1) RY(GeV-1) other theoretical predictions in literature. Here, the masses of the baryonsareinunitsofMeV. FIG. 3: Dependence of the predicted partial width of Y(4630) → Λ Λ¯ onR . TheBelledataareshownintheplotforacomparison. c c Y States PDG[45] Thiswork Ref.[30] Ref.[46] Ref.[47] The black dashed line and the gray band correspond to the central |1S,1/2+(cid:105) 2286.46 2286.1 2265 2286 2286 valueanderrorforthetotalwidthofY(4630)measuredbytheBelle collaboration(Γ=92+41MeV).Thecoloredcurvescorrespondtothe |2S,1/2+(cid:105) 2766.6 2768.5 2775 2769 2766 fourdifferentspinass−i3g2nmentsY ,Y ,Y ,Y respectively. Thesolid 1 2 3 4 |3S,1/2+(cid:105) 3115.0 3170 3130 3112 anddashedcurvescorrespondtothenr =1andnr =2cases. Here, intherightpanelweusetheprimetodistinguish“Y”statesinthe |1P,1/2−(cid:105) 2592.3 2627.6 2630 2598 2591 twocases. |2P,1/2−(cid:105) 2939.3 3006.9 3030 2980 2989 |1D,5/2+(cid:105) 2881.53 2864.9 2910 2880 2879 WefirstcomputethedecaywidthofY(4630)→Λ Λ¯ with c c the n = 1 assignment. The left panel of Fig. 3 shows the r dependence of the calculated width Γ on R within a range Y Y 10 (1 ∼ 6)GeV−1. The colored curves correspond to the four spin states Y which are marked on the figures. As dis- Radialwavefunction 468 dtdchuoiecsmtseteiodndtaawnblteiw,dfotishdroet1hf,,2wo,t3oreh,4fetchYoed(me4Ycp61aa3,y2r0,4em).atohsIdsinsiegtcnhYame(l4cepu6nll3otas0tte)oddno→epcacoΛraitnnicacΛ¯lfiidcnwedsihdwtothhiuethldwptrihbtehee- 2 dataandtheerrorbandof1σ(grayregion)givenbytheBelle 0 collaboration(Γ=92+41MeV). 0.0 0.5 1.0 1.5 2.0 2.5 −32 p(GeV) FortheY3casethefigureshowsthatthevaluesofthecurves are obviously lower than the data Γ = 92+41 MeV. This sup- FIG.2:TheradialwavefunctionofΛcasadiquark-quarksystem. pressioniscausedbytherelativelysmallo−v3e2rlapbetweenthe spinwavefunctionsofinitialandfinalstates(onecanseethe appendix A for some details). Therefore it is concluded that thedatadonotfavorY(4630)tobeagroundstatewithY spin 3 structure. C. Numericalresults Next, as Y(4630) being assigned as the first radial excita- tion state, our numerical results are shown at the right panel Following Ref. [48], in the numerical computations, we of Fig. 3 for all the four spin assignments. The results show adoptthemodelparameterγ=6.3whichisconsideredasuni- thattheY statescanmeetwiththeexperimentaldataaslong 1,2 versalintheQPCmodel. MeanwhiletheRvalueforP-wave asR liesinarangeof1.5∼3GeV−1and/oraround5GeV−1. Y tetraquark in the SHO wave function, which represents the ThevaluescorrespondtotheY stateareslightlylower,how- 4 mean-squareroot(RMS)radius,caneithernotbedetermined ever,theyarestillofthesameorderasthetotalwidth. Again, fromanunderlyingprinciple,soweperformanumericalanal- for Y state, the situation is similar to that for n = 1, the 3 r ysis dependent on R with certain ranges, where R denote computedwidtharemuchbelowthedata. Y Y the R-value of the wave functions for Y(4630) in tetraquark In a brief summary, our numerical results indicate that structure. Since there still exists an ambiguity about the in- within certain regions of the parameter R , the partial width Y nerstructureofY(4630),wecalculatethedecaywidthfortwo of Y(4630) → Λ Λ¯ can be comparable with the Belle data. c c possiblecases: assuming(1)Y(4630)asthegroundstatewith Given the fact that the peak of Y(4630) has only been ob- theradialquantumnumbern =1and(2)thefirstradialexci- served at the invariant spectrum of Λ Λ¯ , one is tempted to r c c tationwithn =2assignments. assume that the Λ Λ¯ mode dominates the decay of Y(4630) r c c 5 Moreover our calculation indicates that this predicted partial beobservedbymuchmoreaccuratemeasurements[49].Sim- width is comparable with the total width of Y(4630). This ilarly,wemayconjecturethatthecolor-re-arrangementwhich consistency supports the assumption that the Y(4630) is a P- proceedsalongsimilarwayshouldhavecomparablerates. wavetetraquarkinthediquark-antidiquarkconfigurationand In fact, such quark exchange mechanism was investigated decays mainly into double charmed baryons. We will make by some authors for meson decays [50–52], but since it is morediscussionsonthisissueinthenextsection. completely induced by the non-perturbative QCD effect, the AlsoY(4630)couldbeineithertheradialgroundstatewith estimate in terms of the present theories cannot be accurate, n =1orthefirstexcitedstatewithn =2.Inotherwords,the or at the best can be valid to the order of magnitude if one r r presentdatacannotruleoutanyofthetwopossibleconfigu- canfindanappropriatemodeltocarryoutnumericalcompu- rations. Sodefinitelyitneedstobestudiedwithmoreexper- tations. imentalinformationinthefuturetodecidethemoreaccurate natureofY(4630),soasthespinstructures. IV. SUMMARY III. DISCUSSIONSONOTHERDECAYMODES To evaluate the hadronic matrix elements which are gov- ernedbythenon-perturbativeQCD,phenomenologicalmod- As discussed in the introduction, beside the dominant elsareneeded. FortheOZI-allowedstrongdecays, theQPC Y(4630) → ΛcΛ¯c, there may exist other decay modes, such model,fluxtubemodel,QCDsumrulesandlatticeQCD,etc. as DD¯, DD¯∗, D∗D¯∗, ψ(2S)π+π−, ψ(2S)η, etc. For instance, havebeensuccessfullyusedtoestimatethedecayrates,even if one considers the both observed Y(4630) and Y(4660) to thoughexceptthelatticecalculationnoneofthemcanbedi- betetraquarkstates[7],Y(4630(4660)) → ψ(2S)π+π− occurs rectlyderivedfromquantumfieldtheory. Weareassuredthat throughaquarkrearrangementprocess. allofthosemodelshavecertainreasonabilityandtheyarein For the tetraquark structure, this decay mode requires a parallelsomehow. Inthiswork,weemployedtheQPCmodel quark-antiquarkrearrangementwhichisalsoacolorexchange tostudythestrongdecayofY(4630)→Λ Λ¯ . c c process.Intheprocessaquarkandanantiquarkwhichbelong First we assume that Y(4630) is a tetraquark which is a to different clusters are switched round to produce the final bound state of a diquark and an anti-diquark. As its mass is states. slightlyabovethethresholdoftwocharmedbaryons,itwould In the figure 4, tracing the diquark (antiquark) flow lines, favorablydecayintoΛ Λ¯ pair,thereforethefactthatY(4630) c c onecandrawaneffectivehadronicFeynmandiagramasadi- isonlyobservedattheinvariantspectrumofΛ Λ¯ , isunder- c c quark (scalar or vector) which brings a color-content (color- standable. triplet 3 or anti-triplet (3¯) is exchanged between the diquark Therecouldbedifferentquantumstructuresforthediquark- [cq]andantidiquark[c¯q¯],andresultsinthefinalstatetobein anti-diquarkboundstate,andwetrytoassignitwithvarious color-singlet. radialquantumnumbersandspinassignmentsandthencalcu- latethedecaywidthofY(4630)→Λ Λ¯ inallpossiblecases. c c The numerical results show that, within certain parameter 1 rangeofR , onecangainproperdecaywidthΓ thatagrees Y Y 3 coupling ψ(2S) withtheexperimentaldataifweassignY(4630)aseitherthe ... radial ground state n = 1 or the first radially excited state r Y46( 3 wnrid=th a2r.e sWuphperreesassedfobryththeecsamsealolfoYve3r,lathpeboetbwtaeiennedthpeasrtpiianl 3 0) wavefunctions,sotheY spinstateisruledout. Ouranalysis 3 providesastrongsupporttothepostulatethatY(4630)isthe ¯3 coupling η... diquark-antidiquarkboundstatewhosemainlydecaychannel 1 shouldbeY(4630)→Λ Λ¯ . c c FIG.4:EffectivediagramforthedecayofY(4630)toψ(2S)η,etc. We are looking forward to getting more information from theBelle-II,LHCbexperiments,especiallywewillpaymore attentionto,suchasDD¯,DD¯∗,D∗D¯∗,ψ(2S)π+π−,ψ(2S)ηetc, decaymodes, whichmayshedmorelightonthestructureof In figure, such processes occur via a hadronic loop, there- Y(4630).Inparticular,wesuspectifthereisamixingbetween fore is suffering from a loop suppression. This Feynman di- the tetraquark and molecular states to result in Y(4630) and agram is similar to the final state interaction where all lines Y(4660),itwouldbeaninterestingpicture. Indeedinthenear corresponding to (no matter inside the loop or outside fi- future,withtheaccumulateddataatvariousaccelerators,our nallyproducedhadrons)color-singlethadrons,thusonlydif- understanding on the XYZ states will be improved and the ference between the quark rearrangement and final state in- observationsofnewstatesareexpected. teraction is their color configurations. But both of them are suppressed. In our another paper, we estimated the rates of Noteadded. Whenwemakechangestoourmanuscript, we Y(4630) → Λ Λ¯ → pp¯,DD¯,DD¯∗,ππ,K+K− etc. through noticethatanotherwork[53]whichsuggeststouseY(4630) c c hadronicrescatteringandfoundthatsuchasfinalstatescould as a window to the landscape of tetraquarks appears, by J. 6 Sonnenshein and D. Weissman, and we cite it at the end of ForspinstateY : 3 thismodifiedmanuscript. 1 A1221 = − (I−1−1+I0−1+I1−1) Acknowledgement 9 1 A21−12 = A−1212 = √ (I−10+I00+I10) We would like to thank Prof. Hai-Yang Cheng for help- 9 2 fulandinspireddiscussions. WewouldalsothankKanChen, 1 Yuan Sun and Hao-Kai Sun who help us with programming A−12−12 = − (I−11+I01+I11) (A4) 9 for the numerical computations. This project is supported by the National Natural Science Foundation of China under ForspinstateY : 4 Grants No. 11375128 No. 1135009, No. 11222547, No. 11175073. XiangLiuisalsosupportedbytheNationalYouth Top-notch Talent Support Program (“Thousands-of-Talents 1 Scheme”). A1221 = √ (I−1−1+3I−10−2I0−1−3I00+7I1−1) 9 5 1 A21−12 = A−1212 = √ (2I−10−3I0−1−4I00 AppendixA:Explicitformulaeforthematrixelements 9 10 − 3I01+6I1−1+2I10) 1 A−21−12 = √ (I11+3I10−2I01−3I00+7I1−1) (A5) 1 (cid:113) 9 5 MMJΛcMJΛ¯c =−√6γ 8EYEΛcEΛ¯cAMJΛcMJΛ¯c (A1) ForspinstateY1: AppendixB:Wavefunctions 1 In this work, we employ the SHO wave functions for A2121 = √ (I−1−1+I0−1+I1−1) Y(4630)astheinputwavefunctions. Forthedecaychannels 3 ofinterest,weneedaP-wavetwo-bodywavefunctionforthe 1 A12−21 = A−1221 =−√ (I−10+I00+I10) Y(4630). 6 Forthetwo-bodywavefunctionwithquantumnumbersn r 1 A−12−21 = √ (I−11+I01+I11) (A2) andl[54] 3 (cid:114) (cid:32) (cid:33) 2R5/2 R2k2 ForspinstateY2: Ψnr=1,l=1(k) = −i2 3π1/4Y1m(k)exp − 2 , (B1) (cid:32) (cid:33) 2 R5/2 R2k2 A1212 = −1(I−1−1+I−10+I00+I1−1) Ψnr=2,l=1(k) = i√15π1/4(5−2k2R2)Y1m(k)exp − 2 , 3 1 (B2) A21−12 = A−1212 = √ (I01+2I1−1+I0−1) 3 2 √ where Y (k) = 3/(4π)(cid:15) ·k is the solid harmonic poly- 1 1m √ −m √ A−12−12 = (I−11−I00−I10−I11) (A3) nomial,with(cid:15)±1 =(±1/ 2,−i/ 2,0)and(cid:15)0 =(0,0,1). 3 [1] G. Pakhlova et al. [Belle Collaboration], “Observation of a Lett.85,61002(2009)[arXiv:0809.1151[hep-ph]]. near-threshold enhancement in the e+e− → Λ+Λ− cross sec- [5] F. K. Guo, J. Haidenbauer, C. Hanhart and U. G. 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