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

0.32 MB·

by  Xuewen Liu

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:[e-mail]
respectively to constitute a baryon-anti-baryon pair. Indeed,
†Electronicaddress:[e-mail]
‡Electronicaddress:[e-mail] thisassociationcanbeviewedasthatduetosoftgluonemis-
§Electronicaddress:[e-mail] 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
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