Study of structures and dynamical decay mechanisms for multiquark systems

0.23 MB·

by  Xuewen Liu

Studyofstructuresanddynamicaldecaymechanismsformultiquarksystems
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
Theinnerstructuresofthemultiquarkstatesareaninterestingsubjectinhadronphysics,generallytheymay
beintetraquarkstateswhicharecomposedofcoloredconstituents,orinmolecularstateswhicharecomposedof
twocolorsinglets,ortheirmixtures.Therefore,themechanismswhichbindtheconstituentsinauniquesystem
andinducethemultiquarkstatestodecaywouldbedifferentinthosecases. Inthiswork,usingthequantum
mechanicsweanalyzethedynamicalmechanismsinducingdecaysofthetetraquarkswhereY(4630)standsas
anexampleforthestudy,wealsocommentonthemolecularstateswithoutmakingnumericalcomputations.
6
PACSnumbers:14.40.Rt,12.39.Pn,11.80.Fv
1
0
2
I. INTRODUCTION Y(4630)hasbeenobservedintheinvariantmassspectraof
r thee+e− → Λ Λ¯ channel[4],anditisidentifiedasa JPC =
p 1−− resonancecwicth mass M = 4634+9 MeV and width Γ =
A To study the inner structures of multiquark states is an in- 92+41MeV. −11
teresting subject in hadron physics, because it is obviously −32
3 Therearemanyalternativeinterpretationsfortheobserved
1 beyondthesimpleqq¯ orqqqtextilewhichwasestablishedby peak[5,6],forexample,theauthorsofRef.[7]considerthat
] Gtuerells-Mofatnhnemmuolrteiqtuhaarnkhsaylsftecmenstuhrayveagboee.nTshtuedmieodlebcyumlaarnsytruauc-- a strong attraction between Λc and Λ¯c bind them together,
h so that Y(4630) may be interpreted as a baryon-antibaryon
thors,incomparisonthetetraquarkstateshavemuchlessbeen
p molecule. Instead, in Refs. [8, 9] Y(4630) is interpreted as
investigated so far. Generally, the tetraquark is suggested to
- a53S charmoniumstate. Also,theY(4630)isconsideredto
p becomposedofadiquarkandanantidiquarkwhichresidein be ind1uced by a threshold effect instead of being a genuine
e
colorantitripletandtriplet,respectively[1–3]. Becauseofthe
h resonance[10].
different color structures, the dynamics for molecular states
[ Among those proposals, the suggestion that Y(4630) is a
and tetraquark would be very different. In fact, in molecular
tetraquark state is more favorable [11–13]. In Ref. [11],
3 states, the two constituents are bound by exchanging color-
Y(4630) is identified as the ground state with its orbital an-
v
6 singletmesonsorbaryonswhichcanbedescribedbythechi- gular momentum L = 1. By reanalyzing the ΛcΛ¯c and
2 ral Lagrangian. Instead, the two constituents in tetraquarks ψ(2S)π+π− spectra, Cotugno et al. suggested that Y(4630)
areboundbydirectgluonexchange. Asforthequarkonium,
2 and Y(4660) [14, 15] could be the same tetraquark state,
0 theinteractionbetweenquarkandantiquarkisrealizedbyex- and is the first radial excitation of the Y(4360) with L =
0 changinggluons,asingle-gluonexchangecomposesthelead-
1[12]. InRef.[13],theauthorsstudiedtheopen-charmdecay
2. ingcontributionwhichresultsinaCoulomb-typeeffectivepo- Y(4630) → Λ Λ¯ by assuming that Y(4630) is a radially ex-
c c
0 tential, however, for the hadron formation, the energy scale citedstateofthediquark-antidiquarkboundstatewithhidden
6 is below ΛQCD, therefore the nonperturbative QCD effects charm. Byanothertheoreticalphysicsgroup[16]Y(4630)is
1 would play an important role which induces a confinement
interpretedasamolecularstatemadeofψ(2S)and f (980).
: pieceinthepotential.Similarly,forthetetraquarkcase,thedi- 0
v Asiswellknown,ingeneralthemultiquarkstatesmaybein
i quarkandantidiquarkinteractbyexchanginggluonsanddefi- tetraquarkstateswhicharecomposedofcoloredconstituents,
X
nitelythesinglegluonexchangeexistsandinduceaCoulomb-
or in molecular states which are composed of two color sin-
r likepotentialwhereasthenonperturbativeQCDeffectsshould
a glets, or their mixtures. Therefore, the mechanisms which
alsobeintroduced. Itisgenerallybelievedthatsuchinterac-
bindthecomponentsinauniquesystemandinducethemul-
tionmaybedescribedbythecolor-fluxmodel[2]. Following tiquark states to decay would be different in those cases. In
thisscenario, wewillstudythedynamicswhichnotonlyre-
this work, using the quantum mechanics we analyze the dy-
sultsin thedifferentstructures ofthe twoconfigurations, but
namical mechanisms inducing decays of the tetraquarks and
also determines their different decay patterns. Because the
themolecularstates,whereY(4630)standsasanexamplefor
newly observed Y(4630) is very likely to be a tetraquark, it
the study. Thus, a numerical analysis on the decay width of
wouldbeanidealplaceforcarryingresearchontheteraquark channel Y(4630) → Λ Λ¯ based on the tetraquark postulate
andmolecularstatesviatheirdecaybehaviors. c c
ismadeandaqualitativediscussionaboutthepossibledecay
processY(4630)→ψ(2S)π+π−whereY(4630)maybeamix-
tureoftetraquarkandmolecularstateispresented.
The paper is organized as follows: after this introduction,
∗Electronicaddress:[email protected]
westudythedecayofY(4630)withthetetraquarkandmolec-
†Electronicaddress:[email protected]
‡Electronicaddress:[email protected] ularinterpretationsinSecsIIandIIIrespectively,thenSecIV
§Electronicaddress:[email protected] isdevotedtoourdiscussionandconclusion.
2
II. TETRAQUARKPICTURE m =minthiswork. Theinteractionpotentialis
2
V(r)=V (r)+V (r), (2)
Inspired by the fact that the Y(4630) decays into charmed oge conf
baryon pair, one is tempted to conjecture this resonance as a
where r is the distance between the diquark and the antidi-
tetraquarkwhichismadeofthediquark-antidiquark[cq][c¯q¯],
quark. The one-gluon-exchange (oge) term V (r), which
where q is a light quark either u or d, [cq] resides in a color oge
playsthemainroleatshortdistances,is[18]
antitripletwhereas[c¯q¯]isinacolortriplet.
Here we take the diquark-antidiquark picture proposed by 4α 32πα
Brodskyetal. [2]thatadiquarkandanantidiquarkarebound Voge(r)=−3 rs + 9m2sS1·S2δ(r), (3)
together by a gluon-flux-tube into a color singlet tetraquark
wheretheconstituentsareseparatedbyasubstantialdistance andtheconfinementpartV (r)takesthelinearform[19]
conf
once they are created. The interaction for the system can be
well described by a generalized Cornell potential [17] since V (r)=br+c, (4)
conf
the constituents (diquark and antidiquark) are treated as two
pointlike color sources in analog to the configuration for an whereS1(2) and−4/3are,respectively,thespinoperatorsand
ordinaryQQ¯ quarkonium. thecolorfactorspecificto3-3¯attraction.bisthestringtension
In terms of the quark pair creation (QPC) model where a andcisaglobalzero-pointenergy. αs isthephenomenologi-
quark-antiquark pair is excited out from vacuum, we calcu- calstrongcouplingconstant.
lated the decay width of Y(4630) → Λ Λ¯ [13]. In that pic- Theδ-functioninEq.(3)isreplacedbyaGaussiansmearing
c c
ture,theQCDvacuumisexcitedandaquark-antiquarkpairis function[20]withafittedparameterσ
created. The quark-antiquark pair would “tear” apart the di-
quarkandantidiquarkduetostrongQCDinteractionbetween δ(r)→(cid:16)√σ (cid:17)3e−σ2r2. (5)
quarkanddiquark(antiquarkandantidiquark). Thenjoining π
the diquark, the created quark becomes a constituent of the
charmedbaryonΛ ,andaswellfortheΛ¯ . The spin wave functions for the Y(JPC = 1−−) state
c c with L = 1 is taken as Y = |0,0,0,1(cid:105) in the basis of
Inparallel,letusconsideranalternativewaytodiscussthe 1 1
productionofΛ Λ¯ intheframeworkofquantummechanics. |Sqc,Sq¯c¯,Stotal,L(cid:105)J=1 [11].
c c HerewetakethevaluesfromRef.[18]: α = 0.5461,b =
In fact, the color flux-tube results in a potential barrier to s
0.1425GeV2,c=0,σ=1.0946GeV.
forbidtheinnerconstituents(eitherthediquarkortheantidi-
By adopting the suggestion given by authors of Refs. [12,
quark) to escape from the bound state. In terms of a mod-
13]thatY(4630)isaradialexcitationstateofP-wave(L=1),
ified flux-tube-induced potential, the decay of Y(4630) may
the(anti)diquarkmassisdeterminedtobe1.878GeVwhichis
occur as the diquark-antidiquark bound system falling apart
viatunnelingthroughtheeffectivepotentialbarrier.Aftertun- closetotheDmesonmass. Thisvalueisalsoconsistentwith
thatcomputedbyusingQCDsumrulesinRef.[21]whereit
nelingoutthebarrier,thediquark(antidiquark)wouldimme-
is1.86±0.05GeV.
diately attract a quark (antiquark) from the vacuum to com-
poseacolorsingletΛ (Λ¯ )andthehadronizationprocessis ThefittedspectraarepresentedinFig.1, andthecharmo-
c c
niumspectracalculatedbyauthorsofRef.[18]arealsoshown
somehow similar to the picture frequently used to study the
inthefigureforaclearcomparison. Inthisframework,wefit
multiparticle production at high energy collision. Then the
thegroundstatetobe4235MeV,andsuchastateisconsistent
transitionprobabilitycanbecalculatedintermsoftheWKB
withtheobservedY(4230)resonance[22]and/orY(4220)res-
(Wentzel-Kramers-Brillouin)approximation.
onance[23,24]whichisalsoconsideredasatetraquark [25].
TheradialwavefunctionofY(4630)isplottedinFig.2.
A. Potentialmodel
First, we employ a nonrelativistic potential model with a
Cornell-like potential where some free parameters are ob-
tained by fitting the mass spectrum of heavy quarkonia and
generalized to the case for tetraquark, then we get the wave B. DecayofY(4630)asatetraquark
functionofY(4630)bysolvingtheSchro¨dingerequation.
The general Hamiltonian of a diquark-antidiquark system The interaction between the constituents in the tetraquark
(i.e. aquarkonium-likesystem)canbewrittenas cannot simply be derived from the quantum field theory yet
andphenomenologicallythedynamicsofthenonperturbative
p2 p2 QCDeffectswhichdeterminestheconfinementbehaviormay
H = 1 +m + 2 +m +V(r), (1)
2m 1 2m 2 be described by the flux-tube model. Moreover, as is well
1 2
known, whenthetensionoftheflux-tubegoesbeyondacer-
where the m (p ) and m (p ) are the masses(3-momenta) of tainbound,namely,thedistancebetweenthediquarkandan-
1 1 2 2
thediquarkcqandantidiquarkc¯q¯ respectively,wetakem = tidiquark gets long enough, the flux-tube will break into two
1
3
Here the interaction between the diquark and the antidi-
quarkatarelativelylargedistanceisdescribedbyamodified
potentialas[29].
1
V(cid:48)(r)=V(r) . (7)
e(r(cid:15)−rr00) +1
FIG. 1: The fitted spectra. We also present the charmonium spec-
traobtainedinRef.[18]. Thedashedlinesstandforthecomputed
masses, and the solid ones are experimental values taken from the
databook[26].
FIG.3: ThedecaymechanismforY(4630). Thedottedlinestands
fortheeigenvalueE,anda,baretheturningpointsatthepotential
barrier. Inthisfigure,weset(cid:15) andr tobe0.03and1.8fmrespec-
0
tivelyforanillustration.
In this scenario, we translate the flux-tube induced con-
finement into the potential barrier, and breaking the tube
corresponds to tunneling through the barrier. The diquark-
antidiquarkboundsystemfallsapartbytunnelingthroughthis
effectivepotentialbarrier,thenishadronizedintoaΛ Λ¯ pair.
c c
TheprocessisgraphicallyshowninFig.3. Bymeansofthe
FIG.2:Theradialwavefunctionφ(r)ofY(4630)withquantumnum-
WKBapproximation,thetransitionprobabilityofthetunnel-
bersn =2,L=1.
r ingprocesscanbecalculated.
Underthisassignment,thetransitionprobabilityisgivenby
strings and at the new ends a quark-antiquark pair is cre- 2(cid:90) b (cid:112)
ated [27, 28]. One can use a step function to describe the T =exp[− 2µ(E−V(cid:48)(r))dr], (8)
(cid:126)
breakingeffectas a
whereµ=m m /(m +m )=m/2isthereducedmassofthe
1 2 1 2
[1−θ(r−r0)]×V(r), (6) [cq]-[c¯q¯] system, a and b are the turning points, as shown in
Fig.3.
where r0 is a parameter corresponding to the strengthening One can obtain the effective velocity v of the motion of
limit of the string at where the probability of the string frag-
a particle with the reduced mass inside the system from the
mentationreachesmaximum.Atypicalscalefornonperturba- averagekineticenergy(cid:104)ψ(r)|p2|ψ(r)(cid:105)whereψ(r)isthewave
tiveQCDisΛ ,thereforeitisnaturaltoconsiderr should 2m
QCD 0 functionobtainedintermsofthepureCornellpotentialwith-
beoforderof∼1/Λ . Justassmearingthedeltafunction,
QCD outthefluxtubebreakingcorrection.Sincethebreakingeffect
weneedalsotosmearthestepfunction. Infact
onlyaffectsthelongdistancebehavior,itdoesnotchangethe
wavefunctionψ(r)andthedecaywidthΓ = v T isdeduced.
1−θ(r−r )=lim 1 , Y 2a
0 (cid:15)→0e(r(cid:15)−rr00) +1
so smearing the step function implies that we keep (cid:15) as a Assuming the Y(4630) as the first radial excited state, we
nonzero free parameter to be determined. In fact, if we do compute the partial decay width of the channel Y(4630) →
not consider breaking of the flux tube, the effective potential Λ Λ¯ .InFig.4,dependenceofthecalculatedpartialwidthΓ
c c Y
isthesameasV(r)giveninEq.(2). Therefore,takingintoac- onthefreeparameter(cid:15) isplotted. Thepurple,blueandcyan
countofthe“breaking”effectdoesnotaffectthecomputation curvescorrespondtothecasesofr =1.5,1.6,1.7fm,respec-
0
ontheY(4630)spectrum. tively. As(cid:15) increases,thepredicteddecaywidthincreasesas
4
III. MOLECULARPICTURE
As is mentioned in the introduction and the above discus-
sion,thehadronicmolecularpictureψ(cid:48)f (980)isanotherpos-
0
siblechoicefortheinnerstructureofY(4630)whichwaspro-
posedbyGuoetal.[16]. Itisnotedthatthemassandwidth
of Y(4630) are consistent with those for the Y(4660) state
(M =4652±10±8MeV,Γ=68±11±1MeV[15])within
error tolerance. By taking into account the Λ Λ¯ final state
c c
interaction,itisfoundthattheY(4630)maybethesamestate
as Y(4660), and the resonance can be a ψ(cid:48)f (980) molecular
0
boundstate.
FIG.4: DependenceofthepartialdecaywidthofY(4630)→Λ Λ¯
c c
on(cid:15),whereY(4630)issupposedtobeinn =2state,theBelledata
r
areshownintheplotforacomparison. Thedashedblacklineand
thegreenbandcorrespondtothecentralvalueanderrorforthetotal
width of Y(4630) measured by the Belle collaboration (Γ = 92+41
−32
MeV[4]). Thepurple,blueandcyancurvescorrespondtothethree
differentr assignmentsrespectively.
0
showninFig.4,whereeachcurveinthefigurecorrespondsto
aspecialr value,anditisnotedthateachcurvestopsatsome
0
valueof(cid:15),becauseatthat(cid:15)value,thediquarkandantidiquark
arenolongerboundbythemodifiedpotential. Itsetsacon-
straint on (cid:15). The dashed black line and the green band are FIG. 5: Diagram illustrating the constituents ψ(2S) and f0(980) of
respectivelythecentralvalueandtheerrorofthetotalwidth Y(4630)residingintheattractivepotentialwellwhichissupposedto
of Y(4630) measured by the Belle collaboration (Γ = 92+41 beahadronicmolecularstate. Theeffectivepotentialisinducedby
−32 theσmesonexchange,seeRef.[30]fordetails.
MeV)[4]. Varyingr whichrepresentstheflux-tubebreaking
0
effectdoesnotchangethegeneralpicture.
Itisnoticedthatthecalculatedpartialdecaywidthchanges
with r , but for any specific value of r , there exists a range
0 0 Nowletusstudythemechanismwhichmayinducethede-
of (cid:15) which allows the predicted width to be consistent with
cay of the molecular state. Because the two constituents in
the experimental data, namely, falls within the error toler-
themolecularstatearecolorsinglets, theydonotinteractby
ance region set by present measurements, which is rather
directly exchanging gluons, but only via exchanging color-
wide. Indeed, even though the calculated decay width of
Y(4630)→Λ Λ¯ issmallerthanthecentralvalueofthemea- singlet mesons, and the leading contribution is coming from
c c the σ(f (600)) exchange between ψ(2S) and f (980) which
suredtotalwidth,onestillcannotconcludethatY(4630)isnot 0 0
doesnotinduceapotentialbarrierasforthetetraquarkcase.
apuretetraquarkyet.
Instead,theinteractionprovidesapotentialwellandthecon-
Our numerical results indicate that the tetraquark picture stituentsareconfinedinthewell,asshowninFig.5. Thuswe
is able to predict the correct decay width of Y(4630), even shouldhaveM = M +M +∆Ewhere∆Eisthe
though not completely confident, we believe that its decay bindingenergyYo(4f6t3h0)emoleψc(2uSl)arstaft0e(9a8n0)droughlyis−30MeV.
mechanism could be considered as the diquark-antidiquark Inthetraditionalframeworkofquantummechanicsitisasta-
system falling apart via tunneling through an effective po- blestructure,i.e. Y(4630)cannotdissolveintoon-shellψ(2S)
tential barrier, and then diquark and antidiquark are respec- and f (980),however,duetothequantumfluctuation, f (980)
tivelyhadronizedintocolorsinglethadrons,andtheΛcΛ¯cpair canju0mpoutthepotentialwelltobecomeanoff-shellv0irtual
shouldbethemainproduct.Thisconclusionisconsistentwith particle. Ifthevirtual f (980)doesnottransitintotworealpi-
0
thatofRef.[13], inwhichasimilarresultisobtainedwithin ons,itwouldfallbacktoitsoriginalstateinsidethemolecule.
theQPCframework. The duration of it being virtual particle can be estimated by
Itisnoteworthythatifthefuturemeasurementindeedcon- the uncertainty principle as ∆τ·|∆E| ∼ (cid:126) = 1 in the natural
2 2
firms a rather large total width which is larger than our pre- unitsystem,thusthevirtualitytime∆τisproportionalto 1
|∆E|
dictionbasedonthepuretetraquarkstructure,apossiblemix- where∆E isthebindingenergy. Obviouslythedecayampli-
ture between tetraquark and molecular state should be taken tudeshouldbeproportionalto∆τ,namelythelargerthebind-
into account and other decay modes such as Y(4630) → ingenergy|∆E|is,theshorter∆τis,andthenthesmallerthe
ψ(2S)π+π−mayoccurwithnon-negligiblefraction. decay probability would be. By this principle, we can write
5
out an effective Lagrangian which induces the decay of the posedtoaccountforitseffects. Forexample,theQPCmodel,
moleculeY(4630) → ψ(2S)+ f∗(980) → ψ(2S)+ππwhere flux-tube model, QCD sum rules and lattice QCD, etc. have
0
the superscript ∗ denotes that f (980) is an off-shell virtual been successfully used to estimate decay rates, even though,
0
meson which later transits into two pions. The effective ver- withtheexceptionofthelatticecalculation,noneofthemcan
texatY(4630)−ψ(2S)f (980)is bedirectlyderivedfromquantumfieldtheorysofar.
0
g ←→ For the QQ¯ systems, the physics picture is clear, even
L= A† ∂ Bµ∂αφ, (9)
|∆E| µ α though a phenomenological model must be embedded to re-
flect the nonperturbative QCD effects and the computation
where A , B and φ correspond to ψ(2S), Y(4630), f (980) schemes are mature. However, for the four-quark states, the
µ µ 0
andgisadimensionlessuniversalcouplingconstant. Bythe innerstructureanddynamicswhichleadsthebindingandde-
equationofmotionitiseasytobereducedintoL(cid:48)whichreads cayofthestatearestillnotwellunderstoodandtherearevari-
as ousproposalsforthem.Inthisworkwestudythedecaymech-
anismsofY(4630)inbothtetraquarkandmoleculepicturesin
g(m2 −m2)
L(cid:48) = B A A†Bµφ. (10) the framework of quantum mechanics. Namely, we use the
|∆E| µ WKBapproximationtocalculatethedecaywidthofY(4630)
asitisassumedtobeatetraquarkstate,andthenqualitatively
The effective coupling given by authors of Ref. [16] is g(cid:48)2 = discuss its decay mechanism as it is postulated as a molecu-
(cid:112) 4π
t4h(eMψψ(cid:48)(cid:48) +anmdff0(9(8908))02),g2(cid:48)|∆hEas|/aµn(cid:48)ewnheergreyµd(cid:48)imisetnhseiorend.uTcheudsmwaesscaonf lqaurasnttautme wflhuecrteuaft0i(o9n8a0n)djubmepcosmouest taheviprtoutaelntpiaarltwicelelladnudeltaotear
0
identifytherelation transitsintotworealpions.
Definitely,alloftheassignmentstotheobservedresonance
g(m2 −m2)
B A =g(cid:48). (11) at 4630 MeV should be tested in the future by more precise
|∆E| measurements. In our other works [13, 31], we study the
casethatifY(4630)isatetraquark,itsfavorabledecaymode
WiththisassumptionanotherdecaymodeofY(4630)could should be Y(4630) → Λ Λ¯ which would overwhelmingly
be Y(4630) → ψ(2S)π+π− for the off-shell f (980) mainly c c
0 dominate its width, but due to the inelastic rescattering pro-
decayingintoπ+π−,asshowninFig.6. cesses between Λ and Λ¯ , some other final states, such as
c c
WiththegivenLagrangian,thedecaywidthwascalculated pp¯, nn¯, D(∗)D¯(∗) and ππ, KK¯ might be produced with mea-
inRef.[16]asΓ(Y → ψ(2S)π+π−) = 8MeV,herewedonot
surable rates, whereas, if Y(4630) is a molecular state, its
repeatitandadvisereaderstorefertothatpaper.
dominant decay mode would be ψ(2S)ππ and due to the de-
cayofψ(2S)andfinalstateinteraction,thepatternofthede-
cay products which will be experimentally measured would
be completely different from the tetraquark case. Thus the
measurementswouldprovidemoreinformationabouttheas-
signmentofY(4630). Wearelyinghopeonthefutureexper-
imentswhichwillbecarriedoutattheBELLEII,BESIIIand
evenLHCbinthecomingyears.
Moreover, we suspect if there is a mixing between the
tetraquarkandmolecularstateswhichresultsinY(4630)and
Y(4660),itwouldbeaninterestingpicture.
FIG.6:DiagramillustratingapossibledecaychanneloftheY(4630)
inthemolecularpicturewhichisY(4630)→ψ(2S)π+π−. Acknowledgments
WesincerelythankProf.Mu-LinYanforhelpfuldiscussion
on the issue of quantum fluctuation. This work is supported
by the National Natural Science Foundation of China under
IV. CONCLUSIONANDDISCUSSION the contract No. 11375128, No. 11135009, No. 11222547
and No. 11175073. Xiang Liu is also supported by the Na-
The hadronic decay is closely associated with the nonper- tionalYouthTop-notchTalentSupportProgram(“Thousands-
turbativeQCD,andalotofphenomenologicalmodelsarepro- of-TalentsScheme”).
6
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