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(1)Title. 中間子と反中間子の分子的状態に起源をもつΥ共鳴状態はあるか?. Author(s). 酒井, 源樹; 平野, 雅宣; 松田, 康夫. Citation. 北海道教育大学紀要. 自然科学編, 61(2): 33-40. Issue Date. 2011-02. URL. http://s-ir.sap.hokkyodai.ac.jp/dspace/handle/123456789/2344. Rights. Hokkaido University of Education.

(2) 北 海 道 教 育 大 学 紀 要(自. 然 科 学 編)第61巻. 第2号. 平 成23年2月 February,2011. JournalofHokkaidoUniversityofEducation(NaturalSciences)Vol.61,No.2. IsThere7'ResonanceOriginatinginaMolecularState ofaMesonandanAntimeson?. SAKAIMotoki,HIRANOMasanobu*andMATSUDAYasuo**. DepartmentofPhysics,AsahikawaCampus,HokkaidoUniversityofEducation,AsahikawaO70-8621 *P. rofessorEmeritus,HokkaidoUniversityofEducation,SapporoOO2-8502. **F. acultyofDairyScience,RakunoGakuenUniversity,EbetsuO69-8501. 中間子 と反 中間子 の分子的状態 に起源 をもつ 丁共鳴状態 はあるか? 酒井. 源 樹 ・平 野. 雅 宣*・ 松 田. 康 夫**. 北海道教育大学旭川校物理学教室 *北海道教育大学名誉教授 **酪農学 園大学酪農学部. ABSTRACT. Althoughi/rresonancehasbeensupposedtobeacc(charmonium)stateinessence,theauthors'peviousanalysisina coupled-channelmodelrevealedthatψ(4040)isbroughtfromaD*D*statebystrongchannelcouplingandcanbeassigned ashavinga3Scharmoniumstate.Toaddressthequestion"Isai/r(3S)-like`Y'statebroughtfromaheavymesonandantiheavymesonsystembychannelcoupling?;'amodelwasappliedtoa7'system.Theresultsoftheanalysisperformedin thisstudyshowedthatT(10865)assignedasSSisexpectedtobebroughtfromaB*B*(continuum)state,verymuchlike ψ(4040).. 11ntroduction. ducethisbehaviorinsingle-channelmodels.1)・2)More. recently,newparticleshavealsobeendiscovered3)makHeavyquarkoniahaveaspecialstatusinhadron. ingheavyquarkoniaphysicsaveryactiveareaofre-. physics.Oneofthereasonswhythey(andTstates. search.TheTfamilyplaysanimportantroleinthestudy. inparticular)areattractivetostudyisthattheircon-. ofexoticstatesandquark-gluonplasma.Thefundamen-. stituentquarkisveryheavyandmaybetreatednon-. taldynamicsQCDofhadronsallowsformsofexistence. relativistically,makingitpossibletoascertainthetrue. otherthanquark-antiquarksystemsformesonsandthree-. potentialamongmanytheoreticallyproposedforms.In. quarksystemsforbaryons.Theseformsofexistenceare. fact,historically,manyradialexcitedS-wave`Y'states. calledexoticstates.Onesuchstaterelatingtomesonsis. havebeenobservedandstudiedforlongperiods.The. knownasatetraquarkstate,andconsistsofadiquark. trendoftheirneighboring-levelsplittingsinradialex-. (gigz)plusananti-diquark(g394).Inrelationtothe. citedstatesshowsnon-smoothbehaviorastheradial. behaviorofT(3S),`Y'(4S)andT(SS),theexistenceof. quantumnumbernincreases.Ithasbeenhardtorepro一. tetraquarkstateshasbeendiscussed.. 33.

(3) SAKAIMotoki,HIRANOMasanobuandMATSUDAYasuo. FortheT(3S),itispointedoutthatthenaivemodel. consideredtobeflavor-independent.Weslightlymod-. doesnotwork.ThepossibilityofitscontainingaD-wave. i且edl2)theirmodelbyintroducingpotentialsbetween. componentisnoted4)anddataforT(3S)→T(3S)π. π. showapeculiardoublepeakstructurefortheπ spectruminT(3S)→T(3S)π. πmass π.Amongseveralideas. proposedtosolvethepuzzleoftheπ. πmassspec-. trumareanumberofproposalsincludingafour-quark. heavymesonsinopenchannels.. Eichtenalsorefinedthismodelwithahigherstring. tensionstrengththanthatchosenoriginally,andshowed. thatthemodelproducesgoodnumericalresultsl3)for. charmoniumstaticpropertiesincludingpartialdecay. resonances)ForT(4S),evidenceofanunexpectedly. width.Wehavealsoshown14)thatourmodelwitha. highrateofhigh-momentumJ/ψproductionhasbeen. leveloftensionhigherthantheoriginalCornellstrength. found,andMaetal.consideredthatthismaybeduetoa. canreproducedataonthemass,totaldecaywidthand. four-quarkstatemixingwiththeT(4S)6)BELLE'sdata partialdecaywidthsofcharmoniumstates.Asurpris-. ontheprocesse+e-→T(1S)π. π,T(2S)π. πnearthepeak. ofT(SS)resonancearefoundtohaveanomalouslyhigh ratescomparedtosimilardipiontransitionbetweenlower Tresonances.Ali,HambrockandAslam7)interpretthe finalstatesarisingfromtheproductionanddecayof `Y'b(10890) ,whichisaboundtetraquarkstate,asbeing differentfromT(SS).. Thenoteworthinessofheavyquarkoniahavebrought. ingandinterestingfeatureofouranalysisisthatψ(3S). isastatebroughtfromaD*D*statebystrongcoupling. withcc-states.Thepossibilityofψ(3S)beingamolec-. ularstateofaheavymesonandananti-heavymeson. hasbeensuggestedbyVbloshin-Okun15)andRujula-. Glashow-Okuni6). Itisthereforeinterestingnotonlytoknowhowour modelworksintermsofthemass,totaldecaywidthand. themmuchattentioninrelationtothephenomenonof. partialdecaywidthsofeach`Y'state,butalsotohave. thequark-gluonplasmaformedinheavyioncollisions.. ananswertothequestion"lsai/r(3S)-likestatebrought. ThemassesanddecaywidthsallowedbytheOkubo-. fromaheavymesonandananti-heavymesonsystem. Zweig-Iizuka(OZI)rule$)fbrsomecharmoniumand. bychannelcoupling?Furthermore,ifsuchastateex-. bottomoniumstatesplayanimportantroleasather-. fists,fromwhichheavymesonandanti-heavymesonsys-. mometerinmeasuringthetemperatureofthisquark-. temdoesitoriginate?"Inthisarticleweoutlinethe. gluonplasma9). studyofTstatesinacoupled-channelframeworkthat. Afewcoupled-channelmodelshavebeenappliedto. u. the`Y'system.Arepresentativeoneusesthe3Pomodel, whichisnecessarytodescribethecouplingoftheQQ. modelisbasedonaneffectivequarkinteractional)intro-. 鱒. effectiveinteractiongivestheconfiningplusCoulomb. 6S 4D. JBB*BSBS55B SBS. 3D 4S ..... 2D. ..... 3S. Σ. ducedbytheCornellgroupshowingQCDfeatures.This. (﹀・∪)羅. channeltotheheavyMM'channel10)Anothertypical. 11.  1D 2S. (....). potentialbetweentheheavyquarkandtheanti-heavy quarkinccchannelsaswellasthechannelcouplingpo-. T-family 1S. tentialsintheoff-diagonalpartsofthepotentialconnectingconfinedccandopendecaychannelsfortheheavy. Fig.1:Spectraofthe`Y'systemandofthelower-lying. mesonandanti-heavymeson.Stringtensionisthemost. bottomed-mesonandanti-bottomed-mesondecaychan-. importantparameterinvolvedinbothpotentials,andis. whilethedottedlinesshowthemodelresults.. 34. nels.Thesolidlinescorrespondtotheobserveddata,.

(4) IsThere7'ResonanceOriginatinginaMolecularStateofaMesonandanAntimeson?. hasalreadybeenappliedtocharmoniumstatesi4>The. nelshavetworelativeorbitalangularmomental=1,3,. observed`Y'statesandtheiropendecaychannelsconsid-. andtheothershavel=1.Theseheavymesonsinthe. eredareshowninFig.1.Asthestringtensionisflavor-. openchannelsaretreatedaselementaryones.. independent,itisquiteinterestingtoseewhetheramodel canreproducethepropertiesofbothcharmoniumand. Theradialpartofourcoupled-channelequationis schematicallywritteninthefollowingform:. bottomoniumstatesatthesametime.Weevaluatetheir チで Ψ(Y). masses,OZI-alloweddecaywidthsandfractions.Then, bystudyingthesepropertiesthroughartificialandgradualweakeningofthechannelcouplingstrength,wecan. _{7-(r)+rV(r)}`1'(r)=ELI'(r) . (1). Here,7"(r)andN(r)arethekineticandpotentialenergy operators,respectively:. ascertaintheirorigin,i.e.,identifythesingle-channelsold2. lutionsfromwhichtheycomebythecoupling.These T(r)= solutionsaresometimesknownasbarestatesolutionsio> 2Briefsummaryofourcoupled-channel. 0. 2オodr2. (2). 1d2 0. 2オdr2. and. framework. Gの+. L(L+1) ノ(Y). 2オOY2. でv(r)=. Inthissection,webrieflyoutlineourcoupled-channel. 1(1+1). (3). Y(わ+2. プTの. オr2. modelfortheheavyquark-antiquarksystem.Forde-. tails,seeRef.12),Yabusakietal.Inthemodel,heavy. whereオorepresentsthereducedmassesofthe(b,b)sys-. quark-antiquarkQQ(Q:heavyquark)channelsarecou-. tem.Thematrix,u-1representsadiagonalmatrixwhose. pledwithheavymeson-antimesonchannels.Asolution. diagonalelementsaregivenbythereducedmasses1/μ. forQQchannelswithoutdecaycouplingisaboundstate. ofthej-thopenchannelasshowninTableI.. freeofanydecay.Asolutionforthelattersystemwith-. ノ. ThepotentialG(r)inthebbchannelisofthestandard. outcouplinginvolveseitheracontinuumstateoranun-. linearplusCoulombictypewithaLorentzvectorcharac-. boundS-matrixpolestatewithalargedecaywidth.The. ter: a. couplingofthesechannelsbringsboundstates,resonant. G(Y)=λr+わ. 一 一,(4) r. states,andpolestatesandcontinuumstates.Thatis,a whereλ,わandαarethestringtension,aconstantpoheavyquark-antiquarkpairstatebelowopendecaychantentialandtheone-gluon-exchangeconstantofQCD,renelsmaintainsaboundstate,oneaboveopen-channel spectively.Thetransitionpotentialf(r)connectsthebb thresholdsobtainsadecaywidthandbecomesaresonant channelsandtheheavymeson-antiheavymesonchanstate,andsomewidepolestatesandcontinuumstatescan nels.Theexplicitfbrmノ(Y)isgiveninRef.12).The becomparativelynarrow.Whenthetotalwidthofsome. ofthesepolestatesisless,theymaybeobservableresonantstates.. TableI:Openchannelsofthemeson-antimesontaken intoaccountinthepresentanalysis. Inthispaper,wedealonlywithJPC=1--Tstates.. bbchannelsandsixopenmeson-antimesonchannels.. 弓 づ. Theformeraretwoconfinedbbchannels,onewithrela-. j(Channelno.) 11 ( ∠. Ourcoupled-channelequationconsistsofconfined. 4 〆 0. L=2.Thelatteropen-channelstatesconsideredare. 5. tiveorbitalangularmomentumL=Oandtheotherwith. Openchannel. BB +B*B)/VG (BB* ... CC. (BSBs +BsB、)ハr B*B*ss. showninTableI.Amongthem,theB*B*andBsBschan一. 35.

(5) SAKAIMotoki,HIRANOMasanobuandMATSUDAYasuo. 102. O lm10 m ら O O 冨. 102. 0. 50. 100. 150. 200. 250. 300. X11. 350. C(GeV-1). Fig.2:Rii(r)inEq.(9)plottedagainstthechannelradiusrinthecaseof`Y'(SS).. Y(r)isthepotentialbetweenaheavymesonandananti-. Alexenergyeigenvalueofthisequation.Toevaluatethe. heavymeson.Thestrengthλalmostwhollydetermines. partialdecaywidths,weusedtheformulaobtainedby. notonlytheQQPartofthecentralpotential甲(りbut. MoiseyevandPeskin.19). alsothestrengthoftheoff-diagonalpotentialf(r).It. AsshowninRef.19),theratioofthepartialwidths. shouldbestressedthatduetotheQCDexpectationthat. 1一ノandrノ. thestrengthλisindependentofthequarkspecies,we. openchannelsjandj'asshowninTableI,respectively,. usethesamevalueadoptedinthepreviousψanalysis.A. isgivenby. `Y'stateisexpressedbyamulti. ・ofdecayforaheavyquarkoniumstateψto. -componentwavefunction. r;/r;,一. Σ1・ll2/Σ1・. チ12・(6). `F(r)asasumofclosedquark-antiquarkcomponentsb(r) ll'. andopenmeson-antimesoncomponentsuノ(Y):. wh・ ・eαli・thec・mpl・x-scal・dflux・mplit・d・. Ψ(・)=Σ. わL(・)+Σ. 喝(・)・(5). ・fth・. ノー. thopen-channelcomponentinanasymptoticregionofr. Lノ,1. whereonlytheoutgoingcomponentexists. Thecoupled-channelequationistransformedby. complexcoordinatescalerotation,17)andtheresultis. 喝(reiθ)肇. ・1φ(reiθ),. thensolvednumerically.Themassspectraandtotal. φ(・e'θ)蕩. ・'k・(・eiθ)・. (7) (g). decaywidthsofψsl2)andTsl8)havepreviouslybeen. studiedusingthisframework.Aspartialdecaywidth. whereφ(re'B)isnormalizedtotheunitcurrentdensity.. analysisrequiresmoredetailedknowledgeoftherelated. Expression(6)isrewrittenintermsofthewavefunctions. wavefunctions,thereliabilityofthepresentframework. 吻. ・・. istestedindetailbythisanalysis.Wesolvethecomplex-. scaletransformed17)coupled-channelequation,andthe. 糎 massandtotaldecaywidthareobtainedusingthecom一. 36. 畿. 給(2ん. β ・)「eiθ ≡塊Rダ 〆・)・ (9).

(6) IsThereY'ResonanceOriginatinginaMolecularStateofaMesonandanAntimeson?. TableIL・MassesM,totaldecaywidthsT,. andpartialwidths∫. ンofdecaytochannelsノ(ノfbrBB,BB*andB*B*).Here,. MandwidthsaregiveninMeV.. Experimentaldata. T(2D) `Y'(4S) T(3D) `Y'(10860). 9460 10023 10161 10355 10580 10865±8. 0 0 0 A U. M. T(1S) T(2S) T(lD) `Y'(3S). rBB. Modelprediction 」 「BB・. ::・. 20.5±2.520.5±2.5. 110±13. <15.215.4±9.0. Inouractualmodelcalculation,weidentifythevalue. 48.4±16.5. M. rBB. FBB・. 0.14. 0.14. 0. 21.4. 16.4. 2.6. 2.3. 35.2. 4.2. 1.6. 12.9. 9460. 0. 10000. 0. 10154. 0. 10315. 0. 10437. 0. 10560 10679 10850. ::・. 0. ysisig>ofTstates.. Rill,(r)foracertainvalueofrwithoneatinfinity.To. Takingthescaleangle8inthecomplexrotation. ascertainthestabilityofthevalue,weplottedthevalplanetobeO.62rad,wesolvedourscaledcoupleduesforallノ. ≠ノ,=DDagainsttheradialdistancer」Fc)rthe. caseofλ=0.2andY=0.30asanexample,asshown. inFig.2.Werealizethatforsomeratiooscillations,. weshouldtaker>200GeV-1forallratiostoobtain. theirasymptoticvalues.Inthefollowingprediction,we. channelequation.Themass,totaldecaywidthandpar-. tialwidthpredictionforTstatesareshowninTableII. The`Y'-statepredictionssatisfactorilyfittheexperimental. data,20>consideringthefactthatourmodelneglectsspin-. dependentforceslikeLSandtensorforces,andthatthe. usetheratioatY'=280GeV-1.Oncethetotaldecay parameterswerenotfine-tuned. widthofaresonantstateisobtainedasanimaginarypart Thecouplingeffectofbb-channelstoopendecay ofthecomplexenergyeigenvalueofthecomplex-scale channelsisrelativelysmallforboundstatesbutlarger coupled-channelequation,partialdecaywidthsareobforresonantstates.Forthemassofa`Y'(4S)state,which tamedfromthevalueofthetotalwidthandequation(6) isthelowestmassresonantstate,thesinglebbchannel foralljandj'. modeltendstogiveratherlargervaluesig>thantheac-. 3Numericalresults. tualdata.However,theeffectofcouplingtoopendecay channelsandtherepulsivemeson-antimesonforceinthe. InthepresentTanalysis,weusedtheparametersλ,. Y,mu,andmS,whichwerefixedasλ=0.20GeV2,. Y=0.30,mu=and=0.33GeV,andms=0.45GeV,. asinourpreviousψanalysis.14)λisaparameterofthe. l-channel,makesitpossibletolowerthesingle-cha皿el. predictionandprovidesavalueig>consistentwiththe data.. Thepredictionsforeachpartialdecaywidtharealso. quarkbindingforce,andisconsidereduniversalamong. listedinTableII.Inourcoupled-channelmodel,aTstate. quarkflavors.TheYparameteristhestrengthofthe. isexpressedasasuperpositionofS,D,BB,BB*and. forceactingbetweenaheavymesonandananti-heavy. B*B*(1=1andl=3)channels,andsoon.Thefractions. meson.Itstemsfromtheinteractionoflightquarksin. fordominantchannelsoftypicalTstatesarelistedinTa-. heavymesons,andwecanassumethatitdoesnotdepend. bleIII.Theseshowthattheusualnominationofthetradi-. ontheheavymesonflavor.Forparameterspropertob-. tional`Y'statesasbeingnL(n:principalquantumnumber;. quark,weuseabb=0.473andmb=5.17GeV,which. L:orbitalangularmomentum)ofbbisappropriate.The. weresetinourpreviousmassandtotaldecaywidthanal一. modelpredictedthat`Y'(3S)isdominantlyS-wave-based,. 37.

(7) SAKAIMotoki,HIRANOMasanobuandMATSUDAYasuo. 10600. 1:11. 10600. 1:11. 11000. 11200. 11000. 11200. 0. 一50. (﹀ΦΣ と 一100. 一150. Er(M2V). Fig.3:Thestrengthparameterfdependenceofthetransitionpotentialonthecomplexenergyeigenvalue(massErand widthT=‐2E,)forresonantsolutionsisshowninthecaseofvectorconfinementcoupling.Here,fisvariedthrough1.O toO.Ogradually.Thepositionsshownby`1('(…)areexperimentaldata.. butinfactitisabout12%D-wavecomponent.Itisnoted. state,i.e.,eitherわ. thatin`Y'(3S)decaytotheT(1S)+2πanalysisassuming. theoriginsofthecoupled-channelsolutions.Theclari-. QCDmultipoleexpansion,a17%D-wavecomponentin. fledsituationforresonantstatesabovetheopen-channel. `Y'(3S)i sobtained.4>Thefactthatthestatecontainsafi-. niteD-wavecomponenthasaneffectonitse+e-decay. 五(r)oru五. のinEq.(5).Thus,wefind. thresholdisdepictedinFig.3.. Whenthecouplingisoff,thelS-4SandlD-3Dstates. width.Theexperimentaldatacanbeanticipatedwithin-. excepttheT(SS)statearefoundtobereducedtotheir. terest.. respectivebbsingle-channelstatesontherealaxisof. thecomplexenergyplane.Werealize,however,thatthe. 4Discussion T(SS)stategoestoaB*B*state,nottoabbstate,when. Asolutiontoourcoupled-channelequationtakesthe. thechannelcouplingisturnedoff.Weconfirmedthat. formofEq.(5).Whenthecouplingisturnedoff,thestate. thesituationisverystableagainstsmalldeviationsofpa-. goestoeitherb(r)oru(r)ofthesingle-channelequations.. rametervalues,forinstancebychangingtheO.3value. Wecallsuchasingle-channelstatetheorigin.Whatare. oftheYparameter.Thus,weunderstandthattheT(SS). theoriginsofthesecoupled-channelstates?Toanswer. statecorrespondstotheψ(38)stateoriginatingfヒoma. thisquestion,wecalculatedthemassesofthebbsingle-. D*D*state.Therelationshipsbetweenthecoupledand. channelsolutionsandthecomplexenergyeigenvaluesof. thecorrespondingbare(single-channel)solutionsfor`Y'. thecoupled-channelequationinwhichthestrengthofthe. statesupto`Y'(10860)aresummarizedinTableIII.. transitionpotentialisgraduallyreducedtozero.When. Aquestionthenarises.Whatisthestatethatturnsout. thecouplingisswitchedoffcompletely,thewavefunction. tobetheSSsolutionofthesinglebbchannelequation. willbereducedtoeitherasingle-channelstateorabare. whentheopen-channelcouplingisswitchedoff?The. 38.

(8) IsThereY'ResonanceOriginatinginaMolecularStateofaMesonandanAntimeson?. TableIII:Fraction(in%)oftheS-ID-waveandthoseofthemeson-antimesonchannelintheTwavefunctions(5).Mass M,baremassMoandtotalwidthTareourpredictionsgiveninMeV. M. state. 0 4 1. 1. 6. 7 0. 1 6. 5. 0.  . 0. 1  . 1. 7. 4. 76. 83.  . 21. 16. 0. 62. 3. 70.  . 12. 17. 4 0. 1.::. 71. 1. 10850. 77. 5. T(10860). 3. 12. 2. 10758. 89. (1=1)(1=3). 2. 10682. 10679. 0.  . 10560. 99. B*i蚕* 0. 10495. 1 5 (∠ 3. 10396. 10437. 4 ユ 0. 10315. T(2D) T(4S) `Y'(3D). 0. 10191. 0. 10042. 10154. 0. 10000. T(lD) `Y'(3S). D-waveBBBB* O 1 2 2 2 2 1 1. 9466. 0. 9460. S-wave 0. T(1S) `Y'(2S). Mo. TableIV:OurmodelpredictionsofTstatesotherthanthoseinTableIII.MassesandwidthsaregiveninMeV. Modelprediction nL. M. `Y'(10860)SS10850. bbprobability 3583%S7%D. Expt.data Mo. M. [uncoupled]. 1.::. B*B*. T(?)4D10961 `Y'(11020)6S11024. 2327%S,68%D. 10936. SS. 5178%S,12%D. 10996. SD. T(??)7311162 T(???)8S11173. 3352%S,45%D. 11163. 6S. 3952%S,40%D. 11217. SD. 10865±8110±13. 11019±879±16. 5Concludingremarks. answeristhe4Dstate;itisbroughtaboutfromthebb. single-channelSSstate.Wealsorealizethatthephysical 6Sstateisbroughtfromthebbsingle-channel4Dstate.. Asimilarsituationwillarizeabovethe6Smassregion.. Thisphenomenonisthoughttoresultfromacrossover. ofthenearbyDandSstatesasseeninFig.3.Itisnoted. thatthiscrossoverphenomenonwasalsorealizedinthe. ψcase;itwasseenbetweenthe2Dand48statesabove the3SstatewithD*D*-channelorigin.Wesummarize. thesituationforTstatesabove`Y'(SS)inTableIV.. Bythepresentanalysisof`Y'statesandtheprevi-. ousanalysisofψstates,wehaveascertainedthatthe. propertiesofheavyquarkoniaarewellreproducedby verystrongcouplingofquark-antiquarkstateswithheavy. meson-anti-heavymesonstates.Theverystrongcou-. plingofanopenmeson-antimesonstatewithheavy. QQstatesproducesanobservablestatewitharathersmall totalwidth.ExamplesofsuchastateareSSforthe`Y'. familyand3Sfbrtheψfamily.Thesesituationsmay. TheexperimentaldataforS-waveTstatessuggestthat. suggestthatsuchastatesimilartoψ(3S)or`Y'(SS)exists,. themassdifferencebetweentheneighboringstatesofS-. eveninlowerradialquantumnumbers,intheordinary. waveradialexcitationshowsnon-monotonicdecreasing. mesonandbaryonfamilies.Accordingly,thenaivepic-. behaviorastheradialquantumnumberincreases.Licht-. tureoftheconstituentquarkmodelhastobechanged. enbergetal.i>triedtoreproducethisbehaviorbychoos-. drastically.. inganintricateformofpotentialbetweenbandb.The. Wecanlearnfromthefractionofeacht/ror`Y'statethat. presentcoupled-channelmodel,ontheotherhand,natu-. thepresentmodelstronglymodifiestheleptonicdecay. rallyexplainsthebehaviorbypreparingtheSSstatefrom. widthresultofthesingle-channelmodel.Besideslep-. anopenB*B*system.Thedataandmodelpredictionsare. tonicdecay,tryingtoobtaincorroborativeevidencefor. comparedinFig.4.. thepresenceofψ(3S)orT(∬)isalsoimportant.Tothis. 39.

(9) SAKAIMotoki,HIRANOMasanobuandMATSUDAYasuo. 5)V.V.Anisovich,D.V.Bugg,A.V.SarantsevandB.S. Zou,Phys.Rev.D51(1995),R4619 6)ErnstMa,JamesPantaleoneandS.UmaSankar,Phys. Rev.D46(1992),463. 7)AhmedAli,ChristianHambrockandM.JamilAslam, Phys.Rev.Lett.104(2010),162001. [ ﹀ ΦΣ ︼ 嵩 ΦO⊂Φ﹂Φヒ一〇 ののoロΣ. 8)S.Okubo,Phys.Lett.5(1963),165. G.Zweig,CERNReportNo.8419/Th(1964),412 (unpublished). 1.Iizuka,K.OkadaandO.Shito,Prog.Theor.Phys.35 (1966),1061. 9)HelmutSatz,Nucl.Phys.A783(2007),249. 10)K.Heikkila,N.A.TorngvistandSeijiOno,Phys.Rev. D29(1984),110. Forrecentanalysisusingtherefined3Pomodel,seeT. Barnes,S.GodfreyandE.S.Swanson,Phys.Rev.D72 (2005),054026. 11)E.Eichten,K.Gottfried,T.Kinoshita,K.D.Laneand T.M.Yan,Phys.Rev.Lett.36(1976),500;Phys.Rev. D17(1978),3090;21(1980),203. E.Eichten,Phys.Rev.D22(1980),1819.. Fig.4:Massdifferencebetweentheneighboringlevels: OMn=M((n+1)S)‐M(nS).MassesaregiveninMeV.. end,studyofthepropertiesoftheseheavyquarkoniaat hightemperaturesmaybebeneficial,sinceconfinement atthesetemperatureswillplayaminimalrole,andwhen theconfiningforcediminishes,stateswithdifferentoriginswillbehavedifferently.. Finally,thisanalysisrevealedthathigherTresonances haveoriginsquitedifferentfromthosecorrespondingto bbstates.Theresultsarestillqualitative,andmustbe refinedtomakethemquantitativebyadjustingthemodel parametervalues. ACKNOWLEDGEMENTS. 12)N.Yabusaki,K.Kato,M.Hirano,M.SakaiandY.MatBuda,Few-BodySystems28(2000),1. 13)E.Eichten,K.LaneandC.Quigg,Phys.Rev.D73 (2006),014014. 14)M.Sakai,Y.Matsuda,M.Hirano,andK.Kato,FewBodySystems46(2009),189. 15)M.B.VoloshinandL.B.Okun,JETPLett.23(1976), 333. 16)A.DeRujula,H.GeorgiandS.L.Glashow,Phys.Rev. Lett.37(1976),398;38(1977),317. 17)J.AguilarandJ.M.Combes,Commun.Math.Phys.22 (1971),269. E.BalslevandJ.M.Combes,Commun.Math.Phys.22 (1972),280. B.Simon,Commun.Math.Phys.27(1972),1. Forareviewofthecomplexscalingmethod,seeY.K. Ho,Phys.Rep.99(1983),1. 18)N.Yabusaki,M.Hirano,K.Kato,M.SakaiandY.Matsuda,Prog.Theor.Phys.106(2001),389. 19)N.MoiseyevandU.Peskin,Phys.Rev.A42(1990), 255. 20)ParticleDataGroup,J.ofPhys.G37(2010),075021.. TheauthorswishtothankProf.K.Kato,Prof.H.Ma一. suiandProf.N.Moiseyevfortheirinterestandvaluable discussioninrelationtothiswork.. REFERENCES. 1)D.B.Lichtenberg,E.Predazzi,R.Roncaglia,M.Rosso andJ.G.Wills,Z.Phys.C41(1989),615. D.B.Lichtenberg,Phys.Rev.D49(1994),6244. 2)N.A.Torngvist,Phys.Rev.Lett.53(1984),878. 3)VincentPoireau,arXiv:hep-ex/0705.3716. 4)SmantraChakravarty,SunMyongKimandPyungwon Ko,Phys.Rev.D48(1993),1212.. 40. (酒井 源樹. 旭 川校 教 授). (平野 雅 宣. 本学 名誉 教 授). (松田康 夫. 酪 農 学 園大 学 非 常 勤 講 師).

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