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(1)Title. ψとΥに対するチャンネル結合模型. Author(s). 平野, 雅宣; 薮崎, 望; 加藤, 幾芳; 松田, 康夫; 酒井, 源樹. Citation. 北海道教育大学紀要. 自然科学編, 50(1): 9-24. Issue Date. 1999-08. URL. http://s-ir.sap.hokkyodai.ac.jp/dspace/handle/123456789/516. Rights. Hokkaido University of Education.

(2) . 北海道教育大学紀要 (自然科学編) 第50巻 第1号. 平成11年8月. i ISc i lof Hokkai do Un iver i )VOL50 ty ofEducat ences Jouma on(Natura s .I , No. Augス t S ,1999. I Mode lfor 少 紅l dT Coupl ed-Chmne. - A one‐open‐channel an副ysis -. 2 yosh iKAT02 M脳sanobu HIRANO1 , , Nozomi YABUSAK1,Ki Yasuo. 4 MATSUDA3and M【otokiSAKA1. 1Phys 8502 ion ido Un i i i ty ofEducat I ‐ r s csLaboratory, Hokka ve Pus ,SaPPoroo02 ,Sapporo ca立 2Di H k k i i i S 0 6 0 0 8 08 h l fS i d U t i G d S i i t ‐ n a n v e r s a o r o r c o o o c e c e o o s onofPhys cs a u a e y v p p , , , 3Depa 0922 i in l l i IEng in tomot tmen tofE1 t ‐ ege ve Eng l ec ro ‐Mechan ca . Co . ,SapPoro062 , Hokkaido Au 4Phys K h i 0 8 0 8 0 K 5 5 h i c fE i H k i U i i d i L b k d t t ‐ s yo uca on cs a oratory, o a o nver , us ro , us ro amPus. 歩 と T に対するチャ ンネル結合模型 平野. 雅 宣1・ 薮 崎. 望2・ 加 藤. 幾 芳2・ 松 田. 康 夫3・酒井. 源樹4. 1北海道教育大学札幌校物理学教室 札幌 0 02 ‐8 50 2 2北海道大学大学院理学研究科物理学専攻 札幌 06 0‐ 08 08 3北海道自動車短期大学電子機械工学科 札幌 0 2 62 ‐0 9 2 4北海道教育大学釧路校物理学教室 釧路 0 85 ‐0 80 5. Abstract. Charmonium ( th Charmed-andbot tomed-meson decaying )and Bottomonium (bb)areCoupled wi Cご ive ly,throughthe pair Creation processes ofl t i states,respec ght quarks‐ fect der ived f ive rom an ef. QCD. Thetrans i ion potent iali t s. l hami inal (decaying Channel tonian- ln 廿 l ) state ef , we assume a. ive f inal state interact ion between 菖leson and ant i t 】menologicalshor - range repul s lneson‐ The pheno massesand decay wid廿l eCoupled‐Char softhe s‐wavestatesof 歩 and T areobtainedbysolving 廿l ・ nel ion‐ Thef inalstateinteract iontogether Wi 1 i ion potent ia1 works we 1inreproducing ththetrans t equat. t he masses and decay Widths ofstates above open Channe1thresho1ds‐. 1 .INTRODUCT1ON. ial modelhasbeensuccessfulinexplainingtheheavy‐quarkonium Thenaivenonre lat ivist iCpotent. ( QQ )masslevelsbelow the 故resholdofdecayinto meson Qyl)andantimeson @①-. However ,abovethe. lthreshold,the naive modelhas to be modi f ied dueto the ef fect of strong OZエ lowed open Channe ーal 1the predict decays ion deviatesfrom the obser▽ed masses and strong decay wi ddh s ー lnsuch a mode caruIot be produced ,. Not ingthatthe heavy QQ stateisCoupledto hea\ 7y mesonstates ,ope止channel. l fra江. fects have been studied in coup1ed-Channe ef eworks for the heavy quarkonium by α. any. (9).

(3) . M‐ HIRAN0, N.YABUSAK1 ,K.KAT0,Y. MATSUDA and M・SAKA工. 10. researchers ‐. They areclass i f iedintotwo groups :a phenomenologicalapproach [1‐9] assuming an ad. ] based on ef [ hoctrans fect l ive quark hami i ion potent ial t tonians ‐ ,and a microscopic approach 10‐13 Compar ingbothapproaches the microscopicapproachseemstoagreeless wel lwi imenta1data thexper , than the pheno ]menological one‐ The KOgut ] was proposed as a phenomenological modelto investigate the l[ 3 ‐Susskind mode l ef fects on the charInoniu叢n spect 1 open-channe rum in ロ e assultーption of a one-open‐channeL. Kogut. lsystem oft コ heconf inedcごchannelandopen DD channelstates andsusskindstudiedthecoupled‐channe in an approximate way・ 1nthis mode l i ionpotent ialbetweenconf inedceandopen DD states t ,thetrans inalstateinteract ly assumed,and a phenomenologicalf ion between D-αIeson and i s phenomenological D‐mesoni ltocharmonium ly, we [4‐5] appl iedthe KOgut stakenintoaccount ‐Susskind mode . Recent. (少 ) and bottomonium system‐. l ef fects by solving 伍e coupled‐channe l (T) scussed the open‐channe , and di. We applied the complex scaling method [14] to the coupled‐channel equation including. decaying statesin open channel ion to the thout any approximat sin orderto solve unbound states wi boundarycondi ion. Asaresul inalstateinteract ion(FSI t t )playsanimportantrole ,wefoundthatthef in thecoupl ing mechani ion ofthe observed quarkonium sm between QQ and M M andintheexplanat ive li i fect rmedthatthe one‐open‐channel mode spect ra‐ VVe al so conf s ef ‐ ln the microscopicapproach せl ions between quarksrespons iblefor massspectrum and , einteract i ion( trans t QQ→. MM)arepostulatedtobegivenby aneffectivequarkinteractionhamiltonian [10‐13]‐. ive interact ial but al The ef fect i ion ion ha i l t tonian gives not only the quark potent ] α 1 so ロle trans ial igated by a few groups [ 13 ] based on the 10 ‐ potent . The spectra of 小 and T have been invest [ m・croscopIc approach ‐ Eichten et aL [10 ,11] and Jaronskiet aL 12] employed the Lorentz vector. ] ining interact ion in the anaiys i type of conf 13 s of雪 身 and T spectra‐ ln addition, Barbour et a1 ‐[ invest igatedthespect ther ofthem could repro- ththe Lorentzscalartype rum of 汐 wi ‐ However ,nei lthreshold. ducetheobserved data abovethe open channe These worksfol lowingt id nottakeinto accountthe FS1in the open he microscopic approach d ion about the important role of the FS1 f rom our previous channel s ‐ Based on our conclus is fects on t he open channel usingt he s paper we elucidate the FS工ef phenomenologicalanalys ,in thi ial ininginteract ion ロロcroscopict i ion potent t rans ‐ Forthe Lorentz property ofconf ,boththe Lorentz ing andthe i ts are di vector and scaiar cases aret reated on thesa ]mefoot rresu1 scussed and Compared ludingthe FS1 ioninc th each other wi ‐ To perform thecoupled‐channelcalculat , weemploythesame ] method asin our previous work [4 ,5 - M[any D 4凸な channels are open as shownin Fig.1,anditis not easy to treat a l lofthesechannels inc ludingthe i ime ththephenomenological r manyparametersatthesa ]met ‐ lnourpreviousstudies wi h l l i i ial model i ion potent tesat trans t si s qui sfactory. There‐ , wefoundthatthe one‐open‐c anne ana ys. fore i ion potent ialand FS1 igate i iveroleofthe microscopictrans tandthe qual tat t , weinvest ,to unders d 値e mass and wid伍 of 少 and T within a one‐open‐channel model e chose the D*D*- ‐ Wァ , an BB- lbecause ofthe fol lowing exper imental propert ies [15 ‐ope法channe charmel as the one. 3S)is 3 1 ) 小( .. lyintothe D*D*channe l 43S ) sjustabove 値e 社l reshold, obse IVedto decay dominant . ,and2)T ( ,whichi hasonly one decay‐mode ofthe BB channe l ‐ l lows ion 工 1 ive t The out l ine oft he hi s paper is as fol ‐ ln sect , we explain our mode and der (10).

(4) . 11. Coupl I Mode lfor 少 and T ‐Channe ed. 多. open‐channe l s. Y. open‐channe l s. . (Geの. (Gev). 4 5 .. 10 5 .. ID 2S------. 3 S-. DD. - . BB B目 BB. lo 2S. 3 5 .. 9 5 ,s .. I S- 3. l FIG‐1 s ‐ observed spectra of 小 and T with some open channe. ion potent ial ion l 工 1 ief ly explain t iona1 11 t i l l l l ・croscopic t ransi l ethod‐ the e co1nputat . ln sect , we br ing method‐and presentthe null led complex scal tsoftheCoupled-channelequation‐ ーericalre so-cal sul 工n section IV, we di he effects of the FS1 on masses and the strong decay width of heavy scuss t ion れ quarkonia. our conclusions are givenin sect. 11 C MODEL . ONE‐OPEN‐CHANNEL SEM-MICROSCOPI. A‐The mode I Anef fect iveinteract ion hami l i tonian,差Z加ご tuentquarksi he sassumedtobecomPri sed oft ,ofconst ion forshortto medium ranges and the conf ion for ining interact one‐gluon exchange(OGE)int eract l . edium tolong ranges: l oGE十 万「CNF 亙ぎ ご= 万r れ. 1 ( ). They are expressed as. 腔 』 お た3 zぬ少郷. 傷鱒‐籾. γがゑ せ}”. ーギ I D}“α ず ( 勿 α)x{ 戸E. ”) ,. -ギ ーD}ぁ 仔}α ず ”) 吻ば)x{ 啄( ,. 2 ( ). 3 ( ). ininginteract ion,wecons ider )runsover quark flavors‐ Asthe Lorentzproperty of 値econf re/ぴ′ whe. 値e vector and scalar casesto be on thesamefoot ing asshownin Eq.(3) . lnthevectorcase , wehave droppedthe γ,ごpartby assuming anonre lat ivist icapproximat ion‐ Thepotent ial )inEqs s y(に′-エー ‐. 2 3 ( )and( )are assumed as 4 G E vo ーガ-工』 ( )=-T. αず. E F 可’. 4 ( ). V腿F (メ ーエ1 );ぇ1ギーエ1+b。&. 5 ( ). ( 1 ) 1.

(5) . M. HIRAN0, N‐YABUSAK1 ,K.KAT0,Y・ MATSUDA and M・SAKA工. 12. ieS iVe 1y‐ ) pr0pert Wheretheindices り and s denotethe Lorentz vector(り) and Sca1ar(S ,reSpect ion for the s‐Wave ofthe QQ channel and for the The two‐channe l coupled Schr6dinger equat i P‐Wave ofthe MM channelis Wr t ten as 6) (. ″ 夢(γ)= E W(γ) ,. ‐ ialenergy operator V(γ )are ic energy operator T(γ)andthepotent Where 旦 = r(γ)+ V ) ‐ Thekinet t ten as wri l. d2. 0. -★Q 』 dγ2 l d2 r⑦- ( 一 麦 湯) , - 2gM メメ 0. の. v ( 申さ ラ メ 湯) ; ,. 8 ) (. ion ず(γ)i s given as andthe Wavefunct wの AS. MM, we take D*D* and BB for 歩 and. 彊) -. ◎. ined ively. The reduced mass ofthe conf respect. ion t l mesons by 厳M‐ The 六γ )i s the transi s represented by ”Q and that of open‐channe channeli iaI Whichi sobtainedf rom Eqs potent .. 2 3 )and( )and Willbediscussedin detailinthenextsubsection‐ (. The potent iaI G (γ )istheinteraction oftheQQ channelobtainedfrom theeffective hamiltonian,Eqs‐ 2 i 3) tten as ( ) and( s wr ,andi. G(“ 一九γ一 昔 偽 讐. 10 ( ). び‐. ‐ low massesto be measuredfrom the open‐channel The constant parameter U isintroducedto al. threshold. ion(FSI inalstateinteract Moreover weassumethefol lowingf )between meson(M)and antimeson ,. (M) : Y(γ)= ”8ーしγ ‐ ,(シ≧○). Thi ‐Susskind [3]. s Fsl has been introduced by Kogut. (11 ). We employ the same Fsl as Kogut ‐. l ionforthepresent mode ialobtainedby a microscopiccalculat ion potent Susskindbut Weadd atransi t lsemimicroscopic model ) (oneope卦chamle . i ion potent iaI i ion ofthetrans t B. Der vat. Wehere derivethetransition potential/(γ)from the quark‐antiquark effectiveinteraction given in Eqs 3 2 ) )and( .( ‐ The Feymoanamplitude 財 for QQ 一 M M iscalculatedfrom the Feynman diagram. hel ightquark corresponding tot i ici ly Wr ten as i t t ion shownin Fig‐2. The ampl tudei sexpl rcreat pai. 財 = -Z (朋Q十一脈) ,. (12). } {死(P2 }y(げ,-p, 朋Qニ Q{薙 ) )rり(p ) ) )r”@, (げ, 3 ,. 13 ) (. ( 2 ) 1.

(6) . lfor 少 and T I Mode Coupl ed ‐Channe. } }V(口4- 比) {”(p2 肌Q=C萄{ガ ) )rり@4 ) )rり(p3 (〆4 ,. Cだ. 13. 14 ( ). 蕊q )縫え◎,. . . l i )quark,九αis Gel ‐Marm’ S racsPinors,Pれrepresentsa momentum ofthe %1山 (ant where “ and り are Di i ionofthe “th(ant )i ix,and じれi )quark‐ The y(ゼ“-pれ 九‐matr s me momentum s mecolor wavefunct 1 0 l ion of Eq 5 4 ) representat )or( .( ‐ Thesymboll represents γ forthe vectorcase ,andlforthe sca ar Case‐. ightquark Pai F1G.2 r Creation am Causedbythel ‐ Decay diagr. ibut ionf Sincethecontr rom 脳窮,wehereshow onlytheprocess rom 肌Qtof(γ)i sthesameasthatf ionintoanorderof ivi icreduct ion ofEq‐(13) wi forcalculat ingtheformer th expans st ‐ ln 値enonrelat. 2 値e quark coupl ing operator UQ 1/c (q)isint roduced as ) ( , 朋Qαz,露ゴ ロQ (q)×.のゎ. 16) (. f ispinorsand qi ing operator ひQ ical ly given (q)i s が,一p, Where 尤 and の are Paul sspeci ‐ Thecoupl as. 17 ( ). 1 8 ( ) 1 9 ) ( ini i l ty,andd t : latthe energies Eqofthecreated where we haveassumedthatt eheaVy quark massisinf ion,weemploy a nonre lat ivi l i t iCapproxi ion βq^)“も: lnat st ghtquarks are equaltoeach other ‐ lnaddi. &=J煽許可北J煽許可~伽?. 20 ( ). Takingthe Four iertransformat ionof 豆Q threspectto q (q) wi ,weobtainthecoordinaterepresenta‐ ions t. F (〆)=c Q玲. お ▽/ ー (〆) ,. 燃). 際 (〆)=c Q曲. お ▽/ 呼 (〆) ,. 鰹). ( ) 1 3.

(7) . 14. M. HIRAN0, N.YABUSAK1 ,K.KAT0, Y‐ MATSUDA and M‐SAKAI. ひかF (. ) 物 -劫W w(〆 ,. =c Q向. 23 ( ). ’ 〆i rom a Q where V(〆) staken as 値ecoordinatef shave already been givenin Eqs.(4)and(5) ,and ioni incethel ightquark pai r(qd)creat sassumedtooccuratalocalpointasshown quarkto a qquarks in Fig.2 rom ‐ Wretake another coordinate r ,f. Qto Q.. Then me coordinate from Q to q i s r‐ 〆-. And “ris def ined asthecoordinatefrom thecenter ofthe massof Qqto thecenter ofthe massofQ q , where ガ ニ 粥Q/(粥Q十 粥一 ions of ion potent ialfor QQ→MM i Thetransit th the wave funct s obtained by folding ぴ(〆) wi lowingform ( imeson(M)and QQ re lat ive mot ion ixelementshavethefol meson(M) see ‐ The matr ,ant Appendix for detai l ) : s. 1防+ 脆 匝 びぢの に〆平 A <M 肺)M”ぁ) , 嫡 逓 加,鳶 の,. 2 4 ( ). where. 乎 A 噸ぢ ( )サー ボγ ) γ , 鹸 J粥 彦 の = 灘 赫 ゑ だ の 』& 身 〆沢* “ ‐. 2 5 ) (. Thetotalan≦醜lar momentum isrepresentedbyノ andi tsz‐componentby 考,and れi sshorthandfor ix element of CQ or the 7 2s‐ state ofhea\w quarkonium‐ Thefactor4 お/9 comesf rom the matr. 箇 of. Eq.( 1 5 )whentheinitialandfinalstatesarecolorsinglets. 尺 isthe wavefunction ofrelative motion ix element wi ibesthe matr between M and M,and s仏 参,たら, 痔,〆) 鋤F金・descr th respectto angular ion off(γ)i momenta・ Thefunct s given as. 戸E F ( の‐ 筈 煮N2ぞ 陀十, ,◎ ÷ - 影) ,. 2 6 ( ). か(か 舟 貫 か2 (÷ ) 陀蕊F ;÷ ;-多2 , ,. 27 ) (. F か の-冬 禽 か2だ影{÷ 凡けす;-影)-, ;-号〆)- 呼 子} ,侍;÷ ,. ㈱. and. 洗. ÷ ,. (凋. ively‐ Here.E(α fortheOGE conf ining vector iningscalar typeterms typeandconf s 廿le ‐ ‐ ;る;×)i ,respect , icalfunct ion luenthypergeometr conf ‐ inalstates ively. Thisi The u and dpair creations contr ibutetothe D*D*and BBf sospin ,respect iedbythespinsof i i f degreeoff tude A, i sspec reedom givesthefactorof 霧 onj(γ) - Thedecay ampl ion potent iali hetransit the meson and anti・neson states. Thi s expressed by s spin dependence on t ing S拓卓ああノ考”)&“& us : ・ , ヂ(γ)= 霧 刃 馬 可 (γ) ,. 30) (. S =S 仏 友 あ あ,孟考, 粥) , ,. 31) (. 2 1き ぎ. where. ( ) 1 4.

(8) . Coupl I Mode lf -Cha l l me or 小 and T ed. 又 ノ容姿ニ ノラ (入 ニム ニ1 ) ; D*meson , ,. 32) (. 刀 /容姿ニ ノヱ (五 ;ん =0 ; Bmeson) , ,. 32) (. 2 き1き. 2 号 ‐ 1き. Thus efactor , せl. 15. ″isforthecharmonium system,and. i tomoniu コ ヒ n one sforthe bot ‐. i ion potent ialcorrespondingt Thetotaltrans t )in o/(γ lschrじdingerequat ion thecoupled-chal ) ule , Eq.(6 ,is given by. (γ)=/oGE (γ)+んSNF(γ) ん, s , ,. 34 ( ). inguish t ining and scalar conf ining types he subscripts り and s dist コ 1 1 1 Where t e L0rentz vector c。nf , ive.y. 工 le to note d ion potent ial has a conspicuous hatリ ヒ ー tis Worthwhi respect e scalar type transit ferencef i ion potent di f ialincludes a para]meter ら′ t ro 1m dhe vect。r one - The scalartypetrans s Which inear potent ial givenin Eq‐( 5) comesfrom theconstantterm in 値el ‐. 11 1 .NUMERICALRESULTS. To solve the coupled‐charme lschrbdingerequat ion i。nal me値od , Weemploy a computat , Eq‐(6) , ledrenormal ized Nu lopedby johnson [16]‐ This memodi theso‐cal imerov me値od deve sof…箪eatuse for。btai l “ ing accurate eigenvalues o i ISch コ rbdingerequat fthe mul ion‐ t -channe Totakeinto accountt ionofdecaying Wavefunct ionsindhe△4D佳 openchanne heboundarycondit l , ing memod [14] to thecoupled-cha・melequat ion‐ ln the complex scal ing We apply thecomplex scal ddhs are obtained as complex eigenvalues ofthe compleX method, resonance energies and decay wi ion scaled schr6dingerequat. 互(β)▼β (γ)= & ヤタ (γ) ‐. 35) (. Thecomplex‐ l ion ヤβ t。nian 甘(β)and Wavefunct scaledhami (γ)areobtainedunder metransforma‐ ion s( t β )γ → γexp(霧)for a radialcoordinate γ:. 1 亨β ‐ 冴βニS(β )亙S(β ) )=S(β )▼‐ , (γ. 36 ( ). Fore igenvaluesofthecomplex‐ ion,‐me ABCtheorem [16] guarantees me scaledschrbdingerequat fol lowing e igenvalue propert ies ) bomld‐ :i state and resonance‐state energies do n。t change due to ing,andi i )theresonanceenergy(E, scal )and Width(r)areobtained asrealandimaginarypartsofthe ing parameter β When β>( complex eigenvalues Ee r/2isindependent of mescal 1 / 2 ) , &=E -Z , e‐g‐. 1せ/2E一 tan冊 ‐ Wァecalculatethe massesand 鉱estrongdecay Widmsupto 鉱e4S‐ statefor 少 andupto me5S- state for T. The parametersforthe QQ charme lareshownin Table. thoughthecoupl ing constant α ‐ A1 in/oGE(γ)dependson 値e quarkf lavors ti ing constantofthe sapproximatedto be 鉱ecoupl ,Q and q ,i. ( ) 15.

(9) . M. HIRAN0,N.YABUSAK1 ,K‐KAT0,Y. MATSUDA and M‐SAKA工. 16. TABLE I . Parameters used in the present ion. calculat. mQ i s the mass of a heavy quark( c. ight quark s the created l and b quark) . mぬdi h i i k 入 t t d ) ( d r ng tension‐ r e s s mass u an qua - / T h tersf 4 3 Thevalueofαi ) r ( a ame or 歩 ep αs s ‐ f U [ ] 1 0 R f t h t excep or . are esameas e.. ~ β ト T. Q姦system, αrQ).. . . . GeV. GeV. GeV2. 1 840 . 5 716 -. 330 0 ‐ 0 330 ‐. 0 183 ‐ Q183. . . GeV 520 0 . 475 0 ・. -1 175 ‐ ー -○ 951 ・. ial The parametersfor 少 arethesame as Eichten etal ‐[10]‐ Theconstantpotent. f inedin Eq‐(10)aredeterminedsoastoreproducethe masses measuredfrom parameters び≠and びr de ion ingtens ively‐ Thestr V IB=10 557 GeV,respect 014 GeV and21 lthresholds theopen‐channe *=4 p . ‐ ,2N1 5)i staken commonly for 歩 and T‐ parameter ぇ in Eq ‐(. l ingle chal口le Wi値 these Parameters ,thes. ionin low ロleopen ロl ions accurately reproduce boundstatesbe resholdbutshow alargedeviat calculat i ted states abovethethreshold exc ‐ ial ん を)f i ion potent d lf fthetrans ln Fig.3 t or 汐 and T systems , whereん , weshow thera ia orm o. )is presentedforthree values of ケ . ( γ. Theradialform of 左(γ) has a ratherstrong ケザdependence ‐. th 故e ionpotent ialん(γ)hasashort The Lorentzvector‐typetransit ‐ rangeradialform in comparlson wi Lorentzscalar・ one 左(γ) - ixed:1)therangeparameterシand2 ion,twofreeparameters mustbef )thestreng故 ln ourcalculat ionto y and ジi t ion Eq・(11 spresent inalstateinteract ) one morefreeparameter わ′ y ofthef sinaddi E 2 . ial inthe case ofthescalar typepotent ‐ ,shownin q‐( 8). i tsinorder Wehavesearchedforthebestf. heexperimentaldataintherange シ;○-.()○・85 Gev, Y =○(}4-O Gev and -わ′sニー-4{)2・9 to reproducet ing valuesofthe る′ igatedby al ter i Gev一. s ofthescalartypeareinvest s ‐ Severalcases for an analys paranleter.. 1 ) (Gev‐. . ) (Gev柵. i ion potent ial FIG・3 t s of 少 and T. The dependence ofthe b′ ・ Trans ia li fthe transition potent tedin the herad ia lfor ・n o s plot paranQeter on t i T h f i i b t t )T e sshown ( ) ( n n n v e c o r : 坊 ec o scalart a 班 ) g ~ 唾 )eforeachcase - ′ = ー10 -2o i b d h d l h d dd T h d d t ine d a r t t e idl n - o ‐ a s e e a n e - a s e e o asasol ‐, . s ‐ , , 1 respect ‐ lar types ing con ively represent f i l “ i G V s a O c n e g and -3 ‐ ‐ , ,. ( 1 ) 6.

(10) . . 17. Coupl I Mode lfor 小 and T ed ‐Channe. lowing 尤2-value: To search for 壮絶 best Para]meter sets, we calculate 廿 i efol 尤. 2 r禦ーr肌 2 ぬ d 』) 十( △r ), △朋. 37 ( ). imental(calculated) mass )aretheexper (肌。α)and r ゅ( α sused where △ 朋 ;△r=1O MeV i z x p ,and 班e ively‐ The 尤2 ining ‐plotsofvector ‐conf statefor T,respect and width of 伍e3S‐ statefor 歩 and me4S- leyinthe interact ioncasesareshownin Fig‐4 ‐ Forboth 少 and T,aval. (a ). (b). 75. ‐. 30. 100. x. (Y,ジ) -ploti sfoundnotonly. /. ‘. 20. X. 墓. ん. . I0. . . ノI 8 . Y メ. 鱒. . . 4 .. . . 2 FIG・4 - )3S(少)and(b)4S(T)by meansofthevector- a P1otofresults of( ・ z (γ,シ) 2 leyi ial typePotent sseen notonlyi n analysesofthevector-tyPe Potential . The ズ ‐val but also ofthescalar‐type potential ‐. TABLE II i zed . The nuumericalresults of 歩 and T are summar ‐ The. l l FS1parameters(Y and ジ)arechosenfrom the 尤2 - va ey . Theresults of. ‘naive“ ばlode1 r in 1 - h nn 1 the‘ ts wi thout open-channe1 effect. a e s g e c a e resul ‘ The”vector“ and‘ ine- he Lorentz property oftheconf scalar“typel r ー eant. lues are given in MeV. ln i mentter nthetransition‐ Va Exp ] ‐[15. na・ve. r. I 1 1ass. r I ーass. scalar. vector I r ーass. r. 立1ass. r. [ set B] わ 4481 54. 4S. 4415. 3S. 4040. 4113. [ set A] α 4438 31 4043 22. 2S. 3686. 3688. 3660. 0. 3687. 0. 1S. 3097. 3097. 3090. 0. 3096. 0. 4289. 311. 4403. 866. 43±15 52±10. 4472. le po T 5S 4S 3S 2S 1S. I0865 110±13 I0580 21±4 I0355 IO023 9460. 10856 10622 10355 10023 9460. e pol. 53. [ setC] c 10849 36 10595 26 10346 0 10019 0 9459 0. [ set D] d 10847 19 10580 24 10349 0 10021 0 9459 0. l0770. 11130. αY =3 4 Gev ジニ0 6 GeV2 ‐ . ‐ , bY =1 8 GeV ジ=○ 5 GeV2 ヶ ; ー2 7 GeV- 1 - - - - , , s CY ニー 8 GeV ジ=○ 7 GeV2 ‐ ‐ ‐ , dY ニー 8 Gev シ=○ 6 GeV2 ろ′ ニ ー1 5 Gev-1 . - ‐ - , , s. ) ( 17. 4039. 708. 626.

(11) . M‐ HIRAN0, N‐YABUSAK1 ,K.KAT0,Y. MATSUDA and M.SAKA1. 18. 6 0 0 0. ÷ 一二 , 三二;; . . . 4 0 0 0. 0. 1. 2 Y[ G … ▽ ] e. 3. 4. 0. . 0. 1. 2 Y[ G v ] e. 3. ー干 ≠. も醐. 2 G V Y{ ] e. 3. 4. ー十三 キ ,。 デー 0. 4. 1. 1. 2 Y[ G V ] e. 3. 4. FIG.5. obtai statesof 小 ned massesand Widthsareshov靴 forthe3S‐and4S‐ ,and h l h f h F S 1 h t t fT t γ o te the4S‐and5S‐ t MM po es a eac s reng wi stateo parameter . l ibeswidth These values of y arechosenf rom theval ey ‐ Thecross . Thebardescr i ionpotent ial t t“)et l imentaldata )thevector represents せ rans ‐ a eexper - ①戸山e ‐ 燐( ′ 1 ‐ i l 一2 V 故 7Ge ‐ T c i ion potenta‐ TheらsPa t~やetrans t ) evector ‐ rameteri s ‐ scalar ‐ i ionpotent ial ial d t )thescalar t“>et t“コ - rans etransitionPotent ・ Theるをparameter ‐ ( d square s l b tate of 少 iamond b i 5 GeV‐1 r r nt4S e e s e n s ‐ a o a s r p s -1 y m ‐ ‐ The d ,3S‐ , * * d ive ly tateofT and BB,respect s and D D ,and5S- ‐ The masses an Widthsof ,4S‐ MM polesbecomelarger whenthestren罫h Y‐isincreasedineach case. lnthecase th 廿i f3S(少)and 4S(T) are overestimated in compar・son wi e of Y=0 ,the massE海 o imentalda ta exper ‐. dヒ セ メ in the vector caSe but al soin the scalar caSe s ofboth bound and ‐ Thecalculated masses and Wi ley of北2 ththeobserveddataexcePtfor tatesalongtheVal resonances ‐Valuesarein good agTeement Wi ion,the ら′ i the y =o case l lbe discussedlater‐ 1nthe Present calculat schosen sparameteri , wh ch Wi 7 GeV as -2 .. l icalresul ts and obse for 汐 and -1 5 GeV-lfor T‐ ln Table l 1red data are . ,theoret. lnarized‐ sunl. ing ofthe continuum ハ江ハ江 states wi t ht he ▽Ve obtained new resonance states caused bythe couP1 t力 quadranto i× Polesinthe4 fl 比 l bound QQstates e . SuchresonancestatescorresPondingtothe S-matr ftheimaginaryPartofthecomP1exPoleenergy ledtheD僅公4Poles comP1exenergyP1ane are herecal ‐ 1 l l日hatDAD僅Polearesonancestatedue i tsreaIPart thi sextremelylargein comParison Wi ,Wecannotca h hd ibe massesand Widths i fet ime i ionso ft heD江DAPoles t tsVery shortl toi . ThePos ,dePend , W ic escr on. 値 d i l di 故 M M (D* ialandthe FSI ingPotent 故ecoupl . ln Fig‐5 ,Wepresent e 少 an T spectra ncu ng e. D* BB )poles Which are ca1cu・ated a1ong the Z2-va1ley for Parameters y and シ. ,. ** Asshownin Fig.5 )polesaPPearin additiontothe3S‐4S statesof 少 and4S‐5S , MM (D D ,BB. ) ( 1 8.

(12) . I9. I Mode lfor 少 and T Coupl ed‐Cha lme. ler mass and a sha thas a smal states ofll 工 rP r. ler h when thestrength Y of FS1becomessmal widt ・. For nof inalstateinteraction case(Y =0 ) sobtainedatanenerきn′lowerthanthe3Sstate ,the M M polei ionallevelhasnotbeen observed yetin ively. Sincesuch an addi t of 少 andthe 4S stateof T,respect. 少 and T spectra, wecannotemploy. γ =0 iabletoemploytherather tseems morerel ‐ Furthermore ,i. large value of y,becausethelarge wi dth of 化l eハ4D江 poleis favorable for having no experimental ive FS1between meson and tsstrongly suppo貸 せlattherepul ts existence s evidence ofi ‐ Theseresul i he experi lneson i ant lnentaldata‐ s necessary to explaint. ′ ■2 Forthe Lorentzscalartypeofthe b i ion potent ial 7 Gev一 and -1 5 Gev-. t ÷ ans - - , wechose ろs= for 少and. lowing way ively,as ment ioned above :As respect ‐ Thesevaluesaredeterminedinthefol. 2 2 l l l l ら′ schanges value, we ca cu ate 尤 -valuesfor γ and y parameters and p otthe mlnlmum 尤 -va uesfor. 少 and T asin Fig.6‐. f l ferentvaluesofも三 arefavorablefor 少( る≦= -2 tisseen 仕l 7 GeV-1 )and atdi ‐. lavor 5 GeV-1 T(かs= -1 ) rom 値evieWPointthattheら′ sindependentoff . However sparameteri - ,f ,we ′ l 2 - i is ケザvalue 3 GeV at廿l th d] choose acommon らs= -2 ecrosspointof尤 -plotsfor 小 and T‐ 帆『 ‐ ,we 30. 20. べ. 10. 、 、 、 、 、 、 0 、 、 ・ 、 ‐ 、 、 、 ▲ 、 ・ 、 ・ 、 、. x ′ ′ ′ 十 ′ ′ 〆. な ノ . o. ′/ ′ ′ ′ x‐捧※-”※/ 15. 2. 、 ・ ◇ 、 ◇、~◇‐◇ 2 5 .. 3. ーる′ s 2 luesfor サ in 歩 and FIG・6 . 尤 va T. The d iamond and crossind i cate 歩 and T cases, respectively. The iate com mon value of る≦ aPProPr 3 s 一2 parameter between 歩 andTi ‐. GeV-1 l ine is a guide ‐ ‐ The dashed forthe reader’s eye.. lobserved data of masses and widths wi can expl thin a range oflo′~20 M【eV‐ ain al. IV.DISCUSSION. Theradialdependence ofthe Lorentzscalar i ion potent typetrans ial t ‐and vector ‐ s are shown in Fig‐3‐ The contr ibut ion f ining interact ion is very dominant he conf imental data rom t ‐ The exper ion potent ial hetransit t ゴ セ ーonlythe 。GEinteract cannot bereproduced byt l コucted wi ion scons , whichi thouttheconf ine ion. lnthescalarcase thetrans i ion potent ialofT has a short and wi t l ・ l entinteract , h rangeform incontrastto 少 from 値e 汐ずdependenceof尤2 -valuesshownin Fig‐6 ‐ However ,the w ole i ion potent ialindicates a ratherlong rangein compar hescalar‐ typetrans t ison wi th the property oft Lorentz vector‐type trans i ion potent ial t i ial wel l type potent slong‐range property ofthescalar‐ . Th. ) ( 9 1.

(13) . M. HIRAN0, N.YABUSAK1 AT0,Y. MATSUDA and M.SAKA工 ‐ ,K.K. 20. ial lythe2S stateof 歩,compared wim theresul tsinthe case reproducesthebound s tate masses ,espec ial ial ional parameter t tyPe Potent type Potent thoughincluding the addi ofthevector - ‐ . Thescalar ,al ial t h the exPerimentaldatathan the Vector‐type potent らも giVes better agreement wi . Resonance po les appear due to t 江公4 decaying channelstates and the ing between the D he coupl. ined QQ channe lstates thoutany ambigui tyby conf ‐ Theenergiesofsuch M M Poles areobtained wi i ions ofthe D ing method 江D互 po t l means ofthe complex scal es strongly dePend on . The energy pos ing 4ハ江 Poles, however, we can see thatthe coupl Parameters ofthe FS1 - From the behavior ofthe ハ ined QQ states wi ththe decaying M M states are almostequivalentin both 少 mechanisms oftheconf ly, whentheハ江D僅 polesappearattheenergyregion abovethe3s stateandt he4sstate and T‐ Name in. 歩 and T,respectively,the coupling resultsin the observed. f ts and decay widths i mass shi s ‐ Th. ion ofthe phenomenologicaI KOgut ing has al he calculat ‐ so been observedint property ofthe coupl Susskind mode l[ ] 4 . ,5 EichtenetaL[ 10 ]alsostudiedthespectroscoPy of 少 with a microscoPictransitionpotential model in whichthepairproduct ion process wastreated nonrelat ivi ical ly ing operator st . They usedthecouPl ion ofthe OGE part ibut 22 r corresponding to Eq‐( )butig1 loredthecont . on the other hand, Barbour. ] employedthe Lorentzscalar‐potentialfortheconfinementpart‐ Theyinvestigated 値e 少 1 3 et al ‐[ 21 23) ingthecoupl ing operator correspondingto Eqs )and( id spectrum us .( . However ,both of 位em d tsfor 少 arecomPared wi ththoseofEichtenetal notdiscussthestrong decay widths ‐ ourresul .[10]. ]in Fig.7. Theyperformed multi‐open‐channelcalculations withouttheFS1in and Barbouretal 1 3 .[ ththe FS. contrastto our one-open-channelones wi ‐. Theresul teof tsobtainedf rom our model ,inspi. ipt imentaldata than ion,give a bet i ive descr ion ofthe exper tat the one-open‐channel calculat ter qual imated the i ionsin whicht he massofthe3S‐stateisoverest stateis stendencyforthe3S‐ rpredict ‐ Thi ion l l 1isswitched of fin our calculat al so obtained when the FS .7 and Tablel ‐ ,asshownin Fig. 5000. 4S. ′樹-器、 、冊 冊- -郷. 纏. 3S 姿擬. 餌日 ( め 〉①夏 )の. め 雨日 ( 〉①呂 )の. 4000. 5ooo. ( ) a. 一 細 、 、 、 --. 4S 厩. S細 4 o 3 o o. 2S - - 一- - ‐ ‐--一 一. 30oo. ①). ‐一 樹-佃- -総-- . - -.三.計′冊‐斑. 2S - - -. 1S 一 ■ 30oo. 4 6 3 Ex 0 1 82 1o 0 0 1 2 1 p . . .〔 . . . Y(GeV). -一. 一. s ■ - --------一 一 i 4 Ex 1 2 63 82 ] o o 0 1 2 1 p . .【 . . . . Y(GeV). FIG,7 i ) vectortype chten et al son of our results wim those ofEi . Compar ,[10]( ] F 1i l b h f [ 2 l T h i S h i d l 1 t t ( t ) n n e r mo e o o s c a a r e e r e sn o s and Barbom et al y p ‐ ‐ ‐ imi lartot imated・ Theseresul h 3s statesareoverest tsares hoseinthe y =。 the ] ユ ュ ,t e - case ‐ . Seete×t. ( ) 2 0.

(14) . 21. I Mode lfor 少 and T Coupl ed ‐Channe. i lresul )Case are tsof 少intheno FS1(Y;0 TABLEII1・ Thenumer ca 1エーo]( )and Barbouretal tort“)e ththose ofEi vec chteneta comPared w1 ‐ l ) Theytookinto accot灯1t [ 12 ho ]( r mode t seby ら’ α 一3O Gev-linthei. - . ‘ id not discussthe decay width many open channels butd ‐ The results of *D*pol T h D i imi larresultsto the i chi s not rs e whi state g e our 3S‐ ves . T h t r f e c o ident i i v and state e sseen underthe3S‐ edbythe experimenti ‐ inementter i ln i bethe Lorentz propenies ofthe conf n scalar types de scr i ion‐ Values are giveni t trans n N[ eV‐. [10]. Exp‐ [15] r ーass I. 4S 3S 2S 1S. r. 4415 43±15 4040 52±10 3686 3097. [12] 江lass. l r lass. r. 4190 3700 3065. 4490 4171 3687 3096. 0‐08 36 0 0. 4015. 0‐72. I E ーass. 窟lass. r. 4625 4225 3684 3095. 4462 4207 3647 3086. 1‐19 75 0 0. 4016. 0‐80. e pol ‐ αY =O GeV ‐ by =○ GeV ろ′ ; ■2 7 GeV-1 ‐ , s. [ set B]わ scalar. ‐[ set A]α vector. *D* polei ‐ 工n the case of no FS1(Y=0 sobtained arotu・dthe D*D*threshold ) . Theoveres ,the D ibedtothe appearance ofthis D*D* polebelow the 3- i ion ofthe3S… t state mass s ascr ・nat state massi ゴ herhand,the D*D*polehaslargel r l assanddecay Width asseenin Fig‐7 - AsshoWnin Fig‐5 ,ontheot i ive force betWeen 11 lneson in the decaying states‐ in the case of the strong repul leson and ant s. ま l fthe3S」 Therefore r ー ーnarkablyi比ー e statel assi sre provedbytakinginto account せ ,thepredicted va ueo FS1 i li h せl h he i FSI s indispensable even in t s . Based on ロコ , we can say t at e s ort range repusve lforexplainingthe observed data [15]. m・croscopic mode. V‐CONCLUS-ON. Wr e analyzed 少 and T. fect due to coupl ings spectra by taki ng into account the open-channel ef. i ialf ionpotent betweentheconf ined QQ andthedecaying NO虹states t rom QQ statesto MM - Thetrans icapproxl ion wi thinanonrelat ivi ivedf ivequark‐ant iquarkinteract - st rom theeffect stateshasbeender ion was performed Wi thint ion‐ A1 thought hetwo-channel mode1by assuJ 〔 ning he present calcu1at mat l d dd i i ftheexper imentaldataforbo l t h theone‐open‐channe ,theobtainedresutsprovi eagoo escrptono. 歩 and T‐. i ion potent ialcan be Theshortcomingsin previousstudiesbased onthe microscopictrans t. ial form of FS.between decaying a improved bytakinginto accountthe phenomenologicalexponent - ‐ h l せ}erangeandstren罫h parameters(し imeson. ln 鉱e present mode meson and an ant ,Y)are c osen , ini in order to reproduce the observed data. Theresul te Fsli tsindicatethatthe f s needed andthe ing mechani f ferss l ight lyf stateof歩 radialform for 少 di rom 値eonefor T‐ Thecoupl smsforthe3S- andthe4s‐ stateofT haveaconunonproperty- Theobserved massesand widthsarereproduced when the MM(D*D* and BB )poles appearin the appropriate position on me complex enerを夢 plane‐ This ing has al so been obtained through previous analyses based on the property of the coupl ‐Susskind model[4 phenomenologicaI KOgut ,5]. VVe al igated which Lorentz property ofthe scalar‐ or vector‐type i s preferable so invest ‐ VVe s not preferable because the short range property of the conclude that the vector‐type property i. ( ) 21.

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