Consistent baryon mapping of quark systems

PHYSICAL REVIEW CVOLUME 50, NUMBER 1JULY 1994 Consis en ba yon mapping o qua k sys ems S. Pi el Ba ol Resea ch Ins i u e, Uni e si y o Delawa e, Newa k, Delawa e 19716 J.M..A ias Depa amen o de Fisica A omica, Molecula yNuclea , Uni e sidad de Se illa, Apdo 10. 65, $1080 Se illa, Spain J.Dukelsky G upo de Fasica Nuclea -Facul ad de Ciencias, Uniee sidad de Salamanca, 5'7008 Salamanca, Spain A. F ank Ins i u o de Ciencias Nuclea es and Iabo a o io de Cue naeaca, Ins i u o de Ecsica, U u 'e sidad Nacional Au ono na de Mexico, Apdo Pos. al 70-5)8, 04510 Megico, Dis i o Fede al, Megico (Recei ed 7Decembe 1993) We p esen anew and consis en mapping o colo less h ee-qua k clus e s on o colo less iple e mions (ba yons) and es i in he con ex o a h ee-colo ex ension o he Lipkin model. Fo sys ems wi h wo iple s ( o which he p oblem can be sol ed wi hou app oxima ion bo h be o e and a e he mapping), we exac ly ep oduce he dynamics o he model o he a ie y o co ela ion s uc u es conside ed. PACS numbe (s): 21.60.Gx, 21.60.Fw, 21.30.+y, 12.39.— x I. INTRODUCTION Es ablishing aconnec ion be ween nuclea physics and /CD has been an a ea o in ense esea ch in he las ew yea s. Cen al o his e o is he goal o isola ing qua k e ec s in nuclei. In ecen yea s, cons i uen qua k models [1]ha e been applied wi h conside able success o nuclea sys ems wi h ew pa icles. While many ques ions emain conce ning he alidi y o such an app oach (e.g., he lack o acon- nec ion o /CD and i s appa en iola ion o he unde - lying physics o chi al symme y and spon aneous chi al symme y b eaking p ocesses), in iew o hese successes i seems wo hwhile o de elop hese models u he , in o de o see whe he hey can p o ide apa ial b idge be ween he physics o /CD and ha o ini e nuclei. A p esen , such models ha e been di ec ly applied o one- and wo-ba yon sys ems only. To ea sys ems wi h la ge numbe s o nucleons, i has p o en necessa y o in- oduce app oxima ions. One possibili y ha has been explo ed is o use he esona ing g oup me hod in he six- qua k p oblem o ex ac an e ec i e nucleon-nucleon in- e ac ion, which is subsequen ly diagonalized in he space o se e al nucleons [2]. Un o una ely, many-qua k e - ec s ha may a ise when mo e han wo nucleons a e p esen will be missed in such an app oach. To inco - po a e hem, we would like o bypass he wo-nucleon p oblem and wo k di ec ly in he space o many qua ks. Since asys em o 3A qua ks will clus e in o A iple s (nucleons) a no mal nuclea densi ies, anecessa y in- g edien in any such app oach is ame hod o handling s ong h ee-body co ela ions in amany-body en i on- men . Recen ly, i has been sugges ed [3— 5] ha mapping me hods [6] migh p o ide ap ac ical means o accom- plishing his. The basic idea is o map colo less h ee- qua k clus e s, which do no sa is y exac e mion an i- commu a ion ules, on o iple e mions (ba yons) ha do. Such amapping leads &om he o iginal mul i- qua k Hamil onian o an e ec i e Hamil onian o hese ba yons, which igo ously inco po a es he physics o he Pauli p inciple a he qua k le el. The i ue o his app oach is no in a educ ion o he deg ees o eedom — he e a e in ac mo e s a es in he mapped space han in he o iginal space — bu a he in he ep esen a ion o he dynamics as asys- em o in e ac ing ba yons. These ba yons con ain as a subspace he physical nucleons (as well as exci ed nu- cleonic s a es) and in e ac wi h one ano he in ways ha should be amenable o he usual e mion many-body echniques [7], e.g.,Ha ee-Fock, Tamm-Danco , and andom phase app oxima ions, B ueckne heo y, e c. Se e al di e en mappings o qua ks o iple e mions ha e been ecen ly discussed. Pi el, Engel, Dukelsky, and Ring [3] (he ea e e e ed o as PEDR) p oposed a wo-s ep p ocedu e, whe eby pai s o qua ks a e 6 s mapped on o diqua k bosons and hen boson- e mion pai s a e mapped on o iple e mions. Despi e he suc- cess o his mapping in ep oducing he dynamics o he es model o which i was applied, i ne e heless has some d awbacks. On he one hand, i does no lead di- ec ly o iple e mions ha a e an isymme ic in hei h ee indices. In addi ion, as o mula ed, i can only be applied o qua k Hamil onians domina ed by wo-body in e ac ions. A oughly he same ime, Nadjako [4] sugges ed an al e na i e mapping ha leads di ec ly o an isymme ic iple e mions and, u he mo e, is ap- plicable o h ee-body in e ac ions. His mapping, how- 0556-2813/94/50(1}/423(12}/$06.00 50 423 1994 The Ame ican Physical Socie y S.PITTEL, J.M. ARIAS, J.DUKELSKY, AND A. FRANK e e , is no app op ia e o sys ems ha a e domina ed by wo-body in e ac ions. Clea ly, wha is needed is a mapping ha consis en ly ea s bo h wo- and h ee- qua k in e ac ions. One such me hod was in ac p oposed soon he e- a e by Meye [5]. Howe e , his mapping does no seem o p ope ly ea wo-body in e ac ions ei he , a leas when a unca ion o colo less iple e mions is imposed. Building on he ideas o Meye , we ha e now succeeded in o mula ing anew ba yon mapping o (col- o less) h ee-qua k clus e s ha seems o sa is y all he desi ed equi emen s. In Sec. II, we b ieHy e iew some gene al ea u es o ba yon Inappings and p esen ou col- o less e sion. Bo h PEDR and Meye es ed hei mappings in he con ex o an exac ly sol able model o qua ks o en e- e ed o as he Bonn qua k shell model (BQSM) [8]. While his model has some a ac i e ea u es, i also has some se ious limi a ions. Pe haps he mos signi ican is ha i does no p oduce spa ially localized colo less h ee-qua k clus e s (i.e.,nucleons) [9]. Thus asecond goal o his wo k has been o de elop an al e na i e qua k model on which o es ou mapping. The model ha we ha e chosen is a h ee-colo ex ension o he well-known Lipkin model [10], which in i s adi ional e sion has been used ex ensi ely o es a ious nuclea many-body echniques. We desc ibe his model in Sec. III and dis- cuss i s algeb aic solu ion o small numbe s o pa icles. Ac ucial componen o his model is ha i admi s, o di Fe en alues o i s pa ame e s, dynamical one-, wo-, and h ee-body co ela ions. In Sec. IV, we apply ou mapping o he h ee-colo Iipkin model o wo iple s and p esen he esul s. The bo om line is ha he mapping seems o wo k pe ec ly, when all colo less ba yon s a es a e included. Fo sys- ems in ol ing ala ge numbe o iple s, his is clea ly no possible. Thus, in Sec. V, whe e we summa ize he p incipal conclusions o ou wo k, we also desc ibe some u u e ex ensions needed o u he es he applicabili y o ou me hods o many- iple sys ems. II. BARYON MAPPINGS OF QUARK SYSTEMS A. P elimina ies aS —gq1aq1S ) 1 (2.2) +abed =g~123 &145 q2 q3gq5d q4c 12345 (2.3) 5d —g6123 6456 'VloQ2513 e I"~4d .( 123456 He e, and in he subsequen analysis, we deno e he colo indices by numbe s and he es by oman le e s. The idea o aba yon mapping is o eplace he sys em o in e ac ing qua ks by an equi alen one o in e ac ing iple e mions. We deno e he c ea ion and annihila ion ope a o s o he esul ing (mapped) space by Al 253, and A12/3 espec i ely. They sa is y he mul i-index an i- commu a ion ela ion (Al 253 A4de e )— b(la2b3c, 4d5e6 ) whe e (2.5) Ou s a ing poin is anon ela i is ic model o con- s i uen qua ks. We deno e he qua k c ea ion and an- nihila ion ope a o s o he model by q,-and q;, espec- i ely. The i s subsc ip deno es he colo quan um numbe and he second deno es all he es . These ope - a o s sa is y he usual e mion an icommu a ion ela ion (gci Ipc&i& )=bci,c'i' =bcc' bii' (2.l) QCD conside a ions sugges ha he qua k Hamil o- nian may include up o h ee-body in e ac ions, all o which a e colo scala s. Such aHamil onian can always be exp essed in e ms o he ollowing colo less ope a o s: b(la2b3c, 4d5e6 ) =bl 4db25, 5ebsc, e +bla, ca&45,e &3c,4d +&la,e b25, 4db3c, 5e ala, 4db2b, e b3c,ee ala, 5eb2b, 4db3c, e ala, e b2b, eeb3c,4d (2.6) The ope a o A1 2&3,,by de ini ion, c ea es aba yon co esponding o h ee qua ks in he s a es la, 26, and 3c. This co espondence is exp essed h ough he e- qui emen ha bo h A1 2g3 and A1 2p3, a e an isym- me ic unde in e changes o hei qua k indices, e.g., A=— A~ e c. la263c 2bla3c~ The space gene a ed by hese ba yon ope a o s is in ac la ge han ha o he o iginal qua ks. This can be seen by conside ing he s a e o wo ba yons, ~la2b3c, 4d5e6 )~=Al 253,A4de, e ~0)~, (2.7) whe e he subsc ip B e e s o s a es in he ba yon space. The s a e (2.7) is an isymme ic unde he in e change o he indices co esponding o any wo qua ks wi hin one o he wo iple s (e.g.,la wi h 2b o 4d wi h 5e) and also unde he in e change o one iple wi h he o he . How- e e , i is no an isymme ic unde he in e change o he quan um numbe s o aqua k in one iple (e.g.,la) wi h hose o aqua k in he o he (e.g.,4d). A ully an isym- me ic wo- iple s a e may be eco e ed by aking an app op ia e linea combina ion o he s a es (2.7). As a consequence, he e is indeed asubse o wo- iple (and likewise many- iple ) s a es ha a e ully an isymme ic and, u he mo e, a e in one- o-one co espondence wi h he s a es o he o iginal qua k space. This is e e ed o as he physical subspace. The e is, howe e , ano he class o s a es ha a e no ully an isymme ic unde in- e change o qua k indices and which he e o e ha e no coun e pa s in he o iginal space; his is e e ed o as he unphysical subspace. The e a e a a ie y o possible ways o ensu e ha he 50 CONSISTENT BARYON MAPPING OF QUARK SYSTEMS physics o he o iginal qua k p oblem is p ese ed un- de he mapping o ba yons. We ollow he app oach o Belyae and Zele insky, whe e ope a o s in he o ig- inal space a e mapped on o ope a o s in he new space so as o p ese e hei (an i)commu a ion ela ions. Im- plemen a ion o his p esc ip ion gua an ees ha all he physics o he o iginal sys em is exac ly p ese ed by he mapping wi hin he physical subspace. Fo amapping o be o p ac ical use, he unphysical s a es mus lie high in ene gy ela i e o physical s a es. O he wise, i will be di icul o disen angle he physical s a es o in e es &om hose ha a e unphysical, pa icu- la ly in he p esence o a ia ional app oxima ions. This places s ingen limi a ions on he kinds o mappings ha should be conside ed o p ac ical applica ions. Fo example, i is s aigh o wa d o w i e down amap- ping o colo less pa icle-hole (p-h) o one-body qua k ope a o s ha p ese es hei commu a ion ela ions: 1 ).q&.nb ~2)A] 23/Alb2 3d (28) 1123cd Since any o he colo less ope a o s o he ype (2.2)— (2.4) can be ew i en in e ms o colo less pa icle-hole qua k ope a o s, i would seem ha we could simply apply (2.8) and achie e ou goal. This is un o una ely no he case. Since apa icle-hole ope a o does no in ol e mo e han one c ea ion and one annihila ion ope a o , i canno (by i sel ) inco po a e any in o ma ion on he qua k Pauli p inciple. As aconsequence, he spec um ha would esul om apu e p-h mapping ( o e mion sys ems) would in a iably ha e unphysical s a es lying below he physical s a es o in e es . To inco po a e qua k Pauli ei ec s (in aphysically use ul way), we mus map di ec ly he mul iqua k c ea ion and annihila ion ope a o s ha appea in he Hamil onian. In wha ollows we adop he Dyson app oach, which leads o aba yon Hamil onian ha is non-He mi ian bu ini e. The non-He mi ici y is adi ec e8ec ion o qua k Pauli e ec s. Ano el ea u e o ou analysis is ha , in con as o ea lie wo k, we ocus on he mapping o colo less ope a o s. This emo es some o he ambigu- i ies ha a ise when he mapping is ca ied ou mo e gene ally. I also leads o amapping ha is ailo ed o physical applica ions in which a unca ion o colo less iple s mus be implemen ed. B.Mapping o colo less one-body ope a o s We begin wi h adiscussion o he colo less one-body ope a o Abo (2.2). To map his ope a o , we can make di ec use o ea lie esul s. Namely, using (2.8), we can exp ess i s ba yon image as A his poin , he image o Agis exp essed in e ms o ba yons wi h colo . We know, howe e , ha i is possi- ble o desc ibe he ele an physics in abasis o colo less ba yons only. Towa ds his end, i is use ul o ca y ou a unca ion o colo less ba yons; his can be done using he p esc ip ion spelled ou by PEDR. The basic idea is o ca y ou acolo -SU(3) coupling and o isola e he piece ha is ully an isymme ic in colo and ully sym- me ic in he o he noncolo indices. Asimple way o implemen his is h ough he eplacemen s A ca.li2j3~ +6123Paij ~1i2j3& M6123~zjk (2.10) The ope a o s A -& and A;jg in oduced he e a e ully symme ic unde he in e change o hei indices. Fu - he mo e, hey sa is y he an icommu a ion ela ion (A;,b, A .„j=—S(ijk,lmn), 1 U~lmn (2.ii) whe e S(ijk, lmn) =b;)b, bb„+b; b,.„bb) +b;„b,(bb +4b, bb +b' b,ibb~+b; b, (2.12) Inse ing (2.10) in o (2.9) and hen ca ying ou an explici sum o e he colo indices (gz2z e&2s — —6), we a i e a he ollowing esul : Abm3 )A ,qAb, g. cd (2.13) As an example, conside amapping o he qua k num- be ope a o Nq =Pz qz qq .Applying (2.13), we ob ain he expec ed esul Nq m3) A ~A b, =3N~, (2.14) whe e N~ is he numbe ope a o o colo less ba yons. C. Mapping o colo less h ee-body ope a o s Nex we conside he colo less h ee-body ope a o Cb,g,yappea ing in (2.4). He e, oo, we can di ec ly use he mapping gi en by Nadjako and Meye , since he se o one-body ope a o s plus he se o h ee-qua k c ea ion and annihila ion ope a o s close unde commu- a ion. The non-He mi ian mappings o h ee-qua k c e- a ion and annihila ion ope a o s equi ed o p ese e his commu a ion algeb a a e [4,5] ~ab =)qj~~qlb ~))A]~2~MAlb2csd ~(2.9) 1123 Cd ql'q2 q3A: ~~3I2j1; (2.i5) q.q-q mA . .+—p p p 1 2j sb 12gsb 2)(4js $'2jsbs +4gs 2~ sby s+A4~5 sbAy. 2~s )A4gb s 456,lmn 1 +- ) 456789,lmnopq 4l5mli 6n7o2j"8p9q3&~4l6n8p~5m7o9q (2.16) 426 S.PIII'EL, J.M. ARIAS, J.DUKELSKY, AND A. FRANK 50 Applying (2.15)— (2.16) o he colo less h ee-body ope a o Cbd and ca ying ou asubsequen unca ion o colo less iple s, we ob ain he ollowing esul : Cab,d, mCb",d,y— —36 Ab,Adey+36) (A „Ab, +A„bA, ,+A „,A~)Agh;Adey . ghi (2.17) Asupe sc ip nh has been included o indica e ha his is anon-He mi ian ba yon image. No e ha acolo less h ee-qua k in e ac ion maps on o colo less one-body plus wo-body in e ac ions only. The h ee-body piece cancels exac ly in he unca ion o colo less ba yons, because o p ope ies o he 6y23 ac o s. D. Mapping o colo less wo-body ope a o s Finally we u n o he wo-body ope a o , o which we canno di ec ly use he esul s o ea lie wo k. Meye p oposed apossible non-He mi ian ba yon image o wo-body ope a o s, based on he use o he Usui ope a o . Howe e , as no ed ea lie , his mapping does no seem o wo k when unca ed o colo less iple s and applied o he es model we p esen la e . Ne e heless, he e is asugges ion in he wo k as o how o build ap ope mapping o colo less wo-body ope a o s, which we now exploi . A he end, we make some ema ks as o why he esul s a e no applicable. As we saw in he p e ious subsec ion, i is possible o map acolo less h ee-body ope a o in non-He mi ian o m. Since he mapping (2.17) ollowed &om an exac p ese a ion o commu a ion ela ions, i is ce ainly legi ima e. Bu he e is ano he way o map acolo less h ee-body ope a o ha is equally legi ima e and which leads o aHe mi ian o m. The h ee-body ope a o Cbd can be ew i en in e ms o he colo less one-body ope a o s A;~ o (2.2) as ollows: Cabcde =AadAbeAc +AadAb Ace +AaeAbdAc +AaeAb Acd +Aa AbdAce +Aa AbeAcd — bbd(AaeAc +Aa Ace) — bbe(AadAc +Aa Acd) —'4 (AadAce+ AaeAcd) 26 d(cA Aaeb +Aa Abe) — 2bce(AadAb +AayAbd) — 2bcj(AadAb, +Aa, Abd) +2(~bc,e Aad +~bc,d Aae +~bc,deAa )(2.18) Applying he mapping o colo less one-body ope a o s (2.13) and ocusing on he one-and wo-body pieces, we ob ain, o he He mi ian image, Cabcde ~Cabcde 3AabcAde 36 )(AghaAbci +AghbAcai +AghcAabi) (AdghAe i +AeghAd i +A ghAdei) . ghi (2.19) No e ha he se e al 8- unc ion e ms in (2.18) do no su i e a e he mapping. They a e exac ly canceled by o he e ms ha'. a ise when he ba yon image is pu in no mal o de . The one-body piece o (2.19) is iden ical o ha gi en in (2.17). The wo-body pa , howe e , is no . One is He mi ian and he o he non-He mi ian. Howe e , bo h a e o mally jus i ied and hus mus be equi alen in he physical subspace. I is easy o show ha ei he o he o ms o he wo-ba yon image o Cb,d, can be ans o med in o he o he by pe o ming he ollowing eplacemen on he wo annihila ion ope a o s: 1 AabcAde j~(AabdAce +AabeAdc +Aab Adec +AadcAbe +AaecAdb +Aa cAdeb 3 +AdbcAae +AebcAda +A bcAdea) (2.20) These obse a ions sugges ap ocedu e o mapping acolo less wo-body in e ac ion. Namely, we i s ans o m i o colo less p-h o m, hen map i using he well-known (and o mally jus i ied) colo less p-h mapping (2.13) and inally ans o m i s wo-body pa o anon-He mi ian o m by ca ying ou he eplacemen (2.20). We now apply his p esc ip ion o he colo less wo-qua k ope a o Bbdo (2.3). T ans o ming i o p-h o m leads o he esul +abed AacAbd +AadAbc ~bdAac ~bcAad (2.21) Mapping his ope a o in colo less p-h o m and w i ing he esul in no mal o de gi es h Babcd ~Babcd =12 )Aaby Acd 9)Aae Abgh(Ace Adgh +Ade Acgh) e gh (2.22) Finally, when we impose he eplacemen (2.20) on he wo-ba yon piece, we a i e a Babcd ~Babcd =12 )AabeAcde +9)Aaey Abgh(AcdeA gh +Ae gAcdh) ee gh (2.23) 50 CONSISTENT BARYON MAPPING OF QUARK SYSTEMS 427 This esul di e s &om he one ha would a ise &om Meye 's mapping supplemen ed by a unca ion o colo - less ba yons, eQec ing ou consis en ea znen o colo . In ac , he non-He mi ian mapping o wo-body op- e a o s p oposed by Meye can be de i ed in much he same way by conside ing agene al h ee-body in e ac ion a he han acolo less one. Such ap ocedu e, howe e , is no unique, since agene al h ee-body ope a o can be ecas as ap oduc o p-h ope a o s in di e en ways. Explici conside a ion o colo less ope a o s emo es his ambigui y. The end esul is ap esc ip ion o ans- o ming &om He mi ian o non-He mi ian images ha p ope ly inco po a es colo and is hus meaning ul when iznplemen ing a unca ion o colo single s. The essen ial esul s o his sec ion, p ope non- He mi ian ba yon images o colo less one-, wo-, and h ee-qua k ope a o s, a e con ained in Eqs. (2.13), (2.23), and (2.17), espec i ely. Al hough he mapping o wo-body ope a o s was no de i ed by explici conside - a ion o commu a ion ela ions, we ha e con i med ha his se o ba yon images does indeed p ese e he com- nu a o [B~s~a, E, g], whe e E, g =+~2s eq2sq~, qz qsg We should also emphasize he e ha his se o map- ping equa ions can be applied o any colo less cons i uen qua k Hamil onian w i en in uncoupled o m. E. Physical con en o he eplacemen p ocedu e Some unde s anding o he eplacemen p ocedu e p o- posed o gene a e anon-He mi ian wo-body image &om aHe mi ian one can be ob ained by s udying he map- ping o acolo less six-qua k s a e, ~abc, de )Q — ——)~»s ~4&s qi~q2sqscq4dqseqs l0)q 123456 (2.24) whe e he subsc ip qindica es as a e in he o iginal qua k space. Mapping his s a e wi h (2.16) and impos- ing a unca ion o colo less ba yons leads o he esul ~abc, de ) p~A&,A&, ~0)a+(A &,Ab, +— A As,p+AQ As„ 3 +A A +A, A +A,„A...+A A, +A..A.,+A.d A...}10) (2.25) The physical s a e no only in ol es he di ec wo- ba yon componen bu also asum o e all nine possible in e changes o he indices o one ba yon wi h hose o he o he (wi h an o e all ac o o s). I is no di icul o con i m ha when we ac ei he wi h he ope a o A~Ap, yon his physical s a e o wi h he eplacemen o m gi en in (2.20) we a i e a exac ly he same esul . Thus he p oposed eplacemen indeed sa is ies he de- si ed c i e ion ha i p oduces he same esul s wi hin he physical subspace. I is also in e es ing o no e he co espondence be ween he di ec and exchange pieces o (2.25) wi h he le and igh hand sides o (2.20). III. THREE-COLOR LIPKIN MODEL I body in e ac ion ha sca e s pai s o pa icles among he wo le els wi hou changing he p alues. This znodel can be sol ed exac ly by using g oup he- o e ical echniques. I is well known ha he se o all possible bilinea p oduc s o med &om a ini e se o c e- a ion and annihila ion ope a o s cons i u es aLie alge- b a. In he case o he Lipkin model, he e a e (20)2such bilinea p oduc s o c ea ion and annihila ion ope a o s (gene a o s), and so he ele an Lie algeb a is U(20). These gene a o s will be deno ed by K,"„, =q „q Since we a e dealing wi h e mions, all he s a es o he sys em belong o he i ep [ln] o he U(20) dynamical g oup. The s uc u e o he p oblem sugges s he decom- posi ion U(20) oU(O) gU(2), (3 1) The h ee-colo Lipkin model is based on he well- known Lipkin model [10], which can be sol ed analy i- cally and has been used ex ensi ely in nuclea physics o es ing many-body app oxima ion me hods. Since many o he cha ac e is ics o he h ee-colo Lipkin model a e al eady in he o iginal one, we e iew i b ieBy he e. The Lipkin model has wo le els, each one 0- old de- gene a e, sepa a ed by an ene gy A. I is assumed ha in he unpe u bed g ound s a e N=0pa icles oc- cupy all he single-pa icle s a es in he lowe le el. The e mion c ea ion and annihila ion ope a o s o he model a e w i en as q „and qz, espec i ely. He e, ois a quan um label which cha ac e izes whe he he pa icle is in he lowe le el, o. =—,o in he uppe one, o=+, and pdis inguishes which o he 0degene a e s a es o ha le el he pa icle occupies. The Hamil onian o he model includes, in addi ion o he one-body e m, a wo- whe e he 0ope a o s K„", =PKP~ gene a e he U(A) algeb a, while he 22 objec s K, =g„K,"„a e he gene a o s o he U(2) algeb a, and commu e wi h he K„,.One can easily e i y ha he Lipkin Hamil onian can be w i en en i ely in e ms o he U(2) gene a o s and hus ha all he s a es belong o ade ini e i educible ep esen a ion o U(2) [o SU(2)]. This in u n implies ha he Hamil onian zna ix can be analy ically e alu- a ed using he well-known angula moxnen um algeb a SU(2). The h ee-colo Lipkin model has amuch iche al- geb aic s uc u e and analy ic solu ions a e co espond- ingly mo e di Bcul o de i e. The model in ol es h ee se s (one o each colo ) o s anda d wo-le el Lipkin models. Again he lowe le els a e assumed o be com- ple ely illed in he unpe u bed g ound s a e, which in 428 S. PITTEL, J.M. ARIAS, J.DUKELSKY, AND A. FRANK 50 his case con ains N=30 pa icles. The c ea ion and annihila ion ope a o s now include alabel i ha ep e- sen s he colo quan um numbe and a e hus w i en as q,.„and qi „, espec i ely. The model Hamil onian now includes one-body, wo-body, and h ee-body in e - ac ions, which sca e pa icles cohe en ly among he le - els, wi hou changing he p alues and main aining all s a es colo less: H=H1+H2+H3, (3.2) (V&+„V&+p && && — ~) ip (3.3) and H2 =Og&1236145(g2+ 13+ 15 P14 P+1. „1.-„~+. ~+. 12345,pg p2 (3.4) X3 ) 123456)pi p2p3 &123&456 g1y+ 'V2+ 13+ 'V6 —95—g4—+14—95—16—13+P 12+ Ql+ )(3.5) In his case he e a e (6O)2 gene a o s K& ", ,— —q „qs „,leading o he Lie algeb a U(6O), and he s uc u e o he model sugges s ha we ca y ou he classi ica ion o s a es in e ms o he chain U(6O) zU(O) U(6) zU(O) 13 U(3) U(2) .(3.6) The O2 ope a o s K", =P,.K,'. ~, gene a e U(O), while he 62 objec s K&, — —gK& ~a e he U(6) gene a o s, in e ms o which we shall ew i e he Hamil onian (3.2)— (3.5) below. We can u he decompose U(6) by con ac ing again o he 22 ope a o s K, =g,.K,',,which gene a e U(2), o o he 32 gene a o s Kl', — —PK& co esponding o U(3). The la e g oup is indeed necessa y in he classi ica ion, since all physically admissible s a es should be colo less; i.e., hey should belong o he (O, O, O) U(3) ep esen a ion [which co esponds o he (A, p) =(0, 0) scala ep esen a ion in Ellio sSU(3) no a ion]. The si ua ion is mo e complex han in he s anda d Lipkin model, howe e , since di e en U(6) ep esen a ions can con ain hese s a es and, mo eo e , o each o hem se e al U(2) ep esen a ions a e connec ed by he Hamil onian. F om (3.2)— (3.5), we see ha he model in ol es h ee pa ame e s, one each o he one-, wo-, and h ee-body in e ac ions. As men ioned be o e, i may be ew i en in e ms o he U(6) gene a o s as H1 —— AJ, ,(3.7) ik (3 8) and 2(J++J)+2J+) K„'+K, "+ +J)K„'+K, "+ ik ik )(K +Kq+ K,". ++K,*+K„'+K, "+ ), (3.9) ilk whe e J„J~,and J,de ined by J, =2(K++ — K), J+ =K+, and J=K+, a e he SU(2) subg oup gene a o s, which oge he wi h he numbe ope a o N=P,.K,'comp ise he U(2) g oup in (3.6). The o he ope a o s in (3.8) and (3.9), namely, g,.&K&+ K,". +and g1& Kl*+K&+K,"+, oge he wi h hei He mi ian conjuga es, clea ly lie ou side SU(2). Using hei commu a ion ela ions wi h he SU(2) gene a o s, we eadily conclude ha hey beha e as ank 2and ank 3 enso s T=T2 and T3 in SU(2), espec i ely. (e) (2) (3) . Using his no a ion, he model Hamil onian acqui es he simple o m H1 — —AJ, ,(3.10) X2 (J2 J2 )+X2 (T(2) +T(2)) (3.11) and 50 CONSISTENT BARYON MAPPING OF QUARK SYSTEMS 429 o which we shall e e hence o h. Since we a e dealing wi h asys em o e mions, he s a es o he model belong o he [ls+] ep esen a ion o U(6O), while he U(O) and U(6) ep esen a ions a e com- pleinen a y; i.e.,once he U(6) ep esen a ion is de e - mined, he U(O) ep esen a ion is ixed. Since he U(3) ep esen a ions a e he colo less ones men ioned be o e, i should be clea ha he basis s a es can be unambigu- ously deno ed by ~[&i h2 li3 h4 h5 hs], ~jm), (3.13) The s a e ~G) has h ee pa icles wi h p=1, h ee wi h p=2,...,and h ee wi h p=O. De ining P=P,.p;, we see ha he s a e ~G) has aunique P alue o 3O(O+1)/2. In gene al, di Fe en U(6) i educible ep esen a ions appea which con ain he U(3) colo less i eps in he e- duc ion U(6) DU(3) U(2) o (3.6). Since he Hamil o- nian (3.10)— (3.12) is buil solely in e ms o U(6) gene - a o s, he ene gy ma ix will sepa a e in o blocks, each co esponding o ade ini e U(6) i ep. Fo 0=1, he g oup analysis is pa icula ly simple. The e is only asingle U(6) i ep, (1,1,1,0,0,0), which con- ains he (1,1,1)(j =3/2) and he (2, 1,0) (j =1/2) U(3) SU(2) i eps. Howe e , only he i s one belongs o ou (colo less) space. Fu he mo e, all s a es ha e P=3. Fo 0=2 he si ua ion is mo e complex in se e al e- spec s. Fi s , as we shall enume a e sho ly, he e a e se e al possible U(6) i eps ha con ain colo less s a es. All con ain s a es wi h P=9, while se e al also con ain s a es wi h o he P alues. These o he s a es, howe e , can be gene a ed om he co esponding P=9s a es by he U(O) aising and lowe ing ope a o s Ki', wi h pgp', and a e hus degene a e in ene gy wi h hem. Fo his eason, we need only conside he s a es wi h P=9 o ully exhaus he spec um. In Table I, we display he ou possible O=2U(6) i eps ha con ain colo less P=9s a es and also indica e he associa ed angula momen a. Asimila analysis can also be ca ied ou o highe 0 alues. TABLE I. Colo less s a es and g oup labels o 0=2. U(6) labels [2,2,2,o,o,o] [2,2,1,1,0,0] [2,1,1,1,1,0] [1,1,1,1,1,1] SU(2) label j=1,3 j=0,2 j=1 j=0 Degene acy 10 6 3 1 whe e [hi, ...,hs] labels he U(6) ep esen a ions, jm a e he SU(2) DSO(2) quan um numbe s, and ais an ex a label ha may be needed o dis inguish ei he epea ed U(6) i educible ep esen a ions (i eps) wi hin he same U(6O) ep esen a ion, epea ed j's wi hin he same U(6) ep esen a ion, o any o he necessa y quan um numbe s. Fo example, he unpe u bed g ound s a e o he model, de ined as he s a e o which all o=—le els a e illed, co esponds o [G) =[O, O, O, 0,0, 0], j=,m=—.(3.14) 30 30 In he ollowing sec ion we es ou mapping p ocedu e o he case o O=2. He e, we illus a e he algeb aic e alua ion o ma ix elemen s o he [2,2,2,0,0,0] O=2 subma ix, o which we will need o conside s a es wi h j=3and j=1(see Table I). Analogous calcula ions ha e also been done o he o he s a es and a e included in he esul s ha we p esen . Be o e p oceeding, we w i e down he commu a ion ela ion o he U(6) gene a o s, which will be used ex- ensi ely in ou analysis: (Kq+~ +Kq )~j,m) =bi,;A ~j,m) . To 6nd he alue o A, we no e ha )(K&++K& )~j,m) =N ~j,m) =6~ j,m), o 0=2. Since all colo s should be equally ep esen ed in acolo less s a e, we a i e a he use ul ela ion (K„'++K„' )~j,m) =2bi,;]j,m) .(3.16) The basic idea o ou analysis is o conside he ac ion o he ope a o s Tz and Tz on he s a es ~j,m) (2) (~) ~3, — 3) and ]1,— 1), namely, T, ~3, -3) =ass ]3, -1)+us 2] 1, -1), (2) (3.17) T( ) i3, — 3) =bs,o]3,0) +bs, 2i1,0), (3.18) ~2"' ll-1) =aiol1+1) +oi,2]3,+1), (3») and &s" ]1-1) =bi,2]3+2) .(3.20) We use ano a ion whe eby he 6 s index in he ex- pansion coe icien s deno es he SU(2) label o he ini- ial s a e and he second gi es he inc emen needed o ob ain he SU(2) label o he inal s a e. I we can de e mine all o he independen aand bex- pansion coe icien s in (3.17)— (3.20), we will ha e e ec- i ely sol ed he p oblem. We can hen use he Wigne - Ecka heo em o de e mine all he ele an educed ma- ix elemen s o T& ~and T& ~and om hem de e mine all o he ma ix elemen s o he Hamil onian o any choice o he model pa ame e s. We now ou line asim- ple algeb aic p ocedu e o e alua ing hese expansion coeKcien s. We 6 s conside he calcula ion o he coe icien s ap- pea ing in (3.17) and (3.18). By using he U(6) commu- a ion ela ion (3.15) epea edly, we ind ha [K„' ', ,K,"*]=Si,„b, ,K,''— bah, ~,K„"' ,.(3.15) We simpli y he no a ion and w i e he [2, 2, 2, 0, 0, 0] se o s a es as ~j,m). We also no e ha he U(3) ope a o s K& — —gK& ac as colo aising o lowe ing ope a o s i kgi. Thus, ac ing on acolo less s a e, hey gi e ze o unless k =i, i.e., 430 S.PITTEL, J.M. ARIAS, J.DUKELSKY, AND A. FRANK 50 [J,T2 ]=(4J, +2)T2 +J+ )[K'+ (K" — K"+)+(K„' — Ki,+)K +), (3.21) which leads o J'T' 'IG) =2T2 ' IG) —2J+ IG) .(3.22) Using he Wigne -Ecka heo em o ela e (1,1[T, '"~3, — 1) o (1,— 1~T, ''~3, — 3) Doing he same o T3, we ob ain (3) J'Ts~'l ~G) =-3J+T,'" ~G) —3J+' [G) (3.23) (3, — 1~JT2 ~3, — 3) =2asQ —4~15 (3.24) To calcula e as Q, we mul iply (3.22) by (3, — 1~and use (3.18). Remembe ing also ha ~G) =~3, — 3), we ind ha and he ea lie esul s o a3 pa3 2, and b3 2, we ind ha 42 'l2' xp5(3.33) F om he aand bcoe Bcien s e alua ed abo e, we can de e mine all he emaining coe Ficien s as well as all pos- sible educed ma ix elemen s o in e es by using he Wigne -Ecka heo em; he esul s a e Applying J o he le and no ing ha asQ — —(3, — 1iT~~ l i3, — 3), we ob ain (3.2s) 0=3iT"i~ =3) =-6 14 5(3.34) G3 p=— 2(3.26) 5 To calcula e as 2, we need he o e lap JV =(G (T2 l) T2 ~G) .Asimple calcula ion using he com- mu a o ela ion (3.15) gi es (3.3s) (3.36) which leads o N' =60 G3 2=12 (3.27) (3.28) 0=3iiT&'& ii& =»=»5(3.37) b3 P12 ~s (3.29) Likewise, o ob ain bs 2, ™1~~ply (3.23) by (1,0~; he esul is 36 b3 5(3.30) Finally, we u n o he las independen coe%cien aqp, which appea s in (3.19). We mus i s in oduce ano he enso ope a o T4 — —(T2 ),whose ac ion on he un- (4) (2) 2 pe u bed g ound s a e ~G) can be eadily shown o be JT~~ ~ ~G) =4J+T2 ~G) — 8J— +Ts ~G) .(3.31) To de e mine ai,Q, we mul iply (3.31) by (1,1~and apply J2 o he le . This leads o 2a3Q(1, 1~T2 ~3, — 1) +2 as 2aiQ 8as,— 2— 8V2 b3, — 2.(3.32) The e emains an unde e mined o e all sign o his co- e Ecien , which can be chosen a bi a ily wi h no change in he inal esul s . Asimila p ocedu e can be used o e alua e b3 pand b3 — 2~In pa icula , o ob ain b3 Qwe mul iply (3.23) by (3,0~, which leads o 15 (3.39) These esul s coupled wi h u he use o he Wigne - Ecka heo em pe mi us o de e mine all ma ix ele- men s o he h ee-colo Lipkin Hamil onian, which be- cause o he small numbe o basis s a es can be easily diagonalized. We ha e limi ed ou algeb aic analysis he e o 0=2, since ha is he case o which we ca y ou mapping es s in he nex sec ion, bu i is possible o use simila algeb aic echniques o la ge alues o O. The me hod, howe e , apidly becomes mo e complica ed o inc eas- ing O. The eason is ha mo e j alues appea o la ge 0and an i e a i e p ocedu e is equi ed o de e mine he ac ion o T2( ~and T3 on p og essi ely smalle angula momen um s a es. As we ha e seen in he 0=2case, each s ep in he i e a i e p ocedu e equi es he in o- duc ion o anew enso ope a o , buil ou o he un- damen al ones. VVe ha e al eady succeeded in ob aining algeb aic esul s o 0=3using simila me hods. Thei gene aliza ion o a bi a y 0is cu en ly being in es i- ga ed . 50 CONSISTENT BARYON MAPPING OF QUARK SYSTEMS 431 IV. TEST OF MAPPING ON THE THREE-COLOR LIPKIN MODEL TABLE II. Numbe o dis inc colo less one- iple s a es .p, spg ~0)S O agiVen O al P=p1 +pg +pg. In his sec ion we apply he colo less ba yon mapping de eloped in Sec. II o he h ee-colo Lipkin model. We ca y ou he analysis and he esul ing compa isons o 0=2only, o which he numbe o ba yons is likewise 2. Diagonaliza ion o he e Fec i e iple Hamil onian can be done exac ly o his case, leading o adi ec es o he mapping. P 4 5 6 No. o s a es 4 6 6 4 A. Cons uc ion o he colo less ba yon space The colo less s a es o he model, a e ca ying ou he mapping, a e cons uc ed in e ms o ba yons wi h quan- ~~ numbe s c ~pq, 0'2@2, and T3p3. As in he p e ious sec ion, wo noncolo quan um numbe s a e needed o speci y he s a e o each o he h ee qua ks ep esen ed by he colo less ba yon. Since all h ee qua ks ha e di e - en colo s, he e is no Pauli es ic ion on hese quan um numbe s. As no ed in he p e ious sec ion, ause ul way o cha - ac e ize s a es o he model is in e ms o he o al P alue, which o asingle ba yon is P=p1+p2 +p3, In Table II, we enume a e he numbe o dis inc colo less one-ba yon s a es o each possible alue o P, anging om P=3— 6(= 30). The wo-ba yon s a es o pa icula in e es a e hose wi h o al P=Pq +P2 ——9. The e a e wo ways o achie e P=9, ei he wi h one iple ha ing P=3 and he o he P=6, o wi h one ha ing P=4and he o he P=5. F om Table II, we see ha he num- be o dis inc wo- iple s a es wi h P=9is 52; 16 ha e (P1,P2) =(3,6) and 36 ha e (P1,P2) =(4, 5). This is signi6can ly la ge han he numbe o P=9s a es in he o iginal qua k model (see Table I), which is 20. The eason is ha he wo- iple space includes bo h physical and unphysical s a es. Acen al heme o ou analysis will be o con6 m ha ou non-He mi ian map- ping no only ep oduces he spec um o physical s a es (as ob ained in he algeb aic analysis o Sec. III) bu also pushes up he unphysical s a es ela i e o he He mi ian (pu e p-h) mapping. B.Mapping he Hamil onian HmH„h — —T„h +V„h, (4.1) whe e The gene al h ee-colo Lipkin Hamil onian, gi en by (3.2)— (3.5), can be mapped ei he in non-He mi ian o He mi ian o m. We will be pa icula ly in e es ed in he non-He mi ian mapping, since i is expec ed o p o ide a mo e p ac ical inco po a ion o qua k Pauli e ec s. How- e e , in wha ollows, we p esen bo h, o see whe he ou expec a ions a e indeed ealized. The non-He mi ian (nh) mapping is implemen ed by using (2.13) o he one-qua k e m, (2.23) o he wo- qua k in e ac ion, and (2.17) o he h ee-qua k in e ac- ion. The esul ing e ec i e Hamil onian Hh o colo - less ba yons is gi en by 36 Tnh =2lA J1' +pq as peag pg +ps asps asps — pq e g ps asps px aspeas ps ) A px ps ps~as X& A )+pl+pgaspg pl pgagps — pl-psagpg +pl+psagpg ) P1PQPS ~S 36y3 A Q2 l~1+Pl+Ps+Ps Pl ps Pg +pc — ps — pg +Pl +Ps+Ps ) P&P&PS (4.2) and Vh — —— +A A )+pqaspsaepe' +pgaspsaepe(' Px PsasPs— 'aepe— aspsaepe +aspsaepeaepe Pl PgasPS) P1~P 3CTS ~Cog ~cs pqc spsaepe — pgc spsc epe1 +p— x+Pgasps aepeaspsaepe aspsaepeaepe +Pc+Psasps)) A+A A 108y3 02 (+Pl c ePec sPs +Pg+PsaePe aspsaspsaepe Pc Pg Ps A AA p1~pe~e ~~e pc e e peas ps pg psaepe aepe— a— spsa— epe +pa+ps+ps ) AA(4 3) The He mi ian (h) mapping is implexnen ed by using (2.13) o he one-body e m, (2.22) o he wo body e m, and (2.19) o he h ee-body in e ac ion. The inal esul o his mapping is