Consistent baryon mapping of quark systems

PHYSICAL REVIEW CVOLUME 50, NUMBER 1JULY 1994 Consistent baryon mapping of quark systems S. Pittel Bartol Research Institute, University of Delaware, Newark, Delaware 19716 J.M..Arias Departamento de Fisica Atomica, Molecular yNuclear, Universidad de Sevilla, Apdo 10. 65, $1080 Sevilla, Spain J.Dukelsky Grupo de Fasica Nuclear -Facultad de Ciencias, Unieersidad de Salamanca, 5'7008 Salamanca, Spain A. Frank Instituto de Ciencias Nucleares and Iaboratorio de Cuernaeaca, Instituto de Ecsica, Uruv'ersidad Nacional Autonorna de Mexico, Apdo Pos.tal 70-5)8, 04510 Megico, Distrito Federal, Megico (Received 7December 1993) We present anew and consistent mapping of colorless three-quark clusters onto colorless triplet fermions (baryons) and test it in the context of athree-color extension of the Lipkin model. For systems with two triplets (for which the problem can be solved without approximation both before and after the mapping), we exactly reproduce the dynamics ofthe model for the variety ofcorrelation structures considered. PACS number(s): 21.60.Gx, 21.60.Fw, 21.30.+y, 12.39.— x I. INTRODUCTION Establishing aconnection between nuclear physics and /CD has been an area of intense research in the last few years. Central to this effort is the goal of isolating quark effects in nuclei. In recent years, constituent quark models [1]have been applied with considerable success to nuclear systems with few particles. While many questions remain concerning the validity of such an approach (e.g.,the lack of aconnection to /CD and its apparent violation of the underlying physics of chiral symmetry and spontaneous chiral symmetry breaking processes), in view of these successes it seems worthwhile to develop these models further, in order to see whether they can provide apartial bridge between the physics of /CD and that of finite nuclei. At present, such models have been directly applied to oneand two-baryon systems only. To treat systems with larger numbers of nucleons, it has proven necessary to introduce approximations. One possibility that has been explored is to use the resonating group method in the sixquark problem to extract an effective nucleon-nucleon interaction, which is subsequently diagonalized in the space of several nucleons [2]. Unfortunately, many-quark effects that may arise when more than two nucleons are present will be missed in such an approach. To incorporate them, we would like to bypass the two-nucleon problem and work directly in the space of many quarks. Since asystem of 3A quarks will cluster into Atriplets (nucleons) at normal nuclear densities, anecessary ingredient in any such approach is amethod for handling strong three-body correlations in amany-body environment. Recently, it has been suggested [3— 5] that mapping methods [6] might provide apractical means of accomplishing this. The basic idea is to map colorless threequark clusters, which do not satisfy exact fermion anticommutation rules, onto triplet fermions (baryons) that do. Such amapping leads &om the original multiquark Hamiltonian to an effective Hamiltonian for these baryons, which rigorously incorporates the physics of the Pauli principle at the quark level. The virtue of this approach is not in areduction of the degrees of freedom — there are in fact more states in the mapped space than in the original space — but rather in the representation of the dynamics as asystem of interacting baryons. These baryons contain as a subspace the physical nucleons (as well as excited nucleonic states) and interact with one another in ways that should be amenable to the usual fermion many-body techniques [7], e.g.,Hartree-Fock, Tamm-Dancoff, and random phase approximations, Brueckner theory, etc. Several different mappings ofquarks to triplet fermions have been recently discussed. Pittel, Engel, Dukelsky, and Ring [3] (hereafter referred to as PEDR) proposed atwo-step procedure, whereby pairs of quarks are 6rst mapped onto diquark bosons and then boson-fermion pairs are mapped onto triplet fermions. Despite the success of this mapping in reproducing the dynamics of the test model to which it was applied, it nevertheless has some drawbacks. On the one hand, it does not lead directly to triplet fermions that are antisymmetric in their three indices. In addition, as formulated, it can only be applied to quark Hamiltonians dominated by two-body interactions. At roughly the same time, Nadjakov [4] suggested an alternative mapping that leads directly to antisymmetric triplet fermions and, furthermore, is applicable to three-body interactions. His mapping, how0556-2813/94/50(1}/423(12}/$06.00 50 423 1994 The American Physical Society S.PITTEL, J.M. ARIAS, J.DUKELSKY, AND A. FRANK ever, is not appropriate for systems that are dominated by two-body interactions. Clearly, what is needed is a mapping that consistently treats both twoand threequark interactions. One such method was in fact proposed soon thereafter by Meyer [5]. However, this mapping does not seem to properly treat two-body interactions either, at least when atruncation to colorless triplet fermions is imposed. Building on the ideas of Meyer, we have now succeeded in formulating anew baryon mapping of (colorless) three-quark clusters that seems to satisfy all the desired requirements. In Sec. II, we brieHy review some general features of baryon Inappings and present our colorless version. Both PEDR and Meyer tested their mappings in the context of an exactly solvable model of quarks often referred to as the Bonn quark shell model (BQSM) [8]. While this model has some attractive features, it also has some serious limitations. Perhaps the most significant is that it does not produce spatially localized colorless three-quark clusters (i.e.,nucleons) [9]. Thus asecond goal of this work has been to develop an alternative quark model on which to test our mapping. The model that we have chosen is athree-color extension of the well-known Lipkin model [10], which in its traditional version has been used extensively to test various nuclear many-body techniques. We describe this model in Sec. III and discuss its algebraic solution for small numbers of particles. Acrucial component of this model is that it admits, for difFerent values of its parameters, dynamical one-, two-, and three-body correlations. In Sec. IV, we apply our mapping to the three-color Iipkin model for two triplets and present the results. The bottom line is that the mapping seems to work perfectly, when all colorless baryon states are included. For systems involving alarger number of triplets, this is clearly not possible. Thus, in Sec. V, where we summarize the principal conclusions of our work, we also describe some future extensions needed to further test the applicability of our methods for many-triplet systems. II. BARYON MAPPINGS OF QUARK SYSTEMS A. Preliminaries aS —gq1aq1S ) 1 (2.2) +abed =g~123 &145 q2 q3gq5d q4c 12345 (2.3) t 5df—g6123 6456 'VloQ2513 ref I"~4d .( 123456 Here, and in the subsequent analysis, we denote the color indices by numbers and the rest by roman letters. The idea of abaryon mapping is to replace the system of interacting quarks by an equivalent one of interacting triplet fermions. We denote the creation and annihilation operators of the resulting (mapped) space by Al t253, and A12/3 respectively. They satisfy the multi-index anticommutation relation (Al 253 A4de ef)— b(la2b3c, 4d5e6f ) where (2.5) Our starting point is anonrelativistic model of constituent quarks. We denote the quark creation and annihilation operators of the model by q,-and q;, respectively. The first subscript denotes the color quantum number and the second denotes all the rest. These operators satisfy the usual fermion anticommutation relation (gci Ipc&i& )=bci,c'i' =bcc' bii' (2.l) QCD considerations suggest that the quark Hamiltonian may include up to three-body interactions, all of which are color scalars. Such aHamiltonian can always be expressed in terms of the following colorless operators: b(la2b3c, 4d5e6f) =bl 4db25, 5ebsc, ef +bla, ca&45,ef&3c,4d +&la,efb25, 4db3c, 5e ala, 4db2b, efb3c,ee ala, 5eb2b, 4db3c, ef ala, efb2b, eeb3c,4d (2.6) The operator A1 2&3,,by definition, creates abaryon corresponding to three quarks in the states la, 26, and 3c. This correspondence is expressed through the requirement that both A1 2g3 and A1 2p3, are antisymmetric under interchanges of their quark indices, e.g., A=— A~ etc. la263c 2bla3c~ The space generated by these baryon operators is in fact larger than that of the original quarks. This can be seen by considering the state of two baryons, ~la2b3c, 4d5e6 f)~=Al 253,A4de, ef ~0)~, (2.7) where the subscript Brefers to states in the baryon space. The state (2.7) is antisymmetric under the interchange of the indices corresponding to any two quarks within one of the two triplets (e.g.,la with 2b or 4d with 5e) and also under the interchange of one triplet with the other. However, it is not antisymmetric under the interchange of the quantum numbers of aquark in one triplet (e.g.,la) with those of aquark in the other (e.g.,4d). Afully antisymmetric two-triplet state may be recovered by taking an appropriate linear combination of the states (2.7). As a consequence, there is indeed asubset of two-triplet (and likewise many-triplet) states that are fully antisymmetric and, furthermore, are in one-to-one correspondence with the states of the original quark space. This is referred to as the physical subspace. There is, however, another class of states that are not fully antisymmetric under interchange of quark indices and which therefore have no counterparts in the original space; this is referred to as the unphysical subspace. There are avariety of possible ways to ensure that the 50 CONSISTENT BARYON MAPPING OF QUARK SYSTEMS physics of the original quark problem is preserved under the mapping to baryons. We follow the approach of Belyaev and Zelevinsky, where operators in the original space are mapped onto operators in the new space so as to preserve their (anti)commutation relations. Implementation of this prescription guarantees that all the physics of the original system is exactly preserved by the mapping within the physical subspace. For amapping to be of practical use, the unphysical states must lie high in energy relative to physical states. Otherwise, it will be difficult to disentangle the physical states of interest &om those that are unphysical, particularly in the presence of variational approximations. This places stringent limitations on the kinds ofmappings that should be considered for practical applications. For example, it is straightforward to write down amapping of colorless particle-hole (p-h) or one-body quark operators that preserves their commutation relations: 1 ).q&.nb ~2)A] 23/Alb2 3d (28) 1123cd Since any ofthe colorless operators ofthe type (2.2)— (2.4) can be rewritten in terms of colorless particle-hole quark operators, it would seem that we could simply apply (2.8) and achieve our goal. This is unfortunately not the case. Since aparticle-hole operator does not involve more than one creation and one annihilation operator, it cannot (by itself) incorporate any information on the quark Pauli principle. As aconsequence, the spectrum that would result from apure p-h mapping (for fermion systems) would invariably have unphysical states lying below the physical states of interest. To incorporate quark Pauli eifects (in aphysically useful way), we must map directly the multiquark creation and annihilation operators that appear in the Hamiltonian. In what follows we adopt the Dyson approach, which leads to abaryon Hamiltonian that is non-Hermitian but finite. The non-Hermiticity is adirect re8ection of quark Pauli effects. Anovel feature of our analysis is that, in contrast to earlier work, we focus on the mapping of colorless operators. This removes some of the ambiguities that arise when the mapping is carried out more generally. It also leads to amapping that is tailored to physical applications in which atruncation to colorless triplets must be implemented. B.Mapping of colorless one-body operators We begin with adiscussion of the colorless one-body operator Abof (2.2). To map this operator, we can make direct use of earlier results. Namely, using (2.8), we can express its baryon image as At this point, the image of Agis expressed in terms of baryons with color. We know, however, that it is possible to describe the relevant physics in abasis of colorless baryons only. Towards this end, it is useful to carry out atruncation to colorless baryons; this can be done using the prescription spelled out by PEDR. The basic idea is to carry out acolor-SU(3) coupling and to isolate the piece that is fully antisymmetric in color and fully symmetric in the other noncolor indices. Asimple way to implement this is through the replacements At ca.li2j3~ +6123Paij ~1i2j3& M6123~zjk (2.10) The operators At-& and A;jg introduced here are fully symmetric under the interchange of their indices. Furthermore, they satisfy the anticommutation relation (A;,b, At.„j=—S(ijk,lmn), 1 U~lmn (2.ii) where S(ijk, lmn) =b;)b, bb„+b; b,.„bb) +b;„b,(bb +4b, bb +b' b,ibb~+b; b, (2.12) Inserting (2.10) into (2.9) and then carrying out an explicit sum over the color indices (gz2z e&2s — —6), we arrive at the following result: Abm3 )At,qAb, g. cd (2.13) As an example, consider amapping of the quark number operator Nq =Pz qz qq .Applying (2.13), we obtain the expected result Nq m3) At~A b, =3N~, (2.14) where N~ is the number operator for colorless baryons. C. Mapping of colorless three-body operators Next we consider the colorless three-body operator Cb,g,yappearing in (2.4). Here, too, we can directly use the mapping given by Nadjakov and Meyer, since the set of one-body operators plus the set of three-quark creation and annihilation operators close under commutation. The non-Hermitian mappings of three-quark creation and annihilation operators required to preserve this commutation algebra are [4,5] ~ab =)qj~~qlb ~))A]~2~MAlb2csd ~(2.9) 1123 Cd ql'q2 q3A: ~~3I2j1; (2.i5) q.q-qt mAt. .+—pt pt pt ttt 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) to the colorless three-body operator Cbdfand carrying out asubsequent truncation to colorless triplets, we obtain the following result: Cab,d,fmCb",d,y— —36 Ab,Adey+36) (A „Ab, +A„bA, ,+A „,A~)Agh;Adey . ghi (2.17) Asuperscript nh has been included to indicate that this is anon-Hermitian baryon image. Note that acolorless three-quark interaction maps onto colorless one-body plus two-body interactions only. The three-body piece cancels exactly in the truncation to colorless baryons, because of properties of the 6y23 factors. D. Mapping of colorless two-body operators Finally we turn to the two-body operator, for which we cannot directly use the results of earlier work. Meyer proposed apossible non-Hermitian baryon image for two-body operators, based on the use of the Usui operator. However, as noted earlier, this mapping does not seem to work when truncated to colorless triplets and applied to the test model we present later. Nevertheless, there is asuggestion in her work as to how to build aproper mapping for colorless two-body operators, which we now exploit. At the end, we make some remarks as to why her results are not applicable. As we saw in the previous subsection, it is possible to map acolorless three-body operator in non-Hermitian form. Since the mapping (2.17) followed &om an exact preservation of commutation relations, it is certainly legitimate. But there is another way to map acolorless three-body operator that is equally legitimate and which leads to aHermitian form. The three-body operator Cbdfcan be rewritten in terms of the colorless one-body operators A;~ of (2.2) as follows: Cabcdef =AadAbeAcf +AadAbf Ace +AaeAbdAcf +AaeAbf Acd +Aaf AbdAce +Aaf AbeAcd — bbd(AaeAcf +AafAce) — bbe(AadAcf +AafAcd) —'4f(AadAce+ AaeAcd) 26 d(cA Aaebf +Aaf Abe) — 2bce(AadAbf +AayAbd) — 2bcj(AadAb, +Aa, Abd) +2(~bc,efAad +~bc,df Aae +~bc,deAa f)(2.18) Applying the mapping of colorless one-body operators (2.13) and focusing on the one-and two-body pieces, we obtain, for the Hermitian image, Cabcdef ~Cabcdef 3AabcAdef 36 )(AghaAbci +AghbAcai +AghcAabi) (AdghAe fi +AeghAdfi +AfghAdei) t.t t tttt ghi (2.19) Note that the several 8-function terms in (2.18) do not survive after the mapping. They are exactly canceled by other terms tha'. arise when the baryon image is put in normal order. The one-body piece of (2.19) is identical to that given in (2.17). The two-body part, however, is not. One is Hermitian and the other non-Hermitian. However, both are formally justified and thus must be equivalent in the physical subspace. It is easy to show that either of the forms of the two-baryon image of Cb,d,fcan be transformed into the other by performing the following replacement on the two annihilation operators: 1 AabcAde j~(AabdAcef +AabeAdcf +Aabf Adec +AadcAbef +AaecAdbf +AafcAdeb 3 +AdbcAaef +AebcAdaf +AfbcAdea) (2.20) These observations suggest aprocedure for mapping acolorless two-body interaction. Namely, we first transform it to colorless p-h form, then map it using the well-known (and formally justified) colorless p-h mapping (2.13) and finally transform its two-body part to anon-Hermitian form by carrying out the replacement (2.20). We now apply this prescription to the colorless two-quark operator Bbdof (2.3). Transforming it to p-h form leads to the result +abed AacAbd +AadAbc ~bdAac ~bcAad (2.21) Mapping this operator in colorless p-h form and writing the result in normal order gives ht Babcd ~Babcd =12 )Aaby Acdf 9)Aaef Abgh(Ace fAdgh +Ade fAcgh) fefgh (2.22) Finally, when we impose the replacement (2.20) on the two-baryon piece, we arrive at Babcd ~Babcd =12 )AabeAcde +9)Aaey Abgh(AcdeA fgh +Ae fgAcdh) eefgh (2.23) 50 CONSISTENT BARYON MAPPING OF QUARK SYSTEMS 427 This result divers &om the one that would arise &om Meyer's mapping supplemented by atruncation to colorless baryons, reQecting our consistent treatznent of color. In fact, the non-Hermitian mapping of two-body operators proposed by Meyer can be derived in much the same way by considering ageneral three-body interaction rather than acolorless one. Such aprocedure, however, is not unique, since ageneral three-body operator can be recast as aproduct of p-h operators in diferent ways. Explicit consideration of colorless operators removes this ambiguity. The end result is aprescription for transforming &om Hermitian to non-Hermitian images that properly incorporates color and is thus meaningful when iznplementing atruncation to color singlets. The essential results of this section, proper nonHermitian baryon images of colorless one-, two-, and three-quark operators, are contained in Eqs. (2.13), (2.23), and (2.17),respectively. Although the mapping of two-body operators was not derived by explicit consideration of commutation relations, we have confirmed that this set of baryon images does indeed preserve the comrnutator [B~s~a, E,fg], where E,fg =+~2s eq2sq~, qzfqsg t We should also emphasize here that this set of mapping equations can be applied to any colorless constituent quark Hamiltonian written in uncoupled form. E. Physical content of the replacement procedure Some understanding of the replacement procedure proposed to generate anon-Hermitian two-body image &om aHermitian one can be obtained by studying the mapping of acolorless six-quark state, ~abc, def)Q — ——)~»s ~4&s qi~q2sqscq4dqseqsfl0)q ttttt 123456 (2.24) where the subscript qindicates astate in the original quark space. Mapping this state with (2.16) and imposing atruncation to colorless baryons leads to the result ~abc, def) p~A&,A&,f~0)a+(A &,Ab,f+— AfAs,p+AQf As„ 3 +A Af+A, fA +A,„A...+A A,f+A..A.,+A.dfA...}10) (2.25) The physical state not only involves the direct twobaryon component but also asum over all nine possible interchanges of the indices of one baryon with those of the other (with an overall factor of s). It is not difficult to confirm that when we act either with the operator A~Ap, yon this physical state or with the replacement form given in (2.20) we arrive at exactly the same result. Thus the proposed replacement indeed satisfies the desired criterion that it produces the same results within the physical subspace. It is also interesting to note the correspondence between the direct and exchange pieces of (2.25) with the left and right hand sides of (2.20). III. THREE-COLOR LIPKIN MODEL I body interaction that scatters pairs of particles among the two levels without changing the pvalues. This znodel can be solved exactly by using group theoretical techniques. It is well known that the set of all possible bilinear products formed &om afinite set of creation and annihilation operators constitutes aLie algebra. In the case of the Lipkin model, there are (20)2such bilinear products of creation and annihilation operators (generators), and so the relevant Lie algebra is U(20). These generators will be denoted by K,"„, =qt„q Since we are dealing with fermions, all the states of the system belong to the irrep [ln] of the U(20) dynamical group. The structure of the problem suggests the decomposition U(20) oU(O) gU(2), (3 1) The three-color Lipkin model is based on the wellknown Lipkin model [10], which can be solved analytically and has been used extensively in nuclear physics for testing many-body approximation methods. Since many of the characteristics of the three-color Lipkin model are already in the original one, we review it brieBy here. The Lipkin model has two levels, each one 0-fold degenerate, separated by an energy A. It is assumed that in the unperturbed ground state N=0particles occupy all the single-particle states in the lower level. The fermion creation and annihilation operators of the model are written as qt„and qz, respectively. Here, ois a quantum label which characterizes whether the particle is in the lower level, o. =—,or in the upper one, o=+, and pdistinguishes which of the 0degenerate states of that level the particle occupies. The Hamiltonian of the model includes, in addition to the one-body term, atwowhere the 0operators K„", =PKP~ generate the U(A) algebra, while the 22 objects K, =g„K,"„are the generators ofthe U(2) algebra, and commute with the K„,.One can easily verify that the Lipkin Hamiltonian can be written entirely in terms of the U(2) generators and thus that all the states belong to adefinite irreducible representation of U(2) [or SU(2)]. This in turn implies that the Hamiltonian znatrix can be analytically evaluated using the well-known angular moxnentum algebra SU(2). The three-color Lipkin model has amuch richer algebraic structure and analytic solutions are correspondingly more difBcult to derive. The model involves three sets (one for each color) of standard two-level Lipkin models. Again the lower levels are assumed to be completely filled in the unperturbed ground state, which in 428 S. PITTEL, J.M. ARIAS, J.DUKELSKY, AND A. FRANK 50 this case contains N=30 particles. The creation and annihilation operators now include alabel ithat represents the color quantum number and are thus written as q,.„and qi „,respectively. The model Hamiltonian now includes one-body, two-body, and three-body interactions, which scatter particles coherently among the levels, without changing the pvalues and maintaining all states colorless: 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 rtttttt &123&456 g1y+ 'V2+ 13+ 'V6 —95—g4—+14—95—16—13+P 12+ Ql+ )(3.5) In this case there are (6O)2 generators K& ", ,— —qt „qs „,leading to the Lie algebra U(6O), and the structure of the model suggests that we carry out the classification of states in terms of the chain U(6O) zU(O) U(6) zU(O) 13 U(3) U(2) .(3.6) The O2 operators K", =P,.K,'. ~, generate U(O), while the 62 objects K&, — —gK& ~are the U(6) generators, in terms of which we shall rewrite the Hamiltonian (3.2)— (3.5) below. We can further decompose U(6) by contracting again to the 22 operators K, =g,.K,',,which generate U(2), or to the 32 generators Kl', — —PK& corresponding to U(3). The latter group is indeed necessary in the classification, since all physically admissible states should be colorless; i.e.,they should belong to the (O, O, O) U(3) representation [which corresponds to the (A, p) =(0, 0) scalar representation in Elliott sSU(3) notation]. The situation is more complex than in the standard Lipkin model, however, since different U(6) representations can contain these states and, moreover, for each of them several U(2) representations are connected by the Hamiltonian. From (3.2)— (3.5), we see that the model involves three parameters, one each for the one-, two-, and three-body interactions. As mentioned before, it may be rewritten in terms of the U(6) generators as H1 —— AJ, ,(3.7) ik (3 8) and 2(J++J)+2J+) K„'+K, "+ +J)K„'+K, "+ ik ik )(Kt+Kq+ K,". ++K,*+K„'+K, "+ ), (3.9) ilk where J„J~,and J,defined by J, =2(K++ — K), J+ =K+, and J=K+, are the SU(2) subgroup generators, which together with the number operator N=P,.K,'comprise the U(2) group in (3.6). The other operators in (3.8) and (3.9), namely, g,.&K&+ K,". +and g1& Kl*+K&+K,"+, together with their Hermitian conjugates, clearly lie outside SU(2). Using their commutation relations with the SU(2) generators, we readily conclude that they behave as rank 2and rank 3tensors T=T2 and T3 in SU(2), respectively. (e) (2) (3) . Using this notation, the model Hamiltonian acquires the simpler form H1 — —AJ, ,(3.10) X2 (J2 J2 )+X2 (T(2) +T(2)) (3.11) and 50 CONSISTENT BARYON MAPPING OF QUARK SYSTEMS 429 to which we shall refer henceforth. Since we are dealing with asystem of fermions, the states of the model belong to the [ls+] representation of U(6O), while the U(O) and U(6) representations are compleinentary; i.e.,once the U(6) representation is determined, the U(O) representation is fixed. Since the U(3) representations are the colorless ones mentioned before, it should be clear that the basis states can be unambiguously denoted by ~[&i h2 li3 h4 h5 hs], ~jm), (3.13) The state ~G) has three particles with p=1, three with p=2,...,and three with p=O. Defining P=P,.p;, we see that the state ~G) has aunique Pvalue of3O(O+1)/2. In general, difFerent U(6) irreducible representations appear which contain the U(3) colorless irreps in the reduction U(6) DU(3) U(2) of (3.6). Since the Hamiltonian (3.10)— (3.12) is built solely in terms of U(6) generators, the energy matrix will separate into blocks, each corresponding to adefinite U(6) irrep. For 0=1, the group analysis is particularly simple. There is only asingle U(6) irrep, (1,1,1,0,0,0), which contains the (1,1,1)(j =3/2) and the (2, 1,0) (j =1/2) U(3) SU(2) irreps. However, only the first one belongs to our (colorless) space. Furthermore, all states have P=3. For 0=2 the situation is more complex in several respects. First, as we shall enumerate shortly, there are several possible U(6) irreps that contain colorless states. All contain states with P=9, while several also contain states with other Pvalues. These other states, however, can be generated from the corresponding P=9states by the U(O) raising and lowering operators Ki', with pgp', and are thus degenerate in energy with them. For this reason, we need only consider the states with P=9to fully exhaust the spectrum. In Table I, we display the four possible O=2U(6) irreps that contain colorless P=9states and also indicate the associated angular momenta. Asimilar analysis can also be carried out for higher 0values. TABLE I. Colorless states and group labels for 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 Degeneracy 10 6 3 1 where [hi, ...,hs] labels the U(6) representations, jm are the SU(2) DSO(2) quantum numbers, and ais an extra label that may be needed to distinguish either repeated U(6) irreducible representations (irreps) within the same U(6O) representation, repeated j's within the same U(6) representation, or any other necessary quantum numbers. For example, the unperturbed ground state of the model, defined as the state for which all o=—levels are filled, corresponds to [G) =[O, O, O, 0,0, 0], j=,m=—.(3.14) 30 30 In the following section we test our mapping procedure for the case of O=2. Here, we illustrate the algebraic evaluation of matrix elements for the [2,2,2,0,0,0] O=2 submatrix, for which we will need to consider states with j=3and j=1(see Table I). Analogous calculations have also been done for the other states and are included in the results that we present. Before proceeding, we write down the commutation relation for the U(6) generators, which will be used extensively in our analysis: (Kq+~ +Kq )~j,m) =bi,;A ~j,m) . To 6nd the value of A, we note that )(K&++K& )~j,m) =N ~j,m) =6~ j,m), for 0=2. Since all colors should be equally represented in acolorless state, we arrive at the useful relation (K„'++K„' )~j,m) =2bi,;]j,m) .(3.16) The basic idea of our analysis is to consider the action of the operators Tz and Tz on the states ~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 anotation whereby the 6rst index in the expansion coefficients denotes the SU(2) label of the initial state and the second gives the increment needed to obtain the SU(2) label of the final state. If we can determine all of the independent aand bexpansion coefficients in (3.17)— (3.20), we will have effectively solved the problem. We can then use the WignerEckart theorem to determine all the relevant reduced matrix elements of T& ~and T& ~and from them determine all of the matrix elements of the Hamiltonian for any choice of the model parameters. We now outline asimple algebraic procedure for evaluating these expansion coeKcients. We 6rst consider the calculation of the coefficients appearing in (3.17) and (3.18). By using the U(6) commutation relation (3.15) repeatedly, we find that [K„' ', ,K,"*]=Si,„b, ,K,''— bah, ~,K„"' ,.(3.15) We simplify the notation and write the [2, 2, 2, 0, 0, 0] set of states as ~j,m). We also note that the U(3) operators K& — —gK& act as color raising or lowering operators if kgi. Thus, acting on acolorless state, they give zero 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 to J'T' 'IG) =2T2 ' IG) —2J+ IG) .(3.22) Using the Wigner-Eckart theorem to relate (1,1[T, '"~3, — 1) to (1,— 1~T, ''~3, — 3) Doing the same for T3, we obtain (3) J'Ts~'l ~G) =-3J+T,'" ~G) —3J+' [G) (3.23) (3, — 1~JT2 ~3, — 3) =2asQ —4~15 (3.24) To calculate as Q, we multiply (3.22) by (3, — 1~and use (3.18). Remembering also that ~G) =~3, — 3), we find that and the earlier results for a3 pa3 2, and b3 2, we find that 42 'l2' xp5(3.33) From the aand bcoefBcients evaluated above, we can determine all the remaining coefFicients as well as all possible reduced matrix elements of interest by using the Wigner-Eckart theorem; the results are Applying Jto the left and noting that asQ — —(3, — 1iT~~ l i3, — 3), we obtain (3.2s) 0=3iT"i~ =3) =-6 14 5(3.34) G3 p=— 2(3.26) 5 To calculate as 2, we need the overlap JV =(G (T2 l)tT2 ~G) .Asimple calculation using the commutator relation (3.15) gives (3.3s) (3.36) which leads to N' =60 G3 2=12 (3.27) (3.28) 0=3iiT&'& ii& =»=»5(3.37) b3 P12 ~s (3.29) Likewise, to obtain bs 2, ™1~~ply (3.23) by (1,0~; the result is 36 b3 5(3.30) Finally, we turn to the last independent coe%cient aqp, which appears in (3.19). We must first introduce another tensor operator T4 — —(T2 ),whose action on the un- (4) (2) 2 perturbed ground state ~G) can be readily shown to be JT~~ ~ ~G) =4J+T2 ~G) — 8J— +Ts ~G) .(3.31) To determine ai,Q, we multiply (3.31) by (1,1~and apply J2 to the left. This leads to 2a3Q(1, 1~T2 ~3, — 1) +2 as 2aiQ 8as,— 2— 8V2 b3, — 2.(3.32) There remains an undetermined overall sign for this coefEcient, which can be chosen arbitrarily with no change in the final results . Asimilar procedure can be used to evaluate b3 pand b3 — 2~In particular, to obtain b3 Qwe multiply (3.23) by (3,0~, which leads to 15 (3.39) These results coupled with further use of the WignerEckart theorem permit us to determine all matrix elements of the three-color Lipkin Hamiltonian, which because of the small number of basis states can be easily diagonalized. We have limited our algebraic analysis here to 0=2, since that is the case for which we carry our mapping tests in the next section, but it is possible to use similar algebraic techniques for larger values of O. The method, however, rapidly becomes more complicated for increasing O. The reason is that more jvalues appear for larger 0and an iterative procedure is required to determine the action of T2( ~and T3 on progressively smaller angular momentum states. As we have seen in the 0=2case, each step in the iterative procedure requires the introduction of anew tensor operator, built out of the fundamental ones. VVe have already succeeded in obtaining algebraic results for 0=3using similar methods. Their generalization to arbitrary 0is currently being investigated . 50 CONSISTENT BARYON MAPPING OF QUARK SYSTEMS 431 IV. TEST OF MAPPING ON THE THREE-COLOR LIPKIN MODEL TABLE II. Number of distinct colorless one-triplet states .p, spg ~0)S fOr agiVen tOtal P=p1 +pg +pg. In this section we apply the colorless baryon mapping developed in Sec. II to the three-color Lipkin model. We carry out the analysis and the resulting comparisons for 0=2only, for which the number of baryons is likewise 2. Diagonalization of the efFective triplet Hamiltonian can be done exactly for this case, leading to adirect test of the mapping. P 4 5 6 No. of states 4 6 6 4 A. Construction of the colorless baryon space The colorless states ofthe model, after carrying out the mapping, are constructed in terms of baryons with quant~~ numbers cr~pq, 0'2@2, and fT3p3. As in the previous section, two noncolor quantum numbers are needed to specify the state of each of the three quarks represented by the colorless baryon. Since all three quarks have different colors, there is no Pauli restriction on these quantum numbers. As noted in the previous section, auseful way to characterize states of the model is in terms of the total P value, which for asingle baryon is P=p1+p2 +p3, In Table II, we enumerate the number of distinct colorless one-baryon states for each possible value of P, ranging from P=3— 6(= 30). The two-baryon states of particular interest are those with total P=Pq +P2 ——9. There are two ways to achieve P=9, either with one triplet having P=3 and the other P=6, or with one having P=4and the other P=5. From Table II, we see that the number of distinct two-triplet states with P=9is 52; 16 have (P1,P2) =(3,6) and 36 have (P1,P2) =(4, 5). This is signi6cantly larger than the number of P=9states in the original quark model (see Table I), which is 20. The reason is that the two-triplet space includes both physical and unphysical states. Acentral theme of our analysis will be to con6rm that our non-Hermitian mapping not only reproduces the spectrum of physical states (as obtained in the algebraic analysis of Sec. III) but also pushes up the unphysical states relative to the Hermitian (pure p-h) mapping. B.Mapping the Hamiltonian HmH„h — —T„h +V„h, (4.1) where The general three-color Lipkin Hamiltonian, given by (3.2)— (3.5), can be mapped either in non-Hermitian or Hermitian form. We will be particularly interested in the non-Hermitian mapping, since it is expected to provide a more practical incorporation of quark Pauli effects. However, in what follows, we present both, to see whether our expectations are indeed realized. The non-Hermitian (nh) mapping is implemented by using (2.13) for the one-quark term, (2.23) for the twoquark interaction, and (2.17) for the three-quark interaction. The resulting effective Hamiltonian Hhfor colorless baryons is given by 36 Tnh =2lAt J1' +pq as peag pg +ps asps asps — pq erg 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~Pf3CTS ~Cog ~cs pqcrspsaepe — pgcrspscrepe1 +p— x+Pgasps aepeaspsaepe aspsaepeaepe +Pc+Psasps)) A+A A 108y3 02 (+Pl crePecrsPs +Pg+PsaePe aspsaspsaepe Pc Pg Ps At AA p1~pe~e ~~e pc ere peas ps pg psaepe aepe— a— spsa— epe +pa+ps+ps ) AA(4 3) The Hermitian (h) mapping is implexnented by using (2.13) for the one-body term, (2.22) for the two body term, and (2.19) for the three-body interaction. The final result of this mapping is