On the use of baryon mappings to derive nuclei from quarks

lleVlsta Mexicana de Física 40, Suplemento 1 (1994) 318-329 On the use of baryon mappings to derive nuclei from quarks S. PITTEL Bartol Researc/¡ Institute, Univer.,ity of Delawarc New07'k, Delaware 19716, USA J. M. ARIAS Departamento de Fúica Atómica, Molecular y Nuclear Univer.,idad de Sevilla, Apartado 1065, 41080 Sevilla, Spain .J. DUKELSKY Grupo de Fí.,ica Nuclear, Facultad de Ciencia., Univenidad de Salamanca, 37008 Salamanca, Spain ANO A. FRANK Instituto de Ciencia., Nucleare., and Laboratorio de Cuernavaca, Instituto de Fúica, UNAM Apartado po.,tal 70-543, 04510 México, D.F., México Received 6 January 1994; accepted 11 April 1994 ABSTRACT. A consistent baryon mapping oC colorless three-quark clustcrs is proposed and tested in thc context oC a three-color extcnsion oC the Lipkin model. The results suggest that baryon mappings may provide a practical means oC deriving nuclei Cromconstituent quark rnodels. RESUMEN. Se propone en Cormaconsistente un mapeo bariónico de agrupamientos de tres quarks acoplados a color cero. El mapeo se pone a prueba en el contexto de una extensión de tres colores del modelo de Lipkin. Los resultados sugieren que los mapeos bariónÍCos podrian dar lugar a métodos prácticos de derivación de la estructura de los núcleos a partir de modelos de quarks constituyentes. PACS: 21.60.Gx; 21.60.Fw; 21.30.+y l. INTROOUCTION The traditional view of nuclear physics is based on the assumption that the QCD interactions that cluster quarks into nucleons completely decouple from the residual interactions between nucleons responsible for nuclear strueture. Recent experiments by the EMC collaboration, however, indicate that the structure of the nucleon is in fact modified in a nuclear tnedium, suggesting that thcrc is some coupling betwcen the nuclcon and nuclear scales. As a consequence, there is currently great interest in trying to develop methods to 318 ON TIIE USE OF BARYON MAPP1NGS... 319 derive the properties of nuclei directly from QCD. Establishing such a link between QCD ami nuclear physics should also help to elucidate where to look for explicit quark effects in l1uclci. At present, it is not feasible to establish this connectiou startiug from QCD. A somewhat less-ambitious starting point is the use of QCD-inspired constituent quark models [1]. Such models have been applied with considerable success to nuclear systems with very few particles, but they have not yet been implemeuted for many-nucleon systems. The reason is that in a nuclear environment quark triplets cluster into spatial!y-Iocalized nucleons, a scenario that canuot be described with existing many-body techniques. In this talk, we explore the use of mappiug methods as a practical means of accomplishiug this. The idea is to mal' colorless three-quark clusters, which do uot satisfy exact fermiou auticommutation rules, onto triplet-fermions (baryons) that do. Such a mapping leads from the original multi-quark hamiltonian to an effective baryon hamiltonian, which rigorously incorporates the physics of the quark Pauli principIe. In addition, since it is a hamiltonian for interactiug fermions, it is amenable to the usual fermion many-body techniques [2]. Several groups have recently addressed the issue of baryon mappings [3-5]. Pittel, Engel, Dukelsky and Ring [3] proposed a two-step mapping that was specifical!y tailored to two-quark interactions. Their mapping cannot be applied, however, to systems with strong three-quark interactions. At roughly the same time, Nadjakov [4] suggested an alternative baryon mapping that should be applicable to systems dominated by three-quark interactions, but not to systems with strong two-quark interactions. \Vhat is needed is a mapping that consistently treats both two- and three-quark interactions. \Ve have now succeeded in developing such a mapping, which we briefly describe in Section 2. In Section 3, we describe a simple model that we have developed to test our consistent mapping. The model we have chosen is a three-color extension of the wel!-known Lipkin model [6], which in its traditional version has been used extensively to test various nuclear many-body techniques. In Section 4, we apply our mapping to this model for two baryons and present the results. The bottom line is that it seems to work perfectly. 2. BARYON MAPPING OF QUARK SYSTEMS 2.1. General remarks Our starting point is a nonrelativistic model of constituent quarks. \Ve denote the quark creation and annihilation operators by qL and qla, respectively. The first subscript denotes the color quantum number and the second al! the resto These operators satisfy the usual fermion anticommutation relation (1) QCD considerations suggest that a realistic quark hamiltonian may include up to threebody interactions, al! of which are color scalars. Such a hamiltonian can always be ex- 320 S. PITTEL ET AL. pressed In terrns of the following colorless one-, two- and three-body operators, respectively: and Aab = L qt.q1b; 1 Babed = L <123 <145 qL q1bq5d q4e; 12345 Cabed,¡ = L <123<456 qt. q~b qt q6¡ q5, q4d. 123456 (2) (3) (4) The <123 quantities are antisyrnrnetric tensors that guarantee the colorless nature of these operators. The idea of a baryon rnapping is to replace this problern by an equivalent one involving triplet-ferrnions or baryons. We denote the creation and annihilation operators of the baryon space by Ala2b3e and Ala2b3e, respectively. They, by definition, satisfy the rnultiindex anticornrnutation relation where {A,a2b3e, A~d5,6¡} = 6(la2b3c, 4d5e6f), (5) 6(la2b3e,4d5e6f) '" 6'a,4d62b,5,63e,6¡ + 6'a,5,62b,6¡63e,4d + 6'a,6¡62b,4d63e,5, -6Ia,4d62b,6¡63e,5, - 6'a,5,62b,4d63e,6¡ - 6Ia,6¡62b,5,63,,4d' (6) Since the operator Ala2b3e (A,a2b3e) creates (annihilates) a baryon that eorresponds to three quarks in the states la,2b and 3e, it is antisyrnrnetric under the interchange of its quark indiees, e.g., Ala2b3e = -A~bla3e' ete. The space generated by these baryon operators is in fact larger than the original quark space. lt ineludes a subset of states that are fully antisyrnrnetric under the interehange of quark indices and in one-to-one eorrespondence with the original states of the quark space; this is referred to as the physieal subspaee. But it also contains states that are not fully antisyrnrnetric under quark interchange. These states, called unphysieal, have no counterparts in the original quark space. They are apure artifact of the rnapping and provide the principal difficulty in developing a practical baryon rnapping. As we will discuss shortly, there are several possible rnappings that can be developed, all of which exactly preserve the physics of the original problern in the physical-baryon subspaee. Where they differ is in their predictions for unphysical states. Clearly, for a rnapping to be of practical use, the unphysical states rnust lie high in energy relative to the physical states. Otherwise, it will be difficult to disentangle the physical sta tes of interest frorn those that are unphysical, particularly in the presence of variational approxirnations. So far, we have not discussed how to guarantee that the physics of the original problern is preserved by the rnapping. Here we follow the Belyaev-Zelevinski prescription, whereby ON TIIE USE OF BARYON MAPPINGS... 321 the mapping is defined so as to exactly preserve the commutation relations for physical operators in the original quark space. There is a simple way to accomplish this. Consider the colorless one-body operator Aab defined in (2.2). The commutator [Aab, Acdl is exactly preserved if this operator is mapped according to (7) Since the operators in (2.3-2.4) can be rewritten in terms of colorless one-body operators, it would seem that we could also apply (2.7) to them and achieve our goal. Unfortunately, this is not the case. Since a one-body operator contains just one creation amI one annihilation operator, it cannot incorporate information on the quark Pauli principie. Thus, when we apply this simple mapping, we find that (i) it reproduces al! quark dynamics in the physical subspace, but (ii) it invariably leads to unphyical states lower in energy than the physical states of interest. To incorporate quark Pauli effects in a practical way, we must mal' the multi-quark creation and annihilation operators that appear in the two- and three-body interactions directly. \Vhat this means is that we must find a mapping that preserves the commutation relations between the colorless one-, two- and three-body operators simultaneous/y. 2.2. Colorless baryons The baryon operators introduced in the previous subsection al! have color. \Ve know, however, that it is possible to describe al! of the relevant physics solely in terms of colorless baryons. The rclevant operators for colorless baryons can be defined according to (8) and (9) They are ful!y symmetric under the interchange of their indices and satisfy the anticommutation relation \vherc {Aabc, A~ef} = ~ S( abe, de!), (10) In what fol!ows, we earry out our mapping directly from the space of colorless quark operators to the space built out of these colorless baryon operators. 322 S. PITTEL ET AL. 2..1. Tile ilennitean baryon mal'ping First we present the results obtained using the simple mapping described earlier, in which the colorless quark operators are first expressed as products of colorless one-body operators Aab and then mapped accortling to (2. i). A subsequent truncation to colorless baryons is then achieved by making the replacements A1a2b3c ---+ f123A~bcl Ala2b3c -+ fl23 A abcl as ,Iiscussed in ReL [31. Implementing the above prescription leads to the following results: Aab -+ A~b = 3 L A~cdAbcd; ,d Bakd ~ B~b,d = 12 L A~bfA,df f -9 LA~efAlgh(A<efAdgh +AdefA,gh); efgh Cab,def ~ C~kdef = 36 A~kAdef - 36 L(A~haAL +A~hbA~ai +A~h,A~bi) 9hi (12) (13) (14) \Ve refer to this as the hermitean baryon mal'ping and inelude a superseript h to distinguish it from the improved nonhennitean baryon mapping to follow. 2.4. The nonhermitean baryon mapl'ing The nonhermitean baryon mapping arises when we require the simultaneous preservation of a/l (anti)eommntation relations. As noted earlier, we expect it to provide a more satisfactory deseription of unphysieal states, through implicit incorporation of quark Pauli erreets. A detailed discussion on how this mapping was obtained can be found in ReL [iJ. Here, we simply present the final results: Aab ~ A~~ = 3 L A~,dAkd; ,d Babcd -t B~kcd = 12 L A~beAcde e +9 L A~efAlgh(A'deAfgh +AefgA,dh); efgh (16) (Ji) ON TIIE USE OF BARYON MAI'I'INGS... 323 een" abedef ~ abedef = 36¡\~bc¡\dcf + 36 L(¡\~ha¡\Li + ¡\~hb¡\~ai + ¡\~hc¡\~bi)¡\ghi¡\dcf' ghi (18) The superscript nh denotes that these are the nonhermitean baryon images. \Ve should emphasize here that the sets of mapping equations given in this and the preceding subsections can be applied to any colorless constituent quark hamiltonian written in uncoupled fOl'ln. 3. TIIE TIIREE-COLOR LII'KIN MOlJEL The three-color Lipkin model is based on the well-known Lipkin model [6]' which has been used extensively in nudear physics to test many-body approximation methods. Since many of the characteristics of the three-color Lipkin model are already in the original one, we first devote a few lines to reviewing it. The Lipkin model has two levels, each l1-fold degenerate, separated by an energy tI.. It is assumed that in the unperturbed ground state, N=11 partides occupy all the singlepartide sta tes of the lower leve!. The fermion creation and annihilation operators of the model are written as q~m and qum respectively, where acharacterizes whether the partide is in the lower level, a = -, or in the upper one. a =+, and m denotes which of the 11 degenerate sta tes of that level the partide occupies. The hamiltonian of the model can be expressed as H = H¡ + H2, tI.", t , H¡ = '2 LJq+mq+m - q-mq-m), m (19) (20) (21 ) In addition to the one-body term, it contains a two-body interaction that scatters pairs of partides among the two levels, without changing their m values. This lllodel can be solved exactly using group theoretical techniqlles for an)' value of 11and any values of the parameters ti. and X. The set of aH possible one-hody operators built from its creation and annihilation operators generates the Lie algebra U(211). The structure of the problem suggests a decomposition U(211) :) U(I1) 0 U(2). (22) The Lipkin hamiltonian can be rewritten solely in terms of the U(2) generators and all the states helong to a definite irreducible representation of U(2) (or 5U(2)). This in turn implies that the hamiltonian matrix can he analytically evaluated using the well known SU(2) angular momentum algehra. 324 s. PITTEL ET AL. Alternatively, the model can be solved using standard shell-model techniques. Here, solutions are limited to O values for which the size of the hamiltonian matrix is tractable. In those cases for which both algebraic and shell-model solutions can be generated, the results are in complete agreement. The three-color 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 this case contains N = 30 partieles. The creation and annihilation operators now inelude a label I that represents the color quantum number and are thus written as qlum and qlum, respectively. The model hamiltonian ineludes one-body, twobody and three-body interactions, all colorless, which scatter partieles coherently among the levels. H = H 1 + H2 + H 3, ~'" t t Hl = 2 L)qI+mqI+m - qI_mql-m), 1m and (23) (24) (25) (26) The model contains three parameters, one for each of the terms in the hamiltonian. Whenever X2 » X3 and ~, the system will be dominated by two-quark correlations. Whenever X3 » X2 and ~, it will be dominated by three-quark correlations. Thus, it has a rich enough structure to make it useful as a test of our proposed mapping. The group structure of this model is significantly more complex than for the usual Lipkin mode!. Now the set of one-body operators generates a Lie algebra U(60), and the structure of the model suggests a elassification of states in tenns of the chain U(60) ::J U(O) <2) U(6) ::J U(O) <2) U(3) <2) U(2). (27) The group U(3) is essential in this elassification, since all physically admissible states should be colorless, i.e., they should belong to the (O, O, O) representation of U(3) (or equivalently the (>',1') = (O, O) scalar representation in ElIiott's SU(3) notation). The complication is that different U(6) representations can contain these states and, moreover, for each of them several U(2) representations are connected by the hamiltonian. Nevertheless, we have succeeded in generating algebraic solutions for this model for both O = 2 ami 3, by appropriate commutator manipulations. As for the usual Lipkin model, shell-model techniques can also be nsed here to obtain exact solutions. Such solutions have now been generated for all n values np to n = 6. ON TIIE USE OF BARYON MAPPINGS.. . 325 4. TEST OF BARYON MAPPING ON TIIE TIIREE-COLOR LIPKIN MODEL Here, we apply the colorless baryon mappings developed in Section 2 to the three-color Lipkin mode!. \Ve carry out the analysis for fl = 2 only, for which the number of baryons is likewise 2. Diagonalization of the effective baryon hamiltonian can be carried out exactly for this case, leading to a direct test of the mapping. The colorless sta tes of the model, following the mapping, can be expressed as The total number of two-baryon states that can be formed is 52. In contrast, the total number of colorless six-quark states in the Lipkin model is 20. That the two-baryon space is larger than the six-quark space was anticipated in our earlier discussion. The two-baryon space ineludes not only physical sta tes (in one-to-one correspondence with the sta tes of the quark model) but unphysical states as well. The general three-color Lipkin hamiltonian (3.5-3.8) can be mapped in either nonhermitean or hermitean form, using the results of the previous section. We will be particularly interested in the nonhermitean mapping, since it is expected to provide a more practical incorporation of quark Pauli effects. However, in the results that follow, we consider both, to see whether our expectations are realized. In order to assess the feasibility of "pushing up" unphysical states with respect to physical states with the nonhermitean mapping, it is important to have a criterio n for distinguishing one from the otheL This can be done by introducing a Majorana operator [8]' analogous to the one used in boson mappings. Consider the square of the quark number operator, (28) This operator can be mapped both in hermitian and nonhermitean formo The Majorana operator is defined as the difference between the two resulting images, (29) Clearly, it has a zero expectation value for all physical states. Equally important, its expectation value for all unphysical sta tes is positive-definite, making it useful as a means of distinguishing physical from unphysical states. In Figures 1-2, we present sorne representative results of oue test calculations for two different choices of the model parameters. In both, we show the algebraic results obtained prior to the mapping (denoted exact) ami the results obtained after the nonhermitean (nh) and hermitean (h) mappings. In the spectra that refer to diagonalization after the mapping, we distinguish physical from unphysical states by using the Majorana operator of (4.2). Physical states are indicated by solid lines and unphysical states by dashed lines. \Ve use a heavy solid line to denote degencratc (or ncarly dcgenerate) solutions, and indicate to the right the number of physical (P) and unphysical (U) states at that energy. \Ve only show the low-energy portions of the spectra corresponding to E<O. 326 S. PITTEL ET AL. Energy -2 1- --2 =2P,2U = 2P,1U- --3U =2U --2U -6 1- -- lP,1U- -10 - -14 - exact nh h - - FIGURE l. Calculaled speclra of lhe lhree-color Lipkin mode! for n = 2, t. = O, X2 = 1 and X3 = O. Only lhe levels wilh E < O are shown. The speclrllm denoled eract refers lo a diagonalizalion of lhe hamillonian in lhe original quark space. Degenerale levels in lhis speclrllm include lo lhe righl the degeneracy. The spectra denoted oh and h rcfer to results obtained following nonhermitean and hermitean triplet-fermion mappings l respectively. Physkal states in the mapped spcctra are denoled by solid lines and unphysical sIal es by dashed lines. Heavy solid lines indicale degenerale (or nearly degenerale) sol111ions;lo lhe righl are given lhe number of physical and unphysical states at that cncrgy.