aa r X i v : . [ h e p - ph ] O c t SU(3) Clebsch-Gordan coefficients at large N c Fl. Stancu a University of Li`ege, Institute of Physics B5,Sart Tilman, B-4000 Li`ege 1, Belgium (Dated: October 21, 2018)
Abstract
It is argued that several papers where SU(3) Clebsch-Gordan coefficients were calculated in orderto describe properties of hadronic systems are, up to a phase convention, particular cases of analyticformulae derived by Hecht in 1965 in the context of nuclear physics. This is valid for irreduciblerepresentations with multiplicity one in the corresponding Clebsch-Gordan series. For multiplicitytwo, Hecht has proposed an alternative which can provide correct 1 /N c sub-leading orders in large N c studies. a E-mail address: [email protected] . INTRODUCTION Since the 1963 classical paper of de Swart [1] where Clebsch-Gordan (CG) coefficientsof SU(3) were derived for the most important direct (or Kronecker) products of irreduciblerepresentations needed in particle physics at that time, namely × , × , × , × and × ¯10 , many authors devoted their papers or parts of them to the derivation of CGcoefficients which were missing in de Swart’s paper. As recalled in the next section thisamounts to derive the corresponding isoscalar factor for each CG coefficient. In 1963 aswell, numerical values for the SU(3) isoscalar factors were published by Edmonds [2]. Moretables were given in 1964 by McNamee and Chilton [3].In recent years the SU(3) flavor group was frequently used to study new hadronic prop-erties and quark systems involving an arbitrary number of quarks as for example in large N c QCD studies. The existing results seemed to be insufficient so that several authors de-rived their own tables. Here we show that some of them are particular cases of the analyticexpressions obtained by Hecht in 1965 in the context of nuclear physics [4].The purpose of this note is twofold: a) To draw attention to Hecht’s work, which maynot be known by particle physicists. Some analytic formulae obtained by Hecht for SU(3)isoscalar factors can straightforwardly be used for particular cases. b) To show that Hecht’sresults for multiplicity two in the direct products are useful in large N c studies, which givea qualitative insight into the structure of baryons. In the large 1 /N c expansion one has firstto analyze formulae at arbitrary N c and afterwards take N c = 3 in applications. II. REMINDER OF SOME SU(3) CG PROPERTIES
In the chain SU (3) ⊃ SU (2) I × U ( I ) Y each SU(3) CG coefficient factorizes into anSU(2)-isospin CG coefficient and an SU(3) isoscalar factor [1] ( λµ ) ( λ a µ a ) ( λ ′ µ ′ ) Y II Y a I a I a Y ′ I ′ I ′ ρ = I I ′ I I a I ′ ( λµ ) ( λ a µ a ) ( λ ′ µ ′ ) Y I Y a I a Y ′ I ′ ρ . (1)where ( λµ ) labels an SU(3) irreducible representation (irrep) and the index ρ distinguishesbetween identical representations occurring in the decomposition of a given direct productwhere the multiplicity of ( λ ′ µ ′ ) = ( λµ ) is larger than one. The highest multiplicity consideredhere is two and in this case a typical example of direct product representations is when2ne takes ( λ a µ a ) = (11), which is the adjoint representation of SU(3), also denoted by itsdimension . The CG series reads( λµ ) × (11) = ( λ + 1 , µ + 1) + ( λ + 2 , µ −
1) + ( λµ ) + ( λµ ) + ( λ − , µ + 2) + ( λ − , µ + 1) + ( λ + 1 , µ −
2) + ( λ − , µ − . (2)The isoscalar factors of SU(3) satisfy an orthogonality relation resulting from the orthogo-nality relations of SU(3) and SU(2) CG coefficients. This is X Y ′′ I ′′ Y a I a ( λ ′′ µ ′′ ) ( λ a µ a ) ( λ ′ µ ′ ) Y ′′ I ′′ Y a I a Y I ρ ( λ ′′ µ ′′ ) ( λ a µ a ) ( λµ ) Y ′′ I ′′ Y a I a Y ′ I ′ ρ = δ λ ′ λ δ µ ′ µ δ Y ′ Y δ I ′ I , (3)and X ( λµ ) ρ ( λ ′′ µ ′′ ) ( λ a µ a ) ( λµ ) Y ′′ I ′′ Y a I a Y I ρ ( λ ′′ µ ′′ ) ( λ a µ a ) ( λµ ) Y ′′ I ′′ Y a I a Y I ρ = δ Y ′′ Y ′′ δ I ′′ I ′′ δ Y a Y a δ I a I a . (4)For completeness, we also recall that the isoscalar factors obey the following symmetryproperties [4] ( λµ ) ( λ a µ a ) ( λ ′ µ ′ ) Y I Y a I a Y ′ I ′ =( − ) ( λ − µ + λ a − µ a − λ ′ + µ ′ + I + I a − I ′ ) ( λ a µ a ) ( λµ ) ( λ ′ µ ′ ) Y a I a Y I Y ′ I ′ . (5)and ( λµ ) ( λ a µ a ) ( λ ′ µ ′ ) Y I Y a I a Y ′ I ′ =( − ) ( µ ′ − µ − λ ′ + λ + Y a )+ I ′ − I vuut dim( λ ′ µ ′ )(2 I + 1)dim( λµ )(2 I ′ + 1) ( λ ′ µ ′ ) ( λ a µ a ) ( λµ ) Y ′ I ′ − Y a I a Y I . (6)where dim( λµ ) = 12 ( λ + 1)( µ + 1)( λ + µ + 2) is the dimension of the irrep ( λµ ) of SU(3). Analternative notation of the isoscalar factors is h ( λµ ) Y I ; ( λ a µ a ) Y a I a || ( λ ′ µ ′ ) Y ′ I ′ i , see Hecht’spaper. 3 II. CALCULATION OF SU(3) CLEBSCH-GORDAN COEFFICIENTS
The usual procedure to calculate CG coefficients is to start from the highest weight basisvector of a representation and use ladder operators, which are U ± , V ± and I ± in SU(3).Their matrix elements were first determined by Biedenharn [5]. Recursion relations amongClebsch-Gordan coefficients are obtained by coupling two states, as in the usual way, likefor the rotation group. These recursion relations contain isoscalar factors.To uniquely define the matrix elements of the ladder operators some phase conventionsmust be made. For the states in the same isomultiplet the standard Condon and Shortleyhas been chosen. Accordingly the non-vanishing matrix elements of I ± are positive. Therelative phases between different isomultiplets were defined by the requirement that thenon-vanishing matrix elements of V ± are real and positive [1] (for the phase convention ofde Swart see [1], Section 10).This procedure has been followed by Kaeding [6] who provided a large number of tablesfor ( λ a µ a ) = (10), (01), (20), (11), (30) and (21) or in dimensional notation , ¯3 , , , and ′ .More recently Hong [7] has derived the isoscalar factors of the direct product of × ,with the purpose of using them to the calculation of baryon magnetic moments and decuplet-to-octet transition magnetic moments. For multiplicity one, all the isoscalar factors areparticular cases of the formulae derived by Hecht [4] in his Table 4, up to a phase convention(see next section).In large N c QCD Cohen and Lebed [8] derived N c dependent SU(3) CG coefficientsrelevant for the coupling of large N c baryons to mesons. They provided extended tables forthe direct products for ( λµ ) = (1 , N c −
12 ) , (3 , N c −
32 ) (7)denoted by ” ” and ” ” respectively and ( λ a µ a ) = (11) denoted by . Their results, atmultiplicity one, up to an overall phase, can directly be reproduced from Hecht’s Table 4.For multiplicity two, for example, ” ” a × → ” ” a they are different at arbitrary N c ,but identical at N c = 3, as compared to those derived here using Hecht’s analytic forms (seenext section).For the same direct products as those of Cohen and Lebed [8] partial tables were previ-ously provided in Ref. [9]. 4he explicit algebraic expressions derived by Hecht [4] for SU(3) isoscalar factors wereintended to nuclear physics applications, in particular to describing rotational states ofdeformed light nuclei from the 2 s − d shell. The deformed nuclei possess collective statesdescribed by Elliott [10, 11] in a model where the SU(3) group is used. Thus the applicationof SU(3) in nuclear physics in 1958 predates the SU(3) classification of elementary particlesof Gell-Mann [12] and Ne’eman [13] in 1961. The basic reason of using SU(3) in nuclearmodels is that intrinsic levels of nuclei can be described by the harmonic oscillator and SU(3)is the symmetry group of the harmonic oscillator in three dimensions (see, for example, Ref.[14] chapter 8). The physical states of a given angular momentum can be obtained by aprojection technique [15].In addition to the isoscalar factors needed for the 2 s − d shell, Hecht had also derivedexplicit expressions for the direct product ( λµ ) × (11), considering such results as beingof interest, not surprisingly, because (11) is the adjoint representation of SU(3). He usedthe standard technique of generating CG coefficients through recursion formulae containingmatrix elements of the SU(3) generators, but introduced a phase convention different fromthat of de Swart. The difference is clearly explained in a footnote of Ref. [4]. In addition,when the irrep ( λµ ) appears twice in the decomposition of the direct product ( λµ ) × (11), seeEq. (2), he introduced the quantum number ρ to label the independent modes of coupling,such as to have non-zero matrix elements of the SU(3) generators for only one state ρ . Then,according to the Wigner-Eckart theorem, the matrix elements of the generators T a of SU(3)are h ( λ ′ µ ′ ) Y ′ I ′ I ′ ; S ′ S ′ | T a | ( λµ ) Y II ; SS i = δ SS ′ δ S S ′ δ λλ ′ δ µµ ′ X ρ =1 , h ( λ ′ µ ′ ) || T (11) || ( λµ ) i ρ ( λµ ) (11) ( λ ′ µ ′ ) Y II Y a I a I a Y ′ I ′ I ′ ρ , (8)where the reduced matrix elements are defined as [4] h ( λµ ) || T (11) || ( λµ ) i ρ = q C (SU(3)) for ρ = 10 for ρ = 2 , (9)in terms of the eigenvalue of the Casimir operator C (SU(3)) = 13 g λµ where g λµ = λ + µ + λµ + 3 λ + 3 µ. (10)5uch a definition is useful for extending the method of calculation of isoscalar factors toother SU(N) groups. It has been applied to the calculation of the matrix elements of SU(6)generators, where one takes into account that SU(3) is a subgroup of SU(6) [17, 18].The correspondence with other notations is ρ = 1 ⇐⇒ ( λµ ) ⇐⇒ ( λµ ) a ,ρ = 2 ⇐⇒ ( λµ ) ⇐⇒ ( λµ ) s . (11)where s and a stand for symmetric and antisymmetric respectively [19, 20]. Historically,following Gell-Mann, in Eq. (11), it is customary to call the symmetric combinations D coupling and the antisymmetric a combinations F coupling (the F and D notation is usedin Ref. [9], for example).Ambiguities in distinguishing the representations at multiplicity larger than one are typ-ical for all groups, including the permutation group [21].Another way to derive Clebsch-Gordan coefficients for SU(3) is based on the tensormethod (for an introduction see, for example, Ref. [14], Sec. 8.10). This method hasbeen used for the Clebsch-Gordan series ” ” × and ” ” × in the systematic analysis oflarge N c baryons [22]. IV. EXAMPLES
Here we wish to demonstrate the usefulness of Hecht’s results, especially for multiplicitytwo, by using Table 4 of Ref. [4]. We use the same table format as that of de Swart becauseit helps in comparing with previous results found in the literature and moreover, it allowseasy checking of the orthogonality relations (3) and (4). We consider two examples relevantfor our purpose.
A. Example 1
The first example, shown in Table I, corresponds to one table obtained by Hong in Ref.[7]. It contains the isoscalar factors for all irreducible representations with Y = 2, I = 2from the decomposition of the direct product × . These are , , s and a in thiscase. 6 ABLE I. Isoscalar factors for the irreducible representations with Y = 2, I = 2 from the decom-position of the direct product × . The first two columns indicate the hypercharge and isospinof and respectively. The phase convention is that of Hecht [4]. Y I ; Y I
81 64 35 s a ,
52 ; 1 , − r r r r ,
32 ; 1 , r r
225 0 − r ,
2; 0 , r r − r r ,
2; 0 , r − r r r Note that one must use the symmetry property (5) to recover the phases for × asin Ref. [7], because here we consider × . For the columns and the absolutevalues are the same as those of Hong. Incidentally column also has the same phases asHong and column has an overall opposite phase. Our results for s and a are entirelydifferent from those of [7] because the definition is different. In applications care must betaken in passing from one convention to another, especially for calculating transition matrixelements. B. Example 2
The second example is exhibited in Table II and corresponds to a table of Cohen andLebed [8], containing isoscalar factors with Y = N c / I = 3/2 from the decompositionof the direct product ” ” × . Cohen and Lebed obtained analytic expressions of theisoscalar factors as a function of N c needed for large N c baryon-meson coupling. Our tablewas obtained as a direct application of Hecht’s Table 4, part of which is reproduced in TableIII of the Appendix, referring to the irrep ” ” with multiplicity 2, denoted here by ” ” a and ” ” s respectively. For completeness, to the three rows listed by Cohen and Lebed wehave added a fourth one, corresponding to Y = N c / − I = 2 and Y = 1, I = 1/2, inorder to check the orthogonality of columns, given by Eq. (3), valid at every N c . Column7 ABLE II. Isoscalar factors for the irreducible representations with Y = N c / I = 3/2 from the decomposition of the direct product” ” × obtained from Table III. Y I ; Y I ”35” ”27” ”10” a ”10” s N c ,
32 ; 0 , s N c + 9) s N c + 1) s N c + 6 N c + 45 − s ( N c − N c + 5)( N c + 6) ( N c + 1)( N c + 9)( N c + 6 N c + 45) N c ,
32 ; 0 , s N c + 9) − s N c + 1) s N c N c + 6 N c + 45 s N c − N c + 5)( N c + 1)( N c + 9)( N c + 6 N c + 45) N c − ,
1; 1 , s N c + 5)16( N c + 9) s N c + 516( N c + 1) − s N c + 5)4( N c + 6 N c + 45) s N c − N c + 1)( N c + 9)( N c + 6 N c + 45) N c − ,
2; 1 , − s ( N c − N c + 9) s N c − N c + 1) s N c − N c + 6 N c + 45) s N c + 5)( N c + 21) N c + 1)( N c + 9)( N c + 6 N c + 45) ” has the same phase for all entries as that of Cohen and Lebed and column ” ” hasopposite phase for all entries. It may happen that the phase conventions of de Swart andHecht coincide sometimes. The column ” ” a ≡ ” ” is entirely different, inasmuch as weuse the definition (9) of Hecht to define the representations with multiplicity 2. We havealso added the column ” ” s ≡ ” ” where the first three entries vanish at N c = 3, asobserved in Ref. [8], but the last entry does not. Such a result may be important for large N c baryon studies [23].In large N c studies the observables are described by operators expressed in terms of SU(6)generators when one considers three flavours, N f = 3. The SU(6) generators are componentsof an irreducible SU(6) tensor operator which span the invariant subspace of the adjointrepresentation denoted here by the partition [21 ], or otherwise by its dimensional notation . Like for any other irreducible representation its matrix elements can be expressed interms of a generalized Wigner-Eckart theorem [16, 17] which factorizes each matrix elementinto products of Clebsch-Gordan coefficients and a reduced matrix element, like in Eq. 8.The notation is as follows. The generic name for every generator is E ia . An irrep of SU(6)is denoted by the partition [ f ]. Then one can write the matrix element of every SU(6)generator E ia as h [ f ]( λ ′ µ ′ ) Y ′ I ′ I ′ S ′ S ′ | E ia | [ f ]( λµ ) Y II SS i = q C [ f ] (SU(6)) S S i S ′ S S i S ′ I I a I ′ I I a I ′ × X ρ =1 , ( λµ ) ( λ a µ a ) ( λ ′ µ ′ ) Y I Y a I a Y ′ I ′ ρ [ f ] [21 ] [ f ]( λµ ) S ( λ a µ a ) S i ( λ ′ µ ′ ) S ′ ρ , (12)where C [ f ] (SU(6)) is the SU(6) Casimir operator eigenvalue associated to the irreduciblerepresentation [ f ], here being the reduced matrix element, followed by the familiar Clebsch-Gordan coefficients of SU(2)-spin and SU(2)-isospin. The sum over ρ contains products ofisoscalar factors of SU(3) and SU(6) respectively. The label ρ is necessary whenever onehas to distinguish between irreps [ f ′ ] = [ f ] with multiplicities m [ f ] larger than one in theClebsch-Gordan series [18] [ f ] × [21 ] = X [ f ′ ] m [ f ′ ] [ f ′ ] . (13)The two values for ρ both in SU(6) and SU(3) reflects the multiplicity problem alreadyappearing in the direct product of SU(3) irreducible representations, as discussed in Sec.9I. It is clear that one must make the sum over ρ in all cases. The large N c behaviour isobtained from the analytic expressions of the isoscalar factors of SU(3) and SU(6). Thisbehaviour is necessary for finding the most dominant contributions in the 1 /N c expansion.Examples of physical interest in baryon spectroscopy for the analytic expressions of SU(6)isoscalar factors can be found in Ref. [23] for [ f ] = [ N c ] and [ f ] = [ N c − , N c behaviour resulting from SU(3) isoscalar factors.For a comparison with Cohen and Lebed [8] let us consider the column ” ” a of TableII alone because the column ” ” s is missing in Ref. [8]. For the first three rows ourisoscalar factors are of order O ( N − c ), O ( N c ) and O ( N − / c ) respectively while from Ref.[8] Table II at Y = N c / I = 1/2, column ” ” a we obtain O ( N c ), O ( N c ) and O ( N − / c )respectively. Thus the large N c behaviour is different from ours for I = 3 / , Y = N c / I = 1 , Y = N c / I = 3 / , Y = N c / I = 0 , Y = 0. For a proper analysis atlarge N c the missing column ρ = 2 equivalent to ” ” s , is necessary as required by Eq.(12), even if some isoscalar factors vanish at N c = 3. By summing up the contributionsfrom ” ” a and ” ” s one would expect a similar answer in any convention, provided theSU(6) isoscalar factors are calculated consistently with those of SU(3). Moreover the case I = 2 , Y = N c / − I = 1 / , Y = 1 is missing in Table II of Ref. [8], at Y = N c / I = 1/2. Therefore, the results of Ref. [8] should be completed with extra rows and columns,whenever necessary, if one wishes to recover a proper large N c behaviour. In the physicalworld of N c = 3 they are sufficient for the exhibited I , Y , I , Y cases.It would be interesting to consider further applications of Hecht’s SU(3) isoscalar factorseither in large N c QCD or in nuclear physics.
Appendix A
In Table III we reproduce part of Table 4 of Hecht’s paper [4] which contains the analyticexpressions of the isoscalar factors h ( λµ ) Y I ; (11) Y I || ( λµ ) Y I i , often used in quark physics.Note that the entry in the column ρ = 2 for Y = 1 , I = 12 , I = I + 1 / λ + µ + 2 − q + 1) has been replaced by ( λ + µ + 2 − q ) and in the denominatorthe bracket ( µ + p − q ) has been replaced by ( µ + p − q + 1). In Table III we have used g λµ ABLE III. Isoscalar factors < ( λµ ) Y I ; (11) Y I || ( λµ ) Y I > of Hecht’s Table 4, p.31 [4] with corrections for the row Y = 1, I = 1 / I = I + 1 / Y I I ( λ ′ µ ′ ) = ( λµ ) ( λ ′ µ ′ ) = ( λµ ) ρ = 1 ρ = 2 − I + 1 / " p + 1)( λ − p )( µ + 2 + p )2 g λµ ( µ + p − q + 1) / [2 g λµ q − µ ( λ + µ + 1)( λ + 2 µ + 6)][( p + 1)( λ − p )( µ + 2 + p )] / [ λ ( λ + 2) µ ( µ + 2)( λ + µ + 1)( λ + µ + 3)2 g λµ ( µ + p − q + 1)] / − I − / " q + 1)( µ − q )( λ + µ + 1 − q )2 g λµ ( µ + p − q + 1) / [2 g λµ p + λ ( µ + 2)( λ − µ + 3)][( q + 1)( µ − q )( λ + µ + 1 − q )] / [ λ ( λ + 2) µ ( µ + 2)( λ + µ + 1)( λ + µ + 3)2 g λµ ( µ + p − q + 1)] / I − λ + µ − p − q [4 g λµ ] / √ λµ ( µ + 2)( λ + µ + 1) − µ ( λ + µ + 1)( λ + 2 µ + 6) p + λ ( µ + 2)( λ − µ + 3) q + 2 g λµ pq [ λ ( λ + 2) µ ( µ + 2)( λ + µ + 1)( λ + µ + 3) g λµ ] / I + 1 0 [2( p + 1)( λ − p )( µ + 2 + p ) q ( µ + 1 − q )( λ + µ + 2 − q ) g λµ ] / [ λ ( λ + 2) µ ( µ + 2)( λ + µ + 1)( λ + µ + 3)( µ + p − q + 1)( µ + p − q + 2)] / I − − [2 p ( λ + 1 − p )( µ + 1 + p )( q + 1)( µ − q )( λ + µ + 1 − q ) g λµ ] / [ λ ( λ + 2) µ ( µ + 2)( λ + µ + 1)( λ + µ + 3)( µ + p − q + 1)( µ + p − q )] / I [3( µ + p − q )( µ + p − q + 2)] / [4 g λµ ] / E λ ( λ + 2) µ ( µ + 2)( λ + µ + 1)( λ + µ + 3) g λµ ( µ + p − q )( µ + p − q + 2)] / I + 1 / " q ( µ + 1 − q )( λ + µ + 2 − q )2 g λµ ( µ + p − q + 1) / [2 g λµ p + λ ( µ + 2)( λ − µ + 3)][ q ( µ + 1 − q )( λ + µ + 2 − q )] / [ λ ( λ + 2) µ ( µ + 2)( λ + µ + 1)( λ + µ + 3)2 g λµ ( µ + p − q + 1)] / I − / − " p ( λ + 1 − p )( µ + 1 + p )2 g λµ ( µ + p − q + 1) / − [2 g λµ q − µ ( λ + µ + 1)( λ + 2 µ + 6)][ p ( λ + 1 − p )( µ + 1 + p )] / [ λ ( λ + 2) µ ( µ + 2)( λ + µ + 1)( λ + µ + 3)2 g λµ ( µ + p − q + 1)] / ABLE IV. Values of λ ′ , µ ′ , p and q needed for Y = N c / I = 3/2 to calculate the isoscalar factorsof ” ” × using Table III. The label ( λ ′ µ ′ ) identifies the irreps of the Clebsch-Gordan series (2)for a given ( λµ ) in the left hand side. The isoscalar factors are presented in Table II. λ ′ µ ′ p q ( λ ′ µ ′ ) ”35” N c −
12 3 N c −
12 ( λ + 1 , µ + 1) ”27” N c + 12 2 N c −
12 ( λ − , µ + 2) ”10” N c −
32 3 N c −
32 ( λµ ) defined by Eq. (10) and E defined by E = λ ( λ + µ + 1) µ ( µ + 2)(2 λ + µ + 6) + 2( λ + µ + 1) µ × [ λ ( λ + 2) − ( µ + 2)( µ + 3)] p − µ ( λ + µ + 1)( λ + 2 µ + 6) p − λ [( µ + 1)( λ + µ + 1)(2 λ + µ + 6) − µg λµ ] q + λ ( µ + 2)( λ − µ + 3) q − λ ( λ + µ + 1)(2 λ + µ + 6) − g λµ ] pq + 2 g λµ ( p q + pq ) . (A1)Table III and the rest Table 4 of Hecht can straightforwardly be applied to a given ( λ ′ µ ′ )with definite values of Y and I , from which one can obtain the integers p and q defined as Y = p + q − λ ′ + µ ′ , I = µ ′ + p − q Y is related to the a quantity called ǫ by ǫ = − Y. (A3)For λ = 3 and µ = N c −
32 the values of λ ′ , µ ′ together with p and q defined by Eqs. (A2)are listed in Table IV.We believe there is no reason to reproduce the full Table 4 of Hecht which contains fourdistinct tables. Acknowledgments
This research was supported by the Fond de la Recherche Scientifique - FNRS, Belgium,12nder the grant 4.4501.05. [1] J. J. de Swart, Rev. Mod. Phys. (1963) 916 [Erratum-ibid. (1965) 326].[2] A. R. Edmonds, Proc. Roy. Soc. A268 (1963) 436.[3] P. S. J. McNamee and F. Chilton, Rev. Mod. Phys. (1964) 1005.[4] K. T. Hecht, Nucl. Phys. (1965) 1 .[5] L. C. Biedenharn, Phys. Lett. (1962) 69; Phys. Lett. (1962) 254. See also G. E. Baird andL. C. Biedenharn, J. Math. Phys. (1963) 1449.[6] T. A. Kaeding, Atom. Data Nucl. Data Tabl. (1995) 233 [nucl-th/9502037].[7] S. T. Hong, J. Korean Phys. Soc. (2015) 2, 158.[8] T. D. Cohen and R. F. Lebed, Phys. Rev. D (2004) 096015.[9] D. Diakonov, V. Petrov and A. A. Vladimirov, Phys. Rev. D (2013) 7, 074030.[10] J. P. Elliott, Proc. Roy. Soc. Lond. A (1958) 128.[11] J. P. Elliott, Proc. Roy. Soc. Lond. A (1958) 562.[12] M. Gell-Mann, Phys. Rev. (1962) 1067.[13] Y. Ne’eman, Nucl. Phys. (1961) 222.[14] F. Stancu, “Group theory in subnuclear physics,” Oxford Stud. Nucl. Phys. (1996) 1,(Oxford University Press, Oxford).[15] K. T. Hecht, ’Collective models’ in ’Selected topics in nuclear spectroscopy’ (ed. B. J. Verhaar),North-Holland, Amsterdam, 1964.[16] N. Matagne and F. Stancu, Phys. Rev. D (2006) 114025.[17] N. Matagne and Fl. Stancu, Nucl. Phys. A (2009) 161.[18] N. Matagne and Fl. Stancu, Phys. Rev. D (2011) 056007.[19] D. B. Lichtenberg, Unitary symmetries and elementary particles , Academic Press, New Yorkand London (1970), Chapter 8.[20] For details on the derivation of SU(3) Clebsch-Gordan coefficients see J. F. Cornwell,
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