SU(6) ⊃ SU(3)xSU(2) and SU(8) ⊃ SU(4)xSU(2) Clebsch-Gordan coefficients
aa r X i v : . [ m a t h - ph ] O c t SU(6) ⊃ SU(3) ⊗ SU(2) and
SU(8) ⊃ SU(4) ⊗ SU(2)
Clebsch-Gordan coefficients
C. Garcia-Recio and L.L. Salcedo
Departamento de F´ısica At´omica, Molecular y Nuclear,Universidad de Granada, E-18071 Granada, Spain
Tables of scalar factors are presented for ⊗ and ⊗ in SU(8) ⊃ SU(4) ⊗ SU(2), andfor ⊗ and ⊗ in SU(6) ⊃ SU(3) ⊗ SU(2). Related tables for SU(4) ⊃ SU(3) ⊗ U(1) andSU(3) ⊃ SU(2) ⊗ U(1) are also provided so that the Clebsch-Gordan coefficients can be completelyreconstructed. These are suitable to study meson-meson and baryon-meson within a spin-flavorsymmetric scheme.
PACS numbers: 11.30.Ly 02.20.Qs 11.30.Hv 21.60.Fw
Contents
I. Introduction II. Representation reductions andClebsch-Gordan series III. Explanation of the tables IV. Method of construction and phaseconventions
Acknowledgments A. Tables of scalar factors of SU(8)
71. SU(8): ⊗
72. SU(8): ⊗ B. Tables of scalar factors of SU(6) ⊗ ⊗ C. Tables of scalar factors of SU(4) ⊗ ⊗ ′ ⊗ D. Tables of scalar factors of SU(3) ⊗ ⊗ ∗ ∗ ⊗ ∗ ⊗ ⊗ ∗ ⊗ ⊗ ⊗ ∗ ⊗ ⊗ ⊗ ∗ ⊗ References I. INTRODUCTION
After the original SU(3) flavor symmetry, SU(6) spin-flavor symmetry was soon introduced to give rise to thequark model [1–3], reaching considerable phenomenolog-ical success. (See reviews in [4–6].) It was shown earlythat chiral symmetry, a powerful tool to extract infor-mation from QCD [7–9], and spin-flavor symmetry werenot in conflict [10]. Recently, spin-flavor has reappearedas a natural classification symmetry of baryons withinthe large N c approach to QCD [11], see e.g. [12, 13].On the other hand, in the framework of unitarized chiralmodels [14–17], spin-flavor symmetry has found an im-portant role not only for three flavors but also for four[18–24]. The reason is that spin-flavor symmetry, in itschiral version, can naturally accommodate the chiral andheavy quark symmetries of QCD. Although the under-lying SU(4) flavor symmetry is severely broken in thekinematics through the quark masses, based on experi-ence on three flavors, it is expected that the breakingshould be mild at the level of interaction amplitudes.In view of this renewed interest in spin-flavor symme-try, we have undertaken a study of the reduction of thespin-flavor group SU(8) under SU(4) ⊗ SU(2), as requiredin the description of meson-meson and baryon-meson in-teractions. Concretely we compute the scalar factors(also called singlet factors) of the reduction SU(8) ⊃ SU(4) ⊗ SU(2) for the products ⊗ (meson-meson)and ⊗ (baryon-meson). These have not been yetcomputed in the literature. It is true that, following theobservation in [25, 26], all scalar factors and Clebsch-Gordan coefficients of special unitary groups follow fromthose of the symmetric group, S n , however, the irre-ducible representations of SU(8) considered here wouldrequire large values of n for which no results are readilyavailable.The prescription of [27] is widely used to define stan-dard bases of irreducible representations of SU(3) [28, 29],however, for four or more flavors the natural prescriptionis that of [30] (see, e.g. [31]). Previous calculations ofscalar factors of SU(6) ⊃ SU(3) ⊗ SU(2) have been doneusing [27], for instance [32] (see also, [33–36]), so we havealso computed the corresponding SU(6) ⊃ SU(3) ⊗ SU(2)scalar factors for ⊗ and ⊗ with the prescrip-tion [30]. We reproduce the results in [32], up to signsdue to the different choice of standard bases, and alsodiffer in the breaking of degeneracies.We present also the related SU(4) ⊃ SU(3) ⊗ U(1)and SU(3) ⊃ SU(2) ⊗ U(1) scalar factors, always withinthe prescription [30]. We check the results in [31][37]and extend them, since we need the product ′ ⊗ in SU(4) which was not computed there, along with thecorresponding new SU(3) ⊃ SU(2) ⊗ U(1) scalar factorsrequired.In section II notation is introduced, in section III weexplain how to use the tables and in section IV themethod of construction of the tables and the way thephases have been fixed is described. The scalar factorsare displayed in the appendices (see table of contents).
II. REPRESENTATION REDUCTIONS ANDCLEBSCH-GORDAN SERIES
For the groups SU(8) and SU(6) we consider the fol-lowing reductionsSU sf (8) ⊃ SU f (4) ⊗ SU J (2) , (1)SU sf (6) ⊃ SU f (3) ⊗ SU J (2) , (2)with SU f (4) ⊃ SU f (3) ⊗ U C (1) , (3)SU f (3) ⊃ SU I (2) ⊗ U Y (1) , (4)SU I (2) ⊃ U I z (1) , (5)SU J (2) ⊃ U J z (1) . (6) The labels sf, f, J , C , I , Y , I z and J z refer to spin-flavor,flavor, spin, charm, isospin, hypercharge, third compo-nent of isospin and third component of spin, respectively.In what follows we shall often drop these labels in thegroups.To be concrete, we often refer to SU(8) although thefollowing remarks apply to the other groups as well withobvious modifications. Let the vector space H R carryan irreducible representation (irrep) R of SU(8), e.g. . This space admits an orthonormal basis, | R ; µ, γ, ζ i ,adapted to the reduction chains above. The label µ denotes the SU(4) ⊗ SU(2) irrep, e.g. , where refers to SU(4) and the subindex refers to 2 J + 1, with J = 0. γ is the degeneracy label needed to distinguish thevarious irreps µ appearing in the reduction of R underSU(4) ⊗ SU(2). Finally, ζ denotes the remaining quan-tum numbers ( ν, I, J, I z , J z , Y, C ), where ν denotes theSU(3) irrep, e.g., . The range of values of the label γ depends on R and µ . That of ζ depends on µ .Having two SU(8) irreducible spaces, H R and H R ,their tensor product can be decomposed under SU(8) as H R ⊗ H R = M R,σ H R,σ . (7)The label σ denotes the degeneracy label of the irrep R in the Clebsch-Gordan (CG) series of R ⊗ R and itsrange of values depends on R , R and R . The CG coeffi-cients then relate the “uncoupled” and “SU(8)-coupled”bases as: | R , R ; R, σ ; µ, γ, ζ i = X µ ,γ ,ζ ,µ ,γ ,ζ (cid:18) R R Rσµ γ ζ µ γ ζ µγζ (cid:19) | R ; µ , γ , ζ ; R ; µ , γ , ζ i , (8) | R ; µ , γ , ζ ; R ; µ , γ , ζ i = X R,σ,µγ,ζ (cid:18) R R Rσµ γ ζ µ γ ζ µγζ (cid:19) | R , R ; R, σ ; µ, γ, ζ i . (9)The symbols between large parenthesis denote the CGcoefficients.The CG coefficients pick up a plus or minus sign, ξ ,under exchange of the states 1 and 2, (cid:18) R R Rσµ γ ζ µ γ ζ µγζ (cid:19) = ξ (cid:18) R R Rσµ γ ζ µ γ ζ µγζ (cid:19) . (10) The phase ξ depends only on R , R , R , σ , µ and γ .On the other hand, the uncoupled basis can be coupledunder SU(4) ⊗ SU(2) | R , µ , γ ; R , µ , γ ; µ, γ ′ , ζ i = X ζ ,ζ (cid:18) µ µ µγ ′ ζ ζ ζ (cid:19) | R ; µ , γ , ζ ; R ; µ , γ , ζ i , (11) | R ; µ , γ , ζ ; R ; µ , γ , ζ i = X µ,γ ′ ,ζ (cid:18) µ µ µγ ′ ζ ζ ζ (cid:19) | R , µ , γ ; R , µ , γ ; µ, γ ′ , ζ i , (12)where γ ′ is the degeneracy label of the irrep µ in the CGseries of µ ⊗ µ . Hence its range of values depends on µ , µ and µ , but not on γ and γ . The CG coefficientsin Eqs. (11,12) are the product of the CG coefficients ofthe reductions chains SU(4) ⊃ SU(3) ⊗ U(1) and SU(2) ⊃ U(1).The relation between the SU(8)-coupled and SU(4) ⊗ SU(2)-coupled basis provides the SU(8) ⊃ SU(4) ⊗ SU(2)scalar factors (SF): | R , R ; R, σ ; µ, γ, ζ i = X µ ,γ ,µ ,γ ,γ ′ (cid:18) R R Rσµ γ µ γ µγγ ′ (cid:19) | R , µ , γ ; R , µ , γ ; µ, γ ′ , ζ i , (13) | R , µ , γ ; R , µ , γ ; µ, γ ′ , ζ i = X R,σ,γ (cid:18) R R Rσµ γ µ γ µγγ ′ (cid:19) | R , R ; R, σ ; µ, γ, ζ i . (14)The symbols between large parenthesis and vertical bardenote the SF.The degeneracy labels are redundant when there is nodegeneracy. Following the standard practice, throughoutthis work, in those cases the label is omitted. This referseither to degeneracy in the irrep reduction ( γ ) or in theCG series ( σ , γ ′ ). As degeneracy labels we use s , a and b (or s , a , b ), in this order. Three is the highest degeneracyencountered. In the general case, the labels s and a donot imply symmetry of the representation. When thereis symmetry (e.g. in ⊗ ) we take the symmetric rep-resentation to be the first one ( s ) and the antisymmetricrepresentation to be second one ( a ). In those cases thephase ξ = ± s and a respectively.There are completely analogous relations for CG andSF of SU(6) ⊃ SU(3) ⊗ SU(2), SU(4) ⊃ SU(3) ⊗ U(1),and SU(3) ⊃ SU(2) ⊗ U(1), and similar comments apply.Because the reduction chains in Eqs. (3-6) are canonicalno degeneracy label γ is needed for SU(4) and SU(3). III. EXPLANATION OF THE TABLES
For SU(8) we consider the following CG series ⊗ = ⊕ s ⊕ ⊕ ⊕ a ⊕ ⊕ ∗ , ⊗ = ⊕ ⊕ ⊕ , (15)and for SU(6) ⊗ = ⊕ s ⊕ ⊕ ⊕ a ⊕ ⊕ ∗ , ⊗ = ⊕ ⊕ ⊕ . (16) The irreps are labeled by their dimension. The labelsused and the corresponding Young tableaux are listed intable I.The tables of SF are presented in the appendices. Therelated SU(4) and SU(3) reductions are also included.The reduction of the irreps in the CG series is obviousfrom the SF tables. Nevertheless, for convenience wehave collected in table II these reductions for SU(8) ⊃ SU(4) ⊗ SU(2) and for SU(6) ⊃ SU(3) ⊗ SU(2).Once again, to be concrete, we discuss the SU(8) case.The tables are expressed in the form of equations, interms of state vectors as in Eq. (13), i.e. a l.h.s withthe SU(8)-coupled stated and a r.h.s. with the SU(4) ⊗ SU(2)-coupled state. From such equations the SF canbe read off. The plus or minus sign between parenthesisdisplayed at the left of the l.h.s. of the equation is thephase ξ defined in Eq. (10). It is put there for the sake ofpresentation only (it is not meant to multiply the vector).Regarding the notation, as compared to Eq. (13), inthe tables, redundant labels have been omitted. Specifi-cally, in the l.h.s. we omit R , R since they are explicitedin the heading of the corresponding subsection, and ζ isalso omitted. The degeneracy labels σ and γ , if required,take the form of a subindex. E.g., | a ; i ( σ = a ),or | ; s , i ( γ = s ).In the r.h.s., γ and γ are not needed for the SU(8)representations R and R considered, namely, or . The labels µ and ζ are also omitted ( µ is explicitedin the l.h.s.). The irreps ( R , µ ) and ( R , µ ) are repre-sented by symbols of the lowest lying particles with thosequantum numbers. In each case, the particle with high- TABLE I: List of Young tableaux and irrep labels.Group irrep label Young tableauSU(8) [ ] [1] ∗ [1 ] [2 , ] [3] [2 , [2 , ] [3 , ] ∗ [3 , ] [4 , ] [5 , ] [4 , , ]SU(6) [1] ∗ [1 ] [2 , ] [3] [2 , [2 , ] [3 , ] ∗ [3 , ] [4 , ] [5 , ] [4 , , ]SU(4) [1] ∗ [1 ] [2 , ] [2 , ′ [3] ′′ [2 ] ∗ [3 , ] [3 , ∗ [3 , ∗ [3 , [4 , ] [5 , ] [4 , , [1] ∗ [1 ] [2] ∗ [2 ] [2 , [3] ∗ [3 ] [3 , ∗ [3 , ′ [4] ∗ [4 , [4 , [5 ,
1] TABLE II: Reduction of SU(8) and SU(6) irreps.SU(8) SU(4) ⊗ SU(2)
63 1
120 20 ′
168 4 ∗ ′
720 1 s , a , ′′ ′′ ′′ ∗
945 1 s , a , ′′ ∗ ∗ s , a , ′′ ∗ ∗ ∗ s , a , ′′ ∗ ′ ′ ′ ∗ ∗ ∗ s , a , b , s , a , b , ′ s , ′ a , ′ s , ′ a , ′ ∗ s , ∗ a , ∗ ∗ ∗ s , a , s , a , SU(6) SU(3) ⊗ SU(2)
35 1
56 8
70 1
189 1 s , a , ∗
280 1 s , a , ∗ ∗ s , a , ∗ ∗ ∗
405 1 s , a , ∗
700 8 ∗ s , a , b , s , a , b , s , a , s , a , ∗ ∗ s , a , s , a , est weight is used . E.g., ( , ) is labeled by Σ and( , ) is labeled by ρ . The list of particle symbols isdisplayed in Table III and there the rows are ordered sothat they run from highest (top) to lowest weight (bot-tom).Further, if the degeneracy label γ ′ is required, it is putas a subindex. E.g. | (Σ , ρ ) s i for γ ′ = s . As a final nota-tional convention, in the r.h.s, when R = R we use alabel S or A to indicate symmetrized or antisymmetrizedstates under exchange of particle labels. E.g., | ( ρ, π ) a i S = 1 √ | ( ρ, π ) a i + 1 √ | ( π, ρ ) a i , | ρ, ω i A = 1 √ | ρ, ω i − √ | ω , ρ i . (17)Hence, Eq. (13) are written in the appendices as( ξ ) | R σ ; µ γ i = X µ ,µ ,γ ′ R R Rσµ µ µγγ ′ ! | ( µ , µ ) γ ′ i . (18)The equations corresponding to Eq. (14) are not givenas they are easily reconstructed from the equations pro-vided (of the type Eq. (13)). E.g., from the tablesfor R ⊗ R = ⊗ with µ = , one finds, for TABLE III: List of particle symbols.label SU(8) SU(6) SU(4) SU(3)
I Y C J ∆
120 56 20 ′ Σ ∗
120 56 20 ′ Ξ ∗
120 56 20 ′ − Ω
120 56 20 ′ − Σ ∗ c
120 20 ′ Ξ ∗ c
120 20 ′ − Ω ∗ c
120 20 ′ − Ξ ∗ cc
120 20 ′
12 13 Ω ∗ cc
120 20 ′ − Ω ccc
120 20 ′ Σ
120 56 20 8 N
120 56 20 8 Ξ
120 56 20 8 − Λ
120 56 20 8 Σ c
120 20 6 Ξ ′ c
120 20 6 − Ω c
120 20 6 − Ξ c
120 20 3 ∗ − Λ c
120 20 3 ∗ Ξ cc
120 20 3
12 13 Ω cc
120 20 3 − ρ
63 35 15 8 K ∗
63 35 15 8 K ∗
63 35 15 8 − ω
63 35 15 8 D ∗
63 15 3 ∗ − D ∗ s
63 15 3 ∗ D ∗
63 15 3
12 13 − D ∗ s
63 15 3 − − ψ
63 15 1 ω
63 35 1 1 π
63 35 15 8 K
63 35 15 8 K
63 35 15 8 − η
63 35 15 8 D
63 15 3 ∗ − D s
63 15 3 ∗ D
63 15 3
12 13 − D s
63 15 3 − − η c
63 15 1 η ′ µ = µ = , | ( ρ, ρ ) s i = r | ; i + r | ; i . (19)As explained, here | ( ρ, ρ ) s i = | ⊗ ; s , i .For SU(6) ⊃ SU(3) ⊗ SU(2) everything is similar. Theparticle symbols in the r.h.s. correspond now to multi- plets ( R , µ ) and ( R , µ ) of (SU(6) , SU(3) ⊗ SU(2)), sofor instance, Σ stands now for ( , ) and ρ stands for( , ).For SU(4) ⊃ SU(3) ⊗ U(1) the SU(3) ⊗ U(1) irrep µ in the l.h.s. is represented as ( ν, C ), where ν is theSU(3) irrep and C is the charm quantum number. NowΣ stands for the (SU(4); SU(3) ⊗ U(1)) irrep ( ; , π stands for ( ; , ρ is not used here. Itfalls in the same representation of SU(4) as π and wechoose to use the pseudoscalars to label the states.)Finally, for SU(3) ⊃ SU(2) ⊗ U(1) the SU(3) ⊗ U(1)irrep µ in the l.h.s. is represented as ( I, Y ), where I is theisospin and Y the hypercharge, and each particle symbollabels a complete isospin-hypercharge multiplet. IV. METHOD OF CONSTRUCTION ANDPHASE CONVENTIONS
The fundamental and antifundamental representationsof SU( n ) can be realized by the ladder operators: E ij | k i = δ ik | j i − n δ ij | k i ,E ij | ¯ k i = − δ kj | ¯ i i + 1 n δ ij | ¯ k i , i, j, k, = 1 , . . . , n, (20)which fulfill the su( n ) algebra relations[ E ij , E kl ] = δ il E kj − δ kj E il , ( E ij ) † = E ji . (21)These operators act the on a tensor product representa-tion in the usual way E ij ( | A i ⊗ | B i ) = ( E ij | A i ) ⊗ | B i + | A i ⊗ ( E ij | B i ) . (22)For SU(8), using the tensor products ∗ ⊗ and ⊗ ⊗ we construct the adjoint, , and symmetric, , rep-resentations, corresponding to mesons and baryons, re-spectively. The CG series of ⊗ and ⊗ areresolved by the standard method of extracting the statewith highest weight and applying ladder operators on itto fill a full highest weight representation. The method isthen repeated recursively for the orthogonal spaces. Thesame technique is applied for the other groups, SU(6),SU(4) and SU(3). As usual higher dimension always im-plies higher weight. The representations are ordered asin table III.In the CG series considered the degeneracy label σ isoften redundant. When needed the symmetric combina-tion, label s , is taken to be of highest weight than theantisymmetric one, label a . A. Flavor groups
To fix the phases within an irrep of a flavor group,SU(4) or SU(3), we adopt the prescription of [30], namely,we demand that the matrix elements between basis statesof the ladder operators of form E ii +1 should be nonneg-ative. This produces a standard basis, by definition.[38]Identifying, as usual, the states | i i , i = 1 , , , u , d , s , c (in this order) this prescription differs form theSU(3) choice in [27].To fix the relative phases between the irreps in theCG series of SU(4), we fix the sign of the state withhighest weight, i.e., corresponding to the highest SU(3)irrep. We do this is the usual way, namely, we order theuncoupled states attending first to µ , if necessary to µ ,and finally to γ ′ . The sign of the highest coupled stateis then chosen so that it has a positive overlap with thehighest uncoupled state. E.g., in ⊗ to give s ,the coupled state with highest weight is | s ; , i . Thehighest uncoupled state is | ( π, π ) s i . Everything is similarfor SU(3). The prescription adopted is just the naturalextension of that in SU(2). B. Spin-flavor groups
For the group SU(8) we are interested in its reductionunder SU(4) ⊗ SU(2). Therefore the prescription in [30],devised for SU( n ) ⊃ SU( n − ⊗ U(1), does not directlyapply. We do apply [30] for the relative phases betweenstates inside each SU(4) ⊗ SU(2) irrep, but the phasebetween two such irreps in the reduction of an irrep ofSU(8) is to be fixed. Also, because the reduction SU(8) ⊃ SU(4) ⊗ SU(2) is not a canonical one, an SU(4) ⊗ SU(2)irrep can appear several times (label γ in Eq. (8)) andthis has also to be fixed. In addition, the phase of eachSU(8) irrep in the CG series is to be settled.There is no widely accepted way of defining standardbases of the irreps of SU(8) ⊃ SU(4) ⊗ SU(2). Ratherthan introducing such general prescription we adopt someconcrete choices in what follows. Everything we say herefor SU(8) can be immediately translated to SU(6).Let us consider the irrep obtained from ∗ ⊗ andthe irrep obtained from ⊗ ⊗ . Their reductionunder SU(4) ⊗ SU(2) can be looked up in table II. Forthem we choose h π ; 1 , | E u, + u, − | ρ ; 1 , i > , h ω ; 0 , | E u, + d, + | ρ ; 1 , i > , h Σ; 1 , | E u, + u, − | Σ ∗ ; 1 , i > . (23)Here we have used the notation | p, I z , J z i where p de-notes a concrete state R, µ, ν, I, J, Y, C (see table III).Also ( u, +) , ( d, +) , . . . , ( c, − ) are the 8 labels i in the lad-der operators E ij of SU(8). The two first relations fixthe phases between , and in , whereas thesecond relation fixes the phase between and ′ in .For all the SU(4) ⊗ SU(2) irreps produced through theproducts ⊗ and ⊗ , we adopt a commonprocedure which is customary and extends that used toresolve the CG series in SU( n ) ⊃ SU( n − ⊗ U(1). This is as follows. The uncoupled states (r.h.s. in Eq. (8) orin Eq. (13)) are given a well defined order. Then the firstcoupled state is taken to have the maximum overlap withthe first uncoupled state. (If this overlap is zero the nextuncoupled state is taken instead, etc.) Next, the secondcoupled state is chosen as the state orthogonal to thefirst one having the maximum overlap with the seconduncoupled state. And so on, recursively.So, for instance, in | ⊗ ; s ; i , | ( ρ, ρ ) a i is thehighest uncoupled state ( ρ being of higher weight than π and ω ) and so its coefficient is positive. In this examplethe label γ was not needed. A more complicated caseis | ⊗ ; ; γ, i with γ = s, a, b . The relevantuncoupled states are, listed from highest to lowest, | ∆ , ρ i , | ∆ , π i , | (Σ , ρ ) s i , | (Σ , ρ ) a i , and | Σ , ω i . Hence, | s , i haspositive overlap with | ∆ , ρ i , | a , i has no overlap with | ∆ , ρ i and positive overlap with | ∆ , π i , and | b , i hasno overlap with | ∆ , ρ i or | ∆ , π i and positive overlap with | (Σ , ρ ) s i .We point out that this prescription, although simpleenough, does not automatically produce standard bases.The bases obtained for a given SU(8) irrep will dependon how that irrep is obtained. For instance, for the obtained from ∗ ⊗ we have fixed the relative phasesbetween , and as in Eq. (23). However, theproduct ⊗ produces two irreps (namely, s and a ). For them the relative phases between , and are fixed instead from the “coupling method” justdescribed. Unfortunately, it turns out to violate the twoinequalities in (23). And the same violation takes placesfor the new irrep generated from ⊗ for thethird inequality. Of course, we could just redefine thenecessary signs within these s , a and , but weprefer to be systematic rather than to enforce standardbases for particular irreps of SU(8).Regarding the order of the uncoupled states, there isa subtlety. In general, the order depends first on µ ,then on µ and then on γ ′ . However, in ⊗ ,we attend first to µ (the baryon state) and then to µ (the meson state). This is necessary to have a welldefined ξ in Eq. (10) in all cases. For instance, in | ⊗ ; ; γ, i we want | π, ∆ i to be of higherweight than | ( ρ, Σ) s i , so that under exchange of labels1 and 2 (Eq. (10)), | ⊗ ; ; a , i maps to | ⊗ ; ; a , i and | ⊗ ; ; b , i mapsto | ⊗ ; ; b , i . Acknowledgments
We thank J. Nieves for discussions. Research sup-ported by DGI under contract FIS2008-01143, Juntade Andaluc´ıa grant FQM-225, the Spanish Consolider-Ingenio 2010 Programme CPAN contract CSD2007-00042, and the European Community-Research Infras-tructure Integrating Activity
Study of Strongly Inter-acting Matter (HadronPhysics2, Grant Agreement no.227431) under the 7th Framework Programme of EU.
Appendix A: Tables of scalar factors of SU(8) | ρ i = | ; i , | π i = | ; i , | ω i = | ; i , | ∆ i = | ; ′ i , | Σ i = | ; i .
1. SU(8): 63 ⊗ (+) | ; i = r | ρ, ρ i − r | ω , ω i − r | π, π i (+) | ; i = r | ρ, ρ i − r | ω , ω i + r | π, π i (+) | ; i = r | ρ, ρ i + r | ω , ω i + r | π, π i (A1)(+) | s ; i = | ρ, π i S ( − ) | a ; i = r | ρ, ρ i − r | ω , ω i ( − ) | ; i = r | ρ, ρ i + r | ρ, π i A + r | ω , ω i ( − ) | ∗ ; i = r | ρ, ρ i − r | ρ, π i A + r | ω , ω i (A2)(+) | ; i = r | ρ, ρ i + r | ω , ω i (+) | ; i = r | ρ, ρ i − r | ω , ω i (A3)(+) | s ; i = r | ( ρ, ρ ) s i + r | ρ, ω i S − r | ( π, π ) s i ( − ) | a ; i = r | ( ρ, ρ ) a i − r | ( π, π ) a i (+) | ; i = r | ρ, ω i S + r | ( π, π ) s i ( − ) | ; i = r | ( ρ, ρ ) a i + r | ρ, ω i A + r | ( π, π ) a i ( − ) | ∗ ; i = r | ( ρ, ρ ) a i − r | ρ, ω i A + r | ( π, π ) a i (+) | ; i = r | ( ρ, ρ ) s i − r | ρ, ω i S + r | ( π, π ) s i (A4)(+) | s ; i = r | ( ρ, ρ ) a i + r | ( ρ, π ) s i S + r | ω , π i S ( − ) | a ; i = r | ( ρ, ρ ) s i + r | ρ, ω i S + r | ( ρ, π ) a i S (+) | ; s , i = r | ( ρ, ρ ) a i − r | ( ρ, π ) s i S + r | ω , π i S (+) | ; a , i = r | ρ, ω i A − r | ( ρ, π ) a i A ( − ) | ; s , i = r | ( ρ, ρ ) s i − r | ρ, ω i S + r | ( ρ, π ) s i A − r | ( ρ, π ) a i S − r | ω , π i A ( − ) | ; a , i = r | ρ, ω i S − r | ( ρ, π ) s i A − r | ( ρ, π ) a i S − r | ω , π i A ( − ) | ∗ ; s , i = r | ( ρ, ρ ) s i − r | ρ, ω i S − r | ( ρ, π ) s i A − r | ( ρ, π ) a i S + r | ω , π i A ( − ) | ∗ ; a , i = r | ρ, ω i S + r | ( ρ, π ) s i A − r | ( ρ, π ) a i S + r | ω , π i A (+) | ; s , i = r | ( ρ, ρ ) a i − r | ( ρ, π ) s i S − r | ω , π i S (+) | ; a , i = r | ρ, ω i A + r | ( ρ, π ) a i A (A5)(+) | ; i = r | ( ρ, ρ ) s i − r | ρ, ω i S ( − ) | ; i = r | ( ρ, ρ ) a i − r | ρ, ω i A ( − ) | ∗ ; i = r | ( ρ, ρ ) a i + r | ρ, ω i A (+) | ; i = r | ( ρ, ρ ) s i + r | ρ, ω i S (A6)(+) | ; ′′ i = r | ρ, ρ i − r | π, π i (+) | ; ′′ i = r | ρ, ρ i + r | π, π i (A7)(+) | ; ′′ i = | ρ, π i S ( − ) | ; ′′ i = r | ρ, ρ i + r | ρ, π i A ( − ) | ∗ ; ′′ i = r | ρ, ρ i − r | ρ, π i A (A8)(+) | ; ′′ i = | ρ, ρ i (A9)( − ) | ; i = r | ρ, ρ i − r | π, π i ( − ) | ∗ ; i = r | ρ, ρ i + r | π, π i (A10)(+) | ; i = r | ρ, ρ i + r | ρ, π i A ( − ) | ; i = | ρ, π i S (+) | ; i = r | ρ, ρ i − r | ρ, π i A (A11)( − ) | ; i = | ρ, ρ i (A12)( − ) | ; ∗ i = r | ρ, ρ i + r | π, π i ( − ) | ∗ ; ∗ i = r | ρ, ρ i − r | π, π i (A13)(+) | ; ∗ i = r | ρ, ρ i − r | ρ, π i A ( − ) | ∗ ; ∗ i = | ρ, π i S (+) | ; ∗ i = r | ρ, ρ i + r | ρ, π i A (A14)( − ) | ∗ ; ∗ i = | ρ, ρ i (A15)(+) | ; i = r | ρ, ρ i + r | π, π i (+) | ; i = r | ρ, ρ i − r | π, π i (A16)( − ) | ; i = r | ρ, ρ i − r | ρ, π i A ( − ) | ∗ ; i = r | ρ, ρ i + r | ρ, π i A (+) | ; i = | ρ, π i S (A17)(+) | ; i = | ρ, ρ i (A18)
2. SU(8): 120 ⊗ (+) | ; ∗ i = r | Σ , ρ i − r | Σ , π i (+) | ; ∗ i = r | Σ , ρ i + r | Σ , π i (A19)0( − ) | ; ∗ i = | Σ , ρ i (A20)(+) | ; i = r | ∆ , ρ i + r | (Σ , ρ ) s i − r | (Σ , ρ ) a i + r | Σ , ω i + r | (Σ , π ) a i (+) | ; i = r | ∆ , ρ i − r | (Σ , ρ ) s i + r | (Σ , ρ ) a i − r | Σ , ω i + r | (Σ , π ) s i − r | (Σ , π ) a i (+) | ; i = r | ∆ , ρ i − r | (Σ , ρ ) s i + r | (Σ , ρ ) a i + r | Σ , ω i− r | (Σ , π ) s i + r | (Σ , π ) a i (+) | ; s , i = r | ∆ , ρ i − r | (Σ , ρ ) s i − r | (Σ , ρ ) a i − r | Σ , ω i− r | (Σ , π ) s i − r | (Σ , π ) a i ( − ) | ; a , i = r | (Σ , ρ ) s i + r | (Σ , ρ ) a i + r | Σ , ω i − r | (Σ , π ) s i− r | (Σ , π ) a i (+) | ; b , i = r | (Σ , ρ ) a i − r | Σ , ω i − r | (Σ , π ) s i + r | (Σ , π ) a i (A21)( − ) | ; i = r | ∆ , ρ i − r | ∆ , π i + r | (Σ , ρ ) s i + r | (Σ , ρ ) a i− r | Σ , ω i ( − ) | ; i = r | ∆ , ρ i + r | ∆ , π i + r | (Σ , ρ ) s i − r | (Σ , ρ ) a i + r | Σ , ω i ( − ) | ; s , i = r | ∆ , ρ i + r | ∆ , π i − r | (Σ , ρ ) s i + r | (Σ , ρ ) a i− r | Σ , ω i (+) | ; a , i = r | ∆ , π i + r | (Σ , ρ ) s i + r | (Σ , ρ ) a i − r | Σ , ω i (+) | ; b , i = r | (Σ , ρ ) s i − r | (Σ , ρ ) a i − r | Σ , ω i (A22)(+) | ; i = | ∆ , ρ i (A23)1( − ) | ; ′ i = r | ∆ , ρ i − r | ∆ , ω i + r | Σ , ρ i + r | Σ , π i ( − ) | ; ′ i = r | ∆ , ρ i + r | ∆ , ω i + r | Σ , ρ i − r | Σ , π i ( − ) | ; ′ s , i = r | ∆ , ρ i + r | ∆ , ω i − r | Σ , ρ i − r | Σ , π i (+) | ; ′ a , i = r | ∆ , ω i + r | Σ , ρ i + r | Σ , π i (A24)(+) | ; ′ i = r | ∆ , ρ i − r | ∆ , ω i − r | ∆ , π i + r | Σ , ρ i (+) | ; ′ i = r | ∆ , ρ i + r | ∆ , ω i + r | ∆ , π i + r | Σ , ρ i (+) | ; ′ s , i = r | ∆ , ρ i + r | ∆ , ω i + r | ∆ , π i − r | Σ , ρ i ( − ) | ; ′ a , i = r | ∆ , ω i − r | ∆ , π i − r | Σ , ρ i (A25)( − ) | ; ′ i = r | ∆ , ρ i − r | ∆ , ω i ( − ) | ; ′ i = r | ∆ , ρ i + r | ∆ , ω i (A26)( − ) | ; ∗ s , i = | Σ , ρ i (+) | ; ∗ a , i = | Σ , π i (A27)(+) | ; ∗ i = | Σ , ρ i (A28)(+) | ; ∗ i = r | Σ , ρ i + r | Σ , π i (+) | ; ∗ i = r | Σ , ρ i − r | Σ , π i (A29)( − ) | ; ∗ i = | Σ , ρ i (A30)(+) | ; i = | ∆ , ρ i (A31)( − ) | ; i = r | ∆ , ρ i − r | ∆ , π i ( − ) | ; i = r | ∆ , ρ i + r | ∆ , π i (A32)(+) | ; i = | ∆ , ρ i (A33)( − ) | ; i = r | ∆ , ρ i − r | Σ , ρ i + r | Σ , π i ( − ) | ; s , i = r | ∆ , ρ i + r | Σ , ρ i − r | Σ , π i ( − ) | ; a , i = r | Σ , ρ i + r | Σ , π i (A34)2(+) | ; i = r | ∆ , ρ i + r | ∆ , π i − r | Σ , ρ i (+) | ; s , i = r | ∆ , ρ i − r | ∆ , π i + r | Σ , ρ i ( − ) | ; a , i = r | ∆ , π i + r | Σ , ρ i (A35)( − ) | ; i = | ∆ , ρ i (A36) Appendix B: Tables of scalar factors of SU(6) | ρ i = | ; i , | π i = | ; i , | ω i = | ; i , | ∆ i = | ; i , | Σ i = | ; i .
1. SU(6): 35 ⊗ (+) | ; i = r | ρ, ρ i − r | ω , ω i − r | π, π i (+) | ; i = r | ρ, ρ i − r | ω , ω i + r | π, π i (+) | ; i = r | ρ, ρ i + r | ω , ω i + r | π, π i (B1)(+) | s ; i = | ρ, π i S ( − ) | a ; i = r | ρ, ρ i − r | ω , ω i ( − ) | ; i = r | ρ, ρ i + r | ρ, π i A + r | ω , ω i ( − ) | ∗ ; i = r | ρ, ρ i − r | ρ, π i A + r | ω , ω i (B2)(+) | ; i = r | ρ, ρ i + r | ω , ω i (+) | ; i = r | ρ, ρ i − r | ω , ω i (B3)3(+) | s ; i = r | ( ρ, ρ ) s i + r | ρ, ω i S − r | ( π, π ) s i ( − ) | a ; i = r | ( ρ, ρ ) a i − r | ( π, π ) a i (+) | ; i = r | ( ρ, ρ ) s i − r | ρ, ω i S − r | ( π, π ) s i ( − ) | ; i = r | ( ρ, ρ ) a i + r | ρ, ω i A + r | ( π, π ) a i ( − ) | ∗ ; i = r | ( ρ, ρ ) a i − r | ρ, ω i A + r | ( π, π ) a i (+) | ; i = r | ( ρ, ρ ) s i − r | ρ, ω i S + r | ( π, π ) s i (B4)(+) | s ; i = r | ( ρ, ρ ) a i + r | ( ρ, π ) s i S + r | ω , π i S ( − ) | a ; i = r | ( ρ, ρ ) s i + r | ρ, ω i S + r | ( ρ, π ) a i S (+) | ; s , i = r | ( ρ, ρ ) a i − r | ( ρ, π ) s i S + r | ω , π i S (+) | ; a , i = r | ρ, ω i A − r | ( ρ, π ) a i A ( − ) | ; s , i = r | ( ρ, ρ ) s i − r | ρ, ω i S + r | ( ρ, π ) s i A − r | ( ρ, π ) a i S − r | ω , π i A ( − ) | ; a , i = r | ρ, ω i S − r | ( ρ, π ) s i A − r | ( ρ, π ) a i S − r | ω , π i A ( − ) | ∗ ; s , i = r | ( ρ, ρ ) s i − r | ρ, ω i S − r | ( ρ, π ) s i A − r | ( ρ, π ) a i S + r | ω , π i A ( − ) | ∗ ; a , i = r | ρ, ω i S + r | ( ρ, π ) s i A − r | ( ρ, π ) a i S + r | ω , π i A (+) | ; s , i = r | ( ρ, ρ ) a i − r | ( ρ, π ) s i S − r | ω , π i S (+) | ; a , i = r | ρ, ω i A + r | ( ρ, π ) a i A (B5)(+) | ; i = r | ( ρ, ρ ) s i − r | ρ, ω i S ( − ) | ; i = r | ( ρ, ρ ) a i − r | ρ, ω i A ( − ) | ∗ ; i = r | ( ρ, ρ ) a i + r | ρ, ω i A (+) | ; i = r | ( ρ, ρ ) s i + r | ρ, ω i S (B6)4( − ) | ; i = r | ρ, ρ i − r | π, π i ( − ) | ∗ ; i = r | ρ, ρ i + r | π, π i (B7)(+) | ; i = r | ρ, ρ i + r | ρ, π i A ( − ) | ; i = | ρ, π i S (+) | ; i = r | ρ, ρ i − r | ρ, π i A (B8)( − ) | ; i = | ρ, ρ i (B9)( − ) | ; ∗ i = r | ρ, ρ i + r | π, π i ( − ) | ∗ ; ∗ i = r | ρ, ρ i − r | π, π i (B10)(+) | ; ∗ i = r | ρ, ρ i − r | ρ, π i A ( − ) | ∗ ; ∗ i = | ρ, π i S (+) | ; ∗ i = r | ρ, ρ i + r | ρ, π i A (B11)( − ) | ∗ ; ∗ i = | ρ, ρ i (B12)(+) | ; i = r | ρ, ρ i + r | π, π i (+) | ; i = r | ρ, ρ i − r | π, π i (B13)( − ) | ; i = r | ρ, ρ i − r | ρ, π i A ( − ) | ∗ ; i = r | ρ, ρ i + r | ρ, π i A (+) | ; i = | ρ, π i S (B14)(+) | ; i = | ρ, ρ i (B15)
2. SU(6): 56 ⊗ ( − ) | ; i = r | Σ , ρ i − r | Σ , π i ( − ) | ; i = r | Σ , ρ i + r | Σ , π i (B16)5(+) | ; i = | Σ , ρ i (B17)(+) | ; i = r | ∆ , ρ i + r | (Σ , ρ ) s i − r | (Σ , ρ ) a i + r | Σ , ω i + r | (Σ , π ) a i (+) | ; i = r | ∆ , ρ i − r | (Σ , ρ ) s i + r | (Σ , ρ ) a i − r | Σ , ω i + r | (Σ , π ) s i − r | (Σ , π ) a i (+) | ; i = r | ∆ , ρ i − r | (Σ , ρ ) s i + r | (Σ , ρ ) a i + r | Σ , ω i− r | (Σ , π ) s i + r | (Σ , π ) a i (+) | ; s , i = r | ∆ , ρ i − r | (Σ , ρ ) s i − r | (Σ , ρ ) a i − r | Σ , ω i− r | (Σ , π ) s i − r | (Σ , π ) a i ( − ) | ; a , i = r | (Σ , ρ ) s i + r | (Σ , ρ ) a i + r | Σ , ω i − r | (Σ , π ) s i− r | (Σ , π ) a i (+) | ; b , i = r | (Σ , ρ ) a i − r | Σ , ω i − r | (Σ , π ) s i + r | (Σ , π ) a i (B18)( − ) | ; i = r | ∆ , ρ i − r | ∆ , π i + r | (Σ , ρ ) s i + r | (Σ , ρ ) a i− r | Σ , ω i ( − ) | ; i = r | ∆ , ρ i + r | ∆ , π i + r | (Σ , ρ ) s i − r | (Σ , ρ ) a i + r | Σ , ω i ( − ) | ; s , i = r | ∆ , ρ i + r | ∆ , π i − r | (Σ , ρ ) s i + r | (Σ , ρ ) a i− r | Σ , ω i (+) | ; a , i = r | ∆ , π i + r | (Σ , ρ ) s i + r | (Σ , ρ ) a i − r | Σ , ω i ( − ) | ; b , i = r | (Σ , ρ ) a i + r | Σ , ω i (B19)(+) | ; i = | ∆ , ρ i (B20)6( − ) | ; i = r | ∆ , ρ i − r | ∆ , ω i + r | Σ , ρ i + r | Σ , π i ( − ) | ; i = r | ∆ , ρ i + r | ∆ , ω i + r | Σ , ρ i − r | Σ , π i ( − ) | ; s , i = r | ∆ , ρ i + r | ∆ , ω i − r | Σ , ρ i (+) | ; a , i = r | ∆ , ω i + r | Σ , ρ i + r | Σ , π i (B21)(+) | ; i = r | ∆ , ρ i − r | ∆ , ω i − r | ∆ , π i + r | Σ , ρ i (+) | ; i = r | ∆ , ρ i + r | ∆ , ω i + r | ∆ , π i + r | Σ , ρ i (+) | ; s , i = r | ∆ , ρ i + r | ∆ , ω i + r | ∆ , π i − r | Σ , ρ i ( − ) | ; a , i = r | ∆ , ω i − r | ∆ , π i − r | Σ , ρ i (B22)( − ) | ; i = r | ∆ , ρ i − r | ∆ , ω i ( − ) | ; i = r | ∆ , ρ i + r | ∆ , ω i (B23)(+) | ; ∗ i = r | Σ , ρ i + r | Σ , π i (+) | ; ∗ i = r | Σ , ρ i − r | Σ , π i (B24)( − ) | ; ∗ i = | Σ , ρ i (B25)( − ) | ; i = r | ∆ , ρ i − r | Σ , ρ i + r | Σ , π i ( − ) | ; s , i = r | ∆ , ρ i + r | Σ , ρ i − r | Σ , π i ( − ) | ; a , i = r | Σ , ρ i + r | Σ , π i (B26)(+) | ; i = r | ∆ , ρ i + r | ∆ , π i − r | Σ , ρ i (+) | ; s , i = r | ∆ , ρ i − r | ∆ , π i + r | Σ , ρ i ( − ) | ; a , i = r | ∆ , π i + r | Σ , ρ i (B27)( − ) | ; i = | ∆ , ρ i (B28)(+) | ; i = | ∆ , ρ i (B29)7( − ) | ; i = r | ∆ , ρ i − r | ∆ , π i ( − ) | ; i = r | ∆ , ρ i + r | ∆ , π i (B30)(+) | ; i = | ∆ , ρ i (B31) Appendix C: Tables of scalar factors of SU(4) | π i = | ; , i , | D i = | ; ∗ , i , | ¯ D i = | ; , − i , | η c i = | ; , i , | Σ i = | ; , i , | Σ c i = | ; , i , | Ξ c i = | ; ∗ , i , | Ξ cc i = | ; , i , | ∆ i = | ′ ; , i , | Σ ∗ c i = | ′ ; , i , | Ξ ∗ cc i = | ′ ; , i , | Ω ccc i = | ′ ; , i .
1. SU(4): 15 ⊗ (+) | ; , i = r | π, π i − r | D, ¯ D i A − r | η c , η c i (+) | s ; , i = r | π, π i + r | D, ¯ D i A + r | η c , η c i ( − ) | a ; , i = | D, ¯ D i S (+) | ; , i = − r | π, π i − r | D, ¯ D i A + r | η c , η c i (C1)(+) | s ; , − i = r | π, ¯ D i A + r | ¯ D, η c i S ( − ) | a ; , − i = r | π, ¯ D i S + r | ¯ D, η c i A ( − ) | ∗ ; , − i = − r | π, ¯ D i S + r | ¯ D, η c i A (+) | ; , − i = − r | π, ¯ D i A + r | ¯ D, η c i S (C2)( − ) | ; , i = | D, D i (C3)( − ) | ∗ ; ∗ , − i = | ¯ D, ¯ D i (C4)(+) | s ; ∗ , i = − r | π, D i A + r | D, η c i S ( − ) | a ; ∗ , i = r | π, D i S − r | D, η c i A ( − ) | ; ∗ , i = r | π, D i S + r | D, η c i A (+) | ; ∗ , i = r | π, D i A + r | D, η c i S (C5)8(+) | ; , − i = | ¯ D, ¯ D i (C6)(+) | ′′ ; , i = | π, D i A ( − ) | ; , i = | π, D i S (C7)(+) | ′′ ; ∗ , − i = | π, ¯ D i A ( − ) | ∗ ; ∗ , − i = | π, ¯ D i S (C8)(+) | ; ∗ , i = | D, D i (C9)(+) | s ; , i = r | ( π, π ) s i − r | π, η c i S + r | D, ¯ D i S ( − ) | a ; , i = r | ( π, π ) a i − r | D, ¯ D i A (+) | ′′ ; , i = r | ( π, π ) s i + r | π, η c i S − r | D, ¯ D i S ( − ) | ; , i = r | ( π, π ) a i + r | π, η c i A + r | D, ¯ D i A ( − ) | ∗ ; , i = − r | ( π, π ) a i + r | π, η c i A − r | D, ¯ D i A (+) | ; , i = − r | ( π, π ) s i + r | π, η c i S + r | D, ¯ D i S (C10)( − ) | ; , i = | π, π i (C11)( − ) | ∗ ; ∗ , i = | π, π i (C12)( − ) | ; , − i = | π, ¯ D i A (+) | ; , − i = | π, ¯ D i S (C13)( − ) | ∗ ; ∗ , i = | π, D i A (+) | ; ∗ , i = | π, D i S (C14)(+) | ; , i = | π, π i (C15)
2. SU(4): 20 ⊗ ( − ) | ∗ ; , i = − r | Σ , π i + r | Ξ c , ¯ D i (+) | ∗ ; , i = − r | Σ , π i − r | Ξ c , ¯ D i (C16)( − ) | ′ ; , i = −| Ξ cc , D i (C17)(+) | ∗ ; , − i = −| Σ , ¯ D i (C18)9(+) | s ; , i = − r | Σ c , D i + r | Ξ c , D i + r | Ξ cc , π i + r | Ξ cc , η c i ( − ) | a ; , i = r | Σ c , D i + r | Ξ c , D i + r | Ξ cc , π i − r | Ξ cc , η c i ( − ) | ′ ; , i = − r | Σ c , D i + r | Ξ c , D i − r | Ξ cc , π i − r | Ξ cc , η c i (+) | ; , i = r | Σ c , D i + r | Ξ c , D i − r | Ξ cc , π i + r | Ξ cc , η c i (C19)( − ) | ∗ ; ∗ , i = r | Σ , D i − r | Σ c , π i − r | Ξ c , π i − r | Ξ c , η c i + r | Ξ cc , ¯ D i (+) | s ; ∗ , i = − r | Σ , D i − r | Σ c , π i + r | Ξ c , π i + r | Ξ c , η c i − r | Ξ cc , ¯ D i ( − ) | a ; ∗ , i = r | Σ , D i + r | Ξ c , π i − r | Ξ c , η c i − r | Ξ cc , ¯ D i (+) | ∗ ; ∗ , i = r | Σ , D i − r | Σ c , π i − r | Ξ c , π i + r | Ξ c , η c i − r | Ξ cc , ¯ D i (+) | ; ∗ , i = r | Σ , D i + r | Σ c , π i + r | Ξ c , π i + r | Ξ c , η c i + r | Ξ cc , ¯ D i (C20)(+) | s ; , i = r | Σ , D i − r | Σ c , π i − r | Σ c , η c i − r | Ξ c , π i + r | Ξ cc , ¯ D i ( − ) | a ; , i = r | Σ , D i + r | Σ c , π i − r | Σ c , η c i − r | Ξ cc , ¯ D i ( − ) | ′ ; , i = r | Σ , D i − r | Σ c , π i − r | Σ c , η c i + r | Ξ c , π i − r | Ξ cc , ¯ D i ( − ) | ∗ ; , i = r | Σ , D i − r | Σ c , π i + r | Σ c , η c i − r | Ξ c , π i − r | Ξ cc , ¯ D i (+) | ; , i = r | Σ , D i + r | Σ c , π i + r | Σ c , η c i + r | Ξ c , π i + r | Ξ cc , ¯ D i (C21)( − ) | ∗ ; ∗ , − i = | Σ , ¯ D i (C22)(+) | ∗ ; ∗ , i = r | Ξ c , D i − r | Ξ cc , π i (+) | ; ∗ , i = r | Ξ c , D i + r | Ξ cc , π i (C23)(+) | s ; , i = r | (Σ , π ) s i − r | (Σ , π ) a i + r | Σ , η c i − r | Σ c , ¯ D i + r | Ξ c , ¯ D i ( − ) | a ; , i = r | (Σ , π ) a i + r | Σ , η c i − r | Σ c , ¯ D i − r | Ξ c , ¯ D i (+) | ∗ ; , i = − r | (Σ , π ) s i − r | (Σ , π ) a i + r | Σ , η c i − r | Σ c , ¯ D i − r | Ξ c , ¯ D i ( − ) | ∗ ; , i = r | (Σ , π ) s i − r | (Σ , π ) a i + r | Σ , η c i + r | Σ c , ¯ D i − r | Ξ c , ¯ D i (+) | ; , i = − r | (Σ , π ) s i + r | (Σ , π ) a i + r | Σ , η c i + r | Σ c , ¯ D i + r | Ξ c , ¯ D i (C24)0(+) | ; , i = | Ξ cc , D i (C25)( − ) | ′ ; , i = r | Σ , π i − r | Σ c , ¯ D i (+) | ; , i = r | Σ , π i + r | Σ c , ¯ D i (C26)( − ) | ∗ ; ∗ , i = | Σ , π i (C27)(+) | ; , − i = | Σ , ¯ D i (C28)( − ) | ∗ ; , i = r | Σ c , D i − r | Ξ cc , π i (+) | ; , i = r | Σ c , D i + r | Ξ cc , π i (C29)(+) | ∗ ; ∗ , i = r | Σ , D i − r | Σ c , π i − r | Ξ c , π i ( − ) | ∗ ; ∗ , i = r | Σ , D i + r | Σ c , π i − r | Ξ c , π i (+) | ; ∗ , i = r | Σ , D i + r | Σ c , π i + r | Ξ c , π i (C30)(+) | ; ∗ , i = | Σ c , π i (C31)(+) | ; , i = | Σ , π i (C32)
3. SU(4): 20 ′ ⊗ ( − ) | ′ ; , i = r | Ξ ∗ cc , D i − r | Ω ccc , η c i (+) | ; , i = r | Ξ ∗ cc , D i + r | Ω ccc , η c i (C33)(+) | ; , i = r | Σ ∗ c , D i − r | Ξ ∗ cc , π i − r | Ξ ∗ cc , η c i + r | Ω ccc , ¯ D i ( − ) | ′ ; , i = r | Σ ∗ c , D i + r | Ξ ∗ cc , π i − r | Ξ ∗ cc , η c i − r | Ω ccc , ¯ D i (+) | ; , i = r | Σ ∗ c , D i + r | Ξ ∗ cc , π i + r | Ξ ∗ cc , η c i + r | Ω ccc , ¯ D i ( − ) | ; , i = r | Σ ∗ c , D i − r | Ξ ∗ cc , π i + r | Ξ ∗ cc , η c i − r | Ω ccc , ¯ D i (C34)(+) | ; ∗ , i = r | Σ ∗ c , π i − r | Ξ ∗ cc , ¯ D i ( − ) | ; ∗ , i = r | Σ ∗ c , π i + r | Ξ ∗ cc , ¯ D i (C35)1(+) | ; ∗ , i = | Ω ccc , D i (C36)(+) | ; , i = r | ∆ , D i − r | Σ ∗ c , π i − r | Σ ∗ c , η c i + r | Ξ ∗ cc , ¯ D i ( − ) | ′ ; , i = r | ∆ , D i + r | Σ ∗ c , π i − r | Σ ∗ c , η c i − r | Ξ ∗ cc , ¯ D i (+) | ; , i = r | ∆ , D i + r | Σ ∗ c , π i + r | Σ ∗ c , η c i + r | Ξ ∗ cc , ¯ D i ( − ) | ; , i = r | ∆ , D i − r | Σ ∗ c , π i + r | Σ ∗ c , η c i − r | Ξ ∗ cc , ¯ D i (C37)( − ) | ; ∗ , i = | Ξ ∗ cc , π i (C38)(+) | ; , i = r | ∆ , π i − r | Σ ∗ c , ¯ D i ( − ) | ; , i = r | ∆ , π i + r | Σ ∗ c , ¯ D i (C39)(+) | ; , i = r | Ξ ∗ cc , D i + r | Ω ccc , π i ( − ) | ; , i = r | Ξ ∗ cc , D i − r | Ω ccc , π i (C40)( − ) | ′ ; , i = r | ∆ , π i + r | ∆ , η c i − r | Σ ∗ c , ¯ D i (+) | ; , i = r | ∆ , π i + r | ∆ , η c i + r | Σ ∗ c , ¯ D i ( − ) | ; , i = − r | ∆ , π i + r | ∆ , η c i − r | Σ ∗ c , ¯ D i (C41)( − ) | ; , − i = | ∆ , ¯ D i (C42)(+) | ; , i = r | Σ ∗ c , D i + r | Ξ ∗ cc , π i ( − ) | ; , i = r | Σ ∗ c , D i − r | Ξ ∗ cc , π i (C43)(+) | ; ′ , − i = | ∆ , ¯ D i (C44)( − ) | ; ∗ , i = | Σ ∗ c , π i (C45)(+) | ; ∗ , i = r | ∆ , D i + r | Σ ∗ c , π i ( − ) | ; ∗ , i = r | ∆ , D i − r | Σ ∗ c , π i (C46)( − ) | ; , i = | ∆ , π i (C47)(+) | ; , i = | ∆ , π i (C48)2 Appendix D: Tables of scalar factors of SU(3)1. SU(3): 3 ⊗ (+) | ; 0 , − i = | ¯ D s , ¯ D s i (D1)( − ) | ∗ ; 0 , i = | ¯ D, ¯ D i (D2)( − ) | ∗ ; , − i = | ¯ D, ¯ D s i A (+) | ; , − i = | ¯ D, ¯ D s i S (D3)(+) | ; 1 , i = | ¯ D, ¯ D i (D4)
2. SU(3): 3 ⊗ ∗ ( − ) | ; 0 , i = r | ¯ D, D i + r | ¯ D s , D s i (+) | ; 0 , i = − r | ¯ D, D i + r | ¯ D s , D s i (D5)(+) | ; , − i = | ¯ D s , D i (D6)(+) | ; , i = | ¯ D, D s i (D7)(+) | ; 1 , i = | ¯ D, D i (D8)
3. SU(3): 3 ∗ ⊗ ∗ ( − ) | ; 0 , − i = −| D, D i (D9)(+) | ∗ ; 0 , i = | D s , D s i (D10)( − ) | ; , i = | D, D s i A (+) | ∗ ; , i = | D, D s i S (D11)(+) | ∗ ; 1 , − i = | D, D i (D12)3
4. SU(3): 6 ⊗ (+) | ; 0 , − i = | Ω c , ¯ D s i (D13)( − ) | ; 0 , i = | Ξ ′ c , ¯ D i (D14)( − ) | ; , − i = r | Ξ ′ c , ¯ D s i − r | Ω c , ¯ D i (+) | ; , − i = r | Ξ ′ c , ¯ D s i + r | Ω c , ¯ D i (D15)( − ) | ; , i = | Σ c , ¯ D i (D16)( − ) | ; 1 , i = r | Σ c , ¯ D s i − r | Ξ ′ c , ¯ D i (+) | ; 1 , i = r | Σ c , ¯ D s i + r | Ξ ′ c , ¯ D i (D17)(+) | ; , i = | Σ c , ¯ D i (D18)
5. SU(3): 6 ⊗ ∗ ( − ) | ; 0 , − i = r | Ξ ′ c , D i + r | Ω c , D s i (+) | ; 0 , − i = − r | Ξ ′ c , D i + r | Ω c , D s i (D19)(+) | ; , − i = | Ω c , D i (D20)( − ) | ; , i = r | Σ c , D i + r | Ξ ′ c , D s i (+) | ; , i = − r | Σ c , D i + r | Ξ ′ c , D s i (D21)(+) | ; 1 , − i = | Ξ ′ c , D i (D22)(+) | ; 1 , i = | Σ c , D s i (D23)(+) | ; , i = | Σ c , D i (D24)4
6. SU(3): 6 ⊗ ( − ) | ; 0 , − i = r | Ξ ′ c , ¯ K i + r | Ω c , η i (+) | ∗ ; 0 , − i = − r | Ξ ′ c , ¯ K i + r | Ω c , η i (D25)(+) | ∗ ; 0 , i = r | Σ c , π i + r | Ξ ′ c , K i ( − ) | ∗ ; 0 , i = − r | Σ c , π i + r | Ξ ′ c , K i (D26)(+) | ∗ ; , − i = | Ω c , ¯ K i (D27)(+) | ∗ ; , − i = r | Σ c , ¯ K i − r | Ξ ′ c , π i + r | Ξ ′ c , η i − r | Ω c , K i ( − ) | ; , − i = r | Σ c , ¯ K i + r | Ξ ′ c , π i + r | Ξ ′ c , η i + r | Ω c , K i ( − ) | ∗ ; , − i = − r | Σ c , ¯ K i + r | Ξ ′ c , π i + r | Ξ ′ c , η i − r | Ω c , K i (+) | ∗ ; , − i = − r | Σ c , ¯ K i − r | Ξ ′ c , π i + r | Ξ ′ c , η i + r | Ω c , K i (D28)( − ) | ∗ ; , i = | Σ c , K i (D29)( − ) | ∗ ; 1 , − i = r | Ξ ′ c , ¯ K i − r | Ω c , π i (+) | ∗ ; 1 , − i = r | Ξ ′ c , ¯ K i + r | Ω c , π i (D30)( − ) | ; 1 , i = r | Σ c , π i − r | Σ c , η i + r | Ξ ′ c , K i ( − ) | ∗ ; 1 , i = r | Σ c , π i + r | Σ c , η i − r | Ξ ′ c , K i (+) | ∗ ; 1 , i = − r | Σ c , π i + r | Σ c , η i + r | Ξ ′ c , K i (D31)( − ) | ∗ ; , − i = r | Σ c , ¯ K i − r | Ξ ′ c , π i (+) | ∗ ; , − i = r | Σ c , ¯ K i + r | Ξ ′ c , π i (D32)(+) | ∗ ; , i = | Σ c , K i (D33)(+) | ∗ ; 2 , i = | Σ c , π i (D34)5
7. SU(3): 8 ⊗ ( − ) | ; 0 , − i = r | ¯ K, ¯ D i − r | η, ¯ D s i (+) | ; 0 , − i = r | ¯ K, ¯ D i + r | η, ¯ D s i (D35)( − ) | ∗ ; 0 , i = | K, ¯ D i (D36)(+) | ; , − i = | ¯ K, ¯ D s i (D37)( − ) | ; , i = r | π, ¯ D i − r | K, ¯ D s i + r | η, ¯ D i ( − ) | ∗ ; , i = r | π, ¯ D i + r | K, ¯ D s i − r | η, ¯ D i (+) | ; , i = r | π, ¯ D i + r | K, ¯ D s i + r | η, ¯ D i (D38)( − ) | ∗ ; 1 , − i = r | π, ¯ D s i − r | ¯ K, ¯ D i (+) | ; 1 , − i = r | π, ¯ D s i + r | ¯ K, ¯ D i (D39)(+) | ; 1 , i = | K, ¯ D i (D40)(+) | ; , i = | π, ¯ D i (D41)
8. SU(3): 8 ⊗ ∗ ( − ) | ; 0 , − i = −| ¯ K, D i (D42)( − ) | ∗ ; 0 , i = r | K, D i + r | η, D s i (+) | ∗ ; 0 , i = − r | K, D i + r | η, D s i (D43)( − ) | ∗ ; , − i = r | π, D i + r | ¯ K, D s i − r | η, D i ( − ) | ; , − i = − r | π, D i + r | ¯ K, D s i − r | η, D i (+) | ∗ ; , − i = − r | π, D i + r | ¯ K, D s i + r | η, D i (D44)(+) | ∗ ; , i = | K, D s i (D45)6(+) | ∗ ; 1 , − i = | ¯ K, D i (D46)( − ) | ; 1 , i = r | π, D s i − r | K, D i (+) | ∗ ; 1 , i = r | π, D s i + r | K, D i (D47)(+) | ∗ ; , − i = | π, D i (D48)
9. SU(3): 8 ⊗ ( − ) | ; 0 , − i = −| ¯ K, ¯ K i (D49)(+) | ; 0 , i = r | π, π i − r | K, ¯ K i A − r | η, η i (+) | s ; 0 , i = − r | π, π i − r | K, ¯ K i A − r | η, η i ( − ) | a ; 0 , i = | K, ¯ K i S (+) | ; 0 , i = − r | π, π i − r | K, ¯ K i A + r | η, η i (D50)( − ) | ∗ ; 0 , i = | K, K i (D51)(+) | s ; , − i = − r | π, ¯ K i A − r | ¯ K, η i S ( − ) | a ; , − i = r | π, ¯ K i S + r | ¯ K, η i A ( − ) | ; , − i = − r | π, ¯ K i S + r | ¯ K, η i A (+) | ; , − i = − r | π, ¯ K i A + r | ¯ K, η i S (D52)(+) | s ; , i = r | π, K i A − r | K, η i S ( − ) | a ; , i = r | π, K i S − r | K, η i A ( − ) | ∗ ; , i = r | π, K i S + r | K, η i A (+) | ; , i = r | π, K i A + r | K, η i S (D53)(+) | ; 1 , − i = | ¯ K, ¯ K i (D54)7(+) | s ; 1 , i = r | π, η i S − r | K, ¯ K i S ( − ) | a ; 1 , i = r | π, π i − r | K, ¯ K i A ( − ) | ; 1 , i = − r | π, π i + r | π, η i A − r | K, ¯ K i A ( − ) | ∗ ; 1 , i = r | π, π i + r | π, η i A + r | K, ¯ K i A (+) | ; 1 , i = r | π, η i S + r | K, ¯ K i S (D55)(+) | ; 1 , i = | K, K i (D56)( − ) | ∗ ; , − i = | π, ¯ K i A (+) | ; , − i = | π, ¯ K i S (D57)( − ) | ; , i = | π, K i A (+) | ; , i = | π, K i S (D58)(+) | ; 2 , i = | π, π i (D59)
10. SU(3): 10 ⊗ (+) | ′ ; 0 , − i = | Ω , ¯ D s i (D60)( − ) | ; 0 , − i = | Ξ ∗ , ¯ D i (D61)( − ) | ; , − i = r | Ξ ∗ , ¯ D s i − r | Ω , ¯ D i (+) | ′ ; , − i = r | Ξ ∗ , ¯ D s i + r | Ω , ¯ D i (D62)( − ) | ; , i = | Σ ∗ , ¯ D i (D63)( − ) | ; 1 , − i = r | Σ ∗ , ¯ D s i − r | Ξ ∗ , ¯ D i (+) | ′ ; 1 , − i = r | Σ ∗ , ¯ D s i + r | Ξ ∗ , ¯ D i (D64)( − ) | ; 1 , i = | ∆ , ¯ D i (D65)( − ) | ; , i = r | ∆ , ¯ D s i − r | Σ ∗ , ¯ D i (+) | ′ ; , i = r | ∆ , ¯ D s i + r | Σ ∗ , ¯ D i (D66)(+) | ′ ; 2 , i = | ∆ , ¯ D i (D67)8
11. SU(3): 10 ⊗ ∗ ( − ) | ; 0 , − i = r | Ξ ∗ , D i + r | Ω , D s i (+) | ∗ ; 0 , − i = − r | Ξ ∗ , D i + r | Ω , D s i (D68)(+) | ∗ ; , − i = | Ω , D i (D69)( − ) | ; , − i = r | Σ ∗ , D i + r | Ξ ∗ , D s i (+) | ∗ ; , − i = − r | Σ ∗ , D i + r | Ξ ∗ , D s i (D70)(+) | ∗ ; 1 , − i = | Ξ ∗ , D i (D71)( − ) | ; 1 , i = r | ∆ , D i + r | Σ ∗ , D s i (+) | ∗ ; 1 , i = − r | ∆ , D i + r | Σ ∗ , D s i (D72)(+) | ∗ ; , − i = | Σ ∗ , D i (D73)(+) | ∗ ; , i = | ∆ , D s i (D74)(+) | ∗ ; 2 , i = | ∆ , D i (D75)
12. SU(3): 10 ⊗ ( − ) | ; 0 , − i = r | Ξ ∗ , ¯ K i + r | Ω , η i (+) | ; 0 , − i = − r | Ξ ∗ , ¯ K i + r | Ω , η i (D76)(+) | ; 0 , i = r | Σ ∗ , π i + r | Ξ ∗ , K i ( − ) | ; 0 , i = − r | Σ ∗ , π i + r | Ξ ∗ , K i (D77)(+) | ; , − i = | Ω , ¯ K i (D78)9(+) | ; , − i = r | Σ ∗ , ¯ K i − r | Ξ ∗ , π i + r | Ξ ∗ , η i − r | Ω , K i ( − ) | ; , − i = r | Σ ∗ , ¯ K i + r | Ξ ∗ , π i + r | Ξ ∗ , η i + r | Ω , K i ( − ) | ; , − i = − r | Σ ∗ , ¯ K i + r | Ξ ∗ , π i + r | Ξ ∗ , η i − r | Ω , K i (+) | ; , − i = − r | Σ ∗ , ¯ K i − r | Ξ ∗ , π i + r | Ξ ∗ , η i + r | Ω , K i (D79)(+) | ; , i = r | ∆ , π i + r | Σ ∗ , K i ( − ) | ; , i = − r | ∆ , π i + r | Σ ∗ , K i (D80)( − ) | ; 1 , − i = r | Ξ ∗ , ¯ K i − r | Ω , π i (+) | ; 1 , − i = r | Ξ ∗ , ¯ K i + r | Ω , π i (D81)(+) | ; 1 , i = r | ∆ , ¯ K i − r | Σ ∗ , π i + r | Σ ∗ , η i − r | Ξ ∗ , K i ( − ) | ; 1 , i = r | ∆ , ¯ K i + r | Σ ∗ , π i + r | Ξ ∗ , K i ( − ) | ; 1 , i = − r | ∆ , ¯ K i + r | Σ ∗ , π i + r | Σ ∗ , η i − r | Ξ ∗ , K i (+) | ; 1 , i = − r | ∆ , ¯ K i − r | Σ ∗ , π i + r | Σ ∗ , η i + r | Ξ ∗ , K i (D82)( − ) | ; 1 , i = | ∆ , K i (D83)( − ) | ; , − i = r | Σ ∗ , ¯ K i − r | Ξ ∗ , π i (+) | ; , − i = r | Σ ∗ , ¯ K i + r | Ξ ∗ , π i (D84)( − ) | ; , i = r | ∆ , π i − r | ∆ , η i + r | Σ ∗ , K i ( − ) | ; , i = r | ∆ , π i + r | ∆ , η i − r | Σ ∗ , K i (+) | ; , i = − r | ∆ , π i + r | ∆ , η i + r | Σ ∗ , K i (D85)( − ) | ; 2 , i = r | ∆ , ¯ K i − r | Σ ∗ , π i (+) | ; 2 , i = r | ∆ , ¯ K i + r | Σ ∗ , π i (D86)0(+) | ; 2 , i = | ∆ , K i (D87)(+) | ; , i = | ∆ , π i (D88) [1] F. Gursey and L. A. Radicati, Spin and unitary spin in-dependence of strong interactions,
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