JOURNAL OF MATHEMATICAL PHYSICS 47, Clebsch-Gordan coefficients for U 8 O 8 SU 3 I. Sánchez Lima and P. O. Hess Instituto de Ciencias Nucle

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1 Clebsch-Gordan coefficients for U(8)O(8)SU(3) I. Sánchez Lima and P. O. Hess Citation: J. Math. Phys. 47, (006); doi: / View online: View Table of Contents: Published by the American Institute of Physics. Related Articles Quantum time of arrival Goursat problem J. Math. Phys. 53, (01) Restricted forms of quantum supergroups J. Math. Phys. 53, (01) Positive spaces, generalized semi-densities, and quantum interactions J. Math. Phys. 53, 0330 (01) Classification, selection rules, and symmetry properties of the Clebsch- Gordan coefficients of symmetric group J. Math. Phys. 53, (01) Quantum deformation of two four-dimensional spin foam models J. Math. Phys. 53, 0501 (01) Additional information on J. Math. Phys. Journal Homepage: Journal Information: Top downloads: Information for Authors: Downloaded 0 Apr 01 to Redistribution subject to AIP license or copyright; see

2 JOURNAL OF MATHEMATICAL PHYSICS 47, Clebsch-Gordan coefficients for U 8 O 8 SU 3 I. Sánchez Lima and P. O. Hess Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, Apdo. Postal , Mexico D.F., Mexico Received 15 March 006; accepted 1 May 006; published online 19 June 006 The group chain U8O8SU3 plays an important role in many particle systems whenever the fundamental particles have eight degrees of freedom. As a particular example, the systems of many gluons and pairs of quark-antiquark are discussed, which can be coupled to a flavor octet. In order to determine the explicit structure of states and decay probabilities, the calculation of the Clebsch-Gordan coefficients CGC of this group chain is indispensable. In this contribution, the polynomial states of the U8 chain are constructed and also the isoscalar factors of the CGC s. Tables of isoscalar factors are presented. The method shown serves as an example for higher rank groups. 006 American Institute of Physics. DOI: / I. INTRODUCTION Clebsch-Gordan coefficients CGC play an important role in physics. The group structure related to it depends on the particular problem to consider. For example, in the case of angular momentum, spin or total spin of a many particle system, the underlying structure is the group SU. Knowing the CGC serves to calculate matrix elements of tensors of SU, which are, for example, related to interactions in a Hamiltonian and/or to transition operators. The fundamental group involved here the SU depends on the basic degrees of freedom of the lowest nontrivial irreducible spin representation irrep, which is spin 1. There are other types of CGC, which are related to different basic degrees of freedom. As a further example we mention the SU3 group. It appears in nuclear physics as the symmetry group of the harmonic oscillator in three dimensions and it defines the fundamental structure of the shell model. 1 Much effort has been involved to obtain these CGC s. The culmination is the work presented in Refs. 4 where up to recoupling coefficients are given, equivalent to the 6-j and 9-j symbols of SU. 5 The SU3 CGC s are also used in particle physics, where the meaning of SU3 is different and rather related to flavor or color. In order to obtain the CGC s, in the example of SU and SU3 the recursion relations, obtained from the algebraic properties of the Lie algebra, were exploited. The number of these recursion relations is given by the number of generators minus the rank of the group. For SU it is while for SU3 it is 6, still a manageable number. However, for groups like U8, of interest here, this number of recursion relations raises to 56. Thus, alternative procedures are called for. Now, why to use an U8 group? The group U8 appears whenever eight degrees of freedom are involved. Of course, the main motivation is the current need of this group in a particular, though very important, area in physics, namely particle physics. The description of many particle state becomes more important since the recognition that a hadron state is not just a three-quark state for baryons, a quark-antiquark state for mesons or a two gluon state for low-lying glueballs, i.e., there is an additional background of pairs of quark-antiquark and gluons. The spin problem of the nucleon 6 is one hint in this direction. It implies the urgent need to obtain the CGC s in higher rank groups. Simple SU3 coupling coefficients, related, for example, to color and/or flavor, will not suffice. The SU3 CGC is expressed in terms of a product of an SU CGC and an isoscalar factor. This will be similar for U8O8SU3, i.e., the CGC will be expressed in terms of already known SU3 CGC s and isoscalar factors. The U8 group also appears in /006/476/063505/19/$ , American Institute of Physics Downloaded 0 Apr 01 to Redistribution subject to AIP license or copyright; see

3 I. Sánchez-Lima and P. O. Hess J. Math. Phys. 47, many gluon systems. Gluons have eight color degrees of freedom 7 and thus the color part can be described by a U8 group. Because the gluon has spin one it has three mathematical spin degrees of freedom to distinguish them from the physical ones which only involve the transverse modes, i.e., a many gluon system can be described by U4U8 U3, 8 where the U3 refers to the spin part and a many gluon state must be in the complete symmetric irreducible representation N of U4. Because N is symmetric, the color and spin part are intimately connected due to complementarity. 9 The U8 group can be reduced to the color SU3 group and further the U3 to the spin SO3 group, with integer spins only. The group reduction is well known and given in Refs. 8 and 10. The construction of states is less known. For symmetric irreps, the first attempts are presented in Refs. 11 and 1, though, the SU3 subgroup considered in Ref. 1 is not the color group. The symmetric irreps play a particular important role in a system of many quark-antiquark pairs, as in the model presented in Refs , where an effective model of QCD is proposed considering many quark and antiquark states. A general classification for many quark and antiquark states in the s-orbital level was given. In order to simplify calculations, pairs of quarkantiquark were mapped to bosons. 16 There are four different types of bosons corresponding to quark-antiquark pairs with flavor, =0,1 and spin S S=0,1 denoted by,s. The cases 0,0 and 0,1 correspond to a one- and three-dimension harmonic oscillator, known from textbooks. The case 1,0 corresponds to the eight-dimensional harmonic oscillator, i.e., to the U8 group with just a symmetric irrep. The last case 1,1 is mathematically identical to the many gluon problem and allows up to three rows in the Young diagram of U8. The same structure appears in any model whose basic ingredients are quark-antiquark and/or gluon pairs. Restricting to symmetric irreps, coupling coefficients are still important in obtaining information on decay properties and for the coupling of two systems. Though, so far we mainly mentioned bosonic systems, the isoscalar factors, calculated in this contribution, also serve for the equivalent fermion pairs. Another motivation is to decide whether the pentaquark 17 1 exists in the model or not. A first estimate within the schematic model, where we have information about the distribution of quarkantiquark pairs in the pentaquark and in the residual particles, indicates that the pentaquark is just the sum of a nucleon and a kaon or at most a molecule in these particles, i.e., the width of the state should be very large and a peak should not be seen, confirmed in part by other experiments and also criticized in Ref. and references therein. If this is the case, it can only be decided through an explicit calculation, using U8 CGC s. Of course, the immediate application to topics in particle physics is only one possible application, used here as an example. Does one also need the explicit form of the states defined by the U8 group? The answer is that they are necessary for the calculation of CGC s. In the calculation of CGC s one usually exploits the algebraic properties of a group obtaining recursion relations, as illustrated in Ref. 5 for the SU group and in Refs. 4 for the SU3 group. However, for higher rank groups these methods get more involved and unpractical. In Refs. 3 and 4 a more practical procedure was proposed for the U5SO5SO3, playing an important role in the geometric model of the nucleus. 1 There, the polynomial expressions of the U5 states were constructed explicitly using elementary tensors in terms of boson creation operators. The Clebsch-Gordan coefficients were obtained by direct calculation of the integrals involving the polynomial states, with the help of algebraic routines. The basic idea for the construction of the polynomials were borrowed from Refs. 5 and 6. In conclusion, the explicit knowledge of the polynomial states is of great use. Thus, a first and important step forward towards the construction of many particle states is the explicit construction of these states in the symmetric irrep of U8. The CGC s are obtained as integrals over a product of three polynomials. The first steps of the procedure were presented in Ref. 7 where the basic ingredients are illustrated for the case of the well investigated group chain SU3SO3. In this contribution, explicit expressions of the polynomial states up to eight particles and tables of U8 isoscalar factors are presented, which are useful for models treating, for example, Downloaded 0 Apr 01 to Redistribution subject to AIP license or copyright; see

4 Clebsch-Gordan coefficients for U8 J. Math. Phys. 47, TABLE I. Content of SU3 irreps for a given seniority. When a SU3 irrep appears more than once multiplicity larger than 1 it is indicated by an upper index., 0 0,0 1 1,1 1,1+, 3 0,0+3,0+0,3+,+3,3 4 1,1+,+4,1+1,4+3,3+4,4 5 1,1+,+4,1+1,4+3,3+5,+,5+4,4 +5,5 6 0,0+3,0+0,3+,+3,3 +6,0+0,6+5, +,5+4,4+ 6,3+3,6+5,5+6,6 7 1,1+,+4,1+1,4+3,3+5,+,5+4,4 +7,1+1,7+ 6,3+3,6+5,5+7,4+4,7+6,6+7,7 8 1,1+,+4,1+1,4+3,3+5,+,5+4,4 +7,1+1,7+ 6,3+3,6+5,5 +8,+,8+7,4+4,7+6,6 +8,5+5,8+ 7,7+8,8 many gluon and fermion pair systems, as outlined above. These tables are of great use in determining the structure and the decay properties of a many particle state, like hadrons or a system of many quarks, antiquarks and gluons, and the quark-gluon plasma. The paper is structured as follows: In the second section we give the classification and the explicit expressions of the states up to eight particles in the symmetric representation of U8O8SU3SU U1 in terms of the polynomials in boson creation and annihilation operators. We show how to obtain states with good seniority. In Sec. III the isoscalar factors are calculated. In Sec. IV conclusions are drawn. II. THE HIGHEST WEIGHT STATES IN SU 3 OF U 8 O 8 SU 3 SU U 1 The particles under consideration have eight degrees of freedom and belong to the SU3 irrep 1,1. For symmetric irreps, the relevant group chain is U8 O8 SU3 U1 SU N 000, Y T, T z, 1 where Y is the hypercharge, T the isospin, and T z its third component. The N is the total number of bosons, the seniority number of bosons not coupled in pairs with flavor 0,0. The CGC s of the chain SU3U1 SU are well known and available. 8,9 The classification of the states, described by the group chain U8SU3, is immediately obtained, using Ref. 10. The reduction of U8 to O8 is obtained recursively: For a given N the possible values of the seniority are =N,N,...,0 or 1. When the content up to a given seniority =N is known, the one for =N+ is obtained, determining the content of the irrep N+ of U8 and subtracting the SU3 content of all seniorities up to =N. The SU3 content of O8 for =0 is 0,0 and for =1 it is 1,1, which can be used as initial conditions. In Table I we give the list up to seniority eight. The generators of U8 are given by Downloaded 0 Apr 01 to Redistribution subject to AIP license or copyright; see

5 I. Sánchez-Lima and P. O. Hess J. Math. Phys. 47, TABLE II. Linearization of the index YTT z to. Y T T z C YTT3 = b b YTTz, where is a short-hand notation for, and is a multiplicity label which is 1 except for the irrep =1,1 where it obtains values 1 and. For this case, the value =1 refers to the antisymmetric coupling and = to the symmetric coupling. 30 The possible values of are 0,0,,, and 1,1 for the symmetric and 3,0, 0,3, and 1,1 1 for the antisymmetric coupling. The index YTT z can be linearized to as defined in Table II, which is of use in the further discussion. The generators of O8 are given by the coupling to the antisymmetric irreps, implying 8 generators for the algebra of O8. The generators of the SU3 subgroup are obtained, restricting the to 1,1 1. The Lie algebra for the generators is given by C,C = 1 sign 0, 0 +sign,+, max sign,+sign,+sign 0, 0 0, 0 max max 0, max 0, 0 0 =1,,, 0, 0 0 max 1 max dim, 8 U 0, 0 1,1,1,1;1,1 0,b b 0 0, The 0, 0 max refers to the multiplicity in the coupling 1,1 1,1 0, 0, the, max to the multiplicity in the coupling 1,1 1,1,, max to,, 0, 0, sign, to the symmetry property of the 1,1 1,1, under exchange of the first two irreps in the GCG symmetric or antisymmetric and U is the U-coefficient, whose definition can be found in Ref. 30. As a particular case, the algebra for the subgroup SU3, which is a subgroup of O8, is given by C 11,1,C 11,1 = 1,1,1,11,1 0 1 b 1,1 b In terms of the standard notation of the SU3 generators, 31 the relation to the C 11,1 of Eq. is T 0 =T z T ± =± 6b 11,1 b01±1, T 0 = 3b 11,1 b010, V ± =± 6b 11,1 b±11/±1/, U ± = 6b 11,1 b±11/1/, 5 Downloaded 0 Apr 01 to Redistribution subject to AIP license or copyright; see

6 Clebsch-Gordan coefficients for U8 J. Math. Phys. 47, Y = b b 11, In the next step we express an arbitrary tensor of U8 in terms of a product of powers of elementary tensors, which are called elementary couplings epd s 5,6 sometimes also called integrity basis. The origin of this name is that any tensor of a given order can be expressed in terms of products of powers in these epd s. The epd s are basic tensors each coupled to the maximum weight in SU3. Because these elementary tensors are of maximum weight in SU3, any product of two or more of these tensors result again in a tensor of maximum weight in SU3, in the same way two tensors with spin j 1 and j with maximal projection, i.e., m 1 = j 1 and m = j, result in a new tensor with spin j= j 1 + j and projection m= j. Products of powers in these epd s generate U8 states with quantum numbers N and, in the maximum weight state of SU3. One must take care that the list of epd s is complete, checking if all irreps of U8 can be written in terms of them. If one state is missing, a further epd must be added. In Ref. 11 a procedure is developed how to obtain these epd s using generating functions. Please consult this reference for more details and further references. This method was applied successfully to various areas of physics, like the geometric collective model 1,5,6 and in Ref. 11 more possible applications are given, like the problem we are interested in this contribution. In Ref. 11 the following epd s sometimes denoted as integrity basis where obtained using the method of generating functions see Eq. 16 of Ref. 11. The epd s are given by A = b 011, B = b b 0,0 000, C = b b 1,1 011, D = b b b 1,1 0,0 000, 6 E = b b b 1,1 3,0 13/3/, F = b b b 1,1 0,3 13/3/, where the coupling of two boson creation operators must be always symmetric, otherwise it would give zero due to the symmetry properties of the CGC. These are expressions obtained by coupling the definite SU3 tensors b to new tensors using the SU3 CGC s. A pedestrian way to look at it, is to ask: How many basic tensors one needs to construct all possible state polynomials which are in the maximum weight in SU3? Table I is here for assistance: For a given number of bosons N the SU3 content is obtained by summing the seniority content for =N,N,..., 0 or 1. For N=1 we need a tensor with the SU3 structure 1,1, which is just A. However, powers of the type A n 1 gives only polynomials with the SU3 irrep labels n 1,n 1. Inspecting Table I we see that for N= the SU3 content is 0,0+1,1 +,, implying that we also need tensors with the SU3 labels 0,0 and 1,1, which gives us the tensors B and C. The irrep, is presented by A. Powers and products ofa, B, and C are still not sufficient: For N=3 the SU3 content is given by 0,0+3,0+0,3+1,1+, +3,3. The states with 3,3,,, and 1,1 are, respectively, presented by A 3, AC, and AB. We note the need of tensors with SU3 labels 0,0, 3,0, and 0,3, which are just the D, E, and F tensors. Continuing to larger N one notes that all other polynomial states are obtained by the product of powers in these basic tensors. An ambiguity, however, arises when the product of the tensors E with F or powers of C 3 are considered. For example, for N=6 and the SU3 irrep 6,3 there are the following possible representations, namely FA 3 and EA 3, though, 6,3 appears only once. This will be discussed in what follows. Downloaded 0 Apr 01 to Redistribution subject to AIP license or copyright; see

7 I. Sánchez-Lima and P. O. Hess J. Math. Phys. 47, A maximum weight state in SU3 is a monomial in these epd s, where according to Ref. 11 the epd C appears only in powers of 0, 1 or, i.e., the C 3 can be expressed as a linear combination in the other epd s. Instead of choosing C 3 as dependent, the E and F can be used as dependent epd s. These two epd s satisfy a relation which permits the appearance of powers in E or F only. The relation of the product EF to the other epd s is given by EF = C A BC 15 A 3 D, 7 i.e., any product of EF can be expressed in terms of the other epd s, implying that either only powers in E or F may appear. This relation can also be used to express third powers of C in terms of the other epd s, as is suggested in Ref. 11. We take the choice 7 as also was done in Ref. 1. Choosing E and F as dependent epd s, there are two types of polynomials, one with +3k, and the other with,+3k, respectively, i.e., E n 5D n 4C n 3B n A n 10, F n 5D n 4C n 3B n A n 10, 8 which now covers all possible irreps of U8, as proven in Ref. 11. Fixing the total number of quanta N, the and for the first case only powers of E appear, we obtain the following relation between the powers of the monomial: N = n 1 +n +n 3 +3n 4 +3n 5, = n 1 + n 3 +3n 5, 9 = n 1 + n 3. These relations are obtained, applying the number operator to the polynomial state, taking into account the order of the tensors in the creation operators and by noting that the tensors A, C are in the maximum weight and transform as 1,1, while E transforms as 3,0, which is also in the maximum SU3 weight. Similar relations hold when only powers of F appear N = n 1 +n +n 3 +3n 4 +3n 5, = n 1 + n 3, 10 = n 1 + n 3 +3n 5. Up to now, the polynomials have no definite seniority, the quantum number of the O8 group. This is achieved requiring that the application of 0,0 B = b b on a polynomial in terms of the monomials of Eq. 8, gives zero no pairs are contained, see, e.g., Refs. 5 and 6. This leads to the polynomials which have a seniority =N. Explicitly, the condition reads B P N=, A,B,C,D,EF0 =0, 1 where the polynomial has the following structure for the case when only powers of E appear P 1 = E /3 n 1 n c n1 n A n 1B n C n 1D N +n 1 n /3, 13 and similar when only powers of F appear Downloaded 0 Apr 01 to Redistribution subject to AIP license or copyright; see

8 Clebsch-Gordan coefficients for U8 J. Math. Phys. 47, TABLE III. List of the polynomial coefficients with definite seniority for =0 until =4. For the coefficients only numerical values are given. They can be expressed in terms of square roots of rational numbers. However, for larger N there are cases where this cannot be done see comments in text, which is the reason for this presentation. The powers of all epd s can be deduced using Eqs. 13 and 14, the value of the SU3 irrep and the numbers of n 1, n listed., n 1 n State No. k c n1 n 0 0, , , , , , , , , , , , , , , , , P = F /3 n 1 n c n1 n A n 1B n C n 1D N +n 1 n /3. 14 Each of these polynomials we will abbreviate by P = n 1 n c n1 n n 1 n, 15 where we omitted the labels, and applied the operators on the vacuum, resulting in a ket-state. Applying from the left n 1 n B, with n 1 +n +1=n 1 +n and the same, on both sides, we arrive at the defining equation for the coefficients c n1 n, n 1 n c n1 n n 1 n +1n 1 n =0. 16 The overlaps n 1 n +1n 1 n in the sum are calculated. How to determine the overlaps is described in more detail in Sec. III. Having obtained the overlaps, the above equation is solved numerically, leading in this way to maximal weight states in SU3 with good seniority and N=. The state with N is depicted below in Eq. 3. In Tables III VIII we give a list of all polynomials up to eight bosons, with the additional restriction of N=. In case there is more than one state, an additional multiplicity index is assigned, called k. Only numerical values are given, though, most of the numbers listed in the tables can be expressed as square roots of rational numbers. However, when more than one state for a given seniority and SU3 irrep, appears, a Schmidt orthogonalization procedure is applied, yielding only simple numbers for the first state but in general real numbers for states with a higher index in k. In calculations within the models of Refs. 8, 13, and 14 no more than six bosons are needed. Up to now, only highest weight states in SU3 were considered. In the highest weight of the irrep, the value of the hypercharge and the third component of the isospin is given by Downloaded 0 Apr 01 to Redistribution subject to AIP license or copyright; see

9 I. Sánchez-Lima and P. O. Hess J. Math. Phys. 47, TABLE IV. The same as in Table III, but now for 5., n 1 n State No. c n1 n 5 5, , , , , , , , , , , , /3 and +/, 31 respectively. A state with a lower weight can be reached via the application of T U + V. When the weight of a state is given by Y,T and the difference to the highest weight is T z, for the third component of the isospin, and +Y for the hypercharge, the exponents,, and fulfill the relations T z = + 1 +, Y =. 17 As can be seen, there is in general more than one possibility. The final state reached has only a definite isospin when or is zero. In general, the construction of the states with definite isospins, for a fixed weight Y and T z must still be performed. For the construction of states with a definite isospin, the overlap of the states TABLE V. The same as in Table III, but now for 6., n 1 n State No. c n1 n 6 6, , , , , , , , , , , , , , , , , , , , , , , Downloaded 0 Apr 01 to Redistribution subject to AIP license or copyright; see

10 Clebsch-Gordan coefficients for U8 J. Math. Phys. 47, TABLE VI. The same as in Table III, for 7., n 1 n State No. c n1 n 7 7, , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , Nk, = T U + V hw = 18 must be determined, where hw is an abbreviation for highest weight and in the last line a short-hand notation for the state is given. This overlap is directly obtained using the commutation properties of the operators involved. The final result is!!!!!! + +! +! =! +! +! min,, k=max0, + k! + k! k! k! k! + + k! + k! k!. 19 Downloaded 0 Apr 01 to Redistribution subject to AIP license or copyright; see

11 I. Sánchez-Lima and P. O. Hess J. Math. Phys. 47, TABLE VII. The same as in Table III. Partial list for 8., n 1 n State No. c n1 n 8 8, , , , , , , , , , , , , , , , , , , , , , , , , , States with definite isospin are obtained by diagonalizing the operator T =T + T +T z T z 1 within the states given above. Note, however, that are not orthogonal with respect to different values of,, and. The overlap is given in 19. The most practical method is to solve numerically the equation T a = TT +1 a, 0 where the coefficients a are the expansion coefficients in the basis. The matrix elements of T + T are obtained through the overlaps +1 +1, 1 while T z is diagonal. The state with definite isospin is then given by YTT N,YTT z = a z N,/, with the constriction on,, and given by Y =+/3+ and T z = / 1 +. Downloaded 0 Apr 01 to Redistribution subject to AIP license or copyright; see

12 Clebsch-Gordan coefficients for U8 J. Math. Phys. 47, TABLE VIII. The same as in Table III. Partial list for 8., n 1 n State No. c n1 n 8 4, , , , , , , , , , , , , , , , , , , , , , , Care must still be taken with respect to the phase. The lower weight states are obtained via the application of T, U +, and V. For example, when only a power of the T operator is applied to the maximal weight state, we arrive at 00 with a positive sign in front and the same isospin as in the maximum weight state. Similar, parting from the highest weight state, the application of only U + or only V leads to a state with definite isospin + / for =0 or + / for =0. Posterior application of T leads to states with the same isospin. Therefore, we require that the component with either or equal to zero, within a state of definite isospin, has also a positive sign in front of 0 for =0 or 0 for =0. In case the numerical program produced states with an opposite sign, we corrected for it by multiplying all coefficients by 1. III. THE ISOSCALAR FACTORS AND CLEBSCH-GORDAN COEFFICIENTS Once the states are obtained, we can proceed in the calculation of the CGC s. Before that, a general remark on the Wigner-Eckart theorem for a given group chain GH is due: When represents the label of the group G and of its subgroup H and T is a tensor, then the matrix elements can be expressed as 9 T 0 0 =, 0 0 T 0, where is a multiplicity index in the coupling 0, the first factor in the sum over is the CGC and the last one the reduced matrix element. In order to distinguish the reduced matrix element of a group G different from SU, often the notation of multiple reduced matrix element is used. Like in SU3 Ref. 30 where the reduced matrix element of SU3 is called triple reduced matrix element. Here we will continue to use this notation for SU3 and for the reduced matrix element of U8O8SU3 the name of quadruple reduced matrix element is 3 Downloaded 0 Apr 01 to Redistribution subject to AIP license or copyright; see

13 I. Sánchez-Lima and P. O. Hess J. Math. Phys. 47, proposed. The sum over the multiplicity index is only applied when the multiplicity is larger than one. Now we will return to the determination of the U8O8SU3 CGC s and its isoscalar factors. In a first step we determine the triple reduced matrix elements with respect to the SU3 group. 30 The equation to solve is N 3 3 k 3 3, 3 Y 3 T 3 T 3z P N1 1 k 1 1, 1 Y 1 T 1 T 1z b N k, Y T T z =, Y T T z ; 1, 1 Y 1 T 1 T 1z 3, 3 Y 3 T 3 T 3z N 3 3 k 3 3, 3 P N1 1 k 1 1, 1 b N k,. 4 Let us suppose that we have the result concerning the first line in Eq. 4. How to obtain it will be explained further below. The SU3 Clebsch-Gordan coefficients are well known 9 and, thus, 4 represents an equation to determine the triple reduced matrix elements. It suffices to use maximal weight states for the polynomial 1 and 3, while the weight of the second polynomial is given by the difference of the weight of the third with the first polynomial. In the next step, the Wigner-Eckart theorem for the U8O8SU3 group chain is used, i.e., N 3 3 k 3 3, 3 Y 3 T 3 T 3z P N1 1 k 1 1, 1 Y 1 T 1 T 1z b N k, Y T T z = N k, Y T T z ;N 1 1 k 1 1, 1 Y 1 T 1 T 1z N 3 3 k 3 3, 3 Y 3 T 3 T 3z N 3 P N1 b N. There is no sum over a multiplicity index because the labels of the U8 irrep is the number N, given here in the notation of a Young diagram, and the multiplication of two symmetric irreps is always free of multiplicities. Comparing Eq. 5 with Eq. 4 we arrive at 5 N k, Y T T z ;N 1 1 k 1 1, 1 Y 1 T 1 T 1z N 3 3 k 3 3, 3 Y 3 T 3 T 3z =, Y T T z ; 1, 1 Y 1 T 1 T 1z 3, 3 Y 3 T 3 T 3z N 3 3 k 3 3, 3 P N1 1 k 1 1, 1 b N k, N 3 P N1 b. 6 N The CGC of the SU3 group are known, thus, it suffices to determine N k,,n 1 1 k 1 1, 1 N 3 3 k 3 3, 3 = N 3 3 k 3 3, 3 P N1 1 k 1 1, 1 b N k, N 3 P N1 b, 7 N which we will denote as isoscalar factors of the group chain U8O8SU3. Equation 7 implies that the triple reduced matrix element with respect to the group chain SU3SU U1 and the quadruple reduced matrix element of U8O8 must be determined. The last is the simplest one. For that, the U8 CGC s are calculated, assuming the seniority k in all polynomials equal to N k then, the CGC is just 1. The polynomials have the form Downloaded 0 Apr 01 to Redistribution subject to AIP license or copyright; see

14 Clebsch-Gordan coefficients for U8 J. Math. Phys. 47, P k = 1 Nk! AN k0, 8 with k=1,,3. The result is N 3 P N 1 N = N 1/ 3!. 9 N 1! N! The triple reduced matrix elements of the group chain SU3SU U1 are obtained through the use of Eq. 4, as already explained. The important part is the calculation of the left-hand side of Eq. 4. In a first step the overlaps n 1 n n 1 n 1, as defined in Eqs. 15 and 16, are determined. For that, we write the explicit form of the epd s in terms of the boson creation operators. To simplify notations, we rename a boson creation operator b by a coordinate x and the corresponding annihilation operator b by a derivative /x. Both satisfy the same commutation relations with this, it is easier to translate it to an algebraic routine, like MATEMATICA, 3 which assists in the evaluation of the overlaps. In terms of this new notation and using the association of the linear index to YTT z see Table II, the epd s of U8 have the following structure: A = x 1, B = 1 8 x 4 x 1 x 8 + x 5 x x 7 +x 3 x 6, C = 3 10 x 3x 5 x 1 x 5, D = x 1 x 6 x 7 + 3x1 x 5 x x 3x 5 x 6 3 x x 5 x x x 4 x 7 3 x x 3 x x 3x 4 x 6 3 x 4 x x 5 3, 30 E = 3 5 x x x 1 x x x 1 x x 1x x 4, F = 3 5 x 1 x x 1x 3 x x 3 x x 1 x 3 x 5. These expressions are obtained using the explicit values of the SU3 CGS s, as obtained in Refs. 4. One can easily verify that they are in the maximum weight of the corresponding SU3 label see Eq. 6. The Hermitian conjugate expressions are obtained by changing x to /x. What we also need is the explicit form of the lowering operators T, U +, and V, in order to obtain a lower weight state in what we call the second polynomial. Starting from Eq. 5 and using the explicit form of the SU3 CGC s, we obtain T = x8 x 4 + x4 x 1 + x 6 x + x 7 x 3, Downloaded 0 Apr 01 to Redistribution subject to AIP license or copyright; see

15 I. Sánchez-Lima and P. O. Hess J. Math. Phys. 47, U + = 3 x x x x 8 1 x x 3 x 7 6 x x 6 x, x 5 x 1 31 V = 1 x 4 x + 3 x 5 + x 8 + x 3 x x 6 x x x x 5 7. x 4 For the overlaps, we must apply these lowering operators to a highest weight state in SU3 of what we called the second polynomial, as given in Eq. 18, using the norm depicted in Eq. 19 and multiply it with what we called the first polynomial in the highest weight state. That is, we must construct P N1 1 k 1 1, 1 Y 1 max T1 max T1 maxx P N k, Y T T z x. This gives us an expression which depends on the x only. On that we must apply the conjugate expression of what we call the third polynomial, which is a pure function of derivatives, i.e. P N3 3 k 3 3, 3 Y 3 max T3 max T3 max/x. In principle, this quite involved calculation can be done by hand. It is advisable to take the assistance of an algebraic routine like MATEMATICA, 3 considering that the number of overlaps to calculate is in the thousands. With this help, we finally obtain the overlap on the left-hand side of Eq. 4. Not all possible matrix elements were calculated but only those with N 1 = 1, N = and, thus, N 3 = 1 + and N 3 3. The corresponding isoscalar factors of Eq. 7 were determined. In order to obtain all other possible isoscalar factors, with N k k k=1, and N 3 =N 1 +N, we note that the corresponding state can be written as N k k k = k +3! 4 N k k / N k k! N k+ k +6 1/! b b N k k / k k, where now k denote all other quantum numbers k, k Y k T k T kz, k k is the state with N k = k and b b is the scalar product between the boson creation operators. Introducing this on the left-hand side of Eq. 5 leads to 3 N P N1 1 1 b N = 1 +3! +3! N 3 3! N ! N 1 1! N! N ! N + +6! 1 + 3! N 3 = 1 +, 3 3 P N1 = b N =,. Using 5, we arrive at the following relations of CGC s:!1/ 33 N,N N = 1 +3! N 1! N 1 1! 1! N ! 1 +! 1 + 3! N!!!1/ N + +6! +3! N! N 3 3! N ! N =,,N 1 = 1, 1 1 N 3 = 1 +, The same holds for the isoscalar factors, which can be verified using Eq. 6. Programs for all the steps mentioned are available and can be handed over on request. In Table IX a partial list of the isoscalar factors for up to eight bosons is given, involving only the SU3 irreps 0,0, 1,1, 3,0, 0,3 and restricting to 1 for 1 we use the symmetric property of the isoscalar factors under permutation of the first two irreps. Irreps with a higher value were calculated, too, but are not tabulated. In total there are about isoscalar factors calculated involving states up to eight particles. Here, we only give a partial list containing, as we think, the isoscalar factors of most interest. These involve the SU3 irreps 0,0, 1,1, 3,0, N 3! Downloaded 0 Apr 01 to Redistribution subject to AIP license or copyright; see

16 Clebsch-Gordan coefficients for U8 J. Math. Phys. 47, TABLE IX. Partial list of isoscalar factors denoted by IF for the group chain U8O8SU3. Only those are listed with k =N k k=1, and 1, thus, the N 3 = 1 +. For this reason, the value of N 3 is not listed. Equation 34 must be applied in order to obtain the isoscalar factors for N 1 1, N and N 3 =N 1 +N. k, 1 k 1 1, 1 3 k 3 3, 3 IF 0 1 0, , , , , , , ,1 1 1, , ,1 1 1, , , , ,0 1 1,1 1 1, ,1 1 1, , ,1 1 1, , ,1 1 1, , ,1 1 1, , ,1 1 1, , ,1 1 1, , ,1 11, , ,1 1 1,1 1 1, ,1 1 1,1 1 1, ,1 1 1, , , , , , , , , , , , , , , , , , , , , ,0 1 1, , ,3 1 1, , ,0 1 1, , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,0 1 1, , ,3 1 1, , , , , , , , , , , , ,1 1 1 Downloaded 0 Apr 01 to Redistribution subject to AIP license or copyright; see

17 I. Sánchez-Lima and P. O. Hess J. Math. Phys. 47, TABLE IX. Continued. k, 1 k 1 1, 1 3 k 3 3, 3 IF 1 1 1, , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,1 1 1, , ,1 1 1, , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,1 1 1, , ,1 1 1, , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , Downloaded 0 Apr 01 to Redistribution subject to AIP license or copyright; see

18 Clebsch-Gordan coefficients for U8 J. Math. Phys. 47, TABLE IX. Continued. k, 1 k 1 1, 1 3 k 3 3, 3 IF 3 1 3, ,1 1 1, , ,1 1 1, , ,1 1 1, , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,1 1 1 and 0,3. For example, when a many quark-antiquark state with one additional gluon is considered, the quark state must be in a color 1,1 irrep as the gluon. Likewise, when one quarkantiquark pair is coupled to a color or flavor octet, the other pair must be in the same state not necessarily for the flavor case. More complicated isoscalar factors are available on request. Next we explain the tests performed in order to assure that the results are of confidence. First of all, we calculated by hand many of the overlap matrix elements of the polynomial No. 3 with the product of the polynomials with Nos. and 1 and compared them to the values obtained via the MATEMATICA code. This we did up to eight bosons choosing arbitrarily the states. Then we checked if the isoscalar factors obtained satisfy the orthogonality condition. We use the orthogonality relation of the U8 CGC s, i.e., i k i i N k,n 1 k 1 1 N 3 3 k 3 3 N k,n 1 k 1 1 N 3 3 k 3 3 = N3,N 3 3, 3 3, 3, 35 where the index i is1or. We arrive at the following condition for the isoscalar factors, given in 7, N k,,n 1 1 k 1 1, 1 N 3 3 k 3 3, 3 i k i i, i N k,,n 1 1 k 1 1, 1 N 3 3 k 3 3, 3 = N3 N k 3 k This orthogonality relation we have checked throughout the range from zero to eight bosons. In total about 350 orthogonality relations, involving several thousand isoscalar factors, were checked. Downloaded 0 Apr 01 to Redistribution subject to AIP license or copyright; see

19 I. Sánchez-Lima and P. O. Hess J. Math. Phys. 47, IV. CONCLUSIONS In this contribution we have constructed the isoscalar factors for the group chain U8O8SU3. Only the totally symmetric irreps N of U8 were taken into account. This is of use in any model/theory which deals with quark-antiquark and/or gluon pairs, as in the model published in Refs The CGC s will be used to calculate transition probabilities and can be used in any other model involving quark-antiquark and/or gluon pairs, like the background in hadron states. The coefficients are also of importance for many gluon systems, restricting to the completely symmetric irreps of U8, which lie at lower energies. The method presented is very practical for high rank groups when only few particular irreps are of interest and thus serves as an example. It is more powerful than traditional methods, which use the whole algebraic structure of the group and all recursion relations possible. For many gluon systems, up to three rows in the Young diagrams are needed, i.e., h 1,h,h 3. Therefore, the next step is to consider two rowed Young diagrams. Steps in this direction are already taken by the authors. We presented the procedure on how to obtain the Clebsch-Gordan coefficients of the chain U8O8SU3U1 SU for symmetric irreducible representations in U8, through the use of isoscalar factors. The importance of the Clebsch-Gordan coefficients of the chain starting with U8 lies not only in the possibility to obtain, via their use, branching ratios of hadron decays involving gluons and quark-antiquark pairs, but it can also be used in any other problem related to a U8 group, i.e., eight degrees of freedom, not necessarily in particle physics, though this was the main motivation. Another area where the U8 group could play a role, though probably still far in the future, is in quantum computing, 33 related to cyclic networks of quantum gates with three-qubits as elementary structure. Trying to describe a system of many three-qubits will require U8 CGS s, though, one must still understand the basic three-qubit structure alone. ACKNOWLEDGMENTS Financial help from DGAPA, Project No. IN10806, and CONACyT is acknowledged. 1 J. M. Eisenberg and W. Greiner, Nuclear Theory I: Nuclear Models North-Holland, Amsterdam, J. D. Vergados, Nucl. Phys. A 111, J. P. Draayer and Y. Akiyama, J. Math. Phys. 14, D. J. Rowe and C. Bahri, J. Math. Phys. 41, A. R. Edmonds, Angular Momentum in Quantum Mechanics Princeton University Press, Princeton, NJ, H.-Y. Cheng, Int. J. Mod. Phys. A 11, K. Huang, Quarks, Leptons and Gauge Fields, nd ed. World Scientific, Singapore, P. O. Hess, S. Lerma, J. C. López, C. R. Stephens, and A. Weber, Eur. Phys. J. C 9, J. P. Elliott and P. G. Dawber, Symmetry in Physics Oxford University Press, New York, 1979, Vols. 1 and. 10 R. López, P. O. Hess, P. Rochford, and J. P. Draayer, J. Phys. A 3, L R. Gaskell, A. Peccia, and R. T. Sharp, J. Math. Phys. 19, and references therein. 1 E. Chacón, in Anales de Física. Monografías, edited by M. A. Olmo, M. Santander, and J. Mateos Guilarte, CIEMAT, Proceedings of the XIX International Colloquium Salamanca, Spain, 199, p S. Lerma, S. Jesgarz, P. O. Hess, O. Civitarese, and M. Reboiro, Phys. Rev. C 67, S. Jesgarz, S. Lerma, P. O. Hess, O. Civitarese, and M. Reboiro, Phys. Rev. C 67, M. Nuñez, S. Lerma, P. O. Hess, S. Jesgarz, O. Civitarese, and M. Reboiro, Phys. Rev. C 70, A. Klein and E. R. Marshalek, Rev. Mod. Phys. 63, T. Nakano et al., Phys. Rev. Lett. 91, V. V. Barmin et al. DIANA collaboration, Phys. At. Nucl. 66, S. Stepanyan et al. CLAS collaboration, Phys. Rev. Lett. 91, J. Barth et al. SAPHIR collaboration, Phys. Lett. B 57, A. E. Asratyan, A. G. Dolgolenko, and M. A. Kubantsev, Phys. At. Nucl. 67, A. R. Dzierba, D. Krop, M. Swat, S. Teige, and A. P. Szczepaniak, Phys. Rev. D 69, D. J. Rowe, P. S. Turner, and J. Repka, J. Math. Phys. 45, D. J. Rowe, Nucl. Phys. A 735, E. Chacón, M. Moshinsky, and R. T. Sharp, J. Math. Phys. 17, E. Chacón and M. Moshinsky, J. Math. Phys. 18, I. Sánchez Lima and P. O. Hess, Rev. Mex. Fis. 5, Y. Akiyama and J. P. Draayer, Comput. Phys. Commun. 5, Downloaded 0 Apr 01 to Redistribution subject to AIP license or copyright; see

20 Clebsch-Gordan coefficients for U8 J. Math. Phys. 47, C. Bahri, D. J. Rowe, and J. P. Draayer, Comput. Phys. Commun. 159, J. Escher and J. P. Draayer, J. Math. Phys. 39, and references therein. 31 W. Greiner and B. Müller, Quantum Mechanics: Symmetries Springer, Heidelberg, MATEMATICA-5 package, Wolfram Research, P. Cabauy and P. Benihoff, Phys. Rev. A 68, Downloaded 0 Apr 01 to Redistribution subject to AIP license or copyright; see