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Fundamental fermion masses from deformed SU

q

**( 2 ) triplets (*)**

B. E. PALLADINOand P. LEALFERREIRA

Instituto de Física Teórica, Universidade Estadual Paulista Rua Pamplona 145, 01405-900 Sa˜o Paulo, SP, Brazil (ricevuto il 15 Gennaio 1997; approvato il 28 Aprile 1997)

Summary. — A spectrum-generating q-algebra, within the framework of SUq( 2 ), as

firstly suggested by Iachello, is studied in order to describe the mass spectrum of three generations of quarks and leptons. The SUq( 2 ) quantum group is a q-deformed

extension of SU( 2 ), where q 4ea_{(with a real) is the deformation parameter. In this}

work, the essential use of inequivalent representations of SUq( 2 ) is introduced. The

inequivalent representations are labelled by ( j , n0), where j 40, d, 1, R and n0is a

positive real number. A formula for the fermion masses Mm( j , n0), with 2jGmGj

is derived. As an example, a possible scheme which corresponds to two triplets ( j 41) associated to up and down quarks is presented here in some detail. They are associated to different values of the deformation parameter, indicating a dependence of the charge Q on the parameter a . The masses of the charged leptons are treated in a similar way. The current results show that some mass relations for quarks and leptons found in the literature can be considered as approximations of the equations obtained in the j 41 representations. The breaking of SUq( 2 ) necessary to describe

the Cabibbo-Kobayashi-Maskawa (CKM) flavor mixing is briefly discussed. PACS 12.15.Fg – Quark and lepton masses and mixing.

PACS 11.30.Na – Nonlinear and dynamical symmetries (spectrum-generating sym-metries).

1. – Introduction

In the algebraic framework of spectrum-generating Lie algebras (SGLA) and associated dynamic symmetries (DS) introduced in physics in the last two decades the problem of classifying the spectra of hadrons [1], nuclei [2] and molecules [3] has achieved considerable success. We wish in this work to address the classification problem of the fundamental fermions (quarks and leptons) of the Standard Model (SM) in a similar algebraic context. As first suggested by Iachello [4], the spectrum of quark masses can be analysed based on a deformed q-algebra, namely that of SUq( 2 ) [5], with

(*) The authors of this paper have agreed to not receive the proofs for correction.

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q, the deformation parameter of the algebra, defined as q 4ea_{, for real a . We}

emphasize that the SUq( 2 ) here assumes the unique role of a classification group, in

the context of a spectrum-generating deformed algebra (SGDA) similar to that of SGLA.

In algebraic language the present approach may be represented by the chain

SUq( 2 ) & SOq( 2 ) .

(1.1)

Taking into account the corresponding Casimir operators of SUq( 2 ) and of its

subalgebra SOq( 2 ) this suggests the mass spectrum

M 4AC

(

SOq( 2 )

)

_{1 BC}

(

SUq( 2 )

)

, (1.2)

where A and B are constants to be determined for each multiplet associated to a irreducible representation (irrep) of SUq( 2 ). In order to describe flavor mixing as

predicted by the SM, additional terms in the right-hand side of eq. (1.2) that break the

SUq( 2 ) symmetry are required. In ref. [4] a term of the form CJy2 (C an additional

parameter) was proposed for that purpose, leading to the CKM matrix elements. This procedure is reminiscent to the breaking of SU( 3 )F symmetry necessary to generate

Gell-Mann–Okubo mass formula in the “eightfold way” [1].

We note that in the present treatment an important role is played by the

non-equivalent (or inequivalent) irreps of SUq( 2 ) [6]. The inequivalent representations

will be labelled by the parameters ( j , n0), where j 40, d, 1, R and n0is a positive real

number. For n0K 0 we recover the usual (“fundamental”) irreps of SUq( 2 ) [5].

Following ref. [6] the following representations were obtained:

.

/

´

a1 Nnb 4 q2n0/4_{[n 11]}1 /2 q Nn 1 1 b , aNnb 4qn0/4_[n]1 /2 q Nn 2 1 b , NNnb 4 (n1n0) Nnb , n 40, 1, 2, R (1.3)

with a Casimir operator

CNnb 4q2n0/2_[n 0]qNnb

(1.4)

and with the square bracket defined as [x]q4

qx/2

2 q2x/2

q1 /2_{2 q}21 /2 .

(1.5)

This definition for the q-numbers coincides with that of Biedenharn in ref. [5] and Iachello in ref. [4]. Notice that a different definition of [x] using q instead of q1 /2 _is

adopted, for example, by Mac Farlane in ref. [5] and by the authors in ref. [6]. In any case, notice that [x]q4 [x]q21.

The mass formula for SUq( 2 ) associated to the chain (1.1) is discussed in the next

section, followed by examples of fittings of quark and lepton masses, made in the spirit of SGDAs. We give emphasis to the triplets representations, with j 41. In sect. 3 we show the connection between the present results and other mass relations found in literature. Section 4 is dedicated to our comments and final remarks.

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2. – Mass formula in the inequivalent representation and corresponding fittings

A mass formula based on the chain (1.1) and associated to the inequivalent representation, eqs. (1.3), is the following:

_{j 1} 1 2

l

2 , (2.2b) with q 4ea.

Applications of the above mass formula for n04 0 have been done in ref. [4]. Two

possibilities arose. First, the case of j 45/2, having in mind the possibility of accommodating the six quarks in a sextet of SUq( 2 ), including the very massive top

quark discovered recently by CDF and DØ groups at the Fermilab Tevatron collider [7]. Another possibility, also suggested in ref. [4], was to separate the top quark as a singlet ( j 40) from the rest, considered as a quintet ( j42), in view of its very high mass. We have been investigating the possibility to fit the quark and lepton families in different multiplets of j in the extended case of n0c0 for both cases of a real

and imaginary. Expressions in the case of a real have been more successful to describe the increase in mass of quarks and leptons within the multiplets due to the dependence of q-numbers on the hyperbolic sine in this case. Thus, for the present work we preferred to apply eq. (2.1) treating the quarks as two separated triplets ( j 41) of

down and up quarks with real a . In this way we could treat the neutrinos and the

charged leptons as two triplets as well. A good fit of the triplets, with the same value of

n0, can be obtained, each one corresponding to a different value of the deformation

parameter a and so to a different value of the charge Q of the triplet.

2 a , (2.3c)

where we have used the fact that [ 1 ]a4 1 , [ 0 ]a4 0 , (a . Masses will be ordered M1E

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And M3, the largest mass of the triplet, does not depend on the value of the parameter

n0 of the inequivalent representation. For n04 0 one has e2an04 1 and the triplet

collapses to a single state with a mass M 4B[3/2]2

a, A 40.

For the j 41 triplets, one obtains

M31 M22 M1

M2

4 1 1Y

1 1e2ay0_Y , (2.4)

where we have defined

Ya4 [ 2 ] [ 1 /2 ]2 a . (2.5)

Equation (2.5) can be written as

Ya_{4 [ 3 ]}2_{a 84a/2}_{2 1 4 N}_{a 84a/2}2 _{( j 41)21 ,}

(2.6)

where we defined the number Na( j) as

Na( j) 4 [2j11]a.

(2.7)

We have chosen to use this equation to guide our fittings of the parameter a . The right-hand side of eq. (2.7) corresponds to the expression for the “q-dimension” of representations. For certain values of a the number Na( j) will be an integer. Thus,

eq. (2.7) determines a set of a’s such that they satisfy the condition dima( j) 4

integer .

For real a the number of states is given by

Na( j) 4

!

m 42j j ema_. (2.8) Equation (2.8) reduces to Na_{( j) 4}

g

!

k 40 j

2 cosh ka

h

_{2 1 ,} for integer j ; (2.9a)

Na_{( j) 4}

g

!

k 41/2

j

2 cosh ka

h

, for half-integer j . (2.9b)

In the case of imaginary a analogous expressions hold, with a replaced by ia in eq. (2.8) and cosh ka replaced by cos ka in eqs. (2.9).

For a 40 one has N( j) 42j11, and the multiplet has the usual number of states found in the non-deformed case. In general, for a real N( j) D2j11 and the number of states increases with a . We may interpret these additional states as “extra” copies of the ( 2 j 11)-states with the masses given by eq. (2.1), each of these multiplets lying on a different vector sub-space with a highest weight mA 42j [8]. If this is so, such multiplet copies would be indistinguishable from the point of view of the classification of elementary particles in the SM.

The states we refer here are “good” states of type II representations, as established in refs. [8]. We note, however, that here we are dealing with a very different situation: while in ref. [8] the authors discuss the problem of spin chains, and thus having a

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TABLE _{I. – Fitting of triplets j 41. Numerical limits on the parameters obtained according to} eq. (2.4). Charged leptons Quarks down Quarks up Neutrinos Y D16.81 Kan0D 2.879 Y D24.95 Kan0D 3.256 (a) Y D129.62 Kan0D 4.872 (b) Y D114.81 Kan0D 4.751 (c)

(a) For md4 10 MeV, ms4 200 MeV, mb4 5 GeV in eq. (2.4). (b) For mu4 5 MeV, mc4 1350 MeV, mt4 175 GeV in eq. (2.4). (c) With the present limits on neutrino masses.

TABLEII. – Fitted parameters for each triplet in two fittings, I and II

Fitting I a A (GeV) B (GeV)

charged leptons down quarks up quarks 2.6339158 3.1335985 4.7790528 20.10514738 20.165 21.325 0.4517393 0.9634722 15.918229

Fitting II a A (GeV) B (GeV)

charged leptons down quarks up quarks 2.6339158 3.1335985 4.7790528 20.10514738 20.16315 21.31768 0.4517393 0.9456125 15.82729

“path” of fixed spins ( j 41/2), for our purposes in the present work we had to analyse the different possibilities to fit the multiplets in irreps for each value of j.

For the numerical analysis we took the following procedure: we have listed all values of a satisfying the condition of eq. (2.7) 4 integer. Then, from that list we could find the appropriate a’s which would fit the experimental masses in the left-hand side of eq. (2.4). For each triplet of quarks and leptons we took the smallest a which would make a fit possible, according to eq. (2.4). Table I shows the numerical constraints on the parameters which led us to the fitting for every triplet.

We emphasize that we have chosen this procedure as an alternative to the usual one of treating a as a free parameter to be fitted. It is a choice we have made, to select the values of a from the list of those which give to the dimension of representation an integer number, instead of treating a as a completely free parameter.

The numerical values of the parameters found in two fittings (called fittings I and II) are shown in table II. They were obtained including the charged lepton sector, also treated as a j 41 triplet. The value of n0 was fixed as n04 1.5513007 from the

charged-lepton sector where the experimental masses are known with greater accuracy. We then considered the same n0for all the triplets.

The parameters given in table II conduct, from eqs. (2.3), to the calculation of the following masses:

In fitting I, for the down quarks:

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and for the up quarks:

mu4 5 MeV , mc4 1330 MeV , mt4 175 GeV .

In fitting II, for the down quarks:

md4 8.54 MeV , ms4 171.69 MeV , mb4 4 .70 GeV .

and for the up quarks:

mu4 4.77 MeV , mc4 1322.45 MeV , mt4 174 GeV .

For charged leptons, in both fittings we set the values

me4 0.5109 MeV , mm4 105.658 MeV , mt4 1771.74 GeV .

The two fittings show that quark masses can be fitted inside the range of their running masses with just a small change in parameters A and B. The greater accuracy on parameters A and B leads to a greater precision on the calculation of masses.

The main difficulty to treat the neutrinos in the same way is the problem of fixing the parameters for the neutrinos triplet, since only upper limits in their masses are known. According to the mass limits of [9] mneE 5.1 eV, nnmE 0.27 MeV, mntE 31 MeV,

eq. (2.4) would indicate that for the neutrinos triplet a D3.0632313 and thus showing that the increasing in neutrino masses within the three generations might be as much accentuated as it is for the case of the quarks up triplet, or even more.

We note that in our approach a larger a describes a larger increase in mass within the triplet.

3. – Connection with other mass relations

It is worthwhile to point out that recently Sirlin [10] suggested an interesting empirical mass formula involving mass ratios for the charged-lepton, up-quark and down-quark sectors. Sirlin’s formula is

m2 m1 4 3

g

m3 m2

h

( 3 /2 ) NQN , (3.1)

where mi(i 41, 2, 3) is the mass of the i-th generation and Q is the electric charge in

units of the electron charge. In the charged lepton sector (NQN41), eq. (3.1) leads to a value of the tau lepton mass in excellent agreement with experiment.

A scaling parameter x 4 (m3/m2)1 /2 may be defined for each sector, giving rise to

the scaling relations

m3 m2 4 x2, (3.2a) m2 m1 4 3 xN3 QN, (3.2b) m3 m1 4 3 xN3 QN 1 2. (3.2c)

One could speculate, of course, to extend the application of these relations to the triplet of neutrinos too. In this case one would get NQN40 leading to m2/m143, etc.

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If we compare eq. (2.1) with the scaling relations, eqs. (3.2), we find, for each sector x2_{1 1 2} 1 3 xN3 QN 4 1 1Y 1 1eay0_Y , (3.3)

which relates quantities of the SUq( 2 ) formalism with the scaling variable x, for each

triplet. In particular, we see the dependence of NQN on the deformation a for fixed n0.

Sirlin [10] has also defined a mass matrix for the charged leptons which depends only on the scaling parameter. It reads

M 4mm

.

`

´

0 ( 3 x3)21 /2 0 ( 3 x321/2 [ 1 2 (3x3)21_] 0 0 0 x2

ˆ

`

˜

. (3.4)

This matrix can be extended to the sector of up and down quarks if we suppose the texture of all matrices is the same. This may be true at the low-energy scale we are involved. Matrices with the texture of (3.4) have been analysed by Fritzsch, Dimopoulos and others [11]. Under the hypothesis of a single texture, Sirlin’s matrix, eq. (3.4), can be rewritten in the generalized form

M 4

.

`

´

0 ma 0 ma m( 1 2a2) 0 0 0 mb

ˆ

`

˜

, (3.5) where we defined a 4 (3xN3 QN₎21 /2_{, b 4x}2 , m 4M2.

The eigenvalues of the mass matrix, eq. (3.5), are

l14 M2x24 M3, (3.6a) l24 M2, (3.6b) l34 2M 2 3 xN3 QN 4 2M1. (3.6c)

Equations (3.6) bring an evident contact with the SUq( 2 ) approach. The left-hand

side of eq. (3.3), which relates Sirlin’s scaling to the SUq( 2 ) formalism, also

corresponds to the trace of the eigenvalues matrix l 4l/M2, namely

Tr l 4 M31 M22 M1

M2

4 b 1 1 2 a24 x21 1 2 1 3 xN3 QN .

(3.7)

The above remarks indicate that Sirlin’s relations work like one-parameter approximations of SUq( 2 ) equations.

From eqs. (3.2) one gets the following values to the scaling parameter: for charged leptons x ` 4.1, for quarks down x 8B5–7, for quarks up x 9B9–11 and for the neutrinos triplet x R B11–12 (estimated from the present upper limits in their masses [9]). We have used these numbers to guide our fittings on the SUq( 2 ), based on

eq. (3.3) relation. It is important to note that the values of x for the two triplets of quarks are not exact. There is a certain flexibility on the determination of x 8 and x 9 due to the running character of their masses.

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TABLEIII. – Solutions of eqs. (3.9) and (3.11) for quarks and leptons triplets. Solution of eq. (3.11) includes the dependence on the angle qM.

Q N3 QN x(eq. (3.9)) x(eq. (3.11)) neutrinos quarks down quarks up charged leptons 0 21 /3 2 /3 21 0 1 2 3 6.4604836 5.0232454 4.4115883 4.1008728 16.603529 5.0232454 10.857609 4.1008728

Sirlin’s relations are excellent for charged leptons, quite good for down quarks, but not so good for quarks up. From eqs. (3.2) for quarks up one gets mt4 mc2/3 mu. Then,

with the quark masses mu4 5 MeV, mc4 1.35 GeV at 1 GeV [12] one gets mt4 119 GeV,

to be compared with mt4 175 GeV (pole mass) or mtB 225 6 75 GeV, at 1 GeV.

Another very accurate mass relation for charged leptons has been derived by Koide [13] based on a U( 3 )-family Higgs potential model and related to a “democratic” family-mixing mass matrix [14]. Koide’s sum rule reads

me1, mm1 mt4

2

3(kme1 kmm1kmt)

2_.

(3.8)

We have generalized this relation using the scaling parameter x, as defined by eq. (3.2a). We rewrite eq. (3.8) as

( 1 13xN3 QN

1 3 xN3 QN 1 2) 44(k3 xN3 QN/2

1k3N3 QN/2 1 1

1 3 xN3 QN 1 1) . (3.9)

This equation gives the results displayed in column 3 of table III for the triplets of quarks and leptons.

Koide [13] pointed out that an analogous relation of eq. (3.8) related to quarks down would work fine too, but not for the up quarks triplet. Koide’s mass formula, eq. (3.8), was extended by Esposito and Santorelli [15] to include both the up quarks and the neutrinos triplets based on a geometrical interpretation of Koide’s relation proposed by Foot [16]. They introduced an angle uMbetween a vector V 4 (km1,km2,km3) and a

vector M of components (km0,km0,km0) f ( 1 , 1 , 1 ) in a orthogonal Cartesian

reference frame of an Euclidean vector space. The angle uM would be given

by [15, 16] cos uM4 MV NmN NVN 4 1 k3 km11km21km3 km11 m21 m3 . (3.10)

For charged leptons and quarks down one has [15] uML4 uMD4 45.000306 0.00120

and for quarks up and neutrinos uMU4 uMN4 50.906 0.20.

With the introduction of the angle uMwe could rewrite eq. (3.9) in the form

( 1 1xN3 QN 1 3 xN3 QN 1 2) 4 2 3 cos2u_{M 21}

(

k3 x N3 QN/2 1k3 xN3 QN/2 1 1 1 3 xN3 QN 1 1

)

. (3.11)

Solutions of eq. (3.11) are displayed in the fourth column of table III. A new result is obtained for the case of neutrinos and quarks up, due to the angle uM. For charged

leptons and quarks down eq. (3.11) reduces to eq. (3.9) and the same value of x is obtained.

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Using eq. (3.3) results of table III can be compared to those of SUq( 2 ). Curiously,

eq. (3.11), which contains the angle uM, gives the results in agreement with SUq( 2 ).

This can be observed by the up quarks sector. Sirlin’s relations, eqs. (3.2), indicated that for up quarks one has x 9B9–11, while the fourth column of table III gives the value 10.857609 for the up quarks triplet, with the inclusion of the angle uMU.

With the knowledge of the numerical value of x for each triplet it is possible to calculate other observables derived from the mass matrix. We have used a single texture for all mass matrices, for charged leptons, quarks down and quarks up, as given by eq. (3.5), with the appropriate value of x for each triplet case. For a discussion about the implications of texture choices for the matrices we refer the reader to ref. [17] (general) and [11] (with the present texture). We note that the present formulation can also lead to the CKM matrix if flavor mixing is introduced by means of a breaking of the

SUq( 2 ) symmetry. A simple estimate already gives good expectations. Consider, for

instance, the Wolfenstein parametrization [18], in which the matrix elements of the CKM matrix are expressed approximately in powers of the Cabibbo angle. One has [18, 19] l BNVusN ` s12, with [18] NVusN `kmd/ms. From eq. (3.2b) we find l B

NVusN ` 1 /k34 x 8. Then, with x 845.023 245 4 we have lA0.257 6. And with the value of

quark masses in the SUq( 2 ) fitting II we find l BNVusN 4 0 .223. These are approximate

results, estimated numerically in the present context, but very close to the averaged experimental value accepted at the moment [19].

4. – Conclusions

We would like to address now to some concluding remarks. From a spectrum-generating deformed algebra (SGDA) based on the simplest of the deformed

u-algebras, that of the SUq( 2 ), we were able to describe the masses of the fundamental fermions of the Standard Model (SM), assuming that the number of quark (lepton) generations in the SM of electroweak and strong interactions based on the gauge group SU( 3 ) 7 SU( 2 ) 7 U( 1 ) is three.

We have discussed here the possibility to understand quark and lepton generations as triplets in the j 41 representations, each of them characterized by a given electric charge Q and corresponding to a different value of the deformation parameter a .

To obtain the mass spectra essential use has been made of inequivalent representations of SUq( 2 ), characterized by a pair ( j , n0). In the fittings the same

value of n0 has been used for all the triplets. We point out that inequivalent

representations of a q-oscillator algebra have been applied in the thermodynamical properties of a q-gas, where n0-dependent physical effects have been found (see

Monteiro and Rodrigues, ref. [6]).

For each triplet j 41 it is possible to fit the increasing of masses adjusting only the two parameters of the mass formula and using the parameter of the inequivalent representation n0, obtained from the charged lepton sector. With the

mass formula of the inequivalent representation the correct result mdD mu was

obtained. (And it can be obtained not only in the case of j 41 representations, but also for a quintet of quarks, with j 42, for appropriate values of A and B, as shown in the appendix).

It was shown that Sirlin’s approach [10] may be considered as an one-parameter approximation of our present SUq( 2 ) treatment. To each triplet is associated a

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scaling parameter x, in Sirlin’s notation. A larger value of a corresponds to a faster increase of masses within the triplet, an increase which is tempered by the presence of n0.

We note that as the parameters A and B are not the same for all the triplets in the j 41 representations we cannot claim we are calculating the quark masses. We are just fitting them, inside the allowed region of their running masses, for the case of j 41 irreps. However, as for the first time one is able to describe the different masses of quarks and leptons, altogether from an algebraic model approach the authors believe it is worth to pursue on the study of SUq( 2 ) representations and

other associated SGDA. Perhaps, the present algebraic context can lead us in the future to visualize the classification of elementary particles on the SM from a different point of view. We invite the reader to investigate this question with us.

Before closing, two important comments are still in order. The first one is related to the description of the anti-fermions in the present context. This turns out to be possible due to the invariance of the SUq( 2 ) mass formalism by the

transformations q Kq21_{, implying a K2a and n}

0K 2n0.

Second, the relevant problem of flavor mixing, especially in connection with the calculation of the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements [19] has not been developed here. In the present scheme, this would require a breaking of

SUq( 2 ) symmetry by appropriate terms. The choice of these terms is not, of course,

unique and may give rise to mass matrices with different symmetry properties. A first choice, made by Iachello in ref. [4] was for a term of the form CJ2

y (where C is

an additional parameter) to be added to eq. (1.2), generating in this way non-diagonal matrix elements, for both up- and down-quark mass matrices, needed to take into account flavor mixing by constructing the appropriate CKM matrix. This subject will be discussed in a forthcoming publication.

Note added in proofs

The invariant operator which leads to eq. (7) is, in the inequivalent representation, of the form C(SUq( 2 ))4 q2n0( [ Jz1 d]2J₂J₁). This operator is a central element, as it commutes

with the generators of the suq( 2 ) algebra. Note that when n0K 0 one obtains the eigenvalue

[ j 1d]2_{. Through an alternative definition of the step operators by J} 14× q

n0/2_a1 1 a2,

J₂4× _qn0/2_a1

2 a1, in the q-analogue construction of the Jordan-Schwinger realization of SU( 2 )

for the case of the inequivalent representation, the same Casimir invariant is obtained, but then the dependence in m in eq. (6) is directly removed. Thus, the appropriate definition seems to depend on the way the vacuum is defined in the associated q-oscillators algebra, for non-equivalent representations. For the present analysis we have made a choice which leads to the mass formula given by eq. (6), convenient to study non-degenerated triplet states, subject of this work. (However, the alternative possibility of a linear Casimir operator of C(SOq( 2 ))

should also be examined.)Further investigations about the possibilities for the classification of states in different multiplets in the inequivalent representation and its relationship with the q-oscillators construction is currently in progress and will be the subject of a forthcoming paper.

* * *

One of us (BEP) is indebted to Prof. F. IACHELLO for a stimulating conversation,

during his recent visit to Sa˜o Paulo. We would also like to thank D. GALETTI for reading the manuscript and A. A. NATALE, C. LIMA and B. M. PIMENTEL ESCOBAR

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for their continuous interest and for useful discussions. Finally, PLF is grateful to the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), Brazil, for the financial support in the form of a Research Fellowship.

AP P E N D I X

Quark masses in the inequivalent representation with j 42

2 a , (A.5)

For A E0, BD0 one can do the identification of these five states with quarks d , u , s , c , b , respectively. For appropriate values of A and B their masses will be fitted in this order, with the correct detail of mdD mu. As suggested in ref. [4] the top quark

can be fitted separately, as a singlet state. Its mass would be given by

M( j 40) 4B0

k

1 2

l

2 a , (A.6)

with B0an arbitrary constant.

The fitting of the quintet states can be done, for instance, with the parameters a 4 1.5667992, n04 2.7027958, A 4 20.170 GeV and B 4 0.3931072 GeV, giving the masses

M14 10 MeV K md, M24 5 MeV K mu, M34 175 MeV K ms, M44 1.0 GeV K mc,

M54 5.135 GeV K mb.

It is interesting to note that for j 42 one has the following mass relations:

M52 M1 M42 M2 4 [ 4 ]a [ 2 ]a 4 ea1 e2a, (A.7) M42 M1 M32 M2 4 [ 3 ]a4 ea1 e2a1 1 . (A.8)

These curious relations connect the ratio of quark mass splittings directly to the deformation parameter a , independent of the value of n0.

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Notice that, from eq. (2.7) of the text, we have [ 3 ]a4 Na( j 41), showing an obvious

connection of the above relations with the dimension of representations for j 41. Inspection of the different possibilities of substitution of quark masses in these relations led us to the unique possible choices of identification of quark states as 1 Kd, 2 Ku, 3 Ks, 4 Kc, 5 Kb or as 1 Ku, 2 Kd, 3 Ks, 4 Kc, 5 Kb, with the top quark necessarily left to be identified as the singlet state.

R E F E R E N C E S

[1] GELL-MANNM., Phys. Rev., 125 (1962) 1067; NE8EMANY., Nucl. Phys., 26 (1961) 222; BARUT A. O., Phys. Lett. B, 26 (1968) 308; BO¨HMA., Phys. Rev. D, 33 (1986) 3358; GELL-MANNM. and NE8EMANY., The Eightfold Way (W. A. Benjamin, Inc., New York) 1964.

[2] ARIMAA. and IACHELLOF., Phys. Rev. Lett., 35 (1975) 1069. [3] IACHELLOF., Chem. Phys. Lett., 78 (1981) 581.

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