Quark and lepton mass formula from a partially conserved induced supercurrent hypothesis

Volume 206, number 4

PHYSICS LETTERS B

2 June 1988

QUARK AND LEPTON MASS FORMULA FROM A PARTIALLY CONSERVED INDUCED SUPERCURRENT HYPOTHESIS H i d e z u m i T E R A Z A W A a n d Masaki YASUE

Institutefor NuclearStudy, Universityof Tokyo, Midori-cho, Tanashi, Tokyo 188, Japan Received 24 December 1987

A mass formula for composite quarks and leptons is derived from a partially conserved induced supercurrent hypothesis. It leads to a sum rule for quark and lepton masses which is m r -rn~, =m,] -m~ for r=p/2 or p/3 where p= 1 or 3, depending on whether quarks and leptons are close to almost Nambu-Goldstone fermions or quasi Nambu-Goldstone fermions. The sum rule is satisfied remarkably well for r= 1/2.

I. Introduction One o f the most difficult p r o b l e m s in c o m p o s i t e models o f quarks and leptons [ 1 ] is how to explain the existing peculiar mass spectrum o f quarks a n d leptons. In order to explain the gross p r o p e r t y o f the mass spectrum in that quark a n d lepton masses are much smaller than their inverse sizes ( > 1 T e V ) , the following three possibilities have been p r o p o s e d so far: quarks and leptons are ( 1 ) almost chiral ferm i o n s [2], ( 2 ) almost N a m b u - G o l d s t o n e ( N - G ) fermions [ 3 ] or ( 3 ) quasi N - G fermions [ 4 ]. The latter two possibilities m a y not be i n d e p e n d e n t since quarks and leptons are taken as almost massless N - G fermions due to spontaneous b r e a k d o w n o f ( a p p r o x i m a t e ) s u p e r s y m m e t r y ( S U S Y ) in ( 2 ) a n d as almost massless superpartners o f N - G bosons due to spontaneous b r e a k d o w n o f ( a p p r o x i m a t e ) flav o r - c o l o r global s y m m e t r y in ( 3 ) . More precisely, let H, S and QA be the h a m i l t o n i a n for the u n k n o w n composite dynamics, the supercharge ( o f N = 1 ) and the f l a v o r - c o l o r charges, respectively. Suppose the most idealistic case in which the h a m i l t o n i a n is b o t h s u p e r s y m m e t r i c a n d f l a v o r color symmetric and in which both the supercharge and the f l a v o r - c o l o r charges c o m m u t e with each other, i.e. [H, S ] = [H, QA] = [S, QA] =0. In the case (2), SUSY is spontaneously b r o k e n by the vacu u m ( 1 0 ) ) but flavor-color s y m m e t r y is not, i.e. S I 0 ) ¢ 0 and QA[0):0 while, in the case ( 3 ) fla-

v o r - c o l o r s y m m e t r y is spontaneously broken but SUSY is not, i.e. S I0 ) = 0 a n d QA I 0 ) ~ 0. In the case ( 2 ) , composite quarks a n d leptons are taken as almost N - G fermions which behave as QAS IO) while, in the case ( 3 ) , they are taken as quasi N - G fermions which behave as SQ n I0 ) . However, it is obvious that both o f these states are o f zero-norm since [S, QA ] = 0 a n d either QA[o) = 0 or S[ 0 ) = 0 . The wishful existence o f almost N - G fermions or quasi N - G fermions, therefore, is i n d u c e d either by an explicit breaking o f SUSY or f l a v o r - c o l o r symmetry, by n o n - c o m m u t a t i v i t y o f their charges, or by a spontaneous b r e a k d o w n o f both SUSY a n d flav o r - c o l o r symmetry, i.e. [H, S] re0, [H, QA] 50, [S, QA ] ¢ 0 or S[ 0 ) ~: 0 and QA Io ) o. One o f the m a i n purposes o f this p a p e r is to clarify the relation between almost and quasi N - G fermions in such a more realistic case. A n o t h e r purpose is to present a general mass formula for quarks and leptons as almost or quasi N - G fermions. Furthermore, a simple sum rule for quark and lepton masses will be derived from the mass formula by assuming an explicit form o f symm e t r y breaking.

2. Partially conserved induced supercurrents Suppose there exist n sets o f N = 1 supermultiplets ~ , = (~)i, Z~) a n d ~ c - - ( o c , zc ) ( i = l ..... n), which represent subquarks, the constituents o f quarks a n d 669

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PHYSICS LETTERSB

leptons, whose masses are (/z~,Mr). Let q/= (Z, zC,)T be the four-component spinor and q = O L + ¢pC*R be the corresponding scalar, where L = ( 1 - Y5)/2 and R = ( 1 + Ys)/2. The supercurrent s u and supercharge S are given by su =i(O~rlt)y~Tu~-tltTuM~

(la)

S = f daxso ,

(lb)

with ( M ) o = M f l o . The second term o f s u arises from the elimination of the F-component of • and Oc, =FtL+FCR, by the equations of motion F = - 0Wt/0~ t and FC= - 0 W*/00 ct (as 7uF~= - qtTgM~ for W= OCMO), where W is the superpotential. The flavor-color currents J~5)u and charges Q~5) are given by - 2 A ~t + itltO <-~uAA rl+ iqco u2A r/c* , l Au = qlYu

(2a)

QA= f d 3 x j A ,

(2b)

JAu=qlTuy52A~/+irlOuY52Aq--lqCOuysAA~lC* ,

Qg = f d3x J~o ,

S Au - (iq*yuy~OV-~l*yuM)2a~+OV(~l*truv2Aql) , (6a) s~ u = (iqtTuy~0 ~_ ?ltTuM) 2 A75 ql + O"(qtau, y52A~u),

(6b)

S A = j d 3 x s A = IS, QA] ,

(6c)

S A = ~ d3XSAo= [S, Q5a] .

(6d)

Alternatively, if flavor-color symmetry is exact (QA = const. ), other supercurrents can be induced as local currents by the commutators S'uA = [ S . ,

QA],

Srs%= [ S u ,

Q~] .

(7a,b)

By using the relations ( 4c ) - ( 4f ), the induced supercurrents and their charges can be written as (8a)

(2c)

S'A= I d3x S'°A: [S, Q A ] ,

(8b)

(2d)

S'sAu= - r l t y u Y s { M , )ta}g¢ ,

(8C)

S'fl = I d3xs;~ = [S, Qgl •

(Sd)

...,

(3a)

Notice that the difference between these two classes of induced supercurrents is essentially a total divergence and that their charges are identical to each other, i.e.

(3b)

s tA u - s Au ~

0*'JJ = -i(q712", M] q/+ n* [2 a,/.rE ] n +r/c[2 A, pZlrlC*),

currents and their charges can be written in terms of subquark fields as

s ~ = - rltyu [M, 24 ] ~u,

where r/c=OcL+0tR and 2A ( A = 1, n z) are the n × n matrices for the U (n) symmetry. They have the following appropriate properties: 0USu= i q t ( M 2 - p2)g/,

2 June 1988

--

(9a)

O~(rltau~2A~) ,

'A - S s Au --- -- O~(qtau~Y52a~ ') , ss~

OuJgu = -i(q775 {2~, M}~,+ q'y5 [2 n, #2lr/

(9b) (9c,d)

(3C)

S ' A = S ",

IS, n] =q/,

(4a)

{S,, ~a] = (i0~q*Yu-rl*M),a,

(4b)

Since there is no difference in taking either one of the induced supercurrents, s~5)u and s~5)u, ,a we shall discuss s~5)u hereafter. For later convenience, the supercurrent is made to be symmetrized under q,--~q/and qc*--,~tc= C(~t) T by adding

- - qC~5 [ ~ A ]A2 ] qC~') ,

[QA, q] = _2Arl,

[Qg, q] = - 2 a ~ q ,

[QA, q/] = _2a~u,

[Qg, ~u] = --;tny~U.

(4c,d) (4e,f)

If SUSY is exact (S= const), supercurrents with the flavor-color quantum numbers can be induced as local currents by the commutators s u - [S, jA],

Sgu = [S, Jgu] •

(5a,b)

By using the relations (4a), (4b), the induced super670

S'(=S~.

C( gu) T = i ( O ~ q C t ) y ~ y u ~ c - q c * (yug/CM)i .

The supertransformation is now operative on r/c and ~,as well as on q and ~: {S,~, ~p} = (iOuq*7~ - q * M ) , ~

,

(lOa)

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PHYSICS LETTERS B

{S~, ~ta}= - [ (i0 uqctTu - M q C t ) C ] , ~ a ,

(10b)

[& q] =V,

[S, qc] =~ tc-

(10c,d)

Similarly, s A and SAU are symmetrized. Let us now introduce the "partially conserved induced supercurrent (PCIS) hypothesis", that is, the divergence of this induced supercurrent be dominated by a fermion field (fA), which represents a quark or lepton, as 0US~_icc, fA or

O~sAu~iCfA~SfA,

(11)

where cf~ is a constant. This constant depends on the fermion mass (rnr~) as cf.=F~,mrA

or

-Ff~m~,

(12)

where Ff. is the"fermion decay constant", depending on whether the fermion is close to an almost N - G fermion or a quasi N - G fermion. The reason for this crucial difference of mass dependences is the following: If the partially conserved flavor-current hypothesis of jA ~ iFb. 0ub A (where b A is a N - G boson and Fb~ is the "decay constant" ) is super-transformed, the u = iFf~ 0~fa, indicating PCIS relation becomes SA~" co, = - F c , m2A. On the other hand, if the partially conserved supercurrent hypothesis of s u ~ - - F 2 ~ ' u f (where f i s a N - G fermion and Ff is the "decay constant") is flavor-color transformed, the PCIS rela2 tion becomes s A ~ - F f. ~'ufA, indicating cf~ = F ~AtufA" This suggests that the term o f - t/tTM1//2AVin the righthand side of eq. (6a) would be dominated by the almost N - G fermion while the remaining term by the quasi N - G fermion. It seems plausible since the former term behaves as - < q t > o m T , u / ~ A ~ / , / (more precisely, ~'u< ~ > o2~q/) while the latter as i < qt > 02A0u~Vif SUSY or flavor-color symmetry is spontaneously broken by the non-vanishing vacuum expectation values (VEV's) of o and o or <¢>o and

<0c>0.

Once we assume the PCIS relation (11 ), it is straightforward to derive a mass formula for quarks and leptons as was made by Dashen for pions as almost N - G bosons [ 5 ]. Let us consider the quantity

d

and a similar one for s¢~. By using the PCIS relation, this can be transformed into

C)A f

d4x

exp (iqx)

(T(fA(x)fA(o)) )0

+ f d4x exp(iqx)6(Xo) ( {kA(X),yA(0) } )0 •

(14) In the limit of q ~ 0 (where the quantity should vanish), we can derive the relation c ~ 4 m ~ , ~ p = ( { S ~ 5 ) , ~ , [S~5)~, AL]})o ,

o,

(13)

(15)

where AL is the symmetry breaking lagrangian density. Thus, we obtain the simple mass formula for quarks and leptons c fA a m -f4' ~ , ~ p = ({ [S~, Q~s)] ' [ [S~, Q~5)], AL]})o

"

(16) For the quasi N - G fermions, Veneziano has derived another type of mass formula, mfA ~ [ QA, [ QA, AWI o=o] ] [6], which can be considered as related to the fermion masses of 02AW/O~?co~ since QA ~ ( 2 % ) 0 / 0 ~ + (...). On the other hand, our mass formula, mfA ~ {~A, [S A, AL ] }, can be interpreted as related to the masses of -02AL/0~,0~v since S A ~ (2A~)0/0~V+ (...). It is readily applicable for any types of SUSY-breaking by simply adding the breaking terms to AL although the other formula can be extended to include SUSY-breaking after some calculations [7 ]. This is why the explicit and spontaneous breaking of SUSY can be simultaneously treated. In order to find an explicit form of the mass formula, suppose that SUSY or flavor-color symmetry is not only broken spontaneously by the appropriate condensations of scalar subquarks but also explicitly broken by the subquark mass terms AL = - gTMq/- t/f# 2r/- r/c/t 2r/c* .

3. Mass formula

q~q~ [ d 4 x e x p ( i q x ) ( T ( s a u ( x ) ~ ( O ) )

2 June 1988

( 17 )

By using the relations ( 4 c ) - ( 4 f ) and ( 1 0 a ) - ( 1 0 d ) for the symmetrized S, the formula (16) can be reduced to, by using (q*~,)*~o= ~qc,, c~,mg I =

( ~9[ 2A, M] ++_M[,~At, M ] + qc*

+ ?/* [)A*, M] + M[2 A, M ] +_r/c*) o ,

( 18 ) 671

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PHYSICS LETTERS B

where [A, B] _ = [A, B] for the "vector" S A and [A, B] + = {A, B} for the "axial" S~. For quarks and leptons specified by fA= fii for (2A9) ~= Cj that behaves as qc~j+~ucrb, i.e. fijL~OC~/jLq-(~C)LOj a n d f~jR~ ~t %R + ( ~Uc) R~C*, the formula ( 18 ) can be further reduced to c2f,j m -fij' = (M~_+Mj)Z(Mj(¢c¢~)o + M ~ ( ¢ c Cj)o) (19) for ( 0 c 0 ) o = ( 0 ' 0 c*) o, where the minus sign is for the "vector" S ~ and the plus sign is for the "axial" S~. By using the explicit dependence of cr. on mf~ (12), we finally obtain the explicit form of the mass formula

2 June 1988

dynamics, i.e. M, AM, ~M<
( p = l , 3) (21a)

f o r p = 1 or 3,

(20)

depending on whether a quark or lepton is close to an almost N - G fermion (p = 1 ) or a quasi N - G fermion (p=3).

for the "vector" S A and m~j~_const..(M~+Mj) 3

( p = l , 3)

(21b)

for the "axial" S A. Obviously, from this follows the mass sum rule for quarks and leptons of the following type:

4. M a s s sum rules

m~23+m~,3= +_(m~2,+_mf~4) The mass formula (20) indicates that the quark or lepton mass (or the cubic m a s s m 3) behaves essentially as (AM)2Mv2AP-5 for the "vector" S n and ( E M ) 2 M v 2 A p-5 for the "axial" Sg, where M, AM, Y~M, v 2 and A are the subquark mass, the mass difference, the mass sum, the VEV's of the scalar condensates (~)cOi)o~V2 and the mass scale of the underlying dynamics, respectively, since the "decay constant" must be determined by the mass scale as F ~ A . The candidate v2 is perhaps either of order A 2 or of order # 2 _ M 2 (the SUSY-breaking parameter). For quasi N - G fermions as superpartners of the N - G bosons, the spontaneous breaking by (0c0~)0 in the SUSY limit is a must; therefore, v2 must be of order A 2. On the other hand, for the almost N - G fermions, it is not necessarily required in the SUSY limit; t h e r e f o r e , u 2 ~A 2 o r / 2 2 - m 2 and the condensation of ( ~ / ) o ~ A 3 perhaps occurs to induce spontaneous SUSY breaking. This mass formula can thus explain as expected that the quark and lepton masses (m) are much smaller than their inverse sizes (which may be of order A), i.e. m <
(r = 1 / 2 , 3 / 2 ) (22a)

for the "vector" S ~, in which I M z - MA <
( r = l / 3 , 1)

(22b)

for the "axial" S~ depending on whether they are close to the almost N - G fermions ( r = 1/2, 1/3) or the quasi N - G fermions ( r = 3/2, 1 ). In order to find whether the existing mass spectrum of quarks and leptons satisfies this type of sum rules, we must take a definite composite model. The simplest such model is the minimal composite model of quarks and leptons. It possesses six color-flavor quantum numbers Oi= (0i, ~L~) as L-handed superfields and O c . = (0c., ~Ri) as R-handed superfields for i = 1..... 6, which consist of a weak isospin doublet of subquarks (wt, w2 ) [ 8 ] and a Pati-Salam SU (4) color quartet of them (Co, cl, c2, c3) [9]. I f O is assigned to (WIR, W2R, COL,ClL, C2L, C3L) and Oc. to (~--~L, w2L, C0R, CIR, C2R, C3R), the first generation of quarks and leptons can be identified as vefl 3, e = f23, ua=f~a+3 and d a = f 2 a + 3 where a is the SU(3) color

Volume 206, number 4 index

(a=l,

2, 3)

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PHYSICS LETTERS B

(more explicitly, VeL(R)~

W:gcoa(A~= (COACOA--V2)

WILC0L(WIRCOR), eL(R)~W2LCOL (W2RC0R), UaL(R)~

WILCaL(WIRCaR)and d~L0~)~W2kC~L (W2RCaR)). Notice that the vanishing (or at least almost vanishing) mass ofv~ requires the equal masses ofw~ and Co, i.e. M~ = M3, for the "vector" SA and the vanishing masses ofw~ and Co, i.e. MI = M3 = 0, for the "axial" S A. Then the mass sum rules (22a), (22b), become in the form of m~=+_(rn~++_m~) ( r = l / 2 , 3/2) and = m ~ m r ( r = 1/3, 1 ). Obviously, since m ¢ < m o < m d , the possible combination of __sign must be fixed as

rn~=m[-m~

.

(23)

For r = 1/2 for the "vector" S A, this mass sum rule is nothing but the one which has been recently found almost empirically by one of the present authors (H.T.) [ 10 ] and which is satisfied by the experimental value of m~ and the estimates of m~ and md. In fact, for m~=0.5110034(14) MeV, m~=4.5_+1.4 MeV and md= 7.9 +_2.4 MeV ~, the left-hand side of eq. (23) is 0.71 MeV ~/2 while the right-hand side is 0.70 ___0.22 MeV w2. Of course, since the estimation of the current quark masses for the first generation is ambiguous [ 12 ], this might indicate a mere coincidence. However, if it is taken seriously (as by us), it may indicate that quarks and leptons (at least) in the first generation are close to the almost N - G fermions. On the other hand, for the "axial" S A, the linear mass sum rule (r = 1 for quasi N - G fermions) has recently been derived by us in a SUSY nonlinear-interaction model of the Nambu-Jona-Lasinio type [ 13 ] although it has no change to be consistent with the observed quark and lepton masses. It is rather difficult to present a dynamical model for generating quarks and leptons as the almost N - G fermions. For a simple example, we examine an instructive model of the a-model type that consists of COo= (or, X~), COA= (OA,Z~) (A= 1, ..., n) and CO0=(Oo, Zo) where (COo, COA) belong to n + l of the flavor-color symmetry SO (n + 1 ). The superpotential is given by

~ See ref. [ 11 ] for a review.

+ "2

((I)0 "31-m~)(I)o(I)~'3c a = l

(¢I)O"}-mA)Iffi)ACOA '

which is invariant under SO (n + 1 )transformations for g = 0 and m~ = mA and under SO (n) transformations for mA = const. The minimum of the potential V=Ei=o.o.~lOW/OOil z is given by (•>o= (~)A>0= 0 and < % > o= - mo that yields a spontaneous SUSY breaking induced by < F ~ > o = - < O W / O ~ ) o = g v 2 and no spontaneous flavor-color symmetry breaking. Since

s~,=

~

i= O,o,A

[i(O"**,),3.o,,z,-~,(OW*lOO*)f¢,]

and s~ = Is u, Q A ] _ ~ i g v Z c ~ for QA of S O ( n + l ) / SO(n) satisfying [QA, CO~]=--icoA and [QA, COB] = icoogABand, therefore, {S,, Xo~}= - e ~ g v 2 and {S A ~, Z~a) = -ie~pgv 2, it is certain that the ZA are the almost N - G fermions transformed from the N - G fermion ~o. Even if the SUSY-breaking mass term ½A#AT~Z4 is present, the masses of the almost N - G fermions 7~ directly calculated a s rnA--m,~+A#A (which can be small even for large mA and mo) are exactly the same as those calculated from the formula (22), which in this case takes the form of 2 --1 e , ~ = ( { S ~ , [S~, AL]}>o with Cm=im~gv z. cAmA From this lesson, we observe that the conditions for the presence of the almost N - G fermions may include: (i) the theory is invariant under transformations of a certain subgroup of G ( S O ( n + 1 ) in the above example, H ( S O ( n ) ) ; (ii) spontaneous SUSY breaking is induced by the H-singlet member (F~,); (iii) the N - G fermion (which is, therefore, H-singlet but not G-singlet) is flavor-color transformed into the almost N - G fermions only by the generators of G/H. For the second and third generations of quarks and leptons, the mass sum rules of this type would not be well satisfied by the experimental values. This does not, however, mean that these quarks and leptons are neither the almost N - G fermions nor the quasi N - G fermions since the validity of this particular form of mass sum rules (22a), (22b), strongly depend on additional assumptions such as the near universality of the subquark masses for the "'vector" case, of the 673

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" d e c a y constants" a n d o f the VEV's o f the scalar condensates. In fact, in s u p e r s y m m e t r i c q u a n t u m chrom o d y n a m i c s , in the absence o f the spontaneous breaking o f S U S Y , ( O c ~ i ) o is known to d e p e n d on a s u p e r s y m m e t r i c mass M, as far as I/~e-M,I <> M, in which case the large mass/1 m a y induce large mass corrections to the fermions. F o r the almost N - G fermions, ( i ) p r o b a b l y less exotics will be present if the almost N - G fermions are i n d u c e d by the f l a v o r - c o l o r t r a n s f o r m a t i o n S~5) for A restricted to be in G / H , as in the instructive m o d e l and (ii) another constraint m a y come from the a n o m a l y matching if ( a p p r o x i m a t e ) chiral symmetries are present [2 ].

5. Summary and discussion We have derived the mass formulas ( 1 6 ) for composite quarks a n d leptons as almost N - G fermions or quasi N - G fermions from the PCIS hypothesis ( 11 ). By assuming an explicit breaking o f S U S Y or flav o r - c o l o r s y m m e t r y due to the subquark masses, we have further d e r i v e d a m o r e explicit form o f the mass formula (20). Further, by assuming the near universality o f the " d e c a y constants", o f the VEV's o f scalar condensations a n d o f the subquark masses for the " v e c t o r " S A, we have finally o b t a i n e d the mass sum rules for quarks and leptons ( 2 2 a ) , ( 2 2 b ) . In the m i n i m a l composite model, the mass sum rules take the forms o f m~ + taro = + (m~ + m~ ) for the "vector" S n a n d m~ - m v e = m ~ - rn~ for the " a x i a l " Sg. If quarks and leptons in the first generation are the almost N - G fermions for the " v e c t o r " S n, the mass sum rule becomes ,,~,~1/2 --m~/2 - m ~ / 2 and it is satisfied r e m a r k a b l y well by the experimental and estim a t e d values for the quark and lepton masses. 674

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The relation between the almost N - G fermion and the quasi N - G fermion has b e c o m e clear. They m a y d o m i n a t e a different part o f the i n d u c e d supercurrents, but, in the divergence o f the current ( l 1 ), leave only a different fermion-mass d e p e n d e n c e in the " d e c a y c o n s t a n t " as in (12), which eventually leads to a different power-dependence in the mass sum rule (20). Since the knowledge o f the almost N - G ferm i o n is poor, a further effort is required to clarify its d y n a m i c a l properties and to see whether the squareroot mass sum rule is in fact realized. Also, the possible existence o f the genuine N - G fermion (call it " s u p e r i n o " ) which is generating the almost N - G ferm i o n s a n d which is f l a v o r - c o l o r neutral is intriguing. It is a massless (or almost massless) neutral fermion o f spin 1/2 which has no interaction except for gravitation as the graviton and the gravitino. Unless it is a b s o r b e d by the gravitino by the Higgs mechanism, it would p r o v i d e an a d d i t i o n a l source for various cosmological problems. The authors would like to thank Professor K. A k a m a for useful discussions.

References [ 1] H. Terazawa, in: Proc. XXII Intern. Conf. on High energy physics (Leipzig, GDR, July 1984), eds. A. Meyer and E. Wieczorek, Vol. 1 (Akademie der Wissenschaften der DDR, Leipzig, 1984) p. 63; M.E. Peskin, in: Proc. 1985 Intern. Symp. on Lepton and photon interactions at high energies (Kyoto, August 1985 ), eds. M. Konuma and K. Takahashi (RIFP, Kyoto, 1986) p. 714; J.C. Pati, in: Superstrings, supergravity and unified theories, eds. G. Furlan et al., ICTP Series in Theoretical Physics Vo. 2 (World Scientific, Singapore, 1986) p. 377. [2 ] G. 't Hooft, in: Recent developments in gauge theories, Proc. NATO Advanced Study Institute (Carg6se, 1979), eds. G. 't Hooft et al. (Plenum, New York, 1980) p. 135. [3]H. Terazawa, Prog. Theor. Phys. 64 (1980) 1763; INS, University of Tokyo preprint INS-Rep.-485 (December 1983 ); in: Flavor mixing in weak interactions, ed. L.L-. Chau (Plenum, New York, 1985 ) p. 655; W.A. Bardeen and V. Vignji6, Nucl. Phys. B 194 (1982) 422; W.A. Bardeen, T.R. Taylor and C.K. Zachos, Nucl. Phys. B 231 (1984) 235. [4] W. Buchmiiller, R.D. Peccei and T. Yanagida, Phys. Lett. B 124 (1983) 67; R. Barbieri, A. Masiero and G. Veneziano, Phys. Lett. B 128 (1983) 179;

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