P H Y S I C A L R E V I E W D V O L U M E 2 1 , K I J h l B E R 1 2

Fermion masses and hierarchy of symmetry breaking

G. C. Branco Carnegie-~Mellon University, Pittsburgh, Pennsylvaniu 15213

H. P. Nilles Stanford Linear Accelerator Center, Stanford University, Stanford, Califon~ia 94305

V. Rittenberg Physikalisches Institut der Univers~tiit Bonn, 0 5 3 Bonn, Wert Germany

(Received 3 March 1980)

W e suggest that the breaking o f a symmetry unifying the families o f fermions occurs in stages. W e consider the total Lagrangian to be invariant under the group SU(2) x U ( l ) x G , where G is a discrete group. The Higgs potential is, however, invariant under SU(2) x IJ(1) x G , where G ~ G . In a first stage G is broken to a subgroup H C G , but H is not contained in G . The u and d quarks are naturally massless at the tree level, and we discuss how they could acquire mass through radiarive corrections.

I . INTRODUCTION

One of the outstanding problems in the unified gauge theor ies of weak and electromagnetic in te r - actions i s the understanding of the fermion m a s s spectrum and the f lavor mixing angles. It has been shown that the unification of the various fermion famil ies through a discrete1 o r continuous2 symme- t r y can lead to calculable Cabibbo-type angles. Motivated by the smal lness of some of the fermion m a s s ra t ios , we suggest that the spontaneous breakdown of the family symmetry occurs in s ta - g e ~ . ~ " More precisely, we a r e envisaging a situa- tion such that in a f i r s t s tage the d i sc re te symme- t r y is broken to one of i t s subgroups, with the heavy fe rmions acquiring m a s s and the light ones remaining mass less . At a second s tage , the sym- met ry would b e fur ther broken and a s a resul t the light fe rmions would acquire mass . The hope i s that i n the c o r r e c t unification, these s teps of symmetry breaking would occur in a natural way.5 In general , i n o r d e r t o obtain a natural hierarchy of symmetry breaking, some r a t h e r s t r i c t condi- tions have to be satisfied. F o r definiteness, con- s ider a gauge theory invariant under G ::SU(2) xU(l ) , where G is a finite d i sc re te group. The constraints of renormalizability and gauge invari- ance may lead to a c lass ica l Higgs potential with a higher symmetry 6 3 G. Consider now the exact effective potential written a s

where z1° a r e the vacuum expectation values of Q i n the t r e e approximation and X stands for the m a t r i c e s of group H. We seek a situation such that

where h v is the change in the vacuum expectation values produced by AV. In o r d e r t o find under what conditions (1.2) and (1.3) may b e satisfied, i t is convenient to consider the following two ca- s e s :

(a) H i s a subpoup of G. In this c a s e the Geor- g i -Pa i s theoremG gives a necessary condition to achieve a hierarchy of symmetry breaking. The theorem s ta tes that if a Lagrangian i s invariant under a (discrete o r colltinouus) symmetry and if the vacuum expectation values respec t th i s sym- m e t r y in the t r e e approximation, then the symme- t r y will s t i l l hold i n higher o r d e r s , provided that a t the t r e e level there a r e no m a s s l e s s s c a l a r meson f ie lds which a r e not Goldstone bosons. A well-known situation where th i s necessary condi- tion can be satisfied i s when the Higgs potential has a l a r g e r (continuous) symmetry than the total Lagrangian, in which c a s e pseudo-Goldstone bo- sons7 occur at the t r e e level. It. is c l e a r that th i s necessary condition is unlikely to be sat isf ied if the unification of fermion famil ies i s done through a d i sc re te group.

(b) H i s not a subgvo?g o-f G . This i s probably V(@) = V(O'(q5) -t &V(@) , (1.1) the mos t interest ing case. Since H i s not a sym-

met ry of the total Lagrangian, quantum c o r r e c - where V(O) is the t r e e approximation to the effec- tions to the effective potential will i n genera l pro- tive potential and A V a r e radiative correct ions. duce a breaking of H, euen if thew? a7.e no mass -

Assume that the minimum of V(O)(@) leaves a sub- less ?ton-Goldstone bosons at the tvee level. In

group H C C invariant : the sequel , we will consider i n d e t a i l a ~ ; example

x ( u O ) = v0 , (1.2) in which C is chosen t o b e the te t rahedra l groups

3418 G . C . B R A N C O , H . P . N I L L E S , A N D V . R I T T E N B E R G - 2 1

T and we show that the f i r s t s t ep of symmetry breaking naturally l eads to a subgroup D, (a di- hedral group) which i s not a subgroup of T.

This paper is organized a s follows: In Sec. I1 we present some genera l resu l t s about pat terns of symmetry breaking and the i r implications f o r the fermion m a s s spectrum and the Cabibbo-type angles. In Sec. I11 we work out i n detail a specific example where the total Lagrangian i s ?' invariant while the Higgs potential has a n 0, symmetry. We analyze the various pat terns of symmetry breaking of 0, and consider i n detail the c a s e in which the tree-approxima.tion minimum has a D, symmetry. It tu rns out that the correspondent m a s s m a t r i c e s f o r the up and down quarks each contain a z e r o eigenvalue. The m a s s l e s s n e s s of the Light quarks i n the t r e e approximation i s thus obtained in a natural way, being the resul t of a par t icular pat tern of symmetry breaking. We fur ther show how a s m a l l perturbation about the D, symmetr ic minimum (and along a part icular i r reducible representation) leads to a rea l i s t i c m a s s spectrum and calculable Cabibbo-type ang- les . Finally we discuss the possibility of obtaining the breaking of D, through radiative correct ions.

Consider the s tandard SU(2) x U ( 1 ) gauge t h e ~ r y , ~ with the famil ies of fe rmions unifled through a f i - nite d i sc re te group G. We will assume that G commutes with SU(2) x U ( l ) , thus ensuring that a l l m e m b e r s of i r reducible representat ions of G a r e identical representat ions of the gauge group. The fe rmions a r e assumed to appear i n n famil ies , where the left-handed components

behave a s SU(2) doi~blets , whereas u,,, ,IiK, and e,, a r e singlets. We fur ther assume that L, and I, a s well a s the right-handed components f o r m n-dirnensio?zal ivreducible representat ions of the horizontal symmetry G. Given the observed f e r - mions, this assumption (which is pr imar i ly based on aesthet ical reasons) r e s t r i c t s the choice of G to groups with three-dimensional (or higher) ir- reducible representations. The Yukawa interac- tion can b e expressed a s

label a takes a s many values a s the number of t imes the representat ion of r$ i s contained in the i r reducible expansion of the product Exu,. After spontaneous symmetry breaking (SSB) the m a s s m a t r i x reads

where v, =(01@,10). The quark m a s s mat r ices a r e diagonalized through biunitary t ransformations UU,I1/lUUU,, U;+M*U~, , and the generalized Cabibho mat r ix is given by C = LJttlJ$. The calculability of f lavor mixing angles crucially depends on the number of independent p a r a m e t e r s i n the m a s s matr ix. The sources of independent p a r a m e t e r s a r e ( i ) the number of independent Yukawa cou- plings, which equals the number of t imes the rep- resentation @ appears in the Clebsch-Gordan ex- pansion of z x u , and Z xd, and (ii) the number of independent vacuum expectation values which de- pends on the pat tern of symmetry breaking. The Higgs potential is constructed t o be G invariant, but s ince i t only contains t e r m s that a r e quadratic and quart ic in $I, i t often has a higher symmetry 8 3 G. The SSB will not, i n general , b reak 8 com- pletely and the minimum of V(@) will s t i l l b e in- variant with respec t t o a nontrivial subgroup H C (not necessar i ly a subgroup of G). In o r d e r that a given subgroup H may b e a n invariance of the minimum of the potential fo r a given @, i t is necessary that the Frobenius decomposition of the induced representat ion of r$ (with respec t to the subgroup H) contains a t l eas t one s inglet s c a l a r representat ion. In mos t cases , one has only one independent vacuum expectation value (VEV) and fur ther s teps of symmetry breaking (obtained, e.g., through the introduction of additional Higgs m u l t i ~ l e t s ) a r e required to have m o r e than one independent VEV. Some of the fea tures of the fe rmion m a s s spectrum and Cabibbo-type m a t r i x can b e deduced on group-theoretical grounds, with- out explicit calculation. F o r simplicity, we will a ssume that Li, u,,, and d,, f o r m equivalent ir-- reducible n-dimensional representat ions r of G, and will consider H to be the la rges t subgroup of 6 which is left invariant a f te r SSB. Assume fur - t h e r that the reducible representat ion r(S) of H in- duced by r decomposes with respec t t o H i n the following way:

- . -

(2.1) where d") denotes the dimension of the represen- C,U = g ~ ~ i C y ; ~ u , , q 3 1 +H.c.

tation and x! ,, d") = n. Then the s t ruc ture of the with s i m i l a r expressions f o r the down quarks and m a s s m a t r i x will depend on the proper t i es of the leptons. In expression (2.1), $ denotes the Higgs d" ) . F o r example, if I? contains I s c a l a r s , i.e., doublets which f o r m a n i rreducible multiplet under d(') = d @ ) = = d " ) = 1, while d" ) # 1 f o r i > I , the G and CF:: a r e Clebsch-Gordan coefficients. The m a s s m a t r i x will b e block diagonal:

2 1 - F E R M I O N M A S S E S A N D H I E R A R C H Y O F S Y M M E T R Y B R E A K I N G 3.119

where A and R a r e Z x 1 and (a - 1) X (x - I) mat r ices , respectively. The two blocks completely decouple, i.e., the general ized Cabibbo angles between the two s e c t o r s vanish. The m a t r i x B can have a special substructure depending on the propert ies of the representat ions d ( l ) , I S L k. If, f o r ex- ample , d( '+') has dimension (n - Z), then B will b e diagonal and the corresponding (n - I ) m a s s e s will be degenerate. It often happens that in the break- down of with one Higgs multiplet, the re is only one independent VEV. In the c a s e of simply reduc- ible groups, a l l fermion m a s s ra t ios will then b e

given by Clebsch-Gordan coefficients, which a r e likely to be unreal is t ic , given the known fermion spectrum. In o rder t o obtain m o r e independent p a r a m e t e r s , without proliferation of Higgs bosons, one has to look f o r not simply reducible groups.

111. A SPECIAL EXAMPLE

In o rder to i l lus t ra te the s tatements of the p re - vious sect ions, we consider the t e t rahedra l g roup T which has been recently proposed by Wyler.l0 We r e s t r i c t ourselves to the c a s e of t h r e e genera- tions (n = 3)) and put L, u,, and d, i n t r iplets . Since 3 x 3 contains the 3 twice, @ i s chosen t o b e a T t r iplet a s well. he most genera l Higgs po- tential with a SU(2) x U(1) x T invariance i s given by

The Yukawa interaction is given by

Cd, =glEi~:;:dhR@I +g2ZiC:;L~lk R @ l + H.c., (3.2)

where c:;; = ( 1 / n ) l c i k l 1 and C:;:= ( l / d H ) ~ ~ ~ ~ , with s i m i l a r expressions fo r the up quarks. The actual symmetry of V ( @ ) tu rns out to b e much l a r g e r and i t can be shown to b e SU(2) x U(1) XO,. The group8 0, is a 48-element subgroup of O(3): 0, = 0 x Ci, where 0 denotes the octahedral group which i s isomorphic t o S,.

We f i r s t investigate the poss ib le minima i n a group-theoretical language. According to our s t rategy, th i s corresponds to looking a t the maxi- m a l subgroups that contain a s c a l a r i n the induced representat ions in one of the four 0, tr iplets . (There is a n ambiguity that these four t r ip le t s coincide with the T t r ip le t on T.) These maximal subgroups and the corresponding Frobenius de- composition t u r n out t o b e (in the notation of Ham- ermesh8)

(i) S, = D, corresponding to 3, - (1,2) , (ii) D, C 0, D, (Z T f o r 3 , - ( 1 , 1 1 , 1 " ) ,

(iii) C, fo r 3 ,-(1, lU,1" ' ) , (3.3)

(iv) D , d 0 f o r - 3 , i - (1 ,1" ,1 ' ) ,

(v) C, @ 0 f o r - 31i - (1, l ' , 1") .

F r o m these r e s u l t s one can read of, that i n a l l possible pat terns of symmetry breaking c o r r e - sponding to a @ t r ip le t , the re is only one scalnv in

1

the decomposition of the subduced representat ions. 'This in turn implies that t h e r e i s only one inde- pendent VEV and i t is then possible, f o r each case , to choose a b a s i s such that the minima of the clas- s ica l potential correspond to (0/@10) = ( v , 0 , O), In a different b a s i s , the nonvanishing VEV will b e related. The expression f o r V ( @ ) i n (3.1) is wri t - t en i n t h e T-triplet b a s i s which coincides with the natural b a s i s f o r the SO(3) triplet. In th i s b a s i s the minima of the potential a r e the following:

(a) (v , 0 , O), (0, v , 0), (0, 0 , u). These solutions have a remaining D, (C,) invariance f o r 3, (3,) - - breaking. T h e r e a r e th ree D, (C,) subgroups of 0 which correspond to these t h r e e solutions. The fourth D, subgroup of 0, which is simultaneously a subgroup of T , cannot b e reached by t r iplet breaking.

(b) (v, v , O), (v , 0 , v ) , (0, v , v) . These solutions correspond to D, symmetr ies which a r e not sub- groups of 0. If the Higgs potential would have had only an 0 symmetry these minima would c o r r e - spond to a C, subgroup. The decomposition of 3, with respec t t o C, contains two s c a l a r s and there - f o r e if i t were not f o r the accidental 0, symmetry , th i s C, subgroup would b e a maximal subgroup and allow f o r two independent VEV.

(c) (v , v , v ) , (-v, v , v ) , and permutations. These minima have a remaining S, symmetry and c o r r e - sporid to pat tern (i).

(d) (v , veiT13, ve'2"'3) and permutations c o r - respond to (v) and have a C, symmetry.

G . C . B R A N C O , I < . P . N I L L E S , A N D V . R I T T E N B E R G 2 1 -

The first stage of symmetry breaking will consider a n example in which a "perturba-

Among the pat terns of symmetry breaking pre - viously considered, we a r e specially in te res ted i n the solution that corresponds t o a D2 invariance. In part icular , we will show that i t is a good can- didate f o r a f i r s t s tage of symmetry breaking. F i r s t of a l l , one has to make s u r e that this f i r s t s tage can be achieved in a natural way, i.e., ~t has to be shown that f o r a range of values of the f r e e p a r a m e t e r s of the Higgs potential the D, solution corresponds indeed to a n absolute minimum of the c lass ica l potential. F r o m (3.1) i t i s s t raightfor- ward to check that (0 /$ /0) = (v ,0 ,0 ) i s a s ta t ionary point of the potential. In o r d e r to find the condi- tions f o r being a minimum, we have t o examine the corresponding Higgs-scalar m a s s matr ix. It tu rns out that a f te r we make the replacement (01$1O) = (v, 0,O) the Higgs-scalar m a s s m a t r i x i s diagonal, with diagonal e lements given by (we de- note $, = R, + LI?)

where we have used that 2vZ(X, + 4h,) = -$. In o r - d e r f o r (v, 0,O) to be a minimum the following con- ditions must be sat isf ied:

Next we will analyze the fermion m a s s matr ix. Using (3.2) and (014 10) = (v, 0,O) one obtains

and s imi la r ly f o r the up quarks. The m a s s ma- t r i x i s diagonal and a l l Cabibbo angles vanish. This i s to be expected f r o m our analysis i n Sec. 11, s ince the decomposition of 3, with respec t t o D, is (1, I t , 1"). Fur thermore , the u and d quarks a r e naturally m a s s l e s s a t th i s stage. We now con- s ider the next s tage of symmetry breaking, i n which D, is further broken. This could i n principle be achieved by ei ther taking into account quantum cor rec t ions to the Higgs potential (if they lead t o a fu r ther breaking), o r m o r e modestly by adding ex t ra Higgs particles. But independently of the mechanism which produces the breaking of D,, we

tion" along a given i rreducible representat ion of D, leads to a calculable Cabibbo mixing matr ix. We r e c a l l that the nontrivial e lements of 0, which leave the vacuum (v, 0,O) invariant a r e

We exchange now t o a b a s i s allowing f o r a direct decomposition into D, representat ions. This is achieved by performing a n/4 rotation i n the 2-3 plane. In the new b a s i s the m a t r i c e s (3.7) a r e given by

The decomposi t iol~ 3, - (1, l ' , 1 ") is t ransparent . Now we a s s u m e that the breaking of D, is along a given i rreducible representat ion, say Lt , i .e. , the new minimum i s given by

(o l$~o)=(v , JZ-€v ,O) , € < < I (3.9)

o r , in the previous bas i s ,

(0I$lO) =(u ,cv , -EV).

Using (3.2) and (3.10), one obtains f o r the fermion m a s s m a t r i x

((g+, +g-2)€2 K-RtE -g-g+c 'Lf(I"Md' = ?i2 I S - g t c g+, +€Zg-2 -g-g+e2 ,

2 2 -g -g tc -g-gte g- +€2g+2 I where we have introduced g, =gl i g,. The eigen- value equation can now b e used to re la te the quark m a s s e s to the p a r a m e t e r s i n the m a s s mat r ix :

where 6' = (812 - g22)/(g12 +gZ2). Assuming now that E << 1 one obtains

21 - F E R h l I O N M A S S E S A N D H I E R A R C H Y O F S Y M M E T R Y B R E A K I N G 3421

Using the ~ o b a y a s h i - ~ a s k a w a " parametr izat ion one obtains f o r the angles

with s i m i l a r expressions f o r s:. The purpose of th i s example was to i l lus t ra te how a s m a l l per - turbation around the D,-symmetric vacuum leads to physically interest ing resul ts .

We now a d d r e s s ourselves t o the following ques- tion: Can quantum cor rec t ions generate th i s sec- ond s tage of symmetry breaking, thus making the smal lness of the parameter E (or equivalently the smal lness of mu, m,) a natural feature of the mod- e l ? We f i r s t note that the D2 symmetry of the minimum (v, 0,O) is not a symmetry of the total Lagrangian. In par t i cu la r , i t can b e easi ly ver i - f ied that the Yukawa interact ions C, a r e not in- variant under (3.7) transformations. This is due to the fact that 2, is only T invariant and the par - t i cu la r subgroup of 0, considered h e r e i s not a subgroup of T. Since D, is not a symmetry of the total Lagrangian, the Georgi-Pais theorem6 does not apply and th i s r a i s e s the hope of generating light quark m a s s e s through radiative corrections. Unfortunately i t t u r n s out that i n the present ex- ample that is not possible. This can be seen in the following way: Among the D, mat r ices in (3.7) only the f i r s t belongs t o T. Together with the identity i t f o r m s a C, subgroup of T , to which the Georgi-Pais theorem applies. F r o m (3.4) i t is apparent that t h e r e a r e no non-Goldstone m a s s l e s s s c a l a r s (I, is the Goldstone boson needed to give m a s s to the neutral vector meson 2) a s long a s one does not choose nonnatural values of t h e A,, and there fore one concludes that the C, symmetry will b e respected by higher-order correct ions. On the other hand th i s C2 invariance is sufficient to guar - antee the f o r m (v, 0,O) f o r the minimum of the Higgs potential.

In view of previous analysis , we examine now the other tree-approximation minima listed i n (3.3). Among these , only the one corresponding to

l eads t o m a s s l e s s u , d quarks i n the t r e e approxi- mation. In th i s c a s e , 0, is broken to a D, subgroup whose nontrivial e lements a r e given by

This c a s e has the interest ing feature that none'of

these m a t r i c e s belongs to T. Before pursuing, we have to investigate if (3.16) can b e obtained in a natural way. The ~ i ~ ~ s - s c a l a r m a s s m a t r i x cor - responding t o (3.16) is given by

a 2 v a2v y = y = - 4 ~ ~ ( h ~ + h ~ ) , a c , ac,

where 2v2(hl +hz+X3) = -$, and the other nondiago- nal e lements vanish. Diagonalizing (3.18) gives the following s c a l a r m a s s e s :

,W1=8v2(h1+h2+h3), 1Wl2=8v2(3h2-h3),

M3= -8v2(X3 + h 4 ) , .1.T4=0, (3.19)

n/l,=4v2(A3- 3h2), ~ ~ = - 4 ~ ~ ( 3 ~ ~ + ~ ~ + 2 h ~ ) .

A s i n the previous case , t h e r e i s one m a s s l e s s s c a l a r which is a Goldstone boson. In o r d e r that (3.16) i s a minimum, a l l the other eigenvalues in (3.19) should be positive. It happends h e r e by ac- cident that th i s cannot b e achieved in a natural way (i.e., fo r a finite range of the f r e e p a r a m e t e r s of the theory) s ince ;ll, ,- 0 and 2W,2 0 lead to1'

- -

F r o m the previous discussion, we conclude that i n the present example i t i s not possible to gener- a t e a hierarchy of symmetry breaking through radiat ive correct ions in a natural way. However, we feel that the possibility of avoiding the Georgi- Pa i s theorem through a f i r s t - s tage breaking into a group which i s not a symmetry of the total La- grangian is sufficiently at t ract ive to deserve f u r - t h e r attention.

ACKNOWLEDGMENTS

We would like to thank Ling Fong Li , A. Pa i s , L. Wolfenstein, and D. Wyler f o r helpful discus- sions. The r e s e a r c h of G.C.B. was in par t sup- ported by the U.S. Department of Energy. H.P.N. thanks F. J. Gilman for the hospitality extended to him a t SLAC. This work was a l so supported i n par t by the U.S. Department of Energy under Contract No. DE-AC03-76SF00515. H.P.N. has received a Max Kade Fellowship.

3422 G . C . H R A Y C O , H . P . N I L L E S , A N D V . R I T T E N B E R G 21 -

or a revlew of the subject, 'cogether with a complete l ~ s t of references, see R. Gatto, University of Geneva Report No. UGVA-DPT 1979/04-199 (unpuhl~shed).

'F. LVilczelr and A. Zee, Phys. Rev. Lett. 42, 421 (1979). 3 ~ h e r e is another motivat~on for hav~ng the breaking of

the discrete symmetry occurring in stages: In rnodels with natural flavor conservation in the IIiggs sector, the choice of generation symmetry is restricted to some special groups (see Ref. 4) and it has been shown that in the t ree approximation one is led to a physically unacceptable mixing matrix (all mixing angles a r e either 0 o r 7r/2). For the difficulties one encounters in actually tnlplement~ng this idea, see G. SegrB and H. \Veldon, Phys. Lctt. 8(iB, 29: (1979).

4 ~ . bar bier^, I<. Gatto, and F. Strocchl, Phys. Lett. 74B, 344 (1978); R . Gatto, G. Morchio, and F. Stroc- - cht, ibid. je, 263 (1979); G. ~ e g ~ 6 and H. Weldoa, Ann. Phys. (N.Y.) 124, 37 (1980).

'H. Georgi and A. Pais, Phys. Rev. D g , 3520 (1977). 6 ~ . Georgi and A. Pais, Phys. Rev. D g , 1246 (1974). 's. Weinberg, Phys. Rev. Lett. 2, 1698 (1972). *We use the notation of Rl. Hamermesh, Grmp Theory

(Addison-Wesley, Reading, Rlassachusetts, 1962). 's. Weinberg, Phys. Rev. Lett. 19, 1264 (1967); A. Sa-

lam, in Elementary Par t ic le Physics, Relativistic Groups and Analyticity (Nobel Symposium No. 8 ) , edit- ed by N. Svartholm (Almqvist and Wiksell, Stockholm, 1968), p. 367.

'OD. Wyler, Phys. Rev. D g , 3369 (1979). "~LI. Kobayashi and K. Maslrawa, Prog. Theor. Phys.

49, 652 (1973). '"en 3h2= h3, i t turns out that the minima correspond-

ing to (v , 0, O), ( l / f l ) ( v , v , O), and (1/16)(v,v,v) a r e degenerate. This is due to the appearance of an SO(3) symmetry which implies that any (vl ,v2,v3) with vI2 +v22+v32=v2 is a minimum.