Quark mass matrices without small parameters

Nuclear Physics B296 (1988) 717-731 North-Holland, Amsterdam

QUARK MASS MATRICES WITHOUT SMALL PARAMETERS* Howard GEORGI and Lisa RANDALLt

Department of Physics, Boston University, Boston, MA 02115', USA David A. KOSOWER

Department of Physics, Columbia University, New York, New York 10027, USA Received 20 July 1987 In the context of composite quarks, leptons and Higgs, we build models of the quark mass matrix in which the only small parameters are powers of flavor gauge couplings. The interesting structure of the mass matrix results from the form of the approximate global symmetries in the models.

1. ~ u ~ o n

The lack of explanation for the bizarre spectrum of quark and lepton masses and quark mixing angles is one of the strongest indications of structure beyond that included in the standard model. Large mass ratios and the existence of small but nontrivial mixing angles can only be satisfactorily explained in a theory whose interactions distinguish among flavors. The apparently random flavor structure might then be understood as the low energy consequence of symmetries which existed at a higher scale. In this paper, we illustrate this possibility in a class of composite models in which the light fermions are composite states interacting via gauged flavor interactions. The symmetries in the theory determine the masses and mixings, which occur at different order in flavor coupling. We thereby reproduce quark mass ratios and mixings, without fine tuning or introductory arbitrary fundamental parameters. However, a model of flavor must not only demonstrate how mass mixing occurs, but must also explain why flavor changing neutral currents do not. Our models all employ the composite Higgs scenario [2], in which the compositeness scale is * Research supported in part by the National Science Foundation under grant # PHY-82-15249, preprint # ' s BUHEP-87-22, HUTP-87/A049. Supported by a Bell Laboratory GRPW Fellowship. * Permanent address, Lyman Laboratory of Physics, Harvard University, Cambridge, MA 02138, USA. 0550-3213/88/$03.50©Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

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separated from the weak scale. This allows us to suppress dangerous flavor gauge particle exchange interactions, but only at the expense of a large fine tuning of the ratio of scales. We find this to be a highly unsatisfactory feature of these models, which we hope to eventually eliminate. We defer further discussion of the flavor changing neutral currents to the conclusion. We first present the idea and specific examples which demonstrate the existence of models with flavor structure which reproduces quark masses and mixings. Some readers may find the models we describe here a bit complicated. We regard them as existence proofs. We have not tried to make them simple and elegant, but rather have chosen them to make the flavor physics we describe explicit. We describe our class of composite models and review the composite Higgs scenario in sect. 2. In sect. 3 we review the mechanism that leads to the approximate global symmetries which are instrumental to our models' success. Sect. 4 illustrates the generation of quark masses in a simple model which is more elegantly implemented in the following section. An alternative model, in which the leading order approximate global symmetry breaking required for mass generation occurs in the masses themselves, is outlined in sect. 6. We conclude with a discussion of flavor changing neutral currents.

2. CompositeHiggs We work in the context of composite models, in which both the low energy fermions and the Higgs are composite states of ferrnions which are strongly bound at the scale A. To determine the low energy interactions, we do not need to know the full strongly interacting theory. We need only to specify the global symmetry group of the high-energy theory, G; the subgroup of G left invariant below the compositeness scale by the strong interactions, H; the weakly gauged subgroup of G, Gw; and the transformation properties of the fermions q~ under the unbroken subgroup H. To construct the low energy effective chiral lagrangian of the composite Goldstone bosons and fermions, we will employ the formalism of Coleman, Wess, Zumino, and Callan [4]. According to the CWZC procedure, one constructs the effective chiral lagrangian from the following matrix function of the Goldstone boson fields: ~ = e i~x°/I° ,

(1)

where X a are the generators of G / H , % are the associated Goldstone boson fields, and f~ are constants of order of magnitude A/4~r, where A is the compositeness scale. Under a rotation by g ~ G, ~ transforms as ~' = g~h-t(~r),

(2)

where h ~ H is defined by the above equation. The fermions transform under the

H. Georgi et aL / Quark mass matrices

719

nonlinearly realized symmetry, H. The CWZC procedure instructs us to build the most general G-invariant function of ~, Lk, the weak-gauge covariant derivative D,, and the weak charges T ~, where the transformation properties of the T ~ are taken to be T " ~ g T ~ g - 1. This incorporates the symmetry breaking due to the weak gauge interactions. For our purposes, we will be concerned only with the potential, and so need not worry about derivatives. This procedure is often most easily implemented by constructing, out of ~ and ~b, fields which transform linearly under the G symmetry. The non-linear realization of the H symmetry ensures that different choices for these linearly-transforming objects will lead to the same matrix elements for physical states. For example, if the fermion representation is such that ~k transforms under a G transformation by

(3) then '/" = $~k will transform linearly under G. If P is an H-invariant matrix, .~ = ~P~* will transform linearly under G. If either G or H is semi-simple (as will be the case in all the models we discuss), it is possible to break up ~ into smaller matrices which transform irreducibly under G and H. Constructing the various independent linearly-transforming objects out of these smaller matrices corresponds to using the different possible choices for the projection matrix P. In the models we present, the fermions are protected from acquiring a mass of order A by a surviving chiral symmetry, a subgroup of H. Of course, at some order in the weak gauge coupling constant g, this surviving symmetry must be broken. In the CWZC language, the symmetry will be reflected in the vanishing of candidate zeroth-order mass terms such as m

f~=~

(4)

and its breaking in the (eventual) appearance of mass terms proportional to a power of g 2, the simplest of which might be g2f'~Ta~"~ffaffl.

(5)

In our models, the situation is somewhat more complicated; the fermions are protected from acquiring a large mass both by the usual electroweak gauge interactions, and by additional chiral symmetries, which are explicitly broken by the gauged flavor interactions we introduce. The electroweak interactions are broken by a composite Higgs [2]; the role of the flavor interactions will be to introduce Yukawa couplings of the fermions proportional to various powers of g2. The additional chiral symmetry will be reflected in the lack of order O(g °) couplings of the fermions to the Higgs. In the models in this paper, the Higgs doublet is a pseudo-Goldstone boson associated with the spontaneous symmetry breakdown of G to H. The potential

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arises from weak gauge exchange, and is determined by the pattern of spontaneous symmetry breakdown. Consider one example of such a model, taking G = SU(3)× U(1)× SU(3)c breaking to H = SU(2) × U(1) x SU(3)c. SU(3)c is the gauged color group, which is unbroken, and can be neglected in our analysis of the potential. Gauge an SU(2) × U(1) × U(1) subgroup of SU(3)× U(1) generated by T2b, Tt and Ta, embedded in U(3) as

/ °°/ / ) /o o) ob

,

0

0

1

0

0

0

1

0

0

0

0

1

,

0 1

0

,

(6)

0

with associated gauge couplings g2, gl, and ga. As in ref. [2], we can demonstrate the existence of the desired minimum in which electroweak symmetry breaking occurs. Following the CWZC procedure, we define the 3 × 3 matrix X = (X2X3) to be the exponential of the broken generators, where X2 is a 3 × 2 matrix and X3 is a 3-dimensional column vector. Notice that under SU(3) rotations, X3 transforms as X3 - - * gx3, where g ~ SU(3), and that the first two components of X3 transform as a weak doublet. For the composite Higgs scenario to work, we need the minimum of the potential to occur when

(x3)

=

0

,

(7)

1 + O(vZ/f z) where v is the electroweak scale, and f is defined as in eq. (1). This is possible when the axial charge (B) is suitably chosen, in which case we obtain a small ratio of v to A. Since in this model, the charge (but not the coupling) is fine tuned, the magnitude of ga need not be related to the electroweak couplings. In the full model which incorporates fermions and flavor, there might be separate chiral factors of G and H associated with the weak interactions (as in the above model), or the weak and flavor gauge interactions can be embedded in a common simple nonabelian factor of G. In either case, since we are interested in the flavor structure, we will suppress the weak interactions in our discussion of the flavor groups. 3. Approximate global symmetries

In our models, fermion masses are protected by approximate global symmetries. A nontrivial fermion mass structure arises because the distinct approximate symmetries are broken at different order in the flavor gauge coupling. In this section, we briefly review the mechanism which leads to approximate global symmetries and

H. Georgi et al. / Quark mass matrices

721

outline the model which was presented in ref. [1]. In the following section, we will demonstrate how approximate symmetries are used to generate interesting mass matrix structure. In a composite model, the global symmetry group, G, the unbroken subgroup, H, the weakly gauged subgroup, and the fermions' transformation under H determine the form of the allowed interaction terms in the potential. Because the Goldstone boson potential would be flat were it not for the G-breaking weak gauge boson exchange contributions, the potential can always be expanded in powers of the weak gauge coupling. Approximate global symmetries (AGS) occur when the lowest order potential and vacuum alignment respect a symmetry which is not a symmetry of the full theory. If models with low energy fermions are to manifest an AGS, the lowest order fermion interactions must also respect this symmetry. In ref. [1], we analyzed the following model, which we will call a tadpole sector: G / H = SU(2) x SU(2)'/U(1)

(8)

in which T[']" generates SU(2)['], and H = U(1) is generated by T 3 + (n + 1)T '3

(9)

(in the sense of eq. (1)) where n is a positive integer. The gauged SU(2) is a subgroup of G generated by Tsa = T " ~ T '". We will demonstrate the presence in this model of a symmetry of the lowest order potential (and vacuum) that is broken by O(a n+l) potential terms. In our models, we will assume that the flavor gauge couplings are all equal to gF, and define e = gg/16~r 2. As in eq. (1), we define the Goldstone boson fields as ~- -- e'"o~ = (ft, - ~ , )

.~[=e'~'"r"=(ff,-ff),

(10)

where ~t = iOEf* and T a and T '~ are the generators of SU(2) and SU(2)'. Due to the gauged SU(2) symmetry, we are free to choose (ft)-- (1).

(11)

The leading order alignment of f[ is then determined by the unique lowest order term in the potential, which is A

, f,t Tgaftft,t T~aft"

(12)

The sign of A, which determines the relative alignment of ft and f[, is assumed to be negative. Therefore,

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H. Georgi et al. / Quark mass matrices

so there exists a global symmetry of the lowest order potential under which ~t and ~/ transform as: ~t ~ e(1/2- r3)°~/,

~; --* e0/2- r'3)o~/.

(14)

However, at higher order, the symmetry is broken. The lowest order term which does not respect the symmetry is

e"+ x~"trg~'~tT;(( ~tTb~'*Tb~')n.

(15)

Eq. (15) contains a tadpole for a linear combination of pion fields. If we maintain our choice of gauge and eliminate this tadpole, the ~' VEV at higher order is (~/> = ( l n ) .

(16)

If we add to this model additional particles, which transform under the same gauged SU(2) symmetry as ~, and ~/, the full theory manifests an AGS iff the lowest order interactions of all the fields preserve a global symmetry. However, the higher order interactions of the additional fields do not respect this symmetry, which is not respected by the higher order ~/alignment. An example with "tadpole" fields and additional fields which transform under a common gauge symmetry is presented in the following section, in which the tadpole sectors break the global symmetries that protect fermion masses. 4. Model 1 We now demonstrate the origin of fermion masses in these models. For simplicity, we restrict our attention to quarks. At lowest order in gauge coupling, the quark masses are protected by global symmetries. The order at which they are broken determines the magnitude of the masses and mixing angles. We demonstrate how this occurs in the following straightforward model. Suppose the global symmetry group G breaks to a subgroup, H, where G = SU(4)L )< SU(4)u X SU(4)d

H = su(a)L x su(a)u x su(a)d.

(17)

The composite fermions transform under the H-symmetry as: l ~ L ~-

(3,1,1),

UR =

(1,3,1),

DR= (1,1,3).

(18)

The following six SU(2) subgroups of Gf are gauged. SU(2)L[u,d]

generated by aLiu,dl ® 1/2,

SU(2)~/u,dl

generated by I ® eLIu,dl/2 ,

when a and T are commuting sets of Panli matrices.

(19)

H. Georgiet al. / Quarkmassmatrices

723

We analyze the potential for this model according to the CWZC procedure. We define "~Ltu,dl= (~3~)Ltu, al = ei'~°r~u'd~, where T ~ are the broken SU(4) generators, the ~3's are 4 × 3 matrices, and the ~'s are 4 dimensional vectors which transform linearly under G. We exploit the SU(2) × SU(2) gauge freedom to parameterize ~ as

The leading order vacuum alignment is determined from the nontrivial O(e) terms in the potential. For appropriate signs of the chiral coefficients, we get:

(~L) = (~u) = (~a) =

•

(21)

Notice that this alignment preserves symmetries of the lowest order potential under which ~Ltu,al transform as: ['] L[u,d] "--) e - 1/2- T['l~tu' ~)~[,] Ltu,d].

(22)

We proceed to the construction of the fermion Goldstone boson Yukawa couplings. The fermions transform under the nonlinearly realized symmetry, H, but we can construct the Yukawa couplings from the product, (~3ff)LEu,dl, which transforms as a four-dimensional representation of SU(4)LEu,d1. To construct a nontrivial Yukawa coupling which is consistent with this transformation, we must incorporate the breaking of these SU(4) groups by their weakly gauged subgroups. A Yukawa coupling, generated by exchange of gauge bosons associated with both the left- and right-handed fermion, first occurs with a coefficient of order 4~re2f. It is "t a t t b~o~ord t b~o3DR. ~L ~r3T~L~L T ~a L ~,tTd

(23)

where we have suppressed the coupling of the composite Higgs. Since (~tTa~)rtu, dl is nonzero only for T~tu,dj=r3Ltu,dl, the O(e 2) down quark masses are zero. This is a consequence of the approximate global symmetry, which is also preserved by the fermion interactions, under which: = ~3~ ~ ei(-1/2- T3)¢~ •

(24)

We conclude that in order to generate masses for the quarks, we will need to break the approximate global symmetries. In this example, we implement this breaking by including separate "tadpole" and "linking" sectors in the model, as we now explain.

724

H. Georgi et al. / Quark mass matrices

The full model (suppressing color and weak interactions) is: G / H = SU(4)L/SU(3)L × SU(4)a/SU(3)d × SU(4)JSU(3)~ × W × L, (25) where W and L contain tadpole and linking sectors which violate the approximate symmetry. There are 6 approximate global symmetries. W contains six tadpole sectors to break each of these symmetries. Each of the six gauged SU(2) groups in these tadpole sectors is identified with one of the six gauged SU(2) groups in G above. For example, one gauged SU(2) group acts on both the Goldstone bosons of a tadpole sector and on ~d- The lagrangian therefore contains terms such as

e~dT~d~JT~,d,

(26)

where ~t'a is the field in the tadpole sector defined in eq. (10). Recall that

(~d) = (el,,d),

(27)

where n a is determined by the U(1) symmetry in the tadpole sector. Therefore, eq. (26) contains a tadpole for the second component of ~d. When we rotate away this tadpole, the ~d field alignment, up to factors of order unity, is

(28) The remaining tadpole sectors introduce tadpoles in the second and third components of ~L, ~u, and ~d" When they are rotated away, the vacuum alignments of these fields are ~ n L[u,d] + k L[u,dl

(~Ltu,d]) =

eku"'d' En L[u,d]

(29)

1 where nLiu,d] and kLtu,dl are determined by the U(1) symmetries in the appropriate tadpole sector. The tadpole sector Goldstone bosons can also couple directly to the fermions. This occurs in the following equation. -"[" a t~" a p p b p r[" b p ~L~L3T~.~L~,Lr~tL~Td ~ d Td~d3DR.

(30)

When we replace the Goldstone boson fields by their VEVs, we discover that the terms in eq. (29) and eq. (30) give the bottom quark a mass of order 4~roenL+nd+2,

725

H. Georgi et al. / Quark mass matrices

where the factor 4~rv follows from dimensional analysis [3] (the analysis in ref. [3] was done for the low-energy chiral lagrangian of QCD, but since it was performed entirely in the context of the low-energy theory, analogous dimensional analysis will apply here), and we have assumed there is no additional e suppression for the composite Higgs (that is the composite Higgs, which has been suppressed for simplicity, is in the same chiral factor as the flavor groups). Notice that although we can do a gauge rotation to eliminate the mass contribution from one of the above two terms, we cannot eliminate both, so the above estimate of the bottom quark mass is correct. The tadpole sectors completely break the accidental symmetries of the low order theory. Alternatively, we could have broken the tensor product of two global symmetries acting on the left- and right-handed fermions to a single symmetry generated by the diagonal sum of the initial two symmetries. L contains four linking sectors which mix the SU(2)[']L and SU(2)[']dtu 1 gauge bosons. Each linking sector is of the form /-1 G / H = I-I SU(2),L × S U ( 2 ) , r / S U ( 2 ) , v . (31) i=1 The ! SU(2) groups generated by T 1 = T{L, T2 = TI~R+ T2~L..... Tt = Tta- 1R are gauged. Below the scale of spontaneous symmetry breakdown, there is a single remaining gauge symmetry which is generated by T1 + .-. + Tt. In the linking sectors in L, the gauged SU(2) generated by T1L is identified with SU(2)L or SU(2)[ and the gauge group generated by Tt_IR is identified with SU(2)u[d ] or SU(2)~id ]. The linking sectors therefore permit construction of mass terms of the form etut", ~L~taLTa~L~(d]Ta~3utdlV[ D ]R,

(32)

and corresponding terms with primed fields. We can now determine the up and down mass matrices. Up to factors of order unity, they are M~dl =

4~rveE%l,

(33)

where

Eu[d] =

d1

nL+l'

kL+l

n R + ]'

d2

kL+ nR+ 2

kR+l

nL+kR+2

,

(34)

d3

with d 1 = min(l + 1', kL+ k R + 2 + 1, n L + n R + 2 + l', kL+ k R + n L + nR+ 2),

d2=min(kL+kR+2, l"), 1' = min(k L + k R + 1, l'),

d3=min(nL+nR+2, l ), 1= min(n L + n a + 1, l).

(35)

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H. Georgi et al. / Quark mass matrices

We take e = ~ which yields: m t = 25 GeV ( = 4~'e 3 x 250 GeV), m b -- 5 GeV, m c = 1 GeV, m s = 200 MeV, m u - m d -~ 8 MeV, 01

=

1,

02 = 03 = ~ .

(36) 5. M o d e l 2

In this section, we briefly describe how the above model could be more elegantly implemented without separate tadpole generating sectors. We again restrict our attention to the flavor structure of the model. The initial symmetry group, G, breaks to a subgroup, H, where G / H is ((SU(6)/SU(3))L × (SU(6)/SU(3))D x (SU(2) x SU(E)')/U(1)F × (SU(4)/SU(3)) X L~ × L~.

(37)

L~ and L~ are the linking sectors defined in the previous section, and the superscript indicates the gauged SU(2) group with which the "left" gauge bosons mix while the subscript indicates the order in gauge coupling of the associated fermion mass. The fermions transform under the preserved global group SU(3)L X SU(3)D × SU(3)u X U(1)F as V'L = (3, 1, 1,2aL + l ( k L - - 2nL)), D R = (1,3,1,2a D+ l ( k D - 2riD) ), UR = (1,1,3,1),

(38)

where n[k]L and n[k]~td J have opposite sign. The U(1)F is chosen to generate a nontrivial vacuum alignment, analogous to that in the previous model. Following the by now standard procedure, we associate a set of .~ fields with each of the factors of G by exponentiating the broken generators. We define: -u

~ __ = (~ 3 ~4 )U--LiD1(~3~'~5~6)Lt01"

(39)

The ~3's are 6 × 3 matrix fields and the ~i for i = 4,5,6 are 6-dimensional vectors which transform under the nonlinearly realized U(1)F symmetry as ~ i ~ g ~ h [ 1,

H. Georgi et aL

/

727

Q u a r k m a s s matrices

where h i is generated by 0i: 03LID ] = - - a L t D ] q- ½ ( 2 n L [ D ] 05L[D ] = aL[D] -- he[D] ,

Oiu = 0

kLtDI),

04L[D] = aLtDI ,

06L[D ] = aL[D] -t- kL[D] -- nL[D] ,

for i = 3,4.

(40)

The constants a l p ] are chosen so that G-invariants constructed using an e-tensor contribute to the potential at high order, and can therefore be neglected. The U(1) charge of the ~ fields of the SU(2) factors (defined as in eq. (10)) are chosen so that S = ~tTa(l~'tTa( ' has unit charge. We assume the lowest order potential is minimized when

(~4}=

(i (0'/ (0'/ ~ ,

{~5)= 0 ,

(~6)=

~ .

(41)

Notice that with this alignment and our embedding of the gauge generators are such that ~4 plays the role of ~ in the previous model. In fact, this model also preserves six AGS's under which: ~L[u,d] ~ eiT['13~L[u,dl(~[u,d])e-iT[']3(~L[u,d]).

(42)

At higher order in gauge coupling, these symmetries are broken. Up quark masses are generated through the linking sectors as before. However, the tadpole sectors are no longer necessary, since the vacuum in the left and down sectors realigns itself at higher order. For example, since S = ~ t z a ( ~ t t z a ( ' has unit charge, ~t4Ta~4~t4Ta~ssn is U(1)F invariant. This induces an O(e n) tadpole for the fourth component of ~4. There are similar terms to determine the full vacuum alignment. When we rotate away the tadpoles, we get: f

F

ek+n

Ek + 2n

ek 4

Ek+n

Ek - n

ek

5

E2n

en

] en

LIDI

"nI

en

1

L[DI

en

6

1

(43)

$k ek-n

ek

L[D]

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H. Georgi et al. / Quark mass matrices

This vacuum alignment results in the same mass matrices we obtained in the previous section, with the exception of the (1, 3) and (3,1) components of the up mass matrix, which occur at one higher order in ~, because there are mass terms constructed with both ~4 and ~6 which contribute to this entry, but a gauge rotation can only eliminate one of these contributions.

6. Model 3

Consider a model of quarks in which the original symmetry group is G = SU(6)L x SU(6)u x SU(6)d × SU(3)c × SU(3) × U(1)

(44)

spontaneously broken down to H = SU(3)L × SU(3)u x SU(3)d × U(1)F × SU(3) c × SU(2) X U(1).

(45)

The U(1)F is a subgroup of the original SU(6)L X SU(6)u x SU(6)d generated by a charge QFL + QF~ + QFd, where QFk for k = L, u or d is a charge in SU(6)k of the ~rm: 'qk 0 0 Q Fk -~ 0 0 0

0 qk 0 0 0 0

0 0 qk 0 0 0

0 0 0 rlk 0 0

0 0 0 0 r2k 0

0 0 0 0 0 r3k

(46)

The LH quarks transform under H as ~kL - (3,1,1, SL, 3, 2, {),

(47)

where s L and ~ are the flavor (F) and electromagnetic U(1) quantum numbers, respectively. The RH quarks transform as: UR-- (1,3,1,Su,3,1, ~),

D R - (1,1, 3, sa,3,1, -- ½).

(48)

One of the crucial elements of the model is the behavior of these fields under the unbroken U(1)F. The other U(1) is simply chosen to reproduce the appropriate electroweak quantum numbers. But the U(1)F quantum numbers further restrict the form of the allowed terms in the effective lagrangian, giving rise to a nontrivial structure of the quark mass matrix. The other crucial element is the structure of the weak gauge group. This must contain SU(3) × SU(2) × U(1), of course, and something to destabilize the SU(2) ×

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H. Georgi et aL / Quark mass matrices

U(1) invariant vacuum (as is sect. 2), but must also contain the flavor gauge group that is directly responsible for building the Yukawa couplings. Here we take the flavor gauge group to be SU(2)L X SU(2)u X SU(2)a, where SU(2)k is a subgroup of SU(6) k under which the 6 transforms irreducibly, as a 6 of SU(2). Our only reasons for choosing this particular gauge group are that it makes the effective lagrangian relatively easy to analyze, and, we hope, to understand. We take the U(1) F charges defined in eqs. (46)-(48) to be those given in the table below:

k

Sk

qk

L u

5

-2

rkl

rk2

rk3

2

--3

-4

-1

d

1

"5

(49)

6 4

-6

As in sects. 3 and 4, we define a set of fields,

.=(~ F, ~1, ~2, ~3 ) Ltu, dl, --Liu,d]

(50)

in terms of which we can describe the vacuum. We find that for a range of parameters in the effective lagrangian, the lowest order vacuum has the form:

(Llu.,j)

=

F 5= ( t2L[u,d].

(i)l, 00,/ /i/ /i/ .

0

(~.tu.,l)

0

0

"

=

.

~j3

(Ltu.d~)

=

(51)

0 0 Assuming that eq. (51) gives the right zeroth order vacuum, we can use the techniques described in the previous two sections to construct the higher order terms that generate the quark mass matrix. The new feature here, aside from the relative simplicity of the flavor gauge couplings and the vacuum structure, is that fact that no additional tadpole or linking sectors are required. Everything except the composite Higgs structure is contained in the fields of eq. (50). The resulting matrix

730

H. Georgi et (11./ Quark mass matrices

of Yukawa couplings has the form:

(52)

7. Conclusions No one yet understands the origin of the perplexing quark and lepton mass hierarchy. It is therefore worthwhile to explore models which might elucidate the nature of flavor. In this paper, we only studied part of the problem, the patterns of spontaneous symmetry breakdown which could reproduce the known flavor structure, without specifying the full strongly interacting theory. We discovered a class of models in which approximate symmetries distinguished the quark flavors. The mass matrices which resulted reproduced the experimentally determined masses and mixings. However, any theory of flavor must also explain why flavor changing neutral currents (FCNC) are suppressed. The model in sect. 6 illustrates the possibility of generating the Cabibbo angle in the up sector, where the FCNC constraint is less severe. In addition, in this model, right-handed angles are small, so FCNC generated by gauge exchange between right-handed quarks is suppressed. We can also trivially modify the above theories to implement a “technigim”-like scenario, in which quark masses and FCNC are suppressed by quark mass ratios (in addition to gauge couplings). However, in the composite models we have studied, the compositeness scale is never reduced by more than a factor of l/&. This is not much suppression. Even with the Cabibbo angle in the up sector and a “ technigim”-like mechanism, the compositeness scale is over fifty times bigger than the weak scale, well beyond experimental detectability. We expect we can only reduce this scale if it is possible to take advantage of the nonabelian flavor symmetry to rotate away FCNC. However, we have not yet discovered how to do this without increasing the number of parameters or scales. Until the mechanism for producing flavor mixing without flavor changing neutral currents in experimentally testable models is found, the flavor problem will remain a mystery. Note added This work was done before the idea of a GIM mechanism for technicolor model discussed in ref. [5]. There seems to be no reason why the mechanism discussed there for suppression of FCNC cannot be extended to composite Higgs models. In the resulting models, the kind of mechanisms we discuss in this paper could be

H. Georgi et al. / Quark mass matrices

731

responsible for preon masses, at a large energy scale, rather than directly producing quark and lepton masses. References [1] [2] [3] [4]

H. Georgi, D. Kosower and L. Randall, Phys. Lett. B, to be published H. Georgi and D. Kaplan, Phys. Lett. 136B (1984) 183 H. Georgi and A. Manohar, Nucl. Phys. B234 (1984) 189 S. Coleman, J. Wess and B. Zumino, Phys. Rev. 177 (1968) 2239; C. Callan et al, Phys. Rev. 177 (1968) 2247 [5] R. S. Chivukula and H. Georgi, Phys. Lett. 188B (1987) 99