Quark mass matrices in a susy composite model

Volume 149B, number 1,2,3

PHYSICS LETTERS

13 December 1984

QUARK MASS MATRICES IN A SUSY COMPOSITE MODEL ZHOU Bang-Rong Department of Modern Physics, China University of Science and Technology, Hofei, Anhui, The People's Republic of China 1 and Stanford Linear Accelerator Center, Stanford University, Stanford, CA 94305, USA and J.L. LUCIO M. 2 Departamento de fisica, CINVESTA V IPN, Apartado Postal 14-740, Mexico D.F. 0 7000, Mexico Received 30 April 1984 Revised manuscript received 30 July 1984

It is shown that the ZI0 symmetry which survivesthe breaking of a combination of the axial UA(1) and the R symmetry in a susy composite model, can lead to the relation tg 0c ~ (rod/ms) 1/2.

Recently it has become apparent [ 1 - 4 ] that the horizontal symmetries, which can help to solve the generation puzzle, could originate in a more fundamental theory; i.e. a theory in which quarks and leptons are considered as composites of more fundamental constituents-preons. For example, a chiral discrete family symmetry appears as the remnant of the UA(1 ) after hypercolor instanton effects are taken into account in the rishon model [I ]; also a continuous (even gauged) family symmetry appears in some threefermion models [4]. The quark mass matrices have already been discussed in the case of a discrete symmetry with encouraging results [2,3,5]. In this note we will study the quark mass matrices within a susy composite model. This model was proposed in ref. [6] but we discuss it here in a slightly different way. The analysis is based on the chiral discrete symmetry Z10 present in the model (see below). The model we consider has the hypercolor gauge group SUH(3 ) and flavor number n = 8. Such a selection is subjected to the asymptotically free constraint on SUH(3) [6]. Thus preons are components of the 1 Permanent address. 2 Supported in part by COSNET, SEP. 152

chiral superfleld [7] ~ a- - a ( ~ ) a containing a complex scalar #~a ( ~ a ) and a left-handed ffa_a (fight-handed tb~a) Weyl spinor (a = 1,2, 3 is the hypercolor index whereas a = 1,2 ..... 8 is the flavor index). The gauge bosons Wb belong, together with the gauginos kb, to the vector superfleld. The global flavor symmetry of the model is G F = SUL(8 ) X SUR(8 ) X Uv(1 ) X U(1)a.f,

(1)

U(1)a.f being associated to the anomaly-free current J Y (1)a'f = -2~us rUA(1) + 4jU(1)R .

(2)

The other anomalous current can be any linear combination otJUA(1) +/~jy(1)R except t~//~ = --5/8. Consequently no definite discrete symmetry survives the anomalous U(1)'s breaking induced by the hypercolor instantons and we have omitted it in (1). In a susy theory, the chiral symmetry in G F will only be partially broken on the scale A n . This has been further strengthened by a recent effective lagrangian approach of susy QCD, there being more flavors than colors [8]. Thus we assume that the hyper. color is able to form the scalar condensates

. .~a' i..~. \ eo•,,,'.d3#',,.,.od..• e \v¢_ ~u+#i~/_ o/+#,i/ = Oh

,

t

= Oh,

13 December 1984

PHYSICS LETTERS

Volume 149B, numbe~ 1,2,3

i = 1,2, 3 (no summation over i ) , i = 4,

(3)

then the symmetry G F will be broken down to H F = SUc(3 ) X SUL(2 ) X SUR(2 ) X Uv(1 ) X U~r(1)X Z10 ,

(4)

[here we have changed the notation ~ = 1,2..... 8 _~ ~ i , ¢~= 1,2 corresponds to the subgroup SUE(2 ) X SUR(2), i = 1,2, 3 to SUc(3 ) and i = 4 to U~r(1)]. We note that the condensate (3) breaks spontaneously U(1)a_ f leaving only the discrete symmetry Z10. This can be seen from table 1 where preon representations under SUn(3 ) X H E are given. We will

not distinguish between the Z10 and U(1)a_ f charges of the particles and therefore a Z10 singlet has U(1)a-f (Z10) charge equal to an integer multiple of 10. Composite fermions come from the three different kinds of configurations (~0~0), (~q~), and (~0~*). They form representations of the flavor group H F listed in table 2. However, the simplest bound states (~b¢*) have been omitted from table 2 because they contain no physical quarks and leptons under the considered flavor symmetry-breaking pattern G F ~ H F. We specify an index for each representation in table 2 which has to satisfy the anomaly consistency equations [9]. In the present case we have two anomaly consistency equations from the three-point functions [SUL(2)] 2Uv(1 ) and [SUL(2)] 2U~(1) as follows:

Table 1 Preons representations under SUH(3) × HF.

q~3-,~1 ~3+, ~1+ ~03-, 01 ~k3+, ~01+ h

SUH(3)

SUc(3)

SUL(2)

SUR(2)

UV(1)

U~r(1)

Zlo

3 3 3 3 8

3,1 3,1 3,1 3,1 1

2 1 2 1 1

1 2 1 2 1

1 1 1 1 0

a, a, a, a, 0

-5/2 5/2 3/2 -3/2 -4

-3t~ -3a -3a -3t~

Table 2 The representations and indices of left-handed composite fermions. Composite fermions (qJ3-) 3 qJ3-(qJ 3+)2, ~ 3+~3+~3-

SUH(3) SUc(3)

SUL(2) SUR(2) UV(1)

U~(1)

1

2

3c~

1+8

1

3

Indices

ZlO

111, llS

9/2 -3/2

121,128

~3-(~a~:) 2

131,138

($1-) 3

1] 1

t-(~ 1+)2 , ~ 1+~1+~1I-(~I :I:) 2

1

1

2

1

3

-9,'.,

~03-(~ I-)2 3-(~ 1+)2, ~ I-~ 3+~ I+,

~ 3+~bl+q~1_, ~ 1+~)1+~b3:~ 3-(q~ 1• ) 2

1

3

2

1

3

--5~

~_(~ ~_)2

3-~ 3+¢ 1+, ¢ 1-(¢ 3+) , ~ 3..~3:~bl :~, ~ ~_(q~:~)2

1

3+6

2

1

-3

o~

-7/2 9/2

!~1

-3/2

I~;I lla

--7/2 9/2

123

--3/2

133

--7/2

i~3, !16

--9/2

123,126

3/2

133,/36

7/2

153

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8X 8 + X~ + 3X 3 - 3X~ - 6X 6 = 4 ,

(5a)

X 1 + 8X 8 - 3X~ - 5X 3 + X~ + 2X 6 = 0 ,

(5b)

X1 +

to the weak gauge bosons as well as to the leptons and quarks. In the following we will concentrate on the quark mass matrix and in particular on the mass relations obtained as a consequence of the Z10 symmetry. So let us consider the HF -invariant, effective lagrangian corresponding to the quark mass term

where

X~' =lli, + 12' i + 13i. '

X i=lli+12i+13i,

(6)

i = I, 3, 6 and 8 are the dimensions of color representations. The meaning of l and l' can be understood from table 2 where only left-handed composite fermions and their indices are listed but the corresponding right-handed composite fermions with opposite indices are implied. We have assumed that exotic particles other than color 6- and 8-plets belonging to electroweak doublets have masses of order AH. Furthermore we assume that the absolute value of each index is less than or equal to one * t and there is a one-toone correspondence between quark and lepton generations. In that case eqs. (5a), (5b) have the following solutions t

X 1 = X 3 =-X 6 =

X I=X 6=1,

I

1,

X 1 = X3 = X8 = 0,

X~=-3,

X~=X 3 =X 8 =0,

(7)

and X i = -X 3 = -X 6 =

"/2M = AH2 [g~x0(CI~Lq#R)(q6Rq6L) +~

-

1,

X 1 = X~ = X 8 = 0.

(8)

*1 For some HF-representatlons , each o f which corresponds to several different configurations o f (~0~0~0) and (qJC¢), the assumption o f Ilkil ~ 1 implies that only one o f the linear combinations o f these configurations could be massless.

-c

c

•

a0(qaLq~R)(q6Rq6L) ] + h.c.,

(9)

where a, 3 = 1,2, 3 are the family indices, = \_D~L]

ifq6L

\D6L

and U, D represent the quark sector with electric 1 charge 2, - ~ respectively. Noting that ~6 = (q6R q6L) as well as its charge conjugate ~6 = ° 2 ~ ° 2 --C C * (q6Rq6L) transforms like (2, 2", 1) under the eventuaUy gauged group SUL(2 ) X SUR(2 ) X X UB_L(1 ) ,2 but they have opposite ZI0 charges *a. The lagrangian (9) must be invariant under the "parity reflection" (L ~- R), this leads to -

gao=gjc~,

All of them correspond to three generations of quarks and leptons plus one generation of color 6-plet quarks (noting that X 1 = 1 can be considered as 111 = 121 = -131 = 1, which means three doublets of colorsinglet fermions with the same Uv(1 ) and U~r(1) charges, thus three generations of leptons in a l e f t right symmetric model, etc.). The three generations carry different Z10 quantum number, for example, for the solutions (7), the Z10 charges associated to the three families of left-handed quarks are (_9, ~,~), a 7 i.e. under Z10 the transformation qL1 exp(--i9~r/10)qL1, etc. On the other hand the Z10 9 3 7 charge of the color 6-plet can be - ~ , ~ or ~. It is supposed that condensates of the color 6-plet can be used as composite Higgs in order to give masses

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13 December 1984

L 0 =~;a .

(10)

We assume that the following exotic color condensates are formed at the scale AC6 ~ 1 TeV [10] (q6Rq6L) =

6

°I

w '

-c c • _ 3 ( ( q 6 R q 6 L ) ) - A~6 (O*

O.).

(11)

These condensates will give masses to the quarks. The mass matrices associated to the U and D type quark sector are given by (Mu)a3 = (A 36/A2 )(gaoo + L o w * ) , (MD)~O = (A~6/A2H)(ga0w +gaoo*).

(12)

Because the non-zero elements ofgaa and ~a3 corre• u For the solutions (7) B - L = IQv - (1/6~x)Q~rand B + L = (l[3a)Q', V ' where QV and QV are generators of U v ( 1 ) and UV(1) respec~vely. . 3 Replacing the charge conjugate couplings (qaLq0R) --C C * × (q6Rq6L) in (9) we can use the HF-invariant couplings qczLqORq6Lq6R, however, it is easy to show that both are equivalent as far as mass matrices are concerned.

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PHYSICS LETTERS

spond to only Zlo-Singlet coupling terms in (9), the form of the mass matrices is dependent on the ZlO charge assigned to the color 6-plet. It is well known that in the I-Iiggs approach [1 1,12] at least two Higgs doublets are needed in order to produce realistic quark mass matrices in the model SUL(2 ) X SUR(2) X UB_L(1). Similarly in the present approach, it is easy to show that no realistic mass matrix appears with only one generation of color 6-plets. In fact, when the ZlO charges of the three generations of quarks are taken as (_9, {, ~_),the q6L with Z10 charge - 9 will give mass matrices with a zero determinant and the q6L with Z10 charge { leads to mass matrices like

(Mu) ~

g'w" 1

(MD)~

f2 o* \0

g33 W

g22 w 0

-1

--

1

= (q6Rq6L),

~62 = (q.62RCI~L),

-1 2 ~12 = (q6Rq6L),

¢21

--=(~2Rq~t),

(14)

and theircharge conjugates.As in the previous case the detailedform of the mass matrix willstrongly depend on the Z chargeassociatedto the color 6-plets The 10 1 2 93 97" Z3107charges of (q6L,q6L) can be (-~, ~), (-~, ~) or (~, ~). However, the former two assignments lead to the mass matrices M U and M D of the form (when the condensates of q6 are taken as real)

b*

d

*

0

,

whose diagonalization, in general, does not give the relation tg 0 c = (maims) 1/2 [13] and we will not consider them here. Thus in the case that the ZlO charges of q~L and q2 L are } and ~ respectively, the effective lagrangian corresponding to the quark mass term is

~ 2w* g22o 0

~I

13 December 1984

"/~M = AH 2 {qlLq2R [fl2(Cl~Rq2L) + g l 2 ( ~ R q l ~ ) * ] • -2 2 ~ , - l c lc *

~33 o*

and furthermore to the following invalid relation:

+ q2LqlR [f~2(q6Rq6L) + g12(C16Rq6L) ]

m~/m b = (m s - md)/(m c - mu) = Iw/ol .

+ q2L q2R [g22(qlR ~L) + f22(cl6~R4~,)* ]

A similar result as to the latter relation also emerges from q6L with Z10 charge ~. Replacing the Z10 charge 9 3 7 of three generations of quarks by (~, -~, - ~ ) (corresponding to the solution (8)) does not lead to different

+ q2Lq3R [h23 (qlR 4L) + t23(q2RCl~L)

consequences.

+

Therefore we have to enlarge the content of composite Higgs particles, which in our case is equivalent to looking for solutions of the anomaly consistency equations containing more than one color exotics. Looking back to eq. (5) we see that the anomaly equations are not consistent with the existence of two generations of color 8-plets or one generation of color 6-plets,plus one generation of color 8-plets but XI=I,

X~=3,

X~ = X 3 = X 8 = 0 ,

+ h23(~q62~) * +'/23 ( ~ q l ~ ) -

*

-1

2

*

(13)

is a solution which corresponds to two generations of ql and q~ besides the three generations of quarks and leptons. Therefore we will have four composite Higgs

1

+ q3Lq2R [h23(q6Rq6L) + t23(q6Rq6L) * -

2

-2

-Ic le

~

*

+ qaLq3R[f33(q6Rq6L)+g33(q6Rq6L) ]}

(15)

Assuming the formation of the exotic color condensates (q6Rq6L)- A~6_

_w i

'

16 A~ (Oil 0 ) ( Rq L )= 6 wi1 ,

X6 = - 2 ,

*] -2

i--1,2, i:/=/=1,2

(16)

and taking the o and w to be real we obtain

MU~

(i a i)(it * b

c*

,

MD "~

* b'

c'*

:/

.

d'/

(17) 155

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As pointed out in ref. [3] the diagonalization of these matrices will lead to the relation tg 0 c ~ (md/ms) 112 -- (mu/mc) 1/2 ,

(18)

,f b < d (or d < b) and lal and Ic I are of the same order. In conclusion, the well-known relation (18) between the Cabibbo angle and the quark mass ratio can arise in a susy composite model in which the quark masses are produced by the condensates of color 6-plet quarks through effective four-ferrnion couplings, and the generations are distinguished by the Zlo chiral symmetry. To calculate all the Kobayashi-Maskawa angles, we must know the effective fermion couplings and the exotic color condensates. However, this seems premature due to our lack of knowledge about bound states and condensates. So far we have been dealing with a left-right symmetric model. Phenomenology requires violation of parity. This could be spontaneously achieved by the following condensates of the color 6-plet quarks (q6L~'taq6Lq6LYUq6L) = 0 , <¢I6RV~q6R¢I6R'y~q6R) =/=0 .

(19)

However, it seems that nothing can be told about the neutrino masses by this pattern of parity violation only. Hence we would rather take another assumption. The above discussion of quark mass matrices is also applicable to the lepton sector. The three generations of leptons, based on the solution (13) and table 2, 9 3 7 have respectively Z10 charges (5, - 5 , - ~ ) (opposite to the quark ones). If we still take the two generations of q6 with left-handed Z10 charge (6, 3 ~), 7 then some lepton mass matrices which are different from those of the quarks will appear. The masses so obtained are Dirac masses of the leptons. Then we further assume [ 14] that the composite neutrinos (for each generation) could form the condensates

(pTc R) ~"M 3

,

(20)

at a mass scale M somewhat lower than A H, as a result

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13 December 1984

of the residual hypercolor forces. Thus the righthanded neutrinos will acquire heavy Majorana masses, mv R ~ M 3 / A 2, and the left-handed neutrinos will have lower masses, mull_ ~ m 2 /(M3 /A 2 ) (~ = e, la, r), by the standard procedure [I 5]. It is easy to verify that the mv~ L so computed are consistent with the experimental limits when we take A H ~ 10 TeV and M > 2 TeV. The realizability of this mechanism is dependent on whether the condensates (20) could form or not. One of the authors (B.R. Zhou) would like to thank C.L. Ong and Duan Yi-Shi for useful conversations and S. Drell for the kind hospitality extended to him during the visit to the SLAC theory group.

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