A natural theory of the K-M mixing angles, weak and strong CP

Volume 160B, number 6

PHYSICS LETTERS

A NATURAL THEORY OF THE K-M MIXING ANGLES, WEAK AND STRONG

17 October 1985

CP

Michael SHIN t Lyman Laboratory of Physics, Harvard University, Cambridge, MA 02138, USA Received 14 June 1985 We present a natural model in which the quark mass matrices are of the symmetric Fritzsch form with the two physically relevant phases o = ~"~ + ,r/2. The model is natural in the sense of 't Hooft and explains all of the known aspects of the weak interaction phenomenology of the K-M matrix (including the weak CP) in terms of the observed quark mass eigenvalues only. The model solves the strong CP problem by the Peccei-Quilm mechanism while the phases o = T = + ~r/2 are generated by spontaneous symmetry breaking. The invisibilityof the axion is more natural than in the existing models and is related to the absence of FCNC and the observed long b-quark lifetime. The key predictions of the model are m t < 80 GeV and 5.0 )< 1 0 - 3 6 R b m 1"03 --~ ue~)/r(b --* ce~) ~<1.46 X 10 - 2 , while the present experimental upper bound on R b is 3 × 10 - 2 . The prediction o n R b is to be tested by experiments at CLEO and CUSB in the near future.

1. Introduction. O n e o f the m o s t m y s t e r i o u s a s p e c t s o f p a r t i c l e physics at the p r e s e n t time is t h e o b s e r v e d f a m i l y (generation) structure of q u a r k s a n d l e p t o n s . A l t h o u g h the existence of the f a m i l i e s itself m a y n o t b e a severe p r o b l e m * l , the p a t t e r n s o f t h e o b s e r v e d values of the f e r m i o n m a s s e i g e n v a l u e s a n d the K - M m i x i n g angles are beyond our comprehension and present a great t h e o r e t i c a l c h a l l e n g e at this time. I n a n a t t e m p t to u n d e r s t a n d the o b s e r v e d [1] K - M m i x i n g angles i n c l u d i n g the strength of the w e a k C P v i o l a t i o n in terms of the o b s e r v e d q u a r k m a s s eigenvalues, we c o n j e c t u r e d in o u r p r e v i o u s a r t i c l e [2] t h a t the u n d e r l y i n g structure of p a r t i c l e p h y s i c s is s u c h t h a t the m a s s m a t r i c e s for the q u a r k s are o f the F r i t z s c h f o r m (see eq. (1)), w i t h t h e t w o p h y s i c a l l y relevant phases, o a n d % b e i n g o = 1" = + ~r/2. T h e s e phases were c o n j e c t u r e d [2] to b e g e n e r a t e d f r o m t h e VEVs o f the c o m p l e x s c a l a r s a n d m o d e l b u i l d i n g was a t t e m p t e d [3, 4] in that direction. H o w e v e r , t h e m o d e l p r e s e n t e d in ref. [3] suffered f r o m the disease of the s t r o n g C P

p r o b l e m , a n d this disease was cured in a subseq u e n t article [4] using the w e l l - k n o w n P e c c e i - Q u i n n m e c h a n i s m [5]. A l t h o u g h the m o d e l "presented in ref. [4] is a p e r f e c t l y a c c e p t a b l e r e n o r m a l i z a b l e one a n d all p h y s i c a l q u a n t i t i e s c a n b e c a l c u l a t e d f r o m the l a g r a n g i a n itself (the m o d e l explains all of the k n o w n a s p e c t s o f the w e a k i n t e r a c t i o n p h e n o m e n o l o g y o f t h e K - M matrix), the m o d e l was n o t a n a t u r a l o n e in the sense of 't H o o f t [6]. T o b e m o r e precise, the soft b r e a k i n g terms of d i m e n s i o n 2, w h i c h w e r e i n t r o d u c e d in our m o d e l [4], were n o t t h e m o s t g e n e r a l ones with given s y m m e t r y , a n d it is o u r w i s h in this article to cure this u n n a t u r a l a s p e c t o f t h e m o d e l . T o d o so, we shall i n t r o d u c e s o m e u n b r o k e n discrete s y m m e t r y (such as Z8) i n t o the l a g r a n g i a n . T h e g e n e r a l strategies t h a t we shall use in o u r m o d e l b u i l d i n g a r e the same as the ones ,2 used in ref. [4], a n d the r e a d e r is advised to refer to o u r p r e v i o u s articles [3,4].

:~2

t Research is supported in part by the National Science Foundation under Grant No. PHY-82/15249. ,1 Some horizontal (family) symmetry group may be present in the next layer of high energy physics.

We shall use the extended survival hypothesis [7] to avoid the flavor changing neutral currents and the Peccei-Quinn mechanism [5] to solve the strong CP problem. To use the ESH, we shall allow complete mass mixings among SU(2)L doublet scalars assuming that there is only one fine tuning (the usual one) to make the weak scale light. 411

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PHYSICS LETTERS

Before going on to discuss model building, we give a brief summary of Fritzsch phenomenology [2] to set up notations for quark mass matrices and to remind the reader of the predictions of the Fritzsch mass matrices with the conjectured phases o =

"r =

0

IA Iei**

0

]

[ale i**

0

[Ble i*B ,

0

IBle i~"

IClei*~J

Md=Mu(A,B,C--A',B',C' ).

(1)

The physically relevant phases which enter in the expression for the K - M matrix are o and ~, with o

-

•

=

(,c

-

,c,)

-

-

• = (@A - ~a,) - ((/)B - *s,)-

(2)

With the conjectured [2] values of o and z, o = r = + ~r/2, the K - M mixing angles are predicted to be [2]

VusI = (md,/m ~+ mime)l~2[1 + O ( m d / m s ) ] , V~bI =

[(ms/mb)'/2-(mJmt) '/2]

× [1 + O(md/m~) ] ,

Vubl/I ~bl = [mu/m c +(md/ms)(ms/mb)3/lgcb[2] 1/2 × [1 + O ( m d / m s ) ] , 8' =

from eq. (3), and this implies 5.0 X 10 .3 ~
-- o =

g ir/2,

(3)

where 8' is the CP violating phase in the Maiani parameterization .3 of the K - M matrix. The t-quark mass is also predicted to be m t < 80 GeV in order to be consistent with the present bounds on the K - M mixing angles. These predictions are not only consistent with the present experimental bounds on the K - M matrix, but also provide a more stringent bound on one of the mixing angles ([Vub D. It is given by 0.052 ~< I Vub/V~b[ ~<0.084,

(4)

ce~)

< 1.46 X 10-2,

(5)

while the present upper bound on R b is [9] R b~3×10

I Vubl-

-2

(90%CL).

(6)

If this prediction on R b is confirmed by future experiments, then the case for a high energy theory which forces the quark mass matrices to have the Fritzsch form with the conjectured phases, o -- r = :t:~r/2, will be very strong. In the following sections of this paper, we will present a natural (in the sense of 't Hooft) model in which the phases o = r = :t: ¢r/2 are generated by the VEVs of complex scalars while the strong CP problem is solved via the Peccei-Quinn mechanism.

2. An SU(2)LXSU(2)RX U(1)n_LX U(1)eQX U(1) x Z 8X CP X P model. The minimal gauge group which can produce the symmetric Fritzsch form of quark mass matrices while containing the successful standard SU(2)L × U(1)r electroweak gauge group is SU(2)L × SU(2)R × U(1)B_ L. In this section, we consider a model with this gauge group where extra U(1)pQ × U(1) global symmetry and discrete symmetry Z s × CP × P are imposed on the lagrangian. To begin with, we introduce the following notations for three generations of quarks (color indices are suppressed). QiL=

[U] :(1/2,0,1/3), D iL

Q J R = [ DU ] j R : ( 0 ' I / 2 ' I / 3 ) '

/--1,2,3, j=1,2,3,

(7)

where the numbers in the parenthesis represent the SU(2)L X SU(2)R X U(1)B_ L quantum numbers. The electric charges are then given by Qem = T3L + T3R + ½ ( B - Z ) .

(8)

In order to generate the tree-level quark masses, we introduce the following scalars: .=

,3 The relation of 8' to 8 (the standard K - M phase) is given in ref. [1] and they are roughly equal to each other for small

412

r(b - , ue~)/r(b - - ,

+~r/2.

The symmetric Fritzsch form [8] for the quark mass matrices are given by

gu=

17 October 1985

[¢P° ~ ]

(1/2,1/2",0),

~j -= r2~Tr2 : ( 1 / 2 , 1 / 2 " , 0),

j=1,2,3, (9)

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PHYSICS LETTERS

17 October 1985

Table 1 Quantum numbers of quarks and scalars in an SU(2)L × SU(2)R × U(1)s_ L × U(1)pQ × U(1) × Z s × CP × P model. Particle

SU(2)L × SU(2)R × U(1)B_ L

U(1)pQ × U(1)

Zs -- 1/4 1/4 0 1/4 - 1/4 0

quarks

Q1L Q2L Q3L Qlg Q2R Q3R

(1/2, 0,1/3) (1/2, 0,1/3) (1/2, 0,1/3) (0,1/2,1/3) (0,1/2,1/3) (0,1/2,1/3)

(1, 0) (0,1) (1,1) ( - 1,0) (0, - 1) ( -- 1, - 1)

scalars

AL AR

(1,0,2) (0,1, 2)

(0,0) (0, 0)

0 0

O1 •2

(1/2,1/2", 0) (1/2,1/2", 0)

(1,1) (1, 2)

0 1/4

0

•3

(1/2,1/2", 0)

(2, 2)

(1/2, 0,1)

(1/2,1/2)

o2 o3 o4 o5 O12

(0,1/2,1) (1/2, 0,1) (0,1/2,1) (0,0,0) (0,0,0) (0,0,0) (0, 0, 0) (0,0,0) (0, 0, 0)

( - 1/2, - 1/2) (1/2,1) ( - 1/2, - 1) (2,2) (2,4) (4,4) ( - 1, 0) (0, - 1/2) (0, 2)

,h

(o,o,o)

(o,o)

0 1/8 - 1/8 0 1/2 0 1/4 - 1/8 1/2 - 1/8

7/2

(0, 0, 0)

(0, 0)

- 1/4

• XlL

XlR X2L X2g Ol

where ¢2 is the 2 x 2 pure imaginary antisymmetric Pauli matrix. Under SU(2)L X SU(2)R, the quarks and the above scalars transform according to Q L j -+ U L Q L j ,

QRj ~ URQRj,

-,,-.

¢,-"

(10)

are:

bkt~k)QjR

+QjR-- ['a**t~t,j i + b/~*t~*k)QiL-

(11)

Due to the imposed P (left-right symmetry), we have

a q = aj~,

bq = b~,

coupled to the quarks and the mass matrices for the quarks are the symmetric Fritzsch form. Their q u a n t u m numbers are shown in table 1. We also introduce additional scalars ( A L ' A R ' X 1 L ' X2L, X1R, X2R, ffl . . . . . 0"5' 0"12' l h ' 1~2)

The most general Yukawa couplings for the quark

~ Y = QiL ( ak¢~k +

0

(12)

and the imposed CP invariance further requires that all a q and b;j are real. N o w we impose the global U(1)pQ X U(1) and the discrete Z s quantum numbers on the quarks and ~j so that only ~j are

to give the desired symmetry breaking pattern, SU(2) L × SU(2) R × U(1)s_ L --, SU(2)L × U(1)y, and to allow complete mass mixings among SU(2) L doublet scalars (to avoid FCNC.) when the scalars develop VEVs. All of the scalars except A L and SU(2)L doublets are supposed to get superlarge VEVs (compared to the weak scale). Table 1 shows the representation content of all scalars and quarks in the model. The notations for the q u a n t u m n u m b e r under Z 8 are such that a particle f with Z s q u a n t u m number q transforms according to f - , exp [q2,ri]f. F r o m the quantum numbers shown in table 1, it is then easy to see that the only non-trivial couplings among scalars present in the lagrangian (hence in - V ) , which are relevant to the determination of the phases of the VEVs and the mass mixings among SU(2)L 413

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PHYSICS LETTERS

doublet scalars, are ,4:

84-(02-03)=0

Tr(¢;,i,:) + h.c., (13,14) Tr(O~~3) +

h.c.,

xfi.¢dXx~> + b.c.,

x~d'dx:~> + h.c.,

(X?LX=dO,> + Xi~.X=~) + h.c., 2 , + h.c., ~1~/2

(19)

~42+ h.c. (20,21,22)

In order to see how these non-trivial couplings present in the lagrangian, with real coefficients (taken to be positive) due to the imposed CP invariance, determine the phases of the VEVs, we introduce the following notations: <(I)j> ~

ei°j

j - - 1,2,3,

0l

83 - (203 + 03) = 0,

(19') 281 - 6~ = 0,

46~ = 0, + 2~r, 4~r.

(20', 21') (22')

Note that eq. (16'), which comes from the term in (16), already determines one of the physically relevant phases in (2),

o

-

•

-=( ~

-

~,)

-

(

~

-

o~,)

--- -q'3 + ~2 = 0,

(24)

while the other phase r is given by

= + ¢r/2. (23)

6 2 - (202 + ~2) = 0,

(13',14') (15')

,4 In addition to the terms given in (13)-(22), there are two more allowed in the lagrangian: Tt(O~t~l)(O2o~> + h.c. and Tr(¢~- ~2)(olo12 ) + h.c., which give redundant information on the relative phases (same as eqs. (13'), (14') and (20')).

414

a n d - 01R + 02rt -- 6s = 0,

(25)

o571~' + h.c.

(26,27)

85 = 6 I.

(26', 27')

Eqs. (27'), (21') and (22') determine 4g 5 to be 0 or ~r and eqs. (26') and (25) determine r to be

j=l,2,

Then eqs. (13)-(22) imply 8 1 - ( 2 0 ~ + 0 ~ ) = 0,

-- 01L "l- 02L -I- 65 = 0

812 --- + ~r/2,

j = 1 ..... 5,12,

a = 1,2.

(18')

0,

=

Then the terms in (26) and (27) imply

--eiO°LtR)[OaLO(R)], %L(R)> 0,

- 02L + 02R + 02 + ~2

- m 2 ( % 2 ) 2 + b.c.,

kjei,l,j '

le~S,, Q/j> = [O/j>leiS;,

(17')

from eqs. (13'), (14'), (17')-(20'). N o w let us introduce the most general terms of dimension 2, which respect U ( 1 ) p Q X Z 8 but break the U(1), into the lagrangian. There are only two such terms and they are:

e i'~j '

[:

(16')

-- 01L -I- 01R + 01 + t~l = 0,

= -~bl + @2 = - (812 + 465),

?0 k; 0 l

kj, k~>O,

and 0 2 - ~3 = 0,

81 - 82 + 812 = 0, (17,18)

~102"0r12 + h . c . ,

17 October 1985

(28)

Thus o = "r = ¢r/2 (or -¢r/2) from eqs. (25) and (28). Therefore, the model generates the desired phases o = r = + ¢r/2 in the symmetric Fritzsch quark mass matrices. The unbroken global U(1)pQ symmetry is still present in the lagrangian and the strong CP is avoided by the well-known Peceei-Quinn mechanism. The model is natural in the sense of 't Hooft since the soft breaking terms introduced in eqs. (26) and (27) are the most general ones (with dimension 2) which respect U(1)pQ x Z s.

3. An SO(IO)X U(1)eex U(1)xZsx CP model. The model considered in the previous section can

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17 October 1985

be naturally embedded in an SO(10) GUT. The imposed left-right symmetry ( P ) is then automatic and all of the mass parameters are naturally of the G U T (or axion) scale. CP will be violated by SSB at the G U T scale and cosmological problems are avoided. The global symmetry to be imposed is U(1)v Q × U(1), where the latter U(1) is to be broken softly by the most general terms of dimension 2. Discrete symmetries to be imposed are Z s × CP, which are not to be broken by the soft breaking terms. Only one fine tuning (the usual one) will be required to make the weak scale light. There will be no problem of FCNC. and the axion is naturally invisible. In table 2, we present the entire particle content of the model. All fermions are taken to be left-handed chiral fields. The only Yukawa couplings allowed by the symmetries of the model are:

Table 2 Particle content of an S0(10) × U(1)VQ× U(1) × Z8 × CP model.

Y1~11~1~2 + Y2~22~3 + y 3 ~ - - ~ 3 + h.c.

(29)

fermions

+hj~xzq~ s + h.c.

(30)

Cz, 42, and ¢3 contain SU(2)L doublets which will get VEVs of the weak scale. The Yukawa couplings in (29) then produce mass matrices of the symmetric Fritzsch form of (1). The X will get super-large VEVs which break SO(10) ~ SU(5) and the terms in (30) will remove the right-handed neutrinos from the low energy theory. There is also a scalar 45 multiplet in the theory whose VEV breaks SU(5) ~ SU(3)c × SU(2)L × U(1)r. In order to see how the physically relevant phases o = "r = ± rr/2 are generated naturally from the model, we first list all the terms allowed by the U(1)VQ × U(1) × Z 8 symmetry, which determine the relative (or absolute) phases of the VEVs (and give complete mass mixings among the SU(2)L doublet scalars). They are 01'0~'(oz) + h.c.,

¢~'¢~'(a2)

+ h.c.,

¢J'0~'(03) + h.c.,

¢~'¢3(a4)

+ h.c.,

Xl(I)l(Xl) + h.c.,

X2~2(X2) + h.c.,

x?x2(o;') + h.c.,

O102"O12+ h.c.,

2 . + h.c., T~1'1~2

~4 + h.c.

(31)

These terms are identical to the ones introduced in (13)-(22). Now we introduce the most general terms of dimension 2, which respect U(1)v Q × Z 8

Particle scalars

S0(10) U(1)pQ X U(1) representation

Zs

~ ~1 ~2 ~3

45 10 10 10

(0, O) (1,1) (1,2) (2,2)

0 0 1/4 0

Xl

16

( - 1/2, -- 1/2)

X2 a]

16 1

a2 a3

1 1

( - 1/2, - 1) (2, 2) (2,4) (4,4)

0

- 1/8 0 1/2 0 1/4

a4

1

(-1,0)

os

1 I

(0, - 1/2) (0,2) (0,O) (0,0)

- 1/8 1/2 - 1/8 - 1/4

a12

Th 7h

1

¢1 1/'2 ¢3 E1

16 16 16 1

(1, 0) (0,1) (1,1) (1/2, - 1/2)

-- 1 / 4 1/4 0 - 1/4

E2 E3

1 1

( - 1/2,1/2) (1/2,1/2)

1/4 0

gauge boson A~

45

(0, 0)

0

1

but break the remaining U(1). There are only two such terms and they are:

--m2(t/12) 2 + h.c.,

a57/~+ h.c.

(32)

These are identical to the ones in (26) and (27) and the terms in (31) and (32) produce the desired phases o = 1- -- ± ~r/2, as was explicitly shown by eqs. (13')-(28) in the previous section. The unbroken global U(1)pQ is still present and strong CP is avoided.

4. Discussion and conclusion. In this article, we have presented a natural (in the sense of 't Hooft) model in which the physically relevant phases a = ¢ = ± ~r/2 in the Fritzsch quark mass matrices are naturally generated by SSB and the strong CP problem is solved by the Peccei-Quirm mechanism. The model explains all of the known aspects of the weak interaction phenomenology of the standard K - M model in terms of the observed quark mass eigenvalues. The model is a modifi415

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cation of the one in our previous article [4]. The only difference is that we have introduced unbroken Z 8 discrete Symmetry as a subgroup of the family group. Two complex singlet scalars (*h, ~/2), which are singlets under global U(1)p o × U(1) but non-singlets under Z8, are introduced. At the same time, we have reduced the number of soft breaking terms and the model became a natural one in the sense of 't Hooft. The softly broken U(1) is to be identified as the weak CP generator. U(1)po is a part of the unbroken family group (U(1)pO × Zs), and the invisibility of the axion is natural and is related to the absence of F C N C and the observed long lifetime of the b-quark (o - • = 0 from the presence of o4), as was first observed in our previous article [4]. We find this a very interesting aspect of the model. (The invisibility of the axion has something to do with the low energy phenomenology such as the absence of F C N C or a long b-quark hfetime!) It is also interesting to speculate that the family structure of quarks and leptons m a y answer some of the other problems which are apparently unrelated to one another (such as the

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17 October 1985

unobserved proton decay and the gauge hierarchy problem) in the future. I would like to thank Howard Georgi for his encouragements.

References [1] M. Shin, Harvard preprint HUTP-84/A024 (1984). [2] M. Shin, Phys. Lett. 145B (1984) 285; Harvard preprint HUTP-84/A070 (1984). [3] H. Georgi, A. Nelson and M. Shin, Phys. Lett. 150B (1985) 306. [4] M. Shin, Phys. Lett. 154B (1985) 205. [5] R. Peccei and H. Quinn, Phys. Lett. 38 (1977) 1440; S. Weinberg, Phys. Lett. 40 (1978) 223; F. Wilczek, Phys. Rev. Lett. 46 (1978) 279. [6] G. 't Hooft, Lecture Carg6se Summer Institute (1979). [7] H. Georgi and S. Dimopoulos,Phys. Lett. 140B (1984) 67. [8] H. Fritzsch, Nucl. Phys. B155 (1979) 189; Phys. Lett. 73B (1978) 317; L.F. Li, Phys. Lett. 84B (1979) 461. [9] M.G.D. Gilchriese, Invited talk Workshopon Experimental and theoretical investigationsof family problems (ITP, Santa Barbara, January 1985).