SO(14) flavor unification

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PHYSICAL REVIEWD VOLUME 26, NUMBER 1 1JULY 1982

SO(14)

flavor unification

K.

S.

Soh

Department ofPhysics, Seoul National University, Seoul 151,Korea Sung Ku Kim

Department ofPhysics, Ewha Women's University, Seoul120,Korea (Received 19April 1982}

An SO(14}grand unification model isproposed tosolve the flavor problem. There are four families ofquarks and leptons, which are divided into three types by a new quantum number. The bare and renormalized Weinberg angle and the proton-decay mass scale are similar tothose

ofthe Georgi-Glashow SU(5}theory. Right-handed neutrinos ofelectrons and muons can be superheavy.

I.

INTRODUCTION

The successes

of

the Weinberg-Salam

SU(2)

x

U(l)

theory

of

weak and electromagnetic interac-tions has spurred model-building activities

of

unify-ing weak, electromagnetic, and strong interactions by a single gauge group. Among the models proposed

so far, the simplest and most successful ones are the

Georgi-Glashow

SU(5)

theory' and its extension to

the

SO(10)

scheme.3

Especially the latter naturally synthesizes the Pati-Salam idea4

of

lepton number as afourth color with the idea

of

grand unification first exhibited in

SU(5).

2 _With _all _{their amazing} _features

the

SU(5)

and the

SO(10)

theories have, however, an important shortcoming: They give no explanation

for the existence

of

more than one family

of

quarks and leptons. This is called the flavor problem. The

purpose

of

this Communication is to offer a new solution

of

the flavor problem while keeping the good

features

of

the

SO(10)

[or

SU(5)]

theory.

there are seven mutually commuting diagonal

ele-ments,

S2k ]2kp k 1p ~ ~ ~ p7 (2)

The

SU(3),

diagonal generators are

(S3

4 S12) and

(S3

6

—

S3

4),

and the

SU(2)

1vthird component is

13=

₂1

(S9

10

—

S7, 8)

(3)

The crucial point

of

our theory lies in the

defini-tion

of

the electromagnetic charge operator Q given by

0 =

3

(S1,

2+S3

4+Ss

6)

+S9

10

1

(4)

Another important point is a new operator

C

defined by

which combines with I3to form the electroweak hy-percharge

&—

=

_g

—

&3=

—

3(S1,2+S3,4+S3,

6)+

—, '

(S7,+$9,

0) 1

(5)

II. DEFINITION OF THE MODEL ₁

2(S13,14 S11,12)

(6)

Webegin with the

SO(14)

Clifford algebra by in-troducing notations. I matrices are defined by tensor

products

of

Pauli and 2

x

2identity matrices:

which will be used for family classification and other

purposes.

l„=o.

I'-"&

_Xg,

Xl«-", 12„,

=gI'-"'X

o-

XI'"

"

(lb)

where kruns from 1to

6.

The tensor products may

be illustrated by two examples:

al

~2)

=0~

Xg&, 0~

Xg

=

0

2

1

Among the 91

SO(14)

generators

S„I=

—

II

I

„I,

, k

At

2l

III. FERMION CONTENTS

The 64-dimensional spinor representation $=

(64)

accomodates four families

of

quarks and leptons.

Their

SU(3),

x

SU(2)

s

x

U(1)

y

x

U(1) c

contents are shown in Table

I.

All fermions are left-handed

(LH).

The

SU(7)

decompositions are also given for convenience.

From the table one can see the role

of

the new quantum number C. First it classfies the fermions by three types.

(C=O:

u,

dcs),

(C=

—

—:

b,

t),

(C =

—:

h,

f)

for quarks and

(C =0:

e,

p„vt,

v2),

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26 RAPID COMMUNICATIONS 341

TABLE

I.

Fermion classification. The SU(7)indices are included forconvenience. a,b,care color indices and idenotes weak index. C,classification charge; Y,hypercharge; Q, electromagnetic charge.

sv(3),

x

sv(2)

Particles

sv(7)

1 2 1 2 1 2 1 2 1 2 1 2 1 2

(3,

2) (3,

i)

1 6 2 3 1 3 2 3 1 3 1 6 2 3 1 3 1 6 1 2 0 1 2

—

₁ 0 1 2 2 1 (

—

3'

—

3) 2 3 1 3 2 3 1 3 2 1 (

—

₃f

—

₃) 2 3 1 3 2 1 (

—

——

_3'3

)

(-

&,o)

—

1 0

(l,

o)

—

₁ 0 (a,o) {u,d)L (c,s)L QL CL dL SL tL bL {gcbc) fL hL {fCIC) (e,v1)L

(I"

v»L e jxL v2LC v3L

(~; v;),

v4L ( ',v )

{ai)

(abi)

{a)'

(a

67)'

{ab)

(a4s)

(a7)

(ah

6)'

(ai7)

(a6)

(ah

7)'

(ai6)

(I

)"

(i67)

singlet (67) (4s)

(abc)'

(6)'

{4s6)'

(I6)

(7)'

(4s7)'

(i7)

(C=

—

2

.

'r,

v3),

(C =

—,: e,v4) for leptons. 1 1

With conserved C, it is possible to suppress flavor-changing neutral currents.

'

The C-breaking mass scale

Mc

should be larger than but could be

of

the same order

of M~,

weak symmetry-breaking

scale.

Secondly, we note that the spinor representation is real and complex with respect to Gw

=

SU

(3),

x

SU(2)

w

x

U(1)

r

and Gc

=SU(3),

x

SU(2)

w

x

_U(l)

_{r x}

U(l)

c,

respectively. Therefore, if

Mc

—

_O(Mw)

the fermions are protected from having superheavy masses. Effectively we follow the

reason-ing behind Georgi's second law

of

grand unification,

"the

representation

of

the LH fermions should be complex with respect to

G~.

"6

For

the first point M~ is better to be safely larger than

M~,

while the second point requires M~as

close as possible to

M~.

In aspecific realization

of

symmetry breaking we find that M~ is afree

parame-ter, and can only be determined experimentally.

IV. CHARACTERISTICS OF THE MODEL

(1)

There are no exotic fermions.

(2)

A single ir-reducible representation includes four families

of

quarks and leptons classified to three types by C.

(3)

The model has left-right symmetry, and lepton number can be viewed as afourth color as proposed by Pati and Salam.4 (4) There are two neutral lep-tons v1,v2 which are singlet under G~. These parti-cles can have superheavy mass. Therefore, it is pos-sible tomake electron and muon neutrinos very light.

(5)

The model is anomaly free automatically and asymptotically free.

(6)

The bare Weinberg angle is sin280w=

_s.

₍₇₎

The C-breaking mass scale must be

O(Mw).

V. SYMMETRY-BREAKING PATTERN

The following scheme

of

symmetry breaking is only one

of

many possibilities. The full stages are

(3)

SO(14)

SO(10)

x

SO(4)

s

SO(6)

x

SO(4)

c

x

SO(4)

s Gc

G~

G,

M14 M10 Mp Mc MH 2ln Mp

=~+

—

(2g-~),

M~

(8)

3 in

+ln

=

₂₄

(2$

—

g)

10 P where

(9)

sin 8@ 11 &em t

ec

(10)

3cos

8~

7l=

11n,

5

—

_sin

_e~

Before plunging into numerical examples, we note

that

21nMGG

W

where we denote the Georgi-Glashow

SU(5)

mass scale by MGt-„.

'

Then we have

2g

—

ri Mp MGGexp

12

(12)

For

a,

=1/128.

5,

sin'8&=0.

20,

n,

=0.

23

(0.15),

we find

(2( —

q)/12

=0.

96

(0.

30).

This gives Mp

=

2.

61 Moo

(1.

35Moo)

3ln

+In

=2.

4

(0.

75)

10 P

where G,

=SU(3),

x

U(1),

.

Bythe renormali-zation-group method the effective coupling constants at

M~

and renormalized Weinberg angles are related

to the mass scales M14, M10, Mp as follows:

VI. DISCUSSIONS

The present model has remarkable similarity to

SU(5)oo

in its phenomenological aspects, which is

not

a

priori manifest at all. The bare steinberg angle is the same and the renormalized angles are very

close to each other. The proton lifetime could be-come

of

the same order and the gauge hierarchy problem is almost identical (Mp/Ms

=

Moo/Ms

)

for both models. Even the magnetic-monopole prob-lems are similarly present.

Group theoretically our model has more affinity to the Pati-Salam model.4 _{At the} _{Mp scale} _{except for}

SO(4)

_c

factor, the

SO(6) x

SO(4)

s is the left-right symmetry and the realization

of

lepton as afourth color.

Proton-decay modes, b-meson decay

characteris-tics, and neutral-current phenomenology are impor-tant subjects to clarify the structure

of

the theory. Mass

of

fermions, mixing angles, and other

theoreti-cal details depend upon the Higgs sector and Yukawa couplings, which deserve further study.

Finally, we note that the group structure at Mp has C-Asymmetry

SO(4)

s

xSO(4)c.

Especially

the opeators I3 and

C

have similar forms [see Eqs.

(3)

and

(6)].

It isalso possible to use

C+=

2

(St3

]4+S~t

t2) instead

of

C. But it isnot so

useful for particle assignment.

More precision experiments and/or higher-energy

(Ms

)

neutral-current experiments' may reveal group structure beyond

SU(2)

s

x

U(1)

r.

Frequently am-bidextrous nature at higher mass scales has been

sug-gested.9 Our theory suggests another symmetric

na-ture,

i.

e.

, C-8'symmetry, which isa new idea that could produce testable predictions because

Mc

—

0(Ms

).

A detailed presentation

of

the systematic search that led to the particular model and other breaking patterns will begiven elsewhere.

In brief, for reasonable values

of

0,

,

n„and

sin28@, we have 2g

—

q

=

0

and Mt4

=

M~p

—

Mp

=

MGG

—

10 GeV.

An important point is that M~ does not appear in

(8)

and

(9),

and therefore can only be determined by

other aspects

of

the model, such as the

neutral-current parameters.

ACKNOWLEDGMENTS

This research has been supported in part by grants from the Ministry

of

Education through the Research Institute

of

Basic Sciences, Seoul National University, and the Korean Science and Engineering Foundation.

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S.

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S.

L.Glashow,

J.

Il-liopoulos, and L.Maiani, Phys. Rev. D 2,1285(1970).

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(1974).

3H. Georgi and D.V.Nanopoulos, Phys. Lett. 82B, 392

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5K. Kang and

J.

E.Kim, Phys. Lett. 64B, 93(1976); S. L.

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J.

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