Flavor changing neutral processes and Bd0−B̄d0 mixing

Volume 219. number 2,3 PHYSICS LETTERS B 16 March 1989

F L A V O R C H A N G I N G N E U T R A L P R O C E S S E S A N D rto ~o M I X I N G 1~ d --Jt~ d

Michae l S H I N , M y r o n B A N D E R and D e n n i s S 1 L V E R M A N

Department of Physics, University of California, Irvine, CA 927] 7, USA

Received 22 December 1988

We propose that the observed Ba°-Ba-° mixing reported by ARGUS and CLEO is due to the tree-level flavor changing neutral coupling of the standard model Higgs scalar, H °, or the Z °, induced by new physics with a mass scale beyond the standard model. The strengths of the flavor changing couplings of H ° and Z ° are shown to be increasing with the masses of the fermion flavors involved. If the observed B°-l] ° mixing is due to the flavor changing coupling of H °, the key predictions are D°-D ° mixing of O(10%) of the present experimental upper limit and BR(la--~e-y) ~ ( 1.1 _+0.6)X 10 ~z, and the mass of the Higgs scalar M, ~_ (200-300) GeV, In case the observed Bd°-Ba-° mixing is due to the flavor changing coupling of Z °, the rare decay mode la--~e e+e is predicted to be observable at any time in the near future with the branching ratio in the neighborhood of the present experimental upper limit, while other predictions include: D°-I) ° mixing of O( 1-10)% of the present upper limit, BR( B ~ t +~t-X ) -~ (8.5 _+ 4.2) X 10 -s, BR(z ~t-~+bt - ) -~ (8.8 _+ 4.8 ) X 10 s, and the branching ratios for the flavor changing decay modes o fZ °, BR(Z°-bs+sl3)x 107-~ (14_+ 7), BR(Z°~tc+ct )X 107-~ (1500_+700)(mJ60 GeV), BR(Z°-~b'l~+bl3 ' ) X I07~ (4800±2300) (rob /50 GeV), BR(Z°~kt-z+ +p+z ) X 107~3.6- + 1.8, and BR(Z°-~'~ ' ~+~"~) X 107~ (1300_+600) (m~/ 40 GeV). These flavor changing branching ratios of Z ° can be tested at LEP with 107 Z°'s. From the observed strength of B~-B ° mixing the scale of new physics can be inferred to be M-~ 250 GeV.

1. Introduction

In our present unde r s t and ing o f par t ic le physics,

there is one po ten t ia l ly i m p o r t a n t p iece o f exper i -

menta l da ta whose exp lana t ion m a y requ i re new

physics b e y o n d the s tandard model . It is the surpris- ingly large o -o Bo-Ba mix ing r epor t ed by the A R G U S

Co l l abo ra t i on [ 1 ], which has been c o n f i r m e d by the

C L E O Col l abora t ion [ 2 ].

A l though such a large mix ing can be unde r s tood

wi th in the s tandard K M mode l by as suming [3] a

large t -quark mass (m, > 100 G e V ) or cer ta in K M

mat r ix e l emen t s in the ne ighbo rhood o f the i r p resent

upper l imits , this will fail to p rov ide the so lu t ion if

the t -quark is d i scove red be low 100 G e V or so in the

near future and the [ [1ub ] e l emen t is m e a s u r e d to be

well be low (e.g. [Vub[/lVcbl--~0.06--0.08 [ 4 ] ) the

present uppe r l imit , ] V, b l / I [~b I ~< 0.2 l [ 5 ]. A n o t h e r reso lu t ion o f the d i l e m m a could c o m e f rom the exis-

tence o f the four th fami ly o f quarks ( a n d l ep tons ) ;

the t ' - q u a r k con t r i bu t ion to the usual box d i ag ram could accoun t for the m a g n i t u d e o f o -o Ba-Bo mixing.

This would requi re special va lues for m,. and for the

four fami ly K M e lemen t VrdVT, b. Whi le there exist

pract ica l ly no expe r imen ta l cons t ra in ts on these pa-

rameters , a recent inves t iga t ion [ 6 ] suggests that this

m a y not be the case. This is because the t ' - q u a r k ex-

change diagrams, which are r equ i red to br ing the

B ° - B ° mix ing to the obse rved level, inev i tab ly en-

hance the CP-v io la t ing a m p l i t u d e in K ° - K ° mixing,

g iv ing too large a con t r i bu t ion [6] to R e e k

( = 1.62X 10-3) .

In this art icle, we propose that this mix ing is due

to t ree- level f l avor changing coupl ings o f the stan-

da rd m o d e l Higgs scalar, H °, or the Z °, i nduced by

new physics b e y o n d the s tandard model . In sect ion

2, we present an i l lus t ra t ive example , where physics

b e y o n d the s tandard mode l could be responsib le for

such f lavor changing neutra l cur ren t processes

( F C N P ) o f the known quarks and lepton s. In sect ion

3, we set up our conven t ions for such coupl ings and

look at the present expe r imen ta l bounds . As men-

t ioned, we assume that the large ~,dn°-~,a~° mix ing is due

to such F C N P and this anchors one o f these cou-

plings. In sect ion 4, we look at va r ious theore t ica l ex-

0 3 7 0 - 2 6 9 3 / 8 9 / $ 03.50 © Elsev ier Science Publ ishers B.V. ( N o r t h - H o l l a n d Physics Publ i sh ing D i v i s i o n )

381

Volume 219, number 2,3 PHYSICS LETTERS B 16 March 1989

pectat ions for the flavor dependence o f these couplings and in section 5 we show how these expectat ions hold up in view of the previously stud- ied bounds. The results are pleasing in that not only are the bounds not seriously violated, but several un- seen reactions are on the verge of observability. These discussions, predict ions for flavor changing decays of the Z °, and speculat ions on the scale of this new physics responsible ['or the FCNP are presented in section 6.

2. An illustrative example: vector singlet model

As a simple illustrative example of the general class of models, in which tree-level neutral f lavor changing couplings of H ° and Z ° between ordinary quarks and leptons are generated through the effect of mixings with heavy exotic fermions, we consider a model ¢~ with an SU (2) c vector singlet of charge - 1/3 quarks, DL and DR, plus the three s tandard families of quarks and leptons.

In the basis of weak-eigenstates dO and d ~ , the mass and the Yukawa couplings of the charge - I / 3 quarks are given by

-- L,- ----- [ d~. ( M d )4 d°N + d~L (y'J)u d°N H ° / x / ~ ]

+h.c. , (1)

where

IYII 3;12 Y)3 .VI4q

Y21 Y22 Y23 Y24 [ ( 2 ) ( y d ) = ) . I 3'32 Y33 ~ 4 ' LoGo ;J ( M d) = ( y d ) v / , j 2 + ( m ' ),

v = - ( \ / /2Gv) ~/3=246 GeV, (3)

with

( M ' ) = 0 0 0 0 0 0

0 0 0

(4)

~ Generalization to the case with a leptonic vector singlet with charge - 1 (EE- and ER ) and/or charge 2/3 quarks (UL and U~) is straightforward. For earlier works on models with exotic heavy fermions, see ref. [ 7 ] and references therein.

In eq. ( 1 ), i, j = 1, 2, 3 correspond to the three families of ordinary d-type quarks (d, s, b) , while i , j = 4 corresponds to the heavy exotic ones, DL and DR. ( M ' ) o feq . (4) is due to the bare mass term M a n d takes the given form without any loss of generali ty ~2. The mass matr ix ( M d) can be diagonal ized by the uni tary matrices VL and Vn,

_ ~[//d ( V L ) t ( M d ) ( V n ) =, ~iag. - (5 )

Defining the mass-eigenstates d;L and din, for i = 1, 2, 3, 4, by

d~,)~ = ( V~) , ;d ,~ , d,~ = ( V{_)vd°~,

d°R -- ( VR)ndjn, d;n - ( V~)vd°n, (6)

the Yukawa couplings of the (standard model) Higgs scalar H ° are given by

- L , ~> =(1~ [ ( V [ ) ( y d ) ( V n ) ] d R H ° / x / 2 + h.c.

=dTkik (Mdiag.)~,d,RH°/v

-- (M/v)d ,L ( VL)], (VR) 4/djRH°+ h.c. (7)

where we have used (yd) = [ ( M ~) _ ( M ' ) ] ( x f2 /v ) , f rom eq. (3) . Thus the flavor changing coupling of H ° is given by

LF.C.H° =y,idT~iLd,RHO+h.c., for i ¢ j , (8)

where

y ' , i = ( M / v ) ( V L ) ' ~ , ( V n ) 4 j , for iCj . (9)

In the basis of weak-eigenstates, the neutral current coupling o f Z ° is

0 /~ LZ°.c. = ( e / s in 0w cos 0w)J~,Z , (10)

where

j o __j;~ _ sinZ0w j~,, ( 1 1 )

0 k ~ 0 =d,L (13),i21,diL

_ ( _ 3 1 ) s i n 2 0 w o,, o (O,L z;,d;L + O°R 7,,d°R), (12)

~2 One may think that the most general form of the matrix (M') has four nonvanishing elements in the last row, i.e. (M')4, ---M~ ¢ 0. However, a unitary rotation in the space of the four dj~ s (j= 1.2, 3, 4), can always bring the matrix (M') to the form given in eq. (4) with M~(IM'II2+IM'2IZ+IM~[ 2 +IM;I 2) ,/2.

382


(t~-) = d i a g . ( - ½, - ½, - 21,0)

= - ½ [ d i a g . ( 1 , 1 , 1 , 1 ) - d i a g . ( 0 , 0 , 0 , 1 ) ] . (13)

In terms of the mass-eigenstates of eq. (6) , eq. ( 11

becomes

J~,', ½d,LT, diL+ l * = -- ~d,L ( VL)4,( VL)4jdjL

-- ( -- } )sin20w (d,L }'/,d,L-4-d,R y/,d,R ). ( 14

Thus, the flavor changing coupling o f Z ° is

Z ° ~ L M 0 Lv.c. = (e/2 sin 0w cos Ow.)god,L7 d iLZ/,,

for i¢=j, (15

with

~ = (VL)]~( VL)4/, for iCj . (16)

The main results ~3 of this section show that the flavor changing couplings o fH ° and Z ° are given by eqs. (8) , (9) , (15), and (16), with strengths proportional to the product of the mixing angles ( VL)], and

( V L . R ) 4 ! of the unitary matrices VL and VR.

3. E x p e r i m e n t a l cons tra in t s on the f lavor c h a n g i n g c o u p l i n g s o f H ° and Z °

In order to discuss the physics of flavor changing couplings of H ° and Z ° we need to know their present experimental constraints. We shall use the notat ion and convent ions described below. The most general form of couplings of the Z ° to ordinary quarks and leptons is

I ' ~ L ~ ~ll~C R i t 0 Lz,,= ~ ~6</.,L/ J,L +goZR? f R ) Z , + h . c . i.]

2 sin Ow cos Ow

- ) --aRC ,,,C ~7O+h.c" (17) ~ N t l d t R / d l R l ~ l t

From hermiticity

= g . and (18) gi, =gq •

~3 Similar tree-level flavor changing couplings ofH ° and Z ° exist for vector doublet models ( ~ , YiJ) and mirror fermion models (gli. g~, Y, ); generalizations of our discussions to these models are straightforward.

The indices i a n d j stand for flavors of the ordinary quarks and leptons. Similarly, the most general form of couplings of the Higgs scalar, H °, to the ordinary quarks and leptons is given by

LHO ~ R , L " 0 = (f, LY,/£~ +f,,Rk,,J;L)H +h.c. i.]

e (F, LY,,£R -- 2 sin 0w cos 0w

- ) +f=)o'~£L) H ° + h.c. , ( 19 )

with

S ) R ^ L * I~L __ I ' )R* =y,/ and (20) .~ 11 -- .~ tl •

The unknown mass of the Higgs scalar, MH, will be expressed in terms of ~(/n, where

MH =-MH/Mz =MH/(92 GeV). (21)

Note that for sin20w=0.23, e / ( 2 sin 0w cos 0 w ) = 0.36. The advantage of expressing all couplings in these units is that the relations

(e /2 sin 0w cos Ow)2/Mz = , ~ G F ,

L . R L , R / R A t 2 Y~j Ykl /~V~H

- - G L , R ~3 L.R i ~/2 --V,: >~: t~-/ sin 0wCOS 0w te/MH

^ - - 2 ~ L . R " L . R =x/2GFMn Yu Y:,-I (22)

simplify the expressions appearmg in the effective la- grangians. We have investigated a variety of FCNP which are likely to provide the most stringent constraints on the flavor changing couplings of H ° and Z°; details will be published elsewhere [8]. Our results are summarized in the first three columns of tables 1 and 2 ~4. Note that the results for Bo_B d o o mixing are not to be taken as a bound, but, in the spirit of this work, as a positive result fixing the parameters coupling the b-quark to the d-quark.

~4 The full expressions, of course, involve the coupling constants in a complicated manner. For example, in the case of H ° ex- change, the constraint from AMK is (1/3~/H )I (J;'~d)2+ (~ )2 -~(6+f l~ ' ~L w2<9.5× ~/2 ) (Y~d) (.9~) I 10-5(10/ilK) , with ilK= [mK/(m,+md ) ]2~ ( 10+ 5). Since we cannot disentan- gle the various parts of this expression, we assume that there are no fortuitous cancellations and present results for a generic coupling f,L.R

_ s t .

383

tao

~

o

Tab

le 1

P

rese

nt e

xper

imen

tal

cons

trai

nts

on t

he f

lavo

r ch

angi

ng c

oupl

ings

(ti

mes

Hig

gs m

ass

fact

or)

of th

e H

iggs

sca

lar,

H °,

and

the

ir p

redi

cted

val

ues

for

p=

1/2

and

p

= 1

by c

q. (

34).

In

the

firs

t ro

w,

fl~

--- [

m~

/ ( m

~ + m

e ) ]

~ -~

( 10

+ 5

). F

or p

bet

wee

n 1 /

2 a

nd 1

, the

pre

dict

ed v

alue

s ar

e be

twee

n th

e va

lues

for

p =

1 / 2

and

p =

1.

,<

t,3

==

Sou

rce

Cou

plin

g E

xper

imen

tal

Pre

dict

ions

: p

= 1

/2

Pre

dict

ions

: p

= 1

up

per

boun

d

g

AM

K=

3.5

2x

I0-~

SG

eV

[5]

~(4~

~ _9

L'R~

d 9.

0×10

-5,e

/10/

/3K

(1

.6-+

0.4

)×1

0

4 (2

.6+

0.7

)x1

0-~

D

°-D

°mix

ing

[9]

aT

/~ ]

0~R

I 7.

2X10

4(

O. 1

6GeV

/fD

) (3

.9±

0.9

)X1

0

4 (1

.6±

0.4

)X1

0

4 B

',I-B

~] m

ixin

g: A

RG

US

, C

LE

O [

1,2]

37

/~'

[)3~a

R I

(9.5

-+2

,3)×

10

4(

0.15

GeV

/f~

) in

put

inpu

t B

R(B

O~

g+

~-X

)~<

I.2

xI0

3

[10]

,?l

~/t~

2 Lv

LsR

I 10

.3(I

V~h

l/0.

043)

~/

fi~(

4.1

± 1

.0)x

l0

3 aT

/fi~

(l.8

_+0.

4)x

10-2

_

~/~

fiL

~R

_ (4

.2+

+_l

.O)×

lO--

\/m

J6O

GeV

(

1.9_

+ 0.

5) (

mJ6

0 G

eV)

- ~

/~1,

9~:~

1 -

(7.5

-+1

.8)×

10

-,

~/m

b/5O

GeV

(5

.9+

l.4

)(m

u/5

0G

eV)

BR

(la-

~e-

y)~

<4

.9×

10

-~

[1

1]

MH

'I))

~R

I 1.

6×10

~

(1.1

-+0

.3)×

10

5

(1.2

+0

.3)×

10

7

6a~,

<2x

lO

~ [1

2]

37/,

~ I.~

¢)7~

I '/

2 0.

38

(1.4

-+0.

3) X

10

-4

(2.0

-+0.

5) X

10

s B

R(l

a ~

e 7)

~<

4.9X

10

~ [1

1]

~(4~

21.v,

~y,."R

~c I

1.9

xl0

-~'

(2.8

±0

.7)X

10

v

(8.4

_+4.

0)X

10-s

8

a~<

3×

10

-s[1

2]

~(lf

i'lfe

~2{~

l '/~

23

.8

(2.0

-+0.

5)X

10

3 (4

.2-+

1.0

)×1

0

3 B

R(-

c-~

p

bt+b

t )~

<2.

9X10

51

11]

37/~

'13)

~; ~

1 35

.3

~/~

(2.0

±0

.5)×

10

-3

~

/~'(

4.2

±1

.0)×

10

3

- ~(

1~ ~

I/'L

R I

-

(3.9

_+ 0

.9)

× 10

-2 tx

/~./

40

GeV

(1

.6-+

0.4)

(m

~/4

0 G

eV)

a:

,.<

Tab

le 2

P

rese

nt e

xper

imen

tal

cons

trai

nts

on t

he f

lavo

r ch

angi

ng c

oupl

ings

of

the

Z °,

and

the

ir p

redi

cted

val

ues

for

p=

1/2

and

p=

1 b

y eq

. (3

4).

For

p b

etw

een

l/2

and

i,

the

pre

dict

ed v

alue

s ar

e be

twee

n th

e va

lues

fo

rp=

1/2

and

p=

1.

ca')

t"

tm

,q

~z

Sou

rce

Cou

plin

g E

xper

imen

tal

Pre

dict

ions

: p

= 1

/2

Pre

dict

ions

: p

= 1

up

per

boun

d

AM

~=

3.5

2×

10

,5 G

eV [

5]

I~iR

I

2.2×

10

-4

(1.3

--+

0.3)

× 1

0 .4

(2.1

--+

0.5)

× 1

0 -5

B

R(K

t-,g

+g

-)=

9.1

×I0

9

[5]

IRe(

~--g

sd)J

~L

1.3

4×

10

-5

(1.3

-+0

.3)×

10

a

(2.1

+0

.5)×

10

-5

D°-

D °

mix

ing

[11

] I~

a)dR

I 7.

9× 1

0 4(

0.16

GeV

/fD

) (3

.0_+

0.7)

X 1

0 -4

(1

.2-+

0.3)

X I

0 4

B',I

-I}~

mix

ing:

AR

GU

S,

CL

EO

[1,

2]

I~aR

I (7

.5_+

1.8)

×10

4(

0.15

GeV

/,/B

) in

put

inpu

t B

R(B

°~g

+p

-X)

~ 1.

2X 1

0 3

[101

I~

'~,~R

I 1.

2× 1

0 2(

L l~

.b] /

0.04

3)

(3.2

+0

.8)

× 10

-3

(1.4

--+

0.3)

X 1

0 2

- I~

.'~l

(3.3

-+0.

8) ×

10

2x/m

,/60

GeV

(1

.5_+

0.4)

(m

,/6

0 G

eV)

- I~

abL'

~I

(5.9

±1

.4)×

10

2

~ (4

.7-+

l.l)

(mb

./5

0G

eV)

BR

(g-~

e-e+

e )~

<l.

0×

10

-J-'

[ll]

r~

a~R

I 2

.3x

10

6

(8.4

_+2.

0)×

10 -

6 (9

.5-+

2.3

)x1

0

8 B

R(z

~

e p+

p.-)

~<

3.3×

10

S[l

l]

b~6~

gRI

3.9

×1

0-2

(1

.1+

_0.3

)×10

-4

(1.6

_+0.

4)×

10 -

5 B

R(g

~

e-7

)~<

4.9

×1

0

" [1

1]

Ig~:

R,~

Rgk

l 7

.5×

10

-6

(1.7

-+0.

7)X

10

v (5

.3__

+2.

4)×

10-S

B

R(z

-~g

-g+

p.-

)~<

2.9

×1

0-5

[ll]

]g

~;R

1

2.9

×1

0

2 (1

.6-+

0.4

)×1

0

3 (3

.3-+

0.8

)×1

0

3 ~

E.R

-

[g~

[

(3.1

-+ 0

.7)

× 1

0-2

\/~

/~

GeV

( 1

.3-+

0.3

) (m

~./

40 G

eV)

Z,

g-

7~

~D


4. Theoretical expectations on the flavor dependence of the flavor changing couplings

In section 2, eq. (9) and eq. (16), we have seen that, in the context of a simple vector singlet model, the flavor changing couplings are proportional to the product of mixing angles ( VL,~ )]i( V~,R )4J. We expect similar results to hold in other models involving heavy exotic fermions. In this section, we consider how such mixing angles ( ( VLR ) 4j'S) should depend on the generation (family) indexj. From experience with KM angles, we expect

t(VLn)41[<<l(Vk.R)421<<l(rk.R)431<<l. (23)

Lighter fermions are expected to have smaller mixing with the heavy exotic ones in order to keep their masses small; too much mixing would spoil this smallness. This may be the reason why the flavor changing neutral processes between the first two lightest families (i.e., d*--~s, e,- ,g) have not been observed thus far, and the GIM [ 13] mechanism has been so successful, since these are the ones that are likely to be the most suppressed in terms of the mixing angles.

To make a reasonable estimate on these mixing angles, we consider the following simple case of 2 × 2 mixing as a guide, which should shed some light on the general relation ~5 between the mixing angles and the mass-eigenvalues. Consider a 2 X 2 real symmet- ric matrix,

( m ) = [ ; ~1 (24,

which is diagonalized by an orthogonal matrix R (0),

I-o 0] R(O)V(m)R(O) - mdiag" ~= m 2

R(O)=- I cos0 sin0 1 - s i n 0 cos 0_] " (25)

Note that the matrix (m) contains two free parameters excluding the overall mass scale, while R(O) has only one. Therefore, given the mass-eigenvalue hierarchy,

r=--ml/m2 << 1, (26)

~5 For a recent review on this subject, see ref. [ 14 ].

the mixing angle 0 will be a function of r and one ad- ditional parameter (say a ) . Choosing the unit of the mass scale such that m2= 1, eqs. (24)and (25) imply

OL - -F

[ -r+(l+r) sin20 - ( l+r) sinOcosO] = - ( l+r) sinOcosO 1 - (1 + r ) sin20 "

(27)

Thus we have

c x = - r + (1 + r ) sin20,

which is equivalent to

r + a sin20= 1 +~" (28)

Now, the eigenvalue equation (trace and determi- nant ofeq. (27)) implies

a - f l 2 ~ - r . (29)

Assuming the mass hierarchy ofeq. (26) is not due to a fine-tuned cancellation between a and /32, one expects a to be at most O(r ) from eq. (29). There- fore, in the absence of a fine-tuned cancellation in r + a in eq. (28), sin 0is expected to be

sin 0-~ 0 ( ~ r ) = O ( x ~ / m 2 ). (30)

If a = 0 for some symmetry reason, then

sin 0 = x / r / O +r)~-~X~l/m2. (31)

I fa fine-tuned cancellation exists in r+ c~, so that their sum is of higher order, i.e., r + c~ = O (r 2 ), then

sin O~_O(ml/m2). (32)

Thus the most natural relation between the mixing angle and the mass ratio is sin 0 - ~ O( m ~ l / m 2 ), and a particularly interesting one is that ofeq. (31 ), where the mixing angle is exactly the square root of the mass ratio. Several examples of this relation of mixing an-

385


gles as square roots of mass ratios already exist in the literature ~6

From the above discussion on the mixing angles and the mass ratios, we see that the most reasonable estimate on ( VL,R)41 is

( VL~)4/--~ (rnJM)", with 1 < p < 1 (33)

and the generation dependence of the flavor changing couplings is expected to be

(m,m,']" ~,j with ½ < p < 1. (34) Y/, I

Moreover, p-~ 1/2 is expected to be more realistic than p ~ 1.

5. Comparison with existing data and predictions for future experiments

Considering iA dl~O-L~d~0 mixing as an anchor for the

FCNP, we use the results of the last section as summarized in eq. (34) to predict the expected values for other coupling constants and compare these with experimental data on positive results or on bounds. The results f o r p = ½ and f o r p = 1 are shown in the last two columns of tables 1 and 2. Before looking at the details, we should discuss two caveats. First, we use eq. (34) to extend the coupling constants, not only to systems made out of charge - 1/3 quarks, but also to those of charge 2/3 quarks, and to leptons. This would be valid if the mass scale responsible for the breaking of the GIM mechanism would be the same for all three of the above systems. Even though this may be unlikely, we do not expect these masses to be orders apart; thus there may be a rescaling by a small factor as we go from group to group. Second, as dis- cussed in footnote 4, we are presenting results for a

"6 The charge - 1/3 quark mass matrix of the form of eq. (24) with ~=0 for the two family case is well known to give the phenomenologically successful relation sin 0(~x//mJm~= ,~./1/20=0.22, where 0c is the Cabibbo angle. A similar mass matrix for the neutrinos gives rise to the well known seesaw mechanism [ 15 ]. For three families of quarks, a generalization of this matrix gives phenomenologically successful relations [4,16] between their mixing angles and masses, where the entire set of KM angles are expressible in terms of the square roots of the quark mass ratios. For the four family case, see ref. [61.

common coupling constant for each process, while the detailed expressions may involve complicated sums of products of left and right handed couplings. Thus we are ignoring possible detailed cancellations or en- hancements. Due to these two caveats, all of our results should be viewed as valid only up to a factor not too different from one~ With these remarks in mind we see that we have no'gross violations of any present experimental bounds. We also note that predictions for several, as yet unobserved processes are close to their present bounds. We shall discuss these in some detail.

In case the observed B °- I ] ° mixing is due to flavor changing couplings of the Higgs scalar H °, table 1 predicts two flavor changing couplings which are slightly below the present experimental upper limit. These are the ones for D ° - D ° mixing and B R ( g - - ~ e - y ) . Predictions on these quantit ies are given in table 3. If we also take eq. (33) literally, the

])L,R required strength of I. bd L (of table 1) for Bd-Bd° -o mixing predicts

M . ~ (2 .4±0 .6 ) [fB/(0.15 G e V ) ] M z

(200-300) GeV, (35)

tbr p = 1/2, and MHM--- (0.5_+0.1)(fB/0.15 GeV) GeV for p = 1. Thus the latter case predicts a rather unrealistic, low value of M < 5 GeV (since we know ~/n > 0.1 ) and thus we conclude that p = 1/2 would be much closex~ to reality than p = 1.

In case the observed ~,dw~-~,o~° mixing is due to the flavor changing coupling of Z °, table 2 predicts several flavor changing couplings of Z ° which may, in the near future, have observable consequences, namely, B R ( g - - , e - e + e - ) , D ° - I ) ° mixing, BR(B ° --,g+g X), B R ( z - - , l a - g + g - ) , B R ( g - - - * e - 7 ) and the flavor changing decay modes ~7 of Z ° which can be tested with the 107 Z°'s expected at LEP. The predictions on these quantities are likewise given in table 3. Again, if one takes eq. (33)literally, one finds from the required strength of I~,iR [ (of table 2) for B ° - B ° mixing

M-~ (275_+66) G e V × (fB/0.15 GeV), (36)

nv These flavor changing Z ° decay mode branching ratios are much larger than the ones expected [ 17 ] in the standard model.

386

Volume 219, number 2,3

Table 3 Summary of the predictions for future experiments.

PHYSICS LETTERS B 16March 1989

Mediating Present experimental boson upper limits

Predictions for future experiments

p = l / 2 p=l

Higgs scalar

H/I

ZII

D°-I) ° mixing: rD~5.6XI0 z3 BR(Ia --*e-y) ~<4.9X 10 ~

BR(la ~ e - e + e - ) ~ l . 0 x l 0 12

D~}-I3 ° mixing: rD~ 5.6X 10 3 BR(B°-*~t+g X)~I .2x I0 3 BR(z --*g ~a+~t )~<2.9X10 5 BR(g ~e-y)~<4.9X 10 ~

Flavor changing Z ° decay mode branching ratios BR ( Z ~ b s + sb) X 10 7 BR(Z°~tc+ct) X 10 7 BR(Z°~b'b +bb ' ) X 10 7 BR(Z°~B ~++la+x ) X I 0 7

BR(Z°~z'z+t'z) × 10 7

D{~-O ° mixing of O(10%) of present tipper limit (1.1_+0.6)×i 0 ~2

any value in the neighborhood of present upper limit D°-I) ° mixing of O( 1-10)% of present upper limit (8.5_+4.2)×10 5 (8.8+4.8))<10 s (2.5_+2.0)×10 14

D°-ID ° mixing of O(0. l%) of present upper limit (1.1_+0.8)×10 ~3

(1.7_+0.8) X 10 ~

O(0.01-0.1 )% of present upper limit (1.6_+0.7)×10 x (3.8_+1.8)×10 v (2.4_+2.0) × l0 -t5

14_+7 270_+ 120 ( 1500_+ 700) (m,/60 GeV) O( 104 ) (4800_+2300)(m~/50GeV) O(105 ) 3.6_+1.8 15_+7 ( 1300_+ 600) (m,/50 GeV) O( 104 )

for p = l / 2 , while p = l gives M---(7.5_+0.9)

GeV×xf fB /0 .15 GeV. The latter value o f M ( = t h e

scale of the heavy exotic fermion masses) is, again,

rather too low to be realistic. It is interesting to note

that the observed strength of B ° - B ° mixing implies

that the values of MH in eq. (35) and of M in eq.

(36) to be O ( v = 2 5 0 GeV) , the scale of the electro-

weak symmetry breaking; this may not be a numeri-

cal coincidence.

6. Summary and conclusion

In this article, we investigated the implicat ions of

the possibility that the observed o -o Bd-B~ mixing is due

to small f lavor changing couplings of the Higgs sca-

lar, H °, or the Z °, induced by new physics at an en-

ergy scale beyond the standard model. The implica-

tions are rich and the predictions for future

experiments are summarized in table 3. Moreover, the scale associated with the new physics, M, a n d / o r

the mass of the Higgs scalar seem to coincide with the

Higgs vacuum expectation value, v= 250 GeV. This

may not be a coincidence but may indicate that this

new region will indeed show up at a mass scale of 250

GeV. Although our discussions were made in the context

of a model of ordinary fermions mixing with heavy,

exotic ones, the general structure of these flavor

changing couplings should be valid in a broader class

of theories ~s. CP violat ion could be included in such

a class of models; as the GIM mechanism is violated,

an electric dipole moment could be induced at the

one-loop level. Likewise these effects would become

stronger with increasing quark mass. We plan to re-

port on these effects in a future publication [ 18 ].

Acknowledgement

We thank Gordon Shaw and Bill Molzon for valu-

able discussions. This research is supported by the

NSF under Grant No. NSF-PHY-8605552.

ns We believe that some composite models of quarks and leptons and some technicolor models can give rise to similar flavor changing couplings.

387


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Flavor changing neutral processes and Bd0−B̄d0 mixing