Bottom quark mass predictions in non-supersymmetric SU (5) unification

Physics Letters B 277 (1992) 130-136 North-Holland

PHYSICS LETTERS B

Bottom quark mass predictions in non-supersymmetric SU (5) unification Paul H. Frampton, James T Lm and Masahlro Yamaguchl lnstttute of FwldPhysws, DepartmentofPhyswsandAstronomy, UniversityofNorth Carolina, ChapelHill, NC 27599-3255, USA Received 3 December 1991

We study the bottom quark to tau lepton mass ratio in a large class of non-SUSY SU (5) models Our general analys~sallows us to rule out most one-Hlggs models which predict too large a value for the mass of the bottom quark We then look at a particular two-Hlggs model and show that the correct bottom quark mass can be achieved by setting the top quark Yukawa coupling close to its mtermedlate fixed point

After the initial proposal o f SU ( 5 ) grand unification [ 1 ] and the calculation o f the unification scale and mixing angle sm20w [2], there were a few years o f lacuna until interest in SU ( 5 ) was relgnlted by the observation [3,4] that the ratio o f the b o t t o m quark and the tau lepton masses, R - - ( m b / m , ) , was also successfully predicted by the theory, in fact, the predicted mass ratio depended, in an interesting way [ 4,5 ], on the assumed n u m b e r o f q u a r k - l e p t o n famlhes, and the observed ratio seemed to require three famlhes F r o m the recent data o f LEP at CERN, we know now that this further prediction o f three families was v i n d i c a t e d These accurate d a t a on the coupling constants ot,(Mz) ( t = 1, 2, 3) o f the s t a n d a r d m o d e l allow us now to r e - e x a m m e the mass ratio mb/m, in SU ( 5 ) In particular, we know that minimal SU (5), in which the first such predictions were made, unfortunately does not consistently unify at a unique MOOT This leads us to investigate n o n - m l m m a l SU ( 5 ) models One case is the m i n i m a l supersymmetrlc s t a n d a r d model ( M S S M ) in which unification works well [ 6 8 ] A n o t h e r case IS the spht-multiplet non-supersymmetric S U ( 5 ) m o d e l [ 9 ] Recently, such SU ( 5 ) models with m o r e asymptotic freedom than the M S S M have been constructed [ 10-12 ] with less proliferation o f unobserved particles These i m p r o v e d models are examples o f n o n - s u p e r s y m m e t n c SU ( 5 ) models which hence merit further study Here, we in130

ltlate such an investigation by calculating mb/m~ in a large class o f these non-SUSY models One o f our key results is that only that subset o f models with at least two Hlggs doublets can a c c o m m o d a t e the observed b o t t o m quark mass In particular, the A B C m o d e l o f ref [ 11 ] contains the requisite extra doublet a n d we estimate the b o t t o m quark mass in it We assume the unification group is SU (5) with the s t a n d a r d m o d e l Hlggs doublet contained in a five-&menslonal representation Such a theory predicts that mb (McuT) = m, (Mova-) and we are Interested in how this equality is r e n o r m a l l z e d when we continue down in energy to the experimentally relevant d o m a i n The actual mass o f the tau lepton is m r = 1 7 8 4 + 0 004 OeV On the other hand, because o f the subtleties involved in Q C D confinement, the experimentally observed mass o f the b o t t o m quark is not well determ i n e d [ 13 ] Potential m o d e l fits to the spectra o f b o t t o m quark mesons [ 14 ] generally yield a value o f 5 0 + 0 2 GeV which should be regarded as a constituent quark mass We assume that the Q C D dressing o f the b o t t o m quark is not large and in this p a p e r we take the value rob(rob) = 4 8 + 0 2 GeV for the current quark mass These masses are on-shell values and become renormallzed according to the well-known renormallZatlon group ( R G ) flow equations O u r procedure, which is very general, is to assume that the unification o f the a ? 1 (#) m the a 7 ~-/2 plane is exact at some # = M o o r To achieve this, it is nec-

0370-2693/92/$ 05 00 © 1992 Elsevier Science Pubhshers B V All rights reserved

Volume 277, n u m b e r 1,2

PHYSICS LETTERS B

27 February 1992

essary to add properly chosen extra matter states (fermlons and/or scalars) Although each addmonal state could enter at a different threshold, we assume they are all degenerate at some common mass M~ satisfymg M z ~ M l < M ~ u T A general such non-SUSY SU (5) may be characterized by three parameters MGUT, MI and the value of the unified gauge coupling Ot~,l~T ~ Or- 1 (MGuT) Note that we do not specify the details of the matter fields but instead make model-independent statements which will enable us to exclude large classes of non-SUSY unifications Earlier analyses [ 12 ] have focused on specific models Because of the uncertainty m the bottom quark mass, it ~s sufficient to work at one-loop order m the R G equations between MI and M~UT, although below Mz it is important to use the full two-loop equatmns since in this low-energy regime a3 (#) is sufficiently large that two-loop effects can be as much as 4% We take the standard definition for the SU (3) × SU (2) X U ( 1 ) coupling constants

which shows that the couplings evolve hnearly in the ot. 1-log/2 plane Since we can ignore threshold effects consistently at the one-loop level, below MI we use the R G equations of the minimal SM, whereas above MI these coefficients take on different values, and the slopes change abruptly Although these b, above MI are determined completely by the particles of the model, we do not specify these new particles exphcltly Instead, we assume perfect unification so that the couphngs meet at a point ( O ~ T , MGUT) In the a/-~-/2 plane (with Motrr> M~) The gauge coupling evolution of these non-SUSY SU (5) models is specified completely at the one-loop level by the three values MI, MOUT and OtE~T The matter content above MI is determined lmphcltly from these parameters by the values of the b, (for # > MI) gaven by

al = ~g'2 / 4rt= 5 a / 3 cos20w,

Since ad&tlonal scalars and fermlons contnbute posmvely to the b,, a model must additionally satisfy the requirement b,>~ (b,)sM for t = 1, 2, 3, which gives the restriction

a2 = g 2 / 4 n = a / s l n 2 O w , or3 = g 2 / 4 x

( 1)

The R G equations determine the evolution of these couplings as the mass scale # is changed At the oneloop level, these equations are d

b,

2

/2 ~t-~/2a, ( /2 ) = -~-~nnot , ( lt ) ,

(2)

where the coefficients, b,, are determined by the parttcle content of the theory For the minimal SM, they are given by

b, = (b,)sM =

ion/ / + NFam

\-11/

4

+

(!)

Nm~,~

\g.

(3)

_

"b "

t ,~sM1

(1 a,(

1

b,

2n

lnf/2 ~,

)

ot

,)

~UT

ln(M~/Mz)

~

- - ~1<

1

O/GUT

at (MGuT)SM

1 =- ot,(Mz~)

(5)

)

( b , ) s . ln{MOuT ~

2rg

~,~)

(6)

where SM denotes values calculated m the mtmmal SM In the SM, the quarks and leptons acquire masses from Yukawa couplings m the Hlggs sector For a general umficatlon model wtth several Hlggses, let n ~ ( a = 1, , NHI~e.~)be the Hlggs doublets Ignoring any mixing between families, the third family Yukawa interaction terms m the lagrang~an are wntten as ot=l

(,;.~q3LH,~bR+).",'f3LH,:R

+2~'q3H*tR) + h c

ot,(/2o)

-Mz

NH~sgs

to yield 1

-

LPYukawa= ~

w h e r e NFa m = 3 and NH,**~= 1 Eq (2) is easily solved

or,(~2)

27r

b, = ln(MouT/Mi)

(4)

(7)

The masses of the bottom, tau and top are g~ven by the vacuum expectation values (VEVs) of the various nIggses, 131

Volume 277, number i,2

mb= E2~(H°),

PHYSICSLETTERSB

27 February 1992

m,= ~2~(H°),

Ot

Ot

(-i)

d,=c~ b) - c ~ ) =

mt = ~, ~-ta ( n , ~o) ,

(8)

c~

where the superscript denotes the neutral component of the Hlggs field The one-loop R G equauons for the Yukawa couphngs are given m a mass independent renormahzatmn scheme by [ 15,16 ] / t ~d- ~ 2 b = - ~ c ~ ) ot,(/t)2g'+ 1/2

+ ~

1

~ ;t~2tP2g,

d 2~ = -

c~)

+__L_ 3 16n z

2~;ttP2~,

d

a

=-

9/2

+ ~

~ ('z,p)~'~' (9a)

(9b)

c(t)

%

~ (2tP)22~' ,

(9c)

where we have assumed that the Yukawa couphngs ;t~ and 2," are small and have ignored their effects on the fl-functlons Throughout MZ<~#~MGuT, the coefficients c(,A) (A=b, r, t) are

~' - \ 8 J '

c~b)=

=

,

(10)

and are independent of the additional particle spectrum In the case of a one-Hlggs model, these R G equatmns slmphfy Let 2A =21 (A=b, r, t) and R(/t) - m b (/z) / m, (/t) = 2 b(/~)/2 ~(/1) Then the relevant R G equations m our analysis can be written as /t ~ In R(/t) = _ ~ 4~ a,(/t) +

D

From (9a), we see that in the one-Hlggs model D=

3 Done-HlggS

=

-

-

When there is more than one Hlggs doublet, the SltUaUon becomes more comphcated However, the analysis slmphfies m the case of a two-Hlggs model with separate Hlggs doublets coupling to the top and bottom In this model, we let 2t = 2 2, 2b=2~ and 2~=;t~ (all other Yukawa couplings vanish) Then the R G equations again reduce down to ( 11 ) where all the coefficients are the same as above with the exception of D = D,wo-H,sss= 1 because the last terms m (9a) and (9b) vanish Using the one-loop gauge fl-funcUons, it is easy to solve the R G equaUons ( 11 ) [ 12 ], m order to determine R ( M z ) m terms o f R (McuT) --=1 and 2t(Mz) Note that, at the one-loop level, R ( M z ) is specified completely by the parameters M~, M~ux, CrS~T and 2 t ( ~ z ) , which enables us to develop a very general analysis In order to compare R (Mz) with experiment, we convert it to a value for mb(mb), the on-shell bottom (current) quark mass Below Mz, the electroweak theory is broken, and evolution of the gauge and Yukawa couplings obeys the SU (3) × U ( 1 ) era theory In the regmn mb<~/z<~Mz, or3 is relatively large, so we include two-loop gluon contributions to the fl-funcUons This amounts to about a 4% effect when compared to the one-loop results The tau lepton mass also runs from the on-shell value to the value at mb HOWever, this comes from the running of the U( 1 )Era coupling which is small Thus we calculate mb(mb) using this approximate two-loop evolution of R(/t) along w~th a one-loop evolution of mr We use the world average values of the couplings determined at the Z ° energy o t - l ( M z ) = 127 9_+0 2 ,

sln20fi-g(Mz) = 0 2333 _+0 0008,

2

ot3(Mz) = 0 113_+0 005 d c, 9/2 # ~ - ~ l n A t ( / z ) = - - ~ - ~ t ~ , ( / ~ ) + l--~-~2A2(/z), where c, =c~ t) and 132

(12)

(11)

(13)

Th~s value for the m~xlng angle, s m 2 0 ~ , is calculated m the MS scheme and comes from a detailed analysis of world data by Langacker and Luo [ 8 ] The

Volume 277, number 1,2

PHYSICS LETTERS B

value of O/3 IS quoted from the world average given by Hebbeker [ 17 ] Experimental hmlts on the proton hfetlme can be used to put a lower bound on MGtn- for reahstlc models The current value is Tv(p--,e + rc°)

>_-3 l × 1 0 3 2 y r

(IMB [ 1 8 ] ) ,

>~26×1032yr

(KamlokandelI[19]),

(14)

which translates to a minimum of about 2 X 101 s GeV for the untficatlon scale using the simple estimate 1

Zp~

2

O/GUT

4 Mcux , mp5

(15)

where rnv=0938 GeV is the proton mass For MGUT>~2× 1015 GeV, we find that the requirement on the addmonal matter, (6), is domxnated by the mequahty for O/F 1 Given the expenmental data for the O/,(Mz), we can calculate m b ( m b ) m a variety of models First we ~gnore the contributions of the top Yukawa couphng to the calculation (or formally set 2 t = 0 ) We have illustrated in fig 1 the calculated values for mb(rnb) as a function of Mcu-r and O/~6x for a phenomenologically interesting intermedmte scale of M×= 1 TeV 40

~

•

i

5 58 59 60 61 62

• 30 -

6 3

i

,

,

,

,

I

,

7

~--

i

4

,

,

i

i

~

i

i

~

/

/

/

mb=6 51/-

. . . . . . . . . . .

I

E0 - .~

09

27 February 1992

We see that the predicted mass increases as new matter is added and O/~'r decreases This is easily understood because the dominant contribution to R comes from the strong mteractlon The minimum value for rnb(mb) m this parameter space occurs m the upper left corner when MGUT has the minimum value 2 × 10 is GeV and a g ~ r takes on the maximum value of O/~v = 38 8 We thus see that mb >~5 6 GeV

for MI = 103 GeV

(16)

So for this set of models, when the Yukawa coupling ,~,t IS ignored, the predicted bottom quark mass is always somewhat too large In the rest of this work, we take this small discrepancy seriously In order to take the effects of the top Yukawa couphng into account, we need to specify the Hlggs sector of the theory Let us first consider a model with only one Hlggs doublet In th~s case, ~,t at the electroweak scale xs given by the relation

mt=~tv/N/~ ,

(17)

where v= 246 GeV is the VEV of the neutral Hlggs scalar For this Hlggs structure D < 0 m (11 ), so 2t contributes to mcrease R ( M z ) As a result, mb gets larger as 2t increases This seems to suggest that the predicted value for m b is always too large m a model with only one Hlggs doublet To make this more evident, we may also look at what happens when Mr is allowed to vary In fig 2 we have plotted the mmlmum possible value of mb(mb) as a function of the intermediate scale for various values of mt m a oneHlggs model We notice that as the mtermedtate scale is ratsed, the mimmum possible value of m b becomes smaller For large MI, all the models approach the mmlmal SU(5) theory (with heavy threshold corrections) and mb approaches the value predicted in that theory From the experimental lower bound on the top quark mass, mt>~89 GeV [20], we see that mb(mb) > 5 5 GeV for all one-H~ggs models w~th mtermediate scales up to 10 ~3 GeV This minimum bound is saturated by the theory with the parameters M1 = 1013 GeV,

10 0 1 5

i

~

i

i

i

i

IL

i0 ~s Moor [GeV]

k

i

i

i

1017

Fig 1 On-shell bottom quark mass calculated m the case ;tt=0 for an intermediate scale of M~ = 1 TeV We take ct3 = 0 113

MGUT = 2 × 1015 GeV, O/~IT = 38 8

(18)

So far, we have used the value O/3= 0 113 m this 133

Volume 277, number 1,2 6

.......~ " . . ' . 1

PHYSICS LETTERS B

........i ........i ........i ........i ........~ ........i ........i ........]

ff

E E ~

5

N

10 a

10 a

10 4

10 5

10 °

10 7

10 a

10 9

1010

1 0 it

[ 0 I~ 1013

M~[GeV] Fig 2 Minimum values of mb(rob) as a function of the mterme&ate scale for vanous values of m, in a one-H]ggs model The dotted hne corresponds to the experimental lower bound mt >189 GeV analysis Since this value lS not well estabhshed, we now consider what happens for different values of or3 The b e h a v i o r o f the m i n i m u m b o t t o m quark mass (which comes from the m o d e l given in ( 1 8 ) ) depends strongly on or3 a n d is roughly h n e a r in the region 0 1 0 0 < a 3 < 0 130 F o r smaller values o f the strong couphng constant, the m i n i m u m for mb ts reduced, and at the one-sigma hm]t, or3 = 0 108, we get a lower b o u n d for the b o t t o m quark mass o f 5 3 G e V Although the b o t t o m quark mass is not that well known, this value is almost certainly too large We thus conclude that with a3 = 0 113 + 0 005, all oneHlggs models o f thts type cannot predict a value o f mb consistent with experiment Even if or3 turns out to be smaller than 0 108, tt is still possible to rule out the one-Hlggs models in most o f p a r a m e t e r space On the other hand, m the case where separate Hlggses couple to the top a n d b o t t o m quarks, the top quark Yukawa coupling tends to reduce the value o f mb because D > 0 and 2t now contributes to R ( M z ) in the opposite direction To be specific, let us consider the two-H]ggs m o d e l m e n t i o n e d above in which H~ (HE) IS the Htggs field responsible for gtvlng a b o t t o m ( t o p ) quark mass Then the top quark mass is given by 134

mt = 2 t v sln (fl/ v/2 ) ,

27 February 1992

(19)

where tan f l = ( H ° ) / ( H ° ) Thus knowledge o f 2t is no longer suffioent to predict m t Although it will turn out that we need a large 2, to predict mb(mb) consistent with experiment, this result is not in conflict with +24 G e V the preferred top quark mass o f m t = 127 -30 [21 ] because o f the sin fl a m b l g m t y The evolution o f R (or mb) m M z ~ l ~ M i depends on the H]ggs mass spectrum W h e n both neutral scalars are hght, that is, a r o u n d Mz, the choice D = ½ m ( 11 ) is v a h d from a r o u n d M z to Mov-r But this IS no longer the case when the second Hlggs is heavy, 1 e has mass MI Assuming that the heavy field decouples from the theory below Mz ~ , the low-energy (# < M[) effective theory ts well described by the one-Hlggs m o d e l with the effective top Yukawa coupling 2t sm fl Although the evolution o f 2 t ( # ) d e p e n d s on whether the second Hlggs is heavy or light, the general b e h a v i o r is the same In both cases The evolution is governed by an intermediate fixed point, 2¢r, a r o u n d the F e r m i scale [ 22 ] F o r 2t ( M z ) close to this fixed point, the Yukawa coupling gets very large at the unification scale Since we expand in powers o f t, perturbatlve unification m u s t break down for sufficiently large ~,t(MGuT) TMs puts an effective upper hm]t on how much mb can be reduced by contributions from the top Yukawa couphng The values for mb (rob) m models wlth a heavy seco n d Hlggs are shown in fig 3 We have chosen the same lnterme&ate scale as before, namely MI = 1 TeV In the Hlggs sector, we have taken s t u f f = 0 5 a n d ,~t(MGuT)= 3 which is a reasonable u p p e r limit At this m t e r m e d l a t e scale, there is little dependence on sin fl C o m p a r i n g this graph with fig 1, we see that the large top Yukawa couphng reduces the calculated mass o f the b o t t o m quark by about 17% This significant reduction means that agreement with experim e n t can be o b t a i n e d m a wide range o f the p a r a m e ter space To be more specific, we now examine the A B C sphtmultlplet m o d e l o f ref [ 1 1 ] In this model, the SM ~ Note that this is a non-trivial assumption If the Hlggs field developing the VEV mixes greatly with its orthogonal direction (which we assume to be heavy) in the mass matrix, then the decouphng does not occur

Volume 277, number 1,2

PHYSICS LETTERS B

27 February 1992

65

40

i

~

,

,

mt~89

49

1

1

,

,

~

,

I

GeV

2 ~ = ~ = =

. . . .

4

5O 51 30

-

\

52

-

5 5

53

\

\

\\

\

~I

\ 1 \

\\

\~ \

\

\

54 rob=55 I

7

1

~

1

I

1

I

1 3

5

~o

~

I

I 5

45

4

1o01 sl

i

i

i

i

i i i ]

i0

i TM

i

i

i

i

i

4

i

I 0 tv

M~ [GeV]

, 50

,

It

I 100

~

t

,

,

~b

~

I

7

,

150

,

,

~ 6

6

,

I 200

,

,

, 250

m t [GeV]

F~8 3 On-shell bottom quark mass m a two-H~g~s model vnth

Fig 4 Constant stuff (dashed) and At(MGuT) (dotted) con-

At(Motrr)= 3 and sm fl= 0 5 for an mtermedmte scale of M~= 1 TeV

tours m the mt,-mt plane for the ABC model The one-sigma l,m~ts for mband mt used m the text are mdmated by the box

is a u g m e n t e d with a second Hlggs doublet a n d ferm l o n s in the B a n d C split representations o f ref. [ 9 ] In terms o f SU ( 3 ) × SU ( 2 ) X U ( 1 ) representations, these addlt, onal fermlons are gwen as B = (3, 1 )2/3 + (3, 1 )-2/3 a n d C = (3, 2)~/3+ (3, 2)_~/3 T h e u m ficatmn p a r a m e t e r s for this m o d e l are

value o f mb(mb) can be obta, ned for a rather large )I/(MGoT) which however does not Invalidate the p e r t u r b a t l v e expansion In the region where the theory reproduces the experimental value o f mb (mb) ~, 5 GeV, one can find a close relation between mt a n d sin fl This can easily be u n d e r s t o o d from ( 1 9 ) i f we recall that 2 t ( M z ) is very close to its fixed point I f mt is e s t i m a t e d to he between 100 a n d 150 GeV, we predict from fig 4 that sin fl~.0 4 - 0 6 Although we have used the A B C m o d e l as a specific example o f a two-Hlggs model, the result that 2t(Mz) ~2cr IS rather general We checked that in a wide range o f the p a r a m e t e r space (MI, MGuT, Ot~,ST), the desirable prediction for mb(mb) Is obt a i n e d for a sufficiently b u t not disastrously large top Yukawa coupling This conclusion is also v a h d in the case when both Hlggses are hght In this case, rnb (mb) lS shghtly smaller than in the heavy Hlggs case for a given 2, (MGuT), b u t a large Yukawa couphng is still necessary While we have concentrated on the mass relations o f the t h i r d family, SU ( 5 ) also p r e & c t s similar relations in the first two famlhes It is well known that these results do not seem to be consistent with experi m e n t One possibility for explaining this dmcrepancy is to consider the a d d l t m n o f n o n - r e n o r m a h z a -

MI -- 1032_+09 G e V , MGUT = 10160+-0 3 G e V , Or&ST = 3 5 2 + 0 6 ,

(20)

a n d were calculated using two-loop gauge fl-functmns (~gnonng Yukawa couphngs) Using (15 ), the corresponding p r e d l c n o n for the partial p r o t o n lifetime p--, e + zc° is 103~ ~-+12 y r F o r the A B C model, we solve (11 ) n u m e n c a l l y using two-loop evolution o f the gauge couplings We assume that the heavier Hlggs has mass M~ a n d below M~ we use the R G equations o f the one-H~ggs m o d e l Using ( 1 9 ) , we can plot mb versus mt for various values o f sin # This is shown in fig 4 along with contours o f constant ;t, (MGuT) Since the initial prediction for mb(mb) IS a b o u t 6 G e V when Y u k a w a couphngs are ignored, we need a large c o n t r i b u t i o n from the t o p quark Yukawa evolution in o r d e r to reduce thin value to agree with e x p e n m e n t A reasonable

135

Volume 277, number 1,2

PHYSICS LETTERS B

ble t e r m s t h a t m a y c o n t r i b u t e to the f e r m l o n masses [ 2 3 ] It Is interesting to n o t e that for a large M ~ m - ~ 1016 G e V , t h e mass c o n t r i b u t i o n s f r o m this effect can be o f the o r d e r MouTMz/Mpl.nck ~ 0 (100) M e V so t h a t this i d e a can a p p l y n o t o n l y to the first f a m i l y b u t also to the s e c o n d o n e In s u m m a r y , we h a v e s h o w n t h a t the Hlggs structure is strongly c o n s t r a i n e d in o r d e r to o b t a i n the correct q u a r k - l e p t o n mass ratio in n o n - S U S Y S U ( 5 ) In particular, the m o d e l m u s t h a v e d i f f e r e n t Hlggs d o u b l e t s c o u p h n g to the t o p a n d b o t t o m q u a r k s A d ditionally, we f o u n d that for these t w o - H l g g s m o d e l s , 2 t m u s t be close to ~ts fixed point, w h i c h g~ves the approxlmaterelatlonmt=2crVSln(fl/x/~) T h e s e interesting features m a y be e x a m i n e d by future experiments T h i s w o r k was s u p p o r t e d in part by the U S D e p a r t m e n t o f Energy u n d e r G r a n t N o D E - F G 0 5 85ER-40219

References [1] H Georgl and S L Glashow, Phys Rev Lett 32 (1974) 438 [2]H Georgl, H R Qumn and S Welnberg, Phys Rev Lett 33 (1984) 451

136

27 February 1992

[ 3 ] M S Chanowltz, J Elhs and M K Galliard, Nucl Phys B 128 (1977) 506 [4] A J Buras, J Elhs, M K Galliard and D V Nanopoulos, Nucl Phys B 135 (1978)66 [ 5 ] D V Nanopoulos and D A Ross, Nucl Phys B 157 ( 1979 ) 273, Phys Lett B 108 (1982)351 [6 ] U Amaldl, W de Boer and H Furstenau, Phys Lett B 260 ( 1991 ) 447 [7] J Elhs, S Kelley and D V Nanopoulos, Phys Lett B 260 (1991) 131 [ 8 ] P Langacker and M Luo, Phys Rev D 44 ( 1991 ) 817 [ 9 ] P H Frampton and S L Glashow, Phys Lett B 131 ( 1983 ) 340 (E),B 135 (1984) 515 [ 10] H Murayama and T Yanagtda, Tohoku University prepnnt TU-370 (May 1991 ) [ 11 ] U Amaldl, W de Boer, P H Frampton, H Furstenau and J T Llu, UNC preprlnt IFP-415-UNC (November 1991 ) [ 12 ] A Glveon, L J Hall and U Sand, LBL prepnnt LBL-31084 (July 1991 ) [ 13 ] J Gasser and H Leutwyler, Phys Rep 87 (1982) 77 [ 14 ] D B Llchtenberg, E Predazzl, R Roncagha and J G Wills, Z Phys C47 (1991) 83 [ 15 ] T P Cheng, E Elchten and L -F L1, Phys Rev D 9 (1974) 2259 [ 16] M T Vaughn, Z Phys C 13 (1982) 149 [ 17 ] T Hebbeker, in Proc Joint Intern Lepton-photon Symp and Europhys Conf on High energy physics (Geneva, JulyAugust 1991 ), to be published [ 18 ] S Seldel et al, Phys Rev Lett 61 (1988) 2522 [ 19] K S Hlrata et al, Phys Lett B 220 (1989) 308 [20] F Abe et al, Phys Rev D 43 ( 1991 ) 664 [21 ] J Ellis and G L Fogh, Phys Lett B 249 (1990) 543 [ 221 C T Hill, Phys Rev D 24 ( 1981 ) 691 [ 231 J Ellis and M K Galliard, Phys Lett B 88 (1979) 315