PHYSICAL REVIEW D VOLUME 28, NUMBER 9 1 NOVEMBER 1983

Left-right mixing in SU(2)L @ SU( 2 IR e U( 1 I, -, gauge models and weak-interaction phenomenology

Amitava Datta Department of Physics, Jadavpur University, Calcutta 700 032, India

Dharmadas Kumbhakar Department of Physics, B. B. College, Asansol 713 303, West Bengal, India

(Received 10 August 1982; revised manuscript received 28 March 1983)

SU(2)L~SU(2)R @U( I ) B - L models with Majorana neutrinos are reexamined neglecting neutrino mixing. Careful analysis shows that values of the left-right mixing angle larger than those obtained by previous authors are permitted by experimental data. As a consequence lower values of charged- and neutral-gauge-boson masses are permissible. The recent bounds on M w from the CERN pp col- lider further restrict the allowed ranges of the parameters obtained from neutral-current data fits.

I. INTRODUCTION

Unified gauge models with left-right symmetry have been discussed by several author^."^ While there is no compelling evidence for the existence of right-handed weak currents, the theoretical interest derives from two main reasons: (1) such models can exhibit a basic left- right (L-R) symmetry which is broken spontaneously by the vacuum, and (2) the additional mass scale (MR -MWR) for the right-handed symmetry breaking may provide alternative to the "great desert" expected in some grand unified theories, thereby leaving open the possibility of some surprises at the C E R N pp collider, ISABELLE, or LEP in the near future.

Interest in such models acquired a new dimension with the suggestion3 that the neutrinos may be Majorana fer- mions with YR superheavy. In such a situation right- handed effects in the leptonic and semileptonic sectors would be kinematically suppressed due to heavy right- handed neutrinos. The success of the conventional V - A theory in these sectors would therefore be retained, even if the right-handed gauge bosons were light ( M W R -MW, ).

Ranges of values of the parameters of this--modelwere obtained by Rizzo and senjanovic4 (RS) by fitting the ex- isting neutral-current data. Similar ranges were obtained by Parida and ~ a ~ c h a u d h u r i ' (PR), who considered the phenomenological L-R-asymmetric model (gL #gR ).

The neutral-current sectors of L-R-symmetric models were also studied by Li and ~ a r s h a k ~ and Barger et al.' These analyses were, however, more general since no specific choice for the Higgs multiplets were assumed. In Ref. 6, the possibility of mixing between Majorana neutri- nos, which was neglected in Refs. 4 and 5, was considered.

In this work, our attention is focused on a particular parameter-the angle (6) that determines the mixing be- tween left- and right-handed gauge bosons. It was shown by Beg, Budny, Sirlin and Mohapatra8 that the L-R mix- ing angle was small. This analysis was done in L-R- symmetric models with massless neutrinos and conse- quently with no kinematical suppression through large mvR. Since the above analysis was based on semileptonic prdiesses, the results of Ref. 8 cannot be automatically ex-

tended to models with heavy right-handed neutrino^.^ On reexamining the data analysis of Ref. 8, we find that the experimental data permits larger values of this mixing an- gle. In particular, the mixing parameter 1 a 1 (to be de- fined latkr) can be as high as 0.1, while in previous analy- ses4sS a smaller upper bound ( / a 1 50.06) was used.

Our analysis neglecting neutrino mixing reveals that while larger mixing is consistent with present neutral- current data, the slight relaxation of the upper bound on

1 a 1 introduces significant changes in the allowed ranges of the physically interesting parameters of the model. In particular, large mixing in general favors a low WR mass. The limitations of this method of determining the L-R mixing based on lepton-polarization measurements in semileptonic processes have been pointed out by previous author^.^ Hence the actual upper bound on 1 a 1 is not ab- solutely reliable. Nevertheless, one must emphasize that significant changes in the physical parameters can result from the small change in I a I .

In doing this analysis we have used recent data on the forward-backward asymmetry parameter in e +e - -,u+,u- (Ref. 10) and atomic parity violation (Ref. 11). We have also reinvestigated the low-mixing-angle sector of the parameters obtained in Refs. 4 and 5 in the light of the new data. We find that some values of the parameters quoted in earlier works (Refs. 4 and 5) are excluded by the new data. If the vector boson of mass 8 1 ? 5 GeV detected in the C E R N pp c ~ l l i d e r ' ~ is identified with W,, further restrictions are imposed on the allowed ranges of the pa- rameters of this model (for example, sin2ow < 0.24).

We have also computed the ratio R I =a"/av in this model, where aV ((74 stands for the total cross section of the neutrino- (antineutrino-) induced deep-inelastic scatter- ing in the charged-current sector. Experimental results in- dicate that this ratio is a constant for a wide range of in- cident neutrino energy.13 This can be understood in a naive quark-parton model with Q'-independent quark- distribution functions. Moreover, the numerical value of this ratio computed in the standard Glashow-Salam- Weinberg modelI4 with V - A charged currents agrees with the experimental result. We have found that the in- troduction of V + A currents with a low MWR and relative- ly large L-R mixing does not change R ] significantly. R

2216 AMITAVA DATTA AND DHARMADAS KUMBHAKAR

however, is not the quantity most sensitive to the presence of right-handed currents. Right-handed currents can be best detected by analyzing the y dependence of the data. This method was suggested by Bigi and re re.^ However, limitations of the approach arising due to QCD contribu- tions beyond the leading-logarithm term and/or unknown contributions from higher-twist terms were pointed out by these authors. Their result (sing < 10%) is therefore more like an order-of-magnitude estimate. We find that our es- timates of the mixing angles are also of the same order of magnitude.

In Sec. 11, we define the parameters of the model, present the analysis for determining the mixing coefficient (a) and the details of neutral-current parameters. In Sec. 111, we have presented the numerical results and discus- sions. In Sec. IV, a brief analysis of charged-current data has been presented.

11. WORKING FORMULAS AND EXPERIMENTAL DATA

A. Parameters of the SU(2)L @ S U ( ~ ) R @Uil), , gauge model

We closely follow the notation of Refs. 4 and 5. In or- der to give masses to the charged and neutral gauge bo- sons, we use the Higgs representation of Ref. 4 with the vacuum expectation values

It is convenient to introduce the parameters

The model and the physical meanings of the parameters have been described by R S ~ in detail. Here we simply note that the limit qR+O and small 7 7 ~ corresponds to the standard Glashow-Salam-Weinberg model.14 The parame- ter z controls the WL , WR mixing.

We define R =gR /gL; the situation R = 1 corresponds to the symmetric model extensively dealt with by R S . ~

The Fermi coupling can be expressed in terms of the pa- rameters of this model from purely leptonic amplitudes and is given by

In this model the left-handed and right-handed gauge bosons get mixed and the mixing angle ,t can be expressed in terms of the above parameters,

On diagonalizing the mass matrices in the usual way, one obtains the four mass eigenvalues M,.,, MW2, M Z , , and M ~ ~ . ~ In obtaining the masses the relations

and

are used. One further introduces the parameters

which is the coefficient of the mixing term in the general charged-current ~ a m i l t o n i a n ~ and

where p is the coefficient of right-handed current^.^

B. Phenomenological analysis of the coefficient of mixing (a)

In all previous the coefficient a was taken to be small ( 1 a 10.06) . This was consistent with the analysis of Bkg, Budny, Sirlin, and ~ o h a ~ a t r a ~ who used a model with L-R symmetry with massless neutrinos and calculated a related parameter by fitting experimental data. In models with superheavy right-handed Majorana neutrinos, the leptonic currents are different. So, it is worthwhile to reexamine their analysis before fixing a.

Following Ref. 8, we write the effective current-current Lagrangian as

In terms of the parameters given in Eqs. (7) and (8), ~ A A

and vAv can in general be written as

In Ref. 8, the ratio ~ ~ ~ / v ~ ~ was fixed by fitting the data of lepton polarization in semileptonic processes. However, since the leptonic current is purely left-handed, in this model the parameter vanishes in the special case of semileptonic processes. Thus

It can be shownR that the lepton polarization in Gamow-Teller transitions is given by

In Ref. 8, the experimental number for the above ratio was taken to be 1.001 t0.008. It is, however, clear from the l i t e r a t ~ r e ' ~ , ' ~ that the quoted number describes the lepton polarization averaged over all transitions including mixed transitions. For pure Gamow-Teller transitions the experimental number for the above ratioI5 is 0.994 + 0.023, which in turn yields -a 20.1 1 . Thus

/ a 2 0 . 1 is consistent with experimental results. In our

28 - LEFT-RIGHT MIXING IN s U ( 2 ) ~ 8 SU( 2 ) ~ 8 U( 1 )B L

analysis we shall use this bound on a which is clearly larger than the bound ( / a / 10.06) previously used.435

C. Neutral-current data

Before discussing the neutral-current parameters we in- troduce the combinations A, B, C, and F defined in Ref. 5 in terms of which the neutral-current parameters can be most conveniently expressed.

The experimentally measured physical parameters of the neutral-current processes and the experimental limits on them are as follows5:

(i) Parity violation in atoms and asymmetry in e-d scattering:

C';+ 1.15Cf=0.49i0.24 (Ref.18) , (13d)

el;+ l . l2Cf=0.15iO.O35 (Ref. 11) . (1 3e)

(ii) Neutrino-hadron scattering:

cL(u)=0.34+0.033 (Ref.20) ,

(iv) Asymmetries in e +e --p+p-:

hAA =0.28 i0 .07 (Ref. 10) . (131)

As in Refs. 5 and 20, we have carried out our analysis at the 90% confidence level. The results are presented in the next section.

111. RESULTS AND DISCUSSIONS

We now proceed to fit the neutral-current data with an increased upper limit on a l . We have chosen to take R 2 , sin20w, a , p L , and 6 as our independent parameters rather than R , sin20w, v L , v R , and z as in earlier

The remaining parameters gL and vR are fixed from Eqs. (3) and (5). For small q L , we have found numerous sets of values of these parameters which con- form with the neutral-current data (see Tables 1-111). Sometimes for fixed R 2 , sin2ew, v L , and a fits with dif- ferent e s are obtained. In these tables we have presented only the fits corresponding to the highest mixing angle 6 for each set of R 2 , sin20w, q L , and a. It should be noted that lower values of 6 correspond to larger M w 7 and MZ2. The change in Mw, with 6 in asymmetric models is shown

in Figs. 1 and 2. e R ( u ) = -0.424k0.026 (Ref.20) , (13g) In the symmetric model (R 2 = 1.0) the following ranges

of the parameters are obtained for a 20.06: e L ( d ) = -0.179k0.019 (Ref.20) , 0.25 2 sin20w 5 0.28, 0.07 j 6 < 0.12. vL larger than 0.05

e R ( d ) = -0.017k0.058 (Ref.20)

(iii) Neutrino-electron scattering:

(13i) is not favored by the data. However, for 7, slightly larger than zero, the lower bounds on M w z and MZ2 decrease considerably. It should also be noted that an increase in

(13j) vL slightly increases both the bounds on sin20w. We ob- tain the following bounds (in GeV) on the gauge-boson

( 13k) masses:

The upper bounds on M W 2 , MZ2, etc., are consequences of by the bounds on sin20w (and 7~ in case of Mz,). It is ap-

restricting 1 a / to rather high values which in turn sets a parent from our fits that much larger values of z (z ~ 0 . 4 ) lower bound on vR for z < 1.0 [see Eq. (7)]. The limits on are permitted by the neutral-current data ( R S ~ restricted M w , and M Z I , on the other hand, are essentially dictated their data analysis to the region O c z 50.2) . The lower

TABLE I. Parameters of the L-R-symmetric model (R '= 1, 7 7 ~ =0) and gauge-boson masses as oh- tained by neutral-current data fit. The asterisks correspond to 7 7 ~ =0.025.

< Mwl Mw2 M ~ , M ~ 2

a sin20w (radian) (GeV) (GeV) (GeV) (GeV) 17 R z

AMITAVA DATTA AND DHARMADAS KUMBHAKAR

TABLE 11. Parameters of the phenomenological asymmetric model (R 2=0.666, v L =0) and gauge- boson masses as obtained by neutral-current data fit. The asterisks correspond to T L =0.05.

< Mwl Mw2 M ~ l Mz2 a sin20w (rad) (GeV) (GeV) (GeV) (GeV) T R z

bounds on MZ2 and M W 2 are considerably smaller than those obtained by RS by fitting the data within the 1.5 o limit. However, these higher values of z do not corre- spond to the highest value of ( for given R 2 , sin20w, v L , a and hence are not presented in the tables. In fact, it is clear from Eqs. (4) and (7) that z decreases as 6 increases with R 2 , sin20w, q L , and a fixed.

It appears from the above bounds on the gauge-boson masses that the allowed values of M w , are somewhat smaller than the recently reported value 8 1 ? 5 G ~ v . ' ~ The symmetric model with / a 1 2 0.06 will be in trouble if this

result is confirmed. We next consider phenomenological asymmetric

models. It was pointed out in Ref. 5 that positivity of Mz, ,2 requires that R 2 2 sin20w/( 1 -sin2ew). For the al-

L,?-

lowed range of sin20w the lower bound on R therefore turns out to be -0.3.

In the asymmetric model with R2=0.667, 1 a 1 20.06, the allowed ranges of the parameters sin20w and ( in- crease: 0.24 < sin20w < 0.28, 0.09 < ( < 0.27. The limits on T ] L are practically the same as those in the symmetric model. We obtain the following bounds:

The corresponding ranges for R '=0.4 are: 0.23 5 sin20w 1 0.27,O. 13 j f j 0.88,

q L =0, 70.1 j M w l g77.1, 87 .0<Mw2 1163.4, 85 .9<Mzl 290.9, 203.5<Mz2 5747.9 ;

qL=0.05, 7 1 . 2 ~ M ~ ~ 1 7 6 . 7 , 80 .9<Mw2<145.1 , 8 7 . 2 j M z 1 < 9 1 . 4 , 171.0 j M z 2 < 6 6 2 . 9 .

TABLE 111. Parameters of the phenomenological asymmetric model (R2=0.4, T L =0) and gauge- boson masses as obtained by neutral-current data fit.

l Mwl Mw2 M ~ l Mz2 a sin2ew (rad) (GeV) (GeV) (GeV) (GeV) TR z

28 - LEFT-RIGHT M I X I N G IN SU(2)L@SU(2)R @U( 1 ), _ L .

FIG. 1. 4 vs MW2 curve for fixed R, V L , a , and sin2OW. in the phenomenological asymmetric model (R2=0.667). The number attached to each curve refers to the value of sin28w.

We have found that the upper bound on z (0.7) allowed by experimental data is considerably larger than that taken by P R (z < 0.2). For R =0.667, the upper bound on M w , is slightly larger than that obtained in the symmetric model, but still below the values indicated by the CERN experiment.12 For ~ ~ = 0 . 4 , the asymmetric model is con- sistent with the results of Ref. 12 even for i a 1 2 0.06. The allowed ranges of the parameters are, however, much restricted. For example, sin20w 20.24 is not allowed, and qL 2 0.05 is not allowed if a 1 > 0.06. The bounds on the gauge-boson masses are (0.19 < g 0.431,

The highest allowed value of 6 (0.43) corresponds to the lower bound on MW2.

Since a large mixing coefficient is allowed, the lower limits on W2 and Z2 masses obtained by us are smaller than those predicted by R S ~ and P R . ~ This result can be clearly understood from Eq. (71, since a is an increasing function of 7,. Thus for a fixed z, larger u implies larger 7 , and hence a decrease in W2 mass. It should also be noted that W , and Z , masses are fairly insensitive to the choice of a.

We next turn our attention to the small-mixing sector of the data. We obtain the following results for a 1 < 0.06:

(i) For R '= 1.0, 0.24 5 sin20w 2 0.28, 0 6 ~ 0 . 0 6 , 0 2 qL 5 0.1, we have

It is found that in most cases M w l turns out to be smaller than the value predicted by the C E R N data. A few results (with sin2ew 50.24, qL =0.0) are barely con-

$ m radians - F I G . 2. < vs MW2 curve for fixed R, T L , a, and sin2Bw in the

phenomenological asymmetric model (R 2=0.41.

sistent with the lower limit of the C E R N data. The corre- sponding ranges for the remaining gauge-boson masses are

(ii) For R 2=0.667, 0.23 5 sin20w 2 0.28, 0 5 6 c 0.13, 0 jqL 50.1, we have

With sin2ew 50.24, q~ =0, quite a few values of M w l are seen to be within the range of the CERN data, though most of them lie close to the lower limit. The gauge-boson masses lie in the following ranges:

76.0 < M w , < 77.6, 143.6 < M w 2 ,

8 5 . 7 < M z l < 8 9 . 2 , 2 5 6 . 9 < M z 2 .

(iii) For R "0.4. 0.22 < sin2ew 50.27, 0 < g < 1.02, 0 5 v L 50.1, we have

7 1 . 4 < M w l 279 .3 , 78 .4<_Mw2 ,

85 .8<Mzl<90 .8 , 170.7<Mz2 .

Imposing the constraint on M w l , we obtain the following ranges:

2220 AMITAVA DATTA AND DHARMADAS KUMBHAKAR - 28

It should be noted that with the limiting case sin2ew=0.24, the highest value of Mw, on both cases (i)

and (ii) just equals the lower limit indicated by the CERN experiment.

Estimates of MZ, and MZ, were given by Li and

~ a r s h a k ~ and Ma, a n d & ~ h i s n a n t . ~ In Ref. 6, the following lower bounds were given (neglecting neutrino mixing): MZ, > 54.3 GeV, MZ, > 73.5 GeV. These lower

bounds, obtained from an analysis without any specific choice of Higgs multiplets, are lower than those obtained by us. However, it appears that for any reasonable choice of Higgs multiplets, the corresponding W , mass will be too small to be consistent with the CERN results. Our

bounds on Z1 mass appear to be more stringent than those obtained by Barger, Ma, and whisnant7 (83 < M Z I < 116 GeV.) The lower limit on Z2 mass ( iMZ2 > 200 GeV) ob-

tained in Ref. 7 is consistent with that obtained by us.

IV. BRIEF DISCUSSION OF CHARGED-CURRENT DATA

We have seen in the previous section that a large mixing angle and correspondingly low charged-boson masses are allowed by the neutral-current data. Here we further in- vestigate whether charged-current data impose any signifi- cant constraint on the mixing angle and the boson masses. The limitations of this method9 discussed earlier should, however, be kept in mind.

In the asymmetric model of pR5 the charged-current cross section for neutrino-hadron deep-inelastic scattering on isoscalar targets 1s given by

o v = ~ ~ c o s 4 g ( 1 +a tanc12[ici2+ t R 2cR2p2tan25)Il +(si2+ ~ ~ ~ s ~ ~ ~ ~ t a n ~ j ~ ~ ~ + ( ~ ~ p ' t a n ' e + If, 1 . (141 1 6 r M w l

Similarly for antineutrino scattering,

- gi4MpE uv= cos4c( 1 +a t a n & ) 2 [ ( ~ L 2 + f~ *cR2p2tan2g)& + ( s L 2 + + ~ ~ s ~ * p * t a n ~ ~ ) & tan^[+ 3 )I,] , (15)

1 6 r M w I

where

I I= s lxq(x)+xs(x)< , )dx , (16a)

IL= $ [ x q ( x ) l C + x s ( x ) ] d x , (16b)

13= s [ x q ( x ) + x c ( x ) ] d x , ( 1 6 ~ )

Fl =I1(q--tq,s--tS) , (16d)

F2:I2(q-fq,s--tS) , ( 16e)

& = I 3 ( q + q , c - - t ~ ) , (16f)

q ( X I = + [ u p ( x ) + d P ( x ) ]

= ~ [ u n ( x ) + d n ( x ) ] , (16g)

and

CL,R = ( C O S ~ ~ )L,R, S L , ~ = (slnec . (17)

Neglecting the sea and other small contrrbutrons, the ra- tro R can be written as

where

p e l -R ( M w , /Mw, I2sec2l ,

and we have taken c~ = C L It is clear from the above equation that even with higher values of 6 obtained in the

I

previous section, R , will not be significantly different from the corresponding result in the standard model. In Table IV, we have presented R , (with and without sea contributions) for different R 2, 6, M w l , Mw, obtained by

neutral-current data analysis. The effects of E-R mixing are found to be rehtively small. The following parton dis- tribution functions have been used at Q 2 = 1.8 G ~ V ~ (Ref. 13):

- t ~ , ( x ) = d , ( x ) = u i x ) = d ( x )

= s ( x ) = r ( x ) = + s ( x / ,

where

A xS(x)=As( 1 - X I .

As in Ref. 13, we have taken

It should be noted that the choice c~ =cR (referred to as

LEFT-RIGHT MIXING IN S U ( ~ ) L @SU(2), @U( 1 ),-, .

TABLE IV. The ratio R 1 =u'/uV for different choices of the parameters R 2, 6, M w l , and M w 2 . For

smaller values of 5, the ratio approaches the well known standard-model result R = f cL2.

R I = u'/uV 6 M w l Mw2 Without With

R (rad) (GeV) (GeV) sea sea

"manifest L-R symmetryw8) may not be compatible with plagued by the inherent ambiguities of strong-interaction low M W 2 as shown by Beall, Bander, and ~ o n i . ~ ' It was physics. Moreover, questionable assumptions like "mani-

subsequently pointed out by Datta and ~ ~ ~ ~ h ~ ~ d h ~ ~ i ~ ~ l ~ " e ~ t " L-R were used in some cases. One that if one allows a strong departure from manifest L-R should, therefore, wait for further refinement of these cal-

symmetry, low M,, consistent with the K ~ - K ~ mass culations before spelling the final verdict on the limits on

difference may still be permitted. However, both these is- sues are afflicted with many theoretical uncertainties and we have worked with the simple choice c~ ~ C R . A de- tailed analysis of effects more sensitive to the presence of right-handed charged currents is plagued with various un- certainties,' as mentioned earlier, and we propose to do it elsewhere.

V. CONCLUSION

In conclusion, we would like to emphasize that values of the mixing angle (6) larger than those obtained by pre- vious authors4s5 are consistent with neutral-current data. L-R-symmetric models with larger 6 appear to be more in- teresting since they can accommodate new gauge bosons with lower masses.

A more careful analysis of the charged-current data with better handling of Q C D and higher-twist effects may provide further interesting information about the parame- ters of this model.

It should, however, be noted that much smaller limits on 6 were obtained by various authors'324 by considering hadronic nonleptonic decays. These calculations are

5 . The possibility of embedding L-R-symmetric models

with low M W R in grand unified theories has been con-

sidered by various It has already been shown that this possibility does not exist for asymmetric rnodek5 This is the origin of the qualifying word "phenomenological" for such models. Regarding the embedding of the symmetric model with low M W R , there exist conflicting v i e ~ s . ~ ' , ~ ~

Note added . After this work was completed a paper [A. Argento et al . , Phys. Lett. m, 245 (1983)l came to our notice which seems to rule out a neutral heavy lepton of any mass in a right-handed doublet with the muon. Impli- cations of this result for the above analysis is being con- sidered.

ACKNOWLEDGMENTS

The authors wish to thank Dr. A. Raychaudhuri for valuable discussions. They acknowledge various research facilities from the S.N. Bose Institute of Physical Sciences, Calcutta University. A critical reading of the manuscript by Dr. P. Ghose is gratefully acknowledged.

'J. C. Pati and A. Salam, Phys. Rev. D 10, 275 (1974); R. N. Mohapatra and J. C . Pati, ibid. 11, 566 (1975); 11, 2558 (1975); G. Senjanovic and R. N. Mohapatra, ibid. l2, 1502 (1975).

2For a review see, for example, R. N. Mohapatra, lectures delivered at the 1981 Winter School on High Energy Physics, Kalapakkam, India, Max Planck Institute, Munich Report No. MPI-PAE/P Th B/81 (unpublished),

3M. Gell-Mann, P. Ramond, and R. Slansky, Rev. Mod. Phys.

S, 721 (1978); Supergravity, edited by P. van Nieuwenhuizen and D. Z. Freedman (North-Holland, Amsterdam, 1979); T. Yanagida, in Proceedings of the Workshop on the Baryon Number of The Universe and Unified Theories, Tsukuba, Japan, 1979, edited by 0. Sawada and A. Sugamoto (KEK, Tsukuba, 1979).

4T. G. Rizzo and G. Senjanovic, Phys. Rev. D 24,704 (1981). 5M. K. Parida and A. Raychaudhuri, Phys. Rev. D 26, 2364

(1982).

2222 AMITAVA DATTA A N D DHARMADAS KUMBHAKAR - 28

6X. Li and R. E. Marshak, Phys. Rev. D 25, 1886 (1982). 7V. Barger, E. Ma, and K. Whisnant, Phys. Rev. Lett. 9, 1589

(1982); Phys. Rev. D 26, 2378 (1982). 8M. A. Beg, R. V. Budny, R. Mohapatra, and A. Sirlin, Phys.

Rev. Lett. 38, 1252 (1977). 91. I. Bigi and J. M. Frere, Phys. Lett. m, 255 (1982). I0B. Adeva et a[., Phys. Rev. Lett. 48, 1701 (1982). "M. A. Bouchiat et al., Phys. Lett. m, 358 (1982). I2G. Arnison et al., Phys. Lett. m, 103 (1983). I3See, for example, A. J. Buras, Rev. Mod. Phys. 2, 199 (1980). I4S. L. Glashow, Nucl. Phys. 2, 579 (1961); S. Weinberg, Phys.

Rev. Lett. 19, 1264 (1967); A. Salam, in Elementary Particle Theory: Relativistic Groups and Analyticity (iVobe1 Symposi- um No. 8), edited by N. Svartholm (Almqvist and Wiksell, Stockholm, 19681, p. 367.

I5G. Klinken, Nucl. Phys. 75, 145 (1966). I6R. E. Marshak, Riazuddin, and C. P. Ryan, Theory of Weak

Interactions in Particle Physics (Wiley-Interscience, New

York, 1969). I7C. Y. Prescott et al., Phys. Lett. m, 347 (1978); B84, 524

(1979). 18L. M. Barkov and M. S. Zolotov, Pis'ma Zh. Eksp. Teor. Fiz.

27, 379 (1978) [JETP Lett. 27, 357 (1978)]. 1 9 c ~ o n t i et al., Phys. Rev. Lett. 42, 343 (1979). 205. E. Kim, P. Langacker, M. Levine, and H. H. Williams, Rev.

Mod. Phys. 53, 21 1 (1981). 21G. Beall, M. Bander, and A. Soni, Phys. Rev. Lett. 48, 848

(1982). 22A. Datta and A. Raychaudhuri, Phys. Lett. m, 392 (1983). 2 3 ~ . Datta and A. Raychaudhuri, Phys. Rev. D 26, 2364 (1982). 24J. F. Donoghue and B. R. Holstein, Phys. Lett. 113B, 383

(1982). 25T. G. Rizzo and G . Senjanovic, Phys. Rev. D 235 (1982). 26N. G. Deshpande and R . J. Johnson, Phys. Rev. D 27, 1165

(1983); V. Barger et al., Oregon Institute for Theoretical Sci- ence Report No. 190, 1982 (unpublished).