On the Kobayashi-Maskawa model with four generations

Volume 156B, number 1,2

PHYSICS LETTERS

13 June 1985

ON THE KOBAYASHI-MASKAWA MODEL WITH FOUR GENERATIONS A.A. A N S E L M , J.L. C H K A R E U L I , N.G. U R A L T S E V and T.A. Z H U K O V S K A Y A

Leningrad Nuclear Physics Institute, Gatchina, Leningrad 188350, USSR Received 13 February 1985

A four-generation version of the Kobayashi-Maskawa model is considered. A convenient parametrization of the mixing matrix is suggested and restrictions on mixing angles are given. The existence of the fourth generation is shown to resolve easily the possible difficulties with ~ and c'/c for K t. decay. The implications of the model for mesons containing b- and c-quarks are discussed. The definite predictions for (AM/F)B0' of the standard scheme can be violated while the charge dilepton asymmetry can increase substantially in the model, both for B° and, more important, for B.,°.

1. At present the Kobayashi-Maskawa (KM) model with three generations of quarks and leptons is examined critically mainly in connection with CP violation in K0-meson decay. The unexpectedly small mixing angles 0 2 and 03 [1] give a rather large prediction for e'/e ~ 1-2% [2] which, maybe, is in conflict with recent experiments [3]. The theoretical magnitude of e can also hardly agree with the experimental value if m t ~ 40 GeV [2]. Future experiments, planned to measure Vcb and Vub , as well as e'/e, could face the standard KM model with serious difficulties. As to the mass difference of K L - KS, the unknown value of long-distance contributions makes it difficult to get any definite theoretical prediction for AmLS [2]. In this letter we consider the simplest generalization of the KM model - the model with four generations. We shall see that the abovementioned difficulties can be removed naturally in this scheme but only if the fourth generation is mixed with the first two generations not weaker than the third one. Note that the existence of the fourth generation does not contradict the cosmological arguments [4]. We shall also confider the implications of the model for the systems D 0 - D 0, B 0 - B 0, B S0 - B 0S ~ and show that the fourth generation can affect B0S - - D0S mixing and likely increases substantially the CP violation in this system. These predictions could be critical for the model. The first discussion known to us of the four-generation case is in ref. [5]. At the time being, much less experimental information was available both about mixing angles and CP violating parameter e'. 2. We shall parametrize the mixing matrix V for n generations in the form 1

V=

n

l-I I] v~j = v23 v12 v , , i=n-l j=i+l = V34V23V24V12V13V14,

n = 3, n =4.

(1)

Here Vii are two-dimensional unitary matrices which mix the ith and thejth generations. One can prove that an arbitrary unitary n X n matrix admits such a representation. In fact, the product u p o n ] in eq. (1) can be taken in an arbitrary order. Of course, the renumbering of generations as well as the overall transposition are also possible. The next step is to get rid of unphysical phases in (1) by a redefinition of the quark field phases. For this end it is convenient to use one of the following representations of the Vt/s:

102

Volume 156B, number 1,2 o

PHYSICS LETTERS o

v j=6v j61j,

o

v j=Ijv i61 j ,

(l

o

v

13 June 1985

o

(2)

j=61jvijI/,

where Vii are orthogonal (real) matrices and I i are the diagonal phase matrices: 1 ...

ciJ... "'" - sij

o

Vii= 0 ...

si1

,

1.

Ii =

ci!

$i

".e!Ot "'I

(3) I

ci/ = cos Oil >1 O, sii = sin Oii >~O. We can now exploit two facts: (i) I k commutes with Vii at i 4=k,/4= k. (ii) I f I k is put to the left (right) of all other matrices it can safely be omitted due to redefinition of the kth up- (down-) quark phase. Thus, one can obtain a set of parametrizations which do not contain the unphysical phases. For n = 3 and n = 4 we choose the parametrization o

o

o

V = V2313(a) V12 V13, o

o

o

V = I'~3414("/)I~231~2414(/3)I3(¢x)I~'12V13 V14,

o

Vi] = Vii(Oil).

(4)

The advantages of such a choice are: (i) The matrix elements of the first line, including Vud = VII and Vus = V12, are real. This simplifies the CP analysis in K0-decays, since the amplitude of a direct K°-decay caused by (ffuX5d) interaction does not contain CP-odd phases in this case. (ii) The available experimental information concerns mostly the Vud, Vus, Vub, and Vcb. This will be shown to fix the values o f s l 2 , S l 3 and s23 directly, rather than the combinations of the angles as it does in the KM parametrization (this advantage also holds for the Maiani parametrization for n = 3 [6]). (iii) The CP-violation parameters in K0-decay e and e' are expressed in terms of two or three phases only, if the mixing angles 0i1 are small. The explicit form of the V matrices for n = 3 and n = 4 is given below. n=3:

c12c13

.

V = IS12¢13c23 - s13s23 el~ ~s12c13s23 + s13c23ela

~S 12

.

~Cl2S ~3

c12C23

-s12s13c23 - C13S23e ~

c12s23

-s12$13s23 + ¢13c23 eia

(5)

n = 4:

c12c13c14

\ ic~

Vii

if3

s12c13c23c14c24 - s13s23clae - c23Slas24e ~ "6a" S12c 13s 23¢ 14C24 c 34 + s 13c 23C14¢ 34 em - s 23s 14$24c 34 e1~ - $12c 13c 14$24s 34 e ry - $ 14c 24s 34 e 1(~ ~) ia i# i'r i(#+'r) \ $12C13S23C14C24$34 +S 13C23C14$34 e -- s 23$ 14$24$34 e +$ 12c 13c 14$24c 34 e +$ 14c 24c 34 e /

--s12 I = l c12c23c24 Vi2 ~ c12s23c24c34 - c12s24s34e~7

(6b)

\ c12s23c24s34 + c12s24c34err ]

103

Volume 156B. number 1,2

PHYSICS LETTERS

---c12s13

13 June 1985

/

-s12s13c23c24 -c13s23 eia

.

, (6c)

V. 3 =~-s12s13s23c24c34 + c13c23c34eia + s12s13s24s34e17 -s12s13s23c24s34 + c13c23s34eia - s12s13s24c34e17/ --c12c13s14

\

-s12c13c23s14c24+s13s23s14e _

.

-c23c14s24 e ,

i~

V'4-1-s12c13s23s14c24c34-s13c23s14c34e _

.

_

s12 t: 13S23S14C24S34

ia

s13c23s14s34e

.

-s23c14s24c34e _

iB +

i7

s12c13s14s24s34e --c14c24s34e

ij3 _

i~, +

i(~+7)

]"

i(13+3,)

s23c14s24s34e s12c13s14s24c34e c14c24c34e

/ (6d)

When the mixing with the fourth generation vanishes, n = 3. At smallsi] , IVi/[ ~8ii + Isiil.

si4 = O, the

matrix V for n = 4 reduces to the matrix V for

3. The available data on quark mixing parameters [7] lead to the constraints: (a)s12, s13, s14, s23 "~ 1. (b)s12 = 0.221 -+ 0.002 [8] , l . (c) s13 < 0.006 [1]. (d) s14 < 0.085; this bound follows from s124 < 1 - [Vud 12 - IVus[2. (e) s23 = 0.044 -+ 0.006 [1]. (The first term s12s13c23c24 in V23 does not exceed 0.0015 and can be neglected.) (f) The estimate IVcs[ > 0.9 [9] implies s24 < 0.4. The angle 034 can be arbitrary as it determines only the mixing of t- and t'-quarks. Some additional constraints on the angles can be obtained from the experimental data on I'(K L ~/a+/a-), AmB/I" B and from Am K. The former are not covered completely by the "unitary restrictions" given above only i f m t , m t, ~> 40 GeV. Thus K L -*/a+/~ - decay analyses [10,2] put the upper limit on the value [s12s24 + s14eiOls14 from 0.04 to 0.01 when mt, m t, range from 40 to 100 GeV. The constraints from K L - K S mass difference are even weaker at mt, m t, > 40 GeV taking into account the unknown possible long-distance contribution to AmLS. 4. Let us write down the expressions for e and e' parameters in KL-decay [11,2] : e

TM

--(efir/4/2 X/-2)z(lm Mbox/Re Mbo x - 2~).

(7)

Here Mbo x is a K 0-g,0 transition amplitude given by the box diagrams; ~ = lm A 0/Re A 0, A 0 is the direct K 0 ~ 2n ( / = 0) amplitude with the pion scattering phase being extracted;z = Re Mbox/Re M K ~. = 2Re Mbox/Am K . At si] '~1 2 m2 sin a [ ~ m t l n - 1 + X s13s23 cos a - - t ~ 2s14s24+ sin~3 Ir mt2"ln ___~X ' m1 + ReMbo x s12 m-2 s12 m2 s12 C C C 2 2 2 m t m t, m t, +p s13s23s14S24s22sin(a +/3) m2( m t2' _ m2 ) In--,m2

lmMb°x2S13S23

s14s24 cos/3 s12

m2 t~] C

(8)

where the factors K, X, ~', X', p are due to the gluon corrections and W-boson propagator effect. Numerically, for m t = 40 GeV, m t, = 60 GeV, K ~ K' ~ 0.5, ), ~ ),' ~ 19 ~ 0.7-0.8. We neglected in eq. (8) both t and t' contributions to Re Mbo x,1 We use the value of Pus extracted from K~3 decays. For s12 the value 0.231 ± 0.003 is often used based on the hyperon semileptonic decay analysis. However, here the SU(3) symmetry violation may lead to rather large uncextainty in s12 (see e.g. ref. [ 81 and references therein). 104

Volume 156B. numL~zr 1.2

PItYSICS LET-I'I~RS

13 June 1985

The value of ~ in eq. (7) equals

= (s13s23/s12) sin ot.H + (s14s24/s12) sin/3.H',

(9)

with H ' "~ H. The estimates of H(H') vary from ~ 0 . 3 up to ~ 1.2 [12] depending on the value of hadronic matrix elements. The ratio e'/e is e__',,, _w_

e

2~

z'ImMbox/ReMbo x - 2 ~ e x p { i [ ( 6 2 - 6 0 ) + l r ] 4 ] ) '

52-60+rt/4"0'

(10)

where w = IA 2]A0 I ~ 1120, A 0,2 are the amplitudes of K 0 ~ 21r decays, 60, 2 are the rrrt scattering phases. Eqs. ( 7 ) - ( 1 0 ) show that the third and the fourth generations contribute additively both to e and e' except for the last term in (8). Therefore, if the fourth generation is mixed to the first and second ones essentially weaker than the third o n e (s14s24 ,~ s13s23), the KM model predictions for e and e'/e remain unchanged. However, already at s14s24 ~ s13s23 the value of ~ and, consequently, e'/e can decrease essentially and even change sign. Since H' "" H, the vanishing of e' implies approximately S13S23 sin c~~ - s 1 4 s 2 4 sin/3, which results in a strong cancellation in e of the logarithmic terms, proportional t o s13s23]s12 and s14s24]s12. However, since the terms "-'m2 for heavy quarks may be of the order of, or greater than the log terms, the value of e may remain the same. In eqs. (8) and (9) only the leading terms in all sii are kept separately in log and quadratic in m t, m t, terms. However, since the si/are experimentally of different orders, the modification of (8) and (9) may prove to be necessary if s14,s24 >~ s13 , s23 , or s34 ~ 1. When s14s34 ~ s13 or s24s34 ~ s23 additional terms appear in eqs. (8) and (9), containing the phase ~,. Thus, the existence of the fourth generation can easily explain both the experimental value of e and any sign of e'/e provided that the fourth generation is mixed not too weakly with the first and second generations. In a number of theoretical speculations the hierarchic structure of the mixing matrix appears [13,14] when the fourth generation is mixed with the first two ones essentially weaker than the third generation. In that case the new quarks do not change the CP-violation in KL-decays. Note, that the possibility of the existence of a new long-lived quark discussed in ref. [15] also assumes such a hierarchic structure, when s34 "- s42, s24 ~ S62. 5. Nevertheless, it seems to us that the structure of the mixing matrix might be quite different from what is often naively expected. For example, let us discuss qualitatively one of the possible patterns for the mixing matrix. For a long time it seemed rather natural that the mixing of the successive generations could have been of the same order: s23 "~ s12 , while s13 ~ s22 . Apparently this pattern fails since IVcbl ~ s23 ~ s122 and [Vubl "" s13 < s]2. We would like to mention that such a pattern can still survive if one supposes that the b-quark actually belongs to the fourth generation whereas a heavier, yet undiscovered, b'-quark belongs to the third one. In this case it might be 2 , s13 ~ S32, in agreement with experiment, and s24 , s34 ~ s12 , 814 ~ s212. The latter values expected that s23 ~ s12 also satisfy the above given restrictions and are still compatible [ 1 1 ] with AmLS for kaons. This "rehabilitated hierarchy" would simply accommodate the CP-nonconservation in K°-mesons. In this case the values of e and e' could obviously be dominated by the contribution of the fourth generation, proportional to sin/3. To meet the experimental magnitude of e it requires sin/3 ~ 10 - 3 . The smallness of e'/e is simply due to the large ratio of quadratic to log terms in (8a). 6. So far we have been discussing CP violation in the K ° - g , 0 system. As is well known [16] it is quite difficult to observe the mixing in D 0 - D 0, both in mass difference and in CP violation because of Am/[" ,~ 1. The fourth generation does not change this conclusion connected particularly with the fact that the D°-meson width is dominated by the Cabibbo-allowed channels while the mass difference is due to the Cabibbo-suppressed transitions. Thus, at m b, ~-- 30 GeV Arn/I ~ < 0.05, with Am/p ~ 0.05 only at the maximal, though hardly probable, S14524 ~-- 0.03. For the BO-B 0 system Am~I" for three generations can be estimated by (see e.g. ref. [16]):

105

Volume 156B, number 1,2

PHYSICS LETTERS

13 Junc 1985

* (__#__)Am ~ 32,r fl~m 2 2t i(VtbVtd ) 2 [ Bsl .. 1.5[ei~sl 2 +s13/s2312-7X'" 10 -2. B m4b [Vcb[2 Zc

(11)

Here we putfB ~ 130 MeV [17], m t = 40 GeV;Bsl ~ 0.12 is the semileptonic branching ratio for the B°-meson; z c ~ 0.38 [1,18] is a suppression factor owing to the phase space, gluon corrections and the bound state effects. For the four generations a modification is to be done in eq. (1 1) ,2 m2 ~-~ - (VtbVtd) ~ ( tbVtd )2+(V~,bVt,d ) 2 m 2 V • 2 V* +2(Vtb td)(Vt'bYt'd) 2_m2t t mt'

m2 1

t'

n._~_2" mt

(12)

To write down expression (12) explicitly according to (6a)-(6d) is very cumbersome. If s14, s24 ~ s13, s23 (as seems to be required by CP violation in K0-decays) and s34 is not too small, then the contribution of the fourth generation to the B 0 mass difference could be of the same order as that of the third one. However, in principle, if the values of Vtd, Vt, d are restricted only by unitarity, the following very weak inequality can be obtained ( A m / 1 0 B < 8(mt,/40 GeV) 2.

(13)

Thus, the experimental data on the same sign dilepton yield on B0B 0 decays r B = (N +÷ + N - -)/iV ÷ - ~ 0.3 [19], or (Am/F)B ~ 0.8, leads to some additional constraint on the combination of the mixing parameters s14 , s24 , s34 together with the phases/3 and 3'. As to the CP-odd B 0 - B 0 mixing, the existence of the fourth generation can substantially change its value. One can obtain the following expression for the charge asymmetry in dllepton production in B 0 decays: N ++ - N - -

a

= --4

Re ~:B =

N ++ + N - 4 2_

FB§ - I m MB ~

m2 Xc('At + Xt,) c rr--Immt 2 ~kt2 +x~kt2, + [ 2 x / ( x - 1 ) ]

Xc = l'ZcbV:d'

Xt = VtbVtd'

m2 Xt2,(x- 1)+ 2 [ x / ( x - 1)1 l n ( x - 1)XtX t, 3~" b I lnx-XtX t, + "~--.--~mt m X2 +xXt2, + [2x/(x - 1)] lnx-~kt~kt,

Xt' = Vt'bVt'd '

x =m~/rn2t .

(14)

For three generations eq. (14) reduces to a=

2 m2 ~mci Xc c 7r-- m--=4,r~2Im mt2 ~kt mt

s e-iS -1 ( + 13 1 1 s12s23 I

6 X 10-3(40 GeV/mt)2.

(15)

For four generations the second term in eq. (14) is likely to dominate. It vanishes at m t = mt,, which is natural since at m t = m t, t - t ' mixing becomes unphysical. At m t ~ m t, the Im in the second term is roughly proportional to s34 (unless s34 is extremely small) and can well be of the order of unity if s34 is not too small. Note that this term depends on the phase 7 as well as on a and/3. Thus, the fourth generation can lead to an essential increase of a up t o a ~ 0.1. The most interesting situation occurs for B°-mesons. As is known [20,21 ], the dependence on mixing angles practically disappears from A m / F in a three-generation standard scheme: eq. (11) with gtd ~ Vts ~ves simply (we assume fB s ~--fB ~ 130 MeV) ( A m / l o B ~ ~ 1.5 (mt/40 GeV) 2.

(16)

,2 Here and in what follows we assume not too large masses for the new quarks: mr' , rnb, < roW, and, hence, do not consider the W-boson propagator effect. We neglect also the gluon corrections. 106

Volume 156B, number 1,2

PHYSICS LETTERS

13 June 1985

If this, relatively reliable, theoretical prediction occurs to be strongly violated experimentally, it might serve as an argument for the existence of extra generations. For four generations (Arn/l")B s is given by

t-2m 2 m2 ~][Am~ ... 32rrJB'-t{(V~bVts)2 + * 2 t' y (Vt,bVt, s) ---2+ 2(VtbVts)(Vt, bVt, s) Bs

/7/41

mt

2

m2

mt' 1 Bsl 2--- 2 In~ m t,-m t rn t [Vcb 12zc

(17)

where it is natural to expect the contribution of the fourth generation to be at least of the same order as that of the third one. CP.odd mixing in the B s - B s system is defined by eq. (14) where ~'c, Xt and Xt, are to be understood in this case as ~kc = V c b V ~ s ,

~kt = V t b V t s ,

Vt, bVt, s.

X t, =

(18)

For three generations the charge asymmetry parameter is very small [20,21 ] :

a ~-- -4rr(m~/m2t) sin o~s12s!3/s23 ~ 3.7 X 10 - 4

(for m t = 40 GeV).

(19)

The situation drastically changes for four generations. Here, as in the case of B 0, one can expect that the last term 2 t2 dominates. It is interesting that in this case a is proportional to sin % Here again it is plausible that Im[...] ~mb/m 1, since ;kt,Dkt ~-- s34(s23s34 + s24e-i't)/(s23

-

s24s34e-i3' ).

(20)

This would lead to a ~ 0.1. At (Am/F)B s ~ 1 one can hope to measure such an asymmetry ,a So far we have discussed the charge asymmetry in the same sign dileptons for B 0 and B 0. One can also consider the total lepton charge asymmetry of all the leptons coming from the decays of the neutral B system produced in a state with the angular momentum L = 1. This asymmetry is expressed also through Re ~ [20,21] :

A = (N + - N-)/(N+ + N - ) ~

-2Re

~[x2/(l

+x2)],

x =

Am~r,

and, therefore, our discussion above is also relevant for this case. Besides the CP~dd mixing one can try to observe CP violation as an effect of a partial width difference of D(B) and D(B) mesons decaying in the same, or charge conjugated, final states. Experiments have been proposed, both independent of the mixing [23], as well as linear in Am/I" [24]. For D-mesons Cabibbo-suppressed decays are the only ones for which it is not hopeless to detect the partial width differences [25]. Here [' - F is determined by a similar combination of the angles as the parameter e' for K L decay (eq. (8)):

F

F ~ (s13s23/s12) sin c~.H+ (slaS24/s12) sm/3"H ,

with ~¢/H' ~ 1. Therefore, it is hardly possible that the fourth generation can enhance the effect F :# F. In the Bmeson case one can imagine two quite different mechanisms leading to the inequality of the partial widths [' and F. One of them is not connected with the mixing and is also present for charged mesons. It requires [25], however, (i) the presence of different (spin, isospin, etc.) structures in the effective lagrangian of the decay; (ii) different CP-odd phases in these structures, and (iii) different phase factors due to final state interaction. The value of the effect depends here heavily on the choice of the decay channel; one can, however, trace some general regularities. The abovementioned different structures can arise, for example, in the penguin diagrams [25] from which one can see quite appreciable CP-odd phase factors in the four-generation case. Another mechanism for F :/: F was proposed in ref. [24] and is based on the interference of the channels B -+ f with B ~ B ~ f for the decay B ~ f, and B ~ f with B ~ B ~ f for the B ~ f decay. Here the effect is proportional • 3 It is possible that at present we have already seen Bs-[~ s mixing in the same sign dimuon production in p~ collisions (see ref.

[221). 107

Volume 15613, number 1,2

PIIYSICS LETIF.RS

13 June 1985

to the value of Am/P discussed above. Besides, it is proportional to a certain CP-odd phase O, the value of • being rather large for the B°-meson, but extremely small for the B°-meson in the standard scheme [20,24]. In the latter case for four generations it can, however, increase essentially up to • ~ 1 due to the phase % The authors are grateful to Ya.I. Azimov and V.A. Khoze for useful remarks.

Note added. After this work was completed we became aware of the preprint by Gronau and Scheckter [26] where the four-generation KM model is also considered. Here, however, the fourth generation of quarks is not discussed from the point of view of the improvement of the description of CP violation in K-meson decay and of the possible consequences for B-meson decays. We are obliged to V.A. Khoze who has drawn our attention to this preprint. References [ 1] J. Lee-Franzini, Talk at the XXII Intern. Conf. on Itigh energy physics (Leipzig, July 1984). [2] See, e.g.A.J. Buras, W. Slominski and H. Steger, Nucl. Phys. B238 (1984) 529, and references therein; N.G. Uraltsev and V.A. Khoze0 Proc. of the XIX LNPI Winter School (Leningrad, 1984); Uspekhi Fiz. Nauk 146 (1985), to be published. [3] K. Nishikawa, Talk at the XXI! Intern. Conf. on tligh energy physics (Leipzig, July 1984); R.K. Adair et al., Talk at the XXII Intern. Conf. on High energy physics (Leipzig, July 1984). [4] See, e.g., V.M. Chechetkin, M.Yu. Khlopov and M.G. Sapozhnikov, Rev. Nuovo Cimento 5 (1982) 456. [5] S.K. Bose and E.A. Paschos, Nucl. Phys. 169B (1980) 384. [6] L. Maiani, in: Proc. Intern. Symp. on Lepton and photon interactions at high energies (llamburg, 1977) (DESY, Hamburg, 1977) p. 867. [7] See, e.g., C. Jarlskog, in: Proc. Intern. Europhysics Conf. on High energy physics (Brighton, July 1983), eds. J. Guy and C. Costain (Rutherford Appleton Lab., UK, 1983) p. 768. [8] H. Leutwyler and M. Roos, Preprint TH.3830-CERN (March 1984). [9] T.M. Aliev, V.L. Eletsky and Ya.l. Kogan, Yad. Fiz. 40 (1984) 823. [10] R.E. Shrock and M.B. Voloshin, Phys. Lett. 87B (1979) 375; F.J. Gilman and J.S. ltagelin, Phys. Lett. 126B (1983) 111 ; L. Bergstr6m et al., Preprint TH.3659-CERN (July 1983). [11] L. Wolfenstein, Nucl. Phys. B160 (1979) 501; C.T. Hill, Phys. Lett. 97B (1980) 275. [12] B. Guberina and R.D. Peccei, Nucl. Phys. B163 (1980) 289; P.H. Ginsparg and M.B. Wise, Phys. Lett. 127B (1983) 265; F.J. Gilman and J.S. llagelin, Phys. Lett. 126B (1983) 111 ; M.B. Voloshin, Preprint ITEP-22 (February 1981). [131 H. Fritzsch and P. Minkowski, Phys. Rep. 73 (1981) 69. [ 14] Z.G. Berezhiani and J.L. Chkareuli, Pis'ma Zh. Eksp. Teor. Fiz. 35 (1982) 494; Yad. Fiz. 37 (1983) 1043. [15] C. Jarlskog, USIP Reports 84-08 and 84-09 (May 1984). [16] L.-L. Chau, Phys. Rep. 95 (1983) 1. [17] T.M. Alley and V.L. Eletsky, Yad. Fiz. 38 (1983) 1537. [18] G. Altarelli, N. Cabibbo and G. Gorbo, Nucl. Phys. B208 (1982) 365. [191 CLEO Collab., P. Avery et al., Cornell Report CLNS 84/612 (1984). [201 I.I. Bigi and A.I. Sanda, Phys. Rev. D29 (1984) 1393. [21 ] A.J. Buras, Preprint MPI-PAE/P'I~-26/84 (April 1984). [22] A. Ali and C. Jarlskog, Preprint "I-H.3896-CERN. [231 A.A. Anselm and Ya.I. Azimov, Phys. Lett. 85B (1979) 72. [241 l.l. Bigi and A.I. Sanda, Nuel. Phys. B123 (1981) 85; A.B. Carter and A.I. Sanda, Phys. Rev. D23 (1981) 1567. [25] Ya.l. Azimov and A.A. Johansen, Yad. Fiz. 33 (1981) 388. [261 M. Gronau and J. Scheckter, Preprint SLAC-PUB-3451 (September 1984).

108