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Testing the family replication-model through B0B 0 mixing
Volume 164B, number 1,2,3
PHYSICS LETTERS
5 December 1985
TESTING THE FAMILY REPLICATION-MODEL THROUGH B ° - B ° MIXING
Amitava D A T T A 1 and Jogesh C. PATI 2 International Centre for Theoretical Physics, Trieste, Italy
Received 19 August 1985
It is observed that the family-replication idea, proposed in the context of a minimal preon-model, necessarily implies a maximal mixing (i.e. AM >> F) either in the B°-B° or the B~-B,~ system, in contrast to the standard model• {)
1. Recently a preon model possessing the desirable features of family replication and an inter-family mass hierarchy has been proposed [1]. In this model the "bare" r(0)-families and a fourth r'(0)-family are formed as the replications of the bare e (0)- and/a (0)families, respectively. The physical particles are, of course, mixed through the mass matrix. The r( °)- and the r'(O)-families, prior to mixing, have a relatively large size of order (1 TeV) -1 , while the e (0)- and the /a(0)-families have a very small size of order ( 10131015 GeV) -1 . The purpose of this note is to point out that the replication mechanism proposed in ref. [1] necessarily predicts - through compositeness alone either a maximal B 0 - B 0 or a maximal BO-BO mixing [i.e. either AM(B0)->> P-(B°), or AM(B0;>> FIB°)], which is much larger than predicted by the standard model. On the other hand, K - - g ° and D 0 - D 0 mixings are predicted to be normal - they are essentially described by the standard model. A large B s0-Bs-0 or B 0 - B 0 mixing would, of course, reflect itself through an enhancement in the production of like-sign dileptons in e - e + and gp collisions, and for the case of B 0 - ~ 0 mixing in T(as) decay as well. Since maximal B 0 _ g 0 mixing (AM >> P) cannot be accommodated within the standard model, even for the B 0s - B-0s system, observation of such a mixing will strongly favour Permanent address: Department of Physics, Jadavpur University, Calcutta 700032, India. 2 Permanent address: Department of Physics. University of Maryland, College Park, MD 20742, USA. 0370-2693•85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
--
0
the family-replication idea while absence of such mixRo_~od system will ing in the B°-B°ss as well as in the ~d disfavour the idea, at least in the context of the minimal model [1]. 2. A few salient features of the model [1] which we need for developing our remarks are these. The model starts off as an N = 1 supergravity model with four left-handed plus four right-handed chiral superfields, each transforming as a fundamental representation N_Nof _ a local metacolour gauge symmetry G M = SU(N). Supergravity and supersymmetry break spontaneously through the formation of the metacolour gaugino condensate [2,3] - i.e. (k-k) ~ A3M. Thereby the gauginos become superheavy of order A M ~ 1013-1015 GeV, and get decoupled. After this decoupling the model consists of four left- plus four right-handed spin-l/2 fields (flavons) f~,Ra'i = (u, d, c, s)~ R plus eight spin-0 fields (chromons) C a d = (r, y, b', £1r', y', b', £ ' ) i - (CI[CII) i, where i runs over the metacolour index 1 to N. With the gauginos decoupled, the interactions of these fields thus define effectively a global symmetry which could be as large as [SU(4)L X SU(4)R] flav°ur X S U ( 8 ) c ° l ° u r . It is assumed that subsequent to a decoupling of the gauginos, the anomaly-free gauge symmetry SU(2)L X SU(2)R X SU(8) c is born as an effective gauge symmetry through the formation of composite gauge particles at the metacolour-scale. Of this gauge symmetry, only the subgroup SU(2)L X U(1)y X SU(3) c X SU(4) H survives dynamical symmetry breaking in173
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volving preonic condensates at-or around the metacolour scale. The symmetries SU(2)L,R treat (u, d)L,R and (c, S)L,R as doublets, while SU(8) c treats the eight chromons as an octet with the subgroups SU(3) c and • (r, y, b) and ( r ', y ,' b ,' ~' ) as triplet • SU(4) n treating and quartet, respectively. The SU(4) H symmetry is assigned to bind the "bare" T- and r'-families as well as to break SU(2)L X U(1)y and chiral symmetry of quarks dynamically. Accordingly, SU(4) H is assigned with a scale parameter A H of order 1 TeV. The dynamics of the model is thus based on just two scale parameters A M and A H - apart from the Planck mass, of course. There are no other arbitrary fundamental parameters as there are no elementary Higgs. The gaugino condensate, following the mechanism of ref. [2], induces soft supersymmetry breaking terms, characterized by the nearly common mass of the gravitino and of the scalar chromons, which is of order (A3M/Mp21)-- 100-300 GeV. It is assumed that the flavons fL,R and the metacolour single t flavon-chromon composites fL,R C*, both possessing spin 112, remain massless at the metacolour scale owing to chiral SU(2)L X SU(2)R-symmetry; while the scalar preons C* and the metacolour singlet CC* composites remain light owing to the protection via supersymmetry and Planck mass as mentioned above. The relevant '~massless" metacolour-singlet composites bound by the metacolour force are listed below: a _ a * a.(0)e,u ~L,R - (f~,,RCI)4~,IH =- ~L,R ' a a "* ~L,R = (f~,R CII)l C,4~I'
g 0 = (CICII)4C,4H"
(1)
The index a runs over (u, d, c, s). These composites are almost point-like with size ~ A ~ 1 ~ (1014 GeV) -1 . The composites ~ carrying colour and flavour are identified with the fermions belonging to the "bare" e- and /a-families. The hypercolour carrying composites ~[ R and (-DOwith spins 1/2 and 0 can bind through the ~ypercolour force to make new spin-l/2 composites of sizes A~ 1 ~ 1 TeV -1 >> A~ 1 : a a * . = ,,(0)~,~' XL,R = (~L,R ~ 0)4C, 1H --AL, R " Except for their relatively large size ~AH 1 , these have 174
5 December 1985
precisely the same flavour-colour-hypercolour and metacolour quantum numbers as the composites ~k. They are, therefore, identified with the fermions of the bare r (0)- and r '(0) -families. ~ Note that the members of the r(0)-family, having constituents (~u,d g * ) carry, in their cores, the same attributes - i.e. fu,dc; -- as those of the e (0)-family. It is in this sense that the bare r(°)-family is regarded as the replication of the bare e(0)-family and the r '(0) of the/fl°)-family. It should be stressed that the r(°)-family is not, in any sense, the familiar radial, orbital or quantum pair-excitation of the e(0)-family, as otherwise the r(0)- and the e(0)-families would have nearly the same size. It is assumed that the hypercolour force breaks chiral symmetry SU(2)L X SU(2)Rdynamically through the formation of hyperfermion condensates ( ~ b R) 4: 0. In conjunction with the effective fourfermion processes
+
+
+
+
,
(3)
which are induced through compositeness, these condensates generate diagonal and off-diagonal masses for quarks and leptons of the four-families. Given that the spin-0 composites g 0 - carrying colour and hypercolour - are light 014 g ~ 100-300 GeV, say), it is presumed that g 0 bind~ to hypercolour gluon V H to make light spin-1 composites g u (= " g 0 V urI'') with a m a s s M g ~ (200-300) GeV, ~--(1/5 - l / 3 ) A q , say, and, for this reason, ~u" exchange dominates at least the four-fermion processes (i) and (ii) listed in (3), and can contribute significantly to the process (iii) ,1. Subject to the assumptions of g u dominance for (i) and (ii) and charge conservation the fourfermion processes in (3) lead to an 8 X 8 mass matrix which splits into two 4 X 4 blocks MUD acting on up and down flavours:
~(e(O),u(o)) x(r'(o))
,
N
i_m ]i=U,D
L rlimiab
(4) I
(l + x)mZabJ
,1 In ref. [1], a ~nplified version with c-/)/~exchange dominating all three processes (i), (ii) and (iii) was presented.
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5 December 1985
Here 60 = x/(1 + x) 2 . Since 6Omax = 1/4, we see that for a value oft/i ~ 1/5 to 1/10, which, is certainly within reason , a , one can naturally obtain a large hierarchy, i.e. (me, r ~ ) ~ 10-3(r~r, r~,,), where the bars denote generic masses for the respective families. Because of the off-diagonal ~X element, it follows that the physical particles are mixtures of the "bare" small and large-size composites ~b(°) and X(°) . For the bare particles, we thus have relations o f the type:
matrix (4). Such corrections can be especially important for the light (e,/a)-families (see remarks later). To obtain the complete mass spectrum one needs to know the structure ofmiab for which a first-principle calculation is not yet at hand. In ref. [1], an interestingansatz*4'i'e'AUb~--(e0 el) and A Db ~_ (e el') K ' ' was suggested which leads to the mass relations: ~ ~ _ ~ ~, ~ 2 ~ ~(0) - ~ "" ( m u / e m ) - (t / t mm ) -- - e and (m d/m - ,) - ( m b / m ~ (e~)2 , while Cabibbo angle 0 c ~ e - e . It is remarkable that with only a few effective parameters characterizingM U and M D , all of which are assigned values in a natural range, e.g. r / ~ 1 / 5 - 1 / 1 0 , w "" 1/10, e "~ - - e ' ~ 1/10 and (gD ~ ')/(gu) ~- 1/8, one very nearly obtains the desired masses of all eight flavours and the correct Cabibbo angle 0c; the only flaw at this • that m ~ u exceeds '~" " " m ~ u and m "~ d are stage Is md. Since both small in magnitude (~10--40 MeV) and both can be negative, however, the induced corrections which have been neglected so far can be important particularly for these light masses and can lead to lind[ > Imul for the physical masses [1]. These induced corrections may even be as large as of order 1 0 0 - 1 0 0 0 MeV. In this case, one can conceive of a situation ,s under which the physical c quark with a mass of about 1.3 GeV is primarily (barring Cabibbo mixing) is what one called qu "" q(u0) before [see discussions following eq. (8) for notations], while the nearly massless physical u quark is primarily what one called "qc "~ q(:) ; likewise the physical qs and qd quarks are primarily qd "~ q(0) and "qs "~ q~0)' re" spectively. We shall refer to this phenomenon as e ~/a interchange. The two alternatives, with and without e ~/a interchange [case (i) and (ii)], are exhibited below (here Cabibbo-mixings are suppressed): case (i), no e ~,/a interchange
q(bO) ~ cos ~ qb + sin ¢ q d '
(qs)physic
m i = . 2 / ~ - ~ b x/M2 = - 2 • 3/M2 ~A i ab--SiX~LCR "r cD-~gilkf/" c-D) ab'
(5)
(a, b) run over (u, c) for i = U and over (d, s) for i = D;
rli = (hi/g 0, where h i and gi denote the effective coupling constants associated with the transitions ~a + @/a* ~ba and ~a + cb t~ ~ xa, respectively. Following recent observation that spontaneous breaking of parity [i.e. SU(2)R] at the metacolour scale can induce large isospin violations in Yukawa couplings [4], we allow h , g and rl to depend on the index i = U or D. The parameterx in eq. (4) denotes contributions to (iii) from mechanisms other than the one involving c/)u exchange ;x can even be of order unity. The simplification brought about by the familyreplication idea is obvious: the same matrix m~b enters into each of the four 2 × 2 blocks in (4). As a result, the 4 × 4-matrix M i, which normally would require, e~g., 10 parameters - if it is real and symmetric - requires at best ,2 5 - i.e.~li,x and three more to describe mab.i Regardless. o f the structure of the condensate matrix rntab, the pattern exhibited in (4) leads to the interesting mass relations: ~ 2 ~ ~ mc) -- (r/UCO)(mt, mt,), 2
~
~
- - (T/D ¢.~)(m b , m b , ) .
q(dO) = --sin ~bqb + cos C q d '
(6) (7)
(8)
( q d ) physical
q 0), TM "~ d
~ q(d0)'
and likewise for (q(0), qT)) and also for the up-quark flavours. Here sin ¢ "~ tan ¢ ~ ~D/(1 + x ) . The states and masses with tildes as a rule signify eigenstates and eigenvalues of the mass matrix (4). These eigenstates would represent the physical particles except for the fact that induced corrections are not included in the
case (ii), with e ~/a-interchange
, 2 Even these five are in principle calculable. , 3 I.e. consistent with Dirac's philosophy o f "naturalness".
J.C. Pati. *s Details of this discussion may be found in ref. [5]•
(qs) phy cai = 7 d ~~
q(d°), (o)
(qd) physical -- qs -- qs "
(9)
, 4 An attempt to derive such an ansatz following the idea o f vacuum alignment is being pursued by M. Cvetic and
175
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Note that the Cabibbo angle 0.c and, therefore, the familiar weak interactions connecting only the (e, p)families will not be altered due to e ~, p interchange. But such an interchange will clearly have important implications on processes connecting (e , /a)- with (r, r')-families. It turns out that such an interchange might in fact be needed to account for the experimental fact that Vbc > Vbu [5]. We shall, therefore, allow for the possibility of such an interchange, while discussing B ° - ~ 0 mixing.
3. BO-B 0 mixing. In the family replication model, since the bare q(b°) is a replication of the bare q(d0), the process q~U, + 7:1(0)~ q(d0) + Cl(b0)involves only _the flayour-conserving preonic transition d + d -+ d + d, and is thus intrinsically allowed inside the cores of the composites. Ordinarily the amplitude for such a process would be strongly damped by AM2, i.e. by the large inverse sizes of the e- and the/a-families. The situation alters drastically, however, owing to mass mixing [see (8)] and also due to the presence of the light c/) u particles (Me/) ~ 2 0 0 - 4 0 0 GeV). First, observe that double q)u exchange ,6 shown in fig. 1 would induce even without the aid of mass mixing, the transition qb + U:ld~ q + Clb' and, therefore, B 0 ~ ~0 for the c a s e (i), or ~ternatively the transition qb + Cts ~ qs + Clb and, therefore, B 0 ~+ ~0 for case (ii). Note that such a graph cannot, however,
5 December 1985
than that of ~ u ' fig" 1 leads to an effective hamiltonJan given by: H e ft = (g2h2/128rr2m~) X [10(T:l/TUqb 12 -- 6(Cl/TU75qb 121 +h.c.
(10)
Here the subscript/denotes the alternatives of d and s for case (i) and (ii). Following standard procedure [6], this defines a heavy B°ri and alight BOL ( j = d or s) with a mass splitting: AM(B]0)fig.1 ~g2h2C ~ f2/MBj[6rtZM2" cl~ is the so-called bag the vacuum saturation fB i is the analog o f f K and Ir decay constants. suggest ,a cB = 0 . 3 - 1 . 0 ,
fBl
TM
(11)
factor which reduces to unity in approximation [6] (VSA), and and f~ which represent the K Present theoretical estimates 0.15-0.3 GeV.
(12)
The preon model parameters can be constrained as follows. The heaviest t' quark has a mass m t, =g2A3f/m~) = ym t,y > 1. From the scale of electroweak breaking, Af ~--300-500 GeV, and thus (g2/m2cl)) = (Ymt)/A3f t> (40y)/(500) 3 GeV - 2 . Since g and h represent effective coupling constants which are generated by strong interactions, it seems safe to assume that neither of them are unnaturally small, i.e.g,h > 0.1 (say). We, therefore, have the conservative bound
induce the flavour non-conserving transitions involving fermions of the first two families only, such as qc + Clu
g2h2/m2Q) >~y(3 X 10 -9 GeV - 2 )
~+ qu + Tic and qd + qs ~-~qs + 7qd- i.e. D ° - D 0 and K ° - K 0 mixings. With the effective mass of ~d being much smaller * 7
Substituting q~min = 0.3, (fBi)min ~ 0.15 and the bound (13), we thus obtain
~6 For purposes of estimating the order of magnitude of the effect, we are treating (/)it as an elementary particle inside the convergent loop of fig. 1, although cOt~has a size A~C. +7 In ref. [1], it is estimated that m(~d)eff ~ 2 GeV at AHC.
qb
qa(,)
Fig. 1. Box diagram for B° _~0 mixing. The subscripts d and s correspond to the two alternatives - eases (i) and (ii), respectively (see text). 176
( y > 1).
Am(B/.)fig.l ~ 2y X 10 -12 GeV.
(13)
(14)
The weight of the experimental data ,9 on B 0 life-time shows [11] r B = (1.5 + 0.5) X 10 -12 sec, i.e. P(B 0) ~ ( 0 . 6 6 - 0 . 3 3 ) X 10 -12 GeV. We thus obtain [Am(B.)fig.1/r(BO)l/=dor s ~ 3y ( y > 1).
(15)
,a The result ~ ~ 0.3 is suggested by bag-model calculations (see e.g. ref. [7] ). A recent calculation by Pith and de Rafael (ref. [8] ) based on chiral perturbation theory also yields c'~ ~ 0.3 for the kaon system. The range of values forfBj reflects the uncertainty in its theoretical estimate in the present literature [7,9,10]. ,9 The present experiments can only measure the average life times of b-flavoured hadrons which include BO, B+ etc. The average value quoted above is from ref. [11].
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The weight of this conclusion regarding large BO - 5 9 mixing [eq. (15)] is enhanced even further by realizin~ that such a mixing is induced in addition to the interfamily mixing exhibited in (8), which implies that the bare q~)~ is in fact a mixture of physical qb and (q])] = d or s with a mixing angle of order * zo ~ ~ 1/ 5 1/10 [see eqs. (8) and (9)]. Now, prior to interfamily mixing, we expect that the hypercolour force would generate a diagonal process with an amplitude of the type 2 2 -(0) (0) -(0) (0) ~ t a l ( g H / A H ) ( q b Flqb ) ( q b Plqb ), where the sum runs over the Dirac covariants;a l ~ O(1), g 2 ,,. 1 - 1 0 , A H "" 1 - 3 TeV (say). Replacing q(O) by the physical fields [see eqs. (8) and (9)], we would thus obtain the four-fermion transition a~tO. + n. ~ , "J-i -x1 + ~lb w i t h / = d or s for cases .11 (i) or (ii), with an'amplitude ~--(cos2~b" sin2(.b) ( g 2 / A 2) ~ ~72(g2H/A2 ) (1/10)2(1)(1-3 T e V ) - 2 ~ 10 -9 GeV - 2 . This is typically at least 40 times larger than the amplitude of fig. 1 given by eq. (10), subject to the bound (13). Comparing with eq. (15), we thus expect, very conservatively [Am(B/)/F(Bj)] preonic ~ 10
( / = d or s).
(16)
Thus the family-replication idea necessarily predicts either a maximal B O - f f 0 [case (i)] or a maximal BO-BOs mixing[case (ii)] with A m >>p. This may now be compared with the predictions of the standard model. We first review the case of B ° - B 0 mixing, which has been discussed by several authors [7,9,10]. One can show that regardless of the Kobayashi-Maskawa (KM) parameters the ratio Am(Bs)/P(Bs) is restricted, i.e. Am(Bs)/P(Bs)~ 1.6 q~ [ 10]. The well-known signature for B0_~0 mixing is
,1o The interfamily mass hierarchy ratios, i.e., ~ e / ~ r and ~t~/~ r, fixes r/to be in this range. e¢ll Similar arguments would also lead to qbqs ~ qsqb for case (B, i.e. ] = d, and to qbqd ~ qdqb for case (ii), i.e. ] = s, with a suppressed amplitude ~g~ r/2 (sin2~')/A~ in each case, where sin2~• is determined ~dependently'l~rom parameters of the mass matrix to be of order 10 -2 . Thus, the family-replication mechanism may induce not only maximal B°a-J~d mixing for case (i) but also rather large Bs0 _~0 mi~ing which is bigger than the standard-model value. For case (ii), maximal Bs0 _~0 mixing may be accompanied by not so insignificant B~-B~ mixing.
5 December 1985
the production of like sign dilepton pairs produced in e+e - or pp collisions ,12. The experimental signal depends sensitively on the quantity - [Am(Bj)/p(Bj)] 2 / ( 2 + [ A m ( B y P ( B . ) I 2}, (17) where ] = d, s. For maximal mixing (i.e. Am >> F) a/ "" 1. The standard model estimate of o s therefore sensitively depends on the parameter q0. For q3 = 0 . 3 - 1 , one obtains ,Sm(Bs)/P(Bs) "" (1.6)(0.3-1.0) and thus o s ~ 0.1 to 0.6 i.e., maximal B 0 - B 0 mixing is highly unlikely in the standard model. Now, coming to B 0d - B-0d mixing, since one can argue that it is suppressed compared to B s0 - B-0s mixing by the ratio I V t j V t s l 2 which (conservatively) is known to be smaller than 1/30, it is clear that Am(B°)/F(B 0) ~ 1, in the standard model. The UA1 group has recently reported preliminary data for the ratio R = (N ++ + N - - ) I N +- , where N ++ and N - - refer to the number of like sign dileptons and N + - those of the unlike sign dileptons. The most recently quoted result is R = 0.47 -+ 0.1 [13]. This value of R , however, includes contributions to the production of like sign dileptons arising from B ° - ] ] ° mixing as well as from cascade decays, i.e. b ~ c + hadrons -+ (e+v + hadrons). The recent Euroject Monte Carlo analysis [14], which allows for both sources, leads to the following theoretical estimates: R ~-- 0.32-0.33 for o = 0 and R ~ 0.53 for o = 1. Given the present error bars on R , it is clear that no finn conclusion on the degree of mixing in the B s0 - B-0s system can be drawn at present. With better statistics and Monte Carlo studies, one hopes that the cascade contribution can be subtracted out, thereby enabling a clear comparison between theory and experiment. The mixing in the B 0d - B-0d system can be studied more easily through the decay of the T(4s) which on energetic grounds can decay to Bd - B 0 and not to B 0s _ ~ o . An experimental upperbound for the quantity Re+ e- = ( N ++ + N - - ) I N +- , where N ++ and N - now correspond to the number of like sign dileptons arising from T(4s) decay due to mixing alone and N +includes unlike sign dileptons coming from B d - i f 0 systems only, already exists in the literature [15]. Theoretical estimates of Re+ e_ is straightforward and one obtains Re+ e_ = a d (defined above). The present experimental upperbound [15] on Re~ ,12 For a review see ref. [12]. 177
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is unfortunately not conclusive. This is essentially due to the difficulties in estimating the number of unlike sign dilepton pairs produced by the decay o f B + - B systems also present in T(4s) decay. It is, however, interesting to note that for p = BR(B 0 --> £ux)/BR(B + -+ ~ux) < 0.6 the present experimental data do not exclude maximal mixing (R~e ~ 1, i.e. o , ~ 1). Thus the precise experimental (or theoreUcal *1~ ) determma " tion o f 0 and/or improvement o f the existing bounds on ReE have the potential o f either vindicating the large B ° - B 0 mixing scenarios [case (i)] or ruling it out. To conclude, the family-replication idea inevitably implies a maximal mixing (i.e. Am >> I') either in the B°-B0ss or the B 0 - B ° system (although not in both) in contrast to the standard model. The thrust of ongoing experiments can thus serve in the near future to either strongly favour or exclude the icIea. •
The authors thank Dr. E. Rademacher for helpful discussions on the experimental situation. One o f them ( A D . ) would like to thank Professor Abdus Salam, the International Atomic Energy Agency and UNESCO for , l a It has recently been observed (ref. [16] ) that in view of non-spectator contributions to B-meson decay one obtains rB+/rB 0 ~ 1.4-1.8 which in turn implies p = 0.550.7.
178
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hospitality at the International Centre for Theoretical Physics, Trieste. The research of J.CJ ). is supported by a grant from the US National Science Foundation.
References [1] J.C. Pail, Phys. Lett. 144B (1984) 375. [2] S. Ferrara, L. Girardello and H.P. Nilles, Phys. Lett. 125B (1983) 457. [3] J.C. Pail and A. Salam, Nuel. Phys. B234 (1984) 223. [4] D. Chang, R.N. Mohapatra, P.B. Pal and J.C. Pati, Maryland University preprint, to be published. [5] A. Datta and J.C. Pail, in preparation. [6] M.K. Galliard and B.W. Lee, Phys. Rev. D10 (1974) 897. [7] I.I. Bigi and A. Sanda, Phys. Rev. D29 (1984) 1393. [8] A. Pith and E. de Rafael, Marseille preprint CPT-85/P. 1768 (1985). [9] A. Ali and C. Jarlskog, Phys. Lett. 144B (1984) 266. [ 10] A.J. Buras, W. Slominski and H. Steger, Nuel. Phys. B245 (1984) 369; L. Wolfenstein, Nucl. Phys. B246 (1984) 45 [ 11 ] R. Klanner, XXIIth Intern. Conf. on High energy physics (Leipzig, GDR, July 1984). [12] E.g., L.-L. Chau, Phys. Rep. 95C (1983) 1, and references therein. [13] E. Rademacher, talk Europhysics Conf. (Bari, Italy, 1985) [14] R. Batley, talk Conf. on Tests of electroweak theories (Trieste, Italy, 1985). [15] P. Avery et al., Phys. Rev. Lett. 53 (1984) 1309. [16] A. Soni, Phys. Rev. Lett. 53 (1984) 1407.
PHYSICS LETTERS
5 December 1985
TESTING THE FAMILY REPLICATION-MODEL THROUGH B ° - B ° MIXING
Amitava D A T T A 1 and Jogesh C. PATI 2 International Centre for Theoretical Physics, Trieste, Italy
Received 19 August 1985
It is observed that the family-replication idea, proposed in the context of a minimal preon-model, necessarily implies a maximal mixing (i.e. AM >> F) either in the B°-B° or the B~-B,~ system, in contrast to the standard model• {)
1. Recently a preon model possessing the desirable features of family replication and an inter-family mass hierarchy has been proposed [1]. In this model the "bare" r(0)-families and a fourth r'(0)-family are formed as the replications of the bare e (0)- and/a (0)families, respectively. The physical particles are, of course, mixed through the mass matrix. The r( °)- and the r'(O)-families, prior to mixing, have a relatively large size of order (1 TeV) -1 , while the e (0)- and the /a(0)-families have a very small size of order ( 10131015 GeV) -1 . The purpose of this note is to point out that the replication mechanism proposed in ref. [1] necessarily predicts - through compositeness alone either a maximal B 0 - B 0 or a maximal BO-BO mixing [i.e. either AM(B0)->> P-(B°), or AM(B0;>> FIB°)], which is much larger than predicted by the standard model. On the other hand, K - - g ° and D 0 - D 0 mixings are predicted to be normal - they are essentially described by the standard model. A large B s0-Bs-0 or B 0 - B 0 mixing would, of course, reflect itself through an enhancement in the production of like-sign dileptons in e - e + and gp collisions, and for the case of B 0 - ~ 0 mixing in T(as) decay as well. Since maximal B 0 _ g 0 mixing (AM >> P) cannot be accommodated within the standard model, even for the B 0s - B-0s system, observation of such a mixing will strongly favour Permanent address: Department of Physics, Jadavpur University, Calcutta 700032, India. 2 Permanent address: Department of Physics. University of Maryland, College Park, MD 20742, USA. 0370-2693•85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
--
0
the family-replication idea while absence of such mixRo_~od system will ing in the B°-B°ss as well as in the ~d disfavour the idea, at least in the context of the minimal model [1]. 2. A few salient features of the model [1] which we need for developing our remarks are these. The model starts off as an N = 1 supergravity model with four left-handed plus four right-handed chiral superfields, each transforming as a fundamental representation N_Nof _ a local metacolour gauge symmetry G M = SU(N). Supergravity and supersymmetry break spontaneously through the formation of the metacolour gaugino condensate [2,3] - i.e. (k-k) ~ A3M. Thereby the gauginos become superheavy of order A M ~ 1013-1015 GeV, and get decoupled. After this decoupling the model consists of four left- plus four right-handed spin-l/2 fields (flavons) f~,Ra'i = (u, d, c, s)~ R plus eight spin-0 fields (chromons) C a d = (r, y, b', £1r', y', b', £ ' ) i - (CI[CII) i, where i runs over the metacolour index 1 to N. With the gauginos decoupled, the interactions of these fields thus define effectively a global symmetry which could be as large as [SU(4)L X SU(4)R] flav°ur X S U ( 8 ) c ° l ° u r . It is assumed that subsequent to a decoupling of the gauginos, the anomaly-free gauge symmetry SU(2)L X SU(2)R X SU(8) c is born as an effective gauge symmetry through the formation of composite gauge particles at the metacolour-scale. Of this gauge symmetry, only the subgroup SU(2)L X U(1)y X SU(3) c X SU(4) H survives dynamical symmetry breaking in173
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volving preonic condensates at-or around the metacolour scale. The symmetries SU(2)L,R treat (u, d)L,R and (c, S)L,R as doublets, while SU(8) c treats the eight chromons as an octet with the subgroups SU(3) c and • (r, y, b) and ( r ', y ,' b ,' ~' ) as triplet • SU(4) n treating and quartet, respectively. The SU(4) H symmetry is assigned to bind the "bare" T- and r'-families as well as to break SU(2)L X U(1)y and chiral symmetry of quarks dynamically. Accordingly, SU(4) H is assigned with a scale parameter A H of order 1 TeV. The dynamics of the model is thus based on just two scale parameters A M and A H - apart from the Planck mass, of course. There are no other arbitrary fundamental parameters as there are no elementary Higgs. The gaugino condensate, following the mechanism of ref. [2], induces soft supersymmetry breaking terms, characterized by the nearly common mass of the gravitino and of the scalar chromons, which is of order (A3M/Mp21)-- 100-300 GeV. It is assumed that the flavons fL,R and the metacolour single t flavon-chromon composites fL,R C*, both possessing spin 112, remain massless at the metacolour scale owing to chiral SU(2)L X SU(2)R-symmetry; while the scalar preons C* and the metacolour singlet CC* composites remain light owing to the protection via supersymmetry and Planck mass as mentioned above. The relevant '~massless" metacolour-singlet composites bound by the metacolour force are listed below: a _ a * a.(0)e,u ~L,R - (f~,,RCI)4~,IH =- ~L,R ' a a "* ~L,R = (f~,R CII)l C,4~I'
g 0 = (CICII)4C,4H"
(1)
The index a runs over (u, d, c, s). These composites are almost point-like with size ~ A ~ 1 ~ (1014 GeV) -1 . The composites ~ carrying colour and flavour are identified with the fermions belonging to the "bare" e- and /a-families. The hypercolour carrying composites ~[ R and (-DOwith spins 1/2 and 0 can bind through the ~ypercolour force to make new spin-l/2 composites of sizes A~ 1 ~ 1 TeV -1 >> A~ 1 : a a * . = ,,(0)~,~' XL,R = (~L,R ~ 0)4C, 1H --AL, R " Except for their relatively large size ~AH 1 , these have 174
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precisely the same flavour-colour-hypercolour and metacolour quantum numbers as the composites ~k. They are, therefore, identified with the fermions of the bare r (0)- and r '(0) -families. ~ Note that the members of the r(0)-family, having constituents (~u,d g * ) carry, in their cores, the same attributes - i.e. fu,dc; -- as those of the e (0)-family. It is in this sense that the bare r(°)-family is regarded as the replication of the bare e(0)-family and the r '(0) of the/fl°)-family. It should be stressed that the r(°)-family is not, in any sense, the familiar radial, orbital or quantum pair-excitation of the e(0)-family, as otherwise the r(0)- and the e(0)-families would have nearly the same size. It is assumed that the hypercolour force breaks chiral symmetry SU(2)L X SU(2)Rdynamically through the formation of hyperfermion condensates ( ~ b R) 4: 0. In conjunction with the effective fourfermion processes
+
+
+
+
,
(3)
which are induced through compositeness, these condensates generate diagonal and off-diagonal masses for quarks and leptons of the four-families. Given that the spin-0 composites g 0 - carrying colour and hypercolour - are light 014 g ~ 100-300 GeV, say), it is presumed that g 0 bind~ to hypercolour gluon V H to make light spin-1 composites g u (= " g 0 V urI'') with a m a s s M g ~ (200-300) GeV, ~--(1/5 - l / 3 ) A q , say, and, for this reason, ~u" exchange dominates at least the four-fermion processes (i) and (ii) listed in (3), and can contribute significantly to the process (iii) ,1. Subject to the assumptions of g u dominance for (i) and (ii) and charge conservation the fourfermion processes in (3) lead to an 8 X 8 mass matrix which splits into two 4 X 4 blocks MUD acting on up and down flavours:
~(e(O),u(o)) x(r'(o))
,
N
i_m ]i=U,D
L rlimiab
(4) I
(l + x)mZabJ
,1 In ref. [1], a ~nplified version with c-/)/~exchange dominating all three processes (i), (ii) and (iii) was presented.
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Here 60 = x/(1 + x) 2 . Since 6Omax = 1/4, we see that for a value oft/i ~ 1/5 to 1/10, which, is certainly within reason , a , one can naturally obtain a large hierarchy, i.e. (me, r ~ ) ~ 10-3(r~r, r~,,), where the bars denote generic masses for the respective families. Because of the off-diagonal ~X element, it follows that the physical particles are mixtures of the "bare" small and large-size composites ~b(°) and X(°) . For the bare particles, we thus have relations o f the type:
matrix (4). Such corrections can be especially important for the light (e,/a)-families (see remarks later). To obtain the complete mass spectrum one needs to know the structure ofmiab for which a first-principle calculation is not yet at hand. In ref. [1], an interestingansatz*4'i'e'AUb~--(e0 el) and A Db ~_ (e el') K ' ' was suggested which leads to the mass relations: ~ ~ _ ~ ~, ~ 2 ~ ~(0) - ~ "" ( m u / e m ) - (t / t mm ) -- - e and (m d/m - ,) - ( m b / m ~ (e~)2 , while Cabibbo angle 0 c ~ e - e . It is remarkable that with only a few effective parameters characterizingM U and M D , all of which are assigned values in a natural range, e.g. r / ~ 1 / 5 - 1 / 1 0 , w "" 1/10, e "~ - - e ' ~ 1/10 and (gD ~ ')/(gu) ~- 1/8, one very nearly obtains the desired masses of all eight flavours and the correct Cabibbo angle 0c; the only flaw at this • that m ~ u exceeds '~" " " m ~ u and m "~ d are stage Is md. Since both small in magnitude (~10--40 MeV) and both can be negative, however, the induced corrections which have been neglected so far can be important particularly for these light masses and can lead to lind[ > Imul for the physical masses [1]. These induced corrections may even be as large as of order 1 0 0 - 1 0 0 0 MeV. In this case, one can conceive of a situation ,s under which the physical c quark with a mass of about 1.3 GeV is primarily (barring Cabibbo mixing) is what one called qu "" q(u0) before [see discussions following eq. (8) for notations], while the nearly massless physical u quark is primarily what one called "qc "~ q(:) ; likewise the physical qs and qd quarks are primarily qd "~ q(0) and "qs "~ q~0)' re" spectively. We shall refer to this phenomenon as e ~/a interchange. The two alternatives, with and without e ~/a interchange [case (i) and (ii)], are exhibited below (here Cabibbo-mixings are suppressed): case (i), no e ~,/a interchange
q(bO) ~ cos ~ qb + sin ¢ q d '
(qs)physic
m i = . 2 / ~ - ~ b x/M2 = - 2 • 3/M2 ~A i ab--SiX~LCR "r cD-~gilkf/" c-D) ab'
(5)
(a, b) run over (u, c) for i = U and over (d, s) for i = D;
rli = (hi/g 0, where h i and gi denote the effective coupling constants associated with the transitions ~a + @/a* ~ba and ~a + cb t~ ~ xa, respectively. Following recent observation that spontaneous breaking of parity [i.e. SU(2)R] at the metacolour scale can induce large isospin violations in Yukawa couplings [4], we allow h , g and rl to depend on the index i = U or D. The parameterx in eq. (4) denotes contributions to (iii) from mechanisms other than the one involving c/)u exchange ;x can even be of order unity. The simplification brought about by the familyreplication idea is obvious: the same matrix m~b enters into each of the four 2 × 2 blocks in (4). As a result, the 4 × 4-matrix M i, which normally would require, e~g., 10 parameters - if it is real and symmetric - requires at best ,2 5 - i.e.~li,x and three more to describe mab.i Regardless. o f the structure of the condensate matrix rntab, the pattern exhibited in (4) leads to the interesting mass relations: ~ 2 ~ ~ mc) -- (r/UCO)(mt, mt,), 2
~
~
- - (T/D ¢.~)(m b , m b , ) .
q(dO) = --sin ~bqb + cos C q d '
(6) (7)
(8)
( q d ) physical
q 0), TM "~ d
~ q(d0)'
and likewise for (q(0), qT)) and also for the up-quark flavours. Here sin ¢ "~ tan ¢ ~ ~D/(1 + x ) . The states and masses with tildes as a rule signify eigenstates and eigenvalues of the mass matrix (4). These eigenstates would represent the physical particles except for the fact that induced corrections are not included in the
case (ii), with e ~/a-interchange
, 2 Even these five are in principle calculable. , 3 I.e. consistent with Dirac's philosophy o f "naturalness".
J.C. Pati. *s Details of this discussion may be found in ref. [5]•
(qs) phy cai = 7 d ~~
q(d°), (o)
(qd) physical -- qs -- qs "
(9)
, 4 An attempt to derive such an ansatz following the idea o f vacuum alignment is being pursued by M. Cvetic and
175
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Note that the Cabibbo angle 0.c and, therefore, the familiar weak interactions connecting only the (e, p)families will not be altered due to e ~, p interchange. But such an interchange will clearly have important implications on processes connecting (e , /a)- with (r, r')-families. It turns out that such an interchange might in fact be needed to account for the experimental fact that Vbc > Vbu [5]. We shall, therefore, allow for the possibility of such an interchange, while discussing B ° - ~ 0 mixing.
3. BO-B 0 mixing. In the family replication model, since the bare q(b°) is a replication of the bare q(d0), the process q~U, + 7:1(0)~ q(d0) + Cl(b0)involves only _the flayour-conserving preonic transition d + d -+ d + d, and is thus intrinsically allowed inside the cores of the composites. Ordinarily the amplitude for such a process would be strongly damped by AM2, i.e. by the large inverse sizes of the e- and the/a-families. The situation alters drastically, however, owing to mass mixing [see (8)] and also due to the presence of the light c/) u particles (Me/) ~ 2 0 0 - 4 0 0 GeV). First, observe that double q)u exchange ,6 shown in fig. 1 would induce even without the aid of mass mixing, the transition qb + U:ld~ q + Clb' and, therefore, B 0 ~ ~0 for the c a s e (i), or ~ternatively the transition qb + Cts ~ qs + Clb and, therefore, B 0 ~+ ~0 for case (ii). Note that such a graph cannot, however,
5 December 1985
than that of ~ u ' fig" 1 leads to an effective hamiltonJan given by: H e ft = (g2h2/128rr2m~) X [10(T:l/TUqb 12 -- 6(Cl/TU75qb 121 +h.c.
(10)
Here the subscript/denotes the alternatives of d and s for case (i) and (ii). Following standard procedure [6], this defines a heavy B°ri and alight BOL ( j = d or s) with a mass splitting: AM(B]0)fig.1 ~g2h2C ~ f2/MBj[6rtZM2" cl~ is the so-called bag the vacuum saturation fB i is the analog o f f K and Ir decay constants. suggest ,a cB = 0 . 3 - 1 . 0 ,
fBl
TM
(11)
factor which reduces to unity in approximation [6] (VSA), and and f~ which represent the K Present theoretical estimates 0.15-0.3 GeV.
(12)
The preon model parameters can be constrained as follows. The heaviest t' quark has a mass m t, =g2A3f/m~) = ym t,y > 1. From the scale of electroweak breaking, Af ~--300-500 GeV, and thus (g2/m2cl)) = (Ymt)/A3f t> (40y)/(500) 3 GeV - 2 . Since g and h represent effective coupling constants which are generated by strong interactions, it seems safe to assume that neither of them are unnaturally small, i.e.g,h > 0.1 (say). We, therefore, have the conservative bound
induce the flavour non-conserving transitions involving fermions of the first two families only, such as qc + Clu
g2h2/m2Q) >~y(3 X 10 -9 GeV - 2 )
~+ qu + Tic and qd + qs ~-~qs + 7qd- i.e. D ° - D 0 and K ° - K 0 mixings. With the effective mass of ~d being much smaller * 7
Substituting q~min = 0.3, (fBi)min ~ 0.15 and the bound (13), we thus obtain
~6 For purposes of estimating the order of magnitude of the effect, we are treating (/)it as an elementary particle inside the convergent loop of fig. 1, although cOt~has a size A~C. +7 In ref. [1], it is estimated that m(~d)eff ~ 2 GeV at AHC.
qb
qa(,)
Fig. 1. Box diagram for B° _~0 mixing. The subscripts d and s correspond to the two alternatives - eases (i) and (ii), respectively (see text). 176
( y > 1).
Am(B/.)fig.l ~ 2y X 10 -12 GeV.
(13)
(14)
The weight of the experimental data ,9 on B 0 life-time shows [11] r B = (1.5 + 0.5) X 10 -12 sec, i.e. P(B 0) ~ ( 0 . 6 6 - 0 . 3 3 ) X 10 -12 GeV. We thus obtain [Am(B.)fig.1/r(BO)l/=dor s ~ 3y ( y > 1).
(15)
,a The result ~ ~ 0.3 is suggested by bag-model calculations (see e.g. ref. [7] ). A recent calculation by Pith and de Rafael (ref. [8] ) based on chiral perturbation theory also yields c'~ ~ 0.3 for the kaon system. The range of values forfBj reflects the uncertainty in its theoretical estimate in the present literature [7,9,10]. ,9 The present experiments can only measure the average life times of b-flavoured hadrons which include BO, B+ etc. The average value quoted above is from ref. [11].
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The weight of this conclusion regarding large BO - 5 9 mixing [eq. (15)] is enhanced even further by realizin~ that such a mixing is induced in addition to the interfamily mixing exhibited in (8), which implies that the bare q~)~ is in fact a mixture of physical qb and (q])] = d or s with a mixing angle of order * zo ~ ~ 1/ 5 1/10 [see eqs. (8) and (9)]. Now, prior to interfamily mixing, we expect that the hypercolour force would generate a diagonal process with an amplitude of the type 2 2 -(0) (0) -(0) (0) ~ t a l ( g H / A H ) ( q b Flqb ) ( q b Plqb ), where the sum runs over the Dirac covariants;a l ~ O(1), g 2 ,,. 1 - 1 0 , A H "" 1 - 3 TeV (say). Replacing q(O) by the physical fields [see eqs. (8) and (9)], we would thus obtain the four-fermion transition a~tO. + n. ~ , "J-i -x1 + ~lb w i t h / = d or s for cases .11 (i) or (ii), with an'amplitude ~--(cos2~b" sin2(.b) ( g 2 / A 2) ~ ~72(g2H/A2 ) (1/10)2(1)(1-3 T e V ) - 2 ~ 10 -9 GeV - 2 . This is typically at least 40 times larger than the amplitude of fig. 1 given by eq. (10), subject to the bound (13). Comparing with eq. (15), we thus expect, very conservatively [Am(B/)/F(Bj)] preonic ~ 10
( / = d or s).
(16)
Thus the family-replication idea necessarily predicts either a maximal B O - f f 0 [case (i)] or a maximal BO-BOs mixing[case (ii)] with A m >>p. This may now be compared with the predictions of the standard model. We first review the case of B ° - B 0 mixing, which has been discussed by several authors [7,9,10]. One can show that regardless of the Kobayashi-Maskawa (KM) parameters the ratio Am(Bs)/P(Bs) is restricted, i.e. Am(Bs)/P(Bs)~ 1.6 q~ [ 10]. The well-known signature for B0_~0 mixing is
,1o The interfamily mass hierarchy ratios, i.e., ~ e / ~ r and ~t~/~ r, fixes r/to be in this range. e¢ll Similar arguments would also lead to qbqs ~ qsqb for case (B, i.e. ] = d, and to qbqd ~ qdqb for case (ii), i.e. ] = s, with a suppressed amplitude ~g~ r/2 (sin2~')/A~ in each case, where sin2~• is determined ~dependently'l~rom parameters of the mass matrix to be of order 10 -2 . Thus, the family-replication mechanism may induce not only maximal B°a-J~d mixing for case (i) but also rather large Bs0 _~0 mi~ing which is bigger than the standard-model value. For case (ii), maximal Bs0 _~0 mixing may be accompanied by not so insignificant B~-B~ mixing.
5 December 1985
the production of like sign dilepton pairs produced in e+e - or pp collisions ,12. The experimental signal depends sensitively on the quantity - [Am(Bj)/p(Bj)] 2 / ( 2 + [ A m ( B y P ( B . ) I 2}, (17) where ] = d, s. For maximal mixing (i.e. Am >> F) a/ "" 1. The standard model estimate of o s therefore sensitively depends on the parameter q0. For q3 = 0 . 3 - 1 , one obtains ,Sm(Bs)/P(Bs) "" (1.6)(0.3-1.0) and thus o s ~ 0.1 to 0.6 i.e., maximal B 0 - B 0 mixing is highly unlikely in the standard model. Now, coming to B 0d - B-0d mixing, since one can argue that it is suppressed compared to B s0 - B-0s mixing by the ratio I V t j V t s l 2 which (conservatively) is known to be smaller than 1/30, it is clear that Am(B°)/F(B 0) ~ 1, in the standard model. The UA1 group has recently reported preliminary data for the ratio R = (N ++ + N - - ) I N +- , where N ++ and N - - refer to the number of like sign dileptons and N + - those of the unlike sign dileptons. The most recently quoted result is R = 0.47 -+ 0.1 [13]. This value of R , however, includes contributions to the production of like sign dileptons arising from B ° - ] ] ° mixing as well as from cascade decays, i.e. b ~ c + hadrons -+ (e+v + hadrons). The recent Euroject Monte Carlo analysis [14], which allows for both sources, leads to the following theoretical estimates: R ~-- 0.32-0.33 for o = 0 and R ~ 0.53 for o = 1. Given the present error bars on R , it is clear that no finn conclusion on the degree of mixing in the B s0 - B-0s system can be drawn at present. With better statistics and Monte Carlo studies, one hopes that the cascade contribution can be subtracted out, thereby enabling a clear comparison between theory and experiment. The mixing in the B 0d - B-0d system can be studied more easily through the decay of the T(4s) which on energetic grounds can decay to Bd - B 0 and not to B 0s _ ~ o . An experimental upperbound for the quantity Re+ e- = ( N ++ + N - - ) I N +- , where N ++ and N - now correspond to the number of like sign dileptons arising from T(4s) decay due to mixing alone and N +includes unlike sign dileptons coming from B d - i f 0 systems only, already exists in the literature [15]. Theoretical estimates of Re+ e_ is straightforward and one obtains Re+ e_ = a d (defined above). The present experimental upperbound [15] on Re~ ,12 For a review see ref. [12]. 177
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is unfortunately not conclusive. This is essentially due to the difficulties in estimating the number of unlike sign dilepton pairs produced by the decay o f B + - B systems also present in T(4s) decay. It is, however, interesting to note that for p = BR(B 0 --> £ux)/BR(B + -+ ~ux) < 0.6 the present experimental data do not exclude maximal mixing (R~e ~ 1, i.e. o , ~ 1). Thus the precise experimental (or theoreUcal *1~ ) determma " tion o f 0 and/or improvement o f the existing bounds on ReE have the potential o f either vindicating the large B ° - B 0 mixing scenarios [case (i)] or ruling it out. To conclude, the family-replication idea inevitably implies a maximal mixing (i.e. Am >> I') either in the B°-B0ss or the B 0 - B ° system (although not in both) in contrast to the standard model. The thrust of ongoing experiments can thus serve in the near future to either strongly favour or exclude the icIea. •
The authors thank Dr. E. Rademacher for helpful discussions on the experimental situation. One o f them ( A D . ) would like to thank Professor Abdus Salam, the International Atomic Energy Agency and UNESCO for , l a It has recently been observed (ref. [16] ) that in view of non-spectator contributions to B-meson decay one obtains rB+/rB 0 ~ 1.4-1.8 which in turn implies p = 0.550.7.
178
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hospitality at the International Centre for Theoretical Physics, Trieste. The research of J.CJ ). is supported by a grant from the US National Science Foundation.
References [1] J.C. Pail, Phys. Lett. 144B (1984) 375. [2] S. Ferrara, L. Girardello and H.P. Nilles, Phys. Lett. 125B (1983) 457. [3] J.C. Pail and A. Salam, Nuel. Phys. B234 (1984) 223. [4] D. Chang, R.N. Mohapatra, P.B. Pal and J.C. Pati, Maryland University preprint, to be published. [5] A. Datta and J.C. Pail, in preparation. [6] M.K. Galliard and B.W. Lee, Phys. Rev. D10 (1974) 897. [7] I.I. Bigi and A. Sanda, Phys. Rev. D29 (1984) 1393. [8] A. Pith and E. de Rafael, Marseille preprint CPT-85/P. 1768 (1985). [9] A. Ali and C. Jarlskog, Phys. Lett. 144B (1984) 266. [ 10] A.J. Buras, W. Slominski and H. Steger, Nuel. Phys. B245 (1984) 369; L. Wolfenstein, Nucl. Phys. B246 (1984) 45 [ 11 ] R. Klanner, XXIIth Intern. Conf. on High energy physics (Leipzig, GDR, July 1984). [12] E.g., L.-L. Chau, Phys. Rep. 95C (1983) 1, and references therein. [13] E. Rademacher, talk Europhysics Conf. (Bari, Italy, 1985) [14] R. Batley, talk Conf. on Tests of electroweak theories (Trieste, Italy, 1985). [15] P. Avery et al., Phys. Rev. Lett. 53 (1984) 1309. [16] A. Soni, Phys. Rev. Lett. 53 (1984) 1407.