PHYSICAL REVIEW D VOLUME 34, NUMBER 1 1 1 DECEMBER 1986

Systematic study of large CP violations in decays of neutral b-flavored mesons

Dongsheng Du* Institute for Advanced Study, Princeton, New Jersey 08540

Isard Dunietz The Enrico Fermi Institute and Department of Physics, University of Chicago, Chicago, Illinois 60637

Dan-di Wu Institute of High Energy Physics, Beijing, People's Republic of China

(Received 1 1 April 1986)

CP-violation effects in partial-decay-rate asymmetries of B: and B: systems are examined within the framework of the Kobayashi-Maskawa model. We concentrate on those hadronic final states into which both B0 and B O can decay. The rephasing-invariant formalism is used in our calcula- tion. We find a quite large asymmetry (-0.6) in some decay modes, such as B ~ - + D + T - , B;-+Df T- , D O I $ , etc., but we still need 10~--10' b6 pairs for testing these effects for 30 signature. In addition, the contribution of the penguin diagrams and the problem of strong-interaction phases of final states are also briefly discussed. We find that for testing CP violations with only the penguin contribution the best decays are B:+I$K~, B;-rI$KSs, which need, for 30 signature, 3 x lo8 b6 pairs.

I. INTRODUCTION

Until now CP violation has been observed only in the KO-EO complex.' As of yet, this effect could be entirely indirect CP violation coming solely from mixing. Only experimental upper bounds exist for a direct CP-violation effect (the E ' / E parameter).

There are many speculations and estimates for large CP-violating effects in the B:-3 2 and B:-8 systems. As we know, the charge asymmetry in semileptonic de- cays, the same-sign dilepton asymmetry,2 is predicted to be very small ( < lo-'). But in nonleptonic decays, the asymmetry may be large due to the interplay of mixing and amplitude interference. Bigi, Carter, and sanda3 were the first to discuss this problem in general. They restrict- ed themselves to the hadronic final states being CP eigen- states (i.e., f =T, here f denotes the final hadronic state, 7 is the CP-conjugate state of f, namely, 17) =CP I f )). Chau and cheng4 discussed the case for f#T under the condition BO- f , 8 O+f, so they have to calculate the de- cay amplitudes explicitly. We should avoid this because we do not know how to calculate the nonleptonic decay amplitude reliabl . sachs5 was the first to discuss the 8 asymmetry of Bd, 3 2 -FD *rT without calculating the decay amplitudes. There are several advantages of Sachs' idea.

(a) The decay amplitudes have a very simple depen- dence on ~obavash i -~askawa (KM) matrix elements.

(b) The decay amplitudes cancel out approximately in the expression of the partial-decay-rate asymmetries, so

avoid calculating the decay amplitudes. Especially we analyzed carefully the possible two-body nonleptonic de- cay channels. We find that the most promising decay channels for large asymmetry are B:-D+T-, B ~ - + D + T - , B;-D0r$. The penguin contributions are also examined. We discuss the two extreme cases: the penguin-dominant case and the one in which penguins are negligible. We find that B:,, - 4 ~ ~ are the best candi- dates for testing CP violation with only the penguin con- tribution. In our calculation, the rephasing invariants6 of the KM matrix are extensively used. That is the only way that the KM elements can enter into physical calculations. We want to stress the fact that as long as we limit our- selves to final states which are strong-interaction eigen- states, the final-state strong-interaction phases cancel out, and we are left with phases coming solely and intrinsically from the KM matrix. This defines a clean and useful testing ground for the three-generation standard model.

The outline of this article is as follows. In Sec. I1 we give the rephasing-invariant formalism. In Sec. I11 we discuss partial-decay-rate asymmetries and the exclusive two-body nonleptonic decays. A different tagging, other than the leptonic tagging of Bigi, Carter, and sanda3 will be advocated. In Sec. IV we discuss the penguin-diagram contributions for the extreme cases. Section V is devoted to the discussion and conclusion.

11. REPHASING-INVARIANT FORMALISM

the estimated asymme&es are not sensitive to the hardly As know, all the physical quantities must be measurable strong-interaction amplitudes and phases. particle-phase independent. In the standard model the

(cl The amplitude interference will lead to the asym- KM matnix7 appears in the charged current metries even when the rl z.z I p /q I = 1 (see the text). . , - - ,

Following Sachs' idea, we studied systematically the asymmetries in nonleptonic decays of B:-$ 2 and B:-8: $u Vqd

systems. We discuss both cases for f =7 and f #T and where

34 3414 - @ 1986 The American Physical Society

SYSTEMATIC STUDY OF LARGE CP VIOLATIONS IN

and the KM matrix is

In the standard KM parametrization we have

However, our quark fields are defined up to arbitrary phases. If we make a phase transformation

where

and I Ui 1 = / Di 1 = l . Then V'=UVD will appear as our new KM matrix. Any physical quantity will not depend on our phase convention. As proved by Jarlskog and W U , ~ the rephasing invariants we can construct from the KM elements are the nine absolute values of KM matrix elements j Fj I ; and the nine hi, defined by

Aia= vjSvky( y tV&) (i,j,k and a , P,y co-cyclic) . (3)

In Appendix A we list all the nine independent quanti- ties hi,. As a trivial consequence we obtain a rephasing- invariant proof that ImAia=t, where t is a unique CP- gauging parameter independent of i and a (Refs. 6 and 8). All physical quantities should be expressed by means of these rephasing invariants.

Take the phase convention as

CP I B O ) = 1 B 0 ) (4)

and assume CPT invariance, then the physical eigenstates are

with eigenvalues = m H , ~ - i yH,L /2 where the sub- script indicates heavy or light, respectively. According to Ref. 9,

Define B: =5a, := b ~ , where a = d or s. Using the box-diagram d~rn inance '~- '~ and assuming ( m t / M~ l2 << 1 we get

where ft=VtbVG and a = d or s for Bd or B,. BM = Br = 1 under the assumption of vacuum insertion. Actually, for B: and B: systems,

By spectator we get

where p= 3.2 in which we include phase-space and QCD corrections.

Therefore 2 2

f ~ , Z B ~ = ~ ~ ] ~ ~ = O . ~ B & [ 100 MeV ] I i f . ] 40 GeV '

The uncertainties in determining our mixing parameter are the decay constant fB =50-200 MeV, bag parame- ter- 1, top-quark mass, and for the Bd system an added uncertainty of KM elements. using1' the b lifetime 1.06-tO.17 psec and the upper limit on r ( b + u )/T(b-+c) <4% (Ref. 19) we obtain for the KM phase convention that

In a later calculation, we shall use, for definiteness,

mt-40 GeV, s l -0.231 , 2

Sz-Sl , s3-0.5s2, Sg-1 .

Our physical states evolve in time by e- imt I BL,H ) with m being the mass matrix. A pure B0 at t =O evolves in time as

whereas a pure 3 O at t =O evolves as

3416 DONGSHENG DU, ISARD DUNIETZ, AND DAN-DI WU

We limit ourselves to the final states that both the pure B0 and B O can decay into

Unlike Bigi, Carter, and Sanda, we do not restrict f to be a CP eigenstate. Denote the CP-conjugate state off by -

Define

It is easy to see that (q/p)x and (p/q)X are particle re- phasing invariants. [Since / B H ) t 1 BL ) in Eq. (5) are physical eigenstates, rephasing6 BO,B O must not change our physical eigenstates. Suppose we rephase I B ~ ) - + ~ ' ~ / B ~ ) , O ) - e a / O ) . Then we must

simultaneously rephase p -e -iBp, q -e -iBq. Also x -+ei(B-8)x. Therefore ( q /p)x -(q /p)x. We still have an overall phase ambiguity of I BH) to / BL ), but that does not concern us here.] Hence even the phase of (q /p )x and ( p /q)X has physical relevance.

Now we are in a position to discuss the partial-decay- rate asymmetries.

111. PARTIAL-DECAY-RATE ASYMMETRY

Since the neutral b-flavored mesons have a short life- time, we will be primarily interested in time-integrated ef- fects. Our CP-violating asymmetry is defined as

where r i ~ :~~~- f ) is the time-integrated partial decay rate of a time-evolved BO(t=O) into the specified final state f.

We will consider in this paper mainly those decays where there is no direct CP violation in magnitude in the pure B0 and B O. Namely, that

It is readily realized how to accomplish requirement (22). ~ a k e ~ ? ~

where G I ,G2 are multiplication of two KM elements, a , 0 are the strong-interaction final-state phases, and a,b denote real amplitudes. Then the CP-conjugated ampli- tude is

It is clear that to obtain / A ( f ) 1 # 1 2i-7) I we need to have two different strong channels (a#B) and further- more Gl#Gz must happen too. Therefore as long as we limit ourselves to decay amplitudes where only one KM combination appears; or alternatively limit ourselves to a decay amplitude where only one strong channel is avail- able we are guaranteed that no direct magnitudinal CP violations occur in the pure amplitudes. Assuming Eq. (22) holds, we get, for our time-integrated asymmetry (211,

where

Assuming box-diagram dominance we obtain5, - I 3

1, / <<l and3.5p11-'3

Now if we limit ourselves further to hadronic final states which can proceed only through one strong-interaction channel we get

since the final-state strong-interaction phases cancel out and we are left only with the complex conjugation of our weak phases (KM elements).

After some lengthy arithmetic we obtain for our asym- metry assuming 1 A (f ) 1 = I Ti(?) I , X =x *, and box- diagram dominance

M [ ( x 1 2 ( l - - a ) + y ~ e h ] - ~ z ~ m h Cr = (28) 1 +a + / x 1 2( 1 -a)+2y ~ e h - M a z Imh '

where

and &,St are defined in Eqs. (7) and (8). We notice that

34 - SYSTEMATIC STUDY OF LARGE CP VIOLATIONS IN . . .

cc/c, is rephasing invariant, and the Im(fc /f, ) part in M ~ ~ ~ G ~ o - ~ , M / B s = ( - ) 5 ~ 1 0 - 4 . the standard KM parametrization is Im( f, /ft ) G -s12(s3/s2)ss for Bs, and Im(fc/f t ) =(s3 /s2)ss t 1 Assuming 02r (Ay /Am l2 << 1 in addition to all the as- for Bd, SO that M is small for B, and Bd cases: sumptions needed for obtaining Eq. (28), we get

For the large asymmetries we will encounter below, the M terms can be neglected up to a few percent. That is to say q/p=c:/ct.

The accessible final state f in quark content via both a B0 and 3 O are presented in Table I for Bd and B, systems. For convenience of later calculation, we list in Table I1 all the expressions for I x / 2 , Imh, and Reh in the KM phase conven- tion.

The conventional wisdom is to tag onto the primary lepton of the accompanying Bihys (Ref. 3). Hence the time- integrated asymmetry3 to be measured is

where N ( f l - ) denotes the number of BsYs8 f l - events integrated over time. The BOB wave function can be charge-conjugation parity ( - even or odd, and 1s given by3

For primary leptons (not from cascade decay) we assume a pure B 0 ( r 6 a ) + l + ~ only, a pure B O( r ba)-1-X only, that 1 A(I+ ) I = 1 x ( I - ) and that Eq. (22) is satisfied, namely, I R / = 1 x I . Then we obtain, for our time-integrated lepton-tagging asymmetry,

For C odd BOB O state, / q/p I #1 will lead to nonzero asymmetry Cfr. In addition to the assumptions leading to Eq. (33) we require also X =x * and box-diagram dominance including o2 << 1 (Refs. 5, 11-13, 10); we get, for the C even case,

At a high-energy e + -e- collider, for example, CERN LEP, we have copious production of 2'. Sitting on the ZO reso- nance, b6 pairs will be produced and hadronized into B:,, B, and B z,, B:. Here, the creation of b-flavored mesons is incoherent, then we can tag on the accompanying charged b-flavored mesons B'. Observing the charge of B: (B, )

0" - would confirm the observed decayed neutral b-flavored meson to be 3: =ba (Bd =bd) or B := bS (B:=~s) structure at t =O. In that case, we can use the asymmetry defined by Eq. (21).

Assuming the ratio of the probabilities of creating qtj pairs from the vacuum

the probabilities3 for producing B,B;, B,B,+, according to Eq. (35) are u(B,B')= $-, and for B ~ B , , % ~ B , ~ , u(B~B') = G. In general, having an asymmetry

DONGSHENG DU, ISARD DUNIETZ, AND DAN-DI WU

TABLE I. The accessible final states f in quark content via both a B0 and B O are presented for Bd and B, systems. The parentheses adjacent to each final state gives the possible decay mechanisms into this particular f . We employ Chau's (Ref. 1) weak-decay classification: spec for spectator; ex for ex- change; pen for penguin. The subscript attached to the exchange and penguin diagrams tells you what vacuum pair has been created. For example, ex, is the exchange diagram with uii pair creation from the vacuum. We also list the corresponding parameters x and h. Note that h is expressed by KM phase invariants and ~ K P K are the corresponding parameters q and p for K O - T I O system.

Case

(quark decay) f h

B:-B 5j system

EuPu (ex, ), PuSs(ex,

~cdd(spec,exd 1, Pciiu (ex, 1,

EcSs (ex, 1, iicCc (exc 1

~ u Z d (spec,exd ) ,~ui iu(ex, )

EuSs (ex, ), FuCc (exc 1

TcSs (ex, 1

EuKs(spec, pen)

gKs( spec, pen)

KOR O

BP-B: system

auiiu(ex, l,nudd(exd)

ZicFs( spec, ex, ), PcPu (ex, 1,

~ c d d ( exd ), iicFc ( exc 1

TuSs(spec,exs ),TuPu (ex, )

~ u d d (exd ),?u??c( ex,

?cad ( exd

- D OKs( spec)

SYSTEMATIC STUDY OF LARGE CP VIOLATIONS IN . . .

TABLE I. (Continued).

Case

(quark decay) f acf, X = A ( f )

h

d.rl( pen, spec)

aIf 7 has some glue in it, the penguin will by far dominate the spectator diagram.

TABLE 11. Expressions for 1 x / 2, Imh, and Reh.

Case / X l 2 Imh Reh

B:-B 2 system

BP-B: system

DONGSHENG DU, ISARD DUNIETZ, AND DAN-DI WU

and desiring a 3a signature (SA = f A), we need

So, if

is the asymmetry, for 3a accuracy, we need the number of b,6 pairs

where a = d or s, E is the inverse of our detection efficiency, and

As a challenge to experimentalists and therefore being op- the form of c,- ,B( f +T), and Cfl are considerably simpli- timistic, we assume the detection efficiency of a Bi to be fied: 50%. However for the other particles we are realistic and - 22 Imh their efficiencies can be read off Table 111. Cf = 2+z2+z2 I x / '

(40) The B&,,,--+ f branching ratios are whenever possible

taken directly from experiment: 2+z2+z21x 1 ~ ( f +J)=B(B,,,,, -+f) 7 (41)

1 +z2 B(B~- -+D-P+) -~% (Ref. 2 1 ) , (39a) for C even,

B(B~-+$K')-0. 1% (Ref. 22) (39b) Cfl = - 42 Imh 2fz2+z4+ I X j 2 ~ 2 ( 3 + ~ 2 ) '

If not yet available we extract them from Eq. (39) with the help of KM elements. Whenever only the exchange diagram leads to the final state we assume the exchange to be significant and thereby get an estimate of this branch- ing ratio again with the help of Eq. (39). Whenever the internal W-emission diagram is being encountered, we do not color suppress it. All that is not unreasonable extra- polating from our accumulating knowledge of D',D',D ' decays from Mark I11 data.23

Now we make numerical estimates for exclusive two- body nonleptonic decays.

A good approximation is setting M =0, w=O. Then

TABLE 111. Detection efficiency for various particles.

Particle Detection efficiency (%)

where Imh, / x 1 can be read off Table 11, and z can be calculated by use of Eqs. (12) and (13). Thus from Eqs. (40), (41), and (371, we can estimate the number of b6 pairs Nb6 for exclusive two-body nonleptonic decays. We list them in Table IV. In these tables we also present some possible two-body hadronic final states, Cf,z, and the branching ratios. Note however that final-state phases have been entirely omitted in constructing these tables. Also, we take the values of the parameters given in Eq. (1 5 ) . The only exception is for 6+iiu;i, iiuS processes for the Bd-Bd system, where, we take S=4S0 to increase the asymmetry. All the values of I x / 2, Imh, and Reh are shown in Table I1 only for quark processes. For final physical states, care must be taken. If the final state is a CP-odd eigenstate, an additional minus sign should be added to A, x, Imh, Reh in Table 11. Also, the relative signz4 of h for a B'+P~P~ pseudoscalar) decay versus BO+ VIP2 ( V1 the excited vector state of P I ) is, neglecting differences in matrix structures,

For example,

when neglecting final-state phases. The sign of ABo-DoKs

34 - SYSTEMATIC STUDY OF LARGE CP VIOLATIONS IN . . . 342 1

and h ~ ~ + ~ ~ ~ L are opposite. However, an open question is how to fix the overall sign of the A's. For completeness, we assume that BO-+DOK, and BO-B OK, are described by the A's in Tables I and 11. Then all the sign of A's for other physical final states will be fixed. In Table IV we still use the A's in Tables I and 11. Thus, for individual exclusive two-body decays, a minus sign might be needed to multiply the A. This essentially will cause a sign flip of

Cf, but not change the number of 66 pairs needed. For details see Appendix B.

Notice that the B, system has the best asymmetry for B,-DOI#J with - 10' b6 pairs needed. If exchange dia- grams prove to be important, B, -D + n- has also a pure AI = $ transition and only needs - 7 x lo6 66 pairs. For the Bd case, even if nature is SO kind as to provide us with a large mixing parameter (Am / y IBd-0. 1, still B d - + $ K s

-0.61

-2X lo-'

TABLE IV. Some possible two-body hadronic final states, corresponding decay diagrams, z = Am / y , Cf, branching ratios, and numbers of b6 pairs.

Asymmetry Case B d ~hys--'f z = A m / y cf B ( BdOPure -+f ) Nb6

1 . 6-iiud P+P- < > 5.6X lo7 n u s 0.1 0.1 1.8 x 4 . 7 ~ 109

K + K - < 10-4 > 5 . 6 x lo7

llKs 6.1 x 3.5 x 10'O 2. i3-+iica D +T- lo-5 3 . 5 ~ 107

F + K - 10-5 3.5 x lo8

*DO 5 x 0.5 x lo9 DO^ 5 x 7 x lo7

3. 6 + ~ u d D -T+ 2 x lo-* 0.5 x 10" F-K+ 2X lo-2 0.5 X 10"

*Do 10-2 1 X 10"

b '.rrO lo-2 1.4X 10"

4, 8 --+TCF, F+F- 8.8x 10" $4 lo-5 1 . 3 ~ 10"

s l ~ f KS 5 x 10-4 2 x lo8

4Ks 5 X 5.3 X lo8 D+D- lo-' 8.8 x lo8 r°Ks 2.5 x 5 . 3 ~ los

V K S 1 0 - ~ 3.3 x lo8 8 - i i c ~ 0.1 0.19 2.2 x lo8 - 6 -Z.US D OKs 0.1 0.04 5 x 1 0 - ~ i x lo9

Asymmetry Case BE phys--tP z = ~ m / y Cf B ( B P ~ ~ ~ + f 1 Nb6

1 . 6 + ~ u d T+T- lo-s 6.7X lo7 iiuS K + K - 0.8 0.38 lo-$ 6.7 X lo7

r°Ks 5 x lo-3 2.0 x lo7 2. sics D O ~ J 2 x 1 0 - ~ 1 . 4 ~ lo7

F f K - 0.8 0.56 2 x 7 x lo7

D +T- 2 x 1 0 - ~ 7 x lo6

D o f f 0 10-4 1 . 4 ~ lo7

3. ~ - + ? U U S 25 Od 5 x lo7 F-K+ 0.8 0.23 2.4X lo8 D -P+ lo-' 2 . 4 ~ lo7 B of f0 5 x 4.8 x lo7

4. 6-CCF 44 3~ lo-' 1.1 x lo9 TCZ WS 2.7 x 1 0 - ~ 9 x lolo

0.8 -0.022 D-D+ 5 x lo-' 4.7 x lo9 F-F+ 2 x 1 0 - ~ 1 . 2 ~ 10"

6 - U C ~ - 0.8 -0.054 5.7X lo-6 3 x 10' 8 - T U ~ D OKs 0.8 -0.013 1.2 x lo9

8-2 bKs 0.8 -0.4 2.1 x lo-6 8 . 4 ~ 10'

3422 DONGSHENG DU, ISARD DUNIETZ, AND DAN-DI WU

requires 2 x 10' bps (with leptonic tagging we encounter here constructive interference; we could increase our asymmetry by a factor of 2). However, most estimates abound predicting smaller3, " - 13' l 5 mixing parameters and for (hm /Y IBd - 10-~--5x we require -4x 10' 66 for Bd-D+r-. (This decay however is plagued with two final-state isospin phases.) Or, again, if exchange is important Bd -Ff K- - 4 X 10' b& pairs are needed. (The detection efficiency of F + is 10% relative to that for D +.I

In Table V we show the maximal value of Cf and Cfl and the corresponding mixing parameter for various pro- cesses. The maximal value of Cf (Cfl) is reached for a

mixing parameter typically of order 1 (0.5). However, the &-+iica process is an exception, due to the large ratio of amplitudes 1 x 1 '= 1756 the maximum is reached at 3X lo-' (2X

If nature chooses a tiny mixing for B O - 8 O, the highly Cabibbo-suppressed process 8-ijc2 might be a good choice to observe CP violation.

- For mixing parameter of order 1, the processes b-ilu~,ZiuS,~c3',,Tc~,ZicS,,TuS, lead to large asymmetries for Bd system (see Table V). And ~ - + i i u ~ , ~ i u S , i i c S , ~ u ~ , lead to large asymmetries for B, system. If mixing in the B, system is large ( z > 5 ) , Cfl is unsuitable due to the z4 dependence in the denominator of Eq. (42), and Cf must

TABLE V. The maximal values of Cfl and C, and the corresponding mixing parameter z. Here s, =0.231, s2=0.05, s,=+s,, 8=9O0but for 6+irua, ?iuFin B~ decays, 6=45".

Bd-Bd system

Process Am z--(max) Cfl (max) Y -

6 - ( i i u ~ , i i u ~ ) 0.58 0.65 6 -+ iica 1.9 x lo-' -0.736 6 -~;u;I 0.8 1 - 2 . 2 0 ~ lo -z Z-(FCF,B;I,S) 0.58 0.531 6-iics 0.33 0.686 8 -?US 0.74 0.362

Process

B,-B, system Am ---- ( max )

Y

Process Bd-Bd system

z(max)

B,-8, system Process z(max)

- Cf (max)

5-+nuZ,aus 0.98 0.393 6 -UCS 0.58 0.583 6 -+CUT 1.30 0.260 6 -FCF, FCJ 1.03 - 2 . 2 2 ~ lo-z 6 - iicd 3 . 4 ~ -0.627 6 - Z U ~ 1.41 - 1.48 x lo-' 6-;I 1.02 -0.407

34 - SYSTEMATIC STUDY OF LARGE CP VIOLATIONS IN . . . 3423

be used. For large B,-B, mixing, 6 - ~ u ~ due to the soft varia-

tion - on z is particularly promising, less so are the b-+a,iiuu$,iiu~ processes, and even less promising is - b -m.

A copious supply of b6 is expected from the Cornell Electron Storage Ring, LEP, and the Stanford Linear Col- lider. There lo5-lo6 e + e - - b b per year seem not im- possible. So, we hope experimentalists will search for these asymmetries.

IV. PENGUIN-DIAGRAM CONTRIBUTIONS

We follow the analysis of penguins of Guberina, Peccei, and ~ u c k l ~ ~ and neglect the absorbtive part of the penguin diagrams. It will be shown that we can neglect the penguin contribution safely for B +$Ks decay .3,

For Bd-+~OKs, the penguin will be shown to dominate over the spectator diagram, however, not extremely. So, this will be a bad CP testing ground. For the extreme case rPn( B~ - T O K ~ ) >> ro+ - ( B~ -TOK~), we ob- tain

where pen denotes penguin contribution, 0+ stands for ordinary-diagram contributions defined in Ref. 26.

Guberina, Peccei, and Ruck1 got for the penguin b-+s decay the additional effective Hamiltonian

where

In the above, K 2 is the momentum transfer carried by the gluon and we assume ~ ~ = m ~ ~ . The mass correction term ln[K 2 / ( ~ 2 + mC2)] has been neglected and Cp -(2-5) X We take Cp -0.03 in later estima- tion. Different ha, yp structure of the penguin operators as compared to O+ operators provides an enhancement in the inclusive rates26>25 over the small penguin coefficient

and for B, system

In the above P(vac+cF) denotes the probability of creat- ing a cF pair out of the vacuum [P(vac-cF) < 1] and T(ZF) stands for Zweig-rule-forbidden penguin transition [T(ZF) < 11. We take the square roots of the above rate ratios to obtain the relative strength between the penguin amplitudes (with KM combination factor) and the Ok operators (with different KM combination factor).

However, for Bd-~OKs, we cannot claim any knowledge of asymmetry. Taking the amusing limit of infinite penguin dominance, we get

then

LP. The ratio of the partial widths for the decay A23 --

had-#~,- A ; ~ . (50) Bd -+K + n ~ generated by penguins and 0+ operators, respectively, is given by In KM phase convention, this means

In the above the inequality arises from the experimental s22+s32cos2~+2s2s3c~ limit (14). We have used the KM parameters in Eq. (15) R e h ~ d + p ~ s = S ~ ' + S ~ ' + ~ S ~ S ~ C ~ (51b) for definiteness.

Applying those ratios (46) to exclusive two-body decays Even in this extreme case, care must be taken to see how we see the mass effects of mC2 versus K* enter. They give rise to

3424 DONGSHENG DU, ISARD DUNIETZ, AND DAN-DI WU - 34

a Vcb V,*, term. For details consult Ref. 25. Back to reali- ty, for B~-+?~ 'K~ , we obtain comparable amplitudes for penguins and the spectator diagram, and hence even the assumption of I A ( B ~ - T O K ~ ) I = 1 x ( B Z - + T O K ~ ) 1 does not hold due to penguins [Gl#G2, see Eqs. (23) and (2411.

The estimates (46) and (48) have been derived from per- turbative OCD calculations. See Ref. 27 for a discussion . of experimentally isolating penguin contributions through exclusive B-meson decays, that will test the predicted penguin strength.

In this presentation we look at final states where the penguin diagram does not contribute to the asymmetry or at worst, if it is present we can safely neglect it. Our ap- proach can be contrasted with Chau and Cheng's analysis4 of CP violation in the neutral B mesons. First, they con- sider final states that cannot be fed simultaneously from BO and B O. So. in that case no interference of A ( B + f ) and A(B- f ) will occur. Such cases we do not consider at all in our pa r. Second, they consider final states into ge - which both B and B O can decay. However they con- strain themselves to the case f =?, and they do not look at final states which arise solely via penguin operators. We, however, discuss later large CP asymmetries predicted in Bd,, +4KS for which only penguins contribute.

It is our belief that to estimate the penguins involves more theoretical uncertainties. And to extract their rela- tive strengths from experiments in the years to come is much more problematic (see Ref. 27) than to obtain the strengths of the spectator and/or the exchange diagrams. The strength of spectator and exchange suffices for the large asymmetries that we obtain. Therefore we avoided the penguin diagrams in this paper or we looked at decays where we can safely neglect them, or we looked at decays where only penguins contribute.

Note that the result for B, -+K + K - in Ref. 4 is

Let us check this. We know that

Taking the representative values Eq. (15) we get for the ratio in Eq. (52):

Hence, the penguins and 0+ amplitudes have the same order of magnitude. If Chau and Cheng still have

1 A ( f ) 1 = I J(f) 1 , assuming penguin dominance they should obtain, up to some mass corrections which intro- duce V,*b Vcs in penguin amplitude,

Hence,

t: Vtb v: Vt?, Vts h=Lx=--- - 1 +mass correction . c t v:6 vts vtb v,*

So, Imh=O. This is in good agreement with Chau and Cheng's original result because Imh=2s2s3s6 is very small.

We must emphasize that for some processes, such as Bd,,-4Ks, only penguins contribute. We can obtain po- tentially large asymmetries. We take only the leading term ~ n ( r n , ~ / ~ ~ ) of Eq. (44) and neglect the mass- correction term l n [ ~ ~ / ( K ~ +mc2)]. The asymmetry for Bd-4Ks involves penguin transition b+s and can reach - 0.1. While the asymmetry for B, -+Ks invokes b +d penguin transition and can reach up to 0.4. In both cases 5X lo8 bb pair is needed for testing these asymmetries. All these results are listed in Table IV.

Now we estimate the branching ratios:

rpen(Bd-f4K~ ) ro+(Bd+D-p+) -

where Eq. (46) has been used. The factor f arises from I (Ks I KO) 1 2. Fps is the phase space, FmtpI is the multi-

plicity factor, p(uac+sF) gives the probability creating an SF pair from vacuum. We put them all together as FpFmt,lp (uac -SF) - 1 to obtain2'

The last number 2.5 X is obtained from

These branching ratios are also listed in Table IV.

V. DISCUSSION AND CONCLUSION

We have used

to derive our asymmetries. Now, criterion (53) is always satisfied when no penguin

diagram contributes to the final state [see discussion around Eq. (22)]. It is also satisfied quite accurately for those processes where penguins are negligible or dom- inant. For B ~ - ? ~ O K ~ Eq. (53) cannot be justified. But Eq. (54) is not always satisfied even when Eq. (53) holds true. The hurdle is the problem of final-state phases. For example, consider B:-D -T+ and 3 :-D+T-. After a lengthy analysis5,28 we have

Similarly we get

34 - SYSTEMATIC STUDY OF LARGE CP VIOLATIONS IN . . . 342 5

where

For ZD+,,.- = x i - f l + to hold, we need the KM stripped from the right-hand side of Eqs. (55) and (56) to be real. That does not happen in general. So in general

The decay B:-T+T- seems promising, since the weak effective Hamiltonian allows only I =0 and I = 1 chan- nels. Because of Bose statistics this final state can proceed through I =0 channel only. Unfortunately, the branching ratio for this decay mode is very small ( 1 0 ~ . For analogous reasoning the decay B:-+K+K- should also have a pure isospin channel. But the rescattering through the multipion intermediate states involves I= 1 channels, so makes Eq. (54) prob- lematic. The decay B,-+D-r+ or B, -+D+n- has only I = channel. These two decays seem promising. The only open question is how much do those virtual inclusive states "D-r+ (pairs of n)" or "D+T- (pairs of r)" intro- duce ~hases to our res~at ter ing.~~ The only drawback is that the branching ratio might be by a factor of 10 small- er than what we optimistically assumed-a scaling argu- ment for exchange diagrams from the observed D de- cays.26

B,+D0q5 suffers from the final-state phases. The eigenvectors of the S matrix is a linear combination of Do# and F+K- and therefore

8, ,ab being final-state phases. The decay B:-+$K~ does not suffer from final-state

phases because of Eq. (B7), although there is a rescatter- ing of D -F++$Ks. For non-CP eigenstates our quanti- tative argument based solely on KM angles turns sadly into just a qualitative one. Further work is definitely needed to elucidate final-state phases. An especially promising way to study CP violation is via inclusive or semi-inclusive channels, due to their larger branching ra- t i o ~ . ~ * However, progress is also hampered due to our lack of understanding of final-state phases.

We want to point out that if the mixing of B:-8 I: is extremely small (even smaller than to observe reasonable B:--+D+~- asymmetry, which needs 2, for good event number Nb6), we still could hope. for asymmetries in the pure decay rate B:-TOK~. Here penguin and 0+ operators have the same order of magnitude but different KM structure. Therefore assuming different final-state strong-interaction phases, a large decay-rate asymmetry r( B~,,--+T~K~) versus r(8dpw,+r0~S) could result. So we have CP violation in magnitude. The same remark might apply to B,-+n+n-. In general, whenever we have two decay channels with different weak and strong phases this remark might apply.

Probably, the best approach, in the event of negligible B O - 8 O mixing, would be to look for partial-decay-rate

differences of the charged b-flavored mesons (i.e., B& - f versus B, -7) (Refs. 29, 30, and 3 1).

Now we come to our conclusion. (a) CP-violation effects in nonleptonic decays of B0

meson can be quite large owing to the mixing and ampli- tude interference. The best decay modes for testing CP violation are B:-D +T-, B;-D+IT-, and B;-DO+.

They need 4X lo7, 7 X lo6, and 10' bg pairs for 30 signa- ture, respectively. For testing CP violation with only penguin contribution, the best modes are B:,~+$K~ which need 5 X 10' b6 for 30 signature.

(b) The problem of the strong-interaction phases of the final states is very difficult and subtle. It needs further investigation.

(c) According to the prediction of the standard KM model, the Cabibbo-Kobayashi-Maskawa-favored decays, 6 - c ~ ~ (e.g., B,-$+, F + F - , etc.), b-+~ud (e.g., B, +DOK~) for B, decays and &-+~ud (e.g., Bd-D-rr+, etc.), 6-+i?cF (Bd -t+bKs) for Bd decays have very large decay rates (so branching ratios) but very small asym- metries. The only exception is the B:+$K~ where the asymmetry could be large, however, probably suppressed by small mixing. If we find a large asymmetry in all the processes mentioned above, new physics will emerge. So it is worthwhile to make the efforts.

ACKNOWLEDGMENTS

We thank M. Chanowitz, Mary K. Gaillard, and the Faculty of LBL for their hospitality where this work be- gan. We would like to thank S. Adler, R. Dashen, F. Dyson, and the Faculty of the School of Natural Science at the Institute for Advanced Study for their hospitality where this work is continuing. We also want to thank L. L. Chau, H. Y. Cheng, and W. Y. Keung, I. I. Bigi, L. Brekke, J. W. Cronin, J. Pilcher, J. L. Rosner, R. G. Sachs, A. I. Sanda, B. Winstein, and R. Winston for many enjoyable discussions. This work was supported in part by DOE Contract Nos. DE-AC02-80ER10587, DE- AC02-82ER40073. and DE-AC02-76ER02220.

APPENDIX A

We list the invariants of the KM matrix:

3426 DONGSHENG DU, ISARD DUNIETZ, AND DAN-DI WU - 34

~ c ~ s ~ ~ s ~ s ~ s ~ for all i , a

(where we have taken c2 -c3 - 1,s2,s3 <<SI ).

APPENDIX B

When dealing with final states that are CP eigenstates, special care must be taken. Assume for simplicity that only one weak phase contributes to our process. Then the claim is that for CP-odd states we obtain - h (not A), for CP-even state we obtain h.

Proof: We have a couple of different strong eigenchan- nels labeled by a . Define

out(f ,a 1 ~ ~ ) ~ ~ = i r ~ e ~ ~ ~

Put

i.e., + for CP even 1 f ) , - for CP odd 1 f ) , for in- stance, ( + ) for f =D+D-, ( - ) for f =$Ks.

Choose the phase convention

Applying CPT onto (Bla) and (Blb), we obtain

In the above, the sign rtr merely reflects whether we deal with CP-even ( + ) or CP-odd (-) eigenstates f.

In reality our decay may proceed via several strong eigenchannels with unknown final-state strong-interaction phases. Now

Assume that only one weak phase enters

where +,k does not depend on a. Then

Eq. (B7) means that x will be essentially a ratio of KM combinations (note here we only have one weak phase, i.e., one KM combination) and the k sign reflects what CP eigenstate we deal with. Because x changes sign for CP- odd eigenstates, h does also. That completes our proof.

For final states that are not CP eigenstates, for in- stance, D+IT- , D*+n- , etc., we have also a sign ambigui- ty in h. In general, for B O - P ~ P ~ and BO- v,P,, owing to the odd relative CP parity, we should have24

where, we have neglected the difference of the strong- interaction phases and the kinematical considerations, and PI ,P2 are pseudoscalars, V1 is just the excited vector counterpart of PI.

'permanent address: Institute of High Energy Physics, Beijing, The People's Republic of China.

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