Nuclear Physics B281 (1987) 41-71 North-Holland, Amsterdam

C P V I O L A T I O N IN H E A V Y F L A V O R DECAYS:

Predictions and search strategies*

I.I. BIGI 1

Stanford I_znear Accelerator Center, Stanford Unwerszty, Stanford, Cahforma, 94305, USA

A I SANDA 2

Rockefeller Unwersay, New York, NY 10021, USA

Received 23 June 1986

We present a comprehensive analysis of the phenomenology relevant for CP violation in exclusive B and D decays Some of the CP asymmetries can be calculated rather rehably in the standard model, while others contain large uncertainties. We point out that studies of inclusive decays will have to deal with partial cancellations occurring between different exclusive channels None of the measurements will be easy Yet a realistic evaluation of the promise the various searches hold out cannot be given at present; first, one has to budd a data base on the production and decay properties of heavy flavor decays. We try to identify the required information.

1. Introduction and summary

M o r e than twenty years have passed since the discovery of CP violation in the

decays o f neutral kaons. Despite strenuous efforts, we cannot claim to possess a

p roven descript ion of this phenomenon, let alone a deeper understanding. We

believe this unfor tuna te situation will not change unless CP violation is found in a different dynamica l system.

The weak decays of bo t tom mesons hold out the promise to allow such a discovery. Some decay modes are expected to exhibit relatively large CP asymme-

tries. Fur thermore , we can have greater confidence in the theoretical t reatment of B

mesons than of kaons. On the other hand, bo t tom mesons are much shorter lived

than kaons; thus, different methods have to be employed to exhibit possible or even expected C P asymmetries.

A n u m b e r o f papers on this subject have appeared in the literature; nevertheless

we think that a comprehensive and updated t reatment of these phenomena is called

* Work supported by the Department of Energy, contract DE-AC03-76SF00515 1 Max-Kade Fellow. 2 Supported in part by the Department of Energy, contract DE-AC02-81ER40033B.

0550-3213/87/$03.50©Elsevler Science Pubhshers B.V (North-Holland Physics Publishang Division)

42 LL Btgt, A.I S a n d a / CP v,olatton

for. By doing so, we hope to clarify some of the considerable confusion that has arisen in parts of the literature. At the same time, we want to suggest further directions in research - both experimental and theoretical - that would be of great benefit in this context.

The paper will be organized as follows: in sect. 1 we present a summary of our analysis on CP violation in bottom and charm decays; the remainder of the paper will be devoted to explaining our findings in more detail. Sect. 2 contains the phenomenology of those CP asymmetries that involve mixing before we employ the standard model to calculate or at least estimate the relevant quantities in sect. 3. In sect. 4 we treat CP asymmetries that can exist even in the absence of mixing, namely those involving final state interactions, and in sect. 5 we list different strategies to search for CP violation in B decays and present numerical estimates. It is obviously premature to give definite numerical predictions when so little is known about branching ratios, etc.; therefore, we discuss in sect. 6 how future data on B production and decays will allow us to sharpen our predictions. At this point we also want to stress that although we phrase our analysis for the case of B decays, it is actually a generic treatment that is easily applied to charm decays as well. We will comment on that at the appropriate places.

CPT invariance implies that CP violation can enter only via complex phases. Thus CP asymmetries can show up only if at least two coherent amplitudes contribute to a given process. There are basically two ways in which this can happen: (a) via particle-antiparticle mixing and (b) via final state interactions.

1.1. MIXING AND CP VIOLATION

Mixing describes a situation where the mass eigenstates are a coherent superposi- tion of a particle and its antiparticle.

The time evolution of a meson that was produced as a B ° (bottom) or ~0 meson, respectively, at time t = 0 is given by [1]

IB°(t)> = g+(t) lB °) + q g _ ( t ) l B ° ) , (1.1)

I§° ( t ) ) = P g - ( t ) l B °) + g ÷ ( t ) l § ° ) , (1.2)

g ± ( t ) = ½exp(- l I ' l t )exp( iml t} (1 ++_ exp( -½AFt )exp( iAmt ) ) , (1.3)

q 1 - e A F = / ' 2 - F1, Am = m 2 - ml , -- = - - . (1.4)

p l + e

iv,, m,, i = 1, 2 are the width and mass of the two mass eigenstates B,. In the

LL Blgl, A L Sanda / CP molatwn 43

following we will set AF = 0 for convenience. As far as CP violation is concerned, no new features are introduced by a F 4= 0; secondly, for B ° mesons one calculates ,:IF << Am with considerable confidence [2] (although this might not hold for D o mesons [3, 4]).

Mixing will manifest itself most clearly via decays of neutral bottom or charm mesons that lead to "wrong sign" leptons [1, 5]:

F(B ° --, d+X) x 2 - ( 1 . 5 )

rB = F(B ° -* d-X) 2 + X 2 '

F(D°-+ d-X) x 2 + y 2 - - - (1.6) rD = F(D ° --*/+X) 2 '

Am AF x = - F - ' Y = 2--Y- " (1.7)

The detailed time evolution of these ratios will be given later, as well as a method that uses nonleptonic decays to search for mixing.

There are three ways in which CP violation can then show up: (a) Semileptonic decays [1]

F(B ° --, ¢+X) - r ~ ° --, c - x ) 1 - [p /q[ 4

r(B ° - . e - x ) + r¢B° ~ r+x) 1 + [p /q[4" (1.8)

(b) Nonleptonic decays. (i) When f is a final state that is common to both B ° and ~o decays (a property

that is then shared by the CP conjugate state fce) one can look for the following asymmetry [6, 7]:

r ( B o . f) _ r ~ o _~ fc . )

F(B o _~ f) + FCBO _. fcp)

2 x p

- 1 + Ipt[ ~ 1 + x - 2 I m q pt '

A(-BO -.+ f) (1.9)

Pt - A (B ° ---' f) "

The detailed time evolution that has been integrated over here will be given later• A simple realization of this scenario is obtained when f is a CP eigenstate as in

0 o 0 + B ° ~ k K s , DDKs , DD or in D --+ Ks~r ,KsP ,K K- ; in this case f c p = ± f • ( - - )

holds. Furthermore, the amplitudes A( B --+ f) are not necessarily suppressed by small KM quark mixing angles in that case which leads to very sizeable asymme-

44 LL Bzgl, A.L Sanda / CP molanon

tries. If f is not a CP eigenstate, e.g., f = D+~r -, then the asymmetry will be reduced by small K M angles I U(b ~ u) × U(d ~ c)] as we will quantify below.

In electromagnetic or strong processes one always produces bottom mesons in conjunction with antibottom hadrons: B°B or BB °. Therefore, one can measure the asymmetry given in eq. (1.9) only via flavor-tagging the other B decay. This can be achieved most simply by observing direct leptons from semileptonic B decays; thus the asymmetry [eq. (1.9)] gets translated into a difference between the ~+f and g-- f cP correlation:

o(BO8 + e+fX) _ o(Bo8 + cfc x)

asymmetry = o(BO 8 + B~ o -~ g+fX) + o(B°B + BB ° --} & f C P x ) "

The details can be found in sect. 2. (ii) If a B°B ° (or D°D °) pair is produced in a p wave configuration as it happens

for e+e - --, B°B ° (for a one-photon intermediate state), then the process

B°B ° -~ ftf2 (1.10)

can occur only due to CP violation if fl and f2 are CP eigenstates with the same CP parity [7,10].

Employing the standard SU(3)c x SU(2)L X U(1) model, one obtains predictions that are summarized in table 1. Two general features are obvious: some of the asymmetries are expected to be really large, yet the accompanying branching ratios for these exclusive modes are small.

This raises the question whether one can search for such asymmetries in inclusive decays, thus increasing the available statistics. This can be done, but only with considerable care for the following reason: if the final state f is a CP eigenstate, then the sign of the asymmetry is determined - as will be shown later - by the CP parity of f. Thus an asymmetry in B ° ~ K s + X will be equal in size, but opposite in sign to the one in B ° ~ K L + X. An analogous effect holds when comparing, say, art asymmetry in B ° ~ D+cr - versus 8 °--} D-or + with one in B °--} D+p - versus 8 ° --} D - p +. Therefore when one performs an analysis of such decays in an inclusive fashion, one takes the risk that the large asymmetries in the various exclusive channels will cancel each other, at least partially.

It follows, then, that when one sums over different channels to increase statistics, one should not do it blindly. Instead, one has to include information on the CP parities of the states one sums over. This is actually an easier task than it might appear at first sight. As an example, consider the decay chain

B o ~ D O M o

~o ~ ~OMO

( K s N ° ) M ° . (1.11)

LL Bzgl, A.L Sanda / CP molanon

TABLE 1 CP asymmetries in B ° decays

45

Estimated C P Decay modes branching ratio asymmetry

A. Semdeptomc

BdB d ---, :±d±X _< 4 × 10 -4 10 -3

Bsn s --+ ,a±d± x 10-3_10 -2 10-4

B Nonleptonic CP pure final state

B d --+ ffK s - 5 × 10 -4 0 02-0 2 --+ tkKs*r ° 10 -3 0 02-0 2 ---, tkK s + X 5 × 10- 3 0 02-0 2 ---, D D K s (0 1 - 2) × 10 -2 0 02-0 2 ---, DD 5 × 10- 3 0.02-0 2

B s ---, ff¢ 10 -3 10-3_0.01 ---, F + F - 3 X 10 -2 10-3-0.01 --+ ~ / 5 × 10 -4 10-3-0.01

B d ~ Ks~r < 10 -4 0 1-0 5 --* Ksp 10 -4 0 1-0 5

CP mixed final state B d ~ E,.:(KsNj)ooMj - O(1%) 0 02-0.2

N,, Mj ~ (pseudoscalar vector, axaal-vector nonet}

non-CP final state B d ~ D % r - 1% 10-3-001

---, D ° K s O(10 -3 ) 10-3-0.01 B~ "-+ F + K - O(10 -3) 0 1-0 5

It is easy to convince oneself that the CP parity of the state (KsN°)M ° is positive for all configurations as long as N and M belong to the pseudoscalar, vector or axial-vector nonet.

1.2. CP ASYMMETRIES VIA FINAL STATE INTERACTIONS

CP asymmetries can emerge also in the absence of mixing, for which the cleanest scenario is provided by charged B (or D) decays [6, 8, 9]. When two different amplitudes contribute to the decay of a bottom hadron B into a final state f, one writes for the matrix element

Mf = (f l .~[ B)

= ( f l .~xIB) + (fI.CP2IB)

= gxMlexp { lax } + g2M2exp { ia2 ) , (1.12)

where M1, M 2 denote the matrix elements for the weak transition operators ~1, -~2

46 I L Bigt, A.L Sanda / CP vtolatwn

with the weak parameters gl, g2 and the strong (or electromagnetic) phase shifts Otl, Ot 2 factored out.

For the CP conjugate decay B ~ fce one then finds

~.f =_ <fCPl.~l~ >

= g~'Mlexp ( iax ) + g~'MEexp ( ia2 } , (1.13)

and therefore

F(B ~ f) - F(B ~ f cP) oc Im g~g2sin(a 1 - a 2 ) M 1 M 2 . (1.14)

From eq. (1.14) one reads off two necessary conditions for such a CP asymmetry to show up:

(i) The weak couplings gl and g2 have to possess a relative complex phase. In the standard model this implies that the transition rates for such modes are suppressed by small KM angles.

(ii) Nontrivial phase shifts a 1 ~ a 2 have to be generated from the strong (or electromagnetic) forces.

Condition (ii) does not, in principle, pose a severe restriction. Since the two amplitudes will in general differ in their isospin structure, a nontrivial phase difference will exist. However, as we will see later on, no reliable estimate of condition (ii) can be given.

Basically there are three ways in which condition (i) can be met: (1) Interplay between the two cascade processes [6]

B - ~ D°(Ks or K - ) + X- B - ~ b ° ( K s or K +) + X -

(1.15)

for which the Feynman diagrams are given in fig. 1. (2) Interplay between quark decay and weak annihilation reactions (see fig. 2) in

B - ~ D°*D -. (3) Penguin contributions generating B- ~ K-oo modes, fig. 3.

u + b ~

LI $

Fig. 1. Two cascade d iag rams con t r ibu t ing to B - ~ D ° K ~ K s K + • -

Fig. 2

LL Btgt, A L Sanda / CP vtolanon

d

- u u u

Quark decay and weak anmhilatlon contnbutton to B- ~ D°*D -.

47

q- u ~

Fxg. 3 Penguin contributions to B- ---, K *r°,K-p °

TABLE 2 CP asymmetnes in charged B decays

Estimated CP Decay modes branching ratio asymmetry

A Interference between different cascades: - O(0 1%) 10-3-0.01 B ± ~ F~,,j(KsN~)KM j

B Interference between quark decay and WA: - O(0 1%) 10-3-0 01 B ± "-* D°*D ±

C. Pengmn processes' B ± ---, K±p ° 10 - 4 0 0 1 - 0 1

Typica l guest imates for C P asymmetries in charged B decays are given in table 2.

The most rel iable can be given for case 1; nevertheless, all the possibilities have to

be analyzed. Again, care has to be applied when summing over different channels.

1 3 c P ASYMMETRIES IN D DECAYS

The whole previous discussion can be applied to D decays in an analogous

fashion. Thus, one should search for a difference in

D O ~ K s N , N c pseudoscalar, vector, ax ia l -nonet , D O ~ K + K - or

D + ~ ~r+p °, K + K °, F + --* KsP ÷ versus its C P conjugate modes .

48 I L Btgt, A.L Sanda / CP ololatton

NO observable effects are predicted by the standard model, yet it is consistent with present phenomenology that asymmetries of order 0.1% up to 1% could show up.

1 4. BUILDING THE DATABASE ON HEAVY FLAVOR STATES

Some of the uncertainties encountered before can be overcome by more experi- mental data. This is obviously true for the branching ratios of the relevant channels, such as

B d ---, ~kKs, lk K srr 's, D D K s D ÷D- , p~,

Bs ~ ~,rl, ~rf , ~¢ , F + F - ,

B- ---* D°*D -, K-*r °, K - p ° .

Of particular interest are the branching ratios for the channels

B ° ~ DOM --, (KsN)DM ,

where N, M are any members of the pseudoscalar, vector or axial-vector nonets. As we have already stressed, the CP asymmetries from these channels will add rather than cancel.

There are other decay modes which are of great indirect help in elucidating the underlying decay mechanisms, such as

B- --* ~kK-, D D K - , D F - , "rB± versus zB0.

At the moment, we still have only fairly rudimentary information on B decays; this will have to change.

The strength of B°-B ° mixing, which is a fascinating phenomenon in its own fight, acts as an important input parameter to the size of CP asymmetries. Its detection would be highly valuable for refining the predictions.

Lastly, although e÷e - annihilation provides the cleanest setting for the produc- tion of heavy flavors, one might have to use hadronic collisions to accumulate the required statistics. Very little is known about production specifics of heavy flavors in hadronic collisions. Again, this has to and will change.

The results summarized above will be explained in more detail in the subsequent sections.

2. CP asymmetries involving mixing

2 1. EXCLUSIVE DECAYS

To obtain a sufficiently large sample of B (and D) mesons, one has to rely on electromagnetic or even strong production processes. In either case, one has to

I L Blgt, A L Sanda / CP vwlatwn

consider pair production. Three cases should then be distinguished: (i) A neutral meson and its antiparticle are produced:

49

B~, B~,

with Bq = (bq). This is the situation encountered just above threshold. (ii) The two neutral mesons are not conjugate to each other:

B ~ + h.c.

(iii) The neutral meson is produced in conjunction with a charged meson or baryon

"Bu BqXb + h.c. , q = d , s .

Subsequently, the bottom hadrons decay. For our later discussion, we distinguish between f o (final states of neutral B mesons) and f • (final states of charged B or A decays).

Case (iii) is the simplest to treat since only one meson can oscillate. Using the definitions for the decay amplitudes

A ((~)o __+ f) = (if), (2.1)

one finds

rate[BO(t)~u __. fof+] o: rate(-B u ~ f+)exp{ - Ft }

X l+cosAmt)lAl2+(1-cosAmt I-~[ 2

--2sin(Amt)ImP A.~* },

ra te [B0( t )B ~ f0 f - ] tx rate(Bu ~ f - )exp{ - F t }

x (l+cosamt)lAI2+(1-cosamt) IAI 2

(2.2)

+2sin( mt)ImP }

50 LL Btgz, A L Sanda / CP oiolatton

Case (ii) presents a more complex situation since both mesons can oscillate, though not into each other. Defining

A((B)d ~ fd)= (A)d, A((B)s--'> fs)= (A)s, (2.4)

one obtains

rate{ [Bs(t~)Bd(td) + Bd(/d)B~(t~) ] ~ fdL}

e x p ( - Gtd) e x p ( - rst~)

x {[1 + cos(Amd/d + Amst~)]lA~Ad + Ad~l 2

+ [1--COs( Amdtd + Amsts)] l P AsAd + q.~Yd[ 2

+ 2sin(Arndtd + amsts)Im( P AsAd + qXsAd)(AdiTs+AsXd)*). (2.5)

Case (i) finally has the added complexity that the two states can mix into each other. Therefore one has to include effects due to quantum statistics and gets accordingly

rate(Bq(t)Bq(')lc=~" fafb)

oc exp(-Fq(t + i)}

× ([1 + cos Amq(t g i)]IA~.4 b -Y AbAT.I 2

+[1-cosAmq(t -T- i ) ] PAaAb-T- qA~Ab 2

for BqBq being in a charge conjugation odd (C = - 1 ) or even ( C - + ) configura- tion, respectively. The notation is analogous to the one used in previous cases:

(A)(Bq fa,b) (-'~) ~ - - - a,b, q = d, s. (2.7)

LL Btgz, A.I Sanda / CP molatton 51

The expressions (2.2), (2.3), (2.5), and (2.6) serve as master equations to describe all types of CP asymmetries that can emerge in the presence of mixing. First we discuss the important features qualitatively.

A difference between CP conjugate reaction rates can be produced by two terms in eqs. (2.2)-(2.6):

(a) Ip/q[ 2 could differ from unity:

p 2 4 Re e (2 o8) - 1 = [ 1 - e l 2 '

and thus [p/q[2@ iq/p[2. Such a difference would exhibit itself most cleanly in semileptonic B decays. For example, one derives easily from eqs. (2.2), (2.3):

o(BO~ + --+ <+ ... f+) - o(BOB - __) <- .. . f - )

aSL = o(BO~+__) e + . . . f ) + o(-B°B --) g ' -+ . . . f - )

[q/pl 2 - Ip/ql 2

[q/pl 2 + Ip/ql 2' (2.9)

where we have used IAI 2= IAI 2 due to CPT invariance. The production of these "wrong sign" leptons of course depends crucially on the strength of mixing; more specifically, the production rate is proportional to (Am~F) 2.

(b) Among the nonleptonic modes there are some that are common to both B ° and ~o decays. As already stated in sect. 1, they can be CP eigenstates - like ~ K s in B a decays, but need not be [11] - like D-~r +. Setting [p/ql 2 = 1 for simplicity and clarity (later we will see that this is actually predicted to hold in the standard model), one derives from eqs. (2.2), (2.3):

ANL(t ) = o(-BO(t)B- ~ f o f - ) _ a(BO(t)B + ~ (fo) ce f+)

o ( B ° ( t ) B - ~ f o f - ) + o(BO(t)B+ __. (fo) cef+)

2 sin A mt I m ( p / q ) A.4*

(1 + cosAmt)lA 12 + (1 - cosAmt)lA 12, (2.10)

with A = A(B ° ---, f 0), .~= A(~0 ~ f 0); (f 0)o ' denotes, as before, the CP conjugate state of f 0.

Two technical remarks might elucidate eq. (2.10): (a) Integrating over all decay times t from zero to infinity one finds

x Am (ANL)t~ I + X 2 ' X= F (2.11)

52 LL Blgt, A.L Sanda / CP wolatton

In other words, this CP asymmetry vanishes both in the limit of small mixing (x --- 0) and large mixing (x >> 1). For x = 1 this mixing factor actually reaches its maximal value of ~- 2"

(b) CP violation resides in the quantity

I m P A A * = IAI 2Im Of, Of= f , (2.12) q

which contains one factor (p/q) coming from mixing, i.e. AB = 2 transitions, and another one (Of) coming from decays, i.e. z~B = 1 transitions. It is exactly this combination that is invariant under changes in the phase convention for 8 o states: 18°) ~ e x p ( - i a } l 8 ° ) leads to the simultaneous transformations ( p / q ) ~ e x p { - i e t } ( p / q ) and Of--> exp(ia}pf, thus leaving (p/q)pf invariant as it has to be for a physical observable.

Closely related asymmetries can be obtained from eqs. (2.5), (2.6). The procedure to follow there has the same basic elements as used in eq. (2.10):

(i) One chooses a decay channel f o that is common to both B ° and 8 o mesons. (ii) One selects a second decay channel which betrays whether the decaying

hadron was, at decay time t, a B ° or 8 °. Semileptonic modes (or decays to charged D mesons) meet this requirement. Thus case (i) can be illustrated by the following example:

o [B° ( t )~° ( i ) [ c = ± ~ f0y- . . . ] _ o B ° ( t ) ~ ° ( i ) [ c = ± ~ (f0) c e ~ . . . .

ANL(t, ~ ) = o[BO( t~o(7 ) l c_± ._ , foe- . . . ] + o B°( t )8° (~) lc=±- - ' ( f 0 ) o ' : + .. .

2sin Am(t d:t) = I m P A A * (2.13)

Ih12+ IAI 2 q "

There is one more way in which CP violation can manifest itself: if B°B ° are produced in a relative p wave, i.e. a C odd configuration as it happens in e+e - ~ B°B ° for one-photon intermediate states, then the reaction

B O S O l c - _ _ . f o f o

can proceed only via CP violation if fl, f2 denote CP eigenstates with the same CP parity. This fact can also be read off from eq. (2.6): using CPT invariance to obtain

A, = exp(i , ,}A, , t = 1,2, (2.14)

LL Blgl, A.L S a n d a / CP vtolatton 53

one derives

rate[B°(t)B°(~)[c=_---, f l f2] ~ exp( - F ( t + ~))IA~I21A21 z

× ([1 + cos A m ( t - i)] [1 - cos(q~z - if1)]

+ [1 - cosAm(t-- i)] [1 - cos(q~2 + t~l)] )

(2.15)

using, as usual, the standard model result [p/q[2 = 1 for simplicity. Again, some technical comments might serve well to elucidate eq. (2.15): If there is only one universal phase q~, i.e. (1)1 = 02 = ~, as in a superweak ansatz of

CP violation, then one finds

ra te [B°( t )B°( [ ) lc= -"-) fifE]

cx exp( - r ( t + [))[AlIEIA212[1 - cos A m ( t - t )](1 - cos2qQ, (2.16)

which behaves like (Am~F) 2 for small mixing. If however, q~t 4= q'2, then one obtains for small mixing

rate[B°( t )B°( f)lc=_ --' fxf2]

cc exp( - r ( t + [)}IAl121A212[1 + cos Am(t - [)] [1 - c o s ( O 2 - ~bx) ] + . . - ,

(2.17)

which depends rather little on Am/['. To summarize our preceding discussion, there are three ways in which CP

violation can manifest itself in B ° decays as one learns from inspecting the master equations (2.2)-(2.6). The driving force can be:

(i) a difference IP/ql 2 _ [ q/P 12 0c Re e, which is measured most directly in semi- leptonic decays;

(ii) an interplay between phases in decay amplitudes and in e leading to a difference between F(B°(t) ~ fo) and F~°( t ) ~ (fo)cp];

(iii) phases in decay amplitudes producing the reaction B°( t~° ( [ ) l c=_~ flf2 where fl, f2 are CP eigenstates with the same CP parity.

2.2. INCLUSIVE STUDIES

So far we have discussed CP asymmetries in exclusive nonleptonic decays. It was already stated in sect. 1 that the branching ratios for the relevant modes are

54 I L Btgt, A.L S a n d a / CP vtolanon

expected to be small, as will be discussed in more detail later on. Then the question arises rather naturally whether one can increase statistics by summing over several channels. We will show now that such semi-inclusive measurements can be done, if great care is applied.

2.2.1. The problem. If the final state fo common to both B ° a n d B ° decays is a CP eigenstate, then its CP parity will determine the sign of the corresponding CP asymmetry:

( f± I£e(za B -- 1)13 o)

P/± = (f+ I ~ ( A B = 1)IB °)

with

( f + I .~ce(aB = 1)IB °)

= _+ ( f + l A O ( A B = 1 ) l B O ) , ( 2 . 1 8 )

CPIf+> = ± I f+) .

Thus summing over final states f with opposite CP parities will create a strong tendency for cancellations. As an extreme example, if one sums over ~kK s and ~kK L decays of ~B)d mesons one is guaranteed to find a vanishing CP asymmetry since

CPI+Ks> =

cPI¢ :L) = ICKL>,

(2.19)

At the same time, the asymmetry in the individual channels (B)d ~ f fKs and (B)a ---' ~k K L is expected to be large as stated in sect. 1.

The same problem of cancellations in inclusive measurements is encountered when f0 is not a CP eigenstate. For a positive asymmetry in, say, B ° ~ D+rr - versus ~ o ~ D-or+ translates into a negative asymmetry in B°-* D+~r+~r ° versus 3 ° ~ D-~r+cr ° (leaving out subtleties due to final state interactions to which we will return in sect. 4):

(D+Tr - I.~e(AB = 1)13 °) = (D-or + I.oq'ce(AB = 1)IB°),

(D+~r-~r°l.~a(AB -- 1)13 °) = - (D-~r+~r°lLa(AB = 1)ltl°), (2.20)

since CP[cr °) = -I~r°) . The extent of the cancellation depends of course on the relative branching ratios of the channels.

2.2.2. Possible solutions. The cleanest way of avoiding these cancellations con- sists of summing only over identified exclusive modes weighting each contribution

LL Btgt, A.L Sanda / CP vtolatton 55

with the appropriate sign. This task will probably be less formidable than it appears at first:

(i) There are wide classes of decay modes that contribute with the same sign to CP asymmetries. For example, one easily determines the CP parity of the final state in

(-) (-)

B ° ~ D ° M

L K s N

to be

CP [ (KsN)M ] = ( - 1) JN +JMCp [M]CP[N] , (2.21)

where J s , JM denote the spin of system N and M, respectively. Thus one finds that the combined CP parity is even as long as N, M are any member of the pseudo- scalar vector or axial-vector nonet.

(ii) M A R K III data [12] show that D decays are dominated by two-body final states (when resonances are included). Thus it appears possible (though not certain) that decays like (B)°~(D)°M ~ ( K s N ) M are actually dominated by final states with even CP parity. Then an inclusive measurement would suffer only mildly from cancellations. This represents one example where more experimental information on B decays will be of crucial help.

2.2.3. D decays. Suffice it to state at this place that this whole phenomenologi- cal analysis can be applied to D O decays in a completely analogous fashion.

3. Predictions from the standard model

We will show that three essential parameters, x = Am~F, Re e, Im(p/q)pf, can be estimated in the standard model with considerable, though by no means absolute, confidence for B d and B s mesons. The situation is much less promising for D O mesons, as will be explained briefly in the end.

31 Am

A m per se does not tell us anything about CP violation, although it certainly is a fascinating phenomenon in its own right. Yet a nonvanishing x = Am/F is required for many CP asymmetries to manifest themselves, as shown in sect. 2.

Since the mass of the B mesons is dearly outside the energy regime where the usual hadronic resonances operate, we can rely on the quark box operator to calculate the effective A B = 2 transition operator; the latter is a function of the top mass m t and various KM mixing angles involving the top quark. None of these parameters is known for sure; yet relying on an ansatz with just three quark-lepton families, we can relate the KM angles involving the top to those involving the

56 LL Bzgl, A L Sanda / CP vwlatwn

bottom quark [19]:

IU(tb) I--- 1, IU(ts) I --- IU(bc) I = 0.05 + 0.01,

IU(td) I--[U(bu) I < 0.008. (3.1)

Eq. (3.1) is based on the information obtained from experimental studies of B decays. Two features should be noted:

(a) Although I U(bc) l and afort iori I U(ts)[ are not well determined one should keep in mind that these uncertainties drop out from A m/I'(Bs).

(b) There exists only an upper limit on I U(td)l; thus we will obtain only an upper limit on Zirn/F(Bd) (unless one insists on the KM scheme to reproduce CP violation as observed in K L decays).

Intrinsic theoretical uncertainties enter when one undertakes to evaluate the on-shell matrix element (B°IL~'(Z~B=2)IB°), which is usually parametrized as follows:

~ ° I (bq)v_A(bq)v_AIB °) = ~RBf2m 2 . (3.2)

R B = 1 represents saturation by vacuum insertion [13]:

(~0[(~q)V_A(bq)v_AI B°)

RB ---- (~01 (gq)V_Ai0)(01 (gq)V_AIB0) • (3.3)

For the problem at hand one is directly interested in the size of the product R j 2 . Nevertheless, we choose to discuss them separately; this corresponds to the usual treatment in the literature. From some theoretical models of the hadronic wavefunc- tions, one has so far extracted only fB, and R B has to be of order one. In principle, at least, fm fr~ and fF can be obtained from measurements of B ~ vv, F--* "rv, D ~ "rv,/~J, or (with more theoretical bias) from the hyperfine mass splittings M(B*) - M(B), M(F*) - M(F), M(D*) - M(D).

In table 3 we have listed the results on fD,V,B,,Bs derived from different theoreti- cal approaches. One can add that recent experimental evidence for M(F*) - M(F) - 140 MeV would suggest fF - 200 MeV.

Table 3 shows a still considerable scattering of theoretical predictions though one should also note that the various predictions have shown a marked tendency over the last few years to converge.

As first shown in ref. [20] (see also ref. [16])

R B = I

under two conditions: (i) one employs a static quark model obeying manifest translational invariance; (ii) one relies on a valence quark description for the mesons.

Dropping condition (ii) will lead to [20,16]

R B > I .

I L Blgt, A.L Sanda / CP vtolanon

TABLE 3 Theorelacal estimates of meson decay constants (m MeV)

57

f~ = 133 MeV fD fF fBd fB~

QCD sum rules [14] 200 190 + 30 210 + 30 nonrelat, potenttals [15] 150 210 125 175 relat potentials [16] 150-230 190-390 100-190 140-230 MIT bag models [17] 100-170 - 190 70-150 140-200 M(1--) - M(0 -+) [18] 11-200 70-130

Giving up condi t ion (i) - as is the case for bag models - yields

R B < I .

Put t ing everything together, we estimate as a reasonable guideline

R~f~ - [ 0.01-0 .02 GeV 2 for S d (3.4)

0 .02-0 .04 GeV 2 for B s "

Assuming m t ~< 40 GeV, one obtains as typical numbers

X(Bd) _.< 0 .03-0 .02 , r (nd ) _.< 0 .1%-4%, (3.5)

x(Bs) - 1 - 4 , r(B~) - 33%-90%, (3.6)

indicat ing sizeable or even large Bs-B s mixing and small, yet possibly observable

Bd-B d mixing. On the other hand, at present one cannot make a precise predict ion

even when m t is known.

32. Ree

As stated before, the charge asymmetry in B°B°---> to-+E-++ X is propor t ional to

Re e

o ( B ° B ° ~ t*-t ~- + X) - o ( B ° B ° ~ f+•+ + X)

aSL = a ( B ° B ° ---> e - f - + X) + o (B°B ° ~ d+e + + X)

Ip/ql 2 - Iq /p l 2 [p/q[2 + [q/p[2

4(1 + lel 2) Im(F12/MI2 )

(1 + lel2) 2 + 4(Re e) 2 1 + ¼1F12/M~212 (3.7)

O n ra ther general grounds, one expects aSL to be small: [F12/M121 is small due to m 2 << m2; more specifically I/'lz/M121 < 0.01 for rn t > 35 GeV. In addition, if one

58 LI. Btgl, A L Sanda / CP molatton

restrictS oneself to the standard model with just three families, one finds

and thus [22]

I m M1---- 7 << << 1, (3.8)

10 3 for B a decays

a sL - 10- 4 for B s decays ' (3.9)

which would be too small for observation. Yet one should keep in mind that "new physics" beyond the standard model could yield sizeable contributions to Mr2 while being insignificant for/12. Then

could hold leading to

(3.10) ImM1--- 2-O([M12 ]

aSL- O(1%). (3.11)

Supergravity, left-right or composite models allow for such scenarios.

3.3. Im(p/q)o f

In calculating Or = [A(B°~ f ) ] / [A(B°~ f)] one has to distinguish between f being a CP eigenstate and otherwise.

(i) f is a CP eigenstate. Two examples are provided by B a ~ ffK s, DD, where the latter mode is Cabibbo suppressed. Comparing them with the CP conjugate channels Bd ~ ~Ks, D+D- one realizes, and this holds for all such decays:

(a) the topologies of the quark diagrams for B d and Ba decays are equivalent; (b) phase space and final state interactions are identical. Therefore the amplitudes for B d ~ f and Ba ~ f can differ at most by a phase:

a ( s f) = lafl = A(-B0~ f) 1. (3.12)

Secondly, IP/q[ --- 1 for Re e = 0. Therefore

P - - & = e x p { i , ) . (3.13) q

LL Bzgt, A L Sanda / CP vtolanon 59

One can consider B d and B s decays due to the transitions b ---, crs, b ---, crd and b ---, ufid and finds

I UbcUsc I

PlOfB q (ebtUq~)2 ( ebcUd~c ) 2 = ie cUd,

( b~l-'* c6sq for ~ bq--, c~dq ~. (3.14)

~, bC t ~ u~dq I

One technical comment is in order here to avoid a possible confusion: the expressions (3.14) per se are not invariant when one changes the phases of the quark fields:

d--*dexp(t~d), s~sexp{irs), (3.15)

leads to

PPfB~ ~pp, exp(2t (gd--8~))"

One has to keep in mind however that the relevant quantity is, strictly speaking, (p/q)pfF where

F = ( K s l ( d s ) ) ( K s l ( g d ) ) * (3.16)

( K s l ( g d ) ) I "

Since

F ~exp(2t(8~--Sd))F

under eq. (3.15) one sees that (p/q)pfF is indeed invariant, as it has to be. Just for simplicity we will suppress F in the following.

It is most convenient to employ the Wolfenstein parametrization [23] of the KM matrix

UKM 1 - I • X h3A(p - i~/)

- ~, 1 - 1~2 ~2 A

A3A(1 - p - zT/) -A2A 1

(3.17)

The B lifetime requires A = 1.1 4- 0.2 and the upper limit on U(bu) translates into

pa + ,/2 < 0.5. (3.18)

60 LL Btgt, A.L Sanda / CP otolation

Insisting on e K being reproduced in the KM ansatz yields, for m t~ 40-60 GeV,

'1 - 0 . 5 ,

1Ol >- 0.5. (3.19)

This information is obviously not sufficient to lead to precise predictions of I m ( p / q ) o t . Yet typical numbers can be obtained when using *! = ½, p = ½ [0] - ½:

b ~ c~s, b --, cEd b ~ u~d

imPp,[ 2./(1 - p) _1[_4] -} _ 2.1[ p - ( p2 + */2)1 q Iaa (i--p)2+*/2 [(1--p)2+*/2](p2+*/2)

20,1 ImPpf O(10-2) p2 + ~12 --I [0] 1

q IB~

-0151

These numbers represent a clear tendency for large CP asymmetries in B d and in those B s decays that are suppressed by the small mixing angle U(bu). Assembling the various elements one finds for the asymmetry defined in eq. (2.10)

o [~d(t)B- ~ f0f-] _ o[Bd(t)B+ ~ fOr+]

AdL(t) = O [-Bd(t)B- ~ fof-] + o [Bd(t)B + ~ fof+]

P = Im--pfsin Amdt , (3.20)

q

which upon integration over all decay times t gives

+xd <A~L>t-- 1 x----~ Im 0 , - 0 . 0 2 - 0 . 2 . (3.21)

Analogously, one finds for (bg) --* (u~d~) transitions

xs P <A~L)t-- 1 + x ~ I m q p~ ~ 0.1-0.5. (3.22)

The smallness of I m ( p / q ) p f for bg--* b~cg is easily understood" on the leading level only the second and third family contribute; thus, on this level, there cannot be a CP asymmetry.

(ii) f is not a CP eigenstate. There are final states common to both B ° and ~o decays which are not CP eigenstates; e.g., [11, 24]

(B)d ~ D-'r+, D+cr-,

(~) ~ F + K - , F - K + .

LI Btgt, A L S a n d a / CP vwlatzon

a

d

d

a

( - ) + + Fig. 4. Decay diagrams for Ba ~ D ~r - , D - ~r

61

The diagrams producing ~B)d ~ D ±~r ~: are shown in fig. 4. From it, one reads off two features that hold in general for such transitions:

(a) Two suppressed KM angles are involved, namely U(bu) together with U(dc) or U(su).

(b) The topology of the relevant quark diagrams is not the same for B ° and ~0 decays. For example, one sees in fig. 4 that the D + is produced in a different way in B d and Bd decays. We do not know to evaluate the relevant matrix elements from first principles; employing a factorization ansatz, one finds

A(Bd - , D+~ -- )

P'= A(B. -~ D+~ -)

- ( 1 - 2 ) ~ 2 ( p - i 7 / ) ,

U(bu) U(dc) fD m4

U(bc) fir (m 2 _ m 2)2

(3.23)

where our ignorance concerning the correct value for fD/f~ is the source of the uncertainty listed above. In addition, there are further uncertainties concerning the validity of the factorization ansatz and the importance of color mixed operators [25]. In any case the ratio pe is not a unit vector in the complex plane nor is (p/q)pf, when f is not a CP eigenstate, and considerable uncertainties surface.

The analogous procedure yields for Bs decays

A ~ s - , F+K -) V(bu)U(sc) fF m' Bs Of= A(Bs-~ F+K -) -- U(bc)U(su) fK (m~ --m2) 2 1 .6 (p - i* / ) . (3.24)

62 I.L Btgz, A.L Sanda / CP molatzon

These estimates for the ratio of matrix elements are entered into the expressions for CP asymmetries in eqs. (2.10), (2.13):

a [ B d ( t ) B + --> ( D + r t - ) f + ] - o [ B d ( t ) B - --) ( D + ~r+)f - ]

AdNL(t) = O [Bd(t)B + =-) (D+rr - ) f +] + o [Bd( t )B- --> (D+vt+)f - ]

2sin Amt )t2~l [ 2 0 ( 1 _ p) _ ( l _ p)2 +,)2] - 1 + c o s A m t ( 1 - p)2+~/2

sin A mt 1 + cos Amt {0.05, --0.03, --0.05} (3.25)

for = p = { ½ , o , -

a [Bd(t)B + -+ (D-~r+)f + ] - a [Bd(t)B---> (D +rr - ) f - ]

A~I ' ( t ) = e [Bd(t)B + -+ (D-er+)f + ] + a [Bd( t )B- -0 (D+rr - ) f -]

2 ) l [ p - (p2 + ~12)1 = - s i n Amt [(1 -- p)2 + ~2] (p2 + ~12)

- - s i n A m t { 0 , { , 4 } , (3.26)

e [Bs(t)B + ---, ( F + K - ) f +] - o [Bs( t )B- --* ( F - K + ) f - ]

ANL(t) = o[Bs( t )B + ---) ( F + K - ) f +] + o [ B s ( t ) B - - - - ) ( F - K +) f - ]

- - 1.6)/sin Zimt - sin Zlmt. (3 .27)

Integrating over the decay times one finally obtains the guestimates shown in table 1.

(iii) f containing a neutral D decay. Those final states f that contain the decay products from (2)0 mesons present a further subtlety: their classification of being a CP eigenstate or not depends on the decay mode of the neutral D meson. For example, the decay

B d ~ D % °

at first sight does not lead to a CP eigenstate. Yet a difference between D O and 2 ° has to be established via its decays. Therefore the decay chain

Bd--, D % °

~ Ks + rr's (3 .28)

LI Btgt, A.L S a n d a / CP molatton 63

leads to a CP eigenstate and is thus treated in complete analogy with subsect. 3.3. (i). The process

B d -o D% 0 /

K - + *r's (3.29)

on the other hand is treated like the modes discussed in subsect. 3.3.(ii).

3.4 CP VIOLATION IN D O DECAYS

The standard model does not predict CP violation in D decays on a level that could be observed. Nevertheless one should search for them in a dedicated fashion.

As has been stated in the literature [3,4], the standard model cannot generate D ° - D ° mixing with a strength exceeding 0.1% appreciably. On the other hand, new physics in the form of genuine flavor changing neutral currents could produce D ° - D ° mixing on the 1% level. The impact of such new physics on CP asymmetries would be two-fold:

(i) There could be a CP asymmetry in semileptonic decays as measured by F(D ° --* d-X) versus / ' (D O --* d+X); our present knowledge allows for a signal on the 1% level.

(ii) Decays of (~)o into CP eigenstates f like Kszr°,Ksp °, etc., offer the best chance for finding CP violation in nonleptonic decays. One should keep in mind that mixing observables like the rate for D°D °---, d-+d-++ X depend on x 2 + y 2, x = A m / F , y = A F / 2 F , whereas the CP asymmetry F(D ° --* f) - F(D ° --* f) is proportional just to x! Thus mixing if it is observed on the 1% level can lead to CP

asymmetries that are also on the 1% level.

4. Final state interactions and CP violation

As explained in sect. 1, CP violation can appear also in the absence of mixing. Other mechanisms have to be invoked then to supply the required two coherent amplitudes. This can happen due to an interplay between two different cascade processes, between quark decay and weak annihilation or due to penguin operators.

41 INTERPLAY BETWEEN DIFFERENT CASCADES

The (Cabibbo disfavored) decay B----, K s K + --- different cascades:

can be produced by two

B---* D°(K - or Ks) + . . .

L K s + . - - ,

B - - -*D°(K - or Ks) + - . .

L K s + . . . . (4.1)

64 I L Bigz, A.L S a n d a / CP vtolatton

From the quark diagrams given in fig. 1, one reads off

A ( B - ~ KsKX,Yj) tx U(bc)V*(su)U*(sc)U(du)A 1

with

+ U(bu) U* (sc) U(sc)V(du)A 2 , (4.2)

AI = (D°KX,l( ?.b )v_s( gU)v_~dB-)(K~Yjl( gC)v_a( ~d )v_AD°),

A= = (-D°KX,l( ~b )v_A ( & )v_~B-><KsY~l( ~s )v_ , ( dU)v_dT)°> .

Therefore

i U(bu)12U(sc)l 2 r(B ':~'--, KsT.'X,Y~)¢c }All 2 1 + [ U(bc)[2[ U(su)lZlg[ 2

V(bu) V(sc) + 2 Re U(bc) U*(su) Re

U(bu) U(sc) ) ( -T- )2 Im U(bc) U* (su) Im t5 , (4.3)

with ff = Az /A 1. Employing again the Wolfenstein parametrization, one gets

F(B ~:r) ~ KstK, JXY)~x [All2{1 + (0 2 + ~i2)D~I 2 + 2p Re~ T- 2~ Im ~}

_< IAll2{1 + ½1~l/+ Reff -Y- I m p } ,

F(B + ~ KsKX,Yj ) - F ( B - ~ KsKX,Yj) Im~

r ( n + ~ KsKX,Yj) + F ( B - ~ KsKX,Yj) -< 1 + ½1~12 + Re~" (4.4)

Final state interactions have to be evoked to generate a nonvanishing imaginary part in the ratio of matrix dements. While we do not know how to estimate Im ff in a reliable way, there exists no reason why it should be particularly tiny. A 10% asymmetry in F ( B + o KsKX,Yj) versus F ( B - ~ KsKX,Yj) is an optimistic, but not unreasonable guestimate.

Eq. (4.4) was written down for a single exclusive decay mode. Each individual mode will however command only a small overall branching ratio. This raises again the question whether statistics can be gained via summing over different exclusive modes. An analysis analogous to the one given in sect. 2 shows that adding excluswe decays in an indiscriminate way will very likely wash out any asymmetry; on the

LL Blgt, A.L Sanda/ CP molatwn

other hand it is a good idea to analyze

65

l , J

where X,, Yj are any member of the pseudoscalar, vector or axial-vector nonet.

4.2. INTERPLAY BETWEEN QUARK DECAY AND WEAK ANNIHILATION

The B decays will receive contributions from quark decay a n d - to a certain degree - from weak annihilation (WA). They can contribute coherently to modes

such as

B u ~ D°*D - , (4.5)

for which the quark diagrams are shown in fig. 2. The KM angles involved certainly do contain the CP violating phase. The question one then has to address concerns the size of the WA contribution.

Recent CLEO data yield for the lifetime ratio of charged and neutral B mesons

[261

-+) 0.5 < ~.(B0---- ~ < 2, (4.6)

i.e. a substantially smaller ratio than that for D mesons. This agrees with the general theoretical expectation that WA is much less significant for B than for D decays. Nevertheless, it is possible (though not assured) that WA contributes significantly to certain special channels such as the one in eq. (4.5), even considering the fact that it requires c6 excitation. Again, the relevant final state interactions are not known. There are two alternative approaches for obtaining a guestimate which use com- pletely different starting points [8]: firstly, one can argue that a rather wide axial resonance with a mass M A - 4.2 GeV affects B u -~ D * D in a significant way; the diametrically opposed view is to rely on asymptotic form factors for W --* D * D as suggested by perturbative QCD. Such arguments sketch a scenario where

F ( B - ~ D°*D - ) - r ( B ÷ - , D ° * D +)

r(B- -* D°*D - ) + r ( B ÷ - . D ° * D +) ~ O(1%) (4.7)

might hold.

4.3 CP VIOLATION AND PENGUIN OPERATORS

Allowing for penguin operators, as shown in fig. 3, opens up more possibilities. They can be calculated with some degree of confidence, since they are short distance operators [27], and they possess both a real and an imaginary part. A relevant

66

channel is

One actually obtains large includes spectator diagram

LL Btgl, A L Sanda / CP wolatton

B u ~ K - p °. (4.8)

CP asymmetries from a penguin calculation which

r ( B - ~ K-p °) - r (B + - . K + po)

r ( B - ~ K-p °) + r(B+ -~ K ÷ po) O(10%). (4.9)

The drawback is that the branching ratios for such modes are expected to be very small as discussed in more detail in sect. 5.

4.4. SUMMARY OF CP VIOLATION IN CHARGED B DECAYS

(i) Very sizeable asymmetries of order 10% could show up in certain decay modes while 1% effects can emerge in others.

(ii) The branching ratios for the relevant exclusive modes are expected to be small, i.e. not exceeding 0.1%. Statistics can be gained by summing over exclusive modes, but this had to be done in a careful and not in an indiscriminate manner.

(iii) Estimates on the expected size of the effects are severly affected by our ignorance of the relevant final state interactions. This implies in turn that nil results cannot be interpreted in an unambiguous fashion.

5. Search strategies

As explained in sect. 3, we can make rather reliable predictions on the size of CP

asymmetries such as [F(B ° ---, f) - F(B ° ---, f ) ] / [ / ' (B ° ~ f) + / . (~0 ___, f)], in particular when f is a C P eigenstate. Yet one needs the branching ratio for the relevant modes to evaluate the prospects for doing such a measurement.

5.1. BRANCHING RATIOS FOR EXCLUSIVE MODES 5.1.1. f is a CP eigenstate. (ct) Hidden charm. As explained before, the decay

B ° ~ f fK s provides a clean lab to search for CP asymmetries. While the inclusive reaction B ~ ~ + X has recently been established with a branching ratio of - 1%, no signal has been seen yet for B ~ ~pK. There exists a direct upper limit [28]

Bl~(Bd-~ ~ o ) <0.9%. (5.1)

Yet with a modicum of theory a much more stringent limit can be obtained. From

LL Blgt, A.L Sanda / CP molatton

BR(B -+ ~ ~kK ~) < 0.07% [28] one infers BR(B d ---> ~kK °) < 0.07% and thus

BR(B d ~ ~kKs) < 3.5 × 10-4.

67

(5.2)

The equally useful channel B°--->~Ks~r ° suffers from a similar suppression: BR(B ° ~ f fK *°) ~< 0.5% [28] implies

BR(B d ---> ~kKsvr °) ~< 10 -3 . (5.3)

Theoretically one expects the branching ratios to be close to these upper limits.

Nothing is known, of course, about the corresponding B s decays; however, one expects similar branching ratios of order 10-3.

( f l ) Open charm. Experimentally nothing is known about B---> DD + X. Thus we can offer only guestimates:

BR(B d --) DDKs) - 0.1%-2%,

BR(B s --) F + F - ) - 3%,

BR(B d ---) D+D -) - 0.5%.

(5.4)

(5.5)

(5.6)

The last mode is formally Cabibbo-suppressed, yet form factor effects could enhance it over naive expectations as it happened for the analogous mode D ° ~ K + K -

(T) Charm cascade. Cascade decays like

B 0 __> ( ~ ) o + ~r's

) K s + vr's (5.7)

lead to CP eigenstates. A typical combined branching ratio is (we give a more complete list later on):

(_)0 ) BR B d ~ D pO~(Kso~)p0 _ 6 × 1 0 -4 . (5.8)

5.1.2. f is not a CP eigenstate. One of the few known branching ratios is

BR(B d --) D+~r - ) - 0.5%. (5.9)

Hence, one estimates

BR(-B d --) D+vr - ) _< 0 (10-5 ) . (5.10)

68 LL Btg,, A.L Sanda / CP vwlatwn

Analogously, one predicts

BR(B s ~ F + K - ) , BR(B s --, F + K -) - O ( 1 0 - 3 ) , (5.11)

BR(B s ~ D ° K s ) - O(10-3) , (5.12)

BR(B s - , D°~) - fewlO -3 , (5.13)

BR(B d "--, D % °) - (1 - 3) × 10 -3 , (5.14)

BR(B d --, D ° K s ) - 10 -4, (5.15)

BR(B± --~ D ±(~)°*)_ 10-3, (5.16)

BR(B ± - , K ±p°, K ±~r °) < 10 -5" (5.17)

5.2. INCLUSIVE SEARCHES

The guestimates given above lead to one general conclusion: those decay modes that should exhibit CP asymmetries of order 10% possess branching ratios of typically at most 10-3. There are modes with somewhat larger branching ratios - yet there the expected asymmetry does not exceed 1% in an appreciable fashion. Since searches for such effects pose very difficult problems, one would like to increase statistics by studying inclusive processes.

It has already been stated that an indiscriminate summation over exclusive modes - such as B ~ ~ + X - will wash out an asymmetry, since its sign will be different for different channels.

If the final state f is a CP eigenstate, then the sign of the asymmetry will depend on the CP parity of f. Therefore we give not only the expected branching ratios for the relevant modes in table 4, but also their CP parities.

This table shows what was said before. Although the overall branching ratios for the relevant exclusive channels do not exceed 10 -3 (apart from some exceptions), there are a large number of such channels. Thus one can entertain the idea of summing the contributions from different modes; this has to be done with care, keeping track of the CP parities of the final states. In particular, one should not sum over modes without discriminating K s against K L.

LL Btgt, A.L Sanda / CP molatlon

TABLE 4

69

E x p e c t e d

M o d e b r a n c h i n g r a t to CP p a n t y

B D ""* tkK s few x 1 0 - 4 -

.--, I k K s , r ° O(10 - 3 ) +

---* D D few )<10 - 3 +

---, D D K s 1 0 - 3 - 2 X 10 - 2 + (mos t ly)

B~ ~ ak~ few )< 10 - 4 +

tbq5 O(10 - 3 ) + (mos t ly)

---, F + F - - 3% +

B d ~ D % r ° ~ f D'/'r0 -- 1% )< B R ( D ° --, f D ) _ ( _ 1)CPrD

_.. DOpO ~ fDp0 -- 0 ( 1 % ) )< B R ( D 0 -'* f D ) _ ( _ 1)CPfD

M e a s u r e d

M o d e b r a n c h i n g r a t io CP p a n t y

D O -- ' K s p ° 0.7% -

-'-* Ks~0 2% -

"--' Ksq~ 0 5% -

--" Ks*r ° 1.1% -

---' KsT1 0 9% - - " p~ " - 1% -

K + K - 0.6% +

--.* pp - 1 5% +

Very similar considerations have to be applied when f is not a CP eigenstate. The CP asymmetry expected for (B)d--* D ±,r ~: changes sign when (B)d--' D ±,r ~:~r ° is considered. Therefore, a blind summation over final states is likely to destroy a signal.

6. Building the data base

Considering how little we know for certain about B decays, it is amazing that we can make all these predictions discussed before. It would be clearly preposterous to treat them as more than guide lines for further studies. Obtaining more experimen- tal information on heavy flavor physics in the future will allow us to refine our predictions in due course. In the following we will list the kind of information needed to achieve such a refinement.

6.1. B R A N C H I N G A N D L I F E T I M E R A T I O S

A few exclusive B decays have been seen, so in the future one can hope to identify many more. Of particular interest are, as explained before,

B d ---> 6 K s , ~kKser °, 6Ks,r+~--, DDKs , DD, D ' D * , D%°, D°p ° . (6.1)

70 LL Btgt, A L Sanda / CP molatton

No B~ decay has been identified. In the present context it would be of most direct interest to find

B s -4 ~,/, if,/,, ~ , F+F - , Do~. (6.2)

There are other decay modes that can shed light on the underlying decay mechanism and the impact of final state interactions:

(i) Observing B -4 Kit would shed light on the significance of penguin operators [271.

(ii) Comparing B°-4 D%°,D°o° with B°-4 D+cr- ,D+o - will tell us something about the impact of final state interactions.

(iii) Most authors argue that B :~, B d and Bs lifetimes should differ very tittle from each other, since weak annihilation reactions are expected to be relatively small (unlike in the D case). It would be very useful to test this either by measuring ~(B 5) versus z(Bd) versus z(B~) directly or via BR(B ± -4 t~+ X) versus BR(B d -4 ~+ X) etc., or via BR(B ± -4 ~ K ±) versus BR(B d -4 ~ o ) or BR(B ± -4 ~kK ± *) versus BR(B ° -4 ,/,K°*).

6,2. B°-B ° MIXING

Our predictions on the strength of B°-B ° mixing can be sharpened considerably once the top quark mass is known to within a few GeV. It would yield a decent prediction for Am(Bs); to do the same for Am(Bd) requires a determination of U(bu).

Of c o u r s e - and preferably- the strength of B°-B ° mLxing can be measured, either via like-sign dileptons or the forward-backward asymmetry of bottom jets in e+e-annihilat ion, or via searching for B°B ° -4 DD + X, DF + X, FF + X.

6 3. PRODUCTION CROSS SECTIONS FOR HEAVY FLAVORS

The cleanest environment is provided by e+e - annihilation on or off the Z ° resonance. Besides direct production of charm and bottom, one should study also the option of indirect production; in particular, top or even toponium decays could serve as B factories.

Hadronic collisions offer much higher yields. Then the question arises whether and how these large yields can harnessed to study delicate phenomena like B°-B °, D ° - D ° mixing and C P violation in these systems. Much more experimental as well as theoretical work is needed before a meaningful answer can be given.

Photoproduction of heavy flavors represents a situation between the two extremes sketched above and holds a great promise for the future.

We gratefully acknowledge useful discussions with J. Bjorken.

LL Btgl, A.I Sanda / CP vwlanon

Note added in proof

T h e r e c e n t p a p e r b y D u n i e t z a n d R o s n e r [29] will be of in te res t to the reader .

71

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