Branching ratios and CP-violating asymmetries in exclusive charged B-meson decays

Physics Letters B 272 (1991) 395-403 North-Holland

PHYSICS LETTERS 13

Branching ratios and CP-violating asymmetries in exclusive charged B-meson decays Hubert Simma Institut J~r Mittelenergiephysik, ETH HOnggerberg, CH-8093 Zurich, Switzerland

and Daniel Wyler lnstitut J~r TheoretischePhysik, UniversiRit Ziirich, Sch6nberggasse 9, CH-8001 Zurich, Switzerland

Received 16 September 1991

Using perturbative methods, branching ratios and CP-violating rate asymmetries for some rare hadronic B meson decays are calculated. The decay of the heavy meson is modeled by the decay of the heavy quark and the exchange of a (fast) gluon with the spectator; the necessary phases are generated by perturbative (penguin) diagrams. The values of the rate asymmetries are of the order of 1.5% for B- --.K - x° and 5% for B- ~ K - K°. The effects of short distance QCD corrections are investigated. The analysis yields improved values for BR(B- ~xn, xK, KK, riD, DD).

1. The B-meson system offers m a n y possibilities to investigate CP violation [ 1 ]. Whereas some effects in neutral B-meson decays are predicted reliably, Q C D effects complicate enormously calculations o f rate asymmetries in charged B decays, in particular those with hadronic final channels. A m o n g the most p r o m i s i n g channels are the decays o f B into charmless hadrons. On the quark level, these c o r r e s p o n d to decays like b--,su~, where tree level and penguin diagrams contribute with similar strength, and to b ~ s d a , b ~ d s g , etc. which are pure penguin decays. It is well known that a non-zero a s y m m e t r y requires two a m p l i t u d e s with different C K M phases and different absorptive parts. On the p e r t u r b a t i v e quark level, these are p r o v i d e d by tree level diagrams and by penguin d i a g r a m s with internal u and c quarks which are p r o p o r t i o n a l to vu= VubV*s and vc= VcbVc*s and which are complex for gluon m o m e n t a q2 >> 4 m 2, 4 m 2. Since the tree level diagram yields a real a m p l i t u d e and is prop o r t i o n a l to vu, the interference can produce an a s y m m e t r y [2]. Also higher-order corrections (e.g. p e n g u i n penguin interference) have been included [ 2 - 4 ]. Clearly, the a s y m m e t r y d e p e n d s strongly on q2; for b--.su0 it is large above, but tiny below 4 m2; in pure penguin m o d e s the situation is reversed. Unfortunately, q2 is not measurable in decays into physical particles; rather an average is taken, weighted with the wavefunctions o f the quarks in the mesons. So far, all exclusive estimates have used the quark m o d e l result and assigned, ad hoc, a " r e l e v a n t " q2 where to evaluate the asymmetry; the results d e p e n d crucially on this value. In this p a p e r we a t t e m p t to i m p r o v e these unsatisfactory estimates by calculating the q2 distribution o f some exclusive decays a n d then fold it with the m o m e n t u m d e p e n d e n c e o f the loop amplitudes. To make this possible at all, some simplified m o d e l for mesons must be used. One o f the p r o b l e m s in heavy particle decays into few light mesons is that the " s p e c t a t o r " quark which does not participate in the quark decay itself must be "accelerated". In p e r t u r b a t i v e Q C D , this can be achieved by an extra gluon coming from the other quarks [ 5 ]. This modifies significantly the quark m o m e n t a c o m p a r e d to the free quark decay a n d m a y be an adequate represen0370-2693/91/$ 03.50 © 1991 Elsevier Science Publishers B.V. All rights reserved.

395

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tation of the true momenta. Therefore we have adopted this picture. Recently, Szczepaniak et al. [6] have implemented it into heavy meson decays. In our present analysis, we extend their work to penguin graphs and to certain non-factorized graphs and apply it to CP asymmetries. We refer the reader to refs. [ 3,4 ] for detailed calculations of quark-level asymmetries and to refs. [ 5,6 ] for a more detailed discussion of treating the extra gluon. Since the rates in ref. [ 6 ] turned out to be rather small, we have also used this model to calculate semileptonic decay rates in order to understand its branching ratios. This will be presented only briefly and we hope to return to it in a future note. 2. In our picture, the mesons are described as a collinear pair of quark and antiquark with longitudinal momentum fractions x and 1 - x , respectively. The hadronic process is written as a convolution of the distribution amplitudes for the quark-antiquark pair inside the meson and the perturbatively calculated amplitude of the corresponding quark process (including the exchange of the extra gluon). For the B and light mesons, respectively, we use the distribution amplitudes SuB(PB, x) =tin(x) (~'B +MB)?5 ®~co,or,

5UL(PL,Y) = ~L(Y)¢L75 @'D¢o,or•

( 1)

This structure of ~UBis a special case of ref. [ 6 ]. It arises from demanding strictly onshell quarks and gauge invariance. The m o m e n t u m dependence is parametrized as in ref. [6 ], ~B(X) = fB

~p(X)

4Nf~o(x) dx'

X2(1--X) 2

~0(x)=

[e2x+(l_x)212,

fL

(bL(y)=~6y(1--y),

(2)

wherefB, fL are the decay constants of the mesons, N = 3 is the number of colors and e is related to the average transversal momentum of the light quark and therefore small; we use 0.05 << e << 0.1. The weak transition originates from tree-level four-quark operators and penguin graphs. We work within the framework of an effective hamiltonian for energies well below Mw and mt [ 7 ]. In this approach, leading QCD short distance corrections are incorporated in a reliable fashion. The coefficients ci of the effective operators Oi and their matrix elements depend on the renormalization scale/t. With a suitable choice of/l, the matrix elements are free of large logarithms; the corresponding values of the coefficients are determined by the renormalization group equation. The starting value for their evolution is obtained by equating the exact theory to the respective matrix element of the effective hamiltonian at a suitable scale, here/1 ~ Mw. Since the matrix elements of the effective theory are divergent, they must be renormalized; in turn the starting values of the coefficients are prescription dependent. We assume that it is sufficient to consider three effective operators, defined as [L = ½ ( 1 -~5) ]

01 = (gyuLb) (aTULu), 02 = (aT, Lb) ([7"Lu) and Op=

([~)uL'½j.ab) ~. (~Tu'½2aq) . q

The matching condition with cl ( M w ) = 0 and c2(Mw) = 1 yields (MS subtraction) (mi2 "~ F~ m2 2 mi2 "~ ( m R "] (~ww)+~( l+ln 2--Fl

cp(Mw)=Fl\~5~w ] -

wwj

where i = u or c for the contributions proportional to v~ or v~, respectively, and F~ is the well known Inami-Lim function F~(x) = - 2 l n x +

x2(15-16x+4x2) l n x + x(18-11x-x2) 6(l-x) 4

12(l-x) 3

In this picture the "penguin" diagram has two parts: an effective local four-quark operator, governed by Cp and 396

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a m o m e n t u m d e p e n d e n t piece [8] which can be viewed as the matrix element o f the operators O1, 2 (governed by Cl,2). This second term will give a b s o r p t i v e parts. As in refs. [3,4], we will calculate the a s y m m e t r y to order a~ - apart from a d d i t i o n a l powers o f the strong coupling due to the extra gluon in all amplitudes. To distinguish them, we will denote the latter ones by &s. The simplest situation is d e p i c t e d in fig. 1 where the usual four-fermion interaction causes the decay a n d the extra gluon provides the m o m e n t u m for "accelerating" the spectator. The gluon can emanate from any o f the fermions entering the weak vertex; fig. 1 illustrates this for the b line. The d i a g r a m s o f fig. 1a for B - - - , K - n ° yield

(3)

TD = --Cl ( Wb-}- WsJc Wu)CF--Cz WbNCF ,

where c, = - 0.2 5, c2 = 1.1 are the Q CD corrected coefficients at # ~ m b [ 7 ] and CF = ½( N 2 _ 1 ) is a color factor. The various w~ contain the meson wavefunctions and the Dirac trace, corresponding to the d i a g r a m where the gluon is e m i t t e d from the fermion line i, a n d are given by Xr[

-/~2 + M.)7~

~=7#L~JKT~'L(xJ~.

Wb =

~.7"1

k2[ (XPB -- k 2 ) 2 - M ~ ]

'

Ws =

T r [ ~y#Lt[YKya(y?K +/~2)y'L7%7~l k~(yPK + k 2 ) 2 '

Wa

Tr{ ~ . y u L [ ( y - 1)~'K --/~2 ] 7. t/JK}'#L t/'t. Yc¢} k 2 [ ( y _ 1)P n -ka] 2 '

=

(4)

where

k2 : ( 1 --z)P~-- ( 1 --x)PB

=

( x - z ) P , ~ - ( 1 --X)PK

is the m o m e n t u m o f the gluon (see also fig. 2 for the definition o f the kinematics). Neglecting the masses o f the outgoing mesons, the c o n t r i b u t i o n from a gluon e m i t t e d from the u quark vanishes in the F e y n m a n - ' t Hooft gauge. Since CB is strongly p e a k e d at x = 1 - e, we keep only the leading order in 1 - x (which is also required if gauge invariance is to hold strictly in our m o d e l ) . This yields for instance 4z-8 Wb ~'~ ( 1 - - X ) ( 1 - x ) 2 C B ( X ) O K ( Y ) O K ( Z )

"

x% b

b

B

PB

B uu

~_" ~ T r

a

o

uu

_u " ~ K

b

-

Fig. 1. Decay of a B meson with gluon corrections to the quark decay. The box stands for the effective four-fermion interactions O~ or 02. Crosses ( X ) indicate all possible lines where the extra gluon can be attached.

( l - X ) PB

Fig. 2. Definition of the various parton momenta. 397

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In exclusive decays where the antiquark from the b decay has the same flavor as the spectator, there is a second class of"exchanged" diagrams as shown in fig. lb TE = - - ¢ I wbNCF--¢2(Wb'k-Ws"[-Wu)CF.

(5)

Annihilation topologies do not contribute to the leading order in 1 - x and are therefore neglected. In the present formalism, this reflects the commonly adopted suppression of annihilation diagrams. Next we consider the matrix elements for the dispersive (real) penguin amplitude. The local and the momentum dependent contributions are summarized in figs. 3a and 3b, respectively. For an internal quark i = u, c we have

[

Wg+Re w~i-

+c2 Re3F~ ~ {

]

(CE-CD)+c2RewaI+ (CE+CD)+Cl Re wal+ Cv,

(6)

where (N2-1) 2 cD-

4N

N2_l ,

CE=----

4N

The matrix element of O2 (with MS subtraction)

JF,(~?)=AFI(~n.2,)-3(l+Inm~) q2

q2

-~-j

(7)

differs from the momentum dependent part of the bsg vertex [ 8 ] 1

q2

JF'(m2,)---4f

q2

ln(1-

u(1-u)

"7"2Umi ( 1 - u ) ) d u

0

by the additional renormalization dependent term. At # ~ mb, leading log short distance QCD corrections yield for the coefficient of the local part [ 9 ] cp(mb) ~ 0.6 [cp(Mw) + 5 ] . In fig. 3b we must also include gluon emission from the internal lines; otherwise the results are not gauge invariant. In particular the second term in the bsg vertex/'u = F~ (q2yu- ~q~,)L of the full theory and in the matrix element of O2 would not vanish by current conservation. These contributions are given in terms of wg=

Tr(~[Jny'u~[JKyaLItIB~) u ) 2 2

klk2

Vai, v

'

w~1_+

- T r ( ~Jnyu~'tKAI~u ~tB y v )

=

2 2 klk2

,

(8)

where the three-gluon vertex V,~u, and the momentum dependent part AI#, of the color-symmetric/antisym-

a

398

b

c

Fig. 3. Penguin amplitude for B - ~ K - n °. In (a) the local part is given; the blob stands for the effective penguin operator Op. (b) and (c) show the momentum dependent piece (see text and ref. [ 8 ] ). The crosses denote again all possible attachments of the hard gluon.

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metric one-particle-irreducible bsgg vertex ~1 can be extracted from ref. [8]. For the local diagrams of fig. 3a these contributions cancel (see ref. [ 8 ] for details). In spite of the different chiral structure of the penguin diagrams (compared to the tree diagram), the weight functions Wb, Ws and w~ are the same as in eq. (4) since the right-handed part of OF does not contribute here. The "exchange" diagrams of fig. 3c can only contribute to decays like B - ~ K - q ' where the neutral particle is an SU ( 3 ) singlet. Thirdly, we have the absorptive part of the O ( as )-penguin diagram which arises from cutting the loop in the graphs of fgs. 3b, 3c. The expression for Im Pi is simply given by taking the imaginary instead of the real part everywhere in eq. (6). In those diagrams where the gluon is emitted from the loop, there are two possibilities to cut; before and after the emission of the gluon, corresponding to the thresholds in the two variables q2=

(xPB -yPK) 2= x ( x - y ) m 2 and k~ = [( 1 -Y)PK +zP~]Z=z( 1- y ) M 2 .

(9)

Because the asymmetry depends crucially on the position o f the thresholds, we always use the exact x dependence of eq. (9) for the imaginary parts of AFt and AI~,~, whereas we can safely neglect the offshellness of the extra gluon, since k~ = O ( 1 - x ) <0. From the distribution amplitudes ofeq. (2), we expect the average values ( q 2 ) - ~ 1: M B2 ( 1 - - 3 e )

and

(10)

( k 2 ) = IaMB2.

Finally, to order o~2 we need those penguin diagrams (with internal c and t quarks), where a vacuum polarization on the gluon line is cut through the inner quarks j --u, d, s, c. For the diagrams shown in fig. 4 we find \~-/

F

Im P~(j) =

1\m2] ]

\mZ]

_ q2 "~ -[¢p+c2ReAFI(~2c)]IImAFI(~.~) [WuCE+Wg(CE--CD) ]+Imw31 ( C E - - C D ) ]

-c2 I m A F , ( ~ { I R e A F , ( ~ z ) w g + R e w 3 , \mj /

] (CE--CD) + R e WAI + ( CE -~-C D ) }

(k2

- c l Im

AFI \m2. ] Rew~z+ Cv.

(11)

Here, Im w3l- arises from the v a c u u m polarization and therefore the internal quark is given by j and not by the quarks of the penguin loop (as in all other cases). In the case of an internal s quark, there are additional diagrams which arise when it is interchanged across the cut with the external s quark. Due to color suppression, these identical particle effects are unimportant numerically in the inclusive calculation [ 4 ], we will therefore neglect them here too. In the above formulas for the amplitudes [eqs. (3), (5), (6) and ( 1 1 ) ] an integration f~ dxf~ dyf~ -~ dz is implicitly assumed. To order a 2 the CP-violating asymmetry for B - ~ K - n ° is given by L

~ Additional terms to Al-, arising from the renormalization of the matrix elements, cancel against those in AFt.

a

b

Fig. 4. Absorptive part of the vacuum polarization corrected penguin graphs. (a) Local part, (b) momentum dependent pieces. The dashed line corresponds to the cut. 399

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a ( B - ~ K - n o ) = F(B ----,K- n °) _ F ( B + ~ K + = o) F(B - - ~ K - n °) + F ( B + ~ K + n °) = 2 I m ( v . v * ) TImPc+RePulmPc+TEq=u,a,s,clmP,~(q)-ImPuRePc I Vu 12 I T+Pul2+lvc 12 IPc 12+2 Re(vuv*)Re[ (T+Pu)P* ]

(12)

and the rate becomes F(B-__.K-rco)

2

5

i -4 GvMB 2n Ivu(TWP")+vcPcl2"

= ~gs

(13)

The factor ~ is due to the presence of the no. For other decays like B - , K K , etc. the tree contributions T = To+ TE may be absent and the CKM factors have to be changed appropriately. Note that eq. (13) is very sensitive to the various parameters and typically yields rates which are one to two orders of magnitude smaller than those of ref. [ 10] (see also ref. [6] ). Therefore we will compare the rates with (MB--MD) 2

F(B__,Deg) = i Vcb 12 24~z G2 3

f

FL(q2)IPD(M2B, M 2, q2)]3 dq2,

(14)

0

where FL is defined in ref. [ 6 ]. 3. We have calculated branching ratios and rate asymmetries, within the single gluon exchange model [ 5,6 ], for some exclusive B-meson decays where penguin diagrams can contribute. The rates ofeq. (13) depend strongly on the parameters of the model, such as gs and ~. Dividing by the total (free quark) decay rate Fro, the resulting branching ratios are considerably smaller than predicted by other calculations [ 10 ]. Since this might be a problem of absolute normalization (which does not affect momentum distributions, etc. ), we have also calculated the exclusive semileptonic decay B-,Deg. We then define the branching ratios of the hadronic decays by B R ( B - ~ K - n °) = F(B--*K-~°) F ( B - , D e g ) .1.8%,

etc.,

(15)

using the experimental value of ( 1.8 _+0.8)% for the branching ratio of B ° - ~ D - e + v [ 11 ]. In this way, the parameter dependence practically disappears and the BR's basically agree with the earlier values of 4 × 10 -4 for B - - - , D - D ° and 6 × 10 -6 for B---,Tt-rt ° from ref. [ 10]. At present, we can test the predictions from eq. ( 15 ) only for the BR of the hadronic decay B - - , D ° n - . Our result of 6 × 10- 3 is about twice as large as the experimental value [ 11 ], indicating good agreement within the uncertainties of the model. The asymmetry follows from the absorptive part of the penguin graph, which strongly depends on the momenta q2 or k~ which flow through it. While in the free quark decay q2 is observable, here the momenta are sampled with a weight determined by the meson wavefunctions. The momentum distributions resulting from some of the weight functions are shown in fig. 5. Compared with the distribution for the inclusive free quark decay, we note that in both cases a large range of momenta is important for averaging the absorptive amplitudes. The exclusive two-body decays considered here cover the region with large momentum transfer, whereas the lower values are expected to be favored by final states with higher multiplicities. Our results are summarized in tables 1 and 2. We note the effects of the leading log short distance QCD corrections, which have not been included so far and which affect likewise exclusive and inclusive decays. In comparison to the previous treatment, their main effect is a reduction of the effective penguin coupling relative to that of the tree. In b ~ s transitions with tree and penguin contributions the O ( a s ) penguin-tree interference increases. However the O ( a~ ) interferences, which after the q2 integration cancel the O ( o~s) contribution [ 3,4 ], decrease. In pure penguin modes the QCD corrections simply decrease the branching ratio and the denominator 400

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2.0

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..... ..

0.8

/

0.4

/

°°o.o'

~ ,

\

"'"""

\

\,,

"'-,.

'1

,,\

'i:o

4mc2

q2

k~2 )

2

M~

Ma

2

MB

Fig. 5. Normalized distribution of the gluon m o m e n t a (q2 respectively k~ ) arising from the weight functions wb (solid line ), wa (dashed line), and in the free quark decay (dotted line).

Table 1 The various amplitudes as defined in the text in units of (12MB)-3~2fBf~f~ for e=0.05. The dominant contribution, coming from wb alone, and the values without QCD corrections are shown separately. For comparison, the various interference terms of the inclusive quark level amplitudes as defined in ref. [ 4 ] are also shown in units of 10- 3 G 2 m ~/( 192 n 3). We use a~ = 0.2, Vu= 0.001 exp ( - i 160 ° ), v~=0.045, MB~ rob= 5.2 GeV and rnc= 1.5 GeV. Exclusive decay

IvuI TD

Ivu[ TE

IvuI Re Pu

v¢Re Pc

IvuI Im Pu

vcIm Pc

vcIm P~

B---,K-n ° wb alone without QCD

11.5 12.0 11.8

3.73 1.38 6.08

-0.335 -0.379 - 0.395

- 17.6 - 19.1 - 19.7

-0.205 -0.193 - 0.193

-3.43 -3.41 - 3.02

2.02 2.16 2.27

Inclusive decay

F( TT)

F( PT)

F( PP)

dce(PT)

A~e(PP)

A~e(P,~T)

Ada(P,~T)

14.9 12.3

-0.89 - 0.95

- 1.11 - 1.13

-3.85 - 3.96

b-~sufi without QCD

0.0004 0.0003

0.001 0.001

0.0017 0.002

o f the a s y m m e t r y . T h e v a l u e s " w i t h o u t Q C D c o r r e c t i o n s " c o r r e s p o n d to c~ = 0, c 2 = 1, cp = Cp ( M w ) a n d / t - - M w in eq. ( 7 ) , b u t using the s a m e v a l u e o f a s ( r o b ) . In the e x c l u s i v e decays, the i n t e r m e d i a t e v a l u e s o f the m o m e n t u m t r a n s f e r are w e i g h t e d m o r e strongly t h a n the l o w e r values, in c o n t r a s t to the i n c l u s i v e case. Since the i m a g i n a r y parts o f the p e n g u i n d i a g r a m s increase w i t h i n c r e a s i n g m o m e n t u m transfer, the a b s o r p t i v e phase is larger for exclusive decays, a n d thus the exclusive a s y m m e t r y is larger. T h e e d e p e n d e n c e is e x p l a i n e d by eq. ( 1 0 ) t o g e t h e r w i t h the o b s e r v a t i o n that Wb, w h i c h c o r r e s p o n d s to a m o m e n t u m t r a n s f e r o f q 2, is d o m i n a t i n g . O f s o m e interest are weakly C a b i b b o - s u p p r e s s e d decays like B - - , D - D ° : there, the rate is g o v e r n e d by a tree a m p l i t u d e p r o p o r t i o n a l to VcbVc*o a n d h e n c e is relatively large, w h e r e a s the a s y m m e t r y , arising t h r o u g h the O ( a s ) p e n g u i n - t r e e i n t e r f e r e n c e , is r a t h e r small. H o w e v e r , since the a b s o r p t i v e part is g e n e r a t e d by the i n t e r n a l u~ pair, it d o e s n o t suffer f r o m a k i n e m a t i c a l threshold. T h e r e f o r e the a s y m m e t r y o f a r o u n d 0.5% is practically i n d e p e n d e n t o f the m o m e n t u m t r a n s f e r i n v o l v e d . In o u r calculation, several d i a g r a m s h a v e b e e n neglected. C o n s i d e r first those w h e r e there is only one extra gluon, a n d w h i c h c o r r e s p o n d to d i a g r a m s a l r e a d y discussed in the q u a r k c a l c u l a t i o n [4 ]. T h e r e , c o n t r i b u t i o n s o f a large class o f a b s o r p t i v e parts cancel in the rate a s y m m e t r y , b e c a u s e the " c u t " state (e.g. fig. 4 w i t h an i n n e r u q u a r k ) is e q u a l to the final state; this is so b e c a u s e in b o t h states all q u a r k c o n f i g u r a t i o n s are s u m m e d over. In the p r e s e n t s i t u a t i o n the final state selects o n l y " t w o - m e s o n l i k e " c o n f i g u r a t i o n s a n d thus an a s y m m e t r y m a y r e m a i n . We h a v e n o t i n c l u d e d all o f these c o n t r i b u t i o n s . T h e y are o f o r d e r a~ ( a p a r t f r o m the extra g l u o n ) a n d 401

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Table 2 Asymmetriesand branching ratios. The rates Fare given in units of 3 i V~bIGvMB/2 z 5 192n3,,~Fto~.We usef~= 130 MeV,fK=fD = 160 MeV, fB= 200 MeV and &~=0.38. The values in parentheses are without QCD corrections, as discussed in the text. Inclusive decay

b-*suQ b-,sdd b-,dss b~dce b~dufa, ddd

Exclusive a [%]

0.6 (0.3) 0.5 (0.4) -5.1 (-4.2) 0.4 (0.4) -0.4

decay

B---*K n° B-~Tt-K° B - ~ K - K° B--,D-D° B - ~ n - n°

a [%]

/"

e=0.05

~=0.1

e=0.05

1.6 (0.8) 0.5 (0.5) -5.4 (-4.6) 0.5 (0.4) 0

1.0 (0.3) 0.7 (0.6) -6.7 (-5.7) 0.5 (0.5) 0

(6.9)<10 -7 ) 3.1 )< 10 -7 (3.9)<10 -7 ) 4.8)<10-8 (6.0)<10 -8 ) 1.3)<10 5 (1.3)<10 -5 )

5.2)< 10 -7

1.6)< 10 -6

BR e=0.1 9.5)< 10-8 (1.2)<10 -7 )

8)< 10 -6

(1)<10 5)

5.4)< 10 -8

5)< 10 -6

(6.7)<10 8) 8.3)<10-9 (1.0)<10 -8 ) 2.2)<10-6 (2.4)<106)

(6)<10-6) 7)<10 7 (9)<10 -7 ) 2)<10 4 (2)<10-4) 2)< 10-5

3.0)< 10 -7

thus should not drastically change the larger asymmetries. As a measure of the expected effect, we can use the quantity T I m P~ ~u ) / I m Pu Re Pc. While it is u n i t y in the inclusive quark level calculation by C P T [ 3 ], it ranges between 1.5 a n d 2 in the exclusive case ( d e p e n d i n g on whether the exchange topology TE contributes in T). Secondly, there are new absorptive parts, when we include two "extra" gluons. Using two (different) values for the strong coupling, as for the quark diagram gluons and &s for the extra gluons, these correspond to terms of order &sas, rather than to the terms of order a 2 discussed so far. The new terms will not be treated here; and we hope to return to them in the future. We note, however, that both types of contributions are irrelevant when only penguin modes are considered (B-~KK, etc.). For a discussion of these modes see also ref. [ 12]. A further uncertainty in B - ~ K - n ° with both tree level a n d penguin graphs, comes from a strong rno dependence; for instance mc = 1.7 GeV reduces the asymmetry to 0.5% (for e = 0.05). Also the dependence on as is non-trivial in the case of B - - * K - n ° ( b ~ su~): choosing as = 0.25 instead of 0.2 decreases the asymmetry to 1.1% (0.3%), while for a s = 0 . 1 5 it increases to 2.1% (0.9%). For pure penguin modes and B - ~ D - D ° all these uncertainties are much smaller or absent. Thus, we conclude that their asymmetry is calculated reliably; as to B - ~ K - n °, errors of factors of 2 are still possible. We note that with the recently advocated large value offB a larger imaginary part of Vub is expected [ 13 ] and the asymmetries will increase correspondingly. We would like to thank S.J. Brodsky, J.-M. G6rard, W. Jaus, M. Simonius, A. Szczepaniak and L. Wolfenstein for helpful discussions.

Note added. After finishing this work, we received a paper where next-to-leading order Q C D corrections are evaluated [ 14]. We have estimated their consequences on our predictions by using the coefficient of 0 4 - ( 1 / N ) 0 3 ~ Or, from ref. [ 14] (we neglect the c o n t r i b u t i o n from 0 4 + ( 1I N ) 0 3 , which vanishes a t / z = M w ) . Compared to the leading order, discussed so far, the next-to-leading effects tend to increase the penguin contributions. As a result, the asymmetry for B - - , K - ~ ° can decrease to ~ 1%, while the other modes considered here should be less sensitive. Next-to-leading effects will be studied in more detail by Fleischer. We thank the authors of ref. [ 14 ] for c o m m u n i c a t i n g their results to us.

402

Volume 272, number 3,4

PHYSICS LETTERS B

5 December 1991

References [ 1 ] I. Bigi et al., in: CP-violation, ed. C. Jarlskog (World Scientific, Singapore, 1989). [2] M. Bander, A. Silverman and D. Soni, Phys. Rev. Lett. 43 (1979) 242. [ 3 ] J.-M. G6rard and W.S. Hou, Phys. Rev. Lett. 62 (1989) 855; Phys. Rev. D 43 ( 1991 ) 2909. [4] H. Simma, G. Eilam and D. Wyler, Nucl. Phys. B 352 ( 1991 ) 367. [5 ] G. Lepage and S. Brodsky, Phys. Lett. B 87 (1979 ) 359; Phys. Rev. D 22 (1980) 2157; R. Field et al., Nucl. Phys. 186 B ( 1981 ) 429. [6 ] A. Szczepaniak, E.M. Henley and S.J. Brodsky, Phys. Lett. B 243 (1990) 287. [ 7 ] M. Shifman, A. Vainshtein and V. Zakharov, Nucl. Phys. B 120 ( 1977 ) 316; F. Gilman and M. Wise, Phys. Rev. D 20 (1979) 2392; B. Grinstein, R. Springer and M. Wise, Phys. Lett. B 202 (1988) 138. [8] See e.g.H. Simma and D. Wyler, Nucl. Phys. B 334 (1990) 283. [9] R. Grigjanis et al., Phys. Lett. B 224 (1989) 209; G. Celia et al., Phys. Lett. B 258 ( 1991 ) 212. [ 10] M. Bauer, B. Stech and M. Wirbel, Z. Phys. C 34 (1987) 103. [ 11 ] Particle Data Group, J.J. Hern~indez et al., Review of particle properties, Phys. Lett. B 239 (1990) 1; M.V. Danilov, talk LP-HEP-91 Conf. (Geneva, 1991 ). [ 12] J.M. G6rard and W.S. Hou, Phys. Lett. B 253 ( 1991 ) 478. [ 13 ] See e.g.M. Schmidtler and K.R. Schubert, Karlsruhe University preprint IEKP-KA-91-4 ( 1991 ). [ 14] A.J. Buras et al., University of Munich preprint TUM-T31-16/91 ( 1991 ).

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