On the rate asymmetries in charged B-decays

Nuclear Physics B352 (1991) 367-384 North-Holland

ON THE RATE ASYMMETRIES IN CHARGED B-DECAYS H. SIMMA Institut für Mittelenergiephysik, ETH-Hänggerberg, 8093 Zürich, Switzerland

G. EILAM Physics Department, Technion, 32000Haifa, Israel

D. WYLER Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA

and

Institut für Theoretische Physik, Universität Zürich, 8001 Zürich, Switzerland

Received 21 September 1990 We recalculate the rate asymmetries in rare B-decays on the quark level to order as, taking into account all gg intermediate states. In contrast to previous investigations, we find the total asymmetries to lie below I%, except for b - d + (dJ, s's) where it is 4% . The asymmetry of differential rates reaches 3% in b --+ suû for large momentum transfer. The complete calculation is outlined and the infrared problems are analyzed. We comment on the decay BS - gg.

1. Introduction CP violation [1] has only been confirmed in processes related to K°-K° mixing [2] and the most promising observation of this effect in the B-system also involves the mixing of the neutral B-mesons (see Bigi et al. in ref. [1]). On the other hand, "direct" CP violation in K-decays has not been established until now; the measurements of e'/,E do not yet preclude a zero value [3]. The discovery of direct CP violation remains to be of great interest and considerable effort has been spent exploiting it in B-decays [1]. In particular, decays of the charged B-mesons, where there is no mixing, have been investigated [4, 5]. A good understanding of CP violation in these decays would considerably deepen our knowledge of this phenomenon. The goal is to calculate and measure CP-violating asymmetries

a(s ---> f) -

r(B --3. f) - r(B --* f) r(B --o f) + r(B

0550-3213/91/$03 .50 U 1991 - Elsevier Science Publishers B.V. (North-Holland)

368

Fig . 1. Contributions to b

H. Simma et al. / Charged B-decays

--1, sun . Dark squares denote the weak interactions . The amplitude from fig . l a contains imaginary pieces due to the loops .

where f is a particular final state (and f its CP conjugate) and T is a total or differential rate. It is already obvious from this definition that a calculation done at the parton level will be difficult to interpret, since a physical state f corresponds to a variety of quark/gluon states. However, as new calculational tools are being developed [6-9] quark-level calculations will become one important ingredient of a complete treatment . Leading decays, involving the large CKM matrix element Vcb , such as b -> cdû, b --3- csc yield no appreciable CP-violating asymmetry on the quark level and involve complicated mesonic rescatterings . The asymmetry is proportional to an interference of the leading amplitude with a strongly suppressed one and hence is inherently small, presumably not larger than 10-3. Semileptonic decays, such as B -* K// could exhibit (cleanly calculable) asymmetries of - 1% [10]. Their small branching ratios of < 10 -6, however, render these otherwise so interesting decays less attractive for CP violation. More promising are hadronic rare decays which involve penguin-loop graphs (effective neutral flavor violation) and, depending on the final state, also CKM suppressed tree-level graphs (see fig. 1). After the work of Bander et al. [4] it became common practice to evaluate the strong rescattering, necessary to produce the asymmetry (see eq. (2 .3)), by means of the imaginary part of the penguin diagram responsible for the decay. (Having in mind the Cutkosky rules for evaluating imaginary parts, we can say that the cc quarks rescatter after being produced at the weak vertex.) To order as, the asymmetry for b -4 su-u is the result of the interference of the cc part of fig. la with fig. 1b. For q2 < 4m2, it vanishes and the asymmetry arises only in higher order in as. For instance, there is an interference of the cc and uû parts of fig. 1a, yielding an asymmetry to order as. Gerard and Hou [4] have discussed this case recently and showed that some new diagrams must be included to satisfy the CPT theorem to order as. A closer look shows that in fact even more diagrams are required . In particular, the treatment

H. Simma et al. / Charged B-decays

369

requires a complete calculation of the process b -4 sgg [11] which was not available to Gerard and Hou. In this paper we carry out an improved calculation of the CP-violating asymmetries. Ours, and the previous calculations, are all done at the quark level; one considers only the decays of the heavy (b-) quark and ignores long distance effects other than simple hadronization . While this should be an adequate procedure for very heavy quarks, it might be less reliable for the case at hand, and, indeed, Wolfenstein [12] recently criticized this approach (which should be adequate for mb -), oo). The main point is that the quark final states (like suû, etc.) are not uniquely related to the physical states. In particular, the overlap between physical states and parton states could be very complicated . Furthermore the simplest calculation approximates strong rescattering by a one-gluon exchange . Although only a detailed and long treatment, along the lines of refs. [6-9] will, in the future, yield completely reliable results, our improved calculation yields a better representation of the rescattering . Since the asymmetry turns out to be sizeable only for large momentum transfers q 2 (from the b-s system), the perturbative calculation is better founded. Indeed, since q 2 > (3 GeV)2, the results are presumably indicative of the true numbers. Our paper is organized as follows: In sect. 2 we give a detailed treatment of the perturbative method of calculating the asymmetries . We exhibit the various contributions and discuss the infrared behavior. In sect. 3 the results are shown and discussed . 2. The CP-violating asymmetry for b -4 f The general b --* s transition amplitude can be written as [4] A=vu A u +vc Ac +vt At = V'uAu + vcAc . Here, the vi are related to the CKM matrix elements by vi =VS Vb

(2.2)

and we have used the unitarity of the CKM matrix, that is vu + v, + vt = 0, to obtain the second line of eq. (2.1). Au,c are the full amplitudes, calculated to first order in the weak and (in principle, to all orders in the strong interactions. The rate asymmetry (1.1) is given, for the process b -4 f, by (

)

211 Uu121Au12

+

2 1VC121A c 1 + 2Re(vu v*)Re(A u A* )}

H. Simma et al. / Charged B-decays

370

The amplitudes A ., can be calculated by Feynman graphs. These are tree graphs (if they contribute) or loop graphs with increasingly higher orders of gS. The imaginary parts of them, so crucial in the calculation of a, obtain only contributions from loops; they are calculated by "cutting" the loops through appropriate intermediate physical states (Cutkosky rules). Accordingly, we write lm Au,c = F

(2.4)

i[ Au,J ,

where j~? i[ A uj is the absorptive part of A,, due to the intermediate state i. Depending on the power of gs considered, only certain states i contribute . We now obtain for the numerator in eq. (1.1) respectively (2.3), 4

.lT = -4Im(vvc) 2m b -4 lm(v L'c

f

E~Vi[AujAc - FjVi[Ac]Au d45f i

i

(2 .4a) (2.4b)

) FAi,

where dOf = S4

pb -

E

n=1

pn

)

n= i

13 n (2-ir) 32

( E:

A

denotes the 11îfparticle phase space measure of the final state and A u, c is the real (or dispersive) part of the amplitudes Au,c . The absorptive parts can be calculated using the Cutkosky rules: '-

A ( b ~ f) =

(27r) 2

4

fA(b -* i) A(i -~ f) d0i .

(2.6)

Hence one obtains for the differential rate difference : dd

-

d~ f

(

2 Tr

)4

2mb

(2 Tr

)4

2

f

( dOi A u(b ~ i) AS(i ~ f) Ac(b ~ f) 11

-Ac(b-+i)AS(i --> f)Au(b-+f)), where Mi

-3,

(2.7)

f) is the dispersive strong amplitude for i --- f (see fig. 2). Note that 3,

f i

Au

A 1 Ac

-(uhc)

Fig. 2 . Schematic representation of the rate difference d; in eq. (2.7) .

H. Simma et aL / Charged B-decays

371

particles outside of the cut loop (for example the s in fig. la) should only counted in the phase space of either i or f. We will include them by convention in the final state f. The integrated rate difference _ dd 4' -~d~ dOf f

satisfies the useful relation (2 df = -If .

.9)

Whether eq. (2.9) holds also in the unintegrated form, depends on f and i and the diagram in question . For the present investigation this question is, numerically, irrelevant . Eq. (2.9) implies

Of course, eq. (2.10) holds in "double differential" form, that is when (2.7) is not integrated over d(Pl and is evaluated at the same point in the intermediate and final phase space. Eq. (2.7) was given in integrated formal form by Wolfenstein [12]. The automatic result (2.10) is a consequence of CPT and insures that the final state need not be included among the intermediate states. This was found by Gérard and Hou for the un state . _ ~ In the following we consider the processes b sun, b ~ sdd, b -~ sss, b sgg, b --* sg and b --+ sung ; the analogous decays where s is replaced by d are treated in the same fashion. All these states have to be included in a complete treatment to order as. The most complicated and interesting case is b --+ sun (b ~ dun). To order as, there is only one contribution to eq. (2.4); the cut through the CE lines in fig. la. Since to order as a huge number of graphs must be included, we introduce some notation (see fig. 3). T(°) denotes the tree-level diagram (fig . lb). T(jt) is a O(as) correction to V°), with an extTR gluon connecting legs i and j as P(2) refers to enumerated in fig. 3. P(2) is the usual penguin graph (an index u, c of the flavor of the internal quark), P'(2), pry(2) are reducible "penguins" with flavorchanging self-energies. An additional gluon, connected to the legs i and j, yields the diagrams Pij4~ etc. P,(,4) denotes a penguin with the vacuum polarization inserted into the gluon propagator . The collection of all these diagrams is required and we discuss them below. We have 65 different topologies with nearly hundred (93) cuts. Some of the diagrams can be viewed as ordinary QCD corrections to known vertices, whose leading logs

37?

X. Sinuna et al. / Charged B-decays w

T(2) 23

P P(4) 14

(4) 15

4--' -'-2 P IM

4~ -'- 2 p11(2)

4) Fig. 3. The classes of contributing diagrams and some examples of the notation used . In P,~, the dark blob stands for the penguin loop of fig . l a ; the light blob for the vacuum polarization of the gluon . can summed . Note, that besides two-particle cuts there are also three-particle intermediate states. At this point one may wonder whether there are other singularities of diagrams which may yield imaginary amplitudes (anomalous thresholds etc.). As argued by Coleman and Norton [13], only intermediate physical states contribute to amplitudes in the physical region . Thus we consider our procedure to be adequate. The intermediate states contributing to charmless final states (up to O(as)) are listed in table 1 . We now discuss some of them for sun. (a) Cuts in T (2): Im T (2) gets contributions from su, su and uû cut states. Since the resulting intermediate states are always sun, their contribution to the rate difference cancels against the corresponding one in P(4), P'(4) and p"(4) (e.g., P2), P24) illustrating eq. (2.10). (b) cc cuts: It is natural to divide the cc cuts into the three classes listed in table 1. The interferences (IM P~ 2) - T(°)), (IM PC 2) - Re P,t, 2)) and (Im P.(r4) - T(O)) (where P,(4) is cut only through the cc-loop in the vacuum polarization) of class I have been evaluated previously [4]. The contributions of class II topologies can be estimated by taking into account the leading-log QCD corrections to the tree-level bsgq-vertex (q = u, c). Since the operator (sbXgq) does not contribute here because of its color structure, we simply have the relation Im P~2) - T(°) + Im Pciâss II ' T(°) + Im P~2) - Re T(2) + higher orders

ci Im

PC

2) -

T(°) ,

where cl is the Wilson coefficient of the operator (gb)(sq) . Although these QCD corrections are (in leading-log approximation) not very significant (since c l = 1 at

H. Simma et al. / Charged B-decays

373

TABLE 1

The intermediate states and their topologies contributing to the total asymmetries up to order ace. The pairs of numbers k1, kl', kl" denote the topologies of Pk, )T(o), Pkl4)T(o) and Pklll(4)T(o), respectively. The superscript 2 indicates that a topology contains the cut twice. If it in Pâ4) has an argument, only the indicated particles are included in the vacuum polarization. Final state f

Intermediate state i

sun

dd

Contributing topologies p(4) ~T( o) d

ss cc

P~s)T(o),13', 35', 36' class I)T(o), P(2)P(2), (C)T(o) .,56,,66 .,55 .,33,35,36 .,16 13,,15 class 11: 11, class III : 22, 24,25,26, 27, 44,55, 56,57,66,

PA4) T(0)

P,(4(g)T(O)9 12,14,17, 23,25, 26,27, 34,37, 57,12',14',17, 23',26, 27', 37',12",14",17", 27" 23,34,37,17,13', 14',16',17, 23', 26',35', 36, 37', 56', 66' 23,34,35,36,37,35', 56' 12,14,15,16,17, 23, 252,262, 34, .,55,562, 572, 66 35,36,37

99 s9 scc

gcc sdd sss sgg sg sung

j = u,c SM

cc

rate differences are related to those of b - sun resp. to b - scc by eq. (2.9)

Included here Yes yes yes leading log no

no no

yes yes yes no no

the Mb scale [15]), their effect can be enhanced by cancellations between the various contributions to the numerator of the asymmetry . The contributions of class III topologies represent O(gs)-corrections to the strong amplitude ÂS(cc ~ U7U) in (IM p(4) - T(o)), which are not available in explicit form [14]. (c) ss cuts. ss cuts occur in the vacuum polarization in the P.(4)-diagrams, as well can be as in Pi3, Pas and P3 6. Their contribution, denoted collectively by taken into account by calculating düû and using eq. (2 .9). (d) Gluon gluon cuts. The main new result of this paper is the proper treatment of the gg cut state, which was previously included only through the graphs of the (P,(,4) - T()) interference . However, it can be shown that the resulting rate differences are not gauge invariant and do not satisfy the asymmetry relation (2.9) for a general choice of the gluon polarization vectors. This is not surprising, as the Cutkosky rules are only meaningful for gauge invariant amplitudes of physical processes . Recently, the process b --* sgg has been calculated [11]. The resulting amplitude is small, due to gauge cancellations among the various contributing graphs. Applying eq. (2.9), the gg contribution to b -* sûu can be easily found from the uû cuts in b -+ sgg using the results of ref. [11]. This yields the total rate

N äimma et al. / Cita

37,4

8-d ays

k

Fig. 4.

e cuts of

p;;!

P

P

a)

b)

discussed in the text . The dark blob stands for the charm-penguin loop of

fig . Ia.

difference &:. Because of the cancellations mentioned, it is much smaller than that from quark intermediate states. As we do not expect it to be large anywhere in phase space, also the differential rate difference due to gg states are small. (e) sg cuts. These cuts arise from diagrams such as P;4,, see fig. 4b. Since they involve the on-shell b -). sg transition, they are expected to be small. This is a well known feature of the lowest-order flavor-changing gluon coupling. We have therefore not included these cuts (see, however, the discussion of the infrared poles). (f) Threeparticle cuts. These are not included in the present work; they imply a rather laborious treatment of the three-particle phase space to which we hope to return in a further publication . In comparison to the two-particle cuts, there is a factor of (327r -')- ' from phase space. However, there is no suppression factor of (167r2)- l from the loop integral, and we cannot a priori exclude a sizeable contribution. Because of the cc threshold in the three-particle cuts, we expect their contribution to be at most of the size of other O(as )-contributions from cc cuts. INFRARED DIVERGENCES

An important issue in our calculation are the infrared (IR) divergences (soft and collinear). Although the KLN-theorem [161 insures their absence, once everything has been evaluated and properly combined, we would like to know how much an incomplete calculation suffers from IR divergences . In particular, we discuss here the IR behavior of the partial rate difference and show a number of amazing cancellations . A convenient method to "visualize" the KLN-theorem is in a graphical way by "Kinoshita graphs", which represent products of Feynman graphs. Often, rate differences are less IR divergent than the rates themselves . This can be clearly seen at the order as level, where the rate difference is finite: All the bremsstrahlungsgraphs needed to absorb IR divergences of loops are obviously real and do not contribute to rate differences . In the general case, bremsstrahlungsgraphs have less internal gluons (to the same order in as) and hence fewer cuts.

H Sîmma et al / Cha

B-decays

Fig. 5. Diagrammatic representation of the interference P0243)TI°). Dotted lines represent the cut to leading to the absorptive part, dashed lines the final state. In order to get IR-finite results, as by the KLN theorem, the sum of all possible configurations of dotted and dashed lines has to be considered. Here we show those which do not cut the cE lines.

In the following we comment mainly on b -+ sun. To order acs, the rate difference obtains contributions from the interference of T(°) with the various P4, amplitudes and of PC2~ with Pû2) or T(2 ). From the discussion of eq. (2.10) we know that (IM Pû2) - Re PC2)) and (Im T(2) - Re PC(2)) do not contribute to the rate differ ence. In (IM PC 2) - Re Pû2 0 no IR problems arise since IM PC2 ~ vanishes below q2 = 4m~. As to the (IM PC2) - Re T(2)~)-interferences, IR divergences from the various loops in T(2) are compensated for by inclusion of the corresponding diagrams for soft bremsstrahlung b -)-suug in the usual manner [16]. However, since we consider the T(2) contributions as QCD corrections and take them into account only through local effective vertices, which have no IR divergences, we do not have to include the soft bremsstrahlung in our calculation. Finally, we need to discuss the (P(4) - T(O)) interferences . To illustrate the various cancellations, we discuss Pte) as an example. Ignoring cuts through cc (these can be dealt with separately as they only arise for q 2 > 4m2), fig. 4 shows the two cuts which contribute to the rate difference. The "vertical" cut through the su lines corresponds to an suu intermediate state which is identical to the final state considered and therefore does not contribute . According to ref. [161 all "cuts", also through final states must be considered ; these are given in the graphs of fig. 5. All particles, except b and c, are taken to be massless. We consider first figs. 5a and 5b and the IR poles coming from 1 2 = 0 (in fig. 4a or 5a) and k 2 = 0 (in fig. 4b or 5b) where p2 * 0. Apart from common factors, we have ô(1 2) (2 .11) . A ~_ bA a ~ S(k2) , 12

k2

376

H. Simma et al. / Charged B-decays

Since (k + 1)2 = V +12 + 2k1 = 0, one gets k2 =0 :

12= -2k1,

1 2 =0 :

k 2 = -2k1 .

(2 .12)

Thus, Aa +Ab

'*,

1 (S(1 2 ) - S(k2 )) . 2kl

(2.l3)

The IR divergence corresponds to kl -+ 0, but then k 2=12= 0, and the parenthesis in (2.13) vanishes, leaving a finite total result. When p2 = 0, we must also include the graphs 5c and 5d . Graph 5c corresponds to the uû cut in b ---> sgg while graph 5d represents the suû cut in b -3- sg. These graphs cure the IR divergence at p2 = 0 because of the antisymmetry relation (2 .9). To see this, we consider first graph 5a. When qk * 0 we have q = p. and k = p (for p2 = 0). This follows easily by going to the rest frame of the two gluons . But then p = 0, and the denumerator in the internal u-propagator in graph 5a vanishes. This, together with the appropriate factor from the loop integration cancels the p2=0 pole in the numerator, leaving a IR-finite result. When qk = 0, this reasoning fails. However, this corresponds to the situation where the two gluons are parallel, and consequently the uû pair runs parallel too. In that cûse, the gg state is degenerate with the ûu state and 5c must be added to obtain a finite result . But because of 4gE = - 4u, when uû and gg are at the same phase space point (see the discussion of eq. (2.9)), the two asymmetries cancel, leaving again a IR-finite result. A similar argument can be made for figs. 5b and 5d. Note that the gs cut is needed for the cancellation of the IR divergences ; but as noted before we expect its effects to be small otherwise . We neglect it in the numerical calculation and take the cancellation of the IR divergences into account by a reasonable cutoff [11] in the gg cuts. When the cut in fig. 4 goes through the cc pair instead of the left gluon one finds analogous cancellations, this time between the gcc and scc cuts. However, to compensate for the p2=0 pole, the degenerate suûg final state must be included and the cancellation of IR divergences involves the familiar bremsstrahlung.

3. Results and discussion We first recall the relevant formulas used in the calculation . Contributions proportional to the small form factor F2 are neglected in the same spirit as we have omitted the sg cuts. The kinematics is given in fig. 6.

377

H. Simma et al. / Charged B-decays

b

Fig. 6. The kinematics of b - suu .

(i) Exclusive* rates

sqq (q = u. c) (a) b --), . d2T a N2 - 1 2 = 24T° NI v 1 2S( TT) - s Re( Flvû )S( PT) de dE' 4e 2N a 2 N 2 -1 S(pp)IF 12 l + ( 4-rr 4N

where T° Vi

m sbGF2 1927r3

7

(3.1)

'

N = 3 (number of colors),

= VS Vb)

(3.2)

and (for m s = 0) S(TT) = 8pkp'klm 4b = 4i(1- 2i), S(PT) = (8pkp'k + 4mgp'p)/mb = 4i(1- 2i) + 4(mq/mb)E', S(PP) = (8pkp'k + 8pkp'k + 8mgp fp)lm,

(3 .3)

= 4E(1 - 2E) + 4E(1- 2E) + 8(mq/mb)E',

are the spin sums from tree-tree (TT), penguin-tree (PT) and penguin-penguin (PP) interference . We note (for m s = m q = 0) 1 1 q4 1 q6 1 fS(TT) de = fS(PT) dE = 2 fS(PP) dE = 6 2 m4b + 3 m6b * Here and in GO "exclusive" is understood at the quark level.

.

(3 .4)

H. Simma et al. / Charged B-decays

378

The form factors F,(q 2) = x --- m~/Mw and z = q2/m?: 1

F,

7 3

36x3

21

_

36x2

L'i Fi(m?, q2)

_

7

1

9 +

_+ 6x

can be written as functions of 1 4

3

- 2x2 +

8

2

3x - 3 )In x

- 4f lu(1 - u)ln[1 - zu(1 - u)] du .

(3 .5)

0

b -), sdd In this case, there is no tree-level contribution . The rate is obtained using (3.1) with (3.3) and setting S(PT) = S(TT) = 0. (b)

(c) b -+ sss Identical particle effects lead to dT as 2 N 2_ 1 1 S = 24To DD I F q 2)12 + S EE I Fi(q r2)12 dE dE 47r ) 4N 2 ( ) '( ( ) I

+

2

Re[Fi (g2)F,(q'2)] S( DE) , (3.6)

2

where S(DD), S(DE) and S(EE) are obtained from eq. (3.3) as follows (ms = 0):

S(DD) =S(PP), S(EE) = S(PP) p'- k = 4Z(1

- 2i) + 4E'(1 - 2E'),

f' -,E

S(DE) = -8pkp'k -

-4F(1 - U),

(3 .7)

and q = p - p', (ü)

q'=p-k .

Exclusive rate differences

We write d

2

dE de

(T(b --+ f) - T(b --+ f) = -4Im (u v v* )

~)

d24;(topol) topol

dE d E'

where "topol" denotes the class of diagrams to which i contributes .

(3 .8) 9

H. Simma et al. / Charged B-decays

379

Then the p(2)T(o)-topology yields for b ~ suû the order as contribution d2 a N2 - 1 (PT) _ - 24To41r S(PT)(-ImFi) . dEdE'd'u 2N

(3 .9)}

From the p(2)p(2) interference we have for b -4 sqq (q = u, d, c) the order contributions 2 d2 N 2X1-q1 4 (PP) = 7124ro 1 S PP Im F'Re F' - F= (3 de de' ° ( 4Tr ) 4N ( ) ' ' '

as

.10)

with q= + 1 and j = c for i = u, or with q = -1 and j = u for i = c. Expressing the momentum dependence of the vacuum polarization (with a quark j = u, d, c, s, . . . inside the loop) in terms of F,, the contribution of a jj cut in P,(4) for b -* suû is given by 2 N 2d 2 d"(PTT) as , ii = -24T 1 S PT Re FI - Ft Im F' . (3.11) dE dE ° 47r 2N ( ) ' ' ' For b -4 sss we have 2 N 2_ 1 1 d2 as S(DD)Im F' q 2 n24To Re F' q 2 - Ft FIt ,a; s= dE dE ( 47r 4N 2 ( '( ) 1( ) 1 + NS(DE)lm Fl(g2)Re[Fi(gr2) _ Fit]

+

( q 2 H q.2, E H E ' )

(3 .l2)

q and j are defined as in eq. (3.10). The rate differences for b -~ sgg can be calculated using the formulas provided in the second paper of ref. [11]. We do not enter here into the details, as this mode is numerically unimportant (see table 2). (iii) Inclusive asymmetries We write a = -4Im(vv~ )4/2T

(3 .l3)

4 =4c"+add+4 CE +djg+ . . . ,

(3 .14)

a d get ( ) b --), s + no charm

F

I, uU

+ rsdd

+Fsss

+ rsgg + . . . .

(3 .15)

ff. Sinmia er al

C

B-d

T% 2 rates a rate differ as defined in eqs. (3.0-0.12). The numbers in parenthesis corValues d to the replacement of s- by d-quarks. The rate differences are equal in the two cases. The values the tens : a, = 0..24 Mw = 81.8 GeV, rnb = 5.2 GeV, me =1.5 GeV, m = 4.5 éV, t'. =0, le -i1 ), V~=0. n ity b

sW (du)

T ) T( RPM

Units

m, =130 GeV

m, =1

" " 10® 'To " " " " "

0.30(6.1) 1.02(-1 .50) 2.13 (0.2l) 12.3 -0.95 -1 .13 -3.% -3.29 -0.0002

0.30(6.1) 1.04(-1 .54) 2.22 (0.22) 12.3 -0.97 -1.16 -4.04 -3.37 -0.0002

10®STo 10® ;T® "

1.79 (0.18) 3.29 -0.95

1.86 (0.l8) 3.37 -0.97

10' $To 10_ 3T0 "

0.75 (0.02) 0.0002 0.47

0.97 (0.03) 0.0002 0.48

10

ajpn

P) ~ ) $ )

- ;TO

GeV

e dots denote the yet not calculated contributions ®Sd, A;: ; ®S etc. In writing down (3.14) extensive use of the asymmetry relation (2.9) has been made which of course eliminates non-cc cuts (see also ref. [121). For the numerical evaluation, we use -®è (PT) +4 '7" (PP) +d(PrT),

B~d =

®c( PP) .

(3 .16)

(b) b -1- s + no charm + no additional strange 4=~u+Add+as @g+Asu + . . ., c

(3.17)

r = rsun + rsdd + rsgg + . . . .

(3.18)

(c) b --), d + no charm Asymmetry and rate are given by (3.14)-(3 .16) with the appropriate substitutions.

T

3

Total rate etries for the considered. The cMunn BR process for m t =130 GeV, using l',, = FO/160. Tine List three columm different values of mt and cl-

Mode b -~ WOO b --~ sdd b -~ ss.s b -" duû b -" ddd b-~ds's b-~s,noc b -> s, no c,'s b --~ d, no c b-~d. no c.s

BR [

m, =1 GeV c,=1 .0

0.6 03 03 0.8 0.03 0.03 1.2 0.9 0.8 0.8

03 0.4 0.4 -0.2 -3.9 -4.3 0.3 03 -0.4 -0.4

mt =1 c,=1.0

V

0.2 0.4 0.4 -0.2 4.0 -4.2 0.3 0-3 -0.4 -0.4

t =1 V c, 1.1 0.5 0.4 0.4 -0.3 -3.9 -4.3 0.4 0.4 -0, -0.5

(d) b - :o no flavor (no strange and no charm) ® =®â" +A

+Ae+®d,&«"(PzT) + . . . ,

=rduir+1ddà+rd gg + . . . .

(3.19) (3 .

)

The results from the numerical evaluation of eqs. (3.1-(3 .12) are given in table 2 for two values of the top-quark mass. Table 3 contains the expected asymmetries for exclusive (on the quark level) and inclusive decays, using eqs. (3.13)-(3.20). Note, that the rate differences for b --> s and b -), d are essentially the same. This is because the calculations are the same and the product v - vc* differs only by a sign in the two cases. Also, it is clear that some rates are the same, such as the PP contribution to b --)' dun and r(b --* dss). Finally, the asymmetry of the differential rates for b --.- sun is given in fig. 7. These results indicate much smaller total asymmetries than most of the values previously published. With branching ratios of exclusive rare decays (such as B -4 KTr) around 10 -5 [15], the observation of asymmetries around 0.5% requires 10 1° 13-mesons. The main reason for the small asymmetries is the near absence of the gg intermediate state due to gauge cancellations [11]. An exception might be the decay b -3, dss, leading to a strange/antistrange particle in the final state. The asymmetry is 4 .3% ; the expected (inclusive) BR about 10 -4. An estimate of the rates of B -+ 7rK+ K- etc. will be most useful. Clearly, an (exclusive) BR of 10-5 would yield accessible asymmetries . Of considerable interest is the asymmetry of differential rates in fig. 7. It is only sizeable for (momentum transfer)' above 4m2 and thus near the value of

H. Simma et al. / Charged B-decays

382

5k

C%J

4 3 2 1 0 2

0.0

0.2

0.4

0.6

0.8

4 .0

q

mb

Fig. 7. Asymmetry of the differential rate for b - suu. The solid line is the sum of all calculated contributions. The dashed line shows the O(a s) contribution, the dotted lines the effects of the dd cuts in the Pnd )T(°) topology (straight line) and of the cc cuts in the PC2 ~PU2) and P,(,4(c))T(°) topology, respectively.

(4.4 GeV)2 above which there seems to be no long-distance activity of the cc system . We thus expect the quark model calculation to yield a reliable estimate . If the spectator does not undergo large momentum transfers, the result points out B --3, K7rr (with large (momentum)2 of the two pion system) as a promising mode . In our calculation, the three-particle cuts have been neglected (see discussion in sect. 2). Because of the cc threshold, they contribute only in the region where order a s is dominating . The other cuts neglected - the sg cuts - are small due to the well known on-shell suppression of the b -+ sg amplitude. The inclusion of QCD-corrected vertices does not change the results of the calculation strongly (see last column of table 3). While the asymmetries for the decays considered are relatively small, it is possible to enhance them by looking at a decay where the imaginary part of an amplitude dominates. An instance where this occurs is the novel decay B S -1 gg (see ref. [17] for B S -3- yy). From ref. [11] we see that the form factor dis obtains a large imaginary part for large (momentum transfer) 2 which here is m Bs. In the notation of ref. [11], the transition matrix element for the color symmetric gluon configuration is 1

Tmv=415(Yl2Bs, 0,(]) (kl -k 2)aeagvQy'YL

~e

A6

( 2

2

H. Simma et al. / Charged B-decays

383

Summing over spin and color we get

2r(BS

With

MB,

= 5.38 GeV,

-~ gg) = 31

fB s

VC

b

2r0 -

N2 -

= 0.15 GeV and

VfB a s dis

1

N

Iai S I

2

MB,

=1 .4, this yields

BR(BS _~. gg) = 10-4 In contrast to the usual penguin induced decay b -), s + x, where there is background from tree-level processes, one might be able to isolate the penguin here by looking at BSdecays without flavor in the final states. Only the highly suppressed annihilation Bs , -3, uû may lead to similar final states. The asymmetry amounts to about 5%, but competes with mixing effects [1]. An important question, as emphasized by Wolfenstein [12] concerns the "longdistance" effects . Although the b-quark is quite heavy compared to the QCD scale A, non-perturbative and other effects of the strong interactions might change our results. This is particularly relevant when interpreting the momentum dependent asymmetry of fig. 7, where one must know how q2 is distributed in the mesonic final state. Possibly, a treatment along the lines of ref. [7] can be attempted . The existence of several hadronic states corresponding to a given set of parton states might destroy their coherence and thus wash out any large asymmetry of several %. The only support for coherent quark states comes from the calculation of the D + - D° lifetime difference [18], however, the quark model explanation is not really established . As to the magnitude of the absorptive part, the loop estimate is presumably quite reasonable . This is supported by the fact that e+e annihilation at large q2 is well represented by the quark picture. Another argument against a quark-level calculation is that the overlap between the final states of the charm-penguin and the tree-level diagram is small [12], because the tree-level produced uû pair is likely to yield several pions. As we have seen, the uû pair is fast in the interesting region. Thus it seems difficult for it to distribute its momentum effectively among many pions . A possible assessment will be the relative size of the rates B ---> K(n-rr), n = 1, 2, . . . . We thank M. Glück for a collaboration in the early stages of this work and M. Simonius, A . Aeppli, L. Wolfenstein, J. Donoghue, W.S. Hou and W. Woolcock for many helpful discussions . H.S. thanks the Bloch-Fonds for support . This research was supported in part by the National Science Foundation under Grant No. PHY89-04035, supplemented by funds from the National Aeronautics and Space Administration, at the University of California at Santa Barbara .

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