

 11 Dec 95

Preprint LMU-21/95

October, 1995

Hadronic Rare B Decays via Exchange or Annihilation Diagrams

Zhi-zhong Xing 1 Sektion Physik, Theoretische Physik, Universit"at M"unchen,

Theresienstrasse 37, D-80333 M"unchen, Germany

Abstract The two-body mesonic B decays induced only via a single W -exchange or annihilation quark diagram, such as B\Gamma u ! D(\Lambda )\Gamma s K(\Lambda )0 and _B0d ! D(\Lambda )+s K(\Lambda )\Gamma , are analyzed in the factorization approximation. We estimate the branching ratios for those transitions into two pseudoscalar mesons, and find them to be negligibly small. The significant effect of final-state rescattering is illustrated by taking B\Gamma u ! D\Gamma _K0 for example.

(Accepted for publication in Phys. Rev. D as a Brief Report)

1Electronic address: Xing@hep.physik.uni-muenchen.de

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As a result of the large data sample of weak B transitions collected on the \Upsilon (4S) resonance by CLEO and ARGUS collaborations [1], the hadronic decays of B mesons appear to be a valuable window for determining the quark mixing parameters, probing the origin of CP violation and investigating the nonperturbative confinement forces. Further experimental efforts towards the above physical goals, such as the program of CLEO III and the construction of KEK and SLAC B-meson factories, are underway.

The dynamics of exclusive hadronic B decays, in particular those via the W -exchange or annihilation quark diagrams, is not yet well understood. The decay rates of W -exchange and annihilation transitions are usually argued to be negligibly small due to the suppression of helicity and (or) formfactors [2], however, a solid justification of this argument is necessary in both theory and experiments. Current data have given upper bounds on some of the W exchange or annihilation decay modes of B mesons, e.g., Br(B\Gamma u ! D\Lambda \Gamma s K0) ! 1:2 \Theta 10\Gamma 3 and Br( _B0d ! D\Lambda +s K\Lambda \Gamma ) ! 1:2 \Theta 10\Gamma 3 [1]. A better understanding of such processes is possible in the near future, with the accumulation of larger data samples.

As a preliminary step towards comprehensive studies of the hadronic rare B transitions via W -exchange and annihilation diagrams, this short note concentrates on the two-body mesonic decays. We first survey all possible decay modes of this nature by use of a complete quarkdiagram scheme, and then estimate branching ratios for those channels into two pseudoscalar mesons. Finally the significant effect of final-state rescattering is illustrated by taking B\Gamma u ! D\Gamma _K0 for example.

According to the topology of lowest-order electroweak interactions with QCD effects included, all two-body mesonic B decays can be graphically described in terms of ten distinct quark diagrams [3] 2. In the assumption of no final-state rescattering or channel mixing, it is possible to survey those "pure" decay modes induced only by a single quark graph. We find that the following neutral B decays occur solely through the W -exchange diagram (see Fig. 1(a) for illustration):

_B0d \Gamma ! D(\Lambda )+s K(\Lambda )\Gamma ; D(\Lambda )\Gamma s K(\Lambda )+ ; _B0s \Gamma ! D(\Lambda )\Gamma M + ; D(\Lambda )0M 0 ; D(\Lambda )+M \Gamma ; _D(\Lambda )0M 0 ; (1)

2Note that the popular six-graph scheme [4] does not include the quark diagram for the color-matched electroweak penguin transitions and those for the decays where one (or both) final-state meson(s) must be the flavor singlet(s). For a detailed discussion, see Ref. [3].

2

where M denotes an I = 1 light unflavored meson like ss, ae or a1 [1]. For two-body charged B transitions, the following decay modes occur only via the annihilation diagram (see Fig. 1(b) for illustration):

B\Gamma u \Gamma ! D(\Lambda )\Gamma _K(\Lambda )0 ; D(\Lambda )\Gamma s K(\Lambda )0 ; B\Gamma c \Gamma ! K(\Lambda )\Gamma K(\Lambda )0 ; K(\Lambda )\Gamma M 0 ; _K(\Lambda )0M \Gamma ; M \Gamma M 0 : (2) In experiments, some of the above processes have been searched for, and upper limits to the branching ratios of _B0d ! D(\Lambda )+s K(\Lambda )\Gamma and B\Gamma u ! D(\Lambda )\Gamma s K(\Lambda )0 have been obtained [1, 5]. Nevertheless, there has not been any preliminary estimation of the decay rate for any of these modes.

The effective weak Hamiltonian responsible for the decays in Eqs. (1) and (2) is given by

Heff = GFp2 VcbV \Lambda uq hc1 (_qu)V \Gamma A(_cb)V \Gamma A + c2 (_cu)V \Gamma A(_qb)V \Gamma Ai + (u , c) + H:c: ; (3) where q = d or s, V represents the Cabibbo-Kobayashi-Maskawa (CKM) matrix, c1 and c2 are two Wilson coefficients at the scale O(mb). Assuming a generic decay mode Bffi(b_ffi) ! X(ff_fl) + Y (fl _fi) as illustrated in Fig. 1, one can factorize its amplitude hXY jHeffjBffii into a product of three terms: the CKM factor, the combination of Wilson coefficients and the matrix element of color-singlet currents. For example,

hD\Gamma _K0jHeffjB\Gamma u i = GFp2 a1 (VubV \Lambda cs) \Omega D

\Gamma _K0

scu (4a)

or

hD\Gamma s K+jHeffj _B0di = GFp2 a2 (VubV \Lambda cd) \Omega D

\Gamma s K+

ucd ; (4b)

where a1 j c1 + c2=3, a2 j c2 + c1=3, and the hadronic matrix elements are obtained from the definition

\Omega XYfffiffi j hXY j( _fffi)V \Gamma Aj0ih0j(_ffib)V \Gamma AjBffii : (5)

Subsequently we treat a1 and a2 as free parameters, in order to phenomenologically accommodate the contribution of color-octet currents which has been neglected in the above naive factorization approximation [6, 7]. The matrix elements \Omega XYfffiffi can be Lorentz-invariantly decomposed in terms of the decay constants and formfactors, however, many difficulties exist in evaluating the relevant annihilation formfactors.

For simplicity and illustration, here we only calculate \Omega XYfffiffi for the case that both X and Y are pseudoscalar mesons. Following the work of Bernab'eu and Jarlskog [8], we obtain

\Omega XYfffiffi = i mX \Gamma mYm

X + mY h(m

X + mY )2 \Gamma m2Bffii fBffi F a+(m2Bffi) ; (6)

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where fBffi is the decay constant of Bffi meson, and F a+(m2B

ffi) is the annihilation formfactor.

The perturbative QCD calculation gives F a+(m2B

ffi) = i16ssffsf

2 Bffi =m2Bffi [9], which is primarily

absorptive. In estimating the branching ratios of Bffi ! XY , we take ffs(mb) = 0:20, a1 = 1:15

and a2 = 0:26 [5]. The central values of meson masses can be found from Ref. [1]. The average lifetimes of Bu, Bd, Bs and Bc mesons are taken to be 1:54 \Theta 10\Gamma 12s, 1:50 \Theta 10\Gamma 12s, 1:34 \Theta 10\Gamma 12s and 0:5 \Theta 10\Gamma 12s respectively [1, 10]. Considering the constraints of unitarity on the CKM matrix V [11], we adopt jVudj = 0:9744, jVcsj = 0:9734, jVusj = jVcdj = 0:22, jVcbj = 0:04 and jVubj = 0:08jVcbj. We also input fBu = fBd = 0:196 GeV, fBs = 0:212 GeV, fBc = 0:48 GeV and MBc = 6:25 GeV [12, 10]. The numerical results are listed in Table 1.

Decay mode Quark diagram CKM factor Wilson factor Branching ratio B\Gamma u ! D\Gamma _K0 annihilation VubV \Lambda cs a1 8:1 \Theta 10\Gamma 9 B\Gamma u ! D\Gamma s K0 annihilation VubV \Lambda cd a1 4:2 \Theta 10\Gamma 10

_B0d ! D+s K\Gamma W -exchange VcbV \Lambda ud a2 6:5 \Theta 10\Gamma 8

_B0d ! D\Gamma s K+ W -exchange VubV \Lambda cd a2 2:1 \Theta 10\Gamma 11 _B0s ! D+ss\Gamma W -exchange VcbV \Lambda us a2 1:2 \Theta 10\Gamma 8 _B0s ! D0ss0 W -exchange VcbV \Lambda us a2 1:2 \Theta 10\Gamma 8 _B0s ! D\Gamma ss+ W -exchange VubV \Lambda cs a2 1:5 \Theta 10\Gamma 9 _B0s ! _D0ss0 W -exchange VubV \Lambda cs a2 1:5 \Theta 10\Gamma 9

B\Gamma c ! K\Gamma K0 annihilation VcbV \Lambda ud a1 6:3 \Theta 10\Gamma 9 B\Gamma c ! ss\Gamma ss0 annihilation VcbV \Lambda ud a1 1:1 \Theta 10\Gamma 7 B\Gamma c ! K\Gamma ss0 annihilation VcbV \Lambda us a1 6:7 \Theta 10\Gamma 6 B\Gamma c ! _K0ss\Gamma annihilation VcbV \Lambda us a1 6:4 \Theta 10\Gamma 6

Table 1: Typical examples of weak B decays into two pseudoscalar mesons via a single W exchange or annihilation diagram.

From Table 1 we observe that all 12 decay modes have negligibly small branching ratios in the context of the factorization approximation and the formfactor model used above. The suppression of decay rates mainly arises from the smallness of F a+(m2B

ffi) and jVubj. For those W exchange induced channels, the smaller Wilson factor a2 also suppresses the decay rates to some

extent (ja2j2 , 7%). Note that \Omega XYfffiffi / (mX \Gamma mY ) comes from the application of a constituent U (2; 2) quark model [8], and this may lead to large suppression and uncertainty if mX and mY are comparable in magnitude (e.g., B\Gamma c ! K\Gamma K0 and ss\Gamma ss0). In comparison with our rough

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results for B\Gamma u ! D\Gamma s K0 and _B0d ! D+s K\Gamma , the existing data give Br(B\Gamma u ! D\Gamma s K0) ! 1:1\Theta 10\Gamma 3 and Br( _B0d ! D+s K\Gamma ) ! 2:4 \Theta 10\Gamma 4 [1, 5].

The above discussions do not take into account the rescattering effect of final states due to strong interactions. Such effects may give rise to mixing of a "pure" decay mode (via a single quark diagram) with others, including those which were not originally coupled to this weak channel. Thus it is necessary to estimate the magnitude of final-state rescattering for those transitions listed in Eqs. (1) and (2). For illustration, here we take B\Gamma u ! D\Gamma _K0 for example to demonstrate that significant channel mixing can completely ruin a "pure" decay mode.

An isospin analysis shows that B\Gamma u ! D\Gamma _K0 may mix under rescattering with B\Gamma u ! _D0K\Gamma and _B0d ! _D0 _K0 [13, 14] 3. Note that _D0 _K0 is a pure I = 1 state, and _B0d ! _D0 _K0 occurs only through a single color-mismatched spectator diagram [3, 14]. In contrast, B\Gamma u ! _D0K\Gamma takes place via both the color-mismatched spectator graph and the annihilation one. Ignoring final-state interactions, the amplitude of _B0d ! _D0 _K0 can be factorized as:

h _D0 _K0jHeffj _B0di = GFp2 a2 (VubV \Lambda cs) \Pi _D

0 _K0

ucs ; (7)

where the hadronic matrix element \Pi _D

0 _K0

ucs is obtainable from the generic formula

\Pi XYfffioe j hXj( _fffi)V \Gamma Aj0ihY j (_oeb)V \Gamma A jBffii

= \Gamma i im2B

ffi \Gamma m

2 Y j fX F

BffiY0 (m2

X)

(8)

for Bffi(b_ffi) ! X(ff _fi)+Y (oe_ffi). Under isospin invariance, one finds \Pi _D

0 _K0

ucs = \Pi

_D0K\Gamma ucs and \Omega

_D0K\Gamma scu =

\Omega D

\Gamma _K0

scu . After taking into account the rescattering effect of D

\Gamma _K0, _D0 _K0 and _D0K\Gamma , the

transition amplitude of B\Gamma u ! D\Gamma _K0 can be written as

hD\Gamma _K0jHeffjB\Gamma u i = GF2p2 (VubV \Lambda cs) hia2 \Pi _D

0K\Gamma

ucs + 2a1 \Omega

D

\Gamma _K0

scu j ei

OE0 \Gamma a2 \Pi _D0K

\Gamma

ucs ei

OE1i ; (9)

where OE0 and OE1 are the strong phases of I = 0 and I = 1 states. It is obvious that contribution of the rescattering term \Pi _D

0K\Gamma

ucs to hD\Gamma _K0jHeffjB\Gamma u i disappears if \Delta OE j OE1 \Gamma OE0 = 0 [14]. As a

result, we find the effect of nonvanishing \Delta OE on the branching ratio of B\Gamma u ! D\Gamma _K0:

R(\Delta OE) j Br(B

\Gamma u ! D

\Gamma _K0)j\Delta OE6=0

Br(B\Gamma u ! D\Gamma _K0)j\Delta OE=0 = 1 \Gamma , sin(\Delta OE) + ,2 sin2 \Delta OE2 !

(10)

3Here a reasonable assumption is that no additional channel mixes with these three modes.

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with

, j \Sigma a2a

1 fififififi

\Pi _D

0K\Gamma

ucs

\Omega D

\Gamma _K0

scu fififififi : (11)

The sign ambiguity of , arises from the unknown relative sign between \Pi _D

0K\Gamma

ucs and \Omega

D

\Gamma _K0

scu . The

size of \Pi _D

0K\Gamma

ucs can be estimated with the inputs F

BdK0 (0) = 0:38 and f

D0 = 0:253 GeV [6, 12].

We approximately obtain , ss \Sigma 18:8. The change of R(\Delta OE) as a function of \Delta OE is numerically illustrated in Fig. 2.

It is clear that the significant rescattering effect (j\Delta OEj * 500) can dramatically enhance the branching ratio of B\Gamma u ! D\Gamma _K0 to the level O(10\Gamma 6). In this case, the transition is indeed dominated by the contribution from _B0d ! _D0 _K0 and B\Gamma u ! _D0K\Gamma . Considering DK scattering via a t-channel exchange of Regge trajectories, Deshpande and Dib have estimated the strong phase shift \Delta OE and obtained tan(\Delta OE) ss \Gamma 0:14 [14]. This result has a two-fold ambiguity: for \Delta OE ss \Gamma 80, R(\Delta OE) deviates only a little from R(0); for \Delta OE ss 1720, R(\Delta OE)=R(0) , 102 may turn out. Note also that R(\Delta OE) is insensitive to the sign of ,, due to the fact j,j ?? 1.

Certainly the above calculation approaches have many uncertainties which are unable to be removed to the limit of our present understanding of the W -exchange and annihilation transitions. Thus the relevant quantitative results might not be trustworthy, but only serve as illustration of the possible qualitative effects. A reliable examination of the true role of W exchange and annihilation quark diagrams playing in different types of hadronic rare B decays deserves further theoretical and experimental efforts.

I am grateful to H. Fritzsch for his warm hospitality. I am also indebted to C.O. Dib for his patient help in understanding Ref. [14] and to Y.Q. Chen for a useful conversation. This research was supported by the Alexander von Humboldt Foundation of Germany.

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References

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Gronau, J.L. Rosner, and D. London, Phys. Rev. Lett. 73, 21 (1994).

[5] T.E. Browder, K. Honscheid, and S. Playfer, in B Decays (2nd edition), edited by S. Stone

(World Scientific, Singapore, 1994).

[6] M. Bauer, B. Stech, and M. Wirbel, Z. Phys. C34, 103 (1987). [7] A.J. Buras, J.M. G'erard, and R. R"uckl, Nucl. Phys. B268, 16 (1986); B. Blok and M.

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[8] J. Bernab'eu and C. Jarlskog, Z. Phys. C8, 233 (1981). [9] G.P. Lepage and S.J. Brodsky, Phys. Lett. B87, 359 (1979). [10] C.H. Chang and Y.Q. Chen, Phys. Rev. D49, 3399 (1994). [11] Z.Z. Xing, Report No. LMU-13/95 (talk presented at the Conference on Production and

Decay of Hyperons, Charm and Beauty Hadrons, Strasbourg, France, September 5 - 8, 1995).

[12] S. Narison, Phys. Lett. B341, 73 (1994); D.S. Hwang and G.H. Kim, Report No.  (to be unpublished).

[13] Y. Koide, Phys. Rev. D40, 1685 (1989). [14] N.G. Deshpande and C.O. Dib, Phys. Lett. B319, 313 (1993).

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*_ -

oe

oe oe ?? ?? ?? ??

b

_ffi Bffi

ff

_fi _fl fl Y

X

(a) b

_ffi Bffi

ff

_fi _fl fl Y

X,ss '`'&-

oe

oe -

oe ^^^^

(b) Figure 1: A graphic description of the two-body mesonic decay Bffi(b_ffi) ! X(ff_fl) + Y (fl _fi): (a) the W -exchange diagram with ffi = d or s, ff = u or c, fi = c or u, and fl = u, d or s; (b) the annihilation diagram with ffi = u or c, ff = d or s, fi = u or c, and fl = u, d or s.

10\Gamma 2 10\Gamma 1

100 10+1 10+2 10+3

\Gamma 2000 \Gamma 1000 00 +1000 +2000 R(\Delta OE)

\Delta OE , ! 0 , ? 0

Figure 2: The change of R(\Delta OE) as a function of the rescattering phase shift \Delta OE.

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