Abstract
We consider the effective Hamiltonian of four quark operators in the Standard Model in the exclusive and quasiinclusive decays of the type , , where contains a single Kaon. Working in the factorization assumption we find that the four quark operators can account for the recently measured exclusive decays and for appropriate choice of form factors but cannot explain the large quasiinclusive rate.
UH51186497
ISUHET9707
July 1997
QuasiInclusive and Exclusive decays of to
A. Datta^{1}^{1}1email: , , X.G. He^{2}^{2}2 and S. Pakvasa^{3}^{3}3
Department of Physics and Astronomy
Iowa State University,
Ames, Iowa 50011
Department of Physics
University of Toronto,
Toronto, Ontario, Canada M5S 1A7
School of Physics
University of Melbourne,
Parkville, Victoria, 3052
Australia
And
Department of Physics and Astronomy
University of Hawaii,
Honolulu, Hawaii 96822
1 Introduction
Recently CLEO has reported a branching ratio for the process at [1]. Upper limits on related exclusive decay modes with or with a mesons in the final states were also obtained. The quasiinclusive process for high momentum was also measured with a high rate [2]
(1) 
where stands for one Kaon and up to 4 pions with at most one .
Several explanations for the large quasiinclusive decay rate have been proposed both within the Standard Model and beyond [3, 4, 5, 6]. However, some of these analyses have underestimated the effect of the effective four quark operators to this process. For example, it has been assumed that this contribution is small based on the smallness of [4]. However, the penguin contribution to this process cannot be neglected. A simple estimate based on the magnitudes of the CKM elements associated with the tree and the penguin diagrams clearly suggests that this process is dominated by penguin contributions if one considers only the four quark effective Hamiltonian. In this paper we calculate the contribution of the effective Hamiltonian of the four quark operators to the exclusive decays and along with the quasiinclusive decay of the .
In the sections which follow, we describe the effective Hamiltonian of four quark operators, our calculation of the the exclusive and quasiinclusive rates.
2 Effective Hamiltonian
In the Standard Model (SM) the amplitudes for hadronic decays of the type are generated by the following effective Hamiltonian [7]:
(2) 
where the superscript indicates the internal quark, can be or quark and can be either a or a quark depending on whether the decay is a or process. The operators are defined as
(3)  
where , and is summed over u, d, and s. are the tree level and QCD corrected operators. are the strong gluon induced penguin operators, and operators are due to and Z exchange (electroweak penguins), and “box” diagrams at loop level. The Wilson coefficients are defined at the scale and have been evaluated to nexttoleading order in QCD. The are the regularization scheme independent values obtained in Ref. [8]. We give the nonzero below for GeV, , and GeV,
(4) 
where is the number of color. The leading contributions to are given by: is given by . The function and
(5) 
All the above coefficients are obtained up to one loop order in electroweak interactions. The momentum is the momentum carried by the virtual gluon in the penguin diagram. When , develops an imaginary part. In our calculation, we use MeV, MeV, MeV, GeV [9, 10] and use .
3 Matrix Elements for , and
The effective Hamiltonian described in the previous section consists of operators with a current current structure. Pairs of such operators can be expressed in terms of color singlet and color octet structures which lead to color singlet and color octet matrix elements. We use the factorization approximation, where one separates out the currents in the operators by inserting the vacuum state and neglecting any QCD interactions between the two currents. The basis for this approximation is that, if the quark pair created by one of the currents carries large energy then it will not have significant QCD interactions. Factorization appears to describe nonleptonic B decays rather well[11] . To accommodate some deviation from this approximation one can treat , the number of colors that enter in the calculation of the matrix elements, as a free parameter though the value of is suggested by experimental data on low multiplicity hadronic decays. In this section we describe the calculation of matrix elements for the exclusive decays , and the quasiinclusive decay .
The and mesons are mixtures of singlet and octet states and of .
(6)  
(7)  
(8) 
where the mixing angle lies between and [10]. We can express the decay constants in terms of octet and singlet decay constants ,
(9)  
(10) 
and similarly one has
(11)  
(12) 
In the and Nonet symmetry limit, the relations MeV and hold. However these symmetries are not exact and we will consider values for and away from this symmetry limit. For example the value of indicates about a 30% SU(3) breaking effect.
3.1 Exclusive Decay
In this section we calculate the rates for the two body noleptonic decay for and .
To evaluate the rates for the exclusive decays we have to calculate matrix element of the type where is a or and , as already described in the previous section, has the form.
(13) 
with , and .
As an input to our calculation we need the form factors defined through
(14)  
where .
The matrix element is similarly described in terms of form factors and .
For the modes we define
(15) 
The form factors in this case are defined as
(16)  
with
where .
3.1.1
The amplitude for can be written as
(17) 
where
with , , , , , , , and is effective number of colors.
In the above equations we have used the quark equations of motion to simplify certain matrix elements. The expression for the amplitude can also be used for by making the necessary changes. It is also straight forward to write down the amplitudes for and decays.
(18) 
where
with , and we will use MeV. A similar expression for can also be obtained.
3.1.2
We give below expressions for the various decays
(19) 
(20) 
(21) 
where , , and , and ,
3.2 Quasiinclusive Decay
The technique to handle quasiinclusive decays have been described in detail in Ref.[12]. We represent the total amplitude as the sum of a “two body” and a “three body” piece. The “two body” amplitude is given by
(22)  
where
with and
The “three body” piece has the form
(23)  
with
Details of the calculations of the branching fraction and decay distributions are given in Ref [12].
4 Results and Discussion
For the exclusive decays to the the inputs to the calculation are the form factors, the values of or alternately the values of and , and the mixing angle. We will try to fit the experimental number for by assuming and . We will take , the effective number of colors. We find that there are several solutions corresponding to different values for the set of parameters that can reproduce the experimental data on and the upper limit of if we use the form factors in Ref[13] ^{4}^{4}4 For the BSW model the form factor for and are approximately same and so we assume this equality for the form factors in Ref [13] where only the form factor is calculated. For the CKM parameters we choose two sets ( and ) and ( and ) [15]. In Table.1 we give the rates for the decays involving a and , and in the final state for and for two sets of the CKM parameters with various values of and . Phenomenological studies involving radiative decays of the and indicate values for and [16]. We show the results of using the form factors in Ref [13] and Ref [14] in the third and fourth column of Table. 1. The entries in Table. 1 correspond to the choice of CKM parameters and for in the final state we also include a second number in the parentheses corresponding to the second set of CKM parameters though the data seem to favor a positive value of . The branching ratio for is suppressed by about a factor of 10 or more relative to while is enhanced relative to by a small amount. This is in qualitative agreement with Ref[17] but we find a smaller enhancement relative to . From the branching ratios in Table. 1, it is not possible to rule out the four quark operator explanation for the large branching ratio in .
Process  Experimental BR [1]  Branching Ratio [13] (BR)  BR [14]  
1.0,1.0  

1.1 , 1.3  

1.1 , 1.5  

1.0,1.0  

1.1 , 1.3  

1.1 , 1.5  
1.0,1.0  

1.1 , 1.3  

1.1 , 1.5  
1.0,1.0  

1.1 , 1.3  

1.1 , 1.5  
1.0,1.0  

1.1 , 1.3  

1.1 , 1.5  
1.0,1.0  

1.1 , 1.3  

1.1 , 1.5  
–  
–  
–  
–  
–  
—–  – 
For the quasiinclusive decay we will use the same parameters as used in the exclusive decays with the second set of the CKM parameters as this set gives the larger rate between the two choices. We plot the decay distribution in Fig. 1 showing the contributions from the effective Hamiltonian of four quark operators. From Fig. 1 we find the contribution from the four quark operators to the branching ratio is around for which is the signal region. This is far too small to account for the signal observed at high mass. This continues to be true even if when all parameters are allowed to vary over a reasonable range.
5 Conclusion
Summarizing, we have calculated the the effects of the effective Hamiltonian of four quark operators to the exclusive and quasiinclusive decays of the B meson to . Our analysis indicate that the exclusive data can be explained by the four quark operators without significant contribution from the mechanism , or from the intrinsic charm content of the . The contribution to the quasiinclusive rate from the four quark operator is not enough to account for the observed signal.
6 Acknowledgments
This work was supported by the United States Department of Energy under contracts DEFG 0292ER40730 and DEFG 0394ER40833 and by the Australian Research Council. We thank T.E. Browder for useful discussions.
References
 [1] S. Anderson et.al. CLEO CONF 9722a, EPS97333.
 [2] D. M. Asner et.al. CLEO CONF 9713, EPS97332.
 [3] D. Atwood and A. Soni, Phys. Lett. B 405, 150 (1997).
 [4] I. Halperin and A. Zhitnitsky, eprint hepph/9705251.
 [5] W.S. Hou and B. Tseng, eprint hepph/9705304.
 [6] F. Yaun, KT. Chao Phys. Rev. D 56, 2495 (1997).
 [7] M. Lautenbacher and P. Weisz, Nucl. Phys. B 400, 37 (1993); A. Buras, M. Jamin and M. Lautenbacher, ibid, 75 (1993); M. Ciuchini, E. Franco, G. Martinelli and L. Reina, Nucl. Phys. B 415, 403(1994).
 [8] R. Fleischer, Z. Phys. C 58 438; Z. Phys. C 62 81; G. Kramer, W. F. Palmer and H. Simma, Nucl. Phys. B 428 77 (1994); Z. Phys. C 66 429, (1994); N.G. Deshpande and X.G. He, Phys. Lett. B 336, 471 (1994).
 [9] J. Gasser and H. Leutwyler, Phys. Rep. 87, 77 (1982).
 [10] K. Hikasa et al. (Particle Data Group), Phys. Rev.D 54, 303 (1996).
 [11] M. Neubert, B. Stech, hepph 9705292.
 [12] T. E. Browder, A. Datta, X.G. He and S. Pakvasa, hepph 9705320.
 [13] R. Casalbouni et al., Phys. Lett. 299, 139 (1993); A. Deandrea, N. Di Bartolomeo, R. Gatto, and G. Nardulli, Phys. Lett. B 318, 549 (1993).
 [14] M. Bauer, B. Stech and M, Wirbel, Z. Phys. C 34, 103 (1987).
 [15] A. Ali and D. London, Nucl. Phys. proc. suppl. 54 A, 29, 1997, also hepph 9607392.
 [16] P. Ball, J.M. Frère and M. Tytgat, Phys. Lett. B 365, 367 (1996).
 [17] H. Lipkin, Contributed paper to Conference on B physics and CP Violation, Honolulu, Hawaii, March 2427, 1997 and references therein.
7 Figure Caption

Fig. 1 This figure shows the decay distribution as a function of the recoil mass .