# Prediction of Pure Annihilation Type B Decays

\addressInstitute of High Energy Physics, CAS, P.O.Box 918(4), Beijing 100039, China

The rare decays , and can occur only via annihilation type diagrams in the standard model. We calculate these decays in perturbative QCD approach. We found that the calculated branching ratio of agree with the data which had been observed in the KEK and SLAC B factories. The decay has a very small branching ratio at O (), due to the suppression from CKM matrix elements. The branching ratio of is of order which may be measured in the near future by KEK and SLAC B factories. The small branching ratios predicted in the standard model make these channel sensitive to new physics contributions.

## 1 Introduction

The generalized factorization approach has been applied to the theoretical treatment of non-leptonic decays for years [1]. It is a great success in explaining many decay branching ratios[2]. The factorization approach (FA) is a rather simple method. Some efforts have been made to improve their theoretical application[3] and to understand the reason why the FA has gone well[4]. One of these methods is the perturbative QCD approach (PQCD), where we can calculate the annihilation diagrams as well as the factorizable and nonfactorizable diagrams.

The rare decays are pure annihilation type decays. In the usual FA, this decay picture is described as meson annihilating into vacuum and the and mesons produced from vacuum then afterwards. To calculate these decays in the FA, one needs the form factor at very large time like momentum transfer . However the form factor at such a large momentum transfer is not known in FA. The annihilation amplitude is a phenomenological parameter in QCD factorization approach (QCDF)[3], and the QCDF calculation of these decays is also unreliable. Here, we will try to use the PQCD approach, to evaluate the decays. By comparing the predictions with the experimental data, we can test the PQCD evaluation of the annihilation amplitude.

A boson exchange causes or , which is usually described by the effective four quark operators, and the additional quarks included in are produced from a gluon. This gluon attaches to any one of the quarks participating in the four quark operator. In the rest frame of meson, and quarks included in each has momenta, and the gluon producing them has . This is a hard gluon. One can perturbatively treat the process where the four quark operator exchanges a hard gluon with quark pair. Therefore the quark picture becomes six-quark interactions. The decay amplitude is then expressed as product of the hard six quark operators and the non-perturbative meson wave functions.

## 2 Framework

PQCD approach has been developed and applied in the non-leptonic meson decays[5, 4, 6] for some time. In this approach, the decay amplitude is separated into soft(), hard(), and harder() dynamics characterized by different scales. It is conceptually written as the convolution,

(1) |

where ’s are momenta of light quarks included in each mesons, and denotes the trace over Dirac and color indices. is Wilson coefficient of the four quark operator. In the above convolution, includes the harder dynamics at larger scale than scale and describes the evolution of local -Fermi operators from , down to the scale , where . describes the four quark operator and the spectator quark connected by a hard gluon whose is at the order of , and includes the hard dynamics characterized by the scale . Therefore, this hard part can be perturbatively calculated, which is process dependent. is the wave function which describes hadronization of the quark and anti-quark to the meson . is independent of the specific processes. Determining in some other decays, we can make quantitative predictions here.

The large double logarithms () on the longitudinal direction are summed by the threshold resummation[7], and they lead to which smears the end-point singularities on . The last term, , contains two kinds of logarithms. One of the large logarithms is due to the renormalization of ultra-violet divergence , the other is double logarithm from the overlap of collinear and soft gluon corrections. This Sudakov form factor suppresses the soft dynamics effectively[8]. Thus it makes perturbative calculation of the hard part applicable at intermediate scale, i.e., scale.

In general, having Dirac indices are decomposed into 16 independent components, , , , , . If the considered meson is or meson, to be pseudo-scalar and heavy meson, the structure and components remain as leading contributions. Then, is written by

(2) |

As heavy quark effective theory leads to , we have only one independent distribution amplitude for B meson. The heavy meson’s wave function can also be derived similarly.

In contrast to the and mesons, for the meson, being light meson, the component remains. Then, meson’s wave function is parameterized as

(3) |

where , , . In decay, only longitudinal polarization of the meson wave function is relevant, which is similar to K meson [9].

## 3 Numerical evaluation

In this section we show numerical results. First for the meson’s wave function, we use the same distribution amplitude as adopted in Ref. [4]. This choice of meson’s wave function is almost a best fit from the , decays. For the meson’s wave function, we assume the same form as meson’s one [11]. The wave functions of the meson are expanded by Gegenbauer polynomials, which are given in Ref. [12].

For the neutral decay , the dominant contribution is the nonfactorizable annihilation diagrams, which is proportional to the Wilson coefficient . The factorizable annihilation diagram contribution is proportional to , which is one order magnitude smaller. For the charged decay , it is the inverse situation.

The propagators of inner quark and gluon in FIG. 1 are usually proportional to . One may suspect that these amplitudes are enhanced by the endpoint singularity around . However this is not the case in our calculation. First we introduce the transverse momentum of quark, such that the propagators become . Secondly, the Sudakov form factor suppresses the region of small . Therefore there is no singularity in our calculation. The dominant contribution is not from the endpoint of the wave function. As a proof, in our numerical calculations, for example, an expectation value of in the integration results in , Therefore, the perturbative calculations are self-consistent.

The predicted branching ratios are [9]

(4) | |||

(5) |

for variation of the input parameters of wave functions. They agree with the experimental observation by Belle[13] and BaBar[14],

(6) | |||||

(7) |

and the experimental upper limit given at % confidence level[15]: . For , the predicted branching ratio is [10] which is still far from the current experimental upper limit [15]: .

Despite the calculated perturbative annihilation contributions, there is also hadronic picture for the decay: through final state interaction. Our numerical results show that the PQCD contribution to this decay is already enough to account for the experimental measurement. It implies that the soft final state interaction is not important in the decay. This is consistent with the argument in Ref. [16]. We expect the same situation happens in other decay channels.

## 4 Conclusion

In two-body decays, the final state mesons are moving very fast, since each of them carry more than 2 GeV energy. There is not enough time for them to exchange soft gluons. The soft final state interaction may not be important. This is consistent with the argument based on color-transparency[16]. The PQCD with Sudakov form factor is a self-consistent approach to describe the two-body meson decays. Although the annihilation diagrams are suppressed comparing to other spectator diagrams, but their contributions are not negligible in PQCD approach[4].

## Acknowledgments

We thank the organizers of the conference for local support. This work is partly supported by National Science Foundation of China under Grant (No. 90103013 and 10135060).

## References

## References

- [1] M. Wirbel, B. Stech, M. Bauer, Z. Phys. C29, 637 (1985); M. Bauer, B. Stech, M. Wirbel, Z. Phys. C34, 103 (1987); L.-L. Chau, H.-Y. Cheng, W.K. Sze, H. Yao, B. Tseng, Phys. Rev. D43, 2176 (1991), Erratum: D58, 019902 (1998).
- [2] A. Ali, G. Kramer and C.D. Lü, Phys. Rev. D58, 094009 (1998); C.D. Lü, Nucl. Phys. Proc. Suppl. 74, 227-230 (1999); Y.-H. Chen, H.-Y. Cheng, B. Tseng, K.-C. Yang, Phys. Rev. D60, 094014 (1999); H.-Y. Cheng and K.-C. Yang, Phys. Rev. D62, 054029 (2000).
- [3] M. Beneke, G. Buchalla, M. Neubert, C.T. Sachrajda, Phys. Rev. Lett. 83, 1914 (1999); Nucl. Phys. B591, 313 (2000).
- [4] Y.-Y. Keum, H.-n. Li and A. I. Sanda, Phys. Lett. B504, 6 (2001); Phys. Rev. D63, 054008 (2001); C.-D. Lü, K. Ukai and M.-Z. Yang, Phys. Rev. D63, 074009 (2001); C.-D. Lü, page 173-184, Proceedings of International Conference on Flavor Physics (ICFP 2001), World Scientific, 2001, hep-ph/0110327.
- [5] C.-H. V. Chang and H.-n. Li, Phys. Rev. D55, 5577 (1997); T.-W. Yeh and H.-n. Li, Phys. Rev. D56, 1615 (1997).
- [6] H.-n. Li, Phys. Rev. D64, 014019 (2001); S. Mishima, Phys. Lett. B521, 252 (2001); E. Kou and A.I. Sanda, Phys. Lett. B525, 240 (2002); C.-H. Chen, Y.-Y. Keum, and H.-n. Li, Phys. Rev. D64, 112002 (2001); C.-D. Lü and M.Z. Yang, Eur. Phys. J. C23, 275 (2002); A.I. Sanda and K. Ukai, Prog. Theor. Phys. 107, 421 (2002); C.-H. Chen, Y.-Y. Keum, and H.-n. Li, Phys. Rev. D66, 054013 (2002); M. Nagashima and H.-n. Li, hep-ph/0202127; Y.-Y. Keum, hep-ph/0209002; hep-ph/0209208(to appear in PRL); hep-ph/0210127; Y.-Y. Keum and A. I. Sanda, Phys. Rev. D67, 054009 (2003); C.D. Lü, M.Z. Yang, hep-ph/0212373, to appear in Eur. Phys. J. C..
- [7] H.-n. Li, Phys. Rev. D66, 094010 (2002).
- [8] H.-n. Li and B. Tseng, Phys. Rev. D57, 443, (1998).
- [9] C.D. Lü, Eur. Phys. J. C24, 121 (2002).
- [10] C.-D. Lü, K. Ukai, hep-ph/0210206, to appear at Eur. Phys. J. C; Y. Li, C.D. Lü, hep-ph/0304288.
- [11] T. Kurimoto, H.-n. Li, and A. I. Sanda, Phys. Rev. D67, 054028 (2003).
- [12] P. Ball, JHEP, 09, 005, (1998); JHEP, 01, 010, (1999).
- [13] Belle Collaboration, P. Krokovny et al., Phys. Rev. Lett. 89, 231804 (2002).
- [14] BaBar Collaboration, B. Aubert et al., hep-ex/0207053.
- [15] Review of Particle Physics, K. Hagiwara et al., Phys. Rev. D66, 010001 (2002).
- [16] G.P. Lepage and S.J. Brodsky, Phys. Rev. D22, 2157 (1980); J.D. Bjorken, Nucl. Phys. B (Proc. Suppl.) 11, 325 (1989); C.-H. Chen and H.-n. Li, Phys. Rev. D63, 014003 (2001).