Dynamically enhanced decays in perturbative QCD
Abstract
We investigate the decays, one of hardly understandable processes among charmless B-meson decays, within the perturbative QCD method. Owing to the dynamically enhanced mechanism in PQCD, we obtain large branching ratios and large direct CP asymmetries: , ; , and . The branching ratios are consistent with experimental data and large direct CP violation effects will be tested by near future experimental measurements in Asymmetric B-factory.
pacs:
13.25.Hw, 11.30.Er, 12.38.Bx, 14.40.NdThe predictive power of the perturbative QCD (pQCD) approach has been demonstrated successfully in exclusive 2-body B-meson decays, especially charmless B-meson processes LiYu ; KLS01 ; LUY ; LuYa ; phiK ; phiKst ; KK ; keum which is based on factorization theorem BS . This is a modified version of the pQCD theory for exclusive processesBL . The idea is to separate hard scattering kernels from a high-energy QCD process, which are calculable in a perturbative way . Nonperturbative parts are organized into universal hadron distribution amplitudes, which can be determined from experimental data. By introducing parton transverse momenta , we can generate naturally the Sudakov suppression effect due to the resummation of large double logarithms
In the pQCD approach, we can predict the contribution of non-factorizable term and annihilation diagram on the same basis as the factorizable one. A folklore for annihilation contributions is that they are negligible compared to W-emission diagrams due to helicity suppression. However the operators with helicity structure are not suppressed and give dominant imaginary values, which is the main source of strong phase in the pQCD approach.
An alternative method to exclusive meson decays is QCD-factorization approach (QCDF)BBNS , which is based on collinear factorization theorem.
For some modes, such as the decays, the difference between the pQCD and QCDF approaches may not be signigicant. As explained in ref.KLS01 , the typical hard scattering scale is about 1.5 GeV. Since the RG evolution of the Wilson coefficients increase drastically as , while that of remain almost constant, we can get a large enhancement effects from both wilson coefficents and matrix elements in pQCD.
In general the amplitude can be expressed as
(1) |
with the chiral factors for pseudoscalar meson and for vector meson. To accommodate the data in the factorization and QCD-factorization approaches, one relies on the chiral enhancement by increasing the mass to as large values about 3 GeV at scale. So two methods accomodate large branching ratios of and it is difficult for us to distinguish two different methods in decays. In addition, the direct CP asymmetries in decays are not large enough to distinguish the two approaches after taking into account the theoretical uncertainties.
However the difference can be detected in the direct CP asymmetry of process because of the different power counting rules and the branching ratios of in modes since there is no chiral enhanced factor in LCDAs of the vector meson. Due to the different power counting rules of the QCDF and pQCD approaches, based on collinear and factorizations, respectively, the vertex correction is the leading source of the strong phase in the former, and the annihilation diagram is in the latter. The strong phases derived from the above two sources are opposite in sign, and the latter has a large magnitude. This is the reason QCDF prefers a small and positive CP asymmetry Ben , while pQCD prefers a large and negative KLS01 ; keum .
We can test whether dynamical enhancement or chiral enhancement is responsible for the large branching ratios by measuring the modes. In these modes penguin contributions dominate, such that their branching ratios are insensitive to the variation of the unitarity angle . According to recent works within QCDFCY , the branching ratio of is including annihilation contributions in real part of amplitudes within QCD-factorization approach. However pQCD predicts phiK with mostly pure imaginary annihilation contributions. For decays, QCDF gets about CKY , but pQCD have phiKst . Because of these relatively small branching ratios for and decays in QCD-factorization approach, they can not globally fit the experimental data for and modes simultaneously with same sets of free parameters and zhu . To expalin large branching ratios of modes, they have to break the universality of free parameter sets with and finally lost the predictive power.
In this letter we investigate the more complicated processes, which contain both tree and penguin contributions, while is a pure penguin process. It is well known that it is very difficult to explain the observed branching ratios using the factorization assumption (FA) BSW and QCDFzhu : the experimantal measurements are much larger than the theoretical predictions.
The reason is as follows. The measured are roughly the same as . However, the penguin operators contribute to the latter, but not to the former. Due to the loss of this important piece of contributions, the predicted become a quarter of the predicted . The same difficulty has been encountered in the decays, where the vector meson is replaced by a meson LuYa . This controversy remains, no matter how the angle is varied WS . Hence, the decays is worth of an intensive study.
We shall evaluate the branching ratios of the following modes,
(2) |
and the CP asymmetries, for instance,
(3) |
as functions of the unitarity angle . It will be shown that penguin and annihilation amplitudes in the decays are greatly enhanced by Wilson evolutin effects. There is also small enhancement from the meson wave functions, which are more asymmetric than the kaon wave funcitons, and from the meson decay constant , which is larger than the kaon decay constant . It turns out that these enhancements compensate the loss of the contributions, and that PQCD predictions are in agreement with the data.
The decay rates of have the expressions,
(4) |
The decay amplitudes for the different modes are written as
(5) | |||||
(6) | |||||
(7) | |||||
(8) | |||||
to which , , , and are identical, respectively, but with the the product () of the CKM matrix elements replaced by ().
Scales | ( GeV) | |||
---|---|---|---|---|
Amplitudes | Re ( GeV) | Im ( GeV) | Re ( GeV) | Im ( GeV) |
-1202.0 | — | -1163.0 | — | |
45.9 | — | 35.6 | — | |
-350.1 | 44.2 | -340.7 | 42.8 | |
20.0 | -20.7 | 15.5 | -14.8 | |
-43.7 | 53.2 | -33.8 | 41.1 | |
3.1 | -4.3 | 2.4 | -3.3 | |
-4.4 | -13.0 | -3.4 | 10.2 | |
-0.3 | 0.0 | -0.3 | 0.0 | |
Br. with ann. | ||||
Br. without ann. |
The detail expression of analytic formulas for all amplitudes ( and ) will be presented elsewhereKL . In the above expressions is the meson decay constant. The notations represent factorizable contributions (form factors), and represent nonfactorizable (color-suppressed) contributions. The subscripts and denote the annihilation and W-emission topology, respectively. The superscript denotes contributions from the penguin (Tree) operators. , associated with the time-like - form factor ( form factor), and are from the operators . The factorizable contribution () is associated with the form factor from the penguin (tree) operators, and () is the corresponding nonfactorizable contribution.
In our numerical analysis, we use GeV, the Wolfenstein parameters , , and for the CKM matrix elements, the masses GeV and GeV, and () meson lifetime ps ( ps) LEP . The angle was extracted from the data of the and decays KLS01 ; keum . With all the meson wave functions given in our previous works, we calculate the contributions from all the topologies as shown in Figs. 2 and 3 in paperKLS01 . The allowed range of B-meson shape parameter, and transition form factors. is determined from the reasonable , and chiral factor for pion,
Modes | CELO | Belle | BaBar | PQCD |
— | ||||
— | — | — | ||
— | — | — |
For example, all amplitudes for the modes are listed in Table 1, whose values are mostly the same magitude for other decay channels, because the difference comes only from electroweak penguin contributions. We show in Table 2 the enhancing effect by comparing the decay amplitudes evaluated at the characteristic hard scales in PQCD and in QCDF. It is also found that the annihilation contributions are sizable in two-body charmless meson decays for the heavy-meson mass around 5 GeV KLS01 and in fact contributed about 60% fraction of the branching ratios, since factorized annihilation penguin contribution has large imaginary part and also the same order of magnitudes in real part as one of the factorized penguin contribution. As expected, the dominant factorizable penguin amplitudes are enhanced by about 30% due to the Wilson evolution, more than the factorizable tree amplitudes are.
The PQCD predictions for the branching ratios, presented in Table 3, are consistent with present experimental data. Here the first uncertainty comes from the allowed ranges of both and , and the second one comes from the uncertainty of .
Our predictions for the CP asymmetries in decays are given in Table 4, which have the same sign as of those in decays. For , CP asymmetries of and become large due to the important imaginary penguin annihilation amplitudes.
At last, the dependence of the branching ratios of and on the angle is shown in Fig.1 and large direct CP asymmetries of the and on the angle is exhibited in Fig.2. The branching ratios of and increase with rapidly, while the and modes are insensitive to the variation of . The increase with is mainly a consequence of the inteference between the penguin contribution and the tree contribution . The dependence on both the shape parameter for the meson wave function and the chiral factor is also shown, which is strong in the and modes, and weak in the other two. The sensitivity is attributed to the fact that the former contain both and , which involve the meson wave function, while the latter contain only .
Modes | case A | case B | case C | PQCD |
---|---|---|---|---|
-19.6 % | -19.2 % | -18.7 % | % | |
-4.4 % | -3.6 % | -2.7 % | % | |
-47.9 % | -43.7 % | -39.7 % | % | |
-10.7 % | -9.6 % | -8.8 % | % |
In this letter in order to explain one of the hardly understandable processes in charmless B-decays, we have investigated the dynamical enhancement effect in the decays within pQCD method. Owing to the dynamical enhancement of penguin contributions at GeV, pQCD predictions for all the modes are consistent with the present experimental data, which is a crucial decay process to distinguish pQCD from other approachs. We also predicted large direct CP asymmetry for about -20% and for about -40%, which can be tested in near future measurements. More detail works will appear elsewhereKL .
We wishes to thank S.J. Brodsky, H.Y. Cheng, H.-n. Li, A.I. Sanda and G. Zhu for helpful discussions. This work was supported by Grant-in-Aid for Scientific Research from Ministry of Education, Science and Culture of Japan.
References
- (1) H-n. Li and H.L. Yu, Phys. Lett. B 353, 301 (1995).
- (2) Y.Y. Keum, H-n. Li, and A.I. Sanda, Phys. Lett. B 504, 6 (2001); Phys. Rev. D 63, 054008 (2001); Y.Y. Keum and H-n. Li, Phys. Rev. D63, 074006 (2001).
- (3) C. D. Lü, K. Ukai, and M. Z. Yang, Phys. Rev. D 63, 074009 (2001).
- (4) C.-D. Lu and M.-Z. Yang, Eur. Phys. J. C 23, 275 (2002)
- (5) C.-H. Chen, Y.-Y. Keum and H.-N. Li, Phys. Rev. D 64, 112002 (2001); S. Mishima, Phys. Lett. B 521, 252 (2001);
- (6) C.-H. Chen, Y.-Y. Keum and H.-N. Li, Phys. Rev. D 66, 054013 (2002).
- (7) C.H. Chen and H-n. Li, Phys. Rev. D 63, 014003 (2001).
- (8) Y.Y. Keum, H-n. Li, and A.I. Sanda, hep-ph/0201103; Y.Y. Keum and A.I. Sanda, hep-ph/0209014; Y.-Y. Keum, hep-ph/0209002; hep-ph/0209208 (Accepted in Phys. Rev. Lett.).
- (9) J. Botts and G. Sterman, Nucl. Phys. B225, 62 (1989); H-n. Li and G. Sterman, Nucl. Phys. B381, 129 (1992).
- (10) G.P. Lepage and S.J. Brodsky, Phys. Lett. B 87, 359 (1979); Phys. Rev. D 22, 2157 (1980).
- (11) M. Beneke, G. Buchalla, M. Neubert, and C.T. Sachrajda, Phys. Rev. Lett. 83, 1914 (1999); Nucl. Phys. B591, 313 (2000).
- (12) M. Beneke, hep-ph/0207228
- (13) H.Y. Cheng and K.C. Yang, Phys. Rev. D 64, 074004 (2001).
- (14) H.Y. Cheng, Y.-Y. Keum and K.C. Yang, Phys. Rev. D 65, 094023 (2002).
- (15) M. Bauer, B. Stech, M. Wirbel, Z. Phys. C 29, 637 (1985); Z. Phys. C 34, 103 (1987).
- (16) D. Du. J. Sun, D. Yang, and G. Zhu, Phys. Rev. D 65, 074001 (2002); Phys. Rev. D 65, 094025 (2002); hep-ph/0209233.
- (17) N.G. Deshpande, X.G. He, W.S. Hou and, S. Pakvasa, Phys. Rev. Lett. 82, 2240 (1999); W.S. Hou, J.G. Smith, and F. Würthwein, hep-ex/9910014.
- (18) Review of Particle Physics, Eur. Phys. J. C 3, 1 (1998).
- (19) Y.-Y. Keum, in preparation.