# Hadronic Penguin B Decays In The Standard

And The Two-Higgs-Doublet Models
^{1}^{1}1Work supported in part by the Department of Energy Grant No.
DE-FG06-85ER40224.

###### Abstract

We study in next-to-leading order QCD hadronic penguin decays in the Standard and two-Higgs-doublet models. Although the gluonic penguin dominates, we find the electroweak contribution non-negligible. In the Standard Model, the branching ratio for is predicted to be in the range . The ranges of branching ratios for , , and are , , and , respectively. The contribution from the charged Higgs boson in two Higgs doublet models depend on , and can be as large as 40%.

###### pacs:

^{†}

^{†}preprint: OITS-538

Rare decays, particularly pure penguin decays, have been subject of considerable theoretical and experimental interest recently[1]. The photonic penguin induced process has been observed by CLEO collaboration[2] and is consistent with the Standard Model (SM) prediction[3]. The gluonic penguin induced decays are expected to be observed very soon. A large number of gluonic penguin induced decay channels were studied in Ref.[4] using effective Hamiltonian in the lowest nonvanishing order. In Ref.[5] the next-to-leading order QCD corrected pure gluonic penguin was used with top quark mass fixed at 150 GeV. In this paper we study the next-to-leading order QCD corrected Hamiltonian in the SM and in two Higgs doublet models, taking particular care to include the full electroweak contributions and find the dependence on and . Using this Hamiltonian we study the cleanest signature of hadronic penguin processes: , , and . The process is particularly recommended because it is free from form factor uncertainties. We find not only that the QCD correction in next-to-leading order are large, but also inclusion of the full electroweak contributions have significant effect on the branching ratio which could reduce the pure gluonic penguin contribution by 30% at the upper range of allowed top quark mass. Our results which have been derived independently, agree with Ref.[6] where only the SM is considered.

gluonic penguin Hamiltonian

The QCD corrected relevant to us can be written as follows[7]:

(1) |

where the Wilson coefficients (WCs) are defined at the scale of ; and are defined as

(2) | |||||

The WCs are obtained by solving the renormalization group equation

(3) |

Here is the column vector , and

(4) |

where is the number of active quark flavours.

The anomalous-dimension matrix and the first term in determine the leading log QCD corrections[8]. The rest of the terms contain information about the leading QED and next-to-leading order QCD corrections. The full matrices for are given in Ref.[7]. The matching conditions of the Wilson coefficients at for the next-to-leading order corrections will be different from the leading order ones. One needs to include one loop current-current corrections for at . The full results for the initial conditions can be found in [7].

The WCs obtained above depend on the renormalization regularization scheme (RS) used. In our calculation we used the naive dimentional regularization scheme. The physical quantities, of course, should not depend on RS provided one handels the hadronic matrix elements correctly. In practice, many of the hadronic matrix elements can only be calculated using factorization method. In our later calculation we will also use this approxmation. Since this approximation does not carry information about the RS dependence, it is better for us to use WCs, , which are RS independent[9]. Here the matrices are obtained from one-loop matching conditions. The matrix for the pure gluonic penguin operators has been given in Ref.[9]. Based on the work of Ref.[9], we have worked out the full matrices for and carried out the calculation using the full matrices.

We also need to treat the matrix elements to one-loop leve for consistency. These one-loop matrix elements can be rewritten in terms of the tree-level matrix elements of the effective operators, and one obtains[6, 10]

(5) |

We have worked out the full matrices . For the processes we are considering only contribute. Expressing the effective coefficients which multiply the matrix elements in terms of , we have

(6) |

The leading contributions to are given by: and . Here is the charm quark mass which we take to be 1.35 GeV. The function is give by

(7) |

In the numerical calculation, we will use which represents the average value and the full expressions for .

Using range of values of and we can calculate the coefficients at . We use as input instead of as in Ref.[9]. In Table 1, we show some sample WCs for the central world average value of [11] and for several values of with .

In the two-Higgs-doublet model, there are new contributions to due to charged Higgs boson. The charged Higgs-quark couplings are given by [12]

(8) |

where ; and are the vacuum expectation values of the Higgs doublets and , which generate masses for down and up quarks, respectively. The parameter depends on the models[12]. The main contributions are from the first term in eq.(8) and we will neglect the contribution from the second term. The charged Higgs contributions to gluonic penguin have been studied by several groups[13]. The leading QCD corrected Hamiltonian has been given in Ref.[14]. We have checked the next-to-leading initial conditions for the WCs at . We find that the inclusion of charged Higgs will not change the initial conditions for , but are changed in the same way as those given in eqs.(32-38) of Ref.[14].

Using in eq.(1), we obtain the decay amplitude for

(9) |

where is the polarization of the particle; with , where is the number of colors. The coupling constant is defined by . From the experimental value for , we obtain .

The decay rate is, then, given by

(10) |

where .

We normalize the branching ratio to the semi-leptonic decay of . We have

(11) |

In the above expression, with , is the phase factor, and is the QCD correction factor in , respectively. We will use , [15], and the approximation . The branching ratio is measured to be [16].

Exclusive decays and

For the exclusive decays, we will use the factorization method. We have

(12) |

We can parametrize the matrix elements as

where is a vector meson particle and its polarization. For , , and for , .

In terms of the form factors defined above, we obtain the decay rates

(14) | |||||

To finally obtain the branching ratios, we will use two sets of form factors obtained by Bauer et. al.[17] and Casalbuoni et. al.[18]. Note that we have used different normalization for the form factors and from those in Refs. [17, 18]. The form factors at are determined by using relativistic quark model in Ref.[17], and by using chiral and effective heavy quark theory in Ref.[18]. The form factors at in Ref.[17] are given by: , , , and . In Ref.[18] the form factors at are: , , , and . In Ref.[18] the form factors at for are also calculated. They are: , , and . In both papers, the dependence of all the form factors were assumed to be of a simple pole type. We will use the pole masses used in Refs.[17, 18]. It is interesting to note that the ratios between the exclusive decays and are independent of the Wilson coefficients. If these ratios can be measured experimentally, they can test the models for the form factors. We obtain

We show in, Fuigure 1, the predictions for the branching ratio in the SM as a function of top quark mass and the strong coupling constant . The QCD corrections turn out to be important which enhance the branching ratios by about 30% compared with those of without QCD corrections. There is a large uncertainty in the branching ratios due to error in . From Figure 1, we see that the error in can induce an uncertainty of a factor 2.

The dominant contribuitons are from the gluonic penguin. There is a very small dependence for the branching ratio calculated without the inclusion of the electroweak penguin contributions. The inclusion of the full electroweak contribuitons have sizeable effects which reduce the branching ratios by about 20% to 30% for the central value of with varying from 100 GeV to 200 GeV. It is clear from Figure 1 that the full contribution has a large dependence.

There may be corrections to the branching ratios predicted by the factorization method. It is a common practice to parameterize the possible new contributions by treating as a free parameter[17, 18, 19]. Using experimental values from non-leptonic decays, it is found that[18], and have the same signs, and and . We see that is close to 1/2. To see the effect of varying , we plot the predictions for the branching ratios for and . The branching ratios for are about 2 times those for .

For the central value of and the central value of GeV reported by CDF[20], the value for is about for . The exclusive branching ratios and are about the same which are if the form factors from Ref.[17] are used. If the form factors from Ref.[18] are used, one obtains , , and .

In Figure 2, we show the ratio of the branching ratios and predicted by the two Higgs doublet model and the SM as a function of for GeV and different values of with . The depence on is small. From Figure 2, we see that the effects of the charged Higgs boson contributions are small for . When increasing , the charged Higgs contributions become important and the effect is to cancel the SM contributions. When becomes very large the charged Higgs boson contributions become the dominant ones. However, using the information from , it is found that for small GeV and GeV, is constrained to be less than 1[21]. For these values, the charged Higgs boson effects on the processes discussed in this paper are less than 10%. For GeV, the charged Higgs boson effects can reduce the hadronic penguin decays by 40% because the range of allowed from is now larger[21]. The effects become smaller for larger .

The analyses carried out in this letter can be generalized to other hadronic decays. We will present the full calculations for the Wilson coefficeints, the full expressions for and other related decays in a forthcoming paper[22].

We thank Buras, McKellar, Fleischer for useful corespondences and thank Lautenbacher for many useful discussions.

## References

- [1] For a review see: , edited by S. Stone, World Scientific, 1992.
- [2] CLEO Collaboration, Phys. Rev. Lett. 71, 674(1993).
- [3] N.G. Deshpande et. al., Phys. Rev. Lett. 59, 183(1987); S. Bertolini, F. Borzumati and A. Masiero, , 180(1987); N.G. Deshpande, P. Lo and J. Trampetic, Z. Phys.C40, 369(1988); C. Dominguez, N. Paver and Riazuddin, Phys. Lett. B214, 459(1988); A. Ovchinnikov and V. Slobodenyuk, Phys. Lett. B237, 569(1990); P.J. O’Donnel and H.K.K. Tung, Phys. Rev.D44, 741(1991); R. Casalbuoni et. al., Phys. Lett. B312, 315(1993).
- [4] N.G. Deshpande and J. Trampetic, Phys. Rev. D41, 895(1990).
- [5] A. Deandrea, et. al., Phys. Lett. B320, 170(1993).
- [6] R. Fleischer, Preprint, TUM-T31-40/93 (Z. Phys. in press).
- [7] A. Buras, M. Jamin, M. Lautenbacher and P. Weisz, Nucl. Phys. B400, 37(1993); A. Buras, M. Jamin and M. Lautenbacher, ibid, 75(1993); M. Ciuchini, E. Franco, G. Martinelli and L. Reina, Nucl. Phys. B415, 403(1994).
- [8] F. Gilman and M. Wise, Phys. Rev. D20, 2392(1979); R. Miller and B. McKellar, Phys. Rept. 106, 169(1984).
- [9] A. Buras, M. Jamin, M. Lautenbacher and P. Weisz, Nucl. Phys. B370, 69(1992).
- [10] R. Fleischer, Z. Phys. C58, 483(1993); G. Kramer, W. Palmer and H. Simma, Preprint, DESY-93-192.
- [11] S. Bethke, in Proceedings of the XXV International Conference on High Energy Physics, Dallas, Texas, August, 1992.
- [12] J. Gunion, H. Haber, G. Kane and S. Dawson, (Addison-Wesley, Redwood City, CA 1990).
- [13] Wei-Shu Hou and R.S. Willey, Phys. Lett. B202,591(1988); Wei-Shu Hou, Nucl. Phys. B326, 54(1989); V. Barger, J. Hewett and R. Phillips, Phys. Rev. D41, 3421(1990); A. Davies, G.C. Joshi and M. Matsuda, Z. Phys. C52,97(1991); A. Davies, T. Hayashi, M. Matsuda and M. Tanimoto, Preprint, AUE-02-93.
- [14] G. Buchalla, A. Buras, M. Harlander, M. Lautenbacher and C. Salazar, Nucl. Phys. B355, 305(1991).
- [15] N. Cabbibo and L. Maiani, Phys. Lett. B79, 109(1978); M. Suzuki, Nucl. Phys. B145, 420(1978); N. Cabbibo, G. Corbe and L. Maiani, Phys. Lett. B155, 93(1979).
- [16] P. Drell, in Proceedings of the XXV International Conference on High Energy Physics, Dallas, Texas, August, 1992.
- [17] M. Bauer, B. Stech and M. Wirbel, Z. Phys. C34, 103(1087).
- [18] A. Deandrea, N. Di Bartolomeo, R. Gatto and G. Nardulli, Phys. Lett. B318, 549(1993).
- [19] N.G. Deshpande, M. Gronau and D. Sutherland, Phys. Lett. B90, 431(1980).
- [20] F. Abe, et al., CDF Collaboration, Preprint, FERMILAB-PUB-94/097-E, CDF/PUB /TOP/PUBLIC/2561.
- [21] J.L. Hewett, Phys. Rev. Lett. 70, 1045(1993); V. Barger, M. Berger, R. Phillips, , 1368(1993).
- [22] N. Deshpande and Xiao-Gang He, in preparation.

(GeV) | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

130 | -0.313 | 1.150 | 0.017 | -0.037 | 0.010 | -0.045 | -0.061 | 0.029 | -0.978 | 0.191 |

174 | -0.313 | 1.150 | 0.017 | -0.037 | 0.010 | -0.046 | -0.001 | 0.049 | -1.321 | 0.267 |

210 | -0.312 | 1.150 | 0.018 | -0.038 | 0.010 | -0.046 | 0.060 | 0.069 | -1.626 | 0.334 |

Figure Captions

Figure 1. as a function of and . The regions between the dashed and solid lines are the branching ratios for varying from 0.111 to 0.125 for and , respectively. The branching ratios increases with .

Figure 2. as a function of , and . The curves 1, 2 and 3 are for equals to 100, 500 and 1000 GeV, respectively.