New Physics Effects on Rare Decays B + u π+ l + l, ρ + l + l in a Top Quark Two-Higgs-Doublet Model

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1 Commun. Theor. Phys. Beijing, China pp c Chinese Physical Society Vol. 50, No., September 15, 2008 New Physics Effects on Rare Decays B + u + l + l, + l + l in a Top Quark Two-Higgs-Doublet Model SONG Hai-Zhen, 1 LÜ Lin-Xia1, and LU Gong-Ru 2 1 Department of Physics, Nanyang Normal University, Nanyang 47061, China 2 Department of Physics, Henan Normal University, Xinxiang 45007, China Received January 11, 2008; Revised April 15, 2008 Abstract Using the form factors from light-cone sum rules, we study the branching ratios and forward-backward asymmetries FBAs of the exclusive decays B + u + l + l and B + u + l + l l = e, µ in the standard model SM and the top quark two-higgs-doublet model T2HDM. From the numerical results, we find that the new physics contributions cannot provide very large enhancement to the branching ratios and the theoretical predictions are in good agreement with the SM ones. The T2HDM ects on FBAs of these decays are small. Precision measurements of the dilepton invariant mass distributions, especially in the lower dilepton mass region, and the FBAs in the decays B + u + + l + l will greatly help in discriminating among the SM and the new physics models. PACS numbers: 1.20.He, Ji, Fr, Nd Key words: new physics ects, rare decay, two-higgs-doublet model, branching ratio, forward-backward asymmetry 1 Introduction Rare B meson decays, induced by flavor-changing neutral current FCNC b sd transitions, provide potentially stringent tests of the standard model SM in flavor physics. FCNC transitions are forbidden at the tree level and they proceed at a low rate via penguin and box diagrams in the SM. In addition, these transitions may also be parametrically suppressed since they depend on the quark-flavor rotation matrix the Cabibbo Kobayashi Maskawa CKM matrix. [1] Thus FCNC decays are relatively rare and potentially ective tools in searching for new physics. Along with the excellent performance of B factory experiments and the undergoing large hadronic collider beauty LHCb and other B meson related experiments, precision measurements of the radiative and semileptonic decays d sd transitions will be achieved. The beststudied radiative decay B X s γ has become the standard candle of the flavor physics and provides strong constraints on parameter space of the new physics models. The FCNC semileptonic b sl + l the inclusive decays B X s l + l and their exclusive modes B K l + l have been measured by BABAR [2] and Belle [] experiments and the measurements agree with the SM estimates at NLO level. Furthermore, the radiative b dγ penguin processes B γ and ωγ have also been measured. [4,5] Along with these progresses, BABAR, Belle and CDF have begun to probe b dl + l decays. Recently, BABAR has made a search for B l + l decays. [6] It is worth while to note that the new upper limit of BB l + l < at 90% C.L. [6] has improved the previous upper limits [7] by four orders of magnitude. The renewed upper limit from BABAR [6] will give powerful constraints on new physics. The physical aspects of the rare B decays B u + + l + l and B u + + l + l, induced by b dl + l, are similar to the b sl + l decays. In the context of the SM, these decays have been studied in a number of papers, [8 15] with different degrees of theoretical emphasis. At present, the partial results in the SM are known to next-to-next leading order NNLO, [16 18] which corresponds to NLO in b sγ. The possible new physics contributions to these decays induced by loop diagrams involving various new particles have been carried out in different models, for example, the two-higgs doublet model, [19,20] the supersymmetric SUSY models, [21 2] the SUSY SO10 grand unification theory. [24] The aim of this paper is to study these decays in the top quark two-higgs-doublet model T2HDM [25] by using the new light-cone QCD sum rules for B form factors [26] and the electro-weak parameters. [27] The phenomenological constraints on the new parameters in the T2HDM and the new particles contributions to the semileptonic B decays have been investigated in Refs. [28] [2]. This paper is organized as follows. In Sec. 2, we will present a brief review for the top quark two-higgs-doublet The project partly supported by National Natural Science Foundation of China under Grant No , the Specialized Research Fund for the Doctoral Program of Higher Education SRFDP under Grant No , and the Special Study Foundation of Nanyang Normal University under Grant No. nynu lvlinxia@sina.com

2 No. New Physics Effects on Rare Decays B + u + l + l, + l + l in a Top Quark Two-Higgs-Doublet Model 697 model. The ective Hamiltonian in the T2HDM and the matrix elements expressed by some form factors are also given. In Sec., we show the basic formula of the observables. In Sec. 4, we give the numerical results of the branching ratios and the normalized forward-backward asymmetries for B + u + l + l and B + u + l + l decays. Moreover, we will compare with the SM predictions. The conclusions are included in the final section. 2 Effective Hamiltonian in T2HDM and Form Factors The top quark two-higgs-doublet model considered here was proposed in Ref. [25] and studied for example in Refs. [28] [2], which is also a special case of the 2HDM of type III. [] In this model, the top quark is much heavier than the other quarks and leptons because it couples to a Higgs doublet with a much larger vacuum expectation value. The Lagrangian density of Yukawa interactions of the T2HDM can be of the following form, [25] L Y = L L φ 1 El R Q L φ 1 Fd R Q L φ1 G1 1 u R Q L φ2 G1 2 u R + H.c., 1 where φ i i = 1, 2 are the two Higgs doublets with φ i = iτ 2 φ i ; and E, F, and G are the generation space matrices; Q L and L L are -vector of the lefthanded quark and lepton doublets; 1 1 diag 1, 1, 0; 1 2 diag 0, 0, 1 are the two orthogonal projection operators onto the first two and the third families respectively. The Yukawa couplings for quarks are given in Ref. [25]. In the T2HDM, there are five Higgs bosons. H ± are charged Higgs bosons, while the CP-even H 0, h 0 and CP-odd A 0 are the so-called neutral Higgs bosons. The ective weak Hamiltonian encompassing b dl + l process has been introduced by the authors of Ref. [20]. Following their notation we write it as: H = 4G F 2 V tbv td 10 i=1 [C i µo i µ + C Qi µq i µ], 2 where V td V tb is the CKM factor, and G F is the Fermi coupling constant. C i and C Qi are the Wilson coicients at the renormalization point µ. O i s i = 1,..., 10 are the operators in the SM and are the same as those given in Refs. [9] and [16], while Q i s come from the diagrams due to the neutral Higgs bosons exchange in T2HDM and can be found in Ref. [20] with sb replaced by bd. The above ective Hamiltonian leads to the following free quark decay amplitude for b dl + l process, [11,12] M = G Fα em 2 2 V tbv ˆq td { 2 C ν 7 ˆm b biσµν 1 + γ 5d lγ µ l + C bγ 9 µ 1 γ 5 d lγ µ l + C bγ 10 µ 1 γ 5 d lγ µ γ 5 l + C Q1 b1 + γ5 d ll + C Q2 b1 + γ5 d lγ 5 l}. Here, L/R 1 γ 5 /2, s = q 2, q = p + + p, where p ± are the four-momenta of the leptons, respectively. We put m d /m b = 0, but keep the leptons massive. The hat denotes normalization in terms of the B-meson mass, m B, e.g. = s/m 2 B, ˆm b = m b /m B. In the SM, the ective Wilson coicients, which enter the decay distributions are written as: [16,17] C 7 = C 9 = C 10 = 1 + α sµ ω 7 A 7 α sµ 4 C0 1 F C 0 2 F A 0 8 F 7 8, α sµ ω 9 A 9 + T 9 h ˆm 2 c, + U 9 h1, + W 9 h0, α sµ 4 C0 1 F C 0 2 F A 0 8 F 9 8, α sµ ω 9 A 10, 6 where ˆm c = m c /m b. The function h ˆm 2 c, originates from the one-loop matrix of the four operators and is given in Ref. [16]. The auxiliary quantities A 7, A 8, A 9, A 10, T 9, U 9, and W 9 are expressed in terms of linear combinations of the Wilson coicients C i µ. [4] The numerical values of the auxiliary quantities and C 1 and C 2 are obtained by running the renormalization group equations and have been calculated for three different scales µ in Table 1 of Ref. [4]. At an arbitrary scale µ, the Wilson coicients have been expanded perturbatively as follows: [16] C i µ = C 0 i µ+ g2 s 4 2C1 i µ+ g4 s 4 4 C2 i µ+. 7 The explicit expressions of the Wilson coicients C 0 i, C 1 i, and C 2 i in SM, which correspond to LO, NLO, and NNLO parts respectively, are given in Refs. [10] and [16]. For the new physics part, only C 1NP i M W are known at present, as given explicitly in Ref. [0]. When we consider the new physics contributions, the Wilson coicients at M W can be written as: C i M W C i M W + α s 4 C1NP i M W. 8 In the T2HDM, the B u l + l decays proceed also via additional loops involving charged and/or neutral Higgs boson exchanges. In Ref. [0], we have given a detailed derivation of the lengthy expressions of the T2HDM corrections to the relevant Wilson coicients M W induced by the loop diagrams involving charged Higgs bosons. The detailed calculation of neutral Higgs boson contributions to the Wilson coicients C Qi i = 1,...,10 and their evolution can be found in Ref. [2]. Exclusive decays B u + + l + l and B u + + l + l are described in terms of matrix elements of the quark C 1NP i

3 698 SONG Hai-Zhen, LÜ Lin-Xia, and LU Gong-Ru Vol. 50 operators in Eq. over meson states, which are described by several independent form factors. For the process B + u + l + l, the non-vanishing matrix elements are q = p B p { p bγ µ d Bp B = f + s p B + p µ m2 B m 2 } q µ + m2 B m 2 f 0 s q µ, s s 9 p bσ µν q ν 1 + γ 5 d Bp B = p bσ µν q ν d Bp B = i{p B + p µ s q µ m 2 B m 2 f T s }. m B + m 10 While for B + u + l + l, related transition matrix elements are defined as p V A µ Bp B = iɛ µm B + m A 1 s + ip B + p µ ɛ A 2 s p B m B + m + iq µ ɛ p B 2m s A s A 0 s + ɛ µνσ ɛ ν p Bp σ 2V s m B + m, 11 p bσ µν q ν 1 + γ 5 d Bp B = iɛ µνσ ɛ ν p Bp σ 2T 1 s + T 2 s{ɛ µm 2 B m 2 ɛ p B p B + p µ } { + T sɛ s } p B q µ m 2 B m 2 p B + p µ, 12 where ɛ µ is polarization vector of the vector meson. Basic Formula for Observables In this section we will give formula for experimental observables, including dilepton invariant mass spectrum and FBA. With Eqs. 12, one can get the decay matrix elements for decays B + u + l + l and B + u + l + l in the following form, M = G Fα em 2 2 V tbv td m B [T 1 µ lγ µ l + T 2 µ lγ µ γ 5 l + S ll], 1 where for B + u + l + l, Tµ 1 = A ˆp µ + B ˆq µ, Tµ 2 = C ˆp µ + D ˆq µ, S = S 1, 14 and for B + u + l + l, T 1 µ = Aɛ µαβ ɛ ˆp α Bˆp β ibɛ µ + icɛ ˆp B ˆp µ, T 2 µ = Eɛ µαβ ɛ ˆp α Bˆp β ifɛ µ + igɛ ˆp B ˆp µ + ihɛ ˆp B ˆq µ, S = i2 ˆm ɛ ˆp B S 2, 15 p = p B + p M, q = p B p M M = + or +, ˆm = m/m B, ˆp = p/m B, and the auxiliary functions in Tµ 1,2 are defined as: A = B = C = D = C 9 f ˆm b 1 + ˆm C 7 f T, 16 C 9 f + 2 ˆm b 1 ˆm C 7 f T, 17 C 10 f +, 18 1 ˆm C 2 10 f 2 ˆm l ˆm b ˆm d C Q 2 f 0, 19 S 1 = 1 ˆm2 ˆm b ˆm d C Q1 f 0, 20 A = ˆm C 9 V + 4 ˆm b C 7 T 1, 21 B = 1 + ˆm C 9 A ˆm b 1 ˆm2 C 7 T 2, 22 1 C = 1 + ˆm C 9 A ˆm b C 1 ˆm 2 7 T + 1 ˆm2 T 2, 2 2 E = C 10 V, 1 + ˆm 24 F = 1 + ˆm C 10 A 1, 25 1 G = 1 + ˆm C 10 A 2, 26 H = 2 ˆm + C 10 A A 0 ˆm ˆm l ˆm b + ˆm d C Q 2 A 0, 27 1 S 2 = ˆm b + ˆm d A 0C Q1, 28 where f = 1 ˆm2 [f 0 f + ], 29 A = 1 + ˆm 2 ˆm A 1 1 ˆm 2 ˆm A 2. 0 The contributions of charged Higgs bosons are translated through the RGE step into modifications of the ective Wilson coicients C 7, C 9, and C 10, while the contributions of neutral Higgs bosons are incorporated in terms of S 1, D, H, and S 2. Note that the contributions of neutral Higgs bosons in D and H are

4 No. New Physics Effects on Rare Decays B + u + l + l, + l + l in a Top Quark Two-Higgs-Doublet Model 699 proportional to the inverse mass of the lepton, and for the case l = e, µ, the ects of neutral Higgs bosons will manifest themselves through these terms. The two kinematic variables and û are chosen to be = ˆq 2 = ˆp + +ˆp 2, û = ˆp B ˆp 2 ˆp B ˆp + 2, 1 which are bounded as 2 ˆm l 2 1 ˆm, 2, 2 û û û with ˆm l = m l /m B. Here the variable û is related to the angle θ between the momentum of the B-meson and that of l + in the center of mass frame of the dilepton l + l through the relation û = û cosθ. û can be written as follows: with û = λ 1 4 ˆm2 l 4 λ λ1, ˆm 2,, = 1+ ˆm 4, ˆm 2,1+. 5 Keeping the lepton mass and integrating over û in the kinematic region given in Eq., we can obtain the differential decay branching ratios for the decays B + u + l + l and B + u + l + l : dbr +, + G 2 F = τ α2 emm 5 B B d Vtb V td 2 ûd +, +, 6 D + = A 2 + C 2 λ û2 + S ˆm 2 l + C 2 4 ˆm 2 l2 + 2 ˆm 2 + Re + C D 8 ˆm 2 l1 ˆm D 2 4 ˆm 2 l, ˆm2 l + E 2 û2 + S ˆm 2 lλ + F 2 λ û2 D + = A 2 λ + 1 [ 4 ˆm 2 B 2 λ û2 + 8 ˆm ˆm2 l + + λ 4 ˆm 2 + [ C 2 λ û2 + G 2 λ û2 1 2 ˆm 2 + [ ReBC 1 ˆm 2 + λ û2 + 4 ˆm 2 l ˆm2 + ] + Re FG ] + 8 ˆm ˆm2 l 1 ˆm 2 λ û2 + ] + 4 ˆm 2 lλ 2 ˆm2 l ˆm 2 + λ[ ReFH Re GH 1 ˆm 2 +] + H 2 ˆm2 l ˆm 2 + λ. 8 The normalized forward-backward asymmetries FBA is defined as +1 A +, + 1 FB = d d cosθd2 Br +, + /dd cosθsigncosθ +1 1 d. 9 cosθd2 Br +, + /dd cosθ According to this definition, we can obtain the explicit expressions of differential FBA for the exclusive decays. B + u + l + l, dafb + d B + u + l + l, da + FB d D + = 2 ˆm l û Re S 1 A, 40 { D + = û [ ReBE + ReAF ] 41 + ˆm l [ Re S 2 B 1 ˆm 2 ˆm + + } ReS 2 C λ]. 42 We can see from Eq. 40 that the FBA of the process B + u + l + l does not vanish when the contributions of neutral Higgs bosons are taken into account. 4 Numerical Results The input parameters that we use in our numerical calculations are listed in Table 1. For the strong coupling constant α s µ we will use the two-loop expression, with α s µ = β 0 = 4 β 0 lnµ 2 /Λ 2 MS [ 1 β 1 β0 2 ln lnµ2 /Λ 2 ] MS lnµ 2 /Λ 2, 4 MS 2f, β 1 = 72 10f 8 f, 44 where f is the number of quark flavors and the term MS denotes the modified subtraction scheme.

5 700 SONG Hai-Zhen, LÜ Lin-Xia, and LU Gong-Ru Vol. 50 Table 1 Input parameters and their assumed errors used in our numerical calculations. The parameter A, λ,, and η are Wolfenstein parameters of the CKM mixing matrix. [27] m e GeV m µ GeV m GeV m GeV m Z GeV α s m Z m W GeV α 1 em 17 m b,pole 4.8 ± 0.2 GeV sin 2 θ W m t,pole 17.8 ± 5 GeV A m B u GeV λ τ B u 1.68 ps ± m c /m b 0.29 ± 0.02 η 0.40 ± For the form factors involving the B, transitions, we will use the recently light-cone QCD sum rules LC- SRs results, [26] which gave an improved result of form factors and provide a parametrization of the q 2 -dependence of form factors in the full physical regime of momentum transfer. In the following numerical data analysis, we also consider the uncertainties induced by F0. With these formulae presented in previous section, we are ready to perform our numerical analysis. We get the SM prediction Br B + u + l + l = 1.96 ± , which is smaller than the result in Ref. [19]. The difference is mainly due to different levels of the Wilson coicients and the updated values for form factors and CKM parameters. Our SM predictions for the above exclusive branching ratios are summarized in Table 2, and the relevant experimental upper limits [6] are listed for comparison. The dominant source of uncertainty comes from the form factors dependence. The experimental upper limits of Br B u + + l + l are just one order magnitude above the SM expectations, which will constrain the parameter space of new physics models. Table 2 The SM predictions at NNLO accuracy for exclusive decays involving the quark transition b dl + l and experimental upper limits of branching ratios for B + u + l + l. [6] SM prediction value Experimental data BrB + u + e + e 1.96 ± < % C.L. [6] Br B + u + µ + µ 1.96 ± < % C.L. [6] Br B u + + e + e BrB u + + µ + µ Now we turn to discuss the new physics contributions. The new physics contributions rely on several free parameters, which are charged Higgs mass m H +, tanβ, ξ, a new CP-violating phase δ, and neutral Higgs masses m H 0, m h 0, m A 0 in the T2HDM. We fix ξ = 1 throughout this paper and consider others as variable parameters. From the experimental data of the radiative decay B X s γ, B 0 B 0 mixing and inclusive decays B X s l + l, we have imposed strong bounds on the parameter space of the T2HDM. [0 2] Here we will consider these constraints in our choice for the free parameters of the T2HDM. We will mainly focus on the parameter space at large tanβ. As a representative example, we choose δ = 0, tanβ = 40, m H + = 500 GeV, m h 0 = 115 GeV, m H 0 = 160 GeV, and m A 0 = 120 GeV if not specified. The new physics corrections to the branching ratio of B + u + l + l and B + u + l + l l = e, µ in the T2HDM are shown in Fig. 1. The solid line corresponds to the central value of SM prediction at partial NNLO level, while the three dashed lines stand for the new physics contributions when uncertainties of form factors induced by F0 are taken into account. From Fig. 1 and the numerical results, one can see that i The theoretical predictions strongly depend on the values of the form factors. ii For these four decays, the T2HDM corrections studied here are small. After the consideration of the uncertainties of form factors, the branching ratio of these decays can be increased 50% at most, which are much smaller than the current experimental upper limits: Br B u + + e + e < , Br B u + + µ + µ < iii Within the considered parameter space of the T2HDM, the T2HDM predictions for decays Br B u + + l + l and Br B u + + l + l l = e, µ are consistent well with the SM.

6 No. New Physics Effects on Rare Decays B + u + l + l, + l + l in a Top Quark Two-Higgs-Doublet Model 701 Figure 2 illustrates the -dependence of the FBAs for B + u + µ + µ and B + u + µ + µ in the SM and the T2HDM without including the uncertainties of form factors since they only provide a few percent change. The solid and dashed lines denote the SM and the T2HDM predictions, respectively. As is known, the FBA of decay modes B + u + l + l vanishes in the SM. If the contributions of neutral Higgs bosons to the FBA for exclusive processes B + u + l + l at large tanβ are significant, they can provide a good place to probe the new physics. From this figure and the numerical calculation, we find that the new physics contributions to the FBA of the decay B u + + e + e are too small to be distinguished from the SM prediction. The new physics ects on FBA of the decay B u + + e + e are in the same way as for B u + + µ + µ. The T2HDM corrections to the FBA of B u + + µ + µ decay are not too large, and those of B u + + µ + µ decay remain basically unaffected. We expect the future B related experiments will give accurate data to distinguish different models. Fig. 1 The m H + dependence of the branching ratios of B + u + l + l and B + u + l + l l = e, µ decays in the SM and the T2HDM. The solid lines and dashed lines show the SM prediction, the theoretical results of the T2HDM, respectively. We here also considered the uncertainties of form factors induced by F0. Fig. 2 The FBA of the decay modes B + u + µ + µ and B + u + µ + µ as a function in the SM and the T2HDM. The solid lines and dashed lines show the SM prediction, the theoretical values of the T2HDM, respectively.

7 702 SONG Hai-Zhen, LÜ Lin-Xia, and LU Gong-Ru Vol Summary In this paper, we have calculated the new physics contributions to the branching ratios and forward-backward asymmetries for exclusive decays B + u + l + l and B + u + l + l l = e, µ in the SM and the T2HDM by employing the form factors of the light-cone QCD sum rules LCSRs results. From the numerical results and the figures, we found that the T2HDM contributions to the branching ratios and forward-backward asymmetries are generally small. The theoretical values agree well with the SM predictions. Currently, people have tried to measure the branching ratios of the decay B + u + l + l l = e, µ and only gave the experimental upper limits. At present, there are no experimental data for B + u + l + l decays. Moreover, large theoretical uncertainties also exist. For example, the form factors can lead to large theoretical errors; and the virtual Oα s corrections F ij appearing in the Wilson coicients at NNLO level are known only for the low dilepton invariant mass region and are switched off in the calculations. Therefore, little experimental data and the large theoretical uncertainties prevent us from accurately testing the T2HDM through studies of the decays B + u + l + l and B + u + l + l at present. We expect the great progress in both the experiment and the theory. References [1] N. Cabibbo, Phys. Rev. Lett ; M. Kobayashi and K. Maskawa, Prog. Theor. Phys [2] B. Aubert, et al., BABAR Collaboration, Phys. Rev. D [] K. Abe, et al., Belle Collaboration, [arxiv:hep-ex/ ]. [4] B. Aubert, et al., BABAR Collaboration, Phys. Rev. Lett [5] D. Mohapatra, et al., Belle Collaboration, Phys. Rev. Lett [6] B. Aubert, et al., BABAR Collaboration, Phys. Rev. Lett [7] A.J. Weir, et al., Phys. Rev. D ; K.W. Edwards et al., CELO Collaboration, Phys. Rev. D [8] N.G. Deshpande, et al., Phys. Rev. Lett [9] B. Grinstein, M.J. Savage, and M.B. Wise, Nucl. Phys. B ; M. Misiak, Nucl. Phys. B ; Nucl. Phys. B E. [10] A.J. Buras and M. Münz, Phys. Rev. D [11] F. Krüger and L.M. Sehgal, Phys. Rev. D ; Phys. Rev. D [12] W. Jaus and D. Wyler, Phys. Rev. D [1] A.F. Falk, M. Luke and M.J. Savage, Phys. Rev. D [14] A. Ali, G. Hiller, L.T. Handoko, and T. Morozumi, Phys. Rev. D ; A. Ali and G. Hiller, Phys. Rev. D ; A. Ali and G. Hiller, Phys. Rev. D [15] G. Buchalla and G. Isidori, Nucl. Phys. B [16] C. Bobeth, M. Misiak, and J. Urban, Nucl. Phys. B [17] H.H. Asatryan, H.M. Asatrian, C. Greub, and M. Walker, Phys. Lett. B ; Phys. Rev. D ; H.H. Asatryan, et al, Phys. Rev. D [18] A. Ghinculov, T. Hurth, G. Isidori, and Y.P. Yao, Nucl. Phys. B ; P. Gambino, M. Gorbahn, and U. Haisch, Nucl. Phys. B [19] T.M. Aliev and M. Savci, Phys. Rev. D [20] Yuan-Ben Dai, Chao-Shang Huang, and Han-Wen Huang, Phys. Lett. B , Erratum-ibid. B ; C. Bobeth et al., Phys. Rev. D ; G. Erkol and G. Turan, Nucl. Phys. B ; S. Schilling et al., Phys. Lett. B [21] E. Lunghi, et al., Nucl. Phys. B ; D.A. Demir, K.A. Olive, and M.B. Voloshin, Phys. Rev. D ; Chao-Shang Huang and Xiao-Hong Wu, Nucl. Phys. B ; S.R. Choudhury, A.S. Cornell, N. Gaur, and G.C. Joshi, Phys. Rev. D [22] A. Ali, P. Ball, L.T. Handoko, and G. Hiller, Phys. Rev. D [2] Yuan-Guo Xu, Ru-Min Wang, and Ya-Dong Yang, Phys. Rev. D ; Qi-Shu Yan, Chao-Shang Huang, Wei Liao, and Shou-Hua Zhu, Phys. Rev. D [24] Wen-Jun Li, Yuan-Ben Dai, and Chao-Shang Huang, Eur. Phys. J. C [25] A. Das and C. Kao, Phys. Lett. B [26] P. Ball and R. Zwicky, Phys. Rev. D ; Phys. Rev. D [27] W.M. Yao, et al., Particle Data Group, J. Phys. G [28] K. Kiers, A. Soni, and G.H. Wu, Phys. Rev. D ; G.H. Wu and A. Soni, Phys. Rev. D [29] K. Kiers, A. Soni, and G.H. Wu, Phys. Rev. D [0] Z.J. Xiao and L.X. Lü, Phys. Rev. D [1] L.X. Lü and Z.J. Xiao, Commun. Theor. Phys. Beijing, China [2] L.X. Lü and Z.J. Xiao, High Energy Phys. Nucl. Phys [] W.S. Hou, Phys. Lett. B ; M. Luke and M.J. Savage, Phys. Lett. B [4] A. Ali, E. Lunghi, C. Greub, and G. Hiller, Phys. Rev. D