B s τ + τ decay in the general two Higgs doublet model

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1 B s τ + τ decay in the general two Higgs doublet model arxiv:hep-ph/00005v3 5 Dec 00 E. O. Iltan Physics Department, Middle East Technical University Ankara, Turkey G. Turan Physics Department, Middle East Technical University Ankara, Turkey Abstract We study the exclusive decay B s τ + τ in the general two Higgs doublet model. We analyse the dependencies of the branching ratio on the model parameters, including the leading order QCD corrections. We found that there is an enhancement in the branching ratio, especially for r tb = ξ N,tt U > case. Further, the neutral Higgs effects are detectable ξ N,bb D for large values of the parameter ξ N,ττ D. address: eiltan@heraklit.physics.metu.edu.tr address: gsevgur@rorqual.metu.edu.tr

2 Introduction The study of rare B-decays is one of the most important research areas in particle physics and there is an experimental effort for studying them at various centers such as SLAC (BaBar), KEK (BELLE), B-Factories, DESY (HERA-B). In the Standard model (SM) they are induced by flavor changing neutral currents (FCNC) at loop level and therefore they are sensitive to the fundamental parameters, like Cabbibo-Kobayashi-Maskawa (CKM) matrix elements, leptonic decay constants, etc. These decays also provide a sensitive test to the new physics beyond the SM, such as two Higgs Doublet model (HDM), Minimal Supersymmetric extension of the SM (MSSM) [], etc. Among the rare B decays, B s l + l process, induced by the inclusive b sl + l decay, is attractive since the only non-perturbative quantity in the theoretical calculation is the decay constant of B s which is reliably known. From the experimental point of view, the measurement of the hadronic decay is easier compared to its inclusive channel. The measurement of upper limit of B s µ + µ [] BR(B s µ + µ ).60 6, () stimulated the study of B s l + l decays. In the literature, this process ([3]-[9]) and its inclusive one B s X s l + l ([0]-[3]) have been investigated extensively in the SM, HDM and supersymmetric model (SUSY). When l = e, µ, the neutral Higgs boson(nhb) effects are safely neglected in the HDM because they enter in the expressions with the factor m e(µ) /m W, m H ±. However, forl = τ, thisfactorisnotnegligibleandnhbeffectscangiveimportantcontribution. In [3], B X s τ + τ process was studied in the HDM and it was shown that NHB effects are sizable for large values of tanβ. Therefore the main observation of these calculations is the enhancement of the branching ratio (BR) of these decays for large tanβ values in the HDM and minimal supersymmetric model (MSSM), especially for l = τ lepton case. In a recent work [4], the inclusive b sτ + τ process has been studied in the general HDM with real Yukawa couplings and it was found that the BR has been enhanced for large values of the parameters ξ D N,ττ and ξ D N,bb. In this work, we study the B s τ + τ decay in the general HDM, so-called model III. Our calculations are based on the results of the work [4] for the inclusive b sτ + τ decay. Here we include NHB effects and make the full calculation using the on-shell renormalization prescription. The investigation of the dependencies of the BR on the model parameters, namely ξ N,bb D and ξ N,ττ D, shows that a large enhancement in the BR is possible. The paper is organized as follows: In Section, we present the leading order (LO) QCD

3 corrected effective Hamiltonian and the corresponding matrix element for the exclusive B s τ + τ decay in the framework of the model III. Section 3 is devoted to the analysis of the dependencies of the BR on the the Yukawa couplings ξ N,bb D, ξ N,ττ D and to the discussion of our results. In Appendices, we present the operators appearing in the effective Hamiltonian and their Wilson coefficients. The B s τ + τ decay in the framework of the model III The general Yukawa interaction in the model III is L Y = η U ij Q il φ U jr +η D ij Q il φ D jr +ξ U ij Q il φ U jr +ξ D ij Q il φ D jr +h.c., () where L and R denote chiral projections L(R) = /( γ 5 ) and φ i for i =,, are two scalar doublets. Here η U,D ij, ξ U,D ij are the Yukawa matrices and, in general, they have complex entries. The choice of scalar Higgs doublets φ = [( 0 v +H 0 ) + ( χ + iχ 0 φ = ( H + H +ih )] ),, (3) with the vacuum expectation values, < φ >= ( 0 v ) ;< φ >= 0, (4) and the gauge and CP invariant Higgs potential which spontaneously breaks SU() U() down to U(): V(φ,φ ) = c (φ + φ v /) +c (φ + φ ) + +c 3 [(φ + φ v /)+φ + φ ] +c 4 [(φ + φ )(φ + φ ) (φ + φ )(φ + φ )] + c 5 [Re(φ + φ )] +c 6 [Im(φ + φ )] +c 7, (5) where constants c i, i =,...,7, permits us to carry the SM particles in the first doublet and the information about the new physics by the second one. The Yukawa interaction L Y,FC = ξ U ij Q il φ U jr +ξ D ij Q il φ D jr +h.c.. (6)

4 describes the Flavor Changing (FC) one beyond the SM. Here, the couplings ξ U,D for the charged FC interactions are ξch U = ξn U V CKM, ξch D = V CKM ξn D, (7) and ξ U,D N = (V U,D L ) ξ U,D V U,D R, (8) where the index N in ξ U,D N denotes the word neutral. Notice that H and H are the mass eigenstates h 0 and A 0 respectively, since no mixing occurs between two CP-even neutral bosons H 0 and h 0 in the tree level, for our choice. The exclusive B s τ + τ process is induced by the inclusive b sτ + τ decay. Therefore we start with the effective Hamiltionian of b sτ + τ decay H eff = αg { F V tb Vts C eff 9 ( sγ µ P L b) τγ µ τ +C 0 ( sγ µ P L b) τγ µ γ 5 τ π } C 7 p ( siσ µνp ν P R b) τγ µ τ +C Q ( sp R b) ττ +C Q ( sp R b) τγ 5 τ, (9) with p = p + p, the sum of the momenta of τ + and τ. Note that, C 7,C 9 and C 0, are the Wilson coefficients normalized at the scale µ and given in Appendix B. The additional Wilson coefficients C Q and C Q are due to the NHB exchange diagrams (see Appendix B). In calculating the H eff, one first integrates out the heavy degrees of freedom, namely t quark, W ±, H ±, H 0, H and H bosons in the present case and then obtain the effective theory. Here H ± denote charged, and H 0, H,H denote neutral Higgs bosons. Note that H and H are the same as the mass eigenstates h 0 and A 0 in the model III respectively. At this stage the QCD corrections are added through matching the full theory with the effective low energy one at the high scale µ = m W and evaluating the Wilson coefficients from m W down to the lower scale µ O( ). In the model III, the neutral Higgs particles bring additional contributions (see eq.(9)) since the mass of τ lepton or related Yukawa coupling ξ D N,ττ enter into the expressions (see [4]). Finally the neutral Higgs boson (NHB) contributions are calculated using the on-shell renormalization scheme to overcome the logarithmic divergences. Using the renormalization condition Γ Ren neutr (p ) = Γ 0 neutr (p )+Γ C neutr = 0, (0) 3

5 the counter term Γ C neutr and then the renormalized vertex function ΓRen neutr (p ) is obtained. Here the phrase neutr denotes the neutral Higgs bosons H 0, h 0 and A 0 and p is the momentum transfer. For the exclusive decay B s τ + τ, H eff is to be taken between vacuum and B 0 s > state as < 0 H eff B 0 s > and this matrix element can be expressed in terms of the B0 s f Bs using < 0 sγ µ γ 5 b B 0 s > = if Bs p µ, decay constant < 0 sγ 5 b B 0 s > = if B s m B s +m s, < 0 sσ µν P R b B 0 s > = 0. () Since p = p +p, the C eff 9 term in eq.(9) gives zero on contraction with the lepton bilinear, C 7 gives zero by eq.() and the C 0 term gets a factor of m τ while the remaining C Q and C Q terms get m Bs, when taking m Bs +m s. Thus the effective Hamiltonian eq. (9) results in the following decay amplitude for B s τ + τ A = G [ Fα π m B s f Bs V tb Vts C Q ττ +(C Q m ] τ C 0 ) τγ 5 τ. () m Bs To calculate the branching ratio we find the square of this amplitude, then sum over the spins of the τ leptons and finally integrate over the phase space. This straightforward calculation gives for the branching ratio of B s τ + τ BR = G F α 64π 3 m3 B s τ Bs f B s V tb V 3 Discussion ts 4 m τ m B s [ ( 4 m τ ) C m Q + C Q m ] τ C 0.(3) B s m Bs In the multi-higgs doublet models, there are many free parameters, such as masses of charged and neutral Higgs bosons and the Yukawa couplings. In the present work we study our process in the general HDM, so called model III. The Yukawa couplings, which are entries of Yukawa matrices can be restricted using the experimental measurements. In our calculations, we neglect all Yukawa couplings except ξ U N,tt, ξ D N,bb, ξ D N,ττ by respecting the CLEO measurement [5], BR(B X s γ) = (3.5±0.35±0.3)0 4. (4) This section is devoted to ξ N,bb D and ξ N,ττ D dependencies of BR for the exclusive decay B s τ + τ, restricting C eff 7 in the region 0.57 C eff due to the CLEO measurement, 4

6 eq.(4) (see [6] for details). In our numerical calculations, we take the charged Higgs mass m H ± = 400GeV and the scale µ =. Further, we use the redefinition ξ U,D = and the input values given in Table (). 4GF ξu,d, Parameter Value m τ.78 (GeV) m c.4 (GeV) 4.8 (GeV) m H 0 50 (GeV) m h 0 80 (GeV) m A 0 80 (GeV) αem 9 λ t 0.04 m t 75 (GeV) m W 80.6 (GeV) m Z 9.9 (GeV) Λ QCD 0.5 (GeV) α s (m Z ) 0.7 sinθ W 0.35 Table : The values of the input parameters used in the numerical calculations. Fig. shows ξ D N,bb dependence of the BR of the decay under consideration for ξ D N,ττ = 00GeV and the ratio r tb = ξ N,bb D <. The BR is restricted to the region bounded by solid ξ N,tt U lines for C eff 7 > 0 or to the small dashed line for C eff 7 < 0. This quantity is sensitive to ξ N,bb D and it increases by an amount %60 in the interval 0 ξ N,bb D 80. Besides, the enhancement compared to the SM case is predicted as beeing %80. The ξ N,bb D dependence of the BR for r tb > is presented in Fig.. For this case, the BR increases considerably even for small values of ξ N,ττ, D which is taken 0GeV, in this calculation. The BR enhances with increasing ξ N,ττ D, especially for Ceff 7 < 0 case. This figure shows that the BR is strongly sensitive to the parameter ξ D N,bb for r tb > and it may get the values four (two) times larger compared to the ones in the SM for C eff 7 < 0 (C eff 7 > 0) even at ξ D N,bb =. Figures (3-4) represent the dependencies of the BR on the parameter ξ D N,ττ for r tb < and r tb > respectively. In r tb < case, the BR increases almost.5 times compared to the one in the SM for large values of ξ D N,ττ, ξ D N,ττ = 500GeV (Fig. 3). However, for r tb >, this enhancement is quite high as shown in Fig. 4. Even for small values of ξ D N,bb and ξ D N,ττ there 5

7 is a possible increase nearly (more than) one order of magnitude compared to the SM case for C eff 7 > 0 (C eff 7 < 0). Now we would like to summarize our results: There isapossible enhancement inthebr attheorder ofmagnitude %50for r tb < in the model III compared to the one in the SM for large values of the model III parameters, ξ N,bb D = 80 and ξ N,ττ D = 500GeV. The BR is not so much sensitive to the model parameters given above. Further, the NHB effects become sizable with increasing values of ξ D N,ττ. For r tb >, there is a considerable enhancement at the one order of magnitude compared to the SM, even for the small values of ξ D N,bb and ξ D N,ττ. In this case, the BR is larger and more sensitive the model parameters for C eff 7 < 0 than the ones for C eff 7 > 0. Note that the enhancement for the increasing values of the ξ D N,ττ BR on the NHB effects. Acknowledgement is due to the dependence of the We would like to thank Liao Wei for his comments about the previous version of this manuscript. 6

8 Appendix A The operator basis The operator basis in the HDM (model III ) for our process is [3, 7, 8] O = ( s Lα γ µ c Lβ )( c Lβ γ µ b Lα ), O = ( s Lα γ µ c Lα )( c Lβ γ µ b Lβ ), O 3 = ( s Lα γ µ b Lα ) ( q Lβ γ µ q Lβ ), O 4 = ( s Lα γ µ b Lβ ) O 5 = ( s Lα γ µ b Lα ) O 6 = ( s Lα γ µ b Lβ ) O 7 = O 8 = O 9 = O 0 = ( q Lβ γ µ q Lα ), ( q Rβ γ µ q Rβ ), ( q Rβ γ µ q Rα ), e 6π s ασ µν ( R+m s L)b α F µν, g 6π s αtαβσ a µν ( R+m s L)b β G aµν, e 6π ( s Lαγ µ b Lα )( τγ µ τ), e 6π ( s Lαγ µ b Lα )( τγ µ γ 5 τ), Q = e 6π ( sα L bα R )( ττ) Q = e 6π ( sα L bα R )( τγ 5τ) Q 3 = g 6π ( sα L bα R ) Q 4 = g 6π ( sα L bα R ) Q 5 = g 6π ( sα Lb β R) Q 6 = g 6π ( sα L bβ R) Q 7 = g 6π ( sα L σµν b α R ) ( q β L qβ R ) ( q β R qβ L ) ( q β Lq α R) ( q β Rq α L ) Q 8 = g 6π ( sα L σµν b α R ) ( q β L σ µνq β R ) ( q β R σ µνq β L ) 7

9 Q 9 = g 6π ( sα L σµν b β R ) Q 0 = g 6π ( sα L σµν b β R ) ( q β L σ µνq α R ) ( q β R σ µνq α L ) (5) where α and β are SU(3) colour indices and F µν and G µν are the field strength tensors of the electromagnetic and strong interactions, respectively. Note that there are also flipped chirality partners of these operators, which can be obtained by interchanging L and R in the basis given above in the model III. However, we do not present them here since corresponding Wilson coefficients are negligible. B The Initial values of the Wilson coefficients. For the sake of completeness we also give the initial values of the Wilson coefficients for the relevant process. In the SM they are [7] C SM,3,...6,, (m W) = 0, C SM (m W ) =, C SM 7 (m W ) = 3x3 t x t 4(x t ) 4 lnx t + 8x3 t 5x t +7x t 4(x t ) 3, C SM 8 (m W ) = 3x t 4(x t ) 4 lnx t + x3 t +5x t +x t 8(x t ) 3, C9 SM (m W ) = B(x sin t )+ 4sin θ W θ W sin C(x t ) D(x t )+ 4 θ W 9,, C SM 0 (m W ) = sin θ W (B(x t ) C(x t )), C SM Q i (m W ) = 0 i =,..,0. (6) The initial values for the additional part due to charged Higgs bosons are C H,...6 (m W) = 0, C H 7 (m W) = Y F (y t ) + XY F (y t ), C H 8 (m W ) = Y G (y t ) + XY G (y t ), C H 9 (m W ) = Y H (y t ), C H 0 (m W) = Y L (y t ), (7) where X = ( ξd N,bb + ξ ) N,sb D V ts V tb, 8

10 Y = m t ( ξ U N,tt + ξ U N,tc and due to the neutral Higgs bosons are [4] Vcs ) Vts, (8) C A0 Q (( ξ U N,tt )3 ) = C A0 Q (( ξ U N,tt) ) = ξ D N,ττ ( ξ U N,tt )3 y t 3π m A 0 m t Θ (z A )(y t ) ((y t )( Θ (z A )+(y t )z A )+Θ (z A )lny t ), CQ A0 ( ξ N,tt) U = g ξd N,ττ ξu N,tt x t 8π m t + ξ N,ττ( ξ D N,tt) U ξd ( N,bb ( yt )lny t 3π m A y 0 t ( y x( z + yt )+ Θ (z A ) Θ 3 (z A ) A +(+ln [ ) Θ (z A )] y t (xy +z A ) ), z A Θ (z A ) m W y t(x ) Θ 3 (z A )+(x y)(x t y t)z A ( ( 4z A m A Θ 0 (z A ) + (x(x t +y t ) y t )z A (x t(x )+(x+)y t )z A Θ 3 (z A ) Θ 3 (z A )+(x y)(x t y t )z A + (y t x t )z A +x t y t (z A ) (4x3 t 7y t 4x t(+y t )+x t (5+8y t + yt z A ))lnx t (x t )(y t )z A (x t ) (x t y t ) + (x t( yt z A ) y t )lny t +4ln [ )) Θ (z A )], (x t y t )(y t ) z A ( CQ A0 ( ξ N,bb) D = g ξd N,ττ ξd N,bb Θ3 (z A ) x t ((x )yy t x t (y t z A )) +ln [ Θ 3 (z A )] 64π m A Θ 0 3 (z A ) z A (y t x t (y t +)+x t( yt z A +)lnx t (x t )(x t y t ) CQ H0 (( ξ N,tt U ) ) = g ( ξ N,tt U ( ) m τ y t 56π m H m Wx 0 t ( z H ( Θ 4 (z H )( x)x t +Θ (z H )x) Θ (z H )Θ 4 (z H ) + (x t( y t )+y t (y t )+x t( yt z A ))lny t (y t )(x t y t ) 4xz H + +4yt+y t (lnyt 3) Θ 4 (z H ) (y t ) 3 + cos θ W (y t )(+x t +y t (x t 3))+y t (y t x t )lny t (y t ) 3 )), ( CQ H0 ( ξ N,tt) U = g ξu N,tt ξd N,bb m τ yt ( x t ) x t +x 64π m t ln [ Θ (z H )] (x t ( 5y t )+y t (+x t))lny t H m 0 t y t z H (y t ) z ( H x x t Θ 5 (y t )+x(x t ( Θ 4 (z H )+Θ 5 (+x y))(y t ) Θ (z H )Θ 4 (z H ) ) + y t ( Θ 5 +yy t ))+( x t Θ 6 (y t ) (+y(y t ))y t )z H + y t((y t )( Θ 4 (z H )+z H ( y t ))+Θ 4 (z H )y t lny t ) cos θ W Θ 4 (z H )(y t ) ), ( 4(x ) CQ H0 (g 4 ) = g4 m τ Θ 7 5π m H m 0 W + xt(xt(4 xt) 3+xt(xt )lnxt) (x t ) 3 cos θ W 4(x(4+x t xty z H ) ) 4(x t( y( x) z H +3+Θ 7 ) 4x)+Θ 7 x t lnθ 7 ) Θ 8 x t Θ 7 + ( x t +xt xt 3 +x 4 t +( 4x t x t +6x3 t )lnx t) 4x (x t ) 3 t (+ln [ ] ) Θ 8 ), 9 ),

11 ξ Q (( ξ N,tt U )3 N,ττ D ) = ( ξ N,tt U )3 y t 64Θ (z h )m h m 0 t π ( +y t ) 3(( +y t)(θ (z h )(y t +) C h 0 + (x )(y t ) z h ) Θ (z h )y t lny t ), Q (( ξ N,tt U ) ) = 3m h π ξ N,ττ ξ D N,bb D ( ξ N,tt U )( Θ (z h )+y t (xy z h ) Θ 0 (z h ) ln [ Θ (z h )]), z h C h 0 + ( y t +( +y t )lny t +y t ( CQ h0 ( ξ N,tt) U = g ξd N,ττ ξu N,tt x t xt (8 9y t ) x 3 t (y t )+y t (5y t 4)+x t ( 4+y t +yt ) 8π m h m 0 t Θ 5 (x t ) y tx t ( x t y t ) 4z h( +x(+x t )) xyx t + z h( y t +x(x t +y t )) z h Θ 5 Θ (z h ) Θ 3 (z h ) + z h(x t (x )+y t (x+)) 4( +x)xx t yt + z h Θ 3 (z h )+(x y)(x t y t )z h (Θ 4 (z h )x t x(x t y t )z h )(Θ 4 (z h )x t y(x t y t )z h ) + ( 4+ (y t ) (x 3 t(3 0y t )+7yt 7x t y t (+y t )+3x t(+4y t +yt)) (x t ) + y tx t ( +y t ) ( Θ 6 +4(x t y t )y t ) ) lnx t z h Θ 5 (x t ) ( 0x t y t (y t )+x t (y t )+yt (4y t 5) xtyt z h Θ 6 )lny t 4ln [ Θ (z h )]), Θ 5 z h where ( CQ h0 ( ξ N,bb D ) = g ξd N,ττ ξd N,bb z h (x t ( yt z h +)+y t x t (y t +))lnx t 64π m Wx t (x t )(x t y t ) + (x t (yt z h )+y t (y t )+x t ( y t ))lny t ln [ Θ 3 (z h )] (y t )(x t y t ) z h y t (x t +y(x )) z h (x t y t (y )+y yt x +x t )+x(yy t +z h (y t )) ) t, (9) Θ 3 (z h ) Θ (ω) = (( y +yy t )ω x(yy t +ω( y t )) Θ (ω) = Θ (ω,y t x t ) Θ 3 (ω) = (x t ( y)+y)y t ω xx t (yy t +ω( +y t )) Θ 4 (ω) = (y( y t )+y t )ω x(yy t +ω( +y t )) Θ 5 = ( +x t )(x t y t )( +y t ) Θ 6 = ( +y)(y( +y t )y t ) Θ 7 = (x t +y( x t ))z h +x(z h x t (y +z h )) x t z h Θ 8 = ( y( x t))z h x(x t (y z h )+z h ) x t z h (0) 0

12 and x t = m t m W, y t = m t m H ±, z H = m t m H 0, z h = m t m h 0, z A = m t m A 0, () The explicit forms of the functions F () (y t ), G () (y t ), H (y t ) and L (y t ) in eq.(7) are given as F (y t ) = y t(7 5y t 8y t ) 7(y t ) 3 + y t (3y t ) (y t ) 4 lny t, F (y t ) = y t(5y t 3) (y t ) + y t( 3y t +) lny 6(y t ) 3 t, G (y t ) = y t( yt +5y t +) yt + 4(y t ) 3 4(y t ) lny 4 t, G (y t ) = y t(y t 3) 4(y t ) + y t (y t ) lny 3 t, [ H (y t ) = 4sin θ W sin θ W L (y t ) = xy t 8 y t (y t ) lny t [ ] 47y y t 79y t +38 t 3y3 t 6y t +4 lny 08(y t ) 3 8(y t ) 4 t, [ xy t ] sin θ W 8 y t + (y t ) lny t. ] () Finally, the initial values of the coefficients in the model III are C HDM i (m W ) = Ci SM (m W )+Ci H (m W ), C HDM Q (m W ) = C HDM Q (m W ) = C HDM Q 3 (m W ) = C HDM Q 4 (m W ) = 0 dx x 0 dy(cq H0 (( ξ N,tt U ) )+CQ H0 ( ξ N,tt U )+CH0 Q (g 4 )+CQ h0 (( ξ N,tt U )3 ) + CQ h0 (( ξ N,tt U ) )+CQ h0 ( ξ N,tt U )+Ch0 Q ( ξ N,bb D )), 0 dx 0 x m τ sin (CQ HDM θ W dy(c A0 Q (( ξ U N,tt) 3 )+C A0 Q (( ξ U N,tt) )+C A0 Q ( ξ U N,tt)+C A0 Q ( ξ D N,bb)) (m W )+C HDM Q (m W )) m τ sin (CQ HDM θ (m W ) CQ HDM (m W )) W C HDM Q i (m W ) = 0, i = 5,...,0. (3) Here, we present C Q and C Q in terms of the Feynmann parameters x and y since the integrated results are extremely large. Using these initial values, we can calculate the coefficients C HDM i (µ) and CQ HDM i (µ) at any lower scale in the effective theory with five quarks, namely u,c,d,s,b similar to the SM case [3, 8, 9, 0]. For completeness, in the following we give the explicit expressions for C eff 7 (µ) and C eff 9 (µ). C eff 7 (µ) = C HDM 7 (µ)+q d (C HDM 5 (µ)+n c C HDM 6 (µ)),

13 + Q u ( m c C HDM (µ)+n c m c C HDM (µ)), (4) where the LO QCD corrected Wilson coefficient C LO,HDM 7 (µ) is given by C LO,HDM 7 (µ) = η 6/3 C7 HDM (m W )+(8/3)(η 4/3 η 6/3 )C8 HDM (m W ) 8 + C HDM (m W ) h i η a i, (5) i= and η = α s (m W )/α s (µ), h i and a i are the numbers which appear during the evaluation [9]. The Wilson coefficient C eff 9 (µ) is : C eff 9 (µ) = C HDM 9 (µ) + h(z,s)(3c (µ)+c (µ)+3c 3 (µ)+c 4 (µ)+3c 5 (µ)+c 6 (µ)) h(,s)(4c 3(µ)+4C 4 (µ)+3c 5 (µ)+c 6 (µ)) (6) h(0,s)(c 3(µ)+3C 4 (µ))+ 9 (3C 3(µ)+C 4 (µ)+3c 5 (µ)+c 6 (µ)). Here the functions h(u,s) are given by with u = mc. by [3] h(u,s) = 8 9 ln µ 8 9 lnu x (7) ( ln x+ iπ ) x, for x 4u < 9 (+x) x / s arctan x, for x 4u >, s h(0,s) = ln µ 4 9 lns+ 4 iπ, (8) 9 Finally, the Wilson coefficient C 0 (µ) is the same as C 0 (m W ) and C Q (µ), C Q (µ) are given C Qi (µ) = η /3 C Qi (m W ), i =,. (9)

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15 ½º ½º ½º ÅÓÐ ÁÁÁ ÅÓÐ ÁÁÁ ¼ ¼ ËÅ ½º ½¼ Ê ½º¾ ½º½ ½ ¼º ¼º ¼º ¾¼ ¼ ¼ ¼ ÆÑ ¼ ¼ ¼ Figure : BR as a function of ξ D N,bb / for ξ D N,ττ = 00GeV in case of the ratio r tb <. º ÅÓÐ ÁÁÁ ÅÓÐ ÁÁÁ ¼ ¼ ËÅ ½¼ Ê ¾º ¾ ½º ½ ¼º ½ ½º¾ ½º ½º ½º ¾ ÆÑ Figure : BR as a function of ξ D N,bb/ for ξ D N,ττ = 0GeV in case of the ratio r tb >. 4

16 ¾º¾ ¾ ÅÓÐ ÁÁÁ ÅÓÐ ÁÁÁ ¼ ¼ ËÅ ½º ½¼ Ê ½º ½º ½º¾ ½ ¼º ¼º ½¼¼ ½¼ ¾¼¼ ¾¼ ¼¼ Æ ¼ ¼¼ ¼ ¼¼ Figure 3: BR as a function of ξ D N,ττ, for ξ D N,bb = 40 in case of the ratio r tb <. ½ ½¾ ÅÓÐ ÁÁÁ ÅÓÐ ÁÁÁ ¼ ¼ ËÅ ½¼ ½¼ Ê ¾ ¼ ½¼ ½ ¾¼ ¾ Æ ¼ ¼ ¼ Figure 4: BR as a function of ξ D N,ττ, for ξ D N,bb = 3 in case of the ratio r tb >. 5