b sτ + τ decay in the two Higgs doublet model with flavor changing neutral currents

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1 b sτ + τ decay in the two Higgs doublet model with flavor changing neutral currents arxiv:hep-ph/000807v2 2 May 200 E. O. Iltan Physics epartment, Middle East Technical University Ankara, Turkey G. Turan Physics epartment, Middle East Technical University Ankara, Turkey Abstract We study the decay width and forward-backward asymmetry of the lepton pair for the inclusive decay b sτ + τ in the two Higgs doublet model with three level flavor changing neutral currents (model III) and analyse the dependencies of these quantities on the model III parameters, including the leading order QC corrections. We found that there is a considerable enhancement in the decay width and neutral Higgs effects are detectable for large values of the parameter ξ N,ττ. address: eiltan@heraklit.physics.metu.edu.tr address: gsevgur@rorqual.metu.edu.tr

2 Introduction Currently, there is an impressive experimental effort for studying rare B-meson decays at SLAC (BaBar), KEK(BELLE), B-Factories, ESY(HERA-B) since these decays are rich phenomenologically. They are induced by flavor changing neutral currents (FCNC) at loop level in the Standard model (SM) and with the forthcoming experiments, it would be possible to test the flavour sector of the SM in a high precision, as well as to reveal the physics beyond, such as two Higgs oublet model (2HM), Minimal Supersymmetric extension of the SM (MSSM) [], etc. Among the rare B decays, B K l + l process has received a great interest since the SM prediction for its branching ratio (Br) is large enough to be measured in the near future. This decay is induced by b sl + l transition at the quark level and in the literature it has been investigated extensively for l = e,µ in the SM, 2HM and MSSM [2]- [5]. When l = e,µ, the neutral Higgs boson (NHB) effects are safely neglected in the 2HM because they enter in the expressions with the factor m e(µ) /m W. However, for l = τ, this factor is not negligible and NHB effects can give important contribution. In [6, 7], B X s τ + τ process was studied in the 2HM and it was shown that NHB effects are sizable for large values of tanβ. In this work, we study the b sτ + τ decay in the general 2HM, so-called model III. We include NHB effects and make the full calculation using the on-shell renormalization prescription. We investigate the dependencies of the differential decay width dγ/ds and the decay width Γ on the scale invariant lepton mass square s and some model III parameters, namely m H ±, ξ N,bb and ξ N,ττ. Further, we calculate the differential (direct) forward-backward asymmetry A FB (s) (A FB ) of the lepton pair in terms of the above parameters. We show that a large enhancement is possible in the decay width of the process b sτ + τ for some values of the model III parameters and NHB effects become considerable for large values of ξ N,ττ. The paper is organized as follows: In Section 2, we present the leading order (LO) QC corrected effective Hamiltonian and the corresponding matrix element for the inclusive b sτ + τ decay. Further, we give the expression for A FB (s) and A FB of the lepton pair. Section 3 is devoted to the analysis of the new Wilson coefficients coming from the NHB effects and the dependencies of dγ/ds, Γ, A FB (s) and A FB on the the Yukawa couplings ξ N,bb, ξ N,ττ, the charged Higgs mass m H ±, the parameter s and to the discussion of our results. In Appendices, we give the explicit forms of the operators appearing in the effective Hamiltonian and the corresponding Wilson coefficients.

3 2 The inclusive b sτ + τ decay in the model III Model III (2HM) permits the flavour changing neutral currents in the tree level and the prize is various new parameters, i.e. Yukawa couplings. These couplings are responsible for the interaction of quarks and leptons with gauge bosons, namely, the Yukawa interaction and in this general case it reads as L Y = η U ij Q il φ U jr +η ij Q il φ jr +ξ U ij Q il φ2 U jr +ξ ij Q il φ 2 jr +h.c., () where L and R denote chiral projections L(R) = /2( γ 5 ), φ k, for k =,2, are the two scalar doublets, Q il are quark and lepton doublets, U jr, jr are the corresponding singlets, η U, ij, and ξ U, ij the interaction is given by are the matrices of the Yukawa couplings. The Flavor changing (FC) part of L Y,FC = ξ U ij Q il φ2 U jr +ξ ij Q il φ 2 jr +h.c.. (2) The choice of φ and φ 2 φ = 2 [( 0 v +H 0 ) + ( 2χ + iχ 0 )] ;φ 2 = 2 ( 2H + H +ih 2 ). (3) with the vacuum expectation values, < φ >= 2 ( 0 v ) ;< φ 2 >= 0, (4) ensures decoupling of the SM and beyond. In eq.(2) the couplings ξ U, for the FC charged interactions are where ξ U, neutral is defined by the expression ξ U, N ξ U ch = ξ neutral V CKM, ξ ch = V CKM ξ neutral, (5) = (V U, L ) ξ U, V U, R. (6) Here the charged couplings appear as linear combinations of neutral couplings multiplied by V CKM matrix elements (see [8] for details). Now we would like to start with the calculation of the matrix element for the inclusive b sτ + τ decay. The procedure is to integrate out the heavy degrees of freedom, namely In all next discussion we denote ξ U, neutral as ξu, N. 2

4 t quark, W ±,H ±,H 0,H, and H 2 bosons in the present case and obtain the effective theory. Here H ± denote charged, H 0, H and H 2 denote neutral Higgs bosons. Note that H and H 2 are the same as the mass eigenstates h 0 and A 0 in the model III respectively, due to the choice given by eq. (3). The QC corrections are done 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 (similar to the models I and II, 2HM) neutral Higgs particles bring new contributions to the matrix element of the process b sτ + τ (see eq.(23)) since they enter in the expressions with the mass of τ lepton or related Yukawa coupling ξ N,ττ. As being different from the model I and II, in the model III, there exist additional operators which are the flipped chirality partners of the former ones. However, the effects of the latter are negligible since the corresponding Wilson coefficients are small due to the discussion given in section 3. Therefore, the effective Hamiltonian relevant for the process b sτ + τ is H eff = 4 G { F V tb Vts C i (µ)o i (µ)+ 2 i i C Qi (µ)q i (µ) }, (7) where O i are current-current (i =,2), penguin (i = 3,...,6), magnetic penguin (i = 7,8) and semileptonic (i = 9,0) operators. Here, C i (µ) are Wilson coefficients normalized at the scale µ and given in Appendix B. The additional operators Q i (i =,..,0) are due to the NHB exchange diagrams and C Qi (µ) are their Wilson coefficients (see Appendices A and B). uring the calculations of NHB contributions, we use the on-shell renormalization scheme to overcome the logarithmic divergences. Taking the vertex function and using the renormalization condition Γ Ren neutr(p 2 ) = Γ 0 neutr(p 2 )+Γ C neutr, (8) Γ Ren neutr(p 2 = m 2 neutr) = 0, (9) we get the counter terms and then calculate Γ Ren neutr (p2 ). Here the phrase neutr denotes the neutral Higgs bosons, H 0, h 0 and A 0 and p is the momentum transfer. III, Now we give the QC corrected amplitude for the inclusive b sτ + τ decay in the model M = 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 2π } 2C 7 p ( siσ µνp ν P 2 R b) τγ µ τ +C Q ( sp R b) ττ +C Q2 ( sp R b) τγ 5 τ 3. (0)

5 Using Eq.(0), the differential decay rate reads as with dγ(b sτ + τ ) ds (s) = C eff Re(C 7 C eff 9 ) α 2 = Br(B X c l ν) 4π 2 f(m c / ) ( s)2 ( ( + )(+2s)+4 C 2t2 7 2 s ( + 2t2 s + 2t2 s ( 4t2 s ) /2 V tb V ts 2 V cb 2 (s),() ) ( + 2 ] )+ C 0 [+2s+ 2 2t2 s s ( 4s) ) / C Q 2 (s 4t 2 )+ 3 2 C Q Re(C 0 C Q 2 ), (2) where s = p 2 /m 2 b, t = m τ/, and f(x) is the phase-space factor given by f(x) = 8x 2 + 8x 6 x 8 24x 4 logx. In the above expression for the differential decay rate, we use the inclusive one since, in the heavy quark effective theory, the leading terms of inclusive decay rates of the heavy hadrons in the / expansion becomes that of the free heavy quark, b-quark in our context. The forward-backward asymmetry A FB of theleptonpairis anotherphysical quantity which can be observed in the experiments and provide important clues to test the theoretical models used. Using the definition of differential A FB A FB (s) = 0 0 dγ dz 0 dγ dsdz dzdsdz dz dγ dsdz + 0 dz dγ dsdz with z = cosθ, where θ is the angle between the momentum of the b-quark and that of τ + in the center of mass frame of the dileptons τ + τ, we get Here, In addition, A FB can be defined as (3) A FB (s) = E(s) (s). (4) E(s) = Re(C eff 9 C 0 s+2c 7C 0 +Ceff 9 C Q t+2c 7 C Q 2 t) (5) A FB = 0 dzdγ 0 dz dzdγ dz. (6) Γ Note that during the calculations of Γ and A FB, we take into account only the second resonance for the L effects coming from the reaction b sψ i sτ + τ, where i =,..,6 and divide the integration region for s into two parts : 4m2 τ m 2 b s, where m ψ2 = 3.686GeV is the mass of the second resonance (see Appendix B for L contributions). 4 s (m ψ ) 2 m 2 b and (m ψ ) 2 m 2 b

6 3 iscussion In the general 2HM model, there are many free parameters, such as masses of charged and neutralhiggsbosonsandthecomplexyukawacouplings, ξ U, ij,wherei,j arequarkflavorindices and these parameters should be restricted using the experimental measurements. Usually, the stronger restrictions to the new couplings are obtained from the analysis of the F = 2 (here F = K,B d,) decays, the ρ parameter and the B X s γ decay. The neutral Higgs bosons h 0 and A 0 give contributions to the Wilson coefficient C 7 (see the appendix of [9] for details) C h 0 7 (m W ) = (V tb V ts) i=d,s,b C A 0 7 (m W) = (V tb V ts ) i=d,s,b ξ N,bi ξ N,is ξ N,bi ξ N,is Q i 8m i, Q i 8m i, (7) wherem i andq i arethemasses andchargesofthedownquarks(i = d, s, b)respectively. These expressions show that the neutral Higgs bosons can give a large contribution to the coefficient C 7 which is in contradiction with the CLEO data [20], Br(B X s γ) = (3.5±0.35±0.32)0 4. (8) Such dangerous terms can be removed by assuming that the couplings ξ N,is (i = d,s,b) and ξ N,db are small enough to be able to reach the conditions ξ N,bb ξ N,is << and ξ N,db ξ N,ds <<. The discussion given above results in the following restrictions: ξ N,ib 0 and ξ N,ij 0, where the indices i,j denote d and s quarks. Further using the constraints [2], coming from the F = 2 mixing, the ρ parameter [8], and the measurement by CLEO Collaboration eq. (8) we get the condition for ξ N,tc, ξ N,tc << ξ U N,tt and take into account only the Yukawa couplings of quarks ξ N,tt U and ξ N,bb. As for ξ N,ττ, we do not consider any constraint and increase this parameter to enhance the effects of neutral Higgs boson. (For further discussion about the restrictions of the model III parameters see [8, 2].) In this section, we study the Wilson coefficients C Q ( ) and C Q2 ( ) coming from NHB effects and s, ξ N,bb and ξ N,ττ dependencies of dγ/ds and Γ for the inclusive decay b sτ + τ, restricting C eff 7 in the region C eff due to the CLEO measurement, eq.(8) (see [2] for details). Our numerical calculations based on this restriction and throughout these calculations, we use the redefinition ξ U, = 4GF 2 ξu,, we take the scale µ = and use the input values given in Table (). 5

7 Parameter Value m τ.78 (GeV) m c.4 (GeV) 4.8 (GeV) αem 29 λ t 0.04 Br (B X c l ν) 0.03±0.0 m t 75 (GeV) m W (GeV) m Z 9.9 (GeV) Λ QC (GeV) α s (m Z ) 0.7 sinθ W Table : The values of the input parameters used in the numerical calculations. In Fig. (2), we present m h 0 dependence of C Q ( ) for C eff 7 > 0, ξ N,bb = 40 (3 ), ξ N,ττ = 5GeV in the case r tb = ξ N,tt U < (r ξ N,bb tb > ). Here C Q ( ) lies in the region bounded by solid lines. For r tb >, m h 0 = 80GeV and m H 0 = 00GeV, the value of C Q ( ) changes between and However for r tb >, we get values, -9 and -2, more than two orders of magnitude larger compared to ones for r tb <, for the same value of m h 0. Since C Q ( ) is directly proportional to ξ N,ττ, its value may further increase with the increasing values of ξ N,ττ. The corresponding 2HM model II value of C Q ( ) can be extracted from [6] as beeing 0.4 for large tanβ, tanβ = 25. Forcompleteness, infigs.3and 4,wegivem H 0 dependenceofc Q ( )andm A 0 dependence of C Q2 ( ), for C eff 7 > 0, ξ N,bb = 40, ξ N,ττ = 5GeV in the case r tb <. As seen from Fig. 4, m A0 dependence of C Q2 ( ) is relatively weaker and for m A0 = 80GeV, C Q2 ( ) is between nearly and For r tb >, C eff 7 > 0, ξ N,bb = 3 and ξ N,ττ = 5GeV, C Q2 ( ) reaches up to the value of The 2HM model II value of C Q2 ( ) is 0.4 for tanβ = 25 [6]. Now we continue the analysis of the measurable quantities Γ and A FB of the process under consideration. In the following, we use the numerical values m H 0 = 50GeV, m h 0 = 80GeV and m A 0 = 80GeV in our calculations. In Fig. 5, we plot the differential Γ of the decay b sτ + τ with respect to the parameter s for ξ N,bb = 40, ξ N,ττ = GeV and charged Higgs mass m H ± = 400GeV in case of the ratio r tb <. Here the differential Γ lies in the region bounded by dashed (small dashed) curves for C eff 7 > 0 (C eff 7 < 0). A small enhancement is possible especially for C eff 7 > 0 case compared to 6

8 the SM (solid curve). Further, the restriction region of the differential Γ for model III becomes narrower with increasing or decreasing values of the parameter s. Fig. 6 is devoted the same dependence of the differential Γ including the long distance (L) effects. Here C eff 7 < 0 case for model III almost coincides with the SM (solid curve). In case of the ratio r tb >, extremely large enhancement, 3 orders larger compared the r tb < case, is reached even for the small values of ξ N,bb (see Fig. 7). Fig. 8 shows ξ N,bb dependence of Γ of the decay under consideration for ξ N,ττ = GeV and chargedhiggsmassm H ± = 400GeV incase oftheratio r tb <. HereΓisalmostnon-sensitive to ξ N,bb. However for r tb > case (Fig. 9) Γ is strongly sensitive to ξ N,bb Γ is 2 orders (3 orders) larger compared to the SM result for C eff ξ N,bb < 2. for C eff 7 > 0. Further, 7 < 0 (C eff 7 > 0) even for Fig. 0 is devoted to the dependence of Γ to the charged Higgs mass m ± H. Γ has a weak dependence (almost no dependence) on m ± H for C eff 7 > 0 (C eff 7 < 0). For completeness, in Figs. and 2 we also present ξ N,ττ values of ξ N,ττ. Sensitivity of Γ to ξ N,ττ dependence of Γ for large increases with the increasing values of this parameter. Γ enhances for extremely large values of ξ N,ττ and this is the contribution due to the NHB effects. For r tb < the NHB effects are small and destructive up to the large values of ξ N,ττ, ξ N,ττ = 800GeV. For Ceff 7 > 0, ξ N,bb = 40 and ξ N,ττ = (00)GeV this effect is at the order of the magnitude %0.(4) of the overall contribution. However, it is positive for r tb > and it becomes considerable with increasing values of ξ N,ττ. For Ceff 7 > 0, the small value ξ N,bb = 3 and ξ N,ττ = (00, 200)GeV, the NHB contribution can reach the magnitude %0.5(7, 26) of the overall contribution. Our results on A FB (s) and A FB for the decay under consideration are presented through the graphs given by Figs In Fig. 3 A FB (s) is shown for ξ N,bb = 40, ξ N,ττ = GeV and charged Higgs mass m H ± = 400GeV in case of the ratio r tb <. Here A FB (s) lies in the region bounded by solid lines for C eff 7 > 0. ashed line presents C eff 7 < 0 case and the SM result coincides with this line. There is possible negative values of A FB (s) due to the L effects. For r tb >, A FB (s) almost vanishes ( 0 4 ). Fig. 4 is devoted to ξ N,bb dependence of A FB for ξ N,ττ = GeV, charged Higgs mass m H ± = 400GeV and r tb <. Here, A FB is not sensitive to ξ N,bb, especially for large values of this parameter. The SM and model III average results for C eff 7 < 0 (C eff 7 > 0) are and (0.325), respectively. A FB is sensitive to the parameter ξ N,bb for its small values in the case where r tb > and C eff 7 < 0 (Fig. 5). The enhancement over the SM is possible for 7

9 ξ N,bb < 0.4, namely A FB can reach the value of The restriction region for A FB is large for this case. However, for C eff 7 > 0, A FB almost vanishes. The NHB effects on A FB is sensitive to the coupling ξ N,ττ as it should be. For r tb < and C eff 7 > 0, the NHB contribution is %0.5 for ξ N,ττ = GeV and %.2 for ξ N,ττ = 00GeV in case the parameter ξ N,bb = 40. Increasing ξ N,ττ causes to the enhancement in the NHB effects. For r tb > and C eff 7 < 0, the NHB effects are negative and it increases the overall result by %0 for ξ N,bb = 0.4 and ξ N,ττ = 00GeV. For r tb > and C eff 7 > 0, the NHB effects to A FB are negligible. Now, we would like to summarize our results. Γ for the process under consideration is at the order of 0 6 for r tb < and C eff 7 > 0 results is greater compared to C eff 7 < 0 one. On the otherhand, for r tb >, there is a considerable enhancement, three order larger compared to the SM case even for small values of ξ N,bb. Further, Γ is not sensitive to ξ N,bb to this parameter is observed for r tb >. for r tb <, however strong sensitivity A FB is not so much sensitive to the model III parameters for r tb <. For r tb >, there is a possible enhancement in the A FB for small values of ξ N,bb, however it becomes negligible with increasing ξ N,bb. The NHB effects becomes important for the large values of the Yukawa coupling ξ N,ττ. Therefore, the experimental investigation of Γ and A FB ensure a crucial test for new physics and also the sign of C eff 7. 8

10 Appendix A The operator basis The operator basis in the 2HM (model III ) for our process is [6, 22, 23] O = ( s Lα γ µ c Lβ )( c Lβ γ µ b Lα ), O 2 = ( 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=u,d,s,c,b q=u,d,s,c,b q=u,d,s,c,b q=u,d,s,c,b ( q Lβ γ µ q Lα ), ( q Rβ γ µ q Rβ ), ( q Rβ γ µ q Rα ), e 6π 2 s ασ µν ( R+m s L)b α F µν, g 6π 2 s αtαβσ a µν ( R+m s L)b β G aµν, e 6π 2( s Lαγ µ b Lα )( τγ µ τ), e 6π 2( s Lαγ µ b Lα )( τγ µ γ 5 τ), Q = e2 6π 2( sα L bα R )( ττ) Q 2 = e2 6π 2( sα L bα R )( τγ 5τ) Q 3 = g2 6π 2( sα L bα R ) q=u,d,s,c,b Q 4 = g2 6π 2( sα L bα R ) Q 5 = g2 6π 2( sα Lb β R) Q 6 = g2 6π 2( sα L bβ R) q=u,d,s,c,b q=u,d,s,c,b q=u,d,s,c,b Q 7 = g2 6π 2( 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=u,d,s,c,b Q 8 = g2 6π 2( sα L σµν b α R ) q=u,d,s,c,b ( q β L σ µνq β R ) ( q β R σ µνq β L ) 9

11 Q 9 = g2 6π 2( sα L σµν b β R ) q=u,d,s,c,b Q 0 = g2 6π 2( sα L σµν b β R ) q=u,d,s,c,b ( q β L σ µνq α R ) ( q β R σ µνq α L ) (9) 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 model III. However, we do not present them here since corresponding Wilson coefficients are negligible. B The Initial values of the Wilson coefficients. The initial values of the Wilson coefficients for the relevant process in the SM are [22] C SM,3,...6 (m W) = 0, C SM 2 (m W ) =, C SM 7 (m W ) = 3x3 t 2x 2 t 4(x t ) 4 lnx t + 8x3 t 5x 2 t +7x t 24(x t ) 3, C SM 8 (m W ) = 3x2 t 4(x t ) 4 lnx t + x3 t +5x2 t +2x t 8(x t ) 3, C9 SM (m W ) = B(x sin 2 t )+ 4sin2 θ W θ W sin 2 C(x t ) (x t )+ 4 θ W 9,, C SM 0 (m W) = sin 2 θ W (B(x t ) C(x t )), C SM Q i (m W ) = 0 i =,..,0. (20) 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 2 F (y t ) + XY F 2 (y t ), C H 8 (m W) = Y 2 G (y t ) + XY G 2 (y t ), C H 9 (m W ) = Y 2 H (y t ), C H 0(m W ) = Y 2 L (y t ), (2) where X = ( ξ N,bb + ξ N,sb Y = m t ( ξ U N,tt + ξ U N,tc 0 ) V ts V tb Vcs ) Vts,, (22)

12 and due to the neutral Higgs bosons are C A0 Q 2 (( ξ U N,tt )3 ) = ξ N,ττ ( ξ N,tt U )3 y t (Θ 5 (y t )z A Θ (z A,y t )), 32π 2 m 2 A m 0 t Θ (z A,y t )Θ 5 (y t ) C A0 Q 2 (( ξ U N,tt )2 ) = ξ N,ττ ( ξ U N,tt )2 ξ N,bb 32π 2 m 2 A 0 z ((y t (Θ (z A,y t ) Θ 5 (y t )(xy +z A )) 2Θ (z A,y t )Θ 5 (y t )ln[ A Θ 5 (y t) Θ (z A,y t )Θ 5 (y t ) ] ) Θ (z A,y t), ( CQ A0 2 ( ξ N,tt) U = g2 ξ N,ττ ξu N,tt x t 2 64π 2 m 2 A m 0 t Θ 5 (x t ) xyx t +2z A Θ (z A,x t ) 2ln[ z AΘ 5 (x t ) Θ (z A,x t ) ] ( (x )x t (y t /z A ) (+x)y t xyx t y t (Θ 6 (x y)(x t y t ))(Θ 3 (z A )+(x y)(x t y t )z A ) x(y t +x t ( y t /z A )) 2y ) ) t, Θ 6 Θ 3 (z A ) CQ A0 2 ( ξ N,bb) = g2 ξ ( N,ττ ξ N,bb x 2 t y t +2y(x )x t y t z A (x 2 t +Θ 6) + x2 t ( y t/z A ) +2ln[ z AΘ 6 64π 2 m 2 A Θ 0 3 (z A ) Θ 6 Θ 2 (z A ) ]), CQ H0 (( ξ N,tt U )2 ) = g2 ( ξ N,tt U ( )2 m τ xt ( 2y)y t + ( +2cos2 θ W )( +x+y)y t 64π 2 m 2 H m 2 0 t Θ 5 (y t ) cos 2 θ W Θ 4 (y t ) + z H(Θ (z H,y t )xy t +cos 2 θ W ( 2x 2 ( +x t )yyt 2+xx tyyt 2 Θ ) 8z H )), (23) cos 2 θ W Θ (z H,y t )Θ 7 ( CQ H0 ( ξ N,tt U ) = g2 ξu N,tt ξ N,bb m τ ( +2cos 2 θ W )y ( t 64π 2 m 2 H m 0 t cos 2 θ W Θ 4 (y t ) + z ) H x t y t Θ 7 Θ 5 (y t ) + x ty t (xy z H ) Θ (z H,y t ) [ ]) Θ5 (y t )z H 2x t ln, Θ (z H,y t ) ( CQ H0 (g 4 ) = g4 m τ x t ( +2x)x t + 28π 2 m 2 H m 2 0 t Θ 5 (x t )+y( x t ) + 2x t( +(2+x t )y) Θ 5 (x t ) 4cos2 θ W ( +x+y)+x t (x+y) + x t(x(x t (y 2z H ) 4z H )+2z H ) cos 2 θ W Θ 4 (x t ) Θ (z H,x t ) + y t(( +x)x t z H +cos 2 θ W ((3x y)z H +x t (2y(x ) z H (2 3x y)))) +2(x t ln [ ] Θ5 (x t )z H +ln Θ (z H,x t ) cos 2 θ W (Θ 3 (z H )+x(x t y t )z H ) [ ] ) x(yt x t )z H Θ 3 (z H ) ), (Θ 5 (x t )+y( x t )y t z H ξ Q (( ξ N,tt U N,ττ( ξ N,tt) U 3 y t ( ) )3 ) = Θ (z 32π 2 m 2 h,y t )(2y )+Θ 5 (y t )(2x )z h, h m O t Θ (z h,y t )Θ 5 (y t ) C h 0

13 C h 0 Q (( ξ U N,tt )2 ) = ξ N,ττ ξ N,bb ( ξ N,tt U ( )2 (Θ5 (y t )z h (y t )(x+y ) Θ (z h,y t )(Θ 5 (y t )+y t ) 32π 2 m 2 h [ O ]) zh Θ 5 (y t ) 2ln, Θ (z h ) Θ (z h )Θ 5 (y t ) ( CQ h0 ( ξ N,tt) U = g2 ξ N,ττ ξu N,tt x t 2( +(2+xt )y) x t(x )(y t z h ) 64π 2 m 2 h m 0 t Θ 5 (x t ) Θ 2(z h ) where and + x(x t(y 2z h ) 4z h )+2z h Θ (z h,x t ) + Θ 9 +y t z h ((x y)(x t y t ) Θ 6 )(2x ) z h Θ 6 (Θ 6 (x y)(x t y t )) +2ln (+x)y t z h xyx t y t +z h ((x y)(x t y t ) Θ 6 ) [ ] zh Θ 5 (x t ) Θ (z h,x t ) + x(y tz h +x t (z h y t )) 2y t z h Θ 2 (z h ) ( CQ h0 ( ξ N,bb ) = g2 ξ N,ττ ξ N,bb yxt y t (xx 2 [ ]]) t(y t z h )+Θ 6 z h (x 2)) zh Θ 6 +2ln, 64π 2 m 2 h z 0 h Θ 2 (z h )Θ 6 Θ 2 (z h ) Θ (ω,λ) = ( +y yλ)ω x(yλ+ω ωλ) Θ 2 (ω) = (x t +y( x t ))y t ω xx t (yy t +(y t )ω) Θ 2(ω) = Θ 2 (ω,x t y t ) Θ 3 (ω) = (x t ( +y) y)y t ω +xx t (yy t +ω( +y t )) Θ 4 (ω) = x+xω Θ 5 (λ) = x+λ( x) Θ 6 = (x t +y( x t ))y t +xx t ( y t ) Θ 7 = (y(y t ) y t )z H +x(yy t +(y t )z H ) (24) Θ 8 = y t (2x 2 (+x t )(y t )+x t (y( y t )+y t )+x(2( y +y t ) + x t ( 2y( y t ) 3y t ))) Θ 9 = x 2 t ( +x+y)( y t +x(2y t ))(y t z h ) x t y t z h (x(+2x) 2y) + y 2 t (x t(x 2 y( x))+(+x)(x y)z h ) ), x t = m2 t m 2 W, y t = m2 t m H ±, z H = m2 t m 2 H 0, z h = m2 t m 2 h 0, z A = m2 t m 2 A 0, The explicit forms of the functions F (2) (y t ), G (2) (y t ), H (y t ) and L (y t ) in eq.(2) are given as F (y t ) = y t(7 5y t 8y 2 t ) 72(y t ) 3 + y2 t (3y t 2) 2(y t ) 4 lny t, 2

14 F 2 (y t ) = y t(5y t 3) 2(y t ) + y t( 3y t +2) lny 2 6(y t ) 3 t, G (y t ) = y t( yt 2 +5y t +2) yt (y t ) 3 4(y t ) lny 4 t, G 2 (y t ) = y t(y t 3) 4(y t ) + y t 2 2(y t ) lny 3 t, [ H (y t ) = 4sin2 θ W sin 2 θ W L (y t ) = xy t 8 y t (y t ) lny 2 t ] [ 47y 2 y t 79y t +38 t 3y3 t 6y t +4 lny 08(y t ) 3 8(y t ) 4 t [ xy t ] sin 2 θ W 8 y t + (y t ) lny 2 t. ], (25) Finally, the initial values of the coefficients in the model III are C 2HM i (m W ) = Ci SM (m W )+Ci H (m W ), C 2HM Q (m W ) = C 2HM Q 2 (m W ) = C 2HM Q 3 (m W ) = C 2HM Q 4 (m W ) = 0 dx x 0 + CQ h0 (( ξ N,tt) U 2 )+C h0 0 dx 0 x m τ sin 2 (CQ 2HM θ W dy(cq H0 (( ξ N,tt U )2 )+CQ H0 ( ξ N,tt U )+CH0 Q (g 4 )+CQ h0 (( ξ N,tt U )3 ) Q ( ξ N,tt)+C U Q h0 ( ξ N,bb)), dy(c A0 Q 2 (( ξ U N,tt )3 )+C A0 Q 2 (( ξ U N,tt )2 )+C A0 Q 2 ( ξ U N,tt )+CA0 Q 2 ( ξ N,bb )) (m W )+C 2HM Q 2 (m W )) m τ sin 2 (CQ 2HM θ (m W ) CQ 2HM 2 (m W )) W C 2HM Q i (m W ) = 0, i = 5,...,0. (26) Here, we present C Q and C Q2 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 2HM i (µ) and CQ 2HM 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, 6, 9, 23]. The Wilson coefficients playing the essential role in this process are C 2HM 7 (µ), C 2HM 9 (µ), C 2HM 0 (µ), C 2HM expressions. Q (µ) and CQ 2HM 2 (µ). For completeness, in the following we give their explicit C eff 7 (µ) = C 2HM 7 (µ)+q d (C 2HM 5 (µ)+n c C 2HM 6 (µ)), (27) where the LO QC corrected Wilson coefficient C LO,2HM 7 (µ) is given by C LO,2HM 7 (µ) = η 6/23 C7 2HM (m W )+(8/3)(η 4/23 η 6/23 )C8 2HM (m W ) 8 + C2 2HM (m W ) h i η a i, (28) i= 3

15 and η = α s (m W )/α s (µ), h i and a i are the numbers which appear during the evaluation [3]. C eff 9 (µ) contains a perturbative part and a part coming from L effects due to conversion of the real cc into lepton pair τ + τ : where and C pert 9 (µ) = C 2HM 9 (µ) C eff 9 (µ) = C pert 9 (µ)+y reson (ŝ), (29) + h(z,s)(3c (µ)+c 2 (µ)+3c 3 (µ)+c 4 (µ)+3c 5 (µ)+c 6 (µ)) 2 h(,s)(4c 3(µ)+4C 4 (µ)+3c 5 (µ)+c 6 (µ)) (30) 2 h(0,s)(c 3(µ)+3C 4 (µ))+ 2 9 (3C 3(µ)+C 4 (µ)+3c 5 (µ)+c 6 (µ)), Y reson (ŝ) = 3 κ πγ(v i τ+ τ )m Vi αem 2 V i =ψ i q 2 m Vi +im Vi Γ Vi (3C (µ)+c 2 (µ)+3c 3 (µ)+c 4 (µ)+3c 5 (µ)+c 6 (µ)). (3) In eq.(29), the functions h(u,s) are given by h(u,s) = 8 9 ln µ 8 9 lnu x (32) ( 2 ln x+ iπ ) x, for x 4u 2 < 9 (2+x) x /2 s 2arctan x, for x 4u2 >, s h(0,s) = ln µ 4 9 lns+ 4 iπ, (33) 9 with u = mc. The phenomenological parameter κ in eq. (3) is taken as 2.3. In eqs. (30) and (3), the contributions of the coefficients C (µ),..., C 6 (µ) are due to the operator mixing. Finally, the Wilson coefficients C Q (µ) and C Q2 (µ) are given by [6] C Qi (µ) = η 2/23 C Qi (m W ), i =,2. (34) 4

16 References [] J. L. Hewett, in Proc. of the 2 st Annual SLAC Summer Institute, ed. L. e Porcel and C. unwoode, SLAC-PUB-652 (994) [2] W. -S. Hou, R. S. Willey and A. Soni, Phys. Rev. Lett. 58 (987) 608. [3] N. G. eshpande and J. Trampetic, Phys. Rev. Lett. 60 (988) [4] C. S. Lim, T. Morozumi and A. I. Sanda, Phys. Lett. B28 (989) 343. [5] B. Grinstein, M. J. Savage and M. B. Wise, Nucl. Phys. B39 (989) 27. [6] C. ominguez, N. Paver and Riazuddin, Phys. Lett. B24 (988) 459. [7] N. G. eshpande, J. Trampetic and K. Ponose, Phys. Rev. 39 (989) 46. [8] P. J. O onnell and H. K. Tung, Phys. Rev. 43 (99) [9] N. Paver and Riazuddin, Phys. Rev. 45 (992) 978. [0] A. Ali, T. Mannel and T. Morozumi, Phys. Lett. B273 (99) 505. [] A. Ali, G. F. Giudice and T. Mannel, Z. Phys. C67 (995) 47. [2] C. Greub, A. Ioannissian and. Wyler, Phys. Lett. B346 (995) 45;. Liu Phys. Lett. B346 (995) 355; G. Burdman, Phys. Rev. 52 (995) 6400: Y. Okada, Y. Shimizu and M. Tanaka Phys. Lett. B405 (997) 297. [3] A. J. Buras and M. Münz, Phys. Rev. 52 (995) 86. [4] N. G. eshpande, X. -G. He and J. Trampetic, Phys. Lett. B367 (996) 362. [5] W. Jaus and. Wyler, Phys. Rev. 4 (990) [6] Y. B. ai, C.S. Huang and H. W. Huang Phys. Lett. B390 (997) 257, C. S. Huang, L. Wei, Q. S. Yan and S. H. Zhu, hep-ph/ [7] H. E. Logan and U. Nierste, Nucl. Phys. B586 (2000) 39. [8]. Atwood, L. Reina and A. Soni, Phys. Rev. 55 (997)

17 [9] T. M. Aliev, and E. Iltan, Phys. Rev. 58 (998) [20] M. S. Alam, CLEO Collaboration, to appear in ICHEP98 Conference (998) [2] T. M. Aliev, and E. Iltan, J. Phys. G25 (999) 989. [22] B. Grinstein, R. Springer, and M. Wise, Nucl. Phys. B339 (990) 269; R. Grigjanis, P.J. O onnel, M. Sutherland and H. Navelet, Phys. Lett. B23 (988) 355; Phys. Lett. B286 (992) E, 43; G. Cella, G. Curci, G. Ricciardi and A. Viceré, Phys. Lett. B325 (994) 227, Nucl. Phys. B43 (994) 47. [23] M. Misiak, Nucl. Phys. B393 (993) 23, Erratum B439 (995) 46. 6

18 ¼º½ ¼º½ ¼º½¾ ɽ Ñ µ ¼º½ ¼º¼ ¼º¼ ¼º¼ ¼º¼¾ ¼ Ñ ¼ Î µ Figure : C Q ( ) as a function of m h 0 for ξ N,bb = 40, ξ N,ττ = 5GeV, m H ± = 400GeV and m H 0 = 00GeV in case of the ratio r tb <. ¼ ¹ ¹½¼ ɽ Ñ µ ¹½ ¹¾¼ ¹¾ ¹ ¼ ¹ ½¼¼ Ñ ¼ Î µ Figure 2: Same as Fig., but for ξ N,bb = 3 and r tb >. 7

19 ¼º¼ ¼º¼ ɽ Ñ µ ¼º¼ ¼º¼ ¼º¼¾ ¼º¼½ ¼ ½¼¼ ½½¼ ½¾¼ ½ ¼ ½ ½ Ñ À ¼ Î µ Figure 3: C Q ( ) as a function of m H 0 for ξ N,bb = 40, ξ N,ττ = 5GeV, m H ± = 400GeV and m h 0 = 80GeV in case of the ratio r tb <. ¼º¼¾ ¼º¼¾¾ ¼º¼¾ ɾ Ñ µ ¼º¼¾ ¼º¼¾ ¼º¼¾ ¼º¼¾¾ ½¼¼ Ñ ¼ Î µ Figure 4: C Q2 ( ) as a function of m A 0 for ξ N,bb = 40, ξ N,ττ = 5GeV and m H ± = 400GeV in case of the ratio r tb <. 8

20 ½¼ µ Î µ ½¼ µ Î µ ¾º ¾ ½º ½ ÅÓÐ ÁÁÁ ÅÓÐ ÁÁÁ ¼ ¼ ËÅ ¼ ½ Figure 5: ifferential Γ as a function of s for ξ N,bb = 40, ξ N,ττ = GeV and m H ± = 400GeV in case of the ratio r tb <. ¾ ½ ÅÓÐ ÁÁÁ ¼ ËÅ ¼ ½ Figure 6: The same as Fig 5, but with L effects. 9

21 ½¼ µ ½¼ µ Î µ ¼¼ ¼ ¼¼¼ ¾¼ ¾¼¼¼ ½¼ ½¼¼¼ ¼ ÅÓÐ ÁÁÁ ¼ ÅÓÐ ÁÁÁ ¼ ¼ ½ Figure 7: The same as Fig 6, but for ξ N,bb = 3 and r tb >. ¼º ¼º ÅÓÐ ÁÁÁ ÅÓÐ ÁÁÁ ¼ ¼ ËÅ ¼º¾ ¼º¾ ¾¼ ¼ Æ Ñ Figure 8: Γ as a function of ξ N,bb r tb <. for ξ N,ττ = GeV and m H ± = 400GeV in case of the ratio 20

22 ½¼ µ Î µ ½¼ µ ½¼¼ ¾¼ ÅÓÐ ÁÁÁ ÅÓÐ ÁÁÁ ¼ ¼ ËÅ ¼ ¼º¾ ½ ½º¾ Æ Ñ ½º ½º ½º ¾ Figure 9: The same as Fig. 8 but for r tb >. ¼º ¼º ¼º¾ ÅÓÐ ÁÁÁ ÅÓÐ ÁÁÁ ¼ ¼ ËÅ ¼º¾ ¼ ¾¼ ¼ ¾¼ ÑÀ Î µ Figure 0: Γ as a function of m H ± for ξ N,bb = 40, ξ N,ττ = GeV in case of the ratio r tb <. 2

23 ½¼ µ Î µ ½¼ µ Î µ ¼º ¼º ¼º¾ ÅÓÐ ÁÁÁ ÅÓÐ ÁÁÁ ¼ ¼ ËÅ ¼º¾ ¼ ½¼¼¼ ½¼ ¾¼¼¼ ¾¼ ¼¼¼ Figure : Γ as a function of ξ N,ττ, for ξ N,bb = 40, m H ± = 400GeV, in case of the ratio r tb <. ¾¼ ½ ½¼ ÅÓÐ ÁÁÁ ÅÓÐ ÁÁÁ ¼ ¼ ¼ ¼ ½¼¼¼ ½¼ ¾¼¼¼ ¾¼ Figure 2: The same as Fig. for ξ N,bb = 0. in case of the ratio r tb >. 22

24 µ ÅÓÐ ÁÁÁ ¼ ËÅ ¼º¾ ¼ ¹¼º¾ ½ Figure 3: ifferential A FB as a function of s for ξ N,bb = 40, ξ N,ττ = GeV and m H ± = 400GeV including L effects in case of the ratio r tb <. ¼º ¼º ¼º ÅÓÐ ÁÁÁ ÅÓÐ ÁÁÁ ¼ ¼ ËÅ ¼º ¼º ¼º ¼º ¼º ¾ ¼º ¾ ¼º ½ ¼º ½ ¾¼ ¼ ÆÑ Figure 4: A FB as a function of ξ N,bb for ξ N,ττ = GeV, m H ± = 400GeV and r tb <. 23

25 ¼º ÅÓÐ ÁÁÁ ÅÓÐ ÁÁÁ ¼ ¼ ËÅ ¼º ¼º¾ ¼º¾ ¼º½ ¼º½ ¼º¼ ¼ ¼º¾ ½ ÆÑ ½º¾ ½º ½º ½º ¾ Figure 5: The same as Fig. 4 but for r tb >. 24