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1 YUMS SNUTP HUPD-9710 DPN B! X q l + l, (q = d; s) and Determination of td ts hep-ph/ (ps created: ) C. S. Kim Department of Physics, Yonsei University Seoul , Korea T. Morozumi y Department of Physics, Hiroshima University Kagamiyama, Higashi Hiroshima, Japan 739 A. I. Sanda z Department of Physics, Nagoya University Chikusa-ku, Nagoya, Japan Abstract We propose a new method to extract j tdj j tsj db ds [B! X dl + l, ]= db from the ratio of the decay distributions dr! ds ds [B X sl + l, ]. This ratio depends only on the KM ratio j tdj j tsj with 15% theoretical uncertainties, if dilepton invariant mass-squared s is away from the peaks of the possible resonance states, J=, 0, and etc. We also give a detailed analytical and numerical analysis on! db ds [B X ql + l, ](q = d; s) and dr ds. address: kim@cskim.yonsei.ac.kr, cskim@kekvax.kek.jp y address: morozumi@theo.phy.sci.hiroshima-u.ac.jp z address: sanda@eken.phys.nagoya-u.ac.jp

2 1 Introduction The determination of the elements of KM matrix is one of the most important issues in the quark avor physics. Moreover, the element td (or ub ) is especially important to the Standard Model description of CP-violation. If it were zero, there would be no CP-violation from the KM matrix elements (i.e. within the Standard Model), and we have to seek for other source of CP violation in K L!. Here we study the ratio td ts. In the Standard Model with the unitarity of KM matrix, ts is approximated by, cb, which is directly measured by semileptonic B decays. There are already several ways to determine td available in the literature: j td j can be indirectly extracted through B d, B d mixing. However, in B d, B d mixing the large uncertainty of the hadronic matrix elements prevents us to extract the element of KM with good accuracy. A better extraction of j td ts j can be made if B s, B s is measured as well, because the ratio f B d B Bd =f Bs B Bs can be much better determined. The determination of j td ts j from the ratios of rates of several hadronic two-body B decays, such as,(b 0! K 0 K 0 )=,(B 0! K 0 ),,(B 0! K 0 K 0 )=,(B 0! K 0 ),,(B +! K 0 K + )=,(B +! K + ), and,(b +! K 0 K + )=,(B +! K + ) has been also proposed in Ref. [1]. td can be determined from K +! + decay within 15% theoretical uncertainties with the branching fraction B10,10 []. Why do we discuss yet another method to determine td? The main reason we are interested in B physics is that this area is very likely to yield information about new physics beyond the Standard Model. We expect that new physics will inuence experimentally measureable quantities in dierent ways. For example, most of us expect that B = transition is more sensitive to new physics than the decay rates. New physics may couple dierently to K mesons compared to B mesons. Therefore, it is essential to determine the KM matrix elements in as many dierent methods as possible. In this paper, we propose another method to determine j tdj j tsj precisely from the decay distributions db ds [B! X sl + l, ]and db ds [B! X dl + l, ], where s is invariant mass-

3 squared of nal lepton pair, l + l,. In the decays of B! X q l + l, (q = d; s), the short distance (SD) contribution comes from the top quark loop diagrams and the long distance (LD) contribution comes from the decay chains due to intermediate charmonium states. Therefore, the former (SD) amplitude is proportional to tq tb, and the latter (LD) proportional to cq cb. If the invariant mass-squared of l + l, is away from the peaks of the charmoniuum resonances (J= and 0 ), the SD contribution is dominant, while on the peaks of the resonances the LD contribution is dominant, and therefore we expect that in the SU(3) symmetry limit the ratio becomes dr ds db ds (B! X dl + l, )= db ds (B! X sl + l, ) 1 dr(s) ds! (1, ) + (s! 1Ge ) ;! 1 (s! m J= ) ; (1) where is sin c, c = Cabibbo angle, and for,, see Eq. (8). In the intermediate region, there is a characteristic interference beteen the LD contribution and the SD contribution, which requires the detailed study of the distributions. By focusing on the region, 1(Ge ) s < m J=, we can study j td ts j from the experimental ratio of the distributions dr ds. We stress the advantages to use the inclusive semileptonic decays: If dilepton invariant mass-squared s is away from the peaks of the possible intermediate states, J=, 0,,!, and etc., the short distance contribution due to top quark loop is dominant. Therefore, we may extract very precisely the combination of j td ts j from the decay distributions within the range 1(Ge ) s<m J= without any theoretical uncertainties from the LD hadronic matrices, unknown KM matrix elements, and etc. Inclusive decay distributions are theoretically well predicted by heavy quark mass expansion away from the phase space boundary. The leading order of the expansion agrees with the parton model result. Furthermore, non-perturbative 1=m b power corrections of QCD can be easily incorporated. This paper is organized as follows. In Section, we present the analytic formulae for db ds [B! X sl + l, ] and db ds [B! X dl + l, ]. We also show in detail the decay distri- 3

4 bution db ds [B! X sl + l, ] including the uncertainties coming from top quark mass m t and the scale of renormalization group. In Section 3, after brief discussion on the limit of KM elements coming from B d,b d mixing and B! X u l, we study the ratio of the decay distributions, and show how we may extract j td ts j. Eective Hamiltonian for b! ql + l, and their dierential decay rates. In this Section, the dierential decay rates for B! X q l + l, (q = d; s) are shown including both LD and SD eects as well as QCD =m b power corrections. (See [3] for b! dl + l,, [4] for b! sl + l,, and [5], [6] for B! X s l + l, including the 1=m b power corrections.) We also show in detail how the dierential decay rate varies by changing the input parameters m t, and the renormalization scale. In Table 1, we summarize all the values of the input parameters used in our numerical calculations of decay rates. We use the central values for those input parameters, unless otherwise specied. The eective Hamiltonian for b! ql + l, (q = d; s) isgiven as # C i O i H eff =, 4G F p tq tb " 10X i=1 + 4G F p uq ub [C 1 (O 1u, O 1 )+C (O u, O )] ; () where ij are the KM matrix elements. The operators are given as O 1 = (q L b L )(c L c L ); O = (q L b L )(c L c L ); O 3 = (q L b L ) O 4 = (q L b L ) O 5 = (q L b L ) O 6 = (q L b L ) O 7 = O 8 = X X q 0 =u;d;s;c;b X q 0 =u;d;s;c;b X q 0 =u;d;s;c;b q 0 =u;d;s;c;b (q 0 L q 0 L); (q 0 L q 0 L); (q 0 R q 0 R); (q 0 R q 0 R); e 16 q (m b R + m q L)b F ; g 16 q T a (m b R + m q L)b G a ; 4

5 O 1 u O u = (q L b L )(u L u L ); = (q L b L )(u L u L ); O 9 = e 16 q Lb l l; O 10 = e 16 q Lb l 5 l; (3) where L and R denote chiral projections, L(R) = 1=(1 5 ). We use the Wilson coecients given in the literature (see, for example, [7]). With the eective Hamiltonian in Eq. (), the matrix element for the decays b! ql + l, (q = d; s) canbe written as M(b! ql + l, ) = G F p tq tb where C e 9q is given by + C e 9q, C 10 (q Lb) l Ll C e 9q + C 10 (q Lb) l Rl,C e 7 qi q q (m ql + m b R) b! l l # ; (4) C e 9q (^s) C 9 ( 1+ s()!(^s) ) + Y SD q (^s)+y LD q (^s) ; (5) where q is the four momentum of l + l,, s = q,and^s = s=m b. The function Y SD q (^s) is the one-loop matrix element ofo 9,andY LD q (^s) is the LD contributions due to the vector mesons mesons J=, 0 and higher resonances. The function!(^s) represents the O( s ) correction from the one-gluon exchange in the matrix element of O 9, and is given in our Appendix. The two functions Y SD q and Y LD q are written as Y SD q (^s) = g(^m c ; ^s)(3 C 1 + C +3C 3 + C 4 +3C 5 + C 6 ), 1 g(1; ^s)(4 C 3 +4C 4 +3C 5 + C 6 ), 1 g(0; ^s)(c 3 +3C 4 ) + 9 (3 C 3 + C 4 +3C 5 + C 6 ), uq ub (3C tq 1 + C )(g(0; ^s), g(^m c ; ^s)) ; (6) tb Y q LD (^s) = 3, cq cb C (0), uq ub (3 C tq tb tq 3 + C 4 +3C 5 + C 6 ) tb X i = (1s);:::; (6s),( i! l + l, ) M i M i, ^sm b, imi, i ; (7) 5!

6 where the function g( ^m c = mc m b ; ^s), g(1; ^s), and g(0; ^s) represent c quark, b quark and u; d; s quark loop contributions, respectively. The two functions g(z; ^s) and g(0; ^s) are given [7] in our Appendix. In Eq. (7), C (0) 3C 1 + C +3C 3 + C 4 +3C 5 + C 6, and the rst two terms are dominant, as can be seen from the Table. It is convenient to write the relevant combinations of KM in terms of the Wolfenstein parametrization. In the following, in addition to A and, wechoose (1,) + and 1 = arg(, cdcb ) asindependent variables, then we have td tb cs cb us ub cd cb ts tb ' A; ts tb ',1 ; ts tb ' O( ) ; td tb ' 3 A(1, + i) ; td tb ', 1 1, + i ; =, q exp(,i 1) ; (1, ) + ud ub td tb ', i 1, + i ; =,1+ q exp(,i 1) : (8) (1, ) + In the SD contribution of B! X s l + l,, the u-quark loop contribution is neglected due to the smallness of the combination us ub compared with cs cb ', ts tb,while in B! X d l + l, the term which is proportional to ud ub is maintained. In the LD contribution, there is a contribution coming from the gluonic penguin amplitude which is proportional to ud ub td tb (3C 3 +C 4 +3C 5 +C 6 ). This contribution is neglected because of the smallness of the Wilson coecient (3C 3 + C 4 +3C 5 + C 6 ) compared with C (0). We adopt =:3 [8] to reproduce the rate of the decay chain B! X s J=! X s l + l,. Note that the data determines only the combination, C (0) =0:88. By combining both SD and LD contributions as well as non-perturbative 1=m b power corrections, the dierential decay rate for B! X q l + l, (q = d; s) becomes db d^s =j tq tb j j cb j B 0 3 ^u(^s; ^m q)((1, ^m q) +^s(1 + ^m q), ^s ) (1, 4^m q +6^m 4 q, 4^m 6 q + ^m 8 q, ^s + ^m q^s + ^m 4 q^s, ^m 6 q^s 6

7 ,3^s, ^m q^s, 3^m 4 q^s +5^s 3 +5^m q^s 3, ^s 4 ) ^ 1 ^u(^s; ^m q ) + 1, 8^m q +18^m 4 q, 16 ^m 6 q +5^m 8 q, ^s, 3^m q^s +9^m 4 q^s, 5^m 6 q^s, 15^s,18 ^m q^s, 15 ^m 4 q^s +5^s 3 +5^m q^s 3, 10^s 4 ^ ^u(^s; ^m q ) # jc e 9 j + jc 10 j ^u(^s; ^m q) (1 + ^m q)(1, ^m q), ( ^m q + ^m 4 q)^s, (1 + ^m q)^s (, 6^m q +4^m 4 q +4^m 6 q, 6^m 8 q +^m 10 q,5^s, 1 ^m q^s +34^m 4 q^s, 1 ^m 6 q^s, 5^m 8 q^s +3^s +9 ^m q^s +9^m 4 q^s +3^m 6 q^s +^s 3, 10 ^m q^s 3 + ^m 4 q^s 3, ^s 4, ^m q^s 4 ) +4,6+^m q +0^m 4 q, 1 ^m 6 q, 14 ^m 8 q +10^m 10 q ^ 1 ^u(^s; ^m q ) +3^s +16^m q^s +6^m 4 q^s,56 ^m 6 q^s, 5 ^m 8 q^s +3^s +73^m q^s ^m 4 q^s +15^m 6 q^s +5^s 3, 6 ^m q^s 3 + # jc e 5^m 4 q^s 3, 5^s 4, 5^m q^s 4 7 j h + 8^u(^s; ^m q ) (1, ^m q) ^ ^u(^s; ^m q ),(1 + ^m q)^s + 4(1, ^m q + ^m 4 q, ^s, ^m q^s) ^u(^s; ^m q ) ^ 1 +4,5+30^m 4 q, 40 ^m 6 q +15^m 8 q, ^s +1^m q^s +5^m 4 q^s, 45 ^m 6 q^s +13^s # ) + ^m q^s +45^m 4 q^s, 7^s 3, 15 ^m q^s 3 ^ Re(C e 9 ) C e 7 ^u(^s; ^m q ) We explain some aspect of the expression, (9): ^s : (9) The branching ratio is normalized by B sl of decays B! (X c ;X u )l l. We separate a combination of the KM factor j tq tbj j cb j normalization factor B 0. The normalization constant B 0 is due to top-quark loop from the B 0 B sl f(^m c )(^m c ) ; (10) where f(^m c ) is a phase space factor, and (^m c ) accounts for both the O( s ) QCD correction to the semi-leptonic decay width and the leading order (1=m b ) power correction, They are given in our Appendix explicitly. The Wilson coecients depend on the top quark mass m t, the renormalization scale, and QCD. Their dependences are studied in Tables, 3 and 4. As the value of ( QCD ) decreases (increases), jc 7 j and C 9 are getting larger, while C 10 is independent on. As m t increases, jc 7 j, C 9 and jc 10 j all become 7

8 larger. The matrix elements also depend on the renormalization scale, and their dependence is partially cancelled by the given dependence of the Wilson coecients. The overall dependence can be studied with the dierential rate. In Figs. 1 and, we show the dependence of the dierential rate on m t and. As the value of m t increases from 166 (Ge) to 184 (Ge), the dierential rate increases about 0% at s 5 (Ge ), as shown in Fig. 1. At around the same region of s, bychanging from.5 (Ge) to 10 (Ge) the dierential rate increases about 0%, as shown in Fig.. This observation is consistent with the result of Ref. [7], in which only the SD contribution has been analysed. The dierential decay rate (9) is not a simple parton model result. It contains non-perturbative 1=m b power corrections, which are denoted in Eq. (9) by the terms proportional to ^ 1 and ^. The parameters ^ 1 and ^ are related to the matrix elements of the higher derivative operators of heavy quark eective theory [5], [9]; B D B h (id) h E B MB 1 =M B m b ^1 ; h,i G h B 6 M B =6M B m b ^ ; (11) where B denotes the pseudoscalar B meson, D is the covariant derivative, and G is the QCD eld strength tensor. 3 The extraction of td ts Presently we can constrain the value of (1,) + from B d, B d mixing, while from B! X u l we obtain some limit on +. Using those given limits, we can show how the ratio 1 dr ds depends on input KM values, (1, ) + and 1 (or ). As is well known, j tb td j are written with M d of B d, B d mixing. It reads as j tb td j = = q A 3 (1, ) + " 6 G F QCD M W M B # 1= 4 1= M d 5 ; (1) B 1= f B d Bd je(x t )j 1= where E(x t ) is the Inami-Lim function [10] of the box diagram, and x t mt m W. It has the value je(x t )j = :58 for m t = 175 (Ge) and it varies from :38 to :78 as 3 8

9 m t varies from 166 (Ge) to 184 (Ge). We use the following values; QCD = 0:55, m t = 175 (Ge), and the experimental constraints M d = 0:474 0:031(ps,1 ) ; ub = cb q + =0:08 0:0 : (13) Then, those constraints (13) are translated into the limits of (1,) + and p + as (1, ) + = 0:85(1 0:15) n 0:=(B 1= B d f Bd (Ge)) o = [0:59 0:09; 1:3 0:0] ; q + = 0:36 0:09 : (14) Here we used B 1= B d f Bd =0: 0:04(Ge) to get the range of (1, ) + [0:5; 1:5]. We also used A =0:041(0:003), and =0:05. In Fig. 3, we show the present limit in the plane of ( 1 ; (1, ) + ). The horizontal axis corresponds to 1 (or ), and the unit is degree. The vertical axis corresponds to (1,) +. The thin dashed line is obtained from the central values of j ub j and the thin solid line is obtained from j td j. The central value corresponds to ( 1 ; (1, ) + ) (0 o ; 0:85). Now let us consider the ratio 1 dr(s). In the SU(3) limit, we expect that this ds ratio approaches to the input value of (1, ) + for the dileptonic invariant masssquared s far below the peak of charmonium resonances. resonances, the ratio becomes 1. In Fig. 4, we show the ratio 1 dr(s) ds If s is on the peak of for two sets of the assumed input values of ( 1 ; (1, ) + ), one of whose (1, ) + is 0.59, and the other is They correspond to their small (or large) values allowed from the present experimental result of B d, B d mixing. The solid curve corresponds to ( 1 ; (1,) + ) = (0 o ; 0:59), and the dot-dashed curve corresponds to (0 o ; 1:33). The ratios for the CP conjugate process B! X q l + l, are also shown, denoted by the long-dashed curve for (1,) + =1:33, and by the dashed curve for (1,) + = 0:59. They are obtained by reversing the sign of 1 in the corresponding B! X q l + l, process; i.e., 1!, 1. They are labeled as (,0 o ; 1:33) and (,0 o ; 0:59) in Figure 4. To summarize the numerical results of Fig. 4 and Section 3: The predicted ratio at low invariant mass region s ' 1 (Ge ) is very near to the our assumed input value of (1,) +, while on the peak of the resonances 9

10 J=, 0, this ratio becomes almost 1, as we expected, i.e. 1 dr(s) ds! 1 (s m J= ;m 0 ;::) ;! (1, ) + (s being away from m J= ;::) : (15) In the intermediate region, there is a characteristic interference between the LD contribution and the SD contribution, which can be only derived from the detailed expression of the distributions, Eq. (9). The value of the ratio does not depend much on whether the decaying particles are B or B at any invariant mass-squared region. This ratio changes only a few % when we change the input parameters, m t and, within the range shown in Table 1. The dependences on m t and are almost cancelled away in the ratio. Therefore, the uncertainties in this ratio due to the input parameters are much smaller than the uncertainties in the dierential decay rate itself, as shown in Tables, 3 and 4 as well as in Figures 1 and. In Fig. 4 the range between the dot-dashed curve and the solid curve corresponds to the value of the ratio 1 dr(s) allowed from the present experimental ds result of B d, B d mixing. Future experimental measurements on the ratio of the branching fractions, B(B! X d l + l, ) and B(B! X s l + l, ), can give much better alternative for determination of j td ts j without any hadronic uncertainties, limited only by experimental statistics. If 10 9 BB pairs are produced, the expected number of the events for B! X d l + l, in the range of 4m < s < 6 Ge (m = muon mass) is about 100 for (1, ) + =0:59, and is about 0 for (1, ) + =1:33. Therefore, the statistical accuracy of (1, ) + determined from this method is about 7% 10% with the expected production of 10 9 BB pairs. We have assumed in our numerical analysis the avor SU(3) symmetry with m d = m s. We estimated the corrections due to SU(3) breaking by varying m d from 0.01 Ge up to m s =0: Ge. And we nd the ratio dr decreases within ds 0: 0:3% for the range 1 <s<9 (Ge ). Therefore, we conclude the SU(3) breaking eect does not aect the extraction of j td ts j at all. 10

11 Acknowledgements TM would like to thank G. Hiller, K. Ochi, T. Nasuno, and Y. Kiyo for correspondence and assistance of numerical computations. The preliminary version of this work was given [11] as a talk by TM at BCONF97, Hawaii, March The work of CSK was supported in part by the CTP, Seoul National University, in part by the BSRI Program, Ministry of Education, Project No. BSRI-97-45, and in part by the KOSEF-DFG large collaboration project, Project No The work of TM was partially supported by Monbusho International Scientic Research Program (No ), and that of AIS and TM was supported also by Grant-in-Aid for Scientic Research on Priorty Areas (Physics of CP violation) from the Ministry of Education and Culture of Japan. 11

12 References [1] M. Gronau and J.L. Rosner, Phys. Lett. B376 (1996) 05. [] G. Buchalla and A.J. Buras, Phys. Rev. D54 (1996) 678. [3] F. Kruger and L.M. Sehgal, Phys. Rev. D55 (1997) 799. [4] C.S. Lim, T. Morozumi and A.I. Sanda, Phys. Lett. B18 (1989) 343; N.G. Deshpande, J. Trampetic and K. Panose, Phys. Rev. D39 (1989) 1461; P.J. O'Donnell and H.K.K. Tung, Phys. Rev. D43 (1991) R067; A. Ali, T. Mannel and T. Morozumi, Phys. Lett. B73 (1991) 505; N. Paver and Riazuddin, Phys. Rev. D45 (199) 978. [5] A. Falk, M. Luke, and M.J. Savage, Phys. Rev. D49 (1994) [6] A. Ali, G. Hiller, L. Handoko, and T. Morozumi, Phys. Rev. D55 (1997) [7] A.J. Buras and M. Munz, Phys. Rev. D5 (1995) 186, and references therein. [8] Z. Ligeti and M.B. Wise, Phys. Rev. D53 (1996) [9] I.I. Bigi, M.A. Shifman, N.G. Uraltsev and A.I. ainshtein, Phys. Rev. Lett. 71 (1993) 496; A.. Manohar and M.B. Wise, Phys. Rev. D49 (1994) 1310; B. Blok, L. Koyrakh, M. Shifman and A.I. ainshtein, Phys. Rev. D49 (1994) 335; T. Mannel, Nucl. Phys. B43 (1994) 396; M. Neubert, Phys. Rev. D49 (1994) 339. [10] T. Inami and C.S. Lim, Prog. Theor. Phys. 65 (1981) 97 [E. 65 (1981) 177]. [11] C.S. Kim, T. Morozumi and A.I. Sanda, talk given at BCONF97, Hawaii, March 1997, hep-ph/ (June 1997). [1] M. Je_zabek and J. H. Kuhn, Nucl. Phys. B30 (1989) 0. [13] C.S. Kim and A.D. Martin, Phys. Lett. B5 (1989) 186; A. Ali and E. Pietarinen, Nucl. Phys. B154 (1979) 519; N. Cabibbo, G. Corbo and L. Maiani, Nucl. Phys. B155 (1979) 93. 1

13 Parameter alue m W 80:6 (Ge) m Z 91:19 (Ge) sin W 0:35 m s 0: (Ge) m d 0:01 (Ge) m c 1:4 (Ge) m b 4:8 (Ge) m t (Ge) 5 +5:0,:5 (Ge) (5) QCD 0:14 +0:066 QED,1 19,0:054 (Ge) s (m Z ) 0:117 0:005 B sl (10:4 0:4) % 1,0:0 (Ge ) +0:1 (Ge ) Table 1: alues of the input parameters used in the numerical calculations of the decay rates. Unless, otherwise specied, we use the central values. C 1 C C 3 C 4 C 5 C 6 C e 7 C NDR 9 C (0) Table : (in Ge) dependence of the Wilson coecients used in the numerical calculations. The valuesof QCD and m t are xed at their central values. 13

14 QCD 5 C 1 C C 3 C 4 C 5 C 6 C e 7 C NDR 9 C (0) Table 3: QCD5 dependence of the Wilson coecients used in the numerical calculations. The values of m t and are xed at their central values. m t C e 7 C NDR 9 C ,0:311 4:153,4: ,0:3066 4:0796,4: ,0:3150 4:38,4:9156 Table 4: m t dependence of the Wilson coecients used in the numerical calculations. The values of QCD5 and are xed at their central values. Appendix A Functions g(z; ^s), g(0; ^s) and!(^s) The functions g(z; ^s) and g(0; ^s) are given as g(z; ^s) =, 8 9 ln(m b with y =4z =^s, and ), 8 9 ln z y, q 9 ( + y) j1, yj " (1, y)(ln 1+p # 1, y 1, p 1, y, i)+(y, 1) arctan 1 p ; (16) y, 1 g(0; ^s) = 8 7, 8 9 ln(m b ), 4 9 ln ^s + 4 i : (17) 9 The function!(^s) represents the O( s ) correction from the one-gluon exchange in the matrix element ofo 9 [1];!(^s) =,, Li (^s), 5+4^s ln ^s ln(1, ^s), ln(1, ^s) 3 3(1 + ^s) ^s(1 + ^s)(1, ^s) 5+9^s, 6^s, ln ^s + 3(1, ^s) (1+^s) 6(1, ^s)(1+^s) : (18) 14

15 B Functions f (^m c ) and ( ^m c ) The phase space function for,(b! X c l) in the lowest order (i.e., parton model) f(^m c )=1, 8 ^m c +8 ^m 6 c, ^m 8 c, 4 ^m 4 c ln ^m c : (19) And (^m c ) accounts for both the O( s ) QCD correction to the semi-leptonic decay width and the leading order (1=m b ) power correction, The function g(^m c )isgiven [1],[13] as (^m c )=1, s(m b ) g(^m c )+ h(^m c) 3 m b and nally the function h(^m c )isgiven [9] as : (0) g(^m c )=(, 31 4 )(1, ^m c) + 3 ; (1) h(^m c )= 1 + h i,9 + 4 ^m f(^m c ) c, 7 ^m 4 c +7^m 6 c, 15 ^m 8 c, 7 ^m 4 c ln ^m c ; () where 1 (or, )and (or ) denote the matrix element of the higher derivative G operators of heavy quark eective theory, as dened in Eq. (11). 15

16 mt=175 mt=175 9 mt=175+9 db/ds * S ( Ge ) Figure 1: Dilepton invariant mass spectrum, db ds (B! X se + e, ) The unit of s is (Ge ). The solid line corresponds to m t = 175 (Ge). The short-dashed line corresponds to m t = The long-dashed line corresponds to m t =175, 9: 16

17 15.0 (db/ds db 0 /ds) / db 0 /ds (%) µ=10 (Ge) µ=.5 (Ge) S ( Ge ) Figure : The % variation of dilepton invariant mass spectrum dened as ( db() db 0 ds )= db 0 ds ; B 0 = B( =5(Ge)). The solid line corresponds to = 10 and the dashed line corresponds to = :5: ds, 17

18 .0 B d B d bar Mixing ub 1.6 (1 ρ) +η φ 1 (degree) Figure 3: The limit on (1, ) + and 1 (or ). The horizontal axis corresponds to 1, and the unit is degree. The vertical axis corresponds to (1, ) +. The thin dashed line and the thin solid line are obtained from the central values of j ub j and j td j respectively. The thick solid lines are obtained from the allowed range of j td j from B d B d mixing. The thick dashed lines are obtained from the allowed range of j ub j. 18

19 1.6 (0, 0.59) (0,1.33) ( 0,1.33) ( 0,0.59) 1.4 dr/ds/λ S ( Ge ) Figure 4: The ratio 1 dr ds versus s (Ge ) with two dierent input values of (1, ) +. The solid curve correspond to ( 1 ; (1, ) + ) = (0 o ; 0:59), and the dot-dashed curve corresponds to (0 o ; 1:33). The ratios for CP conjugate process B! X q l + l, are denoted by the dashed curve for (1, ) + = 1:33, and by the long-dashed curve for (1, ) + =0:59. They are obtained by reversing the sign of 1 in the corresponding B! X q l + l, process; i:e:, 1!, 1. 19