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``Non-factorizable'' terms in hadronic B-meson weak decays


TRI-PP-94-78 September, 1994

arXiv:hep-ph/9409443v1 29 Sep 1994

“Non-factorizable” terms in hadronic B-meson weak decays.
Jo?o M. Soares a
TRIUMF, 4004 Wesbrook Mall, Vancouver, BC Canada V6T 2A3

Abstract
The branching ratios for the hadronic B-meson weak decays B → J/ψK and B → Dπ are used to extract the size of the “non-factorizable” terms in the decay amplitudes. It is pointed out that the solutions are not uniquely determined. In the B → J/ψK case, a 2-fold ambiguity can be removed by analyzing the contribution of this decay to B → Kl+ l? . In the B → Dπ case, a 4-fold ambiguity can only be removed if the “non-factorizable” terms are assumed to be a small correction to the vacuum insertion result. PACS:13.25.Hw,13.25.-k,12.15.Ji

1

Introduction

An increasing sample of B-mesons has been gathered from di?erent experiments in recent times, and will tend to increase sharply in the near future with the advent of B-factories and, possibly, experiments in hadron colliders targeted at B-physics. The major concern in this paper is to clarify how such wealth of data can be used to study some of the aspects that remain unclear in the hadronic weak decays of the B and the other ?avored mesons. The focus shall be on the 2-body decays that proceed through the tree level, Cabibbo favored, quark transitions b → ccs and b → cud. The corresponding e?ective weak Hamiltonian, once QCD corrections have been included, is the sum of two 4-quark operators, that only di?er in the color indices of their quark ?elds and the strength of their Wilson coe?cients. In the calculation of the decay amplitudes, one is faced with the task of evaluating the matrix elements, between the initial and ?nal hadronic states, of those two 4-quark operators. The vacuum insertion (factorization) approximation [1] reduces this problem to that of determining the matrix elements of bilinear quark operators; such matrix elements can then be measured from leptonic and semi-leptonic decays, or calculated in some model for the mesons. Unfortunately, this is a poor approximation for one of the 4-quark operators: due to a mismatch in the color indices, its matrix element has signi?cant “non-factorizable” terms [1], that do not appear in the vacuum insertion approximation. That these terms are important can be seen, for example, in the strong disagreement between the factorization predictions and the observed rates for the color suppressed B or D decays (see, for example, ref. [2]. In the color suppressed decays, the e?ect of the “non-factorizable” terms is enhanced by an accidental cancellation of the terms that are non-zero in the vacuum insertion approximation). The standard procedure [2] in dealing with this discrepancy has been to preserve the vacuum insertion result for the hadronic matrix elements, but to replace the Wilson coe?cients that multiply them by two free parameters. These parameters are then determined from a ?t to the observed values of the branching ratios. For the case of the D- and B-meson decays, the two parameters (for each case) ?t the available data quite well. For the D-mesons, the values of the parameters correspond to dropping the contribution of the color mismatched 4-quark operator altogether [2], which is a procedure that ?nds some theoretical justi?cation in 1/Nc expansion arguments [3]. Quite 1

surprisingly, the recent data on B-meson decays has shown that, in this case, the values of the parameters do not obey the same rule [4]: neglecting the contribution of the color mismatched operator cannot be used as a systematic procedure to obtain the value of the free parameters, as it was the case for the D decays. In view of the failure of the factorization approximation when applied to the color mismatched operator, and the failure of the standard procedure of dropping the contribution of that operator altogether, when applied to the B-meson decays, the “non-factorizable” terms have to be dealt with. Theoretical estimates, based on QCD sum rules, have been presented in refs. [5] and [6], for some decays. Here, I address the more basic question of extracting the size of the “non-factorizable” terms from the data. This program was ?rst advocated by Deshpande, Gronau and Sutherland [7], and it has been recently applied to both D- and B-meson decays by Cheng [8]. In this work, I concentrate on the case of the B decays; where, except for the matrix elements of the operators with a color mismatch, the vacuum insertion approximation can be assumed to work well [9] (in particular, the e?ects of inelastic ?nal state scatterings will be neglected for the B-decays considered in here). Special attention is paid to the ambiguities in the values of the “non-factorizable” terms that are extracted from the data. The results of Cheng [8] are recovered among other possible solutions. Finally, ways of lifting these ambiguities are discussed.

2

The “non-factorizable” terms in B → J/ψK and B → Dπ

The tree level, Cabibbo favored, hadronic weak decays of the B-mesons correspond to the quark transitions b → ccs and b → cud. The decay amplitudes are derived from the e?ective weak Hamiltonian GF ? c c Hef f = √ [Vcb Vcs (C1 (?)O1 + C2 (?)O2 ) 2 ? u u +VcbVud (C1 (?)O1 + C2 (?)O2 )], where
c O1 = cα γ ? (1 ? γ5 )bα sβ γ? (1 ? γ5 )cβ ,

(1)

2

c O2 = sα γ ? (1 ? γ5 )bα cβ γ? (1 ? γ5 )cβ ,

(2)

u c and the operators O1,2 are obtained from O1,2 , replacing s and c by d and u, respectively. The Wilson coe?cients C1 (?) and C2 (?) contain the short distance QCD corrections. In the leading logarithm approximation, they are [10] C1,2 = (C+ ± C? )/2, with

C± (?) =

αs (?) αs (MW )

6γ± 33?2nf

(3)
(5)

(γ? = ?2γ+ = 2; nf is the number of active ?avors). For ΛM S = 200 MeV [11], and at the scale ? = 5.0 GeV, this gives C1 = 1.117 is C2 = ?0.266. (4)

0 For the exclusive decays B ? → J/ψK ? or Bd → J/ψK 0 , the amplitude

A(B → J/ψK) = GF ? c c = ? √ Vcb Vcs (C1 < J/ψK|O1 |B > +C2 < J/ψK|O2 |B >). 2

(5)

c The hadronic matrix element of O2 is calculated in the vacuum insertion (factorization) approximation: c < J/ψK|O2 |B >=< J/ψ|cβ γ? (1 ? γ5 )cβ |0 >< K|sα γ ? (1 ? γ5)bα |B > . (6) c Whereas, for O1 , an additional “non-factorizable” term is included that re?ects the color mismatch in the c-c quark ?elds: c < J/ψK|O1 |B >= 1 < J/ψ|cβ γ? (1 ? γ5 )cβ |0 >< K|sα γ ? (1 ? γ5 )bα |B > = Nc c + < J/ψK|O1 |B >non?f act. .

(7)

The factor 1/Nc corresponds to the projection of the two color mismatched quark ?elds into a color singlet; it is included so that the “non-factorizable”

3

term (that is de?ned by eq. 7) vanishes in the vacuum insertion approximation. The decay amplitude can then be written as GF ? A(B → J/ψK) = ? √ Vcb Vcs Ma, 2 with 1 + X), Nc M ≡< J/ψ|cβ γ? (1 ? γ5 )cβ |0 >< K|sα γ ? (1 ? γ5 )bα |B > a = C2 + C1 ( X≡ (8)

(9) (10)

1 c < J/ψK|O1 |B >non?f act. . (11) M In the BSW model [12], M = 5.84 GeV3 × fψ /(395 MeV). With |Vcb | = 0.038 1.63 psec/τB [13], and fψ = 395 MeV (which corresponds to Γ(J/ψ → e+ e? ) = (5.26 ± 0.37) keV [11]), the branching ratio is B(B → J/ψK) = 1.90|a|2%. From the average of the experimental results [11] B(B ? → J/ψK ? ) = (0.102 ± 0.014)% B(B 0 → J/ψK 0 ) = (0.075 ± 0.021)%, it follows that |a| ? 0.22, and so X ? ?0.29 or 0.10 (14) (the numerical di?erences with respect to the analogous results in ref. [8] correspond to the updated values of the parameters that were used in here). The 2-fold ambiguity in the value of X, which corresponds to the unknown sign of a, cannot be resolved by the branching ratio of B → J/ψK alone. 0 Proceeding in a similar way for the exclusive processes Bd → D + π ? , 0 0 ? 0 ? 0 Bd → D π , and B → D π , the weak decay amplitudes are GF ? A+? = ? √ Vcb Vud M1 a1 , 2 GF ? 1 A00 = ? √ Vcb Vud √ M2 a2 2 2 4 (12)

and

(13)

M2 a2 GF ? ], (15) A0? = ? √ Vcb Vud M1 a1 [1 + M1 a1 2 respectively (non-spectator contributions are very small and they have been neglected). The hadronic matrix elements M1 and M2 are
0 M1 ≡< D+ |cα γ ? (1 ? γ5 )bα |Bd >< π ? |dβ γ? (1 ? γ5 )uβ )|0 >

and

(16)

and M2 ≡ √
0 2 < π 0 |dα γ ? (1 ? γ5 )bα |Bd >< D0 |cβ γ? (1 ? γ5 )uβ |0 > .

(17)

In the BSW model [12], M1 = 1.85 GeV3 and M2 = 2.28 GeV3 (for fD = 220 MeV). As in the previous case, the parameters a1 = C1 + C2 ( and a2 = C2 + C1 ( include the terms X1 ≡ X2 1 u 0 < D+ π ? |O2 |Bd >non?f act. M1 √ 2 u 0 ≡ < D0 π 0 |O1 |Bd >non?f act. , M2 1 + X1 ) Nc 1 + X2 ) Nc (18)

(19)

(20)

where the “non-factorizable” matrix elements are de?ned by 1 u 0 M1 + < D+ π ? |O2 |Bd >non?f act. Nc √ √ 1 u u 0 0 2 < D0 π 0 |O1 |Bd > = M2 + 2 < D0 π 0 |O1 |Bd >non?f act. . (21) Nc
u 0 < D+ π ? |O2 |Bd > =

In order to determine the values of the parameters a1 and a2 , the magnitudes of the decay amplitudes are extracted from the experimental value of the corresponding branching ratios [11],
0 B(Bd → D + π ? ) = (0.30 ± 0.04)% 0 B(Bd → D 0 π 0 ) < 0.048% (90%C.L.) B(B ? → D 0 π ? ) = (0.53 ± 0.05)%,

(22)

5

and compared to the predictions in eq. 15. The later are the amplitudes in the absence of ?nal state interaction e?ects, and so the comparison must be done with care (to be sure, I use the notation A+? , A00 and A0? for the full amplitudes). As for the B → J/ψK case, it is assumed that the e?ects of inelastic ?nal state interaction scatterings are negligible. However, the B → Dπ decays involve two isospin channels, and so an elastic ?nal state interaction phase, δ, can appear between the two isospin amplitudes A3/2 and A1/2 . In general, A3/2 = |A3/2 |


A1/2 = |A1/2 |eiδ ,

(23)

where δ ′ = δ or δ + π (according to the relative sign of the two amplitudes in the absence of ?nal state interactions). The full amplitudes A+? , A00 and A0? are related to the isospin amplitudes A3/2 and A1/2 , in the following way: √ 1 2 +? A = √ A3/2 + √ A1/2 , 3 3 √ 1 2 A00 = √ A3/2 ? √ A1/2 , 3 3 √ 0? A = 3A3/2 . (24) This allows to determine the magnitudes |A1/2 | and |A3/2 |, as well as cos δ ′ , from the experimental results in eq. 22. In particular, it follows that cos δ ′ > 0.77 (25)

The lack of a more precise value for δ ′ is due to the fact that only an upper0 limit exists for B(Bd → D 0 π 0 ). For now, I will take δ ′ = 0 (i.e. the ?nal state interaction phase is either δ = 0 or π). Then, 1 |A3/2 | = √ |A0? | 3 √ 3 1 |A1/2 | = √ (|A+? | ? |A0? |). 3 2

(26)

6

Since the decay amplitudes in the absence of ?nal state interactions are √ 2 1 A+? = √ |A3/2 | ± √ |A1/2 |, 3 3 √ 2 1 A00 = √ |A3/2 | ? √ |A1/2 |, 3 3 √ 0? 3|A3/2 | (27) A = (the two signs correspond to δ ′ = δ or δ + π), it follows that |A+? | = |A+? | for δ = 0; and 2 |A+? | = | |A0? | ? |A+? || 3 A0? |A0? | , = 2 0? |A | ? |A+? | A+? 3 (29) |A0? | A0? = +? , A+? |A | (28)

for δ = π. The predictions of eq. 15, in terms of the parameters a1 and a2 , are replaced on the LHS of these equations; whereas the experimental input from the branching ratios is used for the RHS. If the experimental result of eq. 25 is interpreted as showing a negligible phase shift from the ?nal state interaction e?ects, eq. 28 gives (with |Vcb| as before) |a1 | ? 1.07 1 + 1.23 a2 ? 1.33; a1 (30)

and the size of the “non-factorizable” terms is then X1 ? ?0.17 or X1 ? 7.90 X2 ? ?0.35 (32) (the positive value for X2 corresponds to the result in ref. [8]). Alternatively, the data can be interpreted as showing a maximal phase shift. Then eq. 29 gives a2 (33) |a1 | ? 0.12 1 + 1.23 ? ?11.63; a1 7 X2 ? 0.16 (31)

and so X1 ? 1.70 or X1 ? 2.16 X2 ? 0.52. (35) The 4-fold ambiguity corresponds to the fact that the ?nal state interaction phase can only be determined modulo π, and the sign of a1 cannot be determined from the branching ratios in eq. 22. At this point, a word should be said about the uncertainties in the results that were presented. The derivation of the parameters |a|, |a1 | and |a2 | su?ers from the experimental errors in the branching ratios (in particular, 0 B(Bd → D 0 π 0 ) is still missing), and in |Vcb|. These will improve with more accumulated data; which will also allow to derive the hadronic matrix elements of the bilinear quark operators from the semileptonic branching ratios (see the tests of factorization in ref. [4], for example). At present, the use of the BSW model [12] entails an uncertainty that is hard to quantify. The derivation of the terms X, X1 and X2 su?ers from the additional uncertainty on the Wilson coe?cients: in particular, the value chosen for the scale ? is important [14]. The positive solution for X and the negative solution for X1 are the most sensitive as they involve somewhat delicate cancellations; for ? in the range 4.5 to 5.5 GeV, they oscillate by as much as 25% (whereas the other results vary by not more than 10%). I have taken ? = 5.0 GeV, which is approximately the constituent b-quark mass. It must be stressed that the vacuum insertion result has been assumed to work well for the operators with the correct color assignments (see for example eq. 6), and the inelastic ?nal state scatterings have been deemed negligible. This is necessary in order to be able to derive the size of the “non-factorizable” terms from the data. In ref. [9], it is argued, on the basis of color transparency, that these assumptions are expected to hold for the low-multiplicity B decays. The argument is less reliable for the case of the D decays, and so they have not been considered in here. X2 ? ?0.61 (34)

3

Resolving the ambiguities

8

3.1

The sign of the parameter a, that appears in the B → J/ψK amplitude of eq. 8, can be determined from the interference between the short distance contribution to B → Kl+ l? , and the long distance contribution due to B → KJ/ψ → Kl+ l? . The short distance amplitude is derived from the e?ective c weak Hamiltonian in eq. 1 (the operators O1,2 contribute at the 1-loop level), with the additional electroweak terms: GF ′ ? Hef f = √ Vtb Vts Ci (?)Oi , 2 i=7,8,9 where the operators O7 = O8 O9 e mb sα σ ?ν (1 + γ5 )bα F?ν 8π 2 α sα γ ? (1 ? γ5 )bα lγ? l = 2π α = sα γ ? (1 ? γ5 )bα lγ? γ5 l 2π (36)

“Non-factorizable” terms in B → J/ψK

(37)

contribute to B → Kl+ l? at tree level. For mt = 175 GeV, the Wilson coe?cients in eq. 36, in the leading logarithm approximation, are [17] C7 = 0.326, C8 = ?3.752 and C9 = 4.581. The decay amplitude is 1 GF α ? A = √ Vtb Vts [? < K|sγ ? b|B > (C8ef f ul γ? vl + C9 ul γ? γ5 vl ) 2 2π 1 + < K|siσ ?ν qν (1 + γ5 )b|B > C7 mb 2 ul γ? vl ] (38) q (q ≡ pB ? pK ). The factor C8ef f 4m2 m2 c = C8 ? (3C2 + C1 )g( 2 c , 2 ) + 3agLD q mb (39)

includes the contribution 4 8 8 1 √ 1 4 ? (1 + x) x ? 1 arctan √ g(x, y) = ? ln y + x + θ(x ? 1) 9 9 27 9 2 x?1 √ 1 √ 4 1+ 1?x √ + iπ)θ(1 ? x) (40) ? (1 + x) 1 ? x(ln 9 2 1? 1?x 9

c from the operators O1,2 , and the long distance contribution [18]

gLD =

mV Γ(V → l+ l? ) 3π α2 V =J/ψ,ψ′ q 2 ? m2 + imV ΓV V

(41)

from the J/ψ and ψ ′ resonances. The parameter a that multiplies gLD is the same as in eq. 8 (for simplicity, I have taken the same parameter for both the J/ψ and the ψ ′ resonances, but this is not necessary); the relative sign between the long distance and short distance contributions is well determined [19], up to the sign of a. The hadronic matrix elements are parameterized by < K|sγ ? b|B > = (pB + pK )? f+ (q 2 ) + q ? f? (q 2 ), < K|siσ ?ν (1 + γ5 )b|B > = s(q 2 )[(pB + pK )? q ν ? (pB + pK )ν q ? +i??ναβ (pB + pK )α q β ], (42) where, in the static b-quark limit [20], s = ?(f+ ? f? )/2mB . The modi?ed BSW model [12] gives f+ (q 2 ) = h0 q2 1 ? m2 1 ?
0+

1
q2 (mB +mK )2

, (43)

f? (q 2 ) = ?f+ (q 2 )

mB ? mK , mB + mK

with h0 = 0.379 and m0+ = 5.89 GeV. The di?erential branching ratio is then 1 dΓ 1 mB 5 ? 2 = τB G2 α2 |Vtb Vts |2 ( ) (1 ? z)3 f+ F Γ dz 48 2π ×(|C9 |2 + |C8ef f + 2C7 |2 )

(44)

(z ≡ q 2 /m2 ), where the lepton and kaon masses were neglected. This is B shown in ?g. 1 for a positive and negative. Studying the region of the interference between the short and the long distance contributions will allow to determine the sign of a, and resolve the ambiguity in eq. 14. At present, the necessary sensitivity has not been reached yet, and only an upper limit exists on the non-resonant B → Kl+ l? decays [21].

10

3.2

The fortunate interference that allows to determine the sign of a is quite unique, and no similar e?ect appears for the decays of the type b → cud, that would allow to determine the sign of a1 in eq. 15. As for the ?nal state interaction phase δ in eq. 23, it is known [22] that it should be the same phase that appears in D-π elastic scattering, at the energy Ec.m. = mB . But this is of little use in determining its value. Indeed, it is hard to think of an experimental test that would lift the 4-fold ambiguity in the values of the two “non-factorizable” terms in the B → Dπ decays. On the other hand, it has been assumed throughout the analysis that the vacuum insertion result is a good approximation for the matrix elements of the operators with the correct color assignments. If it is further assumed that the vacuum insertion result should provide a ?rst order approximation for the matrix elements of the color mismatched operators, then the X-terms should not be larger than unity. In particular, the solution for the B → Dπ decays is X1 ? ?0.17 and X2 ? 0.16, as in eq. 31. Although there is no reason to expect that this is so (the arguments in ref. [9], for example, cannot be extended to the case of the “non-factorizable” terms), it should be pointed out that the solutions with small X-terms tend to agree with the values predicted by the theoretical calculations that are presently available. Using QCD sum rules techniques, it has been predicted that X1 ? ?0.33 [5] (and, for the case of B → J/ψK, X is between ?0.30 and ?0.15 [6]). Similar results have been obtained for the “non-factorizable” term in the amplitude for B 0 ? B 0 mixing. The hadronic matrix element in the mixing amplitude is
0 0 Mmix. ≡< Bq |qα γ ? (1 ? γ5 )bα qβ γ ? (1 ? γ5 )bβ |Bq > .

“Non-factorizable” terms in B → Dπ

(45)

Proceeding in a similar way as for the decay amplitudes, a “non-factorizable” term is de?ned by Mmix. = 2(1 + 1 2 )m2 fB B Nc 0 0 + < Bq |qα γ ? (1 ? γ5 )bα qβ γ ? (1 ? γ5 )bβ |Bq >non?f act.

(46)

0 (where < Bq (p)|qγ ? (1 ? γ5 )b|0 >= ?ifB p? ). The deviation from the vacuum insertion result (the ?rst terms on the RHS of eq. 46) is parameterized by

11

the bag parameter BB . Here, BB = 1 + where Xmix. ≡
0 0 < Bq |qα γ ? (1 ? γ5 )bα q β γ ? (1 ? γ5 )bβ |Bq >non?f act. . 2 2m2 fB B

1 1 Xmix. , (1 + Nc )

(47)

(48)

0 0 At present, the magnitude of BB cannot be derived from the data on Bd ?Bd mixing, because one lacks a precise determination of fB and of |Vtd | (for 0 0 Bs ? Bs mixing, the CKM parameter is better known, but the strength of the mixing has not been determined experimentally). Using the lattice (BB = 1.2±0.2) [23] and the QCD sum rules (BB = 1.0±0.15) [24] estimates for the matrix element in the mixing, it follows that

0 ≤ Xmix. ≤ 0.53 respectively.

and

? 0.2 ≤ Xmix. ≤ 0.2,

(49)

4

Conclusion

The size of the “non-factorizable” terms in the amplitude for the decays B → J/ψK and B → Dπ was derived from the experimental value of the corresponding branching ratios. The results can only be determined up to a discrete ambiguity. In the case of B → J/ψK, the 2-fold ambiguity can be lifted by determining the sign of the interference between the short distance contribution to B → Kl+ l? , and the long distance contribution due to B → KJ/ψ → Kl+ l? . A similar ambiguity appears in the case of the B → Dπ decays; because the ?nal state interaction elastic phase between the two isospin amplitudes can only be determined modulo π, the ambiguity becomes 4-fold. Contrary to the previous case, there is no simple way to determine the correct solution experimentally. However, all but one of the solutions indicates large “non-factorizable” terms that would indicate a breakdown of the vacuum insertion approximation when applied to the color mismatched operators. The analysis that was presented can be improved with future experimental results, in particular, with a measurement of the branching ratio 12

0 for Bd → D 0 π 0 . Also, other B-decays, similar to the ones shown in here, can be considered; the ambiguities in the values of the “non-factorizable” terms that are extracted from the data will appear in the same fashion.

I would like to thank A. Buras for a very useful discussion, and H.-Y. Cheng for carefully reading the manuscript and for his comments and corrections. This work was partly supported by the Natural Science and Engineering Research Council of Canada.

References
[1] For a recent review of the factorization procedure in the hadronic weak decays of B-mesons, see, for example, M. Neubert et al., in Heavy Flavors, ed. by A. J. Buras and L. Lindner (World Scienti?c, Singapore, 1992); also, a thorough discussion can be found in H.-Y. Cheng, Int. J. Mod. Phys. A 4, 495 (1989). [2] M. Bauer, B. Stech and M. Wirbel, Z. Phys. C34, 103 (1987). [3] A. J. Buras, J.-M. G?rard and R. R¨ ckl, Nucl. Phys. B268, 16 e u (1986). [4] For recent results, see M. S. Alam et al. (CLEO Collaboration), Phys. Rev. D 50, 43 (1994). [5] B. Blok and M. Shifman, Nucl. Phys. B389, 534 (1993); I. Bigi et al., in B Decays (2nd edition), ed. by S. Stone (World Scienti?c, Singapore, 1994). [6] A. Khodzhamirian and R. Ruckl, Max Planck Inst. report no. MPI-PHT-94-26. [7] N. G. Deshpande, M. Gronau and D. Sutherland, Phys. Lett. B90, 431 (1980); M. Gronau and D. Sutherland, Nucl. Phys. B183, 367 (1981). 13

[8] H.-Y. Cheng, Academia Sinica of Taipei, report no. IP-ASTP11-94, 1994. [9] J. D. Bjorken, in Lectures at the 18th Annual SLAC Summer Institute on Particle Physics, Stanford (1990). [10] M. K. Gaillard and B. W. Lee, Phys. Rev. Lett. 33, 108 (1974); G. Altarelli and L. Maiani, Phys. Lett. B52, 352 (1974). [11] Particle Data Group, Phys. Rev. D 50, 1173 (1994). [12] M. Wirbel, B. Stech and M. Bauer, Z. Phys. C29, 637 (1985); see also M. Neubert et al., in ref. [1], for a modi?ed version that was used in here. [13] A. Ali and D. London, CERN report no. CERN-TH.7398/94, 1994. [14] The strong ? dependence persists, when QCD corrections are included at the next-to-leading order [15] [16]. In ref. [15], it is shown that the higher order corrections must be included as part of a simultaneous expansion in C2 + C1 /Nc and αs , to avoid unphysical results; no large deviation from the LLA result is apparent in the B → J/ψ + Xs decay. A di?erent approach is advocated in ref. [16] for the exclusive decays. Using the NLO results in there would lead to a large renormalization scheme dependence in the “non-factorizable” terms. [15] L. Bergstr¨m and P. Ernstr¨m, Phys. Lett. B328, 153 (1994). o o [16] A. Buras, Max-Planck-Institut report no. MPI-PhT/94-60, 1994. [17] B. Grinstein, M. J. Savage and M. B. Wise, Nucl. Phys. B319, 271 (1989). [18] A. Ali, T. Mannel and T. Morozumi, Phys. Lett. B273, 505 (1991).

14

[19] C. S. Lim, T. Morozumi and A. I. Sanda, Phys. Lett. B218, 343 (1989); P. J. O’Donnell and H. K. Tung, Phys. Rev. D 43, R2067 (1991); N. Paver and Riazuddin, Phys. Rev. D 45, 978 (1992). [20] N. Isgur and M. B. Wise, Phys. Rev. D 42, 2388 (1990). [21] CDF Collaboration, report no. FNAL Conf-94/145; CLEO Collaboration, report no. CLEO Conf 94-4; UA1 Collaboration, Phys. Lett. 262, 163 (1991). [22] A. Kamal, J. Phys. G12, L43 (1986). [23] J. Shigemitsu, talk presented at the Int. Conf. in High Energy Physics, Glasgow, 1994. [24] S. Narison, Phys. Lett. B322, 247 (1994); S. Narison and A. Pivovarov, Phys. Lett. B327, 341 (1994).

15

Figure Caption
Figure 1: Di?erential branching ratio for B → Kl+ l? (z ≡ (pB ? pK )2 /m2 ). B The full line corresponds to the long distance contribution alone, whereas the other curves include the short distance contribution: with a > 0 (dashed line) and a < 0 (dotted line).

This figure "fig1-1.png" is available in "png" format from: http://arXiv.org/ps/hep-ph/9409443v1



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