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Next-to-leading order QCD corrections to the lifetime difference of Bs mesons
29 July 1999
Physics Letters B 459 Ž1999. 631–640
Next-to-leading order QCD corrections to the lifetime difference of Bs mesons M. Beneke a , G. Buchalla a , C. Greub b, A. Lenz c , U. Nierste
d
a
Theory DiÕision, CERN, CH-1211 GeneÕa 23, Switzerland Institut fur ¨ Theoretische Physik, UniÕersitat ¨ Bern, Sidlerstrasse 5, CH-3012 Berne, Switzerland Max-Planck-Institut fur Ring 6, D-80805 Munich, Germany ¨ Physik, Werner-Heisenberg-Institut, Fohringer ¨ d DESY - Theory Group, Notkestraße 85, D-22607 Hamburg, Germany b
c
Received 31 August 1998; received in revised form 28 May 1999 Editor: R. Gatto
Abstract We compute the QCD corrections to the decay rate difference in the Bs –Bs system, D GBs, in the next-to-leading logarithmic approximation using the heavy quark expansion approach. Going beyond leading order in QCD is essential to obtain a proper matching of the Wilson coefficients to the matrix elements of local operators from lattice gauge theory. The lifetime difference is reduced considerably at next-to-leading order. We find Ž D GrG .Bs s Ž f Bsr210 MeV. 2 w0.006 B Ž m b . q 0.150 BS Ž m b . y 0.063x in terms of the bag parameters B, BS in the NDR scheme. As a further application of our analysis we also derive the next-to-leading order result for the mixing-induced CP asymmetry in inclusive b ™ uud decays, which measures sin2 a . q 1999 Published by Elsevier Science B.V. All rights reserved. PACS: 12.38.Bx; 13.25.Hw; 14.40.Nd
1. Introduction The width difference Ž D GrG .B s of the Bs meson CP eigenstates w1x is expected to be about 10–20%, among the largest rate differences in the b-hadron sector w2x, and might be measured in the near future. A measurement of a sizeable Ž D GrG .B s would open up the possibility of novel CP violation studies with Bs mesons w3,4x. In principle, a measured value for D GB s could also give some information on the mass difference D MB s w5x, if the theoretical prediction for the ratio Ž D GrD M .B s can be sufficiently well controlled w6x. Furthermore, as pointed out in Ref. w7x, if non-standard-model sources of CP violation are present in the Bs system, D GB s can be smaller Žbut not
larger. than expected in the standard model. For this reason a lower bound on the standard model prediction is of special interest. The calculation of inclusive non-leptonic b-hadron decay observables, such as D GB s, uses the heavy quark expansion ŽHQE.. The decay width difference is expanded in powers of LQCD rm b , each term being multiplied by a series of radiative corrections in a s Ž m b .. In the case of Ž D GrG .B s, the leading contribution is parametrically of order 16p 2 Ž LQC D r m b . 3. In the framework of the HQE the main ingredients for a reliable prediction Žless than 10% uncertainty. are a. subleading corrections in the 1rm b expansion, b. the non-perturbative matrix elements of local four-quark operators between B-meson states
0370-2693r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 9 . 0 0 6 8 4 - X
M. Beneke et al.r Physics Letters B 459 (1999) 631–640
632
and c. O Ž a s . radiative corrections to the Wilson coefficients of these operators. The first issue has been addressed in Ref. w6x. The hadronic matrix elements can be studied using numerical simulations in lattice QCD. In this letter, we present the next-toleading order QCD radiative corrections to the Wilson coefficient functions for D GB s. In addition to removing another item from the above list and reducing certain renormalization scale ambiguities of the leading order prediction, the inclusion of O Ž a s . corrections is necessary for a satisfactory matching of the Wilson coefficients to the matrix elements to be obtained from lattice calculations. Our results provide the first calculation of perturbative QCD effects beyond the leading logarithmic approximation to spectator effects in the HQE for heavy hadron decays. The consideration of subleading QCD radiative effects has implications of conceptual interest for the construction of the HQE. Soft gluon emission from the spectator s quark in the Bs meson leads to power-like IR singularities in individual contributions, which would apparently impede the HQE construction, because they cannot be absorbed into matrix elements of local operators. It has already been explained in Ref. w8x, how these severe IR divergences cancel in the sum over all cuts of a given diagram, so that the Wilson coefficients of four-quark operators relevant to lifetime differences, such as D GB s, are free of infrared singularities. This infrared cancellation is confirmed by the result of our explicit calculation. Using the HQE to finite order in LQC D rm b rests on the assumption of local quark-hadron duality. Little is known in QCD about the actual numerical size of duality-violating effects. ŽSee, e.g. w9x, for a recent discussion of the issue.. Experimentally no violation of local quark-hadron duality in inclusive observables of the B-meson sector has been established so far. In Ref. w10x it has been shown that for D GB s local duality holds exactly in the simultaneous limits of small velocity Ž LQC D < m b y 2 m c < m b . and large number of colours Ž Nc ™ `.. In this case DG
ž / G
s Bs
GF2 m3b f B2s 4p
f 0.18.
< Vcs Vcb < 2
(
2y4
mc mb
It is interesting that the numerical value implied by the limiting formula Ž1. appears to be quite realistic and is in fact consistent with the results of more complete analyses w6,10x. The duality assumption can in principle be tested by a confrontation of theoretical predictions, based on the HQE, with experiment. This aspect is another major motivation for computing D GB s accurately. It is clear from these remarks that a detailed theoretical analysis of D GB s, and particularly of O Ž a s . corrections, is very desirable, both for phenomenological and for conceptual reasons. In this letter we shall concentrate on the presentation of our results and a brief discussion of their main aspects. Details and an extension of our analysis to other b-hadron lifetime differences will be given in a forthcoming publication w11x.
2. Formalism and next-to-leading order results In the limit of CP conservation the mass eigenstates of the Bs –Bs system are < BH r L : s Ž< Bs : " < Bs :.r '2 , using the convention CP < Bs : s y< Bs :. The width difference between mass eigenstates is then given by D GB s ' GL y GH s y2 G 12 s y2 G 21 ,
Ž 2.
where Gi j are the elements of the decay-width matrix, i, j s 1,2 Ž<1: s < Bs :, <2: s < Bs :.. In writing Ž2. we assumed standard CKM phase conventions w12x. For more information about the basic formulas see for instance w6x Žand references therein.. The decay width is related to the absorptive part of the forward scattering amplitude via the optical theorem w2x. The off-diagonal element of the decaywidth matrix may thus be written as
G 21 s
1 2 MB s
² Bs < T < Bs : .
Ž 3.
The normalization of states is ² Bs < Bs : s 2 EV Žconventional relativistic normalization. and the transition operator T is defined by w2x
t Bs
Ž 1.
T s Im i d 4 x T Heff Ž x . Heff Ž 0 . .
H
Ž 4.
M. Beneke et al.r Physics Letters B 459 (1999) 631–640
Here Heff is the low energy effective weak Hamiltonian mediating bottom quark decay. The component that is relevant for G 21 reads explicitly Heff s
GF
'2
6
Vcb) Vcs
žÝ
/
Cr Q r q C8 Q8 ,
rs1
Ž 5.
Q 1 s bi c j
ž
/
VyA
Ž c j si . VyA ,
Q2 s Ž bi c i . VyA Ž c j s j . VyA ,
Ž 6.
Q3 s Ž bi si . VyA Ž q j q j . VyA , Q 4 s bi s j
ž
/
VyA
Ž q j qi . VyA ,
Ž 7.
Q5 s Ž bi si . VyA Ž q j q j . VqA , Q 6 s bi s j
ž
Q8 s
g 8p 2
/
VyA
Ž q j qi . VqA ,
a m b bi s mn Ž 1 y g 5 . Tiaj s j Gmn .
Ž 8. Ž 9.
Here the i, j are colour indices and a summation over q s u, d, s, c, b is implied. V " A refers to g m Ž1 " g 5 . and S y P Žwhich we need below. to Ž1 y g 5 .. C1 , . . . ,C6 are the corresponding Wilson coefficient functions, which are known at next-to-leading order. We have also included the chromomagnetic operator Q8 , contributing to T at O Ž a s .. ŽNote that for a negative C8 , as conventionally used in the literature, the Feynman rule for the quark-gluon vertex is yiggmT a.. A detailed review and explicit expressions may be found in Ref. w13x. Cabibbo suppressed channels have been neglected in Ž5.. Expanding the operator product Ž4. for small x ; 1rm b , the transition operator T can be written, to leading order in the 1rm b expansion, as Tsy
GF2 m2b 12p
Ž Vcb) Vcs .
2
= F Ž z . Q Ž m 2 . q FS Ž z . QS Ž m 2 .
Ž 10 .
with z s m2crm 2b and the basis of D B s 2 operators Q s Ž bi si . VyA bj s j
ž
/
QS s Ž bi si . SyP bj s j
ž
In writing Ž10. we have used Fierz identities and the equations of motion to eliminate the colour rearranged operators Q˜ s bi s j
žb s / Q˜ s ž b s / žb s / ž
S
with the operators
VyA
/
,
SyP
.
Ž 11 .
633
/
VyA
i j SyP
j i VyA ,
j i SyP
,
Ž 12 .
always working to leading order in 1rm b Žsee below.. The Wilson coefficients F and FS can be extracted by computing the matrix elements between quark states of T in Ž4. ŽFig. 1., as well as those of Q and Q S , to a given order in QCD perturbation theory, and comparing them with Ž10.. ŽThe external b quarks are taken to be on-shell.. This matching procedure factorizes the perturbatively calculable short-distance contributions ŽWilson coefficients. from the long-distance dynamics, parametrized by the non-perturbative matrix elements of local D B s 2 operators. We do not use heavy quark effective theory ŽHQET. to expand these matrix elements explicitly in 1rm b . They are to be understood in full QCD, in accordance with the treatment of 1rm b effects in Ref. w6x. In our discussion we shall first concentrate on the contribution to T from the operators Q1 and Q2 in Ž5.. The penguin operators have small coefficients and are numerically less important. Their effect will be included later on. Thus for the time being only the diagrams D 1 – D 10 in Fig. 1 are considered, while D 11 and D 12 belong to the penguin sector to be discussed below. Working in leading order, the matching calculation for Ž10. has to be performed to zeroth order in a s . At next-to-leading order the coefficients C1,2 have to be taken in next-to-leading logarithmic approximation w14,15x and the O Ž a s . matching corrections to the coefficients in Ž10. have to be computed. At this order the coefficients in Ž10. depend on the renormalization scheme chosen for their evaluation. This scheme dependence is cancelled by the matrix elements of the operators in Ž10., which have to be determined accordingly. The scheme we employ for our result is specified as follows. We use dimensional regularization with anti-commuting g 5 and modified minimal Ž MS . subtraction of ultraviolet singularities ŽNDR scheme.. In addition we project D-dimensional Dirac-structures,
M. Beneke et al.r Physics Letters B 459 (1999) 631–640
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Fig. 1. O Ž a s . corrections to the Bs –Bs transition operator, D1 – D12 . Also shown are the corrections to the matrix elements of local D B s 2 operators, E1 – E4 , required for a proper factorization of short-distance and long-distance contributions. Not displayed explicitly are X X X X X X X X X X E1 , E2 , D1 , D 2 , D5 , D6 , D 7 , D 8 , D10 and D12 , which are obtained by rotating the corresponding diagrams by 1808.
arising at intermediate stages of the calculation, according to the prescriptions Ž D s 4 y 2 ´ .
g m g ag n Ž 1 y g 5 .
ij
gm ga gn Ž 1 y g 5 .
m
™ Ž 16 y 4´ . g Ž 1 y g 5 .
kl
gm Ž 1 y g 5 .
ij
kl
,
Ž 13 . g m g ag n Ž 1 y g 5 .
ij
gn ga gm Ž 1 y g 5 .
m
™ Ž 4 y 8 ´ . g Ž 1 y g5 .
ij
kl
gm Ž 1 y g 5 .
kl
,
Ž 14 . g m g n Ž 1 y g5 .
ij
gm gn Ž 1 y g 5 .
kl
™ Ž 8 y 4´ . w 1 y g 5 x i j w 1 y g 5 x k l y Ž 8 y 8 ´ . w1 y g 5 x i l w1 y g 5 x k j .
Ž 15 .
The projections are equivalent to the subtraction of evanescent operators, defined by the difference of the left- and right-hand sides of Ž13. – Ž15.. The definition of evanescent operators is discussed in great detail in Refs. w15–17x and we use the basis of
w17x for the projection. The prescriptions Ž13. – Ž15. complete the definition of our renormalization scheme. They preserve Fierz symmetry, i.e. the oneloop matrix elements of Q, Q S and Q˜S equal the matrix elements of the operators that one obtains from Q, QS , Q˜S by Ž4-dimensional. Fierz transformations. The projections Ž13. – Ž15. are sufficient if we use the Fierz form of Q1 , Q2 in Ž6. that corresponds to closed charm-quark loops in Fig. 1. Since the renormalization of Q1 , Q 2 respects Fierz symmetry, this choice can always be made. We now give the result for the transition operator Ž10. at next-to-leading order, still neglecting the penguin sector. The coefficients in Ž10. can be written as F Ž z . s F11 Ž z . C12 Ž m 1 . q F12 Ž z . C1 Ž m 1 . C2 Ž m 1 . q F22 Ž z . C22 Ž m 1 . , Fi j Ž z . s FiŽ0. j Ž z. q
a s Ž m1 . 4p
Ž 16 . FiŽ1. j Ž z.
Ž 17.
M. Beneke et al.r Physics Letters B 459 (1999) 631–640
and similarly for FS Ž z .. The leading order functions Ž0. FiŽ0. j , FS,i j read explicitly Ž0. F11 Ž z . s 3'1 y 4 z Ž 1 y z . ,
' FSŽ0. ,11 Ž z . s 3 1 y 4 z Ž 1 q 2 z . ,
Ž 18 .
635
q Ž 2 y 259 z q 662 z 2 ln s y76 z 3 y 200 z 4 . 6z y'1 y 4 z Ž 2 y 255z q 316 z 2 .
Ž0. F12 Ž z . s 2'1 y 4 z Ž 1 y z . ,
' FSŽ0. ,12 Ž z . s 2 1 y 4 z Ž 1 q 2 z . , Ž0. F22 Ž z . s 12 Ž 1 y 4 z .
3r2
Ž 19 .
6z
q8 Ž 3 y 2 z . ln Ž 1 y 4 z . q 29 Ž 127 y 199 z y 75 z 2 .
,
' FSŽ0. ,22 Ž z . s y 1 y 4 z Ž 1 q 2 z . .
Ž 20 .
The next-to-leading order expressions
FiŽ1. j ,
Ž1. FS,i j
are
Ž1. F11 Ž z . s 32 Ž 1 y z . Ž 1 y 2 z . Ž Li 2 Ž s 2 .
qln2s q 12 ln s ln Ž 1 y 4 z . y ln s ln z . q 64 Ž 1 y z . Ž 1 y 2 z .
mb m 2 q 43 '1 y 4 z Ž 5 y 8 z . ln , mb
Ž 23 .
64 2 2 FSŽ1. ,12 Ž z . s 3 Ž 1 y 4 z . Ž Li 2 Ž s .
2 q 128 3 Ž1y4 z .
y 4 Ž 13 y 26 z y 4 z 2 q 14 z 3 . ln s
= Ž Li 2 Ž s . q 12 ln Ž 1 y s . ln s .
q'1 y 4 z 4 Ž 13 y 10 z . ln z
q Ž 1 y 35 z q 4 z 2
y12 Ž 3 y 2 z . ln Ž 1 y 4 z . q 16 Ž 109 y 226 z q 168 z 2 . m2 q 2'1 y 4 z Ž 5 y 8 z . ln , mb
q76 z 3 y 100 z 4 .
Ž 21 .
4ln s 3z
y'1 y 4 z = Ž 1 y 33 z y 76 z 2 .
2 2 FSŽ1. ,11 Ž z . s 32 Ž 1 y 4 z . Ž Li 2 Ž s . 1 2
qln s q ln s ln Ž 1 y 4 z . y ln s ln z . q 64 Ž 1 y 4 z 2 .
4ln z 3z
q32 Ž 1 q 2 z . ln Ž 1 y 4 z . . q 49 Ž 68 q 49 z y 150 z 2 .
1 2
= Ž Li 2 Ž s . q ln Ž 1 y s . ln s . y 16 Ž 4 y 2 z y 7z 2 q 14 z 3 . ln s
y 16'1 y 4 z Ž 1 q 2 z . ln
q'1 y 4 z 64 Ž 1 q 2 z . ln z y 643 '1 y 4 z Ž 1 q 2 z . ln
y48 Ž 1 q 2 z . ln Ž 1 y 4 z . y 83 Ž 1 y 6 z . Ž 5 q 7z . y 32'1 y 4 z Ž 1 q 2 z . ln
m1
y 2'1 y 4 z Ž 17 y 26 z . ln
qln2s q 12 ln s ln Ž 1 y 4 z . y ln s ln z .
= Ž Li 2 Ž s . q 12 ln Ž 1 y s . ln s .
2
ln z
m2 mb
m1 mb m2 mb
,
Ž 24 .
Ž1. F22 Ž z . s 43 Ž 4 y 21 z q 2 z 2 . Ž Li 2 Ž s 2 .
,
Ž 22 .
Ž1. F12 Ž z . s 643 Ž 1 y z . Ž 1 y 2 z . Ž Li 2 Ž s 2 .
qln2s q 12 ln s ln Ž 1 y 4 z . y ln s ln z . q 128 3 Ž1yz . Ž1y2 z . = Ž Li 2 Ž s . q 12 ln Ž 1 y s . ln s .
qln2s q 21 ln s ln Ž 1 y 4 z . y ln s ln z . q 43 Ž 1 y 2 z . Ž 5 y 2 z . = Ž Li 2 Ž s . q 12 ln Ž 1 y s . ln s . y Ž 7 q 13 z y 194 z 2 ln s q304 z 3 y 64 z 4 . 6z
M. Beneke et al.r Physics Letters B 459 (1999) 631–640
636
p2
the D B s 1 Wilson coefficients Ci Ž m 1 . in Ž16. to the considered order in a s . Likewise, the dependence on m 2 is cancelled by the matrix elements of the D B s 2 operators Q and QS . To check this, we note that the scale dependence of matrix elements ² Q : s Ž² Q :, ² QS :, ² Q˜S :.T is given by
Ž 1 y 10 z .
y 3
ln z
q'1 y 4 z Ž 7 q 27z y 250 z 2 .
6z
y4 Ž 1 y 6 z . ln Ž 1 y 4 z .
d
y 181 Ž 115 q 632 z q 96 z 2 . y 2'1 y 4 z Ž 5 y 2 z . ln q 43 '1 y 4 z Ž 2 y 5 z . ln
dln m 2
m1
mb
,
Ž 25 .
32 2 FSŽ1. ,22 Ž z . s y 3 Ž 1 q z . Ž 1 q 2 z . Ž Li 2 Ž s .
qln2s q 12 ln s ln Ž 1 y 4 z . y ln s ln z . q 323 Ž 1 y 4 z 2 . = Ž Li 2 Ž s . q 12 ln Ž 1 y s . ln s . q Ž 1 q 7z q 10 z 2 y 68 z 3 q 32 z 4 . 8p
4ln s 3z
2
Ž1q2 z .
q 3
y'1 y 4 z Ž 1 q 9 z q 26 z 2 .
4ln z 3z
y16 Ž 1 q 2 z . ln Ž 1 y 4 z . q 89 Ž 19 q 53 z q 24 z 2 . y 16'1 y 4 z Ž 1 q 2 z . ln q 323 '1 y 4 z Ž 1 q 2 z . ln
m1 mb
m2 mb
.
Ž 26 .
In these equations we have set Nc s 3 and used x
Li 2 Ž x . s y
H0 dt
ss
1 y '1 y 4 z 1 q '1 y 4 z
ln Ž 1 y t . t ,
zs
,
m 2c m2b
.
as 4p
g 2Ž0. ² Q : ,
Ž 28 .
where
mb
m2
²Q: s y
Ž 27 .
The dependence on the renormalization scale m 1 in Ž21. – Ž26. cancels against the scale dependence of
4 g 2Ž0. s 0 0
0 y28r3 16r3
0 4r3 . 32r3
0
Ž 29 .
The operator Q˜S , which is redundant at leading power in 1rm b , can then be eliminated using Ž35. below and the m 2-independence can be verified. Our results in Ž21. – Ž26. correspond to the use of the one-loop pole mass for the b quark in Ž10.. In z s m2crm 2b there is no difference between the ratios of pole masses and MS masses at next-to-leading order in a s . We next present the contribution from QCD penguins to the transition operator T. This sector has been treated in Ref. w6x in the leading logarithmic approximation. At next-to-leading order the penguin coefficients C3 , . . . ,C6 have to be computed with next-to-leading logarithmic accuracy w18,19x and the diagram D 11 in Fig. 1 must be evaluated. Because the coefficients C3 , . . . ,C6 are small Ž; few per cent., contributions of second order in these coefficients are safely negligible and it is sufficient to calculate only the interference of C3 , . . . ,C6 with C1 and C2 . A consistent way to implement this approximation at NLO is to treat C3 , . . . ,C6 as formally of O Ž a s .. The standard NLO formula Žas reviewed in Ref. w13x. can be used for the penguin coefficients, except that terms of order a s C3 , . . . , a s C6 have to be dropped. Accordingly, only current-current operators Q1 , Q 2 are inserted into the diagrams D 1 – D11. The NLO result thus obtained is manifestly scheme independent and formally of order O Ž C3, . . . ,6 . s O Ž a s .. A further contribution, absent in leading logarithmic approximation, comes from the chromomagnetic operator Q8 in Ž5. and is shown as diagram D 12 of Fig. 1. Since this contribution arises first at O Ž a s ., the
M. Beneke et al.r Physics Letters B 459 (1999) 631–640
lowest order expression is sufficient for C8 . For the NLO penguin contribution we then find Tp s y
GF2 m2b 12p
Ž Vcb) Vcs .
2
P Ž z . Q q PS Ž z . Q S ,
Ž 30 . P Ž z . s '1 y 4 z Ž Ž 1 y z . K 1X Ž m 1 . q 12 Ž 1 y 4 z . K 2X Ž m 1 . q 3 zK 3X Ž m 1 . . q
a s Ž m1 .
Fp Ž z . C22 Ž m 1 . ,
4p
Ž 31 .
PS Ž z . s '1 y 4 z Ž 1 q 2 z . Ž K 1X Ž m 1 . y K 2X Ž m 1 . . y Fp Ž z . s y
a s Ž m1 . 4p 1 9
8 Fp Ž z .
C22
Ž m1 . ,
m1 mb
3C8 Ž m 1 . C2 Ž m 1 .
R 0 ' QS q Q˜S q 12 Q,
Ž 34 .
which can be reduced to an explicitly power-suppressed operator using Fierz transformations and the equations of motion. ŽWe have used this in our calculation to eliminate Q˜S .. At order a s , we find the leading power contribution ² R0 : s
as 4p
ž
m
Ž Nc q 1 . ln
q
2 y Nc
mb
/
Nc
² Q:
m
ž
q 4 Ž Nc q 1 . ln
mb
/
q 2 Ž Nc q 1 . ² QS :
q O Ž 1rm b .
q 23 q 4 z y ln z
q'1 y 4 z Ž 1 q 2 z . ln s q
account in the above result. To be more explicit, consider as an example the operator
Ž 32 .
'1 y 4 z Ž 1 q 2 z .
= 2ln
637
,
Ž 33 . where we defined the combinations K 1X s 2Ž3C1C3 q C1C4 q C2 C3 ., K 2X s 2C2 C4 and K 3X s 2Ž3C1C5 q C1C6 q C2 C5 q C2 C6 .. The explicit m 1-dependence in Fp Ž z . is cancelled by the m 1-dependence of C3 , . . . ,C6 . Note that for the penguin sector the scale and scheme dependence of the matrix elements of Q and QS are effects beyond the considered order and numerically negligible. Beyond leading order in the 1rm b expansion, several further operators contribute to D G , denoted by R i in Ref. w6x. Inspection of the factorized matrix elements of these operators given in Ref. w6x shows that these superficially power-suppressed operators contribute at leading power beyond tree level. This is due to the fact that the operators are defined in QCD Žrather than in HQET. and hence the one-loop matrix elements contain m b explicitly. The leading power piece arises only from loop momenta of order m b and can therefore be subtracted perturbatively. This subtraction is necessary for a complete calculation of the a s correction at leading power and is taken into
Ž 35 .
It is crucial that this relation holds independent of the external state, so that power counting is again manifest after subtracting the right hand side of Ž35. from the matrix element of R 0 . ŽThe procedure discussed here bears some similarity with mixing of higher dimension operators into lower dimension operators in cut-off or lattice regularizations.. These subtractions must be kept in mind when a non-perturbative evaluation of the matrix elements of the R i is combined with the present NLO results. In the factorization approximation of w6x these subtractions correspond to using the pole b quark mass in the expressions for the factorized matrix elements. Indeed, in the Nc ™ ` limit, we find w6x
ž
² Bs < R 0 < Bs : s f B2 MB2 1 y s s , f B2sMB2s
a s Nc 4p
MB2s
Ž mb q ms .
ž
6ln
mb
m
2
/ /
y4 ,
Ž 36 .
up to corrections of order Ž Nc a s . 2 and LQCD rm b . The b-quark mass m b in the second expression of Ž36. is the MS mass at the scale m , which corresponds to our renormalization of the scalar operators Q S , Q˜S . To obtain the third expression we used the 1-loop relation between the pole and MS mass in the large-Nc limit and the fact that MB s y m b, p o l e s O Ž LQC D .. The same result as Ž36. is obtained from the large-Nc limit of Ž35.. ŽIn deriving Ž36. we have
M. Beneke et al.r Physics Letters B 459 (1999) 631–640
638
used the Fierz transform of Q˜S . For the coincidence of Ž36. with Ž35. it is crucial that the choice in Ž15. maintains the Fierz symmetry in the one-loop matrix elements entering Ž34...
3. Discussion The complete expression for D GB s with short-distance coefficients at NLO in QCD is given by DG
ž / G
s
16p 2 B Ž Bs ™ Xen . f B2sMB s m3b
g Ž z . h˜ QCD
Bs
ž
= GŽ z .
8 3
B q GS Ž z .
< Vcs < 2
Ž mb q ms .
/
5 2 3
BS
and
Ž 38 .
We eliminated the total decay rate GB s in favour of the semileptonic branching ratio B Ž Bs ™ Xen ., as done in Ref. w6x. This cancels the dependence of Ž D GrG . on Vcb and introduces the phase space function g Ž z . s 1 y 8 z q 8 z 3 y z 4 y 12 z 2 ln z ,
Ž 39 .
as well as the QCD correction factor w20x 2 as Ž mb . 3p
Ž p 2 y 314 . Ž 1 y 'z .
2
q 32 .
Ž 40 . The latter is written here in the approximate form of w21x. The bag factors B and BS parametrize the matrix elements of Q and QS , ² Bs < Q < Bs : s 83 f B2 MB2 B, s
Ž 41 .
s
² Bs < QS < Bs : s y 53 f B2 MB2 s
s
MB2s
Ž mb q ms .
2
BS .
p
ž
m2 ln
m 2b
q 43
/
.
Ž 43 .
m b Ž m b . s 4.4 GeV ,
m s s 0.2 GeV,
MB s s 5.37 GeV ,
Ž 44 .
f B s s 0.21 GeV ,
B Ž Bs ™ Xen . s 0.104.
Ž 37 .
GS Ž z . s y Ž FS Ž z . q PS Ž z . . .
as
Finally, d 1r m describes 1rm b corrections. The explicit expression for d 1r m in the factorization approximation can be found in Ref. w6x. In the present NLO approximation the b-quark mass that appears in d 1r m is the pole mass as mentioned above. For the numerical evaluation we use the following input parameters Žcentral values.: m b s 4.8 GeV ,
where
h˜ QC D s 1 y
mb Ž m . s mb 1 y
z s 0.085 ,
MB2s
q'1 y 4 z d 1r m ,
GŽ z . s F Ž z . q P Ž z .
The masses m b ' m b Ž m ., m s refer to the MS definition. The relation of m b to the pole mass m b is
Ž 42 .
Ž 45 .
The two-loop expression is used throughout for the QCD coupling a s in the form given in Ref. w13x with LŽ5. M S s 0.225 GeV. The NLO coefficients F Ž z ., FS Ž z . in Ž10. and P Ž z ., PS Ž z . in Ž30. are consistently expanded to first order in a s . We take m 1 s m 2 s m b as central values for the renormalization scales. The dependence of D GB s on m s is marginal and its dependence on z stems almost totally from g Ž z . in Ž39.. The results for the Wilson coefficients are displayed in Table 1. The contribution of QS dominates D GB s since the coefficient of QS is numerically much larger than the one of Q. These coefficients are independent of the scale m 1 , related to the D B s 1 Table 1 Numerical values of the Wilson coefficients G, GS , F, FS , at next-to-leading order in the NDR-scheme with evanescent operators subtracted as described in the text Žfor m 2 s m b .. Leading order results Žsuperscript ‘Ž0.’. are also shown for comparison
m1
m b r2
mb
2 mb
GS syŽ FS q PS . GSŽ0. syŽ FSŽ0. q PSŽ0. . Gs F q P G Ž0. s F Ž0. q P Ž0.
0.743 1.622 0.023 0.013
0.937 1.440 0.030 0.047
1.018 1.292 0.036 0.097
y FS y FSŽ0. F F Ž0.
0.867 1.729 0.042 0.030
1.045 1.513 0.045 0.057
1.111 1.341 0.049 0.103
M. Beneke et al.r Physics Letters B 459 (1999) 631–640
operators, up to terms of next-to-next-to-leading order. As can be seen from Table 1, the residual m 1 dependence is indeed substantially decreased for the coefficient G of Q. The reduction of scale dependence is less pronounced for the coefficient GS of QS . On the other hand, in our renormalization scheme, the central value at NLO is considerably smaller Žby about 30%. than the leading order result. However, due to the scheme dependence of the Wilson coefficient, it is premature to draw definitive conclusions on the size of D GB s without combining the coefficient functions with B and BS , computed in the same scheme. For quick reference we rewrite Ž37. as DG
ž / ž G
fBs
s
Bs
210 MeV
2
/
0.006 B Ž m b .
q0.150 BS Ž m b . y 0.063 ,
Ž 46 .
where the numbers are obtained with our central parameter set. A preliminary lattice evaluation of the relevant bag parameters can be found in Ref. w22x, together with a complete 1-loop lattice-to-continuum matching. We can use their result to obtain a Žconceptually. complete Žbut numerically preliminary. nextto-leading-order result for Ž D GrG .B s. In Ref. w22x a continuum renormalization scheme different from ours for the operators QS and Q˜S is chosen. We computed the relation between the two schemes to O Ž a s . and found BS s B4q q B˜S s B5q q
as 4p
as 4p
Ž B4q q 151 B5q . ,
Ž 47 .
Ž y 356 B4q q 12 B5q . .
Ž 48 .
Here B˜S is the bag parameter related to Q˜S , defined in analogy with Ž42. with Q S ™ Q˜S , BS ™ B˜S , and the numerical factor Žy5r3. ™ Žq1r3.. B4q and B5q are the bag parameters BS and B˜S , respectively, but in the scheme of w22x. From the quoted estimates B4qŽ m 0 . s 0.80 " 0.01, B5qŽ m 0 . s 0.94 " 0.01 w22x Žwhere the errors are statistical only and m 0 s 2.33 GeV. we infer BS Ž m 0 . s 0.81, B˜S Ž m 0 . s 0.87. Using Ž29., and taking the running of m b Ž43. into
639
account, the bag parameter BS at the scale m b is given by BS Ž m b . s BS Ž m 0 . q
as 4p
ln
mb
m0
Žy
20 3
BS Ž m 0 . q 154 B˜S Ž m 0 . .
s 0.75.
Ž 49 .
We also take B Ž m b . s 0.9 from the compilation w23x. Without the 1rm b corrections, Ž46. then becomes Ž D GrG .B s 0.118 Ž f B r210 MeV. 2 . There is a s s "15% error from Žcontinuum. perturbation theory, obtained from varying m 1 between m br2 and 2 m b , and a negligible "1% statistical error from the lattice simulation. In addition the sizeable 1rm b correction, computed in Ref. w6x, has to be included. The estimate y0.063 in Ž46. is obtained using factorization of hadronic matrix elements and has a relative error of at least "20%. As a preliminary result we may therefore write DG
ž / ž G
s
Bs
fBs 210 MeV
2
/Ž
.016 0.054q0 y0 .032 Ž m 1 y dep. .
" ??? Ž latt. syst. . . .
Ž 50 .
We emphasize the preliminary nature of the central value which depends crucially on the estimate B4qŽ m 0 . s 0.80 taken from w22x. There is an unspecified systematic error Žindicated by the question marks. attached to this number, related to the fact that the lattice calculation has been performed in the quenched approximation, at finite lattice spacing and with a ‘‘b-quark’’ mass in the charm quark mass region, without extrapolation to the continuum limit and to realistic b quark masses, respectively. Clearly, for further progress improved lattice determinations of bag parameters, most importantly of BS , are mandatory, and the numerical value for Ž D GrG .B s above has to be regarded in this context. The rather low number for the central value, compared to the leading order analysis of w6x, is a consequence of the fact that 1rm b effects and penguin contributions w6x, as well as NLO QCD corrections, all lead to a reduction of D GBS . This is further reinforced by the small value BS Ž m b . s 0.75 in our example.
M. Beneke et al.r Physics Letters B 459 (1999) 631–640
640
It is interesting to consider the ratio of D GB s to the mass difference D MB s w5,6x, in which the dependence on the decay constant f B s cancels out and the sensitivity to Vcb is considerably reduced. In addition Ž D GrD M .B only depends on the ratio of bag pas rameters. Generalizing the results given in Ref. w6x to include the next-to-leading order QCD corrections we can write DG
ž / DM
s Bs
p m2b VcbVcs 2
=
MW2
ž
8 3
2
Vt sVt b
G Ž z . q 53
1
hB S0 Ž x t . MB2s
Ž mb q ms .
/
q'1 y 4 z d 1r m ,
2
GS Ž z .
BS B
Ž 51 .
where hB is the Žscheme-dependent. next-to-leading order QCD factor entering D MB s w24x. In the usual NDR scheme hB Ž m b . s 0.846. The top-quark mass dependent function S0 Ž x t . s S0 ŽŽ m trMW . 2 . s 2.41 for m t s 167 GeV. In analogy with Ž50. we then have DG .67 s Ž 2.63q0 y1 .36 Ž m 1 y dep. . D M Bs
ž /
" ??? Ž latt. syst. . . P 10y3 . Ž 52 . The next-to-leading order calculation presented in this article can also be applied, by taking the limit z ™ 0, to the mixing-induced CP asymmetry in inclusive Bd Ž Bd . ™ uudd decays w25x. The time-dependent asymmetry is given by AŽ t . s Im j sinD Mt, where the coefficient Im j is a measure of the CKM parameter sin2 a . With next-to-leading order QCD corrections included, the expression for Im j in Eq. Ž25. of w25x is modified to read Im j s y0.12sin2 a
ž
fB 180 MeV
2
/
= 0.14 B q 0.64 BS y 0.07 y0.06
sin a sin Ž a q b . sin b sin2 a
.
Ž 53 .
In this equation the bag factors B and BS Žtaken at m s m b . are the Bd analogues of those defined in Ž41. and Ž42. for Bs . A detailed discussion of CP asymmetries will be presented in Ref. w11x.
Acknowledgements We thank Guido Martinelli for comments on the manuscript.
References w1x J.S. Hagelin, Nucl. Phys. B 193 Ž1981. 123; E. Franco, M. Lusignoli, A. Pugliese, Nucl. Phys. B 194 Ž1982. 403; L.L. Chau, Phys. Rep. 95 Ž1983. 1; A.J. Buras, W. Słominski, H. Steger, Nucl. Phys. B 245 Ž1984. 369; M.B. Voloshin, N.G. Uraltsev, V.A. Khoze, M.A. Shifman, Sov. J. Nucl. Phys. 46 Ž1987. 112; A. Datta, E.A. Paschos, U. Turke, Phys. Lett. B ¨ 196 Ž1987. 382; A. Datta, E.A. Paschos, Y.L. Wu, Nucl. Phys. B 311 Ž1988. 35. w2x I. Bigi et al., in: S. Stone ŽEd.., B Decays, 2nd ed., World Scientific, Singapore, 1994, p. 132, hep-phr9401298. w3x I. Dunietz, Phys. Rev. D 52 Ž1995. 3048. w4x R. Fleischer, I. Dunietz, Phys. Lett. B 387 Ž1996. 361; Phys. Rev. D 55 Ž1997. 259. w5x T.E. Browder, S. Pakvasa, Phys. Rev. D 52 Ž1995. 3123. w6x M. Beneke, G. Buchalla, I. Dunietz, Phys. Rev. D 54 Ž1996. 4419. w7x Y. Grossman, Phys. Lett. B 380 Ž1996. 99. w8x I. Bigi, N. Uraltsev, Phys. Lett. B 280 Ž1992. 271. w9x I. Bigi, M. Shifman, N. Uraltsev, A. Vainshtein, hepphr9805241. w10x R. Aleksan, Phys. Lett. B 316 Ž1993. 567. w11x M. Beneke, G. Buchalla, C. Greub, A. Lenz, U. Nierste, in preparation. w12x Particle Data Group, C. Caso, Eur. Phys. J. C 3 Ž1998. 1. w13x G. Buchalla, A.J. Buras, M.E. Lautenbacher, Rev. Mod. Phys. 68 Ž1996. 1125. w14x G. Altarelli, G. Curci, G. Martinelli, S. Petrarca, Nucl. Phys. B 187 Ž1981. 461. w15x A.J. Buras, P.H. Weisz, Nucl. Phys. B 333 Ž1990. 66. w16x M.J. Dugan, B. Grinstein, Phys. Lett. B 256 Ž1991. 239. w17x S. Herrlich, U. Nierste, Nucl. Phys. B 455 Ž1995. 39. w18x A.J. Buras, M. Jamin, M.E. Lautenbacher, P.H. Weisz, Nucl. Phys. B 370 Ž1992. 69; B 375 Ž1992. 501 ŽA.. w19x M. Ciuchini, E. Franco, G. Martinelli, L. Reina, Nucl. Phys. B 415 Ž1994. 403. w20x N. Cabibbo, L. Maiani, Phys. Lett. B 79 Ž1978. 109; Y. Nir, Phys. Lett. B 221 Ž1989. 184. w21x C.S. Kim, A.D. Martin, Phys. Lett. B 225 Ž1989. 186. w22x R. Gupta, T. Bhattacharya, S.R. Sharpe, Phys. Rev. D 55 Ž1997. 4036. w23x J.M. Flynn, C.T. Sachrajda, hep-latr9710057, to appear in Heavy Flavours Ž2nd ed.., A.J. Buras, M. Lindner ŽEds.., World Scientific, Singapore. w24x A.J. Buras, M. Jamin, P.H. Weisz, Nucl. Phys. B 347 Ž1990. 491. w25x M. Beneke, G. Buchalla, I. Dunietz, Phys. Lett. B 393 Ž1997. 132.
Physics Letters B 459 Ž1999. 631–640
Next-to-leading order QCD corrections to the lifetime difference of Bs mesons M. Beneke a , G. Buchalla a , C. Greub b, A. Lenz c , U. Nierste
d
a
Theory DiÕision, CERN, CH-1211 GeneÕa 23, Switzerland Institut fur ¨ Theoretische Physik, UniÕersitat ¨ Bern, Sidlerstrasse 5, CH-3012 Berne, Switzerland Max-Planck-Institut fur Ring 6, D-80805 Munich, Germany ¨ Physik, Werner-Heisenberg-Institut, Fohringer ¨ d DESY - Theory Group, Notkestraße 85, D-22607 Hamburg, Germany b
c
Received 31 August 1998; received in revised form 28 May 1999 Editor: R. Gatto
Abstract We compute the QCD corrections to the decay rate difference in the Bs –Bs system, D GBs, in the next-to-leading logarithmic approximation using the heavy quark expansion approach. Going beyond leading order in QCD is essential to obtain a proper matching of the Wilson coefficients to the matrix elements of local operators from lattice gauge theory. The lifetime difference is reduced considerably at next-to-leading order. We find Ž D GrG .Bs s Ž f Bsr210 MeV. 2 w0.006 B Ž m b . q 0.150 BS Ž m b . y 0.063x in terms of the bag parameters B, BS in the NDR scheme. As a further application of our analysis we also derive the next-to-leading order result for the mixing-induced CP asymmetry in inclusive b ™ uud decays, which measures sin2 a . q 1999 Published by Elsevier Science B.V. All rights reserved. PACS: 12.38.Bx; 13.25.Hw; 14.40.Nd
1. Introduction The width difference Ž D GrG .B s of the Bs meson CP eigenstates w1x is expected to be about 10–20%, among the largest rate differences in the b-hadron sector w2x, and might be measured in the near future. A measurement of a sizeable Ž D GrG .B s would open up the possibility of novel CP violation studies with Bs mesons w3,4x. In principle, a measured value for D GB s could also give some information on the mass difference D MB s w5x, if the theoretical prediction for the ratio Ž D GrD M .B s can be sufficiently well controlled w6x. Furthermore, as pointed out in Ref. w7x, if non-standard-model sources of CP violation are present in the Bs system, D GB s can be smaller Žbut not
larger. than expected in the standard model. For this reason a lower bound on the standard model prediction is of special interest. The calculation of inclusive non-leptonic b-hadron decay observables, such as D GB s, uses the heavy quark expansion ŽHQE.. The decay width difference is expanded in powers of LQCD rm b , each term being multiplied by a series of radiative corrections in a s Ž m b .. In the case of Ž D GrG .B s, the leading contribution is parametrically of order 16p 2 Ž LQC D r m b . 3. In the framework of the HQE the main ingredients for a reliable prediction Žless than 10% uncertainty. are a. subleading corrections in the 1rm b expansion, b. the non-perturbative matrix elements of local four-quark operators between B-meson states
0370-2693r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 9 . 0 0 6 8 4 - X
M. Beneke et al.r Physics Letters B 459 (1999) 631–640
632
and c. O Ž a s . radiative corrections to the Wilson coefficients of these operators. The first issue has been addressed in Ref. w6x. The hadronic matrix elements can be studied using numerical simulations in lattice QCD. In this letter, we present the next-toleading order QCD radiative corrections to the Wilson coefficient functions for D GB s. In addition to removing another item from the above list and reducing certain renormalization scale ambiguities of the leading order prediction, the inclusion of O Ž a s . corrections is necessary for a satisfactory matching of the Wilson coefficients to the matrix elements to be obtained from lattice calculations. Our results provide the first calculation of perturbative QCD effects beyond the leading logarithmic approximation to spectator effects in the HQE for heavy hadron decays. The consideration of subleading QCD radiative effects has implications of conceptual interest for the construction of the HQE. Soft gluon emission from the spectator s quark in the Bs meson leads to power-like IR singularities in individual contributions, which would apparently impede the HQE construction, because they cannot be absorbed into matrix elements of local operators. It has already been explained in Ref. w8x, how these severe IR divergences cancel in the sum over all cuts of a given diagram, so that the Wilson coefficients of four-quark operators relevant to lifetime differences, such as D GB s, are free of infrared singularities. This infrared cancellation is confirmed by the result of our explicit calculation. Using the HQE to finite order in LQC D rm b rests on the assumption of local quark-hadron duality. Little is known in QCD about the actual numerical size of duality-violating effects. ŽSee, e.g. w9x, for a recent discussion of the issue.. Experimentally no violation of local quark-hadron duality in inclusive observables of the B-meson sector has been established so far. In Ref. w10x it has been shown that for D GB s local duality holds exactly in the simultaneous limits of small velocity Ž LQC D < m b y 2 m c < m b . and large number of colours Ž Nc ™ `.. In this case DG
ž / G
s Bs
GF2 m3b f B2s 4p
f 0.18.
< Vcs Vcb < 2
(
2y4
mc mb
It is interesting that the numerical value implied by the limiting formula Ž1. appears to be quite realistic and is in fact consistent with the results of more complete analyses w6,10x. The duality assumption can in principle be tested by a confrontation of theoretical predictions, based on the HQE, with experiment. This aspect is another major motivation for computing D GB s accurately. It is clear from these remarks that a detailed theoretical analysis of D GB s, and particularly of O Ž a s . corrections, is very desirable, both for phenomenological and for conceptual reasons. In this letter we shall concentrate on the presentation of our results and a brief discussion of their main aspects. Details and an extension of our analysis to other b-hadron lifetime differences will be given in a forthcoming publication w11x.
2. Formalism and next-to-leading order results In the limit of CP conservation the mass eigenstates of the Bs –Bs system are < BH r L : s Ž< Bs : " < Bs :.r '2 , using the convention CP < Bs : s y< Bs :. The width difference between mass eigenstates is then given by D GB s ' GL y GH s y2 G 12 s y2 G 21 ,
Ž 2.
where Gi j are the elements of the decay-width matrix, i, j s 1,2 Ž<1: s < Bs :, <2: s < Bs :.. In writing Ž2. we assumed standard CKM phase conventions w12x. For more information about the basic formulas see for instance w6x Žand references therein.. The decay width is related to the absorptive part of the forward scattering amplitude via the optical theorem w2x. The off-diagonal element of the decaywidth matrix may thus be written as
G 21 s
1 2 MB s
² Bs < T < Bs : .
Ž 3.
The normalization of states is ² Bs < Bs : s 2 EV Žconventional relativistic normalization. and the transition operator T is defined by w2x
t Bs
Ž 1.
T s Im i d 4 x T Heff Ž x . Heff Ž 0 . .
H
Ž 4.
M. Beneke et al.r Physics Letters B 459 (1999) 631–640
Here Heff is the low energy effective weak Hamiltonian mediating bottom quark decay. The component that is relevant for G 21 reads explicitly Heff s
GF
'2
6
Vcb) Vcs
žÝ
/
Cr Q r q C8 Q8 ,
rs1
Ž 5.
Q 1 s bi c j
ž
/
VyA
Ž c j si . VyA ,
Q2 s Ž bi c i . VyA Ž c j s j . VyA ,
Ž 6.
Q3 s Ž bi si . VyA Ž q j q j . VyA , Q 4 s bi s j
ž
/
VyA
Ž q j qi . VyA ,
Ž 7.
Q5 s Ž bi si . VyA Ž q j q j . VqA , Q 6 s bi s j
ž
Q8 s
g 8p 2
/
VyA
Ž q j qi . VqA ,
a m b bi s mn Ž 1 y g 5 . Tiaj s j Gmn .
Ž 8. Ž 9.
Here the i, j are colour indices and a summation over q s u, d, s, c, b is implied. V " A refers to g m Ž1 " g 5 . and S y P Žwhich we need below. to Ž1 y g 5 .. C1 , . . . ,C6 are the corresponding Wilson coefficient functions, which are known at next-to-leading order. We have also included the chromomagnetic operator Q8 , contributing to T at O Ž a s .. ŽNote that for a negative C8 , as conventionally used in the literature, the Feynman rule for the quark-gluon vertex is yiggmT a.. A detailed review and explicit expressions may be found in Ref. w13x. Cabibbo suppressed channels have been neglected in Ž5.. Expanding the operator product Ž4. for small x ; 1rm b , the transition operator T can be written, to leading order in the 1rm b expansion, as Tsy
GF2 m2b 12p
Ž Vcb) Vcs .
2
= F Ž z . Q Ž m 2 . q FS Ž z . QS Ž m 2 .
Ž 10 .
with z s m2crm 2b and the basis of D B s 2 operators Q s Ž bi si . VyA bj s j
ž
/
QS s Ž bi si . SyP bj s j
ž
In writing Ž10. we have used Fierz identities and the equations of motion to eliminate the colour rearranged operators Q˜ s bi s j
žb s / Q˜ s ž b s / žb s / ž
S
with the operators
VyA
/
,
SyP
.
Ž 11 .
633
/
VyA
i j SyP
j i VyA ,
j i SyP
,
Ž 12 .
always working to leading order in 1rm b Žsee below.. The Wilson coefficients F and FS can be extracted by computing the matrix elements between quark states of T in Ž4. ŽFig. 1., as well as those of Q and Q S , to a given order in QCD perturbation theory, and comparing them with Ž10.. ŽThe external b quarks are taken to be on-shell.. This matching procedure factorizes the perturbatively calculable short-distance contributions ŽWilson coefficients. from the long-distance dynamics, parametrized by the non-perturbative matrix elements of local D B s 2 operators. We do not use heavy quark effective theory ŽHQET. to expand these matrix elements explicitly in 1rm b . They are to be understood in full QCD, in accordance with the treatment of 1rm b effects in Ref. w6x. In our discussion we shall first concentrate on the contribution to T from the operators Q1 and Q2 in Ž5.. The penguin operators have small coefficients and are numerically less important. Their effect will be included later on. Thus for the time being only the diagrams D 1 – D 10 in Fig. 1 are considered, while D 11 and D 12 belong to the penguin sector to be discussed below. Working in leading order, the matching calculation for Ž10. has to be performed to zeroth order in a s . At next-to-leading order the coefficients C1,2 have to be taken in next-to-leading logarithmic approximation w14,15x and the O Ž a s . matching corrections to the coefficients in Ž10. have to be computed. At this order the coefficients in Ž10. depend on the renormalization scheme chosen for their evaluation. This scheme dependence is cancelled by the matrix elements of the operators in Ž10., which have to be determined accordingly. The scheme we employ for our result is specified as follows. We use dimensional regularization with anti-commuting g 5 and modified minimal Ž MS . subtraction of ultraviolet singularities ŽNDR scheme.. In addition we project D-dimensional Dirac-structures,
M. Beneke et al.r Physics Letters B 459 (1999) 631–640
634
Fig. 1. O Ž a s . corrections to the Bs –Bs transition operator, D1 – D12 . Also shown are the corrections to the matrix elements of local D B s 2 operators, E1 – E4 , required for a proper factorization of short-distance and long-distance contributions. Not displayed explicitly are X X X X X X X X X X E1 , E2 , D1 , D 2 , D5 , D6 , D 7 , D 8 , D10 and D12 , which are obtained by rotating the corresponding diagrams by 1808.
arising at intermediate stages of the calculation, according to the prescriptions Ž D s 4 y 2 ´ .
g m g ag n Ž 1 y g 5 .
ij
gm ga gn Ž 1 y g 5 .
m
™ Ž 16 y 4´ . g Ž 1 y g 5 .
kl
gm Ž 1 y g 5 .
ij
kl
,
Ž 13 . g m g ag n Ž 1 y g 5 .
ij
gn ga gm Ž 1 y g 5 .
m
™ Ž 4 y 8 ´ . g Ž 1 y g5 .
ij
kl
gm Ž 1 y g 5 .
kl
,
Ž 14 . g m g n Ž 1 y g5 .
ij
gm gn Ž 1 y g 5 .
kl
™ Ž 8 y 4´ . w 1 y g 5 x i j w 1 y g 5 x k l y Ž 8 y 8 ´ . w1 y g 5 x i l w1 y g 5 x k j .
Ž 15 .
The projections are equivalent to the subtraction of evanescent operators, defined by the difference of the left- and right-hand sides of Ž13. – Ž15.. The definition of evanescent operators is discussed in great detail in Refs. w15–17x and we use the basis of
w17x for the projection. The prescriptions Ž13. – Ž15. complete the definition of our renormalization scheme. They preserve Fierz symmetry, i.e. the oneloop matrix elements of Q, Q S and Q˜S equal the matrix elements of the operators that one obtains from Q, QS , Q˜S by Ž4-dimensional. Fierz transformations. The projections Ž13. – Ž15. are sufficient if we use the Fierz form of Q1 , Q2 in Ž6. that corresponds to closed charm-quark loops in Fig. 1. Since the renormalization of Q1 , Q 2 respects Fierz symmetry, this choice can always be made. We now give the result for the transition operator Ž10. at next-to-leading order, still neglecting the penguin sector. The coefficients in Ž10. can be written as F Ž z . s F11 Ž z . C12 Ž m 1 . q F12 Ž z . C1 Ž m 1 . C2 Ž m 1 . q F22 Ž z . C22 Ž m 1 . , Fi j Ž z . s FiŽ0. j Ž z. q
a s Ž m1 . 4p
Ž 16 . FiŽ1. j Ž z.
Ž 17.
M. Beneke et al.r Physics Letters B 459 (1999) 631–640
and similarly for FS Ž z .. The leading order functions Ž0. FiŽ0. j , FS,i j read explicitly Ž0. F11 Ž z . s 3'1 y 4 z Ž 1 y z . ,
' FSŽ0. ,11 Ž z . s 3 1 y 4 z Ž 1 q 2 z . ,
Ž 18 .
635
q Ž 2 y 259 z q 662 z 2 ln s y76 z 3 y 200 z 4 . 6z y'1 y 4 z Ž 2 y 255z q 316 z 2 .
Ž0. F12 Ž z . s 2'1 y 4 z Ž 1 y z . ,
' FSŽ0. ,12 Ž z . s 2 1 y 4 z Ž 1 q 2 z . , Ž0. F22 Ž z . s 12 Ž 1 y 4 z .
3r2
Ž 19 .
6z
q8 Ž 3 y 2 z . ln Ž 1 y 4 z . q 29 Ž 127 y 199 z y 75 z 2 .
,
' FSŽ0. ,22 Ž z . s y 1 y 4 z Ž 1 q 2 z . .
Ž 20 .
The next-to-leading order expressions
FiŽ1. j ,
Ž1. FS,i j
are
Ž1. F11 Ž z . s 32 Ž 1 y z . Ž 1 y 2 z . Ž Li 2 Ž s 2 .
qln2s q 12 ln s ln Ž 1 y 4 z . y ln s ln z . q 64 Ž 1 y z . Ž 1 y 2 z .
mb m 2 q 43 '1 y 4 z Ž 5 y 8 z . ln , mb
Ž 23 .
64 2 2 FSŽ1. ,12 Ž z . s 3 Ž 1 y 4 z . Ž Li 2 Ž s .
2 q 128 3 Ž1y4 z .
y 4 Ž 13 y 26 z y 4 z 2 q 14 z 3 . ln s
= Ž Li 2 Ž s . q 12 ln Ž 1 y s . ln s .
q'1 y 4 z 4 Ž 13 y 10 z . ln z
q Ž 1 y 35 z q 4 z 2
y12 Ž 3 y 2 z . ln Ž 1 y 4 z . q 16 Ž 109 y 226 z q 168 z 2 . m2 q 2'1 y 4 z Ž 5 y 8 z . ln , mb
q76 z 3 y 100 z 4 .
Ž 21 .
4ln s 3z
y'1 y 4 z = Ž 1 y 33 z y 76 z 2 .
2 2 FSŽ1. ,11 Ž z . s 32 Ž 1 y 4 z . Ž Li 2 Ž s . 1 2
qln s q ln s ln Ž 1 y 4 z . y ln s ln z . q 64 Ž 1 y 4 z 2 .
4ln z 3z
q32 Ž 1 q 2 z . ln Ž 1 y 4 z . . q 49 Ž 68 q 49 z y 150 z 2 .
1 2
= Ž Li 2 Ž s . q ln Ž 1 y s . ln s . y 16 Ž 4 y 2 z y 7z 2 q 14 z 3 . ln s
y 16'1 y 4 z Ž 1 q 2 z . ln
q'1 y 4 z 64 Ž 1 q 2 z . ln z y 643 '1 y 4 z Ž 1 q 2 z . ln
y48 Ž 1 q 2 z . ln Ž 1 y 4 z . y 83 Ž 1 y 6 z . Ž 5 q 7z . y 32'1 y 4 z Ž 1 q 2 z . ln
m1
y 2'1 y 4 z Ž 17 y 26 z . ln
qln2s q 12 ln s ln Ž 1 y 4 z . y ln s ln z .
= Ž Li 2 Ž s . q 12 ln Ž 1 y s . ln s .
2
ln z
m2 mb
m1 mb m2 mb
,
Ž 24 .
Ž1. F22 Ž z . s 43 Ž 4 y 21 z q 2 z 2 . Ž Li 2 Ž s 2 .
,
Ž 22 .
Ž1. F12 Ž z . s 643 Ž 1 y z . Ž 1 y 2 z . Ž Li 2 Ž s 2 .
qln2s q 12 ln s ln Ž 1 y 4 z . y ln s ln z . q 128 3 Ž1yz . Ž1y2 z . = Ž Li 2 Ž s . q 12 ln Ž 1 y s . ln s .
qln2s q 21 ln s ln Ž 1 y 4 z . y ln s ln z . q 43 Ž 1 y 2 z . Ž 5 y 2 z . = Ž Li 2 Ž s . q 12 ln Ž 1 y s . ln s . y Ž 7 q 13 z y 194 z 2 ln s q304 z 3 y 64 z 4 . 6z
M. Beneke et al.r Physics Letters B 459 (1999) 631–640
636
p2
the D B s 1 Wilson coefficients Ci Ž m 1 . in Ž16. to the considered order in a s . Likewise, the dependence on m 2 is cancelled by the matrix elements of the D B s 2 operators Q and QS . To check this, we note that the scale dependence of matrix elements ² Q : s Ž² Q :, ² QS :, ² Q˜S :.T is given by
Ž 1 y 10 z .
y 3
ln z
q'1 y 4 z Ž 7 q 27z y 250 z 2 .
6z
y4 Ž 1 y 6 z . ln Ž 1 y 4 z .
d
y 181 Ž 115 q 632 z q 96 z 2 . y 2'1 y 4 z Ž 5 y 2 z . ln q 43 '1 y 4 z Ž 2 y 5 z . ln
dln m 2
m1
mb
,
Ž 25 .
32 2 FSŽ1. ,22 Ž z . s y 3 Ž 1 q z . Ž 1 q 2 z . Ž Li 2 Ž s .
qln2s q 12 ln s ln Ž 1 y 4 z . y ln s ln z . q 323 Ž 1 y 4 z 2 . = Ž Li 2 Ž s . q 12 ln Ž 1 y s . ln s . q Ž 1 q 7z q 10 z 2 y 68 z 3 q 32 z 4 . 8p
4ln s 3z
2
Ž1q2 z .
q 3
y'1 y 4 z Ž 1 q 9 z q 26 z 2 .
4ln z 3z
y16 Ž 1 q 2 z . ln Ž 1 y 4 z . q 89 Ž 19 q 53 z q 24 z 2 . y 16'1 y 4 z Ž 1 q 2 z . ln q 323 '1 y 4 z Ž 1 q 2 z . ln
m1 mb
m2 mb
.
Ž 26 .
In these equations we have set Nc s 3 and used x
Li 2 Ž x . s y
H0 dt
ss
1 y '1 y 4 z 1 q '1 y 4 z
ln Ž 1 y t . t ,
zs
,
m 2c m2b
.
as 4p
g 2Ž0. ² Q : ,
Ž 28 .
where
mb
m2
²Q: s y
Ž 27 .
The dependence on the renormalization scale m 1 in Ž21. – Ž26. cancels against the scale dependence of
4 g 2Ž0. s 0 0
0 y28r3 16r3
0 4r3 . 32r3
0
Ž 29 .
The operator Q˜S , which is redundant at leading power in 1rm b , can then be eliminated using Ž35. below and the m 2-independence can be verified. Our results in Ž21. – Ž26. correspond to the use of the one-loop pole mass for the b quark in Ž10.. In z s m2crm 2b there is no difference between the ratios of pole masses and MS masses at next-to-leading order in a s . We next present the contribution from QCD penguins to the transition operator T. This sector has been treated in Ref. w6x in the leading logarithmic approximation. At next-to-leading order the penguin coefficients C3 , . . . ,C6 have to be computed with next-to-leading logarithmic accuracy w18,19x and the diagram D 11 in Fig. 1 must be evaluated. Because the coefficients C3 , . . . ,C6 are small Ž; few per cent., contributions of second order in these coefficients are safely negligible and it is sufficient to calculate only the interference of C3 , . . . ,C6 with C1 and C2 . A consistent way to implement this approximation at NLO is to treat C3 , . . . ,C6 as formally of O Ž a s .. The standard NLO formula Žas reviewed in Ref. w13x. can be used for the penguin coefficients, except that terms of order a s C3 , . . . , a s C6 have to be dropped. Accordingly, only current-current operators Q1 , Q 2 are inserted into the diagrams D 1 – D11. The NLO result thus obtained is manifestly scheme independent and formally of order O Ž C3, . . . ,6 . s O Ž a s .. A further contribution, absent in leading logarithmic approximation, comes from the chromomagnetic operator Q8 in Ž5. and is shown as diagram D 12 of Fig. 1. Since this contribution arises first at O Ž a s ., the
M. Beneke et al.r Physics Letters B 459 (1999) 631–640
lowest order expression is sufficient for C8 . For the NLO penguin contribution we then find Tp s y
GF2 m2b 12p
Ž Vcb) Vcs .
2
P Ž z . Q q PS Ž z . Q S ,
Ž 30 . P Ž z . s '1 y 4 z Ž Ž 1 y z . K 1X Ž m 1 . q 12 Ž 1 y 4 z . K 2X Ž m 1 . q 3 zK 3X Ž m 1 . . q
a s Ž m1 .
Fp Ž z . C22 Ž m 1 . ,
4p
Ž 31 .
PS Ž z . s '1 y 4 z Ž 1 q 2 z . Ž K 1X Ž m 1 . y K 2X Ž m 1 . . y Fp Ž z . s y
a s Ž m1 . 4p 1 9
8 Fp Ž z .
C22
Ž m1 . ,
m1 mb
3C8 Ž m 1 . C2 Ž m 1 .
R 0 ' QS q Q˜S q 12 Q,
Ž 34 .
which can be reduced to an explicitly power-suppressed operator using Fierz transformations and the equations of motion. ŽWe have used this in our calculation to eliminate Q˜S .. At order a s , we find the leading power contribution ² R0 : s
as 4p
ž
m
Ž Nc q 1 . ln
q
2 y Nc
mb
/
Nc
² Q:
m
ž
q 4 Ž Nc q 1 . ln
mb
/
q 2 Ž Nc q 1 . ² QS :
q O Ž 1rm b .
q 23 q 4 z y ln z
q'1 y 4 z Ž 1 q 2 z . ln s q
account in the above result. To be more explicit, consider as an example the operator
Ž 32 .
'1 y 4 z Ž 1 q 2 z .
= 2ln
637
,
Ž 33 . where we defined the combinations K 1X s 2Ž3C1C3 q C1C4 q C2 C3 ., K 2X s 2C2 C4 and K 3X s 2Ž3C1C5 q C1C6 q C2 C5 q C2 C6 .. The explicit m 1-dependence in Fp Ž z . is cancelled by the m 1-dependence of C3 , . . . ,C6 . Note that for the penguin sector the scale and scheme dependence of the matrix elements of Q and QS are effects beyond the considered order and numerically negligible. Beyond leading order in the 1rm b expansion, several further operators contribute to D G , denoted by R i in Ref. w6x. Inspection of the factorized matrix elements of these operators given in Ref. w6x shows that these superficially power-suppressed operators contribute at leading power beyond tree level. This is due to the fact that the operators are defined in QCD Žrather than in HQET. and hence the one-loop matrix elements contain m b explicitly. The leading power piece arises only from loop momenta of order m b and can therefore be subtracted perturbatively. This subtraction is necessary for a complete calculation of the a s correction at leading power and is taken into
Ž 35 .
It is crucial that this relation holds independent of the external state, so that power counting is again manifest after subtracting the right hand side of Ž35. from the matrix element of R 0 . ŽThe procedure discussed here bears some similarity with mixing of higher dimension operators into lower dimension operators in cut-off or lattice regularizations.. These subtractions must be kept in mind when a non-perturbative evaluation of the matrix elements of the R i is combined with the present NLO results. In the factorization approximation of w6x these subtractions correspond to using the pole b quark mass in the expressions for the factorized matrix elements. Indeed, in the Nc ™ ` limit, we find w6x
ž
² Bs < R 0 < Bs : s f B2 MB2 1 y s s , f B2sMB2s
a s Nc 4p
MB2s
Ž mb q ms .
ž
6ln
mb
m
2
/ /
y4 ,
Ž 36 .
up to corrections of order Ž Nc a s . 2 and LQCD rm b . The b-quark mass m b in the second expression of Ž36. is the MS mass at the scale m , which corresponds to our renormalization of the scalar operators Q S , Q˜S . To obtain the third expression we used the 1-loop relation between the pole and MS mass in the large-Nc limit and the fact that MB s y m b, p o l e s O Ž LQC D .. The same result as Ž36. is obtained from the large-Nc limit of Ž35.. ŽIn deriving Ž36. we have
M. Beneke et al.r Physics Letters B 459 (1999) 631–640
638
used the Fierz transform of Q˜S . For the coincidence of Ž36. with Ž35. it is crucial that the choice in Ž15. maintains the Fierz symmetry in the one-loop matrix elements entering Ž34...
3. Discussion The complete expression for D GB s with short-distance coefficients at NLO in QCD is given by DG
ž / G
s
16p 2 B Ž Bs ™ Xen . f B2sMB s m3b
g Ž z . h˜ QCD
Bs
ž
= GŽ z .
8 3
B q GS Ž z .
< Vcs < 2
Ž mb q ms .
/
5 2 3
BS
and
Ž 38 .
We eliminated the total decay rate GB s in favour of the semileptonic branching ratio B Ž Bs ™ Xen ., as done in Ref. w6x. This cancels the dependence of Ž D GrG . on Vcb and introduces the phase space function g Ž z . s 1 y 8 z q 8 z 3 y z 4 y 12 z 2 ln z ,
Ž 39 .
as well as the QCD correction factor w20x 2 as Ž mb . 3p
Ž p 2 y 314 . Ž 1 y 'z .
2
q 32 .
Ž 40 . The latter is written here in the approximate form of w21x. The bag factors B and BS parametrize the matrix elements of Q and QS , ² Bs < Q < Bs : s 83 f B2 MB2 B, s
Ž 41 .
s
² Bs < QS < Bs : s y 53 f B2 MB2 s
s
MB2s
Ž mb q ms .
2
BS .
p
ž
m2 ln
m 2b
q 43
/
.
Ž 43 .
m b Ž m b . s 4.4 GeV ,
m s s 0.2 GeV,
MB s s 5.37 GeV ,
Ž 44 .
f B s s 0.21 GeV ,
B Ž Bs ™ Xen . s 0.104.
Ž 37 .
GS Ž z . s y Ž FS Ž z . q PS Ž z . . .
as
Finally, d 1r m describes 1rm b corrections. The explicit expression for d 1r m in the factorization approximation can be found in Ref. w6x. In the present NLO approximation the b-quark mass that appears in d 1r m is the pole mass as mentioned above. For the numerical evaluation we use the following input parameters Žcentral values.: m b s 4.8 GeV ,
where
h˜ QC D s 1 y
mb Ž m . s mb 1 y
z s 0.085 ,
MB2s
q'1 y 4 z d 1r m ,
GŽ z . s F Ž z . q P Ž z .
The masses m b ' m b Ž m ., m s refer to the MS definition. The relation of m b to the pole mass m b is
Ž 42 .
Ž 45 .
The two-loop expression is used throughout for the QCD coupling a s in the form given in Ref. w13x with LŽ5. M S s 0.225 GeV. The NLO coefficients F Ž z ., FS Ž z . in Ž10. and P Ž z ., PS Ž z . in Ž30. are consistently expanded to first order in a s . We take m 1 s m 2 s m b as central values for the renormalization scales. The dependence of D GB s on m s is marginal and its dependence on z stems almost totally from g Ž z . in Ž39.. The results for the Wilson coefficients are displayed in Table 1. The contribution of QS dominates D GB s since the coefficient of QS is numerically much larger than the one of Q. These coefficients are independent of the scale m 1 , related to the D B s 1 Table 1 Numerical values of the Wilson coefficients G, GS , F, FS , at next-to-leading order in the NDR-scheme with evanescent operators subtracted as described in the text Žfor m 2 s m b .. Leading order results Žsuperscript ‘Ž0.’. are also shown for comparison
m1
m b r2
mb
2 mb
GS syŽ FS q PS . GSŽ0. syŽ FSŽ0. q PSŽ0. . Gs F q P G Ž0. s F Ž0. q P Ž0.
0.743 1.622 0.023 0.013
0.937 1.440 0.030 0.047
1.018 1.292 0.036 0.097
y FS y FSŽ0. F F Ž0.
0.867 1.729 0.042 0.030
1.045 1.513 0.045 0.057
1.111 1.341 0.049 0.103
M. Beneke et al.r Physics Letters B 459 (1999) 631–640
operators, up to terms of next-to-next-to-leading order. As can be seen from Table 1, the residual m 1 dependence is indeed substantially decreased for the coefficient G of Q. The reduction of scale dependence is less pronounced for the coefficient GS of QS . On the other hand, in our renormalization scheme, the central value at NLO is considerably smaller Žby about 30%. than the leading order result. However, due to the scheme dependence of the Wilson coefficient, it is premature to draw definitive conclusions on the size of D GB s without combining the coefficient functions with B and BS , computed in the same scheme. For quick reference we rewrite Ž37. as DG
ž / ž G
fBs
s
Bs
210 MeV
2
/
0.006 B Ž m b .
q0.150 BS Ž m b . y 0.063 ,
Ž 46 .
where the numbers are obtained with our central parameter set. A preliminary lattice evaluation of the relevant bag parameters can be found in Ref. w22x, together with a complete 1-loop lattice-to-continuum matching. We can use their result to obtain a Žconceptually. complete Žbut numerically preliminary. nextto-leading-order result for Ž D GrG .B s. In Ref. w22x a continuum renormalization scheme different from ours for the operators QS and Q˜S is chosen. We computed the relation between the two schemes to O Ž a s . and found BS s B4q q B˜S s B5q q
as 4p
as 4p
Ž B4q q 151 B5q . ,
Ž 47 .
Ž y 356 B4q q 12 B5q . .
Ž 48 .
Here B˜S is the bag parameter related to Q˜S , defined in analogy with Ž42. with Q S ™ Q˜S , BS ™ B˜S , and the numerical factor Žy5r3. ™ Žq1r3.. B4q and B5q are the bag parameters BS and B˜S , respectively, but in the scheme of w22x. From the quoted estimates B4qŽ m 0 . s 0.80 " 0.01, B5qŽ m 0 . s 0.94 " 0.01 w22x Žwhere the errors are statistical only and m 0 s 2.33 GeV. we infer BS Ž m 0 . s 0.81, B˜S Ž m 0 . s 0.87. Using Ž29., and taking the running of m b Ž43. into
639
account, the bag parameter BS at the scale m b is given by BS Ž m b . s BS Ž m 0 . q
as 4p
ln
mb
m0
Žy
20 3
BS Ž m 0 . q 154 B˜S Ž m 0 . .
s 0.75.
Ž 49 .
We also take B Ž m b . s 0.9 from the compilation w23x. Without the 1rm b corrections, Ž46. then becomes Ž D GrG .B s 0.118 Ž f B r210 MeV. 2 . There is a s s "15% error from Žcontinuum. perturbation theory, obtained from varying m 1 between m br2 and 2 m b , and a negligible "1% statistical error from the lattice simulation. In addition the sizeable 1rm b correction, computed in Ref. w6x, has to be included. The estimate y0.063 in Ž46. is obtained using factorization of hadronic matrix elements and has a relative error of at least "20%. As a preliminary result we may therefore write DG
ž / ž G
s
Bs
fBs 210 MeV
2
/Ž
.016 0.054q0 y0 .032 Ž m 1 y dep. .
" ??? Ž latt. syst. . . .
Ž 50 .
We emphasize the preliminary nature of the central value which depends crucially on the estimate B4qŽ m 0 . s 0.80 taken from w22x. There is an unspecified systematic error Žindicated by the question marks. attached to this number, related to the fact that the lattice calculation has been performed in the quenched approximation, at finite lattice spacing and with a ‘‘b-quark’’ mass in the charm quark mass region, without extrapolation to the continuum limit and to realistic b quark masses, respectively. Clearly, for further progress improved lattice determinations of bag parameters, most importantly of BS , are mandatory, and the numerical value for Ž D GrG .B s above has to be regarded in this context. The rather low number for the central value, compared to the leading order analysis of w6x, is a consequence of the fact that 1rm b effects and penguin contributions w6x, as well as NLO QCD corrections, all lead to a reduction of D GBS . This is further reinforced by the small value BS Ž m b . s 0.75 in our example.
M. Beneke et al.r Physics Letters B 459 (1999) 631–640
640
It is interesting to consider the ratio of D GB s to the mass difference D MB s w5,6x, in which the dependence on the decay constant f B s cancels out and the sensitivity to Vcb is considerably reduced. In addition Ž D GrD M .B only depends on the ratio of bag pas rameters. Generalizing the results given in Ref. w6x to include the next-to-leading order QCD corrections we can write DG
ž / DM
s Bs
p m2b VcbVcs 2
=
MW2
ž
8 3
2
Vt sVt b
G Ž z . q 53
1
hB S0 Ž x t . MB2s
Ž mb q ms .
/
q'1 y 4 z d 1r m ,
2
GS Ž z .
BS B
Ž 51 .
where hB is the Žscheme-dependent. next-to-leading order QCD factor entering D MB s w24x. In the usual NDR scheme hB Ž m b . s 0.846. The top-quark mass dependent function S0 Ž x t . s S0 ŽŽ m trMW . 2 . s 2.41 for m t s 167 GeV. In analogy with Ž50. we then have DG .67 s Ž 2.63q0 y1 .36 Ž m 1 y dep. . D M Bs
ž /
" ??? Ž latt. syst. . . P 10y3 . Ž 52 . The next-to-leading order calculation presented in this article can also be applied, by taking the limit z ™ 0, to the mixing-induced CP asymmetry in inclusive Bd Ž Bd . ™ uudd decays w25x. The time-dependent asymmetry is given by AŽ t . s Im j sinD Mt, where the coefficient Im j is a measure of the CKM parameter sin2 a . With next-to-leading order QCD corrections included, the expression for Im j in Eq. Ž25. of w25x is modified to read Im j s y0.12sin2 a
ž
fB 180 MeV
2
/
= 0.14 B q 0.64 BS y 0.07 y0.06
sin a sin Ž a q b . sin b sin2 a
.
Ž 53 .
In this equation the bag factors B and BS Žtaken at m s m b . are the Bd analogues of those defined in Ž41. and Ž42. for Bs . A detailed discussion of CP asymmetries will be presented in Ref. w11x.
Acknowledgements We thank Guido Martinelli for comments on the manuscript.
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