$B_s^0 \to \eta^{(\prime)} \eta^{(\prime)}$ decays in the pQCD approach
ZJOU-PHY-TH-07-01
NJNU-TH-07-05
Studies of B0s → η(′)η(′) decays in the pQCD approach
Xin Liua∗, Zhen-Jun Xiaob†, Hui-Sheng Wangc
a. Department of Physics, Zhejiang Ocean University,
Zhoushan, Zhejiang 316000, P.R. China
b. Department of Physics and Institute of Theoretical Physics,
Nanjing Normal University, Nanjing, Jiangsu 210097, P.R. China and
c. Department of Applied Mathematics and Physics,
Anhui University of Technology and Science,
Wuhu, Anhui 241000, P.R. China
(Dated: November 18, 2018)
Abstract
We calculate the CP averaged branching ratios and CP-violating asymmetries for B0s → ηη, ηη′
and η′η′ decays in the perturbative QCD (pQCD) approach here. The pQCD predictions for
the CP-averaged branching ratios are Br(B0s → ηη) =
14.2+18.0−7.5
× 10−6, Br(B0s → ηη′) =
12.4+18.2−7.0
×10−6, and Br(B0s → η′η′) =
9.2+15.3−4.9
×10−6, which agree well with those obtained
by employing the QCD factorization approach and also be consistent with available experimental
upper limits. The gluonic contributions are small in size: less than 7% for Bs → ηη and ηη′
decays, and around 18% for Bs → η′η′ decay. The CP-violating asymmetries for three decays
are very small: less than 3% in magnitude.
PACS numbers: 13.25.Hw, 12.38.Bx, 14.40.Nd
∗ liuxin@zjou.edu.cn
† xiaozhenjun@njnu.edu.cn
http://arxiv.org/abs/0704.1027v1
Among various B → M1M2 decay channels ( here Mi refers to the light pseudo-scalar
or vector mesons ), the decays involving the isosinglet η or η′ mesons in the final state are
phenomenologically very interesting and have been studied extensively during the past
decade because of the so-called Kη′ puzzle or other special features [1, 2, 3, 4].
Motivated by the large number of Bs production and decay events expected at the
forthcoming LHC experiments, the studies about the Bs meson decays become more
attractive than ever before. Very recently, some two-body Bs → Miη(′) decays, such as
Bs → (π, ρ, ω, φ)η(′) decays have been studied in Refs. [5, 6] in the perturbative QCD
(pQCD ) factorization approach [7, 8, 9]. In this paper, we would like to calculate the
branching ratios and CP asymmetries for the three B0s → ηη, ηη′ and η′η′ decays by
employing the low energy effective Hamiltonian [10] and the pQCD approach. Besides
the usual factorizable contributions, we here are able to evaluate the non-factorizable and
the annihilation contributions to these decays.
On the experimental side, only the poor upper limit on Br(B0s → ηη) is available now
[11] (upper limits at 90% C.L.):
Br(B0s → ηη) < 1.5× 10−3 , (1)
Of course, this situation will be improved rapidly when LHC experiment starts to run at
the end of 2007.
This paper is organized as follows. In Sec. I, we calculate analytically the related
Feynman diagrams and present the various decay amplitudes for the studied decay modes.
In Sec. II, we show the numerical results for the branching ratios and CP asymmetries
of B0s → η(′)η(′) decays. A short summary and some discussions are also included in this
section.
I. PERTURBATIVE CALCULATIONS
Since the b quark is rather heavy we consider the Bs meson at rest for simplicity. It
is convenient to use light-cone coordinate (p+, p−,pT ) to describe the meson’s momenta:
p± = (p0± p3)/
2 and pT = (p
1, p2). Using the light-cone coordinates the Bs meson and
the two final state meson momenta can be written as
(1, 1, 0T ), P2 =
(1, 0, 0T ), P3 =
(0, 1, 0T ), (2)
respectively, here the light meson masses have been neglected. Putting the light (anti-)
quark momenta in Bs, η
′ and η mesons as k1, k2, and k3, respectively, we can choose
k1 = (x1P
1 , 0,k1T ), k2 = (x2P
2 , 0,k2T ), k3 = (0, x3P
3 ,k3T ). (3)
Then, after the integration over k−1 , k
2 , and k
3 , the decay amplitude for Bs → ηη′ decay,
for example, can be conceptually written as
A(Bs → ηη′) ∼
dx1dx2dx3b1db1b2db2b3db3
C(t)ΦBs(x1, b1)Φη′(x2, b2)Φη(x3, b3)H(xi, bi, t)St(xi) e
−S(t)
, (4)
where ki are momenta of light quarks included in each meson, term Tr denotes the trace
over Dirac and color indices, C(t) is the Wilson coefficient evaluated at scale t, the function
H(k1, k2, k3, t) is the hard part and can be calculated perturbatively, the function ΦM is
the wave function, the function St(xi) describes the threshold resummation [12] which
smears the end-point singularities on xi, and the last term, e
−S(t), is the Sudakov form
factor which suppresses the soft dynamics effectively. We will calculate analytically the
function H(xi, bi, t) for the considered decays in the first order in αs expansion and give
the convoluted amplitudes in next section.
For the two-body charmless Bs meson decays, the related weak effective Hamiltonian
Heff can be written as [10]
Heff =
uq (C1(µ)O
1 (µ) + C2(µ)O
2 (µ))− VtbV ∗tq
Ci(µ)Oi(µ)
, (5)
where Ci(µ) are Wilson coefficients at the renormalization scale µ and Oi are the four-
fermion operators for the case of b → q (q = d, s) transition [5, 10]. For the Wilson
coefficients Ci(µ) (i = 1, . . . , 10), we will use the leading order (LO) expressions, although
the next-to-leading order (NLO) results already exist in the literature [10]. This is the
consistent way to cancel the explicit µ dependence in the theoretical formulae. For the
renormalization group evolution of the Wilson coefficients from higher scale to lower scale,
we use the formulae as given in Ref.[13] directly.
A. Decay amplitudes
We firstly take Bs → ηη′ decay mode as an example, and then extend our study to
Bs → ηη and η′η′ decays. Similar to the B0s → π0η(′) decays in [5], there are 8 type
diagrams contributing to the Bs → ηη
decays, as illustrated in Fig.1. We first calculate
the usual factorizable diagrams (a) and (b). Operators O1,2,3,4,9,10 are (V − A)(V − A)
currents, the sum of their amplitudes is given as
Feη = 8πCFm
dx1dx3
b1db1b3db3 φBs(x1, b1)
(1 + x3)φ
η (x3, b3) + (1− 2x3)rη(φPη (x3, b3) + φTη (x3, b3))
·αs(t1e) he(x1, x3, b1, b3) exp[−Sab(t1e)]
+2rηφ
η (x3, b3)αs(t
e)he(x3, x1, b3, b1) exp[−Sab(t2e)]
. (6)
where rη = m
0/mB; CF = 4/3 is a color factor. The explicit expressions of the function
he, the scales t
e and the Sudakov factors Sab can be found Ref. [5]. The form factors of
Bs to η decay, F
Bs→ηss̄
0,1 (0), can thus be extracted from the expression in Eq. (6).
The operators O5,6,7,8 have a structure of (V − A)(V + A). Some of these operators
can contribute to the decay amplitude in a factorizable way, but others may contribute
after making a Fierz transformation in order to get right flavor and color structure for
FIG. 1: Typical Feynman diagrams contributing to the Bs → ηη′ decays, where diagram (a)
and (b) contribute to the Bs → η form factor FBs→η0,1 .
factorization to work. Such kinds of contributions can be written as
F P1eη = −Feη . (7)
F P2eη = 16πCFm
(f sη′ − fuη′)m2η′
2msmBs
dx1dx3
b1db1b3db3 φBs(x1, b1)
φAη (x3, b3) + rη((2 + x3)φ
η (x3, b3)− x3φTη (x3, b3))
·αs(t1e)he(x1, x3, b1, b3) exp[−Sab(t1e)]
η (x3, b3)− 2(x1 − 1)rηφPη (x3, b3)
·αs(t2e)he(x3, x1, b3, b1) exp[−Sab(t2e)]
. (8)
For the non-factorizable diagrams 1(c) and 1(d), the corresponding decay amplitudes
can be written as
Meη =
dx1dx2 dx3
b1db1b2db2 φBs(x1, b1)φ
η′(x2, b2)
2x3rηφ
η (x3, b1)− x3φη(x3, b1)
·αs(tf)hf (x1, x2, x3, b1, b2) exp[−Scd(tf )]} , (9)
MP1eη = 0, (10)
MP2eη = −Meη . (11)
For the non-factorizable annihilation diagrams 1(e) and 1(f), we find
Maη =
dx1dx2 dx3
b1db1b2db2 φBs(x1, b1)
η (x3, b2)φ
η′(x2, b2)
+rηrη′
(x2 + x3 + 2)φ
η′(x2, b2) + (x2 − x3)φTη′(x2, b2)
φPη (x3, b2)
+rηrη′
(x2 − x3)φPη′(x3, b2) + (x2 + x3 − 2)φTη′(x2, b2)
φTη (x3, b2)
·αs(t3f)h3f (x1, x2, x3, b1, b2) exp[−Sef (t3f )]
η (x3, b2)φ
η′(x2, b2)
+rηrη′
(x2 + x3)φ
η′(x2, b2) + (x3 − x2)φTη′(x2, b2)
φPη (x3, b2)
+rηrη′
(x3 − x2)φPη′(x2, b2) + (x2 + x3)φTη′(x2, b2)
φTη (x3, b2)
·αs(t4f)h4f (x1, x2, x3, b1, b2) exp[−Sef (t4f )]
, (12)
MP1aη =
dx1dx2 dx3
b1db1b2db2 φBs(x1, b1)
(x3 − 2)rηφAη′(x2, b2)(φPη (x3, b2) + φTη (x3, b2))− (x2 − 2)rη′φAη (x3, b2)
(φPη′(x2, b2) + φ
η′(x2, b2))
· αs(t3f)h3f (x1, x2, x3, b1, b2) exp[−Sef(t3f )]
x3rηφ
η′(x2, b2)(φ
η (x3, b2) + φ
η (x3, b2))
−x2rη′φAη (x3, b2)(φPη′(x2, b2) + φTη′(x2, b2))
·αs(t4f )h4f(x1, x2, x3, b1, b2) exp[−Sef (t4f)]
, (13)
MP2aη =
dx1dx2 dx3
b1db1b2db2 φBs(x1, b1)
η (x3, b2)φ
η′(x2, b2)
+rηrη′
(x2 + x3 + 2)φ
η′(x2, b2) + (x3 − x2)φTη′(x2, b2)
φPη (x3, b2)
+rηrη′
(x3 − x2)φPη′(x3, b2) + (x2 + x3 − 2)φTη′(x2, b2)
φTη (x3, b2)
·αs(t3f )h3f(x1, x2, x3, b1, b2) exp[−Sef (t3f)]
η (x3, b2)φ
η′(x2, b2)
+rηrη′
(x2 + x3)φ
η′(x2, b2) + (x2 − x3)φTη′(x2, b2)
φPη (x3, b2)
+rηrη′
(x2 − x3)φPη′(x2, b2) + (x2 + x3)φTη′(x2, b2)
φTη (x3, b2)
·αs(t4f )h4f(x1, x2, x3, b1, b2) exp[−Sef (t4f)]
. (14)
For the factorizable annihilation diagrams 1(g) and 1(h), we have
Faη = F
aη = −8πCFm4Bs
dx2 dx3
b2db2b3db3
η (x3, b3)φ
η′(x2, b2)
+2rηrη′((x3 + 1)φ
η (x3, b3) + (x3 − 1)φTη (x3, b3))φPη′(x2, b2)
·αs(t3e)ha(x2, x3, b2, b3) exp[−Sgh(t3e)]
η (x3, b3)φ
η′(x2, b2)
+2rηrη′((x2 + 1)φ
η′(x2, b2) + (x2 − 1)φTη′(x2, b2))φPη (x3, b3)
·αs(t4e)ha(x3, x2, b3, b2) exp[−Sgh(t4e)]
F P2aη = −16πCFm4Bs
dx2 dx3
b2db2b3db3
x3rη(φ
η (x3, b3)− φTη (x3, b3))φAη′(x2, b2) + 2rη′φAη (x3, b3)φPη′(x2, b2)
·αs(t3e)ha(x2, x3, b2, b3) exp[−Sgh(t3e)]
x2rη′(φ
η′(x2, b2)− φTη′(x2, b2))φAη (x3, b3) + 2rηφAη′(x2, b2)φPη (x3, b3)
·αs(t4e)ha(x3, x2, b3, b2) exp[−Sgh(t4e)]
. (16)
For the Bs → ηη′ decay, besides the Feynman diagrams as shown in Fig. 1 where
the upper emitted meson is the η′, the Feynman diagrams obtained by exchanging the
position of η and η′ also contribute to this decay mode. The corresponding expressions
of amplitudes for new diagrams will be similar with those as given in Eqs.(6-14), since
the η and η′ are all light pseudoscalar mesons and have the similar wave functions. The
expressions of amplitudes for new diagrams can be obtained by the replacements
φAη ←→ φη′ , φPη ←→ φPη′ , φTη ←→ φtη′ , rη ←→ rη′ . (17)
For example, we find that:
Feη′ = Feη, Faη′ = −Faη, F P1aη′ = −F P1aη , F P2aη′ = F P2aη . (18)
Before we write down the complete decay amplitude for the studied decay modes, we
firstly give a brief discussion about the η − η′ mixing and the gluonic component of the
η′ meson. There exist two popular mixing basis for η − η′ system, the octet-singlet and
the quark flavor basis, in literature. Here we use the SU(3)F octet-singlet basis with the
two mixing angle (θ1, θ8) scheme [14] to describe the mixing of η and η
′ mesons. In the
numerical calculations, we will use the following mixing parameters [14]
θ8 = −21.2◦, θ1 = −9.2◦, f1 = 1.17fπ, f8 = 1.26fπ. (19)
In this paper, we firstly take η and η′ as a linear combination of light quark pairs
uū, dd̄ and ss̄, and then estimate the possible gluonic contributions to Bs → η(′)η(′)
decays by using the formulae as presented in Ref. [4]. We found that the possible gluonic
contributions are indeed small.
B. Complete decay amplitudes
For B0s → ηη′ decay, by combining the contributions from different diagrams, the total
decay amplitude can be written as
M(ηη′) =
FeηF2f
η′ + Feη′F
C4 − C5 −
FeηF2f
η′ + Feη′F
C4 − 2C5 −
F P2eη F2 + F
C5 + C6 −
+ (Meη +Meη′)F2F
ξuC2 − ξt
C3 + 2C4 −
− [MeηF1F ′2 +Meη′F ′1F2] ξt
MP2eη +M
2C6 +
MP2eη F1F
F ′1F2
+ (Maη +Maη′)F1F
ξuC2 − ξt
C3 + 2C4 −
− (Maη +Maη′)F2F ′2ξt
MP1aη +M
MP2aη +M
2C6 +
MP2aη +M
−fBs ·
F P2aη + F
C5 + C6 −
, (20)
where ξu = V
ubVus and ξt = V
tbVts, and the relevant mixing parameters and decay con-
stants are
cos θ8 −
sin θ1, F2 = −
sin θ8 +
cos θ1, (21)
F ′1 =
sin θ8 +
cos θ1, F
2 = −
sin θ8 +
cos θ1, (22)
f dη =
cos θ8 −
sin θ1, f
η = −
cos θ8 −
sin θ1, (23)
f dη′ =
sin θ8 +
cos θ1, f
η′ = −
sin θ8 +
cos θ1. (24)
Similarly, the decay amplitudes for B0s → ηη and B0s → η′η′ decay can be obtained
easily from Eq.(20) by the following replacements
f dη , f
η ←→ f dη′ , f sη′ ; F1(θ1, θ8)←→ F ′1(θ1, θ8); F2(θ1, θ8)←→ F ′2(θ1, θ8). (25)
Note that the contributions from the possible gluonic component of η′ meson have not
been included here.
II. NUMERICAL RESULTS AND DISCUSSIONS
In this section, we will calculate the CP-averaged branching ratios and CP violating
asymmetries for those considered decay modes. The input parameters and the wave
functions to be used are given in Appendix A. In numerical calculations, central values
of input parameters will be used implicitly unless otherwise stated.
Using the decay amplitudes obtained in last section, it is straightforward to calculate
the branching ratios. By employing the two mixing angle scheme of η − η′ system and
using the mixing parameters as given in Eq. (19), one finds the CP-averaged branching
ratios for the considered three decays as follows
Br( B0s → ηη) =
14.2+6.6−4.2(ωb)
+16.7
−6.2 (m
× 10−6, (26)
Br( B0s → ηη′) =
12.4+5.7−3.6(ωb)
+17.3
−6.0 (m
× 10−6, (27)
Br( B0s → η′η′) =
9.2+3.0−2.0(ωb)
+15.0
−4.5 (m
× 10−6, (28)
where the main errors are induced by the uncertainties of ωb = 0.50 ± 0.05 GeV, and
0 = [1.49−2.38] GeV (corresponding to ms = 130±30 MeV), respectively. The above
pQCD predictions agree well with those obtained in the QCD facterization approach [2].
As for the gluonic contributions, we follow the same procedure as being used in Ref. [4]
to include the possible gluonic contributions to the Bs → η(′) transition form factors
0,1 and found that the gluonic contributions to the branching ratios are less than
3% for B → ηη decay, ∼ 7% for B → ηη′ decay, and around 18% for B → η′η′ decay.
The central values of the pQCD predictions for Bs → η(′)η(′) decays after the inclusion of
possible gluonic contributions are the following
Br(B0s → ηη) =
13.7+6.4−4.0(ωb)
+16.5
−6.1 (m
× 10−6,
Br( B0s → ηη′) =
11.6+5.3−3.4(ωb)
+16.8
−5.7 (m
× 10−6,
Br( B0s → η′η′) =
10.8+3.7−2.4(ωb)
+16.2
−5.2 (m
× 10−6. (29)
Now we turn to the evaluations of the CP-violating asymmetries of Bs → η(′)η(′) decays
in pQCD approach. For B0s meson decays, a non-zero ratio (∆Γ/Γ)Bs is expected in the
SM [15, 16]. For Bs → η(′)η(′) decays, three quantities to describe the CP violation can
be defined as follows [16]:
AdirCP =
|λCP |2 − 1
1 + |λCP |2
, AmixCP =
2Im(λCP )
1 + |λCP |2
, A∆Γs =
2Re(λCP )
1 + |λCP |2
, (30)
λCP = ηf
V ∗tbVts〈f |Heff |B̄0s〉
ts〈f |Heff |B0s 〉
〈f |Heff |B̄0s〉
〈f |Heff |B0s〉
, (31)
in a very good approximation. Here AdirCP and AmixCP means the direct and mixing-induced
CP violation respectively, while the third term A∆Γs is related to the presence of a non-
negligible ∆Γs. By using the mixing parameters in Eq. (19) and the input parameters as
given in Appendix A, one found the pQCD predictions for AdirCP , AmixCP and Hf
AdirCP (B0s → ηη) =
−0.2± 0.1(γ)± 0.1(ωb)+0.4−0.2(m
× 10−2,
AdirCP (B0s → ηη′) =
+0.6+0.1−0.2(γ)± 0.1(ωb)± 0.3(m
× 10−2,
AdirCP (B0s → η′η′) =
−0.8+0.2−0.1(γ)± 0.1(ωb)± 0.7(m
× 10−2, (32)
AmixCP (B0s → ηη) = [−0.3± 0.1(γ)± 0.2(ωb)± 0.5(m
0 )]× 10−2,
AmixCP (B0s → ηη′) = [−0.8± 0.2(γ)± 0.1(ωb)± 0.2(m
0 )]× 10−2,
AmixCP (B0s → η′η′) =
+1.8+0.3−0.5(γ)± 0.0(ωb)+0.5−0.3(m
× 10−2, (33)
A∆Γs(ηη) ≈ A∆Γs(ηη′) ≈ A∆Γs(η′η′) ≈ 1, (34)
where the dominant errors come from the variations of CKM angle γ = 60◦ ± 20◦, ωb =
0.50± 0.05 GeV and mηss̄0 = [1.49− 2.38] GeV ( corresponding to ms = 130 ± 30 MeV),
respectively. It is easy to see that both the direct and mixing-induced CP violations of the
considered Bs decays are very small in magnitude, and thus almost impossible to measure
them even in the LHC experiments. The above pQCD predictions are also consistent with
the QCDF predictions [1, 2].
In short, we calculated the branching ratios and CP-violating asymmetries of B0s → ηη,
ηη′ and η′η′ decays at the leading order by using the pQCD factorization approach. Besides
the usual factorizable diagrams, the non-factorizable and annihilation diagrams are also
calculated analytically in the pQCD approach. From our calculations and phenomeno-
logical analysis, we found the following results:
• Using the two mixing angle scheme, the pQCD predictions for the CP-averaged
branching ratios are
Br(B0s → ηη) =
14.2+18.0−7.5
× 10−6,
Br(B0s → ηη′) =
12.4+18.2−7.0
× 10−6,
Br(B0s → η′η′) =
9.2+15.3−4.9
× 10−6, (35)
where the various errors as specified previously have been added in quadrature. The
pQCD predictions for the three decay channels agree well with those obtained by
employing the QCDF approach.
• The gluonic contributions are small in size: less than 7% for Bs → ηη and ηη′
decays, and around 18% for Bs → η′η′ decay.
• The direct and mixing-induced CP violations of the considered three decay modes
are very small: less than 3% in magnitude.
Note added: After completion of this paper, the paper in Ref.[18] appeared, and where
a systematic study for the Bs → M1M2 decays in the pQCD factorization approach has
been done. Since different mixing-scheme of η − η′ system have been used, the explicit
expressions of the decay amplitudes of the relevant decays are different in these two papers,
but the numerical predictions for branching ratios and CP violations agree well with each
other. The possible gluonic contributions are estimated here.
Acknowledgments
X. Liu would like to acknowledge the financial support of The Scientific Research
Start-up Fund of Zhejiang Ocean University under Grant No.21065010706. This work was
partially supported by the National Natural Science Foundation of China under Grant
No.10575052, and by the Specialized Research Fund for the Doctoral Program of Higher
Education (SRFDP) under Grant No. 20050319008.
APPENDIX A: INPUT PARAMETERS AND WAVE FUNCTIONS
In this Appendix we show the input parameters and the light meson wave functions to
be used in the numerical calculations.
The masses, decay constants, QCD scale and B0s meson lifetime are
(f=4)
= 250MeV, fπ = 130MeV, fBs = 230MeV,
0 = 1.4GeV, ms = 130MeV, fK = 160MeV,
MBs = 5.37GeV, MW = 80.41GeV, τB0s = 1.46× 10
−12s (A1)
For the CKM matrix elements, here we adopt the Wolfenstein parametrization for the
CKM matrix, and take λ = 0.2272, A = 0.818, ρ = 0.221 and η = 0.340 [11].
For the B meson wave function, we adopt the model
φBs(x, b) = NBsx
2(1− x)2exp
M2Bs x
(ωbb)
, (A2)
where ωb is a free parameter and we take ωb = 0.50± 0.05 GeV in numerical calculations,
and NBs = 63.67 is the normalization factor for ωb = 0.50.
For the distribution amplitudes φAη
, φPη
and φTη
, we utilize the result from the
light-cone sum rule [17] including twist-3 contribution. For the corresponding Gegenbauer
moments and relevant input parameters, we here use a
2 = 0.115, a
4 = −0.015, ρηdd̄ =
0 , η3 = 0.015 and ω3 = −3.0. We also assume that the wave function of uū is the
same as the wave function of dd̄ [3]. For the wave function of the ss̄ components, we also
use the same form as dd̄ but with mss̄0 and fy instead of m
0 and fx, respectively:
fx = fπ, fy =
2f 2K − f 2π . (A3)
These values are translated to the values in the two mixing angle method:
f1 = 152.1MeV, f8 = 163.8MeV,
θ1 = −9.2◦, θ8 = −21.2◦. (A4)
The parameters mi0 (i = ηdd̄(uū), ηss̄) are defined as:
dd̄(uū)
0 ≡ mπ0 ≡
(mu +md)
2M2K −m2π
(2ms)
. (A5)
[1] M. Beneke and M. Neubert, Nucl. Phys. B 675, 333 (2003).
[2] J.F. Sun, G.H. Zhu, D.S. Du, Phys. Rev. D 68 (2003) 054003.
[3] E. Kou, Phys. Rev. D 63, 054027 (2001); E. Kou and A.I. Sanda, Phys. Lett. B 525, 240
(2002).
[4] Y.Y. Charng, T. Kurimoto, and H.N. Li, Phys. Rev. D 74, 074024 (2006).
[5] Z.J. Xiao, X. Liu, and H.S. Wang, Phys. Rev. D 75,034017 (2007).
[6] X.F. Chen, D.Q. Guo and Z.J. Xiao, hep-ph/0701146.
[7] C.-H. V. Chang and H.N. Li, Phys. Rev. D 55, 5577 (1997); T.-W. Yeh and H.N. Li, Phys.
Rev. D 56, 1615 (1997).
[8] H.N. Li, Prog.Part.& Nucl.Phys. 51, 85 (2003), and reference therein.
[9] G.P. Lepage and S.J. Brodsky, Phys. Rev. D 22, 2157 (1980).
[10] G. Buchalla, A.J. Buras, and M.E. Lautenbacher, Rev. Mod. Phys. 68, 1125 (1996).
[11] Particle Data Group, W.-M. Yao et al., J. Phys. G 33, 1 (2006).
[12] H.N. Li, Phys. Rev. D 66, 094010 (2002).
[13] C.-D. Lü, K. Ukai and M.Z. Yang, Phys. Rev. D 63, 074009 (2001).
[14] Th. Feldmann, P. Kroll and B. Stech, Phys. Rev. D 58, 114006 (1998); R. Escribano and
J.M. Frere, J. High Energy Phys. 0506 (2005) 029.
[15] M. Beneke, G. Buchalla, C. Greub, A. Lenz and U. Nierste, Phys. Lett. B 459, 631 (1999).
[16] L. Fernandez, Ph.D Thesis, CERN-Thesis-2006-042.
[17] P. Ball, J. High Energy Phys. 9809, 005 (1998); P. Ball, J. High Energy Phys. 9901, 010
(1999); P. Ball and R. Zwicky, Phys. Rev. D 71, 014015 (2005).
[18] A. Ali, G. Kramer, Y. Li, C.D. Lü, Y.L. Shen, W. Wang, and Y.M. Wang, hep-ph/0703162.
http://arxiv.org/abs/hep-ph/0701146
http://arxiv.org/abs/hep-ph/0703162
	Perturbative calculations
	Decay amplitudes
	Complete decay amplitudes
	Numerical results and Discussions
	Acknowledgments
	Input parameters and wave functions
	References
