# Exclusive rare decay b→ sγ of heavy b baryons

**ABSTRACT** We present an analysis on the exclusive rare radiative decay modes Σb→Σγ and Ξb→Ξγ. The transition form factors which parametrize these processes are calculated using QCD light-cone sum rules. The decay widths we predict are Γ(Σb→Σγ)=(7.21±0.04)×10-18 GeV and Γ(Ξb→Ξγ)=(1.34±0.07)×10-16 GeV. The branching ratio of Ξb→Ξγ is predicted to be Br(Ξb→Ξγ)= (3.03±0.10)×10-4.

**0**Bookmarks

**·**

**62**Views

- [Show abstract] [Hide abstract]

**ABSTRACT:**We present a systematic discussion of Λb→Λ transition form factors in the framework of soft-collinear effective theory (SCET). The universal soft form factor, which enters the symmetry relations in the limit of large recoil energy, is calculated from a sum-rule analysis of a suitable SCET correlation function. The same method is applied to derive the leading corrections from hard-collinear gluon exchange at first order in the strong coupling constant. We present numerical estimates for form factors and form-factor ratios and their impact on decay observables in Λb→Λμ+μ- decays.Physical review D: Particles and fields 01/2012; 85(1).

Page 1

arXiv:1103.0081v1 [hep-ph] 1 Mar 2011

The exclusive rare decay b → sγ of heavy b-Baryons

Yong-Lu Liu, Long-Fei Gan, and Ming-Qiu Huang

Department of Physics, College of Science,

National University of Defense Technology,

Changsha, Hunan 410073, People’s Republic of China

(Dated: March 2, 2011)

We present an analysis on the exclusive rare radiative decay modes Σb → Σγ

and Ξb→ Ξγ. The transition form factors which parameterize these processes are

calculated using QCD light-cone sum rules. The decay widths we predict are Γ(Σb→

Σγ) = (7.21±0.04)×10−18GeV and Γ(Ξb→ Ξγ) = (1.34±0.07)×10−16GeV. The

Branching ratio of Ξb→ Ξγ is predicted to be Br(Ξb→ Ξγ) = (3.03 ± 0.10) × 10−4.

PACS numbers: 14.20.-c, 11.25.Hf, 11.55.Hx, 13.40.-f

I.INTRODUCTION

Heavy flavor physics plays an important role both in the precise test of the standard

model in the relatively high energy region and in the investigation of the hadronization

of quarks at the low energy. Hence, a lot of effort has been paid into it and a large

amount of experimental data have been accumulated [1–4]. Theoretically, much progress

has been made in the heavy flavor meson sector for its comparatively simple structure

while knowledge about baryons appears to be limited. Although much literature have

been provided to decipher these heavy flavor states (such as Refs. [5–8]), a deep under-

standing of them undoubtedly demands the information on the dynamical details which

are encoded in various decay modes [9, 10]. Among these modes, the rare radiative de-

cay processes of the b-baryons are important in that they are not only the ways to study

the Cabibbo-Kobayashi-Maskawa matrix elements Vtsand Vtbwhich are closely attached

with the dynamics inside the baryons, but also the ways to probe new physics beyond the

standard model.

These types of processes (such as b → sγ), which are forbidden at the tree level in

the standard model of electroweak theory, are induced by the flavor-changing neutral cur-

rent (FCNC) of the b-quark. Their amplitudes are dominated by the one-loop diagrams

with a virtual top quark and a W boson, and thus are strongly suppressed by Glashow-

Iliopoulos-Maiani mechanism. The relative b-meson rare radiative decay modes have been

investigated experimentally since the early 1990’s [11–14], while not so many experimental

data are available for the corresponding b-baryon processes. Theoretical studies on the

exclusive processes are available for both b-mesons and b-baryons [15–19], despite the fact

that the dynamics of the b-baryons decays are far less clarified in comparison with that of

Page 2

2

the b-mesons. However, most of the existing literature is about the process Λb→ Λγ, the

branching ratio of which has been predicted to be Br(Λb→ Λγ) ≤ 1.3 × 10−3experimen-

tally [20]. Unfortunately, this decay mode is not expected to be measured easily in the

experiments due to the fact that the final state Λ baryon is of neutral charged, as argued in

Ref. [21]. For this reason, we turn to study the possible decay modes of other octet heavy

baryons Σband Ξb, in which charged final states arise and may be easily tested in experi-

ments. It has been estimated early in the 1990′s that an amount number of b-baryons may

be produced at the c.m. energy level of the LHC [22]. Thus we can expect that these rare

decay modes could be measured by the LHC experiments in the near future, the updated

energy of which is expected to be ∼ 14TeV.

The remainder of this paper is organized as follows. We give an introduction to the

exclusive rare decay mode b → sγ and derive the formula of the decay widths in Sec. II.

Then the light-cone QCD sum rules for the relative transition form factors are derived in

Sec. III. Finally, Sec. IV is devoted to the numerical analysis and a summary is given at

the end of this section.

II.PARAMETRIZATION OF THE TRANSITION FORM FACTORS

In the standard model, the process of the exclusive rare decay b → sγ can be described

by the following effective Hamiltonian [23]:

Heff(b → sγ) = −4GF

√2V∗

tsVtbC7(µ)O7(µ),(1)

with

O7=

e

16 π2¯ s σµν(mbR + msL) b Fµν,(2)

where L/R = (1 ∓ γ5)/2 and Fµνis the field strength tensor of the photon. GF is the

Fermi coupling constant and C7(µ) is the Wilson coefficient at the scale µ. Considering

the general form beyond the standard model, O7can be represented as

e

32 π2mb¯ s σµν(gV+ γ5gA) b Fµν.

O7=

(3)

The decay amplitude is given by the expectation value of the effective Hamiltonian

between the initial and final states at the hadron level

M(Xb→ Xγ) = ?Xγ|Heff|Xb? , (4)

where X stands for the baryon involved in the process.

By considering the Lorentz structure, the contribution of the hadronic part to the pro-

cess, which is written as the hadronic matrix elements, is generally parameterized in terms

of the following form factors:

?Xb(P′)|jν|X(P)? =¯ Xb(P′)[f1γν−f2iσµνqµ+f3qν−(g1γν+g2iσµνqµ+g3qν)γ5]X(P), (5)

Page 3

3

where Xband X are the spinors of the baryons and the weak current jνis defined as

jν(x) = i¯b(x)σµν(1 − γ5)qµs(x).(6)

In fact, form factors f3and g3do not contribute to the process due to the conservation of

the vector current. Therefore, the form factors we need to calculate are f1(g1) and f2(g2),

which can be determined from the QCD light-cone sum rules. It is noted that the processes

are only related to the form factors at the point q2= 0, thus we just consider this case in

the following analysis.

With the form factors defined above, the decay width is represented as

Γ(Xb→ Xγ) =G2

F|VtbV∗

ts|2αem|C7|2m2

32π4

b

?M2

Xb− M2

MXb

X

?3

(g2

Vf2

2+ g2

Ag2

2). (7)

III.LIGHT-CONE SUM RULES FOR THE FORM FACTORS

Now we apply the light-cone QCD sum rule approach to calculate the transition form

factors f1(g1) and f2(g2). The interpolating currents to the heavy baryons are chosen as

jΣb(0) = ǫijk[qi(0)C/ zqj(0)]γ5/ zbk(0) for Σband jΞb(0) = ǫijk[si(0)C/ zbj(0)]γ5/ zqk(0) for Ξb,

respectively. Herein q stands for u or d quark, C is the charge conjugaion matrix, and z is

the vector defined on the light-cone z2= 0. The normalization of these currents is defined

by the parameters fΣband fΞb:

?0|jΣb|Σb(P′)? = fΣb(z · P′)/ zΣb(P′),

?0|jΞb|Ξb(P′)? = fΞb(z · P′)/ zΞb(P′).(8)

In the following part, we will take Σ+

calculating the form factors is the correlation function

b→ Σ+γ as an example. Our starting point for

Tµ= i

?

d4xeiqx?0|jΣb(0)jν(x)|Σ(P,s)? (9)

at q2= 0 and with Euclidean m2

procedure of the light-cone sum rule method, we need to express the correlation function

both phenomenologically and theoretically. By inserting a complete set of intermediate

states and using the definitions (5) and (8), the phenomenological side is represented as

b− P′2of about several GeV2. Following the standard

zνTν(P,q) =2fΣb(z · P′)2

M2

Σb− P′2[f1/ z − f2/ z/ q − g1/ zγ5+ g2/ z/ qγ5]Σ(P) + ...,(10)

where “...” stands for the continuum contributions. The correlation function (9) is con-

tracted by zνto remove contributions proportional to the light-cone vector zνwhich is

subdominant on the light-cone.

Page 4

4

On the other hand, the theoretical side is obtained by contracting the heavy b quarks in

the correlation function and using the distribution amplitudes presented in Ref.[24–26]. To

make the paper self-contained, we present in the Appendix the definition and the explicit

expressions of the distribution amplitudes of Σ and Ξ used in this paper. After assuming

the quark-hadron duality and performing the Borel transformation, we arrive at the final

light-cone sum rule of the form factor f2(0):

fΣbf2(0)e

−

M2

Σb

B =

M2

?1

α30

dα3e

−

s

M2

B

?

B0(α3) +M2

M2

B

B1(α3) −M4

M4

B

B2(α3)

?

−M2α2

α2

30e

−

s0

M2

B

30M2+ m2

b

?

B1(α30) −M2

M2

B

B2(α30) −

d

dα30

α2

α2

30M2B2(α30)

30M2+ m2

b

?

,(11)

where s = (1 − α3)M2+ m2

parameter which is introduced to suppress the contributions from the higher resonances

and the continuum states. Our calculation shows that f1= g1= 0 and f2= g2. In Eq.

(11), the following abbreviations are used for convenience:

?1−α3

0

B1(α3) = (2?V1−?V2−?V3−?V4−?V5)(α3),

b/α3, M is the mass of the final baryon, and M2

Bis the Borel

B0(α3) = dα1V1(α1,1 − α1− α3,α3),

B2(α3) = (−??V1+??V2+??V3+??V4+??V5−??V6)(α3).

The distribution amplitudes with tildes which come from the integration by parts in α3

are defined as

?α3

0

0

?α3

000

(12)

?Vi(α3) =

??Vi(α3) =

dα′

3

?1−α′

?α′

3

dα1Vi(α1,1 − α1− α′

?1−α′′

3,α′

3),

dα′

3

3

dα′′

3

3

dα2Vi(α1,1 − α1− α′′

3,α′′

3). (13)

The same procedure is also carried out to calculate the transition form factors of the

process Ξb→ Ξγ. We obtain the final sum rule as follows:

M2

Ξb

M2

B =

α20

s0

M2

B

?

d

dα20

α2

b

fΞbf2(0)e

−

?1

dα2e

−

s′

M2

B

?

C0(α2) +M2

M2

B

C1(α2) +

M2

α2M2

B

C2(α2) −M4

M4

B

C3(α2)

?

+M2α2

α2

20e

−

20M2+ m2

α2

b

C1(α20) +

1

α20C2(α20) −M2

M2

B

C3(α20)

+

20M2C3(α20)

20M2+ m2

?

,(14)

where s′= (1 − α2)M2+ m2

used:

b/α2, M is the mass of Ξ, and the following abbreviations are

C0(α2) =

?1−α2

0

dα1T1(α1,α2,1 − α1− α2),

Page 5

5

C1(α2) = (2?T1−?T2−?T5− 2?T7− 2?T8)(α2),

C3(α2) = (−??T1+??T2+??T5−??T6+ 2??T7+ 2??T8)(α2).

The functions with tildes are defined as

?α2

0

0

?α2

0

0

C2(α2) = (??T2−??T3−??T4+??T5+??T7+??T8)(α2),

(15)

?Ti(α2) =

??Ti(α2) =

dα′

2

?1−α′

?α′

2

dα1Ti(α1,α′

?1−α′′

0

2,1 − α1− α′

2),

dα′

2

2

dα′′

2

2

dα2Ti(α1,α′′

2,1 − α1− α′′

2). (16)

IV. NUMERICAL ANALYSIS AND THE SUMMARY

Before the numerical evaluation of the sum rules (11) and (14), we need to determine

the input parameters. Two important parameters are the decay constants fΣband fΞb,

which can be calculated with the QCD sum rule approach. Using the same expressions

in Refs. [25] and [26] with the replacements ms→ mbfor fΣband mc→ mbfor fΞb, we

get the estimations fΣb= (6.18 ± 0.03) × 10−3GeV2and fΞb= (3.32 ± 0.46) × 10−3GeV2.

Other input parameters needed in our calculation can be read from Ref. [27]:

mb = 4.8GeV, ms= 0.15GeV, MΣ= 1.189GeV,

MΞ = 1.314GeV, MΣb= 5.729GeV, MΞb= 5.81GeV,

Vts = 0.0403,Vtb= 0.9992,(17)

and

αem= 1/137,GF= 1.166364 × 10−5GeV−2,C7(mb) = −0.31.(18)

An important step in the numerical analysis of the QCD sum rules is to determine the

Borel mass parameter M2

be chosen by demanding that the continuum contribution is subdominant in comparison

with that of the ground state which we are concerned about. Simultaneously, the resulting

form factors should not vary drastically along with the threshold. Thus s0 is generally

connected with the first resonance which has the same quantum numbers as the particle we

care about. Here we fix the threshold s0in the region 39GeV2≤ s0≤ 41GeV2. As for the

Borel parameter M2

efficiently, we also demand that the higher twists contributions are less significant and the

form factors should vary mildly along with it. Our calculation shows that the working

windows can be chosen properly in the region 8GeV2≤ M2

9GeV2≤ M2

Using the distribution amplitudes given in Refs. [25] and [26], we obtain the form factors

at the zero momentum transfer f2(g2)(0) as functions of the Borel parameter M2

Band the continuum threshold s0. The continuum threshold s0can

B, which is introduced to suppress the higher resonance contributions

B≤ 11GeV2for Σb→ Σγ and

B≤ 12GeV2for Ξb→ Ξγ.

B, which

Page 6

6

0

0.05

0.1

0.15

0.2

0.25

0.3

8 8.599.51010.511

MB2(GeV2)

Σ

f2(0 )

0

0.2

0.4

0.6

0.8

1

9 9.510 10.5 11 11.512

MB2(GeV2)

Ξ

f2(0 )

FIG. 1:

39, 40, 41GeV2from the top down.

The dependence of the form factors f2(0)’s on the Borel parameter with s0 =

-1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

8 8.59 9.5 1010.5 11

MB2(GeV2)

Σ

twist3

twist4

twist5+6

f2(0 )

-1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

99.5 1010.5 11 11.512

MB2(GeV2)

Ξ

twist3

twist4

twist5+6

f2(0 )

FIG. 2:

eter with s0= 40GeV2.

The contributions to the form factors f2(0)’s from different twists on the Borel param-

are displayed in Fig. 1 . We have also analyzed the contributions from the distributions

of different twists, which are shown in Fig. 2. The results show that the contributions

of the leading-order and next-to-leading-order twists are dominant while the contributions

from higher twists are suppressed efficiently. This implies that the light-cone expansion is

reasonable in the cases we considered in this paper.

By using of the form factors we have estimated above, the decay widths of the processes

can be easily evaluated with the formula (7), which turn out to be Γ(Σb→ Σγ) = (7.21 ±

0.04) × 10−18GeV and Γ(Ξb→ Ξγ) = (1.34 ± 0.07) × 10−16GeV. Although the mean life

of Ξ−

in Ref. [27] to estimate the branching ratio of the process Ξ−

be Br(Ξ−

of the threshold, the sum rule windows, and the uncertainties in the decay constants fΣb

and fΞb. It is worth noting that errors from other sources are not considered here because

the sum rule method itself brings in an amount of uncertainties (about 20%), which makes

it less significant to take into account the errors of the input parameters.

bhas been estimated experimentally [3, 4, 28, 29], here we use the average value given

b→ Ξ−γ which turns out to

b→ Ξ−γ) = (3.03±0.10)×10−4. The errors in the widths come from the choices

Page 7

7

TABLE I: Decay widths and Branching ratios at different points of mb.

mb(GeV)4.74.84.9

Γ(Σb→ Σγ)(×10−18GeV) 6.92 ± 0.03 7.21 ± 0.04 7.26 ± 0.07

Γ(Ξb→ Ξγ)(×10−16GeV) 0.98 ± 0.04 1.34 ± 0.04 1.75 ± 0.05

Br(Ξ−

b→ Ξ−γ)(×10−4) 2.21 ± 0.08 3.03 ± 0.10 3.96 ± 0.11

We also investigate the sensitivity of the form factors to the variation of mbat different

points mb= 4.7, 4.8, and 4.9GeV. The corresponding predictions for the decay widths

and branching ratios are given in Table I.

In summary, we have investigated the exclusive rare decay processes Σb → Σγ and

Ξb→ Ξγ. The corresponding transition form factors are estimated through the light-cone

QCD sum rule approach and the decay widths of these processes are predicted to be Γ(Σb→

Σγ) = (7.21 ± 0.04) × 10−18GeV and Γ(Ξb→ Ξγ) = (1.34 ± 0.04) × 10−16GeV. We also

estimate the branching ratio of Ξ−

As we can see, our prediction is larger than the theoretical estimations for the branching

ratio of the Λb→ Λγ. Therefore it is reasonable to assume that this mode may be tested

easily, provided that a good source of Σbor Ξbis available in the future experiments, such

as the LHC experiments.

b→ Ξ−γ, which is Br(Ξ−

b→ Ξ−γ) = (3.03±0.10)×10−4.

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China

under Contract Nos.10975184 and 11047117.

Appendix

In the following we give the distribution amplitudes of Σ and Ξ used in the paper. In

general, the distribution amplitudes are defined by the matrix element of the three-quark

operator as

4?0|ǫijkq1i

= V1(?pC)αβ

α(a1z)q2

?γ5X+?

j

β(a2z)q3k

γ(a3z)|X(P)?

?γ5X−?

γ+ V5M2

γ+ V2(?pC)αβ

?γ⊥γ5X−?

?γ⊥γ5X+?

?γ5X+?

γ+V3

2M (γ⊥C)αβ

?γ5X+?

?γ⊥γ5X−?

?γ5X−?

?γ⊥γ5X+?

2pzV6(?zC)αβ

γ

+V4

2M (γ⊥C)αβ

+T1(iσ⊥pC)αβ

+T3M

pz(iσp zC)αβ

2pz(?zC)αβ

γ+M2

?γ5X−?

γ

γ+ T2(iσ⊥pC)αβ

γ+ T4M

pz(iσz pC)αβ

γ

γ

Page 8

8

+T5M2

2pz(iσ⊥zC)αβ

+MT7

2(σ⊥⊥′C)αβ

?γ⊥γ5X+?

?

γ+M2

2pzT6(iσ⊥zC)αβ

γ+ MT8

?γ⊥γ5X−?

?

γ

σ⊥⊥′γ5X+?

2(σ⊥⊥′C)αβ

σ⊥⊥′γ5X−?

γ, (19)

where M is the mass of the baryon X and C is the charge conjugation matrix. Note that

the other Lorentz structures which do not contribute to the calculations are omitted. For

each distribution amplitudes Fi= Vi,Tidefined above, it can be presented as

?

with the relationship 0 < xi< 1,?

baryon momentum on the quarks. The integration measure is defined as

?

0

F(aip · z) =Dxe

−ipz?

i

xiaiF(xi),(20)

i

xi= 1, and xicorresponds to the distribution of the

Dx =

?1

dx1dx2dx3δ(x1+ x2+ x3− 1). (21)

The distribution amplitudes can be expanded with a conformal spin. The detailed

process is referred to Refs. [24–26]. The explicit expressions of the distribution amplitudes

are collected below:

V1(xi) = 120x1x2x3φ0

V3(xi) = 12x3(1 − x3)ψ0

V5(xi) = 6x3φ0

3,V2(xi) = 24x1x2φ0

V4(xi) = 3(1 − x3)ψ0

V6(xi) = 2φ0

4,

4,

5,

5,

6,

T1(xi) = 120x1x2x3φ

′0

3,T2(xi) = 24x1x2φ

T4(xi) = −3

T6(xi) = 2φ

T8(xi) =3

′0

4,

T3(xi) = 6x3(1 − x3)(ξ0

T5(xi) = 6x3φ

4+ ξ

′0

4),

2(x1+ x2)(ξ

′0

6

′0

5+ ξ0

5)

′0

5,

T7(xi) = 6x3(1 − x3)(ξ

′0

4− ξ0

4),

2(x1+ x2)(ξ

′0

5− ξ0

5).(22)

The parameters in the expressions are as follows:

φ0

3= φ0

6= fX,ψ0

4= ψ0

5=1

2(fX− λ1),

5=1

φ0

4= φ0

5=1

2(fX+ λ1),

4=1

6(8λ3− 3λ2),

1

6(12λ3− 5λ2).

φ′0

3= φ′0

6= −ξ0

5=1

6(4λ3− λ2),

φ′0

4= ξ0

φ′0

5= −ξ′0

6λ2,

ξ′0

4 =

(23)

The nonperturbative parameters fX, λ1and λ2are determined with QCD sum rules to be

fΣ = (9.4 ± 0.4) × 10−3GeV2,

λ2 = (4.4 ± 0.1) × 10−2GeV2,

λ1= −(2.5 ± 0.1) × 10−2GeV2,

λ3= (2.0 ± 0.1) × 10−2GeV2

(24)

Page 9

9

for Σ and

fΞ = (9.9 ± 0.4) × 10−3GeV2,

λ2 = (5.2 ± 0.2) × 10−2GeV2,

λ1= −(2.8 ± 0.1) × 10−2GeV2,

λ3= (1.7 ± 0.1) × 10−2GeV2

(25)

for Ξ.

[1] D. Acosta et al. (CDF Collaboration), Phys. Rev. Lett. 96, 202001 (2006).

[2] V. Abazov et al. (D0 Collaboration), Phys. Rev. Lett. 99, 052001 (2007).

[3] T. Aaltonen et al. (CDF Collaboration), Phys. Rev. Lett. 99, 052002 (2007).

[4] T. Aaltonen et al. (CDF Collaboration), Phys. Rev. Lett. 99, 202001 (2007).

[5] D. Ebert, R. N. Faustov, and V. O. Galkin, Phys. Rev. D 72, 034026 (2005); Phys. Lett. B

659, 612 (2008).

[6] Jian-Rong Zhang and Ming-Qiu Huang, Phys. Rev. D 78, 094015 (2008)

[7] R. Roncaglia, D. B. Lichtenberg, and E. Predazzi, Phys. Rev. D 52, 1722 (1995).

[8] N. Mathur, R. Lewis, and R. M. Woloshyn, Phys. Rev. D 66, 014502 (2002).

[9] K. Azizi, M. Bayar, A. Ozpineci, Y. Sarac, Phys. Rev. D 80, 036007 (2009).

[10] M. Q. Huang and D. W. Wang, Phys. Rev. D 69, 094003 (2004).

[11] R. Ammar et al. (CLEO Collaboration), Phys. Rev. Lett. 71, 674 (1993) .

[12] T. E. Coan et al. (CLEO Collaboration), Phys. Rev. Lett. 84, 5283 (2000).

[13] B. Aubert et al. (BaBar Collaboration), Phys. Rev. Lett. 88, 101805 (2002); Phys. Rev. D

70, 112006 (2004) ; Phys. Rev. D 72, 052004 (2005).

[14] M. Nakao et al. (Belle Collaboration), Phys. Rev. D 69, 112001 (2004); P. Koppenburg et

al. (Belle Collaboration), Phys. Rev. Lett. 93, 061803 (2004).

[15] T. Mannel and S. Recksiegel, J. Phys. G 24 (1998) 979.

[16] R. Mohanta, A. K. Gin, M. P. Khanna, M. Ishida, Prog. Theor. Phys. 102, 645 (1999).

[17] H. Y. Cheng, C. Y. Cheung, G. L. Lin, Y. C. Lin, T. M. Yan and H. L. Yu, Phys. Rev. D

51, 1199 (1995); H. Y. Cheng and C. K. Chua, Phys. Rev. D 69, 094007 (2004).

[18] C.K. Chua, X.G. He and W. S. Hou, Phys. Rev. D 60, 014003 (1999).

[19] T. Hurth, Rev. Mod. Phys. 75 (2003) 1159, and references therein.

[20] D. Acosta, et al. (CDF Collaboration), Phys. ReV. D 66, 112002 (2002).

[21] L. Oliver, J. C. Raynal and R. Sinha, arXiv:1007.3632.

[22] A. Fridman and R. Kinnunen, CERN-PPE/93-61, Contribution to the 5th International

Symposium on Heavy Flavor Physics, Montreal, July 1993.

[23] G. Buchalla, A. J. Buras and M. E. Lautenbacher, Rev. Mod. Phys. 68, 1125 (1996).

[24] V. M. Braun, R. J. Fries, N. Mahnke, and E. Stein, Nucl. Phys. B 589, 381 (2000).

[25] Y. L. Liu and M. Q. Huang, Nucl. Phys. A 821, 80 (2009).

[26] Y. L. Liu and M. Q. Huang, Phys. Rev. D 80, 055015 (2009).

Page 10

10

[27] K. Nakamura, et al. (Particle Data Group), J. Phys. G 37, 075021 (2010).

[28] D. Buskulic et al. (ALEPH Collaboration), Phys. Lett. B 384, 449 (1996).

[29] J. Abdallah et al. (DELPHI Collaboration), Eur. Phys. J. C 44, 299 (2005).

[30] T. Aaltonen et al. (CDF Collaboration), Phys. Rev. D 80, 072003 (2009).