# Study of NonleptonicB

Authors: Qin Chang; Xiaohui Hu; Junfeng Sun; Xiaolin Wang; Yueling Yang
Journal: Advances in High Energy Physics
 (2015)

---

## Abstract

Motivated by the powerful capability of measurement for theb hadron decays at LHC and SuperKEKB/Belle-II, the nonleptonic B
-
∗
→
D
D
-
∗, D
ρ
-, D
K
∗
-, π
D
∗, and K
D
∗ decays are studied. With the amplitudes calculated with factorization approach and the form factors evaluated with the BSW model, branching fractions and polarization fractions are firstly presented. Numerically, the CKM-favored B
-
q
∗
→
D
q
D
s
∗
- and D
q
ρ
- decays have branching fractions ~10−8, which should be sought for with priority and firstly observed by LHC and Belle-II. The B
-
q
∗
→
D
q
K
∗ and D
q
ρ decays are dominated by the longitudinal polarization states, while the parallel polarization fractions of B
-
q
∗
→
D
q
D
-
∗ decays are comparable with the longitudinal ones; numerically, f
∥ + f
L
≃ 95% and f
L
:
f
∥
≃
5
:
4. Some comparisons between B
-
q
∗
0
→
D
q
V and their corresponding B
-
q
0
→
D
q
∗
V decays are performed, and the relation f
L
,
∥
(
B
-
∗
0
→
D
V
)
≃
f
L
,
∥
(
B
-
0
→
D
∗
V
) is found. With the implication of S
U
(
3
) flavor symmetry, the ratios R
d
u and R
d
s are discussed and suggested to be verified experimentally.


---

## Body

## 1. Introduction

Theb physics plays an important role in testing the flavor dynamics of Standard Model (SM), exploring the source of C
P violation, searching the indirect hints of new physics, investigating the underling mechanisms of QCD, and so forth and thus attracts much experimental and theoretical attention. With the successful performance of BABAR, Belle, CDF, and D0 in the past years, many B
u
,
d
,
s meson decays have been well measured. Thanks to the ongoing LHCb experiment [1] at LHC and forthcoming Belle-II experiment [2] at SuperKEKB, experimental analysis of B meson decays is entering a new frontier of precision. By then, besides B
u
,
d
,
s mesons, the rare decays of some other b-flavored hadrons are hopefully to be observed, which may provide much more extensive space for b physics.The excited statesB
u
,
d
,
s
∗ with quantum number of n
2
s
+
1
L
J
=
1
3
S
1 and J
P
=
1
- (n, L, s, J, and P are the quantum numbers of radial, orbital, spin, total angular momenta, and parity, resp.), which will be referred to as B
∗ in this paper, had been observed by CLEO, Belle, LHCb, and so on [3]. However, except for their masses, there is no more experimental information due to the fact that the production of B
∗ mesons is mainly through Υ
(
5
S
) decays at e
+
e
- colliders and the integrated luminosity is not high enough for probing the B
∗ rare decays. Moreover, B
∗ decays are dominated by the radiative processes B
∗
→
B
γ, and the other decay modes are too rare to be measured easily. Fortunately, with annual integrated luminosity ~13 a
b
-
1 [2] and the cross section of Υ
(
5
S
) production in e
+
e
- collisions σ
(
e
+
e
-
→
Υ
(
5
S
)
)
=
(
0.301
±
0.002
±
0.039
) nb [4], it is expected that about 4
×
1
0
9
Υ
(
5
S
) samples could be produced per year at the forthcoming super-B factory SuperKEKB/Belle-II, which implies that the B
∗ rare decays with branching fractions ≳
1
0
-
9 are possible to be observed. Besides, due to the much larger production cross section of p
p collisions, experiments at LHC [5, 6] also possibly provide some experimental information for B
∗ decays.With the rapid development of experiment, accordingly, the theoretical evaluations forB
∗ weak decays are urgently needed and worthful. Nonleptonic B
∗ weak decays allow one to overconstrain parameters obtained from B meson decay, test various models, and improve our understanding on the strong interactions and the mechanism responsible for heavy meson weak decay. The observation of an anomalous production rate of B
∗ weak decays would be a hint of possible new physics beyond SM. In addition, the B
∗ weak decay provides one unique opportunity of observing the weak decay of a vector meson, where polarization effects can be used as tests of the underlying structure and dynamics of hadrons. To our knowledge, few previous theoretical works come close to studying B
∗ weak decays. Compared with the B
∗
→
P
P decays, which are suppressed dynamically by the orbital angular momentum of final states, B
∗
→
P
V decays are expected to have much larger branching fractions and hence are generally much easier to be measured. So, in this paper, we will estimate the observables of nonleptonic two-body B
∗
→
P
V weak decay to offer a ready reference.Our paper is organized as follows. In Section2, after a brief review of the effective Hamiltonian and factorization approach, the explicit amplitudes of B
u
,
d
,
s
∗
→
D
u
,
d
,
s
(
∗
)
M decays are calculated. In Section 3, the numerical results and discussions are presented. Finally, we summarize in Section 4.


## 2. Theoretical Framework

Within SM, the effective Hamiltonian responsible for nonleptonicB
∗ weak decay is [7](1)
H
eff
=
G
F
2
∑
q
,
q
′
=
u
,
c
V
q
b
V
q
′
p
∗
∑
i
=
1
2
C
i
μ
O
i
μ
+
V
q
b
V
q
p
∗
∑
i
=
3
10
C
i
μ
O
i
μ
+
h
.
c.,where p
=
d or s, V
q
b
V
q
′
p
∗ is the product of the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements, and C
i are Wilson coefficients, which describe the short-distance contributions and are calculated perturbatively; the explicit expressions of local four-quark operators O
i are(2)
O
1
=
q
¯
α
b
α
V
-
A
p
¯
β
q
β
′
V
-
A
,
O
2
=
q
¯
α
b
β
V
-
A
p
¯
α
q
β
′
V
-
A
,
O
3
=
p
¯
α
b
α
V
-
A
∑
p
′
p
¯
β
′
p
β
′
V
-
A
,
O
4
=
p
¯
α
b
β
V
-
A
∑
p
′
p
¯
β
′
p
α
′
V
-
A
,
O
5
=
p
¯
α
b
α
V
-
A
∑
p
′
p
¯
β
′
p
β
′
V
+
A
,
O
6
=
p
¯
α
b
β
V
-
A
∑
p
′
p
¯
β
′
p
α
′
V
+
A
,
O
7
=
p
¯
α
b
α
V
-
A
∑
p
′
3
2
Q
p
′
p
¯
β
′
p
β
′
V
+
A
,
O
8
=
p
¯
α
b
β
V
-
A
∑
p
′
3
2
Q
p
′
p
¯
β
′
p
α
′
V
+
A
,
O
9
=
p
¯
α
b
α
V
-
A
∑
p
′
3
2
Q
p
′
p
¯
β
′
p
β
′
V
-
A
,
O
10
=
p
¯
α
b
β
V
-
A
∑
p
′
3
2
Q
p
′
p
¯
β
′
p
α
′
V
-
A
,where (
q
¯
1
q
2
)
V
±
A
=
q
¯
1
γ
μ
(
1
±
γ
5
)
q
2, α and β are color indices, Q
p
′ is the electric charge of the quark p
′ in the unit of |
e
|, and p
′ denotes the active quark at the scale μ
~
O
(
m
b
);that is, p
′
=
u, d, c, s, and b.To obtain the decay amplitudes, the remaining and also the most intricate work is how to calculate hadronic matrix elements〈
P
V
|
O
i
|
B
∗
〉. With the factorization approach [8–11] based on the color transparency mechanism [12, 13], in principle, the hadronic matrix element could be factorized as(3)
P
V
O
i
B
∗
=
a
P
J
μ
B
∗
V
J
μ
0
+
b
V
J
μ
B
∗
P
J
μ
0
+
c
P
V
J
μ
0
0
J
μ
B
∗
.Due to the unnecessary complexity of hadronic matrix element〈
V
|
J
μ
|
B
∗
〉 and power suppression of annihilation contributions, we only consider one simple scenario where pseudoscalar meson picks up the spectator quark in B
∗ meson;that is, a
=
1, b
=
0, and c
=
0 in (3) for the moment. Two current matrix elements can be further parameterized by decay constants and transition form factors:(4)
V
p
,
ϵ
q
¯
1
γ
μ
q
2
0
=
f
V
m
V
ϵ
μ
∗
,
P
p
P
q
¯
γ
μ
b
B
¯
∗
p
B
∗
,
η
=
2
V
q
2
m
B
∗
+
m
P
ε
μ
ν
ρ
σ
η
ν
p
P
ρ
p
B
∗
σ
,
P
p
P
q
¯
γ
μ
γ
5
b
B
¯
∗
p
B
∗
,
η
=
i
2
m
B
∗
A
0
q
2
η
·
q
q
2
q
μ
+
i
m
P
+
m
B
∗
A
1
q
2
η
μ
-
η
·
q
q
2
q
μ
+
i
A
2
q
2
η
·
q
m
P
+
m
B
∗
p
B
∗
+
p
P
μ
-
m
B
∗
2
-
m
P
2
q
2
q
μ
,where ϵ and η are the polarization vector, f
V is the decay constant of vector meson, V and A
0,1
,
2 are transition form factors, q
=
p
B
∗
-
p
P, and the sign convention ϵ
0123
=
1. Even though some improved approaches, such as the QCD factorization [14, 15], the perturbative QCD scheme [16, 17], and the soft-collinear effective theory [18–21], are presented to evaluate higher order QCD corrections and reduce the renormalization scale dependence, the naive factorization (NF) approximation is a useful tool of theoretical estimation. Because there is no available experimental measurement for now, the NF approach is good enough to give a preliminary analysis, and so it is adopted in our evaluation.With the above definitions, the hadronic matrix elements considered here can be decomposed into three scalar invariant amplitudesS
1,2
,
3:(5)
P
V
O
i
B
∗
=
ϵ
∗
μ
η
ν
S
1
g
μ
ν
+
S
2
p
B
∗
+
p
P
μ
p
V
ν
m
B
∗
m
V
+
i
S
3
ε
μ
ν
ρ
σ
2
p
B
∗
ρ
p
P
σ
m
B
∗
m
V
,where the amplitudes S
1,2
,
3 describe the s, d, and p wave contributions, respectively, and are explicitly written as(6)
S
1
=
-
i
f
V
m
B
∗
+
m
P
m
V
A
1
,
(7)
S
2
=
-
i
2
f
V
m
B
∗
m
V
2
A
2
m
B
∗
+
m
P
,
(8)
S
3
=
+
i
2
f
V
m
B
∗
m
V
2
V
m
B
∗
+
m
P
. Alternatively, one can choose the helicity amplitudes H
λ (λ
=
0
,
+
,
-),(9)
H
P
V
0
=
-
S
1
x
-
S
2
x
2
-
1
,
H
P
V
±
=
-
S
1
±
S
3
x
2
-
1
,with(10)
x
≡
p
B
∗
·
p
V
m
B
∗
m
V
=
m
B
∗
2
-
m
P
2
+
m
V
2
2
m
B
∗
m
V
.Now, with the formulae given above and the effective coefficientsα
i defined as(11)
α
1
=
C
1
+
C
2
N
c
,
α
2
=
C
2
+
C
1
N
c
,
α
4
=
C
4
+
C
3
N
c
,
α
4
,
E
W
=
C
10
+
C
9
N
c
,we present the amplitudes of nonleptonic two-body B
¯
∗ decays as follows:(i)
ForB
¯
q
∗
→
D
q
D
¯
∗ decays (the spectator q
=
u, d, and s),(12)
A
λ
B
¯
q
∗
⟶
D
q
D
∗
-
=
H
D
D
∗
-
λ
V
c
b
V
c
d
∗
α
1
+
α
4
+
α
4
,
E
W
+
V
u
b
V
u
d
∗
α
4
+
α
4
,
E
W
,
A
λ
B
¯
q
∗
⟶
D
q
D
s
∗
-
=
H
D
D
s
∗
-
λ
V
c
b
V
c
d
∗
α
1
+
α
4
+
α
4
,
E
W
+
V
u
b
V
u
d
∗
α
4
+
α
4
,
E
W
.
(ii)
ForB
¯
q
∗
0
→
D
q
V decays (the spectator q
=
d and s and V
=
ρ
- and K
∗
-),(13)
A
λ
B
¯
q
∗
0
⟶
D
q
ρ
-
=
H
D
ρ
-
λ
V
c
b
V
c
d
∗
α
1
,
A
λ
B
¯
q
∗
0
⟶
D
q
K
∗
-
=
H
D
K
∗
-
λ
V
c
b
V
c
s
∗
α
1
.
(iii)
ForB
¯
∗
→
π
D
∗ decays,(14)
A
λ
B
∗
-
⟶
π
-
D
¯
∗
0
=
H
π
-
D
¯
∗
0
λ
V
u
b
V
c
d
∗
α
2
,
2
A
λ
B
∗
-
⟶
π
0
D
∗
-
=
H
π
0
D
∗
-
λ
V
u
b
V
c
d
∗
α
1
,
2
A
λ
B
∗
-
⟶
π
0
D
s
∗
-
=
H
π
0
D
s
∗
-
λ
V
u
b
V
c
s
∗
α
1
,
-
2
A
λ
B
¯
∗
0
⟶
π
0
D
∗
0
=
H
π
0
D
∗
0
λ
V
c
b
V
u
d
∗
α
2
,
-
2
A
λ
B
¯
∗
0
⟶
π
0
D
¯
∗
0
=
H
π
0
D
¯
∗
0
λ
V
u
b
V
c
d
∗
α
2
,
A
λ
B
¯
∗
0
⟶
π
+
D
∗
-
=
H
π
+
D
∗
-
λ
V
u
b
V
c
d
∗
α
1
,
A
λ
B
¯
∗
0
⟶
π
+
D
s
∗
-
=
H
π
+
D
s
∗
-
λ
V
u
b
V
c
s
∗
α
1
.
(iv)
ForB
¯
∗
→
K
D
∗ decays,(15)
A
λ
B
∗
-
⟶
K
-
D
¯
∗
0
=
H
K
-
D
¯
∗
0
λ
V
u
b
V
c
s
∗
α
2
,
A
λ
B
¯
∗
0
⟶
K
¯
0
D
¯
∗
0
=
H
K
¯
0
D
¯
∗
0
λ
V
u
b
V
c
s
∗
α
2
,
A
λ
B
¯
∗
0
⟶
K
¯
0
D
∗
0
=
H
K
¯
0
D
∗
0
λ
V
c
b
V
u
s
∗
α
2
,
A
λ
B
¯
s
∗
0
⟶
K
+
D
∗
-
=
H
K
+
D
∗
-
λ
V
u
b
V
c
d
∗
α
1
,
A
λ
B
¯
s
∗
0
⟶
K
0
D
¯
∗
0
=
H
K
0
D
¯
∗
0
λ
V
u
b
V
c
d
∗
α
2
,
A
λ
B
¯
s
∗
0
⟶
K
0
D
∗
0
=
H
K
0
D
∗
0
λ
V
c
b
V
u
d
∗
α
2
,
A
λ
B
¯
s
∗
0
⟶
K
+
D
s
∗
-
=
H
K
+
D
s
∗
-
λ
V
u
b
V
c
s
∗
α
1
.In the rest frame ofB
¯
∗ meson, the branching fraction can be written as(16)
B
B
¯
∗
⟶
P
V
=
1
3
G
F
2
2
1
8
π
p
c
m
B
∗
2
Γ
tot
B
∗
∑
λ
A
λ
B
¯
∗
⟶
P
V
2
, where the momentum of final states is(17)
p
c
=
m
B
∗
2
-
m
P
+
m
V
2
m
B
∗
2
-
m
P
-
m
V
2
2
m
B
∗
.The longitudinal, parallel, and perpendicular polarization fractions are defined as(18)
f
L
,
∥
,
⊥
=
A
0
,
∥
,
⊥
2
A
0
2
+
A
∥
2
+
A
⊥
2
,where A
∥ and A
⊥ are parallel and perpendicular amplitudes:(19)
A
∥
,
⊥
=
1
2
A
-
±
A
+
.


## 3. Numerical Results and Discussion

Firstly, we would like to clarify the input parameters used in our numerical evaluations. For the CKM matrix elements, we adopt the Wolfenstein parameterization [22] and choose the four parameters A, λ, ρ, and η as [23](20)
A
=
0.81
0
-
0.024
+
0.018
,
λ
=
0.2254
8
-
0.00034
+
0.00068
,
ρ
¯
=
0.145
3
-
0.0073
+
0.0133
,
η
¯
=
0.34
3
-
0.012
+
0.011
,with ρ
¯
=
ρ
(
1
-
λ
2
/
2
) and η
¯
=
η
(
1
-
λ
2
/
2
).The decay constants of light vector mesons are [24](21)
f
ρ
=
216
±
3
MeV
,
f
K
∗
=
220
±
5
MeV
.For the decay constants of D
(
s
)
∗ mesons, we will take [25](22)
f
D
∗
=
252.2
±
22.3
±
4
MeV
,
f
D
s
∗
=
305.5
±
26.8
±
5
MeV
,which agree well with the results of the other QCD sum rules [26, 27] and lattice QCD with N
f
=
2 [28].Besides the decay constants, theB
∗
→
P transition form factors are also essential inputs to estimate branching ratios for nonleptonic B
∗
→
P
V decay. In this paper, the Bauer-Stech-Wirbel (BSW) model [10] is employed to evaluate the form factors A
1
(
0
), A
2
(
0
), and V
(
0
), which could be written as the overlap integrals of wave functions of mesons [10]:(23)
V
B
∗
→
P
0
=
m
b
-
m
q
m
B
∗
-
m
P
J
B
∗
→
P
,
A
1
B
∗
→
P
0
=
m
b
+
m
q
m
B
∗
+
m
P
J
B
∗
→
P
,
A
2
B
∗
→
P
0
=
2
m
B
∗
m
B
∗
-
m
P
A
0
B
∗
→
P
0
-
m
B
∗
+
m
P
m
B
∗
-
m
P
A
1
B
∗
→
P
0
,
A
0
B
∗
→
P
0
=
∫
d
2
p
⊥
∫
0
1
d
x
φ
P
p
→
⊥
,
x
σ
z
φ
V
1,0
p
→
⊥
,
x
,
J
B
∗
→
P
=
2
∫
d
2
p
⊥
∫
0
1
d
x
φ
P
p
→
⊥
,
x
i
σ
y
φ
V
1
,
-
1
p
→
⊥
,
x
,where p
→
⊥ is the transverse quark momentum, σ
y
,
z are the Pauli matrix acting on the spin indices of the decaying quark, and m
q represents the mass of nonspectator quark of pseudoscalar meson. With the meson wave function φ
M
(
p
→
⊥
,
x
) as solution of a relativistic scalar harmonic oscillator potential [10] and ω = 0.4 GeV which determines the average transverse quark momentum through 〈
p
⊥
2
〉
=
ω
2, we get the numerical results of the transition form factors summarized in Table 1. In our following evaluation, these numbers and 15
% of them are used as default inputs and uncertainties, respectively.Table 1
The numerical results of form factors within BSW model.
Transition
V
(
0
)
A
1
(
0
)
A
2
(
0
)
B
∗
→
D
0.76
0.75
0.62
B
∗
→
K
0.41
0.42
0.35
B
∗
→
π
0.35
0.38
0.30
B
s
∗
→
D
s
0.72
0.69
0.59
B
s
∗
→
K
0.30
0.29
0.26To evaluate the branching fractions, the total decay widths (or lifetimes)Γ
tot
(
B
∗
) are necessary. However, there is no available experimental or theoretical information for Γ
tot
(
B
∗
) until now. Because of the fact that the QED radiative processes B
∗
→
B
γ dominate the decays of B
∗ mesons, we will take the approximation Γ
tot
(
B
∗
)
≃
Γ
(
B
∗
→
B
γ
). The theoretical predictions on Γ
(
B
∗
→
B
γ
) have been widely evaluated in various theoretical models, such as relativistic quark model [29, 30], QCD sum rules [31], light cone QCD sum rules [32], light front quark model [33], heavy quark effective theory with vector meson dominance hypothesis [34], or covariant model [35]. In this paper, the most recent results [33, 35](24)
Γ
B
∗
+
⟶
B
+
γ
=
46
8
-
75
+
73
eV
,
(25)
Γ
B
∗
0
⟶
B
0
γ
=
148
±
20
eV
,
(26)
Γ
B
s
∗
0
⟶
B
s
0
γ
=
68
±
17
eV
,which agree with the other theoretical results, are approximately treated as Γ
tot in our numerical estimate.With the aforementioned values of input parameters and the theoretical formula, we present theoretical predictions for the observables ofB
¯
∗
→
D
D
¯
∗, D
ρ, D
K
∗, π
D
∗, and K
D
∗ decays, in which only the (color-suppressed) tree induced decay modes are evaluated due to the fact that the branching fractions of loop induced decays are very small and hard to be measured soon. Our numerical results for the branching fractions and the polarization fractions are summarized in Tables 2 and 3. In Table 2, the first, second, and third theoretical errors are caused by uncertainties of the CKM parameters, hadronic parameters (decay constants and form factors), and total decay widths, respectively. From Tables 2 and 3, the following could be found:(1)
The hierarchy of branching fractions is clear. (i) The branching fractions ofB
¯
∗
→
π
D
∗ and K
D
∗ decays are much smaller than the ones of B
¯
∗
→
D
D
¯
∗, D
ρ, and D
K
∗ decays, which is caused by the fact that the form factors of B
¯
∗
→
D transition are much larger than those of B
¯
∗
→
π and B
¯
∗
→
K transitions. (ii) For B
¯
∗
→
D
D
¯
∗, D
ρ, and D
K
∗ decays, the hierarchy is induced by two factors: one is the CKM factor (see the third column of Table 2), and the other is Γ
tot
(
B
∗
±
)
>
Γ
tot
(
B
d
∗
0
)
>
Γ
tot
(
B
s
∗
0
) (see (24), (25), and (26)).
(2)
Besides small form factors, theB
¯
∗
→
π
D
∗, K
D
∗ decays are either color suppressed or the CKM factors suppressed. So they have very small branching fractions (see Table 2) and are hardly measured soon. Most of the CKM-favored and tree-dominated B
¯
∗
→
D
D
¯
∗, D
ρ, and D
K
∗ decays, enhanced by the relatively large B
¯
∗
→
D transition form factors, have large branching fractions, ≳
1
0
-
9, and thus could be measured in the near future. In particular, branching ratios for B
¯
q
∗
→
D
q
D
¯
s
∗
- and D
q
ρ decays can reach up to 1
0
-
8 and hence should be sought for with priority and firstly observed at the high statistics LHC and Belle-II experiments.
The numerical results and above analyses are based on the NF, in which the QCD corrections are not included. Fortunately, for the color-allowed tree amplitudeα
1, the NF estimate is stable due to the relatively small QCD corrections [15]. For instance, in B
→
π
π and B
→
D
∗
L decays, the results α
1
(
π
π
)
=
(
1.020
)
L
O
+
(
0.018
+
0.018
i
)
N
L
O [14] and α
1
(
D
∗
L
)
=
(
1.025
)
L
O
+
(
0.019
+
0.013
i
)
N
L
O [15] indicate clearly that the O
(
α
s
) correction is only about 2
% and thus trivial numerically. For the color-suppressed decay modes listed in Table 2, even though the NF estimates would suffer significant O
(
α
s
) correction (about 46
% in B
→
π
π decays, e.g., [36]), they still escape the experimental scope due to their small branching factions <
1
0
-
9 and thus will not be discussed further. In the following analyses, we will pay our attention only to the color allowed tree-dominated B
¯
∗
→
D
D
¯
∗, D
ρ, and D
K
∗ decays.
(3)
For theB
∗
-
→
D
0
D
(
s
)
∗
- and B
¯
∗
0
→
D
+
D
(
s
)
∗
- decays, the S
U
(
3
) flavor symmetry implies the relations(27)
A
B
∗
-
⟶
D
0
D
s
∗
-
≃
A
B
¯
∗
0
⟶
D
+
D
s
∗
-
,
A
B
∗
-
⟶
D
0
D
∗
-
≃
A
B
¯
∗
0
⟶
D
+
D
∗
-
.
Further considering the theoretical predictionΓ
(
B
∗
+
→
B
+
γ
)
/
Γ
(
B
∗
0
→
B
0
γ
)
≈
3 (see (24) and (25)) and assumption Γ
tot
(
B
∗
)
≃
Γ
(
B
∗
→
B
γ
), one may find the ratios(28)
R
d
u
≡
B
B
¯
∗
0
⟶
D
+
D
s
∗
-
B
B
∗
-
⟶
D
0
D
s
∗
-
≃
Γ
B
∗
+
⟶
B
+
γ
Γ
B
∗
0
⟶
B
0
γ
≈
theo.
3
,
(29)
R
d
u
′
≡
B
B
¯
∗
0
⟶
D
+
D
∗
-
B
B
∗
-
⟶
D
0
D
∗
-
≃
R
d
u
,
which are satisfied in our numerical evaluations. Experimentally, the first relation (28) is hopefully to be tested soon due to the large branching fractions. For the other potentially detectable B
¯
d
,
s
∗
0
→
D
D
¯
∗, D
ρ, and D
K
∗ decay modes, with branching fractions ≳
1
0
-
9, the U-spin symmetry implies relations(30)
A
B
¯
∗
0
⟶
D
+
D
∗
-
≃
A
B
¯
s
∗
0
⟶
D
s
+
D
∗
-
,
A
B
¯
∗
0
⟶
D
+
D
s
∗
-
≃
A
B
¯
s
∗
0
⟶
D
s
+
D
s
∗
-
,
A
B
¯
∗
0
⟶
D
+
K
∗
-
≃
A
B
¯
s
∗
0
⟶
D
s
+
K
∗
-
,
A
B
¯
∗
0
⟶
D
+
ρ
-
≃
A
B
¯
s
∗
0
⟶
D
s
+
ρ
-
.
As similar toR
d
u, one also could get the ratio and relation(31)
R
d
s
≡
B
B
¯
∗
0
⟶
D
+
D
∗
-
,
D
+
D
s
∗
-
,
D
+
K
∗
-
,
D
+
ρ
-
B
B
¯
s
∗
0
⟶
D
s
+
D
∗
-
,
D
s
+
D
s
∗
-
,
D
s
+
K
∗
-
,
D
s
+
ρ
-
≃
Γ
B
s
∗
0
⟶
B
s
0
γ
Γ
B
∗
0
⟶
B
0
γ
≈
theo.
2
,
which is also satisfied in our numerical evaluation. So, it is obvious that such ratios,R
d
u and R
d
s, are useful for probing τ
B
∗
0
/
τ
B
∗
± and τ
B
∗
0
/
τ
B
s
∗
0, respectively, and further testing the theoretical predictions of Γ
(
B
∗
+
→
B
+
γ
)
/
Γ
(
B
∗
0
→
B
0
γ
) and Γ
(
B
s
∗
0
→
B
s
0
γ
)
/
Γ
(
B
∗
0
→
B
0
γ
) in various models, such as the results in [29–35].
(4)
Besides branching fraction, the polarization fractionsf
L
,
∥
,
⊥ are also important observables. For the potentially detectable decay modes with branching fractions ≳
1
0
-
9, our numerical results of f
L
,
∥ are summarized in Table 3. For the helicity amplitudes A
λ, the formal hierarchy pattern(32)
A
0
:
A
-
:
A
+
=
1
:
Λ
QCD
m
b
:
Λ
QCD
m
b
2
is naively expected. Hence,B
¯
∗
→
P
V decays are generally dominated by the longitudinal polarization state and satisfy f
L
~
1
-
1
/
m
B
∗
2 [37]. For B
¯
∗
→
D
V
(
V
=
K
∗
,
ρ
) decays, in the heavy quark limit, the helicity amplitudes H
λ given by (9) could be simplified as(33)
H
P
V
0
≃
i
f
V
m
B
∗
-
m
D
m
B
∗
+
m
D
2
2
m
B
∗
A
1
+
m
B
∗
+
m
D
m
B
∗
-
m
D
2
2
m
B
∗
A
2
,
H
P
V
±
≃
i
f
V
m
B
∗
-
m
D
m
B
∗
+
m
D
2
2
m
B
∗
A
1
∓
m
B
∗
+
m
D
m
B
∗
-
m
D
2
2
m
B
∗
V
·
2
m
B
∗
m
V
m
B
∗
2
-
m
D
2
.
The transversity amplitudes could be gotten easily through (19). Obviously, due to the helicity suppression factor 2
m
B
∗
m
V
/
(
m
B
∗
2
-
m
D
2
)
~
2
m
V
/
m
B
∗
~
Λ
QCD
/
m
b, the relation of (32) is roughly fulfilled. As a result, the longitudinal polarization fractions of B
¯
∗
→
D
K
∗ and D
ρ decays are very large (see Table 3 for numerical results).
It should be noted that the above analyses and (33) are based on the case of m
V
2
≪
m
B
∗
2 and thus possibly no longer satisfied by B
¯
∗
→
D
D
¯
∗ decays because of the unnegligible vector mass m
D
∗. In fact, for the B
¯
∗
→
D
D
¯
∗ decays, (9) are simplified as(34)
H
P
V
0
≃
i
f
D
∗
m
B
∗
+
m
D
m
B
∗
2
A
1
+
m
B
∗
2
m
B
∗
+
m
D
m
B
∗
2
-
4
m
D
∗
2
A
2
,
(35)
H
P
V
±
≃
i
f
D
∗
m
B
∗
+
m
D
m
B
∗
2
A
1
∓
m
B
∗
2
m
B
∗
+
m
D
m
B
∗
m
B
∗
2
-
4
m
D
∗
2
V
·
2
m
D
∗
m
B
∗
,
in which, due to(
m
D
∗
2
-
m
D
2
)
≪
m
B
∗
2, the approximation x
=
m
B
∗
2
-
m
D
2
+
m
D
∗
2
/
2
m
B
∗
m
D
∗
≃
m
B
∗
/
2
m
D
∗ is used. Because the so-called helicity suppression factor 2
m
D
∗
/
m
B
∗
~
0.8 is not small, which is different from the case of B
¯
∗
→
D
V decays, it could be easily found that the relation of (32) does not follow. Further considering that H
P
V
± are dominated by the term of A
1 in (35) due to its large coefficient, the relation f
L
(
D
D
¯
∗
)
~
f
∥
(
D
D
¯
∗
)
≫
f
⊥
(
D
D
¯
∗
) could be easily gotten. Above analyses and findings are confirmed by our numerical results in Table 3, which will be tested by future experiments.
(5)
As known, there are many interesting phenomena inB meson decays, so it is worthy to explore the possible correlation between B and B
∗ decays. Taking B
¯
∗
0
→
D
+
ρ
- and B
¯
0
→
D
∗
+
ρ
- decays as example, we find that the expressions of their helicity amplitudes (the former one has been given by (33)) are similar to each other except for the replacements B
¯
∗
↔
B
¯ and D
↔
D
∗ everywhere in (33). As a result, our analyses in item (4) are roughly suitable for B
¯
0
→
D
∗
+
ρ
- decay, and the relation (36)
f
L
,
∥
B
¯
∗
0
⟶
D
+
ρ
-
≃
f
L
,
∥
B
¯
0
⟶
D
∗
+
ρ
-is generally expected. Interestingly, our prediction f
L
NF
(
B
¯
∗
0
→
D
+
ρ
-
)
=
(
88
±
1
)
% is consistent with the result f
L
W
S
B
(
B
¯
0
→
D
∗
+
ρ
-
)
=
87
% [38], which is in a good agreement with the experimental data f
L
exp.
(
B
¯
0
→
D
∗
+
ρ
-
)
=
(
88.5
±
1.6
±
1.2
)
% [39]. The relation equation (36) follows. In addition, the similar correlation as (36) also exists in the other B
∗ and corresponding B decays.Table 2
The CP-averaged branching fractions of nonleptonicB
∗ weak decays.
Decay modes
Class
CKM factors
B
B
∗
-
→
D
0
D
∗
-
T, P, and Pew
λ
3
(
3.9
-
0.2
-
1.1
-
0.5
+
0.2
+
1.3
+
0.7
)
×
10
-
10
B
-
∗
0
→
D
+
D
∗
-
T, P, and Pew
λ
3
(
1.2
-
0.1
-
0.4
-
0.1
+
0.1
+
0.4
+
0.2
)
×
10
-
9
B
∗
-
→
D
0
D
s
∗
-
T, P, and Pew
λ
2
(
1.1
-
0.1
-
0.3
-
0.1
+
0.1
+
0.4
+
0.2
)
×
10
-
8
B
-
∗
0
→
D
+
D
s
∗
-
T, P, and Pew
λ
2
(
3.4
-
0.2
-
1.0
-
0.4
+
0.2
+
1.1
+
0.5
)
×
10
-
8
B
-
s
∗
0
→
D
s
+
D
∗
-
T, P, and Pew
λ
3
(
2.3
-
0.1
-
0.7
-
0.5
+
0.1
+
0.8
+
0.8
)
×
10
-
9
B
-
s
∗
0
→
D
s
+
D
s
∗
-
T, P, and Pew
λ
2
(
6.4
-
0.4
-
1.9
-
1.3
+
0.3
+
2.1
+
2.1
)
×
10
-
8
B
-
∗
0
→
D
+
K
∗
-
T
λ
3
(
7.6
-
0.4
-
1.7
-
0.9
+
0.4
+
1.9
+
1.2
)
×
10
-
10
B
-
s
∗
0
→
D
s
+
K
∗
-
T
λ
3
(
1.5
-
0.1
-
0.3
-
0.3
+
0.1
+
0.4
+
0.5
)
×
10
-
9
B
-
∗
0
→
D
+
ρ
-
T
λ
2
(
1.3
-
0.1
-
0.3
-
0.2
+
0.1
+
0.3
+
0.2
)
×
10
-
8
B
-
s
∗
0
→
D
s
+
ρ
-
T
λ
2
(
2.6
-
0.1
-
0.6
-
0.5
+
0.1
+
0.6
+
0.9
)
×
10
-
8
B
∗
-
→
π
-
D
-
∗
0
C
λ
4
(
3.1
-
0.2
-
0.6
-
0.4
+
0.2
+
0.8
+
0.6
)
×
10
-
14
B
∗
-
→
π
0
D
∗
-
T
λ
4
(
4.6
-
0.4
-
1.2
-
0.6
+
0.4
+
1.4
+
0.9
)
×
10
-
13
B
-
∗
0
→
π
+
D
∗
-
T
λ
4
(
2.9
-
0.2
-
0.8
-
0.3
+
0.2
+
0.9
+
0.5
)
×
10
-
12
B
-
∗
0
→
π
0
D
∗
0
C
λ
2
(
1.2
-
0.1
-
0.3
-
0.1
+
0.1
+
0.4
+
0.2
)
×
10
-
10
B
-
∗
0
→
π
0
D
-
∗
0
C
λ
4
(
4.9
-
0.3
-
1.2
-
0.6
+
0.3
+
1.4
+
0.8
)
×
10
-
14
B
∗
-
→
π
0
D
s
∗
-
T
λ
3
(
1.3
-
0.1
-
0.3
-
0.2
+
0.1
+
0.4
+
0.2
)
×
10
-
11
B
-
∗
0
→
π
+
D
s
∗
-
T
λ
3
(
8.1
-
0.7
-
2.2
-
1.0
+
0.6
+
2.5
+
1.3
)
×
10
-
11
B
∗
-
→
K
-
D
-
∗
0
C
λ
3
(
7.4
-
0.6
-
1.9
-
1.0
+
0.6
+
2.1
+
1.4
)
×
10
-
13
B
-
∗
0
→
K
-
0
D
∗
0
C
λ
3
(
1.7
-
0.1
-
0.4
-
0.2
+
0.1
+
0.5
+
0.3
)
×
10
-
11
B
-
∗
0
→
K
-
0
D
-
∗
0
C
λ
3
(
2.3
-
0.2
-
0.6
-
0.3
+
0.2
+
0.7
+
0.4
)
×
10
-
12
B
-
s
∗
0
→
K
+
D
∗
-
T
λ
4
(
4.3
-
0.4
-
1.1
-
0.9
+
0.3
+
1.2
+
1.4
)
×
10
-
12
B
-
s
∗
0
→
K
0
D
∗
0
C
λ
2
(
3.6
-
0.2
-
0.9
-
0.7
+
0.2
+
1.0
+
1.2
)
×
10
-
10
B
-
s
∗
0
→
K
0
D
-
∗
0
C
λ
4
(
1.4
-
0.1
-
0.3
-
0.3
+
0.1
+
0.4
+
0.5
)
×
10
-
13
B
-
s
∗
0
→
K
+
D
s
∗
-
T
λ
3
(
1.2
-
0.1
-
0.3
-
0.2
+
0.1
+
0.3
+
0.4
)
×
10
-
10Table 3
The polarization fractionsf
L and f
∥ (in the units of percent).
Decay modes
f
L
f
∥
B
∗
-
→
D
0
D
∗
-
54
-
2
+
2
40
-
2
+
2
B
-
∗
0
→
D
+
D
∗
-
54
-
2
+
2
40
-
2
+
2
B
∗
-
→
D
0
D
s
∗
-
52
-
2
+
1
43
-
2
+
2
B
-
∗
0
→
D
+
D
s
∗
-
52
-
2
+
1
43
-
2
+
2
B
-
s
∗
0
→
D
s
+
D
∗
-
54
-
2
+
2
40
-
2
+
2
B
-
s
∗
0
→
D
s
+
D
s
∗
-
52
-
2
+
2
42
-
2
+
2
B
-
∗
0
→
D
+
K
∗
-
85
-
1
+
1
13
-
1
+
1
B
-
s
∗
0
→
D
s
+
K
∗
-
85
-
1
+
1
13
-
1
+
1
B
-
∗
0
→
D
+
ρ
-
88
-
1
+
1
10
-
1
+
1
B
-
s
∗
0
→
D
s
+
ρ
-
88
-
1
+
1
10
-
1
+
1


## 4. Summary

In this paper, motivated by the experiments of heavy flavor physics at the running LHC and forthcoming SuperKEKB/Belle-II, the nonleptonicB
¯
∗
→
D
D
¯
∗, D
ρ, D
K
∗, π
D
∗, and K
D
∗ weak decay modes are evaluated with factorization approach, in which the transition form factors are calculated with the BSW model and the approximation Γ
tot
(
B
∗
)
≃
Γ
(
B
∗
→
B
γ
) is used to evaluate the branching fractions. It is found that (i) there are some obvious hierarchies among branching fractions, in which the B
¯
q
∗
→
D
q
D
¯
s
∗
- and D
q
ρ
- decays have large branching fractions ~
1
0
-
8 and hence should be sought for with priority at LHC and Belle-II experiments. (ii) With the implication of S
U
(
3
) (or U-spin) flavor symmetry, some useful ratios, R
d
u and R
d
s, are suggested to be verified experimentally. (iii) The B
¯
∗
0
→
D
K
∗ and D
ρ decays are dominated by the longitudinal polarization states; numerically, f
L  ~  [80%, 90%]. While the parallel polarization fractions of B
¯
∗
→
D
D
¯
∗ decays are comparable with the longitudinal ones; numerically, f
L
:
f
∥
≃
5
:
4. In addition, comparing with B
→
V
V decays, the relation f
L
,
∥
(
B
¯
∗
0
→
D
V
)
≃
f
L
,
∥
(
B
¯
0
→
D
V
) is generally expected. These results and findings are waiting for confirmation from future LHC and Belle-II experiments.



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*Source: 767523-2015-11-03.xml*