# Υ

Authors: Junfeng Sun; Yueling Yang; Qin Chang; Gongru Lu; Jinshu Huang
Journal: Advances in High Energy Physics
 (2016)

---

## Abstract

Besides the traditional strong and electromagnetic decay modes,Υ
(
n
S
) meson can also decay through the weak interactions within the standard model of elementary particle. With anticipation of copious Υ
(
n
S
) data samples at the running LHC and coming SuperKEKB experiments, the two-body nonleptonic bottom-changing Υ
(
n
S
)
→
B
c
⁎
π
,
B
c
⁎
K decays (n
=
1,2
,
3) are investigated with perturbative QCD approach firstly. The absolute branching ratios for Υ
(
n
S
)
→
B
c
⁎
π and B
c
⁎
K decays are estimated to reach up to about 1
0
-
10 and 1
0
-
11, respectively, which might possibly be measured by the future experiments.


---

## Body

## 1. Introduction

The upsilonΥ
(
n
S
) meson is the spin-triplet S-wave state of bottomonium (bound state consisting of bottom quark b and antibottom quark b
-) with well-established quantum number of I
G
J
P
C
=
0
-
1
-
- [1]. The characteristic narrow decay widths of Υ
(
n
S
) mesons for n
=
1, 2, and 3 provide insight into the study of strong interactions (see Table 1, and note that, for simplicity, Υ
(
n
S
) will denote Υ
(
1
S
), Υ
(
2
S
), and Υ
(
3
S
) mesons in the following content if not specified definitely). The mass of Υ
(
n
S
) meson is below B meson pair threshold. Υ
(
n
S
) meson decays into bottomed hadrons through strong and electromagnetic interactions are forbidden by the law of conservation of flavor number. The bottom-changing Υ
(
n
S
) decays can occur only via the weak interactions within the standard model, although with tiny incidence probability. Both constituent quarks of upsilons can decay individually, which provide an alternative system for investigating the weak decay of heavy-flavored hadrons. In this paper, we will study the nonleptonic Υ
(
n
S
)
→
B
c
∗
P (P
=
π and K) weak decays with perturbative QCD (pQCD) approach [2–4].Table 1
Summary of mass, decay width, on(off)-peak luminosity, and numbers ofΥ
(
n
S
).
Meson
Properties [1]
Luminosity (f
b
-
1) [5]
Numbers (1
0
6) [5]
Mass (MeV)
Width (keV)
Belle
BaBar
Belle
BaBar
Υ
(
1
S
)
9460.30
±
0.26
54.02
±
1.25
5.7 (1.8)
⋯
102
±
2
⋯
Υ
(
2
S
)
10023.26
±
0.31
31.98
±
2.63
24.9 (1.7)
13.6 (1.4)
158
±
4
98.3
±
0.9
Υ
(
3
S
)
10355.2
±
0.5
20.32
±
1.85
2.9 (0.2)
28.0 (2.6)
11
±
0.3
121.3
±
1.2Experimentally,(
1
) over 1
0
8
Υ
(
n
S
) data samples have been accumulated at Belle and BaBar experiments [5]. More and more upsilon data samples will be collected at the running hadron collider LHC and the forthcoming e
+
e
- collider SuperKEKB (the SuperKEKB has started commissioning test run (http://www.kek.jp/en/NewsRoom/Release/)). There seems to exist a realistic possibility to explore Υ
(
n
S
) weak decay at future experiments. (
2
) Signals of Υ
(
n
S
)
→
B
c
∗
π
,
B
c
∗
K decays should be easily distinguished with “charge tag” technique, due to the facts that the back-to-back final states with different electric charges have definite momentum and energy in the rest frame of Υ
(
n
S
) meson. (
3
)  B
c
∗ meson has not been observed experimentally by now. B
c
∗ meson production via the strong interaction is suppressed due to the simultaneous presence of two heavy quarks with different flavors and higher order in QCD coupling constant α
s. Υ
(
n
S
)
→
B
c
∗
π, B
c
∗
K decays provide a novel pattern to study B
c
∗ meson production. The identification of a single explicitly flavored B
c
∗ meson could be used as an effective selection criterion to detect upsilon weak decays. Moreover, the radiative decay of B
c
∗ meson provides a useful extra signal and a powerful constraint (the investigation on the radiative decay of B
c
∗ meson can be found in, e.g., [6], with QCD sum rules). Of course, any discernible evidences of an anomalous production rate of single bottomed meson from upsilon decays might be a hint of new physics.Theoretically, many attractive QCD-inspired methods have been developed recently to describe the exclusive nonleptonic decay of heavy-flavored mesons, such as the pQCD approach [2–4], the QCD factorization approach [7–9], and soft and collinear effective theory [10–13], and have been applied widely to vindicate measurements on B meson decays. The upsilon weak decay permits one to further constrain parameters obtained from B meson decay, and cross comparisons provide an opportunity to test various phenomenological models. The upsilon weak decay possesses a unique structure due to the Cabibbo-Kobayashi-Maskawa (CKM) matrix properties which predicts that the channels with one B
c
(
∗
) meson are dominant. Υ
(
n
S
)
→
B
c
∗
P decay belongs to the favorable b
→
c transition, which should, in principle, have relatively large branching ratio among upsilon weak decays. However, there is still no theoretical study devoted to Υ
(
n
S
)
→
B
c
∗
P decay for the moment. In this paper, we will present a phenomenological investigation on Υ
(
n
S
)
→
B
c
∗
P weak decay with the pQCD approach to supply a ready reference for the future experiments.This paper is organized as follows. Section2 focuses on theoretical framework and decay amplitudes for Υ
(
n
S
)
→
B
c
∗
π
,
B
c
∗
K weak decays. Section 3 is devoted to numerical results and discussion. The last section is a summary.


## 2. Theoretical Framework

### 2.1. The Effective Hamiltonian

Theoretically,Υ
(
n
S
)
→
B
c
∗
π, B
c
∗
K weak decays are described by an effective bottom-changing Hamiltonian based on operator product expansion [19]: (1)
H
e
f
f
=
G
F
2
∑
q
=
d
,
s
V
c
b
V
u
q
∗
C
1
μ
O
1
μ
+
C
2
μ
O
2
μ
+
h.c.
,where G
F
≃
1.166
×
1
0
-
5
G
e
V
-
2 [1] is the Fermi coupling constant; the CKM factors V
c
b
V
u
d
∗ and V
c
b
V
u
s
∗ correspond to Υ
(
n
S
)
→
B
c
∗
π and B
c
∗
K decays, respectively; with the Wolfenstein parameterization, the CKM factors are expanded as a power series in a small Wolfenstein parameter λ
~
0.2 [1]: (2)
V
c
b
V
u
d
∗
=
A
λ
2
-
1
2
A
λ
4
-
1
8
A
λ
6
+
O
λ
7
,
V
c
b
V
u
s
∗
=
A
λ
3
+
O
λ
7
.The local tree operators Q
1,2 are defined as (3)
O
1
=
c
-
α
γ
μ
1
-
γ
5
b
α
q
-
β
γ
μ
1
-
γ
5
u
β
,
O
2
=
c
-
α
γ
μ
1
-
γ
5
b
β
q
-
β
γ
μ
1
-
γ
5
u
α
,where α and β are color indices and the sum over repeated indices is understood.The scaleμ factorizes physics contributions into short- and long-distance dynamics. The Wilson coefficients C
i
(
μ
) summarize the physics contributions at scale higher than μ and are calculable with the renormalization group improved perturbation theory. The hadronic matrix elements (HME), where the local operators are inserted between initial and final hadron states, embrace the physics contributions below scale of μ. To obtain decay amplitudes, the remaining work is to calculate HME properly by separating from perturbative and nonperturbative contributions.


### 2.2. Hadronic Matrix Elements

Based on Lepage-Brodsky approach for exclusive processes [20], HME is commonly expressed as a convolution integral of hard scattering subamplitudes containing perturbative contributions with universal wave functions reflecting nonperturbative contributions. In order to effectively regulate endpoint singularities and provide a naturally dynamical cutoff on nonperturbative contributions, transverse momentum of valence quarks is retained and the Sudakov factor is introduced within the pQCD framework [2–4]. Phenomenologically, pQCD’s decay amplitude could be divided into three parts: the Wilson coefficients C
i incorporating the hard contributions above typical scale of t, process-dependent rescattering subamplitudes T accounting for the heavy quark decay, and wave functions Φ of all participating hadrons, which is expressed as (4)
∫
d
k
C
i
t
T
t
,
k
Φ
k
e
-
S
,where k is the momentum of valence quarks and e
-
S is the Sudakov factor.


### 2.3. Kinematic Variables

The light cone kinematic variables inΥ
(
n
S
) rest frame are defined as follows: (5)
p
Υ
=
p
1
=
m
1
2
1,1
,
0
,
p
B
c
∗
=
p
2
=
p
2
+
,
p
2
-
,
0
,
p
3
=
p
3
-
,
p
3
+
,
0
,
p
i
±
=
E
i
±
p
2
,
k
i
=
x
i
p
i
+
0,0
,
k
→
i
⊥
,
ϵ
1
∥
=
p
1
m
1
-
m
1
p
1
·
n
+
n
+
,
ϵ
2
∥
=
p
2
m
2
-
m
2
p
2
·
n
-
n
-
,
ϵ
1,2
⊥
=
0,0
,
1
→
,
n
+
=
1,0
,
0
,
n
-
=
0,1
,
0
,
s
=
2
p
2
·
p
3
,
t
=
2
p
1
·
p
2
=
2
m
1
E
2
,
u
=
2
p
1
·
p
3
=
2
m
1
E
3
,
p
=
m
1
2
-
m
2
+
m
3
2
m
1
2
-
m
2
-
m
3
2
2
m
1
,where x
i and k
→
i
⊥ are the longitudinal momentum fraction and transverse momentum of valence quarks, respectively; ϵ
i
∥ and ϵ
i
⊥ are the longitudinal and transverse polarization vectors, respectively, and satisfy relations ϵ
i
2
=
-
1 and ϵ
i
·
p
i
=
0; the subscript i on variables p
i, E
i, m
i, and ϵ
i corresponds to participating hadrons; namely, i
=
1 for Υ
(
n
S
) meson, i
=
2 for the recoiled B
c
∗ meson, and i
=
3 for the emitted pseudoscalar meson; n
+ and n
- are positive and negative null vectors, respectively; s, t, and u are the Lorentz-invariant variables; p is the common momentum of final states. The notation of momentum is displayed in Figure 2(a).


### 2.4. Wave Functions

With the notation in [16, 21], wave functions are defined as (6)
0
b
i
z
b
-
j
0
Υ
p
1
,
ϵ
1
∥
=
f
Υ
4
∫
d
k
1
e
-
i
k
1
·
z
ϵ
1
∥
m
1
Φ
Υ
v
k
1
-
p
1
Φ
Υ
t
k
1
j
i
,
0
b
i
z
b
-
j
0
Υ
p
1
,
ϵ
1
⊥
=
f
Υ
4
∫
d
k
1
e
-
i
k
1
·
z
ϵ
1
⊥
m
1
Φ
Υ
V
k
1
-
p
1
Φ
Υ
T
k
1
j
i
,
B
c
∗
p
2
,
ϵ
2
∥
c
-
i
z
b
j
0
0
=
f
B
c
∗
4
∫
0
1
d
k
3
e
i
k
2
·
z
ϵ
2
∥
m
2
Φ
B
c
∗
v
k
2
+
p
2
Φ
B
c
∗
t
k
2
j
i
,
B
c
∗
p
2
,
ϵ
2
⊥
c
-
i
z
b
j
0
0
=
f
B
c
∗
4
∫
0
1
d
k
2
e
i
k
2
·
z
ϵ
2
⊥
m
2
Φ
B
c
∗
V
k
2
+
p
2
Φ
B
c
∗
T
k
2
j
i
,
P
p
3
u
i
0
q
-
j
z
0
=
i
f
P
4
∫
d
k
3
e
i
k
3
·
z
γ
5
p
3
Φ
P
a
k
3
+
μ
P
Φ
P
p
k
3
+
μ
P
n
-
n
+
-
1
Φ
P
t
k
3
j
i
,where f
Υ, f
B
c
∗, and f
P are decay constants of Υ
(
n
S
), B
c
∗, and P mesons, respectively.Considering mass relations ofm
Υ
(
n
S
)
≃
2
m
b and m
B
c
∗
≃
m
b
+
m
c, it might assume that the motion of heavy valence quarks in Υ
(
n
S
) and B
c
∗ mesons is nearly nonrelativistic. The wave functions of Υ
(
n
S
) and B
c
∗ mesons could be approximately described with nonrelativistic quantum chromodynamics (NRQCD) [22–24] and time-independent Schrödinger equation. For an isotropic harmonic oscillator potential, the eigenfunctions of stationary state with quantum numbers n
L are written as [17] (7)
ϕ
1
S
k
→
~
e
-
k
→
2
/
2
β
2
,
ϕ
2
S
k
→
~
e
-
k
→
2
/
2
β
2
2
k
→
2
-
3
β
2
,
ϕ
3
S
k
→
~
e
-
k
→
2
/
2
β
2
4
k
→
4
-
20
k
→
2
β
2
+
15
β
4
,where parameter β determines the average transverse momentum; that is, 〈
n
S
|
k
⊥
2
|
n
S
〉
~
β
2. Employing the substitution ansatz [25], (8)
k
→
2
⟶
1
4
∑
i
k
→
i
⊥
2
+
m
q
i
2
x
i
,where x
i and m
q
i are the longitudinal momentum fraction and mass of valence quark, respectively; then integrating out k
→
⊥ and combining with their asymptotic forms, the distribution amplitudes (DAs) for Υ
(
n
S
) and B
c
∗ mesons can be written as [17] (9)
ϕ
Υ
1
S
v
,
T
x
=
A
x
x
-
exp
⁡
-
m
b
2
8
β
1
2
x
x
-
,
ϕ
Υ
1
S
t
x
=
B
t
2
exp
⁡
-
m
b
2
8
β
1
2
x
x
-
,
ϕ
Υ
1
S
V
x
=
C
1
+
t
2
exp
⁡
-
m
b
2
8
β
1
2
x
x
-
,
ϕ
Υ
2
S
v
,
t
,
V
,
T
x
=
D
ϕ
Υ
1
S
v
,
t
,
V
,
T
x
1
+
m
b
2
2
β
1
2
x
x
-
,
ϕ
Υ
3
S
v
,
t
,
V
,
T
x
=
E
ϕ
Υ
1
S
v
,
t
,
V
,
T
x
1
-
m
b
2
2
β
1
2
x
x
-
2
+
6
,
ϕ
B
c
∗
v
,
T
x
=
F
x
x
-
exp
⁡
-
x
-
m
c
2
+
x
m
b
2
8
β
2
2
x
x
-
,
ϕ
B
c
∗
t
x
=
G
t
2
exp
⁡
-
x
-
m
c
2
+
x
m
b
2
8
β
2
2
x
x
-
,
ϕ
B
c
∗
V
x
=
H
1
-
t
2
exp
⁡
-
x
-
m
c
2
+
x
m
b
2
8
β
2
2
x
x
-
,where x
-
=
1
-
x; t
=
x
-
x
-. According to NRQCD power counting rules [22], β
i
≃
ξ
i
α
s
(
ξ
i
) with ξ
i
=
m
i
/
2 and QCD coupling constant α
s. The exponential function represents k
⊥ distribution. Parameters of A, B, C, D, E, F, G, and H are normalization coefficients satisfying the conditions (10)
∫
0
1
d
x
ϕ
Υ
n
S
i
x
=
∫
0
1
d
x
ϕ
B
c
∗
i
x
=
1
f
o
r
i
=
v
,
t
,
V
,
T
.The shape lines of normalized DAs forΥ
(
n
S
) and B
c
∗ mesons are shown in Figure 1. It is clearly seen that (
1
) DAs for Υ
(
n
S
) and B
c
∗ mesons fall quickly down to zero at endpoint x, x
-
→
0 due to suppression from exponential functions; (
2
) DAs for Υ
(
n
S
) meson are symmetric under the interchange of momentum fractions x
↔
x
-, and DAs for B
c
∗ meson are basically consistent with the feature that valence quarks share momentum fractions according to their masses.Figure 1
The normalized distribution amplitudes forΥ
(
n
S
) and B
c
∗ mesons.
(a)
(b)
(c)Figure 2
Feynman diagrams forΥ
(
n
S
)
→
B
c
∗
π decay with the pQCD approach, including factorizable emission diagrams (a, b) and nonfactorizable emission diagrams (c, d).
(a)
(b)
(c)
(d)Our study shows that only the leading twist (twist-2) DAs of the emitted light pseudoscalar mesonP are involved in decay amplitudes (see Appendix). The twist-2 DAs have the expansion [16],(11)
ϕ
P
a
x
=
6
x
x
-
∑
i
=
0
a
i
C
i
3
/
2
t
,and are normalized as (12)
∫
0
1
ϕ
P
a
x
d
x
=
1
,where C
i
3
/
2
(
t
) are Gegenbauer polynomials: (13)
C
0
3
/
2
t
=
1
,
C
1
3
/
2
t
=
3
t
,
C
2
3
/
2
t
=
3
2
5
t
2
-
1
,
⋮and each term corresponds to a nonperturbative Gegenbauer moment a
i; note that a
0
=
1 due to the normalization condition equation (12); G-parity invariance of the pion DAs requires Gegenbauer moment a
i
=
0 for i
=
1
,
3
,
5
,
….


### 2.5. Decay Amplitudes

The Feynman diagrams forΥ
(
n
S
)
→
B
c
∗
π weak decay are shown in Figure 2. There are two types. One is factorizable emission topology where gluon attaches to quarks in the same meson, and the other is nonfactorizable emission topology where gluon connects to quarks between different mesons.With the pQCD master formula equation (4), the amplitude for Υ
(
n
S
)
→
B
c
∗
P decay can be expressed as [26] (14)
A
Υ
n
S
⟶
B
c
∗
P
=
A
L
ϵ
1
∥
,
ϵ
2
∥
+
A
N
ϵ
1
⊥
,
ϵ
2
⊥
+
i
A
T
ϵ
μ
v
α
β
ϵ
1
μ
ϵ
2
v
p
1
α
p
2
β
,which is conventionally written as the helicity amplitudes [26]: (15)
A
0
=
-
C
A
∑
j
A
L
j
ϵ
1
∥
,
ϵ
2
∥
,
A
∥
=
2
C
A
∑
j
A
N
j
ϵ
1
⊥
,
ϵ
2
⊥
,
A
⊥
=
2
C
A
m
1
p
∑
j
A
T
j
,
C
A
=
i
V
c
b
V
u
q
∗
G
F
2
C
F
N
c
π
f
Υ
f
B
c
∗
f
P
,where C
F
=
4
/
3 and the color number N
c
=
3; the subscript i on A
i
j corresponds to three different helicity amplitudes; that is, i
=
L
,
N
,
T; the superscript j on A
i
j denotes indices of Figure 2. The explicit expressions of building blocks A
i
j are collected in Appendix.


## 2.1. The Effective Hamiltonian

Theoretically,Υ
(
n
S
)
→
B
c
∗
π, B
c
∗
K weak decays are described by an effective bottom-changing Hamiltonian based on operator product expansion [19]: (1)
H
e
f
f
=
G
F
2
∑
q
=
d
,
s
V
c
b
V
u
q
∗
C
1
μ
O
1
μ
+
C
2
μ
O
2
μ
+
h.c.
,where G
F
≃
1.166
×
1
0
-
5
G
e
V
-
2 [1] is the Fermi coupling constant; the CKM factors V
c
b
V
u
d
∗ and V
c
b
V
u
s
∗ correspond to Υ
(
n
S
)
→
B
c
∗
π and B
c
∗
K decays, respectively; with the Wolfenstein parameterization, the CKM factors are expanded as a power series in a small Wolfenstein parameter λ
~
0.2 [1]: (2)
V
c
b
V
u
d
∗
=
A
λ
2
-
1
2
A
λ
4
-
1
8
A
λ
6
+
O
λ
7
,
V
c
b
V
u
s
∗
=
A
λ
3
+
O
λ
7
.The local tree operators Q
1,2 are defined as (3)
O
1
=
c
-
α
γ
μ
1
-
γ
5
b
α
q
-
β
γ
μ
1
-
γ
5
u
β
,
O
2
=
c
-
α
γ
μ
1
-
γ
5
b
β
q
-
β
γ
μ
1
-
γ
5
u
α
,where α and β are color indices and the sum over repeated indices is understood.The scaleμ factorizes physics contributions into short- and long-distance dynamics. The Wilson coefficients C
i
(
μ
) summarize the physics contributions at scale higher than μ and are calculable with the renormalization group improved perturbation theory. The hadronic matrix elements (HME), where the local operators are inserted between initial and final hadron states, embrace the physics contributions below scale of μ. To obtain decay amplitudes, the remaining work is to calculate HME properly by separating from perturbative and nonperturbative contributions.


## 2.2. Hadronic Matrix Elements

Based on Lepage-Brodsky approach for exclusive processes [20], HME is commonly expressed as a convolution integral of hard scattering subamplitudes containing perturbative contributions with universal wave functions reflecting nonperturbative contributions. In order to effectively regulate endpoint singularities and provide a naturally dynamical cutoff on nonperturbative contributions, transverse momentum of valence quarks is retained and the Sudakov factor is introduced within the pQCD framework [2–4]. Phenomenologically, pQCD’s decay amplitude could be divided into three parts: the Wilson coefficients C
i incorporating the hard contributions above typical scale of t, process-dependent rescattering subamplitudes T accounting for the heavy quark decay, and wave functions Φ of all participating hadrons, which is expressed as (4)
∫
d
k
C
i
t
T
t
,
k
Φ
k
e
-
S
,where k is the momentum of valence quarks and e
-
S is the Sudakov factor.


## 2.3. Kinematic Variables

The light cone kinematic variables inΥ
(
n
S
) rest frame are defined as follows: (5)
p
Υ
=
p
1
=
m
1
2
1,1
,
0
,
p
B
c
∗
=
p
2
=
p
2
+
,
p
2
-
,
0
,
p
3
=
p
3
-
,
p
3
+
,
0
,
p
i
±
=
E
i
±
p
2
,
k
i
=
x
i
p
i
+
0,0
,
k
→
i
⊥
,
ϵ
1
∥
=
p
1
m
1
-
m
1
p
1
·
n
+
n
+
,
ϵ
2
∥
=
p
2
m
2
-
m
2
p
2
·
n
-
n
-
,
ϵ
1,2
⊥
=
0,0
,
1
→
,
n
+
=
1,0
,
0
,
n
-
=
0,1
,
0
,
s
=
2
p
2
·
p
3
,
t
=
2
p
1
·
p
2
=
2
m
1
E
2
,
u
=
2
p
1
·
p
3
=
2
m
1
E
3
,
p
=
m
1
2
-
m
2
+
m
3
2
m
1
2
-
m
2
-
m
3
2
2
m
1
,where x
i and k
→
i
⊥ are the longitudinal momentum fraction and transverse momentum of valence quarks, respectively; ϵ
i
∥ and ϵ
i
⊥ are the longitudinal and transverse polarization vectors, respectively, and satisfy relations ϵ
i
2
=
-
1 and ϵ
i
·
p
i
=
0; the subscript i on variables p
i, E
i, m
i, and ϵ
i corresponds to participating hadrons; namely, i
=
1 for Υ
(
n
S
) meson, i
=
2 for the recoiled B
c
∗ meson, and i
=
3 for the emitted pseudoscalar meson; n
+ and n
- are positive and negative null vectors, respectively; s, t, and u are the Lorentz-invariant variables; p is the common momentum of final states. The notation of momentum is displayed in Figure 2(a).


## 2.4. Wave Functions

With the notation in [16, 21], wave functions are defined as (6)
0
b
i
z
b
-
j
0
Υ
p
1
,
ϵ
1
∥
=
f
Υ
4
∫
d
k
1
e
-
i
k
1
·
z
ϵ
1
∥
m
1
Φ
Υ
v
k
1
-
p
1
Φ
Υ
t
k
1
j
i
,
0
b
i
z
b
-
j
0
Υ
p
1
,
ϵ
1
⊥
=
f
Υ
4
∫
d
k
1
e
-
i
k
1
·
z
ϵ
1
⊥
m
1
Φ
Υ
V
k
1
-
p
1
Φ
Υ
T
k
1
j
i
,
B
c
∗
p
2
,
ϵ
2
∥
c
-
i
z
b
j
0
0
=
f
B
c
∗
4
∫
0
1
d
k
3
e
i
k
2
·
z
ϵ
2
∥
m
2
Φ
B
c
∗
v
k
2
+
p
2
Φ
B
c
∗
t
k
2
j
i
,
B
c
∗
p
2
,
ϵ
2
⊥
c
-
i
z
b
j
0
0
=
f
B
c
∗
4
∫
0
1
d
k
2
e
i
k
2
·
z
ϵ
2
⊥
m
2
Φ
B
c
∗
V
k
2
+
p
2
Φ
B
c
∗
T
k
2
j
i
,
P
p
3
u
i
0
q
-
j
z
0
=
i
f
P
4
∫
d
k
3
e
i
k
3
·
z
γ
5
p
3
Φ
P
a
k
3
+
μ
P
Φ
P
p
k
3
+
μ
P
n
-
n
+
-
1
Φ
P
t
k
3
j
i
,where f
Υ, f
B
c
∗, and f
P are decay constants of Υ
(
n
S
), B
c
∗, and P mesons, respectively.Considering mass relations ofm
Υ
(
n
S
)
≃
2
m
b and m
B
c
∗
≃
m
b
+
m
c, it might assume that the motion of heavy valence quarks in Υ
(
n
S
) and B
c
∗ mesons is nearly nonrelativistic. The wave functions of Υ
(
n
S
) and B
c
∗ mesons could be approximately described with nonrelativistic quantum chromodynamics (NRQCD) [22–24] and time-independent Schrödinger equation. For an isotropic harmonic oscillator potential, the eigenfunctions of stationary state with quantum numbers n
L are written as [17] (7)
ϕ
1
S
k
→
~
e
-
k
→
2
/
2
β
2
,
ϕ
2
S
k
→
~
e
-
k
→
2
/
2
β
2
2
k
→
2
-
3
β
2
,
ϕ
3
S
k
→
~
e
-
k
→
2
/
2
β
2
4
k
→
4
-
20
k
→
2
β
2
+
15
β
4
,where parameter β determines the average transverse momentum; that is, 〈
n
S
|
k
⊥
2
|
n
S
〉
~
β
2. Employing the substitution ansatz [25], (8)
k
→
2
⟶
1
4
∑
i
k
→
i
⊥
2
+
m
q
i
2
x
i
,where x
i and m
q
i are the longitudinal momentum fraction and mass of valence quark, respectively; then integrating out k
→
⊥ and combining with their asymptotic forms, the distribution amplitudes (DAs) for Υ
(
n
S
) and B
c
∗ mesons can be written as [17] (9)
ϕ
Υ
1
S
v
,
T
x
=
A
x
x
-
exp
⁡
-
m
b
2
8
β
1
2
x
x
-
,
ϕ
Υ
1
S
t
x
=
B
t
2
exp
⁡
-
m
b
2
8
β
1
2
x
x
-
,
ϕ
Υ
1
S
V
x
=
C
1
+
t
2
exp
⁡
-
m
b
2
8
β
1
2
x
x
-
,
ϕ
Υ
2
S
v
,
t
,
V
,
T
x
=
D
ϕ
Υ
1
S
v
,
t
,
V
,
T
x
1
+
m
b
2
2
β
1
2
x
x
-
,
ϕ
Υ
3
S
v
,
t
,
V
,
T
x
=
E
ϕ
Υ
1
S
v
,
t
,
V
,
T
x
1
-
m
b
2
2
β
1
2
x
x
-
2
+
6
,
ϕ
B
c
∗
v
,
T
x
=
F
x
x
-
exp
⁡
-
x
-
m
c
2
+
x
m
b
2
8
β
2
2
x
x
-
,
ϕ
B
c
∗
t
x
=
G
t
2
exp
⁡
-
x
-
m
c
2
+
x
m
b
2
8
β
2
2
x
x
-
,
ϕ
B
c
∗
V
x
=
H
1
-
t
2
exp
⁡
-
x
-
m
c
2
+
x
m
b
2
8
β
2
2
x
x
-
,where x
-
=
1
-
x; t
=
x
-
x
-. According to NRQCD power counting rules [22], β
i
≃
ξ
i
α
s
(
ξ
i
) with ξ
i
=
m
i
/
2 and QCD coupling constant α
s. The exponential function represents k
⊥ distribution. Parameters of A, B, C, D, E, F, G, and H are normalization coefficients satisfying the conditions (10)
∫
0
1
d
x
ϕ
Υ
n
S
i
x
=
∫
0
1
d
x
ϕ
B
c
∗
i
x
=
1
f
o
r
i
=
v
,
t
,
V
,
T
.The shape lines of normalized DAs forΥ
(
n
S
) and B
c
∗ mesons are shown in Figure 1. It is clearly seen that (
1
) DAs for Υ
(
n
S
) and B
c
∗ mesons fall quickly down to zero at endpoint x, x
-
→
0 due to suppression from exponential functions; (
2
) DAs for Υ
(
n
S
) meson are symmetric under the interchange of momentum fractions x
↔
x
-, and DAs for B
c
∗ meson are basically consistent with the feature that valence quarks share momentum fractions according to their masses.Figure 1
The normalized distribution amplitudes forΥ
(
n
S
) and B
c
∗ mesons.
(a)
(b)
(c)Figure 2
Feynman diagrams forΥ
(
n
S
)
→
B
c
∗
π decay with the pQCD approach, including factorizable emission diagrams (a, b) and nonfactorizable emission diagrams (c, d).
(a)
(b)
(c)
(d)Our study shows that only the leading twist (twist-2) DAs of the emitted light pseudoscalar mesonP are involved in decay amplitudes (see Appendix). The twist-2 DAs have the expansion [16],(11)
ϕ
P
a
x
=
6
x
x
-
∑
i
=
0
a
i
C
i
3
/
2
t
,and are normalized as (12)
∫
0
1
ϕ
P
a
x
d
x
=
1
,where C
i
3
/
2
(
t
) are Gegenbauer polynomials: (13)
C
0
3
/
2
t
=
1
,
C
1
3
/
2
t
=
3
t
,
C
2
3
/
2
t
=
3
2
5
t
2
-
1
,
⋮and each term corresponds to a nonperturbative Gegenbauer moment a
i; note that a
0
=
1 due to the normalization condition equation (12); G-parity invariance of the pion DAs requires Gegenbauer moment a
i
=
0 for i
=
1
,
3
,
5
,
….


## 2.5. Decay Amplitudes

The Feynman diagrams forΥ
(
n
S
)
→
B
c
∗
π weak decay are shown in Figure 2. There are two types. One is factorizable emission topology where gluon attaches to quarks in the same meson, and the other is nonfactorizable emission topology where gluon connects to quarks between different mesons.With the pQCD master formula equation (4), the amplitude for Υ
(
n
S
)
→
B
c
∗
P decay can be expressed as [26] (14)
A
Υ
n
S
⟶
B
c
∗
P
=
A
L
ϵ
1
∥
,
ϵ
2
∥
+
A
N
ϵ
1
⊥
,
ϵ
2
⊥
+
i
A
T
ϵ
μ
v
α
β
ϵ
1
μ
ϵ
2
v
p
1
α
p
2
β
,which is conventionally written as the helicity amplitudes [26]: (15)
A
0
=
-
C
A
∑
j
A
L
j
ϵ
1
∥
,
ϵ
2
∥
,
A
∥
=
2
C
A
∑
j
A
N
j
ϵ
1
⊥
,
ϵ
2
⊥
,
A
⊥
=
2
C
A
m
1
p
∑
j
A
T
j
,
C
A
=
i
V
c
b
V
u
q
∗
G
F
2
C
F
N
c
π
f
Υ
f
B
c
∗
f
P
,where C
F
=
4
/
3 and the color number N
c
=
3; the subscript i on A
i
j corresponds to three different helicity amplitudes; that is, i
=
L
,
N
,
T; the superscript j on A
i
j denotes indices of Figure 2. The explicit expressions of building blocks A
i
j are collected in Appendix.


## 3. Numerical Results and Discussion

In the center-of-mass ofΥ
(
n
S
) meson, branching ratio B
r for Υ
(
n
S
)
→
B
c
∗
P decay is defined as (16)
B
r
=
1
12
π
p
m
Υ
2
Γ
Υ
A
0
2
+
A
∥
2
+
A
⊥
2
.The input parameters are listed in Tables1 and 2. If not specified explicitly, we will take their central values as the default inputs. Our numerical results are collected in Table 3, where the first uncertainty comes from scale (
1
±
0.1
)
t
i and the expression of t
i is given in (A.4) and (A.5); the second uncertainty is from mass of m
b and m
c; the third uncertainty is from hadronic parameters including decay constants and Gegenbauer moments; the fourth uncertainty is from CKM parameters. The followings are some comments:(1)
Branching ratio forΥ
(
n
S
)
→
B
c
∗
π decay is about O
(
1
0
-
10
) with pQCD approach, which is well within the measurement potential of LHC and SuperKEKB. For example, experimental studies have showed that production cross sections for Υ
(
n
S
) meson in p-p and p-Pb collisions are a few μ
b at the LHCb [27, 28] and ALICE [29, 30] detectors. Consequently, there will be more than 1
0
12
Υ
(
n
S
) data samples per a
b
-
1 data collected by the LHCb and ALICE, corresponding to a few hundreds of Υ
(
n
S
)
→
B
c
∗
π events. Branching ratio for Υ
(
n
S
)
→
B
c
∗
K decay, O
(
1
0
-
11
), is generally less than that for Υ
(
n
S
)
→
B
c
∗
π decay by one order of magnitude due to the CKM suppression, |
V
u
s
∗
/
V
u
d
∗
|
2
~
λ
2.
(2)
As it is well known, due to the large mass ofB
c
∗, the momentum transition in Υ
(
n
S
)
→
B
c
∗
P decay may be not large enough. One might naturally wonder whether the pQCD approach is applicable and whether the perturbative calculation is reliable. Therefore, it is necessary to check what percentage of the contributions comes from the perturbative region. The contributions to branching ratio for Υ
(
n
S
)
→
B
c
∗
π decay from different α
s
/
π region are shown in Figure 3. It can be clearly seen that more than 93% (97%) contributions come from α
s
/
π
≤
0.2 (0.3) region, implying that Υ
(
n
S
)
→
B
c
∗
π decay is computable with the pQCD approach. As the discussion in [2–4], there are many factors for this, for example, the choice of the typical scale, retaining the quark transverse moment and introducing the Sudakov factor to suppress the nonperturbative contributions, which deserve much attention and further investigation.
(3)
Because of the relations among massesm
Υ
(
3
S
)
>
m
Υ
(
2
S
)
>
m
Υ
(
1
S
) resulting in the fact that phase space increases with the radial quantum number n in addition to the relations among decay widths Γ
Υ
(
3
S
)
<
Γ
Υ
(
2
S
)
<
Γ
Υ
(
1
S
), in principle, there should be relations among branching ratios B
r
(
Υ
(
3
S
)
→
B
c
∗
P
)
>
B
r
(
Υ
(
2
S
)
→
B
c
∗
P
)
>
B
r
(
Υ
(
1
S
)
→
B
c
∗
P
) for the same pseudoscalar meson P. But the numerical results in Table 3 are beyond such expectation. Why? The reason is that the factor of p
/
m
Υ
(
n
S
)
2 in (16) has almost the same value for n
≤
3, so branching ratio is proportional to factor f
Υ
(
n
S
)
2
/
Γ
Υ
(
n
S
) with the maximal value f
Υ
(
1
S
)
2
/
Γ
Υ
(
1
S
) for n
≤
3. Besides, contributions from α
s
/
π
∈
[
0.2,0.3
] regions decrease with n (see Figure 3), which enhance the decay amplitudes.
(4)
Besides the uncertainties listed in Table3, other factors, such as the models of wave functions, contributions of higher order corrections to HME, and relativistic effects, deserve the dedicated study. Our results just provide an order of magnitude estimation.Table 2
The numerical values of input parameters.
The Wolfenstein parameters
A
=
0.81
4
-
0.024
+
0.023 [1],
λ
=
0.22537
±
0.00061 [1],
Mass, decay constant, and Gegenbauer moments
m
b
=
4.78
±
0.06 GeV [1],
f
π
=
130.41
±
0.20 MeV [1],
m
c
=
1.67
±
0.07 GeV [1],
f
K
=
156.2
±
0.7 MeV [1],
m
B
c
∗
=
6332
±
9 MeV [14],
f
B
c
∗
=
422
±
13 MeV [15]c,
a
1
K (1 GeV) =
-
0.06
±
0.03 [16],
f
Υ
(
1
S
)
=
676.4
±
10.7 MeV [17],
a
2
K (1 GeV) =
0.25
±
0.15 [16],
f
Υ
(
2
S
)
=
473.0
±
23.7 MeV [17],
a
2
π (1 GeV) =
0.25
±
0.15 [16],
f
Υ
(
3
S
)
=
409.5
±
29.4 MeV [17].
cThe decay constant f
B
c
∗ cannot be extracted from the experimental data because of no measurement on B
c
∗ weak decay at the present time. Theoretically, the value of f
B
c
∗ has been estimated, for example, in [18], with the QCD sum rules. From Table 3 of [18], one can see that the value of f
B
c
∗ is model-dependent. In our calculation, we will take the latest value given by the lattice QCD approach [15] just to offer an order of magnitude estimation on branching ratio for Υ
(
n
S
)
→
B
c
∗
P decays.Table 3
Branching ratio forΥ
(
n
S
)
→
B
c
∗
P decays.
Modes
Υ
(
1
S
)
→
B
c
∗
π
Υ
(
2
S
)
→
B
c
∗
π
Υ
(
3
S
)
→
B
c
∗
π
1
0
10
×
B
r
4.3
5
-
0.24
-
0.41
-
0.31
-
0.30
+
0.29
+
0.19
+
0.44
+
0.17
2.2
8
-
0.03
-
0.35
-
0.16
-
0.15
+
0.13
+
0.26
+
0.40
+
0.09
2.1
4
-
0.12
-
0.41
-
0.15
-
0.15
+
0.12
+
0.09
+
0.48
+
0.07
Modes
Υ
(
1
S
)
→
B
c
∗
K
Υ
(
2
S
)
→
B
c
∗
K
Υ
(
3
S
)
→
B
c
∗
K
1
0
11
×
B
r
3.4
5
-
0.21
-
0.35
-
0.27
-
0.25
+
0.23
+
0.13
+
0.38
+
0.13
1.9
1
-
0.09
-
0.31
-
0.15
-
0.14
+
0.11
+
0.07
+
0.36
+
0.07
1.6
5
-
0.21
-
0.33
-
0.13
-
0.12
+
0.09
+
0.08
+
0.40
+
0.05Figure 3
The contributions to branching ratios forΥ
(
1
S
)
→
B
c
∗
π decay (a), Υ
(
2
S
)
→
B
c
∗
π decay (b), and Υ
(
3
S
)
→
B
c
∗
π decay (c) from different region of α
s
/
π (horizontal axes), where the numbers over histogram denote the percentage of the corresponding contributions.
(a)
(b)
(c)


## 4. Summary

Υ
(
n
S
) decay via the weak interaction, as a complementary to strong and electromagnetic decay mechanism, is allowable within the standard model. Based on the potential prospects of Υ
(
n
S
) physics at high-luminosity collider experiment, Υ
(
n
S
) decay into B
c
∗
π and B
c
∗
K final states is investigated with the pQCD approach firstly. It is found that (
1
) the dominant contributions come from perturbative regions α
s
/
π
≤
0.3, which might imply that the pQCD calculation is practicable and workable; (
2
) there is a promiseful possibility of searching for Υ
(
n
S
)
→
B
c
∗
π (B
c
∗
K) decay with branching ratio about 1
0
-
10 (1
0
-
11) at the future experiments.



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*Source: 4893649-2016-06-21.xml*