# Effects of Flavor Dependence on Weak Decays ofJ/ψ and ϒ

Authors: Rohit Dhir; R. C. Verma; Avinash Sharma
Journal: Advances in High Energy Physics
 (2013)

---

## Abstract

We carry out a detailed analysis of effects of flavor dependence of average transverse quark momentum inside a meson onJ/ψ→P and ϒ→Bc transition form factors and two-body weak hadronic decays of J/ψ
and ϒ employing the factorization scheme. We predict the branching ratios of semileptonic and nonleptonic weak decays of J/ψ and ϒ mesons in Cabibbo-angle-enhanced and Cabibbo-angle-suppressed modes.


---

## Body

## 1. Introduction

Due to remarkable improvements of experimental techniques and instrumentation in the recent years, it is expected that more accurate measurements may now be available for rare decays also. The BES collaboration has observed some rare decays including the semileptonic as well as nonleptonic modes [1, 2]. As a result, it has revived the interest in the rare weak decays of  J/ψ(cc-)  into the light quarks, whose branching ratios are expected to be of the order of 10-8 [3–11]. The future experiments [3, 12–14] of Beijing Electron Positron Collider (BES-III) and Large Hadron Collider (LHC) hope to accumulate data for more than 1010 events of J/ψ per year, which would make it possible to measure such rare decays. From the theoretical point of view, such weak decays are particularly interesting because these are expected to explore mechanism responsible for hadronic transitions and are also important for the study of nonperturbative QCD effects. Further, decays of a vector meson involve polarization effects that may help in probing the underlying dynamics and hadron structure. Within the standard model framework, the flavor changing decays of J/ψ and Υ states are also possible, though naively these are expected to have rather lower branching ratios in comparison to their conventional hadronic and radiative decays.Earlier, Verma, Kamal, and Czarnecki (VKC) [6] had given the first estimates of J/ψ→PP/PV weak decay rates using the factorization scheme. VKC employs the Bauer, Stech, and Wirbel (BSW) model to estimate the  J/ψ→P transition form factors ignoring their q2 dependence on their predictions. Further, Sharma and Verma [7] reanalyzed  the J/ψ decays in the same model by including q2 dependence and new values of the form factors and decay constants. The predictions in the earlier work [6, 7] are based on  s-wave dominance for J/ψ→PV. The analysis has also been extended to predict branching ratios of weak decays of Υ based on heavy quark effective theory. Recently, Wang et al. [8, 9] and Shen and Wang [10] have calculated J/ψ→P transition form factors using the QCD sum rules and employing covariant light-front quark model, respectively, to predict the decay rates of J/ψ meson.In the present work, we employ BSW model framework [15–17] to investigate the effects of flavor dependence on J/ψ→P  and Υ→Bc transitions form factors and subsequently on J/ψ→PP/PV and Υ→BcP/BcV decays, caused by possible variation of average transverse quark momentum (ω) inside a meson. In the light of Heavy Quark Symmetry (HQS) [18], we use the dipole q2 dependence for the form factors A0,A2, and V and monopole q2 dependence for the form factor A1 and include contributions from  p-  and  d-waves for J/ψ→PV and Υ→BcP/BcV decays. We also predict branching ratios of weak semileptonic and nonleptonic decays of J/ψ and Υ in Cabibbo-angle-enhanced and Cabibbo-angle-suppressed modes. In support of flavor dependence of the form factors and corresponding branching ratios, we also perform an alternate QCDinspired calculation to obtain ω for heavy quarkonium cc- and  bb-  states.Order of presentation is as follows. In Sections2 and 3, we outline the framework employed for analysing semileptonic and nonleptonic weak decays of J/ψ.   Form factors and branching ratios in BSW model are presented in Section 4. Section 5 deals with  effects of flavor dependence on J/ψ decays. Semileptonic and nonleptonic weak decays of Υ are analyized in Section 6.  Summary and discussion are given in the last section.


## 2. Semileptonic Weak Decays ofJ/ψ

The semileptonic decay amplitudeASL(J/ψ→Plν-) can be expressed as
(1)ASL(J/ψ→P)=GF2VQq*LμHμ,
where
(2)Lμ=ν-(kν)γμ(1-γ5)l(kl),Hμ=〈P|Jμ|J/ψ〉,VQq is the appropriate CKM matrix element for Q→q transition and Jμ is the usual weak V-A current. J/ψ→P matrix element is given by
(3)〈P|Jμ|J/ψ〉=1mJ/ψ+mPεμνρσεJ/ψν(PJ/ψ+PP)ρqσV(q2)-ι(mJ/ψ+mP)εJ/ψμA1(q2)-ιεJ/ψ·qmJ/ψ+mP(PJ/ψ+PP)μA2(q2)+ιεJ/ψ·qq2(2mJ/ψ)qμA3(q2)-ιεJ/ψ·qq2(2mJ/ψ)qμA0(q2),
where εJ/ψ is the polarization vector of J/ψ and PJ/ψ and PP are the four momenta of J/ψ and pseudoscalar meson, respectively, and qμ=(PJ/ψ-PP)μ. A3(q2) is related to A1(q2) and A2(q2) as:
(4)A3(0)=(mJ/ψ+mP)2mJ/ψA1(0)+(mJ/ψ-mP)2mJ/ψA2(0).The total decay width forJ/ψ→Plν-  is the sum of longitudinal and transverse decay widths given by
(5)Γ(J/ψ→Plν-)=∑i=L,±Γi(J/ψ→Plν-),
where the longitudinal decay width (ΓL) is defined as(6a)ΓL(J/ψ→Plν-)=GF2(2π)3|VQq|2×∫ml2q2dq2(q2-ml2q2)2×λ(mJ/ψ2,mP2,q2)24mJ/ψ3×[3ml22q2λ(mJ/ψ2,mP2,q2)A02(q2)+(1+ml22q2)|h0|2]
and the transverse decay width (ΓT=Γ++Γ-) is expressed as
(6b)Γ±(J/ψ→Plν-)=GF2(2π)3|VQq|2∫ml2q2dq2(q2-ml2q2)2×λ(mJ/ψ2,mP2,q2)24mJ/ψ3(1+ml22q2)|h±|2.The helicity amplitudesh0  and  h±  are given by
(7)h0=(mJ/ψ+mP)2mJ/ψ{λ(mJ/ψ2,mP2,q2)(mJ/ψ+mP)2(mJ/ψ2-mP2-q2)A1(q2)-λ(mJ/ψ2,mP2,q2)(mJ/ψ+mP)2A2(q2)},h±={λ(mJ/ψ2,mP2,q2)(mJ/ψ+mP)V(q2)∓(mJ/ψ+mP)A1(q2)λ(mJ/ψ2,mP2,q2)(mJ/ψ+mP)},
where ml is the mass of the lepton, 0≤q2≤qmax2=(mJ/ψ-mP)2, and λ(mJ/ψ2,mP2,q2) = (mJ/ψ2+mP2-q2) − 4mJ/ψ2mP2 is related to the three-momentum of the daughter meson in the rest frame of J/ψ meson by
(8)PX=λ(mJ/ψ2,mP2,q2)2mJ/ψ.


## 3. Nonleptonic Weak Decays ofJ/ψ

### 3.1. Weak Hamiltonian

The QCD modified weak hamiltonian [19] generating the c-quark decays for Cabibbo-angle-enhanced mode (ΔC=ΔS=-1) is given by(9a)HW=GF2VudVcs*{a1(s-c)(u-d)+a2(s-d)(u-c)}+h.c.,
and for Cabibbo-angle-suppressed mode (ΔC=-1,ΔS=0)(9b)HW=GF2VudVcd*[a1{(u-d)(d-c)-(u-s)(s-c)}+a2{(u-c)(d-d)-(u-c)(s-s)}]+h.c.,where q-1q2≡q-1γμ(1-γ5)q2 represents the color singlet V-Acurrent and Vij denote standard Cabibbo-Kobayashi-Maskawa (CKM) mixing matrix elements. a’s are the undetermined coefficients assigned to the effective charge current, a1, and the effective neutral current, a2, parts of the weak Hamiltonian. These parameters are related to the QCD coefficients c1,2 as follows:
(10)a1,2=c1,2+ζc2,1,
where ζ=1/Nc, Nc is the number of colors. Usually ζ is treated as a free parameter to be fixed by the experiment. However, we follow the conventional Nc limit to fix the QCD coefficients a1≈c1 and a2≈c2, where [19]
(11)a1=1.26,a2=-0.51
are obtained on the basis of D→Kπ decays.


### 3.2. Branching Ratios

In the standard factorization scheme, the decay amplitudes ofJ/ψ→PP/PV are obtained by sandwiching the QCD modified weak Hamiltonian (up to the weak scale GF/2×CKMfactor×QCD  coefficient),
(12)〈PX|HW|J/ψ〉~〈X|Jμ|0〉〈P|Jμ†|J/ψ〉,
where X=P,V. Matrix elements [15–17] of the weak currents are defined as(13a)〈P(k)|Aμ|0〉=-ιfPkμ,(13b)〈V(k)|Vμ|0〉=εμ*mVfV.The decay rate formula forJ/ψ→PP decays [20, 21] is given by
(14)Γ(J/ψ→PP)=pc324πmJ/ψ2|A(J/ψ→PP)|2,
where pc is the magnitude of the three momenta of final state meson in the rest frame of J/ψ meson and mJ/ψ denote its mass. In general, the three-momentum pc is defined as
(15)pc=12mJ/ψ[mJ/ψ2-(mP+mX)2][mJ/ψ2-(mP-mX)2]1/2.Thus the decay amplitudeA(J/ψ→PP), say for color-enhanced J/ψ→Ds+π- decay, can be expressed as
(16)A(J/ψ→Ds+π-)=GF2VudVcs*a1×(2mJ/ψ)fπA0J/ψDs(mπ2).For color-suppressed decays the QCD factora1 is replaced by a2.The decay rate ofJ/ψ→PV decays [22, 23], composed of three independent Helicity amplitudes H0 and H±1, is expressed as
(17)Γ(J/ψ→PV)=pc8πmJ/ψ2(|H0|2+|H+1|2+|H-1|2).The amplitudesH0 and H±1 are defined as follows:
(18)H±1=a±c(x2-1)1/2,H0=-ax-b(x2-1),
where
(19)x=mJ/ψ2-mP2-mV22mJ/ψmV,a=mVfV(mJ/ψ+mV)A1(mV2),b=-2mJ/ψmV2fVA2(mV2)(mJ/ψ+mP),c=-2mJ/ψmV2fVV(mV2)mJ/ψ+mP.The coefficientsa, b, and c describe the s-, d-, and p-wave contributions, respectively.


## 3.1. Weak Hamiltonian

The QCD modified weak hamiltonian [19] generating the c-quark decays for Cabibbo-angle-enhanced mode (ΔC=ΔS=-1) is given by(9a)HW=GF2VudVcs*{a1(s-c)(u-d)+a2(s-d)(u-c)}+h.c.,
and for Cabibbo-angle-suppressed mode (ΔC=-1,ΔS=0)(9b)HW=GF2VudVcd*[a1{(u-d)(d-c)-(u-s)(s-c)}+a2{(u-c)(d-d)-(u-c)(s-s)}]+h.c.,where q-1q2≡q-1γμ(1-γ5)q2 represents the color singlet V-Acurrent and Vij denote standard Cabibbo-Kobayashi-Maskawa (CKM) mixing matrix elements. a’s are the undetermined coefficients assigned to the effective charge current, a1, and the effective neutral current, a2, parts of the weak Hamiltonian. These parameters are related to the QCD coefficients c1,2 as follows:
(10)a1,2=c1,2+ζc2,1,
where ζ=1/Nc, Nc is the number of colors. Usually ζ is treated as a free parameter to be fixed by the experiment. However, we follow the conventional Nc limit to fix the QCD coefficients a1≈c1 and a2≈c2, where [19]
(11)a1=1.26,a2=-0.51
are obtained on the basis of D→Kπ decays.


## 3.2. Branching Ratios

In the standard factorization scheme, the decay amplitudes ofJ/ψ→PP/PV are obtained by sandwiching the QCD modified weak Hamiltonian (up to the weak scale GF/2×CKMfactor×QCD  coefficient),
(12)〈PX|HW|J/ψ〉~〈X|Jμ|0〉〈P|Jμ†|J/ψ〉,
where X=P,V. Matrix elements [15–17] of the weak currents are defined as(13a)〈P(k)|Aμ|0〉=-ιfPkμ,(13b)〈V(k)|Vμ|0〉=εμ*mVfV.The decay rate formula forJ/ψ→PP decays [20, 21] is given by
(14)Γ(J/ψ→PP)=pc324πmJ/ψ2|A(J/ψ→PP)|2,
where pc is the magnitude of the three momenta of final state meson in the rest frame of J/ψ meson and mJ/ψ denote its mass. In general, the three-momentum pc is defined as
(15)pc=12mJ/ψ[mJ/ψ2-(mP+mX)2][mJ/ψ2-(mP-mX)2]1/2.Thus the decay amplitudeA(J/ψ→PP), say for color-enhanced J/ψ→Ds+π- decay, can be expressed as
(16)A(J/ψ→Ds+π-)=GF2VudVcs*a1×(2mJ/ψ)fπA0J/ψDs(mπ2).For color-suppressed decays the QCD factora1 is replaced by a2.The decay rate ofJ/ψ→PV decays [22, 23], composed of three independent Helicity amplitudes H0 and H±1, is expressed as
(17)Γ(J/ψ→PV)=pc8πmJ/ψ2(|H0|2+|H+1|2+|H-1|2).The amplitudesH0 and H±1 are defined as follows:
(18)H±1=a±c(x2-1)1/2,H0=-ax-b(x2-1),
where
(19)x=mJ/ψ2-mP2-mV22mJ/ψmV,a=mVfV(mJ/ψ+mV)A1(mV2),b=-2mJ/ψmV2fVA2(mV2)(mJ/ψ+mP),c=-2mJ/ψmV2fVV(mV2)mJ/ψ+mP.The coefficientsa, b, and c describe the s-, d-, and p-wave contributions, respectively.


## 4.J/ψ→P Form Factors in BSW Framework

We employ the BSW [15–17] model for evaluating the meson form factors. In this model, the meson wave function is given by
(20)ψm(pT,x)=Nmx(1-x)exp(-pT22ω2)×exp(-m22ω2(x-12-mq12-mq222m2)2),
where m denotes the meson mass, mi denotes the ith quark mass, and  Nm is the normalization factor. ω is the average transverse quark momentum, ω2=〈pT2〉,  which  is of the order of ΛQCD [18].Expressing the currentJμ in terms of the annihilation and creation operators, the form factors are given by the following integrals:
(21)A0J/ψP(0)=A3J/ψP(0)=∫‍d2pT×∫01‍dx(ψJ/ψ*1,0(pT,x)σZ(0)ψP(pT,x)),(22)V(0)=mq1(J/ψ)-mq1(P)mJ/ψ-mPI,(23)A1(0)=mq1(J/ψ)+mq1(P)mJ/ψ+mPI,
where
(24)I=2∫‍d2pT∫01‍dxx(ψJ/ψ*1,-1(pT,x)iσy(0)ψP(pT,x)),mq1(J/ψ) and mq1(P) denote masses of the nonspectator quarks participating in the quark decay process.  From (21) it is clear that the form factors are sensitive to the choice of ω,  which is treated as a free parameter in the model. We wish to remark that in BSW [15–17] model the form factors are calculated by taking same value of ω=0.40GeV for initial as well as final states. With the quark masses (in GeV),
(25)mu=md=0.35,ms=0.55,mc=1.7,mb=4.9.J/ψ→P form factors thus calculated are presented in rows 2 and 10 of Table 1.Table 1
Form factors ofJ/ψ→P transitions.
Models
A
0
J
/
ψ
D
(
0
)
A
1
J
/
ψ
D
(
0
)
A
2
J
/
ψ
D
(
0
)
V
J
/
ψ
D
(
0
)
This work
ω
=
0.40 GeV
0.40
0.44
−0.23
1.17
Using flavordependent ω
0.55
±
0.02
0.77
-
0.07
+
0.09
0.31
-
0.16
+
0.27
2.14
-
0.11
+
0.15
QCD inspiredω≡αsm
0.54
0.80
0.47
2.21
[7]
0.61
0.68
−0.28
1.82
[8]
0.27
0.27
0.34
0.81
[10]
0.68
0.68
0.18
1.6
Models
A
0
J
/
ψ
D
s
(
0
)
A
1
J
/
ψ
D
s
(
0
)
A
2
J
/
ψ
D
s
(
0
)
V
J
/
ψ
D
s
(
0
)
This work
ω
=
0.40 GeV
0.47
0.55
−0.14
1.26
Using flavordependent ω
0.71
-
0.02
+
0.04
0.94
±
0.07
0.33
-
0.23
+
0.33
2.30
-
0.06
+
0.09
QCD inspiredω≡αsm
0.69
0.96
0.51
2.36
[7]
0.66
0.78
−0.10
1.80
[8]
0.37
0.38
0.35
1.07
[10]
0.68
0.68
0.13
1.8It has been pointed out in the BSW2 model [18] that consistency with the Heavy Quark Symmetry (HQS) requires certain form factors such as F1,A0,A2, and V to have dipole q2 dependence, whereas A1 has monopole q2 dependence, that is,
(26)A0(q2)=A0(0)(1-q2/mP2)2,A1(q2)=A1(0)(1-q2/mA2),(27)A2(q2)=A2(0)(1-q2/mA2)2,V(q2)=V(0)(1-q2/mV2)2
with appropriate pole masses.


### 4.1. Numerical Results

In this section, we present the branching ratios calculated from the form factors (without flavor dependence) obtained in the last section for semileptonic and nonleptonic weak decays ofJ/ψ meson.


#### 4.1.1. Branching Ratios of Semileptonic Decays

Using these form factors, we obtain the branching ratios of semileptonic weak decays ofJ/ψ→Plν-, which are presented in column 2 of Table 2. The branching ratios BR0,BR-,  and BR+for various J/ψ→PV, corresponding to the Helicity amplitudes h0 and h±1, are given in columns 2, 3, and 4 of Table 3. We find that net branching ratios for semileptonic decays, B(J/ψ→D+e-ν-e)=0.23×10-10,  B(J/ψ→Ds+e-ν-e)=3.89×10-10, B(J/ψ→D+μ-ν-μ)=0.22×10-10, and B(J/ψ→Ds+μ-ν-μ)=3.75×10-10, are well below the experimental limits.Table 2
Branching ratios (in the units of10-10) of J/ψ→Plν- decays.
Decays
This work
[8]
[10]
ω
=
0.40 GeV
Using flavordependent  ω
QCD inspiredω≡αsm
J
/
ψ
→
D
+
e
-
ν
-
e
0.23
0.60
-
0.07
+
0.08
0.62
0.073
5.4
J
/
ψ
→
D
s
+
e
-
ν
-
e
3.89
10.4
-
0.75
+
0.90
10.6
1.8
5.6
J
/
ψ
→
D
+
μ
-
ν
-
μ
0.22
0.58
-
0.06
+
0.08
0.60
0.071
5.1
J
/
ψ
→
D
s
+
μ
-
ν
-
μ
3.75
9.93
-
0.65
+
0.95
10.2
1.7
5.6Table 3
Branching ratiosBR0,BR-,  and  BR+  corresponding to the  Helicity amplitudes H0 and H±1 (in the units of  10-10) of J/ψ→Plν- decays.
Decays
ω = 0.40 GeV
Using flavor dependentω
QCD inspiredω≡αsm
B
R
0
B
R
-
B
R
+
B
R
0
B
R
-
B
R
+
B
R
0
B
R
-
B
R
+
J
/
ψ
→
D
+
e
-
ν
-
e
0.14
0.076
0.014
0.31
±
0.04
0.24
-
0.03
+
0.04
0.048
-
0.004
+
0.005
0.31
0.26
0.052
J
/
ψ
→
D
s
+
e
-
ν
-
e
2.65
0.11
0.19
6.42
±
0.40
3.25
-
0.20
+
0.42
0.59
-
0.05
+
0.06
6.40
3.54
0.63
J
/
ψ
→
D
+
μ
-
ν
-
μ
0.14
0.074
0.013
0.30
±
0.04
0.24
-
0.03
+
0.04
0.047
±
0.005
0.30
0.36
0.050
J
/
ψ
→
D
s
+
μ
-
ν
-
μ
2.55
0.10
0.19
6.17
±
0.41
3.26
-
0.25
+
0.33
0.58
-
0.05
+
0.07
6.14
3.44
0.61


#### 4.1.2. Branching Ratios of Nonleptonic Decays

(a)
J
/
ψ
→
P
P Decays: for η and η′ emitting decays, we take the following basis:
(28)η=12(uu-+dd-)sinϕp-(ss-)cosϕp,(29)η′=12(uu-+dd-)cosϕp+(ss-)sinϕp,
where ϕp=θideal-θphysical; we take θphysical=-15.4∘ [24].  With the following decay constants (in GeV) [24–26],
(30)fπ=0.131,fK=0.160,fη=0.133,fη′=0.126.
Branching ratios for the Cabibbo-angle-enhanced decays are B(J/ψ→Ds+π-)=3.32×10-10and B(J/ψ→D0K0)=0.72×10-10(column 2 of Table 4).
(b)
J
/
ψ
→
P
V Decays: using the following decay constants (in GeV) [24–26]:
(31)fρ=0.221,fK*=0.220,fω=0.195,fϕ=0.229,
obtained branching ratios for various J/ψ→PV decays are B(J/ψ→Ds+ρ-)=1.77×10-9 and B(J/ψ→D0K*0)=2.51×10-10 (column 2 of Table 5).Table 4
Branching ratios (in the units of10-10) of J/ψ→PP decays.
Decay
This work
[7]
[8]
[10]
ω = 0.40 GeV
Using flavordependent ω
QCD inspiredω≡αsm
Δ
C
=
Δ
S
=
+
1
J
/
ψ
→
D
s
+
π
-
3.32
7.4
1
-
0.23
+
0.13
7.13
6.12
2.0
2.5
J
/
ψ
→
D
0
K
0
0.72
1.3
9
-
0.14
+
0.10
1.34
1.96
0.36
—
Δ
C
=
+
1
,
Δ
S
=
0
J
/
ψ
→
D
s
+
K
-
0.24
0.53
±
0.02
0.52
0.39
0.16
—
J
/
ψ
→
D
+
π
-
0.15
0.2
9
-
0.03
+
0.02
0.28
0.39
0.08
—
J
/
ψ
→
D
0
π
0
0.012
0.02
4
-
0.003
+
0.002
0.023
0.039
—
—
J
/
ψ
→
D
0
η
0.036
0.07
0
-
0.007
+
0.005
0.067
0.011
—
—
J
/
ψ
→
D
0
η
′
0.002
0.00
4
-
0.0005
+
0.0002
0.004
0.002
—
—
Δ
C
=
+
1
,
Δ
S
=
-
1
J
/
ψ
→
D
+
K
-
0.012
0.023
±
0.002
0.022
—
—
0.5
J
/
ψ
→
D
0
K
-
0
0.002
0.00
4
-
0.001
+
0.000
0.004
—
—
—Table 5
Branching Ratios (in the units of10-10) of  J/ψ→PV decays.
Decay
This Work
[7]
[8]
[10]
ω = 0.40 GeV
Using flavordependent ω
QCD inspiredω≡αsm
Δ
C
=
Δ
S
=
+
1
J
/
ψ
→
D
s
+
ρ
-
17.7
51.1
-
6.0
+
7.6
53.2
25.41
12.6
28
J
/
ψ
→
D
0
K
*
0
2.51
7.61
-
1.2
+
1.6
8.12
7.19
1.54
—
Δ
C
=
+
1
,
Δ
S
=
0
J
/
ψ
→
D
s
+
K
*
-
0.97
2.82
-
0.30
+
0.40
2.96
1.48
0.82
—
J
/
ψ
→
D
+
ρ
-
0.72
2.16
-
0.3
+
0.5
2.28
1.54
0.42
—
J
/
ψ
→
D
0
ρ
0
0.06
0.18
±
0.03
0.19
0.15
—
—
J
/
ψ
→
D
0
ω
0.05
0.16
-
0.02
+
0.03
0.17
0.13
—
—
J
/
ψ
→
D
0
ϕ
0.14
0.42
-
0.07
+
0.08
0.44
0.46
—
—
Δ
C
=
+
1
,
Δ
S
=
-
1
J
/
ψ
→
D
+
K
*
-
0.042
0.13
-
0.03
+
0.02
0.13
—
—
5.5
J
/
ψ
→
D
0
K
-
*
0
0.007
0.021
-
0.003
+
0.004
0.022
—
—
—For the sake of comparison of the relative contributions of the Helicity amplitudesH0 and H±1, we have calculated the corresponding branching ratios BR0,BR-, and BR+for these decays, which are given in column 2 of Table 6.Table 6
Branching RatiosBR0,BR- and BR+ corresponding to the  Helicity amplitudes H0 and H±1 (in the units of 10-10) of  J/ψ→PV decays.
Decay
ω = 0.40 GeV
Using flavor dependentω
QCD inspiredω≡αsm
B
R
0
B
R
+
B
R
-
B
R
0
B
R
+
B
R
-
B
R
0
B
R
+
B
R
-
Δ
C
=
Δ
S
=
+
1
J
/
ψ
→
D
s
+
ρ
-
6.26
2.75
8.74
17.3
-
1.8
+
2.3
7.55
-
0.30
+
0
26.3
-
2.7
+
3.6
17.7
7.97
27.7
J
/
ψ
→
D
0
K
*
0
0.75
0.32
1.45
2.13
-
0.35
+
0.42
0.93
-
0.20
+
0.18
4.56-0.67+0.86
2.25
1.00
4.85
Δ
C
=
+
1
,
Δ
S
=
0
J
/
ψ
→
D
s
+
K
*
-
0.27
0.18
0.52
0.74
-
0.08
+
0.10
0.52
-
0.08
+
0.10
1.56
-
0.35
+
0.42
0.77
0.54
1.65
J
/
ψ
→
D
+
ρ
-
0.027
0.076
0.37
0.77
-
0.12
+
0.15
0.22
-
0.05
+
0.06
1.17
-
0.17
+
0.22
0.81
0.24
1.25
J
/
ψ
→
D
0
ρ
0
0.022
0.066
0.031
0.064
-
0.011
+
0.012
0.018
-
0.004
+
0.005
0.096
-
0.014
+
0.018
0.067
0.019
0.103
J
/
ψ
→
D
0
ω
0.020
0.006
0.028
0.056
-
0.019
+
0.011
0.017
-
0.004
+
0.005
0.088
-
0.013
+
0.022
0.059
0.018
0.094
J
/
ψ
→
D
0
ϕ
0.032
0.022
0.083
0.091
-
0.015
+
0.018
0.063
-
0.013
+
0.018
0.26
-
0.04
+
0.05
0.096
0.068
0.28
Δ
C
=
+
1
,
Δ
S
=
-
1
J
/
ψ
→
D
+
K
*
-
0.012
0.0053
0.024
0.035
-
0.006
+
0.007
0.015
-
0.003
+
0.005
0.075
-
0.011
+
0.014
0.037
0.016
0.080
J
/
ψ
→
D
0
K
-
*
0
0.002
0.004
9
0.006
-
0.001
+
0.001
0.003
-
0.001
+
0.000
0.012
-
0.001
+
0.003
0.006
0.003
0.013


## 4.1. Numerical Results

In this section, we present the branching ratios calculated from the form factors (without flavor dependence) obtained in the last section for semileptonic and nonleptonic weak decays ofJ/ψ meson.


### 4.1.1. Branching Ratios of Semileptonic Decays

Using these form factors, we obtain the branching ratios of semileptonic weak decays ofJ/ψ→Plν-, which are presented in column 2 of Table 2. The branching ratios BR0,BR-,  and BR+for various J/ψ→PV, corresponding to the Helicity amplitudes h0 and h±1, are given in columns 2, 3, and 4 of Table 3. We find that net branching ratios for semileptonic decays, B(J/ψ→D+e-ν-e)=0.23×10-10,  B(J/ψ→Ds+e-ν-e)=3.89×10-10, B(J/ψ→D+μ-ν-μ)=0.22×10-10, and B(J/ψ→Ds+μ-ν-μ)=3.75×10-10, are well below the experimental limits.Table 2
Branching ratios (in the units of10-10) of J/ψ→Plν- decays.
Decays
This work
[8]
[10]
ω
=
0.40 GeV
Using flavordependent  ω
QCD inspiredω≡αsm
J
/
ψ
→
D
+
e
-
ν
-
e
0.23
0.60
-
0.07
+
0.08
0.62
0.073
5.4
J
/
ψ
→
D
s
+
e
-
ν
-
e
3.89
10.4
-
0.75
+
0.90
10.6
1.8
5.6
J
/
ψ
→
D
+
μ
-
ν
-
μ
0.22
0.58
-
0.06
+
0.08
0.60
0.071
5.1
J
/
ψ
→
D
s
+
μ
-
ν
-
μ
3.75
9.93
-
0.65
+
0.95
10.2
1.7
5.6Table 3
Branching ratiosBR0,BR-,  and  BR+  corresponding to the  Helicity amplitudes H0 and H±1 (in the units of  10-10) of J/ψ→Plν- decays.
Decays
ω = 0.40 GeV
Using flavor dependentω
QCD inspiredω≡αsm
B
R
0
B
R
-
B
R
+
B
R
0
B
R
-
B
R
+
B
R
0
B
R
-
B
R
+
J
/
ψ
→
D
+
e
-
ν
-
e
0.14
0.076
0.014
0.31
±
0.04
0.24
-
0.03
+
0.04
0.048
-
0.004
+
0.005
0.31
0.26
0.052
J
/
ψ
→
D
s
+
e
-
ν
-
e
2.65
0.11
0.19
6.42
±
0.40
3.25
-
0.20
+
0.42
0.59
-
0.05
+
0.06
6.40
3.54
0.63
J
/
ψ
→
D
+
μ
-
ν
-
μ
0.14
0.074
0.013
0.30
±
0.04
0.24
-
0.03
+
0.04
0.047
±
0.005
0.30
0.36
0.050
J
/
ψ
→
D
s
+
μ
-
ν
-
μ
2.55
0.10
0.19
6.17
±
0.41
3.26
-
0.25
+
0.33
0.58
-
0.05
+
0.07
6.14
3.44
0.61


### 4.1.2. Branching Ratios of Nonleptonic Decays

(a)
J
/
ψ
→
P
P Decays: for η and η′ emitting decays, we take the following basis:
(28)η=12(uu-+dd-)sinϕp-(ss-)cosϕp,(29)η′=12(uu-+dd-)cosϕp+(ss-)sinϕp,
where ϕp=θideal-θphysical; we take θphysical=-15.4∘ [24].  With the following decay constants (in GeV) [24–26],
(30)fπ=0.131,fK=0.160,fη=0.133,fη′=0.126.
Branching ratios for the Cabibbo-angle-enhanced decays are B(J/ψ→Ds+π-)=3.32×10-10and B(J/ψ→D0K0)=0.72×10-10(column 2 of Table 4).
(b)
J
/
ψ
→
P
V Decays: using the following decay constants (in GeV) [24–26]:
(31)fρ=0.221,fK*=0.220,fω=0.195,fϕ=0.229,
obtained branching ratios for various J/ψ→PV decays are B(J/ψ→Ds+ρ-)=1.77×10-9 and B(J/ψ→D0K*0)=2.51×10-10 (column 2 of Table 5).Table 4
Branching ratios (in the units of10-10) of J/ψ→PP decays.
Decay
This work
[7]
[8]
[10]
ω = 0.40 GeV
Using flavordependent ω
QCD inspiredω≡αsm
Δ
C
=
Δ
S
=
+
1
J
/
ψ
→
D
s
+
π
-
3.32
7.4
1
-
0.23
+
0.13
7.13
6.12
2.0
2.5
J
/
ψ
→
D
0
K
0
0.72
1.3
9
-
0.14
+
0.10
1.34
1.96
0.36
—
Δ
C
=
+
1
,
Δ
S
=
0
J
/
ψ
→
D
s
+
K
-
0.24
0.53
±
0.02
0.52
0.39
0.16
—
J
/
ψ
→
D
+
π
-
0.15
0.2
9
-
0.03
+
0.02
0.28
0.39
0.08
—
J
/
ψ
→
D
0
π
0
0.012
0.02
4
-
0.003
+
0.002
0.023
0.039
—
—
J
/
ψ
→
D
0
η
0.036
0.07
0
-
0.007
+
0.005
0.067
0.011
—
—
J
/
ψ
→
D
0
η
′
0.002
0.00
4
-
0.0005
+
0.0002
0.004
0.002
—
—
Δ
C
=
+
1
,
Δ
S
=
-
1
J
/
ψ
→
D
+
K
-
0.012
0.023
±
0.002
0.022
—
—
0.5
J
/
ψ
→
D
0
K
-
0
0.002
0.00
4
-
0.001
+
0.000
0.004
—
—
—Table 5
Branching Ratios (in the units of10-10) of  J/ψ→PV decays.
Decay
This Work
[7]
[8]
[10]
ω = 0.40 GeV
Using flavordependent ω
QCD inspiredω≡αsm
Δ
C
=
Δ
S
=
+
1
J
/
ψ
→
D
s
+
ρ
-
17.7
51.1
-
6.0
+
7.6
53.2
25.41
12.6
28
J
/
ψ
→
D
0
K
*
0
2.51
7.61
-
1.2
+
1.6
8.12
7.19
1.54
—
Δ
C
=
+
1
,
Δ
S
=
0
J
/
ψ
→
D
s
+
K
*
-
0.97
2.82
-
0.30
+
0.40
2.96
1.48
0.82
—
J
/
ψ
→
D
+
ρ
-
0.72
2.16
-
0.3
+
0.5
2.28
1.54
0.42
—
J
/
ψ
→
D
0
ρ
0
0.06
0.18
±
0.03
0.19
0.15
—
—
J
/
ψ
→
D
0
ω
0.05
0.16
-
0.02
+
0.03
0.17
0.13
—
—
J
/
ψ
→
D
0
ϕ
0.14
0.42
-
0.07
+
0.08
0.44
0.46
—
—
Δ
C
=
+
1
,
Δ
S
=
-
1
J
/
ψ
→
D
+
K
*
-
0.042
0.13
-
0.03
+
0.02
0.13
—
—
5.5
J
/
ψ
→
D
0
K
-
*
0
0.007
0.021
-
0.003
+
0.004
0.022
—
—
—For the sake of comparison of the relative contributions of the Helicity amplitudesH0 and H±1, we have calculated the corresponding branching ratios BR0,BR-, and BR+for these decays, which are given in column 2 of Table 6.Table 6
Branching RatiosBR0,BR- and BR+ corresponding to the  Helicity amplitudes H0 and H±1 (in the units of 10-10) of  J/ψ→PV decays.
Decay
ω = 0.40 GeV
Using flavor dependentω
QCD inspiredω≡αsm
B
R
0
B
R
+
B
R
-
B
R
0
B
R
+
B
R
-
B
R
0
B
R
+
B
R
-
Δ
C
=
Δ
S
=
+
1
J
/
ψ
→
D
s
+
ρ
-
6.26
2.75
8.74
17.3
-
1.8
+
2.3
7.55
-
0.30
+
0
26.3
-
2.7
+
3.6
17.7
7.97
27.7
J
/
ψ
→
D
0
K
*
0
0.75
0.32
1.45
2.13
-
0.35
+
0.42
0.93
-
0.20
+
0.18
4.56-0.67+0.86
2.25
1.00
4.85
Δ
C
=
+
1
,
Δ
S
=
0
J
/
ψ
→
D
s
+
K
*
-
0.27
0.18
0.52
0.74
-
0.08
+
0.10
0.52
-
0.08
+
0.10
1.56
-
0.35
+
0.42
0.77
0.54
1.65
J
/
ψ
→
D
+
ρ
-
0.027
0.076
0.37
0.77
-
0.12
+
0.15
0.22
-
0.05
+
0.06
1.17
-
0.17
+
0.22
0.81
0.24
1.25
J
/
ψ
→
D
0
ρ
0
0.022
0.066
0.031
0.064
-
0.011
+
0.012
0.018
-
0.004
+
0.005
0.096
-
0.014
+
0.018
0.067
0.019
0.103
J
/
ψ
→
D
0
ω
0.020
0.006
0.028
0.056
-
0.019
+
0.011
0.017
-
0.004
+
0.005
0.088
-
0.013
+
0.022
0.059
0.018
0.094
J
/
ψ
→
D
0
ϕ
0.032
0.022
0.083
0.091
-
0.015
+
0.018
0.063
-
0.013
+
0.018
0.26
-
0.04
+
0.05
0.096
0.068
0.28
Δ
C
=
+
1
,
Δ
S
=
-
1
J
/
ψ
→
D
+
K
*
-
0.012
0.0053
0.024
0.035
-
0.006
+
0.007
0.015
-
0.003
+
0.005
0.075
-
0.011
+
0.014
0.037
0.016
0.080
J
/
ψ
→
D
0
K
-
*
0
0.002
0.004
9
0.006
-
0.001
+
0.001
0.003
-
0.001
+
0.000
0.012
-
0.001
+
0.003
0.006
0.003
0.013


## 4.1.1. Branching Ratios of Semileptonic Decays

Using these form factors, we obtain the branching ratios of semileptonic weak decays ofJ/ψ→Plν-, which are presented in column 2 of Table 2. The branching ratios BR0,BR-,  and BR+for various J/ψ→PV, corresponding to the Helicity amplitudes h0 and h±1, are given in columns 2, 3, and 4 of Table 3. We find that net branching ratios for semileptonic decays, B(J/ψ→D+e-ν-e)=0.23×10-10,  B(J/ψ→Ds+e-ν-e)=3.89×10-10, B(J/ψ→D+μ-ν-μ)=0.22×10-10, and B(J/ψ→Ds+μ-ν-μ)=3.75×10-10, are well below the experimental limits.Table 2
Branching ratios (in the units of10-10) of J/ψ→Plν- decays.
Decays
This work
[8]
[10]
ω
=
0.40 GeV
Using flavordependent  ω
QCD inspiredω≡αsm
J
/
ψ
→
D
+
e
-
ν
-
e
0.23
0.60
-
0.07
+
0.08
0.62
0.073
5.4
J
/
ψ
→
D
s
+
e
-
ν
-
e
3.89
10.4
-
0.75
+
0.90
10.6
1.8
5.6
J
/
ψ
→
D
+
μ
-
ν
-
μ
0.22
0.58
-
0.06
+
0.08
0.60
0.071
5.1
J
/
ψ
→
D
s
+
μ
-
ν
-
μ
3.75
9.93
-
0.65
+
0.95
10.2
1.7
5.6Table 3
Branching ratiosBR0,BR-,  and  BR+  corresponding to the  Helicity amplitudes H0 and H±1 (in the units of  10-10) of J/ψ→Plν- decays.
Decays
ω = 0.40 GeV
Using flavor dependentω
QCD inspiredω≡αsm
B
R
0
B
R
-
B
R
+
B
R
0
B
R
-
B
R
+
B
R
0
B
R
-
B
R
+
J
/
ψ
→
D
+
e
-
ν
-
e
0.14
0.076
0.014
0.31
±
0.04
0.24
-
0.03
+
0.04
0.048
-
0.004
+
0.005
0.31
0.26
0.052
J
/
ψ
→
D
s
+
e
-
ν
-
e
2.65
0.11
0.19
6.42
±
0.40
3.25
-
0.20
+
0.42
0.59
-
0.05
+
0.06
6.40
3.54
0.63
J
/
ψ
→
D
+
μ
-
ν
-
μ
0.14
0.074
0.013
0.30
±
0.04
0.24
-
0.03
+
0.04
0.047
±
0.005
0.30
0.36
0.050
J
/
ψ
→
D
s
+
μ
-
ν
-
μ
2.55
0.10
0.19
6.17
±
0.41
3.26
-
0.25
+
0.33
0.58
-
0.05
+
0.07
6.14
3.44
0.61


## 4.1.2. Branching Ratios of Nonleptonic Decays

(a)
J
/
ψ
→
P
P Decays: for η and η′ emitting decays, we take the following basis:
(28)η=12(uu-+dd-)sinϕp-(ss-)cosϕp,(29)η′=12(uu-+dd-)cosϕp+(ss-)sinϕp,
where ϕp=θideal-θphysical; we take θphysical=-15.4∘ [24].  With the following decay constants (in GeV) [24–26],
(30)fπ=0.131,fK=0.160,fη=0.133,fη′=0.126.
Branching ratios for the Cabibbo-angle-enhanced decays are B(J/ψ→Ds+π-)=3.32×10-10and B(J/ψ→D0K0)=0.72×10-10(column 2 of Table 4).
(b)
J
/
ψ
→
P
V Decays: using the following decay constants (in GeV) [24–26]:
(31)fρ=0.221,fK*=0.220,fω=0.195,fϕ=0.229,
obtained branching ratios for various J/ψ→PV decays are B(J/ψ→Ds+ρ-)=1.77×10-9 and B(J/ψ→D0K*0)=2.51×10-10 (column 2 of Table 5).Table 4
Branching ratios (in the units of10-10) of J/ψ→PP decays.
Decay
This work
[7]
[8]
[10]
ω = 0.40 GeV
Using flavordependent ω
QCD inspiredω≡αsm
Δ
C
=
Δ
S
=
+
1
J
/
ψ
→
D
s
+
π
-
3.32
7.4
1
-
0.23
+
0.13
7.13
6.12
2.0
2.5
J
/
ψ
→
D
0
K
0
0.72
1.3
9
-
0.14
+
0.10
1.34
1.96
0.36
—
Δ
C
=
+
1
,
Δ
S
=
0
J
/
ψ
→
D
s
+
K
-
0.24
0.53
±
0.02
0.52
0.39
0.16
—
J
/
ψ
→
D
+
π
-
0.15
0.2
9
-
0.03
+
0.02
0.28
0.39
0.08
—
J
/
ψ
→
D
0
π
0
0.012
0.02
4
-
0.003
+
0.002
0.023
0.039
—
—
J
/
ψ
→
D
0
η
0.036
0.07
0
-
0.007
+
0.005
0.067
0.011
—
—
J
/
ψ
→
D
0
η
′
0.002
0.00
4
-
0.0005
+
0.0002
0.004
0.002
—
—
Δ
C
=
+
1
,
Δ
S
=
-
1
J
/
ψ
→
D
+
K
-
0.012
0.023
±
0.002
0.022
—
—
0.5
J
/
ψ
→
D
0
K
-
0
0.002
0.00
4
-
0.001
+
0.000
0.004
—
—
—Table 5
Branching Ratios (in the units of10-10) of  J/ψ→PV decays.
Decay
This Work
[7]
[8]
[10]
ω = 0.40 GeV
Using flavordependent ω
QCD inspiredω≡αsm
Δ
C
=
Δ
S
=
+
1
J
/
ψ
→
D
s
+
ρ
-
17.7
51.1
-
6.0
+
7.6
53.2
25.41
12.6
28
J
/
ψ
→
D
0
K
*
0
2.51
7.61
-
1.2
+
1.6
8.12
7.19
1.54
—
Δ
C
=
+
1
,
Δ
S
=
0
J
/
ψ
→
D
s
+
K
*
-
0.97
2.82
-
0.30
+
0.40
2.96
1.48
0.82
—
J
/
ψ
→
D
+
ρ
-
0.72
2.16
-
0.3
+
0.5
2.28
1.54
0.42
—
J
/
ψ
→
D
0
ρ
0
0.06
0.18
±
0.03
0.19
0.15
—
—
J
/
ψ
→
D
0
ω
0.05
0.16
-
0.02
+
0.03
0.17
0.13
—
—
J
/
ψ
→
D
0
ϕ
0.14
0.42
-
0.07
+
0.08
0.44
0.46
—
—
Δ
C
=
+
1
,
Δ
S
=
-
1
J
/
ψ
→
D
+
K
*
-
0.042
0.13
-
0.03
+
0.02
0.13
—
—
5.5
J
/
ψ
→
D
0
K
-
*
0
0.007
0.021
-
0.003
+
0.004
0.022
—
—
—For the sake of comparison of the relative contributions of the Helicity amplitudesH0 and H±1, we have calculated the corresponding branching ratios BR0,BR-, and BR+for these decays, which are given in column 2 of Table 6.Table 6
Branching RatiosBR0,BR- and BR+ corresponding to the  Helicity amplitudes H0 and H±1 (in the units of 10-10) of  J/ψ→PV decays.
Decay
ω = 0.40 GeV
Using flavor dependentω
QCD inspiredω≡αsm
B
R
0
B
R
+
B
R
-
B
R
0
B
R
+
B
R
-
B
R
0
B
R
+
B
R
-
Δ
C
=
Δ
S
=
+
1
J
/
ψ
→
D
s
+
ρ
-
6.26
2.75
8.74
17.3
-
1.8
+
2.3
7.55
-
0.30
+
0
26.3
-
2.7
+
3.6
17.7
7.97
27.7
J
/
ψ
→
D
0
K
*
0
0.75
0.32
1.45
2.13
-
0.35
+
0.42
0.93
-
0.20
+
0.18
4.56-0.67+0.86
2.25
1.00
4.85
Δ
C
=
+
1
,
Δ
S
=
0
J
/
ψ
→
D
s
+
K
*
-
0.27
0.18
0.52
0.74
-
0.08
+
0.10
0.52
-
0.08
+
0.10
1.56
-
0.35
+
0.42
0.77
0.54
1.65
J
/
ψ
→
D
+
ρ
-
0.027
0.076
0.37
0.77
-
0.12
+
0.15
0.22
-
0.05
+
0.06
1.17
-
0.17
+
0.22
0.81
0.24
1.25
J
/
ψ
→
D
0
ρ
0
0.022
0.066
0.031
0.064
-
0.011
+
0.012
0.018
-
0.004
+
0.005
0.096
-
0.014
+
0.018
0.067
0.019
0.103
J
/
ψ
→
D
0
ω
0.020
0.006
0.028
0.056
-
0.019
+
0.011
0.017
-
0.004
+
0.005
0.088
-
0.013
+
0.022
0.059
0.018
0.094
J
/
ψ
→
D
0
ϕ
0.032
0.022
0.083
0.091
-
0.015
+
0.018
0.063
-
0.013
+
0.018
0.26
-
0.04
+
0.05
0.096
0.068
0.28
Δ
C
=
+
1
,
Δ
S
=
-
1
J
/
ψ
→
D
+
K
*
-
0.012
0.0053
0.024
0.035
-
0.006
+
0.007
0.015
-
0.003
+
0.005
0.075
-
0.011
+
0.014
0.037
0.016
0.080
J
/
ψ
→
D
0
K
-
*
0
0.002
0.004
9
0.006
-
0.001
+
0.001
0.003
-
0.001
+
0.000
0.012
-
0.001
+
0.003
0.006
0.003
0.013


## 5.   Flavor Dependent Effects onJ/ψ Decays

Sinceω, being a dimensional quantity, may possess some flavor dependence, it may not be justified to take the same value for all the mesons. In our recent work [27, 28], we have investigated such flavor dependence through ω in Bc→P/V form factors and consequently in Bc→PP/PV decay widths, which may be measured in near future experiments.


### 5.1. Flavor Dependence onJ/ψ→P Form Factors

In this section, we investigate effects of the flavor dependence onJ/ψ→P/V decays. Following the prescription of [27, 28], we estimate ω for different mesons from |ψ(0)|2, that is, wave function overlap at the origin, using the following ansatz:
(32)|ψ(0)|2=constant×ω3
which may be justified in light of inverse size of the mesons and dimensionality arguments.  |ψ(0)|2 is extracted from the hyperfine splitting for the meson masses [29],
(33)|ψ(0)|2=9mimj32αsπ(mV-mP),
where mV and mP, respectively, denote masses of the vector and pseudoscalar mesons composed of i andj quarks. Conventionally, meson masses fix quark masses (in GeV) to be mu=md=0.31, ms=0.49, mc=1.7, andmb=5.0 for  αs(mb)=0.19, αs(mc)=0.25,and αs=0.48(for light flavors u,  d, and s). However, uncertainty of  αs, particularly for the light quark sector, may lead to some variation in the quark masses [25, 26, 30–34]. So, in our analysis, we allow the following range in the quark masses:
(34)mu=md=0.31±0.05,ms=0.49±0.05,mc=1.7±0.05,mb=5.0±0.05.Calculated values of|ψ(0)|2 for different mesons are given in column 2 of Table 7. We use the well-measured form factor [35–37] F0DK(0)=0.78±0.04 to fix ωD=0.43±0.03GeV which in turn yields ω for other mesons as given in column 3 of Table 7. We find that all the J/ψ→P transition form factors get significantly enhanced due to the flavor dependence of parameter ω, particularly,  ωJ/ψ=0.71±0.02GeV. The J/ψ→P transition form factors, required for the weak decays under investigation, are found to be increased, which are given in rows 3 and 11 of Table 1. Uncertainties shown in the form factors arise due to variation in the quark masses discussed above.Table 7
|
ψ
(
0
)
|
2 and ω for vector and pseudoscalar mesons.
Meson
|
ψ
(
0
)
|
2  (in GeV3)
Parameterω (in GeV)
QCD inspiredω≡αsm (in GeV)
ρ
(
π
)
0.011
±
0.003
0.33
±
0.03
—
K
*
(
K
)
0.011
±
0.03
0.33
±
0.03
—
D
*
(
D
)
0.026
-
0.004
+
0.006
0.43
±
0.03
—
D
s
*
(
D
s
)
0.041
-
0.005
+
0.007
0.51
-
0.03
+
0.02
—
J
/
ψ
(
η
c
)
0.115
-
0.008
+
0.011
0.71
±
0.02
0.79
B
*
(
B
)
0.033
±
0.005
0.47
-
0.03
+
0.02
—
B
s
*
(
B
s
)
0.053
±
0.006
0.55
±
0.02
—
B
c
(
B
c
*
)
0.281
-
0.0011
+
0.0017
0.96
±
0.02
—
Υ
(
η
b
)
1.850
-
0.04
+
0.03
1.80
-
0.02
+
0.01
1.83Note that the normally uncertainties lie between 3% and 11% for all the form factors except forA2(q2), which is drastically increased from 80% to 100%. These are likely to change theoretical predictions for branching of semileptonic as well as nonleptonic decays. Since the form factors  A1(q2), V(q2) and A2(q2), describe the s-, p-, and d-wave contributions to the nonleptonic decay amplitudes, respectively, the s-d interference term may acquire significance when both A1(q2)and A2(q2)become large. Incidentally, A2(q2)being small, such contribution turned out to be very small as compared to the terms proportional to A1(q2). Therefore, the variations in A2(q2)even of the order of 80% to 100% are not going to affect the branching ratios seriously.


### 5.2. QCD Inspired Calculation ofω

The wave function for the heavy meson like, for example,B meson is known to have a pronounced peak at  x≈x0=mb/MB  (MB being the B meson mass). It is thus obvious that the only region close to the peak position contributes to any degree of significance. This implies a good approximation for parameter ω to be of the order of ΛQCD. This may be easily understood  since 〈pT2〉~ΛQCD2,  ψm(pT,x)→0 if  pT2≫ΛQCD2 and has peak when x→1. Also, the average transverse quark momentum of the heavy quark equals to that of heavy meson such that x→1. In accord with the heavy quark effective theory, it also implies equal velocities of the b-quark and of the B meson up to corrections of order pT/MB. In the light of these arguments, we present another method to determine ω, for instance by looking at the typical inverse size of the system under scrutiny, which is of order ΛQCD for the lighter mesons and the heavy-light ones and is of order meson mass multiplied by strong coupling constant (αs) for the heavy quarkonium (qq-) states. It is interesting to note that following the relation
(35)ω≡αs×mforcc-andbb-
gives larger values of ω,  that is,  ωJ/ψ=0.79andωΥ=1.83, ignoring the small uncertainties in αs for heavier quarks.  A plot for ω thus calculated for various mesons (especially for cc-andbb- states) has interesting pattern with respect to the physical masses of the mesons, which is shown  in Figure 1.Figure 1
Variation of average transverse quark momentum with respect to physical meson masses.It may also be pointed out that Sharma et al. [38] have reported that |ψ(0)|2, which may be extracted from leptonic decays of cc-andbb- vector mesons, cannot simultaneously fit with that needed for their masses. Perhaps similar analysis needs to be carried out for |ψ(0)|2 extracted from form factors and hyperfine splitting.


### 5.3. Branching Ratios Including Flavor Dependent Effects

Different values ofω for initial and final state mesons yield J/ψ→P transition form factors, which are given in Table 1. Using these form factors we calculate the branching ratios of semileptonic decays of J/ψ as presented in column 3 of Table 2. Contributions from the Helicity amplitudes H0 and H±1 are also obtained for the corresponding branching ratios BR0,BR-, and BR+for these decays and are given in column 3 of Table 3.Further, we calculate branching ratios of variousJ/ψ→PP/PV decays, which are listed in column 3 of Tables 4 and 5.  In order to compare flavor dependent effects for different Helicity terms with those obtained at fixed ω=0.40GeV, we present branching ratios BR0,BR-, and BR+in column 3 of Table 6. For the sake of comparison, we also give results of other works in Tables 2, 4, and 5. The following observations are made.  (i)
Branching ratios of all semileptonic as well as nonleptonic decays ofJ/ψ meson get significantly enhanced because of inclusion of flavor dependent effects.(ii)
The enhanced branching ratios for semileptonic decays areB(J/ψ→Ds+e-ν-e)=10.4-0.75+0.90×10-10 and B(J/ψ→Ds+μ-ν-μ)=9.93-0.65+0.95×10-10. Presently, only experimental [1] upper limits are available,
(36)B(J/ψ→D+e-ν-e+c.c.)<1.2×10-5,B(J/ψ→Ds+e-ν-e+c.c.)<4.9×10-5.(iii)
Among the Cabibbo-angle-enhancedJ/ψ→PP decays, we find that branching ratios of dominant decays are  B(J/ψ→Ds+π-)  =  7.41-0.23+0.13×10-10  and B(J/ψ→D0K0)=1.39-0.14+0.10×10-10.(iv)
For Cabibbo-angle-suppressed but color-enhanced modesJ/ψ→Ds+K- and J/ψ→D+π-, we obtain the following ratios based on the naive factorization scheme:
(37)B(J/ψ→Ds+K-)B(J/ψ→Ds+π-)=0.072±0.003,B(J/ψ→D+π-)B(J/ψ→D0K0)=0.21±0.03.(v)
In case ofJ/ψ→PV decays, for the color-enhanced decay of the Cabibbo-angle-enhanced mode, we calculate B(J/ψ→Ds+ρ-)=5.11-0.60+0.76×10-9, which is higher than the branching ratio of J/ψ→Ds+π-. While for color-suppressed decay we obtain B(J/ψ→D0K*0)=7.61-1.2+1.6×10-9. Our analysis also yields
(38)B(J/ψ→Ds+K*-)B(J/ψ→Ds+ρ-)=0.055±0.010,B(J/ψ→D+ρ-)B(J/ψ→D0K*0)=0.28±0.07.


### 5.4. Branching Ratios Using QCD Inspiredω

Using QCD inspired values ofω(≡αsm) for heavy quarkonium cc-andbb- states, the obtained form factors are given in rows 4 and 12 of Table 1. It is interesting to note all the form factors  except A0(0) get significantly enhanced. Correspondingly, branching ratios of semileptonic and nonleptonic decays are calculated, which are presented in column 4 of Tables 2 to 6. We observe the following.(i)
Branching ratios of all semileptonic decays ofJ/ψ meson are nearly the same though marginally enhanced, B(J/ψ→Ds+e-ν-e)  =  10.6×10-10 and B(J/ψ→Ds+μ-ν-μ)=10.2×10-10 in comparison to those obtained in our work including the flavor dependent effects. This happens due to the larger value ω for cc- state, leading to increased overlap between the initial and final state wave functions.(ii)
Branching ratios of all theJ/ψ→PP decays, involving the A0(0) form factor only, are slightly decreased in comparison to the flavor dependent branching ratios predicted in our analysis.(iii)
Branching ratios of all theJ/ψ→PV decays are comprable to our predictions for the flavor dependent case.


## 5.1. Flavor Dependence onJ/ψ→P Form Factors

In this section, we investigate effects of the flavor dependence onJ/ψ→P/V decays. Following the prescription of [27, 28], we estimate ω for different mesons from |ψ(0)|2, that is, wave function overlap at the origin, using the following ansatz:
(32)|ψ(0)|2=constant×ω3
which may be justified in light of inverse size of the mesons and dimensionality arguments.  |ψ(0)|2 is extracted from the hyperfine splitting for the meson masses [29],
(33)|ψ(0)|2=9mimj32αsπ(mV-mP),
where mV and mP, respectively, denote masses of the vector and pseudoscalar mesons composed of i andj quarks. Conventionally, meson masses fix quark masses (in GeV) to be mu=md=0.31, ms=0.49, mc=1.7, andmb=5.0 for  αs(mb)=0.19, αs(mc)=0.25,and αs=0.48(for light flavors u,  d, and s). However, uncertainty of  αs, particularly for the light quark sector, may lead to some variation in the quark masses [25, 26, 30–34]. So, in our analysis, we allow the following range in the quark masses:
(34)mu=md=0.31±0.05,ms=0.49±0.05,mc=1.7±0.05,mb=5.0±0.05.Calculated values of|ψ(0)|2 for different mesons are given in column 2 of Table 7. We use the well-measured form factor [35–37] F0DK(0)=0.78±0.04 to fix ωD=0.43±0.03GeV which in turn yields ω for other mesons as given in column 3 of Table 7. We find that all the J/ψ→P transition form factors get significantly enhanced due to the flavor dependence of parameter ω, particularly,  ωJ/ψ=0.71±0.02GeV. The J/ψ→P transition form factors, required for the weak decays under investigation, are found to be increased, which are given in rows 3 and 11 of Table 1. Uncertainties shown in the form factors arise due to variation in the quark masses discussed above.Table 7
|
ψ
(
0
)
|
2 and ω for vector and pseudoscalar mesons.
Meson
|
ψ
(
0
)
|
2  (in GeV3)
Parameterω (in GeV)
QCD inspiredω≡αsm (in GeV)
ρ
(
π
)
0.011
±
0.003
0.33
±
0.03
—
K
*
(
K
)
0.011
±
0.03
0.33
±
0.03
—
D
*
(
D
)
0.026
-
0.004
+
0.006
0.43
±
0.03
—
D
s
*
(
D
s
)
0.041
-
0.005
+
0.007
0.51
-
0.03
+
0.02
—
J
/
ψ
(
η
c
)
0.115
-
0.008
+
0.011
0.71
±
0.02
0.79
B
*
(
B
)
0.033
±
0.005
0.47
-
0.03
+
0.02
—
B
s
*
(
B
s
)
0.053
±
0.006
0.55
±
0.02
—
B
c
(
B
c
*
)
0.281
-
0.0011
+
0.0017
0.96
±
0.02
—
Υ
(
η
b
)
1.850
-
0.04
+
0.03
1.80
-
0.02
+
0.01
1.83Note that the normally uncertainties lie between 3% and 11% for all the form factors except forA2(q2), which is drastically increased from 80% to 100%. These are likely to change theoretical predictions for branching of semileptonic as well as nonleptonic decays. Since the form factors  A1(q2), V(q2) and A2(q2), describe the s-, p-, and d-wave contributions to the nonleptonic decay amplitudes, respectively, the s-d interference term may acquire significance when both A1(q2)and A2(q2)become large. Incidentally, A2(q2)being small, such contribution turned out to be very small as compared to the terms proportional to A1(q2). Therefore, the variations in A2(q2)even of the order of 80% to 100% are not going to affect the branching ratios seriously.


## 5.2. QCD Inspired Calculation ofω

The wave function for the heavy meson like, for example,B meson is known to have a pronounced peak at  x≈x0=mb/MB  (MB being the B meson mass). It is thus obvious that the only region close to the peak position contributes to any degree of significance. This implies a good approximation for parameter ω to be of the order of ΛQCD. This may be easily understood  since 〈pT2〉~ΛQCD2,  ψm(pT,x)→0 if  pT2≫ΛQCD2 and has peak when x→1. Also, the average transverse quark momentum of the heavy quark equals to that of heavy meson such that x→1. In accord with the heavy quark effective theory, it also implies equal velocities of the b-quark and of the B meson up to corrections of order pT/MB. In the light of these arguments, we present another method to determine ω, for instance by looking at the typical inverse size of the system under scrutiny, which is of order ΛQCD for the lighter mesons and the heavy-light ones and is of order meson mass multiplied by strong coupling constant (αs) for the heavy quarkonium (qq-) states. It is interesting to note that following the relation
(35)ω≡αs×mforcc-andbb-
gives larger values of ω,  that is,  ωJ/ψ=0.79andωΥ=1.83, ignoring the small uncertainties in αs for heavier quarks.  A plot for ω thus calculated for various mesons (especially for cc-andbb- states) has interesting pattern with respect to the physical masses of the mesons, which is shown  in Figure 1.Figure 1
Variation of average transverse quark momentum with respect to physical meson masses.It may also be pointed out that Sharma et al. [38] have reported that |ψ(0)|2, which may be extracted from leptonic decays of cc-andbb- vector mesons, cannot simultaneously fit with that needed for their masses. Perhaps similar analysis needs to be carried out for |ψ(0)|2 extracted from form factors and hyperfine splitting.


## 5.3. Branching Ratios Including Flavor Dependent Effects

Different values ofω for initial and final state mesons yield J/ψ→P transition form factors, which are given in Table 1. Using these form factors we calculate the branching ratios of semileptonic decays of J/ψ as presented in column 3 of Table 2. Contributions from the Helicity amplitudes H0 and H±1 are also obtained for the corresponding branching ratios BR0,BR-, and BR+for these decays and are given in column 3 of Table 3.Further, we calculate branching ratios of variousJ/ψ→PP/PV decays, which are listed in column 3 of Tables 4 and 5.  In order to compare flavor dependent effects for different Helicity terms with those obtained at fixed ω=0.40GeV, we present branching ratios BR0,BR-, and BR+in column 3 of Table 6. For the sake of comparison, we also give results of other works in Tables 2, 4, and 5. The following observations are made.  (i)
Branching ratios of all semileptonic as well as nonleptonic decays ofJ/ψ meson get significantly enhanced because of inclusion of flavor dependent effects.(ii)
The enhanced branching ratios for semileptonic decays areB(J/ψ→Ds+e-ν-e)=10.4-0.75+0.90×10-10 and B(J/ψ→Ds+μ-ν-μ)=9.93-0.65+0.95×10-10. Presently, only experimental [1] upper limits are available,
(36)B(J/ψ→D+e-ν-e+c.c.)<1.2×10-5,B(J/ψ→Ds+e-ν-e+c.c.)<4.9×10-5.(iii)
Among the Cabibbo-angle-enhancedJ/ψ→PP decays, we find that branching ratios of dominant decays are  B(J/ψ→Ds+π-)  =  7.41-0.23+0.13×10-10  and B(J/ψ→D0K0)=1.39-0.14+0.10×10-10.(iv)
For Cabibbo-angle-suppressed but color-enhanced modesJ/ψ→Ds+K- and J/ψ→D+π-, we obtain the following ratios based on the naive factorization scheme:
(37)B(J/ψ→Ds+K-)B(J/ψ→Ds+π-)=0.072±0.003,B(J/ψ→D+π-)B(J/ψ→D0K0)=0.21±0.03.(v)
In case ofJ/ψ→PV decays, for the color-enhanced decay of the Cabibbo-angle-enhanced mode, we calculate B(J/ψ→Ds+ρ-)=5.11-0.60+0.76×10-9, which is higher than the branching ratio of J/ψ→Ds+π-. While for color-suppressed decay we obtain B(J/ψ→D0K*0)=7.61-1.2+1.6×10-9. Our analysis also yields
(38)B(J/ψ→Ds+K*-)B(J/ψ→Ds+ρ-)=0.055±0.010,B(J/ψ→D+ρ-)B(J/ψ→D0K*0)=0.28±0.07.


## 5.4. Branching Ratios Using QCD Inspiredω

Using QCD inspired values ofω(≡αsm) for heavy quarkonium cc-andbb- states, the obtained form factors are given in rows 4 and 12 of Table 1. It is interesting to note all the form factors  except A0(0) get significantly enhanced. Correspondingly, branching ratios of semileptonic and nonleptonic decays are calculated, which are presented in column 4 of Tables 2 to 6. We observe the following.(i)
Branching ratios of all semileptonic decays ofJ/ψ meson are nearly the same though marginally enhanced, B(J/ψ→Ds+e-ν-e)  =  10.6×10-10 and B(J/ψ→Ds+μ-ν-μ)=10.2×10-10 in comparison to those obtained in our work including the flavor dependent effects. This happens due to the larger value ω for cc- state, leading to increased overlap between the initial and final state wave functions.(ii)
Branching ratios of all theJ/ψ→PP decays, involving the A0(0) form factor only, are slightly decreased in comparison to the flavor dependent branching ratios predicted in our analysis.(iii)
Branching ratios of all theJ/ψ→PV decays are comprable to our predictions for the flavor dependent case.


## 6. Semileptonic and Nonleptonic Weak Decays ofΥ

### 6.1.ϒ→Bc Form Factors and Branching Ratios

Using the framework described in Sections4 and 5, we extend our analysis for bottom sector. We obtain the form factor for ϒ→Bc transition at both ω=0.40GeV  and using the flavor dependent ω, which are compared in rows 2 and 3 of Table 8. Note that the form factors get enhanced by several orders of magnitude for flavor dependent ω. This happens due to the increased overlap of Υ and the final state wave functions for flavor dependent ω in comparison to that at fixed ω=0.40GeV as shown in Figures 2 and 3. We wish to remark that the flavor dependent form factors come close to the expectation [7] based on HQET considerations (row 4 of Table 8). Consequently, the branching ratios of semileptonic and nonleptonic weak decays of Υ get significantly enhanced.Table 8
Form factors ofΥ→Bc transitions.
Models
A
0
Υ
B
c
(
0
)
A
1
Υ
B
c
(
0
)
A
2
Υ
B
c
(
0
)
V
Υ
B
c
(
0
)
This work
ω
=
0.40 GeV
0.003
0.003
0.003
0.008
Using flavordependent ω
0.46
±
0.01
0.62
-
0.01
+
0.02
0.38
-
0.09
+
0.06
1.61
±
0.01
QCD inspiredω≡αsm*
0.46
0.62
0.38
1.62
[7]
0.98
1.01
1.01
1.01

*The form factors here have shown no significant change in the values,  therefore, the BRs for the decays involving Υ→Bc form factors for QCD inspired   ω  remain the same as for flavor dependent  ω.Figure 2
The overlap of wave functions forϒ→Bc decays at ωΥ=ωBc=0.40GeV.Figure 3
The overlap of wave functions forϒ→Bc decays at ωΥ=1.80GeV and ωBc=0.96GeV.We also use the QCD inspired method to determineω, that is,  ωΥ=1.83, which match well with our value 1.80 for Υ.  Subsequently obtained form factors are given in row 4 of Table 8, which are largely the same as our results, so we exclude the results for decay rates based on the QCD inspired method for further discussion.


### 6.2. Semileptonic Weak Decays ofϒ

Usingϒ→Bc form factor, appearing in b→c transition, and the decay rate formula given in (5), firstly, we calculate the branching ratios for semileptonic decays of Υ at fixed ω=0.40GeV, which are given in column 2 of Table 9. The predicted branching ratios of semileptonic weak decays of Υ using the flavor dependent effects are given in column 3 of Table 9. We find that the branching ratio of dominant semileptonic decay is B(Υ→Bc+e-ν-e)=1.70×10-10 that is sufficiently enhanced in comparison to (6.28×10-15) obtained using fixed ω. Here also, we give  branching ratios of longitudinal and transverse components (BR0,BR-, and BR+) separately, for the semileptonic decays in Table 10.Table 9
Branching ratios (in the units of10-10) of Υ→Plν- decays.
Decays
This Work
ω
=
0.40 GeV
Using flavordependent ω
Υ
→
B
c
+
e
-
ν
-
e
6.28
×
10
-
5
1.70
-
0.02
+
0.03
Υ
→
B
c
+
μ
-
ν
-
μ
6.24
×
10
-
5
1.69
-
0.02
+
0.04
Υ
→
B
c
+
τ
-
ν
-
τ
9.86
×
10
-
5
0.29
-
0.02
+
0.05Table 10
Branching ratiosBR0,BR-, and BR+ corresponding to the  Helicity amplitudes h0 and h±1 (in the units of 10-10) of Υ→Plν- decays.
Decays
ω = 0.40 GeV
Using flavor dependentω
B
R
0
B
R
-
B
R
+
B
R
0
B
R
-
B
R
+
Υ
→
B
c
+
e
-
ν
-
e
5.50
×
10
-
5
0.46
×
10
-
5
3.10
×
10
-
5
1.40
-
0.02
+
0.03
0.18
-
0.00
+
0.01
0.13
-
0.00
+
0.01
Υ
→
B
c
+
μ
-
ν
-
μ
5.47
×
10
-
5
0.45
×
10
-
5
3.08
×
10
-
5
1.39
-
0.02
+
0.03
0.18
-
0.00
+
0.01
0.12
-
0.00
+
0.01
Υ
→
B
c
+
τ
-
ν
-
τ
0.86
×
10
-
5
0.81
×
10
-
6
0.45
×
10
-
6
0.24
-
0.01
+
0.00
0.03
0.018


### 6.3. Nonleptonic Weak Decays ofϒ

In this section, the analysis is extended toϒ→PP/PV decays. The effective weak Hamiltonian generating the dominant b quark decays involving b→c transition is given by
(39)HWΔb=1=GF2{VcbVud*[a1(c-b)(d-u)+a2(d-b)(c-u)]+VcbVcs*[a1(c-b)(s-c)+a2(s-b)(c-c)]}+h.c.,
for the CKM-favored mode. In our analysis we use a1=1.12,a2=-0.26.Similar toJ/ψ→PP decays, the factorization scheme expresses weak decay amplitudes as a product of matrix elements of the weak currents (up to the scale (GF/2)×(CKMfactor×QCDfactor)) as
(40)A(ϒ→PP)~<P|Jμ|0><P|Jμ|ϒ>.For instance, the decay amplitude for the color-enhanced modeϒ→Bc+π- of the CKM-favored decays is given by
(41)A(ϒ→Bc+π-)=GF2VcbVud*a1fπ(2mϒ)A0ϒ→Bc(mπ2).


#### 6.3.1.ϒ→PP  Decays

Following the procedure employed in Sections3 and 4, we calculate the branching ratios for CKM-favored mode both for fixed ω=0.40GeV and for flavor dependent ω, which are presented in columns 2 and 3, respectively, of Table 11. In addition to the decay constants given in (30) we use fD=0.208GeV and fDs=0.273GeV. The dominant decays in this mode are found to be B(Υ→Bc+Ds-)=0.47×10-10(2.25×10-17) and B(Υ→Bc+π-)=0.16×10-10(7.90×10-16), where values in the parentheses are calculated at fixed ω=0.40GeV. It may be noted that our branching ratios including the flavor dependent effects compare well with the earlier results, B(Υ→Bc+Ds-)=0.76×10-10 and B(Υ→Bc+π-)=0.33×10-10, obtained by Sharma and Verma [7] using HQET considerations.Table 11
Branching ratios (in the units of10-10) of Υ→BcP decays.
Decays
This Work
[7]
ω
=
0.40 GeV
Using flavor dependentω
Δ
b
=
1
,
Δ
C
=
1
,
Δ
S
=
0
Υ
→
B
c
+
π
-
7.90
×
10
-
6
0.16
±
0.01
0.33
Δ
b
=
1
,
Δ
C
=
0
,
Δ
S
=
-
1
Υ
→
B
c
+
D
s
-
2.25
×
10
-
5
0.47
±
0.01
0.76
Δ
b
=
1
,
Δ
C
=
1
,
Δ
S
=
-
1
Υ
→
B
c
+
K
-
6.06
×
10
-
7
0.013
0.024
Δ
b
=
1
,
Δ
C
=
0
,
Δ
S
=
0
Υ
→
B
c
+
D
-
7.25
×
10
-
7
0.015
-
0.00
+
0.01
0.031


#### 6.3.2.ϒ→PV  Decays

Employing the decay rate formula for such decays as discussed in Section3 and following the similar procedure used for J/ψ→PV decays, we determine the decay amplitudes for various  Υ→BcV decays for the CKM-favored modes. We use [25, 26] fD*=0.245GeV and fD*s=fDs=0.273GeV to calculate the branching ratios which are given in Table 12. We observe that the dominant mode is B(Υ→Bc+Ds*-)=1.79×10-10(5.01×10-15), following by B(ϒ→Bc+ρ-)  =  0.65×10-10(2.12×10-15). Here also, the values in the parentheses are those obtained for fixed ω=0.40GeV.  For comparison of the contributions of the Helicity amplitudes H0 and H±1 involved, we have calculated the corresponding branching ratios BR0,BR-, and BR+for various ϒ→PV decays both for ω=0.40GeV and for flavor dependent ω,  which are presented in Table 13.Table 12
Branching ratios (in the units of10-10) of Υ→BcV decays.
Decays
This Work
[7]
ω
=
0.40 GeV
Using flavor dependentω
Δ
b
=
1
,
Δ
C
=
1
,
Δ
S
=
0
Υ
→
B
c
+
ρ
-
2.12
×
10
-
5
0.65
±
0.03
0.88
Δ
b
=
1
,
Δ
C
=
0
,
Δ
S
=
-
1
Υ
→
B
c
+
D
s
*
-
5.01
×
10
-
5
1.79
-
0.04
+
0.09
2.57
Δ
b
=
1
,
Δ
C
=
1
,
Δ
S
=
-
1
Υ
→
B
c
+
K
*
-
1.14
×
10
-
6
0.035
±
0.02
0.050
Δ
b
=
1
,
Δ
C
=
0
,
Δ
S
=
0
Υ
→
B
c
+
D
*
-
2.06
×
10
-
6
0.073
-
0.002
+
0.003
0.11Table 13
Branching ratiosBR0,BR-, and BR+ corresponding to the  Helicity amplitudes H0 and H±1 (in the units of  10-10) of  Υ→BcV decays.
Decays
ω = 0.40 GeV
Using flavor dependentω
B
R
0
B
R
-
B
R
+
B
R
0
B
R
-
B
R
+
Δ
b
=
1
,
Δ
C
=
1
,
Δ
S
=
0
Υ
→
B
c
+
ρ
-
1.82
×
10
-
5
2.48
×
10
-
6
6.53
×
10
-
7
0.53
-
0.01
+
0.02
0.09
-
0.00
+
0.01
0.024
-
0.01
+
0.02
Δ
b
=
1
,
Δ
C
=
0
,
Δ
S
=
-
1
Υ
→
B
c
+
D
s
*
-
1.94
×
10
-
5
2.30
×
10
-
5
7.73
×
10
-
6
0.62
-
0.01
+
0.02
0.88
-
0.02
+
0.04
0.29
-
0.01
+
0.01
Δ
b
=
1
,
Δ
C
=
1
,
Δ
S
=
-
1
Υ
→
B
c
+
K
*
-
9.20
×
10
-
5
1.71
×
10
-
8
4.55
×
10
-
8
0.027
-
0.000
+
0.001
0.007
-
0.001
+
0.001
0.017
Δ
b
=
1
,
Δ
C
=
0
,
Δ
S
=
0
Υ
→
B
c
+
D
*
-
8.75
×
10
-
7
2.28
×
10
-
7
2.97
×
10
-
7
0.027
-
0.001
+
0.001
0.035
-
0.001
+
0.001
0.010
-
0.001
+
0.002


## 6.1.ϒ→Bc Form Factors and Branching Ratios

Using the framework described in Sections4 and 5, we extend our analysis for bottom sector. We obtain the form factor for ϒ→Bc transition at both ω=0.40GeV  and using the flavor dependent ω, which are compared in rows 2 and 3 of Table 8. Note that the form factors get enhanced by several orders of magnitude for flavor dependent ω. This happens due to the increased overlap of Υ and the final state wave functions for flavor dependent ω in comparison to that at fixed ω=0.40GeV as shown in Figures 2 and 3. We wish to remark that the flavor dependent form factors come close to the expectation [7] based on HQET considerations (row 4 of Table 8). Consequently, the branching ratios of semileptonic and nonleptonic weak decays of Υ get significantly enhanced.Table 8
Form factors ofΥ→Bc transitions.
Models
A
0
Υ
B
c
(
0
)
A
1
Υ
B
c
(
0
)
A
2
Υ
B
c
(
0
)
V
Υ
B
c
(
0
)
This work
ω
=
0.40 GeV
0.003
0.003
0.003
0.008
Using flavordependent ω
0.46
±
0.01
0.62
-
0.01
+
0.02
0.38
-
0.09
+
0.06
1.61
±
0.01
QCD inspiredω≡αsm*
0.46
0.62
0.38
1.62
[7]
0.98
1.01
1.01
1.01

*The form factors here have shown no significant change in the values,  therefore, the BRs for the decays involving Υ→Bc form factors for QCD inspired   ω  remain the same as for flavor dependent  ω.Figure 2
The overlap of wave functions forϒ→Bc decays at ωΥ=ωBc=0.40GeV.Figure 3
The overlap of wave functions forϒ→Bc decays at ωΥ=1.80GeV and ωBc=0.96GeV.We also use the QCD inspired method to determineω, that is,  ωΥ=1.83, which match well with our value 1.80 for Υ.  Subsequently obtained form factors are given in row 4 of Table 8, which are largely the same as our results, so we exclude the results for decay rates based on the QCD inspired method for further discussion.


## 6.2. Semileptonic Weak Decays ofϒ

Usingϒ→Bc form factor, appearing in b→c transition, and the decay rate formula given in (5), firstly, we calculate the branching ratios for semileptonic decays of Υ at fixed ω=0.40GeV, which are given in column 2 of Table 9. The predicted branching ratios of semileptonic weak decays of Υ using the flavor dependent effects are given in column 3 of Table 9. We find that the branching ratio of dominant semileptonic decay is B(Υ→Bc+e-ν-e)=1.70×10-10 that is sufficiently enhanced in comparison to (6.28×10-15) obtained using fixed ω. Here also, we give  branching ratios of longitudinal and transverse components (BR0,BR-, and BR+) separately, for the semileptonic decays in Table 10.Table 9
Branching ratios (in the units of10-10) of Υ→Plν- decays.
Decays
This Work
ω
=
0.40 GeV
Using flavordependent ω
Υ
→
B
c
+
e
-
ν
-
e
6.28
×
10
-
5
1.70
-
0.02
+
0.03
Υ
→
B
c
+
μ
-
ν
-
μ
6.24
×
10
-
5
1.69
-
0.02
+
0.04
Υ
→
B
c
+
τ
-
ν
-
τ
9.86
×
10
-
5
0.29
-
0.02
+
0.05Table 10
Branching ratiosBR0,BR-, and BR+ corresponding to the  Helicity amplitudes h0 and h±1 (in the units of 10-10) of Υ→Plν- decays.
Decays
ω = 0.40 GeV
Using flavor dependentω
B
R
0
B
R
-
B
R
+
B
R
0
B
R
-
B
R
+
Υ
→
B
c
+
e
-
ν
-
e
5.50
×
10
-
5
0.46
×
10
-
5
3.10
×
10
-
5
1.40
-
0.02
+
0.03
0.18
-
0.00
+
0.01
0.13
-
0.00
+
0.01
Υ
→
B
c
+
μ
-
ν
-
μ
5.47
×
10
-
5
0.45
×
10
-
5
3.08
×
10
-
5
1.39
-
0.02
+
0.03
0.18
-
0.00
+
0.01
0.12
-
0.00
+
0.01
Υ
→
B
c
+
τ
-
ν
-
τ
0.86
×
10
-
5
0.81
×
10
-
6
0.45
×
10
-
6
0.24
-
0.01
+
0.00
0.03
0.018


## 6.3. Nonleptonic Weak Decays ofϒ

In this section, the analysis is extended toϒ→PP/PV decays. The effective weak Hamiltonian generating the dominant b quark decays involving b→c transition is given by
(39)HWΔb=1=GF2{VcbVud*[a1(c-b)(d-u)+a2(d-b)(c-u)]+VcbVcs*[a1(c-b)(s-c)+a2(s-b)(c-c)]}+h.c.,
for the CKM-favored mode. In our analysis we use a1=1.12,a2=-0.26.Similar toJ/ψ→PP decays, the factorization scheme expresses weak decay amplitudes as a product of matrix elements of the weak currents (up to the scale (GF/2)×(CKMfactor×QCDfactor)) as
(40)A(ϒ→PP)~<P|Jμ|0><P|Jμ|ϒ>.For instance, the decay amplitude for the color-enhanced modeϒ→Bc+π- of the CKM-favored decays is given by
(41)A(ϒ→Bc+π-)=GF2VcbVud*a1fπ(2mϒ)A0ϒ→Bc(mπ2).


### 6.3.1.ϒ→PP  Decays

Following the procedure employed in Sections3 and 4, we calculate the branching ratios for CKM-favored mode both for fixed ω=0.40GeV and for flavor dependent ω, which are presented in columns 2 and 3, respectively, of Table 11. In addition to the decay constants given in (30) we use fD=0.208GeV and fDs=0.273GeV. The dominant decays in this mode are found to be B(Υ→Bc+Ds-)=0.47×10-10(2.25×10-17) and B(Υ→Bc+π-)=0.16×10-10(7.90×10-16), where values in the parentheses are calculated at fixed ω=0.40GeV. It may be noted that our branching ratios including the flavor dependent effects compare well with the earlier results, B(Υ→Bc+Ds-)=0.76×10-10 and B(Υ→Bc+π-)=0.33×10-10, obtained by Sharma and Verma [7] using HQET considerations.Table 11
Branching ratios (in the units of10-10) of Υ→BcP decays.
Decays
This Work
[7]
ω
=
0.40 GeV
Using flavor dependentω
Δ
b
=
1
,
Δ
C
=
1
,
Δ
S
=
0
Υ
→
B
c
+
π
-
7.90
×
10
-
6
0.16
±
0.01
0.33
Δ
b
=
1
,
Δ
C
=
0
,
Δ
S
=
-
1
Υ
→
B
c
+
D
s
-
2.25
×
10
-
5
0.47
±
0.01
0.76
Δ
b
=
1
,
Δ
C
=
1
,
Δ
S
=
-
1
Υ
→
B
c
+
K
-
6.06
×
10
-
7
0.013
0.024
Δ
b
=
1
,
Δ
C
=
0
,
Δ
S
=
0
Υ
→
B
c
+
D
-
7.25
×
10
-
7
0.015
-
0.00
+
0.01
0.031


### 6.3.2.ϒ→PV  Decays

Employing the decay rate formula for such decays as discussed in Section3 and following the similar procedure used for J/ψ→PV decays, we determine the decay amplitudes for various  Υ→BcV decays for the CKM-favored modes. We use [25, 26] fD*=0.245GeV and fD*s=fDs=0.273GeV to calculate the branching ratios which are given in Table 12. We observe that the dominant mode is B(Υ→Bc+Ds*-)=1.79×10-10(5.01×10-15), following by B(ϒ→Bc+ρ-)  =  0.65×10-10(2.12×10-15). Here also, the values in the parentheses are those obtained for fixed ω=0.40GeV.  For comparison of the contributions of the Helicity amplitudes H0 and H±1 involved, we have calculated the corresponding branching ratios BR0,BR-, and BR+for various ϒ→PV decays both for ω=0.40GeV and for flavor dependent ω,  which are presented in Table 13.Table 12
Branching ratios (in the units of10-10) of Υ→BcV decays.
Decays
This Work
[7]
ω
=
0.40 GeV
Using flavor dependentω
Δ
b
=
1
,
Δ
C
=
1
,
Δ
S
=
0
Υ
→
B
c
+
ρ
-
2.12
×
10
-
5
0.65
±
0.03
0.88
Δ
b
=
1
,
Δ
C
=
0
,
Δ
S
=
-
1
Υ
→
B
c
+
D
s
*
-
5.01
×
10
-
5
1.79
-
0.04
+
0.09
2.57
Δ
b
=
1
,
Δ
C
=
1
,
Δ
S
=
-
1
Υ
→
B
c
+
K
*
-
1.14
×
10
-
6
0.035
±
0.02
0.050
Δ
b
=
1
,
Δ
C
=
0
,
Δ
S
=
0
Υ
→
B
c
+
D
*
-
2.06
×
10
-
6
0.073
-
0.002
+
0.003
0.11Table 13
Branching ratiosBR0,BR-, and BR+ corresponding to the  Helicity amplitudes H0 and H±1 (in the units of  10-10) of  Υ→BcV decays.
Decays
ω = 0.40 GeV
Using flavor dependentω
B
R
0
B
R
-
B
R
+
B
R
0
B
R
-
B
R
+
Δ
b
=
1
,
Δ
C
=
1
,
Δ
S
=
0
Υ
→
B
c
+
ρ
-
1.82
×
10
-
5
2.48
×
10
-
6
6.53
×
10
-
7
0.53
-
0.01
+
0.02
0.09
-
0.00
+
0.01
0.024
-
0.01
+
0.02
Δ
b
=
1
,
Δ
C
=
0
,
Δ
S
=
-
1
Υ
→
B
c
+
D
s
*
-
1.94
×
10
-
5
2.30
×
10
-
5
7.73
×
10
-
6
0.62
-
0.01
+
0.02
0.88
-
0.02
+
0.04
0.29
-
0.01
+
0.01
Δ
b
=
1
,
Δ
C
=
1
,
Δ
S
=
-
1
Υ
→
B
c
+
K
*
-
9.20
×
10
-
5
1.71
×
10
-
8
4.55
×
10
-
8
0.027
-
0.000
+
0.001
0.007
-
0.001
+
0.001
0.017
Δ
b
=
1
,
Δ
C
=
0
,
Δ
S
=
0
Υ
→
B
c
+
D
*
-
8.75
×
10
-
7
2.28
×
10
-
7
2.97
×
10
-
7
0.027
-
0.001
+
0.001
0.035
-
0.001
+
0.001
0.010
-
0.001
+
0.002


## 6.3.1.ϒ→PP  Decays

Following the procedure employed in Sections3 and 4, we calculate the branching ratios for CKM-favored mode both for fixed ω=0.40GeV and for flavor dependent ω, which are presented in columns 2 and 3, respectively, of Table 11. In addition to the decay constants given in (30) we use fD=0.208GeV and fDs=0.273GeV. The dominant decays in this mode are found to be B(Υ→Bc+Ds-)=0.47×10-10(2.25×10-17) and B(Υ→Bc+π-)=0.16×10-10(7.90×10-16), where values in the parentheses are calculated at fixed ω=0.40GeV. It may be noted that our branching ratios including the flavor dependent effects compare well with the earlier results, B(Υ→Bc+Ds-)=0.76×10-10 and B(Υ→Bc+π-)=0.33×10-10, obtained by Sharma and Verma [7] using HQET considerations.Table 11
Branching ratios (in the units of10-10) of Υ→BcP decays.
Decays
This Work
[7]
ω
=
0.40 GeV
Using flavor dependentω
Δ
b
=
1
,
Δ
C
=
1
,
Δ
S
=
0
Υ
→
B
c
+
π
-
7.90
×
10
-
6
0.16
±
0.01
0.33
Δ
b
=
1
,
Δ
C
=
0
,
Δ
S
=
-
1
Υ
→
B
c
+
D
s
-
2.25
×
10
-
5
0.47
±
0.01
0.76
Δ
b
=
1
,
Δ
C
=
1
,
Δ
S
=
-
1
Υ
→
B
c
+
K
-
6.06
×
10
-
7
0.013
0.024
Δ
b
=
1
,
Δ
C
=
0
,
Δ
S
=
0
Υ
→
B
c
+
D
-
7.25
×
10
-
7
0.015
-
0.00
+
0.01
0.031


## 6.3.2.ϒ→PV  Decays

Employing the decay rate formula for such decays as discussed in Section3 and following the similar procedure used for J/ψ→PV decays, we determine the decay amplitudes for various  Υ→BcV decays for the CKM-favored modes. We use [25, 26] fD*=0.245GeV and fD*s=fDs=0.273GeV to calculate the branching ratios which are given in Table 12. We observe that the dominant mode is B(Υ→Bc+Ds*-)=1.79×10-10(5.01×10-15), following by B(ϒ→Bc+ρ-)  =  0.65×10-10(2.12×10-15). Here also, the values in the parentheses are those obtained for fixed ω=0.40GeV.  For comparison of the contributions of the Helicity amplitudes H0 and H±1 involved, we have calculated the corresponding branching ratios BR0,BR-, and BR+for various ϒ→PV decays both for ω=0.40GeV and for flavor dependent ω,  which are presented in Table 13.Table 12
Branching ratios (in the units of10-10) of Υ→BcV decays.
Decays
This Work
[7]
ω
=
0.40 GeV
Using flavor dependentω
Δ
b
=
1
,
Δ
C
=
1
,
Δ
S
=
0
Υ
→
B
c
+
ρ
-
2.12
×
10
-
5
0.65
±
0.03
0.88
Δ
b
=
1
,
Δ
C
=
0
,
Δ
S
=
-
1
Υ
→
B
c
+
D
s
*
-
5.01
×
10
-
5
1.79
-
0.04
+
0.09
2.57
Δ
b
=
1
,
Δ
C
=
1
,
Δ
S
=
-
1
Υ
→
B
c
+
K
*
-
1.14
×
10
-
6
0.035
±
0.02
0.050
Δ
b
=
1
,
Δ
C
=
0
,
Δ
S
=
0
Υ
→
B
c
+
D
*
-
2.06
×
10
-
6
0.073
-
0.002
+
0.003
0.11Table 13
Branching ratiosBR0,BR-, and BR+ corresponding to the  Helicity amplitudes H0 and H±1 (in the units of  10-10) of  Υ→BcV decays.
Decays
ω = 0.40 GeV
Using flavor dependentω
B
R
0
B
R
-
B
R
+
B
R
0
B
R
-
B
R
+
Δ
b
=
1
,
Δ
C
=
1
,
Δ
S
=
0
Υ
→
B
c
+
ρ
-
1.82
×
10
-
5
2.48
×
10
-
6
6.53
×
10
-
7
0.53
-
0.01
+
0.02
0.09
-
0.00
+
0.01
0.024
-
0.01
+
0.02
Δ
b
=
1
,
Δ
C
=
0
,
Δ
S
=
-
1
Υ
→
B
c
+
D
s
*
-
1.94
×
10
-
5
2.30
×
10
-
5
7.73
×
10
-
6
0.62
-
0.01
+
0.02
0.88
-
0.02
+
0.04
0.29
-
0.01
+
0.01
Δ
b
=
1
,
Δ
C
=
1
,
Δ
S
=
-
1
Υ
→
B
c
+
K
*
-
9.20
×
10
-
5
1.71
×
10
-
8
4.55
×
10
-
8
0.027
-
0.000
+
0.001
0.007
-
0.001
+
0.001
0.017
Δ
b
=
1
,
Δ
C
=
0
,
Δ
S
=
0
Υ
→
B
c
+
D
*
-
8.75
×
10
-
7
2.28
×
10
-
7
2.97
×
10
-
7
0.027
-
0.001
+
0.001
0.035
-
0.001
+
0.001
0.010
-
0.001
+
0.002


## 7. Summary and Discussion

In this paper, we have predicted the rare semileptonic and nonleptonic weak decays ofJ/ψ and Υ mesons. It may be mentioned that the present work differs from the previous ones [6, 7] based on the BSW framework in three aspects. (i)
Firstly, in the light of HQS based BSW2 model [12], we use the dipole q2 dependence for the form factors A0,A2, and V, while the earlier work [6, 7] used the monopole q2 dependence for these form factors.(ii)
Secondly, the results given in [7] are based on s-wave dominance, while our results take into account the contributions from p- and d-waves also.(iii)
Lastly, we incorporate flavor dependent effects onJ/ψ→P and  ϒ→Bc form factors in our analysis through the different values of ω for initial and final state mesons.Also, to support the case of flavor dependence of the form factors, we use an alternate QCD inspired approach to determine ω for heavy quarkonium cc-andbb- states. The following conclusions are readily drawn out of our analysis.  (A)
Weak decays ofJ/ψ.
(i)
Initially, we calculateJ/ψ→P transition form factors at ω=0.40GeV for all the mesons: among the Cabibbo-angle-enhanced J/ψ→PP decays, branching ratios of dominant decays are calculated to be B(J/ψ→Ds+π-)=3.32×10-10 and (J/ψ→D0K0)=0.72×10-10  for Cabibbo-angle-enhanced mode J/ψ→PV decays, branching ratios of dominant decays are obtained to be B(J/ψ→Ds+ρ-)=1.77×10-9 and B(J/ψ→D0K*-)=2.51×10-10.(ii)
We investigate the effects of possible flavor dependence ofω by determining |ψ(0)|2 from meson masses to fix ω for all the mesons through an ansatz; we, then, calculated J/ψ→P transition form factors which get significantly enhanced as compared to those at fixed ω=0.40; as a result of this, branching ratios of all semileptonic as well as nonleptonic decays are enhanced.(iii)
Among the Cabibbo-angle-enhancedJ/ψ→PP/PV decays, the dominant decays are predicted as B(J/ψ→Ds+π-) = 7.41-0.23+0.13×10-10,  B(J/ψ→D0K0)   = 1.39-0.14+0.10×10-10,  B(J/ψ→Ds+ρ-)  =  5.11-0.60+0.76×10-9, and  B(J/ψ→D0K*0)=7.61-1.2+1.6×10-10.(B)
Weak decays ofΥ:
(i)
It is observed that inclusion of flavor dependent effects throughω significantly enhances the form factors and, consequently, the branching ratios of semileptonic and nonleptonic weak decays of Υ.(ii)
Branching ratio of the dominating semileptonic mode is predicted to beB(Υ→Bc+e-ν-e)=1.70×10-10.(iii)
Among the Cabibbo-angle-enhancedΥ→BcP/BcV decay modes, the dominant ones are predicted as  (Υ→Bc+Ds-)=0.47×10-10, B(Υ→Bc+π-)=0.16×10-10,  B(ϒ→Bc+Ds*-)=1.79×10-10,  and B(ϒ→Bc+ρ-)=0.65×10-10.(iv)
Though the predicted branching ratios in the QCD inspired method of obtainingω are marginally changed, the alternate calculation of ω for heavy quarkonium cc-andbb-  states seems to support the flavor dependent effects on rare weak decays of J/ψ and Υ mesons.(v)
Our predictions also agree well with those obtained by Sharma and Verma [7] using the HQET considerations.It is hoped that these branching ratios would lie in the detectable range and may be measured in future experiments.



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