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arXiv:hep-ph/0610223v3 18 Dec 2006

Study of electromagnetic decay of J/ψ and ψ′to vector and pseudoscalar

Qiang Zhao1,2, Gang Li1

1) Institute of High Energy Physics, Chinese Academy of Sciences, Beijing, 100049, P.R. China and

2) Department of Physics, University of Surrey, Guildford, GU2 7XH, United Kingdom

Chao-Hsi Chang3

3) Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing, 100080, P.R. China

(Dated: February 2, 2008)

The electromagnetic decay contributions to J/ψ(ψ′) → V P, where V and P stand for vector and

pseudoscalar meson, respectively, are investigated in a vector meson dominance (VMD) model. We

show that J/ψ(ψ′) → γ∗→ V P can be constrained well with the available experimental information.

We find that this process has significant contributions in ψ′→ V P and may play a key role in

understanding the deviations from the so-called “12% rule” for the branching ratio fractions between

ψ′→ V P and J/ψ → V P. We also address that the “12% rule” becomes very empirical in exclusive

hadronic decay channels.

PACS numbers: 12.40.Vv, 13.20.Gd, 13.25.-k

The decay of charmonia into light hadrons is rich of information about QCD strong interactions between

quarks and gluons. Due to the flavor change in the c¯ c annihilation, it is also ideal for the study of light

hadron production mechanisms, and useful for probing their flavor and gluon contents, such as the

search for experimental evidence for glueball and hybrid. In the past decade, data for J/ψ decays have

experienced a drastic improvement. We now not only have access to small branching ratio at order of

10−6, but also have much precise measurements of most of those old channels from BES, DM2 and Mark-

III. Such a significant improvement will allow a systematic analysis of correlated channels, from which

we expect that dynamical information about the light hadron production mechanisms can be extracted.

In this work, we will study the electromagnetic (EM) decay of vector charmonia (J/ψ and ψ′) into light

vector and pseudoscalar. From an empirical viewpoint, one can separate the decays of J/ψ(ψ′) → V P

into two classes: i) Isospin conserved channels such as J/ψ(ψ′) → ρπ, K∗¯K, ωη, φη, etc. These are

decays via both strong and EM transitions; ii) Isospin violated channels such as J/ψ(ψ′) → ρη, ρη′, ωπ0,

and φπ, of which the leading decay amplitudes are from EM transitions. In association with the above

separation is the observation that branching ratios for some of those isospin violated channels [1], such as

J/ψ(ψ′) → ρη, ρη′and ωπ0, are compatible with the isospin conserved ones such as ωη′and φη′in J/ψ

decays, and ρπ, ωη, ωη′, φη, φη′in ψ′decays. This observation shows that the EM transition may not be

as small as we thought in comparison with the strong one. Therefore, its roles played in J/ψ(ψ′) → V P

should be closely investigated.

On a more general ground, the decay channel J/ψ(ψ′) → V P has attracted a lot of attention in the

literature due to its property that the characteristic pQCD helicity conservation rule is violated here [2].

As a consequence, the pQCD power suppression occurs in this channel and leads to a relation for the

ratios between J/ψ and ψ′annihilating into three gluons and a single direct photon:

R ≡

BR(ψ′→ hadrons)

BR(J/ψ → hadrons)

BR(ψ′→ e+e−)

BR(J/ψ → e+e−)≃ 12%,

≃

(1)

which is empirically called “12% rule”. However, much stronger suppressions are found in ρπ channel,

i.e. BR(ψ′→ ρπ)/BR(J/ψ → ρπ) ≃ (2.0 ± 0.9) × 10−3, which gives rise to the so-called “ρπ puzzle”.

Namely, there exist large deviations from the above empirical “12% rule” in exclusive channels such as

ρπ and K∗¯K + c.c.

As we know that the “12% rule” is based on the expectation that the charmonium 3g strong decays

are the dominant ones in exclusive decay channels, the large deviations from the “12% rule” in ρπ and

K∗¯K + c.c. naturally imply some underlying mechanisms which can interfere with the 3g decays and

change the branching ratio fractions. Theoretical explanations for the “12% rule” deviations have been

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proposed in the literature [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], but so far none of those

solutions has been indisputably agreed [18, 19]. This makes it necessary to provide a detailed study

of the charmonium EM decays. If compatible strength of the EM transition occurs in some of those

exclusive decay channels, one can imagine that large interferences between the EM and strong transitions

are possible, and they may be one of the important sources which produce large deviations from the

“12% rule” in V P decay channels. Relevant studies can also be found in the literature for understanding

the role played by the EM transitions in J/ψ decays. Parametrization schemes are proposed to estimate

the EM decay contributions to J/ψ → V P in Refs. [20, 21], but a coherent study of J/ψ and ψ′is still

unavailable.

Apart from the above interests, the EM decay of J/ψ → V P is also rich of dynamical information about

the Okubo-Zweig-Iizuka (OZI) rule [22]. The decay of J/ψ → φπ0involves both isospin and OZI-rule

violations. Although only the upper limit, BRexp(J/ψ → φπ0) < 6.4 × 10−6, is given, this will be an

interesting place to test dynamical prescriptions for J/ψ → γ∗→ V P. In Refs. [5, 8], apart from the

EM process, the OZI doubly disconnected processes are also investigated, which however possesses large

uncertainties. In particular, the separation of these two correlated processes is strongly model-dependent.

In this work, we will introduce an effective Lagrangian for V γP couplings, and apply the vector meson

dominance (VMD) model to V γ∗couplings. By studying the J/ψ(ψ′) → γ∗→ V P at tree level, we shall

examine the “12% rule” for those exclusive V P decay channels. Deviations from this empirical rule in

the exclusive decays can thus be highlighted.

For J/ψ(ψ′) → ωη, ωη′, φη, φη′, ρπ and K∗¯K + c.c., the strong and EM decay process are mixed,

and the former generally plays a dominant role. For J/ψ(ψ′) → ρη, ρη′, ωπ, and φπ, the transitions are

via EM processes, of which the isospin conservation is violated. Typical transitions for V1 → V2P are

illustrated by Fig. 1, which consists of three contributions: (a) the process that the pseudoscalar meson

is produced in association with the virtual photon via V1annihilation; (b) the pseudoscalar produced at

the final state vector meson V2vertex; and (c) the pseudoscalar produced via the axial current anomaly.

Note that isospin conservation can be violated in both Fig. 1(a) and (b), with the observation of non-zero

branching ratios for J/ψ → γπ0[1]. At hadronic level, these are independent processes where all the

vertices can be determined by other experimental measurements. This treatment is different from that of

Refs. [5, 8]. Although the OZI disconnected diagram considered in Refs. [5, 8] is similar to our Fig. 1(b),

our consideration of the V γ∗P coupling will allow us to include both OZI and isospin violation effects

which can be constrained by experimental data.

We introduce a typical effective Lagrangian for the V γP coupling:

LV γP=gV γP

MV

ǫµναβ∂µVν∂αAβP (2)

where Vν(= ρ, ω, φ, J/ψ, ψ′...) and Aβare the vector meson and EM field, respectively; MV is the

vector meson mass; ǫµναβis the anti-symmetric Levi-Civita tensor.

The V γ∗coupling is described by the VMD model,

LV γ=

?

V

eM2

fV

V

VµAµ,(3)

where eM2

1 and 0 component of the EM field are both included.

The invariant transition amplitude for V1→ γ∗→ V2P can thus be expressed as:

V/fV is a direct photon-vector-meson coupling in Feynman diagram language, and the isospin

M ≡ MA+ MB+ MC

?

fV 2

=

e

gV 1γP

MV 1

Fa+

e

fV 1

gV 2γP

MV 2

Fb+

e2

fV 1fV 2

gPγγ

MP

Fc

?

ǫµναβ∂µVν

1∂αVβ

2P(4)

where gPγγis the coupling for the neutral pseudoscalar meson decay to two photons; Faand Fbdenote the

form factor corrections to the V γ∗P vertices in comparison with the real photon transition for V → γP;

and Fcis the form factor for P → γ∗γ∗.

The partial decay width can thus be expressed as

Γ(V1→ V2P) =

|pv2|

8πM2

V 1

1

2S + 1

?

|M|2

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3

=

|pv2|3

12π

?

e

fV 2

gV 1γP

MV 1

Fa+

e

fV 1

gV 2γP

MV 2

Fb+

e2

fV 1fV 2

gPγγ

MP

Fc

?2

, (5)

where S = 1 is the spin of the initial vector meson; |pv2| is the three-momentum of the final state vector

meson in the initial-vector-meson-rest frame.

Those three typical coupling constants are determined as follows:

(I) For V → γP decay, following the effective Lagrangian of Eq. (2), we derive the coupling constant:

gV γP=

?12πM2

VΓexp(V → γP)

|pγ|3

?1/2

, (6)

where |pγ| is the three-momentum of the photon in the initial vector meson rest frame; Γexp(V → γP)

is the vector meson radiative decay partial width, and available in experiment.

For ργη′and ωγη′couplings, we determine the coupling constants in η′→ γρ and γω:

gV γη′ =

?4πM2

VΓexp(η′→ γV )

|p′γ|3

?1/2

, (7)

where |p′

(II) The V γ∗coupling is determined in V → e+e−channel. With the partial decay width ΓV →e+e−,

the coupling constant e/fV can be derived:

γ| is the three-momentum of the photon in the η′rest frame.

e

fV

=

?3ΓV →e+e−

4αe|pe|

?1/2

, (8)

where we have neglected the mass of the electron and |pe| is the electron three-momentum in the vector

meson rest frame; αe= 1/137 is the fine-structure constant.

(III) For P → γγ, we adopt the following form of effective Lagrangian:

LPγγ=gPγγ

MP

ǫµναβ∂µAν∂αAβP , (9)

where the coupling constant is normalized to the pseudoscalar meson mass MP. With the partial decay

width Γexp(P → γγ) the coupling constant for real photon in the final state can be derived:

gPγγ= [32πΓexp(P → γγ)/MP]1/2.(10)

It is encouraging that for all the decay channels of J/ψ(ψ′) → γ∗→ V P, the experimental data are

available for determining the above coupling constants: gV γP, e/fV, and gPγγ[1]. We are then left with

the only uncertainty from the form factors due to the exchange of off-shell photons.

We find that without form factors, i.e. Fa = Fb = Fc = 1, the calculated branching ratios for the

isospin violated channels will be significantly overestimated. This is expected due to the large virtual-

ities of the off-shell photons and the consequent power suppressions from the pQCD hadronic helicity-

conservation [2]. Since we think that the non-perturbative QCD effects might have played a role in the

transitions at J/ψ energy1, e.g. in Fig. 1(a) and (b) a pair of quarks may be created from vacuum as

described by3P0model, the pQCD hadronic helicity-conservation due to the vector nature of gluon is

violated quite strongly, thus alternatively, we would like to suggest a monopole-like (MP) form factor

dedicated to the suppression effects:

F(q2) =

1

1 − q2/Λ2,(11)

1For instance such as in the inclusive decay J/ψ → q¯ q, the ‘virtualness’ of one quark in the created quark pair is less

than chiral broken energy scale Λχ∼ 1.0 GeV, thus, the non-perturbative QCD effects must be sizeable in the concerned

decays here.

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where Λ can be regarded as an effective mass accounting for the overall effects from possible resonance

poles and scattering terms in the time-like kinematic region, and will be determined by fitting the data [1]

for J/ψ(ψ′) → ρη, ρη′, ωπ0, and φπ0.

It should be noted that this MP form factor can only partly depict the pQCD power suppression due

to violations of the hadronic helicity conservation when q2≫ Λ2[2], but it is quite consistent with the

VMD framework.

By adopting the MP empirical form factor, we have already assumed that non-perturbative effects

might have played a substantial role in the transitions. In principle it should be tested experimentally

via measuring the coupling the processes J/ψ(ψ′) → Pe+e−and e+e−→ Pe+e−, respectively, when the

integrated luminosity at J/ψ and the suitable energies for e+e−colliders is accumulated enough.

The form factor Fcappearing in Eqs. (4) and (5) can be determined in γ∗γ∗scatterings. A commonly

adopted form factor is

Fc(q2

1,q2

2) =

1

(1 − q2

1/Λ2)(1 − q2

2/Λ2), (12)

where q2

assume that the Λ is the same as in Eq. (11), thus, Fc= FaFb.

Proceeding to the numerical calculations, we first determine the coupling constants, e/fV, gV γP and

gPγγin the corresponding decays, and the results are listed in Tables I-III. It shows that the e/fρcoupling

is the largest one while all the others are compatible. For the gV γP, it is sizeable for light vector mesons

and much smaller for J/ψ and ψ′. Note that there is no datum for ψ′→ γπ available. So we simply put

gψ′γπ= 0 in the calculations. The Pγγ couplings can be well determined due to the good shape of the

data [1].

To examine the role played by the form factors, we first calculate the EM decay branching ratios

without form factors, i.e. F(q2) = 1. It shows that all the data are significantly overestimated by the

theoretical predictions as shown by Tables IV and V. Nonetheless, it shows that without form factors

process (b) is the only dominant transition.

To determine the effective mass Λ in the MP form factor, we consider two possible relative phases

between process (a) and (b) in fitting the data for the isospin violated channels, J/ψ(ψ′) → ρη, ρη′,

and ωπ. We mention in advance that the contributions from process (c) will bring only few percent

corrections to the results. Since the corrections are within the datum uncertainties, we are not bothered

to consider its relative phase to process (a) and (b). In Tables IV and V, the results for process (a) and

(b) in a constructive phase (MP-C) with Λ = 0.616 ± 0.008 GeV, and in a destructive phase (MP-D)

with Λ = 0.65 ± 0.01 GeV are listed. The reduced χ2values are χ2= 4.1 in MP-C and 14.2 in MP-D,

respectively.

With the above fitted values for Λ (MP-C and MP-D), predictions for those isospin conserved channels

in J/ψ → γ∗→ V P and ψ′→ γ∗→ V P are listed in Table IV and V to compare with the experimental

data [1].

There arise some basic issues from the theoretical results.

(I) We find that even though with the form factors, process (b) in Fig. 1 is still the dominant one in most

channels except for ρη′. For most channels the couplings gJ/ψγPand gψ′γP in process (a) are generally

small, and similarly are e2/(fV 1fV 2) and gPγγ in process (c). However, we find that the amplitudes of

process (a) and (b) are compatible in J/ψ → ρη′. As shown in Table IV, large cancellations appear in the

branching ratio when (a) and (b) are in a destructive phase (Column MP-D). This is due to the relatively

large branching ratios for J/ψ → γη′[1]. Such a large difference between these two phases makes the ρη′

channel extremely interesting. The branching ratio fraction will be useful for distinguishing the relative

phases between (a) and (b) in the isospin violated channels. It also highlights the empirical aspect of the

pQCD “12% rule” in exclusive hadronic decays.

We also note that process (a) and (c) do not contribute to K∗¯K+c.c. and ρ+π−+c.c. This turns to be

an advantage for understanding the decay mechanism of J/ψ(ψ′) → γ∗→ ρ+π−+ c.c. and K∗¯K + c.c.,

and should be also an ideal place to test the “12% rule” in exclusive decays.

To be more specific, we analyze first those four isospin violation decays: J/ψ(ψ′) → γ∗→ ρη, ρη′,

ωπ0and φπ0. These decays to leading order are through EM transitions. Transitions of Fig. 1 have

shown how the kinematic and form-factor corrections can correlate with the naive pQCD expected ratio:

Γ(ψ′→ e+e−)/Γ(J/ψ → e+e−), i.e. the “12% rule”, and makes it very empirical.

1= M2

V 1and q2

2= M2

V 2are the squared four-momenta carried by the time-like photons. We

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As an example, for those channels dominated by process (b), the exclusive decays are still approximately

proportional to the charmonium wavefunction at its origin, i.e. |ψ(0)|2, by neglecting the contributions

from process (a) and (c). The branching ratio fraction can be expressed as:

RV P≡

BR(ψ′→ γ∗→ V P)

BR(J/ψ → γ∗→ V P)

≃

BR(ψ′→ e+e−)

BR(J/ψ → e+e−)

|pe|

|p′e|

|p′

|pv2|3

v2|3

F2

F2

b(M2

b(M2

ψ′)

J/ψ), (13)

where |pe| and |p′

meson rest frame, respectively; while |pv2| and |p′

in J/ψ → V2P and ψ′→ V2P, respectively. It shows that the respect of the “12% rule” requires that

the kinematic and form factor corrections cancel each other for all those channels, which however, is not

a necessary consequence of the physics at all. Including the contributions from process (a) and (c) will

worsen the situation.

To see this more clearly, we list the branching ratio fractions for the choice of MP-C (RV P

(RV P

2

) and without form factors (RV P

1

) in Table VI to compare with the data. It shows that without the

form factor corrections, ratio RV P

1

has values in a range of (19 ∼ 21)% for those four channels, which are

larger than the expectation of the “12% rule”.

With the form factors, it shows that RV P

3

has a stable range of (7 ∼ 9)%, while more drastic changes

occur to RV P

2

. For instance, we obtain Rρπ

the data still have large uncertainties. We expect that an improved branching ratio fraction for this

channel will be able to determine the relative phase between process (a) and (b) in our model, and

highlight the underlying mechanism.

The branching ratios for φπ0channel are much smaller than others due to the small φγπ coupling.

This is in a good agreement with the OZI rule suppressions expected in φπ0channel.

(III) For the isospin conserved channels, the EM decay contributions in J/ψ decays turn out to be

rather small in both MP-D and MP-C phases. This is consistent with studies in the literature that

J/ψ → V P is dominated by the 3g transitions. Thus, the deviation of the “12% rule” could be more

likely due to the suppression of the amplitudes in ψ′→ V P (see the review of Ref. [18]).

If we simply apply the relations between the strong and EM transitions parametrized by Ref. [21], the

ratio between charged and neutral channels can be expressed as:

e| are three-momenta of the electron in J/ψ → e+e−and ψ′→ e+e−in the vector

v2| are three momenta of the final state vector mesons

3

), MP-D

2 = 52% which strongly violates the “12% rule”. Unfortunately,

Q ≡BR(ψ′→ K∗+K−+ c.c.)

BR(ψ′→ K∗0 ¯

K0+ c.c.)

≃

[g(1 − s) + e]2

[g(1 − s) − 2e]2,(14)

where g and e denote the strong and EM decay strengths, respectively, and s ≃ 0.1 is a parameter for

the flavor SU(3) breaking. One can see that for e = (−1/3 ∼ −1/2)×g(1−s), we have Q ≃ 0.06 ∼ 0.16,

which is in a good agreement with the data, 0.08 ∼ 0.28 [1].

We can also check the other two correlated relations:

BR(ψ′→ γ∗→ K∗+K−+ c.c.)

BR(ψ′→ K∗+K−+ c.c.)

≃

e2

[g(1 − s) + e]2≃ 0.25 ∼ 1, (15)

corresponding to e = (−1/3 ∼ −1/2) × g(1 − s). This is consistent with the range of BRMP

0.22 ∼ 0.56 and BRMP

D

/BRexp≃ 0.28 ∼ 0.70.

Similarly, for ψ′→ K∗0 ¯

K0+ c.c. we have

C

/BRexp≃

BR(ψ′→ γ∗→ K∗0 ¯

BR(ψ′→ K∗0 ¯

K0+ c.c.)

K0+ c.c.)

≃

4e2

[g(1 − s) − 2e]2≃ 0.16 ∼ 0.44, (16)

corresponding to the same range for e (e = (−1/3 ∼ −1/2) × g(1 − s)). It also turns to be compatible

with BRMP

C

/BRexp≃ 0.10 ∼ 0.15 and BRMP

D

/BRexp≃ 0.12 ∼ 0.18.

For ρπ channel, the above relative phase between the strong and EM transitions can explain the

relatively small branching ratios for ψ′→ ρπ, i.e. the EM amplitude will destructively interfere will the

strong one. With e = (−1/3 ∼ −1/2)g(1− s) ≃ (−1/3 ∼ −1/2)g [21], we have a relation:

BR(ψ′→ γ∗→ ρπ)

BR(ψ′→ ρπ)

≃

e2

(g + e)2= 0.25 ∼ 1, (17)

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which is also consistent with BRMP

The above analysis suggests that the relative phase between the EM and strong transition amplitudes in

ψ′→ ρπ favors 180◦. But due to the large uncertainties with the data, other phases may be possible [24].

Evidently, for those channels of which the branching ratio fractions between ψ′and J/ψ are observed

to deviate from the “12% rule” (see Table VI), such as ρπ and K∗¯K + c.c., they turn to have sizeable

EM contributions as shown by Tables IV and V. Since the J/ψ decays are still dominated by the strong

transitions, the measured branching ratio fractions mostly reflect the interfered effects in ψ′decays over

the strong transitions in J/ψ → V P.

We also analyze EM decay contributions to ψ(3770) → V P, and find they becomes negligibly small

due to the small partial width for ψ(3770) → e+e−and form factor suppressions. In most channels, the

EM transitions contribute to the branching ratios at O(10−8) or even smaller, which is beyond the access

of the present experimental facilities. Therefore, we can conclude that for those measured ψ(3770) → V P

channels the isospin violations must be negligibly small.

This approach can also be applied to the study of the EM decay contributions to φ → ρπ and ωπ.

For the isospin violated ωπ0channel, with the same form factor (MP-C), we find BR(φ → ωπ0) =

1.8 × 10−6and BR(φ → ρπ) = 3.2 × 10−6in contrast with the the experimental data, BRexp(φ →

?

−1.1

quite close in mass, the results should be sensitive to the form factors, and more elaborate treatment for

them are generally required. If we slightly adjust Λ, we can reproduce the data for ωπ0well. For ρπ

channel the EM contributions are found much smaller than the data which suggests the dominance of

strong decays in φ → ρπ.

In summary, we make a systematic analysis of the EM decay contributions to J/ψ(ψ′) → γ∗→ V P

essentially in a VMD model. With the constraint from the available independent experimental data, the

EM contributions in all the V P hadronic channels can be consistently investigated. Our results are in a

good agreement with the data for those isospin violated channels such as ρη, ρη′, ωπ, and φπ, especially

for the ‘MP-C’ model (see Tables IV,V and VI). For most of those isospin conserved channels, we find

that the EM decay contributions can bring sizeable corrections to the ψ′→ V P, which could be a major

source accounting for the branching ratio fraction deviations between the V P exclusive decays of J/ψ

and ψ′. Large difference between the data for ψ′→ K∗0 ¯

possibility. An improved measurement of these two channels will be able to clarify the predominant role

played by the EM transitions.

For the ρπ channel, we find that the EM decay contributions in ψ′decays is compatible with the

experimental data. This highlights that the interferences from the EM transitions can play essential

role in the decay transitions, especially, in the understanding of the abnormally small ratio of BR(ψ′→

ρπ)/BR(J/ψ → ρπ). However, it should be pointed out that the compatible strength of the EM decay

contributions with the experimental data in ψ′→ ρπ may reflect the suppressed strong decay strength. In

this sense, a full understanding of the small branching ratio fraction in ρπ channel will require a coherent

study of the strong decay mechanisms [10, 15, 16]. This is beyond the focus of this work and should be

pursued in further studies.

Finally, we note that a full constraint on this model can be reached by accommodating experimental

measurements of J/ψ(ψ′) → Pe+e−and e+e−→ Pe+e−to derive the couplings for gV1γP and gV2γP

from which the form factor information can be derived. Ambiguities from the empirical form factors can

thus be minimized. Experimental analysis of these processes at BES is thus strongly recommended.

C

/BRexp= 0.18 ∼ 0.39 and BRMP

D

/BRexp= 0.22 ∼ 0.49 [1].

ωπ0) =5.2+1.3

?

× 10−5and BRexp(φ → ρπ + π+π−π0) = (15.3 ± 0.4)% [1]. Note that φ and ω are

K0+c.c. and K∗+K−+c.c. [1] has shown such a

Acknowledgement

The authors thank C.Z. Yuan, B.S. Zou and H.B. Li for useful discussions. Q.Z. acknowledges support

from the U.K. EPSRC (Grant No. GR/S99433/01) and the Institute of High Energy Physics of Chinese

Academy of Sciences. C.H.C is supported by National Natural Science Foundation of China (Grant

No.10547001 and No.90403031).

[1] W. M. Yao et al. [Particle Data Group], J. Phys. G 33 (2006) 1.

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(a)

(b)

γ∗

(c)

V1

V1

γ∗

V1

V2

γ∗

gV2γP

P

V2

P

gPγγ

e

fV2

P

e

fV2

e

fV1

e

fV1

V2

gV1γP

FIG. 1: Schematic diagrams for J/ψ(ψ′) → γ∗→ V P.

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[14] M. Suzuki, Phys. Rev. D 63, 054021 (2001).

[15] J.L. Rosner, Phys. Rev. D 64, 094002 (2001).

[16] P. Wang, C.Z. Yuan, X.H. Mo, Phys. Lett. B 574, 41 (2004).

[17] X. Liu, X.Q. Zeng and X.Q. Li, hep-ph/0606191.

[18] C.Z. Yuan, Plenary talk at Hadron 2005, AIP Conference Proceedings, Vol. 814, p65 (2006).

[19] see a recent review by X.H. Mo, C.Z. Yuan and P. Wang, to appear on High Energy and Nuclear Physics.

[20] H.E. Haber and J. Perrier, Phys. Rev. D 32, 2961 (1985).

[21] A. Seiden, H. F.-W. Sadrozinski and H.E. Haber, Phys. Rev. D 38, 824 (1988).

[22] S. Okubo, Phys. Lett. 5, 1975 (1963); G. Zweig, in Development in the Quark Theory of Hadrons, edited

by D.B. Lichtenberg and S.P. Rosen (Hadronic Press, Massachusetts, 1980); J. Iizuka, Prog. Theor. Phys.

Suppl. 37, 38 (1966).

[23] M. Ablikim et al. [BES Collaboration], Phys. Lett. B 614, 37 (2005).

[24] P. Wang, C.Z. Yuan and X.H. Mo, Phys. Rev. D 69, 057502(2004).

Coupling const. e/fV Values (×10−2) Total width of VBR(V → e+e−)

e/fρ

4.28146.4 MeV(4.70 ± 0.08) × 10−5

e/fω

1.268.49 MeV (7.18 ± 0.12) × 10−5

e/fφ

1.604.26 MeV (2.97 ± 0.04) × 10−4

e/fJ/ψ

1.92 93.4 keV (5.94 ± 0.06)%

e/fψ′

1.17337 keV(7.35 ± 0.18) × 10−3

TABLE I:

PDG2006 [1].

The coupling constant e/fV determined in V → e+e−. The data for branching ratios are from

Page 8

8

Coupling const. gV γP

ValuesBranching ratios

gργη

0.372(2.95 ± 0.30) × 10−4

gργη′

0.302(29.4 ± 0.9)%

gρ0γπ0

0.197(6.0 ± 0.8) × 10−4

gρ±γπ±

0.170 (4.5 ± 0.5) × 10−4

gωγη

0.110(4.9 ± 0.5) × 10−4

gωγη′

0.107(3.03 ± 0.31)%

gωγπ

0.565(8.90+0.27

−0.23)%

(1.301 ± 0.024)%

gφγη

0.213

gφγη′

0.218 (6.2 ± 0.7) × 10−5

gφγπ

0.041(1.25 ± 0.07) × 10−3

gK∗±γK±

0.225 (9.9 ± 0.9) × 10−4

gK∗0γK0

0.340 (2.31 ± 0.20) × 10−3

gJ/ψγη

3.13 × 10−3

(9.8 ± 1.0) × 10−4

gJ/ψγη′

7.61 × 10−3(4.71 ± 0.27) × 10−3

gJ/ψγπ

5.49 × 10−4

(3.3+0.6

−0.4) × 10−5

< 9 × 10−5

gψ′γη

1.63 × 10−3

gψ′γη′

2.26 × 10−3

(1.5 ± 0.4) × 10−4

TABLE II: The coupling constant gV γP determined in V → γP or P → γV . The data for branching ratios are

from PDG2006 [1].

Coupling const. gPγγ

Values Branching ratios

gπγγ

2.40 × 10−3(98.798 ± 0.032)%

gηγγ

9.70 × 10−3

(39.38 ± 0.26)%

gη′γγ

2.12 × 10−2

(2.12 ± 0.14)%

TABLE III:

PDG2006 [1].

The coupling constant gPγγ determined in P → γγ. The data for branching ratios are from

Page 9

9

Decay channels F(q2) = 1 MP-DMP-C Exp. data

ρη6.8 × 10−28.0 × 10−51.6 × 10−4(1.93 ± 0.23) × 10−4

ρη′

3.5 × 10−24.4 × 10−61.5 × 10−4(1.05 ± 0.18) × 10−4

ωπ

0.163.4 × 10−42.8 × 10−4

(4.5 ± 0.5) × 10−4

φπ

4.4 × 10−48.1 × 10−78.3 × 10−7

< 6.4 × 10−6

ρ0π0

2.0 × 10−23.6 × 10−53.9 × 10−5

(5.6 ± 0.7) × 10−3

ρπ

5.2 × 10−21.0 × 10−59.2 × 10−5(1.69 ± 0.15) × 10−2

ωη

5.7 × 10−37.0 × 10−61.3 × 10−5(1.74 ± 0.20) × 10−3

ωη′

4.2 × 10−31.6 × 10−61.5 × 10−5(1.82 ± 0.21) × 10−4

φη

1.1 × 10−22.0 × 10−52.0 × 10−5

(7.4 ± 0.8) × 10−4

φη′

8.3 × 10−31.3 × 10−51.8 × 10−5

(4.0 ± 0.7) × 10−4

K∗+K−+ c.c. 3.4 × 10−27.4 × 10−55.9 × 10−5

(5.0 ± 0.4) × 10−3

K∗0 ¯

K0+ c.c.

7.8 × 10−21.6 × 10−41.3 × 10−4

(4.2 ± 0.4) × 10−3

TABLE IV: Branching ratios for J/ψ → γ∗→ V P without the form factor (F(q2) = 1) and with the monopole

(MP) form factor. Column MP-C corresponds to process (a) and (b) in a constructive phase with an effective

mass Λ = 0.616 GeV. Column MP-D corresponds to (a) and (b) in a destructive phase with Λ = 0.65 GeV. The

data for branching ratios are from PDG2006 [1].

Decay channels F(q2) = 1MP-D MP-CExp. data

ρη1.3 × 10−27.3 × 10−61.5 × 10−5

(2.2 ± 0.6) × 10−5

ρη′

7.3 × 10−32.3 × 10−61.1 × 10−5

(1.9+1.7

−1.2) × 10−5

(2.1 ± 0.6) × 10−5

ωπ

3.1 × 10−23.2 × 10−52.6 × 10−5

φπ

8.7 × 10−58.8 × 10−87.0 × 10−8

< 4 × 10−6

ρ0π0

3.8 × 10−33.9 × 10−63.1 × 10−6

***

ρπ

9.6 × 10−39.8 × 10−67.9 × 10−6

(3.2 ± 1.2) × 10−5

ωη

1.1 × 10−36.5 × 10−71.3 × 10−6

< 1.1 × 10−5

ωη′

8.9 × 10−44.0 × 10−71.2 × 10−6

(3.2+2.5

−2.1) × 10−5

(2.8+1.0

−0.8) × 10−5

(3.1 ± 1.6) × 10−5

φη

2.2 × 10−31.9 × 10−62.0 × 10−6

φη′

1.9 × 10−31.6 × 10−61.8 × 10−6

K∗+K−+ c.c. 6.7 × 10−37.0 × 10−65.6 × 10−6

(1.7+0.8

−0.7) × 10−5

K∗0 ¯

K0+ c.c.

1.5 × 10−21.6 × 10−51.3 × 10−5(1.09 ± 0.20) × 10−4

TABLE V: Branching ratios for ψ′→ γ∗→ V P without the form factor (F(q2) = 1) and with the monopole (MP)

form factor. The notations are the same as Table VI. The stars “***” in ρ0π0channel denotes the unavailability

of the data.

Page 10

10

Decay channels RV P

1

(%) RV P

2

(%) RV P

3

(%) Exp. data (%)

ρη 1999 11.5 ± 5.0

ρη′

21527 23.5 ± 17.8

ωπ

19995.0 ± 1.8

φπ

19 118< 62.5

ρ0π0

19 118 ***

ρπ

191180.2 ± 0.1

ωη

19108< 0.6 ± 0.1

ωη′

21 258 18.5 ± 13.2

φη

2010 104.1 ± 1.6

φη′

2313108.7 ± 5.5

K∗+K−+ c.c.

20990.4 ± 0.2

K∗0K0+ c.c.

20992.7 ± 0.7

TABLE VI: Branching ratio fractions for ψ′→ γ∗→ V P over J/ψ → γ∗→ V P. RV P

RV P

2

and RV P

3

correspond to calculations with MP-D and MP-C form factors, respectively. The last column is

extracted from the experimental date [1]. The stars “***” in ρ0π0channel denotes the unavailability of the data.

1

refers to F(q2) = 1, while