Physics Letters B 670 (2008) 55–60

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Hadronic loop contributions to J/ψ and ψ ′ radiative decays into γ ηc or γ η′c

Gang Li a,c, Qiang Zhao b,a,c,∗a Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, PR Chinab Department of Physics, University of Surrey, Guildford GU2 7XH, United Kingdomc Theoretical Physics Center for Science Facilities, Chinese Academy of Sciences, Beijing 100049, PR China

a r t i c l e i n f o a b s t r a c t

Article history:Received 29 September 2007Received in revised form 20 October 2008Accepted 21 October 2008Available online 24 October 2008Editor: A. Ringwald

Intermediate hadronic meson loop contributions to J/ψ , ψ ′ → γ ηc (γ η′c) are studied apart from

the dominant M1 transitions in an effective Lagrangian approach. Due to the property of the uniqueantisymmetric tensor coupling in V → V P , the hadronic loop transitions provide explicit correctionsto the M1 transition amplitudes derived from the naive “quenched” cc̄ transitions via the couplingform factors. This mechanism interfering with the M1 transition amplitudes naturally accounts for thedeviations from the Godfrey–Isgur model predictions in J/ψ and ψ ′ → γ ηc . It also predicts a smallbranching ratio of ψ ′ → γ η′

c , which can be examined by experimental measurements at BES and CLEO-c.© 2008 Elsevier B.V. All rights reserved.

1. Introduction

Charmonium spectrum and decays of charmonium states are anideal place for studying the strong interaction dynamics in the in-terplay of perturbative and non-perturbative QCD regime. In thepast decades there have been significant progresses on the mea-surement of charmonium spectrum and their decays, which pro-vide important constraints on phenomenological approaches.

As the first charmonium state discovered in the history, J/ψhas been one of the most widely studied states in both experimentand theory. As a relatively heavier system compared with light qq̄mesons, the application of a nonrelativistic potential model (NRmodel) including color Coulomb plus linear scalar potential andspin–spin, spin–orbit interactions, has provided a reasonably goodprescription for the charmonium spectrum [1]. This success is adirect indication of the validity of the naive “quenched” cc̄ quarkmodel scenario as a leading approximation in many circumstances.A relativised version was developed by Godfrey and Isgur [2] (GImodel), where a flavor-dependent potential and QCD-motivatedrunning coupling are employed. In comparison with the nonrela-tivistic model, the GI model offers a reasonably good descriptionof the spectrum and matrix elements of most of the u, d, s, c andb quarkonia [2,4].

On the other hand, there also arise apparent deviations in thespectrum observables which give warnings to a simple qq̄ treat-ment and more complicated mechanisms may play a role. Aspointed out in Ref. [4], the importance of mixing between quarkmodel qq̄ states and two meson continua may produce significant

* Corresponding author at: Institute of High Energy Physics, CAS, and Departmentof Physics, Univ. of Surrey, Guildford GU2 7XH, UK.

E-mail addresses: gli@ihep.ac.cn (G. Li), qiang.zhao@surrey.ac.uk (Q. Zhao).

0370-2693/$ – see front matter © 2008 Elsevier B.V. All rights reserved.doi:10.1016/j.physletb.2008.10.033

effects in the spectrum observables. By including the meson loops,the quark model is practically “unquenched”. This immediatelyraises questions about the range of validity of the naive “quenched”qq̄ quark model scenario, and the manifestations of the intermedi-ate meson loops in charmonium spectrum and their decays. Theseissues become an interesting topic in the study of charmoniumspectrum with high-statistic charmonium events from experiment.An example is the newly identified state X(3872) and a possibleassignment for it as a mixture of cc̄ and D D̄∗ [5,6], or open charmeffects [7].

In the recent years, the intermediate meson loop is investigatedin a lot of meson decay channels [8–16] as one of the importantnon-perturbative transition mechanisms, or known as final stateinteractions (FSI). In particular in the energy region of charmo-nium masses, with more and more data from Belle, BaBar, CLEO-cand BES, it is widely studied that intermediate meson loop mayaccount for apparent OZI-rule violations [11–16] via quark–hadronduality argument [17–19].

In this Letter, we shall study the radiative decays of J/ψ and ψ ′into γ ηc and γ η′

c . In the naive qq̄ scenario, this type of decaysis dominantly via magnetic dipole (M1) transitions which flip thequark spin. For J/ψ → γ ηc and ψ ′ → γ η′

c , where the initial andfinal state cc̄ are in the same multiplet, the spatial wavefunctionoverlap is unity at leading order, while ψ ′ → γ ηc will vanish dueto the orthogonality between states of different multiplets. In thissense, the former decays are “allowed” while the latter is “hin-dered”. However, the inclusion of relativistic corrections from thequark spin-dependent potential will induce a nonvanishing over-lap between states of different multiplets such that the decay ofψ ′ → γ ηc is possible [20–23].

Theoretical studies of the heavy quarkonium M1 transitionswith relativistic corrections are various in the literature [24–31].Relativistic quark model calculations show that a proper choice

56 G. Li, Q. Zhao / Physics Letters B 670 (2008) 55–60

of the Lorentz structure of the quark–antiquark interaction in ameson is crucial for explaining the J/ψ → γ ηc data [27]. Sys-tematic investigation of the M1 transitions in the framework ofnonrelativistic effective field of QCD has been reported in Ref. [30],where relativistic corrections of relative order v2 are included. ForJ/ψ → γ ηc the authors found Γ J/ψ→γ ηc = (1.5 ± 1.0) keV, whichis in a good agreement with the data, but with quite large esti-mated uncertainties from higher-order relativistic corrections. Tak-ing into account the transitions of ψ ′ → γ ηc , the overall results forthe M1 transitions still turn out to be puzzling [32,33]. As studiedby Barnes, Godfrey and Swanson [4] in the NR model and the rela-tivied GI model, although the models provide an overall consistentdescription of most of the existing charmonium states, theoreticalresults for the M1 transition have significant discrepancies com-pared with the experimental data [3]. For example, in both NRand GI model, the predicted partial decay widths for J/ψ → γ ηc

are as large as about two times of the experimental value, whilefor ψ ′ → γ ηc , the theoretical predictions are about one order ofmagnitude larger than the data [3]. For ψ ′ → γ η′

c , although thepredicted partial decay widths 0.17–0.21 keV are smaller than theexperimental upper limit (<0.67 keV), it is possible that the M1transition is very different from the experimental measurement.

Therefore, it is likely that there exist additional mechanismsbeyond the cc̄ transitions. This consideration thus prompts us toexplore possible sources which can contribute to the charmoniumradiative decay and cause deviations from the NR and GI modelpredictions, among which the intermediate meson loop transitionscould be a natural mechanism.

As follows, we first brief the calculations from the NR and GImodels for the M1 transitions, and then introduce the formalismsfor the intermediate meson loop contributions in Section 2. Theresults and discussions will be presented in Section 3.

2. M1 transition in NR and GI model

The detailed study of the M1 transition was given by Barneset al. in Ref. [4], and here we quote their standard formula to in-corporate the intermediate meson loop contributions which is tobe introduced later.

In Ref. [4], the partial decay width via M1 transition is evalu-ated by

ΓM1(n2S+1L J → n′ 2S ′+1L′

J ′ + γ)

= 4

3

2 J ′ + 1

2L + 1δLL′δS,S ′±1

e2c α

m2c

∣∣〈ψ f |ψi〉∣∣2

E3γ

E f

Mi, (1)

where n and n′ are the main quantum number of the initial andfinal state charmonium meson; S (S ′), L (L′) and J ( J ′) are the ini-tial (final) state spin, orbital angular momentum and total angularmomentum. Eγ and E f denote the final state photon and mesonenergy, respectively, while Mi is the initial cc̄ meson mass. |ψi〉and |ψ f 〉 are the spatial wavefunctions of the initial and final statecc̄ mesons, respectively.

In the GI model, phase space factor E f /Mi is not includedthough it is close to unity in many considered cases. In both GIand NR model, a recoil factor j0(kr/2) is included. We quote theresults from Ref. [4] for future comparison.

In order to incorporate the intermediate meson loop contri-butions, we derive the effective V γ P couplings due to the M1transition from Eq. (1) by defining

M f i(M1) ≡ gV γ P

Miεαβμν Pα

γ εβγ Pμ

i ενi , (2)

where Pi and Pγ are four-vector momentum of the initial me-son and final state photon, respectively, and εi and εγ are thecorresponding polarization vectors. From Ref. [4], we know that

Fig. 1. Schematic diagrams for J/ψ → γ ηc via (a) D D̄(D∗), (b) D D̄∗(D∗) and(c) D D̄∗(D) intermediate meson loops. Similar processes occur in ψ ′ → γ ηc andγ η′

c .

Fig. 2. The contact diagrams considered in J/ψ → γ ηc . Similar diagrams are alsoconsidered in ψ ′ → γ ηc(η

′c).

these extracted effective gV γ P couplings for J/ψ,ψ ′ → γ ηc appar-ently overestimate the experimental data. Thus, the introduction ofthe intermediate meson loop contributions, which unquench thenaive cc̄ configurations, is supposed to cancel the M1 transitionamplitudes via destructive interferences.

3. Intermediate meson loop contributions

The inclusion of the intermediate meson loops in meson decayssomehow “unquenches” the naive quark model. A full considera-tion of such an effect requires systematic coupled channel calcula-tions for e.g. the charmonium mass spectrum [34]. An interestingfeature arising from the low-lying charmonia, such as ηc , η′

c , J/ψ ,and ψ ′ , is that their masses are lower than the open charmed me-son decay channels. As a consequence, the lowest open charmedmeson decay channels are expected to be dominant if they indeedaccount for contributions beyond the M1 transitions. This scenarioturns to be consistent with the break-down of local quark–hadronduality, where the leading contributions to the sum over all inter-mediate virtual states are from those having less virtualities.

It should be pointed that intermediate states involving flavorchanges turn out to be strongly suppressed. One reason is becauseof the large virtualities involved. The other is because of the OZIrule suppressions. Therefore, intermediate state contributions suchas ρπ etc., are negligibly small.

Following the above consideration, we thus investigate D D̄(D∗),D D̄∗(D∗) and D D̄∗(D) loops as the major contributions to J/ψ →γ ηc , and ψ ′ → γ ηc , γ η′

c as illustrated in Fig. 1. We stress thatalthough some of the vertices in the loop may violate gauge in-variance, such as J/ψ D D̄ , the overall antisymmetric property isretained for the loops. The loop contributions hence only providecorrections to the V V P coupling strength for the external fields,but not change their antisymmetric tensor structure, no matter Vis a massive vector meson or photon. Apart from the transitions inFig. 1, the contact transitions in Fig. 2 will also contribute to thedecay amplitude. We show that the processes of Fig. 2 are gaugeinvariant by themselves. In brief, due to the property of the an-tisymmetric tensor coupling of V V P , where both V and P areexternal fields here, hadronic loop corrections are guaranteed tobe gauge invariant in this effective Lagrangian approach.

The detailed formulation is given in the following subsections.

G. Li, Q. Zhao / Physics Letters B 670 (2008) 55–60 57

3.1. Intermediate D D̄(D∗) + c.c. loop

The transition amplitude for an initial vector charmonium ( J/ψor ψ ′) decay into γ ηc or γ η′

c via D D̄(D∗) can be expressed asfollows:

M f i =∫

d4 p2

(2π)4

∑D∗ pol

T1T2T3

a1a2a3F

(p2

2

), (3)

where the vertex functions are⎧⎨⎩

T1 ≡ ig1(p1 − p3) · εi,

T2 ≡ ig2m2

εαβμν Pαγ ε

βγ pμ

2 εν2 ,

T3 ≡ ig3(P f + p3) · ε2,

(4)

where g1, g2, and g3 are the coupling constants at the meson in-teraction vertices (see Fig. 1). The four vectors, Pi , Pγ , and P f arethe momenta for the initial vector, final state γ and pseudoscalarmeson, respectively, while four-vector momenta, p1, p2, and p3are for the intermediate mesons, respectively, and a1 = p2

1 − m21,

a2 = p22 − m2

2, and a3 = p23 − m2

3 are the denominators of the prop-agators of these intermediate mesons.

As being studied in Ref. [16], this loop diverges logarithmi-cally. Thus, a form factor to suppress the divergence and take intoaccount the momentum-dependence of the vertex couplings is in-cluded:

F(

p2) =(

Λ2 − m22

Λ2 − p22

)n

, (5)

where n = 1,2 correspond to monopole and dipole form factors,respectively. An empirical argument applied here is that in the P -wave V → V P decay the form factor favors a dipole form. Wehence deduce the loop transition amplitudes with a dipole formfactor.

Substitute the vertex couplings of Eq. (4) into Eq. (3), the inte-gral has an expression:

M f i =∫

d4 p2

(2π)4

×∑

D∗pol

[ig1(p1 − p3) · εi][ ig2m2

εαβμν Pαγ ε

βγ pμ

2 εν2 ][ig3(P f + p3) · ε2]

(p21 − m2

1)(p23 − m2

3)(p22 − m2

3)

× F(

p22

). (6)

With a dipole form factor, we have

M f i ≡ g̃a

Miεαβμν Pα

γ εβγ Pμ

i ενi , (7)

where

g̃a ≡ − g1 g2 g3Mi

m2

1∫0

dx

1−x∫0

dy2

(4π)2

[log

Δ(m1,m3,Λ)

Δ(m1,m3,m2)

− y(Λ2 − m22)

Δ(m1,m3,Λ)

](8)

where the function Δ is defined as

Δ(a,b, c) ≡ −(M2

i − M2f

)(1 − x − y)x + M2

f x2

+ a2(1 − x − y) − (M2

f − b2)x + yc2. (9)

In the intermediate meson exchange loop, coupling g2 canbe determined via the experimental information for D∗0 →D0γ (D̄∗0 → D̄0γ ), i.e.

g22 = 12π M2

D∗|p |3 ΓD∗0→D0γ , (10)

γ

where ΓD∗0→D0γ = (38.1 ± 2.9)% ×Γtot is given by experiment [3].We neglect the contributions from the charged meson exchangeloop since ΓD∗±→D±γ = (1.6 ± 0.4)% × 96 keV is about two ordersof magnitude smaller than ΓD∗0→D0γ .

For coupling constant g1, especially g J/ψ D D̄ , there are severalmethods suggested in the literature including quark model usingheavy quark effective theory approach [35], QCD sum rule [36,37],SU(4) symmetry and vector meson dominance (VMD) model [38].They typically give a value of order of one for g J/ψ D D̄ . In this work,we adopt g J/ψ D D̄ = 7.20 which is consistent with the value fromRef. [35].

For the gD∗ Dηc coupling, we assume

gD∗0 D0ηc= g J/ψ D0∗ D̄0 . (11)

3.2. Intermediate D D̄∗(D∗) + c.c. loop

As shown by Fig. 1(b), the transition amplitude from the inter-mediate D D̄∗(D∗) + c.c. loop can be expressed the same form asEq. (3) except that the vertex functions change to⎧⎪⎪⎪⎨⎪⎪⎪⎩

T1 ≡ i f1Mi

εαβμν Pαi ε

β

i pμ3 εν

3 ,

T2 ≡ i f2m2

εα′β ′μ′ν ′ pα′2 ε

β ′2 Pμ′

γ εν ′γ ,

T3 ≡ i f3M f

εα′′β ′′μ′′ν ′′ pα′′2 ε

β ′′2 pμ′′

3 εν ′′3 ,

(12)

where f1,2,3 are the coupling constants. With a dipole form factorthe integration gives

M f i ≡ g̃b

Miεαβμν Pα

γ εβγ Pμ

i ενi , (13)

where

g̃b ≡ f1 f2 f3

m2M f

1∫0

dx

1∫0

dy

1−x−y∫0

dz (1 − x − y − z)2

(4π)2

(A

Δ21

− B

Δ31

),

(14)

with

A = 1

4

[(1 − x − z

2

)(M2

i − M2f

) + xM2f

],

B = − x

4

(M2

i − M2f

)[z(1 − x)

(M2

i − 3M2f

) − (M2

i − M2f

)],

Δ1 = −xz(M2

i − M2f

) + zm21 + ym2

2 + xm23

+ (1 − x − y − z)Λ2. (15)

In the above equation the intermediate meson masses m1,2,3are from the D D̄∗(D∗) loop, which are different from those inEq. (7). f1,2,3 denotes the corresponding vertex coupling constants.

In the D D̄∗(D∗)+c.c. loop, the coupling constant g J/ψ D∗ D̄ is re-lated to g J/ψ D D̄ by the relation of the heavy quark mass limit [35]:

g J/ψ D∗ D̄ = g J/ψ D D̄/M̃D , (16)

where M̃D corresponds to the mass ratio of MD/MD∗ . Similarly,we have gψ ′ D∗ D̄ = gψ ′ D D̄/M̃D .

For ψ ′ → γ ηc and γ η′c , we assume that gψ ′ D D̄ = g J/ψ D D̄ ,

gD∗0 D0ηc= g J/ψ D∗ D̄ , and gD∗0 D0η′

c= gψ ′ D∗ D̄ , which are consistent

with the 3 P0 model [4]. In Table 1 the values of the coupling con-stants are listed.

3.3. Intermediate D D̄∗(D) + c.c. loop

The transition amplitude from the intermediate D D̄∗(D) + c.c.loop can contribute via charged intermediate meson exchange.

58 G. Li, Q. Zhao / Physics Letters B 670 (2008) 55–60

Table 1The absolute values of coupling constants for the effective vertex interactions. Theirrelative phases are determined by the SU(4) flavor symmetry.

Coupling constants |g J/ψ D D̄ | |gψ ′ D D̄ | |g J/ψ D̄∗ D | |gD∗ D̄ηc| |gψ ′ D̄∗ D | |gD∗ D̄η′

c| |gD∗ Dγ |

Numerical value 7.20 7.20 11.96 11.96 14.23 14.23 6.86

Treating the intermediate mesons as fundamental degrees of free-dom, we eventually neglect the contributions from the non-zeromagnetic moments of the D mesons. The charge-neutral loop isthus suppressed due to the vanishing D0 D̄0γ electric coupling.Therefore, we only consider the charged meson loop contributionsas shown by Fig. 1(c). The transition amplitude can also be ex-pressed in a form as Eq. (3) with the vertex functions⎧⎪⎨⎪⎩

T1 ≡ i f ′1

Miεαβμν Pα

i εβ

i pμ3 εν

3 ,

T2 ≡ i f ′2(p1 − p2) · εγ ,

T3 ≡ i f ′3(P f − p2) · ε3,

(17)

where f ′1,2,3 are the coupling constants and F (p2

2) is the form fac-tor. With a dipole form factor the integration gives

M f i ≡ g̃c

Miεαβμν Pα

γ εβγ Pμ

i ενi , (18)

where

g̃c ≡ f ′1 f ′

2 f ′3

1∫0

dx

1−x∫0

dy2

(4π)2

[log

Δ(m1,m3,Λ)

Δ(m1,m3,m2)

− y(Λ2 − m22)

Δ(m1,m3,Λ)

]. (19)

It is interesting to note that the integral of the D D̄∗(D) + c.c.loop has a similar form as that of D D̄(D∗). However, we ex-pect that contributions from this loop integral will be relativelysuppressed since coupling f ′

2 is taken as the unit charge e =(4παe)

1/2.

3.4. Contact diagrams

The contact diagrams of Fig. 2(a) and (b) (as an example inJ/ψ → γ ηc) arise from gauging the strong J/ψ(ψ ′)D∗D andηc(η

′c)D∗D interaction Lagrangians containing derivatives. The gen-

eral form of the transition amplitude of Fig. 2(a) and (b) can beexpressed as follows:

M f i =∫

d4 p2

(2π)4

∑D∗ pol

T1T2

a1a2F

(p2

2

), (20)

where F (p22) is the form factor as before, and Ti (i = 1,2) are the

vertex functions. For Fig. 2(a), the expressions of Ti (i = 1,2) are:{T1 ≡ h1e

Miεαβμν(εα

γ εβ

i pμ2 εν

2 + Pαi ε

β

i εμγ εν

2 ),

T2 ≡ 2ih2 P f · ε2,(21)

where h1,2 represent the J/ψ(ψ ′)D∗ D̄ and ηc(η′c)D∗ D̄ coupling

constants, respectively, and their values have been given in Table 1.Using the Feynman parameter scheme, the amplitude for

Fig. 2(a) can be reduced to

M f i = 2ih1h2e

Miεαβμν

∫d4 p2

(2π)4

×[εα

γ εβ

i pμ2 pν

f + Pαi ε

β

i εμγ (−Pν

f + Pρf pρ

2 pν2

m22

)](m22 − Λ2)2

(p2 − m2)(p2 − m2)(p2 − Λ2)2

1 1 2 2 2

= 2ih1h2e

Miεαβμν

∫d4l

(2π)4

× [εαγ ε

β

i (l − xP f )μ pν

f + Pαi ε

β

i εμγ (−Pν

f + P f ρ (l−xP f )ρ (l−xP f )ν

m22

)](m22 − Λ2)2

(l2 − Δ2)4

= h1h2e

Miεαβμν Pα

γ εβγ Pμ

i ενi

1∫0

dx

1−x∫0

dy1

(4π)2

(1

3Δ22

− 1

6Δ2

)

(22)

≡ g̃d

Miεαβμν Pα

γ εβγ Pμ

i ενi , (23)

with

Δ2 = x2M2f − xM2

f + xm21 + ym2

2 + (1 − x − y)Λ2, (24)

and

g̃d ≡ h1h2e

1∫0

dx

1−x∫0

dy1

(4π)2

(1

3Δ22

− 1

6Δ2

). (25)

For Fig. 2(b), the vertex functions are:{T1 ≡ ih1

Miεαβμν Pα

i εβ

i pμ2 εν

2 ,

T2 ≡ 2eh2εγ · ε2.(26)

The amplitude can then be reduced to

M f i = −2ih1h2e

Miεαβμν Pα

i εβ

i ενγ

∫d4 p2

(2π)4

× pμ2

(p21 − m2

1)(p22 − m2

2)(p22 − Λ2)2

(27)

= −2ih1h2e

Miεαβμν Pα

i εβ

i ενγ

∫d4l

(2π)4

(l − xPi)μ

(l2 − Δ3)4, (28)

with

Δ3 = x2M2i − xM2

i + xm21 + ym2

2 + (1 − x − y)Λ2. (29)

Note that the integrand has an odd power of the internal mo-mentum, the amplitude will vanish and has no contribution to theV V P coupling.

The above deduction shows that only Fig. 2(a) has nonvanishingcontributions to the transition amplitude. Meanwhile, gauge invari-ance is also guaranteed for the contact diagrams. The divergence ofthe loop integral is eliminated by adding the dipole form factor asin Fig. 1.

4. Results and discussions

Proceed to numerical results from the intermediate meson ex-change loops, the undetermined quantities include the cut-off en-ergy Λ in the dipole form factor and the relative phases amongthose amplitudes. The transition amplitude accommodating the M1and intermediate meson exchange loops, i.e. D D̄(D∗), D D̄∗(D∗),D D̄∗(D), and the contact term, can then be expressed as

M f i = 1

Mi

[gV γ P + g̃aeiδa + g̃beiδb + g̃ceiδc + g̃deiδd

]× εαβμν Pα

γ εβγ Pμ

i ενi , (30)

where gV γ P is a real number and fixed to be positive. Couplingsg̃a , g̃b , g̃c and g̃d , calculated by the loop integrals can be com-plex numbers in principle. In this interested case, since the decaythreshold of the intermediate mesons are above the initial meson( J/ψ and ψ ′) masses, the absorptive part of the loop integrals van-ishes as a consequence. However, there might exist relative phases

G. Li, Q. Zhao / Physics Letters B 670 (2008) 55–60 59

Table 2Radiative partial decay widths given by different processes are listed: Γ NR

M1 and Γ GIM1

are the M1 transitions in the NR and GI model, respectively [4]; ΓTri are inclu-sive contributions from the triangle diagrams (Fig. 1); ΓC are from contact diagrams(Fig. 2); ΓHL denote the inclusive contributions from all the intermediate hadronicloops; while Γall are coherent results including the M1 in the GI model and inter-mediate hadronic loops. The experimental data are from PDG2006 [3]. The resultsare obtained at the cut-off energy Λ = 2.39 GeV.

Initial meson J/ψ(1 3 S1) ψ ′(2 3 S1)

Final meson ηc(1 1 S0) η′c(2 1 S0) ηc(1 1 S0)

Γ NRM1 (keV) 2.9 0.21 9.7

Γ GIM1 (keV) 2.4 0.17 9.6

ΓC (keV) ∼ 0. ∼ 0. 0.04ΓTri (keV) 0.096 0.063 17.91ΓHL (keV) 0.083 0.054 16.20Γall (keV) 1.59 0.032 0.86Γexp (keV) 1.21 ± 0.37 < 0.67 0.88 ± 0.13

among those transition amplitudes. We hence include possible rel-ative phases δa,b,c,d in the above expression.

Note that the loop contributions are supposed to provide can-cellations to the M1 amplitude which is real. We thus simply takeδa,b,c,d = 0 or π . In this way, we have several phase combinationswhich are to be examined in the numerical calculation. Yet thereis still a free parameter Λ to be constrained.

We find that a reasonable constraint on the model can beachieved by requiring a satisfactory of the following conditions:(i) For either constructive (δ = 0) or destructive phases (δ = π ), thesame value of Λ is needed to account for J/ψ → γ ηc , ψ ′ → γ ηc

simultaneously. (ii) The value of Λ is within the commonly ac-cepted region, 1.5–2.5 GeV. (iii) The prediction for ψ ′ → γ η′

cwith the same Λ is well below the experimental upper limit,B R(ψ ′ → γ η′

c) < 2.0 × 10−3 [3].Imposing the above conditions on fitting the Λ parameter, we

obtain Λ = 2.39 GeV as the best fit with δa,b,c,d = π , i.e. contri-butions from the loop integrals provide cancellations to the M1transition amplitudes and there is no need for abnormal relativephases among the intermediate meson exchanges.

The numerical results for the intermediate meson exchangeshave some predominant features. We find that the D D̄(D∗)and D D̄∗(D∗) loops have relatively large contributions while theD D̄∗(D) loop is quite small. The contributions from the contactterm are negligibly small in J/ψ → γ ηc and ψ ′ → γ η′

c , while rel-atively large in ψ ′ → γ ηc .

In Table 2, the fitted branching ratios are listed and comparedwith the GI model M1 transitions. We also list the exclusive con-tributions from the triangle diagrams of Fig. 1 and contact dia-grams of Fig. 2 as a comparison. For J/ψ → γ ηc , we find thatthe magnitude of the meson loop amplitude is smaller than theM1 amplitude, while for ψ ′ → γ ηc , the absolute loop amplitudeturns to be larger than the M1. With Λ = 2.39 GeV, we obtainΓ ( J/ψ → γ ηc) = 1.59 keV which is located at the upper limitof the experimental data, Γexp( J/ψ → γ ηc) = (1.21 ± 0.37) keV[3]. For ψ ′ → γ ηc , we have Γ (ψ ′ → γ ηc) = 0.86 keV, which isagree well with the data, Γexp(ψ ′ → γ ηc) = (0.88 ± 0.13) keV [3].Taking into account the still-large uncertainties with the data forJ/ψ → γ ηc , the inclusion of the intermediate meson loop contri-butions significantly improves the theoretical results.

With the fixed Λ, the partial decay width for ψ ′ → γ η′c

is calculated as a prediction. The pure M1 transition predictsΓ GI

M1(ψ′ → γ η′

c) 0.17 keV [4], while the hadronic loops con-tribute ΓHL(ψ

′ → γ η′c) 0.054 keV. The cancellation from the

hadron loops thus leads to Γall(ψ′ → γ η′

c) 0.032 keV which iswell below the experimental upper limit, 0.67 keV [3]. Note that inall these three channels, the hadronic loop cancellations from thereal part of the amplitudes possess the same relative sign to the

M1 amplitudes. This makes the decay of ψ ′ → γ η′c extremely in-

teresting. As the pure M1 transition still predicts a sizeable partialwidth about 0.17 keV while our hadronic loop cancellation predictsa much smaller value, improved measurement of this quantity willhelp us gain further insights into the decay mechanisms.

For other relative phases, we find that there does not exist acommon value for Λ to fit the data for J/ψ → γ ηc and ψ ′ → γ ηc

simultaneously.To summarize, in this work we have studied the hadronic

meson loop contributions to the J/ψ and ψ ′ radiative decaysinto γ ηc or γ η′

c . In the framework of effective Lagrangian phe-nomenology, the intermediate meson exchange loops provide cor-rections to the leading couplings extracted from potential quarkmodels. In comparison with the NR and GI model, the meson loopcontributions turn to cancel the NR and GI amplitudes. It is inter-esting to see that the meson loop contributions in J/ψ → γ ηc issmaller than the M1 transition in magnitude, while in ψ ′ → γ ηc

the situation is opposite. Note that the pure M1 contribution inψ ′ → γ ηc is about one order of magnitude larger than the exper-imental data, the meson loop contributions turn out to be evenlarger. This mechanism suggests significant cancellations betweenthe M1 and meson loop amplitudes. It raises questions on thenaive qq̄ solution for the meson spectrum, and could be a man-ifestation of the limit of the quenched quark model scenario.

As a prediction from this model, we calculate the partial decaywidth of ψ ′ → γ η′

c . It gives a value about one order of magnitudesmaller than the experimental upper limit. Improved measurementof this decay channel is strongly recommended.

It is interesting to note that our model results are similar tothose from a relativistic quark model calculation by Ebert, Faus-tov and Galkin [27], who find that a proper choice of the Lorentzstructure of the quark–antiquark interaction in a meson is crucialfor accounting for the M1 transition data. In our approach we ex-tract the effective couplings from the NR and relativised GI modeland then combine it with the gauge invariant meson loop correc-tions. The validity of this approach is guaranteed by the propertyof the unique antisymmetric tensor coupling for V V P fields. In theframework of effective Lagrangian phenomenology the correctionsto the leading contributions are introduced as coupling form fac-tors.

Although we also observe strong sensitivities of the hadronicloop contributions to the cut-off energy Λ, the advantage of thisapproach is that the number of parameters is limited. In fact, thereis little freedom for the effective couplings at vertices. By a co-herent study of J/ψ and ψ ′ → γ ηc , we find that the constrainton the Λ value is very tight. Certainly, it should be noted thatour treatment of the relative phases is empirical though the fa-vor of a destructive phase between the M1 transition amplitudesand hadron loops turns to be consistent with what one naturallyexpects. Note that it has been shown in Ref. [23] that a propermodification of the color Coulomb potential strength will simul-taneously account for the branching ratios for J/ψ → γ ηc andψ ′ → γ ηc . It seems to support that the intermediate hadronic me-son loops are responsible, at least partly, for such a modification,and hence break down the naive qq̄ scenario.

The study of non-perturbative effects arising from intermedi-ate meson loops in heavy quarkonium decays has attracted a lotof attention recently. Although such approaches still experiencelarge uncertainties from the divergent behavior of the loop inte-grals, we expect that improved experimental measurements withhigh statistics, such as at BES and CLEO-c, provide more and morestringent constraints on the hadronic loops. Thus, insights into theeffective degrees of freedom within hadrons and their decay mech-anisms can be gained. This requires systematic analysis of bothspectroscopy and coupled channels for which more and more the-oretical efforts are undergoing.

60 G. Li, Q. Zhao / Physics Letters B 670 (2008) 55–60

Acknowledgements

Q.Z. would like to thank T. Barnes, K.T. Chao, Y. Jia and B.S. Zoufor very useful discussions. This work is supported, in part, bythe UK EPSRC (Grant No. GR/S99433/01), National Natural ScienceFoundation of China (Grant Nos. 10675131 and 10491306), and Chi-nese Academy of Sciences (KJCX3-SYW-N2).

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