Revisiting the radiative decays $J/ψ\rightarrow γη^{(\prime)}$ in perturbative QCD
Jun-Kang He Ya-Dong Yang















Revisiting the radiative decays J/ψ→γη^(') in perturbative QCD
    Jun-Kang He[hejk@mails.ccnu.edu.cn]  and
Ya-Dong Yang[yangyd@mail.ccnu.edu.cn]

Institute of Particle Physics and Key Laboratory of Quark and Lepton Physics (MOE), 

Central China Normal University, Wuhan, Hubei 430079, China
    
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In the framework of perturbative QCD, the radiative decays J/ψ→γη^(') are revisited
in detail, where the involved one-loop integrals are evaluated analytically with the light quark masses kept. We have found that
the sum of loop integrals is insensitive to the light quark masses and the branching ratios ℬ(J/ψ→γη^(')) barely depend on the shapes of η^(') distribution amplitudes. With the parameters of η-η^' mixing extracted from low energy processes and J/ψ→γη^(') by means of nonperturbative matrix elements ⟨0|G_μν^aG̃^a,μν|η^(')⟩ based on U_A(1) anomaly dominance argument, we could not give the ratio R_J/ψ in agreement with experimental result. However, using the parameters, especially the mixing angle ϕ=33.5^∘±0.9^∘, extracted from γ^∗γ-η^' transition form factor measured at q^2=112 GeV^2 by BaBar collaboration, we obtain R_J/ψ=4.70 in good agreement with R_J/ψ^exp=4.65±0.21. As a crossing check, with Γ^exp(η^(')→γγ) and our results for J/ψ→γη^('), we get ϕ=33.9^∘±0.6^∘. The difference between the determinations of ϕ is briefly discussed.








§ INTRODUCTION



Since its discovery<cit.>, the J/ψ meson has always been an active topic in particle physics <cit.>. The heavy quarkonium physics might be crucially important to
improve our understanding of quantum chromodynamics (QCD), especially, the interplay of perturbative quantum chromodynamics (pQCD) with nonperturbative QCD. The Okubo-Zweig-Iizuka (OZI)-forbidden radiative decays of J/ψ→γη^(') are expected to proceed predominantly via two virtual gluons which subsequently convert to η^('), with the photon emitted from the initial charm quarks. Such decays have been the subject of many experimental and theoretical studies <cit.>. Theoretically, these decays provide a clean environment to study the conversion of gluons into hadrons. In this respect, the radiative decays J/ψ→γη^(') are of particular interest, since they are also closely related to the issues of η-η^' mixing, which are important ingredients for understanding many interesting phenomena related to the η and η^' mesons.

In the literature, the exclusive radiative decays J/ψ→γη^(') have been studied in different approaches. The decay widths Γ(J/ψ→γη^(')) were calculated by Novikov et al. <cit.>, with the assumption that these decays occur as a consequence of the U_A(1) anomaly and are, therefore, controlled by the nonperturbative gluonic matrix elements ⟨0|G_μν^aG̃^a,μν|η^(')⟩, where G_μν^a is the gluon field strength tensor and
G̃^a,μν=1/2ϵ^μναβG_αβ^a its dual tensor. Then the ratio of the decay widths takes the form <cit.>

    R_J/ψ=Γ(J/ψ→γη^')/Γ(J/ψ→γη)
    =(M_J/ψ^2-m_η^'^2/M_J/ψ^2-m_η^2)^3|⟨0|G_μν^aG̃^a,μν|η^'⟩/⟨0|G_μν^aG̃^a,μν|η⟩|^2,

where the matrix elements ⟨0|G_μν^aG̃^a,μν|η^(')⟩ can be calculated with the QCD sum rules and other approaches. For example, Chao <cit.> and Kuang et al. <cit.> derived the expressions of the matrix elements ⟨0|G_μν^aG̃^a,μν|η^(')⟩
in the large-N_c approach and QCD multipole expansion, respectively.

In the Feldmann-Kroll-Stech (FKS) scheme for η-η^' mixing system <cit.>, the nonperturbative matrix elements ⟨0|G_μν^aG̃^a,μν|η^(')⟩ can also be expressed as function of the phenomenological parameters f_q, f_s, ϕ, and the ratio R_J/ψ reads

    R_J/ψ=tan^2ϕm_η^'^4/m_η^4(M_J/ψ^2-m_η^'^2/M_J/ψ^2-m_η^2)^3,

where ϕ denotes the mixing angle of η-η^' system. Compared R_J/ψ with its experimental
value, the mixing angle is found to be ϕ=39.3^∘±1.0^∘ <cit.>. However, as it had been pointed out that the above equation was calculated in the approximation with the assumption of ground state dominance and neglection of continuum contributions to the dispersion relations <cit.>. It is also noticed that the matrix elements ⟨0|G_μν^aG̃^a,μν|η^(')⟩ induced by the U_A(1) anomaly are a higher twist effect(twist-4) <cit.>. The ⟨0|G_μν^aG̃^a,μν|η^(')⟩ would give the main contributions to the radiative decays J/ψ→γη^('), only in the case of the leading twist contributions were strongly suppressed. Although the leading twist contributions from the gluonic content of η^(') are suppressed by a factor of m_η^(')^2/M_J/ψ^2 <cit.>, the assumption that the matrix elements ⟨0|G_μν^aG̃^a,μν|η^(')⟩ dominate the radiative decays J/ψ→γη^(') could be broken, because the leading twist contributions from the quark-antiquark content of η^(') are not suppressed much at the energy scale of M_J/ψ.

The first pQCD investigation of these decays were carried out by Körner et al. <cit.> about thirty years ago. They took the annihilation of cc̅ quarks to be a short-distance process described by pQCD, and nonperturbative dynamics of the bound states factorized to wave functions. It has been argued that pQCD asymptotic behaviors may be expected at the scale of M_J/ψ <cit.>. However, in this pioneer work <cit.>, the nonrelativistic quark model with the weak-binding approximation, has been taken for both J/ψ and η^('). Whereafter, the nonrelativistic approximation for η^(') wave functions in <cit.>
was improved by Kühn <cit.> with light-cone expansion. However, in the calculation of the loop integrals, the approximation of m_η^(')^2/M_J/ψ^2≈0 was made <cit.>. In our calculation, we would keep m_η^(') and m_u,d,s, which result in our final analytical expression of the loop function is much complicated than the ones in Refs. <cit.>. We notice that our results can reproduce the one in Ref. <cit.> in the limit of m_η^(')^2/M_J/ψ^2→0, and detail comparison is presented in the Appendix. Recently, several groups have revisited J/ψ→γη^(') in the framework of pQCD <cit.>. In works <cit.>, the light-cone distribution amplitudes (DAs) were adopted for η^('). However, in Ref. <cit.>, the decay widths of J/ψ→γη^(') were found to be very sensitive to the light quark masses of η^(') involved in the loop integrals, which is needed to be made clear, since the sensitivities of the loop integrals to light quark masses usually point to the possible infra-red (IR) divergences.

Besides the aforementioned QCD approaches, the radiative decays J/ψ→γη^(') have also been studied with phenomenological models, such as the approach with an effective lagrangian <cit.> and the approach considering the η_c-η^(') mixings <cit.>. Generally, predictions compatible with the experimental measurements could be obtained.

In this paper, we present a detail calculation of these decays in the framework of pQCD. The bound-state property of J/ψ is parameterized by its Bethe-Salpeter (B-S) wave function, while the η and η^' are described by their light-cone DAs. Then the loop integrals are evaluated analytically with the light quark masses kept.

In our calculation, the loop function is found to be insensitive to the light quark masses, which differs from the results in
Ref. <cit.>. Moreover, our results of the ℬ(J/ψ→γη^(')) are also insensitive to the shapes of the light meson DAs. The theoretical uncertainties due to choices
of different η^(') DAs available in the literature are negligible in the prediction for the ratio R_J/ψ, so that, the mixing angle of η-η^' mixing could be reliably extracted.
In addition, the corrections from QED processes J/ψ→γ^∗→γη^(') are also considered in our calculation.

The paper is organized as follows. In section <ref>, we present the formalism for calculating the decay amplitudes of J/ψ→γη^('). Numerical results are presented in section <ref>, and the final part is our summary. The analytical expressions for the dimensionless key function H_0 and discussions of its properties are presented in the Appendix.




§ THE RADIATIVE DECAYS J/Ψ→ΓΗ^(') IN PQCD






 §.§ The contributions of the quark-antiquark content of η^(')







For the quark-antiquark content of η^('), the leading order contributions to the radiative decays J/ψ→γη^(') arise from one-loop QCD processes. The corresponding Feynman diagram is illustrated in Fig. <ref>. There are other five Feynman diagrams from permutations of the photon and gluon legs. Usually, it is convenient to divide the invariant amplitude into two parts. One part describes the
effective coupling between J/ψ, a real photon and two virtual gluons, the other part describes the effective coupling between pseudoscalar meson η^(') and two virtual gluons.
To evaluate these effective couplings, we need to deal with the nonperturbative effects of the mesons. Generally, factorization is employed. For the heavy J/ψ, we still use the weak-binding approximation, and factorize the nonperturbative bound-state effects into its B-S wave function. For the light mesons η and η^', we use their light-cone DAs.

In the rest frame of J/ψ, the amplitude of J/ψ→γ g^∗g^∗ can
be decomposed into hard-scattering part and the B-S wave function of J/ψ <cit.>

    A   =   A^αβμνε_α(K)ϵ_β^∗(k)ϵ_μ^∗(k_1)ϵ_ν^∗(k_2)
       =   -i√(3)∫d^4q_c/(2π)^4Tr[χ(q_c)𝒪̂(q_c)],

where χ(q_c) is the B-S wave function of J/ψ and √(3) is the color factor. K and ε(K) stand for the momentum and the polarization vector of the J/ψ, k, k_1, k_2 and ϵ(k), ϵ(k_1), ϵ(k_2) stand for momenta and polarization vectors of the photon and the gluons, respectively. The momenta of the c and c̅ quarks are parameterized as

    f=K/2+q_c,       f̅=K/2-q_c,

where q_c is the relative momentum between the c and c̅ quarks.
Since the B-S wave function χ(q_c) is sharply peaked when q_c∼ 0 for the heavy J/ψ in the nonrelativistic limit, one may neglect the q_c-dependence in the hard-scattering amplitude 𝒪̂(q_c) in the leading order approximation, then the amplitude can be simplified to

    A   =   -i√(3)∫d^4q_c/(2π)^4Tr[χ(q_c)𝒪̂(0)]
       =   -√(3)∫d^3q_c/(2π)^3Tr[
    ψ(q_c)𝒪̂(0)],

where the Salpeter function is defined as

    ψ(q_c)=i/2π∫dq_c^0χ(q_c).

For the vector meson J/ψ, the Salpeter function with leading order Dirac structures reads<cit.>

    ψ(q_c)=ϕ(q_c^2)(M+K)ε(K),

where ϕ(q_c^2) is a scalar function of q_c^2, and M represents the mass of J/ψ. With the help
of the definition of the S-wave wave function evaluated at the origin <cit.>

    ∫d^3q_c/(2π)^3ϕ(q_c^2)=1/2√(1/4π M)R_ψ(0),

one can obtain <cit.>

    A^αβμνε_α(K)ϵ_β^∗(k)ϵ_μ^∗(k_1)ϵ_ν^∗(k_2)=-1/2√(3/4π M)R_ψ(0)
    Tr[(M+K)ε(K)𝒪̂(0)].

And the hard-scattering amplitude 𝒪̂(0) can be written as

    𝒪̂(0)   =   iQ_ceg_s^2δ_ab/6ϵ^∗(k_2)
     k_2-k-k_1+M/-2 (k+k_1)· k_2ϵ^∗(k)k_2+k-k_1+M/-2 (k+k_2)· k_1ϵ^∗(k_1)
          +(5 permutations of k_1, k_2 and k),

where we have made the nonrelativistic approximation M≈ 2m_c.

The light-cone expansion of the matrix elements of the meson η^(') over quark and antiquark fields reads <cit.>

    ⟨η^(')(p) |q̅_α(x)q_β(y)|0⟩   =   i/4f_η^(')^q(pγ_5)_βα∫du e^i(u̅p· y+up· x)ϕ^q(u)+⋯,

where the high twist terms are omitted. With this definition, one can obtain the coupling of g^∗g^∗-η^(') <cit.>:

    M^μν   =   -i (4πα_s)δ_abϵ^μνρσk_1ρk_2σ
          ×∑_q=u,d,sf_η^(')^q/6∫^1_0duϕ^q(u)
    (1/u̅k_1^2+uk_2^2-uu̅m^2-m_q^2+(u↔u̅)),

where u is the momentum fraction carried by the quark and u̅=1-u, m_q is the mass of the quark(q=u,d,s), m is the mass of η^(').  The light-cone DA is <cit.>

    ϕ^q(u)   =   ϕ_AS(u)(1+∑_n=2,4⋯c^q_n(μ)C_n^3/2(2u-1))

with the asymptotic form of DA ϕ_AS(u)=6u(1-u) and c^q_n(μ) the Gegenbauer moments.
In Table <ref>, we list three models of the DAs discussed in Ref. <cit.>. Their shapes are shown in Fig. <ref>, where c_n^q are evaluated at μ=m_c.









The decay amplitude of J/ψ→γη^(') can be obtained by multiplying the above two couplings, inserting the gluon propagators and performing the loop integrations

    M_T=T^αβε_α(K)ϵ^∗_β(k)=1/2∫d^4k_1/(2 π)^4A^αβμνM_μνε_α(K)ϵ^∗_β(k)i/k_1^2+iϵi/k_2^2+iϵ,

where the factor 1/2 takes into account that the two gluons have already been interchanged both in A^αβμν and M_μν.
Using parity conservation, Lorentz invariance and gauge invariance, it can be proved that

    T^αβ∼ϵ^αβμνp_μk_ν,

so there is only one independent helicity amplitude H_QCD^q <cit.>

    T^αβε_α(K)ϵ^∗_β(k)   =   H_QCD^qh^αβε_α(K)ϵ^∗_β(k),

where

    h^αβ   =   i/p· kϵ^αβμνp_μk_ν.

With the help of the helicity projector <cit.>

    ℙ^αβ   =   1/2h_α^'β^'(-g^αα^'+K^αK^α^'/M^2)(-g^ββ^')
       =   -i/2p· kϵ^αβμνp_μk_ν,

one can obtain the helicity amplitude

    H_QCD^q   =   T^αβℙ_αβ
       =   2Q_c/9√(4πα_e)(4πα_s)^2√(3/π M)R_ψ(0)∑_q=u,d,sf_η^(')^q/MH_q.

The dimensionless function H_q reads

    H_q   =   -1/16π^22/1-x∫duϕ^q(u)I(u)

with x=m^2/M^2. I(u) is the sum of the loop integrals of the six Feynman diagrams for the decays

    I(u)   =   1/i π^2∫d^4q/16(N_1/C_1D_2D_3 D_4D_5+N_2/C_1D_1 D_3 D_4 D_5+N_3/D_1D_2 D_3 D_4 D_5
          +N_4/C_1D_1D_3 D_4 D_5+N_5/C_1D_2D_3D_4 D_5+N_6/D_1D_2D_3D_4 D_5)+(u↔u̅)

with q=k_1-k_2. Here the expressions of the denominators are given by

    C_1   =   1/4[(p-k)^2-M^2]+iϵ,
    
    D_1   =   1/4[(q-k)^2-M^2]+iϵ,
    
    D_2   =   1/4[(q+k)^2-M^2]+iϵ, 
    
    D_3   =   1/4[(q+(u̅-u)p)^2-4m_q^2]+iϵ,
    
    D_4   =   1/4(q+p)^2+iϵ,
    
    D_5   =   1/4(q-p)^2+iϵ,

and the six numerators N_i read

    N_1   =   N_5 = 1/4 k· p [k· q (m^2-p· q)+k· p (q^2-p· q)],
    
    N_2   =   N_4= 1/4 k· p [k· p (p· q+q^2)-k· q (m^2+p· q)],
    
    N_3   =   N_6= 1/4[2 k· p k· q p· q-m^2 k· q^2-q^2 k· p^2].

The third and the sixth terms in I(u) are five-point loop integrals, and the other terms are four-point loop integrals
since the denominator C_1 is independent of the loop momentum q. Before going to calculate the integral, we would present a short analysis of its IR properties. When one of the gluons goes on-shell, i.e., q→ p, the denominators D_2, D_3(with u=1-m_q/m), D_5 tend to zero. Following to the procedure in Ref. <cit.>, we can find that the one-loop integrals in the individual Feynman diagram for J/ψ→γη^(') have soft singularities, which depend on the momentum fraction u, and can make the convolution integral over u become sensitive to the shapes of the η^(') DAs. Moreover, the divergent term due to light quark pole would result in the final numerical results strongly dependent on the light quark masses. However, summing up the six Feynman diagrams, one can obtain

    I(u)   =   1/i π^2∫d^4q/16(m^2-q^2)(k· p q^2-k· q p· q)/4 D_1 D_2 D_3 D_4 D_5+(u↔u̅).

When q= p+λ (λ is a small quantity), the numerator, which arises from the summation of all Feynman diagrams,

    (m^2-q^2)(k· p q^2-k· q p· q)∼λ^2,

and the on-shell propagators

    D_2≃1/2K·λ∼λ,        D_3≃m_q/mp·λ∼λ,        D_5≃1/4λ^2∼λ^2.

For the ultrasoft gluon(λ→ 0), the contributions to the loop integral have the form

    ∫_q= p+λd^4q(m^2-q^2)(k· p q^2-k· q p· q)/ D_1 D_2 D_3 D_4 D_5   ∼   ∫d^4λλ^2/λ^4→ 0.

It means the sum of the loop integrals is IR safe.

With algebraic identities

    q^2=2 (D_4+D_5-m^2/2)=2 (D_1+D_2+M^2/2),
          k· p q^2-k· q p· q=2k· p m^2/M^2+m^2(D_1+D_2)+2k· p M^2/M^2+m^2(D_4+D_5)
    -k· q(D_4-D_5),

the loop function I(u) can be decomposed into a sum of four- and three-point one-loop integrals

    I(u)   =   1/i π^2∫d^4q/16(m^4(M^2-m^2)/M^2+m^21/D_2D_3D_4D_5
     -M^2(M^2-m^2)^2/2(M^2+m^2)1/D_1D_2D_3D_4
          -3(M^2-m^2)+2 k· q/21/D_1D_3D_4-M^2-m^2+2k· q/21/D_2D_3D_4)+(u↔u̅),

which can be analytically calculated with the technique proposed in Ref. <cit.> or the computer program Package-X <cit.>.
Performing the convolution integral between the loop function I(u) and the DA ϕ^q(u), we find that the function H_q in Eq. (<ref>) is very insensitive to the light quark masses. Specifically, the change of the absolute value of the function H_q does not exceed 3% when the value of m_q goes from 0 to 100 MeV for all the three kinds of η^(') DAs in Fig. <ref>. Actually, when the light quark propagator is near its mass shell, i.e., q∼±(u-u̅) p, the factor (k· p q^2-k· q p· q) in the numerator of
Eq. (<ref>) tends to zero and cancels the light quark propagator pole.
In Fig. <ref>, we show the dependence on m_q of the functions H_q^η=H_q|_m=m_η and H_q^η^'=H_q|_m=m_η^' with a asymptotic DA. It is noticed that the one-loop QCD contribution to the decay J/ψ→γπ^0 vanishes as a consequence of the antisymmetrical flavor wave function of π^0, even for m_u≠ m_d, which disagrees with the result in Ref. <cit.>.





For simplicity, we can take the following limit safely

    I_0(u)   =   lim_m_q→ 0 I(u),
    
    H_0   =   -1/16π^22/1-x∫duϕ^q(u)I_0(u),

that is H_q(q=u,d,s)= H_0. The expression of I_0(u) is presented in the Appendix. Numerically,
we find that the loop function I_0(u) is quite steady over the most region of u. In Fig. <ref>, we show the u dependence of I^η_0(u)=I_0(u)|_m=m_η and I^η^'_0(u)=I_0(u)|_m=m_η^' with the range u∈(0,1).



Unlike the result in the limit of m^2/M^2→0 <cit.>, the loop functions I^η_0(u) and I^η^'_0(u) change slowly over the momentum fraction u near the endpoints. As a consequence, the convolution integral between the loop function I_0(u) and the DA becomes insensitive to the shapes of η^(') DAs. For example, the difference between the convolution integral of I_0(u) with a “narrow" DA (model I in Fig. <ref>) and the one with a “broad" DA (model II and III in Fig. <ref>) is less than 2%.

After these analyses, we return to the remaining calculations. The helicity amplitude H_QCD^q in Eq. (<ref>) can be simplified to

    H_QCD^q   =   2Q_c/9√(4πα_e)(4πα_s)^2√(3/π M)R_ψ(0)f_η^(')/MH_0

with the effective decay constants

    f_η^'=f_η^'^u+f_η^'^d+f_η^'^s,       
    f_η=f_η^u+f_η^d+f_η^s.


Besides the one-loop QCD contributions, QED processes J/ψ→γ^∗→γη^(') can also contribute to the decays J/ψ→γη^('). The corresponding Feynman diagrams are depicted in Fig. <ref>,



and the contribution reads

    H_QED   =   -Q_c(4πα_e)^3/2√(3/π M)R_ψ(0)
    ∑_q=u,d,sQ_q^2f_η^(')^q/Mh_q,

where the dimensionless function is

    h_q   =   1-x/2∫duϕ^q(u)(1/u-uu̅x
    -m_q^2/M^2+iϵ+(u↔u̅))

and the Q_q represents the light quark charge.




 §.§ The contributions of the gluonic content of η^(')

The gluonic content of η^(') can contribute to the J/ψ→γη^(') at the tree level. However, such contributions are suppressed by a factor of m^2/M^2 <cit.>. The corresponding Feynman diagram is shown in Fig. <ref>. There are other two Feynman diagrams from permutations of the photon and the gluon legs.



The leading twist in the light-cone expansion of the matrix elements of the meson η^(') over two-gluon fields is <cit.>:

    ⟨η^(')(p)|A_α^a(x)A_β^b(y)|0⟩   =   1/4ϵ_αβρσk^ρp^σ/p· kC_F/√(3)δ^ab/8f_η^(')^g∫du e^i(up· x+u̅p· y)ϕ^g(u)/u(1-u)

with the effective decay constant f_η^(')^g=1/√(3)(f_η^(')^u+f_η^(')^d+f_η^(')^s) and the gluonic twist-2 DA <cit.>

    ϕ^g(u)   =   30u^2(1-u)^2∑_n=2,4⋯c^g_n(μ)C_n-1^5/2(2u-1).

The corresponding contributions can be expressed as

    H_QCD^g   =   2Q_c/9√(4πα_e)(4πα_s)R_ψ(0)/√(π M)f_η^(')^g/MH_g

with

    H_g=∫du ϕ_g(u)/u(1-u)2 x(2 u-1)/1-x^2(1-2 u)^2.

From the Eq. (<ref>), one can find that H_g is proportional to a suppression factor of x=m^2/M^2. Actually, the leading twist gluonic content contributions are almost two on-shell gluons contributions, which are suppressed by the factor m^2/M^2 due to the special form of the Ore-Powell matrix elements as found in Refs. <cit.> years ago.





§ NUMERICAL RESULTS



The decay widths of J/ψ→γη^(') can be expressed as

    Γ(J/ψ→γη^('))=2/31/16π1-x/M|H_QCD^q+H_QCD^g+H_QED|^2.

In order to remove the uncertainties from R_ψ(0), we relate the decay widths Γ(J/ψ→γη^(')) to Γ(J/ψ→ e^+e^-)

    ℬ(J/ψ→γη^('))=Γ(J/ψ→γη^('))/Γ(J/ψ→ e^+e^-)ℬ^exp(J/ψ→ e^+e^-)

with the leptonic decay width <cit.>

    Γ(J/ψ→ e^+e^-)=4α_e^2Q_c^2/M^2|R_ψ(0)|^2(1-16/3α_s)

and its experimental value <cit.>

    ℬ^exp(J/ψ→ e^+e^-)=(5.971±0.032)× 10^-2.

For numerical calculations, all the values of meson masses, Γ_J/ψ and f_π are quoted from the PDG <cit.>. We use the FKS scheme for the η-η^' mixing <cit.>, and then the effective decay constants f_η^(')^q can be parameterized as

    f_η^u(d)   =   f_q/√(2)cosϕ,          f_η^s=-f_ssinϕ, 
    
    f_η^'^u(d)   =   f_q/√(2)sinϕ,          f_η^'^s=f_scosϕ.

For the three phenomenological parameters, i.e., the mixing angle ϕ and the decay constants f_q, f_s, they have been determined in different methods <cit.>. Here we take the up-to-date values from Refs. <cit.> which are recapitulated in Table <ref>.



The values in the first line are extracted from the low energy processes(LEPs) V→η^(')γ, η^(')→ Vγ(V=ρ,ω,ϕ). The second and the third lines are the results extracted with rational approximations for the η^(') transition form factor(TFF) F_γ^∗γη^(')(Q^2). The results of both the first and the second lines are generally consistent with the known FKS results <cit.> where the ratio R_J/ψ with the approximation of nonperturbative matrix elements ⟨0|G_μν^aG̃^a,μν|η^(')⟩ was adopted. While in the third line, the parameters are extracted with rational approximations for the η^' TFF F_γ^∗γη^'(Q^2), which is in accord with the BaBar measurement in the timelike region at q^2=112 GeV^2 <cit.>.

The Gegenbauer moments c^q_2(μ), c^q_4(μ) still have large uncertainty as depicted in Table <ref>. Fortunately, the dimensionless function H_q in Eq. (<ref>) is insensitive to the shapes of the η^(') DAs as we have shown in section <ref>. Furthermore, both H_g and h_q are an order of magnitude smaller than the H_q, therefore, the uncertainty of the Gegenbauer moments impacts our numerical calculations lightly (less than 2% among the results with three different models in Table <ref>). So in the following numerical calculations, we choose the Model I with

    c^q_2(μ_0)=0.10,        c^q_4(μ_0)=0.10,       
    c_2^g(μ_0)=-0.26.


As known, it is very hard to give precise predictions for individual decay width. With the nonperturbative matrix
elements ⟨0|G_μν^aG̃^a,μν|η^(')⟩, the decay widths of J/ψ→γη^(') have been given in Ref. <cit.>

    Γ(J/ψ→γη^('))   =   2^3/5^23^10α_e^3α_s^2/π(M^4/m_c^8)
    (1-m_η^(')^2/M^2)^3|⟨0|G_μν^aG̃^a,μν|η^(')⟩|^2/Γ(J/ψ→ e^+e^-),

where the factor m_c^-8 will bring very large uncertainty. While, in the pQCD approach employed in this paper, there
also exist large uncertainty due to the factor α_s^4(μ).

For comparison, in Table <ref>, we present results of these two methods with m_c=1.5 GeV and
α_s=0.34 as benchmarks. For the results in the second column, the matrix elements ⟨0|G_μν^aG̃^a,μν|η^(')⟩ are evaluated with the updated values of ϕ, f_q and f_s in the
first line of Table <ref>. Generally, one may expect the order of magnitude could be correctly predicted
by the both approaches. However, we find the pQCD estimation of ℬ(J/ψ→γη) is too small to
be comparable to its experimental one. The reason may be due to our choice of the inputs ϕ, f_q and f_s extracted
from low energy processes, or our understanding of η-η^' mixing scheme is incomplete which is beyond the scope
of this paper.


 


Recently, the BaBar collaboration <cit.> has made the measurements of the η and η^'
TFFs at q^2=112 GeV^2, which have challenged the theoretical prediction very much.
It is true that precise theoretical estimation of the TFFs is hard, due to uncertainties in the ϕ, f_q,
f_s, the DAs of η^(') and even the mixing scheme at high energy. In the literature, there are extensive discussions of this hot topic <cit.>. In Ref. <cit.>, Escribano et al. have presented a treatment of η and η^' TFFs
with rational approximations and extracted these parameters in Table <ref>. With these inputs and α_s=0.34, we show our results in Table <ref>. From the Table, we find R_J/ψ is in good agreement with its experimental value R_J/ψ^exp=4.65±0.21 <cit.>, only when we adopt the set of parameter values of η^'TFF.
For other two sets of parameter values, ℬ(J/ψ→γη) is estimated to be too small, which results in
R_J/ψ two orders higher than R_J/ψ^exp.





In Table <ref>, we present results with the contributions due to the gluonic content of η^(') and the one from QED processes J/ψ→γ^∗→γη^('). We can find that such contributions enhance
ℬ(J/ψ→γη) much. However, ℬ(J/ψ→γη) is still far from its
experimental value for both LEPs and ηTFF values of ϕ, f_q and f_s. Only η^'TFF set of parameter
values can give ℬ(J/ψ→γη^(')) and R_J/ψ comparable with their experimental data.





In the remaining part of this section, we would present a determination of ϕ without the input values in Table <ref>. The ratio R_J/ψ in our calculation is

    R_J/ψ=M^2-m_η^'^2/M^2-m_η^2|H_QCD^q+H_QCD^g+H_QED|^2_m=m_η^'/|H_QCD^q+H_QCD^g+H_QED|^2_m=m_η.

Since the scalar functions are insensitive to the shapes of the η^(') DAs, the ratio R_J/ψ mainly depends on the angle ϕ and the ratio f_s/f_q. However, with the help of the ratio

    Γ(η→γγ)/Γ(η^'→γγ)
    =m_η^3/m_η^'^3(5√(2)f_s/f_q-2 tanϕ/5√(2)f_s/f_qtanϕ +2 )^2

and the experimental values <cit.>

    Γ^exp(η^'→γγ)=4.36(14) KeV,    Γ^exp(η→γγ)=0.516(18) KeV,

the ratio R_J/ψ becomes a function which only depends on the mixing angle ϕ. In Fig. <ref>, we show
the dependence of the ratio R_J/ψ on the mixing angle ϕ.



As displayed by the horizontal dashed lines in Fig. <ref> of the experimental measurement R^exp_J/ψ=4.65±0.21 <cit.>, one can find

    ϕ   =   33.9^∘±0.6^∘,

which is in good agreement with η^'TFF result ϕ=33.5^∘±0.9^∘ <cit.>[It is noticed that the predicted η TFF is lim_Q^2→+∞Q^2F_ηγ^∗γ(Q^2)=(0.160±0.024) GeV, which is not in line with the BaBar measurement q^2|F_ηγ^∗γ(q^2)|_q^2=112 GeV^2=(0.229±0.031) GeV, while the predicted η^' TFF, lim_Q^2→+∞Q^2F_η^'γ^∗γ(Q^2)=(0.255±0.004) GeV, is in accord
with the BaBar measurement q^2|F_η^'γ^∗γ(q^2)|_q^2=112 GeV^2=(0.251±0.021) GeV.
More discussions could be found in Ref. <cit.>.], but in clear disagreement with the ϕ=40.6^∘±0.9^∘ <cit.> extracted from low energy processes
with nonperturbative methods. It is noticed that the lattice calculation of the UKQCD collaboration <cit.> indicates a value ϕ^fit∼34^∘, while the ETM collaboration <cit.> gives ϕ in the range of 40^∘∼46^∘.

Last but not least, we would compare our result of the loop function with the one obtained by Kühn <cit.>.
Since, besides the approximation M≈2 m_c, no further approximation is made, so our loop function I_0(u), as shown
in the Appendix, is much more complicated than the one in Ref. <cit.>. However, in the limit of m^2/M^2→0, our I_0(u) is reduced to
the weight function 𝒲(u) in the Eq. (11) of Ref. <cit.>.






§ SUMMARY


In this paper, we have revisited the radiative decays J/ψ→γη^(') in detail in the framework of pQCD. Comparing to the pioneer work <cit.>, we do not take the weak-binding approximation for the final mesons and evaluate the involved one-loop integrals analytically with the light quark masses kept. Moreover, we also consider the contributions from the QED processes and the gluonic contents of η^(').
Different from the results obtained in Ref. <cit.>, our numerical results are insensitive to the light quark masses.
In addition, we find that the ℬ(J/ψ→γη^(')) are insensitive to the shapes of the η^(') DAs.

Using three sets of values for ϕ, f_q and f_s, namely LEPs <cit.>, ηTFF <cit.> and η^'TFF <cit.>, we have presented our numerical
results in Table <ref> and <ref>, where ℬ(J/ψ→γη) is too
small to be comparable with the experimental one for the parameters extracted from LEPs and η TFF. Only with the values extracted from η^' TFF, which is in accord with the BaBar measurement in the timelike region at q^2=112 GeV^2 <cit.>, our results of ℬ(J/ψ→γη^(')) and R_J/ψ are
in good agreement with the experimental data.

As a crossing check, we use our calculation of R_J/ψ and Γ^exp(η^(')→γγ)
as inputs to extract the value of ϕ, and find ϕ=33.9^∘±0.6^∘, which is in good agreement with the η^'TFF one ϕ=33.5^∘±0.9^∘ remarkably. However, such a small ϕ differs too much from
the ϕ=40.6^∘±0.9^∘ extracted from low energy processes and J/ψ→γη^(') with
nonperturbative matrix elements ⟨0|G_μν^aG̃^a,μν|η^(')⟩ due to U_A(1) anomaly
dominance argument. The difference may arise from g^∗g^∗-η^(') TFF used in our calculation, in like manner,
γ^∗γ-η^' TFF in the extraction of ϕ from the BaBar measurement of q^2|F_η^'γ^∗γ(q^2)|_q^2=112 GeV^2. Anyhow, the physics under the difference
is interesting and certainly worth further investigations.





§ ACKNOWLEDGEMENTS

This work is supported by the National Natural Science Foundation of China under Grant Nos. 11675061, 11775092 and 11435003.






§ APPENDIX: THE DIMENSIONLESS FUNCTION H_0


The dimensionless function H_0 reads

    H_0=-1/16π^22/1-x∫^1_0duϕ^q(u)I_0(u).1

Since the H_0 is insensitive to the shapes of the η^(') DAs,
we simply take the asymptotic DA ϕ^q(u)=6 u (1-u) in the following discussion.
The analytical expression of the one-loop function I_0(u) can be expressed as

    I_0(u)= g_1(ξ)+g_2(ξ)+g_3(ξ)+g_4(ξ)+g_5(ξ)+g_6(ξ)+(ξ→-ξ)2

with ξ=1-2u and

    g_1(ξ)=   2 x [1+3 (1-x) ξ-x ξ^2]/(1-x) (1+ξ) (1-x ξ)[Li_2(1-x ξ/2-x-x ξ)+Li_2(-(1-2 x) (1-x ξ)/x (1+ξ-2 x ξ))
        +Li_2(1+ξ+2 x^2 ξ (1+ξ)-x (1+4 ξ+ξ^2)/x ξ (-2+x+x ξ)-iξϵ)-Li_2(1-x ξ/1+ξ-2 x ξ)
        -Li_2(-(1-2 x) (2-x-x ξ)/x (1+ξ-2 x ξ))-Li_2(1+x^2 ξ (1+ξ)-x (1+2 ξ)/x ξ (-2+x+x ξ)-iξϵ)
        -Li_2(1+ξ-2 x ξ/2-x-x ξ)+Li_2(2-x-x ξ/1+ξ-2 x ξ)
    ],
    
    g_2(ξ)=   -2 (1-x)/(1+x) (1-ξ) (1-x ξ)[-4 Li_2(x)+2 Li_2(2 x-1)-2 Li_2(-ξ+xξ+x ξ^2)
        +2 Li_2(ξ-x ξ+x ξ^2)-ln^2 x-2 i πln x+4 i πln(2-x-x ξ/1+ξ-2 x ξ)],
    
    g_3(ξ)=   2 x [(1+x^2 ξ) (3-ξ) -x (5-2 ξ+ξ^2)]/(1-x^2) (1-ξ) (1-x ξ)[-Li_2(-(1-x (2-ξ)) ξ/(1-x (1-ξ)) (1-ξ))
        -Li_2(-1-x (2-ξ)/x^2 (1-ξ))+Li_2(x^2 ξ/1-x (1-ξ))+2Li_2(x (2-ξ)-1/x (1-ξ))],
    
    g_4(ξ)=   -8x/(1+x) (1-ξ^2)[2 i πln(2-x-x ξ/1+ξ-2 x ξ)-2 Li_2(3x-1/2 x)+Li_2(1-3 x/2-4 x)
        - Li_2(-1+x (2+ξ)/x^2 (1+ξ))- Li_2(x^2/2 x-1)+ Li_2(3 x-1/2 x^2)+ Li_2(x^2 ξ/-1+x+x ξ)
        +2 Li_2(-1+x (2+ξ)/x (1+ξ))- Li_2(ξ (1- 2x-xξ)/(1+ξ) (1-x-x ξ)) ],
    
    g_5(ξ)=   1/3(1-x) (1-ξ)[ (2- 3x+xξ)[π^2-3 ln^2 x-12Li_2(x)-6 i πln x ]
        +12 iπ (2-x-x ξ) ln2-x-x ξ/1+ξ-2 x ξ],
    
    g_6(ξ)=   2(1+xξ)ln[(1-ξ)(1+x ξ)]/1-x+xξ+4 (x-1)ln(2-2x)/(1-2x)(1-ξ)
        +2(1+ξ)(1-xξ)/(1-ξ)(1-x-xξ)ln[(1-xξ)(1+ξ)].3

In the above expressions, the dilogarithm function Li_2(x) and the logarithm function ln(x) without “i ϵ" prescription are defined as

    Li_2(x)   = lim_ϵ→ 0^+Li_2(x-i ϵ),
    ln(x)   = lim_ϵ→ 0^+ln(x+i ϵ).4


In the limit of x=m^2/M^2→ 0,

    g_1(ξ)   →    0,
    
    g_2(ξ)   →1/1-ξ(2 ln^2 x+ 4 i πln x  +π ^2/3-8 i πln2/1+ξ+4 Li_2(-ξ)- 4Li_2(ξ)),
    
    g_3(ξ)   →    0,
    
    g_4(ξ)   →  0,
    
    g_5(ξ)   →1/1-ξ(-2 ln^2 x-4 i πln x +2 π ^2/3+8 i πln2/1+ξ)
    
    g_6(ξ)   →2 /1-ξ(-ln 4 +(1-ξ) ln(1-ξ)+(1+ξ) ln(1+ξ)),5

and

    I_0(u)   →8/1-ξ^2(π^2/4-ln 2-ξ/2ln(1-ξ/1+ξ)+1/2ln(1-ξ^2)+ξLi_2(-ξ)-ξLi_2(ξ)),6

which reproduces the weight function 𝒲(u) in the Eq. (11) of the work by Kühn <cit.>.
Performing the convolution integral between the loop function I_0(u) and the asymptotic DA,
we display the mass dependence of H_0 in Fig. <ref>. For comparison, we also present the dimensionless function H^K obtained in Ref. <cit.>.



As known, the contribution of on-shell gluons in the partial wave 0^-+ is suppressed by a factor of m^2/M^2 <cit.>, and our results for the absorptive part of the function H_0, which represents the contribution of on-shell gluons, indeed follow this character in the limit of m^2/M^2→0. However,
the dispersive part of the function H_0, which represents the contribution of virtual gluons, is not suppressed in the limit of m^2/M^2→0. While the matrix elements ⟨0|G_μν^aG̃^a,μν|η^(')⟩ are twist-4 effects which are suppressed by the factor of m^2/M^2 relative to the leading twist terms. That is the reason
why the mass dependence of the decay widths obtained in this work differs from that obtained in Ref. <cit.> by an additional factor M^4/m^4.
In additions, when the factor m^2/M^2 is not very small, the suppression for the absorptive part is no longer operative. For example, the dimensionless function H_0^η=H_0|_m=m_η is dominated by its dispersive part, while the dispersive part and the absorptive part of the H_0^η^'=H_0|_m=m_η^' are comparable.







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