Scalar charmonium and glueball mixing in $e^+ e^-\to J/\psi X$

We study the possibility of the scalar charmonium and glueball mixing in $e^+ e^-$ annihilation at $\sqrt{s}=10.6$ GeV. The effects can be used to explain the unexpected large cross section ($12\pm 4$ fb) and the anomalous angular distribution ($\alpha= -1.1^{+0.8}_{-0.6}$) of the exclusive $e^+e^-\to J/\psi\chi_{c0}$ process observed by Belle experiments at KEKB. We calculate the helicity amplitudes for the process $e^+ e^- \to J/\psi H(0^{++})$ in NRQCD, where $H(0^{++})$ is the mixed state. We present a detailed analysis on the total cross section and various angular asymmetries which could be useful to reveal the existence of the scalar glueball state.


Introduction
J/ψ-charmonium productions, such as e + e − → J/ψη c and e + e − → J/ψχ c0 . The angular distributions of all those processes with single virtual photon exchange can be parametrized as by using an asymmetry parameter α (−1 ≤ α ≤ 1). For e + e − → J/ψη c and e + e − → J/ψχ c0 , α = 1 and 0.25 respectively [19], while for the J/ψ-G 0 production mode, α is estimated to be −0.85 in NRQCD [9]. We note here that preliminary result on the J/ψ angular distribution in the J/ψ-χ c0 region is α = −1.1 +0.8 −0.6 , which is more consistent with α ∼ −1 rather than α ∼ 0.25 [3]. Conservatively, the present Belle data [3] suggest that α should be negative. This unexpected angular distribution leads us to consider further the possibility of the scalar charmonium χ c0 and glueball G 0 mixing. In this Letter, we will focus on the mixing effects on various angular distributions which could be useful to reveal the existence of the scalar glueball state.

Formulae
Let us start with the angular momentum and parity considerations. The J/ψ-scalar(H) associated production via a virtual photon in the J P C notation is 1 −− → 1 −− + 0 ++ . If we denote by L the orbital angular momentum between J/ψ and H, the angular momentum conservation tells L = 0, 1, 2. Conservation of parity gives −1 = 1 × (−1) × (−1) L , and hence L should be even. We are left with L = 0 or 2, that is, only S-wave and D-wave J/ψ-H are allowed. Now we present the helicity amplitudes for the J/ψ-χ c0 and J/ψ-G 0 productions separately in NRQCD. One of the crossed Feynman diagrams for e + e − → J/ψχ c0 is shown in Fig. 1(a), and that for e + e − → J/ψG 0 is shown in Fig. 1(b). We adopt the standard covariant projection formalism for the color singlet productions [20]. If a charmonium carries the momentum P , then the charm quark and anti-quark within the charmonium are on-shell and carry momenta P/2 − p and P/2 + p, where p is the relative momentum of the quark pair and satisfies p · P = 0. The amplitudes for the charm-pair production are furthermore projected into a certain spin component by the projection operator After expanding the amplitudes in p and convoluting them with the wave functions at the origin, one obtains the helicity amplitudes for S-wave and P -wave charmonia, such as J/ψ and χ c0 . For the glueball production as shown in Fig. 1(b), we follow the method of Ref. [9] and extract the leading-twist contribution by expanding the gluon momenta in the light-cone direction and requiring the gluon pair to be collinear. We define the light-cone direction as n µ = (1, 0, 0, 1), while the J/ψ momentum direction is alongn = (1, 0, 0, −1). The formation of the scalar glueball G 0 from the gluon pair is described by a collinear wave where φ 0 is the distribution wave function for G 0 and x is the lightcone momentum fraction. We denote the G 0 momentum by k = (k + = k ·n, k − = k · n, 0 ⊥ ), and further represent the transverse momentum of a gluon inside G 0 as k ⊥ . By integrating over k ⊥ and transforming from the momentum to the coordinate space, the distribution wave function φ 0 (x) is obtained as Here F 0 αβ = [−g αβ + (n αnβ +n α n β )/2]/ √ 2 is the scalar projector. G − a is the gluon field along the light-cone direction and a is the color index.
Under the above approximations, the helicity amplitudes for J/ψ-χ c0 production are given by, and those for J/ψ-G 0 production are where j is the electron-positron current and ǫ is the polarization vector of J/ψ. σ and λ denote the helicities of the electron-positron current (along the electron beam direction) and J/ψ, respectively. Φ(0) and Φ ′ (0) are the wave functions at the origin for J/ψ and χ c0 . In the center-of-mass frame, the amplitudes are reduced to where the coefficients a χ,g and b χ,g are for the longitudinally and transversely polarized J/ψ amplitudes respectively. We find for J/ψ-χ c0 production, and for J/ψ-G 0 production. Here r = 4m c / √ s and C F = (N 2 c − 1)/(2N c ) = 4/3 is the color factor. These results are consistent with those in Refs. [9,19,21].
It is natural to expect the mixing between the pure states of χ c0 and G 0 if they carry the same quantum numbers J P C = 0 ++ and have small mass difference. The pure charmonium and glueball states, χ c0 and G 0 , and the mass eigenstates H 1 , H 2 are then linked by where ξ is the mixing angle. The mass-squared matrix for χ c0 and G 0 has the following form and is diagonalized as We assume that the observed χ c0 state is H 1 , and hence M 1 = M(χ c0 ) observed = 3.415 GeV [22]. Because H 1 properties observed in charmonium radiative decays are consistent with the pure χ c0 assumption, we expect the mixing angle to be small sin 2 ξ ≤ 0.1. The mixing angle ξ is real if the transition matrix element ∆ is real.
The mixing angle can be non-negligible (sin 2 ξ ∼ 0.1) if |∆| ∼ |M 2 χ − M 2 g |. The helicity amplitudes for J/ψ-H 1 and J/ψ-H 2 productions keep the forms in Eq. (6) and the coefficients a χ,g and b χ,g have to be replaced by a H 1 ,H 2 and b H 1 ,H 2 respectively. With Eq. (9), we have In the sin ξ = 0 limit, H 1 is a pure χ c0 , while H 2 is a pure glueball.

Results
For J/ψ productions in e + e − annihilation, the mixing effects can be measured via the angular distributions of J/ψ and its leptonic decays. According to the helicity amplitudes in Eq.
(6), the triple angular distribution for the process e + e − → J/ψH → l + l − H (H denotes H 1 or H 2 ) is given by where θ * and φ * are the polar and azimuthal angles of l − in the J/ψ rest frame. The polar angle θ * is measured from the J/ψ momentum direction in the e + e − collision rest frame, and φ * is measured from the scattering plane. In Eq. (12), B is the branching fraction B(J/ψ → l + l − ) = 5.9% for l = e, µ.
With respect to the mass difference of the mass eigenstates H 1 and H 2 , we meet with two different cases: (1) If the mass difference is large enough, only H 1 has been identified as the χ c0 resonance. (2) It is also possible that the mass difference is quite small and both H 1 and H 2 contribute to the observed resonance peak. Therefore we will consider these two cases separately.
In the first case, we rename H 1 as H. By integrating over θ * and φ * and comparing with Eq. (1), we obtain where σ T and σ L are the cross sections for the transversely and longitudinally polarized J/ψ productions, and σ = σ T +σ L is the total cross section. One can obtain the cos θ * distribution by integrating over θ and φ * in Eq. (12), and obtain dσ/d cos θ * ∼ 1 + α * cos 2 θ * . It is easy to find α * = α according to Eq. (12). The interference term 2Re(a * H b H ) can be measured through the combination of the θ, θ * and φ * distributions. We introduce a new quantity α off as the normalized interference term Both α and α off are bounded in the interval [−1, 1]. Now we present our numerical results based on the input parameter values e 2 /4π = 1/129.6, g 2 s /4π = 0.26, m c = 1.5 GeV, and √ s = 10.58 GeV. The value of the J/ψ wave function at the origin can be obtained from Γ(J/ψ → e + e − ), |Φ(0)| 2 = 0.0336 GeV 3 in the leading order of NRQCD [19,6,7]. The χ c0 wave function at the origin Φ ′ (0) is estimated from the observed width Γ(χ c0 → γγ) [19]. Since we assume that the observed state is a mixture H, the partial width is actually proportional to | cos ξΦ ′ (0)| 2 . Here we assume that only the charmonium content of H contributes to the decay χ c0 → γγ at the leading order in α s . Therefore we obtain | cos ξΦ ′ (0)| 2 = 0.0117 GeV 5 . It should be noted that the partial widths and the observed radiative transition rates can be affected at several percent level for the mixing angle of the order of sin 2 ξ ∼ 0.1.
The upper bound of the glueball wave function |I 0 | is obtained from the radiative Υ decay. The CUSB Collaboration reported a 90%-C.L upper limit for the branching ratio Br(Υ → γH) which is about 0.01%-0.15% for the scalar boson (H) mass between 2 and 8.5 GeV [23]. The upper bound is obtained in terms of the CUSB mass resolution, 20 MeV. As discussed in Ref. [9], if the H decay width Γ H is larger than the resolution, the upper bound should be loosen by a factor of Γ H /(20MeV). In Ref. [9], the authors considered the pure glueball production process, e + e − → J/ψG 0 , and they assumed the glueball decay width should be less than about 100 MeV, which is the full width at half maximum of the 'χ c0 ' peak in the Belle fit to the J/ψ recoil mass distribution. They found the upper bound for the glueball wave function |I 0 | 2 < 5.8 × 10 −3 GeV 2 . Accordingly, the rate of the cross sections of the pure glueball and charmonium productions σ J/ψG 0 /σ J/ψχ c0 which is proportional to |I 0 | 2 /|Φ ′ (0)| 2 , is less than 0.72. Similar upper bound applies in our case, except that the radiative Υ decay rate constrains | sin ξI 0 | 2 instead of |I 0 | 2 , since Υ can decay to the glueball content of H only, at the leading order in α s . In the e + e − → J/ψH process, because the contributions from the charmonium and glueball contents are proportional to cos 2 ξ and sin 2 ξ, respectively, the upper bound of Ref. [9] leads to the bound (sin 2 ξσ J/ψG 0 )/(cos 2 ξσ J/ψχ c0 ) < 0.72. We therefore define the following quantity to present our numerical results. Here Sign sin ξ I 0 Φ ′ (0) determines the sign of the interference terms. The ξ-dependence of our results enters only through the rate R. For simplicity, we assume that the wave functions Φ ′ (0) and I 0 are real.
The magnitude of the parameter R can be as large as unity if H 1 is the only gluon-rich state which contribute to the radiative Υ decay. It is, however, unlikely that the H 2 mass lies outside of the reach of the CUSB search [23]. If the H 2 mass is in the region 2-8.5 GeV, the radiative Υ decay rate should constrain | cos ξI 0 | 2 , and therefore gives an upper bound on |R| in Eq. (15) of the order of tan 2 ξ(Γ H 2 /100 MeV) × 0.72. If sin 2 ξ < 0.1 and if the H 2 width Γ H 2 is below 500 MeV, the allowed range of R is reduced to |R| ≤ 0.4.
In Fig. 2, we show the cross section for e + e − → J/ψH in the interval −1 < R < 1. We find the cross section from the contribution of the χ c0 content, cos 2 ξ σ J/ψχ c0 , is 2.7 fb, much smaller than 12 ± 4 fb which we estimate from the preliminary Belle data [24]. We note that the contribution from the interference term can significantly increase the cross section in the region −1 < R < 0, and decrease the cross section for 0 < R < 1. For instance, at R = −0.1, the total cross section is about 4.3 fb, while sin 2 ξ σ J/ψG 0 = |R| cos 2 ξ σ J/ψχ c0 = 0.27 fb. The interference term makes up the difference 4.3 − 2.7 − 0.27 = 1.3 fb of the total cross section, which is about five times greater than the direct contribution from the glueball content. At R = −0.4, the total cross section is about 6.5 fb among which 2.7 fb comes from the interference contribution. While at R = −1, the total cross section can be as large as 9.6 fb (4.2 fb from the interference contribution), which is comparable with the central value of the experimental data.
Now we turn to the second case in which both of the two mass eigenstates H 1 and H 2 are hidden in the 'χ c0 ' resonance peak of the J/ψ recoil mass distribution in the Belle data [3]. In this case, one can obtain the total cross section and asymmetries by summing over the triple angular distributions in Eq. (12) for H 1 and H 2 . By using the relation between the coefficients a H 1 ,H 2 , b H 1 ,H 2 and a χ,g , b χ,g in Eq. (11), we find that From the above relations, we can find that the sum of the contributions from the two mixed states H 1 and H 2 is essentially identical to the sum of the contributions from the pure χ c0 and G 0 states. The sum of the cross sections for J/ψ-χ c0 and J/ψ-G 0 productions is simply to be σ J/ψχ c0 + σ J/ψG 0 = (1 +R)σ J/ψχ c0 , whereR = σ J/ψG 0 σ J/ψχ c0 .
For the leading order estimate of σ J/ψχ c0 = 2.7 fb, the sum of the cross sections can be as large as 5.4 fb forR = 1, which is to be compared with our estimate of 12 ± 4 fb [24]. The asymmetry α for this case is given by and α off is expressed as where a χ,g and b χ,g follow from Eqs. (7) and (8). They are plotted againstR in Fig. 4. One can find α = 0.25 and α off = −0.97 by settingR = 0 for the J/ψ-χ c0 production, and α = −0.85 and α off = −0.52 for the J/ψ-G 0 production in theR → ±∞ limit, which are the same as those for the previous case. α falls off and α off goes up with increasingR.R > 0.20 gives α < 0 and α off > −0.87. AtR = 1, α = −0.41 and α off = −0.70.
In this paper, we studied the disagreement between the experiment and the NRQCD prediction for the J/ψ-χ c0 associated production, in the production cross section and the J/ψ angular distribution. We introduced a charmonium-glueball mixing mechanism to explain this discrepancy and propose to study various angular distributions to explore the possible mixing effects. Our results may facilitate the present and future experimental measurements to resolve possible glueball contents in the charmonium resonances.