Glueball hunting in $e^+e^- \to J/\psi\to \phi f_0$

Building on recent work by Brodsky {\it et al.}, we advocate searching for glueball degrees of freedom in $e^+e^- \to J/\psi\to \phi f_0$ at CLEO-c and BES.

Brodsky, Goldhaber, and Lee [1] have proposed a novel approach to producing (scalar) glueballs in e + e − annihilation to account for the anomalously large cross sections for J/ψ + η c , χ c0 , and η c (2S) observed at Belle [2].They made a pQCD estimate of the cross section for e + e − → γ * → J/ψG 0 at √ s = 10.6 GeV, and found it to be similar to the exclusive charmonium-pair production e + e − → J/ψh for h = η c and χ c0 .Further, since γ * → (cc)(cc) and γ * → (cc)(gg) were of the same nominal order, they suggested that some portion of the anomalously large signal observed by Belle in e + e − → J/ψX may actually be due to the production of J/ψG 0 .This is an interesting idea theoretically but has a potential limitation phenomenologically.As presented, the work of Ref. [1] applies when M G ≃ M J/ψ ≃ 3 GeV.However, Lattice QCD [3,4] and phenomenological studies [5,6] suggest a much smaller mass scale for the lightest scalar glueball M G ≃ 1.5 GeV.Some analyses suggest even lower glueball masses [7,8].Taking into account these factors, we anticipate that the mass scale for the lightest scalar glueball is smaller than 3 GeV.
Thus we consider here the application of the work of Ref. [1] to the scenario of a scalar glueball in the O(1) GeV mass region.The analysis of Ref. [1] allows one to rescale the kinematics such that instead of a 3-GeV glueball recoiling against a J/ψ, we may consider a 1-GeV glueball recoiling against a φ.Also, rescaling the c.m. energy by a factor of three brings one to the kinematic region of interest currently at BES and to CLEO-c.
In Ref. [1], the mass scale is introduced via the mass ratio r = 4m c / √ s, where m c = M J/ψ /2 is the charm quark mass.By choosing the glueball mass M G = M J/ψ , the phase space factor for J/ψG 0 production cancels in the branching ratio fraction of J/ψG 0 to J/ψη c .As a result, the s-dependence of the branching ratio fraction will be embedded in r apart from strong couplings and nonperturbative factors determined through the quarkonium decay in its rest frame.Due to this feature, given that M G = M (q q) , the branching ratio fraction of q h qh → γ * → (q q)G 0 to q h qh → γ * → (q q)(q q) scales in terms of r apart from a constant, where q h denotes a heavy quark.
First, we examine the process γ * → φ(gg) in parallel to γ * → J/ψ(gg).An important argument of Ref. [1] is that the decays of γ * → (q q)(gg) and γ * → (q q)(q q) are the same order (see Fig. 1(a)-(b)).The ratio of γ * → φG 0 to γ * → µ + µ − can be estimated by applying Eq. ( 7) of Ref. [1]: where e s and m s = M φ /2 are the s quark's charge and mass.The gluon distribution factor |I 0 | 2 was assumed to be a function of the glueball's J P C and to scale with mass, so we adopt the same form as Ref. [1].Φ ee 0 is a phase space factor: For these reduced energies we adopt the running coupling constant α s ∼ 0.33 at √ s = 3.1 GeV as a guide, and assume also α G s = α η ′ s = 0.33 in analogy with the treatment of Ref. [1].The matrix element O 1 φ is given by the radial wavefunction of the ss in the φ at the origin R(0) by analogy with the case of cc: , where f φ is the decay constant of the φ meson.For a glueball mass M G ≃ 1 to 1.7 GeV, by analogy with Ref. [1], we would compare φG with φη ′ or any of φf 0 (980), φf 0 (1370), φf 0 (1500), φf 0 (1700), which would be clear if η ′ = η(ss) and f 0 = f 0 (ss).However, in practice, the probability of ss in η ′ is about 1/2.The scalars are even non-trivial.The f 0 (980) may be a K K molecule, or a q 2 q2 state [9,10].In either picture it is not simply related to the ss content of interest to us.The f 0 (1370), (1500), (1700) are believed to be mixtures of G 0 , ss and nn, so it is not possible to normalize the φG 0 to these in a meaningful way [13].Thus we compare φG 0 to φ(ss), where (ss) is an effective ideal ss state with the same mass as η ′ .
To proceed, the rescaling feature of Eq. (1) (i.e.Eq. ( 7) of Ref. [1]) should be examined.In Fig. 2, we present the calculations of the branching ratio R φG0 and R J/ψG0 in terms of r to show the rescaling features between the φ-glueball and J/ψ-glueball production in quarkonium decay via virtual photons.The quantity r is in the range of 0 < r < 1, which corresponds to the physical region √ s > 4m q .For the ideal condition that the phase space factor is cancelled out, the rescaling feature is shown by the constant fraction (dotted curve in Fig. 2(a)) between the J/ψ-glueball and φ-glueball production ratios.The ratio reflects the difference of the factors m c |I 0 | 2 / O 1 ηc and m s |I 0 | 2 / O 1 φ , which denote the ratios of the square of the glueball wavefunctions at their origins compared to these of the produced quarkonia.Note that in these two cases the kinematics in terms of r are quite similar as indicated by the arrows.In Fig. 2(b), we also present the calculation including the contributions from the non-cancelling phase space factors with m c = 1.4 GeV and M G = M ηc .The phase space factor causes deviations from the exact rescaling between the solid and dashed curves as r → 1, but is negligible in most of the kinematical regions.
Apart from the EM transition, the other important process in J/ψ → φ(ss) is via intermediate gluons, i.e.J/ψ → 3g → φη ′ (ss).We can thus express the ratio between the EM decay and strong decay of J/ψ as: br J/ψ→3g→φ(ss) br J/ψ→γ * →φ(ss) = br J/ψ→3g br J/ψ→γ * br φη ′ →3g br φη ′ →γ * . ( For an ideal flavor singlet F , the ratio for its coupling to gluons and a virtual photon γ * can be written as where σ F summarises the flavor dependence of the gluon coupling to the final state configuration, and e F is the charge factor of the quarks.For the ratio of gluon and photon coupling to the initial J/ψ and ss, we then have where we have assumed flavor independence of the quark-gluon coupling.With the experimental values, br J/ψ→3g = 0.877 ± 0.005 and br J/ψ→γ * = 0.17 ± 0.02 [11], we have  GeV for these two reactions.

FIG. 2 :
FIG. 2:Branching ratio fractions multiplied by 10 4 for φ-glueball (solid) and J/ψ-glueball (dashed) production via virtual photons, respectively.The dotted curve is the ratio of the solid to the dashed, of which the stable value shows the validity of rescaling the kinematics.The arrows denote the locations of r corresponding to the c.m. energies of √ s = 3.1 GeV and 10.6