Gulon-pair-Creation Production Model of Strong Interaction Vertices

By studying the $\eta_c$ decays exclusively to double glueballs, we introduce a model to mimic phenomenologically the gluon-pair-vacuum interaction vertices, namely the $0^{++}$ model. Based on this model, we study glueball production in $\eta_c$ decay, explicitly $\eta_c \to f_0(1500)\eta(1405)$. Among them $f_0(1500)$ is well-known scalar glueball candidate and $\eta(1405)$ is thought of a candidate for pseudoscalar glueball. We discuss the possibility of finding these light glueballs in their production via the $0^{++}$ model. We also discuss the heavier glueball production in $\eta_b$ decays, which might be detectable in the LHCb and Belle-II experiments.


I. INTRODUCTION
According to the theory of strong interaction, the Quantum Chromodynamics (QCD) [1], gluons have self-intraction, which suggests in some sense the existence of glueball. The search for glueballs has experienced a long history, however the existence evidence is still vague. Being short of reliable glueball production and decay mechanisms makes the corresponding investigation rather difficult. Another hurdle hindering the glueball searching lies in the fact that usually glueballs mix heavily with the quark states.
In this paper, we discuss the glueballs production in η c decay by introducing a model for the gluon-pair-vacuum interaction vertices, namely the 0 ++ model, as shown in the the Fig.(1). We assume the gluon pair is created homogeneously in space with equal probability. Comparing to the 3 P 0 model [27][28][29][30][31][32][33][34][35], which models the quark-antiquark pair creation in the vacuum, we formulate an explicit vacuum gluon-pair transition matrix and estimate the strength of the gluon-pair creation. Employing the 0 ++ model, we then investigate the η c decay to scalar and pseudoscalar glueballs. Based on our knowledge about glueballs, we choose f 0 (1500) and η(1405) as scalar and pseudoscalar glueball candidates respectively. The possibility of finding η c → f 0 (1500)η(1405) is discussed.
The rest of the paper is arranged as follows. After the introduction, we construct a model for gluon-pair-vacuum interaction vertices in Sec.II. In Sec.III, we evaluate the process η c → f 0 (1500)η(1405). Last section is remained for summary and outlooks.

II. CONSTRUCTION OF THE 0 ++ MODEL
In quantum fields theory, the physical vacuum is the ground state of energy and a large number of particle fields fluctuate in it. Therefore, there are certain probabilities of quark pairs and gluon pairs with vacuum quantum numbers being induced from the vacuum by energy fluctuation. It is reasonable to conjecture that gluon pairs would be created with equal amplitude in space, just like the quark-antiquark pairs do in 3 P 0 model. Created from the vacuum, the gluon pairs hence possess the quantum numbers J PC = 0 ++ , and so the 0 ++ model is named.
Following we investigate glueball production in η c decays by means of the 0 ++ model, as shown in the Fig.(1). The transition amplitude of η c exclusively decay to double glueballs for instance can be formulated as Here, G 1 and G 2 represent glueballs, g s is the strong coupling constant, γ g is a constant with energy dimension representing the strength of gluon pair creation from the vacuum which can be extracted by fitting to the experimental data, η ρσ is known as the Minkowski metric, q i is the quark field with color index, A µ a is for the gluon field with Lorentz superscript and color subscript, t a is the Gell-Mann matrix of SU c (3) group. The δ cd η ρσ A ρ c A σ d term is employed to produce the gluon pair from the vacuum.
Inserting the completeness relation G |G G| = 2E G into Eq.(1), we get where |G is the shorthand notation for two gluons g 1 and g 2 emitted from η c and the phase space integration is hidden in |G state, as shown in the following Eq.(6). T 1 is the transition operator for Gg 3 g 4 → G 1 G 2 , where g 3 g 4 represent the gluon pair created from the vacuum and T 2 is the transition operator for η c → g 1 g 2 . The transition matrix T 1 can be decomposed as follows(see Fig.(2) for illustration): where T vac is the vacuum-gluon pair transition amplitude, I i are identity matrices indicating the propagations of g 1 and g 2 . Considering g 3 and g 4 are created in vacuum and gluon spin equals 1, the spin third-components of two gluons thus have three different combinations, that is |m s 3 , m s 4 may take |1, −1 , |0, 0 , or | − 1, 1 . The total spin state of the vacuum produced gluon pair, |S , M S , is a singlet, and can be formulated as The T vac can be expressed as Here, k 3 and k 4 represent 3-momenta of gluons g 3 and g 4 , a † 3c and a † 4d are creation operators of gluons with color indices, and Y ℓm (k) ≡ |k| ℓ Y ℓm (θ k , φ k ) is the ℓth solid harmonic polynomial that gives the momentum-space distribution of the produced gluon pair.
The state |G obviously possesses the same quantum numbers of |η c , i.e. J PC G = 0 −+ , thus can be expressed as where k 1 and k 2 represent 3-momenta of gluons g 1 and g 2 , ψ n G L G M L G (k 1 , k 2 ) is the spatial wave function with n, L, S , J the principal quantum number, orbital angular momentum, total spin and the total angular momentum of |G , respectively. χ 12 is the corresponding spin state, and can be expressed as |S G M S G . L G M L G S G M S G |J G M J G represents the Clebsch-Gordan coefficient and for |G state it reads 1m; 1 − m|00 . Various K with different subscripts represent the corresponding 3-momenta. The normalization conditions write Similarly we may have expressions for G 1 and G 2 states. Equipped with the gluon-to-glueball transition operator T 1 and expressions for initial and final states, we can now calculate the transition matrix element Here the momentum space integral The glueball and |G state wave functions are conjectured to be in harmonic oscillator (HO) form where k is the relative momentum between two gluons inside the states, N nL is the normalization coefficient and P(k 2 ) is a polynomial of k 2 [31]. χ 13 S G M S G χ 34 00 which denotes the spin coupling can be expressed by Wigner's 9 j symbol [29] χ 13 Here, s i is spin of the gluon g i with i = 1, 2, 3, 4, and by which the decay width for process η c → G 1 G 2 can be readily calculated through [31]: Here, M JL = In this section, we estimate the scalar and the pseudoscalar glueballs production in η c decays via the 0 ++ model, taking f 0 (1500) and η(1405) as the corresponding candidates, named G 1 and G 2 respectively. The quantum numbers of the states involve in the process are presented in Table I, where |G and |η c are the same in quantum number. In Eq.(10), the color contraction gives a number 8, and for scalar glueballs, the spin and orbital angular momentum coupling leads the C-G coefficient to be 00; 00|00 = 1. Therefore, Eq.(10) in this situation turns to The spin coupling χ 13 00 χ 24 1−M 2 |χ 12 1−M 0 χ 34 00 in Eq.(13) is in Wigner's 9 j symbol, which is a representation of 4-particle spin coupling and can be expanded as series of 2-particle spin couplings represented by Wigner's 3 j symbol [29]. The Wigner's 3 j and 9 j symbols read and respectively. Applying to Eq.(13), the spin coupling term then reduces to In Every term in above equation can be evaluated by normal C-G coefficient. That is: After inserting above ingredients into Eq.(20), we get the corresponding spin couplings:    Substituting above spin couplings into Eq.(17), T 1 will be reduced to × 11, 1 − 1|00 11, 1 − 1|00 I 1,0,1 (K) + 10, 10|00 10, 10|00 I 0,0,0 Provided the ground state dominates in our calculation, we may simply take the principal quantum numbers n 0 , n 1 and n 2 to be 1. The wave function ψ then turns to where k M , k ±1 = ∓(k x ± ik y )/ √ 2 and k 0 = k z , are the spherical components of vector k.
Simplifying Eq.(36), we now have Here, in the η c center of mass frame, K G = K η c = 0 and K G 1 = −K G 2 = K. Applying which to Eqs. (37) and (38), the spatial wave functions now write where R 0 , R 1 and R 2 are the most probable radii of η c , f 0 (1500) and η(1405), respectively. After performing the integration, one may notice that the states M = 1 and M = −1 give null contribution, i.e. I 1,0,1 = I −1,0,−1 = 0, and Given δ 3 (K G − K G 1 − K G 2 )I ≡ I 0,0,0 and considering of Eqs. (14), (35) and (44), we have M 000 The most probable radius R of the HO wave function can be estimated through R = 1/α, where α = µω/ . Here, µ denotes the reduced mass, ω is the angular frequency of harmonic oscillator which is given by E in = (2n + L + 3/2) ω, with E in being the glueball inner energy, n the radial quantum number, and L the orbital angular momentum. Note, the inner energy equals to the glueball mass. As discussed in Refs. [37,38], here we also take the effective mass of the constituent gluon to be 0.6 GeV, which yields µ = 0.3 GeV for glueballs. In our calculation, the masses of η c , f 0 (1500) and η(1405) are obtained from PDG [39], i.e. M η c = 2.98 GeV, M 1 = 1.50 GeV and M 2 = 1.41 GeV. In η c center-of-mass system, the momenta of f 0 (1500) and η(1405) are fixed to be 0.32 GeV, and hence their total energies are known, as given in Table II. Taking into account above discussion and input values, we can readily get I = 0.41 GeV −3/2 and M 000 1 = 0.11γ g , and in case of L0JM J G |J G M J G = L0J0|00 = 0000|00 = 1, and  The calculation of the process η c → gg is quite straightforward. In leading order of perturbative QCD, there are only two Feynman diagrams, as shown in Fig.3. Their decay amplitudes read: iM µν,ab is Γ total = 31.8 ± 0.8 MeV, and hence the branching ratio of η c → f 0 (1500)η(1405) process is about Similar as η c , its upsilon family partner η b may also exclusively decay to scalar and pseudoscalar glueballs, which can be evaluated by 0 ++ model. From lattice QCD calculation [2][3][4][5]24], there are scalar and pseudoscalar glueball candidates with masses 1.75 GeV and 2.39 GeV, respectively. With the same procedure performed in above for η c , we can readily get Γ η b →G 0 ++ G 0 −+ = 0.19 +0. 12 −0.07 MeV and Br η b →G 0 ++ G 0 −+ = 1.90 +3.27 −1.10 × 10 −2 .

IV. SUMMARY
In this paper, we discuss the glueball exclusive production in pseudoscalar quarkonium decays by introducing a 0 ++ model, which is employed to mimic phenomenologically the gluon-pair-vacuum interaction vertices. It is assumed that gluon pair is created homogeneously in space with equal probability. By virtual of the 3 P 0 model, we formulate an explicit vacuum gluon-pair transition matrix and estimate the strength of the gluon-pair creation. We then calculate the η c to f 0 (1500) and η(1405) decay width, where f 0 (1500) and η(1405) are supposed to be scalar and pseudoscalar glueball candidates respectively, and find the decay width and branching ratio are 50.73 keV and 1.80 × 10 −3 . From the hint of lattice QCD calculation, suppose there are scalar and pseudoscalar glueball candidates with masses 1.75 GeV and 2.39 GeV, we estimate as well the η b to these states' decay width and branching ratio, i.e. 0.19 MeV and 1.9 × 10 −2 . Noticing branching ratios are not too small, we think these processes deserve to be investigated in BESIII, BelleII, LHCb and other experiments.
We acknowledge that the estimation of gluon-pair-vacuum coupling is quite premature and hence the calculation for pseudoscalar quarkonium exlusive decay to glueballs. Refinement of the 0 ++ model still needs a lot of tedious works. Due to the importance of glueball physics, various mechanisms of glueball decay and production are deserved to explore.