Production of a tensor glueball in the reaction $\gamma\gamma\rightarrow G_2\pi^0$ at large momentum transfer

We study the production of a tensor glueball in the reaction $\gamma\gamma\rightarrow G_2\pi^0$. We compute the cross section at higher momentum transfer using the collinear factorisation approach. We find that for a value of the tensor gluon coupling of $f^T_g\sim 100$~MeV, the cross section can be measured in the near future by the Belle~II experiment.


Introduction
The hadronic states made up only from gluons are known as glueballs [1]. The spectrum of such states have been computed using lattice QCD techniques, see e.g. [2][3][4][5]. However, identifying glueballs in experiment is not an easy task because they have the same quantum numbers as quark-antiquark mesons. At present time there are few scalar candidates which can be associated with the scalar glueballs, see e.g. [6][7][8] and references therein. Much less is even known about tensor glueballs. The lattice calculations predict the mass of the lowest 2 ++ glueball state to be around 2.3 − 2.4 GeV. There are some experimental evidences that such state have may been seen in various processes [9][10][11].
In this Brief Report we would like to study the production of tensor 2 ++ glueballs in two-photon collisions at high momentum transfer. Despite the fact that the cross section of such process is relatively small it can potentially be observed at high luminosity e + e − colliders like Belle II. The production of glueballs in the reaction γγ → Gπ 0 has already been studied long time ago in Refs. [12][13][14]. However the tensor glueball was considered only in Ref. [14] but within the specific approach where the glueball state is described as a weakly bound state of two non-relativistic gluons. In the present work the production amplitude is computed using the QCD factorisation approach [16][17][18] . The coupling of quarks and gluons to the final mesonic states is described by the distribution amplitudes (DAs) describing the momentum fraction distribution of partons at zero transverse separation in a two-particle Fock state. Our calculation is presented in Sec.2 and results for the cross section are shown in Fig.4. Concluding remarks are given in Sec.3.

Calculation
The production amplitude for the γγ → Gπ 0 process can be described in terms of the helicity amplitudes The cross section of the process is given by [15] dσ γγ [π 0 G 2 ] d cos θ = 1 64π Figure 1: Kinematics of the process γ(q 1 )γ(q 2 ) → π(k)f 2 (p).
where we used |A ++ | = |A −− | and |A −+ | = |A +− |, and where m denotes the glueball mass. The square of the amplitudes in (2) implies the sum over polarisations λ of the glueball: We choose the momenta as γ(q 1 )γ(q 2 ) → π(k)G 2 (p) and consider the center mass system (cms) k+p = 0 with the pion and glueball momenta directed along z-axis, see The light-cone expansion of the particle momenta in the region with s ∼ −t ∼ −u Λ QCD reads where θ is the scattering angle in the cms, see Fig.1, and where we neglected the small power suppressed terms.
In the region s ∼ −t ∼ −u Λ QCD the amplitudes can be computed in terms of convolution integrals of the hard coefficient function with the mesonic distribution amplitudes. The typical diagrams are shown in Fig.2. The blobs in Fig.2 denote the light-cone matrix elements which define the DAs of the outgoing mesons. The pion DA is defined as where we assume for the flavor structureqq =ūu −dd. The pion DA has normalisation 1 0 dy φ π (y) = 1 so that the pion decay constant f π is defined as The light-cone matrix element of tensor glueball can be defined in the similar way as for the tensor meson 2 ++ , see, e.g. Ref. [20]. In general case there are three light-cone matrix elements which define two gluon DAs and one quark DA. In the quark case the distribution amplitudes is defined as where we assume for the quark flavorsqq =ūu +dd. The polarisation tensor e (λ) αβ is symmetric and traceless, and satisfies the condition e (λ) αβ p β = 0 1 . The polarisation sum is given by where M µν = g µν − p µ p ν /m 2 and the normalization condition reads e and describes transition into the tensor glueball with helicity λ = 0. The normalization of the quark DA is given by Therefore the constant f q is defined as a matrix element of the local operator where D µ is the covariant derivative.
The gluon DAs are defined as where g ⊥ µν = g µν − (n µnν + n νnµ )/2 and the short notation e (λ) The distribution amplitudes φ T g (x) and φ S g (x) are symmetric with respect to the interchange of x ↔ 1 − x and describe the momentum fraction distribution of the two gluons having the same and the opposite helicity, respectively.
The constants f T g and f S g are defined through the matrix element of the local two-gluon operator: Using these definitions one can compute all necessary diagrams some of which are shown in Fig.2.
The glueball with the tensor polarisation λ = ±2 is only produced if the colliding photons have the same helicities 2 where α is the electromagnetic coupling, N c denotes the number of colors. The convolution integral I ++ g (η) reads where we assume thatx ≡ 1 − x. The factor (1 − η 2 ) −1 which is explicitly shown in Eq.(15) cancels after summation over polarisation λ in the compution |A ++ | 2 defined in (3). Therefore the η-dependence of the cross section (2) is completely defined by the convolution integral I ++ g (η). The production of a glueball with λ = 0 (scalar polarisation) is described by the amplitude A +− which reads where In order to write the convolution integrals in this form we used the symmetry properties of the DAs with respect to interchange x → 1 − x. The hard coefficient functions for various processes with gluons have also been computed in Ref. [19]. We have checked up to a general factor that our results shown in Eqs. (15) and (17) are in agreement with the corresponding hard kernels in Ref. [19]. In order to make numerical estimates we need to specify models for the DAs and provide numerical values for the low energy glueball couplings. In the following we suppose that the states f 2 (2300) and f 2 (2340) which have been recently observed by the Belle [10] and BESIII [11] collaborations are good candidates to be tensor glueball. In our numerical estimates we use the following models of DAs. For pion we take φ π (y) 6yȳ + 6a 2 (µ)yȳC with the second moment a 2 (µ = 1GeV) = 0.20.
This value is close to many phenomenological estimates and lattice QCD result [21]. For the glueball DAs we take the simplest asymptotic models Let us first consider the properties of the convolution integrals I +± g,q (of Eqs. (16), (18) and (19)). In Fig.3 we show the values of the convolution integrals as a function of cos θ. Below we assume that factorisation works reasonably for such values of θ where |u|, |t| ≥ 2.5 GeV 2 . This region corresponds to the inner area between the two vertical lines on the plots in Fig.3. One can easily see that in the vicinity The integrals I +− g,q vanish at θ = 90 o therefore we can conclude that at least around this point the dominant contribution will be given by the amplitude A ++ which describes the production of the glueball in the tensor polarisation.
The values of the couplings f T,S g and f q are not known. It is natural to assume that the glueball state strongly overlaps with the gluon wave function and the value of the gluon couplings are relatively large and can be of the same order as the quark coupling f q ∼ 100 MeV for quark-antiquark mesons, i.e. f S g ∼ f T g ∼ 100 MeV. For the glueball quark coupling f q we consider the different scenarios with f q f g and f q ∼ f g corresponding to the small and to the large quark-antiquark component, respectively. Such scenarios will be described by the following numerical values The evolution of these coupling is the same as the evolution of the corresponding coupling for the tensor meson f 2 (1270) except for flavor mixing and can be found in Ref. [20]. Let us notice that the tensor gluon DA φ T g does not mix under evolution with quark contributions and therefore it describes the genuine gluon component of the glueball wave function.
The numerical estimates show that the value of the cross section is practically saturated by the contribution from the amplitude A ++ describing the production of a glueball in the tensor polarisation. The contribution of the amplitude |A +− | is always about two orders of magnitude smaller for all numerical values of the couplings f q and f g shown in Eqs.(25) and (26). Therefore we can conclude that the contribution with |A +− | does not provide significant numerical impact. Hence the cross section is only sensitive to the value of tensor coupling f T g . This can also be seen, for instance, from the analysis of the decay G 2 → φφ which can be used for identification of the glueball state.
In Fig.4 we show the cross section as a function of cos θ at fixed values of energy s. In the numerical calculations we take n f = 3 and α s (m 2 τ ) = 0.297. The cross section are shown for the energy values s = 13 GeV 2 and 16 GeV 2 . The factorisation scale is fixed to be µ 2 = 3.2 GeV 2 and µ 2 = 4 GeV 2 , respectively. The values of cos θ correspond to the restriction |t|, |u| ≥ 2.5 GeV 2 . We obtain that for  Figure 5: Comparison of the glueball cross section (the dashed line is the same as in Fig.4 but scaled by factor 4 ) and data for the π 0 π 0 cross section for s = 13 GeV 2 . The data are taken from Ref. [22] In Fig.5 we show the glueball cross section for f T g (1 GeV) = 100 MeV and s = 13 GeV 2 in comparison with the cross section data for γγ → π 0 π 0 for s = 13.3 GeV 2 . The data are taken from Ref. [22]. For convenience the glueball cross section is scaled by a factor 4. From this picture one can conclude that the measurement of γγ → G 2 π 0 cross section requires a larger luminosity which will be achieved in the Belle II experiment.

Conclusions
We calculated the amplitudes and cross sections for the production of a tensor glueball in the reaction γγ → G 2 π 0 . We obtained that for the value of the low energy coupling f T g 100 MeV the cross section is dominated by the contribution describing the production of a glueball in the tensor polarisation. A corresponding measurements allow one to constrain the value of the tensor coupling f T g . We expect that the corresponding cross section can be observed in the upcoming higher statistic Belle II experiment.