$\chi_{c2}$ tensor meson transition form factors in the light front approach

We continue our work on the light-front formulation of quarkonium $\gamma^* \gamma$ transition form factors, extending the formalism to $J^{PC} = 2^{++}$ tensor meson states. We present an analysis of $\gamma^* \gamma \to \chi_{c2}$ transition amplitude and the pertinent helicity form factors. Our relativistic formalism is based on the light-front quark-antiquark wave function of the quarkonium. We calculate the two-photon decay width as well as three independent $\gamma^* \gamma$ transition form factors for $J_z = 0,1,2$ as a function of photon virtuality $Q^2$. We compare our results for the two-photon decay width to the recently measured ones by the Belle and BES III collaborations. Even when including relativistic corrections, a very small $\Gamma(\lambda = 0)/\Gamma(\lambda = 2)\sim10^{-3}$ ratio is found which is beyond present experimental precision. We also present the form factors as a function of photon virtuality and compare them to the sparse experimental data on the so-called off-shell width. The formalism presented here can be used for other $2^{++}$ mesons, excited charmonia or bottomonia or even light $q \bar q$-mesons.


I. INTRODUCTION
The production of C-even quarkonia in γ * γ fusion processes keeps providing us with important information on their structure [1][2][3][4][5][6][7][8].While untagged e + e − cross sections give access to the decay width of quarkonia into γγ pair, in single tagged collisions, transition form factors involving one virtual and one real photon can be measured.
Here, we continue our work on the light-front formulation of γ * γ * → χ transition form factors for a given meson state χ.We have already presented the formalism for computing the γ * γ * transition amplitudes to 0 ± , 1 + charmonia using light-front cc wave functions (LFWFs) [9][10][11][12].We adopt two different approaches to the LFWFs.In the first one, they are obtained from the radial wavefunctions in a potential model, supplemented by a Melosh-transform of the relevant spin-orbit structure.The second is based on direct solutions of the bound-state problem formulated on the light-front (LF).Here, convenient tables of the wave function from the Basis Light Front Quantization (BLFQ) approach of Refs.[13,14] are available in the literature [15].
In this work, we wish to extend the formalism to the γ * γ * → χ c2 transition amplitude based on the quarkonium LFWF.For this purpose, we focus on the form factors describing such a coupling for one real and one spacelike virtual photon as a function of the photon virtuality.Only very sparse data are available on this process at the moment, while in principle experiments such as Belle can provide such data in the future.Recently, the Belle collaboration has measured the radiative decay width [3], where they select two quasi-real photon collisions in no-tag mode.
The paper is organised as follows.First, we discuss how the current transition matrix elements for one virtual photon are related to the LFWF.In the next section, we derive the corresponding form factors.We present numerical results for the transition form factors, also in the non-relativistic approximation using cc wave function obtained by solving the Schrödinger equation.We then compare our results for the radiative decay width to available measurements.

II. TRANSITION MATRIX ELEMENTS FOR ONE REAL AND ONE VIRTUAL PHOTON
As in our recent work on the 1 ++ states [12], we start with formulating the γ * γ → 2 ++ process in a Drell-Yan frame, in which one of the photons carries vanishing light-front plus momentum (for notation, see Fig. 1).The relevant four-momentum transfer satisfies q 2 2 = − q 2 2⊥ , and we approach the on-shell limit for this photon by letting its transverse momentum go to zero q 2⊥ → 0 for a meson in an external electromagnetic field.The process can therefore be viewed as a dissociation of an incoming virtual photon in an external electromagnetic field.We chose the polarization vector of the latter such that we project on the light-front plus component of the current.This choice of the frame and the current is the preferred one for the evaluation of electroweak transition currents of hadrons, as it is free from parton-number changing transitions, and instantaneous (in LF time) fermion exchanges [16].The pertinent helicity amplitudes are then related to matrix elements of the FIG.
1.An example diagram for one virtual photon transition, with , q ⊥1 = 0), LF-plus component of the current as Here, σ(σ) denotes the (anti)quark polarization, and in what follows we will represent the helicities ±σ/2 and ±σ/2 by ↑ and ↓.The fine structure constant is α em = e 2 /(4π), e f is the electric charge of quark with flavour f and with mass m f .The derivative operator ( q 2⊥ • ∇ k ⊥ ) is acting on the LFWF of the transverse Ψ γ T σσ or longitudinal Ψ γ L σσ photon, we do not explicitly display the photon polarization λ.
The explicit form of the photon LFWFs reads (see e.g.Ref. [17]) where m f is (anti)quark mass, and z = k + /q + is the light front momentum fraction of photon carried by the quark and (1 − z) by the antiquark.Here, we defined Inserting the photon LFWFs into Eq.(1),we obtain for the transverse photon with helicity λ = +1: and for the incoming longitudinal photon Now we wish to perform the azimuthal angle integration.To this end, we note that In addition to these angular dependencies, also the LFWF depends on the azimuthal angle ϕ of k ⊥ .Indeed, our LFWFs which acts on the WFs as so that we can isolate the ϕ dependence as As a result, We can now straightforwardly perform the angular integration: In the same manner, we obtain for the transitions of the longitudinal photon: The procedure for obtaining the LFWFs for the spin-two state is described in Appendix A.
Now, we wish to express our results for the transition amplitudes in the Drell-Yan frame through the invariant transition form factors commonly used in the literature.For definiteness, here we use the form factors introduced in Ref. [18], while for different conventions, see e.g.Ref. [19,20].
We start from the parametrization of the covariant amplitude for the process γ * (q where Here, the four momenta of photons satisfy q2 1 = −Q 2 , Q 2 ≥ 0, and q 2 2 = 0. We now match the form factors defined above to the transition amplitudes calculated in the LF formalism by expressing them as Introducing the light-like vectors n ± µ , which full fill conditions n we write the photon momentum as Further, we define the polarization of the incoming photon and outgoing meson in the LF notation, for the photon: and for the tensor meson: where We have denoted the four-momentum of the tensor meson as P µ = q 1µ + q 2µ , and notice, that P + = q + 1 .Above M denotes the mass of the tensor meson, and P 2 = M 2 .Now, we can move to the transition amplitudes in the Drell-Yan frame (see Fig. 1).

M(+1 →
Combining these expressions with our results for the matrix elements, we obtain the three independent transition form factors: From this representation of the transition form factors we can distinguish the ingredients related to spin-singlet ( ψλ ′ ↑↓ (z, k ⊥ )− ψλ ′ ↓↑ (z, k ⊥ )), as well as spin-triplet ( ψλ ′ ↑↓ (z, k ⊥ )+ ψλ ′ ↓↑ (z, k ⊥ )).Using the formulas given in the table in Appendix A1 of Ref. [18], these form factors can be also related to helicity amplitudes in the γ * γ c.m. frame.
In Fig. 2 we present transition form factors for one real and one virtual photon as a function of the photon virtuality.In the numerical calculation, we use light-front wave functions obtained for different cc potentials from literature as in Ref. [12] or [21].There is a relatively large spread of the results, similar to what was observed for γ * γ → χ c1 [12].

A. NRQCD limit
It is instructive to derive the transition form factors in the limit of nonrelativistic (NR) motion of quarks in the bound state.To reach the NR limit, we should expand the integrand around the z = 1/2 and k ⊥ = 0, i.e. thus, In the Melosh transform formalism described in Appendix A, the LFWF can be related to the NR radial WF, u 1 (k).After the NR expansion, all FFs will be proportional to the integral where R ′ (0) is the derivative of the (spatial) radial WF at the origin, which we obtain as in Ref. [10].As a result, the transition form factors take the form: Above, M stands for the mass of χ c2 (1P), and N c is the number of colors.In the NR limit, the mass of the meson should be understood as M = 2m f .These results fully agree with those obtained previously in [19,20].In Fig. 3, we present similar results for the non-relativistic approach with M = 2m f , see Eqs. ( 25) -( 27).We hope that in the near future, such form factors will be extracted by the Belle collaboration.So far, only Γ γγ (Q 2 ) as defined by the Belle collaboration was measured.
The radiative decay width is described by two contributions from J z = 2 (F TT,2 ), and J z = 0 (F TT,0 ): Therefore, we can neglect the J z = ±1 contribution related to F LT in no-tag mode.Nevertheless, one would expect the cross-section σ TT (J z = 0) to be considerably smaller than σ TT (J z = ±2).In further calculation we take M χ c2 = 3.556 GeV [22].The form factor at the on-shell point F TT,2 (0) in the non-relativistic limit leads to the following expression: Furthermore, as can be seen from Eq.(25), in the NR limit we have F TT,0 (0) = 0, so that we need to consider only the contribution from J z = 2 for the radiative decay width: In Tab.I we show the values of transition form factors F TT,0 and F TT,2 for Q 2 = 0.In the fully relativistic calculation, we find that at Q 2 = 0 the F TT,0 does not vanish, but gives a negligibly small contribution.The corresponding widths Γ γγ (λ = 0) and Γ γγ (λ = ±2) in keV are shown in Tab.II.Indeed, the decay width for λ = 0 is three orders of magnitude  smaller than that for λ = ±2.We also show the ratios of the different helicity contributions to the width.For the NR limit, where the λ = 0 contribution vanishes, we show the result for λ = ±2 for two different approximations.
The BES III Collaboration measured the ratio between two-photon partial widths, for the χ c2 helicity λ = 0 and λ = 2 [6]: which is a straightforward confirmation that the helicity-zero component is strongly suppressed.We predict the ratio of the order of 10 −3 .The BES III precision is not sufficient to measure the small ratios predicted in this work.
V. FORM FACTOR γγ * → 0 ++ We now want to compare the two-photon decay width for 0 ++ and 2 ++ states.To make the comparison more transparent we reformulate the results of [10] using the same setup in the Drell-Yan frame as in Sec.II.Now we have The helicity amplitude with the transverse photon polarization e T µ = (0, 0, e ⊥ (λ)) is obtained as and F TT (Q 2 ) is a function invariant under Lorentz transformation.

TABLE III. Radiative decay widths obtained in the LFWF approach and the ratio
The definition of off-shell widths that we were using comes from writing the γ * γ crosssection for photons as (i, j ∈ T, L) [24] For the case of one off-shell photon, we have that the kinematical factor √ X = 1 2 (M 2 + Q 2 ).Further, N T = 2, N L = 1, and J is the spin of the resonance of mass M and total decay width Γ.By BW(W 2 , M 2 ) we denote the Breit-Wigner distribution, which in the narrow width limit becomes Now, the TT and LT cross sections are obtained from the c.m.-frame helicity amplitudes as [25] Using the formulas in Ref. [18], we relate our FFs to the helicity amplitudes, and obtain ) for χ c0 (on the l.h.s.) and χ c2 (on the r.h.s.) compared to the Belle data [2].In the NRQCD approach, we took M = M χc .
for the TT case: and, for LT: Comparing to Eq. (39), with N T = 2, J = 2, W = M, we derive the off-shell widths For Q 2 = 0 this agrees with the formula for the two-photon decay width; see Eq. (28).For the LT case, we obtain Let us now turn to the Q 2 -dependence of the single-tag cross-section, which we write as: The factor two appears because each of the lepton can emit the off-shell photon.In the narrow-width approximation, we therefore have with the effective off-shell width defined as Off-shell widths are convention-dependent, and to compare to the experimental data from Ref. [2], we note that the Belle collaboration writes which means, that Then the cross-section for χ c2 can be written as: In the case of χ c0 we have only one form factor, which has transverse contribution F TT .According to Ref. [18] the cross-section for scalar meson has the form: Therefore, the off-shell width for χ c0 is: In Fig. 4 we present the off-shell decay width for χ c0 (l.h.s.) and χ c2 (r.h.s.).We show on the y-axis caption due to the difference between our definition and the one used by the Belle collaboration.The existing data are not sufficient to judge which potential model works better.Future Belle data could provide valuable information on this issue.

VII. CONCLUSIONS
In the present paper, we have extended our light-front formulation to a formalism of photon transition form factors to the case of γγ * → 2 ++ couplings (helicity form factors) in terms of the light-front quark-antiquark wave functions of the meson.We have presented detailed formulae for F TT,0 , F TT,2 as well as F LT form factors expressed in terms of the light-front wave functions.To obtain light-front wave functions, we use methods discussed previously in Ref. [10] for five different cc potential models, see also Appendix A. In addition, we have used the light-front wave functions from the Basis Light Front Quantization approach of [13,15].
The two-photon decay width is smaller than the value measured by the Belle collaboration.This can be caused by too approximate cc wave functions and/or higher Fock components in the χ c2 wave function and requires further studies which go beyond the scope of the present letter.We find the Γ(λ = 0)/Γ(λ = ±2) ratio of the order of 10 −3 , which is in agreement with the current experimental precision.
We also have shown helicity form factor results for one real and one virtual photon as a function of the photon virtuality.We have obtained a large spread of the results for different potentials.The form factor results are ready to be verified e.g. by the Belle collaboration in single-tag e + e − collisions.Furthermore, we have defined and calculated the so-called Q 2dependent off-shell diphoton width and compared it to the Belle data.It is rather difficult to conclude on the consistency of the model with rather low statistics of the available Belle data.
The polarization vector can now be either longitudinal e = n, or transverse, e = e ⊥ .Some simplifications occur in either case.Let us start with the longitudinal case: For the transverse polarization, we obtain Here, we have used that

FIG. 2 .
FIG.2.Three transition form factors within the LFWF approach: on the l.h.s.-F TT,0 (Q 2 ), on the r.h.s.-F TT,2 (Q 2 ), in the middle -F LT (Q 2 ).Here line denoted as BLFQ is a result obtained with the set of 32 expansion terms of light front wave function from the database[15].

TABLE II .
Helicity decomposition of the two-photon decay width of χ c2 (1P).