Two-photon transitions of charmonia on the light front

We investigate the two-photon transitions $H_{c\bar c} \to \gamma^*\gamma$ of the charmonium system in light-front dynamics. The light-front wave functions were obtained from solving the effective Hamiltonian based on light-front holography and one-gluon exchange interaction within the basis light-front quantization approach. We compute the two-photon transition form factors as well as the two-photon decay widths for S- and P-wave charmonia, $\eta_c$ and $\chi_{cJ}$ and their excitations. Without introducing any free parameters, our predictions are in good agreement with the recent experimental measurements by BaBar and Belle, shedding light on the relativistic nature of charmonium.


I. INTRODUCTION
Charmonium is an intriguing system with entangled scales Λ qcd α s m c c 2 m c c 2 [1]. Typical estimates put the average velocity of the quarks v 2 c ∼ 0.3. Thus, there may be large relativistic corrections for observables sensitive to short-distance physics. The two-photon transition of charmonium, viz. H cc → γγ, is one of the leading examples (see Refs. [2][3][4][5] for reviews), where predictions based on non-relativistic dynamics appear deficient. Typical symptoms include the slow convergence in non-relativistic effective theories (e.g., NRQCD [6,7]), large differences among various non-relativistic potential model calculations and discrepancies with the experimental measurements [8,9]. Driven by recent progress on experimental measurements, the call for fully relativistic approaches is clear.
Lattice computation of the two-photon transition with off-shell photons is particularly challenging, as demonstrated by the large discrepancy between theoretical predictions and the PDG value, since the energetic virtual photon is not an eigenstate of the QCD Hamiltonian [10]. Nevertheless, strides have been made to access the twophoton width [11][12][13][14][15], and transition form factors [11,12]. Calculations from other relativistic approaches, notably DSE/BSE, have also been reported in the literature and appear successful [16].
In this work, we report the calculation of the twophoton transition form factors (TFFs) of charmonium in a light-front Hamiltonian approach. This approach solves for the light-front wave functions (LFWFs) directly from a low-energy relativistic effective Hamiltonian for QCD [17]. The advantage of this approach is evident from the short-distance z 2 ∼ 1/Q 2 Λ −2 qcd behavior of the transition amplitude d 4 xe iq·z 0|T {J µ (z)J ν (0)}|H(p) , where the leading contribution comes from the light-cone * mliy@jyu.fi distribution amplitude (LCDA) φ P [18,19], This result can be extended to the full range of Q 2 by using the LFWFs ψ P (x, k ⊥ ) [18], where the hard kernel T H is known to the next-to-leading order [20]. The LFWFs and the associated LCDAs play a dominant role in exclusive processes [18,19,21]. The LFWFs adopted in this work were previously obtained in a light-front Hamiltonian approach to the charmonium spectra [22,23].
We compute the two-photon widths and TFFs of the pseudoscalar η c , scalar χ c0 , axial vector χ c1 , and tensor χ c2 and compare with the recent experimental data wherever available [24][25][26][27]. The same LFWFs have been used to compute the decay constants, radiative transitions, semi-leptonic transitions, form factors, (generalized) parton distributions and cross sections of diffractive vector meson production with reasonable agreement over a remarkable wide range of experimental measurements and theoretical predictions [22,[28][29][30][31][32][33]. All these results, as well as those of the present work, represent predictions of the original model Hamiltonian for charmonia without adjusting its parameters [22].
A comparative summary of our results for the combined predictions of the charmonium masses and dilepton (for vectors) or diphoton (for the rest) widths is shown in Fig. 1. The plotted results are provided in Table I. The dilepton width Γ ee and the diphoton width Γ γγ probe the similar physics since both quantities are proportional to the wave function at origin in the nonrelativistic quark model. Note that the diphoton width of χ c1 vanishes due to the Landau-Yang theorem [34,35].
The helicity amplitudes can be expressed in terms of the LFWFs as (see Fig. 2), where, ψ ss/γ and ψ ss/H are the photon and meson LFWFs, respectively. The former can be computed using light-front perturbation theory [8,18,36]. The momenta of the quark before and after the photon emission are, Note that only the light-front 3-momenta need to be specified since partons are always on their mass shells. Similarly, the momenta of the antiquark before and after the photon In the single-tag case, only one photon is on-shell and the helicity amplitude H λ1λ2;λ can be extracted from the transverse current J ⊥ , similar to the M1 transition investigated in Ref. [28].

A. ηc → γγ
The two-photon transition amplitude of a pseudoscalar can be parametrized by a single form factor [37], For the important case where one of the photons is on-shell, it is useful to define a single-variable TFF F P γ (q 2 ) ≡ F P γγ (−q 2 , 0) = F P γγ (0, −q 2 ). F P γ (q 2 ) is related to the two-photon width, The matrix element associated with ( (5)) has the structure of pseudoscalar-vector-vector (PVV) coupling, where each vector meson is substituted by a virtual photon. Using the techniques developed in Ref. [28] for the M1 transition form factor V P →V γ , we obtain a LFWF representation from the helicity amplitude H 0,±;0 (viz. matrix element of J ⊥ ) [8], where m f is the quark mass, N C = 3, and e f = 2/3. In the above expression for the single-tag TFF, terms like q 2⊥ · k ⊥ vanish. Note that this decoupling is not generally held when both photons are off-shell. Alternatively, one can also extract the TFF from the helicity amplitude H ±,0;0 , i.e., matrix element of J + . This choice leads to a slightly different expression, and the difference is expected to vanish in the NR limit [38]. As a numerical example, the result differs from Eq. (7) at most 6% over the range of Q 2 from 0 to 60 GeV 2 for the η c TFF shown in Fig. 3(a) . At large Q max{Λ qcd , m f }, Eq. (7) reduces to the celebrated partonic interpretation of Brodsky-Lepage [18], where, φ is the (normalized) pseudoscalar LCDA, and f P is the pseudoscalar decay constant. Their relations with the LFWFs are [22], This expression at Q 2 = 0 implies (2m f ≈ M P ), However, this is only accurate in the non-relativistic limit. At small Q 2 , these two equations (10) & (11) clearly overestimate the full light-front prediction (7), as we will see later. For charmonia, the effect is substantial. A further dilemma is that, for pseudoscalar quarkonia η c , the decay constant f ηc cannot be unambiguously extracted from the experimental measurement. The BLFQ prediction f ηc = 0.42 (7) GeV [22] is in good agreement with model-dependent value f ηc = 0.335(75) GeV extracted from the experiment [39] as well as with predictions from Lattice QCD and DSE/BSE calculations [40][41][42][43].
The two-photon TFF of η c is measured by BaBar collaboration [24]. The BaBar data can be well described by a monopole fit with a pole mass Λ 2 = 8.5 ± 0.6 ± 0.7 GeV 2 ≈ M 2 J/ψ . This corroborates the vector meson dominant (VMD) model with the nearest vector mesons.
The BaBar data along with the monopole fit are shown in Fig. 3(a). The BLFQ prediction (7) (7) whereas the BLFQ/DA prediction employs (10). For ηc(1S) the result is compared with calculations from DSE/BSE [16]. A monopole fit with pole mass Λ = M ψ(2S) is provided for ηc(2S) as a reference (see texts). agreement with the BaBar data. Following the analysis in Ref. [22] for the decay constants, our calculation uses the N max = 8 results, which corresponds to a UV resolution µ uv ≈ κ √ N max = 2.8 GeV. The basis sensitivity shown as uncertainty band is estimated as the difference between the N max = 8 and N max = 16 results.
A DSE/BSE result is included for comparison [16]. Two recent Lattice calculations are less successful due to the difficulty to represent the photon with large virtuality [12,13]. More recent Lattice calculations instead focus on the on-shell amplitude.
Note that we present the TFF F P γ (Q 2 ) instead of the normalized TFF F P γ (Q 2 )/F P γ (0) to avoid the propagation of errors from F P γ (0) present in both theory and experiment. The value F P γ (0) ∝ √ Γ P →γγ can also be compared from Fig. 3(a), where we add the PDG value combining measurement of the two-photon width from various processes [27]. The prediction using the LCDA ("BLFQ/DA") are also included in Fig. 3(a) for comparison. At Q 2 = 0, the prediction clearly overshoots the BLFQ prediction as expected.
The on-shell two-photon width from PDG as well as from selected theoretical predictions are collected in Table I. A comparison of selected recent predictions with quantified uncertainties is shown in Fig. 4. Without adjusting any parameter, our predictions appear very competitive with other theoretical approaches.
There is no measurement of the η c (2S) TFF at the present. Thus, our results Fig. 3(b) are predictions. F η c γ (0) can be accessed via other processes and are also shown in Fig. 3(b). Our prediction is again in good agreement with the PDG value [27]. The monopole fit with a pole mass Λ = M ψ(2S) is depicted for comparison.
In the literature, one of the major sources of uncertainty is the quark mass m f . In various models, it is known that both the shape of the TFF F P γ (Q 2 )/F P γ (0) and the two-photon width Γ P →γγ are sensitive to the value of the quark mass m f , in opposing directions. When confronted by the experimental data, the former, dictated by the pole mass Λ 2 ≈ k 2 ⊥ +m 2 f x(1−x) , typically favors a lower value m f ∼ 1.3 GeV close to the current quark mass whereas the latter favors a larger value close to the effective quark mass. Indeed, (11) is more accurate if m 2 f → m 2 f + k 2 ⊥ . In our calculation, the effective quark mass m f = 1.57 GeV is determined from the mass spectroscopy, leaving no room for parameter manipulation. Therefore, it is remarkable that our LFWFs can reproduce the two-photon TFF and the two-photon width simultaneously.

B. χc0 → γγ
The transition amplitude of this process can be parametrized by two TFFs [9,37,47], With this definition, the two-photon width is, If one photon is off-shell, as in "single tagged" experiments, the two-photon width Γ χc0→γ * γ can be solely described by TFF F S 1 . Thus, it is convenient to introduce a single-variable TFF F Sγ (q 2 ) = F S 1 (−q 2 , 0) = F S 1 (0, −q 2 ). Its relation with the single-tag two-photon width Γ χc0→γ * γ is,  Table I for more comparisons of the widths.
The LFWF representation of the TFF is, Similar to the pseudoscalar case, the approximate result using the LCDA representation reads, where the scalar meson LCDA is defined as, and is normalized by Here, we only kept the leading-twist contribution.
The Belle collaboration provided the first measurement of the TFF F χc0γ (Q 2 ), albeit with limited statistics [25].  [8,9] 5.2-21 3.1-8. The extracted data are shown in Fig. 5. A recent result from DSE/BSE is also shown for comparison [16]. The TFF at Q 2 = 0 is accessed from the two-photon decay width Γ S→γγ from various processes as compiled by PDG [27]. Our predictions using the LFWFs (15) as well as using the LCDA (17) are both in agreement with the experimental data despite the low statistics. The BLFQ/DA prediction is scaled by F Sγ (0) obtained from BLFQ. The value is also in agreement with that extracted from the two-photon decay width Γ S→γγ (see Table I).
The reduced width is related to the TFFs F A 2 and F A 3 as, where F Aγ (q 2 ) = F A 2 (−q 2 , 0)/M A = −F A 3 (0, −q 2 )/M A . The LFWF representation of this TFF is, Using the BLFQ LFWFs, the reduced two-photon width of χ c1 (1P ) is predicted to beΓ χc1→γγ = (3.0 ± 0.5) keV. Our BLFQ calculation further predicts an excited pure cc axial vector meson with the mass M χ c1 = 3.948(31)(17) GeV. The two-photon width of this state is predicted to beΓ χ c1 →γγ = (3 ± 1) keV, a value significantly above the recent Belle measurement for the 2P candidate χ c1 (3872), suggesting that the state possesses a large portion of non-cc component [26].
The two-photon decay width of the tensor χ c2 is measured by various experiments as compiled by PDG, and the average value is, Γ χc2→γγ = 0.56(5) keV [27]. Physically, this process is determined by two helicity amplitudes, The LFWFs representation of the helicity amplitudes are, From these expressions, we obtain the width Γ χc2→γγ = 0.70 (13) keV, consistent with the PDG value 0.56(5) keV. Similarly, we can make a prediction for the 2P tensor as pure cc state. The value Γ χ c2 →γγ = 0.58 (25) keV is consistent with the PDG lower bound Γ χc2(3930) > 0.17 keV for the 2P candidate χ c2 (3930).
The Belle collaboration provided the first measurement of the single tagged width Γ χc2→γ * γ (Q 2 ), albeit with limited statistics [25]. The data are compared with our result in Fig. 6, where our BLFQ prediction is in good agreement with the experimental measurements.

E. Asymptotic limit
The large Q 2 asymptotic behavior of the TFFs can be computed from perturbative QCD [37], Figure 7 shows Q 2 F Hγ as a function of Q 2 up to large Q 2 from various approaches. The BLFQ results (solid blue) and the BLFQ/DA results (dashed orange) are computed with fixed scale µ = µ uv ≈ κ √ N max . For the pseudoscalar, their agreement at large Q 2 is excellent as expected. For large Q 2 , the evolution of the DAs may not be negligible. We thus evolve the LCDA using the ERBL evolution [18]. The evolved results (BLFQ/DA, µ = Q, green dotted) show some small deviation from the fixed scale results. However, up to Q 2 = 500 GeV 2 all results are below the pQCD asympotic limit, confirming the long-standing observation that the convergence to the pQCD asympotic limit is very slow [19].

III. SUMMARY
In this work, we investigated the two-photon transitions of heavy quarkonia in the light-front approach using wave functions directly computed from an effective Hamiltonian inspired by light-front holography and onegluon exchange from light-front QCD. We computed the two photon decay widths and the transition form factors of pseudoscalar, scalar, axial vector and tensor mesons. The results are in good agreement with the available experimental measurements. This is significant since all results reported here are pure predictions -no parameter fitting is performed to obtain these results. We also make predictions for two-photon processes yet to be measured.
The success of our work and a similar success of the DSE/BSE calculation imply the relativistic nature of the charmonium system. For describing observables sensitive to the short-distance physics at ∼ α s M cc , the relativistic formulation is required. The light-front formalism is intrinsically relativistic and has the further advantage that it is directly related to the partonic picture at large momentum transfer.
In combination with the successes in charmonium spectroscopy as well as other observables, e.g., decay constant, radiative transitions, leptonic transitions, electromagnetic form factors, and parton distributions, our work provides a unified framework to describe the relativistic structure of the charmonium system. We also compare the results with those computed from the extracted light-cone distribution amplitudes, laying the foundation for applications to exclusive processes at high energy.
would also like to thank M. Masuda for providing the original data on the two-photon transition form factor of χ c0 and χ c2 . This work is supported by the New faculty start-up fund of the University of Science and Technology of China. Y. Li