The doubly virtual $(\pi^0,\eta,\eta')\to\gamma^*\gamma^*$ transition form factors in the light-front quark model

We report our investigation on the doubly virtual TFFs $F_{{\rm P}\gamma^*}(Q^2_1,Q^2_2)$ for the ${\rm P}\to\gamma^*(q_1)\gamma^*(q_2) \;({\rm P}=\pi^0,\eta,\eta')$ transitions using the light-front quark model (LFQM). Performing a LF calculation in the exactly solvable manifestly covariant Bethe-Salpeter (BS) model as the first illustration, we used $q^+_1=0$ frame and found that both LF and manifestly covariant calculations produce exactly the same results for $F_{{\rm P}\gamma^*}(Q^2_1,Q^2_2)$. This confirms the absence of the LF zero mode in the doubly virtual TFFs. We then mapped this covariant BS model to the standard LFQM using the more phenomenologically accessible Gaussian wave function provided by the LFQM analysis of meson mass spectra. For the numerical analyses of $F_{{\rm P}\gamma^*}(Q^2_1,Q^2_2)$, we compared our LFQM results with the available experimental data and the perturbative QCD (pQCD) and the vector meson dominance (VMD) model predictions. As $(Q^2_1, Q^2_2)\to\infty$, our LFQM result for doubly virtual TFF is consistent with the pQCD prediction, i.e. $F_{{\rm P}\gamma^*}(Q^2_1, Q^2_2)\sim 1/(Q^2_1 + Q^2_2)$, while it differs far from the result of VMD model which behaves $F^{\rm VMD}_{{\rm P}\gamma^*}(Q^2_1, Q^2_2)\sim 1/(Q^2_1 Q^2_2)$. Our LFQM prediction for $F_{\eta'\gamma^*}(Q^2_1,Q^2_2)$ shows an agreement with the very recent experimental data obtained from the BaBar collaboration for the ranges of $2


I. INTRODUCTION
The meson-photon transitions such as P → γ ( * ) γ ( * ) (P = π 0 , η, η ) with one or two virtual photons have been of interest to both theoretical and experimental physics communities since they are the simplest possible bound state processes in quantum chromodynamics (QCD) and play a significant role in allowing both the low-and high-energy precision tests of the standard model.
In particular, both singly virtual and doubly virtual transition form factors (TFFs) are required to estimate the hadronic light-by-light (HLbL) scattering contribution to the muon anomalous magnetic moment (g − 2) µ . The HLbL contribution is in principle obtained by integrating some weighting functions times the product of a single-virtual and a doublevirtual TFFs for spacelike momentum [1][2][3]. The singlevirtual TFFs have been measured either from the spacelike e + e − → e + e − P process in the single tag mode [4][5][6] or from the timelike Dalitz decays P →¯ γ [7][8][9][10][11][12] where (2m ) 2 ≤ q 2 ≤ m 2 P . The timelike region beyond the single Dalitz decays may be accessed through the e + e − → Pγ annihilation processes, and the BaBar Collaboration [13] measured the timelike F η ( ) γ TFFs from the reaction e + e − → η ( ) γ at an average e + e − center of mass energy of √ s = 10.58 GeV. Very recently, the BaBar Collaboration [14] measured for the first time the double-virtual γ * (q 1 )γ * (q 2 ) → η TFF F η γ * (Q 1 1 , Q 2 2 ) in the spacelike(i.e. Q 2 1(2) = −q 2 1(2) > 0) kinematic region of 2 < Q 2 1 , Q 2 2 < 60 GeV 2 by using the e + e − → e + e − η process in the double-tag mode as shown in Fig. 1. It is very interesting to note that the measurement of F Pγ * (Q 1 1 , Q 2 2 ) at large Q 2 1 and Q 2 2 distinguishes the predictions of the model inspired by perturbative QCD(pQCD) [15,16], , from those of the vector meson dominance (VMD) model [17][18][19], 1/(Q 2 1 Q 2 2 ), while both models predict the same asymptotic dependence F asy Pγ (Q 2 , 0) ∼ 1/Q 2 as Q 2 → ∞. The low-energy behavior of the TFF for the doubly virtual π 0 → γ * γ * transition was recently investigated within a Dyson-Schwinger and Bethe-Salpeter (BS) framework [20]. In our previous analysis [21], we explored the TFF F Pγ (Q 2 , 0) for the single-virtual P → γ * γ (P = π 0 , η, η ) transition both in the spacelike and timelike region using the light-front quark model (LFQM) [22][23][24][25][26]. In particular, we presented the new direct method to explore the timelike region without resorting to mere analytic continuation from space-to time-like region. Our direct calculation in timelike region has shown the complete agreement with not only the analytic continuation result from spacelike region but also the result from the dispersion relation between the real and imaginary parts of the form factor.
The paper is organized as follows. In Sec. II, we discuss the TFFs for the doubly virtual P → γ * γ * transitions in an exactly solvable model first based on the covariant BS model of (3+1)-dimensional fermion field theory to check the existence (or absence) of the LF zero mode [27][28][29][30] as one can pin down the zero mode exactly in the manifestly covariant BS model [31][32][33][34][35]. Performing both the manifestly covariant calculation and the LF calculation, we explicitly show the equivalence between the two results and the absence of the zeromode contribution to the TTF. The η − η mixing scheme for the calculations of the (η, η ) → γ * γ * TFFs is also introduced in this section. In Sec. III, we apply the self-consistent correspondence relations (see, e.g., Eq. (35) in [34]) between the covariant BS model and the LFQM and present the standard LFQM calculation with the more phenomenologically accessible model wave functions provided by the LFQM analysis of meson mass spectra [22,25]. In Sec. IV, we present our numerical results for the (π 0 , η, η ) → γ * γ * TFFs and compare them with the available experimental data. Summary and discussion follow in Sec. V.

II. MANIFESTLY COVARIANT MODEL
The TFF F Pγ * for the doubly virtual P(P) → γ * (q 1 )γ * (q 2 ) (P = π 0 , η, η ) transition is defined via the amplitude T as follows: where P is the four momenta of the pseudoscalar meson, q 1(2) and ε 1(2) are the momenta and polarization vectors of two virtual photons 1 and 2, respectively. This process is illustrated by the one-loop Feynman diagram in Fig. 2(a) and Fig. 2(b), which represent the amplitudes of the virtual photon with momenta q 1 being attached to the quark and antiquark lines, respectively. While we shall only discuss the amplitude shown in Fig. 2(a), the total amplitude should of course include the contribution from the process in Fig. 2(b) as well.
In the exactly solvable manifestly covariant BS model, the covariant amplitude T in Fig. 2 (a) is obtained by the following momentum integral where N c is the number of colors and e Q(Q) is the quark (antiquark) electric charge. The denominators N p j (= p 2 j − m 2 Q + iε)( j = 1, 2) and N k (= k 2 − m 2Q + iε) come from the intermediate quark and antiquark propagators of mass m Q = mQ carrying the internal four-momenta p 1 = P − k, p 2 = P − q − k, and k, respectively. The trace term S in Eq. (2) is obtained as For theqq bound-state vertex function H 0 = H 0 (p 2 1 , k 2 ) of the meson, we simply take the constant parameter g in our model calculation. The covariant loop is regularized properly with this constant vertex.
Using the Feynman parametrization for the three propagators 1/(N p 1 N k N p 2 ), we obtain the manifestly covariant result by defining the amplitude in Fig. 1 with the physical meson mass M.
For the LF calculation in parallel with the manifestly covariant one, we use the q + 1 = 0 frame, where we take In this frame, the Cauchy integration of Eq. (2) over k − in Fig. 2

(a) yields
where x is the LF longitudinal momentum fraction defined by k + = (1 − x)P + and the LF (Pqq)-vertex function is the ordinary LF valence wave function with being the invariant mass. Note here that the pole of N k = 0 is taken for the Cauchy integration to get Eq.(6). The primed momentum variables are defined by We confirmed numerically that Eq. (5) exactly coincides with the manifestly covariant result given by Eq. (4). This verifies that the LF result obtained from the q + 1 = 0 frame is immune to the LF zero-mode contribution which could have been the additional contribution right at p + 1 = p + 2 = 0 if exists. The LF zero-mode involves the nonvalence wave function vertex discussed in our previous works [21,34]. The Lorentz invariance of the TFF is complete in this work without any issue from the LF zero-mode.
Since the amplitude of Fig. 2(b) gives the same numerical values as that of Fig. 2(a), we obtain the total result as I

III. APPLICATION OF THE LIGHT-FRONT QUARK MODEL
In the standard LFQM [22][23][24][25][26][37][38][39] approach, the wave function of a ground state pseudoscalar meson as a qq bound state is given by where φ R is the radial wave function and R λλ is the spin-orbit wave function with the helicity λ (λ ) of a quark (antiquark). For the equal quark and antiquark mass m Q = mQ, the Gaussian wave function φ R is given by and β is the variational parameter fixed by our previous analysis of meson mass spectra [22,25,26]. The covariant form of the spin-orbit wave function R λλ is given by and it satisfies ∑ λλ R † λλ R λλ = 1. Thus, the normalization of our wave function is given by In our previous analysis of the twist-2 and twist-3 DAs of pseudoscalar and vector mesons [33][34][35] and the pion electromagnetic form factor [34], we have shown that standard LF (SLF) results of the LFQM is obtained by the replacement of the LF vertex function χ in the BS model with the Gaussian wave function φ R as follows [see, e.g., Eq. (35) in [34]] For (η, η ) → γ * γ * transitions, making use of the η − η mixing scheme, the flavor assignment of η and η mesons in the quark-flavor basis η q = (uū + dd)/ √ 2 and η s = ss is given by [36] Using this mixing scheme and including the electric charge factors, we obtain the transition form factors F Pγ * (Q 2 1 , Q 2 2 ) for P → γ * γ (P = π 0 , η, η ) transitions as follows While the quadratic (linear) Gell-Mann-Okubo mass formula prefers φ 44.7 • (φ 31.7 • ) [40], the KLOE Collaboration [41] extracted the pseudoscalar mixing angle φ by measuring the ratio BR(φ → η γ)/BR(φ → ηγ). The measured values are φ = (39.7 ± 0.7) • and (41.5 ± 0.3 stat ± 0.7 syst ± 0.6 th ) • with and without the gluonium content for η , respectively. In this work, however, we use φ = 37 • ± 5 • to check the sensitivity of our LFQM.
For sufficiently high spacelike momentum transfer (Q 2 1 , Q 2 2 ) region, our LFQM result for F πγ * (Q 2 1 , Q 2 2 ) can be approximated in the leading order (LO) as follows where C π = ( √ 2/3) f π and f π is the pseudoscalar meson decay constant and φ 2;π (x) is the twist-2 pion DA in our LFQM given by [33][34][35] Our result for φ 2;π (x) can be found in Ref. [23]. As one can see from Eq. (15), while the singly virtual TFF F πγ * (Q 2 , 0) above some intermediate values of momentum transfer is known to be quite sensitive to the shape of DA, the doubly virtual TFF is not sensitive to the shape of DA since the am- is finite at the end points of x, i.e. x = 0, 1.

IV. NUMERICAL RESULTS
In our numerical calculations within the standard LFQM, we use the model parameters (i.e. constituent quark masses m Q and the gaussian parameters β QQ ) for the linear confining potentials given in Table I, which were obtained from the calculation of meson mass spectra using the variational principle in our LFQM [22,23,25]. The analysis for singly virtual TFFs F Pγ (Q 2 , 0) can be found in our previous work [21].
η γ * with the statistical, systematic and model uncertanties. We note that the error estimates for F (η,η )γ * (Q 2 1 , Q 2 2 ) in our LFQM results come from the choice of η − η mixing angle φ = (37 ± 5) • . We note for the F η γ * (Q 2 1 , Q 2 2 ) that our LFQM result and the experimental data are compatible to each other and the agreement between the two appears fairly reasonable within a rather large uncer- The transition form factors F (π,η,η )γ * (Q 2 1 , Q 2 2 ) (in unit of 10 3 GeV −1 ) for some (Q 2 1 , Q 2 2 ) (in unit of GeV 2 ) values compared with the experimental data [14] for F Exp. The two dimensional plot for 2Q 2 F πγ * (Q 2 , Q 2 ) in the symmetric limit (Q 2 = Q 2 1 = Q 2 2 ) for 0 < Q 2 < 50 GeV 2 region compared with the pQCD LO and the VMD model predictions.
tainty of data.
η γ * (Q 2 1 , Q 2 2 ) include the statistical, systematic, and model uncertainties. As one can see from Fig. 5, our LFQM results for F (π,η,η )γ * (Q 2 1 , Q 2 2 ) show the same behavior as the pQCD predictions. However, our LFQM predictions are quite different from the VMD model predictions since the two models have different power behaviors of (Q 2 1 , Q 2 2 ) as we discussed before. While the data for F η γ * (Q 2 1 , Q 2 2 ) measured from the BaBar [14] agree with the pQCD and our LFQM predictions, they show a clear disagreement with the VMD model predictions.
We then mapped the exactly solvable manifestly covariant BS model to the standard LFQM following the same correspondence relation given by Eq. (11) between the two models that we found in our previous analysis of two-point and three-point functions for the pseudoscalar and vector mesons [33,34]. This allowed us to apply the more phenomenologically accessible Gaussian wave function provided by the LFQM analysis of meson mass spectra [22][23][24][25][26] to the analysis of the doubly virtual F Pγ * (Q 2 1 , Q 2 2 ). For the (η, η ) → γ * γ * transitions, we used the η − η mixing angle φ in the quark-flavor basis varying the φ values in the range of φ = (37 ± 5) • to check the sensitivity of our LFQM.