Doubly virtual $\mathbf{(}{\pi }^0,\eta ,{\eta }^{\prime }\mathbf{)}\to {\gamma }^{*}{\gamma }^{*}$ transition form factors in the light-front quark model

We report our investigation on the doubly virtual transition form factors (TFFs) $F_{\mathrm{P}{\gamma }^{*}}(Q_1^2,Q_2^2)$ for the $\mathrm{P}\to {\gamma }^{*}(q_1){\gamma }^{*}(q_2)(\mathrm{P}={\pi }^0,\eta ,{\eta }^{\prime })$ 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 use the $q_1^{+}=0$ frame and find that both LF and manifestly covariant calculations produce exactly the same results for $F_{\mathrm{P}{\gamma }^{*}}(Q_1^2,Q_2^2)$. This confirms the absence of the LF zero mode in the doubly virtual TFFs. We then map 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 $F_{\mathrm{P}{\gamma }^{*}}(Q_1^2,Q_2^2)$, we compare our LFQM results with the available experimental data and the perturbative QCD (pQCD) and vector meson dominance (VMD) model predictions. As $(Q_1^2,Q_2^2)\to \infty$, our LFQM result for doubly virtual TFF is consistent with the pQCD prediction, i.e., $F_{\mathrm{P}{\gamma }^{*}}(Q_1^2,Q_2^2)\sim 1/(Q_1^2+Q_2^2)$, while it differs greatly from the result of the VMD model, which behaves as $F_{\mathrm{P}{\gamma }^{*}}^{\mathrm{VMD}}(Q_1^2,Q_2^2)\sim 1/(Q_1^2{Q}_2^2)$. Our LFQM prediction for $F_{{\eta }^{\prime }{\gamma }^{*}}(Q_1^2,Q_2^2)$ shows an agreement with the very recent experimental data obtained from the BABAR Collaboration for the ranges of $2<(Q_1^2,Q_2^2)<60{\mathrm{GeV}}^2$.

Figure 1

The diagram for the $e^{+}{e}^{-}\to e^{+}{e}^{-}\mathrm{P}$ process.

Figure 2

One-loop Feynman diagrams that contribute to $\mathrm{P}\to {\gamma }^{*}{\gamma }^{*}$. (a) and (b) represent the amplitudes of the virtual photon attached to the quark and antiquark lines.

Figure 3

The three-dimensional plots for $(Q_1^2+Q_2^2)F_{\pi {\gamma }^{*}}(Q_1^2,Q_2^2)$ obtained from Eq. (14) (upper panel) compared with the VMD result (lower panel) for the range of $0<(Q_1^2,Q_2^2)<10{\mathrm{GeV}}^2$.

Figure 4

The two-dimensional plot for $2Q^2{F}_{\pi {\gamma }^{*}}(Q^2,Q^2)$ in the symmetric limit ($Q^2=Q_1^2=Q_2^2)$ for the $0<Q^2<50{\mathrm{GeV}}^2$ region compared with the pQCD LO and the VMD model predictions.

Figure 5

The three-dimensional plots for $(Q_1^2+Q_2^2)F_{\eta {\gamma }^{*}}(Q_1^2,Q_2^2)$ (upper panel) and $(Q_1^2+Q_2^2)F_{{\eta }^{\prime }{\gamma }^{*}}(Q_1^2,Q_2^2)$ (lower panel) obtained from Eq. (14) with $\varphi =37°$ for the range of $0<(Q_1^2,Q_2^2)<10{\mathrm{GeV}}^2$.

Figure 6

Our LFQM results for $F_{(\pi ,\eta ,{\eta }^{\prime }){\gamma }^{*}}(Q_1^2,Q_2^2)$ (black circles) compared with the pQCD LO (open squares) and NLO (filled squares) predictions; the VMD predictions (blue circles); and the experimental data [14] (triangles, with error bars including the statistical, systematic, and model uncertainties).