Spacelike and timelike form factors for the $({\pi }^0,\eta ,{\eta }^{\prime })\to {\gamma }^{*}\gamma$ transitions in the light-front quark model

We investigate the $({\pi }^0,\eta ,{\eta }^{\prime })\to {\gamma }^{*}\gamma$ transitions both for the spacelike region and the timelike region using the light-front quark model (LFQM). In particular, we present the new direct method to explore the timelike region without resorting to mere analytic continuation from the spacelike region to the timelike region. Our direct calculation in timelike region shows the complete agreement not only with the analytic continuation result from the spacelike region but also with the result from the dispersion relation between the real and imaginary parts of the form factor. For the low energy regime, we compare our LFQM results of the transition form factors (TFFs) for the low timelike momentum transfer region and the slope parameters at $q^2=0$ with the recent experimental data from the Dalitz decays of $({\pi }^0,\eta ,{\eta }^{\prime })$. For the high energy regime, we incorporate the QCD factorization in our LFQM to examine the asymptotic behavior of TFFs both for the spacelike region and the timelike region. We compare our results with the available experimental data.

Figure 1

One-loop Feynman diagrams that contribute to $P\to {\gamma }^{*}\gamma$. The single covariant Feynman diagram (a) is the same as the sum of the two LF time-ordered diagrams (b) and (c), respectively.

Figure 2

The exactly solvable BS model calculation of the normalized $F_{\pi \gamma }(q^2)$ for both space- and timelike regions ($-2\le q^2\le 3$) [${\mathrm{GeV}}^2$]: (a) and (b) represent the contributions from Figs. 1 and 1, respectively, for $0<\alpha <1$ case. (c) shows the normalized $F_{\pi \gamma }(q^2)/F_{\pi \gamma }(0)$ compared with the dispersion relation.

Figure 3

Transverse momentum dependent distribution amplitude (TMDA) ${\psi }_{\pi }(x,{\mathbf{k}}_{\perp })$ (left panel) for the pion in the form of ${\psi }_{\pi }(x,y)$ [see Eq. (27)] and the corresponding two-dimensional contour plot (right panel).

Figure 4

Transverse momentum dependent distribution amplitude (TMDA) ${\psi }_{s\overline{s}}(x,{\mathbf{k}}_{\perp })$ (left panel) for the $s\overline{s}$ sector in the form of ${\psi }_{s\overline{s}}(x,y)$ [see Eq. (27)] and the corresponding two-dimensional contour plot (right panel).

Figure 5

(a) The the normalized $\pi \to {\gamma }^{*}\gamma$ transition form factor $F_{\pi \gamma }(Q^2)/F_{\pi \gamma }(0)$, and (b) $|Q^2{F}_{\pi \gamma }(Q^2)|$ for both timelike ($q^2=-Q^2>0$) and spacelike ($q^2=-Q^2<0$) momentum transfer regions. The data are taken from [6, 7, 13, 18, 19] and [13].

Figure 6

(a) The normalized $\eta \to {\gamma }^{*}\gamma$ transition form factor $|F_{\eta \gamma }(Q^2)/F_{\eta \gamma }(0)|$, and (b) $|Q^2{F}_{\eta \gamma }(Q^2)|$ for both timelike ($q^2=-Q^2>0$) and spacelike ($q^2=-Q^2<0$) momentum transfer region. The dot-dashed, solid, and dashed lines are results obtained from the mixing angles with ${\varphi }_{\eta -{\eta }^{\prime }}=32°$, 37° and 42°, respectively, and the data are taken from [6, 7, 9, 11, 12, 20].

Figure 7

(a) The normalized ${\eta }^{\prime }\to {\gamma }^{*}\gamma$ transition form factor $|F_{{\eta }^{\prime }\gamma }(Q^2)/F_{{\eta }^{\prime }\gamma }(0)|$, and (b) $|Q^2{F}_{{\eta }^{\prime }\gamma }(Q^2)|$ for both timelike ($q^2=-Q^2>0$) and spacelike ($q^2=-Q^2<0$) momentum transfer region. The same line codes are used as in Fig. 6 and the data are taken from [6, 7, 14, 20].