$\eta \text{−}\gamma$ and ${\eta }^{\prime }\text{−}\gamma$ transition form factors in a nonlocal NJL model

We study the $\eta$ and ${\eta }^{\prime }$ distribution amplitudes (DAs) in the context of a nonlocal ${\mathrm{SU}(3)}_L\otimes {\mathrm{SU}(3)}_R$ chiral quark model. The corresponding Lagrangian allows us to reproduce the phenomenological values of pseudoscalar meson masses and decay constants, as well as the momentum dependence of the quark propagator arising from lattice calculations. It is found that the obtained DAs have two symmetric maxima, which arise from new contributions generated by the nonlocal character of the interactions. These DAs are then applied to the calculation of the $\eta \text{−}\gamma$ and ${\eta }^{\prime }\text{−}\gamma$ transition form factors. Implications of our results regarding higher twist corrections and/or contributions to the transition form factors originated by gluon-gluon components in the $\eta$ and ${\eta }^{\prime }$ mesons are discussed.

Figure 1

Quark distribution amplitudes for the $\pi$, $\eta$ and ${\eta }^{\prime }$ mesons. Left panel: $\pi$ DA, ${\varphi }_{\pi }(x)$. Central panel: $\eta$ DAs, ${\varphi }_{\eta }^{(q_{\ell })}(x)$ (solid line) and ${\varphi }_{\eta }^{(q_s)}(x)$ (dashed line). Right panel: ${\eta }^{\prime }$ DAs, ${\varphi }_{{\eta }^{\prime }}^{(q_{\ell })}(x)$ (solid line) and ${\varphi }_{{\eta }^{\prime }}^{(q_s)}(x)$ (dashed line). The dotted lines correspond in all cases to the asymptotic limit ${\varphi }_{\mathrm{asym}}(x)=6x(1-x)$.

Figure 2

Theoretical $\pi \text{−}\gamma$, $\eta \text{−}\gamma$ and ${\eta }^{\prime }\text{−}\gamma$ transition form factors and experimental results from CELLO, CLEO, BABAR, Belle and L3 Collaborations. Solid lines correspond to the case $c=d=0$, while dashed lines correspond to the values of $c$ and $d$ in Table 7. Short-dashed lines show the NLO asymptotic QCD behavior (see text), and dotted lines indicate the QCD asymptotic limits. In the case of the ${\eta }^{\prime }\text{−}\gamma$ TFF, the gray region corresponds to a change in $a_{{\eta }^{\prime }n}^{(q_i)}$ coefficients within a range of 15%. Notice that in all graphs we have used a logarithmic scale for $Q^2$.

Figure 3

Theoretical curves and experimental results for the ${\eta }^{\prime }\text{−}\gamma$ TFF. The full line corresponds to the NLO result with no gluon contributions discussed in Sec. 3, and the gray region indicates the indetermination in the corresponding Gegenbauer coefficients. The dashed line is the TFF obtained when the contributions of gluons are fitted. Short-dashed and dotted lines correspond to the NLO asymptotic behavior of the TFF (see discussion in Sec. 3) and the asymptotic limit, respectively. Notice the usage of a logarithmic scale for $Q^2$.