Transition form factors ${\gamma }^{*}\gamma \to \eta$ and ${\gamma }^{*}\gamma \to \eta \prime$ in QCD

We update the theoretical framework for the QCD calculation of transition form factors ${\gamma }^{*}\gamma \to \eta$ and ${\gamma }^{*}\gamma \to {\eta }^{\prime }$ at large photon virtualities including full next-to-leading order analysis of perturbative corrections, the charm quark contribution, and taking into account $SU(3)$-flavor breaking effects and the axial anomaly contributions to the power-suppressed twist-four distribution amplitudes. The numerical analysis of the existing experimental data is performed with these improvements.

Figure 1

The experimental data on ${\gamma }^{*}\gamma \to {\pi }^0$ [3, 4, 42] (open symbols) compared with the nonstrange component of the eta meson transition FF, ${\gamma }^{*}\gamma \to |{\eta }_q⟩$ (filled symbols), from the combination of BABAR and CLEO measurements [1, 42] on $\eta$ and ${\eta }^{\prime }$ production in the FKS mixing scheme, Eqs. (22) and (23).

Figure 2

Box diagrams contributing to the gluon coefficient function.

Figure 3

Transition form factors ${\gamma }^{*}\gamma \to \eta$ (left panels) and ${\gamma }^{*}\gamma \to {\eta }^{\prime }$ (right panels) [1, 42] compared to the LCSR calculation with three models of the leading-twist DAs specified in Table 2. Asymptotic values at large photon virtualities (50) corresponding to the central values of the FKS parameters in Eq. (26) are shown by the horizontal dashed lines. The dark blue shaded areas correspond to uncertainties of the calculation due to the choice of the LCSR parameters $M^2$ and $s_0$, factorization scale $\mu$, and the higher-twist parameters $h^{(q,s)},{\delta }^{2(q,s)}$; see text. The light blue areas are obtained by adding the uncertainties in the FKS parameters, Eq. (26).

Figure 4

Same as in Fig. 3 for the first model of the leading-twist DAs specified in Table 2 except for the normalization parameter of the gluon DA $c_2^{(g)}$ which is set to zero. The shaded area in light green shows the effect of the variation of this parameter in the range $c_2^{(g)}=\pm 0.5$.

Figure 5

Same as in Fig. 3 using a logarithmic scale in $Q^2$. The calculation uses the first model of the leading-twist DAs specified in Table 2. The timelike data point [2] at $|Q^2|=112{\mathrm{GeV}}^2$ is shown by red stars for comparison.

Figure 6

The anomalous contribution to the twist-four DA ${\psi }_{4M}^{(s)}(u)$ (a33) compared to the asymptotic leading-twist DA $6u(1-u)$ (dashed line).