Can Nonrelativistic QCD Explain the $\gamma {\gamma }^{*}\to {\eta }_c$ Transition Form Factor Data?

Unlike the bewildering situation in the $\gamma {\gamma }^{*}\to \pi$ form factor, a widespread view is that perturbative QCD can decently account for the recent BABAR measurement of the $\gamma {\gamma }^{*}\to {\eta }_c$ transition form factor. The next-to-next-to-leading-order perturbative correction to the $\gamma {\gamma }^{*}\to {\eta }_{c,b}$ form factor, is investigated in the nonrelativistic QCD (NRQCD) factorization framework for the first time. As a byproduct, we obtain, by far, the most precise order-${\alpha }_s^2$ NRQCD matching coefficient for the ${\eta }_{c,b}\to \gamma \gamma$ process. After including the substantial negative order-${\alpha }_s^2$ correction, the good agreement between NRQCD prediction and the measured $\gamma {\gamma }^{*}\to {\eta }_c$ form factor is completely ruined over a wide range of momentum transfer squared. This eminent discrepancy casts some doubts on the applicability of the NRQCD approach to hard exclusive reactions involving charmonium.

Figure 1

Sample Feynman diagrams for ${\gamma }^{*}\gamma \to c\overline{c}({}^1{S}_0^{(1)})$.

Figure 2

The transition form factor $F(Q^2)$ normalized to $F(0)$ in variation with $Q^2$: ${\gamma }^{*}\gamma {\eta }_c$ (upper); ${\gamma }^{*}\gamma {\eta }_b$ (lower). The solid dots with error bars represent the BABAR measurements [8]. The dotted curve stands for the LO NRQCD predictions, the (blue) dashed curve for the NLO predictions, while the solid curve for the NNLO predictions. The uncertainty bands associated with the NNLO predictions are obtained by varying ${\mu }_R^2$ from $\frac{1}{2}(Q^2+m^2)$ to $2(Q^2+m^2)$. There are two distinct NNLO bands, the upper (green) one corresponds to ${\mu }_{\mathrm{\Lambda }}=m$, the lower (red) one to ${\mu }_{\mathrm{\Lambda }}=1\mathrm{GeV}$. We suppress the (red) ${\mu }_R$-uncertainty band for ${\eta }_b$ since it is too narrow to be visible.