Can NRQCD 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 \textsc{BaBar} measurement of $\gamma\gamma^*\to \eta_c$ transition form factor. The next-to-next-to-leading order (NNLO) perturbative correction to the $\gamma\gamma^*\to \eta_{c,b}$ form factor, is investigated in the 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 NRQCD approach to hard exclusive reactions involving charmonium.