Can NRQCD explain the $γγ^* \to η_c$ transition form factor data?

Abstract

Unlike the bewildering situation in the $γγ^*\to π$ form factor, a widespread view is that perturbative QCD can decently account for the recent \textsc{BaBar} measurement of $γγ^*\to η_c$ transition form factor. The next-to-next-to-leading order (NNLO) perturbative correction to the $γγ^*\to η_{c,b}$ form factor, is investigated in the NRQCD factorization framework for the first time. As a byproduct, we obtain by far the most precise order-$α_s^2$ NRQCD matching coefficient for the $η_{c,b}\to γγ$ process. After including the substantial negative order-$α_s^2$ correction, the good agreement between NRQCD prediction and the measured $γγ^*\to η_c$ 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.

6 pages, 3 figures and 1 table; adding Eqs.(8) and (9) as well as some references, correcting errors in Table 1, updating Fig.3 to include the "light-by-light" contributions, devoting a paragraph to discuss why our strategy of interpreting the NNLO corrections is justified; Accepted by PRL