Transition form factors of the pion in light-cone QCD sum rules with next-to-next-to-leading order contributions

The transition pion-photon form factor is studied within the framework of Light-Cone QCD Sum Rules. The spectral density for the next-to-leading order corrections is calculated for any Gegenbauer harmonic. At the level of the next-to-next-to-leading (NNLO) radiative corrections, only that part of the hard-scattering amplitude is included that is proportional to the $\beta$-function, taking into account the leading zeroth-order harmonic. The relative size of the NNLO contribution in the prediction for the form factor $F^{\gamma^{*}\gamma\pi}(Q^2)$ has been analyzed, making use of the BLM scale-setting procedure. In addition, predictions for the form factor $F^{\gamma^{*}\rho\pi}$ are obtained that turn out to be sensitive to the endpoint behavior of the pion distribution amplitude, thus providing in connection with experimental data an additional adjudicator for the pion distribution amplitude. In a note added, we comment on the preliminary high-$Q^2$ BaBar data on $F^{\gamma^{*}\gamma\pi}$ arguing that the significant growth of the form factor between 10 and 40 GeV$^2$ cannot be explained in terms of higher-order perturbative corrections at the NNLO.