  Using the approach based on conformal symmetry we calculate the two-loop
coefficient function for the axial-vector contributions to two-photon processes
in the $\overline{\rm MS}$ scheme. This is the last missing element for the
complete next-to-next-to-leading order (NNLO) calculation of the the pion
transition form factor $\gamma^\ast\gamma\to \pi $ in perturbative QCD. The
corresponding high-statistics measurement is planned by the Belle II
collaboration and will allow one to put strong constraints on the pion
light-cone distribution amplitude. The calculated NNLO corrections prove to be
rather large and have to be taken into account. The same coefficient function
determines the contribution of the axial-vector generalized parton
distributions to deeply-virtual Compton scattering which is investigated at the
JLAB 12 GeV accelerator, by COMPASS at CERN, and in the future will be studied
at the Electron Ion Collider EIC.
