Virtuality Distributions in Application to $\gamma \gamma^* \to \pi^0$ Transition Form Factor at Handbag Level

We outline basics of a new approach to transverse momentum dependence in hard processes. As an illustration, we consider hard exclusive transition process $\gamma^* \gamma \to \pi^0$ at the handbag level. Our starting point is coordinate representation for matrix elements of operators (in the simplest case, bilocal ${\cal O}(0,z)$) describing a hadron with momentum $p$. Treated as functions of $(pz)$ and $z^2$, they are parametrized through virtuality distribution amplitudes (VDA) $\Phi (x, \sigma)$, with $x$ being Fourier-conjugate to $(pz)$ and $\sigma$ Laplace-conjugate to $z^2$. For intervals with $z^+=0$, we introduce the transverse momentum distribution amplitude (TMDA) $\Psi (x, k_\perp)$, and write it in terms of VDA $\Phi (x, \sigma)$. The results of covariant calculations, written in terms of $\Phi (x, \sigma)$ are converted into expressions involving $\Psi (x, k_\perp)$. Starting with scalar toy models, we extend the analysis onto the case of spin-1/2 quarks and QCD. We propose simple models for soft VDAs/TMDAs, and use them for comparison of handbag results with experimental (BaBar and BELLE) data on the pion transition form factor. We also discuss how one can generate high-$k_\perp$ tails from primordial soft distributions.