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We outline basics of a new approach to transverse momentum dependence in hardprocesses. As an illustration, we consider hard exclusive transition process$\gamma^* \gamma \to \pi^0$ at the handbag level. Our starting point iscoordinate representation for matrix elements of operators (in the simplestcase, bilocal ${\cal O}(0,z)$) describing a hadron with momentum $p$. Treatedas functions of $(pz)$ and $z^2$, they are parametrized through virtualitydistribution amplitudes (VDA) $\Phi (x, \sigma)$, with $x$ beingFourier-conjugate to $(pz)$ and $\sigma$ Laplace-conjugate to $z^2$. Forintervals with $z^+=0$, we introduce the transverse momentum distributionamplitude (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 $\Psi (x, k_\perp)$. Startingwith scalar toy models, we extend the analysis onto the case of spin-1/2 quarksand QCD. We propose simple models for soft VDAs/TMDAs, and use them forcomparison of handbag results with experimental (BaBar and BELLE) data on thepion transition form factor. We also discuss how one can generatehigh-$k_\perp$ tails from primordial soft distributions.

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