Virtuality and Transverse Momentum Dependence of Pion Distribution Amplitude

We describe basics of a new approach to transverse momentum dependence in hard exclusive processes. We develop it in application to the transition process \gamma^* \gamma -> \pi^0 at the handbag level. Our starting point is coordinate representation for matrix elements of operators (in the simplest case, bilocal 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.