Virtuality distributions in application to γγ^⁎→π⁰ 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 γ^⁎γ→π⁰ 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 z², they are parametrized through virtuality distribution amplitudes (VDA) Φ(x,σ), with x being Fourier-conjugate to (pz) and σ Laplace-conjugate to z². For intervals with z⁺=0, we introduce the transverse momentum distribution amplitude (TMDA) Ψ(x,k_⊥), and write it in terms of VDA Φ(x,σ). The results of covariant calculations, written in terms of Φ(x,σ) are converted into expressions involving Ψ(x,k_⊥). 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_⊥ tails from primordial soft distributions