Virtuality and transverse momentum dependence of the 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 γ*γ → π 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) Φ(x,σ), with x being Fourier-conjugate to (pz) and σ Laplace-conjugate to z 2 . 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. Furthermore, we discuss how one can generate high-k tails from primordial soft distributions.