Virtuality distributions in application to γγ⁎→π0γγ⁎→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→π0γ⁎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)O(0,z)) describing a hadron with momentum p  . Treated as functions of (pz)(pz) and z2z2, they are parametrized through virtuality distribution amplitudes (VDA) Φ(x,σ)Φ(x,σ), with x   being Fourier-conjugate to (pz)(pz) and σ   Laplace-conjugate to z2z2. For intervals with z+=0z+=0, we introduce the transverse momentum distribution amplitude (TMDA) Ψ(x,k⊥)Ψ(x,k⊥), and write it in terms of VDA Φ(x,σ)Φ(x,σ). The results of covariant calculations, written in terms of Φ(x,σ)Φ(x,σ) are converted into expressions involving Ψ(x,k⊥)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⊥k⊥ tails from primordial soft distributions.