On high $Q^{2}$ behavior of the pion form factor for transitions $γ^{\ast} γ\to π^{0}$ and $γ^{\ast} γ^{\ast} \to π^{0}$ within the nonlocal quark-pion model

The behavior of the transition pion form factor for the processes $\gamma ^{\star}\gamma \to \pi ^{0}$ and $\gamma ^{\star}\gamma ^{\star}\to \pi ^{0}$ at large values of space-like photon momenta is estimated within the nonlocal covariant quark-pion model. It is shown that in general the coefficient of the leading asymptotic term depends dynamically on the ratio of the constituent quark mass and the average virtuality of quarks in the vacuum; and kinematically on the ratio of photon virtualities. The kinematical dependence of the transition form factor allows us to obtain the relation between the pion light-cone distribution amplitude and the quark-pion vertex function. The dynamical dependence indicates that the transition form factor $\gamma ^{\star}\gamma \to \pi ^{0}$ at high momentum transfer is very sensitive to the size of nonperturbative fluctuations in the QCD vacuum.