Cross-link relations between $\pi$ and $\rho$-meson channels and the QCD vacuum

We discuss cross-link relations between the $\pi$ and $\rho$-meson channels emerging from two different descriptions of the QCD vacuum: Instanton physics and QCD sum rules with nonlocal condensates (NLC). We derive in both schemes an intriguing linear relation between the $\pi$ and the $\rho^\|$-meson distribution amplitudes in terms of their conformal coefficients and work out the specific impact of the scalar NLC in these two channels. Using a simple model with Gaussian decay of the scalar NLC, we are able to relate it to the moments of the pion non-singlet parton distribution function measurable in experiment -- a highly nontrivial result. The implications for the pion and the $\rho^\|$-meson DAs entailed by the obtained cross-link relations are outlined in terms of two generic scenarios.