Dispersive derivation of the pion distribution amplitude

We derive the dependence of leading-twist pion light-cone distribution amplitude (LCDA) on a parton momentum fraction $x$ by directly solving dispersion relations for moments with inputs from operator product expansion (OPE) corresponding correlation function. It is noticed that these must be organized into those Gegenbauer coefficients first in order to avoid ill-posed problem appearing conversion coefficients. Given values various condensates OPE, we find solution LCDA, which stable expansion, exists. Moreover, summing contributions up 18 polynomials smooth, and can well approximated function proportional $x^p(1-x)^p$ $p\approx 0.45$ at scale $\mu=2$ GeV. Turning off condensates, get asymptotic form, independent $\mu$, LCDA as expected. then solve different $\mu=1.5$ GeV condensate this demonstrate result consistent one obtained evolving 1.5 That is, our formalism compatible QCD evolution. The strength above approach goes beyond analyses limited only few conventional sum rules highlighted. precision results improved systematically including higher-order higher-power terms OPE.