Dispersive derivation of the pion distribution amplitude

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