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We derive the dependence of the leading-twist pion light-cone distributionamplitude (LCDA) on a parton momentum fraction $x$ by directly solving thedispersion relations for the moments with inputs from the operator productexpansion (OPE) of the corresponding correlation function. It is noticed thatthese dispersion relations must be organized into those for the Gegenbauercoefficients first in order to avoid the ill-posed problem appearing in theconversion from the moments to the Gegenbauer coefficients. Given the values ofvarious condensates in the OPE, we find that a solution for the pion LCDA,which is stable in the Gegenbauer expansion, exists. Moreover, the solutionfrom summing contributions up to 18 Gegenbauer polynomials is smooth, and canbe well approximated by a function proportional to $x^p(1-x)^p$ with $p\approx0.45$ at the scale $\mu=2$ GeV. Turning off the condensates, we get theasymptotic 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 thecondensate inputs at this $\mu$, and demonstrate that the result is consistentwith the one obtained by evolving the Gegenbauer coefficients from $\mu=2$ GeVto 1.5 GeV. That is, our formalism is compatible with the QCD evolution. Thestrength of the above approach that goes beyond analyses limited to only thefirst few moments of a LCDA in conventional QCD sum rules is highlighted. Theprecision of our results can be improved systematically by includinghigher-order and higher-power terms in the OPE.

Dispersive derivation of the pion distribution amplitude