  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.
