  The gauge-invariant, nonlocal, dynamical quark model is proved to generate
the minimal vertices which satisfy the Ward-Takahashi identities. In the chiral
limit, the momentum-dependent quark self-energy results in a flat-like form
with some end point delta-funtions for the light-cone pion distribution
amplitudes, similarly found in the Nambu Jona-Lasino model with constant
constituent mass. The leading order nonzero pion and current quark masses
corrections lead concave type asymptotic-like form modifications to twist-2
pion distribution amplitude with end point pillars and twist-3 tensor pion
distribution amplitude above the flat-like form backgrounds. A by-product of
our investigation shows that the variable u appearing in pion light-cone
distribution amplitudes is just the standard Feynman parameter in the Feynman
parameter integrals; also chiral perturbation works well for these amplitudes.
