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

Minimal Ward-Takahashi Vertices and Pion Light Cone Distribution Amplitudes from Gauge Invariant, Nonlocal, Dynamical Quark Model