Approximation of the Schwinger--Dyson and the Bethe--Salpeter Equations and Chiral Symmetry of QCD

The Bethe–Salpeter equation for the pion in chiral symmetric models is studied with a special care to consistency with low-energy relations. We propose a reduction of the rainbow Schwinger–Dyson and the ladder Bethe–Salpeter equations with a dressed gluon propagator. We prove that the reduction preserves the Ward–Takahashi identity for the axial-vector current and the PCAC relation.