The constraint equation of the vertex function of quantum anomaly and Dyson–Schwinger equations in massless QED2 theory

In this paper, we calculate the quantum anomaly for the longitudinal and the transverse Ward–Takahashi (WT) identities for vector and axial-vector currents in QED2 theory by means of the point-splitting method. It is found that the longitudinal WT identity for vector current and transverse WT identity for axial-vector current have no anomaly while the longitudinal WT identity for axial-vector current and the transverse WT identity for vector current have anomaly in QED2 theory. Moreover, we study the four WT identities in massless QED2 theory and get the result that the four WT identities together give the constraint equation of the vertex function of quantum anomaly. At last, we discuss the Dyson–Schwinger equations in massless QED2 theory. It is found that the vertex function of the quantum anomaly has corrections for the fermion propagator and Schwinger model.