Quantum anomaly of the transverse Ward–Takahashi relation for the axial-vector vertex

We study the possible quantum anomaly for the transverse Ward–Takahashi relations in four dimensional gauge theories based on the method of computing the axial-vector and the vector current operator equations. In addition to the well-known anomalous axial-vector divergence equation (the Adler–Bell–Jackiw anomaly), we find the anomalous axial-vector curl equation, which leads to the quantum anomaly of the transverse Ward–Takahashi relation for the axial-vector vertex. The computation shows that there is no anomaly for the transverse Ward–Takahashi relation for the vector vertex.