Transverse Ward–Takahashi identity for the fermion-boson vertex in gauge theories

The transverse Ward–Takahashi (W-T) identity for the fermion-gauge-boson vertex in gauge theories is derived by calculating the curl of the time-ordered product of the three-point function including the vector current operator. This provides the constraint on the transverse part of the vertex. By combining the transverse and normal (longitudinal) W-T identity, we obtain the full vector-vertex in terms of the axial-vector and tensor vertices.