Transverse Ward-Takahashi Relation for the Fermion-Boson Vertex Function in 4-dimensional QED

I present a general expression of the transverse Ward-Takahashi relation for the fermion-boson vertex function in momentum space in 4-dimensional QED, from which the corresponding one-loop expression is derived straightforwardly. Then I deduce carefully this transverse Ward-Takahashi relation to one-loop order in d-dimensions, with $d = 4 + \epsilon$. The result shows that this relation in d-dimensions has the same form as one given in 4-dimensions and there is no need for an additional piece proportional to $(d-4)$ to include for this relation to hold in 4-dimensions. This result is confirmed by an explicit computation of terms in this transverse WT relation to one-loop order. I also make some comments on the paper given by Pennington and Williams who checked the transverse Ward-Takahashi relation at one loop order in d-dimensions.