Abstract
Some time ago Takahashi derived the so-called transverse relations relating Green's functions of different orders to complement the well-known Ward–Green–Takahashi identities of gauge theories by considering wedge rather than inner products. These transverse relations have the potential to determine the full fermion–boson vertex in terms of the renormalization functions of the fermion propagator. He and Yu have given an indicative proof at one-loop level in four dimensions. However, their construct involves the fourth-rank Levi–Civita tensor defined only unambiguously in four dimensions exactly where the loop integrals diverge. Consequently, here we explicitly check the proposed transverse Ward–Takahashi relation holds at one-loop order in d-dimensions, with d = 4 + ε.
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