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We compute one-loop electron-photon vertex with fully off-shell externalmomenta in an arbitrary covariant gauge and space-time dimension. There existnumerous efforts in literature where one-loop off-shell vertex is calculated byemploying the standard first order Feynman rules in different covariant gaugesand space-time dimensions of interest. The tensor structure which decomposesthis three-point vertex into the components transverse and longitudinal to thephoton momentum gets intertwined in this first order formalism. TheWard-Takahashi identity is explicitly invoked to untangle the two pieces andthe results are expressed in a preferred basis of twelve spin-amplitudes. Wepropose a novel approach based upon an efficient combination of the first andsecond order formalisms of quantum electrodynamics to compute this one-loopvertex. Among some conspicuous advantages is the fact that this less knownsecond order formalism separates the spin and scalar degrees of freedom of anelectron interacting electromagnetically. More noticeably, the longitudinal andtransverse contributions naturally disentangle from the onset in our approach.Moreover, this decomposition leads to identities between one-loop scalarFeynman integrals with higher powers in the propagators and shifted space-timedimensions that can be used to prove the Ward-Takahashi identity at one-looporder without the need to evaluate any Feynman integral. Additionally, thisnatural decomposition allows us to establish the gauge-independence of thePauli form factor through explicit cancellations of scalar Feynman integralsthat depend on the gauge parameter. These cancellations naturally lead to acompact expression for the Pauli form factor in arbitrary dimensions. Wherevernecessary and insightful, we make comparisons with earlier works.

One-loop fermion-photon vertex in arbitrary gauge and dimensions: a novel approach