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In this work we study the renormalization of the electrodynamics of spin 1/2fermions in the Poincar\'e projector formalism which is second order in thederivatives of the fields. We analyze the superficial degree of divergence ofthe vertex functions of this theory, calculate at one-loop level the vacuumpolarization, fermion self-energy and \gamma-fermion-fermion vertex functionand the divergent piece of the one-loop contributions to the\gamma-\gamma-fermion-fermion vertex function. It is shown that these functionsare renormalizable independently of the value of the gyromagnetic factor gwhich is a free parameter of the theory. We find a photon propagator and arunning coupling constant \alpha (q^2) that depend on the value of g. Themagnetic moment form factor contains a divergence associated to g whichdisappears for g=2 but in general requires the coupling g to be renormalized. Asuitable choice of the renormalization condition for the magnetic form factoryields the one loop finite correction \Delta{g}=g\alpha/2\pi. For a particlewith g=2 we recover results of Dirac theory for the photon propagator, therunning of \alpha (q^2) and the one-loop corrections to the gyromagneticfactor.

Renormalization of the QED of second order spin 1/2 fermions