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The approach based on the Nakanishi integral representation of n-legtransition amplitudes is extended to the treatment of the self-energies of afermion and an (IR-regulated) vector boson, in order to pave the way forconstructing a comprehensive application of the technique to both gap- andBethe-Salpeter equations, in Minkowski space. The achieved result, namely a6-channel coupled system of integral equations, eventually allows one todetermine the three K\"all\'en-Lehman weights for fully dressing thepropagators of fermion and photon. A first consistency check is also provided.The presented formal elaboration points to embed the characteristics of thenon-perturbative regime at a more fundamental level. It yields a viable tool inMinkowski space for the phenomenological investigation of strongly interactingtheories, within a QFT framework where the dynamical ingredients are madetransparent and under control.

Fermion and Photon gap-equations in Minkowski space within the Nakanishi Integral Representation method