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

TL;DR

This paper extends the Nakanishi integral representation method to fermion and photon self-energies in Minkowski space, enabling a non-perturbative analysis of strongly interacting quantum field theories.

Contribution

It develops a 6-channel coupled integral equation system for fermion and photon propagator dressing functions using Nakanishi representation in Minkowski space.

Findings

Derived a consistent set of integral equations for fermion and photon self-energies.

Established a framework for non-perturbative analysis in Minkowski space.

Provided initial checks confirming the formalism's validity.

Abstract

The approach based on the Nakanishi integral representation of n-leg transition amplitudes is extended to the treatment of the self-energies of a fermion and an (IR-regulated) vector boson, in order to pave the way for constructing a comprehensive application of the technique to both gap- and Bethe-Salpeter equations, in Minkowski space. The achieved result, namely a 6-channel coupled system of integral equations, eventually allows one to determine the three K\"all\'en-Lehman weights for fully dressing the propagators of fermion and photon. A first consistency check is also provided. The presented formal elaboration points to embed the characteristics of the non-perturbative regime at a more fundamental level. It yields a viable tool in Minkowski space for the phenomenological investigation of strongly interacting theories, within a QFT framework where the dynamical ingredients are made transparent and under control.