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A minimal truncated set of the integral Dyson-Schwinger equations, inMinkowski spacetime, that allows to explore QED beyond its perturbativesolution is derived for general linear covariant gauges. The minimal setincludes the equations for the fermion and photon propagators, thephoton-fermion vertex, and the two-photon-two-fermion one-particle-irreduciblediagram. If the first three equations are exact, to build a closed set ofequations, the two-photon-two-fermion equation is truncated ignoring thecontribution of Green functions with large number of external legs. It is shownthat the truncated equation for the two-photon-two-fermion vertex reproducesthe lowest-order perturbative result in the limit of the small couplingconstant. Furthermore, this equation allows to define an iterative procedure tocompute higher order corrections in the coupling constant. The Ward-Takahashiidentity for the two-photon-two-fermion irreducible vertex is derived andsolved in the soft photon limit, where one of the photon momenta vanish, in thelow photon momenta limit and for general kinematics. The solution of theWard-Takahashi identity determines the longitudinal component of thetwo-photon-two-fermion irreducible vertex, while it is proposed to use theDyson-Schwinger equation to determine the transverse part of this irreduciblediagram. The two-photon-two-fermion DSE is solved in heavy fermion limit,considering a simplified version of the QED vertices. The contribution of thisirreducible vertex to a low-energy effective photon-fermion vertex is discussedand the fermionic operators that are generated are computed in terms of thefermion propagator functions.

Looking at QED with Dyson-Schwinger equations: basic equations, Ward-Takahashi identities and the two-photon-two-fermion irreducible vertex