The Dyson-Schwinger equations and the solution of QED: exploring the two-photon-two-fermion irreducible vertex

Abstract

A minimal truncated set of Dyson-Schwinger equations that allow exploring the non-perturbative regime of QED is derived for a general linear covariant gauge. This minimal set includes the propagators, the the photon-fermion, and the two-photon-two-fermion vertices. If the equations for the first three quantities are exact, to build a closed set of equations, the last one is truncated ignoring further higher-order Green functions. We also show that the truncated equation reproduces the lowest order perturbative result for the two-photon-two-fermion vertex in the limit of the small coupling constant. Then, the two-photon-two-fermion irreducible vertex is studied by combining the corresponding Ward-Takahashi identity with the Dyson-Schwinger equation. The longitudinal part of this irreducible vertex is computed from the Ward-Takahashi identity, which is solved in the low energy limit when one of the photons has zero energy (soft photon limit). For this kinematical configuration, the general expression for the vertex is derived and its form factors related to the photon-fermion vertex and fermion propagator. Contents I. Introduction and Motivation II. Quantum Electrodynamics -definitions and generating functionals III. The Dyson-Schwinger Equations A. The fermion gap equation B. The Dyson-Schwinger equation for the photon propagator C. A Dyson-Schwinger equation for the photon-fermion vertex IV. Ward-Takahashi Identities V. Renormalization VI. The two-photon-two-fermion one-particle irreducible vertex A. The longitudinal component of Γ µν from the Ward-Takahashi identity 1. Solving the contracted Ward-Takahashi identity 2. Low momenta solution of the Ward-Takahashi identity 3. Tensor basis for Γ µν and the solution of the Ward-Takahashi Idendity VII. The Dyson-Schwinger equation for Γ µν A. Exploring further the DSE for the two-photon-two-fermion vertex VIII.

Renormalized Equations of Motions For Scalars and Fermions in the 2pi Formalism

“…The procedure to obtain the counterterms for O(N ) symmetric theories has been explored in [21][22][23][24][25]. Progress has also been with regard to renormalizability with the inclusion of abelian gauge bosons and consistency with Ward identities [5,[26][27][28][29], as well as the inclusion of fermions [30]. A systematic approach for the renormalization of the 2PI action been presented in [31] focusing on scalars.…”