Non-perturbative solutions of the SD equation for an Abelian gauge field theory

In this article we present a mechanism in which we find new non-perturbative solutions of quantum electrodynamics in four dimensions. Two non-perturbative solutions are found for approximate Schwinger-Dyson equations. The mass ratio of the three solutions (one is ordinary solution) is approximately 1 : (8π/α) : 8π/α, which is close to the realistic mass ratio of e, μ and τ. e-mail: [email protected] 1 In this article, we will find new non-perturbative solutions for quantum electrodynamics in continuous four dimensional space-time: L(x) = − 1 4 FμνF νμ + ψ̄(i6∂ − e6 A−m)ψ, (1) where Fμν = ∂μAν − ∂νAμ. The Schwinger-Dyson(SD) equation for the electron two point function S(p) is − i( 6p−m)S(p) + ie ∫ dk (2π) γS(p− k)Λ(p− k, p)S(p)Dμν(k) = 1, (2) where Dμν(k) is the photon two-point function. Λ μ is the 1PI three point function. The SD equations itself are non-perturbative.[1] If we could solve the SD equations non-perturbatively, we would understand non-perturbative phenomena. We must discuss gauge fixing before solving the equations. We consider lattice gauge theories to be more natural regularization of gauge symmetry, because we need not to fix gauge. If we force to introduce gauge fixing into a lattice gauge theory, only Landau gauge is allowed as will be seen below. Therefore we prefer Landau gauge also in the continuous theory. At first, let us recall the derivation of non-Landau gauge, especially Rξ-gauge, in continuum. Following Dirac’s prescription for constrained system and FaddeevPopov’s argument, the path integral formula is