Analytical and numerical study of the Schwinger-Dyson equation with four-fermion coupling

The dressed ladder Schwinger-Dyson equation for the fermion self-energy Σ(p²) with an additional four-fermion coupling term is studied both analytically and numerically. We limit our analysis to asymptotically free theories with a large hierarchy between the confinement scale Λ_c and the ultraviolet cutoff Λ. The dependence of the dynamical mass Σ₀≡Σ(0) and the fermion condensate 〈Ψ̅ Ψ〉 on the four-fermion coupling strength λ is derived. It is found that there exists a critical four-fermion coupling strength λ₂, such that in the region λ<λ₂ the dynamical mass Σ₀ is of the order of the confinement scale Λ_c, while in the region λ>λ₂ the dynamical mass Σ₀ is of the order of the cutoff Λ. In the region λ<λ₂, the condensate 〈Ψ̅ Ψ〉 picks up an enhancement factor λ₂/(λ₂-λ), and in the limit λ→λ₂, reaches the order of Λ²Λ_c. In the region λ>λ₂, the condensate 〈Ψ̅ Ψ〉 is of the order of Λ³. The application of the equation to technicolor theories is also discussed.