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

The dressed ladder Schwinger-Dyson equation for the fermion self-energy $\Sigma \left(p^2\right)$ 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 ${\Lambda }_c$ and the ultraviolet cutoff $\Lambda$. The dependence of the dynamical mass ${\Sigma }_0\equiv \Sigma \mathrm{}\left(0\right)\mathrm{}$ and the fermion condensate $〈\overline{\Psi }\Psi 〉$ on the four-fermion coupling strength $\lambda$ is derived. It is found that there exists a critical four-fermion coupling strength ${\lambda }_2$, such that in the region $\lambda <{\lambda }_2$ the dynamical mass ${\Sigma }_0$ is of the order of the confinement scale ${\Lambda }_c$, while in the region $\lambda >{\lambda }_2$ the dynamical mass ${\Sigma }_0$ is of the order of the cutoff $\Lambda$. In the region $\lambda <{\lambda }_2$, the condensate $〈\overline{\Psi }\Psi 〉$ picks up an enhancement factor $\frac{{\lambda }_2}{({\lambda }_2-\lambda )}$, and in the limit $\lambda \to {\lambda }_2$, reaches the order of ${\Lambda }^2{\Lambda }_c$. In the region $\lambda >{\lambda }_2$, the condensate $〈\overline{\Psi }\Psi 〉$ is of the order of ${\Lambda }^3$. The application of the equation to technicolor theories is also discussed.