Coupled Dyson-Schwinger Equations and Effects of Self-Consistency
  Using the $\sigma -\omega$ model as an effective tool, the effects of self-consistency are studied in some detail. A coupled set of Dyson-Schwinger equations for the renormalized baryon and meson propagators in the $\sigma -\omega$ model is solved self-consistently according to the dressed Hartree-Fock scheme, where the hadron propagators in both the baryon and meson self-energies are required to also satisfy this coupled set of equations. It is found that the self-consistency affects the baryon spectral function noticeably, if only the interaction with $\sigma$ mesons is considered. However, there is a cancellation between the effects due to the $\sigma$ and $\omega$ mesons and the additional contribution of $\omega$ mesons makes the above effect insignificant. In both the $\sigma$ and $\sigma -\omega$ cases the effects of self-consistency on meson spectral function are perceptible, but they can nevertheless be taken account of without a self-consistent calculation. Our study indicates that to include the meson propagators in the self-consistency requirement is unnecessary and one can stop at an early step of an iteration procedure to obtain a good approximation to the fully self-consistent results of all the hadron propagators in the model, if an appropriate initial input is chosen. Vertex corrections and their effects on ghost poles are also studied. 
S. S. Wu, H. X. Zhang, Y. J. Yao (Jilin University, Changchun, China)