Renormalization of the broken-symmetry two-dimensional $\mathrm{SU}\mathrm{}\left(N\right)\mathrm{}$ $\sigma$ model and ${\left(\overline{\psi }\psi \right)}^2$ model

We derive the renormalized Schwinger-Dyson equations for the two-dimensional $\mathrm{SU}\mathrm{}\left(N\right)\mathrm{}$ $\sigma$ model with $N$ ${\psi }_i$ and one $\sigma$ and the ${\left({\overline{\psi }}^i{\psi }^i\right)}^2$ model with spontaneously broken symmetry $〈\sigma 〉\ne 0$. We study the $\sigma$ model in the composite limit when the $\sigma$ wave-function renormalization constant $Z_3$ and the $4-\sigma$ vertex renormalization constant $Z_4$ are zero and find that the renormalized Green's functions of the $\sigma$ model become identical to those of the ${\left(\overline{\psi }\psi \right)}^2$ model. In the composite limit of the $\sigma$ model the renormalized coupling constants $g\overline{\psi }\psi \sigma$ and ${\lambda }_{4\sigma }$ are related and become calculable in terms of the spontaneously generated fermion mass. We obtain the renormalized gap equation for ${M_{\sigma }}^2〈{\sigma }_r〉$ which determines the minimum of the effective potential and discuss the Callan-Symanzik equations for the broken-symmetry sector of these models. We show that the $Z_3=Z_4=0\mathrm{}$ limit of the $\sigma$ model is obtained at the fixed point $\beta =0\mathrm{}$ of the Callan-Symanzik equations.