Renormalization of theσmodel at finite temperature

The temperature dependence of a many-body theory with the dynamics defined by the relativistic linear $\sigma$ model is studied. The model has SU(2) × SU(2) chiral symmetry with fermions belonging to a (½,0) + (0,½) representation interacting with the $\sigma$ and $\pi$ mesons belonging to the (½, ½) representation of the chiral symmetry group. The dimensional-regularization technique together with the renormalization procedure of 't Hooft is used to reproduce the well-known result that the counterterms of the symmetric theory remove the divergences of the theory with spontaneous symmetry breaking, without however the need for invoking auxiliary fermion fields. Renormalizability is maintained at finite temperatures by the cancellation of temperature-dependent infinities which appear at the two-loop level. This is shown explicitly for the ground-state expectation value of the scalar $\sigma$ field at the two-loop level. When the symmetry is explicitly broken by the term $f_{\pi }{m_{\pi }}^2\sigma$ the symmetry of the original Lagrangian is never restored. In the absence of such a term a symmetry change with temperature is realized and the persistence of the Goldstone mode up to a critical temperature $T_c$, above which the original symmetry is restored, is verified. Thus below $T=T_c$ the low-energy theorems of current algebra associated with the existence of the Goldstone pions would be valid except that all parameters of the theory develop finite, temperature-dependent, corrections. A parallel discussion for density dependence of the symmetry is included. All calculations are done in the real-time formalism for the thermodynamic Green's functions.