GROSS-NEVEU MODEL AS A Z3=0 LIMIT OF THE 2-DIMENSIONAL SU(N)-SIGMA MODEL

We study the renormalization of the two-dimensional SU(N) σ model (Nψ's, 1σ) Lσ=ψ̄(iγ-gσ)ψ+(12)(μσ)2-(12) μ02σ2-(14)λσ4 and its limit as the σ becomes purely composite in the first two orders of mean-field perturbation theory (MFPT). The limit is attained by demanding the vanishing of the σ wave function and λ-coupling renormalization constants Z3, Z4, and in that limit MFPT becomes the 1N expansion of the Gross-Neveu (GN) model L=ψ̄iγψ-G02(ψ̄ψ)2. The three-parameter super-renormalizable σ model goes over to the one-parameter renormalizable GN model smoothly without the need for introducing new counterterms to this order. The symmetry ψ→γ5ψ, σ→-σ is necessarily broken in both models to this order and the N fermions pick up a common mass. We demonstrate the feasibility of implementing MFPT beyond lowest order and develop techniques applicable to the four-dimensional problem. Numerical values of the corrections to gR (ψ̄ψσ coupling), λR (σ4 coupling), σR and the effective potential are calculated to provide an estimate of the validity of the 1N expansion. We verify Schonfeld's result that to second order the σ particle remains at the ψ̄ψ threshold in the GN model. © 1979 The American Physical Society.