Confinement in the 3-dimensional Gross-Neveu model

We consider the N-components 3-dimensional massive Gross-Neveu model compactified in one spatial direction, the system being constrained to a slab of thickness L. We derive a closed formula for the effective renormalized L-dependent coupling constant in the large-N limit, using bag-model boundary conditions. For values of the fixed coupling constant in absence of boundaries λ ≥ λc ≃ 19.16, we obtain ultra-violet asymptotic freedom (for L → 0) and confinement for a length L(c) such that 2.07m-1 < L(c) ≲ 2.82m-1, m being the fermionic mass. Taking for m an average of the masses of the quarks composing the proton, we obtain a confining legth which is comparable with an estimated proton diameter.