Boundary behaviour of the four-point function 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 renormalized L-dependent four-point function at vanishing external momenta in the large-N limit (the effective coupling constant), using bag-model boundary conditions. For values of the fixed coupling constant in absence of boundaries λ⩾λc≃19.16, we obtain small-distance asymptotic freedom (for L→0) and a singularity 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” length L(c)p which is comparable with an estimated proton diameter.