Boundary behaviour of the four-point function in the 3-dimensional Gross-Neveu model

Abstract

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 $λ\geq λ_c \simeq 19.16$, we obtain small-distance asymptotic freedom (for $L \to 0$) and a singularity for a length $L^{(c)}$ such that $2.07 m^{-1} < L^{(c)} \lesssim 2.82 m^{-1}$, $m$ 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.