Confinement in the 3-dimensional Gross-Neveu model

TL;DR

This paper analyzes the confinement phenomenon in a 3D Gross-Neveu model with boundaries, deriving an effective coupling and identifying a confinement length scale comparable to the proton diameter.

Contribution

It provides a closed-form expression for the L-dependent coupling in the 3D Gross-Neveu model with boundary conditions, revealing confinement behavior.

Findings

Demonstrates asymptotic freedom as L approaches zero.

Identifies a confinement length scale between 2.07 and 2.82 inverse fermion mass.

Confirms the confinement length is comparable to proton size.

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 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 $\lambda \geq \lambda_c \simeq 19.16$, we obtain ultra-violet asymptotic freedom (for $L \to 0$) and confinement 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 legth $L^{(c)}_p$ which is comparable with an estimated proton diameter.