Confinement in the 3-dimensional Gross-Neveu model at finite temperature

Abstract

We study the N-component (2+1)-dimensional Gross-Neveu model bounded between two parallel planes separated by a distance L at finite temperature (T ). We obtain a closed expression for the large-N effective coupling constant g = g (L, T, λ). Different behavior depending on the magnitude of the fixed coupling constant λ is found to lead to a "critical" value λ c . If λ < λ c , only short-distance and/or high-temperature asymptotic freedom is found. For λ ≥ λ c one also observes spatial confinement, which is destroyed by temperature effects. We find a confining length, L c 1.61 f m, that is close to the proton charge diameter (≈ 1.74 f m) and a deconfining temperature, 138 MeV , which is comparable to the estimated value of ≈ 200 MeV for hadrons.Keywords: Gross-Neveu model; Four-point function; Spatial confinement Effective models in quantum field theories have been employed over the last decades in trials to obtain clues about the behavior of strongly interacting particles. Among them, the Gross-Neveu model [1], dealing with the direct four-fermions interaction, has been analyzed at finite temperature as an effective model for QCD and for superconducting systems (see for instance Ref. [2,3]). Calculation of the effective potential of the φ 4 theory at finite temperature has also been performed [4].Recently, we have studied the N-components tridimensional Gross-Neveu model at zero temperature, bounded between two parallel planes [5]. A closed expression was derived for the large-N effective coupling constant g(L, λ) as a function of the distance L between the planes. From this result, the behavior of g(L, λ) depending on the magnitude of the free space fixed coupling constant λ was found, such that for small λ, the model presents asymptotic freedom at short distances. On the other hand, for large enough values of λ both spatial confinement and short distance asymptotic freedom are simultaneously present. In this context, the analysis of the effect of temperature is crucial, since it affects the confinement properties. The main objective of the present Letter is to study the spatial confinement and thermal deconfinement properties of the Gross-Neveu model.We recall that even though it is perturbatively nonrenormalizable for dimensions D > 2, the massive GrossNeveu model in Euclidean tridimensional (3-D) space has been shown to exist and has been explicitly constructed in the large-N [6]. A decisive physical point that brings consistency to this derivation is a theoretical result [7] supporting the idea that perturbatively non-renormalizable models do exist and have a physical meaning. For the N = 1 case, some operators can be made more relevant in the low energy region if the fermionic field is minimally coupled to the Chern-Simons field [8,9].We consider the N-component 3-D massive Gross-Neveu model, in the large-N limit, compactified along the imaginarytime axis and also along one of the spatial directions. From a physical point of view, in terms of a generalized Matsubara formalism [10], the model is intended to...