Phase transition in the massive Gross-Neveu model in toroidal topologies

Abstract

We use methods of quantum field theory in toroidal topologies to study the $N$-component $D$-dimensional massive Gross-Neveu model, at zero and finite temperature, with compactified spatial coordinates. We discuss the behavior of the large-$N$ coupling constant ($g$), investigating its dependence on the compactification length ($L$) and the temperature ($T$). For all values of the fixed coupling constant ($\lambda$), we find an asymptotic-freedom type of behavior, with $g\to 0$ as $L\to 0$ and/or $T\to \infty$. At T=0, and for $\lambda \ge {\lambda }_c^{(D)}$ (the strong coupling regime), we show that, starting in the region of asymptotic freedom and increasing $L$, a divergence of $g$ appears at a finite value of $L$, signaling the existence of a phase transition with the system getting spatially confined. Such a spatial confinement is destroyed by raising the temperature. The confining length, $L_c^{(D)}$, and the deconfining temperature, $T_d^{(D)}$, are determined as functions of $\lambda$ and the mass ($m$) of the fermions, in the case of $D=2,3,4$. Taking $m$ as the constituent quark mass ($\approx 350MeV$), the results obtained are of the same order of magnitude as the diameter ($\approx 1.7fm$) and the estimated deconfining temperature ($\approx 200MeV$) of hadrons.

14 pages, 10 figures, 1 table, to appear in Phys. Rev. D

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