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 ($\ensuremath{\lambda}$), we find an asymptotic-freedom type of behavior, with $g\ensuremath{\rightarrow}0$ as $L\ensuremath{\rightarrow}0$ and/or $T\ensuremath{\rightarrow}\ensuremath{\infty}$. At $T=0$, and for $\ensuremath{\lambda}\ensuremath{\ge}{\ensuremath{\lambda}}_{c}^{(D)}$ (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 $\ensuremath{\lambda}$ and the mass ($m$) of the fermions, in the case of $D=2,3,4$. Taking $m$ as the constituent quark mass ($\ensuremath{\approx}350\text{ }\text{ }\mathrm{MeV}$), the results obtained are of the same order of magnitude as the diameter ($\ensuremath{\approx}1.7\text{ }\text{ }\mathrm{fm}$) and the estimated deconfining temperature ($\ensuremath{\approx}200\text{ }\text{ }\mathrm{MeV}$) of hadrons.