Asymptotic freedom and the thermal Lee model

Abstract

Dimensional regularization is used to investigate the renormalization of the fermion mass, wave function, and coupling constant in the V-sector of a thermal Lee model in arbitrary space–time dimensions D and temperature T. A closed expression of the fermion mass renormalization shows that it diverges at the high-T limit and replicates a familiar form at the zero-T limit. Corresponding expressions of the wave function renormalization and the renormalized coupling constant vanish at the high-T limit, and resume respective customary forms at the zero-T limit. Likewise, the intrinsic scattering amplitude vanishes at the high-T limit, and reduces to the original amplitude at the zero-T limit. The 1D theory is especially useful for schematically graphing expressions that show thermal effects on the probability, the bare coupling constant, and the bare mass. Bifurcation of the bare parameters is a prominent feature in these graphs for particular choice of input. The β and γ coefficients of the Callan–Symanzik equation are calculated in closed form, and correlations between the β coefficient and the probability are in evidence for dimensions less than, or greater than 4. Schematic graphs incorporating Laurent expansions of the β coefficient, the probability, and the fermion-mass show the dependency of these functions upon the physical parameters in the theory for specific values of D. The thermal model is found to be asymptotically free for D < 4, and for odd D ≥ 5 at the high-T limit.