The thermal coupling constant and the gap equation in the $\lambda\phi^{4}_{D}$ model

TL;DR

This paper investigates the finite-temperature behavior of the O(N)-symmetric λφ^4 model in various dimensions using resummation methods, revealing how the mass and coupling constant evolve with temperature and deriving a formula for the critical temperature.

Contribution

It introduces a combined approach using composite operator formalism and Dyson-Schwinger equations to analyze thermal properties in arbitrary dimensions, providing new insights into phase transition behavior.

Findings

Thermal squared mass remains positive and increases with temperature.

Thermal coupling constant decreases then increases in D=3, and approaches a constant in D=4.

Derived a general formula for the critical temperature consistent with known results.

Abstract

By the concurrent use of two different resummation methods, the composite operator formalism and the Dyson-Schwinger equation, we re-examinate the behavior at finite temperature of the O(N)-symmetric $\lambda\phi^{4}$ model in a generic D-dimensional Euclidean space. In the cases D=3 and D=4, an analysis of the thermal behavior of the renormalized squared mass and coupling constant are done for all temperatures. It results that the thermal renormalized squared mass is positive and increases monotonically with the temperature. The behavior of the thermal coupling constant is quite different in odd or even dimensional space. In D=3, the thermal coupling constant decreases up to a minimum value diferent from zero and then grows up monotonically as the temperature increases. In the case D=4, it is found that the thermal renormalized coupling constant tends in the high temperature limit to a constant asymptotic value. Also for general D-dimensional Euclidean space, we are able to obtain a formula for the critical temperature of the second order phase transition. This formula agrees with previous known values at D=3 and D=4.