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

摘要

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