The Effective Potential for Composite Operator in the Scalar Model at Finite Temperature.pdf

a r X i v : h e p - t h / 0 1 0 2 0 7 2 v 1 1 3 F e b 2 0 0 1 The Effective Potential for Composite Operator in the Scalar Model at Finite Temperature G.N.J.An?an?os1,2 and N.F.Svaiter1 1Centro Brasileiro de Pesquisas F??sicas-CBPF Rua Dr.Xavier Sigaud 150, Rio de Janeiro, RJ 22290-180 Brazil E-mail: nfuxsvai@lafex.cbpf.br 2Laborato?rio Nacional de Computac?a?o Cient??fica-LNCC Av. Getu?lio Vargas, 333 - Quitandinha - Petro?polis, RJ 25651-070 Brazil E-mail: gino@lafex.cbpf.br Abstract We discuss the ?4 and ?6 theory defined in a flat D-dimensional space-time. We assume that the system is in equilibrium with a thermal bath at temperature β?1. To obtain non-perturbative result, the 1/N expansion is used. The method of the composite operator (CJT) for summing a large set of Feynman graphs, is developed for the finite temperature system. The ressumed effective potential and the analysis of the D = 3 and D = 4 cases are given. 1 Introduction The conventional perturbation theory in the coupling constant or in h? i.e., the loop expansion can only be used for the study of small quantum corrections to classical results. When discussing quantum mechanical effects to any given order in such an expansion, one is not usually able to justify the neglect of yet higher order. In other words, for theories with a large N dimensional internal symmetry group, there exist another perturbation scheme, the 1/N expansion, which circumvents this criticism. Each term in the 1/N expansion contains an infinite subset of terms of the loop expansion. The 1/N expansion has the nice property that the leading-order quantum corrections are of the same order as the classical quantities. Consequently, the leading order which adequately characterizes the theory in the large N limit preserves much of the nonlinear structure of the full theory. In the next section we derive the effective action to leading order in 1/N in D-dimensional space-time and consequently the effective potential. It is known