High temperature phase transitions in two-scalar theories with large N techniques

We consider a theory of a scalar one-component field f coupled to a scalar N-component field χ. Using large N techniques we calculate the effective potential in the leading order in 1N. We show that this is equivalent to a resummation of an infinite subclass of graphs in perturbation theory, which involve fluctuations of the χ field only. We study the temperature dependence of the expectation value of the f field and the resulting first and second order phase transitions.