J ul 2 00 1 Temperature phase transition and an effective expansion parameter in the O ( N ) - model

The temperature phase transition in the N-component scalar field theory with the spontaneous symmetry breaking is investigated in the pertur-bative approach.The second Legendre transform and consideration of the gap equations in the extrema of the free energy functional is used. The super daisy and beyond resummations are applied. In the super daisy approximation a first order phase transition is determined. For large N new effective expansion parameter, (1/N) 1/3 , is found that gives possibility to account for other types of graphs perturbatively. Some certain class of diagrams beyond the super daisy one which corresponds to next-to-next-to-leading order in 1/N is calculated. It is shown that it does not alter qualitatively the phase transition. In the limit N goes to infinity the phase transition is of second order. The comparison with other approaches is done.