Super daisy resummations and an effective expansion parameter in the O(N)-model near the phase transition

The temperature induced phase transition in the N-component scalar field theory is investigated within the approach combining the second Legendre transform with the consideration of the gap equations in the extrema of the free energy functional. Performing the super daisy resummations a weak first-order phase transition is found. In the limit of N goes to infinity it turns into a second order one. For large N, near the transition temperature an effective expansion parameter, (1/N)1/3, is derived. This opens a possibility to justify the perturbative approach. Comparison with other methods is done.