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The temperature phase transition in scalar $\phi^4(x)$ field theory withspontaneous symmetry breaking is investigated in a partly resummed perturbativeapproach. The second Legendre transform is used and the resulting gap equationis considered in the extrema of the free energy functional. It is found thatthe phase transition is of first order in the super daisy as well as in acertain beyond super daisy resummations. No unwanted imaginary parts in thefree energy are found but a loss of the smallness of the effective expansionparameter near the phase transition temperature is found in both cases. Thismeans an insufficiency of the resummations or a deficit of the perturbativeapproach.Comment: 14 page

Phase transition in scalar ϕ4-theory beyond the super-daisy resummations

Phase transition in scalar ϕ⁴-theory beyond the super-daisy resummations