Optimized Perturbation Theory at Finite Temperature

An optimized perturbation theory (OPT) at finite temperature T, which resums higher order terms in the naive perturbation, is developed in O(N) phi^4 theory. It is proved that (i) the renormalization of the ultra-violet divergences can be carried out systematically in any given order of OPT, and (ii) the Nambu-Goldstone theorem is fulfilled for arbitrary N and for any given order of OPT. The method is applied for the O(4) sigma model to study the soft modes associated with the chiral transition in quantum chromodynamics. Threshold enhancement of the spectral functions at finite T in the scalar and pseudo-scalar channels is shown to be a typical signal of the chiral transition.