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 $O(N) {\ensuremath{\varphi}}^{4}$ theory. It is proved that (i) the renormalization of the ultraviolet divergences can be carried out systematically in any given order of OPT and (ii) the Nambu-Goldstone theorem is satisfied for arbitrary $N$ and for any given order of OPT. The method is applied for the $O(4) \ensuremath{\sigma}$ 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 pseudoscalar channels is shown to be a typical signal of the chiral transition.

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