**9**, 245-259

## Effective potential of the O(N) linear sigma-model at finite temperature

**Y. Nemoto ^{1,2}
- K. Naito^{1,3} - M. Oka^{1}**

^{1} Department of Physics, Tokyo Institute of Technology Meguro, Tokyo 152-8551 Japan

^{2} Yukawa Institute for Theoretical Physics,
Kyoto University, Kyoto 606-8502 Japan^{}

^{3} Radiation Laboratory, the Institute of Physical and
Chemical Research (RIKEN) Wako, Saitama 351-0198 Japan^{}

nemoto@yukawa.kyoto-u.ac.jp

Received: 21 June 2000

Communicated by W. Weise

**Abstract**

We study the *O*(*N*) symmetric
linear sigma-model at finite temperature as the low-energy
effective models of quantum chromodynamics (QCD) using the
Cornwall-Jackiw-Tomboulis (CJT) effective action for composite operators.
It has so far been claimed that the Nambu-Goldstone theorem is not
satisfied at finite temperature in this framework unless the
large-*N* limit in the *O*(*N*) symmetry is taken.
We show that this is not the case.
The pion is always massless below the critical temperature,
if one determines the propagator within the form such that the
symmetry of the system is conserved,
and defines the pion mass as the curvature of the effective potential.
We use a regularization for the CJT effective
potential in the Hartree approximation, which is analogous to the
renormalization of auxiliary fields.
A numerical study of the Schwinger-Dyson equation and the
gap equation is carried out including the thermal and quantum loops.
We point out a problem in the derivation of the sigma meson mass without
quantum correction at finite temperature.
A problem about the order of the phase transition in this approach
is also discussed.

**PACS**

11.30.Rd Chiral symmetries
- 11.10.Wx Finite-temperature field theory
- 12.39.Fe Chiral Lagrangians

Copyright Società Italiana di Fisica, Springer-Verlag 2000