Variational Approach to the Chiral Phase Transition in the Linear Sigma Model

Abstract

The chiral phase transition at finite temperature is investigated in the linear sigma model, which is regarded as a low energy effective theory of QCD with three momentum cutoff, in the variational method with the Gaussian approximation in the functional Schrödinger picture. It is shown that the Goldstone theorem is retained and the meson pair excitations are automatically included by taking into account the linear response to the external fields. It is pointed out that the behavior of chiral phase transition depends on the three-momentum cutoff, which leads to the careful treatment of the problem.Subject Index: 233 §1. IntroductionIt is well known that the chiral symmetry plays an essential role in the quark and hadronic world governed by the quantum chromodynamics (QCD). Especially, the spontaneous chiral symmetry breaking leads to the existence of the pion, 1) which plays an important role in the nuclear and hadron physics. One of the recent interests in the hadron physics is the chiral symmetry restoration in the context of the BNL Relativistic Heavy Ion Collider (RHIC) and/or the CERN Large Hadron Collider (LHC). 2) It is believed that the chiral symmetry is restored in the high temperature and/or high baryon density, which might or may be created in the above mentioned experiments.The finite temperature phase transition in the field theoretical treatment was investigated in the context of the phase transition in the early universe. 3) Systematically, the field theoretical treatment for the systems at finite temperature was developed in the early stage. 4), 5) Further, another approach 6) to the finite temperature systems was devised with the variational technique in terms of the functional Schrödinger picture with Gaussian approximation 7), 8) for the bosonic field theory. In this paper, we intend to adopt a slight different variational approach to the bosonic field theory at finite temperature.As for the low energy region of the strong interaction, the low energy effective theories of QCD give very useful tool until now to understand the physics of hadrons, especially, based on the chiral symmetry, while the lattice QCD simulation is developed for the problems related to the chiral symmetry. In the low energy region of QCD, the pion and its chiral partner, namely sigma meson, are important ingredients. As an effective theory of QCD, the linear sigma model 9) gives a powerful tool to understand the many properties of low energy hadron physics, which contains pions and sigma meson paying attention to the chiral symmetry.

Study of hadrons using the Gaussian functional method in the O (4) linear σ model

“…We solve then the Bethe-Salpeter Eqs. ( 25) and ( 32 We compare our results with those of Tsue and Matsuda [8]. In their calculation, they dropped the contribution of σ-σ excitations for the physical σ mass in the BS equation, by saying that the threshold energy of σ-σ excitations is much larger than the value of the cut-off momentum.…”

“…Roh and Matsui studied the chiral phase transition at finite temperature in the one-loop level effective potential [23]. The O(4) linear σ model was studied by Tsue and Matsuda in the line similar to the present study [8]. There are renewed interests on this subject recently raised by the improved CJT method [17,19] and it is important to study further the GF+BS method on the order of phase transition and compare with the improved CJT approach.…”

“…This problem was studied by several authors in the GF framework [3][4][5][6][7][8]. In order to recover the Nambu-Goldstone theorem, we have to solve the Bethe-Salpeter (BS) equations (or four-point Green function SD equations) [9,10].…”

“…The resulting dressed mass of the Nambu-Goldstone boson is, however, not zero, even when the chiral symmetry is spontaneously broken in the chiral limit. This problem was studied by several authors in the GF framework [3][4][5][6][7][8]. In order to recover the Nambu-Goldstone theorem, we have to solve the Bethe-Salpeter (BS) equations (or four-point Green function SD equations) [9,10].…”