The linear sigma model at a finite isospin chemical potential

Abstract

The effect of finite isospin chemical potential to the effective masses of the mesons at finite temperature is investigated in the framework of the O(4) linear sigma model with explicit chiral symmetry breaking. We present a mechanism to include the isospin chemical potential in the model. By using the Cornwall-Jackiw-Tomboulis method of composite operators, we obtain a set gap equations for the effective masses of the mesons and get the numerical results in the Hartree approximation. We find that the introduction of the chemical potential only affects the mass of the charged pions and sigma, while there is almost NO effects on the mass of neutral pions.

On the symmetry improved CJT formalism in the O(4) linear sigma model

“…(14) can be obtained without any assumption in the large-N approximation in the framework of the usual CJT formalism, therefore, the large-N approximation in the O(N ) model implies that the pions should be massless in the symmetry broken phase also. In other words, once we take the constraint (14), the well-known results in the large-N approximation could be copied. Using the definition in the framework of the symmetry improved CJT formalism, we can rewrite the finite temperature effective potential V ef f as a function of ϕ in the thermal Hartree approximation by extending the mass-gap equations (11) from φ → ϕ while…”

“…However, in comparison, we can find that the constraint proposed in Eq. (14) can be automatically realized in the case of large-N approximation in the O(N ) linear sigma model [6] [7], whereas, it should be artificially introduced in the framework of the symmetry improved CJT formalism [28]. Moreover, the stationarity equation in Eq.…”

“…(13) and the constraint (14) do not satisfy simultaneously in the case of the symmetry improved CJT formalism in the O(4) linear sigma model, this implies that the effective potential in Eq. (8) is inevitably needed to be reconstructed according to the constraint (14) when leaving the gap equations (10) untouched. This is the underlying reason why we have to redefine the effective potential V ef f as in the equation (17).…”

“…As a quite popular model, the linear sigma model [3] for the phenomenology of QCD has been composed to describe the vacuum structure with incorporating chiral symmetry and its spontaneous breaking. The model can be used to describe a restoration of the chiral symmetry at finite temperature in the Hartree-Fock or Hartree approximation [1,[4][5][6][7][8][9][10][11][12] within the Cornwall-Jackiw-Tomboulis (CJT) formalism [13], also at finite isospin chemical potential [14][15][16][17][18]. However, except the case in the larger-N approximation [5][6] [7], these kinds of studies suffer from two major drawbacks.…”

Functional renormalization for chiral andUA(1)symmetries at finite temperature

“…The linear sigma model is considered as a good laboratory for various approximation methods like mean field and the Cornwall-Jackiw-Tomboulis (CJT) approximation. In the frame of this model, the σ and π properties have been deeply investigated in vacuum and hot medium [15,16]. The model is also extended to include fundamental represented quarks coupled to the mesons in Yukawa form.…”