Gapless Hartree-Fock resummation scheme for the $O(N)$ model

A modified self-consistent Hartree-Fock approximation to the $\lambda {\varphi }^4$ theory with spontaneously broken $O(N)$ symmetry is proposed. It preserves all the desirable features, like conservation laws and thermodynamic consistency, of the self-consistent Schwinger-Dyson scheme generated from a 2PI functional, also known as the $\Phi$-derivable scheme, while simultaneously respecting the Nambu-Goldstone theorem in the chiral-symmetry broken phase. Various approximate resummation schemes are discussed.

Figure 1

Schematic behavior of the solution in the vicinity of the $T_1$ (left panel) and $T_2$ (right panel) points.

Figure 2

Meson masses (left panel) and $\varphi$ (right panel) as functions of temperature for the $m_{\pi }=0$ case. Stable and metastable branches are presented by solid lines and dashed lines, respectively.

Figure 3

Pressure (left panel) and pressure difference between stable and metastable branches (right panel) as functions of temperature for $m_{\pi }=0$ case. The pressure for the stable and metastable branches are presented by solid lines and dashed lines, respectively.

Figure 4

Meson masses (left panel) and $\varphi$ (right panel) as functions of temperature for $m_{\pi }=139\mathrm{MeV}$ case (stable and metastable branches presented by solid lines and dashed lines, respectively).

Figure 5

Pressure (left panel) and pressure difference between stable and metastable branches (right panel) as functions of temperature for $m_{\pi }=139\mathrm{MeV}$ case (stable and metastable branches presented by solid lines and dashed lines, respectively).