Effects of renormalizing the chiral SU(2) quark-meson model

We investigate the restoration of chiral symmetry at finite temperature in the SU(2) quark-meson model, where the mean field approximation is compared to the renormalized version for quarks and mesons. In a combined approach at finite temperature, all the renormalized versions show a crossover transition. The inclusion of different renormalization scales leave the order parameter and the mass spectra nearly untouched but strongly influence the thermodynamics at low temperatures and around the phase transition. We find unphysical results for the renormalized version of mesons and the combined one.

Figure 1

1-PI loops contributing to the effective potential in Eq. (42), i.e. Eq. (43) without ${\mathrm{\Omega }}_2$ from Eq. (44).

Figure 2

(a) Two-loop Hartree contributions, Eq. (44), to the CJT effective potential (${\mathrm{\Omega }}_2$). The solid red line corresponds to $G_{\sigma }$, whereas the dashed blue line corresponds to $G_{\pi }$. The right-most diagram stands for the last term in Eq. (43). (b) Tadpole contribution to the self-energy, obtained by cutting a line.

Figure 3

$\sigma$ condensate as a function of temperature for zero chemical potential without (denoted as “th.”) and with vacuum contribution (denoted as “vac.”) for three different values of the vacuo sigma meson mass $m_{\sigma }^{\mathrm{vac}}$ shown in the upper panel. The lower panels show the in-medium masses of the $\sigma$ and the $\pi$.

Figure 4

Pressure, divided by $T^4$ as a function of temperature for zero chemical potential without (denoted as “th.”) and with vacuum contribution (denoted as “vac.”) for three different values of the initial vacuum sigma meson mass $m_{\sigma }^{\mathrm{vac}}$ shown in the upper plot. The lower plot shows the entropy density $s$ divided by $T^3$ as a function of the temperature. The SB limit represents the Stefan Boltzmann limit.

Figure 5

$\sigma$ condensate in the combined approach as a function of temperature for zero chemical potential without (denoted as “th.”) and with quark vacuum contribution (denoted as “vac.”) for three different values of the initial vacuo sigma meson mass $m_{\sigma }^{\mathrm{vac}}$ shown in the upper panel. The lower panels show the masses of the sigma and the pion as a function of temperature. The value of the quark renormalization scale has been chosen to be $\mathrm{\Lambda }=1033\mathrm{MeV}$.

Figure 6

Negative of the potential, i.e. the pressure, divided by $T^4$ as a function of temperature without (denoted as “th.”) and with vacuum contribution (denoted as “vac.”) for three different values of the initial vacuo sigma meson mass $m_{\sigma }^{\mathrm{vac}}$ shown in the upper panel. The lower panel depicts the entropy density $s$ as a function of temperature. Some curves clearly show maxima and minima.

Figure 7

Vacuum parameters $\lambda$, $m^2$, and $H$, normalized to their respective tree-level value ($\lambda \simeq 16.64$, $m^2=-122683{\mathrm{MeV}}^2$ and $H=1.75\times {10}^6{\mathrm{MeV}}^3$), as a function of the quark renormalization scale $\mathrm{\Lambda }$. The cross marks the tree-level values.

Figure 8

$\sigma$ condensate as a function of temperature for the renormalization scales $\mathrm{\Lambda }=1033\mathrm{MeV}$ and $\mu =0$ (Sec. 7a), $\mathrm{\Lambda }=900$, 1000, 1100 MeV at $\mu =m_{\sigma }/\sqrt{e}$ held fixed, and for $\mathrm{\Lambda }=\mu =1033\mathrm{MeV}$ at $m_{\sigma }=550\mathrm{MeV}$ is shown in the upper panel. The lower panel shows the sigma and the pion mass spectrum.

Figure 9

Pressure, divided by $T^4$ as a function of temperature for the renormalization scale choices $\mathrm{\Lambda }=1033\mathrm{MeV}$ and $\mu =0$ (Sec. 7a), $\mathrm{\Lambda }=900$, 1000, 1100 MeV at $\mu =m_{\sigma }/\sqrt{e}$ held fixed, and for $\mathrm{\Lambda }=\mu =1033\mathrm{MeV}$ at $m_{\sigma }=550\mathrm{MeV}$, is shown in the upper panel. The lower panel shows the entropy density $s$ divided by $T^3$ as a function of temperature.