On-shell parameter fixing in the quark-meson model

The quark-meson model is often used as an effective low-energy model for QCD to study the chiral transition at finite temperature $T$ and baryon chemical potential ${\mu }_B$. The parameters in the quark-meson model can be found by expressing them in terms of the sigma mass $m_{\sigma }$, the pion mass $m_{\pi }$, the constituent quark mass $m_q$ and the pion decay constant $f_{\pi }$. In practice, this matching is done at tree level, which is inconsistent once loop effects of the effective potential are taken into account. We show how to properly perform the matching in the quark-meson model by using the on-shell and the minimal subtraction renormalization schemes relating the physical masses and the pion decay constant to the running mass parameter and couplings. We map out the phase diagram in the ${\mu }_B–T$ plane and compare our results with other approximations.

Figure 1

One-loop self-energy diagrams for the sigma particle.

Figure 2

One-loop self-energy diagrams for the pion.

Figure 3

Counterterm for the two-point functions for the sigma and pion.

Figure 4

Tadpole diagram for the sigma particle and the counterterm.

Figure 5

Vacuum effective potential normalized to $f_{\pi }^4$ as a function of $\mathrm{\Delta }$ for $m_{\sigma }=600\mathrm{MeV}$. Dashed line is the tree-level potential and the solid line is the one-loop effective potential in the large-$N_c$ limit.

Figure 6

Vacuum effective potential normalized to $f_{\pi }^4$ as a function of $\mathrm{\Delta }$ for $m_{\sigma }=800\mathrm{MeV}$. Dashed line is the tree-level potential and the solid line is the one-loop effective potential in the large-$N_c$ limit.

Figure 7

The phase diagram in the $\mu –T$ plane for $m_{\sigma }=600\mathrm{MeV}$. A dashed line indicates a second-order transition, while a solid indicates a first-order transition. The red dot shows the tricritical point. The blue solid line is phase boundary in the no-sea approximation.

Figure 8

The phase diagram in the $\mu –T$ plane for $m_{\sigma }=800\mathrm{MeV}$. A dashed line indicates a second-order transition, while a solid indicates a first-order transition. The blue solid line is the phase boundary in the no-sea approximation.