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 we take loop effects of the effective potential into account. We show how to properly perform the matching in the quark-meson model by using the on-shell and the modified 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.